Abstract: A period of cosmological inflation, driven by one or more scalar fields, is still the best candidate to solve various problems in the standard Big Bang model. We want to find as many observational constraints on the inflationary quantities as possible, in order to pin down the correct inflation model, which in its turn will provide information about the underlying high-energy theory. Fortunately the non-Gaussianity of inflationary perturbations, as encoded in the bispectrum (or 3-point correlator), has become an important additional way of distinguishing between models, going beyond the linear Gaussian perturbation quantities of the power spectrum.

This habilitation thesis provides a review of my work on both the theoretical and the observational aspects of these non-Gaussianities. In the first part a formalism is described, called the long-wavelength formalism, that provides a way to compute the non-Gaussianities in multiplefield inflation. Applications of this formalism to various classes of models, as well as its extensions, are also treated. In the second part an estimator is described, called the binned bispectrum estimator, that allows the extraction of information about non-Gaussianities from data of the cosmic microwave background radiation (CMB). It was in particular one of the three estimators applied to the data of the Planck satellite to provide the currently best constraints on primordial non-Gaussianity. Various extensions of the estimator and results obtained are also discussed.

Chapter 1

Introduction

The main subject of most of my research after my PhD thesis is the non-Gaussianity of cosmological fluctuations. Here non-Gaussianity refers to the distribution of the fluctuations evaluated at different spatial coordinates in the universe (but for a given time). When all equations are linearized, it turns out that the distribution is necessarily Gaussian. The non-Gaussianity is caused by non-linear corrections. Roughly speaking my non-Gaussianity research can be divided into two different parts. The first is the development of a formalism, the long-wavelength formalism, to compute the non-Gaussianity as produced in inflation models, in particular multiple-field inflation models, and the application of this formalism to study those models. The second is the development of an estimator, the binned bispectrum estimator, to extract information about this inflationary non-Gaussianity from cosmic microwave background (CMB) data in an optimal way, its numerical implementation, and its application to the data of the Planck satellite.

In this introductory chapter we will set the stage by going into some detail about (multiplefield) inflation, linear cosmological fluctuations, and the CMB, all of which are required as background knowledge for my research. In chapter 2 and associated appendices we will then describe the first part of my post-PhD research as defined above (non-Gaussianity in multiplefield inflation), while in chapter 3 and associated appendices we will do the same for the second part (non-Gaussianity in the CMB). Some conclusions are presented in chapter 4.

My papers on these subjects are of two different types: papers in which the long-wavelength formalism (for the first part) and the binned bispectrum estimator (for the second part) are developed, and papers which contain applications or specific extensions of the formalism/estimator. The latter type are relatively self-contained papers (especially once the formalism/estimator is known), while the former type corresponds to a succession of papers over many years, in which the formalism/estimator was continuously further refined. For the former type, just reading all the papers in order is not necessarily the most efficient way to learn about them, as certain parts discussed in older papers were improved upon in newer papers. Hence in this thesis we provide a description of the formalism and estimator in their final state and a summary of their derivation as synthesized from several papers, so that it is in principle not necessary to look at the original papers (except for the reader interested in all the details). On the other hand, for the self-contained papers of the second type (applications and specific extensions), we provide only a brief summary in the main text of some of them, and then refer to the appendices where these selected papers have been included almost verbatim.

Inflation

Main ideas

The observation by Hubble that the universe is expanding, naturally led Lemaître and Gamow to propose the hot Big Bang theory, the idea that the universe was smaller and hotter in the past. 1 The confirmed predictions of nucleosynthesis (a cosmic origin for the light elements by nuclear fusion in the hot early universe) and the cosmic microwave background radiation (a uniform radiation left over from the hot early stages of the universe) then led to the general acceptance of this theory. However, some issues remained. In the early 1980s it was discovered that several of these can be solved by introducing a period of inflation. In its simplest form inflation is a period in the very early universe where the total energy density of the universe was dominated by the potential energy of a scalar field, called the inflaton. This potential energy is (almost) constant, either because the inflaton field is trapped in a false vacuum behind a potential barrier, or (in most models) because the potential is very flat and the field rolls down very slowly. As a consequence this potential energy plays the role of a cosmological constant and the universe expands exponentially.

Such a brief period of enormous expansion in the very early universe solves many problems of the standard Big Bang theory: why the universe is so homogeneous on the largest scales despite the absence of causal contact between the different regions (horizon problem), why the universe is so flat (flatness problem), why we see no magnetic monopoles and other topological defects left over from phase transitions at very high energies (monopole/topological defect problem), etc. These problems are all solved because of the very different relation between the energy density ρ and the scale factor a for a cosmological constant (ρ ∝ a 0 ) as compared to for matter (ρ ∝ a -3 ) or for radiation (ρ ∝ a -4 ). This means for example that any topological defect (ρ ∝ a -1 , a -2 , a -3 depending on the type: domain walls, cosmic strings, or magnetic monopoles, respectively) or the term corresponding to the curvature of the universe (ρ ∝ a -2 ) loses energy density faster than the dominant component during cosmological constant domination, so that topological defects or spatial curvature will naturally disappear during inflation. During radiation or matter domination it is the opposite, which is the origin of those problems. It turns out that all these problems can be solved if inflation lasts for at least 60 e-folds (i.e. the universe expands by at least a factor e 60 during inflation). 2 At least in slow-roll models of inflation this is generally not hard to achieve, which means that as far as these issues are concerned, there is not much of an observational constraint on inflation models.

While the energy scale of inflation is currently unknown, theoretical arguments generally put it around the Grand Unified Theory (GUT) scale of about 10 15 -10 16 GeV. These arguments are simple: in order to solve for example the flatness problem, you want inflation to happen as early as possible (otherwise a closed universe might already have recollapsed before inflation could start, for example). On the other hand, one does not want to rely on unknown quantum gravity physics at the Planck scale (10 19 GeV). Hence the slightly lower GUT scale is a better option, where we also have many theoretical scalar field candidates. In addition, observational constraints from Planck and Bicep2-Keck on the tensor-to-scalar ratio (see section 1.2.2) have by now put an upper limit on the inflationary energy scale of about 1.7 • 10 16 GeV (95 % CL) [Planck 2018 XX]. This corresponds to a tiny fraction of a second after the Big Bang (about 10 -37 s if we assume radiation domination before inflation).

Soon after inflation was initially proposed by Guth, Linde, Starobinsky and others in the 1 There will be almost no references in this introductory chapter. In my opinion the basics of cosmology, single-field slow-roll inflation, and the CMB have now become part of the "general knowledge of physics" that is taught in undergraduate and graduate courses and can be found in any book on cosmology, see e.g. [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Peter | Primordial Cosmology. Oxford Graduate Texts[END_REF]. Just as one does not cite the original papers when using basic elements of general relativity or quantum mechanics, I feel there is no longer a need to cite the original paper by Guth when talking about inflation, for example. This chapter is just my way of presenting that general knowledge, to remind the reader who might not use it on a daily basis. There will only be some limited references in the parts of this chapter that I feel are not (yet) part of the general knowledge, like multiple-field inflation.

2 While 60 is the number generally quoted in the literature, the exact number depends on how much the universe expanded after inflation up until the current time, which in its turn depends on the energy scale of inflation and the reheating temperature. Given observational constraints, the value 60 is approximately the upper limit (of the minimum amount of inflation required), but it could easily be somewhat lower. For simplicity we will always use the number 60 in the text, but it should be understood that the actual number could be different.

early 1980s to solve the above problems of the standard Big Bang theory, it was realized that inflation also solves another important problem in cosmology: it provides a source for the small density fluctuations that later through gravitational collapse are the seeds of all the structure in the universe: galaxies, clusters, etc. These small fluctuations are observed very precisely in the cosmic microwave background radiation, and so CMB observations allow us to put much more stringent observational constraints on the various inflation models than just having at least 60 e-folds of inflation. In addition, the fact that inflation can solve all these problems, including one for which it was not even initially designed, is generally considered as a very strong point in favour, and puts the bar high for any alternative theory.

The way inflation generates these fluctuations is as follows. The small-scale quantum vacuum fluctuations that appear all the time during inflation as in any other quantum field theory, and that would in a Minkowski background simply disappear again as usual, now are immediately stretched to very long wavelengths before they can disappear again, because of the enormous expansion during inflation. There is a characteristic length scale in an expanding universe, which is the Hubble length H -1 , or in comoving units (aH) -1 (H = ȧ/a being the Hubble parameter, where the overdot here, unlike in the rest of this thesis, denotes a derivative with respect to cosmic time). 3 During inflation this is also called the horizon, although that is strictly speaking not correct (it would only be the event horizon in an exact De Sitter universe). The equations for fluctuations during inflation have a very different behaviour in what is generally called the sub-horizon (or sub-Hubble) limit, where the fluctuations have wavelengths much smaller than the Hubble length, than in the opposite super-horizon limit. Where on sub-horizon scales we have oscillations, on super-horizon scales we get a growing (or constant with the proper choice of variables) mode and a decaying mode, and the latter rapidly disappears. With the disappearance of the decaying mode, the fluctuations lose their quantum nature ("squeezing") and effectively become classical fluctuations.

Later, long after inflation, when these density fluctuations re-enter the horizon, they will start growing under the influence of gravity and form the seeds of structure formation. Note that in comoving units, where the wavelength (or wave number k) of a fluctuation mode is constant, the comoving Hubble length (aH) -1 rapidly decreases during inflation (ln[(aH) -1 ] ∝ -t, with t the number of e-folds); hence the horizon exit of the fluctuations. But after inflation, during radiation (ln[(aH) -1 ] ∝ t) and matter (ln[(aH) -1 ] ∝ 1 2 t) domination, it grows, so that the fluctuations will eventually re-enter the horizon, see figure 1.1. To be precise we will now define the moment during inflation when k = aH as the time of horizon exit of that mode k, with k > aH corresponding to the sub-horizon region and k < aH to the super-horizon region. In particular the time of horizon exit of the modes that are observable in the CMB will play an important role in what follows. As these are the largest wavelengths in our observable universe, they left the horizon approximately 60 e-folds before the end of inflation. That specific time will be denoted by t * .

The proper way of treating linear fluctuations in single-field inflation (with in particular gauge issues being non-trivial) was worked out during the 1980s and culminated in the review paper by [START_REF] Mukhanov | Theory of cosmological perturbations[END_REF] in 1992. In the rest of this work we will look in particular at multiple-field inflation. Given that scalar fields are quite common in high-energy theories, for example the superpartners of fermion fields in supersymmetric theories or the scalar fields that naturally appear in Kaluza-Klein compactifications of theories with extra dimensions, it is a natural progression to extend inflation theory beyond a single field and consider the impact of having more than one scalar field.

Background equations

Having discussed the main ideas of inflation in words, we will now give the corresponding equations, as far as they will be required to understand the research that is the subject of this thesis. The following subsections are based on [Groot Nibbelink and Van Tent, 2000[START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF]; [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF], 2004]. An extensive list of references to earlier works on linear fluctuations in multiple-field inflation can be found at the beginning of section 4.1 of [ [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF]. With the exception of [START_REF] Sasaki | A General analytic formula for the spectral index of the density perturbations produced during inflation[END_REF][START_REF] Nakamura | The Spectrum of cosmological perturbations produced by a multicomponent inflaton to second order in the slow roll approximation[END_REF], those earlier works were generally limited to specific models, usually with only two fields and standard kinetic terms.

On the matter side we assume a very general multiple-field inflation model with an arbitrary number of scalar fields φ A (where A labels the different fields) and a potential W (φ A ) with arbitrary interactions. We also allow for the possibility of a non-trivial field manifold with field metric G AB . The matter Lagrangean density then is

L m = - 1 2 g µν ∂ µ φ A G AB ∂ ν φ B -W (φ A ). (1.1)
On the metric side we take as background a standard flat Friedmann-Lemaître-Robertson-Walker metric, describing a homogeneous and isotropic universe in expansion with scale factor a(t):

ds 2 = -N 2 (t) dt 2 + a 2 (t) dx 2 .
(1.2)

As any curvature will rapidly become sub-dominant during inflation (exactly the reason why inflation solves the flatness problem), taking a flat metric to describe the last 70 or so e-folds of inflation that we are interested in is a good approximation. The lapse function N (t) encodes our choice of time coordinate. The most common choices are cosmic/comoving time (N = 1), conformal time (N = a), and the number of e-folds t ≡ ln a (with N = 1/H). In most of this work we will use the number of e-folds t as time coordinate, and denote derivatives with respect to this time with overdots. 4 The Hubble parameter of the universe is denoted by H(t). 5In terms of the number of e-folds, the background field equation for φ A and the Friedmann equation for H take the following form:

D t φA + (3 -) φA + G AB W B H 2 = 0, H 2 = κ 2 W 3 - . (1.3)
Here κ 2 ≡ 8πG and the index on W denotes a derivative with respect to the fields: W B = W ,B = ∂W/∂φ B . The quantity is a short-hand notation of which the physical interpretation will be discussed in the next section. It is defined as

≡ - Ḣ H = κ 2 2 φA G AB φB ≡ κ 2 2 φ2
(1.4) (where the second equality follows from the Friedmann equation for Ḣ, which we have not given explicitly here but which is easily deduced). Here we have defined φ as the length of the vector with components φA .

In (1.3) we also encounter for the first time the covariant space-time derivative D µ (of which we indicate the temporal component as D t ): This is a derivative with respect to the space-time coordinates x µ , but it is covariant with respect to the field manifold (Γ A BC is the Christoffel symbol defined from the field metric G AB in the usual way). In fact, the quantity A ... ... is a scalar with respect to space-time but can carry any number of field indices and derivatives with respect to φ A . Explicitly for the D t φA in (1. 3) this means that it is equal to φA + Γ A BC φB φC . In the case of a trivial field metric all covariant derivatives are of course equal to normal derivatives.

D µ A A.

Slow-roll parameters and basis choice

If the potential is almost flat and the field slowly rolls down, certain terms in the equations will be small compared to others. To quantify this we can introduce a set of slow-roll parameters. It is important to keep in mind that the introduction of these parameters is not yet an approximation: the equations are still completely exact and the slow-roll parameters can be considered as just a short-hand notation. It only becomes an approximation (the slow-roll approximation) if we then say that some of these parameters are small and start neglecting certain terms. We will do that in certain later sections, but not here. The first slow-roll parameter is defined in (1.4). It will be small if the kinetic energy of the fields is small compared to their potential energy. The other slow-roll parameters are vectors in field space and can be defined as follows with n ≥ 2 (in terms of the number of e-folds):

η (n)A ≡ 1 H n φ (HD t ) n-1 H φA . (1.6)
The most important ones are for n = 2 (simply called η A ) and n = 3 (called ξ A ). For example, for η A the expression above becomes

η A = 1 φ D t φA -φA .
(1.7)

The components of η A will be small if the corresponding field component rolls slowly, with its acceleration being small compared to its velocity (see below for a more precise statement).

As we have a multi-dimensional field space, we need a basis, and a very useful orthonormal basis was first proposed in [Groot Nibbelink and Van Tent, 2000] 6 (with some minor refinements later added in appendix A of [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]). In this basis the basis vectors are not constant, but defined with respect to the field trajectory, which allows us to easily distinguish between effectively single-field effects and truly multiple-field effects. The basis is defined as follows (note that e.g. η is the vector with components η A ). The first basis vector e 1 is the unit vector in the direction of the field velocity. Next, the direction of the basis vector e 2 is given by the direction of that part of the field acceleration that is perpendicular to e 1 . This orthogonalization process is then continued with higher-order time derivatives, until a complete basis is found.

Using the η (n) defined above we can define the basis vectors via an iterative procedure as

e n ≡ η (n) -n-1 i=1 η (n) i e i η (n) n (1.8)
for n ≥ 2, with e 1 ≡ φ/ φ and η (n) i ≡ e i • η (n) , where the inner product is defined using the metric G AB . There is an arbitrariness in the choice of sign of the basis vectors, which in the original definition was fixed by choosing η (n) n to be non-negative by taking the absolute value:

η (n) n ≡ η (n) - n-1 i=1 η (n) i e i .
(old definition) (1.9)

While being a perfectly valid choice analytically, this choice does mean that certain basis vector components and slow-roll parameters make sudden sign flips when one or more fields are oscillating, and that is hard to deal with numerically. Hence we later [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] proposed a different choice for η

(n)
n , which is identical except for the overall sign, and which eliminates the sudden sign flips:

η (n) n ≡ -ε A 1 •••An e A 1 1 • • • e A n-1 n-1 η (n) An , (new definition) (1.10)
where ε is the fully antisymmetric symbol. From the fact that η = n i=1 η (1.11) so that this choice means that the basis has a definite handedness. Note that in the case where the fields do not oscillate, the two definitions have the same overall sign (hence the choice of the minus sign). To have the expressions for the time derivative of the basis vectors and of the η

(n) i e i it immediately follows that ε A 1 •••An e A 1 1 • • • e An n = -1,
(n)
n unchanged, we see that we also need the relation

ε A 1 •••An e A 1 1 • • • e A n-1
n-1 e An n+1 = 0 (1.12) to be satisfied. Then all results and expressions developed with this basis are unchanged when going from the old to the new definition.

An interesting consequence of these relations, including the orthogonality relation e m • e n = δ mn , (1.13) is that for the cases of two and of three fields we have sufficient conditions to write all basis vectors in terms of e 1 , without knowing anything about the dynamics. For two fields we have e 2 = (e 2 1 , -e 1 1 ), (1.14) with (e 1 1 ) 2 + (e 2 1 ) 2 = 1. The expressions for three fields can be found in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF].

Having defined this basis, we can now look at the components of the various vectors in this basis. For example, for the η A slow-roll parameters we define the parallel and perpendicular components as follows:

η ≡ η A e 1A , η ⊥ ≡ η A e 2A .

(1.15)

For most vectors the use of the symbol ⊥ only makes sense in the case of two-field inflation, as otherwise there would be many perpendicular directions. However, by construction η A only ever has components in the e 1 and the e 2 directions, so that the symbol η ⊥ is unambiguous.

The parameters η and η ⊥ will be small if the components of the field acceleration parallel and perpendicular to the field velocity, respectively, are small compared to the field velocity. 7 The parameter η ⊥ is quite fundamental to anything concerning multiple-field inflation: as long as it is negligible we are in an effectively single-field situation, but as soon as it becomes significant we have truly multiple-field effects. With a trivial field metric a non-zero η ⊥ automatically means a curved field trajectory. However, it should be noted that in the case of a non-trivial field metric even a straight field trajectory could have a non-zero η ⊥ due to the Γ A BC terms in its definition.

In the context of the slow-roll approximation, , η , η ⊥ are called first-order slow-roll parameters, while the components of ξ are second-order slow-roll parameters. Now one might wonder about the fact that we call η ⊥ a slow-roll parameter, given that the actual slow-roll approximation (in the spirit of a field slowly rolling along its trajectory) would only require , η and higher-order parallel slow-roll parameters to be small, and say nothing about the perpendicular parameters. However, in many cases where we need to make the slow-roll approximation in order to make analytical progress, it turns out that we also need to assume that the perpendicular parameters are small (and even χ, defined below). Hence we will call all these parameters slowroll parameters, and assume all of them to be small in the slow-roll approximation (sometimes adding the word "strong" to be explicit). It should also be noted that for the models that we have studied explicitly, it is anyway not possible to have a large η ⊥ while η stays small.

For later use we will define the following quantities: where the m indices denote the components of the basis. The A i indices on W denote multiple derivatives with respect to the fields φ A i , which in the case of a non-trivial field metric should be taken as covariant derivatives (e.g. W AB = W ,A;B = W ,AB -Γ C AB W ,C ). In order to distinguish explicit components of these two different quantities, indices like 1 and 2 will indicate components in the basis defined above (e.g. W21 ), while field indices like φ and σ (in the case of inflation with fields φ and σ) will be used to indicate components in terms of the original fields (e.g. Wσσ ). From the definition of the slow-roll parameters and using the field equation (1.3) and its derivative,

D 2 t φA + 3(1 -)D t φA -2 (3 + η ) φA + G AB W B;C φC H 2 = 0, (1.17) one can show that η = -3 -3 W1 , η ⊥ = -3 W2 , ξ = -3 W11 + 3 -3η , ξ 2 = -3 W21 -3η ⊥ ξ 3 = -3 W31 . (1.18)
We also introduce the parameter χ ≡ W22 + + η .

(1.19)

Despite its similarity to the expressions for the components of ξ, the parameter χ is a firstorder slow-roll parameter and not a second-order one. The reason is that within the slowroll approximation cancellations occur in the right-hand sides of (1.18), making the slow-roll parameters on the left-hand side one order smaller than the individual terms on the right-hand side. However, no such cancellation occurs in (1.19).

We can compute the time derivatives of the basis vectors and find:

D t e i = η (i+1) i+1 η (i) i e i+1 - η (i) i η (i-1) i-1 e i-1
(1.20)

(where the second term should be omitted in the case i = 1). Explicitly for the first two basis vectors this means:

D t e 1 = η ⊥ e 2 , D t e 2 = ξ 3 η ⊥ e 3 -η ⊥ e 1 .
(1.21)

For the derivatives of the slow-roll parameters we get:

˙ = 2 ( + η ), η = ξ + (η ⊥ ) 2 + ( -η )η , η⊥ = ξ 2 + ( -2η )η ⊥ , χ = η + 2 χ -(η ) 2 + 3(η ⊥ ) 2 + ξ + 2 3 η ⊥ ξ 2 + W221 + 2 ξ 3 η ⊥ W32 , ξ = -3 W111 + 2η ⊥ ξ 2 + (2 -3)ξ + 9 η + 3(η ) 2 + 3(η ⊥ ) 2 , ξ2 = -3 W211 -η ⊥ ξ + (2 -3)ξ 2 + 9 η ⊥ + 6η η ⊥ -3η ⊥ χ + (ξ 3 ) 2 η ⊥ , ξ3 = -3 W311 + (2 -3)ξ 3 -3η ⊥ W32 - ξ 2 ξ 3 η ⊥ , (1.22)
or more generally D t η (n) = η (n+1) + ((n -1)η )η (n) .

(1.23)

Linear perturbations

As mentioned at the beginning of the section, it is in particular the fact that inflation also explains the origin of the cosmological fluctuations, for which it was not explicitly constructed, that has made the theory so convincing. In addition it is these fluctuations that allow inflation models to be observationally constrained. So we will now add small inhomogeneous fluctuations to the homogeneous background. As observations tell us these fluctuations were very small initially, linearizing all equations is a good first approximation, and the resulting equations and solutions will be discussed in this subsection (as this is only an introduction and not the main subject of this thesis, only a summary is presented, for details see [Groot Nibbelink andVan Tent, 2000, 2002; [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF], 2004] as in the previous subsections, or [START_REF] Mukhanov | Theory of cosmological perturbations[END_REF] for the single-field case). Of course, with CMB observations having become as precise as they are with WMAP and Planck, we can now put constraints on the even smaller second-order perturbations. That is the main topic of this thesis and will be treated in the next chapters.

In this subsection only we will not use the number of e-folds as time coordinate, but conformal time τ defined by a lapse function N = a. In terms of conformal time the scale factor is an overall multiplicative factor of the background metric (i.e. the metric is a conformal transformation of the Minkowski metric, hence the name). It turns out that in particular the sub-horizon equations will be easier to solve in terms of this time coordinate. We denote a conformal time derivative by a prime.

The scalar fields are perturbed as follows:

φ A full (τ, x) = φ A (τ ) + δφ A (τ, x) (1.24)
with δφ φ. For the metric we write:

g full µν (τ, x) = a 2 -1 0 0 δ ij + a 2 -2Φ B ,j B ,i 2(E ,ij -Ψδ ij ) + a 2 0 S j S i F i,j + F j,i + a 2 0 0 0 h ij . (1.25)
Here the first term is the background metric, the second contains the scalar perturbations encoded in four scalar functions Φ, Ψ, E, B, the third contains the vector perturbations encoded in two divergenceless vectors S, F, and the last term contains the tensor perturbations given by the symmetric transverse traceless tensor h ij . Two of the scalar and two of the vector degrees of freedom are gauge degrees of freedom related to the choice of coordinates. It is not hard to construct combinations of the scalar degrees of freedom (including the scalar field perturbation δφ A ) that are invariant under coordinate transformations, and similarly for the vector degrees of freedom (the tensor h ij is already gauge-invariant). It turns out that working with these gauge-invariant combinations is equivalent to choosing the longitudinal (B = E = 0) and vector (F i = 0) gauges, so that is what we will do. Up to linear order the scalar, vector and tensor perturbations decouple. We will not consider vector perturbations in this thesis, as they are absent if there is no vector matter source, and we will only consider scalar fields as matter sources. Moreover, even if vector perturbations were sourced at some point, they decay afterwards. Tensor perturbations (gravitational waves), on the other hand, are always present, even in the absence of tensor matter sources, as the two tensor degrees of freedom represent the two physical degrees of freedom of the metric and are hence directly sourced by metric quantum fluctuations. However, there is no difference between the tensor perturbations in single-field and in multiple-field inflation, and so they are not so interesting from our point of view, and we will rarely mention them. Our main interest will be the scalar perturbations.

By plugging the above expressions for the field and the metric into the Einstein equations, we obtain equations of motion for the various quantities. A first result (that follows from the off-diagonal ij part of the Einstein equation) is that Φ = Ψ. Hence there is in the end only one metric scalar degree of freedom. And even that is not physical, but is only present because it is sourced by one of the physical scalar degrees of freedom of the inflaton field (the e 1 component of the vector δφ A to be precise). Hence it makes sense to combine the metric and matter scalar quantities into a single quantity, and it turns out that with the following combination the equations of motion simplify significantly:

q A ≡ a δφ A + Ψφ H e A 1 , (1.26)
where H ≡ a /a = aH. This combination is gauge-invariant (even if δφ A and Ψ would not have been made gauge-invariant individually). It is the multiple-field generalisation of the singlefield Sasaki-Mukhanov variable. It satisfies the following equation of motion (after switching to spatial Fourier modes q A k (τ )):

D 2 τ q A k + k 2 + H 2 Ω A B q B k = 0 (1.27)
with the "mass matrix" (1.28) where as before W A B should be read as a covariant derivative: W ,A ;B = G AC (W ,C -Γ D BC W ,D ), and R A CDB is the curvature tensor of the field manifold. In the sub-horizon limit, where k 2 H 2 , the second term between the parentheses of (1.27) can be neglected, and the equation looks like a simple harmonic oscillator. Moreover, at the level of the Lagrangean we also find the proper normalization factor for a harmonic oscillator. The covariant derivative can be dealt with by rewriting q A as a vector in the special basis defined in the previous subsection, q m = e m A q A . The kinetic term D τ q A k G AB D τ q B k in the Lagrangean then becomes (q m k + HZ mn q n k )(q m k + HZ mp q p k ). Here the anti-symmetric matrix Z mn is defined as Z mn ≡ e m A D t e A n = e m A D τ e A n /H, which means in particular that Z 21 = -Z 12 = η ⊥ . Hence we know how to quantize q A , and the initial conditions that would have been undetermined in a classical setting are now almost completely determined by 1) imposing the standard commutation relation between q A and its canonical momentum, and 2) choosing the vacuum as the minimum of the inflationary Hamiltonian. The resulting sub-horizon solution is:

Ω A B ≡ W A B H 2 - 2 κ 2 R A CDB e C 1 e D 1 -(2 -)δ A B -2 (3 + )e A 1 e 1 B + e A 1 η B + η A e 1 B ,
q m k (τ ) = 1 √ 2k U mn (τ )a † n k + h.c. (1.29)
where a † n k and a n k are the standard quantum creation and annihilation operators and h.c. denotes the Hermitian conjugate. The matrix U mn is a unitary matrix that contains the time dependence exp(-ikτ ), a time-dependent rotation matrix due to the Z mn terms in the Lagrangean, and an undetermined constant unitary matrix that is the only part that would still need to be determined from the initial conditions. However, its explicit form is unimportant, as it will drop out from all observables computed from q m .

In the super-horizon region (k 2 H 2 ) it is the k 2 term in (1.27) that can be neglected. It turns out that instead of oscillating solutions we have a growing and a (very rapidly) decaying solution here. In the following we will always neglect the decaying mode on super-horizon scales (although it must be taken into account for a proper matching to the sub-horizon solution). In between the sub-horizon and the super-horizon regions there is a transition region where k ∼ H. If we take this region small enough, we can assume the slow-roll parameters to be constant and then we can solve the differential equation with constant coefficients exactly in terms of a Hankel function (of a matrix-valued order). Matching this solution to the sub-horizon solution is simple using the asymptotic expansion of the Hankel function. In [Groot Nibbelink and [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF] this was worked out in detail up to next-to-leading order in slow roll, and it was matched to the solution in the super-horizon region to finally obtain the complete solution at the end of inflation. However, in the next chapter in this thesis another way of obtaining the super-horizon solution that also works beyond linear order will be detailed, and the only input that it requires is the linear leading-order slow-roll solution at (or rather slightly after) horizon crossing. Hence in this introductory section we will only give that expression (for the growing mode): .30) where all unitary factors that have no impact on the observables have been omitted.

q m k = aH * √ 2k 3 a † m k + h.c. ( 1
In the super-horizon region it is more convenient to work with a different variable than q A , namely ζ A defined as

ζ A = - κ a √ 2 q A . (1.31)
It is the multiple-field generalisation of the single-field curvature perturbation 8 , and it is just a simple rescaling of q A . When taking components of ζ A in the basis defined in the previous subsection, the ζ 1 component is called the adiabatic (or curvature) mode, which is the only one present in single-field inflation, while the ζ m with m ≥ 2 are called isocurvature (or entropy) modes. The latter can only exist in the case of inflation with multiple fields. The main convenience of ζ A is that its adiabatic component satisfies a very simple equation on super-horizon scales after the decaying solution has disappeared:

ζ 1 = 2Hη ⊥ ζ 2 .
(1.32)

From this equation we can read off the well-known result that in single-field inflation the adiabatic mode ζ 1 is constant on super-horizon scales (after the decaying solution has disappeared).

In multiple-field inflation, on the other hand, ζ 1 can evolve on super-horizon scales, being sourced by ζ 2 (and only ζ 2 , independent of how many isocurvature modes there are). The equations for the isocurvature modes ζ m with m ≥ 2 are more complicated, but have the property that they are independent of the adiabatic mode. To conclude we rewrite the linear leading-order slow-roll solution at horizon crossing (1.30) in terms of ζ A , which will be an input for the equations in the next chapter: .33) where by convention we have kept the minus sign, even though other unitary factors were omitted as they have no impact on any observables. We conclude this section with a few more remarks regarding the isocurvature modes. In the universe after inflation, the total isocurvature mode is generally defined in terms of the total pressure and the total energy density and their perturbations as (see e.g. [ [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF], 2004])

ζ m k = - κH * 2 √ k 3 * a † m k + h.c. ( 1
S = 1 4 δp -c 2 s δρ p -c 2 s ρ , (1.34)
where c 2 s is the sound speed squared, defined as c 2 s = p /ρ . The total isocurvature mode is the specific combination of all the isocurvature modes that is the source term for the adiabatic mode. In the case of only photons (with p γ = ρ γ /3) and cold dark matter (with p c = 0), it reduces to .35) Alternatively this can be written in terms of number densities and their perturbations as S = δn c /n cδn γ /n γ , showing that isocurvature perturbations involve relative perturbations in particle number densities between species (while the adiabatic mode can be viewed as perturbations in the total energy density or total particle number density, although this statement, unlike the one for the isocurvature perturbations, is gauge-dependent). It is clear that this S must be related to the quantity ζ 2 defined above. It turns out that the relation is (1.36) where we have expressed everything in terms of slow-roll parameters to make the identification during inflation simpler (alternatively, /( + η ) can be written as (p + ρ)/(pc 2 s ρ)).

S = δρ c ρ c - 3 4 δρ γ ρ γ . ( 1
S = - 1 2 + η η ⊥ ζ 2 ,

The cosmic microwave background radiation 1.2.1 Introduction

Initially the universe was hot enough for protons and electrons, after they had formed, to remain free (ionized). As free electrons scatter photons very efficiently (Thomson scattering), photons had a very short mean free path in the early universe. However, when the temperature in the universe decreased to about 3000 K, which happened about 380 000 years after the Big Bang, protons and electrons combined into neutral hydrogen atoms, which is called recombination.9 Suddenly the universe became effectively transparent to photons. Looking out into the universe, and hence back in time because of the finite speed of light, we will hit this surface in time before which the universe was opaque. Hence it seems as if we are surrounded by a spherical surface in space beyond which we cannot look, and which radiates as a black body. That surface is called the last-scattering surface, and the radiation is the cosmic microwave background radiation (CMB), where "microwave" reflects the fact that through the expansion of the universe those photons have nowadays a temperature of only 2.725 K, which corresponds to microwaves. As illustrated quite nicely by figure 1.2 from the COBE satellite, at a first glance the CMB appears to be completely isotropic. This is actually one of the motivations for inflation, as in a pure radiation/matter-dominated universe the CMB would come from regions that have never been in causal contact, and so it would be very unlikely for the CMB to be this isotropic (this is called the horizon problem). Looking more closely, we see that at the level of 10 -3 K there is a dipole component, caused by the movement of the Earth with respect to the Hubble flow. Finally, at the level of 10 -5 K we see fluctuations in all multipoles (in terms of a spherical harmonic decomposition). Of course the latter are now measured at much higher resolution by the Planck satellite, see figure 1.3. These are the fluctuations that were presumably generated by inflation, as explained in the previous section, and that at later times formed the seeds of structure formation by gravitational collapse. Because these fluctuations are so small, perturbation theory works very well, which is one reason why the analysis of CMB data is simpler than the analysis of large-scale structure data. Linear perturbation theory is a very good approximation, but because of the high precision of current CMB data we even have access to second-order corrections, the so-called non-Gaussianities that will be at the centre of this thesis.

As will be explained in the next section, the primordial fluctuation power spectrum in terms of ζ has a very simple, almost flat, shape. For fluctuations that were still super-horizon at recombination, this translates into a flat CMB temperature power spectrum as well, as there is no evolution on super-horizon scales in the standard picture. For fluctuations that had already re-entered the horizon before recombination, and started to evolve again, the situation is more complicated. Before recombination the universe was filled with a plasma of tightly coupled photons and baryons, which evolves in a landscape of gravitational wells and hills as described by the fluctuations of ζ. The competition between gravity pulling the plasma into the gravitational wells and the radiation pressure of the photons pushing the plasma out of the wells then creates acoustic oscillations in the plasma. In the CMB we see a snapshot of those oscillations at the time of recombination. Certain specific wavelengths will have been right at a maximum or a minimum of the oscillation. Those correspond to peaks in the power spectrum (as the power spectrum is roughly speaking the square of the fluctuations, both a maximum (compression) and a minimum (rarefaction) correspond to a peak). Other wavelengths will have been close to the mean value of the oscillation, which correspond to the troughs of the power spectrum. The position and relative height of the peaks is quite sensitive to the various cosmological parameters (like the amount of dark matter and baryons in the universe, and the fact if the universe is open, closed or flat). This is the reason why the CMB is such a gold mine for precision cosmology.

So in the CMB temperature power spectrum (see figure 1.6 in the next section) we first see an approximately flat region (called the Sachs-Wolfe plateau) at the smallest multipoles, corresponding to the largest scales, which were still outside the horizon at recombination. Because of late-time effects of the dark energy on the gravitational potential through which the CMB photons travel towards us, this plateau has been slightly tilted (integrated Sachs-Wolfe effect, ISW). Then, at intermediate multipoles we see a series of peaks, caused by the acoustic oscillations of those scales that re-entered the horizon during the plasma era before recombination. Finally, at the largest multipoles, we see an exponential decay, called Silk damping. This is a combination of two effects that smear out the fluctuations on the smallest scales: the fact that recombination was not instantaneous, so that the last scattering surface has a finite thickness, and the fact that before recombination the mean free path of the photons was not exactly zero, so that they could diffuse out of gravitational wells (sometimes in the literature only the latter effect is called Silk damping).

The CMB is also polarized, see figure 1.4, and its polarization provides us with additional information. Thomson scattering of an electromagnetic wave off an electron creates linear polarization perpendicular to both the incoming and outgoing wave vectors. If the incoming radiation is isotropic, then a second, perpendicular, incoming wave will provide the other linear polarization direction, so that the outgoing wave is again unpolarized. However, if the incoming radiation is anisotropic, in particular if it has a quadrupole moment, then the two linear polarization directions will not have the same amplitude, so that a net linear polarization remains for the outgoing wave, see figure 1.5. This is how the temperature anisotropies at the lastscattering surface lead to linear polarization of the CMB. Note that it is only the last scattering that is relevant, the effect from earlier scattering will be averaged out by consecutive scatterings. Normally linear polarization is described by the Q and U Stokes parameters. As these are not invariant under rotations, in CMB physics we often use an alternative description in terms of E (gradient) and B (curl) polarization, which are invariant under rotations at the price of being non-local (see next section for precise definitions). Another advantage of the description in terms of E and B is that scalar perturbations can only create E-polarization, not B. Hence the primordial B-polarization signal will be a clear probe of the inflationary tensor perturbations. In the end the CMB provides us with in principle four power spectra: the temperature power spectrum T T , the E-polarization power spectrum EE, the B-polarization power spectrum BB, and the T E cross spectrum (because of parity conditions the T B and EB cross spectra are [START_REF] Hu | A CMB polarization primer[END_REF]]. zero). The Planck satellite has given us very precise measurements of the T T , T E, and EE spectra. A measurement of the BB spectrum is still waiting for a future mission.

Finally, let us mention that in addition to the CMB anisotropies that we consider here, there exist also CMB spectral distortions. CMB anisotropies assume that the CMB spectrum is a perfect black-body spectrum, just with a slightly different temperature in each direction. Spectral distortions on the other hand are deviations from a black-body spectrum. The COBE satellite confirmed that the CMB spectrum is a black-body spectrum to high precision. However, theory does predict deviations at some very small level. As no CMB mission since COBE has had a spectrometer on board, and no proposed CMB spectrometer mission has been accepted by the various space agencies to this date, there are currently no high-precision CMB spectral distortion data, and we will not consider spectral distortions in this thesis.

Power spectrum

As the inflationary fluctuations are quantum fluctuations, they are random. Hence we cannot predict the precise distribution of temperature fluctuations (or galaxies) on the sky. What inflation does predict are statistical properties of that distribution, in particular its various correlation functions. The most important of those is the two-point correlation function, or its Fourier transform, the power spectrum. If we look at the CMB on the spherical last scattering surface, the power spectrum is instead the spherical harmonic transform of the angular twopoint correlation function. If the distribution were Gaussian, then all information would in fact be contained in the power spectrum, with all odd-point correlation functions being zero while all even-point correlations functions could be expressed in terms of the power spectrum. The distribution would be Gaussian if the generation and evolution of the fluctuations were linear, and as we have said before, even though that is not true, it is a good approximation because the fluctuations are so small.

The power spectrum P ζ (k) of the linear adiabatic curvature perturbation ζ 1 is defined such that 0|ζ 1 (x)ζ 1 (y)|0 = dk k sin(k|x -y|) k|x -y| P ζ (k), (1.37) or alternatively such that 10

0|ζ 1 (k 1 )ζ 1 (k 2 )|0 = 2π 2 k 3 1 δ 3 (k 1 -k 2 )P ζ (k 1 ).
(1.38)

Inserting the solution (1.33) gives us the following power spectrum on super-horizon scales:

P ζ (k) = κ 2 H 2 * 8π 2 * .
(1.39) Note however that (1.33) was only the solution for ζ at horizon-crossing. Hence this expression for the power spectrum is only correct if ζ 1 did not further evolve on super-horizon scales. This would be the case in single-field inflation, but not in multiple-field inflation, where a correction term has to be added. This will be worked out in section 2.2.2. The dependence on k of P ζ (k) only comes via the dependence of H and on t * , the time of horizon crossing of the mode k defined by k = aH. As both H and evolve slowly in slow-roll inflation, this is a weak dependence. Because of this weak dependence we can expand P ζ (k) around a pivot scale k 0 as

P ζ (k) = A s (k/k 0 ) ns-1 (1.40)
with the amplitude

A s ≡ P ζ (k 0 ) = κ 2 H 2 0 8π 2 0 (1.41)
and spectral index

n s -1 ≡ ∂ ln P ζ ∂ ln k k=k 0 = -4 0 -2η 0 (1.42)
(the subscript s refers to scalar, as this is the power spectrum of the scalar perturbations). The final equality for n s can be found using the definition of and its derivative (1.22). Again, it must be stressed that these expressions are only valid for single-field inflation, the corrections in the multiple-field case will be treated in section 2.2.2. So in this way we have reduced the power spectrum from a full function of k to just two numbers (which can be extended with higher-order derivatives at the pivot scale, the first of which is called the running of the spectral index).

In addition to scalar fluctuations, inflation also predicts the presence of tensor fluctuations. The derivation of the tensor power spectrum is very similar to, but simpler than, the one of the scalar power spectrum, and we will only give the final result here:

P t (k) = 2κ 2 H 2 * π 2 . (1.43)
This result is independent of the number of scalar fields, and hence the same for single-field and multiple-field inflation. As for the scalar power spectrum we can expand it around a pivot scale and express it in terms of just two numbers, the tensor amplitude and the tensor spectral index:

A t = 2κ 2 H 2 0 π 2 , n t = -2 0 (1.44)
(for historical reasons the tensor spectral index is defined without the -1 that the scalar spectral index has). The main difference between the scalar and the tensor amplitudes is that A t only 10 Various definitions of the power spectrum circulate in the literature. Instead of this so-called dimensionless power spectrum, people also consider the so-called dimensionful power spectrum, defined as in (1.38) but without the factor 2π 2 /k 3 . In that case for example equation (1.52) would have a factor 2k 2 /π instead of 4π/k. In addition, the chosen normalization of Fourier transforms also plays a role. We have chosen to define both the Fourier transform and its inverse with a factor (2π) -3/2 . If instead one takes the Fourier transform to have nothing and the inverse transform to have (2π) -3 , then an additional factor (2π) 3 must be introduced on the right-hand side of (1.38), without any impact on equation (1.52) however.

depends on H 0 and not on 0 , which means that a measurement of A t would directly give the inflationary energy scale. Instead of A t the tensor-to-scalar ratio r is more commonly used:

r ≡ A t A s = 16 0 = -8n t .
(1.45)

The last two equalities are only valid in the single-field case (as A s changes in the multiple-field case). The last equality r = -8n t is called the single-field consistency relation. As A s is larger in multiple-field inflation, it becomes an inequality r ≤ -8n t in the more general case.

To convert these scalar and tensor primordial power spectra into the CMB power spectrum, we first need to properly introduce the temperature fluctuations:

∆T (θ, ϕ) T 0 ≡ T (θ, ϕ) -T 0 T 0 = ∞ =2 + m=-
a T m Y m (θ, ϕ).

(1.46)

Here T 0 = 2.725 K is the average temperature of the CMB, the Y m are the spherical harmonics, and a T m are the temperature mode coefficients that encode the properties of the temperature fluctuations. The division by T 0 is not always performed, in which case the a m would have the dimension of temperature instead of being dimensionless. By definition temperature fluctuations do not have a monopole ( = 0) component. The reason we also do not consider the dipole ( = 1) component, is because it is too contaminated by the much larger dipole due to the Earth's movement, as discussed in the previous section.

To properly define the polarization fluctuations, let us consider for a moment a single monochromatic electromagnetic plane wave travelling in a given direction, which without loss of generality we will take to be the z direction. It has the following electric field: -kz) .

E(t, x) = a 1 e iθ 1 a 2 e iθ 2 0 e i(ωt
(1.47)

Instead of a 1 , a 2 , θ 1 , θ 2 we can also use the 4 Stokes parameters: .48) which are not independent but satisfy the identity

I ≡ a 2 1 + a 2 2 , Q ≡ a 2 1 -a 2 2 , U ≡ 2a 1 a 2 cos(θ 2 -θ 1 ), V ≡ 2a 1 a 2 sin(θ 2 -θ 1 ), ( 1
I 2 = Q 2 + U 2 + V 2
(reflecting the fact that the overall phase of the wave is unimportant, so that there were only 3 independent quantities: a 1 , a 2 and θ 2θ 1 ). I gives the total intensity, Q the horizontal/vertical linear polarization, U the ±45 • linear polarization, and V the right/left-handed circular polarization. Note that the choice of sign of V is related to the usual ambiguity in the circular polarization convention, but as circular polarization is not produced by Thomson scattering and is absent in the CMB, we do not need to worry about that here and can forget about V . Because of the identity

I 2 = Q 2 + U 2 + V 2
, it is clear that a single wave always has some polarization state, it cannot be unpolarized (Q = U = V = 0). When we mention unpolarized radiation, we are talking about a superposition of multiple waves, and the averaged values of the Stokes parameters.

As explained in the previous section, instead of using Q and U to describe the CMB's polarization, CMB physicists often prefer using E and B defined by

Q(θ, ϕ) ± iU (θ, ϕ) T 0 = - ,m (a E m ± ia B m ) ±2 Y m (θ, ϕ) (1.49)
with ±2 Y m spin-weighted spherical harmonics of spin ±2. As for temperature fluctuations, the division by T 0 is performed to create dimensionless a m coefficients, which is not always done.

11 The E and B fields are then defined from their a m coefficients using normal spherical harmonics:

E(θ, ϕ) T 0 = ,m a E m Y m (θ, ϕ) (1.50)
and similarly for B. The publicly available HEALPix code 12 [START_REF] Gorski | HEALPix -A Framework for high resolution discretization, and fast analysis of data distributed on the sphere[END_REF] can be used to compute all these types of spherical harmonic transforms, and perform other useful operations on a pixelized sky as well as display the resulting maps.

The power spectra of the CMB are defined by: (1.51) where p 1 , p 2 = T, E, B. The relation between the power spectrum of ζ and of the CMB is:

a p 1 m a p 2 * m = C p 1 p 2 δ δ mm ,
C p 1 p 2 = 4π dk k P ζ (k)g p 1 (k)g p 2 (k).
(1.52)

In this case p 1 , p 2 = T, E only, as the scalar perturbations do not contribute to B polarization.

Here both the evolution effects (like the acoustic peaks) and the projection effects (going from 3D Euclidean space to a 2D spherical surface) are encoded in the radiation transfer functions g p (k). These functions are the solution of complicated systems of Boltzmann equations describing the interactions between all types of particles, and must be determined numerically. Fortunately public codes to compute them exist. We have always used CAMB 13 [START_REF] Lewis | Efficient computation of CMB anisotropies in closed FRW models[END_REF]] in our research. The contribution of the tensor perturbations to the different CMB power spectra is given by a similar expression as (1.52), just with the tensor power spectrum P t instead of P ζ , and with different radiation transfer functions.

To determine the power spectrum observationally via its definition (1.51) would require performing an ensemble average over a large number of skies. Obviously we only have one sky that we can measure. The only averaging that we can do to determine an estimate Ĉ of the real C (assuming ergodicity) is over the m:

Ĉ = 1 2 + 1 + m=- |a m | 2 .
(1.53)

As the number of m per is limited, especially for low , this leads to a fundamental statistical error on our determination of C , called cosmic variance, that would be present even in the case of an ideal experiment with infinite resolution and no noise. It is given by

Var(C ) = 2 2 + 1 C 2 .
(1.54)

The T T , T E, and EE CMB power spectra (as well as the lensing potential power spectrum) as measured by the Planck satellite are shown in figure 1.6. The red dots with error bars are the Planck measurements, with the error bars including the contribution from cosmic variance explained above. In fact, the uncertainties of the T T spectrum are dominated by cosmic variance, rather than by noise or foreground residuals, at all scales below about = 1800 -a scale at which the CMB information is essentially exhausted within the framework of the ΛCDM model. 14 The T E spectrum is about as constraining as the T T one, while the EE spectrum i.e. the time-averaged flux), and that is their usual dimension (and how they are measured in practice). In addition they are sometimes measured as flux per unit solid angle (radiance) or as flux per unit solid angle and per unit frequency/wavelength (spectral radiance), and the latter is the case for CMB experiments. Finally, to convert this measured spectral radiance into a temperature, which is the dimension of the I, Q, and U maps produced by CMB experiments, Planck's law giving the spectral radiance of a black body is used, Bν = (2hν 3 /c 2 )[exp(hν/(kBT )) -1] -1 . However, this relation is linearized for small temperature fluctuations around the average temperature T0, leading to the linear relation (with constant of proportionality depending on T0) used to convert spectral radiance into temperature.

12 http://healpix.sourceforge.net/ 13 http://camb.info/ 14 The ΛCDM model is the currently widely accepted model of the standard Big Bang cosmology, because it is the simplest model that is in agreement with all data. It is named after its two most important (but also most mysterious) componenents, the dark energy or cosmological constant indicated by Λ and the cold dark matter (CDM), but also includes ordinary matter and radiation. , the E mode of polarization (bottom left) and the lensing potential (bottom right, which we do not discuss in this thesis). The quantity D is defined as ( + 1)T 2 0 C /(2π). The red dots with error bars (that include cosmic variance) indicate the measurements, while the blue curve is the best-fitting ΛCDM model. The anisotropy power spectra use a standard binning scheme (which changes abruptly at = 30) and are plotted here with a multipole axis that goes smoothly from logarithmic at low to linear at high . Figure from [Planck 2018 I].

still has a sizeable contribution from noise, and the potential to be improved by a future CMB mission. The blue line is the best-fit ΛCDM model, which is characterized by only 6 parameters: two are the scalar amplitude and spectral index A s and n s defined above, while the other four are the the baryon and cold dark matter densities, the size of the sound horizon at last scattering, and the amount of reionization of the hydrogen gas in the universe due to the emergence of the first stars (long after recombination and the creation of the CMB). Other parameters, like the tensor-to-scalar ratio r defined above or the curvature of the universe, are compatible with zero and hence not required to describe the basic ΛCDM model. Yet other parameters, like the age of the universe and the dark energy density, are not independent and can be derived from the other six. The best-fit values of all these parameters can be found in [Planck 2018 VI]; here we give A s = (2.100 ± 0.030) × 10 -9 and n s = 0.9649 ± 0.0042.

Everything we discussed in this section is based on the CMB power spectrum, which would contain all information if the fluctuations were Gaussian. However, even though the fluctuations are small and linearizing all equations is a good first approximation, it is not exact. Gravity is inherently non-linear, and most inflation models also introduce non-linearities. Hence non-Gaussianity will always be present at some level, the question is just if it is observable or not. Once we admit that the perturbations are non-Gaussian, there is in principle information in all correlation functions. However, given that the a m are small, it makes sense that generically the next correlation function, the three-point correlation function, will be where non-Gaussianity can most easily be measured and hence will give the tightest constraints if it is not detected, as each successive correlation function will be smaller and smaller. Hence we focus in this thesis exclusively on the three-point correlator, the Fourier or spherical harmonic transform of which is called the bispectrum. 15 Just like we introduced the amplitude A s of the power spectrum, we will also introduce a bispectrum amplitude parameter called f NL . For precise definitions of both the bispectrum and f NL we refer to section 2.2.2.

The Planck mission

After the discovery of the CMB by Penzias and Wilson in 1964(paper [Penzias and Wilson, 1965] published in 1965;Nobel Prize in 1978), its temperature anisotropies were first detected by NASA's COsmic Background Explorer (COBE) satellite 16 . COBE was launched in 1989 and operated until 1993. It consisted of three instruments: FIRAS (Far-InfraRed Absolute Spectrophotometer) to measure the spectrum of the CMB, which to this date remain the most accurate measurements of the CMB black-body spectrum and its temperature, DIRBE (Diffuse InfraRed Background Experiment) to map dust emission, and DMR (Differential Microwave Radiometer) to map the temperature anisotropies of the CMB at three frequencies: 31.5, 53, and 90 GHz. The detection of these anisotropies by DMR was first announced in 1992 [COBE Collaboration, 1992], with further publications based on more data following in the years after. The full four-year data was published in 1996 [COBE Collaboration, 1996]. COBE's FIRAS and DMR principal investigators, John Mather and George Smoot, were awarded the Nobel Prize in 2006.

COBE was followed by NASA's Wilkinson Microwave Anisotropy Probe (WMAP) satellite17 that was launched in 2001 and operated until 2010. Its instrument was similar to COBE DMR, but with 45 times higher sensitivity and 33 times higher angular resolution, and with five frequency channels, at 23, 33, 41, 61, and 94 GHz. First-year results were published in 2003 [WMAP Collaboration, 2003a] and are often credited with starting the era of highprecision cosmology. More releases followed, with the final nine-year results published in 2012 [WMAP Collaboration, 2013]. WMAP also provided the first maps of the E-mode polarization anisotropies of the CMB (the first detection of CMB polarization anisotropies had been made by the DASI (Degree Angular Scale Interferometer) telescope at the South Pole in 2002 [START_REF] Kovac | Detection of polarization in the cosmic microwave background using DASI[END_REF]). As WMAP, unlike COBE, was passively cooled, the experiment had no intrinsic restrictions on its lifetime, and it continued to run until 2010, when the launch of the Planck satellite made it obsolete.

The third, and so far last18 , generation of CMB satellites is ESA's Planck satellite19 . It was launched in 2009 and operated until 2013. It contained two instruments: LFI (Low Frequency Instrument), based on radiometers like COBE and WMAP and with three frequency channels (30,44,and 70 GHz), and HFI (High Frequency Instrument), with bolometric detectors and six frequency channels (100,143,217,353,545,and 857 GHz). Planck also had approximately 10 times higher sensitivity and 2.5 times higher resolution than WMAP. Beam size decreases with frequency, and the beam size of Planck's 217 GHz channel is 4.9 arcmin (as compared to 13 arcmin for WMAP's 94 GHz channel). Planck has the highest sensitivity in its 143 and 217 GHz channels, at 0.55 and 0.78 µK CMB deg, respectively [Planck 2018 I]. A visual comparison regarding resolution between COBE, WMAP and Planck can be found in figure 1.7.

Its broad frequency coverage gave Planck an unprecedented capability of component separation, i.e. using the frequency dependence of various astrophysical foregrounds to identify them and remove them from the CMB signal. Roughly speaking, 100, 143, and 217 GHz are the main CMB channels, with the LFI channels serving to determine the low-frequency foregrounds (most importantly galactic synchrotron radiation) and the three highest HFI channels serving to determine the high-frequency foregrounds (mainly galactic thermal dust radiation). The actual four component separation methods used in the Planck analyses (Commander, Nilc, Sevem, and Smica, see [Planck 2018 IV] for details and references) are more sophisticated than this simple description implies though. The Planck collaboration was much larger than the COBE and WMAP ones, which allowed Planck to use multiple independent pipelines for the various aspects of its analyses, like the four component separation methods mentioned above, for extra robustness of its final results. As will be discussed later in this thesis, there were also four different bispectrum estimator pipelines (based on three distinct estimators).

HFI required active cooling down to 0.1 K using the dilution of 3 He into 4 He. After 2.5 years (5 full-sky scans) the 3 He supply was exhausted (it lasted almost twice as long as the nominal lifetime), leading to the end of HFI for scientific purposes. LFI on the other hand only required passive cooling down to 20 K, and continued functioning for another 1.5 years (3 additional full-sky scans) until it was decided to switch the experiment off in 2013. There were three data releases: the first in 2013 [Planck 2013 I], based on one year of data and temperature-only, the second in 2015 [Planck 2015 I], which included all data for both temperature and polarization (with the exception of < ∼ 40 polarization data which were still considered to be insufficiently cleaned from systematic effects at that time), and a final legacy release in 2018 [Planck 2018 I] containing everything and with an improved treatment of in particular polarization data. Each release was accompanied by a suite of papers discussing all primary science, while so-called intermediate papers on other, mostly astrophysical, topics were released in between.20 Unlike the first two releases, which were contractual obligations with respect to ESA, the third release was an initiative of the principal investigators and hence not as tightly constrained. Because of this, as well as the reduced manpower, since by this time many people had left the collaboration to move on to other projects, the primary science papers of the final release were not all published on the same date, but spread out over a year, from mid-2018 till mid-2019.

The work in Planck was organised in so-called working groups, each of which was in charge of a specific part of the analysis, often associated with one of the main publications. For the non-Gaussianity working group, most relevant for the work in this thesis, we required several products produced by other working groups (most of which had their own prerequisites). In the first place we needed of course the cleaned CMB sky maps, produced by the component separation working group from the raw sky maps of the different frequency channels (at an effective 5 arcmin resolution). In order to compute error bars, the linear correction term (see chapter 3), and for data validation purposes, we also required simulations. The so-called FFP (Full Focal Plane) simulations were produced several times (as indicated by different version numbers) over the course of the Planck data analysis years, including more and more effects and hence becoming more and more realistic over the years. The final version, called FFP10 [Planck 2018 II;Planck 2018 III;Planck 2018 IV] and used in the 2018 data analysis, consisted of a set of CMB-only map realisations (including the effects of gravitational lensing, satellite scanning, and beam asymmetries), and a set of noise-plus-systematics realisations (of which the input also included a fixed CMB and foreground realisation, which was subtracted at the end, so that any sky-signal distortion effects are included as well). These were then passed through the component separation pipelines in the same way as the real sky map. The required beam transfer functions and confidence sky masks (for temperature and polarization) were also provided by the component separation working group. From the simulations we could determine the CMB power spectrum and noise power spectrum to be used in the estimator weights. Finally we required the values of various cosmological parameters to determine the theoretical bispectrum templates, which were provided by the power spectrum likelihood and parameters working group. Conversely, the results produced by the non-Gaussianity working group were used to test for example the quality of the component separation products. This illustrates the important interactivity between the different working groups in the Planck collaboration.

Apart from space missions, there have also been many ground-based and balloon-borne CMB experiments. The DASI telescope that first measured the CMB E-polarization was mentioned above. In 2000 the BOOMERanG (Balloon Observations Of Millimetric Extragalactic Radiation ANd Geophysics) and MAXIMA (Millimeter Anisotropy eXperiment IMaging Array) balloonborne experiments were the first to detect the first peak in the temperature power spectrum [Boomerang Collaboration, 2000;[START_REF] Hanany | MAXIMA-1: A Measurement of the cosmic microwave background anisotropy on angular scales of 10 arcminutes to 5 degrees[END_REF]. Most ground-based CMB experiments are either located in the Atacama Desert in Chile or at the South Pole, due to the good atmospheric conditions (very low humidity) in those places. With no new CMB satellite data expected before 2030, it is quite possible that the next great CMB discovery (for example a detection of the primordial B-polarization) will be made from the ground. However, by their nature ground or balloon experiments cover only a small fraction of the sky, which makes them less suitable for the detection of (local) non-Gaussianity, where the correlation between very small and very large scales is essential, see chapter 3.

Chapter 2

Non-Gaussianity in multiple-field inflation

This chapter treats my work regarding non-Gaussianity in multiple-field inflation. These are my papers in that subject area:

• [Rigopoulos et al., 2006a] -Introduction of the long-wavelength formalism to compute non-linear fluctuations in multiple-field inflation on super-horizon scales.

• [Rigopoulos et al., 2005] -Application to single-field inflation. Introduction of a parametrization to visually represent the momentum dependence of the non-Gaussianity parameter f NL .

• [Rigopoulos et al., 2006b] -Derivation of the explicit long-wavelength equations for the second-order perturbations for multiple-field inflation and their formal solution. Explicit analytic solutions for f NL in the case of constant slow-roll parameters (stronger assumption than usual slow roll).

• [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF]] -Further refinement of the system of equations and their formal solution. Explicit numerical result for f NL for the double quadratic potential.

• [Tzavara and Van Tent, 2011] -The general analytic solution for f NL for two-field inflation is further worked out until just one integral remains. Explicit analytic slow-roll solutions in the case of certain classes of potentials are derived and compared to the exact numerical results. An example of a potential that gives large non-Gaussianity is presented.

• [Tzavara and Van Tent, 2012] -Several remaining technical issues of the long-wavelength formalism are sorted out, in particular regarding gauge invariance of the adiabatic and isocurvature perturbations at second order. The exact cubic action in terms of the gaugeinvariant adiabatic, isocurvature en tensor perturbations is derived, valid at all scales.

• [Tzavara and [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF]] -An investigation of the momentum dependence of the bispectrum and f NL in two-field inflation. Introduction of two non-Gaussianity spectral indices, related respectively to rescaling and squeezing of the momentum triangle.

• [START_REF] Tzavara | Covariant second-order perturbations in generalized two-field inflation[END_REF] -Two-field inflation with non-standard kinetic terms is considered. Several technical and definition issues regarding background and perturbation quantities in such models are sorted out. The derivation of the exact cubic action of [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF]] is extended to this case.1

• [Jung and [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF]] -An alternative formulation of the integral term from [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] is found, which allows further analytic progress for the expression of f NL in the general two-field case. An extensive analytic study of the non-Gaussianity in two-field sum potentials with standard kinetic terms is given, both within and beyond the slow-roll approximation, showing amongst other things how much the parameter space where non-Gaussianity can be large is limited by the Planck constraint on the spectral index.

The first four papers are collaborations with Paul Shellard and Gerasimos Rigopoulos. The next four are collaborations with Eleftheria Tzavara (as well as Shuntaro Mizuno for the last of those four). The final one is a collaboration with Gabriel Jung.

In section 2.1 the long-wavelength formalism is defined and explained, based on [Rigopoulos et al., 2006a]. In section 2.2 the resulting equations for the perturbations at first and second order and for f NL , as well as their general solutions, are summarized and discussed, based on [Rigopoulos et al., 2006b[START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF][START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF]. The (long) calculations to derive some of the results are given in appendix 2.4. Finally, the papers [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF][START_REF] Tzavara | Second-order cosmological perturbations in two-field inflation and predictions for non-Gaussianity[END_REF]Van Tent, 2012, 2013], which are more self-contained extensions or applications of the formalism, are briefly summarized in section 2.3 and included in full in the appendices 2A-2C.

The long-wavelength formalism

Introduction

As discussed in the previous chapter, it is a well-established fact that the universe on large scales exhibits a high degree of uniformity. During most cosmological eras and for a large span of length scales, it can be well approximated by a Friedmann-Robertson-Walker (FRW) spacetime with inhomogeneities described as small linear perturbations around the highly symmetric background. This picture has proved particularly relevant for the early universe, as the smallness of the cosmic microwave background (CMB) temperature anisotropies indicates. An extrapolation of this observational fact suggests that the use of linear theory would also be justified during inflation when the perturbations leading to the CMB anisotropies are thought to have been created. Until about the middle of the first decade of this century, almost all studies of the generation and evolution of perturbations in inflation invoked the use of linear perturbation theory as described in the first chapter. In principle it offers a tremendous simplification of the task of studying the true inhomogeneous spacetime.

However, even when attention is focused on the inflationary era, linear theory cannot be the whole picture. Since gravity is inherently non-linear and the potential of the inflationary model is likely to be interacting, some small non-linearity will be endemic to the perturbations. Given the accuracy of the WMAP and in particular the Planck CMB observations, it became worthwhile to investigate whether this non-linearity could be observationally relevant. The characteristic signatures of non-linear effects are deviations of the primordial fluctuations from Gaussian statistics. In order for this non-Gaussianity to be calculated one needs to go to second order in perturbation theory or develop a fully non-linear approach.

The issue of calculating non-linearity and the consequent non-Gaussianity in the primordial universe started attracting increasing attention from the middle of the first decade of this century, although some earlier attempts to calculate it can also be found in the literature [START_REF] Salopek | Stochastic inflation and nonlinear gravity[END_REF][START_REF] Gangui | The Three point correlation function of the cosmic microwave background in inflationary models[END_REF][START_REF] Yi | Inflationary Stochastic Dynamics and the Statistics of Large-Scale Structure[END_REF]. A tree-level calculation with a cubic action for the perturbations was performed in [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] for the case of slow-roll single-field inflation (with a similar slow-roll calculation for more general single-field Lagrangeans given in [Seery and Lidsey, 2005b]). At the level of the equations of motion, various authors pursued perturbation theory to second order [START_REF] Acquaviva | Second order cosmological perturbations from inflation[END_REF][START_REF] Rigopoulos | On second order gauge invariant perturbations in multi-field inflationary models[END_REF][START_REF] Noh | Second-order perturbations of the Friedmann world model[END_REF][START_REF] Malik | Evolution of second-order cosmological perturbations[END_REF][START_REF] Enqvist | Non-Gaussian perturbations in hybrid inflation[END_REF]Enqvist et al., 2005a;[START_REF] Vernizzi | On the conservation of second-order cosmological perturbations in a scalar field dominated Universe[END_REF]Lyth and Rodriguez, 2005a], with [Bartolo et al., 2004a] providing a review of these techniques. Although interesting results can be obtained at second order, full exploration of the system of equations suffers from great computational complexity. On the one hand, the perturbation equations tend to be rather cumbersome to derive. On the other hand, gauge-invariant variables, which have proved very useful for computations and the interpretation of results in linear theory, are not as simple as their first-order counterparts when second-order perturbations are considered.

That was the state of the art when we published our paper [Rigopoulos et al., 2006a] in early 2005. In that paper we took a different viewpoint on the study of non-linear perturbations during inflation. We used combinations of spatial gradients to construct variables describing the deviation from a spatially uniform spacetime and derived equations for these variables on superhorizon scales. These variables are defined non-perturbatively and are invariant under changes of the time coordinate on such scales. They were first used in [START_REF] Rigopoulos | The Separate Universe Approach and the Evolution of Nonlinear Superhorizon Cosmological Perturbations[END_REF], where a non-linear generalisation of the familiar adiabatic conservation law of linear theory was derived. Later the authors of [Langlois and Vernizzi, 2005a] used similar combinations of gradients in the context of the covariant formalism [START_REF] Ellis | Covariant and gauge invariant approach to cosmological density flunctuations[END_REF] to also derive this conservation law. They showed the relation of these simple, yet fully non-linear, variables to others defined in second-order perturbation theory. An equivalent conservation law was derived in [Lyth et al., 2005] without using such gradient variables. At roughly the same time, also in early 2005, another approach to the study of non-linear perturbations was published in [Lyth and Rodriguez, 2005b;[START_REF] Allen | Non-gaussian perturbations from multi-field inflation[END_REF], based on the so-called δN formalism of [START_REF] Sasaki | A General analytic formula for the spectral index of the density perturbations produced during inflation[END_REF].

In order to include the continuous influx of sub-horizon perturbations to the long-wavelength system, source terms are added to the long-wavelength equations. Thus, we arrive at a set of fully non-linear equations which include both matter and metric perturbations. For actual multiple-field calculations it is more convenient to use the explicit field basis [Groot Nibbelink and Van Tent, 2000[START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF] described in the previous chapter. Since the evolution equations are fully non-linear on long wavelengths, numerical simulations can be performed without the need for analytic approximations. All of this is described in detail in section 2.1, based on [Rigopoulos et al., 2006a]. As discussed in the next section 2.2, based on [Rigopoulos et al., 2006b[START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF][START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF], a perturbative analytic approach can be applied giving results to second order. At first order, this perturbative expansion is equivalent to the well-known linear gauge-invariant perturbation theory. At second order, however, it is much simpler than the corresponding second-order approaches pursued before 2005. The full calculation up to and including the computation of the bispectrum and its amplitude parameter f NL is given in that section 2.2.

Starting from these pioneering works in 2005, and strengthened from the end of 2007 when [Yadav and Wandelt, 2008] claimed a detection of local non-Gaussianity in the WMAP data, primordial non-Gaussianity became a hot topic in the literature. This popularity ended in 2013, when the Planck satellite did not detect any primordial non-Gaussianity [Planck 2013 XXIV] and hence refuted the earlier claim. But of course even a non-detection puts important constraints on inflationary models, and so non-Gaussianity, despite the loss of its "hot topic" status, has remained an important research subject. References to many inflationary non-Gaussianity papers can be found in sections 2A.1, 2B.1, and 2C.1, while we give a few more early ones here: [Bernardeau andUzan, 2002, 2003;Enqvist et al., 2005b;Vaihkonen, 2005;[START_REF] Hattori | Non-Gaussianity in multi-field stochastic inflation with the scaling approximation[END_REF][START_REF] Kolb | Non-gaussianity from broken symmetries[END_REF][START_REF] Calcagni | Non-Gaussianity in braneworld and tachyon inflation[END_REF][START_REF] Bartolo | Non-Gaussianity of Large-Scale Cosmic Microwave Background Anisotropies beyond Perturbation Theory[END_REF][START_REF] Barnaby | Nongaussian and nonscale-invariant perturbations from tachyonic preheating in hybrid inflation[END_REF][START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Langlois | Nonlinear perturbations of cosmological scalar fields[END_REF][START_REF] Weinberg | Quantum contributions to cosmological correlations[END_REF][START_REF] Weinberg | Quantum contributions to cosmological correlations. II. Can these corrections become large?[END_REF]Seery and Lidsey, 2005a;Chen et al., 2007a]. Finally, some selected references to papers from 2019 on non-Gaussianity in inflation models to show that the field is still active: [START_REF] Fumagalli | Hyper-Non-Gaussianities in Inflation with Strongly Nongeodesic Motion[END_REF][START_REF] Ozsoy | Squeezed tensor non-Gaussianity in non-attractor inflation[END_REF][START_REF] Bolis | Non-Gaussianity from Entanglement During Inflation[END_REF][START_REF] Mcaneny | New Shapes of Primordial Non-Gaussianity from Quasi-Single Field Inflation with Multiple Isocurvatons[END_REF][START_REF] Garcia-Saenz | Revisiting non-Gaussianity in multifield inflation with curved field space[END_REF][START_REF] Achúcarro | Origin of ultra-light fields during inflation and their suppressed non-Gaussianity[END_REF][START_REF] Bjorkmo | Mild Non-Gaussianities under Perturbative Control from Rapid-Turn Inflation Models[END_REF][START_REF] Fujita | Primordial Tensor Non-Gaussianity from Massive Gravity[END_REF].

Non-linear equations

The starting point of what became known as the long-wavelength formalism (also sometimes referred to as the RSvT formalism, after its authors) is the long-wavelength approximation [Salopek andBond, 1990, 1991;[START_REF] Comer | Growth or decay of cosmological inhomogeneities as a function of their equation of state[END_REF][START_REF] Deruelle | Long wavelength iteration of Einstein's equations near a space-time singularity[END_REF][START_REF] Parry | Solving the Hamilton-Jacobi equation for general relativity[END_REF][START_REF] Khalatnikov | Comment about quasiisotropic solution of Einstein equations near cosmological singularity[END_REF]: the fact that on super-horizon scales, spatial gradients can be neglected with respect to time derivatives. For any quantity F (t, x) constructed out of metric and matter variables typically we will have ∂ i F = O(F/L) and ∂ t F/N = O(HF ). From this we see that for length scales L 1/(aH), i.e. super-horizon scales, we can expect

1 a ∂ i F 1 N ∂ t F .
In practice this means that the second-order spatial gradients in the various equations of motion can be neglected. 2Using the long-wavelength approximation, the most general form of the metric can be written as

ds 2 = -N 2 (t, x) dt 2 + a 2 (t, x) dx 2 . (2.1)
The Hubble parameter is defined as H ≡ ∂ t a/(N a). Here we have kept the lapse function N that encodes the choice of time coordinate (or in other words the choice of time slicing for the spacetime) but fixed part of the gauge by setting the shift, i.e. the g 0i component of the metric, to zero. For a proof that this can always be done in the long-wavelength approximation, see e.g. [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF] included in appendix 2B. In principle the second term should also contain a factor h ij (t, x) describing the tensor perturbations, which beyond linear order no longer decouple from the scalar perturbations. However, as was shown already in [START_REF] Salopek | Stochastic inflation and nonlinear gravity[END_REF], in the long-wavelength approximation (after decaying modes have disappeared) the (non-linear) tensor perturbations are non-dynamical on super-horizon scales (i.e. h ij (x) does not depend on t) and hence uninteresting for our purposes, so that we set h ij (x) = δ ij for simplicity. The fully non-linear metric (2.1) looks identical to the background metric (1.2), but the difference is that N and a are now inhomogeneous functions of time and space, and not just functions of time alone.

On the matter side we assume the same very general multiple-field inflation model as in section 1.1.2, so that the Lagrangean matter density is given by (1.1). We also define the field velocity with respect to proper time, which is also the conjugate momentum of the field,

Π A ≡ G AB ∂ t φ B N . (2.2)
However, while our general formalism was indeed set up for this general case, in most of our explicit calculations we always took a trivial field metric G AB = δ AB . The only exception is our paper [START_REF] Tzavara | Covariant second-order perturbations in generalized two-field inflation[END_REF], where we looked at a Lagrangean even more general than (1.1) and were interested in the new effects due to the non-standard kinetic terms. For simplicity, we will from now on take G AB = δ AB , and refer the interested reader to the original papers for the more general expressions (which are quite similar, just with covariant derivatives instead of normal derivatives, and an additional curvature tensor of the field manifold next to the second derivative of the potential). Similarly, while the formalism can in principle deal with an arbitrary number of fields, in explicit calculations we always limited ourselves to the case of two fields. Hence in this thesis we will always assume two fields. As explained in chapter 1 and below, adding a second field to a single-field inflation model allows for all kinds of new effects due to the perturbations no longer necessarily being constant on super-horizon scales. On the other hand, no such qualitative changes are expected when adding a third field to a two-field model.

The Einstein and field equations for these metric and matter ingredients are in the longwavelength approximation [START_REF] Salopek | Nonlinear evolution of long wavelength metric fluctuations in inflationary models[END_REF]:

∂ t H = - κ 2 2 N Π B Π B , (2.3)
∂ t Π A = -3N HΠ A -N W A , (2.4
)

H 2 = κ 2 3 1 2 Π B Π B + W , (2.5
)

∂ i H = - κ 2 2 Π B ∂ i φ B , (2.6)
where W B ≡ ∂ B W ≡ ∂W/∂φ B and κ 2 ≡ 8πG = 8π/m 2 pl . We will also use the notation Π ≡ Π B Π B . Just like the metric, the first three equations look identical to the background equations of motion, but it should be kept in mind that H, φ A and Π A are here fully non-linear functions that depend on both time and space, and not just homogeneous background functions depending only on time. This explains the notion of "separate-universe approach": it seems as if on super-horizon scales at each spatial point x the universe evolves as an independent FRW universe from some initial condition. The only coupling between those separate universes comes from the constraint equation (2.6), and from the fact that the initial conditions are not arbitrary, but set up by the sub-horizon evolution where everything is coupled.

Gradient variables

To look at perturbations in such a fully non-linear setup, it makes sense to define gradient variables, as first advocated by [START_REF] Ellis | Covariant and gauge invariant approach to cosmological density flunctuations[END_REF], since by taking a gradient we remove the homogeneous background part, but keep the full inhomogeneous perturbation. In particular the following combination of metric and matter gradients is a key variable in our work:

ζ A i (t, x) ≡ Π A Π ∂ i ln a - H Π ∂ i φ A . (2.7)
When linearized, this is exactly the spatial gradient of the ζ A defined in (1.31). When taking components in the basis defined in (1.8), as we will do in the next section, the first (parallel) component is called the adiabatic perturbation, and the second (perpendicular) one the isocurvature perturbation. The fully non-linear variable ζ A i is gauge-invariant, which in this context means it is invariant under changes of time slicing (choices of N (t, x)) within the longwavelength approximation. It is also gauge-invariant when linearized. Initially we thought that this meant the variable was gauge-invariant to any order in perturbation theory, but that turned out to be wrong: at second order it is not. In [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] we computed the gauge correction required to make contact with the usual non-Gaussian observables for the adiabatic perturbation, and in [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF] (included in appendix 2B) we definitively settled all issues regarding gauge invariance at second order.

From the equations of motion (2.3-2.6), the following equation can be derived for ζ A i (the derivation is mostly straightforward but can be found in [Rigopoulos and Shellard, 2005;Rigopoulos et al., 2006a]; see also the derivation of (1.27) in [Groot Nibbelink and [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF][START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF]):

∂ 2 t ζ A i - ∂ t N N -2N H 3 2 + + η ∂ t ζ A i + (N H) 2 Ξ A B ζ B i = 0 , (2.8) with Ξ A B ≡ W A B H 2 + 3 + 3η + 2 2 + 4 η + (η ⊥ ) 2 + ξ δ A B -2 (3 + ) Π A Π Π B Π + Π A Π η B + η A Π B Π (2.9)
and the various slow-roll parameters defined in section 1.1.3. It should be stressed that the slowroll parameters should just be viewed as short-hand notation here. No assumption about their size has been made and the equations are exact (no slow-roll approximation). This equation looks linear, and seems indeed identical to the first-order perturbation equation written in terms of ζ A . However, as before this is not correct, as the coefficients are functions of the fully inhomogeneous H, φ A and Π A (and N , but that one is fixed by making an explicit choice of time slicing). For this reason the system must be closed with a set of constraint equations expressing the spatial gradient of those quantities in terms of ζ A i and its time derivative. The explicit form of these constraint equations will be given later, in (2.24). To solve (2.8) it is convenient to rewrite this second-order differential equation as two first-order differential equations:

   ∂ t ζ A i -θ A i = 0 ∂ t θ A i - ∂ t N N -2N H 3 2 + + η θ A i + (N H) 2 Ξ A B ζ B i = 0 (2.10)
where the velocity θ A i defined by the first equation should now be viewed as an independent variable.

Source terms

The final ingredient required to complete the system are the initial conditions, which come from the sub-horizon regime. To add these explicitly to the equations we proceed as follows (motivated by the stochastic picture for the generation of inflationary perturbations [START_REF] Starobinsky | Stochastic De Sitter (inflationary) stage in the early universe[END_REF][START_REF] Nakao | Stochastic Dynamics of New Inflation[END_REF][START_REF] Kandrup | Stochastic Inflation as a Time Dependent Random Walk[END_REF][START_REF] Salopek | Stochastic inflation and nonlinear gravity[END_REF][START_REF] Stewart | The Stochastic dynamics of chaotic inflation[END_REF][START_REF] Casini | Stochastic approach to inflation. 2. Classicality, coarse graining and noises[END_REF][START_REF] Winitzki | Effective noise in stochastic description of inflation[END_REF][START_REF] Afshordi | Super Hubble nonlinear perturbations during inflation[END_REF][START_REF] Matarrese | Influence of superhorizon scales on cosmological observables generated during inflation[END_REF][START_REF] Geshnizjani | Coarse-grained back reaction in single scalar field driven inflation[END_REF]). First we note that since ζ A i and θ A i , solutions of (2.10), are valid only on long wavelengths, we can view them as smoothed long-wavelength versions of the exact quantities, smoothed using some window function. In Fourier space this can be written as

ζ A (k) = W(k)ζ A ex (k)
, and a similar expression for θ A (k). The window function W, which we will discuss further below, filters out short wavelengths (large k) below a certain appropriately chosen smoothing length (which depends on time). Secondly, while we do not have the fully non-linear equation of motion on all scales, we do know the equation of motion on all scales in linear perturbation theory. And as noted before, it looks identical to the long-wavelength non-linear equation, just with all coefficients replaced by their homogeneous background version (and after we add the usual second-order spatial gradient term). So viewing for a moment (2.10) as a linear perturbation equation valid on all scales (after adding the gradient term), we can explicitly apply the smoothing as defined above to it (most easily done in Fourier space). Mostly this will just give back the same equation, but with the exact linear curvature perturbation replaced by its long-wavelength smoothed version. However, there will be one extra term in each equation involving the unsmoothed variable and ∂ t W, that comes from pulling the window function through the time derivative. The final step then is to put those extra terms on the right-hand side of the equations, and return to the long-wavelength non-linear case by viewing all coefficients again as fully inhomogeneous functions of space and time (and removing the second-order spatial gradient term). On the left-hand side of the equation this is exact, for the reasons explained above. The assumption here is that this procedure is also correct on the right-hand side. The final result is:

   ∂ t ζ A i -θ A i = S A i ∂ t θ A i - ∂ t N N -2N H 3 2 + + η θ A i + (N H) 2 Ξ A B ζ B i = J A i (2.11)
where the source terms S A i and J A i are given by

S A i (t, x) ≡ d 3 k (2π) 3/2 ∂ t W(t, k) ζ A lin (t, k, x) ik i e ik•x + c.c. , (2
.12) 2.13) where c.c. denotes the complex conjugate and ζ A lin and θ A lin are the full, non-smoothed solutions from linear perturbation theory, that is, incorporating short-wavelength information. The fact that they depend on x as well as on k represents the fact that all background quantities in these solutions have been made inhomogeneous.

J A i (t, x) ≡ d 3 k (2π) 3/2 ∂ t W(t, k) θ A lin (t, k, x) ik i e ik•x + c.c. , (
The source terms describe the continuous influx of short-wavelength modes as they cross the horizon and enter the long-wavelength regime. In this way the physical initial conditions are set up dynamically and added explicitly to the equations. The mathematical initial conditions of the equations are then simply that the smoothed long-wavelength variables ζ A i and θ A i should be zero at early times when all the modes are sub-horizon, represented schematically by

lim t→-∞ ζ A i = 0, lim t→-∞ θ A i = 0. (2.14)
While the equations with the source terms are exact at the linear order, it is only an assumption that by making all background quantities inhomogeneous in the source terms, they would also be exact at the fully non-linear level. The fact that this procedure works for the left-hand side of the equation is no proof. As explained in the next section, when these source terms are expanded to second order, one does not find the correct result and a correction must be added. While discouraging, this does not necessarily mean that the source term cannot be correct fully non-linearly, as something similar occurs with the gauge invariance of ζ A i as explained above. Anyway, this question remains unanswered as for the rest of our research we never needed the fully non-linear expression, only its perturbed version up to first and second order.

Coming back to the window function, we define the smoothing length as R = c/(aH), where c is a small number. The length R separates short wavelengths from long wavelengths. It is chosen a bit larger than the comoving Hubble length so that for wavelengths larger than R we are already in the long-wavelength regime. For typical situations, where slow roll holds at horizon crossing (but can be broken later on), a value of c = 3 is good enough. R clearly depends on time, as a and H do. For the window function the requirements are that for short wavelengths (kR 1) it goes to zero, while for long wavelengths (kR 1) it goes to one. In our original papers we took (half of) a Gaussian window function:

W(t, k) = exp(-k 2 R 2 /2).
Starting from [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] we simplified this to a step function (see below for the expression). Of course the final results should not depend on the explicit form of the window function, something that we verified explicitly.

The choice of the time slicing, and hence the lapse function N , to fix the final gauge freedom, is somewhat related to the window function. In our first papers we chose a time variable t = ln(aH), which corresponds to N H = 1/(1 -). This choice was motivated by the fact that with this choice, the smoothing length R and hence the window function W depend only on time and not on space. It also means that horizon crossing of a mode k is uniform throughout space. For the Gaussian window function this gave ∂ t W = (kR) 2 W(k). However, starting from our paper [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] we switched to the number of e-folds as time variable:

t = ln a ⇔ N H = 1 (2.15)
as this simplifies the calculations. Given that (2.6) can be rewritten as

∂ i ln H = ζ A i Π A /Π
in this gauge, we see that if slow roll holds at horizon crossing, the spatial dependence of H (and hence of R) is of higher order in slow roll, so that it can be neglected during horizon crossing. Slow roll at horizon crossing is both motivated by observations of the CMB and a required assumption for the long-wavelength formalism in order for the decaying mode to disappear rapidly after horizon crossing. From now on we will always assume this choice of time slicing, and indicate time derivatives with respect to this t by a dot. The time derivative of the step function window function that will be used in all calculations is given by .16) This will naturally lead to a time defined by the relation aH = kc/ √ 2, which we call t * , to appear in our relations, where c ≈ 3 is the numerical value introduced above. However, because of the assumption of slow roll at horizon crossing, all quantities vary very slowly at that time, so that in the end the results do not depend on the value of c and in the final results one might just as well take t * to be defined by the usual horizon crossing condition aH = k.

Ẇ(t, k) = δ(kR/ √ 2 -1). ( 2
To obtain the linear solutions required for the source terms, one can either solve the linear perturbation equation exactly numerically, or use the analytical slow-roll solution determined in [Groot Nibbelink and [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF], which to leading order in slow-roll is given in (1.33). The latter contains quantum creation and annihilation operators âA † (k) and âA (k), inherited from the quantum nature of the fluctuations. As is well known, on super-horizon scales when the decaying mode can be neglected, the fluctuations become effectively classical. One can just as well describe them as classical stochastic fluctuations by replacing the quantum creation and annihilation operators with Gaussian stochastic quantities α A (k) that satisfy

α A (k)α * B (k ) = δ A B δ 3 (k -k ) and α A (k)α B (k ) = 0
, where . . . denotes an ensemble average. The main purpose of this replacement is numerics: the stochastic quantities allow for the numerical creation of random realizations. 3 In the analytical work described in the next sections there is no need for this replacement, and starting from [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] we no longer used the stochastic quantities, but returned to using the quantum creation and annihilation operators.

Non-Gaussianity

Perturbative expansion

To make analytic progress with the system of non-linear equations to compute the bispectral non-Gaussianity, it makes sense to expand the system of equations to second order. The resulting equations and their general solutions are described in this section, based on [Rigopoulos et al., 2006b[START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF][START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF]. It should be noted that when we started our project, the standard way to compute (single-field) non-Gaussianity in the literature (with the important exception of Maldacena [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]) was to simply generalize the usual linear perturbation calculations by expanding the metric and fields to second order (with second-order gravitational potentials etc.) and then derive equations of motion for the second-order perturbations by pushing these through the Einstein and field equations as in e.g. [START_REF] Acquaviva | Second order cosmological perturbations from inflation[END_REF]. This led to huge calculations and expressions, prone to mistakes, where it is hard to identify the proper gauge-invariant variables and interpret the various contributions. On the contrary, we started from the fully non-linear equations of motion, which look almost identical to the linear ones, and derived the second-order equations of motion for the proper gauge-invariant variables in a very simple way from those.

Before making the perturbative expansion, we do a few more manipulations on the longwavelength system (2.11). First we switch to the multiple-field basis defined in (1.8), using the indices m, n, . . . to specify components in that basis. As a reminder, m = 1 specifies the component along the field trajectory, and m = 2 the perpendicular one. For perturbations that means that m = 1 is the adiabatic perturbation and m = 2 the isocurvature one. 4 As mentioned before, we restrict ourselves to two fields and from now on use the explicit time slicing N H = 1 where the time coordinate is the number of e-folds. We also combine the components of ζ m i and θ m i into a vector with components v i a , and the sources S m i and J m i into a vector b i a : (2.17) with the indices a, b, . . . taking the values 1,2,3 (as a reminder, the indices i, j are spatial indices as our variables are spatial gradients). The reason that we have not included the adiabatic velocity θ 1 i (and its corresponding source) in this vector is that this component can be eliminated from the equations using the exact (i.e. fully non-linear) long-wavelength identity .18) This important relation was already given and discussed at the linear level in (1.32), 5 but is valid even fully non-linearly. 6 The system then becomes

v i a ≡ (ζ 1 i , ζ 2 i , θ 2 i ) a , b i a ≡ (S 1 i , S 2 i , J 2 i ) a ,
θ 1 i = 2η ⊥ ζ 2 i . ( 2
vi a (t, x) + A ab (t, x)v i b (t, x) = b i a (t, x) (2.19)
with the matrix A given by

A =   0 -2η ⊥ 0 0 0 -1 0 3χ + 2 2 + 4 η + 4(η ⊥ ) 2 + ξ 3 + + 2η   .
(2.20)

For a curved manifold with a nontrivial field metric the term

-(2 /κ 2 )e A 2 R ABCD e B 1 e C 1 e D
2 should be added to the A 32 component, with R A BCD the curvature tensor of the field manifold. Again, we stress that no slow-roll approximation has been made.

To solve the master equation (2.19) analytically, we expand the system as an infinite hierarchy of linear perturbation equations with known source terms at each order. To second order we obtain v(1) i a + A (0)

ab (t)v (1) i b = b (1) i a (t, x), (2.21) v(2) i a + A (0) ab (t)v (2) i b = -A (1) ab (t, x)v (1) i b + b (2)
i a (t, x) , (2.22) where

v i a (t, x) = v (1) i a (t, x) + v (2)
i a (t, x), and

A ab (t, x) = A (0) ab (t) + A (1) ab (t, x) = A (0) ab + ∂ -2 ∂ i (∂ i A ab ) (1) ≡ A (0) ab (t) + Ā(0) abc (t)v (1) c (t, x). (2.23)
Here we have denoted v

(1)

c ≡ ∂ -2 ∂ i v (1)
i c and ∂ i A ab is computed using the constraint equations (derived from the basic equations (2.3)- (2.6) and the definition of ζ A i ):

∂ i ln H = ζ 1 i , e m A ∂ i φ A = - √ 2 κ ζ m i , e A m ∂ i Π A = - H √ 2 κ θ m i + η ζ m i -η ⊥ ζ 2 i δ m1 + (η ⊥ ζ 1 i + ζ 2 i )δ m2 . ( 2

.24)

4 It should be noted that θ m i is defined as θ m i ≡ ∂t(ζ m i ) and hence is not just the basis vector em A applied to θ A i but includes a correction term related to the time derivative of the basis vector. Explicitly one finds θ m i = em Aθ A i + η ⊥ εmnζ n i with εmn the anti-symmetric Levi-Civita symbol. 5 Remember that (1.32) was given in terms of conformal time, which explains the presence of the H in that equation.

6 If one derives an equation for θ1 i from (2.11) in the same way as for the other components in (2.19) (taken without the source term that provides the initial conditions coming from the sub-horizon region and that only acts around horizon-crossing), see e.g. [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF], then it is easy to see that (2.18) is a solution of that equation, using (1.22). In [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] it was also shown (in a rather involved way) that it is the only consistent solution in the long-wavelength formalism where we have neglected decaying terms.

The explicit expression of the object Ā(0) abc can be found in (2.63). W was defined in (1.16). The explicit form of the first-order source term can be computed as b (2.25) using the slow-roll solution for the linear perturbation at horizon crossing (1.33), and where

(1) i a (t, x) = d 3 k (2π) 3/2 Ẇ(k, t)X (1) am (k, t)â † m (k) ik i e ik•x + c.c.,
X (1) am = - κH 2k 3/2 √   1 0 0 1 0 -χ   (2.26)
(which because of the window function will only be evaluated close to t * where the approximation is valid). To find the last line of this matrix, for the time derivative of ζ 2 , expression (1.33) is not enough, as one needs the time dependence of the linear solution around horizon crossing up to first order in slow roll. We have not given that more complicated expression in this thesis, but it was first derived in [Groot Nibbelink and [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF] and can also be found in [Rigopoulos et al., 2006b].

Regarding the second-order source term b

(2)

i a , in our initial papers [Rigopoulos et al., 2005[Rigopoulos et al., , 2006b] ] we computed it as the perturbation of the source term b

(1) i a in a similar way as in (2.23), so by taking the spatial gradient and using the constraint equations. As explained in the previous section, unlike the rest of the non-linear equation, the validity of the source term beyond linear order was only a conjecture. We found the resulting source term at second order to be unsatisfactory. In the first place it depended on the details of the window function. Secondly, in [Rigopoulos et al., 2005], where we studied the single-field limit, we found that while of the correct order of magnitude, the exact momentum dependence did not agree with the result by Maldacena [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]. In [Rigopoulos et al., 2006b], where we looked at the multiplefield case with constant slow-roll parameters, we found that the contribution coming from b

(2) i a was negligible compared to the contribution from -A

(1) ab v

(1) i b (which is the true super-horizon contribution that does not depend on the details of the window function, and which is absent in single-field inflation). Because of these reasons, we decided to simply neglect the b

(2) i a term in [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF]. The issue was finally settled definitively in [Tzavara and Van Tent, 2011] and [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF], where b

(2) i a was computed from the exact cubic action for ζ m i (and it was in fact different from the perturbation of b

(1) i a ). The term does indeed always give negligible non-Gaussianity and is unimportant for our studies, but its inclusion allows for perfect agreement with e.g. the single-field result of Maldacena. Its expression is b .27) (Here c.c. is an abuse of notation, as there are three additional terms for each line, given the multiplication of two complex operators.) The key elements for its derivation can be found in [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF], included in appendix 2B. The term with L abc is a local term that comes from the redefinition of the perturbations required to remove terms proportional to the equation of motion from the cubic action, as explained in that paper. The term with N abc is a non-local term that comes from the gauge transformation between the uniform energy density gauge and the flat gauge (see remarks in the next section). The explicit expressions for L abc and N abc can be found in (2.64).

(2) ia = d 3 k (2π) 3/2 d 3 k (2π) 3/2 Ẇ(max(k , k)) × L abc (t)X (1) bm (k , t)X (1) cn (k, t)â † m (k )â † n (k)i(k i + k i )e i(k +k)•x +N abc (t)X (1) bm (k , t)X (1) cn (k, t)â † m (k )â † n (k)ik i e i(k +k)•x + c.c. . ( 2

Solution for power spectrum and bispectrum

Looking at equations (2.21) and (2.22) we see that they both have the same structure: a linear first-order differential equation with an inhomogeneous source term. In fact it is easy to show that this structure is valid to any order in the perturbative expansion, with the source term at order (α) computable from quantities up to order (α -1), so that an iterative scheme is possible. The system is ideal for solving using a Green's function: i a at each order using the different source terms as in (2.28). More details about these Green's functions can be found in appendix 2.4.1.

v (α) i a (t, x) = t -∞ dt G ab (t, t ) b(α) i b (t , x), ( 2
The first-order solution can then be written as

v (1) ia (t, x) = d 3 k (2π) 3/2 v am (k, t)â † m (k)ik i e ik•x + c.c., (2.30)
introducing the quantities v am given by

v am (k, t) = G ab (t, t * )X (1) bm (t * ), (2.31)
defined for t ≥ t * , with the dependence on k entering through t * . We have not included the explicit first-order indication (1) on v am to lighten the notation, and since that quantity is always first order. It will also be useful to define a "reduced" version of v am , indicated by a bar, without the prefactor of the X

(1) bm (t * ) (see (2.26)):

vam ≡ v am /γ * , γ * ≡ - κH * 2k 3/2 √ * .
(2.32)

Remembering that v ia = (ζ 1 i , ζ 2 i , θ 2
i ) for a = 1, 2, 3 it is now easy to compute the linear power spectrum. Since at first order ζ i is gauge-invariant and a total gradient, it is trivial to get rid of the spatial gradient (the i index). For the adiabatic component we find (see (1.38) for the definition of the power spectrum) .33) For the spectral index, defined as n s = 1 + d ln P ζ /d ln k, we find ) and W 221 * the third derivative of the potential, projected along the indicated directions of the basis and evaluated at t * (see the definition (1.16)). Finally, the tensor-to-scalar ratio r is given by

P ζ (k, t) = k 3 2π 2 (v 1m ) 2 = κ 2 H 2 * 8π 2 * [1 + (v 12 ) 2 ]. ( 2
n s -1 = 1 1 - * -4 * -2η * + 2 v12 1 + (v 12 ) 2 -2η ⊥ * + χ * v12 (2.34) +G 13 (t, t * ) -W221 * + 2 2 * + (η * ) 2 + (η ⊥ * ) 2 + 3 * (η * -χ * ) -2η * χ * + χ 2 * v 12 1+(v _ 12 ) 2 (v _ 12 ) 2 1+(v _ 12 ) 2
r = 16 * 1 + (v 12 ) 2 .
(2.35)

These expressions should be compared to (1.39), (1.42), and (1.45), which were computed for the single-field case, where there is no evolution of ζ on super-horizon scales.8 An important conclusion can be drawn from the expression of the spectral index. Given that G 13 (t, t * ) ≈ v12 /3 as we will later show in (2.61), the relevant factors to study are v12 /(1 + v2 12 ) and v2

12 /(1 + v2 12 ), which are shown in figure 2.1. We see that they are never larger than unity in absolute value and are in fact of order unity unless v12 ≈ 0, which is when multiple-field effects are negligible and which is not interesting from the point of view of this paper.9 So barring any fine-tuned cancellations between terms, the observed value of n s allows us to conclude that slow roll is a good approximation at horizon crossing with all first-order slow-roll parameters at t * at most of order 10 -2 . However, it is certainly possible for slow roll to be broken afterwards.

At second order we have to do an additional step. As ζ m i is not gauge-invariant at second order, we have to think if the flat gauge that we picked (where a does not depend on space) will give us the correct observable, which should be a gauge-invariant quantity by definition. In addition, in the flat gauge at second order, ζ m i is not a total gradient, which complicates things. A careful investigation of gauge invariance at second order, performed in [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF] included in appendix 2B, shows that it is only in the uniform energy density gauge that the expression for ζ m (2) i coincides with the gauge-invariant expression and hence gives the correct observable. Moreover, in that gauge the adiabatic component is a total gradient, so that getting rid of the i index is again trivial. Hence we have to include in our calculations a gauge-correction term to convert the result for ζ m (2) i from the flat gauge to the uniform energy density gauge. For the adiabatic component this correction to be added is 2η

⊥ ζ 1 (1) ζ 2 (1) i . By combining the different permutations of 0|ζ 1 (2) (k 1 )ζ 1 (1) (k 2 )ζ 1 (1) (k 3 )|0 of
the Fourier components of the linear and second-order adiabatic solutions (first subtracting the average of ζ 1 (2) (t, x) to get rid of the divergent part and not forgetting the gauge correction explained above), we find the bispectrum B ζ defined as:

10 0|ζ 1 (k 1 )ζ 1 (k 2 )ζ 1 (k 3 )|0 (2) ≡ (2π) -3/2 δ 3 (k 1 + k 2 + k 3 )B ζ (k 1 , k 2 , k 3 ).
(2.36)

From this we can then finally define the (local) f NL parameter that is generally used as a measure of (local bispectral) non-Gaussianity: 11 .37) Putting all the elements together and doing some calculations that are specified in appendix 2.4.2, one finally obtains:

- 6 5 f NL ≡ B ζ (k 1 , k 2 , k 3 ) 2π 2 k 3 1 P ζ (k 1 ) 2π 2 k 3 2 P ζ (k 2 ) + (k 2 ↔ k 3 ) + (k 1 ↔ k 3 ) . ( 2
12 - 6 5 f NL = -2v 2 12 [1 + (v 12 ) 2 ] 2 g iso + g sr + g int , (2.38)
where

g iso = ( + η )(v 22 ) 2 + v22 v32 , g sr = - * + η * 2v 2 12 + η ⊥ * v12 2 - 3 2 * + η * -χ * + η ⊥ * v12 , g int = - t t * dt 2(η ⊥ ) 2 (v 22 ) 2 + ( + η )v 22 v32 + (v 32 ) 2 -G 13 (t, t )v 22 Ξv 22 + 9η ⊥ v32 .
(2.39)

Here we have defined

Ξ ≡ 12η ⊥ χ -6η η ⊥ + 6(η ) 2 η ⊥ + 6(η ⊥ ) 3 -2η ⊥ ξ -2η ξ ⊥ - 3 2 ( W211 + W222 ).
(2.40)

The explicit time dependence of all functions has been omitted, except for G 13 since it depends on two times. Although no slow-roll approximation has been used on super-horizon scales, we did assume slow roll to hold at horizon crossing, in order to use the analytic linear solution (2.26).

The parameter f NL is weakly momentum dependent, as most of the momentum dependence of B ζ is equal to the momentum dependence of the power spectrum squared and is divided away in the definition (2.37). The (weak) momentum dependence of f NL is often neglected in the literature, but strictly speaking the above expression is only valid for equal momenta. We pointed this out in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF], where we also gave the expression for f NL with general momenta, and studied it further in [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF], included in appendix 2C. As a reminder, v12 is the contribution of the isocurvature mode to the adiabatic mode, which according to (2.18) only builds up if 1) the field trajectory makes a turn (η ⊥ = 0) and 2) an 10 In the literature one often sees a factor (2π) 3 in front of the bispectrum (as well as in front of the power spectrum). This is due to a different definition of the Fourier transform. We use the convention where both the Fourier transform and its inverse have a factor (2π) -3/2 .

11 The left-hand side is called -6 5 fNL and not simply fNL because it was originally defined in terms of the gravitational potential Φ and not ζ as Φ = ΦL + fNL(Φ 2 L -Φ 2 L ) [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF]. During recombination (matter domination) the two are related by ζ = -5 3 Φ. Moreover, when computing the bispectrum divided by the three permutations of the power spectrum squared using this expression of Φ, one obtains 2fNL due to the two ways the two ΦL inside the second-order solution can be combined with the two linear solutions to create the power spectrum. Together these two effects explain the factor -6/5 (see also footnote 5 in chapter 3). In the papers before [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] we used a slightly different definition of fNL which was larger by a factor -18/5, i.e. the definition would have +fNL/3 on the left-hand side.

12 It should be noted that this is only the part of fNL that comes from the three-point correlator of two firstorder perturbations and one second-order perturbation (expressed as products of two first-order ones), sometimes called f (4) NL in the literature (see e.g. [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF]), which is the only contribution on super-horizon scales. It does not include the so-called intrinsic non-Gaussianity f

(3) NL due to interaction terms in the cubic action, which only play a role before and at horizon crossing and are necessarily slow-roll suppressed in models with standard kinetic terms.

isocurvature mode is present. It remains constant after the isocurvature mode has disappeared and/or the field trajectory has become straight. Obviously it will be zero in the single-field case. On the other hand, v22 and v32 are directly proportional to the isocurvature mode and will both go to zero when the isocurvature mode disappears. We will in all our work assume that this is the case by the end of inflation, so that the g iso contribution to f NL vanishes there. If we relaxed our assumption of the isocurvature mode going to zero by the end of inflation, it would be easy to get huge non-Gaussianity at the end of inflation from the g iso term (and indeed such papers exist, see e.g. [START_REF] Byrnes | Conditions for large non-Gaussianity in two-field slow-roll inflation[END_REF]), but it would be meaningless from an observational point of view, since one would have to follow its evolution explicitly through the rest of the evolution of the universe to get a prediction for the observable. Only when the isocurvature mode has gone to zero and we have returned to an effectively single-field situation, will the adiabatic perturbation remain necessarily constant on super-horizon scales, so that we can directly extrapolate the result at the end of inflation to observables in the CMB.

In the single-field limit, a small slow-roll suppressed part of g sr is all that survives and it gives back the f (4) NL part of the usual single-field result of Maldacena [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]. All the other terms of g sr are also slow-roll suppressed, since they are proportional to slow-roll parameters at horizon crossing. (It is easy to check that the various functions of v12 can never become large, independent of the value of v12 .) Hence persistent large non-Gaussianity can only come from the integrated contribution g int , making this the most important term to study. Unfortunately it is also the most complicated one, because of the integral with the integrand depending explicitly on both t and t.

In [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] we showed that this integrated contribution can be computed explicitly if we make two assumptions: that we can use the slow-roll approximation and that we have a potential from a specific class, like a sum or product potential. In particular we showed that the integrated contribution vanishes, in the slow-roll approximation, for any product potential (this proof was later repeated in [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF] which is included in appendix 2A) as well as for any simple equal-power sum potential, a proof that we reproduce here in the next subsection. We also found an explicit model giving an f NL of order unity (which we call large non-Gaussianity, as it is two orders of magnitude larger than for standard single-field inflation; this model was revisited in [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF] and so can be found in appendix 2A).

In [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF], we went further and investigated sum potentials in much greater detail, finding the regions of parameter space where large non-Gaussianity is possible. In addition we found a way to rewrite the expression for g int that allowed us to draw conclusions even beyond the slow-roll approximation. As [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF] is included in appendix 2A and summarized below in section 2.3.1, we conclude here by giving that different expression for g int :

g int (t) = A G 12 (t, t * ) + B G 13 (t, t * ) + t t * dt P (t ).
(2.41)

The derivation is given in appendix 2.4.3 and is based on deriving a differential equation for ġint in closed form, solving it and then integrating one more time. Note that instead of G 12 we can also use v12 . Here A and B are constants to be determined from the initial conditions at horizon crossing (where slow roll is a good approximation and P (t) and its integral can be computed explicitly in certain classes of potentials) and P (t) is the particular solution of the complicated differential equation (2.79). The reason this expression containing an unknown function is useful, is that it can be shown that under certain conditions that are valid even beyond the slow-roll approximation, the term involving P is negligible, see appendix 2A.

Example: equal-power sum potentials

While a much more detailed treatment of the g int contribution to f NL based on (2.41) for general sum potentials is given in appendix 2A, we conclude this section with a simple application of (2.39) for g int to equal-power sum potentials in the slow-roll approximation, originally given in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]. We also know from appendix 2.4.1 that G x3 = G x2 /3 in the slow-roll approximation. Using these results and dropping higher-order terms in slow roll, (2.39) reduces to

g iso = ( + η -χ)(v 22 ) 2 , g sr = - * + η * 2v 2 12 + η ⊥ * v12 2 - 3 2 * + η * -χ * + η ⊥ * v12 , (2.44
)

g int = t t * dt (v 22 ) 2 2η ⊥ -η ⊥ + ( +η -χ)χ 2η ⊥ + G 12 η ⊥ χ-2η η ⊥ - 1 2 ( W211 + W222 ) .
It also proves useful to rewrite g int in a different way using integration by parts. We use the slow-roll version of relation (2.55), 2η ⊥ = -d dt G 12 (t, t ) + χG 12 (t, t ), to do an integration by parts, leading to

g int = v12 -η ⊥ * + ( * + η * -χ * )χ * 2η ⊥ * + t t * dt G 12 (v 22 ) 2 2η ⊥ χ - ( + η -χ)χ 2 2η ⊥ -2η η ⊥ - 1 2 ( W211 + W222 ) + d dt -η ⊥ + ( + η -χ)χ 2η ⊥ . (2.45)
Using the slow-roll version of the relations (1.22),

ξ = 3 η + (η ) 2 + (η ⊥ ) 2 -W111 and ξ ⊥ = 3 η ⊥ + 2η η ⊥ -η ⊥ χ -W211 , (2.46)
as well as the other time derivatives of the slow-roll parameters in (1.22), we can derive that

d dt -η ⊥ + ( + η -χ)χ 2η ⊥ = 1 2η ⊥ -χ 3 + ( + η )χ 2 -4 η + (η ⊥ ) 2 χ (2.47) +4 2 η + (η ) 2 -(η ⊥ ) 2 + η (η ⊥ ) 2 -( + η -χ) W111 + 2η ⊥ + ( + η -χ)χ η ⊥ W211 + ( + η -2χ) W221 .
Inserting this into expression (2.45) for g int and including the remaining terms in the expression for f NL we finally obtain .48) This is an alternative result for f NL in the slow-roll approximation. Equation (2.48), as well as (2.44), is characterized by the same features as the result of the exact formalism (2.39). We can easily distinguish the pure isocurvature v22 term, which we assume to vanish before the end of inflation in order for the adiabatic mode to be constant after inflation, as well as the terms evaluated at the time of horizon crossing, which are expected to be small. Any remaining non-Gaussianity at recombination has to originate from the integral. We will now further work out this expression for the case of the quadratic sum potential, as well as for a more general equal-power sum potential.

- 6 5 f NL (t) = -2(v 12 ) 2 [1 + (v 12 ) 2 ] 2 ( + η -χ)(v 22 ) 2 - * + η * 2v 2 12 - η ⊥ * v12 2 + ( * +η * -χ * )χ * 2η ⊥ * v12 - 3 2 * + η * -χ * + η ⊥ * v12 + t t * dt G 12 (v 22 ) 2 2 η η ⊥ -χ+ +η - (η ⊥ ) 2 η + 1 2 ( W211 -W222 - χ η ⊥ W221 ) - + η -χ 2η ⊥ W111 -W221 - χ η ⊥ W211 . ( 2

Quadratic potential

The quadratic potential has been widely examined in the past and it is known that it cannot produce large non-Gaussianity (see for example [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF]). Here we use our results to analytically explain why. While the quadratic potential is a special case of the more general equal-power sum potential treated below, it is still interesting to discuss it separately in a different way. We start by deriving the result that for a quadratic two-field potential

W (φ, σ) = 1 2 m 2 φ φ 2 + 1 2 m 2 σ σ 2 within slow roll, χ = d dt ln η ⊥ η .
(2.49)

Working out the right-hand side, using (1.22), we find

χ = 2 + η - (η ⊥ ) 2 η - ξ η + ξ ⊥ η ⊥ . (2.50)
Inserting the relations (2.46) (with the third derivatives of the potential equal to zero, since we have a quadratic potential) this reduces to

χ = + η - (η ⊥ ) 2 η .
(2.51)

It can be checked that this result does indeed satisfy the general equation for the time derivative of χ (1.22) within the approximations made, and the remaining integration constant is fixed by realizing that this result has the proper limit in the single-field case. This concludes the proof of (2.49).

Since the third-order potential derivatives as well as the first term of the integral in (2.48) are identically zero, we find that for a quadratic potential the integral completely vanishes in the slow-roll approximation and no persistent large non-Gaussianity is produced. Numerically we find that even for large mass ratios, where during the turn of the field trajectory slow roll is broken, the integral is still approximately zero; see appendix 2A for a detailed discussion.

Potentials of the form

W = αφ p + βσ p
For a potential of the form W (φ, σ) = αφ p + βσ q (2.52)

we can work out explicitly the form of the integrand in equation (2.48). We have to use the slow-roll version of the background field and Friedmann equations as well as of the slow-roll parameters to easily find after substitution that gint = -t t * αβp 4 (y-1)φ p-3 σ py-3 y(py-1)φ 2 + (p-1)σ 2 α 2 φ 2p σ 2 +β 2 y 2 φ 2 σ 2py 2 2κ 4 (αφ p + βσ py ) 4 (α(p -1)φ p σ 2βy(py -1)φ 2 σ py ) 2 dt , (2.53) where y ≡ q/p. We use the tilde here to indicate that this is not exactly the same g int as before, since we have split off a part using integration by parts, see (2.45).

From this expression we can derive an important result: for y = 1, i.e. p = q, we immediately see that the integral is zero. This means that no persistent non-Gaussianity can be produced after horizon exit for potentials of the form W (φ, σ) = αφ p + βσ p , at least within the slowroll approximation (with our usual assumption of vanishing isocurvature modes at the end of inflation). This generalizes the result for the two-field quadratic potential to any two-field monomial sum potential with equal powers.

Summary of additional results

Having discussed the long-wavelength formalism in detail, as synthesized from several papers, we can now start looking at applications and further extensions. As these were generally published in self-contained papers, these papers have simply been added verbatim to the appendices of this chapter, removing only their conclusions and those sections that contain material already covered elsewhere in this thesis. In this section we provide a summary of those papers, based on their conclusions. The papers in question are [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF]] (section 2.3.1 and appendix 2A) containing specific applications of the formalism, [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF] (section 2.3.2 and appendix 2B) containing a proper treatment of gauge invariance at second order and a derivation of the exact cubic action of the second-order perturbations, and [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF]] (section 2.3.3 and appendix 2C) discussing momentum dependence of the bispectrum and f NL .

Explicit solutions

In the article [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF], reproduced in appendix 2A, we discussed the levels of non-Gaussianity produced in two-field inflation with a sum potential13 W (φ, σ) = U (φ) + V (σ) and standard kinetic terms. We looked both at the case where the (strong) slow-roll approximation is valid throughout inflation (meaning that all slow-roll parameters, even the perpendicular ones, are small), and at the case where slow roll is broken during the turn of the field trajectory. An important assumption in our models is that we impose that the isocurvature mode that is present during inflation (and whose interaction with the adiabatic mode on super-Hubble scales generates the non-Gaussianity) has disappeared by the end of inflation. In that case the super-Hubble adiabatic mode is constant after inflation and we can extrapolate the results at the end of inflation directly to the time of recombination and observations of the CMB without knowing any details about the evolution of the universe in between. Without this assumption it would be much easier to create large non-Gaussianity, simply by ending inflation in the middle of the turn, but the result at the end of inflation would be meaningless from the point of view of CMB observations without a proper treatment of the transition at the end of inflation and the consecutive period of (p)reheating.

We highlighted the tension between a large f NL (of order unity or more) and the current observational bounds on the spectral index n s , both being linked to the second derivative of the potential V σσ , where σ is the sub-dominant field at horizon crossing and until the turn of the field trajectory. We evaluated these tensions (within the slow-roll approximation) for monomial potentials, where it would otherwise be easy, with some fine-tuning, to reach the requirements for a large f NL . We showed that a large part of the parameter space for f NL of order unity is simply forbidden because of the constraints on n s . However, we found that these constraints are very sensitive to the value of n s : if the lower bound were only smaller by 0.02 (n s of order 0.94), the situation would be completely different and most of the parameter space would be allowed. This analysis of the monomial potential also revealed that the duration of inflation after horizon-crossing is important: a value around fifty e-folds is much more constraining than the usual sixty e-folds. This also indicates that in the rare working models, the turn of the field trajectory occurs near the end of inflation. This raises several issues, the main one being that at that time, slow-roll parameters generally stop to be small compared to one and the slow-roll approximation does not work anymore. Moreover, if the turn occurs too close to the end of inflation, the isocurvature mode may not have time to vanish. By studying turns where the slow-roll parameter is still small compared to one we avoid this problem: the time needs to increase to one and end inflation can give enough time for the isocuvature mode to vanish.

The natural continuation of this study was to consider what would happen if we abandoned the slow-roll approximation during the turn and allowed the slow-roll parameters η and η ⊥ to become large there. On the other hand, we still assume that remains small during the turn, for several reasons: because of the issue regarding the vanishing of the isocurvature mode mentioned above, because we saw numerically in the models we looked at that this was a good approximation, and because this approximation allowed us to derive some very interesting analytical results (a potential period of large right before the turn was taken into account though). We identified two different types of models where such a turn can happen, shown in figure 2A.3. Substituting the slow-roll expression for ġint into (2.79) (of which (2.41) is the full solution), we were able to show (using simple comparisons of the different terms of the differential equation) that it is also a very good approximation even if the slow-roll parameters η and η ⊥ become large during the turn. The main idea is the following: as long as the slow-roll approximation is valid, we can compute the particular solution (see (2.41)) explicitly, while when it is broken, we can show that the particular solution becomes negligible, even though we cannot compute an analytic expression for it in that case (the fact that remains small is a crucial ingredient in this proof). For the homogeneous solution we have an analytic expression that is valid everywhere. We were also able to show that adding the slow-roll particular solution to the homogeneous solution in the regions where the exact particular solution is negligible does not introduce a significant error, which means that we do not have to perform an explicit matching of the solutions at each transition between a slow-roll and a non-slow-roll region.

This led us to the conclusion that, within the context of the models studied and the assumptions mentioned above, the slow-roll expression for f NL is a very good approximation for the exact value, even in models where η and η ⊥ become large during the turn of the field trajectory and break slow roll. Hence the implications of this expression for having large non-Gaussianity, discussed in the context of the slow-roll approximation, mostly apply to this case as well. In particular, the constraints due to the spectral index n s remain very important. A two-field sum potential with large f NL requires a lot of fine-tuning (and we showed explicitly in the section with numerical examples how to construct such a model). Reducing the error bars on the measurements of the spectral index could even shrink the parameter region of these models where f NL is of order unity more than reducing the error bars on f NL .

Gauge invariance and cubic action

In the paper [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF], reproduced in appendix 2B, we settled some previously unresolved issues concerning gauge invariance at second order in inflation with more than one field. Although the gauge-invariant curvature perturbation defined through the energy density had been known for many years, the energy density is not the quantity that is used in calculations of inflationary non-Gaussianity. These use the scalar fields present during inflation instead of their energy. We found this gauge-invariant quantity in terms of the fields and discovered that it contains a non-local term unless slow-roll is assumed.

We also managed to make contact between gauge transformations and the redefinitions of the curvature and isocurvature perturbations occurring in the third-order action. Since [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] it was known that the redefinition of the curvature perturbation in the action, introduced to remove terms proportional to the first-order equations of motion, corresponds to its gauge transformation. However, these terms appeared at first sight to be absent in the flat gauge which would have had as a consequence the absence of quadratic contributions of first-order curvature perturbations at horizon crossing in this gauge, and hence a gauge dependence of the related horizon-crossing non-Gaussianity. We extended the calculation for both gauges to second order and proved that in both of them the contributions are the same. The difference is that, in our perturbative approach, in the uniform energy-density gauge a part of these contributions is due to the first-order corrections and the other part to the second-order fields, while in the flat gauge they are all due to the second-order fields.

In addition to the adiabatic one, we also found the gauge-invariant isocurvature perturbation defined in terms of the scalar fields, by studying the relevant fully non-linear spatial gradient (2.7), first defined in [Rigopoulos et al., 2006a]. Usually isocurvature perturbations are studied in terms of the pressure of the fields. However, we found a definition using the fields themselves that demonstrates the orthogonality of this quantity to the curvature perturbation. While rewriting the action, these isocurvature perturbations appear naturally in the form we have defined them, thus showing that this quantity is the relevant one to use during inflation.

In order to achieve the above we computed the exact cubic action for the adiabatic and isocurvature perturbations, going beyond the slow-roll or super-horizon approximations (in the appendix we also gave the tensor part of the action). Previously there was no alternative to imposing the slow-roll condition at horizon crossing in order to calculate the non-Gaussianity. This was because the only two-field action available was that of the fields given in [Seery and Lidsey, 2005a], thus demanding slow-roll at horizon crossing in order to be able to use the longwavelength formalism or the δN formalism to find the curvature perturbation bispectrum. The action we provided here can be used directly with the in-in formalism [START_REF] Weinberg | Quantum contributions to cosmological correlations[END_REF] in order to calculate the exact non-Gaussianity beyond any restrictions, slow-roll or super-horizon.

Momentum dependence

In the paper [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF], included in appendix 2C, we studied the scale dependence of the local non-Gaussianity parameter f NL for two-field inflationary models. Multiplefield models with standard kinetic terms do not exhibit the strong scale dependence inherent in models that produce equilateral non-Gaussianity at horizon-crossing through quantum mechanical effects. Nevertheless, they are not scale independent in general and the interesting question is whether we can profit from their scale dependence in order to observationally acquire more information about inflation.

We calculated f NL using the long-wavelength formalism. This constrained us to assume slow roll at horizon-crossing and hence the relevant quantities at that time should not vary much, including the scale dependence of f NL for any shape of the momentum triangle. Indeed we confirmed that by introducing the conformal spectral index n K that measures the tilt of f NL for triangles of the same shape but different size (K is a variable proportional to the perimeter of the momentum triangle). For the quadratic model with mass ratio m φ /m σ = 9 we found n K 0.018, pointing to an almost scale-invariant f NL .

We also studied the scale dependence of f NL while varying the shape of the triangle and keeping its perimeter constant. f NL exhibits the opposite behaviour of the full bispectrum, i.e. it decreases the more squeezed the triangle is (the momentum dependence of the bispectrum is dominated by that of the products of power spectra, not by that of f NL ). This variation is not related to horizon-crossing quantities, but rather to the fact that the more squeezed the isosceles triangle under study is, the smaller the correlation of its two scales. We quantified this effect by introducing the shape spectral index n ω , which for the quadratic model with m φ /m σ = 9 is n ω -0.03 and has a running of about 20% (ω is defined as the ratio of the two different sides of an isosceles momentum triangle).

All our calculations were done numerically in the exact background, assuming slow roll only at horizon crossing, not afterwards. Nevertheless, semi-analytical expressions can be easily produced by directly differentiating f NL . If we do assume slow roll, we showed that we can even simplify these expressions further and find analytical formulas for the final value of f NL and its spectral indices n K and n ω , if the integral in f NL and the isocurvature modes vanish by the end of inflation, which is the case for example for any equal-power sum model.

We used the two-field quadratic potential in our numerical calculations. This potential is easy to examine and allows for simplifications in the relevant expressions. Although its final non-Gaussianity is small, O( k ), its general behaviour should not be different from other multiplefield inflationary models with standard kinetic terms, in the sense that the scale dependence of f NL should always depend on horizon-exit quantities and the evolution of the transfer functions during the turning of the fields. Indeed we checked that for the potential (2C.31) studied in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF], able to produce f NL ∼ O(1), the results remain qualitatively the same, although the values of the spectral indices are smaller due to the very slow evolution of the background at horizon crossing in that model.

Although the effect of the magnitude of the triangle on f NL had been considered before, analytical and numerical estimates were not available before this paper. In addition, it was the first time that the dependence of f NL itself (instead of the power spectra in the bispectrum) on the shape of the momentum triangle was studied. Using the long-wavelength formalism we managed to study the two different sources of momentum dependence, i.e. the slow-roll parameters at horizon crossing and the evolution of the transfer functions, and to understand the role of each for the two different triangle deformations that we studied. In summary, the later a momentum mode exits the horizon, the larger the slow-roll parameters are at that time and the larger f NL tends to be. In contrast, the final value of v12k and the initial value of G 22k k , the two transfer functions that are the most important for f NL , are smaller the later the scale exits, which results in decreasing values of f NL . These two opposite effects manifest themselves in the two different deformations we studied. When keeping the shape of the triangle constant and varying its size, it is the slow-roll parameters at horizon crossing that play the major role in f NL and result in an increasing f NL for larger K. When changing the shape of f NL , it is the correlation between the isocurvature mode at different scales, G 22k k , that has the most important role, resulting in decreasing values of f NL when squeezing the triangle (i.e. increasing ω).

We verified that the spectral indices of f NL (n K and n ω ), which we introduced to describe the effect of the two types of deformations of the momentum triangle, provide a good approximation over a wide range of values of the relevant scales. In the models we studied their values are too small to be detected by Planck, given that f NL itself cannot be big (or it would have been detected by Planck). Models that break slow roll at horizon crossing could in principle have larger spectral indices, but in order to study such models one would need to go beyond the long-wavelength formalism. Such models could be studied using the exact cubic action derived in [START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF]].

Appendix: details of the calculations

This section contains three appendices. The first discusses in more detail the Green's functions that are central to solving the equations of motion for the fluctuations, determined using the long-wavelength formalism, at each order in a perturbative expansion. The second and third appendices give some of the calculations to derive the main results in section 2.2. As these calculations tend to be rather involved, those appendices can be skipped by the more casual reader.

Green's functions

The functions G xy (t, t ) (with x, y = 1, 2, 3 and t ≥ t ) are Green's functions introduced to solve the first-order perturbation equations (and then the same functions also serve to solve the second-order equations). They satisfy the following differential equations (simply (2.29) This means that v12 * = 0, v22 * = 1 and v32 * = -χ * . The functions vx2 satisfy the same differential equation (2.54) in terms of t as the G x2 .

In the general case, these equations cannot be solved analytically. Hence, to go further, we will focus on the case t = t * and we assume that at horizon-crossing the slow-roll approximation is valid for at least a few e-folds. This means that during these few e-folds, the different slowroll parameters, which evolve slowly, can be considered as constants at the lowest order. Under these conditions, the differential equation (2.56) takes the form: g(t) + A 33 ġ(t) + A 32 g(t) = 0, (2.58) where g can be either G 22 , G 23 or v22 , differing only in initial condition. Here, A 32 and A 33 are now constants. The solution of this equation is:

g(t) = 1 λ --λ + (λ -g 0 -ġ0 )e λ + t + (-λ + g 0 + ġ0 )e λ -t , (2.59)
where

λ + = 1 2 -A 33 + (A 33 ) 2 -4A 32 , λ -= 1 2 -A 33 -(A 33 ) 2 -
4A 32 and g 0 , ġ0 are the initial values of g and ġ. In the slow-roll regime, |A 32 | 1 while A 33 ≈ 3. The direct consequence is that |λ + | 1, which implies that the e λ + t mode does not change much in a few e-folds, while λ -≈ -3, which means that the other mode decays exponentially and can be neglected after a few e-folds (three is sufficient).

For two different sets of initial conditions, the ratio between the solutions becomes: (2.54). However, one does not obtain the same factor of proportionality (2.60) with a simple integration of (2.59) because of the constant of integration. On the other hand, from (2.54) we know these functions stay small compared to one before the turn of the field trajectory, because η ⊥ is negligible compared to other slow-roll parameters. During the turn, while η ⊥ is of the same order as other slow-roll parameters or even larger, they can become large. We will see later (in chapter 2A) that typical and interesting values of v12 are larger than order unity. Hence, the only relevant part of the integral is after the beginning of the turn. To compute it, one can just integrate the first equation of (2.54) starting at the beginning of the turn instead of at horizon-crossing. Moreover, once the turn has started, we know that the relations of proportionality between G 22 * , G 23 * and v22 are already established, which means that from (2.54) the same relations exist between Ġ12 * , Ġ13 * and v12 on the only relevant part of the integration interval. Then the common factor is conserved by the integration. During the turn, (2.60) becomes valid for the Green's functions G 12 * , G 13 * and v12 . In particular this is true for the final values of these functions. If these functions stay negligible during the turn, or vanish at the end, the result does not hold. However, as already mentioned, this case is not interesting as multiple-field effects will play no role. To summarize, the explicit proportionality relations are:

g 1 g 2 = ġ1 ġ2 = λ -g 1 0 -ġ1 0 λ -g 2 0 -ġ2 0 , ( 2
G 12 * G 13 * = G 22 * G 23 * = G 32 * G 33 * = -λ -≈ 3 and v12 G 12 * = v22 G 22 * = v32 G 32 * = λ -+ χ * λ - ≈ 1. (2.61)
2.4.2 Derivation of expression (2.38) for f NL

In the case of equal momenta, equation (2.37) reduces to

- 6 5 f NL = -v 1m (t)v 1n (t) (v 1m (t)v 1m (t)) 2 t -∞ dt G 1a (t, t ) Āabc (t )v bm (t )v cn (t ) -2η ⊥ (t)v 2m (t)v 1n (t) -G 1a (t, t * )(L abc * + N abc * )v bm (t * )v cn (t * ) .
(2.62)

We remind the reader that indices l, m, n take the values 1 and 2 (components in the twofield basis), while indices a, b, c, . . . take the values 1, 2, and 3 (labeling the ζ 1 , ζ 2 , and θ 2 components). The v 1m v 1n in the numerator comes from the product of two first-order solutions, while the term between the braces comes from the second-order solution. The (v 1m v 1m ) 2 in the denominator comes from the division by the power spectrum squared. The first and last term between the braces correspond to the first and last term on the right-hand side of (2.22), respectively. The second term is the gauge correction explained in the main text above (2.36).

The explicit form of the object Ā is: (2.63) while the rest of the elements are zero. The explicit form of the objects L and N at horizon crossing (where slow roll holds) is:

Ā121 = 2 η ⊥ -4η η ⊥ + 2ξ ⊥ , Ā122 = -6χ -2 η -2(η ) 2 -2(η ⊥ ) 2 , Ā123 = -6 -2η , Ā321 = -12 η -12(η ⊥ ) 2 -6 χ -8 3 -20 2 η -4 (η ) 2 -12 (η ⊥ ) 2 +16η (η ⊥ ) 2 -6 ξ -12η ⊥ ξ ⊥ + 3( W111 -W221 ), Ā322 = -24 η ⊥ -12η η ⊥ + 24η ⊥ χ -12 2 η ⊥ + 8(η ) 2 η ⊥ + 8(η ⊥ ) 3 -8 ξ ⊥ -4η ξ ⊥ + 3( W211 -W222 ), Ā323 = 12η ⊥ -4 η ⊥ + 8η η ⊥ -4ξ ⊥ , Ā331 = -2 2 -4 η + 2(η ) 2 -2(η ⊥ ) 2 -2ξ , Ā332 = -4 η ⊥ -2ξ ⊥ , Ā333 = -2η ⊥ ,
L 111 * = * + η * , L 122 * = - * + η * -χ * , L 211 * = η ⊥ * , L 222 * = η ⊥ * , L 112 * + L 121 = 2η ⊥ * , N 112 * + N 121 * = -2η ⊥ * , L 212 * + L 221 * = 2 * + η * -χ * , N 212 * + N 221 * = χ * , (2.64)
with the other elements of N abc being zero. A slow-roll approximation which expresses θ 2 in terms of ζ 2 has been used: θ 2 = -χζ 2 . This means in particular that the subscripts a, b, c only take the values 1 and 2, but not 3. However, for consistency in the notation, we will define here all entries of L abc and N abc to be equal to zero if one or more of the indices are equal to 3.

To make the expressions a bit shorter, we will drop the time arguments inside the integrals, but remember that for the Green's functions the integration variable is the second argument. One can show that Āab1 = -∂ t A ab and hence do an integration by parts, with the result

t -∞ dt G 1a Āabc v bm v cn = 2η ⊥ v 2m v 1n + t -∞ dt A ab d dt [G 1a v bm v 1n ] + t -∞ dt G 1a Āabc v bm v cn , (2.65)
where the index c does not take the value 1. Here we used that the linear solutions v am are zero at t = -∞ (by definition), that the Green's function G 1a (t, t) = δ 1a , and that A 1b = -2η ⊥ δ b2 (exact). We see that the first term on the right-hand side exactly cancels with the gauge correction (the second term in (2.62)) that is necessary to create a properly gauge-invariant second-order result. We start by working out the second term on the right-hand side of (2.65). We find

I ≡ t -∞ dt A ab d dt [G 1a v bm v 1n ] = γ 2 * t -∞ dt A ab d dt [G 1a vbm v1n ] = A ab * G 1a (t, t * )v bm * v 1n * + γ 2 * t t * dt A ab [G 1d A da vbm v1n -G 1a A bd vdm v1n -G 1a vbm A 1d vdn ] = A ab * G 1a (t, t * )v bm * v 1n * -γ 2 * t t * dt A ab A 1d G 1a vbm vdn , (2.66)
where, as before, a subscript * means that a quantity is evaluated at t * . Using the explicit form of the matrix A (2.20) and the solutions v am (2.31) this becomes

I = γ 2 * δ m2 δ n1 -2η ⊥ * + χ * G 12 (t, t * ) + A 32 * G 13 (t, t * ) -χ * A 33 * G 13 (t, t * ) (2.67) +γ 2 * δ m2 δ n2 t t * dt 2η ⊥ v22 -2η ⊥ v22 -G 12 v32 + A 32 G 13 v22 + A 33 G 13 v32 .
Realizing that A 32 v22 + A 33 v32 = -d dt v32 we can do an integration by parts:

I = γ 2 * δ m2 δ n1 -2η ⊥ * + χ * G 12 (t, t * ) + A 32 * G 13 (t, t * ) -χ * A 33 * G 13 (t, t * ) -δ n2 2η ⊥ * χ * G 13 (t, t * ) (2.68) + γ 2 * δ m2 δ n2 t t * dt 2η ⊥ -2η ⊥ (v 22 ) 2 + -2G 12 + A 33 G 13 + η⊥ η ⊥ G 13 v22 v32 + G 13 (v 32 ) 2 .
To this result we have to add the final term on the right-hand side of (2.65). We call the sum of these two terms J:

J γ 2 * ≡ I γ 2 * + t t * dt G 1a Āabc vbm vcn , (2.69) which is J γ 2 * = δ m2 δ n1 -2η ⊥ * + χ * G 12 (t, t * ) + A 32 * G 13 (t, t * ) -χ * A 33 * G 13 (t, t * ) -δ m2 δ n2 2η ⊥ * χ * G 13 (t, t * ) (2.70) +δ m2 δ n2 t t * dt Ā122 -4(η ⊥ ) 2 + Ā322 G 13 (v 22 ) 2 + Ā333 + 2η ⊥ G 13 (v 32 ) 2 + Ā123 -4η ⊥ G 12 + ( Ā323 + Ā332 + 2η ⊥ A 33 + 2 η⊥ )G 13 v22 v32 .
From (2.63) we obtain

Ā333 + 2η ⊥ = 0, Ā323 + Ā332 + 2η ⊥ A 33 + 2 η⊥ = 18η ⊥ -4 η⊥ , Ā123 = -2A 33 + 2 + 2η , Ā122 -4(η ⊥ ) 2 = -2A 32 + 2 ˙ + 2 η , (2.71)
so that we can write

J γ 2 * = δ m2 δ n1 -2η ⊥ * + χ * G 12 (t, t * ) + A 32 * G 13 (t, t * ) -χ * A 33 * G 13 (t, t * ) -δ m2 δ n2 2η ⊥ * χ * G 13 (t, t * ) +δ m2 δ n2 t t * dt 2(v 22 ) 2 d dt ( + η ) + 2v 22 d dt v32 -4 η ⊥ G 12 + η⊥ G 13 1 2 d dt (v 22 ) 2 +2( + η )v 22 v32 + Ā322 G 13 (v 22 ) 2 + 18η ⊥ G 13 v22 v32 . (2.72)
Doing integrations by parts on the three terms in the third line we obtain

J γ 2 * = δ m2 δ n1 -2η ⊥ * + χ * G 12 (t, t * ) + A 32 * G 13 (t, t * ) -χ * A 33 * G 13 (t, t * ) +2δ m2 δ n2 -η ⊥ * χ * G 13 (t, t * ) -( * + η * ) + χ * + η ⊥ * G 12 (t, t * ) + η⊥ * G 13 (t, t * ) + ( + η )(v 22 ) 2 + v22 v32 +2δ m2 δ n2 t t * dt -2(η ⊥ ) 2 (v 22 ) 2 -( + η )v 22 v32 -(v 32 ) 2 + 9η ⊥ G 13 v22 v32 + 1 2 Ā322 + 2η ⊥ + 2 η⊥ A 33 + 2η ⊥ A 32 G 13 (v 22 ) 2 .
(2.73)

By computing the derivatives of the slow-roll parameters we find

Ā322 + 2η ⊥ + 2 η⊥ A 33 + 2η ⊥ A 32 (2.74) = 24η ⊥ χ -12η η ⊥ + 12(η ) 2 η ⊥ + 12(η ⊥ ) 3 -4η ⊥ ξ -4η ξ ⊥ -3( W211 + W222 ).
We now drop boundary terms that are second order in the slow-roll parameters at horizon crossing, since it would be inconsistent to include them given that the linear solutions used at horizon crossing are only given up to first order. Then the result is

J γ 2 * = δ m2 δ n1 (-2η ⊥ * + χ * v12 ) + 2δ m2 δ n2 - * -η * + χ * + η ⊥ * v12 + ( + η )(v 22 ) 2 + v22 v32 + 2δ m2 δ n2 t t * dt -2(η ⊥ ) 2 (v 22 ) 2 -( + η )v 22 v32 -(v 32 ) 2 + 9η ⊥ G 13 v22 v32 + 12η ⊥ χ -6η η ⊥ + 6(η ) 2 η ⊥ + 6(η ⊥ ) 3 -2η ⊥ ξ -2η ξ ⊥ - 3 2 ( W211 + W222 ) G 13 (v 22 ) 2 . ( 2
= -2(η ⊥ ) 2 (v 22 ) 2 -( + η )v 22 v32 -(v 32 ) 2 + 2η ⊥ t t * dt v22 G 23 Ξv 22 + 9η ⊥ v32 , (2.76) gint = 2 ξ ⊥ + η ⊥ ( -2η ) t t * dt v22 G 23 Ξv 22 + 9η ⊥ v32 + 2η ⊥ t t * dt v22 G 33 Ξv 22 + 9η ⊥ v32 + (v 22 ) 2 3( + η )χ + 2 3 + 6 2 η + 4 (η ) 2 + 12η (η ⊥ ) 2 + ( + η )ξ -4η ⊥ ξ ⊥ + v22 v32 3 + 3η + 6χ + 3 2 + 8 η + 3(η ) 2 + 3(η ⊥ ) 2 + ξ + (v 32 ) 2 (6 + + 3η ), (2.77)
...

g int = -(3η ⊥ -η ⊥ + 6η η ⊥ -2ξ ⊥ ) t t * dt v22 G 33 Ξv 22 + 9η ⊥ v32 + 9 η ⊥ + 6η η ⊥ -6η ⊥ χ -3ξ ⊥ -3 W211 + 2 η ⊥ -8 η η ⊥ + 6(η ) 2 η ⊥ -6(η ⊥ ) 3 -4η ⊥ ξ + (3 -2η )ξ ⊥ t t * dt v22 G 23 Ξv 22 + 9η ⊥ v32 + (v 22 ) 2 32η η ⊥ ξ ⊥ -60(η ) 2 (η ⊥ ) 2 -36η (η ⊥ ) 2 -4(η ) 2 ξ -3η ξ -12(η ) 2 χ -9η χ + 6(η ⊥ ) 2 ξ + 12η ⊥ ξ ⊥ + 6(η ⊥ ) 2 χ + 12(η ⊥ ) 4 -6ξ χ -4(ξ ⊥ ) 2 -18χ 2 -3η W111 -3 W111 + 9η ⊥ W211 + 3η W221 + 3 W221 -3η ⊥ W222 + η ξ -33η χ + 14η 3 -4(η ) 2 2 -6η 2 -12(η ) 3 -8η ⊥ ξ ⊥ -12(η ⊥ ) 2 2 -36(η ⊥ ) 2 + 5ξ 2 -3ξ -9χ 2 -9χ + 6 4 -6 3 + v22 v32 -12η (η ⊥ ) 2 -24η χ -12(η ) 3 -21(η ) 2 -9η + 4η ⊥ ξ ⊥ -15(η ⊥ ) 2 -9ξ -54χ -3 W111 + 6 W221 + 14η 2 -21(η ) 2 -57η + 3(η ⊥ ) 2 +9ξ + 6χ + 9 3 -30 2 -9 + (v 32 ) 2 -12(η ) 2 -39η + 6(η ⊥ ) 2 + 4ξ + 6χ + 3η + 3 2 -15 -36 . (2.78)
Taking the specific combination of the three expressions above that eliminates all the terms with integrals then gives the differential equation

(η ⊥ ) 2 ... g int + η ⊥ 3η ⊥ -η ⊥ + 6η η ⊥ -2ξ ⊥ gint + (η ⊥ ) 2 -12 + 6χ + 6(η ) 2 + 6(η ⊥ ) 2 + 4ξ + η ⊥ 3 W211 -8η ξ ⊥ + 2(ξ ⊥ ) 2 ġint = K 22 (v 22 ) 2 + K 23 v22 v32 + K 33 (v 32 ) 2 ,
(2.79) with K 22 , K 23 , K 33 given by where A and B are integration constants to be determined from the initial conditions and P (t) is a particular solution of the equation. This expression can then be integrated to give equation (2.41), using the fact that g int (t * ) = 0 to eliminate the additional integration constant. Note that instead of 2η ⊥ G 22 we can also use 2η ⊥ v22 as independent homogeneous solution, which integrates to v12 .

K 22 = -18(η ⊥ ) 2 χ 2 + 2(η ) 2 (η ⊥ ) 2 ξ -6η η ⊥ ξ ⊥ χ + 6(η ) 2 (η ⊥ ) 2 χ -6(η ⊥ ) 2 χξ -2(η ⊥ ) 4 ξ -6(η ⊥ ) 4 χ -18 η (η ⊥ ) 2 χ + 12 2 η (η ⊥ ) 2 + 12 (η ) 2 (η ⊥ ) 2 -6 η ⊥ ξ ⊥ χ -12 2 (η ⊥ ) 2 χ -12 (η ⊥ ) 4 -3η (η ⊥ ) 2 W111 -3 (η ⊥ ) 2 W111 + 3(η ⊥ ) 3 W211 + 3η (η ⊥ ) 2 W221 + 3 (η ⊥ ) 2 W221 -3(η ⊥ ) 3 W222 -2η η ⊥ ξ ξ ⊥ + 6 η (η ⊥ ) 2 ξ -12 2 η η ⊥ ξ ⊥ + 20 3 η (η ⊥ ) 2 + 28 2 (η ) 2 (η ⊥ ) 2 -12 η (η ⊥ ) 4 + 12 (η ) 3 (η ⊥ ) 2 -2 η ⊥ ξ ξ ⊥ -8 (η ) 2 η ⊥ ξ ⊥ + 4 2 (η ⊥ ) 2 ξ -4 3 η ⊥ ξ ⊥ -4 (η ⊥ ) 3 ξ ⊥ + 4 4 (η ⊥ ) 2 -12 2 (η ⊥ ) 4 , K 23 = -36(η ⊥ ) 2 χ -6 η (η ⊥ ) 2 -12 2 (η ⊥ ) 2 -6(η ⊥ ) 4 -6 (η ⊥ ) 2 χ + 6(η ) 2 (η ⊥ ) 2 -6η η ⊥ ξ ⊥ + 6η (η ⊥ ) 2 χ -6(η ⊥ ) 2 ξ -6 η ⊥ ξ ⊥ -12η ⊥ χξ ⊥ -3(η ⊥ ) 2 W111 -3η η ⊥ W211 -3η ⊥ W211 + 6(η ⊥ ) 2 W221 -2η ⊥ ξ ξ ⊥ -2η (ξ ⊥ ) 2 + 2(η ) 2 η ⊥ ξ ⊥ + 2η (η ⊥ ) 2 ξ -2(η ⊥ ) 3 ξ ⊥ -8 η η ⊥ ξ ⊥ + 18 (η ) 2 (η ⊥ ) 2 + 24 2 η (η ⊥ ) 2 + 4 (η ⊥ ) 2 ξ -6 2 η ⊥ ξ ⊥ -6(η ⊥ ) 4 + 6 3 (η ⊥ ) 2 -2 (ξ ⊥ ) 2 , K 33 = -18(η ⊥ ) 2 -6 (η ⊥ ) 2 + 6η (η ⊥ ) 2 -12η ⊥ ξ ⊥ -3η ⊥ W211 + 6 η (η ⊥ ) 2 + 2 2 (η ⊥ ) 2 + 2η η ⊥ ξ ⊥ -2 η ⊥ ξ ⊥ -2(ξ ⊥ ) 2 . ( 2

Non-Gaussianity in two-field inflation beyond the slow-roll approximation

This appendix contains sections 1, 3, 4, 5 and appendix B (incorporated into the main text as section 2A.2.5) of [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF], of which a summary was provided in section 2.3.1. This paper was written in collaboration with Gabriel Jung. Like for the other papers in the next appendices, the conclusions of the paper are not reproduced here, as their relevant parts were used in section 2.3. In addition, for this specific paper, section 2 and appendix A are also not reproduced here, as they were incorporated into the general description of the long-wavelength formalism in section 2.2. We use the long-wavelength formalism to investigate the level of bispectral non-Gaussianity produced in two-field inflation models with standard kinetic terms. Even though the Planck satellite has not detected any primordial non-Gaussianity, it has tightened the constraints significantly, and it is important to better understand what regions of inflation model space have been ruled out, as well as prepare for the next generation of experiments that might reach the important milestone of ∆f local NL = 1. We apply the long-wavelength formalism to the case of a sum potential and show that it is very difficult to satisfy simultaneously the conditions for a large f NL and the observational constraints on the spectral index n s . In the case of the sum of two monomial potentials and a constant we explicitly show in which small region of parameter space this is possible, and we show how to construct such a model. Finally, the general expression for f NL also allows us to prove that for the sum potential the explicit expressions derived within the slow-roll approximation remain valid even when the slow-roll approximation is broken during the turn of the field trajectory (as long as only the slow-roll parameter remains small).

2A.1 Introduction

The theory of inflation [START_REF] Starobinsky | A New Type of Isotropic Cosmological Models Without Singularity[END_REF][START_REF] Guth | The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems[END_REF][START_REF] Linde | Chaotic Inflation[END_REF] describes a period of rapid and accelerated expansion which takes place in the very early universe. It solves several issues of the pre-inflationary standard cosmology like the horizon and the flatness problems. More remarkably, inflation also gives an explanation for the origin of the primordial cosmological perturbations which are the seeds of the large-scale structure in the universe observed today.

The Cosmic Microwave Background radiation (CMB) is an almost direct window on these primordial fluctuations and its temperature and polarization anisotropies have been observed by several missions. The most recent results come from the Planck satellite [Planck 2015 XIII;Planck 2015 XX;Planck 2015 XVII] 1 , which, like its predecessors, found no disagreement with the basic inflationary predictions: the distribution of primordial density perturbations is almost but not exactly scale-invariant and it is consistent with Gaussianity. The main information is encoded in the power spectrum which is the Fourier transform of the two-point correlation function of CMB temperature/polarization fluctuations. The most interesting observable from the point of view of inflation is the spectral index n s that describes its slope, or in other words the deviation from exact scale invariance.

The Planck satellite also significantly improved the constraints on any potential deviations from a Gaussian distribution (i.e. on non-Gaussianity) [Planck 2015 XVII]. Primordial non-Gaussianity is generally parametrized by the amplitude parameters f NL of a number of specific bispectrum shapes that are produced in generic classes of inflation models. The bispectrum is the Fourier transform of the three-point correlator and in the case of standard single-field slowroll inflation it is known to be unobservably small [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]. However, this result does not hold in more general situations and many extensions of that simple case have been proposed with different predictions for non-Gaussianity, meaning that observations can in principle be used to constrain them. 2 For example, models with higher derivative operators based on the Dirac-Born-Infeld action [START_REF] Alishahiha | DBI in the sky[END_REF][START_REF] Silverstein | Scalar speed limits and cosmology: Acceleration from D-cceleration[END_REF][START_REF] Mizuno | Lorentz boost and non-Gaussianity in multi-field DBIinflation[END_REF][START_REF] Mizuno | Primordial non-Gaussianity from the DBI Galileons[END_REF][START_REF] Tzavara | Covariant second-order perturbations in generalized two-field inflation[END_REF] can produce large non-Gausianity of the so-called equilateral type. Another possibility is to consider multiple fields during inflation, which adds isocurvature perturbations to the usual adiabatic perturbation. The isocurvature perturbations can interact with the adiabatic one on super-Hubble scales (while in single-field inflation the adiabatic perturbation is constant on super-Hubble scales) which can lead to socalled local non-Gaussianity. In this case non-Gaussianity can be generated long after inflation as in the curvaton scenario [START_REF] Lyth | Generating the curvature perturbation without an inflaton[END_REF]Bartolo et al., 2004b;[START_REF] Enqvist | Non-gaussianity in curvaton models with nearly quadratic potential[END_REF][START_REF] Ichikawa | Non-Gaussianity, Spectral Index and Tensor Modes in Mixed Inflaton and Curvaton Models[END_REF][START_REF] Malik | A numerical study of non-gaussianity in the curvaton scenario[END_REF][START_REF] Sasaki | Non-gaussianity of the primordial perturbation in the curvaton model[END_REF][START_REF] Kobayashi | Spectator field models in light of spectral index after Planck[END_REF][START_REF] Enqvist | Mixed Inflaton and Spectator Field Models after Planck[END_REF][START_REF] Byrnes | Comprehensive analysis of the simplest curvaton model[END_REF][START_REF] Vennin | Inflation with an extra light scalar field after Planck[END_REF][START_REF] Hardwick | Constraining Curvatonic Reheating[END_REF], or directly after inflation during (p)reheating [START_REF] Zaldarriaga | Non-Gaussianities in models with a varying inflaton decay rate[END_REF][START_REF] Lyth | Generating the curvature perturbation at the end of inflation[END_REF][START_REF] Bernardeau | Modulated fluctuations from hybrid inflation[END_REF][START_REF] Barnaby | Nongaussianity from Tachyonic Preheating in Hybrid Inflation[END_REF]Enqvist et al., 2005a;[START_REF] Jokinen | Very large primordial non-gaussianity from multi-field: application to massless preheating[END_REF][START_REF] Elliston | General analytic predictions of two-field inflation and perturbative reheating[END_REF]. However, in this paper we will be interested in the case where this local non-Gaussianity is produced on super-Hubble scales during inflation. Since we will only talk about local non-Gaussianity in the rest of this paper, f NL should always be understood as f local NL . A large amount of work has been done to study if observably large non-Gaussianity can be produced during multiple-field inflation. This involves studying the large-scale evolution of the perturbations which can be done using different formalisms, the δN formalism [START_REF] Starobinsky | Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations[END_REF][START_REF] Sasaki | A General analytic formula for the spectral index of the density perturbations produced during inflation[END_REF]Lyth and Rodriguez, 2005b] being the most popular but the long-wavelength formalism [Rigopoulos et al., 2006a[START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Tzavara | Second-order cosmological perturbations in two-field inflation and predictions for non-Gaussianity[END_REF]Van Tent, 2011, 2012;[START_REF] Tzavara | Second-order cosmological perturbations in two-field inflation and predictions for non-Gaussianity[END_REF] offering an interesting alternative. Many results have been obtained for two fields, a number sufficient to highlight multiple-field effects (some of them have then been generalized to more fields). In the slow-roll approximation, the sum-separable [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF] as well as the product-separable potential [START_REF] Choi | Spectral Running and Non-Gaussianity from Slow-Roll Inflation in Generalised Two-Field Models[END_REF] have been solved analytically, while more general separable potentials have been studied in [START_REF] Meyers | Non-Gaussianities in Multifield Inflation: Superhorizon Evolution, Adiabaticity, and the Fate of fnl[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]. The solution beyond slow-roll for Hubble-separable models was given in [START_REF] Byrnes | Non-Gaussianity beyond slow roll in multi-field inflation[END_REF][START_REF] Battefeld | On Non-Gaussianities in Multi-Field Inflation (N fields): Bi and Tri-spectra beyond Slow-Roll[END_REF]. Different conditions for large non-Gaussianity have been found [Elliston et al., 2011b,a] depending on whether the isocurvature modes have vanished before the end of inflation or not, the latter case requiring a proper treatment of the reheating phase to be sure that the results actually persist until the time of recombination and the CMB, which is generally not done. The scale dependence of the bispectrum is also an important topic of study of the last few years. Different aspects have been to keep the text and references from the original paper.

studied, like the computation of the bispectrum in the squeezed limit, the scale-dependence of f NL or the possible observational effects [Byrnes et al., 2010a;[START_REF] Byrnes | General formula for the running of fNL[END_REF][START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF][START_REF] Kenton | The squeezed limit of the bispectrum in multi-field inflation[END_REF][START_REF] Byrnes | The hemispherical asymmetry from a scale-dependent inflationary bispectrum[END_REF][START_REF] Kenton | The Separate Universe Approach to Soft Limits[END_REF]. Another related subject that has received much attention in recent years is the study of features in the effective inflaton potential or kinetic terms (like changes in the sound speed for the inflaton interactions), possibly due to the presence of massive fields, which lead to correlated oscillations in the power spectrum and the bispectrum [START_REF] Chluba | Features and New Physical Scales in Primordial Observables: Theory and Observation[END_REF][START_REF] Achucarro | Features of heavy physics in the CMB power spectrum[END_REF][START_REF] Flauger | Productive Interactions: heavy particles and non-Gaussianity[END_REF][START_REF] Achucarro | Correlating features in the primordial spectra[END_REF][START_REF] Hotchkiss | Non-Gaussianity from violation of slow-roll in multiple inflation[END_REF][START_REF] Achucarro | Inflation with moderately sharp features in the speed of sound: Generalized slow roll and in-in formalism for power spectrum and bispectrum[END_REF]. Two codes [START_REF] Dias | Numerical evaluation of the bispectrum in multiple field inflation -the transport approach with code[END_REF][START_REF] Mulryne | PyTransport: A Python package for the calculation of inflationary correlation functions[END_REF] for numerical evaluation of the bispectrum have been recently released.

The first aim3 of the paper is to understand if it is possible to have large non-Gaussianity while staying within the slow-roll approximation. Since Planck has excluded the possibility of large local non-Gaussianity (of order 10), the reader might wonder what the interest is of looking for models with large non-Gaussianity. However, it is very important in order to understand if Planck actually ruled out any significant parts of the multiple-field model space, or if these models generically predict small non-Gaussianity. Moreover, with large non-Gaussianity in this paper we often mean an f NL of order 1, which has not yet been ruled out by Planck but which might be observable by the next generation of experiments.

For explicitness we assume a two-field sum potential (with standard kinetic terms), where explicit analytical results within the slow-roll approximation are possible (and have been derived before). In particular this question was studied within the δN formalism by the authors of [Elliston et al., 2011b,a], who concluded that with enough fine-tuning an arbitrarily large f NL is possible. However, apart from rederiving those results in another formalism, the new ingredient here is that we take into account the constraints from Planck on the other inflationary observables, in particular n s . And it turns out that satisfying the observational constraints on n s while having a large f NL and staying within the slow-roll approximation is very hard. In the case of a sum of two monomial potentials and a constant we explicitly work out the region of the parameter space (in terms of the powers of the two potentials) where this is possible. Note that we assume everywhere that the isocurvature mode has disappeared by the end of inflation. Otherwise it would be easy to get large non-Gaussianity by ending inflation in the middle of a turn of the field trajectory, but we feel that in that case the results at the end of inflation would be meaningless, since they could not be extrapolated to the time of recombination and the CMB without properly treating the end of inflation and the consecutive period of (p)reheating.

The second aim of the paper is to understand the, at first sight very surprising, numerical observation that even in the case where the slow-roll approximation is broken during the turn of the field trajectory, the analytical slow-roll expression for f NL is often still a very good approximation of the final exact result. It turns out that we can understand this using the new formulation of the long-wavelength formalism. In that formulation f NL is given by a differential equation and the solution can be written as the sum of a homogeneous and a particular solution. As we will show, the homogeneous solution can be given analytically in an exact form (without any need of the slow-roll approximation), while the particular solution is negligible exactly in the regions where slow roll is broken and we cannot compute it analytically.

This paper is organized as follows. Section 2A.2 treats the slow-roll results mentioned in aim one above. It uses increasing levels of approximation. First, the slow-roll approximation is discussed. Then we add the hypothesis that the potential is sum-separable to solve the Green's function equations and to obtain simple expressions for the observables. Finally they are applied to the specific class of monomial potentials, where the effects of the spectral index constraint on the region of the parameter space where f NL is large are computed. Some results about product potentials are given at the end of the section. In section 2A.3, we keep the sum-separable potential hypothesis to compute f NL beyond the slow-roll approximation. Two different types of generic field trajectories with a turn are discussed. We show that in the end the slow-roll expression from the previous section also gives a very good approximation of the exact result for f NL in this case. Section 2A.4 contains several specific examples to illustrate the different results of the paper. The method to build a monomial potential that produces a large f NL while satisfying all constraints is detailed, while some examples from existing literature are also discussed. Each time we compare the exact numerical results in the long-wavelength formalism to the approximated analytic expressions derived in this paper.

2A.2 Slow roll

In this section, we use several consecutive levels of approximations to simplify the main expressions of the long-wavelength formalism from section 2.2. We start by applying only the (strong) slow-roll approximation to general two-field potentials. This means that all slow-roll parameters, including η ⊥ and χ, are assumed to be small, which is a stronger approximation than the standard slow-roll approximation where only parallel slow-roll parameters are assumed to be small. Then, in the next subsection, we focus on sum-separable potentials where the Green's functions can be computed as well as the different observables. Afterwards, in the next two subsections, we specialize to the case of monomial sum potentials. The final subsection contains some results for general product potentials in slow roll.

2A.2.1 General case

We apply the slow-roll approximation to all relevant equations, starting by the slow-roll parameters themselves. Using the field equation, we obtain explicit expressions for the basis components. We then perform a first-order slow-roll expansion on the second line of (1.18) to obtain η and η ⊥ . For ξ and ξ ⊥ we proceed in a similar way on (1.22). The results are:

e 1A = -WA , η = -W11 , η ⊥ = -W21 , ξ = 3 η + (η ) 2 + (η ⊥ ) 2 -W111 , ξ ⊥ = 3 η ⊥ + 2η η ⊥ -η ⊥ χ -W211 . (2A.1)
The same slow-roll expansion applied to the differential equations for the Green's functions (2.54) and (2.55) gives:

d dt G 22 (t, t ) + χ(t)G 22 (t, t ) = 0, (2A.2) G 32 (t, t ) = -χ(t)G 22 (t, t ), G x3 (t, t ) = 1 3 G x2 (t, t ). (2A.3)
For the observables, from (2.34) we get:

n s -1 = -4 * -2η * + 2 v12 1 + v2 12 -2η ⊥ * + v12 χ * (2A.4)
and for the different terms of f NL in (2.38):

g iso = ( + η -χ)v 2 22 , g sr = - * + η * 2v 2 12 + η ⊥ * v12 2 - 3 2 * + η * -χ * + η ⊥ * v12 . (2A.5)
For g int , the slow-roll approximation is not sufficient to compute the integral. However, we can simplify the differential equation ( 2.79) to:

η ⊥ gint -η ⊥ ( -2η -χ) + ξ ⊥ ġint = K sr (v 22 ) 2 , (2A.6) with K sr = η η ⊥ ξ + 3(η ) 2 η ⊥ χ -3η η ⊥ χ 2 -(η ) 3 η ⊥ + η (η ⊥ ) 3 -η ξ ⊥ χ -η ⊥ ξ χ -(η ⊥ ) 2 ξ ⊥ + ξ ⊥ χ 2 + η η ⊥ W221 -2η ⊥ χ W221 + η ⊥ W221 -(η ⊥ ) 2 W222 + 4 η η ⊥ χ + 2 η η ⊥ -4 η ⊥ χ 2 + 3 2 η ⊥ χ -2 (η ⊥ ) 3 -ξ ⊥ χ + η ⊥ χ 3 + η ⊥ ξ . (2A.7)
To obtain (2A.6), several steps have to be followed. First, on the right-hand side of ( 2.79), one can use (2A.3) to eliminate v32 . Then one sees that the lowest-order terms (the first of each K in (2.80)) cancel each other. The remaining terms are one or two orders higher than the ones which cancel, so that in the leading-order slow-roll aproximation we only have to keep those one order higher. On the left-hand side of the equation, we also use the fact that a time derivative adds an order in slow roll, so that ... g int is one order higher in slow-rol than gint . Hence, we see that the ... g int term disappears completely from the equation. Finally, it is possible to substitute the second line of (2A.1) into the two sides of (2.79) to eliminate W111 and W211 , and after simplifying the common factor 3η ⊥ the result is given in (2A.6).

This equation can be solved for certain classes of potentials. We will look at the simple case of a sum potential, which was solved initially in [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Battefeld | Non-Gaussianities in Multi-field Inflation[END_REF] and discussed in detail in [START_REF] Byrnes | Conditions for large non-Gaussianity in two-field slow-roll inflation[END_REF]Elliston et al., 2011a,b]. The case of a product potential is treated briefly at the end of the section.

2A.2.2 Sum potential

A sum potential has the form

W (φ, σ) = U (φ) + V (σ). (2A.8)
An immediate consequence of this form is that all mixed derivatives of the potential are zero. Using this and by writing out W11 , W22 , W21 (defined in (1.16)) explicitly in terms of Wφφ , Wσσ , Wφσ and using the normalization of the basis e

(ξ ⊥ + 3η ⊥ ) W221 = (ξ ⊥ + 3η ⊥ ) W111 -(ξ + 3χ -6 ) W211 , (ξ ⊥ + 3η ⊥ ) W222 = (ξ ⊥ + 3η ⊥ ) W211 -(ξ + 3χ -6 ) W221 . (2A.12)
Note that these equations are general and not only slow-roll. After a first-order slow-roll expansion, they become:

η ⊥ W221 = η ⊥ W111 -(χ -2 ) W211 , η ⊥ W222 = η ⊥ W211 -(χ -2 ) W221 . (2A.13)
We use this to rewrite the right-hand term of (2A.6) as

K sr = 2 -3 2 η ⊥ + 3(η ) 2 η ⊥ -3(η ⊥ ) 3 + η ⊥ χ -3η η ⊥ χ + η ⊥ ξ + ξ ⊥ -η ξ ⊥ . (2A.14)
Then, one can show that a particular solution of this equation is ġint = 2 ( + ηχ)v 2 22 , which can be integrated into g int = v2

22 - * . We also know that ġint * = -2(η ⊥ * ) 2 + ( * + η *χ * )χ * from (2.76) and the initial conditions of the Green's functions. Combining this particular solution with the homogeneous solution, we get the full solution for ġint and then g int after integration, in agreement with the known result from [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]:

ġint = 2 ( + η -χ)(v 22 ) 2 - e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * η ⊥ v22 , g int = v2 22 - * -η ⊥ * - 1 2η ⊥ * ( * + η * -χ * )(χ * -2 * ) v12 , = v2 22 - * - e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * 2e 1φ * e 1σ *
v12 .

(2A.15)

Here the first two terms on the last line are the particular solution, and the last term the homogeneous solution. It is possible to show that the particular solution and the homogeneous solution are generally of the same order during inflation (this is discussed later in section 2A.3). However, we are only interested in the final values of the observables n s and f NL . As discussed before, the only large contribution in f NL can come from g int , if we suppose isocurvature modes vanish before the end of inflation, which means in terms of Green's functions that v22 and v32 vanish while v12 becomes constant. Hence in that case, the integrated particular solution is also slow-roll suppressed and only the homogeneous solution matters at the end of inflation.

From now on, the different expressions for the observables are only given at the end of inflation.

For every other parameter (like the Green's functions and the slow-roll parameters), if they are evaluated at the end of inflation, it is indicated by the subscript e.

Using the result (2A.15) with v22e = 0, we can write:

- 6 5 f NL = η ⊥ * - 1 2η ⊥ * ( * + η * -χ * )(χ * -2 * ) 2(v 12e ) 3 (1 + (v 12e ) 2 ) 2 + O(10 -2 ) = e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (v 12e ) 3 (1 + (v 12e ) 2 ) 2 + O(10 -2 ). (2A.16)
This depends on the final value of the Green's function v12 , which describes the contribution of the isocurvature mode to the adiabatic mode. Without computing it, it is possible to determine a necessary condition for f NL to be of order unity or larger. Indeed it is easy to show that, for any value of v12e :

(v 12e ) 3 (1 + (v 12e ) 2 ) 2 ≤ 3 3/2 16 ≈ 0.325. (2A.17)
If the slow-roll approximation is valid at horizon-crossing, which is the main assumption in the computation of f NL , we expect that Ṽσσ * and Ũφφ * are of order slow-roll (small compared to one). Then, the only possibility to get f NL of order unity is that one of the basis components is negligible at horizon-crossing. This means one of the fields is dominating at that time, by definition we choose it to be φ. Hence, at horizon-crossing e 2 1φ * ≈ 1 and e .18) This has to be large to have f NL non-negligible, which means that the second-order derivative V σσ * is large compared to the first-order derivative V σ * . Hence around σ * , the potential is very flat in the σ direction. In terms of slow-roll parameters, this means that

= √ 2 * κ V σσ * V σ * . ( 2A
|η ⊥ * | < ∼ |( * + η * - χ * )(χ * -2 * )|.
For the usual slow-roll order values of 10 -2 , η ⊥ * is at most of order 10 -4 .

(v 12e ) 3 ((1+(v 12e ) 2 )
2 as a function of v12e and its two upper limits 0.325 and 1/v 12e .

Another useful limit is: v3

12e (1 + (v 12e ) 2 ) 2 < 1 v12e , (2A.19)
which becomes a very good approximation if |v 12e | > 4 . These two limits are shown explicitly in figure 2A.1. From (2.54), if v12e is of order unity, this implies that at some time there was a turn of the field trajectory where both the isocurvature mode and η ⊥ are non-negligible. This turn is then a necessary condition of large non-Gaussianity. Still using the slow-roll approximation, we can go further by computing the Green's functions. From (2A.10), we get: At the end of inflation, when the fields reach the minimum of the potential, Z tends to zero. Obviously, this can only happen if there is a turn of the field trajectory at some time after horizon-crossing to make both fields evolve. Moreover, if e 2 1σ * 1 (necessary condition for f NL of order unity), Z * = V * e 2 1φ * . We then obtain, using (2A.1):

χ = 2 + η ⊥ e 2 1φ -
v12e = - V * e 1φ * W * e 1σ * = sign(e 1φ * ) √ 2 * κV * V σ * . (2A.24)
With a small enough e 1σ * , it is easy to obtain v12e larger than four or five. In figure 2A.1, this places us on the right where

(v 12e ) 3 (1+(v 12e ) 2 ) 2 ≈ 1 v12e
. The consequence for the potential V is that κV * V σ * . Substituted into (2A.16), in the case where the slow-roll parameters factor is large, we obtain:

- 6 5 f NL ≈ Ṽσσ * e 1φ * e 1σ * 1 v12e = Ṽσσ * e 1φ * e 1σ * W * e 1 σ * -V * e 1φ * = - V σσ * κ 2 V * . (2A.25)
This directly shows that f NL is of order unity when the second derivative of V * and V * itself are of the same order, while its first-order derivative V σ * is small compared to the two previous quantities because of (2A.18) and (2A.24), a result already highlighted in [Elliston et al., 2011a,b]. Larger f NL is a priori possible, but requires a fine-tuning of the model. Moreover, the sign of f NL is the sign of V σσ * . A negative f NL corresponds to a potential in the form of a ridge at t * , where σ * is very close to the maximum for the potential to be flat enough in the σ * direction, while a positive f NL corresponds to a valley potential.

In the same limit of large v12e , the spectral index takes the form

n s -1 = -4 * -2η * + 2χ * = -2 * + 2 Ṽσσ * . (2A.26)
The spectral index is close to 1, hence Ṽσσ * = Vσσ * κ 2 W * is at most of order 10 -2 . If it is smaller, this requires a fine-tuning of * . If f NL is of order unity, then V * W * is also of order 10 -2 . To summarize, at horizon-crossing, the conditions are

U * V * and |U φ * | |V σ * |.
The second-order derivative V σσ * is not negligible and can be either smaller, equal or larger than U φφ * but it is not hugely larger or smaller. To be precise, we make a quite general assumption that

|V σσ * U 2 φ * | |U φφ * V 2 σ * | and |V σσ * V 2 σ * | |U φφ * U 2 φ * |.
With these different assumptions for the potential, the expressions for the slow-roll parameters and basis vectors become:

= 1 2κ 2 U φ U 2 , η = - U φφ κ 2 U + , η ⊥ = - V σ U φ U φφ -V σσ κ 2 U , e 1φ = -sign(U φ ), e 1σ = - V σ κ √ 2 U , χ = V σσ κ 2 U + + η = U φ V σ η ⊥ + 2 .
(2A.27)

At horizon-crossing, the situation is very close to single-field inflation. In the slow-roll regime, by definition everything evolves slowly, hence a legitimate question is to ask when these conditions will stop to be valid. In fact, they will break at the turn of the field trajectory. At that time V σ stops to be negligible compared to U φ (or equivalently, e 1σ is not small compared to one). As already discussed, the turn is mandatory to have v12e large enough. However, they will also break if V stops to be negligible compared to U , this happens when the field φ is near the minimum of its potential. In this second case, we know the slow-roll approximation will also stop to be valid because is becoming large (similarly to single-field inflation). Hence, if this happens before the turn, as the slow-roll approximation is not valid anymore, we lose the analytical results for the Green's functions and f NL . We have to check if the turn can occur before the first field reaches the minimum of its potential, or in simple terms, is it possible to have f NL of order unity without breaking the slow-roll approximation? To be able to make progress in answering that question, we will consider a specific class of two-field sum potentials, where both U and V are monomial plus a possible constant.

2A.2.3 Monomial potentials

Using the results of the previous section, we want to analytically study inflation between horizoncrossing and the beginning of the turn of the field trajectory. The idea is that the slow-roll approximation is broken when the dominating field φ gets close to the minimum of its potential, and we want to verify if the turn can occur before that time. This means that the form of the potential does not need to describe the end of inflation.

We know that V (σ) has to be very flat around σ * , hence we can use an expansion in σ keeping only the largest term to write:

V (σ) = C + β(κσ) m , (2A.28)
where C, m and β are constants. Here m > 1, while β can be either positive or negative.

Because of the expansion in σ, this potential is in fact quite general. Depending on the sign of β, the potential either corresponds to a ridge where σ * is near the local maximum C (β < 0) or to a valley with σ * near the minimum (β > 0). For the potential U , there are many possibilities, we choose to focus on a monomial potential:

U (φ) = α(κφ) n , (2A.29)
with α > 0 and n > 1.

We redefine the fields as being dimensionless: φ = κφ and σ = κσ and we will omit the tildes in the redefined fields. Using the expressions for the slow-roll parameters given at the end of the previous section (2A.27), we have:

= n 2 2 1 φ 2 , η = - n(n -2) 2 1 φ 2 = - n -2 n , η ⊥ = - mβ nα 2 σ m-1 φ 2n-1 n(n -1)αφ n-2 -m(m -1)βσ m-2 = - mβ n 2n α 2 2 n-1 2 σ m-1 n n-1 (n -1)2 1-n 2 α n 2 + 1 2 -m(m -1)βσ m-2 n-1 2 , χ = 1 α 1 φ n nαφ n-2 + m(m -1)βσ m-2 = 2 n + m(m -1)β n n α 2 n/2 n/2 σ m-2 . (2A.30)
It is useful to express the slow-roll parameters as a function of instead of φ because increases after horizon-crossing, at least until the turn, and with we know exactly when the slow-roll approximation stops to be valid. and η are of the same order except in the case of n = 2 where η is of order 2 as can be checked with a second-order calculation. The next step is to use the conditions that f NL should be of order unity and n s should be within the observational bounds to constrain the free parameters of this potential. With this form of V , we have the useful relation:

(m -1)V σ = σV σσ . (2A.31)
We know that |e 1σ * | 1 and substituting (2A.31) into the expression for e 1σ in (2A.27), we can write:

e 1σ = - Ṽσσ √ 2 σ m -1 . (2A.32)
Combining this with the contraints on the spectral index (2A.26) which imply that * and Ṽσσ * are both of order 10 -2 at most, this imposes σ * to be small compared to 1. Applying these constraints due to the observables to the potential gives:

V σσ * W * = m(m -1)βσ m-2 * αφ n * + C + βσ m * ∼ O(10 -2 ), V σσ * V * = m(m -1)βσ m-2 * C + βσ m * ∼ O(1). (2A.33)
Within the limit σ * 1, we learn from these equations that αφ n * m(m-1)βσ m-2 * ∼ C +βσ m * . We also need to determine the slow-roll parameters at t * , which requires to know φ * . One way to determine this is to know the amount of inflation due to each field between horizoncrossing and the end of inflation. We can start by solving the field equation:

φ = - n φ , (2A.34) n 2 3 4 5 6 10 3
Ṽσσ * -7.7 ± 3 -3.5 ± 3 0.7 ± 3 4.8 ± 3 9 ± 3 Table 2A.1: Constraints from the spectral index on Ṽσσ * for different n with N φ = 60.

which integrates immediately to:

φ(t) = φ * 1 - t N φ , (2A.35) with N φ = φ 2 *
2n the slow-roll approximation of the number of e-folds due to φ after horizoncrossing.

The potential is known only before the turn of the field trajectory, especially for V if it is an expansion of some more complicated function. This means that we do not know the value of N φ , however it is in the range of a few to 60 e-folds. We will test different values. Nevertheless, in the simplest cases N σ (number of e-folds due to σ) is small compared to N φ . As a simple argument here, we consider the case where σ falls off a ridge, so that V * ≈ C. If V keeps the same form almost until the end of inflation, the minimum of the potential (V = 0) corresponds approximately to

σ e = (-C/β) 1/m ∼ [m(m -1)] 1/m σ 1-2/m *
, using the second part of (2A.33). For m = 2, this is of order 1, for larger m it becomes smaller (only m close to 1 is problematic). In a pure monomial potential like U without the constant term, having φ * of order unity would imply that N φ is itself of order unity. V is a bit different because of the constant term, however once σ starts to fall at a non-negligible pace (the turn), it becomes quite similar and σ goes from σ * negligible to σ e of order unity. Hence this also corresponds to N σ of order unity which can be neglected in the total number of e-folds compared to N φ . Note this is not a general proof, just a plausible argument to claim that N φ is the dominant contribution. We can also see that σ e becomes larger if V σσ * /V * in (2A.33) becomes smaller. Hence the fact that N σ is small is linked to having f NL of order unity or more.

The parameter * is related to the value of N φ , hence for these models where N σ N φ , the value of * is directly fixed by the total number of e-folds after horizon-crossing:

* = n 4N φ . (2A.36)
When * is fixed, we can use the spectral index formula (2A.26) to constrain Ṽσσ * :

Ṽσσ * = n s -1 2 + * . (2A.37)
Using n s = 0.968 ± 0.006 from the Planck data, table 2A.1 shows the constraints for integer values of n. Note that for n ≥ 5, the second-order derivative has to be positive. According to (2A.16), we also know that:

- 6 5 f NL < 0.65 η ⊥ * - 1 2η ⊥ * ( * + η * -χ * )(χ * -2 * ) , (2A.38)
which gave the estimation of η ⊥ * of order 10 -4 to get f NL of order unity. We can neglect the first η ⊥ * which is already a few orders of magnitude smaller than the single-field slow-roll typical value of f NL . Then we obtain:

- 6 5 f NL η ⊥ * < 0.325 ( * + η * -χ * )(χ * -2 * ) . (2A.39)
We can rewrite the right-hand side term:

( * + η * -χ * )(χ * -2 * ) = -Ṽσσ * 2(1 -n) n * -Ṽ 2 σσ * . (2A.40) n 2 3 4 5 6 -6 5 f NL η ⊥ *
6.8 × 10 -5 5.0 × 10 -5 2.6 × 10 -5 6.7 × 10 -5 1.2 × 10 -4

Table 2A.2: Upper bounds from the spectral index on -6 5 f NL η ⊥ * for different n with N φ = 60. This is largest for Ṽσσ * = n-1 n * , which corresponds to n s = 1 -1 2N φ ≥ 0.992 which is outside of the observed value. The maximum of the absolute value in (2A.39) will then be given by the upper or the lower bound on n s (because in the interval of the observed value for n s it can change sign). Table 2A.2 gives the numerical constraints on -6 5 f NL η ⊥ * for integer values of n. We observe that the maximum value for η ⊥ * is two orders of magnitude smaller than * for f NL of order unity. Moreover this limit is quite strong since the factor 0.325 (2A.17) is a limit which asks some fine tuning to be reached. This factor can easily be ten or a hundred times smaller. Hence, in most cases η ⊥ * will be a lot smaller than this limit. To summarize, we know * once we fix N φ . We then determine Ṽσσ * using * and the observational constraints on n s . This leads to an upper bound for |η ⊥ * | by imposing a value for f NL . However, to see when the turn exactly happens, we need to know the full evolution of η ⊥ , not just its initial value. For this, some work needs to be done on the expression for η ⊥ given in (2A.30), where we can eliminate unknown quantities (like the parameters of the potential) by using the expressions for the slow-roll parameters at horizon crossing:

* = n 2 2 1 φ 2 * , Ṽσσ * = m(m -1)βσ m-2 * αφ n * . (2A.41)
It is then straightforward to compute:

Ṽσσ = V σσ κ 2 U = Ṽσσ * σ σ * m-2 * n/2 , η ⊥ = η ⊥ * σ σ * m-1 * n/2 2 n-1 n 1/2 -Ṽσσ -1/2 2 n-1 n 1/2 * -Ṽσσ * -1/2 * . (2A.42)
As already discussed, we want to express the time dependence in terms of which is directly related to φ. However, the expression for η ⊥ also depends on σ, and while a bound for its initial value at horizon-crossing can be given using (2A.30) and the bounds on Ṽσσ * and η ⊥ * , we need to know how it evolves with time. For this we solve the field equation:

σ = - mβ α σ m-1 φ n . (2A.43)
Inserting the solution (2A.35) for φ into the equation for σ we find the following differential equation:

dσ σ m-1 = - mβ α 1 φ n * dt (1 -t/N φ ) n/2 . (2A.44)
We see that we need to consider the special cases m = 2 and n = 2 separately. We start with the most general cas m = 2 and n = 2, where (with σ * the initial value of σ):

σ = σ * 1 + m(2 -m) n(2 -n) β α σ m-2 φ n-2 * 1 - t N φ 1-n/2 -1 1 2-m = σ * 1 + 1 2 m -2 m -1 n n -2 Ṽσσ * n/2 * n/2-1 - n/2-1 * 1 2-m . (2A.45)
In the case m = 2 and n = 2, we have:

σ = σ * 1 + m(2 -m) 4σ 2-m * β α ln 1 - t N φ 1 2-m = σ * 1 + 1 2 2 -m m -1 Ṽσσ * * ln * 1 2-m , (2A.46)
while for m = 2 and n = 2:

σ = σ * exp 2β α φ 2-n * n(2 -n) 1 - t N φ 1-n/2 -1 = σ * exp n 2(2 -n) Ṽσσ * n/2 * n/2-1 - n/2-1 * . (2A.47)
Inserting these expressions into (2A.42) gives the ratio η ⊥ /η ⊥ * . In the last case m = 2 and n = 2, these equations take a nicer form:

σ = σ * 1 - t N φ β 2α = σ * 1 - t N φ Ṽσσ * 2 * = σ * * -Ṽσσ * 2 * , η ⊥ η ⊥ * = * -Ṽσσ * 2 * + 3 2 .
(2A.48)

2A.2.4 Discussion

In figure 2A.2, we use the expressions of the previous section to determine the regions of the parameter space of m and n where a turn of the field trajectory might happen before the end of the slow-roll regime. For this we want to verify when multiple-field effects start to play a role or, in terms of slow-roll parameters, we want to find when η ⊥ becomes of the same order as . We choose and not η because η is of the same order as for most cases except if n ≈ 2 when it is much smaller. First, we choose the maximum value of |η ⊥ * | possible for | -6 5 f NL | = 1 using the range of values for Ṽσσ * determined from the spectral index. Then we compute the maximum value of |η ⊥ | when = 0.1. We choose this value of because this is already close to the end of inflation and the slow-roll approximation starts to break down after that point. Moreover, if the turn starts after this time, it is possible that there is not enough time for the isocurvature modes to decay. Finally, we plot the regions of the parameter space of m and n where η ⊥ is at least as large as at that time, meaning there is a turn of the field trajectory. We also assume that N φ = 60. These are the default values for the parameters f NL , N φ and . Next we vary them to test the validity of these choices. We also explore the effects of a future improvement of the spectral index measurements.

The main conclusion of figure 2A.2 is that for most m and n, the turn cannot happen before the end of the slow-roll regime, except in the top left part of the figures (small n and large m). For example, the simple quadratic case m = 2 and n = 2 (indicated by a small cross) is excluded.

The first figure shows that obviously the space of allowed parameters decreases if we want f NL to be larger. In fact, imposing a larger f NL is the same as imposing a smaller η ⊥ * . This does not change the evolution of η ⊥ , only its initial condition, so that it will be harder to reach a final value of order .

In the second figure, we explore the effects of an improvement of the measurements of the spectral index by comparing the Planck result n s = 0.968 ± 0.006, with the accuracy expected with a CORE-like experiment where the error bar would be of order ∆n s = 0.0015. We also add the case where the error bar becomes negligible. We see that the region where f NL is at least of order unity is strongly dependent on the spectral index. Decreasing the error bars on n s decreases the parameter region where f NL is of order unity. We will see later that in fact it is the lower bound of n s which matters. If a more accurate measurement would shift the central value of n s , so that its lower bound would be slightly smaller than for Planck, then the size of the top-left region in this plot would increase. This is not indicated in the figure to keep the plot from being too busy, but n s = 0.94 is sufficient to allow most of the parameter region in the figure (m > 2 and n < 7).

The third plot shows the effect of the parameter N φ . We do not know exactly the total duration of inflation; the usual value is between 50 and 60 e-folds. Moreover, we cannot be sure that N σ can be neglected, which means that N φ is not necessarily the full duration of inflation after horizon-crossing. In this figure, we observe that the surface of the top left region diminishes for smaller N φ . In fact, for N φ smaller than 45 e-folds, it vanishes completely. The smaller N φ , the harder it will be to build a model where f NL is large.

The last figure is here to help to determine at what time the turn can occur. In the other figures, the only condition was before the end of the slow-roll regime. However, this regime is valid for most of the time after horizon-crossing. We can see that simply reducing by a factor two reduces a lot the allowed parameter region. This means that having a turn a few e-folds after horizon-crossing is extremely hard to have or even impossible. Most of the time the turn will happen near the end of slow-roll.

To explain these different behaviours, we first need to discuss Ṽσσ * . It is determined from the spectral index and * using equation (2A.37) which contains two terms: 1 2 (n s -1) which is negative and larger in absolute value for the lower bound on the spectral index, and * which is positive and can be either smaller or larger than the first term. A small * corresponds to small n and/or large N φ . This means that in each of the four figures, the left (small n) corresponds to a negative Ṽσσ * , while Ṽσσ * is positive on the right (large n). The transition happens between n = 4 and n = 5 for N φ = 60 for example. If we decrease N φ , this value decreases and the transition is shifted to the left. The same happens if we increase the lower bound on the spectral index. In every figure this transition is indicated by a dashed vertical line. The sign of Ṽσσ * is important because this corresponds to the form of the potential V at horizon-crossing. If it is positive we have a valley, while a negative value describes falling off a ridge. Now that we have seen the role of the other parameters on Ṽσσ , we have to explain the different regions by looking at the equations for the evolution of the ratio η ⊥ /η ⊥ * for the different cases. In the valley case ( Ṽσσ * > 0), σ has to decrease to the minimum at σ = 0. However, because the potential has to be very flat at horizon-crossing, we start close to the minimum. Even if σ reaches its minimum before φ, η ⊥ does not have the time to become large because in η ⊥ , the decrease of σ is opposed by the increase of . Hence, there is no allowed parameter region to the right of the dashed vertical line in the figures.

In the region of negative Ṽσσ * , the situation is the opposite: σ increases to fall from the almost flat ridge where it started. Hence in η ⊥ we have the effect of both and σ increasing. After inserting σ for the different cases into (2A.42), the only dependence on m appears in the ratio (m -2)/(m -1) which tends to 1 when m increases. This explains the asymptotic behaviour which appears on the right-hand side of the allowed region.

Looking at the different expressions for σ, we also see that the largest Ṽσσ * in absolute value makes σ increase the fastest. This implies that the lower bound on the spectral index is the most important to obtain Ṽσσ * . When n decreases, larger (in absolute value) Ṽσσ * are possible, which explains why smaller m are allowed. But in σ there are also terms which decrease when n becomes smaller and which compensate this effect, which is why for even smaller n the minimum required value of m starts to increase again.

Below equation (2A.27), the difficulty, or at least the high level of fine-tuning, needed for a model where f NL is of order unity or more in slow-roll has been highlighted. Here, we showed explicitly that this is even impossible most of the time for simple monomial potentials. However, some examples exist, when m > 4 and n < 4 generally. We also showed that N φ has to be close to the total number of e-folds after horizon-crossing which should be as large as possible given other constraints (around 60 e-folds), which implies that the turn of the field trajectory is quick. This also means that slow-roll parameters like * and η * are exactly the same as in the purely single-field case. However, the observables n s and f NL are different. Adding a second field which is responsible for the non-negligible f NL can help some single-field models which were not working well given the Planck constraints on n s to go back into the allowed range of parameters. However, this asks a lot of fine-tuning of the potential of the second field. For f NL to be of order unity or more, this asks even more fine-tuning as only the lowest spectral index values will work. This also means that the improvement of the spectral index measurements expected with a satellite like CORE would seriously constrain the possiblity of having a large f NL , especially if the central value of the spectral index moves closer to the upper bound from Planck.

+ + |-6 5 f NL |>0.2 |-6 5 f NL |>1 |-6 5 f NL |>5 1
We have also seen that in the cases that do work, most of the time the turn is near the end of the slow-roll period. This means that and the other parameters are already of order 0.1 at the start of the turn. Then parameters like η and η ⊥ can easily become of order 1 or more during the turn when things are getting more violent. The slow-roll approximation is then broken anyway. If the turn happens a bit later, we can expect that isocurvature modes will not have enough time to vanish before the end of inflation (this does not exclude the existence of some cases where they vanish in time, but only a numerical study of such examples is possible). Finally, we can imagine a case where the turn has not started when φ reaches the minimum of its potential. If this happens, there is a period of large (which would be the end of inflation in the single-field case). Again, during this period the slow-roll approximation is no longer valid. Therefore, these different situations show the need to understand what happens if the very useful slow-roll approximation is not sufficient. This is the topic of the next section, after a brief excursion to product potentials in the final subsection of this section.

2A.2.5 Product potential

In this subsection, we study the case of product potentials, which take the form W (φ, σ) = U (φ)V (σ). This case was solved analytically in [START_REF] Choi | Spectral Running and Non-Gaussianity from Slow-Roll Inflation in Generalised Two-Field Models[END_REF]. Here, we show that the slow-roll version of the g int equation (2A.6) takes a simple and nice form which is easy to deal with.

As for the sum-separable case, we start by using the specific form of the potential to find some new relations concerning its derivatives without assuming any approximation. A simple one is

W φσ = W φ Wσ W
which links the second-order mixed derivative of the potential to the firstorder ones. Then using the field equation (1.3) and the definitions of , η and η ⊥ given in (1.4) and (1.6), this relation can be rewritten in terms of slow-roll parameters:

(3 -) Wφσ = 2 3 e 1φ e 1σ (η + 3) 2 -(η ⊥ ) 2 + η ⊥ (η + 3)(e 2 1φ -e 2 1σ ) . (2A.49)
We also need the generalized version of (2A.9), valid for any two-field potential, which is:

e 1φ e 1σ ( W11 -W22 ) = (e 2 1φ -e 2 1σ ) W21 + Wφσ . (2A.50)
Combining the two previous equations and using (1.18), we obtain:

e 1φ e 1σ -3χ -ξ + χ -2 2 -4 η + 3 ξ -2(η ) 2 + 2(η ⊥ ) 2 = (e 2 1φ -e 2 1σ ) 3η ⊥ + ξ ⊥ + η ⊥ -3 ξ ⊥ + 2η η ⊥ . (2A.51)
Similar computations can be done for the third-order derivatives

W φφσ = W φφ Wσ W and W φσσ = WσσW φ W
to show that:

(3 -) Wφφσ = -2 (η + 3)e 1σ -η ⊥ e 1φ Wφφ , ( 3
-) Wφσσ = -2 (η + 3)e 1φ + η ⊥ e 1σ Wσσ .
(2A.52)

Finally, using the definitions of W221 and W222 in terms of third-order derivatives and basis components, substituting them into (2A.51) and (2A.52) and performing a first-order expansion in terms of slow-roll parameters gives:

W221 = -η -χ + (η ) 2 -2η χ + χ 2 + (η ⊥ ) 2 -ξ + χ η ⊥ ξ ⊥ , W222 = -ξ ⊥ -η ⊥ ( -2η + 2χ) - χ η ⊥ -2 2 -3 η + (η ) 2 -ξ -χ -2η χ + χ 2 - χ η ⊥ 2 ξ ⊥ .
(2A.53)

These equations can then be used to simplify the right-hand side of (2A.6), and one easily finds that in fact the right-hand side completely vanishes. Hence, the slow-roll solution consists only of the homogeneous solution and using the initial condition ġint * = -2(η ⊥ * ) 2 +( * +η * -χ * )χ * (from the slow-roll approximation of (2.76)) we find:

g int = -η ⊥ * - 1 2η ⊥ * ( * + η * -χ * )χ * v12 = - e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * 2e 1φ * e 1σ * v12 . (2A.54)
The most important thing to note here is that the second expression has exactly the same form as the homogeneous part of the sum potential case in (2A.15), without the particular solution.

As discussed there, it is that term which can give a large contribution to f NL . The natural question is then if the situation is the same for the product potential. The similarity of the expressions makes it possible to use exactly the same method to answer this question as for the treatment of the sum potential. First, we define:

gint = -2(v 12 ) 2 (1 + (v 12 ) 2 ) 2 g int = e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (v 12 ) 3 (1 + (v 12 ) 2 ) 2 , (2A.55)
which is the entire term depending on g int in f NL (2.38). As for the sum potential, the only possibility of having this expression larger than order slow-roll is to have one field dominating at horizon crossing: e 2 1φ * ≈ 1 e 2 1σ * . But at the same time, it is required that v12 is at least of order unity (and at least four to obtain the largest f NL ). The main difference with the sum potential case comes in fact from the expression for v12 . In the slow-roll approximation, it is possible to solve the Green's function equations. The computation is similar to the sum potential case and is detailed in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] where it is shown that: . These expressions are quite different from (2A.23) for the sum potential. At horizon crossing, e 2 1φ * ≈ 1 meaning S * ≈ 1. For the value of S at the end of inflation there are two different situations. As discussed several times in this paper, we want that v22 goes to zero at the end of inflation to get rid of the isocurvature mode, meaning that the situation is far closer to single-field inflation at the end of inflation than at horizon crossing. Hence, if at the end φ also dominates (same direction of the field trajectory), |e 1σ | |e 1σ * |. This means that S -S * ≈ e 2 1σ * , which leads to the fact that v12 is small compared to 1. In that case g int cannot give a large f NL . However, if σ dominates at the end of inflation (different direction of the field trajectory), we have:

v12 = -1 e 1φ * e 1σ * , (2A.57)
which is large compared to 1. We can then use that which is of order slow roll. Hence also in this case f NL is small. This is in agreement with the known conclusion that a product potential cannot give a large f NL in the slow-roll approximation with vanishing isocurvature mode at the end of inflation [START_REF] Byrnes | Conditions for large non-Gaussianity in two-field slow-roll inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF].

(v 12 ) 3 (1+(v 12 ) 2 ) 2 ≈ 1 v12 if |v

2A.3 Beyond the slow-roll regime

The previous section showed that it is difficult to have f NL not be slow-roll suppressed in the slow-roll regime. Is the situation the same if we leave this regime for a short period? Here we discuss different cases where this can happen and we will show that like in the slow-roll situation, only the homogeneous part of the solution of (2.79) is relevant once isocurvature modes have vanished. This means we will use the same quasi-single-field initial conditions at horizon-crossing as above (2A.27):

V * U * and |V σ * | |U φ * | while |V σσ * U 2 φ * | |U φφ * V 2 σ * | and |V σσ * V 2 σ * | |U φφ * U 2 φ * |.

2A.3.1 Two kinds of turns

We identified two different cases, illustrated in figure 2A.3, where the slow-roll approximation stops to be valid during the turn. In the figure, the main differences of the two situations are highlighted. With potentials of a quite similar form, we have the possibility for two different trajectories depending on the direction before and after the turn. In the previous section, the importance of the parameters and η ⊥ to study the turn has been highlighted. Graphically they are useful to determine when the turn occurs and when the slow-roll regime is broken. The first case is the one studied in the previous section. We determined that for a simple monomial potential, if the turn is possible before φ reaches the minimum of its potential, it is more likely to happen in the last few e-folds when slow-roll parameters are already of order 10 -1 , at the limit of the slow-roll approximation. Then, during the turn, η parameters may become of order unity or more, which completely invalidates the idea of an expansion in terms of small slow-roll parameters. The turn is still early enough to have η ⊥ small again at the end of inflation to make the isocurvature mode vanish. In this case the direction of the field trajectory is the same before and after the turn. This is compatible with a monomial potential where we established that N σ has to be small compared to N φ and the turn is then short.

In the second case, perpendicular terms are still negligible when becomes of order 10 -1 . Then, like in single-field inflation, continues to grow. This is the end of the slow-roll regime. From (1.22) we see that this makes η also become large (in absolute value) and a maximum of is reached when η = -. A short time after that point, starts to decrease very fast as the η term dominates in ˙ . A large η also has an effect on the perpendicular parameter η ⊥ which has been negligible until then. It is possible that η ⊥ becomes large and that the turn will occur after a few e-folds at most if f NL is of order unity. Hence, it is possible to have the turn starting with 1. This is also motivated by the assumption of isocurvature modes vanishing before the end of inflation. Indeeed, this requires a turn not too close to the end of inflation ( = 1) which is the case if is small compared to one during the turn. In this type of turn, the direction is not the same before and after. Before the turn φ is dominating but also near the minimum of its potential, while σ is still at a local maximum. Inflation ends when φ is still near its minimum but σ is also evolving towards its own minimum.

In both theses cases, we established that the slow-roll approximation can be broken. We know that solving the equations without any approximation is not possible, even in the simple case of a sum potential. However, we have also seen that is small at the start of the turn simply because of the assumption of vanishing isocurvature modes. Moreover, in ˙ (1.22), there is a factor in front. This means that when is small, cannot evolve very fast and will stay small during a short period like the turn, unless the turn is very sharp with η parameters becoming very large. Hence during the turn, except in the most extreme cases we do not treat, we still have that is small compared to one which will play an important role in this section.

In the first type of turn, this hypothesis of small has the important consequence that the slow-roll approximation is in fact broken only for the field σ. Indeed, in the field equation (1.3), each field can only affect the other through H which evolves slowly if 1. Hence, even if σ starts to evolve fast, it is only a small perturbation for φ which continues to evolve slowly during and after the turn until near the end of inflation when ≈ κ 2 φ2 /2 becomes of order unity. Hence, the derivatives of φ of order two and more are negligible. This can be used to simplify the slow-roll parameter expressions from (1.6), keeping only the terms which are larger than order slow-roll:

η = σ σ φ2 + σ2 , η ⊥ = - σ φ φ2 + σ2 , ξ = ... σ σ φ2 + σ2 and ξ ⊥ = - ... σ φ φ2 + σ2 . (2A.59)
Using this, a direct computation gives useful relations between the parallel and perpendicular parameters of the same order:

e 1φ η = -e 1σ η ⊥ and e 1φ ξ = -e 1σ ξ ⊥ . (2A.60)
In the second type of turn, the slow-roll approximation is broken for the two fields, so that these relations are then not valid. However, there is also an important approximation we can make in this case. Before the turn, the slow-roll approximation is broken during the period of large . Having large for some time also means that H decreases a lot during that period. This means that during the turn, we have:

H 2 H 2 * . (2A.61)
A brief remark about the end of inflation is necessary. We use the common definition that the period of inflation finishes when = 1. However, in the second type of turn, can be larger than 1 for a very small number of e-folds before the turn. A more complete definition of the end of inflation is then that = 1 with U U * and V V * , which ensures that the second field as well had time to evolve.

The main tool in this section is the differential equation ( 2.79) which we will call the g int equation. We have already solved it during the period of slow-roll which goes from horizoncrossing to the turn or to of order 1. We also know the exact homogeneous solution of the full equation. The only remaining work is to understand what happens to the particular solution beyond the slow-roll approximation. We will each time follow the same method. First we discuss each equation in the more general case, only supposing that η and η ⊥ are large while 1. Then, when needed to go further, we will study separately each case using (2A.60) or (2A.61) depending on the type of turn considered.

2A.3.2 Green's functions

Beyond the slow-roll regime, we have to solve the second-order differential equation (2.56) to compute the Green's functions (recalling that v22 (t) and G 22 (t, t * ) obey the same equation). We assume that the solution has the form v22 ∝ f e 1φ e 1σ , similar to the slow-roll case (2A.23). One motivation is that, during the turn, the dominant term will be (η ⊥ ) 2 and this is canceled by this form of solution. Substituting this into (2.56), we find a differential equation for the function f : In the general case, an analytical solution cannot be found. However, if we take a solution of the form f = H α by inspiration from the slow-roll solution (because that is the form of the solution until the moment when the slow-roll regime is broken), (2A.62) becomes:

e 1φ e 1σ f + 2η ⊥ (e
e 1φ e 1σ α ln H + α2 (ln H) 2 -α α ln H + α -2 + (3 + + 2η ) ln H +(α -2)(α -1) 2 -4(α -1) η -3(α -2) + 2η ⊥ (e 2 1σ -e 2 1φ ) [ α ln H + (1 -α) ] = 0. (2A.64)
There are two interesting values for α which are 1 and 2. They can be linked to the two regimes already discussed previously where the slow-roll approximation is not valid.

We can see directly that the lowest order term in slow-roll is canceled by α = 2 as expected. Moreover, the 2 term also vanishes with this value. This means that when becomes larger while the other parameters are still small compared to 1, f = H 2 is still a good approximation. This is exactly what happens at the end of the slow-roll regime just before the second type of turn, when the first field is near the minimum of its potential. Then, the complete solutions for the Green where N is a constant used to satisfy the continuity of v22 . If we call the time when this solution becomes better than the previous one t 0 , we have N = H 2 * H(t 0 ) e 1φ * e 1σ * . We cannot directly compute v12 in this regime. However, is supposed to be very small compared to 1 which means that H is almost a constant. We can then write H(t) = H 0 + δH(t) where δH(t) is only a small correction. Taking the square of this expression and doing a firstorder slow-roll expansion gives δH

(t) = 1 2 H 2 -H 2 0 H 0 . Then it is easy to deduce H(t) = 1 2 H 2 +H 2 0 H 0 .
Substituting this into (2A.67), we can perform the integration and we get:

v12 = - H 2 0 H 2 * S -S 0 4e 1φ * e 1 σ * + Z/2 + Z 0 /2 -Z * W * e 1φ
r.h.s. = H N 2 2(η ⊥ ) 4 ( + 3η ) + e 2 1φ e 2 1σ (η ⊥ ) 4 -18 -14 -36η -2(η ⊥ ) 3 ξ ⊥ + 2η (ξ ⊥ ) 2 +(η ⊥ ) 2 -3 W111 -18 -6 2 -24 η + 18(η ) 2 + 6ξ + 2 2 η +10 (η ) 2 + 12(η ) 3 + 2 ξ + 12η ξ + 3η ⊥ η W211 +η ⊥ ξ ⊥ 6 -6η -2 η -10(η ) 2 -2ξ + e 1φ e 1σ (e 2 1σ -e 2 1φ ) -6(η ⊥ ) 5 -4η (η ⊥ ) 2 ξ ⊥ +(η ⊥ ) 3 -6 + 18η + 2 2 + 12 η + 18(η ) 2 + 6ξ . (2A.69)
At first sight, this expression does not look simpler than the original one. However, it has an important new feature which is the factor in front of the whole expression. In fact, in the computation every term without cancels. Recalling that the main assumption we made is that is small during the turn, this indicates that r.h.s. might be negligible during the turn, which means that only the homogeneous solution (which is known) is needed. In the rest of this section we will show that this is indeed the case. First we have to figure out compared to what r.h.s. has to be negligible. One way to answer this question is to use what we already know about the solution: the slow-roll expression given in (2A.15) which we write as ġint = P sr + h sr with:

P sr = 2 ( + η )(v 22 ) 2 + 2 v22 v32 , h sr = - e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * η ⊥ v22 . (2A.70)
Here we used that v32 = -χv 22 in the slow-roll regime. P sr corresponds to the particular solution while h sr is the homogeneous part. We will study these two parts of the solution in the two next subsections to see how they evolve beyond the slow-roll regime. In the subsection after that, we will discuss why they are sufficient to solve the g int equation even beyond the slow-roll approximation. We start by focusing on this homogeneous solution.

2A.3.4 Fate of the slow-roll homogeneous solution

As already discussed before, the homogeneous slow-roll solution is also a homogeneous solution of the full second-order equation. Hence, we can use it and substitute it into (2.79). Then we look at each term (order1 ∝ ḣsr , order2 ∝ ḧsr and order3 ∝ ... h sr ) individually and not at the total sum because that is obviously zero. We want to show that these terms are large compared to r.h.s., so that, during the turn, r.h.s. is only a small correction which can be neglected to get a good approximation of g int . To compute the three left-hand side terms, we use the same steps as in deriving (2A.69) to get: As for the size of and H 2 * /H 2 , this depends on the type of turn. For the first type, is still of order slow-roll but it can be easily larger than Ṽσσ * by an order of magnitude. However H 2 * /H 2 is also larger than one. Moreover, if had enough time to increase since horizon-crossing, the situation is the same for H 2 * /H 2 because H decreases faster if is larger. During a few dozens of e-folds with of order slow-roll, it can also increase by an order of magnitude. This means that both terms will be of the same order during the turn in this case, or at least that neither of them is hugely smaller or larger than the other. For the second type of turn, the situation is different. During the turn, is again of order slow-roll so it is not hugely larger than Ṽσσ * . However, because of the period of large , we know that H 2 * /H 2 1 from (2A.61). Hence the factor in front of order1, order2 and order3 is large compared to the one in r.h.s. in this case.

order1 = - H N e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (e 2 1σ -e 2 1φ ) -6(η ⊥ ) 4 -2(η ⊥ ) 3 ξ ⊥ +e 1φ e 1σ 6(η ⊥ ) 5 + (η ⊥ ) 3 6(η ) 2 + 2ξ -8η ξ ⊥ (η ⊥ ) 2 + 2η ⊥ (ξ ⊥ ) 2 + 3(η ⊥ ) 2 W211 order2 = H N e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (e 2 1σ -e 2 1φ ) (η ⊥ ) 4 -3 + -6η + 2(η ⊥ ) 3 ξ ⊥ +e 1φ e 1σ (η ⊥ ) 3 6η -2 η + 12(η ) 2 + (η ⊥ ) 2 ξ ⊥ -3 + -10η + 2η ⊥ (ξ ⊥ ) 2 order3 = H N e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (e 2 1σ -e 2 1φ ) (η ⊥ ) 4 -3 -+ 6η -4(η ⊥ ) 3 ξ ⊥ +e 1φ e 1σ 6(η ⊥ ) 5 + (η ⊥ ) 3 -6η + 2 η -6(η ) 2 + 2ξ + (η ⊥ ) 2 ξ ⊥ 3 -+ 2η +3(η ⊥ ) 2 W211 . ( 2A
Next we focus on the second part of each expression, which is the part inside the braces and which is a complicated expression depending on basis components and slow-roll parameters. We start with some comments on the factors e 1φ e 1σ and e 2 1σe 2 1φ . By definition of the basis, e 1φ e 1σ goes from -1 2 to 1 2 and e 2 1σe 2 1φ from -1 to 1 and when one is at an extremum, the other one vanishes. When one vanishes, the leftover slow-roll parameter terms are similar in the different expressions. It is also not possible to have both of them small compared to one at the same time, hence the term in r.h.s. without a factor depending on the basis is not an issue. Hence, we can forget about these basis component factors which cannot change the conclusion.

The different expressions depend on all the first and second-order slow-roll parameters, except χ, W221 and W222 which have been eliminated using the relations specific to sum potentials (2A.10) and (2A.11). The first step is to study the cancellations of the left-hand side terms. An obvious one is when η ⊥ vanishes because it multiplies every term in (2A.71); the homogeneous solution vanishes in that case. It also multiplies every term in r.h.s. except the one term 2η (ξ ⊥ ) 2 . However ξ ⊥ is also small when η ⊥ becomes small. During the turn of the field trajectory, it is usual that the slow-roll parameters oscillate, hence η ⊥ can vanish several times. At those times our hypothesis that r.h.s. is much smaller than the other terms is not valid and we cannot neglect the particular solution. However, we will show in a later subsection that we have a way of dealing with this. Apart from this vanishing of η ⊥ , there is no other possibility to cancel order1, order2 and order3 simultaneously. Indeed the expressions contain similar terms, but with opposite signs or different numerical constants.

Once we know there are no cancellations in the left-hand side terms (apart from the moments when η ⊥ = 0), we can compare their expressions to r.h.s. and verify they are of the same order. As the expressions contain terms up to order five in slow-roll parameters, two cases have to be differentiated. First, the slow-roll parameters can be of order unity. Then the powers do not matter and most of the terms have to be taken into account. We remark that the terms are similar on each side of the equation, and that the numerical constants are also of the same order, so that r.h.s. cannot be very large compared to the other expressions in this case. However, the slow-roll parameters can also become larger than order unity and this situation requires more discussion. An important remark is that when the slow-roll approximation is broken, the slow-roll cancellations in (1.18) disappear which means that ξ and ξ ⊥ are of order a few times η and η ⊥ respectively, and not of order (η ) 2 and (η ⊥ ) 2 . Using the expressions for η and η⊥ in (1.22), we can see that when |η | is at a maximum, |η ⊥ | has to be of the same order because the only possibility to cancel the largest term (η ) 2 in the derivative expression is to have (η ⊥ ) 2 of the same order. However, when |η ⊥ | is at a maximum, we can see in a similar way that |η | must be of the order of a few at most.

Then we can study what happens if the perpendicular parameters are the largest (near the maximum of |η ⊥ |). If η is only a few, the dominant terms in r.h.s. and the order1,2,3 are the ones in (η ⊥ ) 5 and (η ⊥ ) 4 (or the equivalent (η ⊥ ) 3 ξ ⊥ ). The same terms exist in all the different expressions meaning the part inside the braces has to be of the same order in general. If, on the other hand, the parallel parameters are the largest, there is a term in (η ⊥ ) 2 (η ) 3 in r.h.s. which does not exist in the other expressions. However, as discussed a few lines earlier, η ⊥ is also of the same order as η at that time. Using this, the dominant terms are actually of order (η ⊥ ) 5 . Again we find similar terms inside the braces for the different expressions which have to be of the same order. Finally, the only term in r.h.s. that has no equivalent in the other expressions is (η ⊥ ) 2 W111 . This term, which is only of order three, can never be dominant because W111 cannot be large enough to make this term a lot larger than the order five ones because this parameter is also in the derivative of ξ (see (1.22)).

Hence, we have established that the terms inside the braces are of the same order in the general case for each expression in (2A.69) and (2A.71). This is exactly the situation for the second type of turn where the only hypothesis not used (2A.61) has no consequence for the terms inside the braces. However, for the first type of turn, the relations (2A.60) between the parallel and perpendicular slow-roll parameters of the same order can change the result. To verify this, we substitute them into (2A.69) and (2A.71). We also introduce the notation with {} in subscript, meaning we consider only the terms inside the braces. The computation gives:

r.h.s. {} = -e 2 1φ e 2 1σ (η ⊥ ) 2 3 W111 + 6 2 + 18 -6η ⊥ ξ ⊥ -e 1φ e 3 1σ 3(η ⊥ ) 2 W211 + (η ⊥ ) 3 -2 2 -12 -e 1φ e 1σ (η ⊥ ) 3 2 2 -6 + 2e 2 1φ (η ⊥ ) 4 , order1 {} = e 1σ e 1φ 4η ⊥ (ξ ⊥ ) 2 + 12(η ⊥ ) 5 + 6(η ⊥ ) 2 W211 + e 3 1σ e 1φ -4η ⊥ (ξ ⊥ ) 2 -6(η ⊥ ) 2 W211 + e 2 1σ 4(η ⊥ ) 3 ξ ⊥ -24(η ⊥ ) 4 + 4(η ⊥ ) 3 ξ ⊥ + 12(η ⊥ ) 4 , order2 {} = 12 e 1σ e 1φ (η ⊥ ) 5 -48 e 3 1σ e 1φ (η ⊥ ) 5 -2e 1φ e 1σ 2η ⊥ (ξ ⊥ ) 2 + (η ⊥ ) 2 ξ ⊥ ( -3) -2e 2 1σ 14(η ⊥ ) 3 ξ ⊥ + (η ⊥ ) 4 (4 -12) + 4(η ⊥ ) 3 ξ ⊥ -6(η ⊥ ) 4 + 2(η ⊥ ) 4 , order3 {} = e 1σ e 1φ (η ⊥ ) 2 (ξ ⊥ (2 -6) -6 W211 ) -24(η ⊥ ) 5 + e 3 1σ e 1φ 48(η ⊥ ) 5 + (η ⊥ ) 2 (6 W211 + ξ ⊥ (6 -2 )) + e 2 1σ 24(η ⊥ ) 3 ξ ⊥ + 8(η ⊥ ) 4 -8(η ⊥ ) 3 ξ ⊥ + (η ⊥ ) 4 (-2 -6). (2A.74)
We can directly see that the higher order terms in r.h.s. {} have disappeared but are still present in the left-hand side terms. Moreover, most of the remaining terms in r.h.s. {} are now proportional to , which makes them even smaller. Finally, the divisions by the basis components e 1φ and e 1σ , which are smaller in absolute value than one, only appear in order1 {} , order2 {} and order3 {} . All these observations leads to the conclusion that r.h.s. {} is in fact small compared to left-hand side terms for the first type of turn.

To summarize the results of the subsection, we have established that r.h.s. is negligible compared to order1, order2 and order3. With the first type of turn, this is due to the cancellations of the dominant terms in r.h.s. due to the relations between the parallel and the perpendicular parameters which exist in that case. For the second type of turn, this is simply due to the factor in front of r.h.s. which is smaller than the one in order1,2,3 because H 2 H 2 * . This means that even if the slow-roll approximation is broken, if the initial condition of that period is the slow-roll homogeneous solution, then the right-hand side of (2.79) can be neglected. This is illustrated in figure 2A.4 which displays |r.h.s.|, |order1| and |order2| (obviously order3 is not needed because it is minus the sum of the two others) for the potentials of each type of turn that are studied in section 2A.4. This figure (with a logarithmic scale) shows that r.h.s. is always several orders of magnitude smaller than the others during the turn (except at the times where η ⊥ crosses zero, which will be discussed in a later subsection).

From this section we learn that the homogeneous solution, which is known, is sufficient to solve (2.79) during the turn when the slow-roll approximation is broken (large η and η ⊥ ) as long as remains small, since the particular solution is negligible.

2A.3.5 Fate of the slow-roll particular solution

In the previous subsection, we showed that we only need the homogeneous solution of the g int equation during the turn when the slow-roll approximation is broken. However, this does not mean that we can forget about the particular solution completely. It is still required during the slow-roll evolution before and after the turn as we will show explicitly in this section (and potentially during the turn when η ⊥ crosses zero, see next subsection) and hence plays a role in principle in the determination of the integration constants in the various regions. In fact, to avoid having to perform an explicit matching at every transition it would be very convenient if we could just add the slow-roll particular solution to the homogeneous solution everywhere. We will come back to this point in the next subsection. As a preliminary we will in this subsection investigate the behaviour of the slow-roll particular solution P sr before and during the turn. We start by comparing P sr to the homogeneous solution in the different regimes. First, we focus on the slow-roll regime using the Green's functions determined in (2A.23) when the slow-roll particular solution can be written as: H 2 to compare, because e 2 1φ * ≈ 1 and by definition Ṽσσ = V σσ /(3H 2 ) and Ũφφ = U φφ /(3H 2 ). As a reminder, we want to see if P sr can be negligible compared to h sr during the slow-roll regime. First, we look at the terms inside the parentheses which both contain second-order derivatives of the potential. Then, for our models where neither of the derivatives is negligible compared to the other at horizon crossing, we can expect that in general this remains true later, at least up to the turn (it can change during the turn, but at that time the slow-roll approximation is broken and these expressions are not valid as we will discuss later in this subsection). So we conclude that the terms between parentheses in the two expressions are in general of a comparable order (the basis components in P sr can make it smaller, but not a lot smaller). If there is a difference between the two expressions, it has to come from the remaining factors, which means we have to compare to Ṽσσ * H 2 * H 2 like in the previous section. As discussed there, these have to be of the same order because in the slow-roll regime H is still of the same order as H * . There is one exception which corresponds to models where is extremely small compared to η even in the slow-roll regime (Starobinsky-like inflation for example), so that is also small compared to Ṽσσ * (in that case, there would be a similarity with the beyond-slow-roll situation studied in this section where η , η ⊥ as well). But apart from those specific models, this leads to the conclusion that in general both the particular solution and the homogeneous solution have to be taken into account during the slow-roll regime.

P sr = -2 e 2 1φ Ṽσσ
As shown in subsection 2A.3.2, the slow-roll expressions for the Green's functions are also valid in a region of large , which occurs just before a turn of the second type. The same expressions as in the previous paragraph can be used, however ξ ⊥ can no longer be neglected in h sr . On the other hand, there is no reason for ξ ⊥ to become much larger than the other term between the parentheses (which is η ⊥ ) either, given that we are still before the turn, so that in the end the conclusion about the terms between parentheses from the previous paragraph still holds. As for the other factors, both the homogeneous and the particular solutions will grow because becomes of order unity, which makes H 2 * /H 2 large compared to 1. However, at the end of this period will decrease and becomes of order slow roll again, but the ratio H 2 * /H 2 will stay large. This means that the slow-roll particular solution finishes the period of large being small compared to the slow-roll homogeneous solution. We will show below that this is fully consistent with the result for the second type of turn (the type that has a period of large right before the turn) that the slow-roll particular solution is negligible during the turn. We continue by considering the behaviour of the slow-roll particular solution during the different types of turn. Obviously, it is not an actual solution at that time, but we want to know if it would cause any problems if we were to simply add it to the solution. Again, we follow the same method using the Green's function expressions given in (2A.67) to write: ) and Ṽσσ * η ⊥ to compare. Again the two expressions have a similar form, excluding the factor HH 0 /H 2 * . As discussed in the previous subsection, Ṽσσ * is typically of order 10 -2 and hence cannot be much smaller than which is of order slow-roll. During the turn, the terms depending on the η parameters are also of the same order, except in the rare case when η ⊥ vanishes. Finally, the only large difference can come from the factor in front in the slow-roll particular solution. The two types of turn give different results. In the first type where is of order slow-roll since horizon-crossing, H and H 0 are not much smaller than H * . Then the factor is not much smaller than one. Moreover, the cases where it is the smallest are also the cases where has increased the most (and can then be larger than Ṽσσ * by an order of magnitude), so that these two effects compensate each other. Hence, the slow-roll particular solution is then typically of the same order as the homogeneous solution during the turn. In the second type of turn, the situation is different, indeed H and H 0 are of the same order and we know that H 2 H 2 * . This means that this time P sr is small and negligible during the turn compared to h sr , fully consistent with the result that P sr has become very small during the period of large just before the turn, as shown above.

P sr = H 2 H
More must be said about the slow-roll particular solution during a turn of the first type and we will now show that it becomes in fact proportional to the homogeneous solution of (2.79).

To show this, we substitute P sr in the left-hand side of (2.79), using the Green's function expressions from (2A.67), the sum potential relations from (2A.10) and (2A.11) to eliminate χ, W221 and W222 , and also the relations between parallel and perpendicular parameters (2A.60). We then compare the three terms of the equation corresponding to the three different orders of derivative (called term1, term2 and term3)4 to their sum (called l.h.s.):

l.h.s. {} = e 2 1φ e 2 1σ (η ⊥ ) 2 -6 W111 -12 2 + 12 η ⊥ ξ ⊥ + e 1φ e 3 1σ (η ⊥ ) 3 4 2 + 24 -6(η ⊥ ) 2 W211 + e 1φ e 1σ (η ⊥ ) 3 12 -4 2 + 4e 2 1φ (η ⊥ ) 4 , term1 {} = -12e 1φ e 1σ (η ⊥ ) 5 + e 3 1φ e 1σ -4η ⊥ (ξ ⊥ ) 2 -6(η ⊥ ) 2 W211 + e 4 1σ 4(η ⊥ ) 3 ξ ⊥ -24(η ⊥ ) 4 + 36e 2 1σ (η ⊥ ) 4 -4(η ⊥ ) 3 ξ ⊥ -12(η ⊥ ) 4 , term2 {} = -12e 1φ e 1σ (η ⊥ ) 5 + e 2 1φ (η ⊥ ) 4 (6 -2 ) -4(η ⊥ ) 3 ξ ⊥ + e 1φ e 3 1σ 48(η ⊥ ) 5 + 36(η ⊥ ) 3 + e 2 1φ e 2 1σ 28(η ⊥ ) 3 ξ ⊥ + 12η ⊥ ξ ⊥ + (η ⊥ ) 4 (20 -24) + (η ⊥ ) 2 6 2 -18 + e 3 1φ e 1σ 4η ⊥ (ξ ⊥ ) 2 + (η ⊥ ) 2 ξ ⊥ (6 -6) + (η ⊥ ) 3 2 2 -6 , term3 {} = e 1φ e 3 1σ -48(η ⊥ ) 5 + (η ⊥ ) 2 (ξ ⊥ (6 -6) -12 W211 ) + (η ⊥ ) 3 6 2 -18 + e 1φ e 1σ 24(η ⊥ ) 5 + (η ⊥ ) 2 (6 W211 + ξ ⊥ (6 -6 )) + (η ⊥ ) 3 18 -6 2 + e 2 1σ -32(η ⊥ ) 3 ξ ⊥ + (η ⊥ ) 2 -6 W111 -18 2 + 18 + (η ⊥ ) 4 (-26 -6) + e 4 1σ 24(η ⊥ ) 3 ξ ⊥ + (η ⊥ ) 2 6 W111 + 18 2 -18 + 20(η ⊥ ) 4 + 8(η ⊥ ) 3 ξ ⊥ + (η ⊥ ) 4 (6 + 6). (2A.78)
The discussion of these expressions is very similar to the one for (2A.69) and (2A.71). We use again the subscript {} to indicate that we have left out an overall factor (cf. (2A.69) and (2A.71)), which is here the same for all four expressions. We can see that in l.h.s. {} the higher order terms like (η ⊥ ) 5 have disappeared. Moreover, most of the terms in l.h.s. have an extra factor of , which is not the case for the other expressions. This implies that the sum of the three terms is much smaller than the individual terms of (2.79) with the slow-roll particular solution. Hence this function is in fact an approximated solution of the homogeneous equation during a turn of the first type when the slow-roll approximation is broken.

If P sr becomes a homogeneous solution it means that it has to be proportional to a linear combination of the two previously determined exact independent homogeneous solutions η ⊥ v22 and η ⊥ G 22 * . However, using (2.61), these independent solutions have in fact become proportional before the turn. Hence, we simply have that P sr and h sr are proportional. Using (2A.77) and (2A.60), we rewrite the particular solution as:

P sr = - H 2 H 2 0 e 1φ e 1σ H 4 * e 2 1φ * e 2 1σ * η ⊥ e 2 1φ . (2A.79)
We find the same factor η ⊥ as in the homogeneous solution (2A.77), but also another factor He 2 1φ . Hence, the proportionality is true only if He 2 1φ is constant during the turn. This happens if e 2 1φ ≈ 1, in that case φ is dominating meaning that and H are purely slow-roll and are almost constant during a short turn. At first, the idea of φ dominating during the turn might seem odd. However, we recall that this does not have to be during the whole turn, but only when η ⊥ and η are large enough to break the slow-roll approximation. Looking at the form of trajectory in the top left plot of figure 2A.3, the only period when φ dominates is in fact at the end of the turn when σ is oscillating around its minimum. This can also be verified with the explicit examples of the next section (see figures 2A.6 and 2A.8). Here, we can observe that η ⊥ becomes large only after the period when e 1σ was not negligible (the turn).

Different behaviours of the slow-roll particular solution depending on the type of turn have been highlighted in this subsection. In the next subsection we will discuss how these results can be used to solve the differential equation (2.79) beyond the slow-roll regime.

2A.3.6 Solution of the g int equation

As usual, we will discuss separately the two types of turn, but we start by reminding the reader about the main result of the previous subsections. The solution of (2.79) is known until the end of the slow-roll regime and it is composed of a homogeneous solution and a particular solution that both have to be taken into account. When η and η ⊥ become large, during the turn, only the homogeneous solution (which is exact and does not depend on any slow-roll approximation) is needed to solve the equation. The difficulty is then to ensure the continuity of the solution at the transition between the two regimes. In fact, after the turn, there may also be another period of slow-roll before the end of inflation, and during the turn the slow-roll parameters can oscillate and vanish for a short time, which could lead to a very brief restoration of the slow-roll conditions. So in the end there might be many transitions and it would be very inconvenient if we had to perform an explicit matching of the solutions at each of them. Fortunately, there is another option as we will now show. Finally, we also recall that the slow-roll particular solution evolves differently depending on the type of turn. In the first type, it becomes proportional to the homogeneous solution of (2.79), while in the second type it becomes negligible compared to the homogeneous solution.

It is then easy to see that the case of the first type of turn is most simply treated by keeping the full slow-roll solution at all times. Indeed, at the moment when the slow-roll regime ends and the turn starts, the solution should become only homogeneous, and that is exactly the case because the slow-roll particular solution becomes a homogeneous solution at that time. Continuity at the transition is then automatic, without the need for any explicit matching. Then, if later during the turn or at the end of the turn the slow-roll approximation is re-established, continuity is also ensured since the same solution works on both sides of the transition. Note that if η ⊥ vanishes, from (2A.60), η has to be of order slow-roll, meaning that the slow-roll approximation is indeed restored during these brief moments.

The second type of turn deserves a longer discussion. Indeed, we do not know the full particular solution during the period of large just before the turn but we know two things: the slow-roll particular solution vanishes (but it is not an exact particular solution at that time) and the right-hand side of (2.79) can be neglected once this period has finished, because r.h.s. is negligible at the start of the turn. These two ingredients are sufficient to prove that the particular solution during the period of large vanishes, even without having its explicit form. To stay general, we write the particular solution P as P = P sr + Ah + P ⊥h , where A is a constant, h the homogeneous solution, and P sr the slow-roll particular solution. P ⊥h is the function that, when inserted into (2.79), gives those right-hand side terms that are not given by P sr , and which is zero when these terms vanish (in other words, it does not contain a homogeneous solution). We know that over the course of the period of large , P sr vanishes (see (2A.77)). The right-hand side of (2.79) vanishes during that period too, which means that P ⊥h has to vanish by definition. The only remaining term could then be the one proportional to the homogeneous solution, but it has to be zero because of the matching conditions at the start of the period of large . Indeed at the end of the slow-roll regime, the particular solution is simply P sr while P ⊥h has to be zero, because the terms of higher order in slow-roll are still negligible and will grow only later during that period of large . The function h is not zero at the transition, hence A has to be. Without knowing the exact formula for P , we can conclude that it vanishes during that period of large . Hence, at the start of the turn, the solution is simply the slow-roll homogeneous solution.

During the second type of turn, keeping the slow-roll particular solution, even if it is not a particular solution of the exact equation at that time, only induces a negligible error, but it solves any potential issues with matching to later slow-roll periods. When η ⊥ vanishes, η can be larger than order slow-roll in this type of turn. This is not an issue because then the parameters evolve very fast, meaning that a very short time before η ⊥ vanishes, the particular solution is still negligible compared to the homogeneous solution, and the same a very short time after. Moreoever, one can verify that at the exact time when η ⊥ = 0, the particular solution is 1 2 P sr and we know that this function is negligible during the rest of the turn. Then it is possible to add this particular solution to the full solution only for these very short periods (without using matching conditions, because at the time of the matchings it it is negligible). It is also important to remember that in the end we are interested in the integrated g int , and when η ⊥ vanishes, the right-hand side of (2.79) is also very small compared to its value a short time before or after (because every term contains η ⊥ except one which also becomes small), meaning this particular solution is also small at that time compared to its usual value during the turn. In the integral it is then negligible. In fact, when η ⊥ vanishes, the only thing that happens is that the whole solution almost vanishes (but the particular solution does not vanish at that exact same time), but because the homogeneous solution is zero, it cannot be large compared to the particular solution for once.

To summarize, we have shown that for both types of turn, the slow-roll solution of (2.79) is sufficient to solve this equation even beyond the slow-roll regime, under the condition that stays of order slow-roll during the turn. Of course, knowing the solution ġint which is given in (2A.70) is not sufficient, we also have to integrate it. But the computation is exactly the same as in the slow-roll case even if the slow-roll approximation is not valid, meaning that g int has again the same form: Once the form of g int is known, it is possible to compute f NL at the end of inflation:

g int = v2 22 - * - e 2 1φ
- 6 5 f NL = e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ * (v 12e ) 3 (1 + (v 12e ) 2 ) 2 + O(10 -2 ). (2A.81)
This expression has the same form as the slow-roll one (2A.16), the difference is hidden in the Green's functions which have been computed in subsection 2A.3.2. The same discussion of this expression as in section 2A.2 holds and the conclusions are the same, see (2A.18). Like in that section, we use the limit (2A.19) which is a good approximation when |v 12e | > 4. Then the only remaining step is to study the value of v12 at the end of the turn using (2A.68), when the slow-roll approximation is valid again, which is equal to v12e .

As usual, we need to distinguish the two types of turn because they have different initial and final conditions. In the first case, the turn occurs early which means that U 0 V 0 (as defined before, the subscript 0 indicates that the function is evaluated at t 0 when the slow-roll approximation stops to be valid). However because there is a turn, we cannot neglect e 1σ0 anymore. We can then write S 0 = e 2 1φ0e 2 1σ0 = 1 -2e 2 1σ0 and Z 0 ≈ -U 0 e 2 1σ0 . Moreover, before the turn we are still in slow-roll, meaning that 0 1 and we can use the slow-roll expression H 2 0 = κ 2 U 0 /3. At the end of the turn, the situation is similar to single-field inflation in the direction φ meaning that Z ≈ 0 and S ≈ 1. Inserting this into (2A.68), we obtain:

v12e = U 0 W * 2e 2 1σ0 4e 1φ * e 1σ * + -U 0 e 2 1σ 0 2W * e 1φ * e 1σ * + -V * W * e 1φ * e 1σ * = -V * W * e 1φ * e 1σ * . (2A.82)
This is exactly the same limit as in the slow-roll situation. Hence for this first type of turn, we get the same result:

- 6 5 f NL = - V σσ * κ 2 V * . (2A.83)
The implications of this result were already discussed in section 2A.2.

In the second type of turn, the situation is slightly different. Firstly, the slow-roll approximation is not valid at the time t 0 , at the end of the period of large . Moreover, at that time we are still in a single-field case (φ dominates), hence S 0 ≈ 1 and Z 0 ≈ -V 0 ≈ V * (because even if U 0 is not zero, it cannot be large compared to V 0 because we are near the moment when φ reaches the minimum of U ). After the turn, the single-field situation is now in the σ direction, hence S ≈ -1. At the end of inflation, the situation is:

v12e = - H 2 0 2H 2 * e 1φ * e 1 σ * + -Z * 2W * e 1φ * e 1 σ * = -3 κ 2 H 2 0 -Z * 2W * e 1φ * e 1 σ * . (2A.84)
Substituting this into f NL , we obtain:

- 6 5 f NL = - 2V σσ * 3H 2 0 + κ 2 V * . (2A.85)
However, we can add that H 2 0 > κ 2 W 0 /3 because is not negligible (equality in the slow-roll case). Moreover,

W 0 = U 0 + V 0 ≈ U 0 + V * > V * .
We can then write:

|v 12e | > -V * W * e 1φ * e 1σ * , (2A.86)
which has an immediate consequence for f NL :

- 6 5 f NL < - V σσ * κ 2 V * . (2A.87)
In this case, the value of f NL is smaller than the slow-roll result. However, it is easily of the same order because U 0 and V 0 are of the same order while even if 0 = 1, it only changes the factor between H 2 0 and κ 2 W 0 from 1/3 to 1/2. So in the end we have derived the rather surprising result that in the class of models considered (two-field sum potentials), the slow-roll expression for f NL gives a very good approximation of the exact result, even in the case where the slow-roll approximation breaks down during the turn. Allowing for the break-down of slow-roll does however increase the region of the parameter space where large non-Gaussianity can occur compared to the results shown in figure 2A.2, because we no longer have the constraint that the turn has to happen before the end of the slow-roll regime.

2A.4 Numerical examples

Here, we provide several explicit examples to illustrate the different results of the previous sections. We also show how to explicitly construct a model that produces f NL of order unity while satisfying all observational constraints.

2A.4.1 Double quadratic potential

The double quadratic potential has the form:

W (φ, σ) = 1 2 m 2 φ φ 2 + 1 2 m 2 σ σ 2 . (2A.88)
It has been studied and discussed in many papers, see e.g. [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]]. However, it is always a good introductory example. Without taking into account the exact constraints of the monomial potential yet, we keep the main idea that the second field has a negligible effect at the time of horizon-crossing. This can be achieved by taking m 2 Van Tent, 2011]: m φ = 20m σ and m σ = 10 -5 κ -1 . As initial conditions, we use φ i = 13κ -1 and σ i = 13κ -1 , while their derivatives φi and σi are determined by the slow-roll approximation.

In figure 2A.5 we show how the various relevant quantities evolve during the turn of the field trajectory. First, one can see clearly when the turn occurs: η ⊥ becomes large and e 1σ becomes of the same order as e 1φ . We also see that this example corresponds to the second type of turn where φ reaches the minimum of its potential and is of order unity before the turn. Another remark is that the second-order parameters ξ and ξ ⊥ do not give new information compared to the first-order parameters η and η ⊥ , at least not by eye. However, in this model the two most important constraints and goals, concerning the two observables n s and f NL , are not achieved. The spectral index, which is 0.92, is clearly outside the bounds from the Planck observations. f NL is slow-roll suppressed and far from the goal of f NL of order unity. Moreover, v12e is only -1.5 which is smaller in absolute value than the value 4 needed to use the approximations (2A.25) and (2A.26) for f NL and n s .

The main result of the previous section was the validity of the slow-roll expressions in cases beyond slow-roll, like this one, at least to give an estimation of the Green's functions. Hence, we can use this approximation to compute v12e to see how the situation can be improved. Using dimensionless fields, (2A.27), the slow-roll expression for v12e given by (2A.24) becomes:

v12e = - V * W * e 1φ * e 1σ * = - √ 2 V * V σ * = - σ * φ * . (2A.89)
This shows that v12e can be increased only by changing the initial conditions. Assuming that now we have v12e large enough, f NL takes the form:

- 6 5 f NL = - V σσ * V * = 2 σ 2 * . (2A.90)
The value of f NL becomes smaller if we increase σ * . Hence, it is impossible to increase both v12e and f NL at the same time. One can also verify there is no optimal value of σ * where f NL would be larger than order slow-roll, meaning that this potential cannot produce large persistent non-Gaussianity. Instead of looking directly at f NL , we could also have used the conclusion that for a monomial potential N σ ∝ σ 2 has to be of order unity to have f NL large, which requires here to decrease σ * and v12e . The solution is then to add an extra parameter in the potential.

2A.4.2 How to build a monomial potential model that produces f NL of order unity

The form of the potentials we are interested in is:

W (φ, σ) = α(κφ) n + C + β(κσ) m +λ(κσ) m , (2A.91)
which is the one studied in section 2A.2.3. There is an extra term with m > m inside the parentheses to complete the model (i.e. make sure it has a minimum) and we will choose it to be negligible until after the turn. Hence this does not change the different expressions determined for a monomial potential. A first step is to choose the value of m and n using figure 2A.2 to be in the region where f NL of order unity is possible. α can be put as an overall factor of the whole potential, hence it does not count in the number of parameters. φ * is given by N φ * ≈ 60 and this also determines * because it only depends on φ * . Once * is known, it is possible to determine σ * , β and C using the three constraints we have (f NL , n s and v12e ) as follows.

We can start by choosing the value of f NL and (2A.25) takes the following form for a monomial potential:

- 6 5 f NL = - V σσ * κ 2 C . (2A.92)
Using (2A.26) and the lower bound on the spectral index n s = 0.962, as this is the easiest way to get a large f NL , we have:

V σσ * = κ 2 U * n s -1 2 + * . (2A.93)
Finally, we need v12e > 4. Using the slow-roll expression for v12e in (2A.24), (2A.27) and (2A.31), we get:

v12e = - V * W * e 1φ * e 1σ * = - √ 2 * κC V σ * = - √ 2 * m -1 σ * κC V σσ * . (2A.94)
A last step is to determine λ, this is done using the fact that the minimum of the potential has to be zero. Then it is possible to verify if the last term is really negligible at horizon-crossing, if not it is possible to increase m to decrease it because σ * is small compared to one. We will now apply this to two different potentials with a turn of the first type.

2A.4.3 First type of turn

First example: n = 2 and m = 4

This first example corresponds to the case where the turn occurs early enough to have a trajectory with the same direction before and after the turn, see the top right plot of figure 2A.3 for an illustration of the field trajectory. The potential is:

W (φ, σ) = αφ 2 + C + βσ 4 + λσ 6 , (2A.95) with α = 1 2 κ -2 , C = 4 27 β 3
λ 2 , β = -12.5, and λ = -4 3 βκ 2 . The intial conditions are φ i = 16κ -1 and σ i = 0.09κ -1 and, as usual, φi and σi are determined by the slow-roll approximation.

With this, it is possible to obtain an analytical estimate of the observables. First, we need to compute φ * and σ * , using the solutions of equations (2A.34) and (2A.44) determined for monomial potentials. These solutions were computed assuming that φ * and σ * were the initial conditions, one has just to replace them by φ i and σ i here. This quick computation gives:

φ * = 15.2κ -1 and σ * = 0.092κ -1 . (2A.96)
Using these values and the different expressions (2A.92), (2A.93) and (2A.94), we obtain:

v12e = - 2 φ * 3 σ * κC 12βσ 2 * = 3.52, n s = 1 - 4 φ 2 * -2 12βσ 2 * κ 2 αφ 2 * = 0.961, - 6 5 f NL = - 12βσ 2 * κ 2 C = 1.2.
(2A.97) In these calculations, there are different approximations. First we use the monomial expressions to compute φ * and σ * (we refer the reader to section 2A.2.3 for the details, but they require the slow-roll approximation and a quasi single-field situation, at least until horizon-crossing). Second, we use the limit of large v12e to compute the observables, the validity of this limit is explained in detail in section 2A.2 as well. Hence, an error of order slow-roll (at horizoncrossing) is expected compared to the exact numerical results, which can be larger here since v12e is a little smaller than four.

Figure 2A.6 contains the same plots as shown for the double quadratic potential except that we have removed the plot of ξ and ξ ⊥ which does not provide any additional information, and added a plot of g int . The different analytical predictions in (2A.97) are reasonable estimations of the different parameters but the difference is larger than expected, especially on the new plot concerning g int . This plot displays both the exact numerical gint and its analytical prediction from (2A.80) (more precisely, the analytical form of the approximated solution, with the different parameters determined numerically), using the definition: As one can see, both curves have a similar form, but there is a difference of around 15%. The reason is that the turn occurs late with ≈ 0.2 when it starts. This value is already too large to have the slow-roll approximation working perfectly, but not enough for it to totally break down. In fact, this problem is quite general with the monomial potential because the turn has to occur late to get f NL of order unity, as we have shown.

gint = - 2(v 12 ) 2 (1 + (v 12 ) 2 ) 2 g int . (2A.98)
However, if we forget momentarily about the observational constraint on the spectral index, only for one example to illustrate the validity of the analytical expressions, it is possible to have the turn occuring earlier. The second set of values is: .6 but for the new parameters, shows that is of order 10 -2 during the turn, which is in the domain of validity of the main hypothesis 1. During the turn, η ⊥ is of order 10 at most, which shows that the slow-roll regime is broken. As expected, analytical predictions are now a very good estimation. However, the spectral index is 0.917, which is outside the observational bounds. This example is also used in the previous section in figure 2A.4 to illustrate that r.h.s. is several orders of magnitude smaller than the left-hand side terms of (2.79).

α = 1 2 κ -2 , C = 4

Second example: Axion

The next example is the axion-quartic model originally introduced in [START_REF] Elliston | Large trispectrum in two-field slow-roll inflation[END_REF] and discussed more recently in [START_REF] Dias | Numerical evaluation of the bispectrum in multiple field inflation -the transport approach with code[END_REF]. The potential is:

W (φ, σ) = 1 4 gφ 4 + Λ 4 1 -cos 2πσ f , ( 2A
σ i = f 2 -10 -3 κ -1 . Defining σ = f 2 -σ,
we have σ κ -1 . This will stay true until the turn, hence it is possible to perform an expansion of the potential in terms of this small parameter.

At first order, we have cos

2πσ f = -cos 2πσ f = -1 + 1 2 2πσ f
2 which substitued into the potential gives:

W (φ, σ ) = 1 4 gφ 4 + 2g 25 2π 2 κ -4 - 1 2 g 25 f 2 κ -4 (σ ) 2 . (2A.102)
This is a monomial potential with n = 4 and m = 2, hence in the region of parameters where the spectral index constraints cannot be satisfied. This is verified by computing the analytical predictions like for the previous example. The fields at horizon-crossing are:

φ * = 21.8κ -1 and σ * = -1.1 × 10 -3 κ -1 . (2A.103) which leads to: v12e = - 8κ φ * σ * f 2π 2 = -8.4, n s = 1 - 16 φ 2 * -8 25 2 κ 2 f 2 φ 4 * = 0.944 and - 6 5 f NL = 2π 2 .
(2A.104) This model gives f NL of order ten, however the spectral index is lower than the Planck constraints.

Figure 2A.8 confirms these results. Again in this model the turn occurs very late and there is a shift between the prediction and the exact result even if is still small enough during the turn. Moreover, η and η ⊥ stay smaller than one during the turn, but χ, which is displayed on the same plot, becomes large. This is another regime than the ones studied in subsection 2A.3.2. This has a direct impact on the Green's functions because χ appears in (2.56) which explains the difference between the slow-roll prediction for v12e and the exact value. However, one interesting point is that the analytical form of g int stays valid. This case of large χ when other slow-roll parameters are small is not common and is due here partially to the fact that Ṽσσ * is too large to respect the Planck constraint (because as discussed in section 2A.2, χ * = * + η * + Ṽσσ * ).

2A.4.4 Second type of turn m = 2 and n = 2

Figure 2A.2 shows that a turn of the first type respecting observational constraints is not possible for a monomial potential with n = 2 and m = 2. However, if we do not keep the constraint that the turn must start before the end of the slow-roll regime, this model can have a turn of the second type. This example was published originally in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] and is here adapted to be in agreement with the latest Planck constraints. See the second line of plots in figure 2A.3 for an illustration of the field trajectory. The potential has the form:

W (φ, σ) = αφ 2 + C + βσ 2 + λσ 4 , (2A.105) with α = 20κ -2 , C = β 2 4λ , β = -9κ -2
and λ = 2. The initial conditions are φ i = 18κ -1 and σ i = 0.01κ -1 with φi and σi determined by the slow-roll approximation. At horizon-crossing, we have:

φ * = 14.9κ -1 and σ * = 0.011 × 10 -3 κ -1 . (2A.106)
Substituted into (2A.92), (2A.93) and (2A.94) this gives:

v12e = - 2κ φ * σ * C 2β = 6.9, n s = 1 - 4 φ 2 * + 4β κ 2 αφ 2 * = 0.974 and - 6 5 f NL = - 2β κ 2 C = 1.8.
(2A.107) Figure 2A.9 confirms that in this example the turn occurs after the field φ reaches the minimum of its potential. The Green's function v12e is larger than the slow-roll value, hence f NL is a little smaller than expected. This is in agreement with the discussion of the second type of turn at the very end of section 2A.3. However, this does not have any impact on the spectral index because the dependence on v12e disappears when it is larger than 4. Hence, this model is allowed by the Planck constraints.

A non-monomial example

This last example is in the vein of the previous one in terms of the form of the field trajectory. However, there are several supplementary terms to show the validity of some analytical results beyond simple monomial potentials. The model has the following potential:

W (φ, σ) = 1 4 λ φ 4 + σ 4 + m 4 -2m 2 φ 2 -2m 2 σ 2 + ν(m -φ) 3 + W 0 , (2A.108)
with λ = 1200, ν = 100κ -1 , m = 2κ -1 and W 0 = 1 4 λm 4 . The initial conditions are φ i = 25κ -1 and σ i = 0.05κ -1 . We cannot use the monomial potential equations to determine φ * and σ * , however the slow-roll estimation of f NL does not require them:

- 6 5 f NL = 4 κ 2 m 2 = 1.
(2A.109) Figure 2A.10 shows a similar behaviour as for the previous example. Again f NL is smaller than its slow-roll prediction. The reason is still the same, the period of large makes v12e larger by a factor of order unity than in the slow-roll approximation and the direct consequence is that f NL is reduced by the same factor.

2B.1 Introduction

The concept of inflationary curvature perturbations was first invoked in order to explain the primordial fluctuations that source the CMB anisotropy and structure formation [START_REF] Guth | Fluctuations in the New Inflationary Universe[END_REF][START_REF] Hawking | The Development of Irregularities in a Single Bubble Inflationary Universe[END_REF][START_REF] Starobinsky | Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations[END_REF][START_REF] Bardeen | Spontaneous Creation of Almost Scale -Free Density Perturbations in an Inflationary Universe[END_REF]]. The inflationary paradigm has been observationally tested for more than 10 years and its prediction for an almost scale invariant spectrum of the first-order curvature perturbations has been verified by the data of many experiments (see for example [WMAP Collaboration, 2011]). Since the definition of perturbations depends in general on the gauge choice, a gauge-invariant definition of the cosmological perturbations is of vital importance to make contact with physical observables, which are obviously gauge-invariant. That was investigated in detail in [START_REF] Bardeen | Gauge Invariant Cosmological Perturbations[END_REF] and later in [START_REF] Mukhanov | Theory of cosmological perturbations[END_REF]. In the mean time the need for more observational quantities than just those based on linear perturbation theory has become clear, in order to break the degeneracy of the immense number of inflationary models. One of the most fruitful has proven to be the non-Gaussian characteristics of the perturbations. This has led to the development of new methods to study the combination of the scalar field and metric perturbations, the one sourcing the other, in a gauge-invariant way beyond first order. It was not until 2003 that Malik and Wands in [START_REF] Malik | Evolution of second-order cosmological perturbations[END_REF]] defined the gaugeinvariant quantity at second order that reduces to the curvature perturbation in the uniform energy-density gauge. In [START_REF] Malik | Gauge-invariant perturbations at second order: Multiple scalar fields on large scales[END_REF] the super-horizon equations of motion of these quantities were derived (but see also [START_REF] Noh | Second-order perturbations of the Friedmann world model[END_REF] for a gauge-ready formulation of the perturbations and their equations). Another way to deal with perturbations at second order is the space gradients approach first introduced in [Ellis and [START_REF] Ellis | Covariant and gauge invariant approach to cosmological density flunctuations[END_REF]] and specifically the gradient of the fully non-linear curvature perturbation used in [Rigopoulos et al., 2006a] and defined by the same authors in [Rigopoulos and Shellard, 2005]. The advantage of the method is that when the space gradients are expanded to first order they are automatically gauge-invariant. Some years later the gradient of the curvature perturbation was redefined in a covariant way in [Langlois and Vernizzi, 2005b]. In this paper it was shown that when expanded to second order, this quantity reduces to the gauge-invariant curvature perturbation defined in [START_REF] Malik | Evolution of second-order cosmological perturbations[END_REF]] plus a gauge transformation term.

In this paper we generalize the definition of the gauge-invariant curvature perturbation (or the gradient of the relevant fully non-linear quantity) in terms of the energy density to a definition in terms of the scalar fields present during inflation and study the consequences of this change at second order. Since the scalar fields are the principal quantities during inflation, it makes more sense to use these as a starting point, especially in the case of multiple-field inflation. Our original need for such a definition was to find the horizon crossing contribution to the second-order curvature perturbation in terms of the first-order ones in the long-wavelength formalism [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]. Indeed, such a definition helps to fully understand and potentially generalize the two formalisms used to compute non-Gaussianity during inflation, i.e. the long-wavelength formalism [Rigopoulos et al., 2006a;[START_REF] Rigopoulos | Gauge invariance and non-Gaussianity in Inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] and the δN formalism [START_REF] Starobinsky | Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations[END_REF][START_REF] Sasaki | A General analytic formula for the spectral index of the density perturbations produced during inflation[END_REF][START_REF] Sasaki | Super-horizon scale dynamics of multi-scalar inflation[END_REF]Lyth et al., 2005;Lyth and Rodriguez, 2005b], where instead of the energy density, the values of the fields themselves are used.

In the case of multiple-field inflation not only an adiabatic curvature perturbation is produced, but also one or more isocurvature perturbations. We would like to provide the same type of study for the isocurvature perturbation as for the curvature one, using the generalized quantity defined in [Rigopoulos et al., 2006a], and deduce from that the second-order gaugeinvariant analogue. This definition of the isocurvature perturbation makes direct contact with the scalar fields during inflation (instead of using their pressure), which we find more useful during the period of scalar field domination of the universe. It has a simple physical meaning, that is the combination of the fields that remains orthogonal to the field trajectory, as opposed to the adiabatic perturbation that is parallel to the field trajectory (and proportional to the energy density).

On a related subject, Maldacena in [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] found the third-order action for the first-order adiabatic perturbation in a single-field dominated universe, both in the flat gauge and in the uniform energy-density gauge. In order to rewrite the action in a gauge-invariant form starting from the uniform energy-density gauge, he needed to introduce a redefinition of the first-order perturbations, hence changing their ground state. This redefinition corresponds to part of the second-order gauge-invariant curvature perturbation and contributes to the local non-Gaussianity. His work was followed by [Chen et al., 2007b] introducing general kinetic terms, [Seery and Lidsey, 2005a] for two fields and [Langlois et al., 2008a;[START_REF] Arroja | Non-gaussianity from the bispectrum in general multiple field inflation[END_REF][START_REF] Gao | Primordial Non-Gaussianities of General Multiple Field Inflation[END_REF] for multiple-field models with a generalized kinetic term. In [Seery and Lidsey, 2005a] the treatment of two fields in the flat gauge showed that no field redefinitions occur (see also [START_REF] Rigopoulos | Gauge invariance and non-Gaussianity in Inflation[END_REF]). Nevertheless, the absence of redefinitions in this case does not mean the absence of local non-Gaussianity, because the action was computed in terms of the scalar fields and not in terms of the adiabatic and isocurvature perturbations. This means that in the method of [Seery and Lidsey, 2005a] the δN formalism or the long-wavelength formalism is needed to compute the final non-Gaussianity, which requires that the slow-roll approximation is imposed at horizon crossing.

Here we generalize the above results and write the two-field action in terms of the gaugeinvariant perturbations themselves, both in the uniform energy-density gauge and the flat gauge in order to compare the results. We expand the calculation to include second-order perturbations and tensor modes, and study the various contributions that occur. Hence we derive the exact third-order action, going beyond the slow-roll or the super-horizon approximation. We thus provide the missing tool that will enable people to calculate non-Gaussianity, using the in-in formalism [START_REF] Weinberg | Quantum contributions to cosmological correlations[END_REF], beyond these standard approximations used by both the longwavelength formalism and the δN formalism. This paper is organized as follows. In section 2B.2 we provide the gauge-invariant definitions and conventions for the metric and the field perturbations, along with the description of the space-time of the universe, using the ADM formalism. In the first part of section 2B.3 we study the gauge-invariant curvature and isocurvature perturbations in terms of the fields, while in the second part we make the connection to the fully non-linear spatial gradients of the relevant quantities. In the whole of section 2B.3 we use the long-wavelength approximation to keep the calculations short and tractable, but we present the generalization of the results beyond this approximation in the first appendix in section 2B.5. In section 2B.4 we construct the exact cubic action, going beyond the long-wavelength approximation, to find the redefinitions of the perturbations and compare their contributions to the gauge-invariant quantities found in section 2B.3. To keep the main text more accessible, many of the details of the calculations have been moved to the appendices in section 2B.5. Finally we summarize the results at the end of section 2B.4.

2B.2 Preliminaries

In this section we give the basic elements required for the calculations in this paper. We start by summarizing in the first subsection the ADM formalism, along with the definitions of the cosmological quantities, the slow-roll parameters and the field basis we use. In the second subsection we provide the conventions of cosmological perturbation theory and clarify different approaches in the literature.

2B.2.1 The ADM formalism

We will consider a universe filled with two scalar fields with a trivial field metric. The generalization to more fields and a non-trivial field metric is conceptually straightforward (see [Rigopoulos et al., 2006a]), but involves more complicated expressions and calculations. The energy-momentum tensor for the two fields ϕ A (A, B = 1, 2) is

T µν = δ AB ∂ µ ϕ A ∂ ν ϕ B -g µν 1 2 δ AB g λκ ∂ κ ϕ A ∂ λ ϕ B + W , (2B.1)
where W is the field potential. We will denote the homogeneous part of the fields by φ A . The Einstein summation convention is assumed throughout this paper. We shall work in the ADM formalism and write the metric g µν as

ds 2 = -N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt), (2B.2)
where N is the lapse function and N i the shift. The action takes the form [START_REF] Misner | Statistical properties of dust far-infrared emission[END_REF]]

S = 1 2 d 4 x √ h -2 N W +κ -2 N -1 (E ij E ij -E 2 )+ N Π2 +κ -2 N R (3) -N h ij ∂ i ϕ A ∂ j ϕ A , (2B.3)
where 3) is the intrinsic 3-curvature, the tensor E ij (proportional to the extrinsic curvature

κ 2 ≡ 8πG = 8π/m 2 pl , h is the determinant of the space metric h ij , R (
K ij = -N -1 E ij ) is E ij = 1 2 ḣij -∇ i N j -∇ j N i (2B.4)
and Π is the length of the canonical momentum of the fields

ΠA = ( φA -N i ∂ i ϕ A )/ N . (2B.5)
Variation of the action with respect to N and N i gives the energy and momentum constraints

κ -2 R (3) -2W -κ -2 N -2 (E ij E ij -E 2 ) -Π2 -h ij ∂ i ϕ A ∂ j ϕ A = 0, (2B.6) ∇ j 1 N (E j i -Eδ j i ) = κ 2 ΠA ∂ i ϕ A , (2B.7)
where ∇ j denotes the covariant derivative with respect to the space metric and E is the trace of E ij . Following [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] we decompose the space metric as

h ij = a(t) 2 e 2α(t,x) e γ ij (t,x) , ∂ i γ ij = 0, γ i i = 0. (2B.8)
From now on contravariant tensors should be understood as T j = η ij T i , where η ij = diag(1, 1, 1), since in the calculations we are showing we have already taken into account explicitly the h ij part of the initial contravariant tensors. The generalized Hubble parameter is defined as

H ≡ E 3 N . (2B.9)
We use the bar for the lapse function, the Hubble parameter and the canonical momentum to distinguish these fully non-linear quantities from their background values N (t), H(t) = ȧ/(aN ) and Π(t) = φ/N , respectively. In this paper we will use as time variable the number of e-folds, meaning that ȧ = a, so that the background value of the lapse function N (t) is just 1/H(t).

The background field equation and the background Einstein equations are

ΠA = -3HN Π A -N W A , Ḣ = - κ 2 2 N Π 2 , H 2 = κ 2 3 Π 2 2 + W , (2B.10)
with W A ≡ ∂W/∂φ A . We construct an orthonormal basis e A m in field space, consisting of e A 1 ≡ Π A /Π, parallel to the field velocity, and e A 2 parallel to the part of the field acceleration perpendicular to the field velocity [Groot Nibbelink and [START_REF] Van Tent | Cosmological Inflation with Multiple Fields and the Theory of Density Fluctuations[END_REF]. The m = 1 component of physical quantities describes the single-field (adiabatic) part, while the m = 2 component captures the multiple-field (isocurvature) effects. One can show that for the two-field case the basis vectors are related through [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]]

AB e A 1 e B 2 = -1, (2B.11)
where AB is the antisymmetric tensor. The background slow-roll parameters then take the form

(t) ≡ - Ḣ N H 2 , η (t) ≡ e 1A ΠA N HΠ , η ⊥ (t) ≡ e 2A ΠA N HΠ , χ(t) ≡ W 22 3H 2 + + η , ξ (t) ≡ e 1A ΠA N 2 H 2 Π - Ṅ N 2 H η , ξ ⊥ (t) ≡ e 2A ΠA N 2 H 2 Π - Ṅ N 2 H η ⊥ , (2B.12)
where W mn ≡ e A m e B n W ,AB . Throughout this paper the indices m, n will indicate components in the basis defined above, taking the values 1 and 2, while i, j are spatial indices and A, B are indices of the original fields. In a slow-roll approximation one can think of η being related to W 11 , η ⊥ to W 21 and χ to W 22 . The ξ parameters are second-order slow-roll parameters (in a slow-roll approximation they are related to the third derivatives of the potential). However, we emphasize that we have not made any slow-roll approximations; the above quantities should be viewed as short-hand notation and can be large. We also give the time derivatives of the background slow-roll parameters and of the unit vectors,

ėA 1 = N Hη ⊥ e A 2 , ėA 2 = -N Hη ⊥ e A 1 , ˙ = 2N H ( + η ), η = N H ξ + (η ⊥ ) 2 + ( -η )η , η⊥ = N H ξ ⊥ + ( -2η )η ⊥ , χ = N H η + 2 χ -(η ) 2 + 3(η ⊥ ) 2 + ξ + 2 3 η ⊥ ξ ⊥ + √ 2 κ W 221 3H 2 , ξ = N H - √ 2 κ W 111 H 2 + 2η ⊥ ξ ⊥ + 2 ξ -3ξ + 9 η + 3(η ⊥ ) 2 + 3(η ) 2 , ξ⊥ = N H - √ 2 κ W 211 H 2 -η ⊥ ξ + 2 ξ ⊥ -3ξ ⊥ + 9 η ⊥ + 6η ⊥ η -3η ⊥ χ , (2B.13)
where W lmn ≡ e A m e B n e C l W ,ABC .

2B.2.2 Second-order perturbations and gauge transformations

In the context of perturbation theory around an homogeneous background any quantity Ā can be decomposed into an homogeneous part and an infinite series of perturbations as 2B.14) where the subscripts in the parentheses denote the order of the perturbation. Up to first order the scalar part of the space metric element of (2B.8) is equal to

Ā(t, x) = A(t) + A (1) (t, x) + 1 2 A (2) (t, x) + . . . , (
h ij = a 2 (t)(1 + 2α (1) )δ ij . (2B.15)
When one wants to expand perturbation theory up to second order there are two choices found in the literature: either expand (2B.15) as Malik and Wands do in [START_REF] Malik | Evolution of second-order cosmological perturbations[END_REF]] to find

h ij = a 2 (t)(1 + 2α (1) + α (2) )δ ij (2B.16)
or expand directly the space part of (2B.8) as for example Lyth and Rodriguez do in [Lyth and Rodriguez, 2005a] to find

h ij = a 2 (t)(1 + 2α (1) + 2α 2 (1) + α (2) )δ ij .
(2B.17)

We will take this second approach and use the exponent of the perturbation in our calculations.

Since perturbations depend on the gauge choice we make, we need to construct quantities that are invariant under gauge transformations. Under an arbitrary second-order coordinate transformation

x µ = xµ + β µ (1) + 1 2 β µ (1),ν β ν (1) + β µ (2) , (2B.18)
the perturbations of a tensor transform as [START_REF] Bruni | Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond[END_REF]]

A (1) = Â(1) + L β (1) A, A (2) = Â(2) + L β (2) A + L 2 β (1) A + 2L β (1) Â(1) , (2B.19)
where L β is the Lie derivative along the vector β

(L β A) µ 1 µ 2 ... ν 1 ν 2 ... = β κ ∂ κ A µ 1 µ 2 ... ν 1 ν 2 ... -∂ κ β µ 1 A κµ 2 ... ν 1 ν 2 ... -• • • + ∂ ν 1 β κ A µ 1 µ 2 ... κν 2 ... + . . . . (2B.20)
Note here that spatial gradients, having vanishing background values, are automatically gaugeinvariant at first order, while at second order they transform as

∂ i A (2) = ∂ i Â(2) + 2L β (1) ∂ i Â(1) . (2B.21)

2B.3 Super-horizon gauge transformations

In this section we derive first and second-order super-horizon gauge-invariant combinations. We study these during an inflationary period and thus, though we start from the energy-density definitions of the perturbations, we naturally end up with field definitions for the gauge-invariant perturbations. Our goal is to find the second-order adiabatic and isocurvature perturbations in terms of the first-order ones and the slow-roll parameters.

In this section we restrict ourselves to super-horizon calculations for simplicity, though in the next section we will abandon this approximation and study the full action of the cosmological perturbations. However, in the first appendix in section 2B.5 we present the generalization of the results of this section beyond the long-wavelength approximation. We note that the longwavelength (or super-horizon) approximation is equivalent to the zeroth order space gradient approximation and is valid once the decaying mode has disappeared (which happens rapidly if slow-roll holds during horizon exit), even if there is a subsequent non slow-roll phase.

In the super-horizon regime one can choose to work in the time-orthogonal gauge where N i = 0 (proof for that choice is given in the next section) and employ the long-wavelength approximation to simplify calculations. The latter boils down to ignoring second-order spatial derivatives when compared to time derivatives. As a consequence the traceless part of the extrinsic curvature quickly decays and can be neglected [START_REF] Salopek | Stochastic inflation and nonlinear gravity[END_REF]. Hence the space part of the metric can be described by

h ij = a(t) 2 e 2α(t,x) δ ij . (2B.22)
The field and Einstein equations in that case are identical to (2B.10), but now the quantities involved are fully non-linear. Additionally the momentum constraint (2B.7) can be written as [Rigopoulos et al., 2006a]

∂ i H = - κ 2 2 ΠA ∂ i ϕ A . (2B.23)

2B.3.1 Gauge-invariant quantities

The well-known first-order adiabatic gauge-invariant curvature perturbation has the form

ζ 1(1) ≡ α (1) - N H ρ ρ (1) , (2B.24)
where ρ is the energy density. The subscript without parentheses corresponds to the first component in our basis, which is exactly the adiabatic component, while the subscript between parentheses denotes the order in the perturbation series. Notice that in the literature it is common to work with cosmic time, i.e. N = 1, while the space part of the metric is decomposed using a quantity ψ = -α (not to be confused with the ψ introduced in the appendices), so that the first-order curvature perturbation becomes in that case -ζ 1(1) = ψ (1) + (H/ ρ)ρ (1) . Here we choose to work with the number of e-folds as time variable so that the first-order curvature perturbation is

ζ 1(1) = α (1) - ρ (1) ρ . (2B.25)
The gauge-invariant combination (2B.24) is calculated via the requirement that it coincides with the curvature perturbation ζ 1(1) = α(1) in the uniform energy-density gauge where ρ(1) = 0. From now on tilded quantities will denote the uniform energy-density gauge, while hatted quantities will denote the flat gauge. One has to use the gauge transformations (2B.19) for a scalar (here the energy density and the logarithm of the space dependent scale factor α) and require that in the uniform energy-density gauge the first-order energy perturbation is zero (for details see the appendix). That way one can determine the first-order time shift and hence find the gauge-invariant combination corresponding to the curvature perturbation. Notice that in the flat gauge, i.e. α(1) = 0,

ζ 1(1) = -ρ (1) / ρ.
Keeping in mind the expansion (2B.17), one can repeat the above considerations at second order. We find that for super-horizon scales (where we neglect second-order space derivatives when compared to second-order time derivatives) the second-order gauge-invariant adiabatic perturbation takes the form (see the appendix)

1 2 ζ 1(2) ≡ 1 2 α(2) = 1 2 α (2) - 1 2
ρ (2) ρ + ρ(1) ρ (1) ρ2 - ρ (1) ρ α(1) - 1 2 ρ 2 (1) ρ2 ρ ρ . (2B.26)
If we chose the second gauge to be flat, i.e. α(i) = 0, we find

1 2 ζ 1(2) = 1 2 α(2) = - 1 2 ρ(2) ρ + ζ1(1) ζ 1(1) + 1 2 ρ2 (1) ρ2 ρ ρ . (2B.27)
During inflation we find it more useful to work directly with the fields and not their energy density, since both the long-wavelength formalism and the δN formalism make use of the field values to compute f NL . Using the fields, the first-order adiabatic perturbation becomes

ζ 1(1) = α(1) = α (1) - H Π e 1A ϕ A (1) , (2B.28)
since the energy-density constraint ρ(1) = 0 is equivalent to e 1A φA (1) = 0. The detailed calculation is shown in the second appendix in section 2B.5. Notice that in (2B.28) we have kept the lapse function N arbitrary, as we will also do in all definitions hereafter, but in our calculations H/Π is just 1/ φ for the choice N = 1/H.

The second-order calculation turns out to be more complicated. The details are given in the second appendix. Here we give the result for the gauge-invariant adiabatic perturbation in the uniform energy-density gauge and in the flat gauge:

1 2 ζ 1(2) = 1 2 α(2) = 1 2 Q1(2) + + η 2 ζ 2 1(1) -ζ 2 2(1) -η ⊥ ζ 1(1) ζ 2(1) + ζ1(1) ζ 1(1) -∂ -2 ∂ i ζ2(1) ∂ i ζ 2(1) , (2B.29)
where we introduced the auxiliary quantities

Q m(i) ≡ - H Π e mA ϕ A (i) (2B.30)
and the new combination [Rigopoulos et al., 2006a]

ζ 2(1) ≡ - H Π e 2A ϕ A (1) = Q 2(1) , (2B.31)
that represents the isocurvature perturbation to first order. Usually the isocurvature perturbation is described in terms of the gradient of the pressure of the matter content of the universe, as for example in [Langlois and Vernizzi, 2005b]. Here we choose to characterize it in terms of the fields themselves and the vector e 2A . The latter indicates we are dealing with a purely multiple-field effect and hence it is an appropriate quantity to use during the inflationary period to describe the non adiabatic perturbations. Starting from the long-wavelength definition of the pressure p = Π2 /2 -W , one can show that the gradient of the isocurvature perturbation defined in [Langlois and Vernizzi, 2005b] is equal to

Γ i ≡ ∂ i p - ṗ ρ ∂ i ρ = -2η ⊥ Π2 ζ 2i , (2B.32)
where ζ 2i is the fully non-linear gradient of the isocurvature perturbation (for more details see the next subsection) and η⊥ the fully non-linear generalization of η ⊥ , i.e. as it is defined in (2B.12) but with barred quantities [Rigopoulos et al., 2006a]. So our definition of ζ 2 agrees with the pressure definition of the isocurvature perturbation. The next logical step would be to define the second-order isocurvature perturbation ζ 2(2) . However, the above equation shows that there is a non-trivial relation between ζ 2 and the pressure p, involving the non-linear quantities η⊥ and Π, which makes a derivation using the methods of this subsection rather complicated.

For that reason we prefer to find ζ 2(2) in an easier way in the next subsection using gradients. Notice that unlike in the original definition of ζ 1(2) in terms of ρ (2) , a non-local term appears in (2B.29) when one uses the fields instead of the energy density, because of (2B.83). The time derivatives of the fully non-linear gradients of the perturbations (see next subsection) were found in [Rigopoulos et al., 2006a]. Expanding to first order these yield

ζ1(1) = 2η ⊥ ζ 2(1) (2B.33)
for the adiabatic perturbation and ζ2(1) = -χζ 2(1) (2B.34)

for the isocurvature perturbation, the latter valid only in the slow-roll regime. Then we find

1 2 ζ 1(2) = 1 2 α(2) = 1 2 Q1(2) + + η 2 ζ 2 1(1) - + η -χ 2 ζ 2 2(1) + η ⊥ ζ 1(1) ζ 2(1) , (2B.35)
i.e. without a non-local term. However, we will not use the slow-roll approximation in this paper.

2B.3.2 The gradient of the perturbations

As an alternative to the ζ m defined in the previous section, one can use the gradient quantity ζ 1i along with the isocurvature analogue ζ 2i , both defined in [Rigopoulos et al., 2006a] and later in [Langlois and Vernizzi, 2005b] in a covariant way, to construct a gauge-invariant quantity. These gradient quantities (not gauge-invariant to all orders) are given by .36) where now ēmA represents the fully non-linear super-horizon version of the orthonormal basis vectors, e.g. ē1A = ΠA / Π, with ΠA = φA / N since we are working in the super-horizon regime.

ζ mi = δ m1 ∂ i α -H Π ēmA ∂ i ϕ A , ( 2B
Notice that the basis vectors still obey (2B.11) as was shown in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF].

ζ mi is by construction gauge-invariant at first order, since it has no background value: it is just the gradient of the gauge-invariant ζ m(1) defined before.

Expanding to second order we find for the adiabatic perturbation

1 2 ζ 1i(2) = 1 2 ∂ i α (2) + Q 1(2) - 1 φ2 φA (1) ∂ i ϕ A(1) -2 1 φ e 1A φA (1) ∂ i Q 1(1) . (2B.37)
In the uniform energy-density gauge this gives (see the second appendix in section 2B.5) .38) while in the flat gauge where ∂ i α = 0 and ζ 1(1) = Q1(1) , we find

1 2 ζ1i(2) = 1 2 ∂ i α(2) , ( 2B
1 2 ζ1i(2) = ∂ i 1 2 Q1(2) + + η 2 ζ 2 1(1) -ζ 2 2(1) -η ⊥ ζ 1(1) ζ 2(1) + ζ1(1) ζ 1(1) -ζ2(1) ∂ i ζ 2(1) -ζ 1(1) ∂ i ζ1(1) , (2B.39)
where we used the basis completeness relation and (2B.84) to rewrite the terms. ζ1i(2) in the uniform energy-density gauge (2B.38) coincides with the gradient of the gauge-invariant secondorder adiabatic perturbation. However, by comparing (2B.29) to (2B.39) we see that in the flat gauge ζ1i( 2) is the gradient of the gauge-invariant curvature perturbation ζ 1(2) expressed in the flat gauge plus a new non-local term. This is in agreement with the findings in [Langlois and Vernizzi, 2005b]. This new term is nothing else but the gauge transformation of ζ 1i (2) . A quantity with zero background value as ζ i is, transforms as (2B.21). One can check, using the gauge transformations (2B.19) for ρ and for α and requiring that ρ(1) = 0, that starting from a flat gauge and transforming to the uniform energy-density gauge the time shift is

T (1) = ζ 1(1)
(see the first appendix), so that

1 2 ζmi(2) = 1 2 ζmi(2) + ζ 1(1) ζmi(1) . (2B.40)
Next we try to find the second-order gauge-invariant part of the isocurvature perturbation ζ 2i by expanding (2B.36),

1 2 ζ 2i(2) = 1 2 ∂ i Q 2(2) + 1 φ2 BA φB(1) ∂ i ϕ A(1) -2 1 φ e 1A φA (1) ∂ i ζ 2(1) , (2B.41) or equivalently, 1 2 ζ 2i(2) = 1 2 ∂ i Q 2(2) - 1 φ e 2A φA (1) ∂ i Q 1(1) - 1 φ e 1A φA (1) ∂ i ζ 2(1) , (2B.42)
where we used (2B.11) to express e 2 in terms of e 1 . The first-order uniform energy constraint

∂ i Q1(1) = 0 alone implies 1 2 ζ2i(2) = ∂ i 1 2 Q2(2) - η ⊥ 2 ζ 2 2(1) + ζ1(1) ∂ i ζ 2(1) , (2B.43)
where we used (2B.90) for the two last terms in (2B.42). For the flat gauge ∂ i α = 0 we find using (2B.84)

1 2 ζ2i(2) = ∂ i 1 2 Q2(2) + η ⊥ 2 ζ 2 1(1) -ζ 2 2(1) + ( + η )ζ 1(1) ζ 2(1) + ζ2(1) ζ 1(1) + ζ1(1) ∂ i ζ 2(1) -ζ 1(1) ∂ i ζ2(1) . (2B.44)
We notice that the term in the second line corresponds again to a gauge transformation familiar from the curvature perturbation case studied earlier (2B.40). In the first appendix we verify that the rest of the expression is a gauge-invariant quantity corresponding to the one in (2B.43). Indeed this expression is gauge-invariant beyond the long-wavelength approximation as shown in that appendix.

We conclude that the gradients of the perturbations are in some sense equivalent to the perturbations themselves, since both allow for the definition of gauge-invariant second-order adiabatic and isocurvature quantities. However, since the gradients are defined using fully non-linear quantities, their equations of motion can be treated more easily, as was shown in [Rigopoulos et al., 2006a].

2B.4 The cubic action

An alternative way to calculate the second-order gauge-invariant quantities and reconsider their meaning, is to compute the third-order action for the interacting fields. Maldacena [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] was the first to perform that calculation for a single field, in the uniform energy-density gauge. In this way he managed to find the cubic interaction terms due to non-linearities of the Einstein action as well as of the field potential, which among other consequences change the ground state of the adiabatic perturbation ζ 1(1) . This change can be quantified through a redefinition of the form [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]

ζ 1(1) = ζ 1c(1) + + η 2 ζ 2 1(1) , (2B.45)
where ζ 1c is the redefined perturbation. One sees that the correction term of the redefinition coincides with the surviving quadratic term of the single-field limit of the transformation (2B.29), taking into account that the super-horizon adiabatic perturbation is constant in that case. In [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] the curvature perturbation was considered a first-order quantity, while the second-order curvature perturbation was not taken into account, since its contribution in the uniform energy-density gauge is trivial: it introduces a redefinition of the form

ζ 1(1) + ζ 1(2) /2 = ζ 1c(1)
(for proof, see later in this section). Seery and Lidsey [Seery and Lidsey, 2005a] performed the same calculation for the multiple-field case in the flat gauge in terms of the scalar fields ϕ A and not of the adiabatic and isocurvature perturbations ζ m . They found no redefinitions, but as mentioned before their results would have to be supplemented by the δN formalism (with its associated slow-roll approximation at horizon-crossing) to say anything about the non-Gaussianity of the gauge-invariant perturbations ζ m .

In this section we generalize the above calculations to second order in the expansion of the curvature perturbation in both the uniform energy-density gauge and the flat gauge. Doing so we compute the full form of the third-order action. The latter not only consists of the cubic interactions of the first-order curvature perturbations, but also of lower order interaction terms of the second order quantities. We first perform the calculation relevant to the first-order quantities and then add the second-order effects. In this section we only present the scalar part of the action, but in the appendices the tensor part can be found as well. We emphasize that in this section we no longer make the long-wavelength approximation, so that the results are valid at any scale.

2B.4.1 The second-order action

We start by performing our calculation in the gauge e 1A φA (1) = 0. This constraint reduces to the uniform energy-density gauge outside the horizon, which is why we will continue to refer to the tilded gauge as the uniform energy-density gauge. From now on we drop the explicit subscript (1) on first-order quantities. We will keep this part brief since its results are already known, but we give the basic elements of the calculation in the third appendix in section 2B.5. The second-order action takes the form

S2 = d 4 xL 2 = d 4 x -a 1 H (∂ζ 1 ) 2 + (∂ζ 2 ) 2 + a 3 H ζ2 1 + ζ2 2 -4η ⊥ ζ1 ζ 2 + 2χ ζ2 ζ 2 (2B.46) +a 3 H 2 κ W 221 3H 2 -2 2 -(η ) 2 + 3(η ⊥ ) 2 + 2 3 η ⊥ ξ ⊥ -3 (η -χ) + 2η χ ζ 2 2 ,
where L 2 is the second-order Lagrangian. While we have started from an action describing the evolution of the fields α and e 2A φA we have now constructed an action in terms of the adiabatic and isocurvature perturbations ζ 1 and ζ 2 . The equations of motion that ζ 1 and ζ 2 obey are (δL 2 /δζ m being a short-hand notation for the relevant variations of the Lagrangian)

δL 2 δζ 1 = -2a 3 H ζ1 + 3 + + 2η ζ1 -2η ⊥ ζ2 + 2(-ξ ⊥ -2 η ⊥ -3η ⊥ )ζ 2 + 2a H ∂ 2 ζ 1 = - d dt (2a 3 H∂ 2 λ) + 2a H ∂ 2 ζ 1 = 0, (2B.47) δL 2 δζ 2 = -2a 3 H ζ2 + 3 + + 2η ζ2 + 2η ⊥ ζ1 + (ξ + 2 2 + 4 η + 3χ)ζ 2 +2a H ∂ 2 ζ 2 = 0,
where

∂ 2 λ = ζ1 -2 η ⊥ ζ 2 (
for the reason of introducing λ see the appendix). Thus we have found the evolution equations for the first-order adiabatic and isocurvature perturbations. Their super-horizon limit coincides with the equations derived in [Rigopoulos et al., 2006a] for the gradient of the perturbations, since up to first order ζ mi(1) = ∂ i ζ m(1) . One can show that the first-order energy constraint, which outside the horizon reduces to

ζ1 -2η ⊥ ζ 2 = 0, (2B.48)
is the first integral of the super-horizon part of the first equation of (2B.47), i.e. without the space gradient. In fact it was shown in [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF] that this is the case at all orders. In the same paper it was found that assuming the slow-roll limit, ζ2 = -χζ 2 is the super-horizon first integral of the equation for ζ 2 , which can be easily verified.

While working in the flat gauge we find the same action (2B.46) (see the appendix). So the curvature perturbations ζ m satisfy to first order the same equations in both gauges as expected, due to the gauge invariance of ζ m (or equivalently the gauge invariance of the action).

2B.4.2 The third-order action

In this section we compute the third-order action. Again we present only the final results, while in the fourth appendix in section 2B.5 we give the intermediate steps of the calculation. In the same appendix we also give the tensor-scalar part of the action. The scalar cubic action in the uniform energy-density gauge due to the first-order perturbations ζ m takes the form

S3(1) = S 3(1) -d 4 x δL 2 δζ m f m (2B.49)
with

f 1 = + η 2 ζ 2 1 -η ⊥ ζ 1 ζ 2 + ζ1 ζ 1 - 1 4a 2 H 2 (∂ζ 1 ) 2 -∂ -2 ∂ i ∂ j (∂ i ζ 1 ∂ j ζ 1 ) + 1 2 ∂ i λ∂ i ζ 1 -∂ -2 ∂ i ∂ j (∂ i λ∂ j ζ 1 ) , f 2 = ( + η )ζ 1 ζ 2 + ζ2 ζ 1 + η ⊥ 2 ζ 2 1 . (2B.50)
The exact form of S 3(1) can be found in the appendix or equivalently it is the cubic part of (2B.59). The reason for introducing S 3(1) without the tilde will become clear below.

The terms proportional to δL 2 /δζ m , i.e. the first-order equations of motion, can be removed by a redefinition of ζ m [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] and lead to a change in the ground state of the perturbations. This works as follows. The cubic terms of the action (i.e. S3(1) ) are not affected by the redefinition, because the redefinition always involves terms proportional to ζ 2 m , which would give quartic and not cubic corrections. It is only the second-order terms (i.e. S2 ) that change. Indeed one can show that under a redefinition of the form ζ m = ζ mc + f m , the secondorder action changes as S 2 = S 2c + (δL 2 /δζ m )f m . These new terms cancel out the relevant terms coming from the cubic action (remember that the total action up to cubic order is the sum of the second and third-order action) and we are left with S3(1) = S 3(1) (ζ mc ).

(2B.51)

If we repeat the same calculations for the flat gauge (see the appendix), performing several integrations by part, we find that Ŝ3(1

) = S 3(1) (ζ m ). (2B.52)
This is a consequence of the action staying invariant under a gauge transformation. Nevertheless if one associates the redefinition appearing in the uniform energy-density gauge to a change in the ground state of ζ m , it would mean that directly after horizon crossing, when super-horizon effects have not yet been switched on, the second-order contribution to ζ 1 would be zero for the flat gauge and non-zero for the uniform energy-density gauge. In terms of non-Gaussianity, this can be restated as: the non-Gaussianity present after horizon-crossing is different for the two gauges. Indeed if one was to calculate the three-point functions for the above action, one would need to perform two steps. First, change to the interaction picture, where it can be proved that the interaction Hamiltonian up to and including cubic order is just H int = -L int , where L int are the cubic terms of the Lagrangian, and compute the expectation value ζ c ζ c ζ c as in [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]. Second, take into account that the fields have been redefined as ζ = ζ c +λζ 2 c . Then the three-point correlation function can be written as

ζζζ = ζ c ζ c ζ c + 2λ[ ζ c ζ c ζ c ζ c + cyclic].
(2B.53)

These new terms, products of the second-order correlation functions, are only present in the uniform energy-density gauge and not in the flat gauge, if we restrict ourselves to S 3(1) .

In order to cure this bad behaviour we need to add to the above results the effect of the second-order fields. We find (see the appendix)

S3(2) = d 4 x δL 2 δζ 1 Q1(2) 2 + ζ 1(2) 2 + δL 2 δζ 2 Q2(2) 2 . (2B.54)
Since all terms in S3( 2) are proportional to δL 2 /δζ m , S3(2) only contains redefinitions of ζ m . Notice that the second-order lapse and shift functions do not appear in the final action, since these two are multiplied by a factor equal to the energy and momentum constraint equations (2B.92). On the other hand, the second-order field perturbations are dynamical variables that obey second-order equations of motion that cannot be set to zero in the action. The singlefield limit of this action is just the term proportional to ζ 1(2) , since Qm(i) = 0 identically in that case for the uniform energy-density gauge. The term proportional to ζ 1(2) in S3( 2) , along with the terms proportional to λ in S3(1) , originate from the contribution of N i in the action. The latter vanish outside the horizon since then ∂ 2 λ coincides with the super-horizon energy constraint and hence is identically zero. So if we were to study only the quadratic contributions of the first-order perturbations outside the horizon, we would be allowed not only to ignore the tensor parts of the metric [START_REF] Salopek | Stochastic inflation and nonlinear gravity[END_REF], but also work in the time-orthogonal gauge

N i = 0.
Coming back to the redefinition, its final form, including the tensor parts (see the appendix), is

ζ 1 = ζ 1c - ζ 1(2) 2 - Q1(2) 2 + ζ1 ζ 1 + + η 2 ζ 2 1 -η ⊥ ζ 1 ζ 2 - 1 4a 2 H 2 (∂ζ 1 ) 2 -∂ -2 ∂ i ∂ j (∂ i ζ 1 ∂ j ζ 1 ) + 1 2 ∂ i λ∂ i ζ 1 -∂ -2 ∂ i ∂ j (∂ i λ∂ j ζ 1 ) - 1 4 ∂ -2 ( γij ∂ i ∂ j ζ 1 ), ζ 2 = ζ 2c - Q2(2) 2 + ζ2 ζ 1 + η ⊥ 2 ζ 2 1 + ( + η )ζ 1 ζ 2 . (2B.55)
Finally we perform the above calculations for the flat gauge and find the action

Ŝ3(2) = d 4 x δL 2 δζ 1 Q1(2) 2 + δL 2 δζ 2 Q2(2) 2 . (2B.56)
The redefinitions in the flat gauge take the simple form

ζ 1 = ζ 1c - Q1 (2)
2

ζ 2 = ζ 2c - Q2 (2) 2 .
(2B.57)

We want to write these redefinitions as well as the action itself in terms of gauge-invariant quantities and compare them. We would also like to compare with the definitions of the secondorder gauge-invariant perturbations found in the first appendix and section 2B.3. After using the second-order uniform energy constraint (2B.89) and the uniform energy gauge definition of ζ 2(2) (2B.43) we can rewrite (2B.55) as

ζ 1 + ζ 1(2) 2 = ζ 1c + ζ1 ζ 1 + + η 2 ζ 2 1 -ζ 2 2 -η ⊥ ζ 1 ζ 2 -∂ -2 ∂ i ζ2 ∂ i ζ 2 - 1 4 ∂ -2 ( γij ∂ i ∂ j ζ 1 ) - 1 4a 2 H 2 (∂ζ 1 ) 2 -∂ -2 ∂ i ∂ j (∂ i ζ 1 ∂ j ζ 1 ) + 1 2 ∂ i λ∂ i ζ 1 -∂ -2 ∂ i ∂ j (∂ i λ∂ j ζ 1 ) ζ 2 + ζ 2(2) 2 = ζ 2c + ζ2 ζ 1 + ζ 2 ζ1 + η ⊥ 2 ζ 2 1 -ζ 2 2 + ( + η )ζ 1 ζ 2 -∂ -2 ∂ i ζ 2 ∂ i ζ1 . (2B.58)
When comparing the first equation of (2B.58) with (2B.76), we see that we recover (2B.57).

The same is true for the isocurvature part of the redefinition: comparing the second equation of (2B.58) with (2B.44), we recover the redefinition for ζ 2 (2B.57). Hence the two redefinitions are the same, as is necessary for the action to be gauge-invariant. Notice that the single-field limit of (2B.58) is

ζ 1 + ζ 1(2) /2 = ζ 1c + ( + η )ζ 2
1 /2 in agreement with the total redefinition found in the uniform energy-density gauge.

Equation (2B.58) is the implicit definition of the redefined, gauge-invariant ζ mc . One can see that up to and including second order, it is a function of only the combination ζ m(1) + ζ m(2) /2. One can also notice that the purely second-order perturbation ζ m(2) does not occur explicitly in the cubic action (see e.g. (2B.59) below). Hence one could in principle consider the quantities ζ m(2) (and similarly Q m( 2) ) as auxiliary quantities and try to avoid introducing them in the first place, but consider the quadratic first-order terms directly as a correction to the first-order perturbations, as is done for the single-field case in [START_REF] Rigopoulos | Gauge invariance and non-Gaussianity in Inflation[END_REF]. While the calculations would be roughly equivalent, we have chosen not to follow this route for two reasons. In the first place it seems conceptually simpler to us to expand the perturbations and the action consistently up to the required order, and more logical to view quadratic first-order terms as a correction to a second-order quantity than to a first-order one. Secondly, in the multiple-field case (as opposed to the single-field case), one would have to introduce the second-order quantities at some intermediate steps anyway in order to find the correct non-linear relation between the Q m and ζ m (which is derived from the second-order gauge transformation performed in the first appendix).

So in the end we have managed to find the source of the non-Gaussianity present at horizon crossing due to first-order perturbations and identify it with the quadratic terms of (2B.58).

With source here we mean the second-order perturbation that, when contracted with two firstorder perturbations, gives the bispectrum. The super-horizon limit of (2B.58) was derived and used in our previous paper [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF], but here we have not only generalized the result, but also have obtained a much better understanding of the gauge issues. Equation (2B.58) is gauge-invariant, as it should be. Additionally, the redefinition of the perturbations that we perform is essential not only to simplify calculations but also to find the gauge-invariant form of the action itself. We clearly see that the quadratic corrections in the flat gauge seem to be zero if one takes into account only the first-order fields. In that gauge all of the second-order contributions are hidden in the second-order fields as opposed to the uniform energy-density gauge where part of the quadratic contributions is attributed to the redefinition of the first-order ζ m and the rest of them lie in the second-order field.

2B.4.3 Summary

Let us summarize the results of this section. Cosmological gauge-invariant perturbations should obey a gauge-invariant action. Using first-order perturbations the action up to third order is the same in the uniform energy-density gauge and the flat gauge only after a redefinition of ζ m in the uniform energy-density gauge ζ m = ζ mc 1 + f m1 (2B.50) (the subscript 1 indicating the use of only first-order perturbations) and takes the form

S = Ŝ(ζ m ) = S(ζ mc 1 ) = d 4 x a H ( ζ 1 -1) (∂ζ 1 ) 2 + (∂ζ 2 ) 2 + d 4 x a 3 H (1 + ζ 1 ) ζ2 1 + ζ2 2 -2∂ i λ ζ2 ∂ i ζ 2 + ζ1 ∂ i ζ 1 -2( + η )ζ 2 ∂ i λ∂ i ζ 2 + 4η ⊥ ζ 2 ∂ i λ∂ i ζ 1 + 1 2 ζ 1 ∂ i ∂ j λ∂ i ∂ j λ -∂ 2 λ 2 +2 ζ2 χζ 2 + ζ 1 ( + η )ζ 2 + η ⊥ ζ 1 + ζ1 -4η ⊥ ζ 2 + ζ 2 1 (3η + 2 ) +ζ 2 1 2 κ W 211 H 2 -2 η ⊥ + η η ⊥ + ξ ⊥ + 3η ⊥ ζ 2 + 2 κ W 111 3H 2 -ξ + 3η -(η ⊥ ) 2 -(η ) 2 ζ 1 +ζ 2 2 2 κ W 221 3H 2 -2 2 -(η ) 2 + 3(η ⊥ ) 2 + 2 3 η ⊥ ξ ⊥ -3 (η -χ) + 2η χ + 2 κ W 221 H 2 + -3(η ⊥ ) 2 + ( + η ) 2 + 3 (χ --η ) ζ 1 + 2 κ W 222 3H 2 ζ 2 (2B.59)
where we have kept the notation ∂ 2 λ = ζ1 -2 η ⊥ ζ 2 in order to mark clearly the terms that vanish outside the horizon, namely the terms proportional to λ along with the terms involving second-order space derivatives. This is one of our main results. We managed to compute the cubic action for adiabatic and isocurvature perturbations in the exact theory, beyond any superhorizon or slow-roll approximation. Its single-field limit coincides with the action computed by Maldacena in [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] or by Rigopoulos in [START_REF] Rigopoulos | Gauge invariance and non-Gaussianity in Inflation[END_REF]. Let us examine the implications of this action. Forgetting about the redefinition of the perturbations in the uniform energy-density gauge, the form of the action is gauge-invariant. One can use it to easily calculate the non-Gaussianity related to the interaction terms as is explained in detail in [Seery and Lidsey, 2005b;[START_REF] Weinberg | Quantum contributions to cosmological correlations[END_REF]. This is known in the literature as f

N L , the parameter of non-Gaussianity related to the three-point correlation function of three first-order perturbations, which is only non-zero in the case of intrinsic non-Gaussianity.

However, taking into account the need for a redefinition of the perturbations in the uniform energy-density gauge, one might worry that the action is not actually gauge-invariant. The action in the uniform energy-density gauge before the redefinition has extra terms that are proportional to the second-order equations that the perturbations obey. This means that when calculating the non-Gaussianity in the uniform energy-density gauge, one not only has contributions due to the the interaction terms in the cubic action, but also ones due to the redefinition of ζ m , which contribute as explained in (2B.53). They are part of what is known in the literature as f (4) N L , the parameter of non-Gaussianity related to the three-point correlation function of a second-order perturbation (in terms of products of first-order ones) and two first-order perturbations, which reduces to products of two-point functions of the first-order perturbations.

This would mean that the non-Gaussianity calculated in the two gauges would not be the same due to the lack of any redefinition in the flat gauge. However, if one takes into account only the corrections coming from first-order perturbations, the redefinition associated to the second-order perturbation is not complete as one can check by comparing the super-horizon version of the adiabatic part of (2B.50) with (2B.29). As we showed, the solution of this issue is to include second-order fields since they also contribute to the cubic action. As one would expect these do not change the action itself, so that (2B.59) still holds. The effect of the new terms is to redefine the perturbations in both gauges. It should be noted that, if one had incorporated all quadratic first-order terms (found by a second-order gauge transformation as in the first appendix) directly as a correction to the first-order perturbations, one would have found the two contributions S 3(1) and S 3(2) together and hence there would have been no initial discrepancy between the two gauges. However, we explained at the end of the previous subsection our reasons for proceeding in this way. So in any case we finally obtain

S = Ŝ(ζ mc ) = S(ζ mc ), (2B.60)
where ζ mc is given in (2B.58). Now the two redefinitions as well as the action itself are the same for the two gauges, hence the action is truly gauge-invariant and the f (4)

N L , related to the products of first-order ζ m in the redefinitions, is the same in the two gauges.

This exact action allows one to compute f (4)

N L without the need for the slow-roll approximation at horizon crossing that is essential for both the long-wavelength formalism and the δN formalism: the long-wavelength formalism needs slow-roll at horizon crossing in order to allow for the decaying mode to vanish rapidly, while the δN formalism requires it in order to ignore the derivatives with respect to the canonical momentum. Additionally, up to now only the slow-roll field action [Seery and Lidsey, 2005a] (and not the action of the ζ m themselves) was known, so in order to compute the non-Gaussianity at horizon crossing one had to use the long-wavelength or δN formalism to transform to ζ m and hence one was in any case required to make the assumption of slow-roll, even if the exact action for the fields would have been known. It will be interesting to investigate models that do not satisfy the conditions for the long-wavelength or δN formalism using the action (2B.59).

In order to connect the redefinitions to some previously derived results in the literature we assume the super-horizon and slow-roll approximations. The super-horizon approximation is already assumed in (2B.58) and it can be supplemented by the condition ζ1 = 2η ⊥ ζ 2 . The slow-roll assumption translates into ζ2 = -χζ 2 . Then the quadratic part of the redefinitions, relevant to f (4) N L , takes the form

ζ 1 = ζ 1c + + η 2 ζ 2 1 + η ⊥ ζ 2 ζ 1 - + η -χ 2 ζ 2 2 ζ 2 = ζ 2c + η ⊥ 2 ζ 2 1 + ( + η -χ)ζ 1 ζ 2 + η ⊥ 2 ζ 2 2 . (2B.61)
The redefinitions in this form were used in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] to find the second-order source term of the evolution equations for the super-horizon perturbations. Their contribution to the super-horizon f (4)

N L was calculated in that paper using the long-wavelength formalism. In the equal-momenta case it was shown to be

- 6 5 f (4) N L,h.c. = * + η * + η ⊥ * v12 1 + (v 12 ) 2 , (2B.62)
where the index * indicates the time when the scale exits the horizon and v12 is essentially a transfer function showing how the isocurvature mode ζ 2 sources the adiabatic mode ζ 1 (see [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF] for details, where this term is part of what is called g sr ). Directly after horizon crossing or equivalently in the single-field limit, when v12 = 0, this reduces to the well-known result by Maldacena -6/5f

(4)

N L = * + η * .

2B.5 Appendices

This section contains the four appendices of the paper.

2B.5.1 Gauge transformations

From the infinite number of possible gauge-invariant combinations, we choose to work with quantities constructed from the energy density and the logarithm of the space dependent scale factor α. We will consider a gauge transformation β (i) = (T (i) , q (i) ) from the hatted gauge to the tilded gauge, where for the moment both gauges are taken to be arbitrary (not yet the flat and uniform energy density gauge). Notice though, that the space part of the transformation is not relevant outside the horizon, since when introduced in the relations below, it is connected to a second-order space derivative [START_REF] Malik | Evolution of second-order cosmological perturbations[END_REF]. Within the super-horizon approximation, we find using (2B.19)

ρ(1) = ρ(1) + ρT (1) , ρ(2) = ρ(2) + ρT (2) + T (1) 2 ρ(1) + ρ Ṫ(1) + ρT (1) , α(1) = α(1) + T (1) , α(2) = α(2) + T (2) + T (1) 2 α(1) + Ṫ(1) . (2B.63)
We want to construct a gauge-invariant quantity that reduces to α (i) in the uniform energydensity gauge, which we now identify with the tilded gauge so that ρ(i) = 0. This way we find

T (1) = - ρ(1) ρ , T (2) = - ρ(2) ρ + ρ(1) ρ(1) ρ2 (2B.64)
and obtain

ζ 1(1) ≡ α(1) = α(1) - ρ(1) ρ , 1 2 ζ 1(2) ≡ 1 2 α(2) = 1 2 α(2) - 1 2 ρ(2) ρ + ρ(1) ρ(1) ρ2 - ρ(1) ρ α(1) - 1 2 ρ2 (1) ρ2 ρ ρ . (2B.65)
Notice that the initial hatted gauge is still arbitrary, but if one was to associate it with the flat gauge α(i) = 0, then the time shift would become T (1) = ζ 1(1) . Next, we derive the exact gauge-invariant adiabatic and isocurvature perturbations, going beyond the super-horizon approximation. We use (2B.19) for the scalar fields and the space part of the metric tensor (2B.8), and find φA

(1) = φA (1) + φA T (1) (2B.66)

α(1) δ ij + 1 2 γ(1)ij = α(1) δ ij + 1 2 γ(1)ij + T (1) δ ij + ∂ i ∂ j q (1) + 1 2 ∂ i q ⊥ (1)j + ∂ j q ⊥
(1)i . (2B.67)

Here we followed [START_REF] Bruni | Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond[END_REF] and split the component i of the space shift as q i = ∂ i q +q ⊥ i , where ∂ i q ⊥ i = 0. We choose the uniform energy-density gauge, defined by e 1A φA (1) = 0 and use (2B.66) to find the first-order time shift to be T (1) = -e 1A φA (1) / φ. Then the trace and the traceless part of (2B.67) give

α(1) = α(1) + T (1) + 1 3 ∂ 2 q (1) (2B.68) γ(1)ij = γ(1)ij + 2 ∂ i ∂ j - 1 3 δ ij ∂ 2 q (1) + ∂ i q ⊥ (1)j + ∂ j q ⊥ (1)i . (2B.69)
In order to make the definition of the super-horizon adiabatic perturbation at first order (2B.63) to agree with (2B.68), we choose q (1) = 0 (any choice of q (1) is a gauge-invariant quantity, but only q (1) = 0 corresponds with the adiabatic perturbation ζ 1(1) in the literature). Note that while working with the super-horizon approximation, no such choice needs to be made, and q (1) remains arbitrary in that case. Similarly, we also assume that q ⊥ (1)i = 0, so we find

ζ 1(1) = α(1) = α(1) - 1 φ e 1A φA (1)
and

γ (1)ij ≡ γ(1)ij = γ(1)ij . (2B.70)
Using (2B.66), one easily finds that the isocurvature perturbation at first order is gauge-invariant since e 2A φA = 0

ζ 2(1) = - 1 φ e 2A φA (1) = - 1 φ e 2A φA (1) . (2B.71)
We now fix the hatted gauge to be the flat one, α(i) = 0, in order to lighten the calculations. This implies that T (1) = ζ 1(1) .

At second order we find

φA (2) = φA (2) + T (2) φA + ζ 1(1) ζ1(1) φA + ζ 1(1) φA + 2 φA (1) , (2B.72)
where we have omitted a term proportional to q (1) , which as mentioned above is chosen to be zero. At second order we choose the gauge

1 2 Q1(2) = +η 2 ζ 2 2(1) +∂ -2 ∂ i ζ2(1) ∂ i ζ 2(1) , see (2B.89),
that reduces to the uniform energy-density gauge on super-horizon scales (for the definition of Q 1( 2) in (2B.87) and details about that gauge choice, the reader can refer to the next appendix). Using (2B.84) we find from (2B.72)

T (2) = Q1(2) + ( + η ) ζ 2 1(1) -ζ 2 2(1) + ζ1(1) ζ 1(1) -2η ⊥ ζ 1(1) ζ 2(1) -2∂ -2 ∂ i ζ2(1) ∂ i ζ 2(1) . (2B.73)
Before turning to the adiabatic perturbation, let us prove that the first line of (2B.44) is a gauge-invariant quantity corresponding to the one in (2B.43). This is true in the exact theory, beyond the long-wavelength approximation, for q (1) = 0. Multiplying (2B.72) with -e 2A /(2 φ), noticing that e 2A φA = 0, and using (2B.84), one finds

1 2 Q2(2) = 1 2 Q2(2) + η ⊥ 2 ζ 2 1(1) + ζ2(1) ζ 1(1) + ( + η )ζ 1(1) ζ 2(1) (2B.74)
and indeed by comparing the total gradient of (2B.43) and (2B.44) we see that it corresponds to the second-order gauge-invariant isocurvature perturbation.

For the second-order adiabatic perturbation we need to perform the gauge transformation (2B.19) of the space part of the metric tensor between the uniform energy-density gauge and the flat gauge, α (2)

δ ij + 1 2 γ(2)ij = 1 2 γ(2)ij + T (2) δ ij + ∂ i ∂ j q (2) + 1 2 ∂ i q ⊥ (2)j + ∂ j q ⊥ (2)i + ζ 1(1) γ(1)ij +∂ i ζ 1(1) ∂ j λ + ∂ j ζ 1(1) ∂ i λ + ζ 1(1) ζ1(1) δ ij - 1 a 2 H 2 ∂ i ζ 1(1) ∂ j ζ 1(1) , (2B.75)
where we substituted T (1) = ζ 1(1) and T (2) is given in (2B.73). In order to find ζ 1(2) we take the trace and subtract the ∂ -2 ∂ i ∂ j of (2B.75) to eliminate the terms proportional to q (2) and obtain the result

1 2 ζ 1(2) = 1 2 α(2) = 1 2 Q1(2) + ζ1(1) ζ 1(1) + + η 2 ζ 2 1(1) -ζ 2 2(1) -η ⊥ ζ 1(1) ζ 2(1) -∂ -2 ∂ i ζ2(1) ∂ i ζ 2(1) - 1 4 ∂ -2 ( γ(1)ij ∂ i ∂ j ζ 1(1) ) - 1 4a 2 H 2 (∂ζ 1(1) ) 2 -∂ -2 ∂ i ∂ j (∂ i ζ 1(1) ∂ j ζ 1(1) ) + 1 2 ∂ i λ∂ i ζ 1(1) -∂ -2 ∂ i ∂ j (∂ i λ∂ j ζ 1(1) ) . (2B.76)
This is the second-order adiabatic gauge-invariant perturbation in the exact theory, the generalization of (2B.29).

2B.5.2 Super-horizon calculations

In this appendix we give the detailed calculations of section 2B.3. An important property of the long-wavelength assumption is that outside the horizon the uniform energy density can be recast in terms of the fields at least at first order: one can show that the exact 0i-Einstein equation (2B.23) outside the horizon can be rewritten as

∂ i ρ = -3 H ΠB ∂ i ϕ B .
(2B.77)

Again ρ denotes the fully non-linear energy density ρ = Π2 /2 + W . Expanding (2B.77) to first order and using the background equations to prove that ρ = -3Π 2 N H, (2B.78) one can show that outside the horizon

ρ (1) ρ = 1 ΠN e 1A ϕ A (1) (2B.79) so that ζ 1(1) = α(1) = α (1) - H Π e 1A ϕ A (1) . (2B.80)
and thus the energy-density constraint ρ(1) = 0 is equivalent to e 1A φA (1) = 0. Unfortunately the nice property described by (2B.79) does not hold anymore at second order. After expanding up to second order, combining the second-order Einstein equations and using the completeness relation of the field basis, one can show that (note that the zeroth order lapse function is taken from now on to be N (t) = 1/H(t))

ρ (2)
ρ -

1 φ e 1A ϕ A (2) = 1 φ e 1A ϕ A (1) 2 (η -) + 1 φ2 A, (2B.81) with 1 2 ∂ i A = η ⊥ e 2B ϕ B (1) e 1A ∂ i ϕ A (1) + e 2B ∂ i ϕ B (1) e 2A φA (1) , (2B.82)
where we used (2B.13) to simplify the expressions in terms of the slow-roll parameters. We see that the purely second-order contribution of ρ (2) is recast as a second-order contribution of ϕ A (2) , some quadratic first-order terms and a non-local term arising essentially from the 0i-Einstein equation. In the flat gauge ∂ i α = 0 this non-local term can be written as

1 2φ 2 ∂ i  = ∂ i η ⊥ ζ 1(1) ζ 2(1) + + η 2 ζ 2 2(1) + ζ2(1) ∂ i ζ 2(1) , (2B.83)
where when needed we employed the following useful relations (valid for the flat gauge beyond the long-wavelength approximation):

- 1 φ e 1A φA (1) = ζ1(1) + ( + η )ζ 1(1) -η ⊥ ζ 2(2) , - 1 φ e 2A φA (1) = ζ2(1) + ( + η )ζ 2(1) + η ⊥ ζ 1(2) , (2B.84)
derived by differentiating the adiabatic perturbation and the new combination

ζ 2(1) ≡ - H Π e 2A ϕ A (1) , (2B.85)
that represents the isocurvature perturbation to first order. Putting everything together in (2B.27) we find the second-order gauge-invariant curvature perturbation in the flat gauge to be

1 2 ζ 1(2) = 1 2 α(2) = 1 2 Q1(2) + + η 2 ζ 2 1(1) - + η 2 ζ 2 2(1) -η ⊥ ζ 1(1) ζ 2(1) + ζ1(1) ζ 1(1) -∂ -2 ∂ i ζ2(1) ∂ i ζ 2(1) , (2B.86)
where we defined the auxiliary quantities

Q m(i) ≡ - H Π e mA ϕ A (i) . (2B.87)
We turn now to the calculations relevant to the gradient of the perturbations. In the uniform energy-density gauge we can use (2B.77) to find the constraints 1) , (2B.89) so that when inspecting (2B.37) we see that

∂ i ρ(1) = -3 φA ∂ i φA (1) = 0 or ∂ i Q1(1) = 0, (2B.88) ∂ i ρ(2) = - 3 2 φA ∂ i φA (2) -3 φA(1) ∂ i φA (1) = 0 or 1 2 ∂ i Q1(2) = 1 φ2 φA(1) ∂ i φA (1) = + η 2 ∂ i ζ 2 2(1) + ζ2(1) ∂ i ζ 2(
1 2 ζ1i(2) = 1 2 ∂ i α(2) .
To derive the last equality in (2B.89) we used the completeness relation of the field basis along with the following relations valid beyond the long-wavelength approximation:

- 1 φ e 1A φA (1) = -η ⊥ ζ 2(1) , - 1 φ e 2A φA (1) = ζ2(1) + ( + η )ζ 2(1) , (2B.90)
derived by differentiating the definition of the isocurvature perturbation and the first-order uniform energy-density gauge constraint e 1A φA (1) = 0.

2B.5.3 Second-order action calculation

In order to rewrite the action we first need to calculate the extrinsic curvature. To do that we decompose N

= 1/H + N 1 , N i = ∂ i ψ + N i ⊥ , where ∂ i N i ⊥ = 0.
From now on we drop the explicit subscript (1) on first-order quantities and set κ 2 = 1 to lighten the notation (notice though that the final results remain unchanged when we restore κ 2 = 8πG, since all κ 2 are absorbed in when rewriting the fields in terms of ζ m ). We start by performing the calculation in the gauge e 1A φA = 0, (2B.91) which we call uniform energy-density gauge, since the above constraint reduces to zero energy perturbation outside the horizon. We first use the energy and momentum constraint (2B.6), (2B.7) to find that to first order

Ñ1 = α H = ζ1 H , Ñ i ⊥ = 0, ψ = - 1 a 2 ζ 1 H 2 + λ, ∂ 2 λ = ζ1 -2 η ⊥ ζ 2 . (2B.92)
It turns out that we do not need to calculate the shift or the lapse function to higher order, since in the action those terms are multiplied by constraint relations and hence vanish. We start by working out the scalar part of the action. Keeping in mind the gauge constraint (2B.91) we perturb (2B.3) to second order

S2 = 1 2 d 4 x a 3 e 3 α ( 1 H + Ñ1 + Ñ2 2 ) -2W -2W A φA -W AB φA φB +H(1 -H Ñ1 + H 2 Ñ 2 1 -H Ñ2 2 ) -6(1 + α) 2 + φ2 + 2 φA φA + φ2 -ae α ( 1 H + Ñ1 )2 (∂ α) 2 + 2∂ 2 α + 1 H ∂ i φA ∂ i φA , (2B.93)
where we have omitted a total derivative with respect to ψ. We then use the background Einstein and field equations to eliminate some terms and find that the term proportional to Ñ2 vanishes. Now the second-order action can be written as

S2 = 1 2 d 4 x a 3 H ζ1 -4 φA φA + φ2 ζ1 + 9ζ 2 1 (-6 + φ2 ) - 1 H 2 W AB φA φB -36ζ 1 ζ1 + φ2 -a 1 H 2 (∂ζ 1 ) 2 + ∂ i φA ∂ i φA . (2B.94)
The terms of (2B.94) proportional to φA can be recast in terms of the curvature perturbations by applying the completeness property of the field basis and (2B.90), so that after integrating by parts and using Ḣ = -H the action can be written as

S2 = d 4 x a 3 H ζ2 1 + ζ2 2 -4η ⊥ ζ1 ζ 2 -3(χ --η )ζ 2 2 + 2( + η )ζ 2 ζ2 +(η ⊥ ) 2 ζ 2 2 + ( + η ) 2 ζ 2 2 -a 1 H (∂ζ 1 ) 2 + (∂ζ 2 ) 2 , (2B.95)
or after further integration by parts as

S2 = d 4 x -a 1 H (∂ζ 1 ) 2 + (∂ζ 2 ) 2 + a 3 H ζ2 1 + ζ2 2 -4η ⊥ ζ1 ζ 2 + 2χ ζ2 ζ 2 (2B.96) +a 3 H 2 κ W 221 3H 2 -2 2 -(η ) 2 +3(η ⊥ ) 2 + 2 3 η ⊥ ξ ⊥ -3 (η -χ)+2η χ ζ 2 2 .
We can reach the same result while working in the flat gauge

∂ i α = 0. One can prove that N1 = -ζ 1 /H, N i ⊥ = 0 and ∂ 2 ψ = ∂ 2 λ = ζ1 -2 η ⊥ ζ 2 .
The ψ terms cancel out and the second-order action takes the form

Ŝ2 = 1 2 d 4 x a 3 H φ2 - 1 H W AB φA φB + N1 (-2W A φA -2H 2 φA φA ) + N 2 1 H 3 (-6 + φ2 ) -a 1 H ∂ i φA ∂ i φA . (2B.97)
Using the definition of ζ m , along with the background equations (2B.10), (2B.12) and (2B.84) this can be rewritten as (2B.96).

The second-order tensor part of the action in both gauges takes the form

S 2γ = d 4 xL 2γ = 1 2 d 4 x a 3 4 H( γij ) 2 - a 4H (∂ k γ ij ) 2 , (2B.98)
where L 2γ is the second-order Lagrangian for the tensor modes. We also give the equation of motion of the gravitational waves

δL 2γ δγ ij = - 1 4 d dt (a 3 H γij ) + 1 4 a H ∂ 2 γ ij = 0, (2B.99)
which we are going to use in the next section. In this paper we will not discuss the evolution and physics of gravitational waves, but at linear order this is a standard subject in the literature, for a discussion see for example [START_REF] Misner | Statistical properties of dust far-infrared emission[END_REF].

2B.5.4 Third-order action calculation

In order to compute S 3 we follow the same procedure starting from the uniform energy-density gauge. Notice that Ñ3 will multiply (-2W + 6H 2 -Π 2 ) in exact analogy with Ñ2 in S 2 , so it vanishes. Moreover, the overall factor multiplying Ñ2 is the first-order energy constraint (2B.92), so it can be consistently set to zero as well.

We start by computing the cubic action of the first-order curvature perturbations up to Ñ1 involving only scalar quantities

S3(1) = 1 2 d 4 x a 3 e 3 α 1 H + Ñ1 -2W -2W A φA -W AB φA φB - 1 3 W ABC φA φB φC +H (1 -H Ñ1 + H 2 Ñ 2 1 -H 3 Ñ 3 1 ) -6(1 + α) 2 + φ2 + 2 φA φA + φ2 + ∂ i ∂ j ψ∂ i ∂ j ψ -(∂ 2 ψ) 2 (1 -H Ñ1 ) -4∂ i ψ∂ i ζ 1 ∂ 2 ψ -2 φA ∂ i ψ∂ i φA -ae α ( 1 H + Ñ1 ) ∂ i φA ∂ i φA + 4∂ 2 ζ 1 + 2(∂ζ 1 ) 2 . (2B.100)
After using the background equations and the definitions of the perturbations, (2B.100) takes the form

S3(1) = d 4 x a 3 e 3ζ H (1 -ζ1 ) ζ2 1 + ζ2 2 + 2( + η ) ζ2 ζ 2 + (η ⊥ ) 2 +( + η ) 2 ζ 2 2 -2η ⊥ ζ 2 ζ1 -2 η ⊥ ζ 2 ζ1 -3(1 + ζ1 ) (χ --η )ζ 2 2 + 2 κ W 222 3H 2 ζ 3 2 -2∂ i ψ∂ i ζ 1 ∂ 2 ψ + 1 2 ∂ i ∂ j ψ∂ i ∂ j ψ -(∂ 2 ψ) 2 (1 -ζ1 ) -2 ∂ i ψ ( + η )ζ 2 ∂ i ζ 2 + ζ2 ∂ i ζ 2 -a 1 H (ζ 1 + ζ1 ) 2∂ 2 ζ + (∂ζ 1 ) 2 + (∂ζ 2 ) 2 . (2B.101)
By performing integrations by parts in (2B.101) we find

S3(1) = d 4 x a 3 H ζ 1 ( ζ2 1 + ζ2 2 ) -2 ζ1 ∂ i λ∂ i ζ 1 -2 ζ2 ∂ i λ∂ i ζ 2 -2 η +(η ) 2 +(η ⊥ ) 2 ζ 1 ζ 2 ζ2 +(3 η ⊥ + ξ ⊥ )ζ 2 1 ζ2 + 2( η ⊥ + ξ ⊥ )ζ 1 ζ 2 ζ1 -2 + 2 η + (η ) 2 + (η ⊥ ) 2 ζ 2 2 ζ1 + 2 2 + 3 η -(η ) 2 -(η ⊥ ) 2 + ξ ζ 2 1 ζ1 + 2 κ W 222 3H 2 ζ 3 2 + 2( + η )ξ ⊥ + η ⊥ 2 (3 + ) + 6η -ξ -3χ ζ 2 1 ζ 2 + -4 2 + 6 + 12 η + η (9 + 8η ) + 8(η ⊥ ) 2 + 2ξ -3χ + 2 κ W 221 H 2 -3(η ) 2 -3(η ⊥ ) 2 -2η ξ -2η ⊥ ξ ⊥ ζ 2 2 ζ 1 -2( + η )ζ 2 ∂ i λ∂ i ζ 2 +4η ⊥ ζ 2 ∂ i λ∂ i ζ 1 + 1 2 ζ 1 (∂ i ∂ j λ∂ i ∂ j λ -(∂ 2 λ) 2 ) + a 2 1 H ζ 1 (∂ζ 1 ) 2 +(∂ζ 2 ) 2 - δL 2 δζ 1 + η 2 ζ 2 1 -η ⊥ ζ 1 ζ 2 + ζ1 ζ 1 - 1 4a 2 H 2 (∂ζ 1 ) 2 + 1 4a 2 H 2 ∂ -2 ∂ i ∂ j (∂ i ζ 1 ∂ j ζ 1 ) + 1 2 ∂ i ζ 1 ∂ i λ - 1 2 ∂ -2 ∂ i ∂ j (∂ i λ∂ j ζ 1 ) - δL 2 δζ 2 ( + η )ζ 1 ζ 2 + ζ2 ζ 1 + η ⊥ 2 ζ 2 1 (2B.102)
where δL 2 /δζ m are the first-order equations of motion. We can further integrate by parts the rest of the action to simplify it and prove that it takes the form of the flat gauge action (2B.106), as expected since the action should be gauge-invariant. The terms involving λ along with the terms with space gradients vanish outside the horizon in the long-wavelength approximation, since λ is equal to the first-order super-horizon energy constraint (2B.48).

Finally we include the second-order fields. The extra terms in the action are

S3(2) = 1 2 d 4 x a 3 e 3ζ 1 ( 1 H + Ñ1 )(-W A φA (2) -W AB φA (2) φB )+H(1-H Ñ1 )( φA φA (2) + φA φA (2) ) -H φA ∂ i ψ∂ i φ(2) + 2H ζ1(2) ∂ 2 ψ -a 1 H -∂ i ζ 1 ∂ i ζ 1(2) + ζ1 ∂ 2 ζ 1(2) + ∂ i φA ∂ i φA(2) , (2B.103)
where ϕ A without a subscript always denotes the first-order perturbation. After performing integrations by parts we find

S3(2) = d 4 x δL 2 δζ 1 ζ 1(2) 2 + Q1(2) 2 + δL 2 δζ 2 Q2(2) 2 . (2B.104)
Next, we perform the same calculation for the flat gauge, starting from

Ŝ3(1) = 1 2 d 4 x a 3 1 H + N1 -2W -2W A φA -W AB φA φB - 1 3 W ABC φA φB φC +H(1 -H N1 + H 2 N 2 1 -H 3 N 3 1 ) -6 + φ2 + 2 φA φA + φ2 + 4∂ 2 ψ +∂ i ∂ j ψ∂ i ∂ j ψ -(∂ 2 ψ) 2 -2∂ i ψ φA ∂ i φA -2∂ i ψ φA ∂ i φA -a N1 ∂ i φA ∂ i φA , (2B.105)
again taking into account that N2 multiplies the first-order energy constraint and thus we set it to zero. We find using the definition of ζ m , along with (2B.10), (2B.12) and (2B.84)

Ŝ3(1) = d 4 x a 3 H ζ 1 ( ζ2 1 + ζ2 2 ) -2 ζ2 ∂ i λ∂ i ζ 2 -2 ζ1 ∂ i λ∂ i ζ 1 +2 ( + η )ζ 1 ζ 2 ζ2 + 2 η ⊥ ζ 2 1 ζ2 + (3η + 2 )ζ 2 1 ζ1 + 2 κ W 211 H 2 -2 ( η ⊥ + η η ⊥ + ξ ⊥ + 3η ⊥ ) ζ 2 1 ζ 2 + 2 κ W 222 3H 2 ζ 3 2 + 2 κ W 221 H 2 + -3(η ⊥ ) 2 + ( + η ) 2 + 3 (χ --η ) ζ 2 2 ζ 1 + 2 κ W 111 3H 2 -ξ + 3η -(η ⊥ ) 2 -(η ) 2 ζ 3 1 -2( + η )ζ 2 ∂ i λ∂ i ζ 2 + 4η ⊥ ζ 2 ∂ i λ∂ i ζ 1 + 1 2 ζ 1 (∂ i ∂ j λ∂ i ∂ j λ -(∂ 2 λ) 2 ) + a 2 H ζ 1 (∂ζ 1 ) 2 + (∂ζ 2 ) 2 . (2B.106)
Finally we include the second-order fields. The surviving terms in the action are

Ŝ3(2) = 1 2 d 4 x a 3 ( 1 H + N1 )(-W A φA (2) -W AB φA (2) φB ) + H(1 -H N1 )( φA φA (2) + φA φA (2) ) -H φA ∂ i ψ∂ i φA(2) -a 1 H ∂ i φA ∂ i φA(2) (2B.107)
and they can be rewritten as

Ŝ3(2) = d 4 x δL 2 δζ 1 Q1(2) 2 + δL 2 δζ 2 Q2(2) 2 . (2B.108)
In the last part of this appendix we consider the tensor scalar part of the action. There will be no contributions from the second-order fields, since these cancel due to γ ij being transverse. We start from the action for two scalar and one tensor modes in the uniform energy-density gauge

Sζζγ = d 4 x a H -2γ ij ∂ i ζ1 ∂ j ζ 1 -γ ij ∂ i ζ 1 ∂ j ζ 1 + γ ij ∂ i ζ 2 ∂ j ζ 2 + 1 2 a 3 H -(3ζ 1 -ζ1 ) γij ∂ i ∂ j ψ + ∂ k γ ij ∂ i ∂ j ψ∂ k ψ , (2B.109)
which after integrations by parts becomes

Sζζγ = d 4 x a 3 H 2 γij ∂ i ζ 1 ∂ j λ + 1 4 ∂ 2 γ ij ∂ i λ∂ j λ + a H γ ij ∂ i ζ 1 ∂ j ζ 1 + ∂ i ζ 2 ∂ j ζ 2 + δL 2γ δγ ij 1 a 2 H 2 ∂ i ζ 1 ∂ j ζ 1 -(∂ i ζ 1 ∂ j λ + ∂ j ζ 1 ∂ i λ) + δL 2 δζ 1 1 4 ∂ -2 ( γij ∂ i ∂ j ζ 1 ) . (2B.110)
In the flat gauge one finds directly after substitution into (2B.3) the first line of (2B.110), so that there are no redefinitions.

Finally we calculate the part of the action consisting of one scalar and two tensor modes, starting from the uniform energy-density gauge:

Sζγγ = 1 2 d 4 x a 3 H 1 4 (3ζ 1 -ζ1 )( γij ) 2 - 1 2 γij ∂ k γ ij ∂ k ψ - a 4H (ζ 1 + ζ1 )(∂ k γ ij ) 2 (2B.111) or equivalently Sζγγ = d 4 x -ζ 1 γij δL 2γ δγ ij + a 3 H 8 ζ 1 ( γij ) 2 - 1 4 γij ∂ k γ ij ∂ k λ + a 8H ζ 1 (∂ k γ ij ) 2 . (2B.112)
In the flat gauge we find directly

Ŝζγγ = 1 2 d 4 x a 3 H 4 ζ 1 ( γij ) 2 - 1 2 γij ∂ k γ ij ∂ k λ + a 4H ζ 1 (∂ k γ ij ) 2 . (2B.113)
The three tensor modes action does not contain any redefinitions. For details the reader may look in [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF].

and Koyama, 2010], thus leading people to consider an extra field in order to achieve smaller values of the parameter f NL of non-Gaussianity, see e.g. [START_REF] Renaux-Petel | Combined local and equilateral non-Gaussianities from multifield DBI inflation[END_REF].

Non-Gaussianity of the squeezed type can be found naturally in multiple-field models of inflation [START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF][START_REF] Bernardeau | Non-Gaussianity in multi-field inflation[END_REF], due to the sourcing of the adiabatic mode by the isocurvature components outside the horizon. For single-field models this is obviously impossible due to the absence of isocurvature modes. There has been much study of two-field models [Seery and Lidsey, 2005a;[START_REF] Kim | Nflation: Non-Gaussianity in the horizon-crossing approximation[END_REF][START_REF] Battefeld | Non-Gaussianities in Multi-field Inflation[END_REF]Battefeld and Battefeld, 2007;Langlois et al., 2008b;[START_REF] Cogollo | On the Issue of the zeta Series Convergence and Loop Corrections in the Generation of Observable Primordial Non-Gaussianity in Slow-Roll Inflation. Part I: The Bispectrum[END_REF][START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Renaux-Petel | Combined local and equilateral non-Gaussianities from multifield DBI inflation[END_REF][START_REF] Peterson | Non-Gaussianity in Two-Field Inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF], being the easiest to investigate, in the hope of finding a field potential that can produce local non-Gaussianity large enough to be measurable in the near-future. It proves to be non-trivial to sustain the large non-Gaussianity produced during the turn of the fields until the end of inflation.

Non-Gaussianity produced at horizon-crossing is known to be momentum-dependent. The scale dependence of the equilateral f NL produced for example from DBI inflation [Langlois et al., 2008a;[START_REF] Arroja | Non-gaussianity from the bispectrum in general multiple field inflation[END_REF][START_REF] Mizuno | Lorentz boost and non-Gaussianity in multi-field DBIinflation[END_REF][START_REF] Cai | Inflation with multiple sound speeds: a model of multiple DBI type actions and non-Gaussianities[END_REF][START_REF] Senatore | The Effective Field Theory of Multifield Inflation[END_REF], has been examined both theoretically [START_REF] Chen | Running non-Gaussianities in DBI inflation[END_REF][START_REF] Khoury | Rapidly-Varying Speed of Sound, Scale Invariance and Non-Gaussian Signatures[END_REF][START_REF] Byrnes | Non-Gaussianity beyond slow roll in multi-field inflation[END_REF][START_REF] Leblond | Simple Bounds from the Perturbative Regime of Inflation[END_REF] and in terms of observational forecasts [START_REF] Loverde | Effects of Scale-Dependent Non-Gaussianity on Cosmological Structures[END_REF][START_REF] Sefusatti | Constraining Running Non-Gaussianity[END_REF]. In this paper we are going to study the scale dependence of local-type models that has not been studied as much. Squeezed-type non-Gaussianity, produced outside the horizon, is usually associated with a parameter of non-Gaussianity f local NL that is local in real space, and therefore free of any explicit momentum dependence, defined through

ζ(x) = ζ L (x)-(3/5)f local NL (ζ L (x) 2 -ζ L (x) 2 )
, where ζ L is the linear Gaussian part. Nevertheless, calculations of f NL for several types of multiple-field models (see e.g. [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Byrnes | Conditions for large non-Gaussianity in two-field slow-roll inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]) show that there is always a momentum dependence inherited from the horizon-crossing era, which can in principle result in a tilt of f NL . When a physical quantity exhibits such a tilt one usually introduces a spectral index, as for example in the case of the power spectrum. The observational prospects of the detection of this type of scale dependence of local f NL were studied in [START_REF] Sefusatti | Constraining Running Non-Gaussianity[END_REF]. Only recently spectral indices for f NL were defined in [Byrnes et al., 2010a,b;[START_REF] Byrnes | General formula for the running of fNL[END_REF], keeping constant the shape of the triangle or two of its sides, within the δN formalism. Note, however, that most theoretical predictions have considered equilateral triangles for simplicity, even though the local-type configuration is maximal on squeezed triangles. If one were to calculate a really squeezed triangle, then f local NL acquires some intrinsic momentum dependence due to the different relevant scales, as was shown in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF].

It is both these effects we want to study in this paper: on the one hand the tilt of f NL due to the background evolution at horizon-crossing and on the other hand the impact of the shape of the triangle on f NL . In order to do that in a concrete way, such that these effects do not mix, we define two independent spectral indices, each one quantifying different deformations of the momentum triangle. Moreover, having an exact expression of f NL for an isosceles triangle, we are able to study and understand for the first time the origin of both types of momentum dependence of f NL . We also provide analytical estimates for the quadratic model (which actually hold for any equal-power sum model) that we use in this paper to illustrate our results.

The paper is organised as follows. In section 2C.2 we present the long-wavelength formalism results and discuss the sources of scale dependence in the expression for f NL . We also introduce two spectral indices, able to quantify the effects of different triangle deformations. In section 2C.3 we study the scale dependence for triangles of constant shape but of varying size, which is mainly due to horizon-crossing quantities, while in section 2C.4 we study the scale dependence related to the shape of the triangle.

2C.2 Sources of scale dependence

In addition to the power spectrum we can gain more information from the CMB by studying the Fourier transform of the three-point correlation function,

ζ 1k 1 ζ 1k 2 ζ 1k 3 ≡ (2π) -3/2 δ 3 ( s k s )B ζ (k 1 , k 2 , k 3 ), (2C.1)
where B ζ is the bispectrum. Because of the overall δ-function we see that the vectorial sum of the three k-vectors has to be zero. In other words, the three k-vectors form a triangle. The amplitude of the bispectrum can provide additional constraints on the slow-roll parameters of a given type of inflationary model. The profile of the bispectrum, i.e. the shape of the momentum triangle, gives information on the type of the inflationary model itself. For example, models with higher-order kinetic terms produce a bispectrum of the equilateral type (see e.g. [START_REF] Komatsu | Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background[END_REF]), mainly due to quantum interactions at horizon crossing. By equilateral type we mean a bispectrum that becomes maximal for equilateral triangles. On the other hand, canonical multiple-field inflation models predict a bispectrum of the local type. This arises from nonlinearities of the form

ζ 1 = ζ 1L -(3/5)f NL (ζ 2 1L -ζ 1L 2 ) (ζ 1L
being the first-order adiabatic perturbation) that are created classically outside the horizon, leading to a bispectrum of the form

B ζ (k 1 , k 2 , k 3 ) = - 6 5 f NL 2π 2 k 3 1 P ζ (k 1 ) 2π 2 k 3 2 P ζ (k 2 ) + (k 2 ↔ k 3 ) + (k 1 ↔ k 3 ) , (2C.2)
where f NL is usually assumed to be constant. This bispectrum becomes maximal for a squeezed triangle, i.e. a triangle with two sides almost equal and much larger than the third one. As we will discuss in the rest of the paper, f NL is not actually a constant, but depends on the size and shape of the momentum triangle.

In order to study the dependence of the non-Gaussianity on the shape of the triangle, instead of using k 1 , k 2 , and k 3 we will use the variables introduced in [Rigopoulos et al., 2005;[START_REF] Fergusson | The shape of primordial non-Gaussianity and the CMB bispectrum[END_REF],

K = k 1 + k 2 + k 3 2 , γ = k 1 -k 2 K , β = - k 3 -k 1 -k 2 2K , (2C.3)
which correspond to the perimeter of the triangle and two scale ratios describing effectively the angles of the triangle. They have the following domains: 0

≤ K ≤ ∞, 0 ≤ β ≤ 1 and -(1 -β) ≤ γ ≤ 1 -β, see figure 2C.1.
As one can check from the above equations, the local bispectrum becomes maximal for β = 1 and γ = 0, or β = 0 and γ = ±1, i.e. for a squeezed triangle. In this paper we always assume k 1 = k 2 , dealing only with equilateral or isosceles triangles (note that the relation k 1 = k 2 is satisfied by definition for both equilateral and squeezed triangles). The two scales of the triangle

k 3 ≡ k ≤ k ≡ k 1 = k 2 can

2C.2.1 Long-wavelength results

In this paper we use the long-wavelength formalism to study the parameter of non-Gaussianity f NL and its scale dependence. The non-Gaussianity parameter for an isosceles triangle of the form [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]] to be

k 1 = k 2 ≡ k ≥ k 3 ≡ k was found in
- 6 5 f NL = -2v 12k /[1 + (v 12k ) 2 ] 1 + (v 12k ) 2 + 2 γ 2 k γ 2 k [1 + (v 12k ) 2 ] v12k g sr (k , k )+g iso (k , k )+g int (k , k ) +2 γ 2 k γ 2 k v12k g sr (k , k)+g iso (k , k)+g int (k , k) , (2C.5)
where f NL = f NL (t; t k , t k ) depends on t k and t k , denoting the horizon-crossing times of the two scales k and k of the triangle, respectively. This result is exact and valid beyond the slow-roll approximation after horizon-crossing. All quantities appearing in this formula will be explained below.

The quantity v12 is a transfer function showing how the isocurvature mode (denoted by the subscript 2) sources the adiabatic component ζ 1 . In the following two more transfer functions will appear, namely v22 and v32 , showing how the isocurvature mode sources the isocurvature component ζ 2 and the velocity of the isocurvature component θ 2 ≡ ζ2 , respectively. va2 is a function of the horizon-exit time t k of the relevant perturbation of scale k and it also evolves with time t, at least during inflation. In (2C.5) as well as in the formulas that follow, va2k ≡ va2 (t, t k ). The indices a, b take the values 1, 2, 3, indicating respectively the adiabatic perturbation ζ 1 , the isocurvature perturbation ζ 2 , and the isocurvature velocity θ 2 .2 va2 comes from the combination of the Green's functions G a2 and G a3 of the system of equations for the super-horizon perturbations (for the system of equations that the Green's functions obey see (2.54)

): va2 (t, t k ) = G a2 (t, t k ) -χ k G a3 (t, t k ). (2C.6)
The quantity γ k in (2C.5) is defined as γ k ≡ -κH k /(2k3/2 √ k ) (and is not related to the γ defined in (2C.3)).

Except for the overall factor, f NL has been split into three contributions: g sr , g iso and g int . 3 g sr is a term that is slow-roll suppressed, since it depends only on horizon-exit quantities, where by assumption slow-roll holds,

g sr (k 1 , k 2 ) = η ⊥ k 1 G 22k 1 k 2 v12k 1 2 - 1 v12k 2 - G 22k 1 k 2 2v 12k 1 + 3χ k 2 4 G 33k 1 k 2 - 3 2 ( k 1 + η k 1 )G 22k 1 k 2 + χ k 1 4 2 v12k 1 v12k 2 + G 22k 1 k 2 - k 1 + η k 1 2( v 12 ) 2 (2C.7) + G 13 (t, t k 1 ) 2 3 (χ k 1 G 22k 1 k 2 -χ k 2 G 33k 1 k 2 ) 2v 12k 1 + G 32k 1 k 2 3 + k 1 + 2η k 1 2v 12k 1 + η ⊥ k 1 - 3 4 G 32k 1 k 2 - 1 2 G 12k 1 k 2 k 1 + η k 1 + 2η ⊥ k 1 - χ k 1 2 (1 + v12k 1 ) + k 1 + η k 1 v12k 1 .
Here we introduce some new notation,

( v 12 ) 2 ≡ v12k 1 v12k 2 , ( v 22 ) 2 ≡ v22k 1 v22k 2 , ( v 32 ) 2 ≡ v32k 1 v32k 2 , v 22 v 32 ≡ 1 2 (v 22k 1 v32k 2 + v22k 2 v32k 1 ),
and also

G abk 1 k 2 ≡ G ab (t k 1 , t k 2 )
. Moreover, we assume k 1 ≥ k 2 . g sr is the only term from which a (small) part survives in the single-field limit, i.e. in the limit where v12 = 0 at all times. For the equilateral case k = k the two last lines of g sr are zero,4 since the Green's functions satisfy

G ab (t, t) = δ ab . (2C.8)
The contribution g iso is a term that survives as long as the isocurvature modes are alive,

g iso (k 1 , k 2 ) = ( + η )( v 22 ) 2 + v 22 v 32 . (2C.9)
If at the end of inflation these are non-zero, f NL can still evolve afterwards and we cannot be sure that its value survives until today. Finally, g int is given by

g int (k 1 , k 2 ) = - t t k 1 dt 2(η ⊥ ) 2 ( v 22 ) 2 +( +η ) v 22 v 32 +( v 32 ) 2 -G 13 (t, t ) v 22 (Ξ v 22 +9η ⊥ v 32 ) (2C.10) with Ξ ≡ 12η ⊥ χ -6η η ⊥ + 6(η ) 2 η ⊥ + 6(η ⊥ ) 3 -2η ⊥ ξ -2η ξ ⊥ -2 1 κH 2 (W 211 + W 222 ), (2C.11)
where W mnl = W ,ABC e mA e nB e lC . It is from this integrated effect that any large, persistent non-Gaussianity originates, if we consider only models where the isocurvature modes have vanished by the end of inflation. For the analytical approximations that we will provide (in addition to the exact numerical results), it is useful to note that within the slow-roll approximation g int can be rewritten as

g int (k 1 , k 2 ) = v12k 1 G 22k 1 k 2 -η ⊥ k 1 + ( k 1 + η k 1 -χ k 1 )χ k 1 2η ⊥ k 1 + gint (k 1 , k 2 ), (2C.12)
where gint is another integral that is identically zero for the two-field quadratic model, or even more generally for any two-field equal-power sum model (see section 2.2.3 for details).

2C.2.2 Discussion

Inspecting (2C.5) ones sees that there are two sources of momentum dependence for f NL : the slow-roll parameters at horizon-crossing and the Green's functions G ab or their combinations va2 . In order to study their impact we shall use the quadratic model

W = 1 2 m 2 φ φ 2 + 1 2 m 2 σ σ 2 , (2C.13)
with m φ /m σ = 9. The procedure to follow is to solve for the background quantities and then for the Green's functions in order to apply the formalism. The quadratic model's Green's functions can be found numerically, or even analytically within the slow-roll approximation, which is valid for a small mass ratio like the one we chose here. However, all our calculations in this paper are numerical and exact, without assuming the slow-roll approximation after horizon crossing. We only use the slow-roll approximation after horizon crossing for the analytical approximations that we provide (e.g. eq. (2C.24)) and sometimes to clarify the physical interpretation of results (e.g. the use of (2C.15) below to explain the behaviour of v12 ). Inflation ends at t f defined as the time when f = 1. From now on a subscript f will denote quantities evaluated at the end of inflation. We also define the scale that exited the horizon 60 e-folds before the end of inflation as k 60 and use it as a reference scale, around which we perform our computations (k 60 being the scale that corresponds to the text books' minimal necessary amount of inflation).

In figure 2C.2 we plot the first-order slow-roll parameters for a range of horizon-crossing times around k 60 . While the heavy field rolls down its potential, the slow-roll parameters increase, reflecting the evolution of the background. This implies that f NL , which is in general proportional to the slow-roll parameters evaluated at t k and t k , should increase as a function of k and k . This can easily be verified for the initial value of f NL,in at t = t k , which according to (2C.5) with v12k = 0 takes the value

- 6 5 f NL,in = k + η k + 2 γ 2 k γ 2 k G 12k k 1 + 2 γ 2 k γ 2 k [1 + (G 12k k ) 2 ] η ⊥ k G 22k k . (2C.14)
Apart from the slow-roll parameters the other source of momentum dependence for f NL lies in the Green's functions and particularly how their time evolution depends on the relevant horizon-crossing scale. The two main quantities that we need to study in order to understand their impact on f NL are the transfer functions v12 and v22 . This is due to the fact that v32 is slow-roll suppressed and the rest of the Green's functions appearing in (2C.5) can be rewritten in terms of v12 and v22 within the slow-roll approximation (for details, see section 2.4.1). In particular As was discussed above, v12 describes how the isocurvature mode sources the adiabatic one, while v22 describes how the isocurvature mode sources itself. By definition v12 (t k , t k ) = 0 and v22 (t k , t k ) = 1 at horizon crossing, since no interaction of the different modes has yet occurred (see also (2C.6) and (2C.8)). For the transfer functions of the adiabatic mode one finds that v11 = 1 and v21 = 0, since the curvature perturbation is conserved for purely adiabatic perturbations and adiabatic perturbations cannot source entropy perturbations. In order to better understand the role of the transfer functions, we can use the Fourier transformation of the perturbations (2.30) along with these last identities, to find

G a3 = G a2 /3, G 32 (t, t k ) = -χ(t)G 22 (t,
ζ 1 (t) = d 3 k (2π) 3/2 γ k v1m â † m (k)e ik•x = ζ 1 (t k ) + v12 (t, t k )ζ 2 (t k ), ζ 2 (t) = d 3 k (2π) 3/2 γ k v2m â † m (k)e ik•x = v22 (t, t k )ζ 2 (t k ), (2C.16)
where ζ m with m = 1, 2 are the first-order adiabatic and isocurvature perturbation.

Let us start by discussing the time evolution of v12 . Each one of the curves on the left-hand side of figure 2C.3 corresponds to the time evolution of v12 for a different horizon-exit scale. At t = t k , i.e. when the relevant mode k exits the horizon, v12k = 0 since the isocurvature mode has not had time to affect the adiabatic one. Outside the horizon and well in the slow-roll regime of the sole dominance of the heavy field, the isocurvature mode sources the adiabatic one and the latter slowly increases. As time goes by, the heavy field rolls down its potential and the light field becomes more important. During this turning of the field trajectory, the slow-roll parameters suddenly change rapidly, with important consequences for the evolution of the adiabatic and isocurvature mode. The transfer function v12k grows substantially during that era because of the increasing values of η ⊥ in (2C.15) as well as the growing contribution of v22k , to become constant afterwards when the light field becomes dominant in an effectively single-field universe. Note that the earlier the mode exits the horizon, the smaller is the final v12k . This is opposite to the behaviour of the initial value, just after horizon-crossing, when the earlier the scale exits the horizon the more has its adiabatic mode been sourced by the isocurvature one at a given time t, and hence the larger is its v12k . This can be understood by the evolution equations of v12k and v22k in (2C.15), showing that v12k is sourced by v22k , which itself is a decreasing function of time, at least during eras when the universe is dominated by a single field (see the right-hand side of figure 2C.3). If the equation (2C.15) for v12k did not depend on η ⊥ , the v12k curves would never cross each other since they would be similar and only boosted by their horizon-crossing time shift. It is the increasing value of η ⊥ that results in the larger values of v12k for larger k.

On the right-hand side of figure 2C.3 we show the evolution of v22 . According to (2C.15) v22 and hence the isocurvature mode evolves independently from the adiabatic mode. At horizoncrossing t = t k , the transfer function v22k = 1. Once outside the horizon, the isocurvature mode decays due to the small but positive value of χ (defined in (1.19)). During the turning of the fields the slow-roll parameters evolve rapidly, thus leading to first an enhancement of v22k and then a diminution due to the varying value of χ in (2C.15). As can be seen from the right-hand side plot in figure 2C.2, during the turning χ first becomes negative and then positive. After the turning of the fields, the remnant isocurvature modes again decay and (for this model) at the end of inflation none are left. The parameter χ plays a crucial role in the evolution of the isocurvature mode. It represents effectively the second derivative of the potential in the 22 direction. Before the turning of the fields the trajectory goes down the potential in the relatively steep φ direction, which means that W 22 then corresponds to the relatively shallow curvature in the direction of the light field σ and hence χ is small. After the turning the trajectory goes along the bottom of the valley in the σ direction and W 22 corresponds with the large curvature of the potential in the perpendicular direction, leading to large values of χ. The negative values of χ during the turn come from the contribution of η (see (1.19)).

Instead of looking at the tranfer functions, using (2C.16) one can also construct more physical quantities from the operators ζ m and hence from the vm2 , namely the ratios of the adiabatic and isocurvature power spectrum to the total power spectrum:

R 1 ≡ ζ 1 ζ 1 ζ 1 ζ 1 + ζ 2 ζ 2 = 1 + (v 12 ) 2 1 + (v 12 ) 2 + (v 22 ) 2 , R 2 ≡ ζ 2 ζ 2 ζ 1 ζ 1 + ζ 2 ζ 2 = (v 22 ) 2 1 + (v 12 ) 2 + (v 22 ) 2 .
(2C.17)

These are plotted on the left-hand side of figure 2C.4 as a function of the number of e-foldings for different scales. One can clearly see that both ratios start as equal to 1/2 when the scale exits the horizon, while afterwards the adiabatic ratio R 1 increases to reach 1 at the end of inflation and the isocurvature R 2 decreases to reach 0, for this particular model. During the turning of the fields we see that the temporary increase in the isocurvature mode due to the negative value of χ is reflected in R 2 , while the adiabatic R 1 necessarily has the opposite behaviour.

On the right-hand side of figure 2C.4 we plot the time evolution of the spectral index of the power spectrum. The spectral index measures by construction the tilt of the power spectrum for different horizon-crossing scales and hence it depends on the horizon-crossing slow-roll parameters. For multiple-field models the power spectrum evolves during inflation even after horizon-crossing, and so does the spectral index. During the turning of the fields the spectral index increases, to remain constant afterwards. The earlier a scale exits the horizon the less negative is its spectral index n s -1. This implies that the power spectrum itself decreases faster for larger horizon-crossing scales. This is due to the fact that except for the factor 1 + (v 12k ) 2 in the expression for the power spectrum there is also an inverse power of k (see (2.33)).

2C.2.3 Spectral indices

Finally let us discuss the scale dependence of the local f NL in terms of the relevant spectral indices. Equation (2C.5) for an isosceles triangle implies that

- 6 5 f NL = 1 (2π 2 ) 2 f (k , k ) + 2( k k ) 3 f (k , k) P ζ (k ) 2 + 2( k k ) 3 P ζ (k )P ζ (k) , (2C.18) where f (k , k) = -2(k k) 3 γ 2 k γ 2 k v12k v12k g sr (k , k) + g iso (k , k) + g int (k , k) . (2C.19)
For an arbitrary triangle configuration this is generalized as

- 6 5 f NL = 1 (2π 2 ) 2 k 3 3 f (k 1 , k 2 ) + perms. k 3 3 P ζ (k 1 )P ζ (k 2 ) + perms. . (2C.20)
The local f NL depends on a two-variable function f (k 1 , k 2 ), with k 1 ≥ k 2 . This is due to its super-horizon origin, which yields classical non-Gaussianity proportional to products of two power spectra. Hence one expects that the scale dependence of f NL can be expressed in terms of only two spectral indices, characterizing the function f . Notice that this is particular to the local case. In general the bispectrum cannot be split as a sum of two-variable functions and one anticipates that three spectral indices would be needed.

The next issue to be resolved is which are the relevant spectral indices for f . The naive guess would be f

(k 1 , k 2 ) = f (k 1,0 , k 2,0 )(k 1 /k 1,0 ) ñk 1 (k 2 /k 2,0 ) ñk 2 .
We tested this parametrization and we did not find good agreement with the exact value of f . Instead of that, we found that f is best approximated by keeping either the shape or the magnitude of the triangle constant. This statement can be expressed as

f (k 1 , k 2 ) = f 0 K K 0 ñK ω ω 0 ñω , ( 2C
k 2 = 1 + β 2(1 -β) . (2C.23)
The last equality is valid only for the isosceles case γ = 0 (see (2C.3)). We dropped the -1 of the power spectrum spectral index definition to follow the definitions in [Byrnes et al., 2010b,a]. We added a tilde to indicate that these spectral indices are defined for the function f , not yet for the full f NL . In the next two sections we are going to examine the scale-dependence of f NL , changing the magnitude and the shape of the triangle separately, and verify assumption (2C.21).

2C.3 Changing the magnitude of the triangle

In this section we shall study the behaviour of f NL for triangles of the same shape but different size, see the left-hand side of figure 2C.1. In figure 2C.5 we plot the time evolution of f NL for equilateral triangles (the result would remain qualitatively the same for any isosceles triangle) of perimeter K = (3/2)k 60 × 10 (top curve), K = (3/2)k 60 (middle curve) and K = (3/2)k 60 /10 (bottom curve). The later the relevant scale exits the horizon the larger is its initial f NL as explained in the previous section. f NL grows during the turning of the fields due to isocurvature effects as described by (2C.9) and (2C.10), but by the end of inflation, when isocurvature modes vanish, it relaxes to a small, slow-roll suppressed value (see e.g. [START_REF] Vernizzi | Non-gaussianities in two-field inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]). In figure 2C.6 we plot the final value of f NL (left) and the final value of the bispectrum (right) for equilateral triangles, varying K for values around K = (3/2)k 60 , within the Planck satellite's resolution (k /k ∼ 1000). The later the scale exits the horizon, i.e. the larger K, the larger is the final value of f NL and of the bispectrum. The final value of f NL can be found analytically for the quadratic model within the slow-roll approximation. By the end of inflation v22,f = 0 so that (2C.9) vanishes, while (2C.10) can be further simplified to give some extra horizon-crossing terms and a new integral gint that is identically zero for the quadratic potential (see (2C.12) and section 2.2.3). For simplicity we give here the final value of f NL for equilateral triangles,

f NL,eq,f (k) = 3 (v 12k ) 2 k + η k -χ k + η ⊥ k v12k + (v 12k ) 3 η ⊥ k - k +η k -χ k χ k η ⊥ k + k + η k 1 + (v 12k ) 2 2
.

(2C.24) This formula is actually valid for any two-field model for which isocurvature modes vanish at the end of inflation and for which gint = 0, like for example equal-power sum models. Inspecting the various terms, it turns out that although v12k tends to decrease the value of f NL,f as a function of k, it is the contribution of the horizon-crossing slow-roll parameters that wins and leads to an increase of the parameter of non-Gaussianity for larger horizon-crossing scales. Note that The relative change of the final value of f NL arbitrarily normalized to one at the smallest value of K on the figure, as a function of K for equilateral triangles (ω = 1), calculated exactly (solid curve), using the analytical approximation (2C.24) (dashed red curve) and using the shape index (2C.29) (dotted black curve). Right: The logarithm of the final value of the exact bispectrum, similarly normalized, as a function of K for equilateral triangles (ω = 1). Both figures are for the quadratic model (2C.13) with mass ratio m φ /m σ = 9. The vertical dashed line corresponds to K = (3/2)k 60 .

for equilateral triangles K is simply 3k/2.

We turn now to the spectral index n K . Using (2C.22) with (2C.4) and assuming that γ = 0 and β = const., we can express ñK in terms of the horizon-crossing time derivatives as

ñK (t; t k 1 , t k 2 ) = ∂ ln f ∂t k 1 1 1 -k 1 + ∂ ln f ∂t k 2 1 1 -k 2 . (2C.25)
Then f NL takes the form

- 6 5 f NL = 1 (2π 2 ) 2 f (k 0 , k 0 ) K K 0 ñK (t k 0 ,t k 0 ) + 2ω 3 f (k 0 , k 0 ) K K 0 ñK (t k 0 ,t k 0 ) P ζ (k 0 ) 2 k k 0 2(ns(t k 0 )-1) + 2ω 3 P ζ (k 0 )P ζ (k 0 ) k k 0 ns(t k 0 )-1 k k 0 ns(t k 0 )-1 .
(2C.26) Note that the ratios k /k 0 = k/k 0 = K/K 0 , since β = const.

The above formula can be simplified in the limit of squeezed-triangle configurations, as well as in the equilateral limit. When one takes the squeezed limit ω 3 1 (note that this would be true for β > ∼ 2/3), one finds:

- 6 5 f NL = 1 (2π 2 ) 2 f (k 0 , k 0 ) P ζ (k 0 )P ζ (k 0 ) K K 0 ñK (t k 0 ,t k 0 )-ns(t k 0 )-ns(t k 0 )+2 ≡ - 6 5 f NL,0 K K 0 n K (t k 0 ,t k 0 ) , (2C.27) where n K (t; t k , t k ) ≡ d ln f NL d ln K = ∂ ln f NL ∂t k 1 1 -k + ∂ ln f NL ∂t k 1 1 -k . (2C.28)
For the equilateral case ω = 1, (2C.26) becomes

- 6 5 f NL = 1 (2π 2 ) 2 f (k 0 , k 0 ) P ζ (k 0 ) 2 K K 0 ñK (t k 0 ,t k 0 )-2ns(t k 0 )+2 ≡ - 6 5 f NL,0 K K 0 n K (t k 0 ,t k 0 ) . (2C.29)
The conformal spectral index n K measures the change of f NL due to the overall size of the triangle, namely due to a conformal transformation of the triangle. For an isosceles triangle this is conceptually sketched on the left-hand side of figure 2C.1, but it can be generalized for any shape. n K coincides with the n f NL of [Byrnes et al., 2010b,a] and grossly speaking it describes the tilt of f NL due to the pure evolution of the inflationary background (note that for an equilateral triangle this statement would be exact).

On the left-hand side of figure 2C.7 we plot the time evolution of the conformal spectral index for an equilateral ω = 1 and an isosceles ω = 5/2 triangle that exited the horizon at three different times, namely for K = (3/2)k 60 (solid curve), K = (3/2)k 60 /10 (dashed curve) and K = (3/2)k 60 × 10 (dotted curve). We plot the ω = 5/2 case only to demonstrate that the results remain qualitatively the same; we shall study the effect of different triangle shapes in the next section. The characteristic peaks that n K exhibits during the turning of the fields are inherited from the behaviour of f NL at that time and it is a new feature that is absent in the time evolution of the power spectrum spectral index n s -1 (see the right-hand side of figure 2C.4).

In the context of the long-wavelength formalism we are restricted to work with the slow-roll approximation at horizon exit, so that the slow-roll parameters at that time should be small and vary just a little. This should be reflected in the initial value of the spectral index, which should be O( k ). The earlier the scale exits, e.g. the dashed curve, the smaller are the slow-roll parameters evaluated at horizon crossing and hence the smaller is the initial n K . Indeed, using the definition (2C.28) with (2C.14) for the initial value of f NL,in , we find for equilateral triangles

n K,in = 2 2 k + 3 k η k + (η ⊥ k ) 2 -(η k ) 2 + ξ k k + η k , (2C.30)
which confirms the above statement. We notice that the initial, horizon-crossing, differences between the values of n K for the different horizon-crossing scales mostly disappear by the end of inflation, after peaking during the turning of the fields. The final value of the spectral index is plotted on the right-hand side of figure 2C.7 and is smaller than its initial value. It exhibits a small running of O(10%) within the range of scales studied, inherited from the initial dispersion of its values at horizon-crossing. To verify that n K describes well the behaviour of f NL , we have plotted the approximation (2C.29) in figure 2C.6 where it can be compared with the exact result. We have also verified this for other inflationary models, including the potential .31) studied in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF], able to produce f NL of O(1). The final value of the spectral index in that model is two orders of magnitude smaller than the value for the quadratic model. This is related essentially to the fact that for the potential (2C.31) the turning of the fields, and hence the slow-roll breaking, occurs near the end of inflation. This means that at the horizon-crossing times of the scales of the triangle, slow-roll parameters change very slowly and as a result the initial variation of f NL is much smaller than the one for the quadratic potential. As a consequence, the final tilt of f NL will be smaller. By differentiating (2C.24) and using (2C.25) we can find the final value of n K for equilateral triangles in the slow-roll approximation, assuming that isocurvature modes have vanished for an equal-power sum potential (for which the gint contribution is zero, see (2C.12) and section 2.2.3): .32) where W221 = ( √ 2 /κ)W 221 /(3H 2 ). We have checked this approximation and we find good agreement with the exact conformal index for equilateral triangles.

W = b 0 -b 2 σ 2 + b 4 σ 4 + a 2 φ 2 , ( 2C
n K,eq,f = -4 v12k (v 12k χ k -2η ⊥ k ) 1 + (v 12k ) 2 - 1 f NL,eq,f (1 + (v 12k ) 2 ) 2 -2 2 k -3 k η k + (η k ) 2 + 5(η ⊥ k ) 2 -ξ k + 3v 12k η ⊥ k (3 k + 6η k -5χ k ) -ξ ⊥ k + 3(v 12k ) 2 W221k + 4(η ⊥ k ) 2 -2( k + η k -χ k )( k + 2χ k ) + (v 12k ) 3 η ⊥ k χ k 3 2 k -2 W221k + 4 k η k + 3(η k ) 2 -8(η ⊥ k ) 2 + η χ k -3χ 2 k + ξ k ( k + η k -χ k ) + η k ( 2 k -(η k ) 2 ) + W221k ( k + η k ) + (η ⊥ k ) 2 (2 k + 5η k ) -ξ ⊥ k η ⊥ k + ( k + η k + χ k )χ k η ⊥ k , ( 2C

2C.4 Changing the shape of the triangle

After studying triangles with the same shape but varying size in the previous section, we now turn to the scale dependence of f NL for triangles of the same perimeter but different shape, Although it is during the peak that the variation of f NL for different shapes is more prominent, its final value is also affected. On the left-hand side of figure 2C.9 we plot the value of f NL at the end of inflation for triangles of perimeter K = (3/2)k 60 , normalised by its value for the equilateral case (ω = 1), as a function of ω. The deviation of the values is small since it is related to horizon-exit slow-roll suppressed quantities. Within the long-wavelength formalism (or the δN formalism) slow roll at horizon crossing is a requirement. Nevertheless, the important conclusion here is that f NL decreases when the triangle becomes more squeezed. This can be attributed to the fact that the more squeezed is the triangle, the more the fluctuation ζ k is frozen and behaves as part of the background when scale k crosses the horizon. As a result the correlation between k and k becomes less and the resulting non-Gaussianity is smaller (see also the discussion below equation (2C.33).)

An analytical formula can be found when applying the slow-roll approximation to expression (2C.5) at the end of inflation, when isocurvature modes have vanished. We perform an integration by parts in the integral (see (2C.12); as before gint = 0). More precisely, assuming that we are really in the squeezed limit k k , the ratio γ 2 k /γ 2 k becomes very large and we can ignore the equilateral terms that depend only on k and not also on k. We also assume that the decaying mode has vanished to simplify the expressions for the Green's functions (see the discussion in section 2C.2). G 12k k can be set to zero as one can see in figure 2C.3 (since it is basically equal to v12 and only involves times at the very left-hand side of the figure). Moreover, the same figure shows that v12k /v 12k ≈ 1 (in the formula these ratios are always multiplied by slow-roll parameters, so that the deviation from 1 would be like a second-order effect), so that we find in the end

f NL,sq,f = G 22k k f NL,eq,f (k ) + 1 -G 22k k 1 + (v 12k ) 2 2 k + η k + 2η ⊥ k v12k -χ k (v 12k ) 2 , (2C.33)
where f NL,eq,f is given in equation (2C.24). The only quantity in the above expression that depends on the shape of the triangle is G 22k k , so it must be G 22k k that is responsible for the decreasing behaviour of f NL,sq,f . Indeed, increasing ω for a constant perimeter K of the triangle means increasing the interval t kt k and hence decreasing the value of G 22k k (see the right-hand side of figure 2C.3, since in the slow-roll regime v22 = G 22 ). This means that the interaction of the two modes becomes less important. In the complete absence of isocurvature modes G 22k k = 0 and f NL,sq,f takes its minimal value. It is only the isocurvature mode that interacts with itself and the greater is the difference between the two momenta the less is the interaction. Notice that the single-field limit of this result would correspond to G 22k k = 0 and v12 = 0. The decrease of f NL for more squeezed triangles seems contradictory to the well-known fact that the local bispectrum is maximized for squeezed configurations. In order to clarify this subtle point, we stress that the left-hand side of figure 2C.9 is essentially the ratio of the exact bispectrum to the bispectrum assuming f NL as a constant (2C.2) and hence the products of the power spectrum cancel out. We also plot on the right-hand side of figure 2C.9 the final value of the bispectrum (2C.2), normalised by the value of the bispectrum for equilateral triangles with K = 3k 60 /2. Although f NL is maximal for equilateral triangles, the bispectrum has the opposite behaviour, since it is dominated by the contribution of the products of the power spectrum, which leads to an increased bispectrum for the more squeezed shape. At the same time though we show that there is a small contribution of f NL itself, leading to smaller values of the bispectrum when compared to a bispectrum where f NL is assumed to be constant.

In order to quantify the above results, we examine the shape index ñω (2C.22), assuming K = const and γ = 0,

ñω = ∂ ln f ∂t k 1 1 -k 1 1 + 2ω - ∂ ln f ∂t k 1 1 -k 2ω 1 + 2ω . (2C.34)
In terms of ñω , f NL takes the form

- 6 5 f NL = 1 (2π 2 ) 2 f (k 0 , k 0 ) + 2ω 3 f (k 0 , k 0 ) ω ω 0 ñω(t k 0 ,t k 0 ) P ζ (k 0 ) 2 k k 0 2(ns(t k 0 )-1) + 2ω 3 P ζ (k 0 )P ζ (k 0 ) k k 0 ns(t k 0 )-1 k k 0 ns(t k 0 )-1 , (2C.35) where k /k 0 = (1 + β)/(1 + β 0 ) ∝ 2ω/(ω + 1 2 ) and k/k 0 = (1 -β)/(1 -β 0 ) ∝ 1/(ω + 1 2
). This can be further simplified in the squeezed region ω 1 to find

- 6 5 f NL = 1 (2π 2 ) 2 f (k 0 , k 0 ) ω ω 0 ñω(t k 0 ,t k 0 ) P ζ (k 0 )P ζ (k 0 ) ω(ω 0 + 1 2 ) ω 0 (ω+ 1 2 ) ns(t k 0 )-1 ω 0 + 1 2 ω+ 1 2 ns(t k 0 )-1 ≡ - 6 5 f NL,0 ω ω 0 ñω(t k 0 ,t k 0 )+ns(t k 0 )-1 ≡ - 6 5 f NL,0 ω ω 0 nω(t k 0 ,t k 0 ) (2C.36) with n ω ≡ d ln f NL d ln ω = ∂ ln f NL ∂t k 1 1 -k 1 1 + 2ω - ∂ ln f NL ∂t k 1 1 -k 2ω 1 + 2ω . (2C.37)
The shape index n ω describes the change of f NL due to the relative size of the two scales, namely due to how squeezed the triangle is, while keeping K constant (see the right-hand side of figure 2C.1). We studied different squeezed triangle configurations with constant K, varying ω from ω = 5/2 to ω = 1000. On the left-hand side of figure 2C.10 we plot the time evolution of the shape index. The negative values of the index signify the decrease of f NL as expected. As one can see from the figure, for the more squeezed triangle (ω = 1000) the initial value of n ω seems to depend solely on the shape of the triangle and not on its magnitude, and even for the less squeezed triangle (ω = 5/2) the initial dependence on K is negligible. We can find the analytical

n ω,in = 1 1 + 2ω n K,in + 4ω 1 + 2ω G 22k k (η ⊥ k ) 2 k + η k . (2C.38)
For G 22k k = 0, which corresponds to the squeezed limit, n ω,in is proportional to the initial shape index for equilateral triangles times a factor depending on the shape, which also becomes very small in the squeezed limit. Super-horizon effects, and especially the turning of the fields, result in a separation of the curves of n ω of the same shape for different values of K, due to the dependence of the evolution of v12k on the scale k . The turning of the fields increases the absolute value of n ω , which is the opposite of the behaviour of the conformal index n K . The shape index depends on the transfer function v12k (see (2C.39) for an analytical approximation). The smaller K, the less does the final value of v12k change with respect to its initial value (see figure 2C.3) and hence the less the shape index is affected. Notice that the slow-roll parameters at horizon-crossing have the opposite behaviour: the smaller K, the smaller they are. Even though n ω also depends on the slow-roll parameters, it is v12k that most affects its evolution.

On the right-hand side of figure 2C.10 we plot the value of the shape index at the end of inflation. It exhibits a running of about 20% within the range of scales studied, somewhat larger than the conformal index. We have analytically computed the shape spectral index for models with gint = 0 and with final g iso,f = 0 and give the result in (2C.39).

The dotted curve in the plot on the left-hand side of figure 2C.9 shows the final value of f NL approximated as a simple power law according to (2C.36). Within the range of validity of our approximation ω 3

1 it describes the exact result very well. We have also studied the shape spectral index for the potential (2C.31). Similarly to n K , its value is two orders of magnitude smaller than the value for the quadratic potential, but the parametrization of f NL in terms of the shape index is in good agreement with the exact result for a larger range of ω > ∼ 3/2.

We repeat the calculation that we did at the end of the previous section for n K here for the shape index n ω (2C.37), differentiating the squeezed f NL (2C.33). Where needed we use the slow-roll approximation

G 32k k = -χ k G 22k k and G 23k k = G 22k k /3. The result is: n ω,sq,f = 1 f NL,sq,f (1 + (v 12 ) 2 ) 2 1 1 + 2ω 2G 22k k v12k 1 + (v 12 ) 2 4 -(ω + (2 + ω)v 12k ) χ k η ⊥ k × η ⊥ k + v12k 3( k + η k ) -2χ k + (v 12k ) 2 η ⊥ k - ( k + η k -χ k )χ k η ⊥ k -G 22k k 2(η ⊥ k ) 2 -v12k ξ ⊥ k -η ⊥ k (11 k + 14η k -8χ k ) + (v 12k ) 3 η ⊥ k χ k 3 2 k -2 W221k + 4 k η k + 3(η k ) 2 -7(η ⊥ k ) 2 -2χ 2 k -k χ k + ξ ( k + η k -χ k ) + η k ( 2 k -(η k ) 2 ) + W221k ( k + η k ) + (η ⊥ k ) 2 (2 k + 5η k ) -ξ ⊥ k η ⊥ k + ( k + η k + χ k )χ k η ⊥ k + (v 12k ) 2 2 W221k -6 2 k + (η k ) 2 + 9(η ⊥ k ) 2 -ξ k -9η k χ k + 8χ 2 k -k (7η k + 5χ k ) - 1 1+(v 12k ) 2 (η k ) 2 + 3(η ⊥ k ) 2 -2 2 k -3 k η k -ξ k -2(v 12k ) 3 ξ ⊥ k + η ⊥ k ( k -2η k -5χ k ) + (v 12k ) 4 W221k + ( k -η k )η k + 3(η ⊥ k ) 2 + ξ k + 2χ k ( k + χ k ) -2v 12k ξ ⊥ k + η ⊥ k (5 k + 2η k + 3χ k ) + (v 12k ) 2 W221k -2 k ( k + η k ) + 5(η ⊥ k ) 2 -(3 k + 3η k + χ k )χ k (2C.39)
We have checked this approximation and we find good agreement with the exact shape index for ω > ∼ 3.

Chapter 3

Non-Gaussianity in the CMB

After having discussed in chapter 2 how non-Gaussianity is produced during multiple-field inflation and how we can calculate it, we will describe in this chapter how we can observe this and other primordial non-Gaussianity in the cosmic microwave background radiation (CMB). These are my papers in this subject area:1

• [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF] -The main idea and first implementation of the binned bispectrum estimator to determine f NL from CMB data are explained (temperature-only and for an ideal sky). The local bispectrum template is studied in detail and two other estimators are introduced, for detecting the acoustic oscillations in the bispectrum.

• [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF]] -Many refinements (e.g. linear correction, inpainting, polarization, more templates, joint analysis) of the binned bispectrum estimator are treated, revealing it in its final form as used for the Planck analyses. The smoothed binned bispectrum as a tool for non-parametric non-Gaussianity searches is introduced. 2

• [Langlois and Van Tent, 2011, 2012] -Two papers (the first short paper highlighting the main ideas, the second longer paper giving all the details) developing and studying the bispectrum templates for isocurvature non-Gaussianity, in temperature and polarization. Forecasts for Planck are also given. Constraints in the context of an explicit class of multiple-field inflation models are discussed.

• [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF] -The bispectra of various galactic foregrounds are determined and studied, and used to verify that no significant foreground contamination remains in the cleaned Planck maps.

• My work on the binned bispectrum estimator [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF][START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF][START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF] is mainly a collaboration with Martin Bucher (first two) and Benjamin Racine (last two). Gabriel Jung was a major collaborator on the last one. Carla Carvalho was involved in the initial stages of the work for [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF]. The work on isocurvature non-Gaussianity is a collaboration with David Langlois. The Planck primordial non-Gaussianity papers are of course credited to the whole Planck collaboration, but I am one of the main authors, having co-written with Michele Liguori at least one third of each paper. The CORE paper was co-written by many members of the CORE collaboration, with Michele Liguori and myself in charge of the chapter on non-Gaussianity. In section 3.1, after a general introduction, we discuss the theoretical bispectrum templates that we searched for in our analyses, both primordial and foreground bispectra. In section 3.2 all aspects of the binned bispectrum estimator are treated. Both of these sections are mostly based on [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF], with some input from [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF]. Further extensions and applications of the estimator (including the very important Planck results on primordial non-Gaussianity), described in the self-contained papers [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF][START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF]Planck 2018 IX], are briefly summarized in section 3.3 and included in full in the appendices 3A-3C.

Introduction and theoretical bispectrum templates

Introduction

A fundamental question of observational cosmology is whether the primordial cosmological perturbations were precisely Gaussian, or whether small departures from exact Gaussianity can be detected at a statistically significant level and then characterized. Here the qualification 'primordial' is essential because our goal is to probe the new physics at play in the very early universe. However, it is also important to study the non-Gaussianity that was subsequently imprinted at late times through known processes, in particular the nonlinear dynamics of gravitational clustering, in order to 'decontaminate' the primordial non-Gaussianity. Observations of the cosmic microwave background anisotropies in temperature and polarization are particularly well-suited to addressing this fundamental question, as they provide a clean probe of the initial conditions because most of the CMB anisotropy was imprinted well before nonlinear effects became important.

Non-Gaussianity manifests itself in odd n-point correlation functions or in the connected even n-point correlation functions, from which the trivial part expressible as combinations of two-point correlation functions has been subtracted away. The extent of departures from Gaussianity can be characterized by ratios of higher-order correlation functions and the appropriate combination of two-point correlation functions [Bernardeau et al., 2002]. The evolution of the primordial fluctuations of the inflaton field, involving both gravity and hydrodynamics, leads to CMB anisotropies whose statistical properties are related to those of the primordial fluctuations. Consequently, by studying higher-order correlation functions of the CMB anisotropies, we can detect and characterize any primordial non-Gaussianity. The lowest order such statistic is the bispectrum, or three-point correlation function in Fourier space. The bispectrum has been shown to be an optimal statistic for measuring non-Gaussianity in the sense that the signal-to-noise squared of the non-Gaussianity estimator based on the three-point correlation function dominates over all higher-order estimators [START_REF] Babich | Optimal estimation of non-Gaussianity[END_REF]. Consequently it would also be significantly easier to constrain.

As discussed in detail in the previous chapters, while Gaussianity is a good first approximation for inflationary fluctuations, inflation does not predict exact Gaussianity no matter what model of inflation is assumed. It cannot be modeled by a free field theory because at a minimum the gravitational sector is nonlinear. Additional nonlinearity will of course also arise from other sources, such as for example from the nonlinearity of the inflationary potential. The departures from Gaussianity predicted within the framework of standard single-field slow-roll inflation were calculated by Maldacena [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF] and by [START_REF] Acquaviva | Second order cosmological perturbations from inflation[END_REF] and are indeed unobservably small. The bispectrum of standard single-field inflation can be viewed as a superposition of the local shape and the equilateral shape (see next subsection for definitions), both terms, however, being slow-roll suppressed [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF][START_REF] Fergusson | The shape of primordial non-Gaussianity and the CMB bispectrum[END_REF][START_REF] Senatore | Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data[END_REF]. In particular, for the squeezed triangle configurations in single-field inflation the non-Gaussian signal would be proportional to the tilt of the power spectrum and thus a detection would imply a strong deviation from scale invariance [Creminelli and Zaldarriaga, 2004b;[START_REF] Cheung | On the consistency relation of the 3-point function in single field inflation[END_REF]. However, many other inflation models have been introduced in the literature that can produce non-negligible non-Gaussianity. For example, models where multiple scalar fields play a role during inflation (as was discussed in detail in chapter 2) or where inflation starts in an excited vacuum state (see e.g. [START_REF] Holman | Enhanced Non-Gaussianity from Excited Initial States[END_REF]). In some string-based models, as well as in some modified gravity or effectivefield theories, the kinetic part of the inflaton Lagrangian can be non-standard, leading to novel bispectral signatures (see e.g. [START_REF] Alishahiha | DBI in the sky[END_REF][START_REF] Arkani-Hamed | Ghost inflation[END_REF][START_REF] Green | Trapped Inflation[END_REF]). Deviations from the pure slow-roll phase in the inflaton potential can also produce oscillations in the bispectrum, see e.g. [Chen et al., 2007a[START_REF] Chen | Generation and Characterization of Large Non-Gaussianities in Single Field Inflation[END_REF]. For more details and references, see e.g. [START_REF] Liguori | Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure[END_REF][START_REF] Chen | Primordial Non-Gaussianities from Inflation Models[END_REF] or [Planck 2013 XXIV;Planck 2015 XVII;Planck 2018 IX] for a review, the latter included in appendix 3C. Many references were also given in the previous chapters.

After horizon crossing, non-linearities in both the gravitational and hydrodynamical evolution of the baryon-photon fluid prior to recombination as well as higher orders in the gravitational potential during recombination, can generate non-Gaussianity [START_REF] Pyne | Higher order gravitational perturbations of the cosmic microwave background[END_REF]Bartolo et al., 2004a;Creminelli and Zaldarriaga, 2004a;[START_REF] Nitta | CMB anisotropies at second order III: bispectrum from products of the first-order perturbations[END_REF][START_REF] Bartolo | On the non-Gaussianity from Recombination[END_REF][START_REF] Pitrou | The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations[END_REF][START_REF] Su | CMB Bispectrum from Non-linear Effects during Recombination[END_REF][START_REF] Huang | Cosmic Microwave Background Bispectrum from Recombination[END_REF]. Other sources of non-primordial non-Gaussianity include secondary anisotropies such as weak lensing via the cross-correlation with the unlensed CMB arising from the integrated Sachs-Wolfe effect [START_REF] Smith | Algorithms for bispectra: Forecasting, optimal analysis, and simulation[END_REF][START_REF] Hanson | CMB lensing and primordial non-Gaussianity[END_REF][START_REF] Mangilli | Non-Gaussianity and the CMB Bispectrum: confusion between Primordial and Lensing-Rees Sciama contribution?[END_REF][START_REF] Lewis | The shape of the CMB lensing bispectrum[END_REF] or the Sunyaev-Zel'dovich effect [START_REF] Goldberg | Microwave background bispectrum. 2. A probe of the low redshift universe[END_REF], as well as foregrounds such as dust, galactic synchrotron radiation and unresolved point sources. Finally there are also instrumental effects, see e.g. [START_REF] Donzelli | Impact of the 1/f noise and the asymmetric beam on non-Gaussianity searches with Planck[END_REF]. These effects contribute spurious non-Gaussian signals, thus biasing the measurement of the primordial signal. It is therefore important to develop tools to isolate the primordial signal from the contaminants.

Under the assumption of statistical isotropy, the bispectrum of the map of a scalar quantity reduces to a function of three multipole numbers B 1 2 3 , where the bispectrum is symmetric under permutations and vanishes unless the -triplet satisfies the triangle inequality [START_REF] Luo | The Angular bispectrum of the cosmic microwave background[END_REF]. If we include polarization, which in turn can be decomposed into E and B components, then the bispectrum needs to be generalized to B p 1 p 2 p 3 1 2 3 , where p 1 , p 2 , p 3 = T, E (we will not consider B-polarization in this thesis). If we insist on exploiting the highest possible spectral resolution of the CMB maps (not necessarily the best idea), then the number of reduced bispectral coefficients that can be measured is huge, scaling with max as 3 max , and the individual coefficients are too contaminated by noise to be useful in detecting bispectral non-Gaussianity. A major and unavoidable contribution to this noise arises from cosmic variance -that is, from the departures from zero of B p 1 p 2 p 3 1 2 3 that would occur even if the underlying stochastic process were exactly Gaussian. While Gaussianity requires that the expectation value of the bispectral coefficients, calculable only in the limit of an infinite number of sky realizations, vanishes, the value calculated for any single sky realization will include fluctuations about this expectation value. For this reason, in order to make any meaningful detection of bispectral non-Gaussianity in the data, it is necessary to combine, in one way or another, many measured bispectral coefficients in order to make the signal stand out over the noise.

There are basically two situations to be considered. If we have a simple parametric model for the expected pattern of bispectral non-Gaussianity (generally parameterized by an amplitude called f NL ), then an optimal estimator can be constructed by summing the observed bispectral coefficients over 1 , 2 , 3 using inverse variance weighting. Another situation to be considered involves non-parametric reconstruction of the bispectrum, where we do not have a specific template in mind, but want to smooth the bispectrum in order to reduce the noise and see whether there is a broad signal that stands out over the noise at a statistically significant level. This latter approach is particularly relevant for studying the bispectral properties of foregrounds, for which a theory of the expected shape of the bispectral non-Gaussianity is lacking.

Combining the bispectral coefficients is not only required from the physical point of view (to obtain statistically significant results), but also computationally: computing O( 3 max ) bispectral coefficients for each map is not feasible in practice. A natural solution, motivated by the second case mentioned above as well as the observation that many of the templates of the first case are very smooth, is to bin the bispectrum in harmonic space. This is the basis of the binned bispectrum estimator [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF][START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF] that is the subject of this chapter (see also [START_REF] Casaponsa | Exploring local fNL estimators based on the binned bispectrum[END_REF] for an independent investigation of the binned bispectrum estimator, and [START_REF] Santos | Multiple methods for estimating the bispectrum of the Cosmic Microwave Background with application to the MAXIMA data[END_REF] for a first rudimentary flat-sky estimator based on a binned bispectrum applied to the MAXIMA data). The binned bispectrum estimator has established itself as one of the three main bispectrum estimators used successfully for the official analysis of the Planck data in all three releases [Planck 2013 XXIV;Planck 2015 XVII;Planck 2018 IX]. The other two are the KSW estimator [START_REF] Komatsu | Measuring primordial non-Gaussianity in the cosmic microwave background[END_REF][START_REF] Yadav | Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background[END_REF]Yadav et al., , 2008] ] and the modal estimator [START_REF] Fergusson | General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f NL[END_REF][START_REF] Fergusson | The CMB Bispectrum[END_REF][START_REF] Fergusson | Efficient optimal non-Gaussian CMB estimators with polarisation[END_REF], and we will now briefly describe the main methodological differences between these three estimators. In addition, other bispectrum estimators exist, based on wavelets (e.g. [START_REF] Curto | On the optimality of the spherical Mexican hat wavelet estimator for the primordial non-Gaussianity[END_REF]), needlets (e.g. [START_REF] Lan | The needlets bispectrum[END_REF]), and Minkowski functionals (e.g. [START_REF] Ducout | Non Gaussianity and Minkowski Functionals: forecasts for Planck[END_REF]) (see [Planck 2013 XXIV] for more complete references).

The KSW estimator (separable template fitting; named after its authors Komatsu, Spergel and Wandelt) is based on the observation that if the primordial bispectrum template is separable as a function of k 1 , k 2 , k 3 (or alternatively the CMB bispectrum template is separable as a function of 1 , 2 , 3 modulo a possible overall integral over r, the radial distance towards the surface of last scattering), then the terms in the optimal estimator for f NL can be reordered as a product of terms depending only on k 1 , 1 , terms depending only on k 2 , 2 , and terms depending only on k 3 , 3 (within an overall integral over r). This significantly reduces the computational cost (by effectively replacing a three-dimensional integral and sum by the product of three one-dimensional integrals and sums), at the cost of losing the ability for full bispectrum reconstruction. The KSW estimator is fast, but only works for separable templates and can only be used for the first case mentioned above (template fitting). 4The modal estimator builds on the idea of the KSW estimator by first expanding the theoretical bispectrum templates and the bispectrum of the map in a basis of separable templates, the so-called modes. (For the Planck 2015 and 2018 analyses two pipelines were used, one with a basis of 600 polynomials, and the other with 2000, both augmented with a few special modes to speed up convergence in the squeezed limit or for the standard shapes.) The coefficients of the individual modes are then computed using the KSW technique. In this way one can in principle treat any bispectrum template, separable or not, as well as reconstruct the full bispectrum of the map. These advantages come at the cost of often needing a large number of modes for sufficient convergence, which can become computationally heavy.

The binned bispectrum estimator does not use the KSW technique and keeps the full threedimensional sum. The required computational reduction comes from reducing the number of terms in the sum by binning the bispectrum in harmonic space, as will be discussed in detail in this chapter. In this way one can do both template fitting (with templates that do not need to be separable) and full bispectrum reconstruction as mentioned above. Moreover, the estimator is very fast when applied to a map, has a convenient modular structure (which means for example that one can analyze an additional template without having to rerun the map), and gives the dependence of f NL on as a free bonus. The possible drawback is that the method works only for bispectra that are relatively smooth (or have rapid oscillations only in a limited -range) in order for a limited number of bins (about 50-60 in practice) to suffice.

The basic output of the binned bispectrum estimator is a binned, or coarse-grained, pseudobispectrum (see section 3.2.1, and section 3.2.4 for the numerical implementation). Here 'pseudo' indicates that full-sky spherical harmonic transforms have been applied to a masked sky, so that the recovered a m coefficients are in fact a convolution of the real CMB multipole coefficients with the multipole coefficients of the mask. How one corrects for the artefacts of the mask will be discussed in detail in section 3.2.3. Below we shall almost always assume the presence of a mask but will omit the qualification 'pseudo'. The coarse-grained pseudo-bispectrum can be combined with a library of theoretical templates by means of an inner product that generates optimally matched filters, see section 3.2.2. It turns out that with a modest number of bins, the loss of information compared to an unbinned analysis is negligible. One can thus determine the f NL parameter for various templates, but one can also construct other estimators, for example to look for the acoustic peaks in the bispectrum (see [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF]).

The binned bispectrum can also be used to carry out a non-parametric, model-independent, analysis, see section 3.2.5. In such an analysis the binned bispectrum can be smoothed to search for a serendipitous statistically significant signal of bispectral non-Gaussianity in the CMB for which templates have not yet been proposed, or to characterize the bispectral properties of foregrounds without a well-motivated theoretical template. The smoothing complicates the statistical analysis of the significance of any non-Gaussian features because it introduces correlations between neighbouring bins, but we developed a method to address this complication. Another application of the smoothed bispectrum method is presented in appendix 3C.

Before discussing the binned bispectrum estimator in detail in section 3.2, we first discuss in the rest of this section what types of bispectra could be present in the CMB. We will consider both primordial bispectra predicted by inflation, which we want to detect, and foreground bispectra, which we have to remove in order to measure the primordial ones.

Standard primordial and foreground templates

Since the temperature and E-polarization fluctuations in the CMB are assumed to originate in density fluctuations produced in the early universe, likely during inflation, the predicted values of the bispectrum of the maps can be expressed in terms of the primordial bispectrum (2.36)) of the (adiabatic) curvature perturbation ζ and the radiation transfer functions g p (k) introduced in section 1.2.2. One finds (see e.g. [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF])

B ζ (k 1 , k 2 , k 3 ) (defined in
B p 1 p 2 p 3 ,th 1 2 3 = h 2 1 2 3 2 π 3 ∞ 0 k 2 1 dk 1 ∞ 0 k 2 2 dk 2 ∞ 0 k 2 3 dk 3 g p 1 1 (k 1 )g p 2 2 (k 2 )g p 3 3 (k 3 )B ζ (k 1 , k 2 , k 3 ) × ∞ 0 r 2 dr j 1 (k 1 r)j 2 (k 2 r)j 3 (k 3 r) (3.1)
where the j are spherical Bessel functions. The quantity h 1 2 3 is defined as (see also (3.19)):

h 1 2 3 = (2 1 + 1)(2 2 + 1)(2 3 + 1) 4π 1 2 3 0 0 0 . (3.2)
Expression (3.1) includes the non-Gaussianity amplitude parameter f NL inside B ζ , but later on we will mostly consider f NL an unknown parameter to be determined from the data and use in our estimator the theoretical bispectrum template B th with f NL set equal to 1.5

Many inflation or other early universe models predict a primordial bispectrum that can be approximated by one (or a combination) of only a few distinct shapes in momentum space (see e.g. [Babich et al., 2004;[START_REF] Fergusson | The shape of primordial non-Gaussianity and the CMB bispectrum[END_REF]). Hence it makes sense to search for these canonical shapes. However, it should be kept in mind that these shapes are only approximations, and with sufficient sensitivity and resolution the difference between slightly different templates that all fall within the same approximate category can be resolved. Inflation models can also produce shapes that are very different from the canonical shapes, for example with localized features or oscillations. See [Planck 2018 IX] for an overview of all the different shapes that were tested using the Planck 2018 data, as well as more complete references. The purpose of this chapter is not to give an exhaustive list of templates, but to describe the methodology of the binned bispectrum estimator, providing only the most important templates as examples.

The most well-known primordial bispectrum type is the so-called local bispectrum [START_REF] Gangui | The Three point correlation function of the cosmic microwave background in inflationary models[END_REF],

B loc ζ (k 1 , k 2 , k 3 ) = - 6 5 f loc NL 2π 2 k 3 1 P ζ (k 1 ) 2π 2 k 3 2 P ζ (k 2 ) + (2 perms) , (3.3)
with the primordial (adiabatic) power spectrum P ζ defined in (1.38). It is called local because in real space it corresponds to the local relation Φ et al., 2000;[START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF] where the subscript G denotes the linear (Gaussian) part. Squeezed configurations where one k (or ) is much smaller than the other two contribute the most to the local bispectrum. The local bispectrum shape is typically produced in multiplefield inflation models on superhorizon scales (see (2.37)), or by other mechanisms that act on superhorizon scales, such as curvaton models (see e.g. [Bartolo et al., 2004b]).

(x) = Φ G (x) + f loc NL (Φ 2 G (x) -Φ G 2 ) [Verde
The two other canonical primordial shapes are the equilateral and orthogonal templates. The equilateral bispectrum is dominated by equilateral configurations where all k's (or 's) are approximately equal, and is typically produced at horizon crossing in inflation models with higher-derivative or other non-standard kinetic terms (or rather, the equilateral bispectrum is a separable approximation to the bispectrum produced in such models, see [START_REF] Creminelli | Limits on non-Gaussianities from WMAP data[END_REF]). It is given by

B equ ζ (k 1 , k 2 , k 3 ) = 18 5 f equ NL [P (k 1 )P (k 2 ) + (2 perms)] + 2 P 2/3 (k 1 )P 2/3 (k 2 )P 2/3 (k 3 ) -[P (k 1 )P 2/3 (k 2 )P 1/3 (k 3 ) + (5 perms)] , (3.4)
where we have defined P (k) ≡ 2π 2 k 3 P ζ (k) for notational simplicity. The orthogonal bispectrum [START_REF] Senatore | Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data[END_REF] has been constructed to be orthogonal to the equilateral shape in such a way that the bispectrum predicted by generic single-field inflation models can be written as a linear combination of the equilateral and orthogonal shapes. It gets its main contribution from configurations that are peaked both on equilateral and on flattened triangles (where two k's are approximately equal and the third is approximately equal to their sum), with opposite sign, and is given by .5) power spectrum this is invisible, as the factor 25/9 in the power spectrum (of ζ compared to Φ) is canceled by two factors -3/5 in the radiation transfer functions of ζ compared to Φ. For the bispectrum, however, it is not invisible, as B ζ is proportional to P 2 ζ as shown below, while there are only three radiation transfer functions in (3.1). This is why the non-Gaussianity amplitude parameter fNL has to be multiplied by a factor -3/5 when considering bispectra in terms of ζ, see also footnote 11 in chapter 2. Here we use the definition in terms of ζ preferred by theorists, for consistency with the previous chapters. However, in our paper [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF] as well as in the Planck papers, the definition in terms of Φ is used.

B ort ζ (k 1 , k 2 , k 3 ) = 5 f ort NL 3[P (k 1 )P (k 2 ) + (2 perms)] + 8 P 2/3 (k 1 )P 2/3 (k 2 )P 2/3 (k 3 ) -3[P (k 1 )P 2/3 (k 2 )P 1/3 (k 3 ) + (5 perms)] . ( 3
It should be noted that the orthogonal shape is not at all orthogonal to the local shape (as sometimes incorrectly stated in older literature). It has a large correlation (about 40-50%) with the local shape at Planck resolution (see section 3.2.2 and table 3.1).

In addition to these three shapes, it is also interesting to look for non-primordial contaminant bispectra, either to study these foregrounds or to remove them. Regarding extra-galactic foregrounds, in the first place a bispectrum will be produced by diffuse extra-galactic point sources. These can generally be divided into two populations: unclustered and clustered sources. The former are radio and late-type infrared galaxies, while the latter are dusty star-forming galaxies constituting the cosmic infrared background (CIB). Secondly, gravitational lensing of the CMB will produce a bispectrum that mimics the local shape, because there is a correlation between the lenses that produce modifications to the CMB power spectrum on small scales and the integrated Sachs-Wolfe effect on large scales (both are due to the same mass distribution at low redshift).

The unclustered sources can be assumed to be distributed according to a Poissonian distribution, and hence have a white noise power spectrum (i.e., with an amplitude independent of ). Then their bispectrum has a very simple theoretical shape [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF]:

B unclust 1 2 3 = h 2 1 2 3 b ps (3.6)
where b ps , the amplitude of the unclustered point source bispectrum, is the parameter that can be determined in the same way as the f NL parameters for the primordial templates. Like most foregrounds, but unlike primordial signals, the amplitude depends on the frequency channel, which allows a multi-frequency experiment like Planck to (partially) clean these contaminants from its maps. The above relation is valid both in temperature and in polarization. However, since not all point sources are polarized, the amplitude b ps is not the same in temperature and polarization, with the difference depending on the mean polarization fraction of the point sources. Without taking into account that fraction, it would not make sense to look at the mixed T T E and T EE components of its bispectrum, nor to try to determine b ps jointly from temperature and polarization maps. In practice for Planck the contribution from polarized point sources is negligible (see [Planck 2015 XVII]), so that we might as well consider it a temperature-only template.

The clustered point sources (CIB) have a more complicated bispectrum. A simple template that fits the data well was established in [START_REF] Lacasa | Non-Gaussianity of the cosmic infrared background anisotropies -I. Diagrammatic formalism and application to the angular bispectrum[END_REF] (see also [Planck 2015 XVII]):

B CIB 1 2 3 = h 2 1 2 3 b CIB (1 + 1 / break )(1 + 2 / break )(1 + 3 / break ) (1 + 0 / break ) 3 q , (3.7)
where the index is q = 0.85, the break is located at break = 70, and 0 = 320 is the pivot scale for normalization. In addition, b CIB is the amplitude parameter to be determined. As for the unclustered point sources, it depends on the frequency. The CIB is found to be negligibly polarized, so that the above template is only used in temperature.

The theoretical shape for the lensing-ISW bispectrum was worked out in [START_REF] Goldberg | Microwave background bispectrum. 2. A probe of the low redshift universe[END_REF][START_REF] Smith | Algorithms for bispectra: Forecasting, optimal analysis, and simulation[END_REF][START_REF] Lewis | The shape of the CMB lensing bispectrum[END_REF] and is given by

B p 1 p 2 p 3 ,lensISW 1 2 3 = h 2 1 2 3 C p 2 φ 2 C p 1 p 3 3 f p 1 1 2 3 + C p 3 φ 3 C p 1 p 2 2 f p 1 1 3 2 + C p 1 φ 1 C p 2 p 3 3 f p 2 2 1 3 + C p 3 φ 3 C p 1 p 2 1 f p 2 2 3 1 + C p 1 φ 1 C p 2 p 3 2 f p 3 3 1 2 + C p 2 φ 2 C p 1 p 3 1 f p 3 3 2 1 . (3.
8) Here C T φ and C Eφ are the temperature/polarization-lensing potential cross power spectra, while the CMB power spectra C T T , C T E , C EE should be taken to be the lensed T T , T E, EE power spectra. The functions f p 1 2 3 are defined by

f T 1 2 3 = 1 2 [ 2 ( 2 + 1) + 3 ( 3 + 1) -1 ( 1 + 1)] , f E 1 2 3 = 1 2 [ 2 ( 2 + 1) + 3 ( 3 + 1) -1 ( 1 + 1)] 1 2 3 2 0 -2 1 2 3 0 0 0 -1 , (3.9)
if 1 + 2 + 3 is even and 1 , 2 , 3 satisfy the triangle inequality, and zero otherwise. Using some mathematical properties of the Wigner 3j-symbols we find that, under the same conditions as above, the ratio of the two Wigner 3j-symbols can be computed explicitly as

1 2 3 2 0 -2 1 2 3 0 0 0 -1 = [ 2 ( 2 + 1) -1 ( 1 + 1) -3 ( 3 + 1)][ 2 ( 2 + 1) -1 ( 1 + 1) -3 ( 3 + 1) + 2]
- 2 1 ( 1 + 1) 3 ( 3 + 1) 4( 1 -1) 1 ( 1 + 1)( 1 + 2)( 3 -1) 3 ( 3 + 1)( 3 + 2) -1

2 .

(3.10)

Note that there is no unknown amplitude parameter in front of this template: its f NL parameter should be unity.

Apart from the extra-galactic templates provided here, one also has to take into account galactic contaminants, although these should in principle be absent in the cleaned maps due to the combination of component separation and masking. Unfortunately no theorerical templates exist for those shapes, but the binned bispectrum estimator also allows for determining and using numerical templates. The study of galactic bispectrum contaminants is the subject of our paper [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF], included in appendix 3B.

Isocurvature non-Gaussianity

The generalization to the case where non-Gaussian isocurvature components are present in addition to the standard adiabatic component was treated in [START_REF] Langlois | Correlated adiabatic and isocurvature perturbations from double inflation[END_REF]Van Tent, 2011, 2012]. The second of those papers, which contains the complete treatment, is included in appendix 3A. For convenience we summarize the resulting template here. In fact this boils down to the joint analysis of a number of additional templates.

We make two simplifying assumptions: we consider only the local shape (because that is the shape typically produced by multiple-field inflation, and we require multiple fields in order to produce isocurvature modes) and assume the same spectral index for the primordial isocurvature power spectrum and the isocurvature-adiabatic cross power spectrum as for the adiabatic power spectrum (to limit the number of free parameters). In that case the primordial bispectrum can be written as .11) where I, J, K label the different modes (adiabatic and isocurvature). As a reminder, P (k) =

B IJK (k 1 , k 2 , k 3 ) = f I,JK NL P (k 2 )P (k 3 ) + f J,KI NL P (k 1 )P (k 3 ) + f K,IJ NL P (k 1 )P (k 2 ), ( 3
2π 2 k 3 P ζ (k).
The meaning of and the reason for the tilde on fNL is explained below. The invariance of this expression under the simultaneous interchange of two of these indices and the corresponding momenta means that f I,JK NL = f I,KJ NL , explaining the presence of the comma, and reducing the number of independent fNL parameters (from 8 to 6 in the case of two modes). Inserting this expression into (3.1), where

g p 1 1 (k 1 )g p 2 2 (k 2 )g p 3 3 (k 3 ) should be replaced by I,J,K g p 1 I 1 (k 1 )g p 2 J 2 (k 2 )g p 3 K 3 (k 3
), finally leads to the result .12) where B p 1 p 2 p 3 I,JK Here we use the notation ( 1 2 3 ) ≡ [ 1 2 3 + 5 perms]/3! and it should be kept in mind that the i and p i are always kept together (so the p i are also permuted in the same way).

B p 1 p 2 p 3 1 2 3 = I,J,K f I,JK NL B p 1 p 2 p 3 I,JK 1 2 3 , ( 3
1 2 3 = 3h 2 1 2 3 ∞ 0 r 2 dr α p 1 I ( 1 (r)β p 2 J 2 (r)β p 3 K 3 ) (r), ( 3
The tilde on fNL indicates that we have explicitly defined f NL parameters here in terms of the adiabatic curvature perturbation ζ = e 1A ζ A and the total isocurvature perturbation S instead of the gravitational potential Φ (in the following we will always assume the presence of just a single isocurvature mode in addition to the adiabatic one, so that the total isocurvature perturbation is simply equal to the only isocurvature perturbation). The relation between S and the isocurvature mode e 2A ζ A is given in (1.36). As explained in footnote 5 in this chapter and footnote 11 in chapter 2, there is a factor between the fNL and the usual f NL defined in terms of Φ, which is equal to -6/5 for the purely adiabatic mode (see also (3.3)). This comes from the fact that ζ = -5Φ adi /3. To compute the factors for the other modes, one has to use the fact that S = -5Φ iso . The final factors for the six modes (ζ, ζζ), (ζ, ζS), (ζ, SS), (S, ζζ), (S, ζS), and (S, SS) are then -6/5, -2/5, -2/15, -18/5, -6/5, and -2/5, respectively. 6 7 As this factor is different for each fNL , it is not feasible to write (3.11), with the power spectra in terms of ζ, in terms of the f NL without a tilde, as we could do in (3.3) for example. Hence the required introduction of the tilded quantities, which are also used in [Langlois and Van Tent, 2012] included in appendix 3A.

We can conclude that including the possibility of isocurvature non-Gaussianity in our investigations means that we have to replace the single local adiabatic bispectrum template by the family of templates (3.13), each with their individual fNL parameter. In particular, if we assume the presence of only a single isocurvature mode in addition to the adiabatic one (i.e. one of cold dark matter, neutrino density, or neutrino velocity), we have six local fNL parameters to determine instead of just one, and these should always be estimated jointly (see section 3.2.2). For more details, see appendix 3A.

The binned bispectrum estimator

This section contains a detailed presentation of all aspects of the binned bispectrum estimator that we developed. It is based mostly on [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF], with some input from [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF].

Binned bispectrum

As discussed in section 1.2.2, a map M p of the CMB temperature or E-polarization fluctuations can be decomposed into spherical harmonics according to

M p (Ω) = ,m a p m Y m (Ω). (3.15)
Here Ω is the solid angle on the sky and the p label refers to either temperature (T ) or Epolarization (E), as we will not consider B-polarization in this thesis. In the following paragraph we will omit the explicit polarization indices, in order to lighten the notation.

The full bispectrum on the celestial sphere consists of cubic combinations of the spherical harmonic coefficients of the form

B m 1 m 2 m 3 1 2 3 = a 1 m 1 a 2 m 2 a 3 m 3 , ( 3

.16)

6 There is a sign mistake in the relation between fNL and fNL given in [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF] (corrected in appendix 3A), which led to a sign mistake in these factors given in [Planck 2015 XVII]. This mistake was corrected in the revised version of [Planck 2018 IX]. However, as [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF] only studies quadratic quantities (error bars, Fisher matrices) and [Planck 2015 XVII;Planck 2018 IX] do not use the fNL but only the fNL, this mistake has no consequences for those papers.

7 To fully understand these factors, one particularity that is not well documented must be noted. When writing the isocurvature β S (omitting polarization indices) from (3.14) in terms of the gravitational potential Φ, it would seem logical to convert the g S P ζ into g Φ iso PΦ adi . However, for historical reasons it is actually converted into g Φ iso PΦ iso , where PΦ iso = 1 9 PΦ adi when P ζ = PS, and the fNL without tilde (as given in e.g. [Planck 2015 XVII;Planck 2018 IX]) are defined with respect to that β iso .

the expectation values of which may be calculated for a given theory. However, under the assumption of statistical isotropy, these expectation values are not independent and can be reduced to quantities depending only on 1 , 2 , and 3 . We may define a manifestly rotationallyinvariant reduced bispectrum, called the angle-averaged bispectrum, in terms of integrals of triple products of maximally filtered maps so that (3.17) where the maximally filtered map is defined as

B 1 2 3 = dΩ M 1 (Ω) M 2 (Ω) M 3 (Ω),
M (Ω) = + m=- a m Y m (Ω). (3.18)
Using the expression for the Gaunt integral8

G m 1 m 2 m 3 1 2 3 = dΩ Y 1 m 1 (Ω) Y 2 m 2 (Ω) Y 3 m 3 (Ω) (3.19) = (2 1 + 1)(2 2 + 1)(2 3 + 1) 4π 1 2 3 0 0 0 1 2 3 m 1 m 2 m 3 ≡ h 1 2 3 1 2 3 m 1 m 2 m 3
we obtain

B 1 2 3 = h 1 2 3 m 1 ,m 2 ,m 3 1 2 3 m 1 m 2 m 3 B m 1 m 2 m 3 1 2 3 . (3.20) As a consequence of the Wigner-Eckart theorem, B m 1 m 2 m 3 1 2 3
is proportional to

1 2 3 m 1 m 2 m 3 .
Using this fact combined with the Wigner-3j-symbol identity (3.21) which holds whenever 1 , 2 , 3 satisfy the triangle inequality (| 1 -2 | ≤ 3 ≤ 1 + 2 and permutations) and the parity condition ( 1 + 2 + 3 = even), we find that

m 1 ,m 2 ,m 3 1 2 3 m 1 m 2 m 3 1 2 3 m 1 m 2 m 3 = 1,
B m 1 m 2 m 3 1 2 3 = h -1 1 2 3 1 2 3 m 1 m 2 m 3 B 1 2 3 . (3.22)
Again, this equality is only valid when the triangle inequality and parity condition are respected, otherwise both B 1 2 3 and h 1 2 3 are zero.9 Because the angle-averaged bispectrum B p 1 p 2 p 3 1 2 3

(restoring its polarization indices, and which we will call simply "bispectrum" in the rest of the thesis) is symmetric under the simultaneous interchange of its three multipole numbers 1 , 2 , 3 and its three polarization indices p 1 , p 2 , p 3 , it is sufficient to consider only the subspace 1 ≤ 2 ≤ 3 . It should be noted, however, that once we have both temperature and polarization, imposing this condition means that we no longer have the freedom to rearrange the polarization indices, so that for example the T T E, T ET , and ET T combinations correspond to three distinct bispectra.

To compute the observed bispectrum with the maximum possible resolution, we would evaluate the integral over the sky of triple products of maximally filtered observed sky maps, as in (3.17). (In practice this integral is evaluated as a sum over pixels.) The total number of triplets would be O(10 7 ) for a WMAP or O(10 9 ) for a Planck temperature map. But we can also use broader filters for the integral in (3.17), with very little loss of information because a modest resolution in suffices for many physically motivated templates for which the predicted B p 1 p 2 p 3 1 2 3 varies slowly with its arguments. We end up having to compute only O(10 4 ) bin triplets, leading to an enormous reduction in the computational resources required. We divide the -range [ min , max ] into subintervals denoted by ∆ i = [ i , i+1 -1] where i = 0, . . . , (N bins -1) and N bins = max + 1, so that the filtered maps are (3.23) and we use these instead of M p in the expression for the bispectrum (3.17). The observed binned bispectrum is

M p i (Ω) = ∈∆ i + m=- a p m Y m (Ω),
B p 1 p 2 p 3 ,obs i 1 i 2 i 3 = 1 Ξ i 1 i 2 i 3 dΩ M p 1 ,obs i 1 (Ω)M p 2 ,obs i 2
(Ω)M p 3 ,obs i 3

(Ω) (3.24) where Ξ i 1 i 2 i 3 is the number of triplets within the (i 1 , i 2 , i 3 ) bin triplet satisfying the triangle inequality and parity condition selection rules. Because of this normalization factor, B p 1 p 2 p 3 i 1 i 2 i 3 may be considered an average over all valid B p 1 p 2 p 3 1 2 3 inside the bin triplet. As for the power spectrum, there will be a fundamental statistical uncertainty in the bispectrum, called cosmic variance, due to the fact that we want to determine an ensemble average (to compare with inflationary predictions for example) but we can only measure one sky. The only averaging we can do is over the m indices. This cosmic variance is in addition to other sources of uncertainty, e.g. due to the finite resolution and noise of the experiment under consideration, due to foreground residuals, etc. To compute the variance we start by considering only the temperature bispectrum. The covariance of the bispectra B 1 2 3 and B 4 5 6 equals the average of the product minus the product of the averages. Under the assumption of weak non-Gaussianity the calculation simplifies significantly. In that case one can neglect the average value of the bispectra, and the average of the product,

B 1 2 3 B 4 5 6 = h 1 2 3 h 4 5 6 × m 1 ,m 2 ,m 3 , m 4 ,m 5 , m 6 1 2 3 m 1 m 2 m 3 4 5 6 m 4 m 5 m 6 a 1 m 1 a 2 m 2 a 3 m 3 a * 4 m 4 a * 5 m 5 a * 6 m 6 (3.25)
(using the fact that B is real so that B = B * ), can be rewritten as the product of three power spectra C ≡ a m a * m using Wick's theorem: (3.26) using obvious shorthand to denote the other permutations of δ-functions. Due to the δ-functions, the covariance matrix is diagonal, so we need to consider only the (diagonal) variance of B 1 2 3 . We use the identity (3.21) and the fact that for even parity of 1 + 2 + 3 the columns of the Wigner 3j-symbol can be permuted to obtain

a 1 m 1 a 2 m 2 a 3 m 3 a * 4 m 4 a * 5 m 5 a * 6 m 6 = C 1 C 2 C 3 [δ 1 4 δ 2 5 δ 3 6 δ m 1 m 4 δ m 2 m 5 δ m 3 m 6 + (14)(26)(35) + (15)(24)(36) + (15)(26)(34) + (16)(24)(35) + (16)(25)(34)],
Var(B 1 2 3 ) = g 1 2 3 h 2 1 2 3 C 1 C 2 C 3 ≡ V 1 2 3 (3.27)
with g 1 2 3 equal to 6, 2, or 1, depending on whether 3, 2, or no 's are equal, respectively, and h defined in (3.2). Similarly the variance of the binned bispectrum

B i 1 i 2 i 3 = (Ξ i 1 i 2 i 3 ) -1 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 B 1 2 3 is given by Var(B i 1 i 2 i 3 ) = g i 1 i 2 i 3 (Ξ i 1 i 2 i 3 ) 2 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 h 2 1 2 3 C 1 C 2 C 3 ≡ V i 1 i 2 i 3 (3.28)
with g i 1 i 2 i 3 equal to 6, 2, or 1, depending on whether 3, 2, or no i's are equal, respectively. The δ-functions in (3.26) lead here to conditions of equality on the bins, since due to the sum over all 's inside a bin, δ a b will always give 1 if a and b are in the same bin, and 0 if not.

With the noise and beam smoothing present in a real experiment, (3.28) becomes

V i 1 i 2 i 3 = g i 1 i 2 i 3 (Ξ i 1 i 2 i 3 ) 2 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 h 2 1 2 3 (b 2 1 C 1 + N 1 )(b 2 2 C 2 + N 2 )(b 2 3 C 3 + N 3 ) (3.29)
where b is the beam transfer function and N the instrument noise power spectrum. This expression is exact only for an axisymmetric beam and isotropic noise; otherwise it is an approximation (because the beam and noise properties would include off-diagonal matrix elements). For a Gaussian beam, the beam transfer function is typically specified by the full width at half maximum θ FWHM (in radians), so that b = exp -1 2 ( + 1)θ 2 FWHM /(8 ln 2) . A pixel window function w to account for pixelization effects is combined with the beam transfer function according to b → w b .

For bispectral elements including both T and E, the variance is replaced by the covariance matrix in polarization space, whose expression without binning is

Covar(B p 1 p 2 p 3 1 2 3 , B p 4 p 5 p 6 1 2 3 ) = g 1 2 3 h 2 1 2 3 ( C 1 ) p 1 p 4 ( C 2 ) p 2 p 5 ( C 3 ) p 3 p 6 ≡ V p 1 p 2 p 3 p 4 p 5 p 6 1 2 3 , (3.30)
where

C = (b T ) 2 C T T + N T b T b E C T E b T b E C T E (b E ) 2 C EE + N E . (3.31)
Here noise uncorrelated in T and E has been assumed. Similarly, for the binned case

Covar(B p 1 p 2 p 3 i 1 i 2 i 3 , B p 4 p 5 p 6 i 1 i 2 i 3 ) = g i 1 i 2 i 3 (Ξ i 1 i 2 i 3 ) 2 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 h 2 1 2 3 ( C 1 ) p 1 p 4 ( C 2 ) p 2 p 5 ( C 3 ) p 3 p 6 ≡ V p 1 p 2 p 3 p 4 p 5 p 6 i 1 i 2 i 3 . (3.32)
Some subtleties arise in the derivation of equation (3.30). The covariance matrix is in principle an 8 × 8 matrix, given that there are 8 independent polarized bispectra T T T , T T E, T ET , T EE, ET T , ET E, EET , and EEE. As mentioned before, note that for example T T E and T ET are not the same: each polarization index p i is coupled to a multipole index i , and cannot be exchanged due to the restriction 1 ≤ 2 ≤ 3 that we will always impose in order to reduce computation time. A naive calculation of this 8 × 8 matrix appears to lead to a more complicated expression in the case of equal 's that is not proportional to g 1 2 3 . However, one should treat the cases where two or three 's are equal separately. For example, when 2 = 3 , one can exchange the last two polarization indices and one finds that T T E = T ET and ET E = EET . Hence in that case there are only 6 independent bispectra, and the covariance matrix is 6 × 6. Similarly, when all three 's are equal, the covariance matrix is 4 × 4.

However, it turns out that as far as computing f NL is concerned, when evaluating the sum in (3.36), properly treating the special cases where 's are equal by reducing the dimension of the covariance matrix and bispectrum vector, the final result is identical to the following calculation: taking the covariance matrix to be the 8 × 8 matrix as computed in the case of all 's unequal, multiplying it by g 1 2 3 , and then computing the sum in (3.36) directly without treating the cases of equal 's separately. This second computation is much more convenient from a practical point of view. Finally it can be shown that the latter expression of the covariance matrix can be rewritten as the separable product involving only 2 × 2 matrices in (3.30).

Similarly it can be shown that the variances of the combinations B T 2E ≡ T T E+T ET +ET T and B T E2 ≡ T EE + ET E + EET used for the smoothed bispectrum (see section 3.2.5) are also recovered correctly when using (3.30) or (3.32). Here one should use of course that Var(B T 2E ) = Var(T T E) + Var(T ET ) + Var(ET T ) + 2 Covar(T T E, T ET ) + 2 Covar(T T E, ET T ) + 2 Covar(T ET, ET T ), and similarly for Var(B T E2 ). So in the end, while one should remember the caveats regarding (3.30) and (3.32) in the case of equal 's or i's, for the practical purposes of this thesis they can be used without any problem.

f NL estimation on an ideal sky

We start by considering the case where we have only temperature. In order to estimate f NL using a template B th 1 2 3 , the estimator fNL = B th,exp , B obs B th,exp , B th,exp (3.33) is constructed using the inner product .34) This definition satisfies the mathematical axioms of an inner product as long as bin triplets with infinite variance are excluded from the sum. The theoretical bispectrum for the experiment is related to the theoretically predicted infinite angular resolution bispectrum by the relation

B A , B B no binning = 1 ≤ 2 ≤ 3 B A 1 2 3 B B 1 2 3 V 1 2 3 . ( 3
B th,exp 1 2 3 = b 1 b 2 b 3 B th,f NL =1 1 2 3
. For the binned estimator the template is first binned as B th,exp

i 1 i 2 i 3 = ( 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 B th,exp 1 2 3 )/Ξ i 1 i 2 i
3 and then the above estimator can be used with the binned version of the inner product:

B A , B B binned = i 1 ≤i 2 ≤i 3 B A i 1 i 2 i 3 B B i 1 i 2 i 3 V i 1 i 2 i 3 . (3.35)
One sees that the above estimator is of the form fNL ∝ [(B th ) 2 /V ][B obs /B th ] (where from now on we drop the explicit "exp" label). Since V i 1 i 2 i 3 is the theoretical estimate of the variance of B obs i 1 i 2 i 3 in the approximation of weak non-Gaussianity, the estimator is inverse variance weighted: B obs /B th is an estimate of f NL based on a single bin triplet, and all these estimates are combined, weighted by the inverse of their variance, V /(B th ) 2 . The proportionality factor 1/ B th , B th is the normalization of the weights and gives the theoretical (Gaussian) estimate for the variance 10 of the total estimator fNL . This is the same as saying that B th , B th is the χ 2 or (S/N ) 2 of the estimator in the case f NL = 1.

The generalization of the f NL estimator to include polarization in the case without binning was worked out in [START_REF] Yadav | Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background[END_REF]. In that case the inner product (3.34) should be replaced by

B A , B B no binning = 1 ≤ 2 ≤ 3 p 1 ,p 2 ,p 3 , p 4 ,p 5 , p 6 B p 1 p 2 p 3 ,A 1 2 3 (V -1 ) p 1 p 2 p 3 p 4 p 5 p 6 1 2 3 B p 4 p 5 p 6 ,B 1 2 3 , (3.36)
which involves the inverse of the covariance matrix given in (3.30). Computing this inverse simply implies inverting the three 2 × 2 matrices C given in (3.31).

Deriving an equivalent expression for the binned estimator is straightforward, as long as one keeps in mind that one should first bin the elements of the covariance matrix (since that corresponds to the covariance matrix of the binned bispectrum) and only afterwards compute the inverse. Trying to bin directly the elements of the inverse covariance matrix (or one divided by these elements) is incorrect and leads to wrong results (in particular for bins where C T E crosses zero). So in the end the generalization of the binned bispectrum estimator to include polarization is given by the prescription that the inner product (3.35) should be replaced by

B A , B B binned = i 1 ≤i 2 ≤i 3 p 1 ,p 2 ,p 3 , p 4 ,p 5 , p 6 B p 1 p 2 p 3 ,A i 1 i 2 i 3 (V -1 ) p 1 p 2 p 3 p 4 p 5 p 6 i 1 i 2 i 3 B p 4 p 5 p 6 ,B i 1 i 2 i 3 , (3.37)
10 If we have independent quantities yi with variances vi and define the inverse-variance weights as wi = (1/vi)/( j 1/vj), then the variance of the weighted mean i wiyi is

i w 2 i vi = ( i vi/v 2 i )/( j 1/vj) 2 = 1/( j 1/vj).
involving the inverse of the binned covariance matrix given in (3.32). However, since the multiplication with h 2 1 2 3 in combination with the binning couples the three C matrices in (3.32) together, the covariance matrix can only be inverted as a full 8 × 8 matrix that is no longer separable in . Fortunately this non-separability is irrelevant for the binned bispectrum estimator.

We can quantify how much the estimator variance increases due to binning, compared with an ideal estimator without binning:

R ≡ Var( f ideal NL ) Var( f binned NL ) = B th , B th binned B th , B th no binning . (3.38)
R is a number between 0 and 1. The closer R is to 1, the better the binned approximation for the template under consideration. To show that 0 ≤ R ≤ 1 we need to rewrite (3.38) in terms of a single inner product definition. It can be checked straightforwardly that the binned inner product of the theoretical bispectrum can be rewritten as the exact inner product (no binning) of the bispectrum template defined below:

B th , B th binned = B bin , B bin no binning , (3.39)
where B p 1 p 2 p 3 ,bin

1 2 3 ≡ 1 Ξ i 1 i 2 i 3 g i 1 i 2 i 3 g 1 2 3 p 4 ,p 5 ,p 6 , p 7 ,p 8 , p 9 V p 1 p 2 p 3 p 4 p 5 p 6 1 2 3 (V -1 ) p 4 p 5 p 6 p 7 p 8 p 9 i 1 i 2 i 3 B p 7 p 8 p 9 ,th i 1 i 2 i 3 (3.40)
with (i 1 , i 2 , i 3 ) the bin triplet that contains the -triplet ( 1 , 2 , 3 ). 11 In addition it is simple to show that B bin , B bin no binning = B bin , B th no binning .

Now we can rewrite R as R = B bin , B bin no binning B th , B th no binning = B bin , B th no binning B th , B th no binning = ( B bin , B th no binning ) 2 B th , B th no binning B bin , B bin no binning .

(3.43) From the first expression, given that x, x ≥ 0 for an inner product, we see that R ≥ 0. And the last expression implies that R ≤ 1 using the Cauchy-Schwarz inequality.

If more than one of the above bispectrum shapes are expected to be present in the data, then a joint estimation of the different f NL parameters is required. For this the Fisher matrix

F IJ = B I , B J , (3.44)
where I, J label the theoretical shapes (for example local and equilateral), is a crucial quantity.

The optimal estimation of the f NL parameter for shape I is given by

f I NL = J (F -1 ) IJ B J , B obs . (3.45)
The estimate of the variance of f I NL is (F -1 ) II . If, on the other hand, the f I NL parameters would have been estimated independently using (3.33) (as if there is only one bispectrum shape present, but it is unknown which), then their variance is given by 1/F II .

11 This result follows from the identity (for any function u 1 2 3 ) Table 3.1: Correlation coefficients between the theoretical templates of section 3.1.2, as defined in (3.46). The numbers are computed using the characteristics of the Planck experiment and are for temperature. We see a large correlation between local and orthogonal and between local and lensing-ISW. Equilateral and orthogonal are mostly uncorrelated, and the correlation between the point source templates and the primordial ones is negligible.

1 ≤ 2 ≤ 3 1 g 1 2 3 u 1 2 3 = 1 6 1 , 2 , 3 u 1 2 3 = 1 6 i 1 ,i 2 ,i 3 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 u 1 2 3 = i 1 ≤i 2 ≤i 3 1 gi 1 i 2 i 3 1 ∈∆ 1 2 ∈∆ 2 3 ∈∆ 3 u 1 2 3 . ( 3
Another useful quantity to define is the symmetric correlation matrix 3.46) giving the correlation coefficients between any two bispectrum shapes. By construction -1 ≤ C IJ ≤ +1, with C IJ = -1, 0, +1 meaning that the two shapes are fully anti-correlated, uncorrelated, or fully correlated, respectively. Note that one could also define a correlation matrix using the inverse of the Fisher matrix instead of the Fisher matrix itself in (3.46). That would give us the correlation of the f NL parameters, while (3.46) represents the correlation of the templates.

C IJ ≡ F IJ √ F II F JJ (
As an example we show the correlation coefficients between the templates of section 3.1.2 in table 3.1. Suppose that we had only two shapes with non-zero correlation, but the amplitude of the second was fixed by theory (as is the case for example for the lensing-ISW template that has no unknown amplitude parameter). If the theory was fully trusted, it would be a shame to do a joint estimation, with the associated increase in variance. In that case the influence of the second shape on the first is more properly treated as a known bias that can be subtracted without increasing the variance. The size of the bias can be found from (3.45), by using the second equation (I = 2) to eliminate B (2) , B obs from the first equation (I = 1). After expressing the elements of the inverse Fisher matrix in terms of the elements of the Fisher matrix, the resulting equation for the first f NL parameter simplifies to:

f (1) NL = 1 F 11 B (1) , B obs - F 12 F 11 f (2) NL , (3.47)
the second term being the bias correction. Here f

NL is the known f NL parameter of the second shape, most likely equal to one if the known amplitude was included in the template (as is the case for example for the lensing-ISW template). The variance of f (1) NL is not influenced by the bias correction and remains equal to 1/F 11 , the same as for a single shape. This result can easily be generalized to more than two shapes.

To conclude this section, we mention that in addition to studies of f NL as presented here, or non-parametric studies as presented later in section 3.2.5, one can also construct other types of parametric estimators. For example, in [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF] we constructed two types of estimators to look for specific features in the CMB bispectrum related to the acoustic oscillations. At that time (before Planck) there were hints from WMAP that f loc NL might be quite large, 30-50. If that had been true, these estimators would have had a high signal-to-noise on the Planck data. Unfortunately, Planck ruled out such a high value of f loc NL , which makes these particular estimators mostly irrelevant, at least as far as the Planck data is concerned.

Extensions for a realistic sky

The definition of the bispectrum in (3.17) or (3.24) assumes a rotationally invariant CMB sky and that the bispectral expectation values have even parity (as a consequence of the parity invariance of the underlying stochastic process, which we assume here). Because of rotational invariance, the m dependence of the expectation can be factored out, and the reduced bispectral coefficients depending only on the provide a lossless compression of the data concerning the bispectrum. However, in a real experiment, as opposed to idealized observations of the primordial sky, two sources of anisotropy arise that break rotational invariance and require corrections to the bispectrum estimation to avoid spurious results.

The first is anisotropic superimposed instrument noise, due to for example an anisotropic scanning pattern of the satellite. The second is anisotropy introduced by a mask needed to remove the brightest parts of our galaxy and the strongest point sources. These two anisotropic 'contaminants', unlike for example foreground contaminants, cannot be removed by cleaning and must be accounted for in the analysis. They can mimic a primordial bispectrum signal. For example, due to an anisotropic scanning pattern of the experiment, certain (large-scale) areas of the sky may have less (small-scale) noise than other areas. This correlation between large and small scales produces a contaminant bispectrum that peaks in the squeezed limit (bispectrum configurations with one small and two large ones). That is also where the primordial local shape has its main signal. Since the CMB and the noise are uncorrelated, the effect will average out to zero in the central value of the bispectrum over a large number of maps (no bias), but it will increase the variance. And while an unbiased estimator will find the correct central value when averaged over a large number of maps, a larger variance does mean that there is more chance to find a value far from the true one when applied to a single map.

These contaminants can be mitigated by subtracting from the cubic expression of the observed bispectrum given in (3.17) or (3.24) a linear correction term, as shown in [START_REF] Creminelli | Limits on non-Gaussianities from WMAP data[END_REF]Yadav et al., 2008], that is,

B p 1 p 2 p 3 ,obs i 1 i 2 i 3 → B p 1 p 2 p 3 ,obs i 1 i 2 i 3 -B p 1 p 2 p 3 ,lin i 1 i 2 i 3 .
(3.48)

'Cubic' and 'linear' here mean cubic and linear in the observed map, respectively. The linear correction term is

B p 1 p 2 p 3 ,lin i 1 i 2 i 3 = dΩ M p 1 ,obs i 1 M p 2 ,G i 2 M p 3 ,G i 3 +M p 2 ,obs i 2 M p 1 ,G i 1 M p 3 ,G i 3 + M p 3 ,obs i 3 M p 1 ,G i 1 M p 2 ,G i 2 , (3.49)
where the average is over Gaussian CMB maps with the same beam, (anisotropic) noise, and mask as the observed map. A detailed derivation of the linear correction term can be found in the first appendix of [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF], which paper is included in appendix 3B of this thesis, see section 3B.4. The linear correction is hugely significant for the local shape as explained above, very significant for orthogonal (due to the large correlation with local), and insignificant for equilateral. While adding the linear correction term completely solves the issue related to anisotropic noise, it turns out that for the issue related to the mask we need an additional ingredient to make our bispectrum estimator optimal again. The region near the galactic plane and around extragalactic point sources, where reliable subtraction of contaminants is not possible, must be masked to prevent contamination of the primordial bispectrum. Masking introduces a number of problems for estimating the bispectrum because the process of filtering maps is nonlocal. If we naively analyze a masked map in which the masked pixels are set to zero -or better yet, set equal to the average value of the unmasked part of the map -by filtering it, say with a high-pass filter, we would observe a deficit of small scale power around the edges of the mask. A filter in frequency space moves around the small scale power in real space. The power is smeared, so that if there is no small scale power in the 3.2. The galactic mask is the f sky = 0.80 galactic mask included in the Planck 2013 release, and the point source mask is the one based on the Planck LFI 30 GHz channel from the 2013 release (with a 4σ threshold level), which has f sky = 0.96. The combined mask has f sky = 0.77. masked region, power from the unmasked region escapes into the masked region without there being a compensating flux returning from the masked region. Another edge effect tending to increase the small scale power around the border of the unmasked region results if there is a jump discontinuity. Such a discontinuity contains spurious small scale power that bleeds into the unmasked region after filtering. It is therefore important to introduce artificially the right amount of small scale power into the masked region and to avoid spurious jumps in the maps so that the two fluxes cancel after filtering.12 This process of filling in the masked regions is also known as 'inpainting'.

Before showing quantitatively how masking affects the determination of f NL , we first have to determine what the effect on the error bars would be if we had none of these problems, but only less data due to the reduced fraction of the sky. When the bispectrum is determined according to (3.24), it should be multiplied with a factor 1/f sky to correct for the partial sky coverage [START_REF] Komatsu | Measuring primordial non-Gaussianity in the cosmic microwave background[END_REF], where f sky is the fraction of the sky that is left unmasked. In practice this is done automatically when the integral is replaced by a sum over the pixels: the product of maps is summed over all unmasked pixels, divided by the number of unmasked pixels, and multiplied by 4π. In addition, the partial sky coverage increases the variance of the estimator, the theoretical estimate of which becomes 1/(f sky B th,exp , B th,exp ). The factor of 1/f sky can easily be understood given that the variance of a quantity determined from N data points scales as 1/N and here the number of data points roughly corresponds to the number of observed pixels on the sky. If the mask is not too large, this simple prescription for the variance works quite well.

To illustrate quantitatively the problems encountered in determining f NL with a mask, we applied a series of tests to simulated CMB maps as described in Table 3.2. The masks used are shown in Fig. 3.1 while the details of the simulations are described in the caption of the Table .  We find that when missing data in the masked regions are naively replaced with the average of the unmasked part of the map (the "no filling in" lines in the Table ), the estimates of f NL are unbiased but have much larger variance than expected, at least in temperature. The expected increase in the standard deviation is only a factor 1/ f sky (i.e., 1.12 for the galactic mask and 1.02 for the point sources) and in particular for the point source mask we observe wider error bars in temperature for all three shapes. Including the linear correction term (3.49) in the estimator reduces this effect to some extent, but in temperature this is clearly not enough. The effect of the point source mask on the local shape is exacerbated when the holes are smaller. 3.2: Importance of filling in and of the linear correction term for determining f NL in the presence of a mask and anisotropic noise. The results are based on a set of 100 Gaussian CMB simulations at Healpix resolution n side = 2048 with power spectrum according to the Planck 2013 release values. The simulations include smoothing by a 5 arcmin FWHM Gaussian beam and noise based on a white noise power spectrum with amplitude 1.5 × 10 -17 for temperature and 6 × 10 -17 for E polarization (in units made dimensionless by dividing by the CMB mean temperature T 0 = 2.7255 K). Where relevant the noise has been made anisotropic by modulating it using the hit count map of the Planck 143 GHz channel of the 2013 release. The maps are analyzed with the binned bispectrum estimator using 54 bins and max = 2500, and 100 maps were used for the linear correction term. The masks used are shown in Fig. 3.1.

No linear correction

For example, replacing the 2013 Planck LFI 30 GHz point source mask with the 2013 Planck HFI 100 GHz channel mask (with a 5σ threshold level), which has a much smaller beam and hence smaller holes (f sky = 0.99), increases the "no filling in, no linear correction" error bars for the local shape from 9.2 to 29.5 (while the error bars for equilateral and orthogonal become smaller). These results demonstrate the need for a suitable filling in of the missing data in the masked regions of the temperature map, in particular for the point source mask.

The simplest inpainting method is diffusive inpainting, which despite its simplicity worked extremely well and was subsequently adopted by the other Planck bispectrum estimators (KSW and modal) as well. It became the common method in all Planck releases. After filling the masked regions with the average of the unmasked part of the map as above, we fill each masked pixel with the average value of its neighbouring pixels and this procedure is iterated. We found that 2000 iterations sufficed for the Planck maps. One can implement the iterative procedure in two different ways: compute the average of the neighbours on the current iteration (Gauss-Seidel method, where some of the neighbour pixels will already have been updated and others not) or on the previous iteration kept in a buffer (Jacobi method, where all neighbour pixels will be on the previous iteration). While the Gauss-Seidel implementation is anisotropic, we found that this has no impact on the results, while on the contrary the faster convergence of that implementation is an advantage. This scheme solves a discretized version of Laplace's equation for the pixels where there is no data with the boundary of the unmasked region providing Dirichlet boundary data. (See [START_REF] Bucher | Filling in cosmic microwave background map missing data using constrained Gaussian realizations[END_REF] for a discussion of how this scheme is related to constrained random Gaussian realizations for filling in the missing data.) While this sort of 'harmonic averaging' is simple to implement and dulls the sharp edges, it appears at first glance not to remedy the problem of missing small-scale power described above, as the resulting maps have clearly visible bald spots, see Fig. 3.2. However, unlike apodization which only dulls the edges, the diffusive filling-in scheme does create small-scale structure inside close to the boundary of the mask. Given that during harmonic transforms it is the wavelength of the modes that determines how far they propagate, this is exactly what we need: the short wavelengths can only propagate small distances and hence need only be reconstructed close to the edges.

After masking, filling in, and filtering the maps, we mask them once again before integrating over products of maps. The masked region is never directly used in the calculation of the bispectrum, but the filling in is crucial to avoid the influence of the masked region spreading out over the sky when filtering the maps, as explained above. In addition, the average of the filtered maps outside the mask is subtracted to remove any monopole. If this is not done, smallscale power (whose origin is from the two-point function) will combine with this monopole to masquerade as (local) bispectral power, and this 'aliasing' can be a large effect.

Other more sophisticated inpainting techniques include nonlinear methods based on sparsity (see [START_REF] Abrial | Morphological Component Analysis and Inpainting on the Sphere: Application in Physics and Astrophysics[END_REF][START_REF] Abrial | CMB data analysis and sparsity[END_REF][START_REF] Perotto | Reconstruction of the CMB lensing for Planck[END_REF]) or constrained Gaussian realizations [START_REF] Bucher | Filling in cosmic microwave background map missing data using constrained Gaussian realizations[END_REF]. Alternatively, and even better for bispectrum determination, one can perform a full inverse covariance weighting (Wiener filtering) of the maps (see e.g., [START_REF] Smith | Optimal limits on f local N L from WMAP 5-year data[END_REF][START_REF] Elsner | Efficient Wiener filtering without preconditioning[END_REF]). However, these methods do not appear necessary, as a combination of diffusive inpainting and the linear correction term leads to results that are effectively optimal for the temperature maps (meaning they cannot be distinguished from the optimal results within the error bars). See also [START_REF] Gruetjen | Using inpainting to construct accurate cut-sky CMB estimators[END_REF] for an investigation of the impact of inpainting on masked CMB temperature maps. For E polarization the situation is even simpler, at least at the Planck resolution and sensitivity. Not even diffusive inpainting is required. Just applying the linear correction term appears sufficient. However, as a precaution we also applied diffusive inpainting to the Q and U maps for the Planck analysis.

Table 3.2 also highlights the importance of the linear correction term if there is anisotropic noise. While there is hardly any impact for the equilateral shape and no bias for any shape, for the local shape the error bars simply explode when we add anisotropic noise to the map, both for temperature and for the E-polarization mode. However, including the linear correction term suffices to recover the same error bars as in the ideal case. As can be seen from ( 3.49), the linear correction to the bispectrum of a given map, and hence to the f NL parameters via (3.33), involves the average over a large number of Gaussian maps. In Fig. 3.3 we show the histogram of the individual contributions of 199 Gaussian maps to the linear correction part of f NL for one of the maps from the "no mask, anisotropic noise" case of Table 3.2. The corresponding mean values are for T : local 165 ± 0.5, equilateral -4 ± 8, orthogonal -40 ± 4, and for E: local -415 ± 3, equilateral -20 ± 24, orthogonal 119 ± 11. As expected we see a hugely significant linear correction for local, a very significant correction for orthogonal (due to the large correlation with local), and no significant correction for equilateral. The error bars on the linear correction term for a single map are much smaller (in this case of 199 maps about a factor 7) than the error bars on the values of the different f NL parameters determined from 100 maps in Table 3.2, indicating that we have used enough maps to determine the linear correction.

Implementation of the estimator

A significant advantage of the binned bispectrum estimator is that it divides the bispectral analysis and determination of f NL into three separate parts, the first two of which are completely independent. The first, slow, part is the computation of the raw binned bispectrum of the map under consideration, including its linear correction. The second, much faster, part involves the computation and subsequent binning of the theoretical bispectrum templates one wants to test and of the expected bispectrum covariance. Finally, the third, extremely fast, part (that runs in less than about a minute) is where the different analyses (for example for different templates) are carried out using the raw binned bispectrum from part 1 and the quantities from part 2 as an input. In the case of f NL determination, this last part corresponds to the evaluation of the sum over the bins and polarization indices in the inner product (3.37) used in (3.33). This approach has several advantages. Firstly, the full (binned) bispectrum is a natural output of the code and can be studied on its own without a particular template in mind. Such an analysis is the subject of the next subsection. Secondly, there is no need for the bispectrum template to be separable, since nowhere in the method does the need arise to split up the template into a separable form. Thirdly, once the bispectrum of a map has been computed, modifications to the theoretical analysis (like for example testing additional templates) is fast, since there is no need to rerun the observational part (which consumes by far the most time). This is in contrast with competing estimators such as the KSW estimator, where the theoretical and observational steps are mixed together (a separation is instead made in terms of 1 , 2 , 3 ), so that the full code has to be rerun for any new template. Fourthly, with the binned bispectrum estimator the dependence of f NL on is obtained almost for free, simply by leaving out bins from the sum when computing the final inner product. In particular this has been used to study the dependence on min and max in the Planck analysis. Finally, the binned bispectrum estimator compares favourably to the other estimators in terms of speed: it is very fast on a single map.

The only disadvantage of this method is that the templates that can be studied accurately have to be reasonably smooth, or if not then any rapid changes should be limited to a small part of -space, in order for the template to be well approximated by a binned template with a not too large number of bins. 13 For most primordial and foreground templates studied so far, this is not a problem. Moreover, even for templates that do not satisfy this criterion, the binned bispectrum estimator could still perform quite well. For example, among the templates discussed in sections 3.1.2 and 3.1.3, only the lensing-ISW template cannot be easily binned. For a typical Planck binning the overlap is of the order of 60-70% (as opposed to 95% or higher for all the other templates considered). Nevertheless the binned bispectrum estimator gives unbiased results even for this template, with error bars that are only slightly widened.

The code has been written mainly in Python, using some routines written in C. It is run on the computers of the Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (CC-IN2P3) in Lyon, France14 and any explicit remarks about computing time refer to that system.

Theoretical part

The theoretical part of the code consists of two steps: first determining the unbinned theoretical bispectrum and power spectrum, and second, computing from these spectra the binned bispectrum templates B th i 1 i 2 i 3 and the inverse of the binned covariance matrix (V -1 ) p 1 p 2 p 3 p 4 p 5 p 6 i 1 i 2 i 3 , see (3.33) and (3.37). This also requires experimental inputs in the form of the beam transfer function b and the noise power spectrum N .

The first step is in some sense not really part of the estimator code. We have a code to compute all the bispectra discussed in sections 3.1.2 and 3.1.3, but in principle an explicitly computed theoretical bispectrum from any source could be used here. In our code we use the radiation transfer functions g p I (with p the polarization index and I the isocurvature index) computed with CAMB (slightly modified to write them to file, since these are not a normal output of CAMB) to compute the primordial templates (3.1). For separable templates, this is a fast calculation, since the triple integral over k 1 , k 2 , k 3 becomes a product of single integrals. For non-separable templates a brute force calculation is much slower, but while one might look for smarter ways to compute such bispectrum templates, it should not be forgotten that (for a given cosmology) for use in the binned bispectrum estimator, a template has to be computed only once. Hence even a slow calculation might be acceptable. While this code can also compute the power spectra from the radiation transfer functions according to (1.52), in practice we use the power spectra computed by CAMB. These power spectra are used in the covariance matrix and some foreground bispectrum templates. The primordial bispectra are precomputed only on a grid (with ∆ increasing to about 10 at high ). This is denser than the binning, and thus accurate enough for the smooth local, equilateral, and orthogonal templates.

The second step involves the binning of the bispectrum templates and the covariance matrix. The calculation of the covariance matrix from the power spectra as well as the calculation of the foreground templates is done directly in this step. As was seen in section 3.1.2, the foreground templates are simpler to compute than the primordial templates, since there are no integrals, so there is no need to precompute them, the required values can be computed in real time while binning. As for the precomputed primordial templates, since these have been precomputed only on a grid, other values are computed by three-dimensional linear interpolation. While we developed a tetrahedral integration scheme to speed up the calculation of all binned quantities, as described in [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF], we later moved away from using it. Given that the theoretical computation is much faster than the observational computation, there is no point in making additional approximations to speed it up. Performing an exact calculation of the binned quantities (where the quantities are explicitly computed for each value of 1 , 2 , 3 and then summed over the bin) is fast enough. We can thus also directly compute the overlap between the binned and the exact template using (3.38).

The final output of this step consists of two files: one containing the binned theoretical bispectrum for all requested shapes, polarizations and isocurvature components; and another containing the inverse of the binned covariance matrix for all polarization components. In addition the exact Fisher matrix (3.44) (without binning) is produced to allow for the estimation of the accuracy of the binning approximation using (3.38).

Choice of binning

The choice of binning is an important part of the implementation. In theory the idea is very simple: one chooses the binning that makes the overlap parameter R defined in (3.38) as close to one as possible. In practice this is not so simple, since both the number of bins and all the bin boundaries are free parameters. Fortunately R does not depend strongly on the exact binning choice. Moreover, one does not need R = 1 to obtain results statistically indistinguishable from the exactly optimal results. For example, even with R = 0.95, which is about the lowest overlap for any of the templates considered in the Planck analysis (except for lensing-ISW), the increase in the standard deviation is only 2.6%. This should be compared to the 5% uncertainty in the standard deviation due to its determination from 200 maps. Note that the code allows the use of separate binnings for the T -only, the E-only, and the full T +E analyses, although for reasons related to time a single binning was used for the Planck analysis.

We developed three optimization tools: one that checks which bin boundary can be removed with the smallest decrease of R (reducing the number of bins by one), one that checks where a bin boundary can be added with the largest increase in R (increasing the number of bins by one; the bin boundary is added in the exact centre of an existing bin), and one that tries moving all the bin boundaries by a given amount (relative to the size of the bin) and tells for which bin this increases R the most (leaving the number of bins unchanged). For all of them one can indicate which shapes and polarizations (meaning T and/or E) should be taken into account. These three tools are then used iteratively to optimize the binning used as starting point, until no more significant improvements are obtained (as defined by a certain threshold in the change of R). The starting point is arbitrary. For example a simple log-linear binning (with bin sizes increasing logarithmically at low , up to a certain value of , after which the binning becomes linear) or a binning that has already been partially optimized in another way can be used. The latter could for example be done using the method described in [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF], which can provide a good starting point. (That method produces suboptimal binnings and can benefit significantly from the procedure described here.) While this method can likely be optimized further, for the Planck analysis the binning obtained in this way produces effectively optimal results.

Observational part

The observational component of the code consists of two parts: one to compute the cubic part of the bispectrum of the map according to (3.24), and the other to compute the linear correction according to (3.49). First the map is fully prepared, which can be as simple as reading an existing map and doing the masking and filling in, or involve the creation of CMB and noise realizations. It is then saved in the form of a m 's for later use with the linear correction term, or for reproducibility in the case of generated random realizations.

The maps are then filtered according to (3.23). This leads to some practical issues that had to be resolved, since in principle we need to hold twice (T and E) 50-60 maps (one for each bin) of Planck resolution (Healpix resolution parameter n side = 2048) in memory for this calculation. However, our computer system had a limit of 16 GB per processor (after the 2015 Planck analysis this was even reduced to only 10 GB), which makes this impossible. We managed to save space in two ways. In the first place, while all the preprocessing of the maps is done in double precision, the final filtered maps are only kept in single precision, which saves a factor two in memory. Tests have shown that this has no significant impact on the final results for f NL . Secondly, it is unnecessary to use n side = 2048 precision for the maps that contain only low-bins. Hence the filtered maps of bins up to about = 400 are produced at n side = 512, and those between about 400 and 800 at n side = 1024, which saves a factor of sixteen and four, respectively, in memory for those maps (the number of pixels in the map is 12 n 2 side ), as well as speeding up the final computation where three maps have to be multiplied and summed (see (3.24)). Polarization maps are never higher resolution than n side = 1024. Using the nested Healpix15 format, it is easy to multiply maps of different n side together.

We have developed two different ways of computing the linear correction term of a map. In the first method, which was used for the Planck analyses in 2013 and 2015 and in the analyses presented in the previous subsection (from [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF]), each job treats one of the Gaussian maps (see (3.49)), which is preprocessed and filtered as above, and the filtered maps are held in memory. Then a filtered map of only the first bin of the observed map is created and all required sums of products involving that map are computed. Next this process is repeated for the second bin of the observed map, etc. 16 The final result of this job is a temporary file with a linear correction term computed with just one Gaussian map. Once all jobs have finished (with the results for the other Gaussian maps), the results are summed and averaged to obtain the final linear correction term for the map. This whole process (preprocessing the map and computing the cubic and linear terms) for a single map at Planck resolution for all T + E (including mixed) components takes a few hours, which is quite fast compared to other bispectrum estimators. (Computing the theoretical part is much faster and requires only a single job, so can easily be done on the side.) With this method one can simply add more Gaussian maps to the linear correction term at a later stage if required, and investigate its convergence as a function of the number of Gaussian maps. However, this first method of computing the linear correction term scales very badly with the number of observed maps. Since the object

M p 1 ,G i 1 M p 2 ,G i 2 in (3.49)
is too large to compute directly and save to file, if one has a set of similar maps (for example to compute error bars), the linear correction term has to be recomputed for each map in the same way as above, making this a very slow process.

For this reason we later developed another way to compute the linear correction term, which was used for the Planck analysis in 2018. This second method is based on the observation that while the object

M p 1 ,G i 1 M p 2 ,G i 2
(consisting of 6612 maps for a full T + E calculation in the case of 57 bins) is too large to handle, saving it in the form of a m 's is doable. Moreover, we make use of the fact that when multiplying several masked maps together (all with the same mask), it is enough if only one of the maps is masked. Hence if the observed map in (3.49) is properly masked, the Gaussian maps can be left unmasked (since the Gaussian maps are based on simulations, they are full-sky maps). This has the advantage that no filling-in needs to be performed on these maps, which would otherwise be required before conversion to a m 's, as explained in section 3.2.3. By limiting the number of considered bins per job in such a way that both the filtered maps for those bins and all the product maps involving those bins can be kept in memory at the same time, one job can compute the full average for the considered bins by treating one Gaussian map after the other. Only at the end are the final maps converted to a m format and written to disk. This precomputation for the linear correction term can be run with a modest number of jobs (about 100) in a reasonable amount of time (less than a day for 200 maps). Once the precomputation has finished, the linear correction for any map can be quickly computed using (3.49). Each job reads in a number of product maps (i.e. for certain values of i 1 and i 2 ; the number being determined by memory considerations), and converts them back to pixel space. They are then multiplied with the filtered observed maps as explained above for the first method. The main difference is that the results are now for the full average of all the Gaussian maps, instead of for a single one. Another (small) advantage of this second method is that at this step we only need to multiply two maps together and not three. Once all jobs are finished, the temporary files containing results for different i 1 -i 2 bins are combined to get the full linear correction for the observed maps. While this second method with precomputation is slower if one is only interested in a single map, its much better scaling with the number of maps makes it by far the preferred method when dealing with a set of maps, for example to compute error bars.

The final result of this part are two files for each map, one containing the binned cubiconly bispectrum of the map and the second its linear correction, both containing all requested polarization components. These can then be combined with the results from the theoretical part to compute f NL according to (3.33), which takes less than a minute even when producing convergence plots and dependence on as well, or be studied directly without the assumption of a theoretical template, as discussed in the next subsection.

Non-parametric bispectrum studies

The previous subsections described how the binned bispectrum of a map can be analysed parametrically by computing the f NL parameters corresponding to a selection of theoretically motivated templates. But one advantage of the binned bispectrum estimator is that the full (binned) three-dimensional bispectrum is a direct output of the code, which can be studied non-parametrically, thus searching for any deviations from Gaussianity even when no suitable template is available. Here we start by describing the smoothing procedure that must first be applied to the binned bispectrum in order to enhance the signal-to-noise of any possible non-Gaussian features, which otherwise would remain hidden in the noise. After that we will discuss the statistical analysis subsequently applied to this smoothed binned bispectrum to assess the statistical significance of any non-Gaussian features appearing as extreme values.

We first normalize the binned data by dividing by the square root of the expected bin variance, so that each bin triplet in the absence of a bispectral signal would have noise obeying a normalized Gaussian distribution. Thus for the bin triplets for which there is data, we define ). This projection entails a loss of information but allows the same analysis to be used as for T T T , as described below.

B T T T i 1 i 2 i 3 = B T T T i 1 i 2 i 3 V T T T T T T i 1 i 2 i 3 , B EEE i 1 i 2 i 3 = B EEE i 1 i 2 i 3 V EEEEEE i 1 i 2 i 3 , B T 2E i 1 i 2 i 3 = B T 2E i 1 i 2 i 3 Var(B T 2E ) i 1 i 2 i 3 , B T E2 i 1 i 2 i 3 = B T E2 i 1 i 2 i 3 Var(B T E2 ) i 1 i 2 i 3 . ( 3
Only bin triplets containing 's that satisfy both the parity condition and the triangle inequality contain data. However, among the bin triplets containing data, we noticed that some triplets systematically produced outliers. It turned out that these bin triplets contained very few valid -triplets (for example, the hypothetical bin triplet ([50, 100], [50,100], [200,300]) would contain only one valid -triplet (100,100,200), since the triangle inequality imposes that 3 ≤ 1 + 2 ). While the theoretical variance calculation is exact, the computation of the observed bispectrum using Healpix spherical harmonic transforms contains some numerical inaccuracies, so that the bispectrum in points outside the triangle inequality is not zero but contains leakage.17 For bin triplets like the above example with many -triplets violating the triangle inequality, a significant mismatch between the theoretical and the actual standard deviation of the bispectrum in that bin is observed. The obvious solution is to remove such bin triplets from the data. Moreover, the statistical analysis described below assumes that bin triplets contain many valid -triplets in order for Gaussian statistics to apply to the noise from cosmic variance, which constitutes another reason to exclude such triplets. After some experimentation, we adopted the selection criterion that the ratio of valid -triplets to the ones satisfying only the parity condition (but not the triangle inequality) in a bin triplet should be at least 1%, finding this a good threshold for rejecting systematic outliers. The results are insensitive to the precise threshold used. For the Planck binning with 57 bins (which is used for the results in this section), this criterion excluded 293 out of 13020 bin triplets.

If we were looking for a sharp bispectral feature of a linewidth narrow compared to the binwidth, there would be no motivation to smooth. We would simply examine the statistical significance of the extreme values of the renormalized binned bispectrum described above, taking into account the look-elsewhere effect. However, for broad features, as are likely to arise from galactic foregrounds, smoothing increases statistical significance by averaging over and thus diminishing the noise. One approach would be to use binning with a range of bin widths, but this approach has the disadvantage that the statistical significance for detecting a feature depends on how it is situated relative to the neighbouring bin boundaries. Instead we rather smooth using a Gaussian kernel and renormalize so that in the absence of a signal the single pixel distribution function is again a unit Gaussian. For a Gaussian kernel K σ bin of width σ bin , we have

B p 1 p 2 p 3 ,smoothed i 1 i 2 i 3 = i 1 i 2 i 3 K σ bin (i 1 -i 1 , i 2 -i 2 , i 3 -i 3 )B p 1 p 2 p 3 i 1 i 2 i 3 (3.51)
where the Gaussian smoothing kernel

K σ bin (∆i 1 , ∆i 2 , ∆i 3 ) = 2πσ bin 2 -3/2 exp - 1 2 ∆i 1 2 + ∆i 2 2 + ∆i 3 2 σ bin 2 (3.52)
is used. Numerically the kernel is applied in the Fourier domain.

Without boundaries this smoothing and renormalization procedure would be straightforward. However, near the boundary the Gaussian smoothing kernel would extend into the region where there is no data. To minimize boundary effects, we first extend the fundamental domain (where i 1 ≤ i 2 ≤ i 3 ) to the five identical domains obtained by permuting (i 1 , i 2 , i 3 ) and pad with zero data beyond the boundaries of this extended domain as well as for triplets inside the domain for which there is no data. The smoothing causes power to leak out into the zero padded regions, and to correct for this leakage, we construct a mask consisting of ones in the domain of definition and zeros outside. After smoothing the signal-to-noise bispectrum B, we renormalize by dividing by the mask that has undergone the same smoothing procedure. For the bin triplet statistic to be a Gaussian of unit variance, we generate 1000 Monte Carlo realizations going through the same procedure and compute the variance, with which we divide our smoothed renormalized bispectrum.

The result using different smoothing lengths is illustrated in Fig. 3.4 as two-dimensional slices showing B as a function of 1 and 2 for a fixed bin in 3 . With the colour scale used in Fig. 3.4, both dark red and dark blue represent extreme values with small p values if Gaussianity is assumed, and thus suggest the presence of statistically significant bispectral non-Gaussianity. A correct analysis of the significance would also take into account the look-elsewhere effectthat is, that the small probability to exceed, calculated for a fixed bin, is too small because it does not reflect that an improbable value could have occurred in any of a number of bins. The analysis of this issue is complicated by the correlations between the bins that result from the smoothing, an issue analyzed next.

In the absence of smoothing, we face the following statistical problem. We have a binned bispectrum that has been rescaled so that we have N bins and the bispectrum value in each bin x i , where i = 1, . . . , N , has a probability distribution function well approximated by a normalized Gaussian distribution. Moreover, values in different bins are almost statistically independent. The quadratic correlation vanishes, but some of the higher-order joint correlations do not precisely vanish, a feature that we shall neglect here. The corrections to Gaussianity and to statistical independence are suppressed when is large and when there are many -triplets containing data in a bin. Thus we have the distribution function

p(x 1 , . . . , x N ) = (2π) -N/2 exp - 1 2 N i=1 x i 2 , (3.53)
and since we are interested in extreme values, we define two new derived statistics

X min = min(x 1 , . . . , x N ); X max = max(x 1 , . . . , x N ), (3.54)
and accordingly define the p-values p min (X) = P (X min < X); p max (X) = P (X max > X) (3.55) where X min and X max are the derived random variables defined above. If either of these pvalues are extremely small, then we have evidence of bispectral non-Gaussianity directly in the unsmoothed binned bispectrum, and this p-value can be converted into a σ for the normal distribution using the inverse error function as is customary.

For this simple unsmoothed case it is not hard to give the probability distribution function for the extreme value statistics X min and X max . Given the (complementary) cumulative distribution function for the normal distribution (integrating from right to left) (3.56) the analogous distribution for the maximum extreme value for N variates is given by

Φ(x) = 1 √ 2π +∞ x dt exp - 1 2 t 2 ,
Φ max (X max ; N ) = 1 -1 -Φ(X max ) N (3.57)
and we may straightforwardly obtain an analogous expression for the case of the minimum value. (Below we shall only give results for the case of the maximum.) For X 1 we obtain an approximation to Φ max (X max ; N ) by inserting the following expression [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] ln Φ(X) ≈ -X 2 2 + ln(X) + 1 2 ln(2π) (3.58) into (3.57).

When we consider extreme values of multivariate Gaussian distributions with correlations, there is, as far as we know, no way of obtaining an analytic result for the extreme value distribution for Φ max (X max ). After the smoothing described above is applied, the probability distribution defined in (3.53) must be replaced with 3.59) where the correlation matrix C has all ones on the diagonal, but also a lot of positive off-diagonal elements as the result of the smoothing process, rather than all zeros away from the diagonal.

p(x) = (2π) -N/2 exp - 1 2 x T C x (
It is these off-diagonal elements that prevent us from solving analytically for the extreme value statistic probability distribution function.

Instead we postulate an Ansatz to approximate the cumulative distribution function (CDF) of the extreme value statistic Φ max , which has one adjustable parameter N eff , the effective number of independent bins, which will be smaller than the actual number of bins N as the result of the smoothing. The Ansatz states that the CDF given in (3.57) (and approximated using (3.58)) holds where N has been replaced with N eff . For a given level of smoothing, we fit N eff to the tail of the CDF, which has been determined empirically by Monte Carlo simulations. We then assess the quality of the approximation, in particular in the tail of the distribution where X is very large, which is the range of values of particular interest here. It should be stressed that we do not need a good approximation to the entire CDF. It suffices to have an approximation that works well asymptotically, in the extreme tail of the distribution where p-values cannot feasibly be obtained by Monte Carlo methods. Thus the Ansatz serves as an asymptotic approximation for the tail of the distribution.

To demonstrate the validity of our Ansatz in a simplified context very similar to the case of interest, we generate a three-dimensional periodic cubic lattice filled with 64 3 independent realizations of a normal Gaussian random variable. This cube is then smoothed using a Gaussian smoothing kernel with widths σ smoothing = 2, 5, and 10. The smoothed cube is rescaled so that the variable at each lattice point has unit variance. For each smoothing width, the extreme value statistic (maximum) is taken for 10 6 realizations and only the greatest 5% of the extreme values are retained. Fig. 3.5 (left) shows the empirical CDF for the extreme values, which are compared to the functional form of the Ansatz for the best-fit values of N eff according to the approximation given in (3.58).

The above discussion demonstrates that an extreme value distribution for N eff independent Gaussian variates can be used to approximate the distribution for N bin variates with correlations due to smoothing. However, the geometry of the allowed bins is complicated and must be taken into account. Thus simulations must be carried out to determine N eff for a given smoothing scale in the more realistic case of the actual bins used for the bispectrum analysis. We show that the Ansatz is still a good approximation also in this case in Fig. 3.5 (right). We generate random numbers in the domain of definition of the binned bispectrum, and smooth it as for the real data. Restricting to those bin triplets that contain enough valid data (see the discussion above), these simulations are a good approximation to a Gaussian CMB map's bispectrum.

We now illustrate this method by applying it to a realistic situation. We add a point source map to a simulated Gaussian CMB map with anisotropic noise, generated as described in the caption of Table 3.2. The point source simulation was created with the Planck Sky Model, at 143 GHz, with a beam with a FWHM of 5 arcmin, and contains faint infrared sources, as described in [START_REF] Delabrouille | The pre-launch Planck Sky Model: a model of sky emission at submillimetre to centimetre wavelengths[END_REF], and faint radio sources with the improved parameters described in [Planck 2015 XII]. The galactic and point source masks were applied as described in Section 3.2.3.

The binned bispectrum of this map was evaluated applying the linear correction and the filling-in procedure, and the f NL 's were determined individually for each of the templates described in Section 3.1.2. The unclustered point source contribution was detected with high significance in this contaminated map: b ps = (64.8 ± 0.8) × 10 -29 . This signal is much stronger than the one detected in the cleaned Planck maps, but of the same order of magnitude as the forecast at 217 GHz (see [Planck 2015 XVII]). No statistically significant detection of a nonzero f NL was obtained for the other templates, with the exception of the CIB template. But the CIB bispectrum has significant overlap with the unclustered point source bispectrum (see Table 3.1), so this result is not surprising. The nonzero result for b CIB disappears in a joint analysis of the unclustered point source and CIB templates. Finally we smooth the bispectrum with a few different values of the smoothing length, namely σ smoothing = 1, 2, 3.

Apart from studying the contaminated bispectrum, we can also try to remove the estimated point source contribution from the measured binned bispectrum, simply by subtracting the corresponding smoothed template (3.6) with the measured amplitude. We can then check if there are remaining non-Gaussian features in this cleaned bispectrum using the method described above. Table 3.3 gives the maximum bin values for the bispectrum before and after the template cleaning is applied. The minima are not given because the inclusion of the point sources tends to gives a positive bispectral contribution. The p-values were calculated using simulations of 10 6 Gaussian realizations and fitting the CDF for N eff to the empirical distribution. We found that the smoothing lengths σ smoothing = 1, 2, and 3 correspond to N eff = 8664, 1948, and 588 respectively. We see that a highly statistically significant detection is found using the above procedure on the uncleaned bispectrum. We also observe that the template cleaning procedure is successful; however, some detectable unsubtracted residual remains. This residual, however, has little overlap with the known theoretically motivated primordial templates.

Summary of applications and results

Having discussed the binned bispectrum estimator in detail, we can now start looking at further extensions, and in particular at applications and results. As in chapter 2, these self-contained papers have been added verbatim to the appendices of this chapter, removing only their conclusions. Those conclusions are used to provide a summary of the papers in this section. The papers in question are [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF]] (section 3.3.1 and apppendix 3A) containing the extension to isocurvature non-Gaussianity, [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF]] (section 3.3.2 and appendix 3B) with an application to galactic foregrounds, and [Planck 2018 IX] (section 3.3.3 and appendix 3C) with the Planck 2018 primordial non-Gaussianity analysis.

Isocurvature non-Gaussianity

In the paper [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF], reproduced in appendix 3A, we systematically investigated the angular bispectra generated by initial conditions that combine the usual adiabatic mode with an isocurvature mode, assuming local non-Gaussianity. We studied successively the four types of isocurvature modes, namely cold dark matter (CDM), baryon, neutrino density and neutrino velocity isocurvature modes. In each case, the total bispectrum can be decomposed into six elementary bispectra and we estimated the expected uncertainties on the corresponding coefficients, which are extensions of the usual purely adiabatic f NL parameter, in the context of the Planck data of the cosmic microwave background radiation (CMB). As we showed, the results for baryon isocurvature can be obtained from a simple rescaling of the CDM isocurvature results, but the others are distinct. At the time of this paper (2012), Planck results were not yet available, and in the 2013 Planck release, due to lack of time, isocurvature non-Gaussianity was not studied. Because this release did not include polarization, and large improvements for the isocurvature non-Gaussianity results were expected with polarization as explained in this paper, this was a logical decision. However, starting from the 2015 Planck release, results for isocurvature non-Gaussianity have been included. The 2018 results can be found in appendix 3C.

In the squeezed limit, where one multipole is much smaller than the other two (which then have to be almost equal due to the triangle inequality), we showed that the six elementary bispectra factorize as a function of the small times the power spectrum as a function of the large . Since the squeezed limit components dominate the bispectrum for local non-Gaussianity, we were able, using this factorization, to give simple explanations for the various interesting results that we observed.

By enlarging the space of initial conditions, one obviously expects a larger uncertainty on the purely adiabatic f NL coefficient. Interestingly, this uncertainty is increased only by a factor 2 for CDM and baryon isocurvature modes, whereas it increases by a factor 6 in the neutrino density isocurvature case and by a factor 5 in the neutrino velocity isocurvature case. This can be explained by the fact that the CDM isocurvature power spectrum decreases much faster with than the adiabatic and neutrino isocurvature ones. As we showed, this means that the uncertainties on f ζ,ζζ NL and f ζ,ζS NL in the case of CDM isocurvature continue to improve as one increases the number of available multipoles, while the other four saturate at a much lower . As a consequence the first two can be determined much more accurately than the other four, and are only weakly correlated with them. This small correlation also means that it was important to look at the data with the full f NL estimator, and not just the adiabatic one, as a large CDM isocurvature non-Gaussianity could have been hiding behind a small adiabatic signal.

We showed that the E-polarization often plays a crucial role in reducing the uncertainties. In the CDM isocurvature case, polarization improves slightly the precision on the coefficients f ζ,ζζ NL and f ζ,ζS NL , but the precision of the other four coefficients improves by a factor of order five. Polarization is also very important for some of the parameters in the neutrino velocity case; the uncertainty on the purely isocurvature f S,SS NL , for example, improves by a factor 8 when polarization is included. Again we were able to explain these results from the behaviour of the power spectra, using the factorization of the squeezed bispectrum.

The decomposition of the bispectrum into six elementary bispectra and the CMB constraints on the six f (i) NL parameters do not depend on any assumptions about the specific model of the early universe. We only assumed that (possibly correlated) primordial adiabatic and isocurvature modes are produced, with a primordial bispectrum of local type and with power spectra that all have the same shape. Note that neither of these assumptions appears to be essential; they were only made for simplicity.

If, however, one does consider an explicit early universe model, there are often relations between the six parameters, and an observational detection or constraint could then be used to check for such a relation and put constraints on the parameters of the model. We discussed a general class of models with two scalar fields, where only one of the fields generates both the isocurvature perturbations and all non-linearities. We also considered a specific implementation of this general model, where the two fields are an inflaton and a curvaton. In this model, a CDM isocurvature mode is produced and the six f (i) NL parameters only depend on two model parameters. In some ranges of the model parameters, the isocurvature mode is subdominant in the power spectrum but provides observable non-Gaussianity that can dominate the usual adiabatic non-Gaussianity. Looking for these new angular shapes in the CMB data would thus provide interesting information on the very early universe.

Galactic foregrounds

In the paper [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF], included in appendix 3B, we used the binned bispectrum estimator to determine the bispectra of different galactic foreground maps, as produced by the Commander component separation method from Planck 2015 data (rescaled to amplitudes representative for the 143 GHz Planck channel). These galactic foreground bispectra were then used as templates for other runs of the binned bispectrum estimator applied to various types of maps: simulations, raw sky maps, and cleaned CMB maps.

This paper serves different purposes. In the first place it is a proof of concept. The possibility to determine the (binned) bispectrum of any map is a clear advantage of the binned bispectrum estimator, and was used in the official Planck releases to present the bispectrum of the observed CMB. The fact that any provided bispectrum, not only if an analytical template is known but also simply any numerical bispectrum, can be used as template in the binned bispectrum analysis pipeline, had also long been presented as an advantage of the method. In fact the possibility of combining these two advantages to do an analysis as presented here was already mentioned in the original paper of the binned bispectrum estimator [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF] and was one of the motivations for developing it in the first place, but had until this paper never been worked out explicitly. This paper proved that this idea also works in practice.

Secondly, this paper shows and discusses the bispectra of the various galactic foregrounds, which is an interesting result in itself, even if for the purposes of the paper it was only an intermediate step. We found that the dust, the free-free and the anomalous microwave emissions have very squeezed bispectra (similar to the local shape, but with an opposite sign). The smallscale fluctuations of the dust radiation are stronger in the large-scale dust clouds, so small-scale and large-scale fluctuations are correlated (and a similar explanation is valid for the other foregrounds). The synchrotron map as provided is different, as its bispectrum is more similar to the equilateral shape, but we were able to show that at least a large part of this effect is due to a residual contamination by unresolved extra-galactic point sources. At 143 GHz (the most important Planck frequency for CMB analysis) only the dust really contaminates the CMB signal, the other foregrounds giving contributions that are orders of magnitude smaller.

An issue with the numerical templates we determined is that they also depend on the mask applied to the foregrounds and contain the characteristics of the experiment like the beam and the noise. We showed that the choice of the mask is very important because the foregrounds are localized in the galactic plane close to the galactic mask, so a small change of mask could give a large difference of bispectrum. This means the same mask should be used for determining the template as for the final analysis. It should be pointed out that for the purpose of studying the non-Gaussianity of galactic foregrounds as goal in itself, the bispectrum is likely not the best tool: a pixel-space based statistic to take into account the localized nature of these foregrounds would seem more logical. However, our main purpose was to investigate the impact these galactic foregrounds have on the determination of primordial f NL parameters in a bispectrum analysis.

The third and final result of this paper is the f NL analysis of real sky maps, both raw and cleaned CMB, with these galactic bispectrum templates, where we investigated in particular if any observable galactic residuals remained in the cleaned CMB map and if a joint analysis of primordial and galactic templates improved the determination of the primordial f NL . But before we did that analysis we obviously first tested and validated our methodology and our new analysis pipeline on simulations. These tests were based on Gaussian realizations of the CMB to which we added noise simulations and a known amount of dust. We showed that both with isotropic and with anisotropic noise we managed to detect the expected amount of dust in our maps. However, we also showed that, to do a joint analysis with the primordial and the dust shapes, the usual choice of bins, while acceptable, can be improved. With more bins at low one can better discriminate between the templates that peak in the squeezed configuration (local and dust especially). We also discussed the effects of the (small) breakdown of the weak non-Gaussianity approximation that occurs when we add the full dust map to the CMB simulations (i.e. the expected amount of dust in raw-sky observations). The main consequence is that the real error bars become larger than the Fisher forecasts.

The testing and validation having been successful, we then used the numerical galactic templates on the cleaned SMICA CMB map of the 2015 Planck release. Fortunately, we did not detect any residual of the dust. The error bars for the dust and local shapes increased in the joint analysis with the usual binning, again because of the choice of bins that was not optimal to differentiate them. Finally, we applied the foreground templates to the raw sky map at 143 GHz and the binned bispectrum estimator succeeded in detecting the dust in it at the expected level (the intensities of the other foregrounds at 143 GHz being too small to detect even if they were present in the map).

The work presented in this paper can be extended in several ways. The additional functionality built into the binned bispectrum estimator code to use numerical bispectra as templates opens new possibilities, and allows us to include the template of any component of which a map exists in our bispectrum and f NL analyses. It would also be interesting to further study the galactic bispectra, or their non-Gaussianity in general, together with an expert on galactic astrophysics, to see if they can be physically understood. This could maybe lead to building an analytical template for these bispectra that can also be used by other bispectrum estimator codes. Finally, the analysis of this paper obviously had to be repeated on the final 2018 Planck data. This was indeed done, although for temperature only, as presented in section 6.3.1 of [Planck 2018 IX], reproduced in appendix 3C. The hope is that the improved treatment of the polarization maps in that release will make an extension to E-polarization of this analysis viable as well.

Planck 2018 results

In the paper [Planck 2018 IX], included in appendix 3C, we presented constraints on primordial non-Gaussianity (NG), using the Planck full-mission CMB temperature and E-mode polarization maps. Compared to the Planck 2015 release, the low-(4 ≤ < 40) polarization multipole range was this time also included.

Our analysis produced the following final results (68 % CL, statistical): f local NL = -0.9 ± 5.1; f equil NL = -26 ± 47; and f ortho NL = -38 ± 24. These results are overall stable with respect to our constraints from the 2015 Planck data. They show no real improvement in errors, despite the additional polarization modes. This is due to a combination of two factors. Firstly, the local shape, which is most sensitive to low-modes and where one would naively expect an improvement, is actually less sensitive to polarization than the equilateral and orthogonal shapes. This means that in the end none of the three shapes are very sensitive to low-polarization modes. Secondly, the temperature and polarization simulations used to determine the errors had a more realistic but slightly higher noise level than in the previous release.

On the other hand, the quality of polarization data shows a clear improvement with respect to our previous analysis. This was confirmed by a large battery of tests on our data set, including comparisons between different estimator implementations (KSW, Binned, and two Modal estimators) and foreground-cleaning methods (SMICA, SEVEM, NILC, and Commander), studies of robustness under changes in sky coverage and multipole range, and an analysis of the impact of noise-related systematics. While in our previous release we had cautioned the reader to take polarization bispectra and related constraints as preliminary, in light of these tests we consider our results presented here based on the combined temperature and polarization data set to be fully reliable. This also implied that polarization-only, EEE bispectra could now be used for independent tests, which led to primordial NG constraints at a sensitivity level comparable to that of WMAP from temperature bispectra, and yielding statistical agreement.

As in the previous analyses, we went beyond the local, equilateral, and orthogonal f NL constraints by considering a large number of additional cases, such as scale-dependent feature and resonance bispectra, running f NL models, isocurvature primordial NG, and parity-breaking models. We set tight constraints on all these scenarios, but did not detect any significant signals.

On the other hand, the non-primordial lensing bispectrum was detected with an improved significance compared to 2015, excluding the null hypothesis at 3.5 σ. The amplitude of the signal is consistent with the expectation from the Planck best-fit cosmological parameters, further indicating the absence of significant foreground contamination or spurious systematic effects. We also explicitly checked for the presence of various non-primordial contaminants, like unclustered extragalactic point sources, CIB, galactic thermal dust, and the thermal SZ effect, but apart from the first, none of these were detected. The small amount of remaining point-source signal in the cleaned maps had no impact on our other constraints because of its negligible correlations.

We updated our trispectrum constraints, finding g local NL = (-5.8 ± 6.5) × 10 4 (68 % CL, statistical), while also constraining additional shapes, generated by different operators in an effective field-theory approach to inflation.

In addition to estimates of bispectrum and trispectrum amplitudes, we produced modelindependent reconstructions and analyses of the Planck CMB bispectrum. Finally, we used our measurements to obtain constraints on early-universe scenarios that can generate primordial NG. We considered, for example, general single-field models of inflation, curvaton models, models with axion fields producing parity-violating tensor bispectra, and inflationary scenarios generating directionally-dependent bispectra (such as those involving vector fields).

In our data analysis efforts, which started with the 2013 release, we achieved a number of crucial scientific goals. In particular we reached an unprecedented level of sensitivity in the determination of the bispectrum and trispectrum amplitude parameters (f NL , g NL ) and significantly extended the standard local, equilateral, and orthogonal analysis, encompassing a large number of additional shapes motivated by a variety of inflationary models. Moreover, we produced the first polarization-based CMB bispectrum constraints and the first detection of the (non-primordial) bispectrum induced by correlations between CMB lensing and secondary anisotropies. Our stringent tests of many types of non-Gaussianity are fully consistent with expectations from the standard single-field slow-roll paradigm and provide strong constraints on alternative scenarios. Nevertheless, the current level of sensitivity does not allow us to rule out or confirm most alternative scenarios. It is natural at this stage to ask ourselves what should be the f NL sensitivity goal for future cosmological experiments. A number of studies has identified f NL ∼ 1 as a target. Achieving such sensitivity for local-type NG would enable us to either confirm or rule out a large class of multi-field models. A similar target for equilateral, orthogonal, and scale-dependent shapes would allow us to distinguish standard slow-roll from more complex single-field scenarios, such as those characterized by higher-derivative kinetic terms or slow-roll-breaking features in the inflaton potential (see e.g. [Alvarez et al., 2014;CORE Collaboration, 2018] and references therein). With this aim in mind, the challenge for future cosmological observations will therefore be that of reducing the f NL errors from this paper by at least one order of magnitude.

Isocurvature modes in the CMB bispectrum

This appendix contains the full paper [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF], except for the conclusions that were used as a summary in section 3.3.1. It was written in collaboration with David Langlois.

We study the angular bispectrum of local type arising from the (possibly correlated) combination of a primordial adiabatic mode with an isocurvature one. Generically, this bispectrum can be decomposed into six elementary bispectra. We estimate how precisely CMB data, including polarization, can enable us to measure or constrain the six corresponding amplitudes, considering separately the four types of isocurvature modes (CDM, baryon, neutrino density, neutrino velocity). Finally, we discuss how the model-independent constraints on the bispectrum can be combined to get constraints on the parameters of multiple-field inflation models.

3A.1 Introduction

Inflation is currently the best candidate to explain the generation of primordial perturbations, but many of its realizations remain compatible with the present data. One can hope that future data will enable us to find additional information in the primordial perturbations that could help to discriminate between the various mechanisms that can have taken place in the very early universe.

In this respect, it is important to test the adiabatic nature of the primordial perturbations. Since single-field inflation predicts only adiabatic perturbations, the detection of a fraction of an isocurvature mode in the cosmological data would rule out the simplest models of inflation. By contrast, multiple-field inflation could easily account for the presence of isocurvature modes [START_REF] Linde | Generation of Isothermal Density Perturbations in the Inflationary Universe[END_REF], which can even be correlated with the adiabatic component [START_REF] Langlois | Correlated adiabatic and isocurvature perturbations from double inflation[END_REF][START_REF] Langlois | Correlated mixtures of adiabatic and isocurvature cosmological perturbations[END_REF].

As shown in [START_REF] Bucher | The General primordial cosmic perturbation[END_REF], the most general primordial perturbation is a priori a linear combination of the usual adiabatic mode with four types of isocurvature modes, respectively the Cold Dark Matter (CDM), baryon, neutrino density and neutrino velocity isocurvature modes. The existence, and amplitude, of isocurvature modes depends on the details of the thermal history of the universe. Various scenarios that can lead to observable isocurvature modes have been discussed in the literature (double inflation [START_REF] Silk | Double Inflation[END_REF][START_REF] Polarski | Isocurvature perturbations in multiple inflationary models[END_REF][START_REF] Langlois | Correlated adiabatic and isocurvature perturbations from double inflation[END_REF], axions [START_REF] Seckel | Isothermal Density Perturbations in an Axion Dominated Inflationary Universe[END_REF], curvatons [START_REF] Linde | Nongaussian isocurvature perturbations from inflation[END_REF][START_REF] Lyth | Generating the curvature perturbation without an inflaton[END_REF]Moroi and[START_REF] Moroi | Effects of cosmological moduli fields on cosmic microwave background[END_REF]Lyth and Wands, 2003]).

In parallel to the possible presence of isocurvature modes, another property that could distinguish multiple-field models from single-field models is a detectable primordial non-Gaussianity of the local type. So far1 , the WMAP measurements of the CMB anisotropies [WMAP Collaboration, 2011] have set the present limit f local NL = 32 ± 21 (68 % CL) [and -10 < f local NL < 74 (95 % CL)] on the parameter f local NL that characterizes the amplitude of the simplest type of non-Gaussianity, namely the local shape. Similarly to isocurvature modes, a detection of local primordial non-Gaussianity would rule out all inflation models based on a single scalar field, since they generate only unobservably small local non-Gaussianities [Creminelli and Zaldarriaga, 2004b]. Scenarios with additional scalar fields, such as another inflaton (see e.g. [START_REF] Byrnes | Review of local non-Gaussianity from multi-field inflation[END_REF][START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]), a curvaton [Lyth et al., 2003] or a modulaton [START_REF] Dvali | A new mechanism for generating density perturbations from inflation[END_REF]Kofman, 2003;[START_REF] Langlois | Primordial perturbations and non-Gaussianities from modulated trapping[END_REF], which can produce detectable local non-Gaussianity, would then move to the front stage.

Isocurvature modes are usually investigated by constraining the power spectrum of primordial perturbations with CMB or large-scale structure data (see e.g. [START_REF] Bean | Constraining Isocurvature Initial Conditions with WMAP 3-year data[END_REF][START_REF] Sollom | Cold Dark Matter Isocurvature Perturbations: Constraints and Model Selection[END_REF][START_REF] Mangilli | Isocurvature modes and Baryon Acoustic Oscillations[END_REF][START_REF] Li | Cold Dark Matter Isocurvature Perturbations: Cosmological Constraints and Applications[END_REF][START_REF] Kawasaki | Differentiating CDM and Baryon Isocurvature Models with 21 cm Fluctuations[END_REF][START_REF] Kasanda | The sensitivity of BAO Dark Energy Constraints to General Isocurvature Perturbations[END_REF][START_REF] Valentino | Future constraints on neutrino isocurvature perturbations in the curvaton scenario[END_REF][START_REF] Valiviita | Constraints on scalar and tensor perturbations in phenomenological and two-field inflation models: Bayesian evidences for primordial isocurvature and tensor modes[END_REF]). However, isocurvature modes could also contribute to non-Gaussianities as discussed in several works [START_REF] Bartolo | Non-Gaussianity from inflation[END_REF][START_REF] Kawasaki | Non-Gaussianity from isocurvature perturbations[END_REF]Langlois et al., 2008b;[START_REF] Kawasaki | A General Analysis of Non-Gaussianity from Isocurvature Perturbations[END_REF][START_REF] Hikage | Limits on Isocurvature Perturbations from Non-Gaussianity in WMAP Temperature Anisotropy[END_REF][START_REF] Kawakami | Non-Gaussianity from Isocurvature Perturbations : Analysis of Trispectrum[END_REF][START_REF] Langlois | General treatment of isocurvature perturbations and non-Gaussianities[END_REF][START_REF] Langlois | Primordial Trispectrum from Isocurvature Fluctuations[END_REF][START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF]. Moreover, there exist models [START_REF] Langlois | General treatment of isocurvature perturbations and non-Gaussianities[END_REF] where isocurvature modes, while remaining a small fraction at the linear level, would dominate the non-Gaussianity. As shown in [START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF], these CDM isocurvature modes would be potentially detectable via their non-Gaussianity in the CMB data such as collected by Planck. Non-Gaussianity can thus be considered as a complementary probe of isocurvature modes.

In the present work, we refine and extend our previous analysis [START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF] by considering all types of isocurvature modes, not only the CDM isocurvature mode. We analyse the bispectrum generated by the adiabatic mode together with one of the four isocurvature modes. The total angular bispectrum can be decomposed into six distinct components: the usual purely adiabatic bispectrum, a purely isocurvature bispectrum, and four other bispectra that arise from the possible correlations between the adiabatic and isocurvature mode. Because these six bispectra have different shapes in -space, their amplitude can in principle be measured in the CMB data and we have computed, for each type of isocurvature mode, the associated 6 × 6 Fisher matrix to estimate what precision on these six parameters could be reached with the Planck data. We also show that the inclusion of polarization measurements improves the predicted precision of some isocurvature non-Gaussianity parameters significantly.

The elementary bispectra discussed above depend only on the adiabatic and isocurvature transfer functions and are thus independent of the details of the generation mechanism. Now, by assuming a specific class of inflationary models, one obtains particular relations between the six bispectra, which can be used as consistency relations for the model or to constrain the model parameters. We illustrate this in the context of curvaton-type models, generating adiabatic and CDM isocurvature perturbations.

The outline of the paper is the following. In the next section, we present the various isocurvature perturbations and discuss their impact on the CMB angular power spectrum. The following section is devoted to the angular bispectrum and its decomposition into six elementary bispectra. We then discuss the observational prospects to detect these elementary local bispectra in the future data, distinguishing the various isocurvature modes. Finally, we consider models where primordial perturbations are generated by an inflaton and a curvaton, and show that the amplitudes of all six bispectra depend on only two coefficients, which can be constrained from the data.

These various modes lead to different predictions for the CMB temperature and polarization. Let us consider for instance the temperature anisotropies, which can be decomposed into spherical harmonics:

∆T T = m a m Y m . (3A.7)
The multipole coefficients a m can be related linearly to any of the primordial modes. The precise correspondance can be computed numerically and written in the form a I m = 4π(-i)

d 3 k (2π) 3 X I (k) g I (k) Y * m ( k) , (3A.8)
where g I (k) is the transfer function associated with the corresponding primordial perturbation (g I (k) depends also on the various cosmological parameters).

For each type of perturbation, the angular power spectrum is thus given by

C I = a I m a I * m = 2 π ∞ 0 dk k 2 g I (k) 2 P I (k), (3A.9)
where we have introduced the primordial power spectrum P I (k) defined by

X I (k 1 )X I (k 2 ) ≡ (2π) 3 δ(k 1 + k 2 )P I (k 1 ) . (3A.10)
For our purposes, the crucial point is that the transfer functions associated with isocurvature perturbations are very different from the adiabatic transfer function. Moreover, each isocurvature mode leads to a specific signature that enables one to distinguish it from the other isocurvature modes. The only exception are the CDM and baryon isocurvature modes which give exactly the same pattern, up to the rescaling: .11) where the parameters Ω b and Ω c denote, as usual, the present energy density fractions, respectively for baryons and CDM (note however that these two modes can in principle be discriminated via other effects, see e.g. [START_REF] Holder | On Possible Variation in the Cosmological Baryon Fraction[END_REF][START_REF] Gordon | Forecasted 21 cm constraints on compensated isocurvature perturbations[END_REF][START_REF] Kawasaki | Differentiating CDM and Baryon Isocurvature Models with 21 cm Fluctuations[END_REF][START_REF] Grin | Compensated Isocurvature Perturbations and the Cosmic Microwave Background[END_REF]).

S b = ω bc S c , ω bc ≡ Ω b Ω c , ( 3A
More generally, when we also allow for possible correlations between the modes and include E-polarization, the angular power spectra are given by .12) where I and J label the isocurvature mode and α and α the polarization (i.e. either T (temperature) or E (polarization)). The primordial power spectrum P IJ (k) is now defined by

C IJ; αα = 2 π ∞ 0 dk k 2 g I;α (k)g J;α (k)P IJ (k), ( 3A
X I (k 1 )X J (k 2 ) ≡ (2π) 3 δ(k 1 + k 2 )P IJ (k 1 ) , (3A.13)
which generalizes the previous definition to include the presence of correlations between the modes, which corresponds to the situation where P IJ with different I and J does not vanish. All this is illustrated in Fig. 3A.1 and 3A.2, where we have plotted the angular power spectra for all the various modes (Fig. 3A.1) and the isocurvature cross power spectra where one of the components is adiabatic (Fig. 3A.2), assuming the same primordial power spectrum for all. As one can see from these figures, the CDM (and baryon) isocurvature mode decreases much faster with than the other modes. In fact it turns out that if one multiplies the CDM isocurvature power spectrum by 2 ( + 1) 2 instead of ( + 1), it falls off roughly in the same way as the other modes at large , as illustrated in Fig. 3A.3. This figure also nicely shows the relative phases of the acoustic peaks for the different modes. The amplitude and spectral index of the primordial power spectrum, as well as the cosmological parameters, on which the transfer functions depend, correspond to the WMAP7-only best-fit parameters (WMAP7-only best-fit parameters are used for all the figures and explicit computations in this paper). 3A.1, except that the horizontal axis is now linear, and that the CDM isocurvature spectrum is multiplied by 10 -42 ( + 1) 2 instead of ( + 1).

When the "primordial" perturbation is a superposition of several modes, the multipole coefficients depend on a linear combination of the "primordial" modes,

a m = 4π(-i) d 3 k (2π) 3 I X I (k) g I (k) Y * m ( k). (3A.14)
(Here we have once again omitted the polarization indices, as we will do in most of the equations of the paper, in order to improve readability.) As a result, the total angular power spectrum is now given by

C = a m a * m = I,J 2 π ∞ 0 dk k 2 g I (k)g J (k)P IJ (k). (3A.15)
We infer from CMB observations that the "primordial" perturbation is mainly of the adiabatic type. However, this does not preclude the presence, in addition to the adiabatic mode, of an isocurvature component, with a smaller amplitude. Precise measurement of the CMB fluctuations could lead to a detection of such an extra component, or at least put constraints on its amplitude. For example, constraints on the CDM isocurvature to adiabatic ratio, .16) based on the WMAP7+BAO+SN data, have been published for the uncorrelated and fully correlated cases (the impact of isocurvature perturbations on the observable power spectrum indeed depends on the correlation between adiabatic and isocurvature perturbations, as illustrated in [START_REF] Langlois | Correlated mixtures of adiabatic and isocurvature cosmological perturbations[END_REF]). In terms of the parameter a ≡ α/(1 + α), the limits given in [WMAP Collaboration, 2011] are 2 a 0 < 0.064 (95%CL), a 1 < 0.0037 (95%CL) , (3A.17) respectively for the uncorrelated case and for the fully correlated case.

α = P Sc P ζ , ( 3A

3A.3 Generalized angular bispectra

In this section, we turn to non-Gaussianities, including both adiabatic and isocurvature modes.

3A.3.1 Reduced and angular-averaged bispectra

The angular bispectrum corresponds to the three-point function of the multipole coefficients:

B m 1 m 2 m 3 1 2 3 ≡ a 1 m 1 a 2 m 2 a 3 m 3 . (3A.18)
Substituting the expression (3A.14) into the angular bispectrum, one can write it in the form

B m 1 m 2 m 3 1 2 3 = G m 1 m 2 m 3 1 2 3 b 1 2 3 , (3A.19)
where the first, purely geometrical, factor is the Gaunt integral

G m 1 m 2 m 3 1 2 3 ≡ dΩ Y 1 m 1 (Ω) Y 2 m 2 (Ω) Y 3 m 3 (Ω) . (3A.20)
The second factor, usually called the reduced bispectrum, is given by 3A.21) which depends on the bispectra of the primordial X I :

b 1 2 3 = I,J,K 2 π 3 3 i=1 k 2 i dk i g I 1 (k 1 )g J 2 (k 2 )g K 3 (k 3 )B IJK (k 1 , k 2 , k 3 ) × ∞ 0 r 2 drj 1 (k 1 r)j 2 (k 2 r)j 3 (k 3 r) , (
X I (k 1 )X J (k 2 )X K (k 3 ) ≡ (2π) 3 δ(Σ i k i )B IJK (k 1 , k 2 , k 3 ) . (3A.22)
The reduced bispectrum (3A.21) is the sum of several contributions, corresponding to different values of the indices I, J and K that vary over the range of modes included in the primordial perturbations. This expression thus generalizes the purely adiabatic expression given in [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF].

It is also useful to define the angle-averaged bispectrum

B 1 2 3 ≡ m 1 ,m 2 ,m 3 1 2 3 m 1 m 2 m 3 B m 1 m 2 m 3 1 2 3 = (2 1 + 1)(2 2 + 1)(2 3 + 1) 4π 1 2 3 0 0 0 b 1 2 3 , (3A.23)
where the second relation is obtained by substituting (3A.19) and by using the identity

m 1 ,m 2 ,m 3 1 2 3 m 1 m 2 m 3 G m 1 m 2 m 3 1 2 3 = (2 1 + 1)(2 2 + 1)(2 3 + 1) 4π 1 2 3 0 0 0 . (3A.24)

3A.3.2 Non-Gaussianities of local type

To proceed further, one must make some assumption about the functional dependence of the bispectra B IJK (k 1 , k 2 , k 3 ) in Fourier space. This corresponds to the so-called "shape" of the bispectrum [Babich et al., 2004], which has been discussed at length in the literature in the purely adiabatic case where the B IJK reduce to the single bispectrum B ζζζ . In the present work, we consider the simplest form of non-Gaussianity, namely the local shape. In the purely adiabatic case, it is defined by (3A.25) in physical space, where the factor -3/5 appears because f NL was originally defined with respect to the gravitational potential Φ = -(3/5)ζ, instead of ζ. The subscript L here denotes the linear part of the perturbation, which is assumed to be Gaussian. In Fourier space, this leads to a bispectrum that depends quadratically on the power spectrum:

ζ(x) = ζ L (x) - 3 5 f NL ζ L (x) 2 -ζ L 2 ,
B ζζζ = fNL [P ζ (k 2 )P ζ (k 3 ) + P ζ (k 3 )P ζ (k 1 ) + P ζ (k 1 )P ζ (k 2 )] , fNL ≡ - 6 5 f NL . (3A.26)
In the present context where we assume the presence of an isocurvature mode in addition to the dominant adiabatic mode, the simplest extension of (3A.26) is to assume that all the generalized bispectra B IJK (k 1 , k 2 , k 3 ) can be written as the sum of terms quadratic in the adiabatic power spectrum (note that this implicitly assumes that the power spectrum of the isocurvature mode and the isocurvature cross power spectrum, if non-vanishing, have the same spectral dependence as the adiabatic one). However, in contrast with (3A.26) where all terms share the same coefficient, as a consequence of the invariance of the bispectrum under the exchange of momenta, this is no longer the case for the generalized bispectra when the indices I, J and K are not identical. What the definition (3A.22) implies is simply that the bispectra are left unchanged under the simultaneous change of two indices and the corresponding momenta (e.g. I and J, k 1 and k 2 ). This leads to the decomposition To keep track of this symmetry, we separate the first index from the last two indices with a comma.

B IJK (k 1 , k 2 , k 3 ) = f I,JK NL P ζ (k 2 )P ζ (k 3 ) + f J,KI NL P ζ (k 1 )P ζ (k 3 ) + f K,IJ NL P ζ (k 1 )P ζ (k 2 ) , ( 3A

3A.3.3 Link with multiple-field inflation

It is instructive to show that our definition of generalized local non-Gaussianity is the natural outcome of a generic model of multiple-field inflation. Indeed, allowing for several light degrees of freedom during inflation, one can relate, in a very generic way, the "primordial" perturbations X I (defined during the standard radiation era) to the fluctuations of light primordial fields φ a , generated at Hubble crossing during inflation, so that one can write, up to second order, .29) where the δφ a can usually be treated as independent quasi-Gaussian fluctuations, i.e.

X I = N I a δφ a + 1 2 N I ab δφ a δφ b + . . . ( 3A
δφ a (k) δφ b (k ) = (2π) 3 δ ab P δφ (k) δ(k + k ) , P δφ (k) = 2π 2 k -3 H * 2π 2 , (3A.30)
where a star denotes Hubble crossing time. The relation (3A.29) is very general, and all the details of the inflationary model are embodied by the coefficients N I a and N I ab . Substituting (3A.29) into (3A.22) and using Wick's theorem, one finds that the bispectra B IJK can be expressed in the form B IJK (k 1 , k 2 , k 3 ) = λ I,JK P δφ (k 2 )P δφ (k 3 )+λ J,KI P δφ (k 1 )P δφ (k 3 )+λ where it is implicitly assumed that the coefficients N I a are weakly time dependent so that the scale dependence of A 2 can be neglected. One can notice that the first index is related to the second-order terms in the decomposition (3A.29), while the last two indices come from the first-order terms.

Except in the last section devoted to a specific class of early universe models, all our considerations will simply follow from our assumption (3A.27) and thus apply to any model leading to this local form, whether based on inflation or not.

3A.3.4 Decomposition of the angular bispectrum

After substitution of (3A.27) into (3A.21), the reduced bispectrum can finally be written as

b 1 2 3 = I,J,K f I,JK NL b I,JK 1 2 3 , (3A.35)
where each contribution is of the form While we have omitted the polarization indices, the reader should keep in mind that each transfer function carries, in addition to the isocurvature index, a polarization index, and hence the same is true for α and β . As a consequence, the bispectrum has three polarization indices that we do not show. The purely adiabatic bispectrum, usually the only one considered, can be expressed as

3 b I,JK 1 2 3 = 3 ∞ 0 r 2 dr α I ( 1 (r)β J 2 (r)β K 3 ) (r), ( 3A
b ζ,ζζ 1 2 3 = 3 ∞ 0 r 2 drα ζ ( 1 (r)β ζ 2 (r)β ζ 3 ) (r) = ∞ 0 r 2 dr α ζ 1 (r)β ζ 2 (r)β ζ 3 (r) + α ζ 2 (r)β ζ 3 (r)β ζ 1 (r) + α ζ 3 (r)β ζ 1 (r)β ζ 2 (r) . (3A.39)
The functions α ζ (r) and β ζ (r) are plotted respectively in Fig. 3A.4 and Fig. 3A.5. We have considered both the temperature and polarization transfer functions. The radial distance r can be expressed as the speed of light times the difference in conformal time between now and the time in the past we consider. In the figures we have chosen some sample values around the time of last scattering, which corresponds to r ≈ 14100 Mpc using the WMAP7-only best-fit parameters.

Since we consider local non-Gaussianity, the main contribution to the bispectrum comes from the squeezed limit, i.e. one of the multipole numbers is much smaller than the other two. To simplify the analysis, let us assume that 1 2 = 3 ≡ L. One finds that, in this limit,

(r) β ζ 2 (r)β S 3 (r) + β ζ 3 (r)β S 2 (r) (3A.41) +α ζ 2 (r) β ζ 3 (r)β S 1 (r) + β ζ 1 (r)β S 3 (r) + α ζ 3 (r) β ζ 1 (r)β S 2 (r) + β ζ 2 (r)β S 1 (r) .
Since b ζ,ζS 1 2 3 and b ζ,Sζ 1 2 3 cannot be distinguished, we will always consider the sum of the two, and similarly for b S,ζS 1 2 3 and b S,Sζ 1 2 3 . In summary, after integration over r of these various combinations of α and β functions, we obtain six independent bispectra, for each type of isocurvature mode. To illustrate the typical angular dependence of these bispectra, we have plotted them as functions of 3 , for fixed values of 1 and 2 , respectively in the CDM isocurvature case (Fig. 3A.12), in the neutrino density isocurvature case (Fig. 3A.13) and in the neutrino velocity case (Fig. 3A.14). We plot only the pure temperature (TTT) and pure polarization (EEE) bispectra, but of course one also has all the polarization cross bispectra. As mentioned before, the (ζ, ζS) curve corresponds to twice b ζ,ζS 1 2 3 since we consider the sum of b ζ,ζS 1 2 3 and b ζ,Sζ 1 2 3 , which cannot be distinguished. The same applies to the (S, ζS) curve. As usual, the bispectra for the baryon isocurvature mode are deduced from the CDM bispectra by the appropriate rescalings. Bispectra neutrino density Bispectra neutrino velocity

3A.4 Observational prospects

As we will see in the next section, one can envisage early universe scenarios that generate significant isocurvature non-Gaussianity, which could dominate the purely adiabatic component, even if the adiabatic mode is dominant in the power spectrum as required by observations. This is why it is important to assess how precisely one can hope to measure and to discriminate the various isocurvature bispectra in the future.

The most general analysis would require to consider simultaneously all five possible modes, which corresponds to a total of n 2 (n + 1)/2 = 75 coefficients, taking into account the symmetry (3A.28). In order to simplify our analysis, we will consider separately the various isocurvature perturbations. In other words, we will assume that the primordial perturbation is the superposition of a dominant adiabatic mode and of a single isocurvature mode. In this case, the total bispectrum is characterized by six parameters, which we now denote f

(i) , b 1 2 3 = f ζ,ζζ b ζ,ζζ 1 2 3 + 2 f ζ,ζS b ζ,ζS 1 2 3 + f ζ,SS b ζ,SS 1 2 3 + f S,ζζ b S,ζζ 1 2 3 + 2 f S,ζS b S,ζS 1 2 3 + f S,SS b S,SS 1 2 3 = (i) f (i) b (i) 1 2 3 , (3A.42)
where the index i varies between 1 to 6, following the order indicated in the upper line. Note that, because of the factor 2 in front of f ζ,ζS and f S,ζS , we define b

(2)

1 2 3 ≡ 2b ζ,ζS 1 2 3 and b (5) 1 2 3 ≡ 2b S,ζS
1 2 3 whereas there is no such factor 2 for the other terms.

3A.4.1 The Fisher matrix

To estimate these six parameters, given some data set, the usual procedure is to minimize

χ 2 = (B obs - i f (i) B (i) ), (B obs - i f (i) B (i) ) . (3A.43)
For an ideal experiment (no noise and no effects due to the beam size) without polarization, the scalar product is defined by

B, B ≡ 1 ≤ 2 ≤ 3 B 1 2 3 B 1 2 3 σ 2 1 2 3 . (3A.44)
The bispectrum variance in that case is given by

σ 2 1 2 3 ≡ B 2 1 2 3 -B 1 2 3 2 ≈ ∆ 1 2 3 C 1 C 2 C 3 (3A.45)
in the approximation of weak non-Gaussianity, where

∆ 1 2 3 ≡ 1 + δ 1 2 + δ 2 3 + δ 3 1 + 2 δ 1 2 δ 2 3 . (3A.46)
The best estimates for the parameters are thus obtained by solving j B (i) , B (j) f (j) = B (i) , B obs , (3A.47) while the statistical error on the parameters is deduced from the second-order derivatives of χ 2 , which define the Fisher matrix, given in our case by j) . (3A.48)

F ij ≡ B (i) , B (
The Fisher matrix is a symmetric matrix, which can be determined by computing the 21 different scalar products between the six elementary bispectra.

(ζ, ζζ)

(ζ, ζS) (ζ, SS) (S, ζζ) (S, ζS)
(S, SS) 3.9 (2.5) × 10 -2 4.5 (3.6) × 10 -2 2.3 (2.1) × 10 -4 2.4 (1.6) × 10 -4 6.9 (4.3) × 10 -4 5.3 (3.1) × 10 -4 -7.1 (6.0) × 10 -2 5.3 (3.8) × 10 -4 3.8 (2.1) × 10 -4 11 (7.4) × 10 -4 8.8 (5.5) × 10 -4 --28 (6.4) × 10 -5 16 (3.7) × 10 -5 33 (9.5) × 10 -5 11 (5.0) × 10 -5 ---15 (3.0) × 10 -5 22 (5.8) × 10 -5 7.5

(3.2) × 10 -5 - - - - 5.1 (1.6) × 10 -4 2.4 (1.0) × 10 -4 - - - - - 21 (8.3) × 10 -5
Table 3A.1: Fisher matrix for the CDM isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the Fisher matrix components obtained without including the polarization.

For a real experiment, and if E-polarization is included as well, the above equations remain valid, except that the definition of the scalar product has to be replaced by a more complicated expression (see e.g. [START_REF] Yadav | Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background[END_REF]):

B (i) , B (j) ≡ αβγα β γ 1 ≤ 2 ≤ 3 1 ∆ 1 2 3 B (i) αβγ 1 2 3 (C -1 1 ) αα (C -1 2 ) ββ (C -1 3 ) γγ B (j) α β γ 1 2 3 , (3A.49)
where α, β, γ, α , β , γ are polarization indices taking the two values T and E. The covariance matrix C (a matrix in polarization space) is given by

C = b 2 C T T + N T b 2 C T E b 2 C T E b 2 C EE + N E , (3A.50)
where b is the beam function and N the noise power spectrum. We assumed the same beam function for temperature and polarization detectors, as well as no correlated noise, but the generalization is straightforward. In the calculation of the covariance matrix we only take the adiabatic power spectrum, since from observations we know that the isocurvature contribution to the power spectrum must be very small. For each type of isocurvature mode, we have computed the corresponding Fisher matrix by extending the numerical code described in [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF] to include isocurvature modes and E-polarization, according to the expressions presented above. We have taken into account the noise characteristics of the Planck satellite [Planck scientific programme], using only the 100, 143, and 217 GHz channels, combined in quadrature. Our computation goes up to max = 2500 and uses the WMAP-only 7-year best-fit cosmological parameters [WMAP Collaboration, 2011].

From the Fisher matrix, one can compute the statistical uncertainty on each of the parameters: .51) This takes into account the correlations between the various bispectra. By contrast, if one assumes that the data contain only a single elementary bispectrum, for example the purely adiabatic one, then the corresponding statistical error is .52) One can also determine the correlations between any two bispectra:

∆ f i = (F -1 ) ii . ( 3A
∆ f i = 1 √ F ii (single parameter) . ( 3A
C ij = (F -1 ) ij (F -1 ) ii (F -1 ) jj . (3A.53)

3A.4.2 CDM isocurvature mode

Our results for this mode have already been presented elsewhere [START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF]], but we discuss here in more detail the peculiarities of the corresponding Fisher matrix, which is

(ζ, ζζ) (ζ, ζS) (ζ, SS) (S, ζζ) (S, ζS) (S, SS) 1.
-0.84 (0.92) 0.18 (0.21) 0.003 (-0.12) -0.15 (0.17) 0.13 (0.18) -1.

-0.15 (0.18) -0.007 (+0.13) 0.13 (0.14) -0.16 (0.16) --1.

-0.069 (+0.24) -0.80 (0.97) 0.58 (0.92) ---1.

-0.42 (0.43) 0.29 (0.39) ----1.

-0.82 (0.98) -----1.

Table 3A.2: Correlation matrix for the CDM isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the correlations obtained without including the polarization (the absence of sign in the parentheses means that it is unchanged with respect to the value taking into account the polarization).

given in Table 3A.1. One can immediately notice the intriguing fact that the coefficients of the upper left 2 × 2 submatrix, corresponding to the purely adiabatic component and the correlated (ζ, ζS) component, are typically two orders of magnitude larger than all the other coefficients.

The correlation matrix, defined in (3A.53) and given in Table 3A.2 shows that the first two bispectra are strongly (anti-)correlated while their correlation with the four other bispectra is weak.

From this Fisher matrix, one finds that the 68 % error on the parameters f i is given by4 .6, 7.1, 160, 150, 180, 140} . (3A.54) For ease of readability, we have written 160 instead of 1.6 • 10 2 , etc., but we are not claiming more than two digits of significance. We also remind the reader that in the purely adiabatic case, our f 1 , i.e. the (ζ, ζζ) component, is -6/5 times the standard f NL . One sees that the first two uncertainties are typically one order of magnitude smaller than the last four.

∆ f i = (F -1 ) ii = {9
It is also interesting to estimate how much the inclusion of the polarization data in the analysis improves the precision of the non-linear parameters. The components of the Fisher matrix when the polarization is not taken into account can be read between the parentheses in Table 3A.1. One notices that whereas the coefficients of the first two lines are reduced by a factor inferior to two, the other coefficients are significantly suppressed when one removes the polarization data. As a consequence, one finds that the uncertainties on the parameters without polarization, given by ∆ f i = (F -1 ) ii = {17, 11, 980, 390, 1060, 700} (no polarization), (3A.55) increase by less than a factor two for the first two parameters, whereas the increase is much bigger for the four other ones. The evolution of these uncertainties as a function of the cut-off max is shown in Fig. 3A.15, both for the case where temperature and E-polarization data are used and for the case where only temperature data is included. One can see that the curves for temperature-only typically look bumpier than the curves that include polarization as well. This can be explained as follows. First, unlike the power spectrum, the bispectrum is an alternating function, so that for certain regions in 1 2 3 space it is zero or close to zero, and the contribution to the determination of f NL , which is quadratic in the bispectrum, is then negligible in these regions. Second, as one can see for example in Fig. 3A.12, the acoustic peaks of the polarization bispectrum are out of phase with the ones of the temperature one, so that including polarization neatly fills in the holes in 1 2 3 space and leads to a smoother determination of f NL , as first pointed out by [START_REF] Komatsu | Measuring primordial non-Gaussianity in the cosmic microwave background[END_REF]. Our results can be understood by the following analysis in the squeezed limit (based on [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF]), assuming that 1 ≡ 2 ≈ 3 ≡ L. In this limit, the bispectra defined in (3A.36) can be decomposed as

b I,JK LL = ∞ 0 r 2 dr α I (r)β J L (r)β K L (r) + α I L (r)β J (r)β K L (r) + α I L (r)β J L (r)β K (r) . (3A.56)
The first term is subdominant, since, like the power spectrum, |β L | decreases as 1/(L(L + 1)) (or even faster as 1/(L 2 (L + 1) 2 ) for large L), as can be seen for example in Fig. 3A.7. The last term, for instance, is explicitly given by

2 π 3 k 2 1 dk 1 k 2 2 dk 2 k 2 3 dk 3 g I L (k 1 )g J L (k 2 )g K (k 3 ) P ζ (k 2 )P ζ (k 3 ) ∞ 0 r 2 drj L (k 1 r)j L (k 2 r)j (k 3 r) ,
(3A.57) where the last Bessel function oscillates slowly while the first two oscillate very rapidly. This leads to a cancellation of the radial integral unless k 1 is very close to k 2 . We find that the above expression can thus be approximated by

2 π 3 k 2 1 dk 1 k 2 2 dk 2 k 2 3 dk 3 g I L (k 1 )g J L (k 2 )g K (k 3 )P ζ (k 2 )P ζ (k 3 ) 5δ(k 1 -k 2 ) k 1 k 2 ∞ 0 dr r j (k 3 r) ≈ 2 π 2 k 2 dk g I L (k)g J L (k)P ζ (k) 10 π k 2 3 dk 3 g K (k 3 )P ζ (k 3 ) ∞ 0 dr r j (k 3 r) ≡ G IJ (L) H K ( ). (3A.58)
The δ(k 1k 2 ) is explained above, but together with the 1/(k 1 k 2 ) also motivated by the closure relation for spherical Bessel functions,

∞ 0 r 2 j L (k 1 r)j L (k 2 r)dr = πδ(k 1 -k 2 )/(2k 1 k 2 )
. The 1/r follows from a dimensional analysis, and the 5 has been determined heuristically by comparing with the exact bispectrum: the ratio is approximately 5 and only weakly dependent on the small . The full squeezed bispectrum (3A.56) is thus approximated by

b I,JK LL ≈ G IJ (L) H K ( ) + G IK (L) H J ( ) . (3A.59)
Assuming all primodial power spectra to be equal, the functions G IJ are the angular (cross) power spectra plotted in Fig. 3A.1 and 3A.2. The function H K ( ) is simply an integral over Figure 3A.17: The absolute value of the squeezed bispectrum (multiplied by L(L + 1)) with 1 = 2 and 2 = 3 ≡ L plotted as a function of L. Both the exact bispectrum and (twice) the approximation given in (3A.58) are plotted, for the purely adiabatic mode and the purely CDM isocurvature mode. The figure on the left shows the temperature-only TTT bispectra, the one on the right the pure polarization EEE bispectra. Note that the configuration is only squeezed in the case of large L; for small L the approximation should not be used.

β K (r):

H K ( ) = 5 ∞ 0 dr r β K (r) (3A.60)
and is shown for small in Fig. 3A.16. The good agreement of our approximation with the exact squeezed bispectrum is shown in Fig. 3A.17. In the squeezed limit, one thus finds that only the first two elementary bispectra, (ζ, ζζ) and (ζ, ζS), depend on G ζζ (L). The four others depend on G ζS (L) and/or G SS (L). The large L limit of G ζS (L) and G SS (L) are strongly suppressed with respect to G ζζ (L), which explains why the uncertainty on the first two non-Gaussianity parameters can be reduced by probing high multipoles (the bispectrum there is still sufficiently large compared to the noise) while the uncertainty on the four other ones saturates as shown in Fig. 3A.15. One can even understand why the curve for the (ζ, ζS) mode is below the one for (ζ, ζζ): their dominant terms both depend on the same G ζζ (L), but different H( ), and |H S ( )| > |H ζ ( )|. Finally, one can understand why including polarization helps much more for the uncertainty on e.g. the (S, SS) mode than for the (ζ, ζζ) mode. As one can see from Fig. 3A.15, it is in particular in the region 50 ≤ ≤ 200 that the distance between the two (S, SS) curves increases compared to the distance between the two (ζ, ζζ) curves. A quick look at Fig. 3A.1 shows that in that region of multipole space the TT CDM isocurvature power spectrum (i.e. G SS (L)) becomes very small compared to the TT adiabatic spectrum (i.e. G ζζ (L)), but the EE CDM isocurvature spectrum still remains comparable to the EE adiabatic one.

It is also instructive to compare (3A.54) with the uncertainties 3.7, 60, 82, 44, 69} (single parameter) (3A.61) obtained by ignoring the correlations, or, equivalently, by assuming that only one parameter is nonzero. In particular, the contamination of the purely adiabatic signal by the other shapes induces an increase of the uncertainty, but only by a factor 2, which is rather moderate.

∆ f i = 1/ F ii = {5.0,
Assuming that the adiabatic and isocurvature modes are uncorrelated implies that only the purely adiabatic and isocurvature bispectra are relevant. The corresponding two-parameter Fisher matrix, which is the submatrix of F (ij) with entries F 11 , F 66 and F 16 , leads to uncertainties on f (1) and f (6) that are almost identical to the corresponding single-parameter errors.

Finally, let us note that if the observed bispectrum is mainly purely isocurvature with amplitude f (6) , a naive analysis using only the purely adiabatic estimator would lead to an apparent adiabatic coefficient 3A.62) thus hiding the isocurvature signal with larger amplitude.

f (1) = F 16 F 11 f (6) 10 -2 f (6) , (

3A.4.3 Baryon isocurvature mode

The Fisher matrix for the baryon isocurvature mode can be easily deduced from the CDM Fisher matrix. Indeed, as discussed earlier, the CDM and baryon isocurvature transfer functions are identical up to a rescaling by ω bc introduced in (3A.11). Consequently, the α and β functions are simply rescaled:

α S b (r) = ω bc α Sc (r), β S b (r) = ω bc β Sc (r) . (3A.63)
The rescaling of the various bispectra will thus depend on the number of S indices, i.e.

b I,JK 1 2 3 (S b ) = (ω bc ) p b I,JK 1 2 3 (S c ), (3A.64)
where the power p is the number of S among the indices {IJK}. In summary, all coefficients of the baryon isocurvature Fisher matrix can be deduced from Table 3A.1 by using the rescaling

F S b ij = N i N j F Sc ij (no summation), N i = {1, ω bc , ω 2 bc , ω bc , ω 2 bc , ω 3 bc }, (3A.65)
where, in our computation, ω bc = 0.2036. The parameter uncertainties can also be deduced from the CDM results via the rescalings 6, 35, 4000, 720, 4300, 16600} . (3A.66) Except for the purely adiabatic coefficient, we thus find that the uncertainties on all the other coefficients are significantly larger than the uncertainties obtained in (3A.54) in the CDM case, simply because the elementary bispectra have a smaller amplitude than their CDM counterparts. By contrast, the correlation matrix, which is independent of the normalization of the bispectra, is exactly the same as in the CDM case.

N i : ∆ f i (S b ) = ∆ f i (S c )/N i . One thus obtains: ∆ f i = {9.
(ζ, ζζ)

(ζ, ζS) (ζ, SS) (S, ζζ) (S, ζS)
(S, SS) 3.9 (2.5) × 10 -2 3.6 (2.6) × 10 -2 5.6 (4.1) × 10 -3 7.9 (5.1) × 10 -3 8.8 (6.1) × 10 -3 2.2 (1.6) × 10 -3 -3.8 (2.9) × 10 -2 6.3 (4.8) × 10 -3 7.4 (5.2) × 10 -3 9.2 (6.7) × 10 -3 The evolution of the uncertainties as a function of max is shown in Fig. 3A.18. As in the CDM case, the (ζ, ζζ) and (ζ, ζS) non-Gaussianity parameters can be determined more accurately than the other four, although the difference is not as big as for CDM. Unlike for CDM, all parameters gain about the same from the inclusion of polarization, and all uncertainties continue to decrease when higher multipoles are probed, since the neutrino density isocurvature power spectrum does not decrease as steeply as the CDM isocurvature one. The correlation matrix is given in Table 3A.4. If one assumes the parameters to be independent, one finds ∆ f i = {5.0, 5.1, 30, 24, 20, 69} (single parameter). (3A.69)

One sees that the increase of the uncertainties due to the correlations is much more important here than for CDM, due to the larger correlations between the various modes.

3A.4.5 Neutrino velocity isocurvature mode

For a neutrino velocity isocurvature mode, we have obtained the Fisher matrix in Table 3A.5. One notices that, including polarization, all entries are roughly of the same order of magnitude, but without polarization, they vary a lot. The corresponding uncertainties on the six non-Gaussianity parameters are (taking into account the correlations) Table 3A.6: Correlation matrix for the neutrino velocity isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the correlations obtained without including the polarization (the absence of sign in the parentheses means that it is unchanged with respect to the value taking into account the polarization).

∆ f i = {25,
The evolution of the uncertainties as a function of max is shown in Fig. 3A.19. One sees that in this case the difference between the first two and the other four uncertainties (when including polarization) is much smaller than for CDM or neutrino density isocurvature (a factor of about 3 compared to a factor of about 15 in the CDM case). In particular, the latter four can be determined more accurately than in the case of CDM or neutrino density isocurvature. However, the improvement due to polarization is much more important than in the neutrino density case, in particular for the (ζ, ζS), the (ζ, SS) and the (S, SS) parameters. One can understand why the (ζ, ζS) mode, for example, gains much more from polarization than the (ζ, ζζ) mode with a similar argument as the one presented for CDM isocurvature. The dominant contributions to these modes both depend on G ζζ (L) (defined in (3A.58)), but for (ζ, ζζ) this is multiplied by H ζ ( ) and for (ζ, ζS) by H S ( ). And as one can see from Fig. 3A.16, the ratio H S /H ζ for neutrino velocity isocurvature increases enormously when one passes from temperature to polarization (remember that it is the lowest values of that contribute most to the squeezed configuration).

The correlation matrix is given in Table 3A.6. If one assumes the parameters to be independent, one finds ∆ f i = {5.0, 3.7, 11, 13, 8.7, 19} (single parameter). (3A.72)

Hence one sees that the correlations in the case of neutrino velocity isocurvature are more important than for CDM, but less than for neutrino density.

3A.5 Constraints on early universe models

In the previous section, we have studied how to obtain constraints on the six non-linearity coefficient f (i) N L without assuming any particular relation between them. In the context of an early universe model, or in a class of models, one can go further and use the results of the previous section to obtain some constraints on the parameters of the model itself.

3A.5.1 General analysis

As we have seen earlier, isocurvature perturbations require the existence of at least two degrees of freedom in the early universe. So, for simplicity, let us focus on models with two scalar fields, φ and σ, such that isocurvature perturbations are generated only by the fluctuations of σ and all non-linearities are also dominated by their contribution from σ. This means that we have Using the general expressions (3A.32-3A.34), one easily finds that the coefficients are interdependent and can be written in the form

ζ = N ζ φ δφ + N ζ σ δσ + 1 2 N ζ σσ δσ 2 , S = N S σ δσ + 1 2 N S σσ δσ 2 . ( 3A
f I,ζζ N L = µ I Ξ 2 , (3A.74) f I,ζS N L = ε ζS µ I α 1/2 Ξ 3/2 , (3A.75) f I,SS N L = µ I α Ξ, (3A.76)
where we have introduced the contribution of σ in the adiabatic power spectrum, .77) and the isocurvature to adiabatic ratio,

Ξ ≡ (N ζ σ ) 2 (N ζ φ ) 2 + (N ζ σ ) 2 , ( 3A
α ≡ (N S σ ) 2 (N ζ φ ) 2 + (N ζ σ ) 2 . (3A.78)
Note that the extraction of an isocurvature component in the power spectrum would fix α while, in principle, Ξ could be determined from observations by measuring both the bispectrum and the trispectrum coefficients, since they satisfy consistency relations [START_REF] Langlois | Primordial Trispectrum from Isocurvature Fluctuations[END_REF] For given values of α and Ξ, the uncertainties on the two parameters µ ζ and µ S are determined from the "projected" Fisher matrix

F IJ = i,j π (i)I F ij π (j)J (3A.80) with π (i)ζ = {Ξ 2 , ε ζS α 1/2 Ξ 3/2 , αΞ, 0, 0, 0}, π (i)S = {0, 0, 0, Ξ 2 , ε ζS α 1/2 Ξ 3/2 , αΞ} . (3A.81)
From this 2 × 2 Fisher matrix, one can easily deduce the expected uncertainties on the two parameters µ ζ and µ S , by using the analog of (3A.51).

3A.5.2 Illustrative example

In [START_REF] Langlois | General treatment of isocurvature perturbations and non-Gaussianities[END_REF][START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF] we have studied a class of models which produces perturbations of the above type. In these models, σ is a curvaton which decays into radiation and CDM. Since part of the CDM can have been produced before the decay of the curvaton, one can introduce as a parameter the fraction of CDM created by the decay as .82) where the Ω's represent the relative abundances just before the decay and γ c is the fraction of the curvaton energy density transferred into CDM. The second relevant parameter, .83) quantifies the transfer between the pre-decay and post-decay perturbations [START_REF] Langlois | General treatment of isocurvature perturbations and non-Gaussianities[END_REF] (one finds ζ after γ = (1r)ζ before γ at the linear level). As shown in [START_REF] Langlois | General treatment of isocurvature perturbations and non-Gaussianities[END_REF][START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF], one can derive the "primordial" perturbations ζ and S as expansions, up to second order, in terms of δσ, which yield the coefficients N I σ and N I σσ . Using these results and assuming r 1, one obtains ) and α = 9 Ξ(f /r) 2 . The coefficients f S,IJ N L , which depend on µ S , are thus enhanced with respect to the coefficients f ζ,IJ N L in the latter case. As discussed in more detail in [START_REF] Langlois | Hunting for Isocurvature Modes in the CMB non-Gaussianities[END_REF], the above results show that a small isocurvature fraction in the power spectrum can, for certain parameter values, be compatible with a dominantly isocurvature bispectrum detectable by Planck (e.g. α 10 -2 and r f c 10 -8 yields f S,SS NL 5 × 10 3 ).

f ≡ γ c Ω σ Ω c + γ c Ω σ , ( 3A
r ≡ 3(1 -γ c ) Ω σ (1 -γ c Ω σ )(4 -Ω σ ) , ( 3A
µ ζ = 3 2r , µ S = 9 2r 2 (f (1 -2f ) -r) , (

3B.1 Introduction

The exploration of the CMB as a source of high-precision information on cosmology and as the best (if somewhat opaque) window on the primordial universe started in earnest with the first WMAP release in 2003 [WMAP Collaboration, 2003b]. The Planck satellite with its three releases in 2013, 2015, and 2018 [Planck 2013 I;Planck 2015 I;Planck 2018 I] raised the game to unprecedented levels of precision. Still, the amount of information we have about the primordial universe, and in particular on the period of inflation, remains very limited. Apart from looking for possible new observables, it is also very important to work as much as possible on the observables that we do have, from both ends: from the observational side to improve estimators and data cleaning to get as precise a value as we can, and from the theoretical side to improve (inflationary) predictions so that we can draw the theoretical consequences from the observations. Microwave observations are contaminated by astrophysical foregrounds, which can be extra-galactic or galactic in origin. In order to improve the quality of the data used for the CMB analyses, these foregrounds are first removed as much as possible by component separation methods, which produce so-called cleaned CMB maps, although these still contain foreground residuals at some level. Some of the most important inflationary observables are the non-Gaussianity parameters f NL . Non-Gaussianity means that not all information about the CMB is contained in its twopoint correlation function / power spectrum, as would be the case for a Gaussian distribution. The lowest-order deviation from Gaussianity will lead to a non-zero bispectrum, the Fourier transform (or spherical harmonic transform on the celestial sphere) of the three-point correlation function. Standard single-field slow-roll inflation produces an unobservably small non-Gaussian signal [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF][START_REF] Acquaviva | Second order cosmological perturbations from inflation[END_REF], but other inflation models predict larger amounts. Moreover, different types of models predict differently shaped bispectra, and we can look for the presence of any of them. The amplitude of each is parametrized by its own f NL parameter. Some of the most important bispectrum templates are the local shape [START_REF] Gangui | The Three point correlation function of the cosmic microwave background in inflationary models[END_REF] (typically produced by multiple-field inflation models) and the equilateral [START_REF] Creminelli | Limits on non-Gaussianities from WMAP data[END_REF] and orthogonal [START_REF] Senatore | Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data[END_REF] shapes (typically produced by single-field models with non-standard kinetic terms), see [START_REF] Chen | Primordial Non-Gaussianities from Inflation Models[END_REF] for a review. So far there is no detection of a primordial f NL value inconsistent with zero, but this null detection with precise error bars has led to the exclusion of inflation models that predict too much non-Gaussianity [Planck 2013 XXIV;Planck 2015 XVII].

In order to extract any information about primordial bispectral non-Gaussianity from the CMB data, given that this information is primarily parametrized in the form of the f NL amplitude parameters of the different bispectrum shapes, we need an estimator for f NL . This estimator should be unbiased as well as optimal (or effectively optimal given the accuracy of the experiment), which means it has the smallest variance theoretically possible, to extract the primordial f NL from real data contaminated by astrophysical foreground residuals and experimental effects like noise. In addition the estimator implementation should be fast enough to make data analysis possible in practice. Three such estimators were developed and used for the official Planck analysis: the KSW estimator [START_REF] Komatsu | Measuring primordial non-Gaussianity in the cosmic microwave background[END_REF][START_REF] Yadav | Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background[END_REF]Yadav et al., , 2008] ] (this was the only one used for the official WMAP analysis as well, as the other ones did not yet exist at that time), the binned estimator [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF][START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF], and the modal estimator [START_REF] Fergusson | General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f NL[END_REF][START_REF] Fergusson | The CMB Bispectrum[END_REF][START_REF] Fergusson | Efficient optimal non-Gaussian CMB estimators with polarisation[END_REF]. All three are based on the same theoretical exact f NL estimator, but differ in the approximations made in their implementations to make them fast enough for practical use. In addition to being f NL estimators, the binned and modal estimators also allow for the determination of the full bispectrum of the data.

In this paper we will use the binned bispectrum estimator to determine the bispectra of various galactic foregrounds (dust, free-free, anomalous microwave emission (AME), synchrotron), and then use those bispectra as templates to determine the corresponding f NL parameters in other maps. The aim of the paper is threefold. In the first place it is a proof of concept. In fact, the ability to determine bispectra from maps and then use them as templates was one of the original motivations for developing the binned bispectrum estimator, but so far this potential ability had not yet been put to the test in practice. Secondly, it is interesting to study the bispectra of these galactic foregrounds as an aim in itself, and see how they correlate with the primordial templates.1 Finally, in the third place we want to test if any detectable galactic non-Gaussianity remains in the cleaned Planck CMB maps. The quality of these maps has been tested in many different ways, mostly using the power spectrum, but also by seeing if primordial f NL measurements remained optimal, and they passed these tests. Still, it is good to also test for the presence of non-Gaussian galactic residuals directly.

The fact that we restrict ourselves to galactic foregrounds is because for the most important extra-galactic foregrounds templates already exist in analytic form [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF][START_REF] Lacasa | Non-Gaussianity of the cosmic infrared background anisotropies -I. Diagrammatic formalism and application to the angular bispectrum[END_REF] (theoretically or heuristically determined), and can for example be used to compute biases [START_REF] Hill | Foreground Biases on Primordial Non-Gaussianity Measurements from the CMB Temperature Bispectrum: Implications for Planck and Beyond[END_REF]. But no such templates exist for galactic foregrounds. One of the advantages of the binned bispectrum estimator is that it does not necessarily require templates in analytic form, but can also deal with a numerical binned template. We restrict ourselves in this paper to temperature maps only. A preliminary exploration of the polarization maps and simulations of the 2015 Planck release showed that these were not yet sufficiently accurate to make a similar analysis in polarization meaningful.

The paper is organized as follows. In sections 3B.2 and 3B.3, our data analysis results using the binned bispectrum estimator on data from the 2015 Planck release are presented. In section 3B.2, several galactic foregrounds are studied at the bispectral level, with some additional results in the second appendix in section 3B.4. The newly determined templates from these foregrounds are then applied to several CMB maps (Gaussian simulations and real data) in section 3B.3. One of the early questions we had, was if we should include a linear correction term in the foreground bispectrum templates. To answer that, we had to look at the theoretical derivation of the linear term and its assumptions, and found that it makes no sense to add a linear correction for highly non-Gaussian bispectra. We include a full derivation in the first appendix in section 3B.4.

3B.2 Galactic foregrounds

In this section, we study several galactic foregrounds with the binned bispectrum estimator. As discussed previously, when studying non-Gaussianity in CMB data maps, the usual method is to compare the observed bispectrum to different theoretical shapes using the inner product (3.35). The determined parameters f NL simply indicate to what extent these shapes are present in the data. Usually this method is applied to several shapes which have analytical expressions and it includes primordial non-Gaussianity (generated during inflation) but also late-time bispectra (generated after recombination) like extra-galactic foregrounds. However, when observing the CMB, the main source of contamination is our own galaxy and there is no equivalent theoretical expression to describe the non-Gaussianity of galactic foregrounds yet. There are many techniques to clean the maps from the presence of different galactic foregrounds (see [Planck 2015 IX;Planck 2015 X;Planck 2015 XXV] for a review) and CMB analyses at the bispectral level are generally performed on these clean maps. In this section, we use the fact that an analytical formulation of theoretical shapes is not mandatory for use with the binned bispectrum estimator, allowing us to examine these foregrounds too. Indeed, to use the inner product (3.35), one only needs the numerical binned theoretical bispectrum. This means that in principle, the binned bispectrum of any map determined numerically could be used as theoretical template for the analysis of another map under the condition that the binning is the same. In this way we determine templates using the maps of different galactic foregrounds from the 2015 Planck release obtained by the Commander component separation technique [START_REF] Eriksen | Power spectrum estimation from high-resolution maps by Gibbs sampling[END_REF][START_REF] Eriksen | Joint Bayesian component separation and CMB power spectrum estimation[END_REF]]. 2In section 3B.3 we will use these new numerical templates on the CMB cleaned maps studied in [Planck 2015 XVII]. To be more precise, we will use the SMICA [Cardoso et al., 2008] CMB map from the 2015 Planck release. We will also study the raw 143 GHz map, which is the dominant frequency channel in the SMICA map (see figure D.1 of [Planck 2015 IX]). While it is the best channel to observe the CMB (best combination of a low noise level and a good resolution), that is not the case for the different foregrounds (at least if the goal was to study the physics of these foregrounds). Nevertheless, here we only need to estimate their eventual contamination to the CMB signal. At that frequency, the CMB dominates the sky after masking the brightest parts (galactic plane and strong point sources). For this, we use the temperature common mask of the Planck 2015 release, which is a combination of the masks of the different component separation methods [Planck 2015 IX]. In section 3B.2.3, we will discuss the influence of the Commander (Planck 2015), using the common mask. On the left, the map is normalized using histogram equalization to highlight all the regions containing dust. On the right, the scale is linear, this shows that most of the signal is coming from the galactic plane near the mask. The range of the scale is also different.

mask on the different foregrounds by using a smaller one (Commander mask). Finally, another important choice is the binning which was determined using the ratio R defined in (3.38) to be optimal for the primordial shapes. It is true that this criterion has nothing to do with the galactic foregrounds, but our ultimate goal is to determine the primordial shapes optimally, not the galactic ones. To illustrate the method, we start by studying the case of thermal dust.

3B.2.1 Thermal dust

Above 100 GHz, the strongest contamination from galactic foregrounds is due to small dust grains (∼ 1 µm or smaller) present in the interstellar medium. This dust plays an important role in galactic evolution (chemistry of interstellar gas, etc., see the textbook [START_REF] Draine | Physics of the interstellar and intergalactic medium[END_REF] for example), but it also has a large influence on astrophysical observations. Indeed dust grains are heated by the UV starlight they absorb, so they emit a thermal radiation (infrared) in the frequency range of CMB experiments. This emission is well described by a modified blackbody model also called greybody (see [Planck 2013

I(ν) = Aν β d B ν (T d ), (3B.1)
where B ν describes Planck's law, T d ∼ 20K is the mean temperature and β d ∼ 1.5 is the free emissivity spectral index. Figure 3B.1 shows the map of the galactic thermal dust at 143 GHz, before and after applying the common mask. As mentioned before, we are interested in the contribution of the foregrounds in a CMB analysis (where a mask is always used to hide the galactic plane). Hence, the map on the right is the most important here because it is the actual contribution of dust that could be seen in a CMB analysis. In the following, we will be interested in the power spectrum and the bispectrum of this map. As expected, most of the signal comes from the galactic plane, and it is strongest close to the mask. Because of the dust localization, this emission is very non-Gaussian [START_REF] Misner | Statistical properties of dust far-infrared emission[END_REF] and anisotropic (and this is also the case for the other galactic foregrounds studied in the next subsection). The bispectrum is not the best tool to describe such a localized non-Gaussianity (an estimator in pixel space would be better). However, we are only interested in the impact of this galactic foreground on the primordial shapes. This requires us to be careful with the different expressions of the binned bispectrum estimator, mostly derived using the weak non-Gaussianity approximation (see the first appendix in section 3B.4). Concerning the observed bispectrum of the dust map, which is exactly what we need to make a dust template, it is still defined by (3.24) (divided by f sky and only integrated over the non-masked part of the sky), without the linear correction terms which are not justified here. Before describing the dust bispectrum, it is interesting to examine the power spectra of the dust and CMB maps shown in figure 3B.2. It is clear that at 143 GHz, the CMB dominates except for the largest scales (smallest ) where the dust power spectrum has a sawtooth pattern. We can see that it is smaller (up to an order of magnitude) for each odd up to ∼ O(20). This is in fact due to the symmetry of the masked map in figure 3B.1 around the galactic plane when viewed on the largest scales. Because of this symmetry, the temperature is an even function of the angle θ (with the usual Ω = (θ, ϕ), where θ describes the latitude position), using the simple approximation that the mask can be seen as a band with all the dust signal on the border. The spherical harmonics Y 0 also have a similar symmetry around the galactic plane so they are the main contribution when decomposing in harmonic space. However, the Y 0 are even in θ only for even and they are odd for odd, so the odd terms have to be small. For the same reasons similar effects are expected in the dust bispectrum as far as large scales are concerned.

Moving on to the dust bispectrum, we use 2D-slices where the multipoles 1 and 2 go from 2 to 2500 but 3 is in a chosen bin, in order to make it easy to visualize. Figure 3B.3 shows a slice ( 3 ∈ [1092, 1149]) of the binned dust bispectrum compared to the local shape in the case f local NL = 1. 3 If we compare the bispectrum amplitudes, it is clear that the dust is several orders of magnitude larger than the local shape. Moreover, as expected, acoustic oscillations present in both the CMB power spectrum and the local theoretical bispectrum are not there in the case of thermal dust.

However, the plots of figure 3B.3 are not well suited to describe quantitatively the non-Gaussian nature of these shapes. As in the case of the power spectrum, which peaks at low if we do not multiply by the factor ( + 1), the CMB bispectrum as well as the dust bispectrum have a strong dependence. This means that we should use an adapted function of to highlight the true nature of a bispectral signal. A good choice is to use signal-to-noise plots [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF] as shown in figure 3B.4: the bispectrum is divided by the square root of the variance of the map computed using the power spectrum, see (3.29) (divided by f sky ). It is important to note that this is different from the correlation coefficients (3.46) that we discuss later in this section where the variance of the cleaned CMB map is used. In this kind of plots, non-Gaussianity is simply represented by values large compared to O(1).

Figure 3B.4 shows the bispectral signal-to-noise ratio for three different slices of the dust template (on the left), but also for a cleaned CMB map which we will study in detail in section 3B.3 (on the right). It is now obvious that the dust map is very non-Gaussian and that indeed its bispectrum peaks in the squeezed configuration. This effect can be seen in the top plot (low 3 ) but also on the left (low 1 ) and on the bottom (low 2 ) of the other plots. A squeezed configuration is expected when there are correlations between small-scale and largescale effects. There is a simple physical explanation for the origin of these correlations. The large clouds of dust (i.e. large-scale fluctuations) have the highest intensity where they are the thickest along the line of sight. Moreover, the brightest parts have stronger fluctuations (smallscale), see [START_REF] Misner | Statistical properties of dust far-infrared emission[END_REF] for a discussion, so the small-scale fluctuations are modulated by the large-scale ones which corresponds to a squeezed bispectrum.

The squeezed signal present in both the dust and the local shapes is a good indication that they are correlated. This can be verified in table 3B.1 which gives the correlation coefficients between the dust and the standard shapes computed using (3.46) in the context of a CMB analysis (more details in section 3B.3), so the denominator of the inner product is the CMB bispectrum variance. There is an anti-correlation between the dust and local shapes (60 %) because they have opposite signs (this anti-correlation was pointed out in [Yadav and Wandelt, 2008]). The local shape is itself correlated to the other primordial shapes (see table 3.1). However, this does not mean that the dust template has to be correlated to them too. And indeed, the dust and equilateral shapes are uncorrelated because the latter does not peak in the squeezed configuration. The correlations between local and dust (squeezed) do not come from the same multipole triplets as the correlations between local and equilateral (acoustic peaks). However, the orthogonal and dust shapes are a little correlated (around 15 %), because the orthogonal bispectrum in the squeezed limit is large. The dust bispectrum template is very weakly correlated to extra-galactic foreground templates like unclustered point sources and CIB, but anti-correlated to lensing-ISW (which is known to be highly correlated to the local shape). An alternative representation of the bispectra of the different shapes, which shows in which regions of multipole space they dominate, is given in the second appendix in section 3B.4.

3B.2.2 Other foregrounds

Apart from dust, there exist other foregrounds which have a greater effect at low frequencies, of which we will study three here. In this section we use maps produced by the Commander method to separate foregrounds, but this time in addition to the Planck data, observations from WMAP between 23 and 94 GHz [WMAP Collaboration, 2013] and a 408 MHz survey map [START_REF] Haslam | A 408 MHz all-sky continuum survey. II. The atlas of contour maps[END_REF] were also used to determine them. They have a lower resolution (n side = 256) and a larger beam (60' FWHM Gaussian beam). For the sake of comparison of these foregrounds with the dust we discussed in the previous section, we will also use here a dust map with the same characteristics.

In the case the dust grains rotate rapidly (in addition to their thermal vibrations), they can produce a microwave emission which probably corresponds to the anomalous microwave emission (AME) [START_REF] Leitch | An Anomalous component of galactic emission[END_REF][START_REF] Draine | Electric dipole radiation from spinning dust grains[END_REF]], large at low frequencies.

Dust is not the only component responsible for the contamination of the CMB signal; some interactions of electrons with the interstellar medium can also generate emissions. On the one hand, ultra-relativistic electrons (cosmic rays) spiraling in the galactic magnetic fields radiate. This synchrotron emission can be described by a power law ν β with β -3 indicating that indeed, this radiation is significant at low frequencies [START_REF] Haslam | A 408 MHz all-sky continuum survey. II. The atlas of contour maps[END_REF]. On the other hand, electrons can be slowed down by scattering off ions. This generates the free-free emission [START_REF] Dickinson | Towards a free-free template for CMB foregrounds[END_REF], also called bremsstrahlung.

The frequency dependence of the foregrounds and the CMB signal can be seen in figure 51 of [Planck 2015 X]. As discussed, the synchrotron, the free-free and the spinning dust (AME) emissions dominate at low frequencies. The dust thermal emission is the main contribution at high frequencies and is of the same order as the CMB at 143 GHz (this of course depends on the choice of mask).

Figure 3B.5 shows the contributions of all these foregrounds at 143 GHz. Similarly to the dust in the previous section, they are all localized in the galactic plane. Moreover, we can see that the dust signal has a higher intensity and therefore is the dominant foreground contribution at 143 GHz. The same hierarchy can be seen in the power spectra, as shown in figure 3B.6. It is clear that at 143 GHz, the contributions of AME, synchrotron and free-free are negligible compared to the CMB (remember that the brightest parts of the sky are masked). Note that because of the low resolution of the map and the 60 arcmin beam, the range of multipoles is a lot smaller than in the previous section ( max = 300 here). This also means that we were able to use smaller bins for the binned bispectrum estimator. We simply took the usual binning, with each bin split into three when possible (two otherwise).

The same behaviour is of course present in the bispectra (i.e. the templates) where the dust dominates everything. However, as discussed in the previous section, it is more interesting to study the bispectral signal-to-noise to study the form of these bispectra. Figure 3B.7 shows these bispectra for three different slices of 3 . Free-free, dust and AME peak in the squeezed configuration (but for AME, the signal is so low that it could be only noise). An argument similar to the dust case described in the previous section can explain this bispectral configuration. We can also verify this in table 3B.2 where we have computed the correlation coefficients of these shapes with the ones previously introduced. As expected, the dust, free-free and AME bispectra are anti-correlated to the local shape (and for the other shapes see the previous section, the discussion is similar) and are correlated between themselves (they share the squeezed configuration). For a visual representation that helps to understand the correlations, see the second appendix in section 3B.4.

The case of synchrotron is different. The signal seems to be larger for three "high" values of , so it is similar to the equilateral shape. This is also the typical shape produced by unresolved point sources and by the CIB. Indeed, the synchrotron is correlated (around 40 %) to the point sources and CIB shapes as well as to equilateral and orthogonal (around 30 %). However, it is also correlated to the other foregrounds (more than 30 %), meaning that the synchrotron bispectrum also peaks in the squeezed limit, as shown in the bottom left plot of figure 3B.7, even if it is not at all its dominant part. Physically that makes sense because we expect a squeezed signal for similar reasons as the other foregrounds. The simplest explanation for the equilateral shape is a contamination of the map by point sources and this possibility is mentioned in [Planck 2015 X]. To verify it, we performed the simple test of subtracting the unresolved point sources bispectral template (of which the amplitude was determined using the estimator (3.33)) from the bispectrum of the synchrotron map. The cleaned bispectrum is shown in figure 3B.8 where one can see that the left plot (showing the squeezed part of the bispectrum) has not changed from the one of figure 3B.7, while the other two are much less non-Gaussian (but not perfectly cleaned either). This is also illustrated in table 3B.3, where the correlation of the synchrotron bispectrum with the local shape increases (to around 15 %) and becomes of the same order as for the other foreground bispectra, while the anomalous correlation with the equilateral, point sources and CIB templates vanishes. From now on, when we mention the synchrotron bispectrum, it will be the one cleaned from the unresolved point sources contamination.

3B.2.3 Noise and masks

The main source of anisotropy in the foreground maps are the foregrounds themselves as they are mostly present in the galactic plane, but we still need to examine the influence of the other sources discussed in section 3.2.3. We start by the noise, which for the CMB has a large effect at high . Hence, it is sufficient to look at the best resolution dust map. Figure 3B.9 shows the noise power spectrum of the dust map evaluated using half-mission maps. Even at high , it seems that it is small compared to the signal. Hence, we will not discuss it further in this paper.

The choice of mask should also be examined more carefully. That is why here we compare our previous results obtained with the common mask (f sky = 0.776) to those obtained with the mask provided by the Commander component separation method which is slightly smaller (f sky = 0.822) and of course fully included in the common mask. Figure 3B.10 shows these two masks in the high and low resolution cases.

Figure 3B.11 shows the power spectra of the different foregrounds with these two masks and highlights the large difference between the two cases. The reason for this difference is quite obvious because the masks have been constructed to hide most of the foregrounds, so with a smaller mask, there is a lot more of the foregrounds to detect. Moreover, as they are anisotropic, both the amplitude and the form are different depending on the mask, this is especially true Table 3B.4: Correlation coefficients between the bispectral templates determined using the common mask and the Commander mask for each foreground.

for the synchrotron signal (as we will explain below). This is also checked for the bispectra as shown in figure 3B.12. With the Commander mask (the smallest one), all the signals are a lot more non-Gaussian. To verify that is not only a difference of amplitude, we have at our disposal a useful tool: the correlation coefficients defined in (3.46). For each foreground, we have computed the correlation between the templates determined using the two masks. The results are given in table 3B.4. For the dust, free-free and AME emissions, the templates are correlated (above 80 %) and indeed we can see that the bispectra peak in the squeezed configuration as discussed previously. However, the fact that the correlation is not 100 % shows that the difference is not only the amplitude.

However, for synchrotron the situation is more complicated, like in the power spectrum case. Indeed the new template is very different from the one in figure 3B.8 and it is confirmed by the low correlation between the two synchrotron templates determined with the two different masks. To understand this result, it is interesting to examine directly the data map with the Commander mask in figure 3B.13. One can see that there are a few pixels where the intensity is ten times larger than with the common mask (where they are hidden). The influence of this very small region dominates the power spectrum and the bispectrum because the transition is so important. It could be modeled as a Heaviside step function, the Fourier transform of which is a sinc function, meaning that these two pixels have a large influence over the whole multipole space and we can see oscillations as expected in both the power spectrum (there is a minimum at ≈ 240) and the bispectrum (there are three regions of negative bispectrum with positive bispectrum around them on the plot for 3 ∈ [255, 267]).

In conclusion, the choice of mask has a large influence on the templates we are determining because of the localization of the foregrounds in the galactic plane. This means that when we apply these templates to other maps in the next section, it is mandatory to use the same mask at every step. From now on, we will exclusively use the common mask. 1.4 ± 1.0 1.0 ± 0.5 1.00 ± 0.20 Joint 0 ± 14 -5 ± 75 -1 ± 39 0.0 ± 2.6 0.0 ± 1.4 1.01 ± 0.24 Dust 75%, 70 bins (expected f dust NL = 0.42) Indep -30 ± 8 11 ± 66 41 ± 36 0.6 ± 0.9 0.4 ± 0.5 0.42 ± 0.12 Joint 0 ± 9 1 ± 70 -2 ± 42 0.0 ± 2.6 0.0 ± 1.4 0.42 ± 0.13 Dust 0%, 70 bins (expected f dust NL = 0) Indep -0.1 ± 0.5 -1.7 ± 6.1 -3.1 ± 3.4 -0.03 ± 0.09 -0.01 ± 0.05 0.001 ± 0.003 Joint -0.3 ± 0.7 -1.6 ± 6.4 -4.2 ± 4.1 -0.15 ± 0.26 0.06 ± 0.13 0.001 ± 0.003 Table 3B.6: Determination of f NL for the local, equilateral, orthogonal, point sources, CIB and dust shapes using a set of 100 Gaussian simulations of the CMB with isotropic noise to which we added a known amount of dust (the high-resolution dust map of section 3B.2.1 multiplied by a factor 1 or 0.75, or no dust at all). The analysis is performed using 57 bins or 70 bins and the error bars are given at 1σ. For the reason behind the much smaller error bars in the 0 % dust case, see the main text.

3B.3.1 Gaussian simulations

For the first tests, we constructed a set of 100 Gaussian simulations of the CMB obtained using the best fit of the CMB power spectrum from the 2015 Planck release [Planck 2015 XI] at the resolution n side = 2048. There are several reasons to use these simulations instead of the observed CMB map. First, it is important to check the validity of this new use of the binned bispectrum estimator with a large number of maps. Moreover, even the cleaned CMB map still contains contamination from extra-galactic foregrounds and the ISW-lensing. Here, these effects are not present. However, we need the Gaussian realizations of the CMB to have the characteristics of the SMICA CMB map. Hence, we smoothed the maps using a 5 arcmin FWHM Gaussian beam and we added noise based on the noise power spectrum of the SMICA CMB map (moreover, our choice of bins is optimal only if this noise is present in the maps, because it diminishes the weights of the bins at high following (3.29)). In this section, we will discuss two different cases for the noise. First, we will assume it has an isotropic distribution in pixel space. In the second case we will make it anisotropic by modulating it in pixel space using the hit-count map corresponding to the scanning pattern of the Planck satellite. Finally, we add some dust to these maps using the high-resolution dust map at 143 GHz discussed in section 3B.2.1. Every analysis presented in this section uses the common mask introduced in section 3B.2, see figure 3B.10. The determination of the amplitude parameters is performed using the binned bispectrum estimator, including a linear correction term to the bispectrum as discussed in section 3.2.3. In practice, the linear correction term is computed using Gaussian simulations of the analyzed maps with the same characteristics (beam, noise, mask). We use the average power spectrum of our 100 maps (CMB + dust) to generate the maps necessary for the computation of the linear correction. Here we use 80 maps for the linear correction. We have verified that this number is sufficient to detect squeezed bispectra like the local and the dust shapes to high precision. The first analysis is performed with the same choice of 57 bins as in the 2015 Planck analysis [Planck 2015 XVII] which was shown to be optimal to determine the primordial shapes, using multipoles from min = 2 to max = 2500 (remember that our analysis is temperature only). We add the dust map to the simulations of the CMB, thus the expected value of the f NL for the dust template is 1. We also determine the amplitude parameters f NL for the primordial shapes, the point sources and CIB bispectra in both the independent and the joint case.

Results are given in table 3B.6. First, we see that we detect the expected amount of dust with a good accuracy. We also observe that the shapes correlated to the dust template (see table 3B.1), because they also peak in the squeezed configuration, are strongly detected in the independent case. However, in the joint analysis all the non-Gaussianity of the maps is attributed to the dust, with only a small impact on the error bars of the primordial shapes, meaning that this test is successful. However, this choice of bins is only optimized to detect the primordial bispectra and not the dust. Then, it is important to verify if the results can be improved by adding a few bins at very low (below 30) to better measure the dust contribution. This can be seen in the second appendix of section 3B.4 where we observe that only the very low 1 are important for the template (it is more squeezed than the local shape). Figure 3B.14 can also be used to highlight this effect. It shows the convergence of f NL when using a smaller multipole interval to determine f NL . In the two top plots, we can see that if we exclude the very low (below 30) both the local and orthogonal f NL are consistent with 0. If we exclude the region of multipole space where the dust template is the strongest, there is no detection of the primordial shapes, even in an independent analysis. The two bottom plots are interesting as they show that the determination of f NL for the dust template is very stable when increasing min or decreasing max . Note however that the error bars on f dust NL increase a lot if we use min > 30. This is visible with the dashed blue lines which correspond to the 68% confidence intervals.

There is another important effect in the dust template when a very large scale (small ) is concerned: the sawtooth pattern in the dust power spectrum (see figure 3B.2) is also expected in the dust bispectrum for the same reason (the only large harmonic coefficients describing the dust at low are the a 0 with even). In principle, it could be used to differentiate between the dust and the local shapes, but this effect is hidden if the bins are large because it is averaged over several 's, thus providing another motivation to add some bins at low . One issue when adding bins is that the memory constraints on the computer system we use limit us to a number of bins between 50 and 60 at most at the Planck resolution when including the polarization too.

Here we can use 70 bins, because we only look at the temperature data and because we only add bins for the largest scales, where it is possible to downgrade the resolution of the filtered maps. With this new binning, the correlation coefficient between the local and dust shapes becomes -0.48 (instead of -0.60 for 57 bins, see table 3B.1). So indeed adding a few bins at low helps to differentiate these squeezed shapes. The results of the same test with 70 bins are also given in table 3B.6. In the independent analysis, the amount of local non-Gaussianity and its error bar decreases which is consistent with the fact that the dust and the local templates are easier to differentiate with the new binning. However, in the joint analysis there is no clear difference, except that the different central values are now very close to the expected values.

We also have to note that the approximation of weak non-Gaussianity, which is needed for the validity of the linear correction of the bispectrum to take into account the effects of the mask here, starts to break down when we observe a local shape at more than 6σ (independent case). This is why it is important to verify how a similar analysis works with a smaller amount of dust in the map. Hence, with the same choice of 70 bins, we perform two other tests with the 100 CMB simulations. For one we multiply the dust map by a factor 0.75 before adding it to the CMB realizations and the expected value of f NL is then 0.75 3 ≈ 0.42. For the other test, we use the the Gaussian CMB maps without adding dust, to verify that we do not detect any bispectral shape. These results are also given in table 3B.6 and are exactly as expected.

Note that the error bars for the case of the CMB only are roughly one order of magnitude smaller than for the rest. The reason is that we made a distinction between the standard deviation (square root of the variance) and the standard error (standard deviation divided by the square root of the number of maps, so divided by 10 here). The standard error gives the expected error on the determination of the mean value of f NL with our sample of 100 Gaussian maps. The standard deviation gives the 1σ interval in which we would detect f NL if we study one map. It is clear that the standard error has to be used in the CMB-only analyses because we determine the mean value of each f NL from a sample of 100 maps. However, when we add dust to these maps, the situation is different because we only have one realization of the dust so the standard error cannot be used. We are however very conservative by using the standard deviation, the real error bars on the mean values of the different f NL are probably between the standard error and the standard deviation (the more dust in the map, the closer to the standard deviation it will be). However, the fact that for the two amounts of dust with 70 bins the central values in the joint analysis are so close to the expected values is an indication that the error bars are likely overestimated for these two cases (the results with dust would still be correct if we divided the standard deviation by 10 to obtain the standard error, which is not true with 57 bins). We can illustrate the breakdown of the weak non-Gaussianity approximation using the variance of the bispectrum. Indeed, we have at our disposal a theoretical prediction for the variance, given in (3.29), that scales as the power spectrum cubed and for which the derivation relies on the weak non-Gaussianity approximation. However, we can also directly compute the variance of the bispectrum from our 100 maps, which we call here observed variance. Figure 3B.15 shows the distribution of the ratio of the observed variance over the theoretical variance for the three different amounts of dust in two different configurations. First, we examine this ratio over the whole triplet space (on the left) where there is no difference between the three cases and the values are distributed around 1 as expected. That is logical because the non-Gaussianity of the dust is very localized in multipole space; the bispectrum is large only in the very squeezed configuration. This is why on the right we consider only the triplets where one is very small (in the first five bins i.e. ≤ 13) and the two others large (in the last 30 bins, i.e. ≥ 742). Adding or removing a few bin triplets here does not change the results. Here we can see that if there is more dust (in red), there are several values which strongly deviate from one. This effect is even more obvious when we examine the mean and the standard deviation of these distributions, which are given in table 3B.7. When considering the full space of multipole triplets, there is no significant difference between the three cases. However, when we examine only the squeezed part of the bispectrum, the standard deviation increases slightly with a small amount of dust (75 %), and is three times larger for 100 % dust compared to the CMB-only case. Hence, the weak non-Gaussianity approximation stops being valid, but not enough to invalidate the results (only a few bin-triplets deviate strongly). However, if we were to add even more dust, we would have to take this effect into account.

In addition to looking at the variance of the bispectrum itself, we can also investigate the variance of the f NL parameters with regard to the validity of the weak non-Gaussianity Table 3B.8: Fisher and observed standard deviations on f local NL and f dust NL (independent analysis) determined from 100 Gaussian simulations of the CMB with isotropic noise to which we added a known amount of dust (100 %, 75 % or 0 % of the dust map of figure 3B.1) using 70 bins. approximation. Every error bar given in table 3B.6 was computed from the observed variance of the set of 100 maps. However, we can also compute Fisher error bars from the theoretical prediction of the variance and they are given in table 3B.8 for the three cases studied in this section for the local and dust shapes. We can see that for both, the more non-Gaussian the map is, the more important is the difference between Fisher and observed error bars. This is related to the breakdown of the weak non-Gaussianity approximation. For a local |f NL | of around 70 (corresponding to 100 % dust), the difference is a factor 2 between the two kinds of error bars. The difference is larger for the dust template, where for this case the observed error bars are four times larger than the Fisher forecasts. For both templates, when there is no dust (so purely Gaussian maps), the observed error bars agree with the Fisher forecasts up to the expected precision (the relative error in the standard deviation is 1/ 2(N -1), which is 7 % for 100 maps).

As explained before, adding noise realizations with the correct power spectrum to the CMB simulations is necessary for the optimization of the binning and to make the simulations more realistic. However, the real instrument noise does not have an isotropic distribution in pixel space because some parts of the sky are observed more often than others, as shown in figure 3B.16. Without a linear correction, the anisotropic noise also gives a large squeezed contribution to the bispectrum for the usual reason: small-scale fluctuations are larger (more noise) in the large-scale regions which are less observed and vice versa. This is why we also verify the previous results with an anisotropic distribution of the noise following the scanning pattern of the Planck satellite. The results are given in table 3B.9. Here we use only the best choice of bins (70 bins) and the results are given for the same three amounts of dust as in table 3B.6. Results are very similar with isotropic and with anisotropic noise for the three cases; each time we detect successfully the amount of dust we added to the maps.

With these different tests, we have proven that the binned bispectrum estimator can be used to detect a galactic foreground shape that we determined numerically. It works well with the amount of dust that is expected at 143 GHz, the dominant frequency channel in the cleaned CMB map. The next logical step is then to use the template on real data. In the three first lines, the ISW-lensing shape is considered as the others. In the last three, the ISW-lensing bias is subtracted. The joint analysis is performed with and without the dust template. The binning consists of 57 bins.

3B.3.2 CMB analyses

The previous tests have shown that detecting the dust is possible when there is a large amount of it. We can now apply the dust template to a real CMB analysis. Here, we follow the analysis of the Planck 2015 paper [Planck 2015 XVII] (note that we only study the temperature bispectrum, while for Planck the polarization was also taken into account). We use a set of 160 simulation maps for the computation of the error bars and the linear correction. The power spectrum is the best-fit cosmological model from the 2015 Planck analysis. This time, we also include the ISW-lensing shape in the analysis because it is present in the data. The amplitude of this template is known, so it can be used to subtract the bias (see equation (3.47)) from the bispectral non-Gaussianity of the map. Results are given in table 3B.10. 4 We include the results with and without taking into account the ISW-lensing bias, and we perform two different joint analyses for comparison, with and without the dust. As expected, there is no detection of the primordial shapes or the dust. However, it is important to note that the error bars of the local and dust shapes in the joint analysis increase because these shapes are correlated. Similarly to the previous section, one way to improve the situation would be to find a binning that is optimal for both shapes.

We also performed a similar analysis on a low resolution cleaned CMB map (n side = 256) with a 60 arcmin FWHM Gaussian beam to look for the other foreground templates with the usual choice of bins. Results are given in table 3B.11. Because of the resolution and the beam, we only analyze multipoles in the interval [2,300], which is the reason for the very large error bars. As that would leave only few bins from the original binning, we split all the bins below = 300 into three (where possible, two otherwise), which gives 39 bins in total. Moreover, we did not subtract the ISW-lensing bias as its contribution is small compared to the error bars. The point sources and the CIB are not given in the table because they were not observed here. The results are consistent with zero non-Gaussianity in the map. But the new foreground shapes (AME, free-free and synchrotron) have very large error bars and even if they were present in the map, it would not be possible to detect them.

3B.3.3 Raw sky

After applying the foreground templates to the cleaned CMB map that is not supposed to contain any galactic foreground (which we confirmed), it is also interesting to study the raw 143 GHz Planck map. Again, we had to generate Gaussian simulations of this map to compute the linear correction. For this, we used the power spectrum of the map and we determined the Table 3B.11: Determination of f NL of some primordial and all galactic templates in the cleaned SMICA CMB map at low resolution n side = 256 with a 60 arcmin FWHM Gaussian beam from the 2015 Planck release. Because of the low max = 300, the analysis is performed using 39 bins. noise power spectrum using the half-mission maps. The power spectra are shown in figure 3B.17. The noise is modulated in pixel space using the hit-count map of figure 3B.16 to make it anisotropic. The beam of this map can be approximated by a 7.3 arcmin FWHM Gaussian beam.

Results are given in tables 3B.12 and 3B.13. We detected the expected amount of dust since f dust NL = 1 is within the 1σ error bars in both the independent and joint analyses. For the other foregrounds, the situation is similar to the previous section: the error bars are far too large for a detection (the synchrotron and AME shapes are not given here because the error bars are many orders of magnitude larger than the expected quantity in the map). To determine error bars, we did not have good simulations of the data map but we had of course Fisher forecasts. We made the simple but reasonable hypothesis that the factor between the real error bars and the Fisher ones due to the breakdown of the weak non-Gaussianity approximation is the same as for the anisotropic case in section 3B.3.1. Then it was easy to determine error bars for the primordial and the dust shapes that are a bit larger than in table 3B.9. However, for the CIB and the point sources, which are not the main object of study here, the situation is different because they were not present in the Gaussian simulations, so we could not apply this method. Hence we only give Fisher error bars for those two shapes, but because of the strong detection we know that they are underestimated. This is not an issue because they are uncorrelated to the local and the dust shapes. For the low-resolution case in table 3B.13 we only have Fisher error bars for all shapes. In conclusion, the method also works correctly when applied to a raw sky map.

3B.4 Appendices

This section contains the two appendices of the paper.

3B.4.1 Derivation of the variance of the bispectrum and the linear correction

In this appendix, we recall the derivation of the variance of the bispectrum to explain the role of the linear correction. By definition, the variance is given by Var(B obs 1 2 3 ) = (B obs 1 2 3 ) 2 -B obs

1 2 3 2 ≡ V 1 2 3 . (3B.2)
In the weak non-Gaussianity regime, the average value of the bispectrum is negligible. This leaves us with computing the mean value of the product of two bispectra. (3B.3) One can use Wick's theorem for Gaussian fields to reduce the six-point correlation function to the sum of fifteen products of two-point correlation functions:

• 6 terms: each is paired with an element of the other triplet like in M 1 ( Ω)M 4 ( Ω ) M 2 ( Ω)M 5 ( Ω ) M 3 ( Ω)M 6 ( Ω ) ;

• 9 terms: the rest (example: M 1 ( Ω)M 2 ( Ω) M 3 ( Ω)M 4 ( Ω ) M 5 ( Ω )M 6 ( Ω ) ).

We will explicitly compute the contribution of the six first terms below. But first, we show that in the isotropic case, the nine last terms are zero. For this, we use the addition theorem, and we find the announced result that these terms vanish.

Concerning the six first terms, we will also explicitly compute only the given example, but the correct permutations to obtain the five other terms will be in the final result. Substituting (3.18) into the integral and using the fact that the maps are real, one obtains

S 2 ×S 2 d Ωd Ω M 1 ( Ω)M 4 ( Ω ) M 2 ( Ω)M 5 ( Ω ) M 3 ( Ω)M 6 ( Ω ) = C 1 C 2 C 3 δ 1 4 δ 2 5 δ 3 6 m 1 ,m 2 ,m 3 S 2 d ΩY 1 m 1 ( Ω) Y 2 m 2 ( Ω)Y 3 m 3 ( Ω) × S 2 d Ω Y * 1 m 1 ( Ω )Y * 2 m 2 ( Ω )Y * 3 m 3 ( Ω ) . (3B.8)
where one can recognize the Gaunt integral (3.19). Substituting it here and using the identity relation

m 1 m 2 m 3 1 2 3
m 1 m 2 m 3 2 = 1, (3B.9) and the fact that the columns of Wigner 3j-symbols can be permuted when the parity condition is respected, one can find that the 6 terms give B 1 2 3 B 4 5 6 = h 2 1 2 3 C 1 C 2 C 3 δ 1 4 δ 2 5 δ 3 6 + δ 1 4 δ 2 6 δ 3 5 + δ 1 5 δ 2 4 δ 3 6

+ δ 1 5 δ 2 6 δ 3 4 + δ 1 6 δ 2 4 δ 3 5 + δ 1 6 δ 2 5 δ 3 4 .

(3B.10)

Hence the variance is

V 1 2 3 = g 1 2 3 h 2 1 2 3 C 1 C 2 C 3 (3B.11)
with g 1 2 3 equal to 6, 2, or 1, depending on whether 3, 2, or no 's are equal, respectively.

Anisotropic case

As explained in section 3.2.3, with observational data from an actual experiment we cannot use the isotropy assumption. This would lead to a large increase of the variance (3B.11), because the nine terms described above are no longer zero. However, it is possible to show that adding the simple linear correction given in (3.49) to the cubic term of the angle-averaged bispectrum solves this issue. To verify this, we will derive the variance similarly to the previous section, the main difference being that integrations are performed on S 2 \ M instead of S 2 . Again, in the weak non-Gaussianity regime, we only have to compute the average of the product of two bispectra B obs 1 2 3 ( Ω)B obs 4 5 6 ( Ω ) and there are three types of terms:

• 1 term: product of the two cubic terms: M 1 M 2 M 3 M 4 M 5 M 6 (this is the only term present in the isotropic case);

• 6 terms: product of a linear term with a cubic term, e.g. -M 1 M 2 M 3 M 4 M 5 M 6 ;

• 9 terms: product of two linear terms, e.g. M 1 M 4 M 2 M 3 M 5 M 6 .

We have seen how the first term gives 15 contributions if we use Wick's theorem to transform the six-point correlation function into a combination of products of three two-point correlation functions. We have also seen that in the isotropic case, only the six terms where each multipole among ( 1 , 2 , 3 ) is coupled with an element of the other triplet ( 4 , 5 , 6 ) are non-zero. The same can be done for the four-point correlation function and each combination of a linear with a cubic term will give three terms, hence a total of eighteen terms. Note also that each term derived from the linear correction contains necessarily two 's of the same triplet that are coupled (it is in the definition of the linear term). Hence, it means that they cannot cancel the six terms of the isotropic case. The new terms (i.e. the terms that are not present in the isotropic case) are nine from the six-point correlation function, nine from the product of two linear terms, and eighteen from terms with the four-point correlation functions (with a minus sign) and it is then easy to check that they exactly cancel each other. So finally where we use an obvious shorthand notation to indicate the other permutations of filtered maps.

B
It is important to note that we recover the same variance as in the isotropic case without a linear term (except for the integration interval). This proves that the estimator with this linear correction is optimal, with the assumption that C m, m is diagonal (thus equation (3B.5) is valid here). However, the integration interval for the two integrals is S 2 \ M. In the f sky approximation, which many tests have shown to be a good approximation, we calculate the integrals as if the interval were the full sky S 2 , and then add appropriate factors of f sky at the end to compensate for the partial sky. Then, performing the same last steps of the calculation as before, the variance is given by .13) when including the effect of the beam and the noise and has a form similar to the isotropic case (3B.11). The factor of 1/f sky can easily be understood given that the variance of a quantity determined from N data points scales as 1/N and here the number of data points roughly corresponds to the number of observed pixels on the sky.

V 1 2 3 = g 1 2 3 h 2 1 2 3 f sky (b 2 1 C 1 + N 1 )(b 2 2 C 2 + N 2 )(b 2 3 C 3 + N 3 ), ( 3B

3B.4.2 Weights of bispectral shapes

In this appendix, we give another representation of the different bispectra discussed in this paper (primordial shapes, ISW-lensing, extra-galactic and galactic foregrounds) well-suited to understand the correlation coefficients given in e.g. tables 3.1, 3B.1 and 3B.2. The weight of a single multipole configuration ( 1 , 2 , 3 ) of a bispectral shape B 1 2 3 is defined by [START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF] where B obs /B can be viewed as an f NL estimator based on just a single -triplet. These equations are the same for bin-triplets (i 1 , i 2 , i 3 ). Figures 3B.18, 3B.19, and 3B.20 show the weights of the different theoretical and numerical shapes discussed in this paper at both high and low resolution, with the usual choice of 57 bins. Instead of using a few slices of 3 like in section 3B.2, we summed over 3 . It has the advantage that now the whole bispectrum is used in one figure, but of course we lose the information about the variation of the bispectrum as a function of 3 . A larger weight means that the region of multipole space is more important for the template. Conversely, a large observed non-Gaussianity in that region of multipole space means that it is more likely to be that particular shape.

w 1 2 3 = 1 B, B (B 1 2 3 ) 2 V 1 2 3 . ( 3B
In this kind of plot, shapes that peak in squeezed configurations will have a colored band/line at the bottom of the figure (low 1 ). As expected it is present for the different foregrounds for very low 1 (< O(20)), including synchrotron (which was not visible in figure 3B.7). As expected, the characteristic line of a squeezed bispectrum can be seen for the local and the ISW-lensing shapes, but also for the orthogonal shape (which explains why it is somewhat correlated to the foregrounds).

Shapes that peak in equilateral configurations have a large weight along the diagonal black line of these plots, when the three 's are of the same order. The primordial equilateral shape is the strongest for three low 's, while the point sources and the CIB are more non-Gaussian at higher multipoles. It is easy to see the correlation between the point sources and the synchrotron bispectra which peak when the three 's are over 150. An additional remark is necessary about the orthogonal shape. Indeed by definition it is orthogonal to the equilateral shape (uncorrelated), which is not visible in these figures because they both have similar acoustic peaks. It is an effect of the sum over 3 which hides the differences of these bispectra.

[Tzavara and Van Tent, 2012] and [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF]. The former paper contains an investigation of gauge issues at second order and a derivation of the full cubic action of the adiabatic and the isocurvature perturbations. It definitively settles a number of previously open questions regarding the long-wavelength formalism. Although this has not yet been followed up on, this full cubic action should make it possible to compute the non-Gaussianity of any two-field inflation model without any approximation.1 The latter paper investigates the issue of the scale-dependence of f NL and defines two spectral indices for the bispectrum, one related to the size of the momentum triangle and the other to its shape.

The second part of this thesis is dedicated to the binned bispectrum estimator and its applications. It was developed in collaboration with Martin Bucher and later also Benjamin Racine (and with Carla Carvalho being involved in the initial stages of the project). The final estimator and its implementation were published in [START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF]. The detailed description and derivation of the binned bispectrum estimator in this thesis is based mainly on that paper.

The binned bispectrum estimator is motivated by the observation that most bispectrum templates we are interested in are relatively smooth in harmonic space. This means that we lose very little information if we pursue a binned data analysis with moderately sized bins inspace. On the other hand, the computational gains of this approximation are enormous, turning a completely unfeasible computation into a fast estimator. The binned bispectrum estimator determines the full three-dimensional bispectrum of a map, binned in harmonic space. This binned bispectrum can then be combined with a library of theoretical bispectrum templates to determine the f NL parameters. Alternatively, the binned bispectrum of the map can also be used directly for non-parametric (blind) non-Gaussianity searches, by smoothing it. The binned bispectrum estimator is very fast, has a convenient modular structure (which means for example that one can analyze an additional template without having to rerun the map), and gives the dependence of f NL on as a free bonus. It accepts many different types of templates, even non-separable theoretical templates or templates that have been determined numerically from a map.

Various other aspects of the estimator are also discussed in detail in this thesis: the standard theoretical templates, the extension to include polarization, analyzing templates jointly or including a bias, the linear correction to deal with anisotropic noise, and how to determine the statistical significance of features in the smoothed binned bispectrum. We also look at how to deal with the presence of a galactic and a point source mask. The method of diffusive inpainting that we developed for this was later adopted by all other bispectrum estimators as well. Finally we discuss the choice of binning and some issues of the implementation of the estimator as a computer code, for example related to memory limitations.

In the paper [START_REF] Langlois | Isocurvature modes in the CMB bispectrum[END_REF], written in collaboration with David Langlois and included as an appendix chapter, we investigate the local bispectrum arising from the (possibly correlated) combination of the adiabatic mode with an isocurvature mode (which can be CDM, neutrino density or neutrino velocity). We show that this leads to six different f NL parameters, and estimate how precisely CMB data can constrain these parameters, emphasizing in particular the importance of polarization. The actual analysis was later performed in the Planck primordial non-Gaussianity papers, confirming our estimates. 2 The second appendix chapter of this part contains [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF], written in collaboration with Gabriel Jung and Benjamin Racine. We numerically determine and study the bispectra of galactic foregrounds and use these bispectra as templates with the binned bispectrum estimator to show that the cleaned Planck maps do not have any detectable galactic foreground residuals.

The binned bispectrum estimator is one of the three bispectrum estimators used for the official Planck analyses of 2013, 2015 and 2018. The third and final of these [Planck 2018 IX] is included in the final appendix chapter of this thesis, as the crowning achievement of a massive amount of work over the last 10 years. While unfortunately no primordial non-Gaussianity has been detected, very tight constraints are put on many different bispectrum templates, including also the isocurvature non-Gaussianity mentioned above. On the other hand, some non-primordial non-Gaussianity templates are detected, in particular the lensing bispectrum.

A large number of validation tests are performed, increasing our confidence in the results. The consequences for various inflation models are worked out. It is likely that this paper will remain the definitive reference for non-Gaussianity constraints for the next 10 years, until the results from the LiteBIRD satellite or the ground-based CMB-S4 experiments come in (and even those will still need to be combined with the Planck data for the best results).

Figure 1

 1 Figure 1.1: A figure showing the horizon exit of a cosmological fluctuation with comoving wave number k during inflation, and its subsequent horizon re-entry long after inflation.

  .. B... = ∂ µ A A... B... + Γ A CD A D... B... + . . . -Γ D CB A A... D... -. . . ∂ µ φ C .(1.5)

2 ,

 2 Wm 1 ...mn = WA 1 ...An e A 1 m 1 • • • e An mn ,(1.16)

Figure 1 . 2 :

 12 Figure 1.2: The CMB temperature monopole, dipole, and other multipoles, as measured by the COBE satellite. Figure from https://lambda.gsfc.nasa.gov/product/cobe/.

  Figure 1.3: The CMB temperature fluctuations with ≥ 2 as measured by the Planck satellite (2018 release). The gray outline shows the extent of the confidence mask due to (mainly) galactic foregrounds. Figure from [Planck 2018 I].

  Figure 1.4: The 2018 Planck map of the polarized CMB anisotropies, shown as rods whose direction and length represent the direction and amplitude of the polarized CMB. The coloured background is the map of temperature anisotropies, smoothed to 5 degrees. Figure from [Planck 2018 I].

Figure 1

 1 Figure 1.5: A figure showing how a quadrupole anisotropy in the incoming radiation at last scattering leads to a linear polarization of the CMB. Figure from [Hu and White, 1997].

Figure 1 . 6 :

 16 Figure1.6: Planck 2018 CMB power spectra for temperature (top), the temperaturepolarization cross-spectrum (middle), the E mode of polarization (bottom left) and the lensing potential (bottom right, which we do not discuss in this thesis). The quantity D is defined as ( + 1)T 2 0 C /(2π). The red dots with error bars (that include cosmic variance) indicate the measurements, while the blue curve is the best-fitting ΛCDM model. The anisotropy power spectra use a standard binning scheme (which changes abruptly at = 30) and are plotted here with a multipole axis that goes smoothly from logarithmic at low to linear at high . Figure from[Planck 2018 I].

Figure 1

 1 Figure 1.7: The three CMB satellites COBE, WMAP, and Planck, with a visual comparison of their resolutions on a 10-square-degree patch of the CMB sky. Image credit: NASA/JPL-Caltech/ESA.

  Figure 2.1: v12 /(1 + v2 12 ) and v2 12 /(1 + v2 12 ) as a function of v12 .

  Figure 2A.1:

Figure 2A. 2 :

 2 Figure 2A.2: The regions of the parameter space of m and n where the turn of the field trajectory can occur before the end of the slow-roll regime. In the top left figure, these regions are determined for several values of f NL : |-6 5 f NL | larger than 5 (green), 1 (blue) or 0.2 (red). In the top right figure, we explore different error bars of the spectral index: the Planck constraint n s = 0.968 ± 0.006 (red), n s = 0.968 ± 0.0015 in blue (CORE-like experiment) and negligible error bars in green. On the bottom left, different values of N φ are represented: 50 (green), 55 (blue) and 60 (red). The last plot changes the constraint on to check when the turn can occur: = 0.05 (green), 0.075 (blue) and 0.1 (red). The dashed vertical lines indicate the change of sign of Ṽσσ * , which depends on n, N φ and n s : it is necessarily positive on the right-hand side of this line. The small cross highlights that the double quadratic potential (m = n = 2) is excluded in all plots.

Figure 2A. 3 :

 3 Figure 2A.3: Illustrations of the two different types of turn where the slow-roll approximation is broken. On the left, the field trajectory is displayed in black on the potential while on the right the slow-roll parameters and η ⊥ are shown for a typical example. The top correspond to what we call the first type, while the plots at the bottom show the second type. Explicit examples of both are given in section 2A.4.

t

  Figure 2A.4: This plot displays |r.h.s.| in blue (thick line), |order1| in red and |order2| in green (dashed) during the turn of the field trajectory for the potentials (2A.95) (on the left) and (2A.105) (on the right), which correspond to the first and second types of turn, respectively. Note the logarithmic scale.

φ m 2 σ

 2 Figure 2A.5: The exact numerical solutions for the different interesting parameters (basis components, slow-roll parameters, Green's functions, the spectral index and f NL ) during the turn for the double quadratic potential (2A.88).

  Figure2A.6: The exact numerical solutions for the different interesting parameters (basis components, slow-roll parameters, Green's functions, the spectral index and f NL ) during the turn for the first example of a monomial potential (2A.95) with n = 2 and m = 4. The last figure shows both the exact numerical solution for gint and its analytical approximation. The horizontal purple dash-dot lines are the analytical predictions for v12e , n s and f NL .

27 β 3 λ 2

 32 Figure 2A.7, which contains the same plots as figure2A.6 but for the new parameters, shows that is of order 10 -2 during the turn, which is in the domain of validity of the main hypothesis 1. During the turn, η ⊥ is of order 10 at most, which shows that the slow-roll regime is broken. As expected, analytical predictions are now a very good estimation. However, the spectral index is 0.917, which is outside the observational bounds. This example is also used in the previous section in figure2A.4 to illustrate that r.h.s. is several orders of magnitude smaller than the left-hand side terms of (2.79).

  .101) with g = 10 -10 , Λ 4 = 25 2π 2 gκ -4 and f = κ -1 . The initial conditions are φ i = 23.5κ -1 and

  Figure 2A.8: Same as figure 2A.6 but for the quartic-axion potential (2A.101).

  Figure 2A.9: Same as figure 2A.6 but for the monomial potential with n = 2 and m = 2 (2A.105).

  be expressed in terms of the new parameters β and K as k = (1condition k ≤ k means that we only have to study acute isosceles triangles 1/3 ≤ β ≤ 1.

Figure

  Figure 2C.1: The types of deformations of the momentum triangle we are considering. Left: Conformal transformation of the triangle. In the γ, β plane this corresponds to a constant (red) point. Right: Keeping the perimeter of the triangle K constant we change the shape of an isosceles γ = 0 triangle, moving along the bold red line for k 1 = k 2 ≥ k 3 .

Figure 2C. 2 :

 2 Figure 2C.2: Left: The relative change of the horizon-crossing first-order slow-roll parameters (solid curve), η (dot-dashed curve), η ⊥ (dotted curve) and χ (dashed curve) at t k as a function of the ratio k/k 60 of the horizon-exit scale to the scale that left the horizon 60 e-folds before the end of inflation, for the model (2C.13) with mass ratio m φ /m σ = 9. Right: The evolution of the first-order slow-roll parameters (solid curve), η (dot-dashed curve), η ⊥ (dotted curve) and χ (dashed curve) as a function of the number of e-foldings N -N 60 for the time interval around the turning of the fields, for the same model.

Figure 2C. 3 :

 3 Figure 2C.3: The evolution of the transfer functions v12 (left) and v22 (right) as a function of the number of e-foldings N -N 60 for the time interval around the turning of the fields and for different horizon exit scales, varying from top to bottom as k 60 × 10 (dotted curve), k 60 (solid curve) and k 60 /10 (dashed curve), for the model (2C.13) with mass ratio m φ /m σ = 9.

Figure

  Figure 2C.4: Left: The time evolution of the adiabatic R 1 and the isocurvature R 2 ratios of power spectra as a function of the number of e-foldings N -N 60 for scales k 60 /10 (dashed line), k 60 (solid line) and k 60 × 10 (dotted line). Right: The time evolution of the spectral index as a function of the number of e-foldings N -N 60 for scales k 60 /10 (dashed line), k 60 (solid line) and k 60 × 10 (dotted line). Both plots are made for the model (2C.13) with mass ratio m φ /m σ = 9.

  Figure 2C.5: The time evolution of f NL in terms of the number of e-foldings N -N 60 around the time of the turning of the fields, for equilateral (ω = 1) triangles with K = (3/2)k 60 (solid curve), K = (3/2)k 60 /10 (dashed curve) and K = (3/2)k 60 × 10 (dotted curve), for the model (2C.13) with mass ratio m φ /m σ = 9.

  Figure 2C.6: Left: The relative change of the final value of f NL arbitrarily normalized to one at the smallest value of K on the figure, as a function of K for equilateral triangles (ω = 1), calculated exactly (solid curve), using the analytical approximation (2C.24) (dashed red curve) and using the shape index (2C.29) (dotted black curve). Right: The logarithm of the final value of the exact bispectrum, similarly normalized, as a function of K for equilateral triangles (ω = 1). Both figures are for the quadratic model (2C.13) with mass ratio m φ /m σ = 9. The vertical dashed line corresponds to K = (3/2)k 60 .

  Figure 2C.7: Left: The time evolution of the conformal index n K (2C.25) around the time of the turning of the fields, for triangles with ω = 1 (black curves) and ω = 5/2 (red curves, below the ω = 1 curves), with perimeter K = (3/2)k 60 (solid curve), K = (3/2)k 60 /10 (dashed curve) and K = (3/2)k 60 × 10 (dotted curve). The three points on the left and on the right correspond to the analytical values of the index as calculated from (2C.30) and (2C.32) respectively. Right: The final value of the conformal index n K (2C.25) for triangles with ω = 1 (black curve) and ω = 5/2 (red dashed curve) as a function of K. Both figures are for the model (2C.13) with mass ratio m φ /m σ = 9.

  Figure 2C.8: The time evolution of f NL as a function of the number of e-foldings N -N 60 around the time of the turning of the fields, for a triangle with ω = 1 (solid curve), ω = 5/2 (dotted curve) and ω = 1000 (dashed curve), all with fixed perimeter K = 3/2k 60 , for the model (2C.13) with mass ratio m φ /m σ = 9.

Figure

  Figure 2C.9: Left: The final value of f NL normalised by the equilateral f NL as a function of ω for triangles with K = (3/2)k 60 calculated exactly (black curve), using the analytical approximation (2C.33) (dashed red curve) and using the shape index (2C.36) (dotted black curve). Right: The logarithm of the final value of the exact local bispectrum normalised by the local bispectrum computed on an equilateral triangle as a function of ω for triangles with K = (3/2)k 60 (black curve) and the same quantity assuming f NL scale independent (red dashed curve). Both figures are for the quadratic model (2C.13) with mass ratio m φ /m σ = 9.

  Figure 2C.10: Left: The time evolution of the shape index n ω (2C.37) around the time of the turning of the fields, for constant K = (3/2)k 60 (black curves) and K = (3/2)k 60 /10 (red curves, above the K = (3/2)k 60 curves at the end), and for the shapes ω = 5/2 (solid curve) and ω = 1000 (dashed curve). The points on the left correspond to the analytical values of the index as calculated from (2C.38), while the points on the right correspond to the values of the index as calculated from (2C.39) for ω 1. Right: The final value of the shape index n ω (2C.37) for constant K = (3/2)k 60 (black) and K = (3/2)k 60 /10 (dashed red) as a function of ω. Both figures are for the model (2C.13) with mass ratio m φ /m σ = 9.

  .13) withα p I (r) ≡ 2 π k 2 dk j (kr) g p I (k), β p I (r) ≡ 2 π k 2 dk j (kr) g p I (k) P (k). (3.14)

Figure 3 .

 3 Figure 3.1: The galactic (left) and point source mask (right) used in Table 3.2. The galactic mask is the f sky = 0.80 galactic mask included in the Planck 2013 release, and the point source mask is the one based on the Planck LFI 30 GHz channel from the 2013 release (with a 4σ threshold level), which has f sky = 0.96. The combined mask has f sky = 0.77.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: The top two panels show a zoom of a Gaussian CMB map with holes from the 2013 Planck HFI 100 GHz point source mask, before (left) and after diffusive filling in (right) with 2000 iterations. The bottom left panel shows what filling in the galactic mask looks like, with the bottom right panel showing the difference between the maps with and without filling in. The units are dimensionless (∆T /T 0 ).

  .50) For the mixed T and E components we analyzed only the combinations B T 2E ≡ T T E + T ET + ET T and B T E2 ≡ T EE + ET E + EET , with corresponding variance Var(B T 2E ) = Var(T T E) + Var(T ET ) + Var(ET T ) + 2 Cov(T T E, T ET ) + 2 Cov(T T E, ET T ) + 2 Cov(T ET, ET T ), and similarly for Var(B T E2

Figure 3 . 5 :

 35 Figure 3.5: Fitting the extreme values of the smoothed bispectrum. The horizontal axis gives the p-value and the vertical axis the maximum bin deviation in number of σ. The solid curves show the CDFs for the smoothed 64 3 cubes (figure on the left) for σ smoothing = 2, 5, and 10(respectively from top to bottom blue, green and red) and for the more realistic situation (figure on the right) for σ smoothing = 1, 2, and 3 as described in the main text. We show only the 5% most extreme values, since we are interested in the extreme tail of the distribution. The dashed curves show non-linear least square fits to these curves using the Ansatz where a distribution with a smaller number N eff degrees of freedom is used to approximate the distribution with correlations between bins. Here (from top to bottom in the figure on the left) N eff = 61900, 3709, and 351 are used to represent the 64 3 = 2.6 × 10 5 degrees of freedom with various degrees of correlation as the result of the smoothing. For the figure on the right the numbers are (from top to bottom) N eff = 8664, 1948, and 588 to represent the original 12727 bin-triplets. The fit provides a good approximation, especially at the small p-values that are of the greatest interest for this application.

  Figure3A.1: Angular power spectra (multiplied by ( + 1)) for the temperature (left) and polarization (right) obtained from purely adiabatic or purely isocurvature initial conditions. The amplitude and spectral index of the primordial power spectrum, as well as the cosmological parameters, on which the transfer functions depend, correspond to the WMAP7-only best-fit parameters (WMAP7-only best-fit parameters are used for all the figures and explicit computations in this paper).

Figure 3A. 2 :

 2 Figure 3A.2: The isocurvature cross power spectra (multiplied by ( +1) and in absolute value) with one component fixed to be the adiabatic one. The figure on the left shows temperature (TT), the one on the right polarization (EE).

Figure 3A. 3 :

 3 Figure3A.3: Same angular power spectra for the temperature and the polarization as in Fig.3A.1, except that the horizontal axis is now linear, and that the CDM isocurvature spectrum is multiplied by 10 -4 2 ( + 1) 2 instead of ( + 1).

2

 2 dk j (kr) g I (k) P ζ (k) . (3A.38)

Figure

  Figure 3A.8: The neutrino density isocurvature α S νd (r) as a function of for temperature (left) and polarization (right) (for the same values of r as in Fig. 3A.4).

Figure

  Figure3A.9: The neutrino density isocurvature ( +1)β S νd (r) as a function of for temperature (left) and polarization (right) (for the same values of r as in Fig.3A.4).

Figure

  Figure 3A.10: The neutrino velocity isocurvature α Sνv (r) as a function of for temperature (left) and polarization (right) (for the same values of r as in Fig. 3A.4).

Figure

  Figure 3A.11: The neutrino velocity isocurvature ( + 1)β Sνv (r) as a function of for temperature (left) and polarization (right) (for the same values of r as in Fig. 3A.4).

0 r 2

 02 Figure 3A.12: Plot of |b I,JK 1 2 3 | in the CDM isocurvature case, as functions of 3 , for 1 = 2 = 1300. The figure on the left shows the temperature-only TTT bispectra, the one on the right the pure polarization EEE bispectra.

Figure

  Figure 3A.13: Plot of |b I,JK 1 2 3 | in the neutrino density isocurvature case, as functions of 3 , for 1 = 2 = 1300. The figure on the left shows the temperature-only TTT bispectra, the one on the right the pure polarization EEE bispectra.

Figure 3A. 14 :

 14 Figure 3A.14: Plot of |b I,JK 1 2 3 | in the neutrino velocity isocurvature case, as functions of 3 , for 1 = 2 = 1300. The figure on the left shows the temperature-only TTT bispectra, the one on the right the pure polarization EEE bispectra.

  Figure3A.15: Evolution of the f NL parameter uncertainties as one increases the cut-off max , for the CDM isocurvature mode. The six thinner curves describe the situation if only temperature data is used, while for the six thicker curves both temperature and E-polarization data are included.

  Figure 3A.16: The function ( + 1)H K plotted as a function of ≤ 30 for different values of K. The figure on the left shows temperature (T), the one on the right polarization (E).

|

  Squeezed bispectrum EEEζ,ζζ exact ζ,ζζ approx. S c ,S c S c exact S c ,S c S c approx.

2 N

 2 similar to the purely adiabatic relation τ N L = f non-Gaussianity of the adiabatic and isocurvature perturbations, respectively, while ε ζS denotes the relative sign of ζ and S: ε ζS = 1 if they have the same sign, ε ζS = -1 otherwise. Interestingly, f I,ζζ N L and f I,SS N L share the same sign as µ I , whereas f I,ζS N L can have a different sign. The hierarchy between the coefficients f I,JK N L , I being fixed, depends on the relative amplitude of α and Ξ: f I,ζζ N L dominates if Ξ α, whereas f I,SS N L dominates if Ξ α.

  3A.84) while ε ζS = sgn(fr) and α finds µ ζ = 3/(2r), µ S = -9/(2r) and α = 9Ξ, with ε ζS = -1 and the amplitudes of non-Gaussianities depend only on the parameter r. By contrast, in the opposite regime r f c 1, µ ζ = 3/(2r), µ S = 9f /(2r 2

  Figure 3B.1: Unmasked (left) and masked (right) maps of thermal dust at 143 GHz from Commander(Planck 2015), using the common mask. On the left, the map is normalized using histogram equalization to highlight all the regions containing dust. On the right, the scale is linear, this shows that most of the signal is coming from the galactic plane near the mask. The range of the scale is also different.

Figure 3B. 2 :

 2 Figure 3B.2: This figure shows the dust (at 143 GHz) and the CMB (Planck 2015 best fit) temperature power spectra (including the beam and the noise for both) as a function of the multipole .

  still well within the observational bounds, but is very large compared to the predictions of standard slow-roll single-field inflation f local NL ∼ O(10 -2 ).

Figure 3B. 3 :

 3 Figure 3B.3: Left: observed binned bispectrum of the thermal dust as a function of the multipoles 1 and 2 for 3 ∈ [1092, 1149]. Right: theoretical bispectrum for the local shape in the case f local NL = 1 for the same bin of 3 . Note the difference of colour scale (both for maximal and minimal values).

Figure

  Figure 3B.4: Left: bispectral signal-to-noise of the thermal dust as a function of the multipoles 1 and 2 for three different bins of 3 . Right: same for the CMB map studied later in section 3B.3 for the same bins of 3 . Note the factor 10 difference in colour scale.

  Figure 3B.5: Masked maps of the different galactic foregrounds we consider in this paper (dust, free-free, AME, synchrotron) at 143 GHz derived using the Commander method from Planck 2015 data. Note the different colour scales.

  Figure 3B.6: The power spectra of the different foregrounds (dust, free-free, synchrotron and AME) and of the CMB at 143 GHz (including the 60 arcmin beam and the noise) as a function of the multipole .

Figure

  Figure 3B.7: Bispectral signal-to-noise of the different foregrounds for 3 ∈ [4, 6], 3 ∈ [71, 79] and 3 ∈ [255, 267]. Note the different colour scales.

Figure

  Figure 3B.8: Bispectral signal-to-noise of the synchrotron after subtracting the unresolved point sources contaminationfor 3 ∈ [4, 6], 3 ∈ [71, 79] and 3 ∈[255, 267]. Note the different colour scales.

Figure

  Figure 3B.9: Dust and noise power spectra for the map studied in section 3B.2.1 as a function of .

Figure 3B. 10 :

 10 Figure 3B.10: The different masks used in sections 3B.2 and 3B.3. On the left, the high resolution versions (n side = 2048) of the Commander mask (f sky = 0.822) in black only and of the common mask (f sky = 0.776) in black and orange. On the right, the masks have a low resolution (n side = 256) and have been obtained by degrading the resolution of the masks on the left. In the process their size has slightly increased (f sky = 0.804 and f sky = 0.745). This effect is easily visible for the point sources.

  Figure 3B.11: Power spectra of the different foregrounds using the common mask (solid line, f sky = 0.776) and the Commander mask (dashed line, f sky = 0.822). On the left, the high resolution dust map (n side = 2048) and on the right, the low resolution foreground maps (n side = 256). Dust Dust (low resolution) Free-free AME Synchrotron 0.90 0.85 0.88 0.91 0.11

Figure

  Figure 3B.12: Bispectral signal-to-noise of the different foregrounds for the same three bins of 3 as figures 3B.3 and 3B.7 using the Commander mask. Note the different colour scales.

  Figure 3B.14: Convergence of f NL in the independent analysis of the 100 Gaussian CMB simulations + dust map as a function of min for the local (top left), orthogonal (top right) and dust (bottom left) shapes, and as a function of max for the dust shape (bottom right). The blue dots correspond to the values determined by the binned bispectrum estimator when min (or max for the fourth plot) is inside the corresponding bin. The 68% confidence interval is given by the blue dashed lines. The horizontal red dashed line corresponds to the determined value of f NL using the whole multipole interval (from 2 to 2500 with 57 bins).

  Figure 3B.15: Distribution of the ratio of the observed variance over the theoretical prediction for the three different amounts of dust (0 % in blue, 75 % in green and 100 % in red). On the left, all the valid -triplets are included while on the right only very squeezed -triplets are shown (one small multipole 1 ≤ 13 and two large ones 2,3 ≥ 742).

  Figure 3B.17: Power spectrum of the raw 143 GHz map as well as the estimated noise power spectrum.

2 d

 2 Ωd Ω M 1 ( Ω)M 2 ( Ω)M 3 ( Ω)M 4 ( Ω )M 5 ( Ω )M 6 ( Ω ) .

S 2 ×S 2 dS 2 ×S 2 dS 2 ×S 2 d

 222222 where P is a Legendre polynomial, and the fact that the maps are real, to compute the two-point correlation functionM ( Ω)M ( Ω ) = m=m =-a m a * m Y m ( Ω)Y * m ( Ω ) = C δ 2 + 1 4π P ( Ω • Ω ). (3B.5)Then we can perform the integration of our example term (and the eight others follow the same computation):Ωd Ω M 1 ( Ω)M 2 ( Ω) M 3 ( Ω)M 4 ( Ω ) M 5 ( Ω )M 6 Ωd Ω P 1 (1)P 3 ( Ω • Ω )P 5 (1) (3B.6)This integral can be solved using well-known properties of Legendre polynomials. First, we have P (1) = 1 and then we can use the integral Ωd Ω P ( Ω • Ω ) = 0, (3B.7)

  .14)It is the inverse of the variance of the ratio of the observed and theoretical bispectra divided by B, B which is the denominator of the estimator for f NL and normalizes the sum of the weights to one. In other words,

  .60) which is a constant. Hence, G 22 * (defined as G 22 (t, t * )), G 23 * and v22 become proportional after a few e-folds of slow-roll. Then, after a few more e-folds of inflation, the approximation of constant slow-roll parameters stops to be valid and we can no longer consider A 32 and A 33 to be constants. However, by this time the proportionality between G 22 * , G 23 * , v22 and their derivatives G 32 * , G 33 * , v32 has been established, and because of the linearity of the differential equation (2.56), they will stay proportional until the end of inflation.The case of G 12 , G 13 and v12 is a little trickier. With η ⊥ being a constant, these functions are the primitives of G 22 , G 23 , v22 according to

  is easy to show that f = H 2 to find the slow-roll result (2A.23). During this initial period of slow roll, having a first-order equation as a very good approximation means that the second mode needed to solve the full equation rapidly becomes negligible. Once slow roll is broken, we only need to study how the remaining mode evolves.

	2 1σ -e 2 1φ ) + (3 + + 2η )e 1φ e 1σ	ḟ
	+ 2 η

⊥ (e 2 1σe 2 1φ ) + (6 + 2 2 + 4 η )e 1φ e 1σ f = 0. (2A.62) In the slow-roll regime, a first-order expansion of this equation gives e 1φ e 1σ ḟ + 2 e 1φ e 1σ f = 0, (2A.63) and then it

  Hence, this is a good solution when η ⊥ and η are large but is small compared to 1, hence during the turn. The solutions are then,

	's functions are:
	v22 =	H 2 e 1φ e 1σ H 2 * e 1φ * e 1σ * v22 = , v32 = H 2 H N e 1φ e 1σ , v32 = H 2	H N	-e 1φ e 1σ + η ⊥ (e 2 1σ -e 2 1φ ) ,	(2A.67)

* e 1φ * e 1σ * -2 e 1φ e 1σ + η ⊥ (e 2 1σe 2 1φ ) . (2A.65) The same integration as in the slow-roll case works to compute v12 : v12 = Z -Z * W * e 1φ * e 1σ * , (2A.66) with Z previously introduced in (2A.22).

The other interesting value α = 1 cancels every second-order term in the equation.

  of the basis to eliminate one of the factors. Having terms with these factors permits us to use equations (2A.10) and (2A.11) to eliminate the slow-roll parameters χ, W221 and W222 . Finally, we obtain:

	e 2 1φ e 2 1σ , e 1φ e 1σ (e 2 1σ -e 2 1φ ) or (e 2 1σ -e 2 1φ ) 2 . We use the relation (e 2 1σ -e 2 1φ ) 2 = 1 -4e 2 1φ e 2 1σ coming from the normalization
	* e 1 σ *	,	(2A.68)
	with S ≡ e 2 1φ -e 2 1σ .

2A.3.3 The g int equation during the turn

A first use of the Green's functions during the turn computed in the previous section is to insert them into (2.79) to simplify the right-hand side of the equation: r.h.s. ≡ K 22 (v 22 ) 2 +K 23 v22 v32 + K 33 (v 32 ) 2 . After this step, every term of r.h.s. has one factor depending on the basis components:

  .71)We separate our equations into parts easier to compare. We start by comparing the factors in front of the braces of each expression in (2A.71) and (2A.69) which are:

	H N	e 2 1φ * Ṽσσ * -e 2 1σ * Ũφφ * e 1φ * e 1σ *	and	(	H N	) 2 .	(2A.72)
		Ṽσσ *	H 2 * H 2	and	.			(2A.73)

After simplifying the common factor H/N and inserting N = H 2 * e 1φ * e 1σ * /H 0 from (2A.67), we use the quasi single-field initial conditions at horizon-crossing to write (2A.72) as

The discussion about the spectral index from around equation (2A.26) is still valid, because the only difference from the slow-roll regime is the value of v12e , but for a large enough value (larger than four) the dependence on v12e in

(2.34)

disappears and (2A.26) can be used. Hence, Ṽσσ * is typically of order 10 -2 , or at least not hugely smaller.

  In the slow-roll approximation (neglecting the higher-order term ξ ⊥ in h sr ), we end up with e 1φ e 1σ e 2 1φ V σσ + e 2 1σ U φφ and Ṽσσ * e 1φ e 1σ (V σσ -U φφ ) H 2 *

	+ e 2 1σ Ũφφ	H 4 e 2 1φ e 2 1σ H 4 * e 2 1φ * e 2 1σ *	=	2 3	e 2 1φ V σσ + e 2 1σ U φφ e 1φ e 1σ	H 2 e 1φ e 1σ H 4 * e 2 1φ * e 2 1σ *
	H 2 * e 2 1φ * e 2 1σ *	= -e 2 1φ 1 3	ξ ⊥ H 2 *	H 2 e 1φ e 1σ H 4 * e 2 1σ * 1φ * e 2	.
							(2A.76)

. (2A.75) Doing the same for the homogeneous part using the quasi single-field initial conditions recalled at the beginning of section 2A.3, as well as (1.18), we get:

h sr = -e 2

1φ * Ṽσσ * η ⊥ H 2 e 1φ e 1σ * Ṽσσ * e 1φ e 1σ ( Ṽσσ -Ũφφ ) -

  t k ) and hence G a2 ≈ va2 . Note that except for the era of the turning of the fields, the slow-roll assumption is a good approximation during inflation in this particular model. The slow-roll evolution equations for v12k and v22k are

	d dt	v12k = 2η ⊥ v22k	and	d dt	v22k = -χv 22k .	(2C.15)

  [Planck 2013 XXIV; Planck 2015 XVII; Planck 2018 IX] -The official Planck primordial non-Gaussianity papers, accompanying the three data releases in 2013, 2015, and 2018. A very complete analysis of the Planck CMB data regarding bispectral non-Gaussianity, as well as the consequences for inflation models. • [CORE Collaboration, 2018] -Detailed forecasts for what the proposed CORE satellite (Cosmic ORigins Explorer; finally not accepted by ESA) could do regarding constraints on inflation from both the power spectrum and the bispectrum of the CMB.3

  .41)

		Local Equil Ortho LensISW UnclustPS	CIB
	Local	1	0.21	-0.44	0.28	0.002	0.006
	Equilateral		1	-0.05	0.003	0.008	0.03
	Orthogonal			1	-0.15	-0.003	-0.001
	Lensing-ISW				1	-0.005	-0.03
	Unclustered point sources					1	0.93
	CIB point sources						1

  The fit provides a good approximation, especially at the small p-values that are of the greatest interest for this application.

	Smoothing		Before		After
	length	template cleaning template cleaning
	(σ smoothing ) X max p-value	X max p-value
	1 2	33.4 51.2	e -553 e -1308	5.1 3.6	1.5 × 10 -3 0.15
	3	59.2	e -1751	4.4	3.3 × 10 -3

Table 3 .

 3 3: Maximum bin values and associated p-values for the bispectrum of a simulated Gaussian CMB + point sources map before and after cleaning of the point source contribution, for various values of the smoothing length. See main text for a more detailed explanation.

  K,IJ P δφ (k 1 )P δφ (k 2 ) , (3A.31) (the summation over scalar field indices a, b, c and d is implicit), which are symmetric under the interchange of the last two indices, by construction. Since the adiabatic power spectrum is given by Pζ = (δ ab N ζ a N ζ b )P δφ ≡ AP δφ ,

				(3A.33)
	one obtains finally (3A.27) with	f I,JK NL	= λ I,JK N L /A 2 ,	(3A.34)
	with the coefficients
		λ I,JK ≡ δ ac δ bd N I ab N J c N K d	(3A.32)

  Table3A.3: Fisher matrix for the neutrino density isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the Fisher matrix components obtained without including the polarization.3A.4.4 Neutrino density isocurvature modeFor a neutrino density isocurvature mode, we have obtained the Fisher matrix in Table3A.3. Unlike the case of CDM isocurvature, here the difference between the different entries in the Fisher matrix is smaller, although the coefficients in the upper left 2 × 2 submatrix are still about one order of magnitude larger than the others. Also in contrast to the CDM isocurvature case, we see that all coefficients increase about equally when polarization is included.

							2.5 (1.8) × 10 -3
	-	-	11 (8.1) × 10 -4	12 (8.5) × 10 -4	1.6 (1.1) × 10 -3 4.4 (3.1) × 10 -4
	-	-	-	1.8 (1.1) × 10 -3 2.0 (1.3) × 10 -3 5.0 (3.2) × 10 -4
	-	-	-	-		2.5 (1.6) × 10 -3 6.8 (4.4) × 10 -4
	-	-	-	-		-	2.1 (1.2) × 10 -4
	The corresponding uncertainties on the six non-Gaussianity parameters are (taking into
	account the correlations) 5	∆ f i = {28, 36, 190, 150, 240, 320}.		(3A.67)
	When using temperature only, the uncertainties increase to
		∆ f i = {58, 75, 540, 340, 720, 950}	(no polarization).	(3A.68)

Table 3A .

 3A 4: Correlation matrix for the neutrino density isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the correlations obtained without including the polarization (the absence of sign in the parentheses means that it is unchanged with respect to the value taking into account the polarization). Evolution of the f NL parameter uncertainties as one increases the cut-off max , for the neutrino density isocurvature mode. The six thinner curves describe the situation if only temperature data is used, while for the six thicker curves both temperature and E-polarization data are included.

	(ζ, ζζ) 1. -----Figure 3A.18: (ζ, ζζ) 1. -----	(ζ, ζS) -0.83 (0.82) 0.42 (0.31) -0.73 (0.67) 0.26 (0.17) -0.16 (0.09) (ζ, SS) (S, ζζ) (S, ζS) (S, SS) 1. -0.58 (0.45) 0.58 (0.55) -0.31 (0.22) 0.25 (0.20) -1. 0.03 (0.35) -0.45 (0.69) 0.19 (0.41) --1. -0.76 (0.81) 0.58 (0.64) ---1. -0.84 (0.86) ----1. 10 1 10 2 10 3 max 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Errors on fNL for neutrino density isocurvature ζ,ζζ ζ,ζS nd ζ,S nd S nd S nd ,ζζ S nd ,ζS nd S nd ,S nd S nd (ζ, ζS) (ζ, SS) (S, ζζ) (S, ζS) (S, SS) -0.50 (0.42) 0.08 (0.11) -0.82 (0.66) 0.34 (0.34) -0.13 (+0.05) 1. 0.36 (0.39) -0.60 (0.80) 0.25 (-0.18) -0.26 (0.35) -1. 0.37 (0.44) -0.51 (+0.07) -0.29 (0.75) --1. 0.25 (-0.10) -0.74 (0.61) ---1. -0.46 (0.36) ----1.
		22, 85, 81, 77, 71}.	(3A.70)
	When using temperature only, the uncertainties increase to
		∆ f i = {51, 120, 460, 180, 190, 570}	(no polarization).	(3A.71)

  Evolution of the f NL parameter uncertainties as one increases the cut-off max , for the neutrino velocity isocurvature mode. The six thinner curves describe the situation if only temperature data is used, while for the six thicker curves both temperature and E-polarization data are included.

	10 6	Errors on fNL for neutrino velocity isocurvature
	10 5
	10 4
	10 3
	10 2	ζ,ζζ ζ,ζS nv
	10 1	ζ,S nv S nv S nv ,ζζ S nv ,ζS nv S nv ,S nv S nv
	10 0	10 1	10 2	10 3
			max
	Figure 3A.19:
				.73)

Table 3B

 3B

	Local Equilateral Orthogonal Lensing-ISW Point sources CIB
	Dust -0.6	0.004	0.15	-0.34	0.054	0.083

.1: Correlation coefficients between the standard theoretical templates and the observed dust bispectrum computed using the characteristics of the Planck experiment (temperature).

Table 3B .

 3B 2: Correlation coefficients between the standard theoretical templates and the observed foreground templates (low resolution) computed using the characteristics of the Planck experiment (temperature).

		Local Equilateral Orthogonal Lensing-ISW Point sources CIB
	Dust (low resolution) -0.14	0.0097	0.087	-0.036	0.0083	0.012
	Free-free	-0.44	-0.045	0.43	0.043	0.069	0.11
	AME	-0.23	0.032	0.052	-0.051	0.033	0.037
	Synchrotron	-0.057	0.33	0.29	0.051	0.44	0.38
			Dust (low resolution) Free-free AME Synchrotron
	Dust (low resolution)	1	0.24	0.28	0.56
	Free-free			1	0.37	0.32
	AME				1	0.32
	Synchrotron					1

Table 3B .

 3B 3: Correlation coefficients between the synchrotron bispectrum cleaned from the point sources contamination and the templates of table 3B.2 computed using the characteristics of the Planck experiment (temperature).

	Local Equilateral Orthogonal Lensing-ISW Point sources CIB
	Cleaned synchrotron -0.14	0.025	0.13	-0.022	0.059	0.033
		Dust (low resolution) Free-free AME Synchrotron
	Cleaned synchrotron	0.62	0.32	0.34	0.92

Table 3B

 3B .7: Means and standard deviations of the distributions of the ratio of the observed variance over the theoretical prediction, shown in figure3B.15, for the three amounts of dust, including the full bispectrum or only a very squeezed part of it.

		0 % dust	75 % dust	100 % dust
		Full Squeezed Full Squeezed Full Squeezed
	Mean	1.09	0.97	1.08	0.97	1.08	1.11
	Standard deviation 0.19	0.16	0.19	0.21	0.20	0.51
			100 % dust 75 % dust 0 % dust
	Local
	Fisher		6.6	6.4	5.6
	Observed	14	7.7	5.2
	Dust
	Fisher		0.05	0.04	0.031
	Observed	0.20	0.12	0.030

  LocalEquilateral Orthogonal P.S./10 -29 CIB/10 -27 Dust/10 -2 Lensing-ISW No ISW-lensing bias subtraction Table3B.10: Determination of f NL for the local, equilateral, orthogonal, point sources, CIB, dust and ISW-lensing shapes in the cleaned SMICA CMB map from the 2015 Planck release.

	Indep	8.7 ± 5.5	8 ± 67	-34 ± 33	9.6 ± 1.0	4.6 ± 0.5	-0.8 ± 3.8	0.59 ± 0.29
	Joint	6 ± 8	-21 ± 69	-3 ± 38	7.3 ± 2.7	1.2 ± 1.4	-2.2 ± 5.2	0.57 ± 0.31
	Joint \ dust 4.2 ± 6.7	-15 ± 68	-6.6 ± 37	7.2 ± 2.7	1.3 ± 1.4		0.55 ± 0.31
	ISW-lensing bias subtracted
	Indep	1.2 ± 5.5	6 ± 67	-8 ± 33	9.6 ± 1.0	4.6 ± 0.5	-4.0 ± 3.8
	Joint	-5 ± 8	-16 ± 69	1 ± 38	7.1 ± 2.7	1.3 ± 1.4	-3.5 ± 5.1
	Joint \ dust 1.0 ± 6.3	-7 ± 68	-5 ± 37	7.0 ± 2.7	1.4 ± 1.4

Table 3B .

 3B LocalEquilateral Orthogonal P.S./10 -29 CIB/10 -27 12: Determination of f NL in the raw 143 GHz map at high resolution n side = 2048 from the 2015 Planck release. The analysis is performed with the usual choice of 57 bins. For the details on the error bars for the primordial and the dust shapes, see the main text. The CIB and point sources error bars are Fisher forecasts.

								Dust
	Indep Joint	-61 ± 13 13 ± 18	22 ± 71 -37 ± 81	-12 ± 39 -81 ± 47	90 ± 4 115 ± 9	28 ± 1 -11 ± 3	1.09 ± 0.25 1.08 ± 0.32
				Local	Equilateral Orthogonal	Dust	Free-free
		Indep Joint	-41 ± 54 232 ± 198 -33 ± 94 543 ± 536	277 ± 177 0.88 ± 0.28 39 ± 50 112 ± 388 0.85 ± 0.39 -9 ± 69

Table 3B .

 3B 13: Determination of f NL in the raw 143 GHz map at low resolution n side = 256 from the 2015 Planck release. The analysis is performed with 39 bins. The error bars are all Fisher forecasts.

  obs 1 2 3 B obs 4 5 6 = d Ωd Ω [ M 1 M 4 M 2 M 5 M 3 M 6 + (14)(26)(35) +(15)(24)(36) + (15)(26)(34) + (16)(24)(35) + (16)(25)(34)] ,

	(3B.12)

In this whole thesis natural units will be used, in which c = h = kB = 1.

In the literature the use of the symbol N for the number of e-folds is more common, with t being reserved for cosmic time. As in this work cosmic time will not be used and N is already used for the lapse function, we denote the number of e-folds by t.

In the case of other time coordinates, the expansion information of the universe is encoded in a while H is directly derived from it using H ≡ ∂ta/(N a). However, when using the number of e-folds as time coordinate, a is a trivial function, and the expansion information is encoded in H, which can in this case not be derived from a.

A version of this basis in the case of two fields only and a trivial field metric was independently proposed in[START_REF] Gordon | Adiabatic and entropy perturbations from inflation[END_REF].

This remark is exact when acceleration in terms of cosmic time is considered. When using the number of e-folds as time coordinate, as we do here, there is a correction term as seen in (1.7). However, that correction disappears for η ⊥ .

In the single-field case, and in a gauge where the field perturbation is zero (e.g. uniform energy density gauge), it is easy to see from (1.26) and (1.31) that ζ = -Ψ (using also that √ 2 = κφ /H). And Ψ is related to the intrinsic spatial curvature on hypersurfaces of constant conformal time as(3) R = 4(∇ 2 Ψ)/a 2 , hence the name curvature perturbation for ζ.

This is a much lower temperature than one would naively expect from the ionisation energy of hydrogen of 13.6 eV. The reason is that there are about

9 times more photons in the universe than protons and electrons. Hence even just the high-energy tail of a lower temperature photon distribution can keep all hydrogen ionized.

The I, Q, and U defined in (1.48) have the dimension of electric field squared. In fact, by multiplying them with the constant 1 2 cε0 they get the dimension of flux (I is then exactly the intensity of a plane wave,

Looking at the linear solution (1.33) for ζ it is easy to see why the three-point correlator would be zero in the linear case: one would be computing the vacuum expectation value of an odd number of creation and annihilation operators. Once you include second-order terms, however, you will be able to take one second-order ζ with two creation/annihilation operators and two linear ζ with one each, for a total of four, which can have a non-zero vacuum expectation value.

https://lambda.gsfc.nasa.gov/product/cobe/

Initially called MAP, it was renamed in 2003 in honour of David Wilkinson, a member of its science team who died in 2002. https://lambda.gsfc.nasa.gov/product/map/current/

Despite many proposals in Europe and in the US, the only future CMB satellite proposal that has been accepted by a space agency so far is the Japanese LiteBIRD satellite (http://litebird.jp/eng/), with a tentative launch date of 2028. LiteBIRD stands for "Lite (Light) satellite for the studies of B-mode polarization and Inflation from cosmic background Radiation Detection".

https://www.cosmos.esa.int/web/planck

Because of the large size of the Planck collaboration, a two-tier system regarding authorship of the papers was used. People who had contributed enough to the collaboration as a whole (e.g. by performing tests of general usefulness or producing products or creating tools for general use, and not just working on their own scientific analysis) obtained the status of Planck Scientist and could sign all Planck collaboration papers, while other collaboration members could only sign the papers they had contributed to directly. It was also decided to keep the order of authors strictly alphabetical for all papers, so that without inside information it is impossible to identify the main authors of a given paper.

The work from this paper is not included in this thesis, for two reasons: in order not to make this thesis even longer, and because this paper is basically only the first step in a generalization of our formalism to inflation models with general kinetic terms, which has so far not been followed up on.

More formally this can be viewed as taking only the leading-order terms in the so-called spatial gradient expansion (see[START_REF] Salopek | Nonlinear evolution of long wavelength metric fluctuations in inflationary models[END_REF][START_REF] Comer | Growth or decay of cosmological inhomogeneities as a function of their equation of state[END_REF][START_REF] Giovannini | Gradient expansion(s) and dark energy[END_REF][START_REF] Tanaka | Gradient expansion approach to nonlinear superhorizon perturbations[END_REF] and references therein). While very plausible, the validity of this approximation beyond linear order was only assumed at the time of our first papers. In a later paper[START_REF] Tzavara | Gauge-invariant perturbations at second order in two-field inflation[END_REF], included in appendix 2B, we showed by an exact calculation at second order that at least up to and including second order the approximation is valid on super-horizon scales once the decaying perturbation mode can be neglected. If slow roll holds at horizon crossing, this happens within a few e-folds.

The fully non-linear equations of motion for ζ A i and θ A i , with the constraint equations for the coefficients to close the system and the explicit stochastic expression for the source terms, is amenable to direct numerical solution on a multi-dimensional spatial grid to give fully non-linear realizations of ζ A i (t, x). The methodology for this is briefly described in[Rigopoulos et al., 2006a].

To be precise, the Green's function is actually defined as the solution of (2.29) with δ(t -t ) on the right-hand side instead of zero. The solution is then a step function times what we call the Green's function. This step function has been taken into account by changing the upper limit of the integral in (2.28) from ∞ to t.

The factor 1/(1 - * ) is not present in (1.42) because that expression was given only to leading order in slow roll.

The factor v12/(1 + v2 12 ) also goes to zero for |v12| → ∞. However, while this term in (2.34) would then be compatible with a large η ⊥ * , that is forbidden by the other terms.

For comparison we also looked at the case of a product potential. As was shown before, in that case one cannot get large non-Gaussianity at all in the slow-roll approximation and with a vanishing isocurvature mode at the end of inflation.

Obviously, with the

Planck release, these are no longer the most recent results. However, I have decided

It has been pointed out[START_REF] Tanaka | Dominance of gauge artifact in the consistency relation for the primordial bispectrum[END_REF][START_REF] Pajer | The Observed Squeezed Limit of Cosmological Three-Point Functions[END_REF] that the finite size of the observable universe leads to gauge corrections, which have to be taken into account to convert the inflationary bispectrum to actual observations. Indeed in single-field inflation the squeezed limit of the bispectrum vanishes identically for a local observer today. In multiple-field inflation, on the other hand, these corrections are also of order 1 -ns[START_REF] Tada | Squeezed Bispectrum in the δN Formalism: Local Observer Effect in Field Space[END_REF] and hence are expected to be negligible in the case of large fNL.

In fact the paper[START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF]] had an additional aim, the continuation of the work on the longwavelength formalism, but as that part of the paper has been incorporated into section 2.2 and is not reproduced here, I have removed the corresponding paragraph from this introduction.

These are the same terms we called order1,2,3 before, however now with the particular solution substituted and not the homogeneous one.

Again I have decided to keep the original text and references from the paper, which dates from

2012-13.

Due to the exact relation θ1 = 2η ⊥ ζ2, there is no need to consider the velocity of the adiabatic perturbation θ1 as an additional variable[START_REF] Rigopoulos | Quantitative bispectra from multifield inflation[END_REF].

In[Tzavara and Van Tent, 2011] we had also a fourth contribution g k , denoting the terms that vanish for an equilateral triangle. Here we have incorporated these terms in gsr (the last two lines), since they are also slow-roll suppressed.

In addition, in the equilateral case k = k the γ k ratios in (2C.5) reduce to 1 and the two terms in the brackets of (2C.5) become identical (apart from the factor 2). See(2.38-2.39) for the full expression of fNL in the equilateral case.

An additional paper[Montandon et al.,

2021] was published recently in collaboration with Guillaume Patanchon and Thomas Montandon. In that paper we perform a joint analysis of the power spectrum and the bispectrum of the CMB to improve the constraints on isocurvature modes. We analyze both the existing Planck data and make forecasts for future experiments. As that work was not yet completed when this thesis was written, it is not included.2 This paper (from 2015) was long delayed because of work for Planck: most of these refinements (except the generalization to polarization) were already in place when work for the first Planck analysis started at the end of 2011.3 Despite the fact that CORE was not accepted, these analyses remain useful in the context of other future missions. However, in order not to make this thesis even longer, this work is not included.

The skew-C extension[START_REF] Munshi | A New Approach to Probing Primordial Non-Gaussianity[END_REF] of the KSW estimator allows the determination of a so-called bispectrum-related power spectrum, which contains the contribution to fNL (for a given shape) of all triangles with one side equal to .

Both the bispectrum (3.1) and the power spectrum (1.52) can be defined either in terms of the gravitational potential Φ or in terms of the curvature perturbation ζ, related by ζ = -5 3 Φ during recombination. For the

In our papers[START_REF] Bucher | Detecting Bispectral Acoustic Oscillations from Inflation Using a New Flexible Estimator[END_REF][START_REF] Bucher | The binned bispectrum estimator: template-based and non-parametric CMB non-Gaussianity searches[END_REF] we defined a quantity N instead of h, equal to the square of h. However, the quantity h given in(3.2) is by now more commonly used in the literature.

In the literature one often encounters the reduced bispectrum b 1 2 3 instead of the angle-averaged bispectrum B 1 2 3 , defined as b 1 2 3 = B 1 2 3 /h 2 1 2 3 , but we will not consider it in this thesis.

Large-scale modes are much less affected by the mask. Since these modes extend out over large parts of the sky, they can be reconstructed reasonably accurately even when some parts of the sky are missing. Furthermore, edge effects are also less important for a mask with larger holes. Consequently for a high-resolution experiment like Planck the use of inpainting algorithms has turned out to be absolutely crucial, while for the lower resolution WMAP experiment, which moreover had larger error bars, less care was required.

There are indications that the binned bispectrum estimator might even perform well for oscillating templates that do not satisfy these criteria. For the so-called constant feature model[Planck 2015 XVII] with a primordial bispectrum proportional to sin(ω(k1 + k2 + k3) + ϕ)/(k1k2k3) 2 , taking ϕ = 0 and ω = 100, we find an overlap of 94% for T -only with the standard Planck binning (i.e. not optimized for this template). This will be investigated in more detail in the future.

http://cc.in2p3.fr

http://healpix.sourceforge.net

In an earlier version of the code these filtered maps of the observed map, which are also produced during the cubic calculation, were saved to disk at that time, and then read in here. However, the required I/O turned out to make this actually slower than when these filtered maps are recreated on the fly, which also has the advantage of using much less disk space.

This results because the pixelization breaks the spherical symmetry as must be the case with any pixelization of the sphere.

Once more, I have decided to keep the original text and references from the paper, which dates from before Planck and even from before the final WMAP release.

Our notation differs from that of[WMAP Collaboration, 2011]: our a corresponds to their α and our fully correlated limit corresponds to their fully anti-correlated limit, because their definition of the correlation has the opposite sign (see also[WMAP Collaboration, 2009] for a more detailed discussion).

We use the standard notation: ( 1 2 3) ≡ [ 1 2 3 + 5 perms]/3!.

The tiny differences in the 3rd,

5th, and 6th value compared to[Langlois and Van Tent, 2011] are due to small improvements in the computer code.

While we were finalizing our manuscript, we became aware of the work[START_REF] Kawakami | Non-Gaussian isocurvature perturbations in dark radiation[END_REF], where the authors also investigate neutrino density isocurvature non-Gaussianity. Their numbers for the uncertainties are very similar to ours (note that they use the six non-Gaussianity parameters that we introduced in [Langlois andVan Tent, 2011] but in a different ordering), although they use a different selection of Planck channels.

It should be noted that for the purpose of studying the non-Gaussianity of a galactic foreground in itself, the bispectrum would probably not be the best tool. Due to their localized (non-isotropic) nature, an approach in pixel space instead of harmonic space would seem more logical. Minkowski functionals, for example, have been used to study galactic synchrotron radiation in the context of

21-cm line studies[START_REF] Rana | Non-Gaussianity of diffuse Galactic synchrotron emission at 408 MHz[END_REF]. However, in this paper we are primarily interested in seeing how much these foregrounds contaminate a determination of primordial non-Gaussianity, which is in general isotropic and for which the bispectrum is then an optimal tool.

https://pla.esac.esa.int

The difference between the values in this table and those in the Planck paper[Planck 2015 XVII], in particular for equilateral, is mainly due to our use here of a slightly different mask (the preferred temperature mask from[Planck 2015 IX] instead of the slightly extended mask used in[Planck 2015 XVII]).

The generalization of this full cubic action to the case of non-standard kinetic terms was computed in[Tzavara et al.,

2014], but that work is not included in this thesis.2 Recently the paper[START_REF] Montandon | Isocurvature modes: joint analysis of the CMB power spectrum and bispectrum[END_REF] was published, where we look in even more detail at the Planck constraints on isocurvature modes, by performing a joint analysis of the power spectrum and the bispectrum, but that work is not part of this thesis.

Appendix 2B

Gauge-invariant perturbations at second order in two-field inflation This appendix contains the full paper [Tzavara and Van Tent, 2012], except for the conclusions that were used as a summary in section 2.3.2. It was written in collaboration with Eleftheria Tzavara.

We study the second-order gauge-invariant adiabatic and isocurvature perturbations in terms of the scalar fields present during inflation, along with the related fully non-linear spatial gradient of these quantities. We discuss the relation with other perturbation quantities defined in the literature. We also construct the exact cubic action of the second-order perturbations (beyond any slow-roll or super-horizon approximations and including tensor perturbations), both in the uniform energy-density gauge and the flat gauge in order to settle various gauge-related issues. We thus provide the tool to calculate the exact non-Gaussianity beyond slow-roll and at any scale.

Appendix 2C

Momentum dependence of the bispectrum in two-field inflation This appendix contains the paper [Tzavara and [START_REF] Tzavara | Momentum dependence of the bispectrum in two-field inflation[END_REF], except for the conclusions that were used as a summary in section 2.3.3. In addition, most of section 2 of the paper has also been removed (and the remainder combined with section 3), as it only summarized equations that have been given earlier in this thesis. The paper was written in collaboration with Eleftheria Tzavara.

We examine the momentum dependence of the bispectrum of two-field inflationary models within the long-wavelength formalism. We determine the sources of scale dependence in the expression for the parameter of non-Gaussianity f NL and study two types of variation of the momentum triangle: changing its size and changing its shape. We introduce two spectral indices that quantify the possible types of momentum dependence of the local type f NL and illustrate our results with examples.

2C.1 Introduction

The study of inflationary non-Gaussianities and their impact on the cosmic microwave background has been an important subject of cosmological research in recent years. In nine years of WMAP data [WMAP Collaboration, 2013] and one year of data from the Planck satellite [Planck 2013 XXIV] no primordial non-Gaussianity of the local and equilateral types (see below) was observed, and constraints have tightened considerably. Next year's Planck release 1 is expected to put even tighter constraints on those types of non-Gaussianity, as well as investigate many additional types (which differ in their momentum dependence). The importance of the current constraints and a future possible detection or further improvements of the constraints lies in the fact that they allow us to discriminate between different classes of models of inflation, since these predict different types and amounts of non-Gaussianity.

There are basically two distinct types of non-Gaussianity that are most important from the point of view of inflation: the equilateral type produced at horizon-crossing, which has a quantum origin and is maximal for equilateral triangle configurations [START_REF] Creminelli | Limits on non-Gaussianities from WMAP data[END_REF], and the local type produced outside the inflationary horizon due to the existence of interacting fields. The latter is maximized for squeezed triangles, i.e. isosceles triangles with one side much smaller than the other two [START_REF] Komatsu | Acoustic signatures in the primary microwave background bispectrum[END_REF]Babich and Zaldarriaga, 2004]. The first type is known to be slow-roll suppressed for single-field models with standard kinetic terms and trivial field metric [START_REF] Maldacena | Non-Gaussian features of primordial fluctuations in single field inflationary models[END_REF]. On the other hand, some models with non-standard kinetic terms coming from higher-dimensional cosmological models are known to produce non-Gaussianity of the equilateral type so large that it is not compatible with WMAP and Planck observations [START_REF] Alishahiha | DBI in the sky[END_REF][START_REF] Silverstein | Scalar speed limits and cosmology: Acceleration from D-cceleration[END_REF][START_REF] Mizuno | Lorentz boost and non-Gaussianity in multi-field DBIinflation[END_REF]

3A.2 Isocurvature perturbations

In this section, we recall the definition of isocurvature modes in the context of linear cosmological perturbations. At the time of last scattering, the main components in the universe are the CDM (c), the baryons (b), the photons (γ) and the neutrinos (ν). All these components are characterized by their individual energy density perturbation δρ i and their velocities V i (as well as higher momenta of their phase space distribution functions, which we do not discuss here; see [START_REF] Ma | Cosmological perturbation theory in the synchronous and conformal Newtonian gauges[END_REF] for details). The "primordial" perturbations for each Fourier mode k are usually defined on super-Hubble scales, i.e. when k aH, deep in the radiation dominated era.

The most common type of perturbation is the adiabatic mode, characterized by

which means that the number of photons (or neutrinos, or CDM particles) per baryon is not fluctuating. In terms of the energy density contrasts (δ ≡ δρ/ρ), the above condition is expressed as

where the 3/4 factor, for photons and neutrinos, comes from the relation ρ ∝ n 4/3 for relativistic species.

Assuming adiabatic initial conditions is natural if all particles have been created by the decay of a single degree of freedom, such as a single inflaton, and, so far, the CMB data are fully compatible with purely adiabatic perturbations. However, other types of perturbations can be included in a more general framework. In addition to the adiabatic mode, one can consider four distinct types of so-called isocurvature modes [START_REF] Bucher | The General primordial cosmic perturbation[END_REF]: the CDM isocurvature mode S c , the baryon isocurvature mode S b , the neutrino density isocurvature mode S νd and the neutrino velocity isocurvature mode S νv . At zeroth order in kτ , where τ is the conformal time, the first three are characterized, respectively, by

while the corresponding velocities tend to zero. As for the neutrino velocity isocurvature mode, it is characterized by non vanishing "initial velocities",

where V γb is the common velocity of the photon-baryon plasma (the photons and baryons are initially tightly coupled via the Thomson scattering off free electons) and N ν is the number of species of massless neutrinos. The above relation between the two velocities ensures that they exactly cancel in the momentum density, while the energy densities satisfy the adiabatic condition (3A.2).

In the following, each mode will be characterized by its amplitude: the curvature perturbation on constant energy hypersurfaces, ζ, for the adiabatic mode, and S c , S b , S νd and S νv for the four isocurvature modes. These variables will be denoted collectively as X I . In the context of inflation, a necessary condition for these isocurvature modes to be created is that several light degrees of freedom exist during inflation. Moreover, since the adiabatic and isocurvature modes can be related in various ways to these degrees of freedom during inflation, one can envisage the existence of correlations between these modes. the integrand in (3A.39) is dominated by (twice)

, whereas the first term is negligible: one can see from Fig. 3A.4 and 3A.5 that β ζ 1 (r) is much larger (in absolute value) than α ζ 1 (r). This is true both for temperature and polarization (one should also keep in mind that the β's have been multiplied by ( + 1) in the plots, which makes them look much larger at large ).

The purely isocurvature bispectrum has exactly the same structure as (3A.39), but with the functions α S (r) and β S (r) replacing α ζ (r) and β ζ (r). Moreover, the shapes of α S (r) and β S (r) depend on the type of isocurvature mode: Fig. 3A.6 and Fig. 3A.7 correspond to the CDM isocurvature mode, Fig. 3A.8 and Fig. 3A.9 to the neutrino density isocurvature mode and Fig. 3A.10 and Fig. 3A.11 to the neutrino velocity isocurvature mode. The functions α S (r) and β S (r) for the baryon isocurvature mode can be deduced from the CDM isocurvature functions by a simple rescaling, according to (3A.11):

The other bispectra depend on a mixing of the adiabatic and isocurvature functions. For example, one finds Table 3A.5: Fisher matrix for the neutrino velocity isocurvature mode. Only the upper half coefficients are indicated, since the matrix is symmetric. The value between parentheses corresponds to the Fisher matrix components obtained without including the polarization.

Appendix 3B

The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation

This appendix contains the paper [START_REF] Jung | The bispectra of galactic CMB foregrounds and their impact on primordial non-Gaussianity estimation[END_REF], with the exception of its section 2 (which only contains a summary of the binned bispectrum estimator as presented earlier in this thesis) and of its conclusions, used in the summary in section 3.3.2. This paper was written in collaboration with Gabriel Jung and Benjamin Racine. We use the binned bispectrum estimator to determine the bispectra of the dust, free-free, synchrotron, and AME galactic foregrounds using maps produced by the Commander component separation method from Planck 2015 data. We find that all of these peak in the squeezed configuration, allowing for potential confusion with in particular the local primordial shape. Applying an additional functionality implemented in the binned bispectrum estimator code, we then use these galactic bispectra as templates in an f NL analysis of other maps. After testing and validating the method and code with simulations, we show that we detect the dust in the raw 143 GHz map with the expected amplitude (the other galactic foregrounds are too weak at 143 GHz to be detected) and that no galactic residuals are detected in the cleaned CMB map. We also investigate the effect of the mask on the templates and the effect of the choice of binning on a joint dust-primordial f NL analysis. 2 × 10 -10 -2 × 10 -11 -1 × 10 -13 7 × 10 -14 1 Table 3B.5: Determination of f NL for the local, equilateral, orthogonal, point sources, CIB and dust shapes using the high-resolution dust map studied in section 3B.2.1. The only error bars at our disposal are Fisher forecasts; they are not indicated because for every case given here, they are several (at least three) orders of magnitude smaller than the determined values for f NL in the independent case, which makes them many orders of magnitude larger than the non-dust values in the joint analysis.

3B.3 Data analyses

The aims of this section are twofold. First, we want to verify that the numerical templates we just determined can be used in the context of a CMB data analysis. The first basic test to check this is to apply the template to the map it comes from. The expected answer for the amplitude parameter of this specific shape is then obviously f NL = 1. Moreover, if we perform a correlated analysis with other shapes like the primordial ones, their own f NL has to be negligible. Indeed, that is what we observe (see table 3B.5) and we can now discuss more interesting tests based on CMB maps. It is important to recall that the galactic foregrounds are highly anisotropic while most other shapes have an isotropic origin (primordial, lensing-ISW or extra-galactic foregrounds). These galactic numerical templates also contain mask and noise effects, but we will show that it is not an issue. For this, we ran a series of tests with the simple idea of artificially adding dust to the maps containing CMB realizations (simulations, but also the observed data) to check that we indeed detect the right amount of dust and that it has no impact on the other shapes.

Then, we will focus on the second aim which is to analyze the CMB map from the 2015 Planck data. We will apply the numerical templates to the cleaned SMICA CMB map [Planck 2015 IX], both at low and high resolutions, for which we expect not to detect any galactic foregrounds. Finally we will perform a similar analysis on raw sky observations at 143 GHz. Table 3B.9: Determination of f NL for the local, equilateral, orthogonal, point sources, CIB and dust shapes using a set of 100 Gaussian simulations of the CMB with anisotropic noise to which we added a known amount of dust (the high-resolution dust map of section 3B.2.1 multiplied by a factor 1 or 0.75, or no dust at all). The analysis is performed using 70 bins and the error bars are given at 1σ. Appendix 3C

Summary and conclusion

In this thesis my work on non-Gaussianity in cosmology has been presented. It has two main parts: 1) the development of the long-wavelength formalism to compute the non-Gaussianity produced in multiple-field inflation models and the application of this formalism to study those models; and 2) the development of the binned bispectrum estimator to extract information about non-Gaussianity from CMB data and its application to the Planck data.

The development of the long-wavelength formalism was a long process that took place over many years, first in collaboration with Gerasimos Rigopoulos and Paul Shellard, and later with Eleftheria Tzavara. The formalism reached its final form in [START_REF] Tzavara | Bispectra from two-field inflation using the long-wavelength formalism[END_REF]. Based on the work from all my papers on this subject, this thesis provides a detailed description and derivation of the long-wavelength formalism.

The formalism is based on the long-wavelength approximation valid on super-horizon scales (which boils down to neglecting second-order spatial gradients in the equations of motion) and a smart choice of gauge-invariant variables constructed from spatial gradients. In this way the fully non-linear equations of motion of the fluctuations on super-horizon scales are determined, and can be used for numerical simulations. The set of equations is completed with source terms describing the continuous influx of sub-horizon perturbations into the long-wavelength system. For analytic results a perturbative approach is required, and in this thesis the full second-order calculation for two-field inflation of the (local) bispectrum and its amplitude parameter f NL is given, the final result being (2.38).

This expression for f NL contains three types of terms: terms that are slow-roll suppressed, terms that are directly proportional to the isocurvature mode, and an integrated contribution. The isocurvature terms can easily become very large, but have the practical disadvantage that with an isocurvature mode still present at the end of inflation, the adiabatic perturbation will continue to evolve on super-horizon scales even after inflation. Computing any observational predictions for the CMB would then require to follow their evolution during (p)reheating and afterwards, a major challenge. Hence we always assume these terms to have disappeared by the end of inflation, so that the only potentially detectable non-Gaussianity comes from the integrated contribution.

Some study of the consequences of this integrated expression for non-Gaussianity predictions of two-field inflation models (and classes of models) was included in earlier papers, but a much more complete analysis was presented in [START_REF] Jung | Non-Gaussianity in two-field inflation beyond the slow-roll approximation[END_REF] in collaboration with Gabriel Jung. That paper is included in one of the appendix chapters of this thesis. After first deriving an alternative expression for the integrated contribution to f NL , we investigate the class of twofield sum potentials in great detail. We show that the region in parameter space where f NL is large enough to be detected in the near future, but that is compatible with the current Planck constraint on the spectral index, is quite limited, although we do manage to construct such a model. We also extend our analytic results to models where slow roll is broken during the turn of the field trajectory, and show why the slow-roll conclusions remain valid even in that case.

The other two papers included in the appendix chapters of the first part of the thesis are