David Wands

Local non-Gaussianity from inflation

The non-Gaussian distribution of primordial perturbations has the potential to reveal the physical processes at work in the very early Universe. Local models provide a well-defined class of non-Gaussian distributions that arise naturally from the non-linear evolution of density perturbations on super-Hubble scales starting from Gaussian field fluctuations. I describe the δN formalism used to calculate the primordial density perturbation on large scales and then review several models for the origin of local primordial non-Gaussianity, including the cuvaton, modulated reheating and ekpyrotic scenarios.

Introduction

The common presumption that primordial density perturbations have a Gaussian distribution is a powerful simplifying assumption that allows one to specify all the properties of the distribution once the two-point correlation function is known in real space, or equivalently the power spectrum in Fourier space. In particular the threepoint and connected higher moments of the distribution vanish. On the other hand the statement that a distribution is non-Gaussian opens up an infinite array of possibilities. This has led to an assortment of empirical tests for non-Gaussianity of the primordial perturbations. By contrast there are relatively few non-Gaussian distributions that are motivated by theoretical models for the origin of structure in the early universe. While one can argue that a Gaussian distribution could describe density perturbations arising from a wide range of possible sources, any detection of deviations from a Gaussian distribution predicted by a specific theoretical model would be strong evidence in support of that model.

Vacuum fluctuations in light, weakly-coupled scalar fields during a period of inflation (defined as accelerated expansion, ä > 0) in the very early universe provide a natural origin for an almost Gaussian distribution of vacuum fluctuations on large scales. Fluctuations in a free quantum field on small scales with comoving wavenumber, k, are swept up to scales much larger than the comoving Hubble scale, H -1 /a = 1/ ȧ, which shrinks during inflation. On super-Hubble scales (k < aH) damping drives the fluctuations into a squeezed state and they can effectively be treated as a classical random distribution [1]. Models of slow-roll inflation in the early universe are driven by canonical fields whose self-interactions are small relative to the Hubble scale and hence their distribution on scales close to the Hubble scale remains well described by a Gaussian distribution with vanishing third-and higher-order connected functions [2,3].

As inflation continues to stretch the perturbations up to scales far larger than the Hubble scale, spatial gradients are expected to become negligible and the we may treat the evolution locally as effectively "separate universes" [4,5] whose initial conditions are set by the Gaussian distribution of the scalar field values during inflation. This motivates the study of local models of primordial non-Gaussianity where the distribution of primordial density perturbations, ζ, can be described by a local function of one or more Gaussian random fields, ζ(δϕ I ). A linear function of Gaussian fields is itself a Gaussian, so a non-Gaussian distribution implies non-linearity and the study of the non-linear cosmological perturbations. This local model for non-Gaussianity turns out to be a very good description of non-Gaussianity in some simple physical models for the origin of structure in the very early universe.

Conventionally the primordial perturbations are characterised by the Newtonian potential in the matter-dominated era, Φ = (3/5)ζ. In Fourier space we define the power spectrum and bispectrum as

Φ k 1 Φ k 2 = (2π) 3 P Φ (k 1 )δ 3 (k 1 + k 2 ) , (1)
Φ k 1 Φ k 2 Φ k 3 = (2π) 3 B Φ (k 1 , k 2 , k 3 )δ 3 (k 1 + k 2 + k 3 ) , (2)
and the amplitude of the bispectrum relative to the power spectrum is then given by the dimensionless parameter

f NL (k 1 , k 2 , k 3 ) = B Φ (k 1 , k 2 , k 3 ) 2 [P Φ (k 1 )P Φ (k 2 ) + P Φ (k 2 )P Φ (k 3 ) + P Φ (k 3 )P Φ (k 1 )] . (3)
In a perturbative expansion, the second-order expression for the Newtonian potential as a local function of a single Gaussian field, φ, is given by [6] Φ

(x) = φ(x) + f NL φ 2 (x) -φ 2 + . . . . (4)
In this special case f NL is, by construction, a constant parameter independent of spatial position or scale. However in more general local models the ratio of the bispectrum to the power spectrum given by Eq. ( 3) may be taken as a definition of f NL which is then scale and shape-dependent. Note that following Komatsu and Spergel [6] we adopt a sign convention for the Newtonian potential Φ, and hence f NL , which is the opposite of that used by, for example, Mukhanov et al [7] and Maldacena [2].

In section 2 I review the δN -formalism that is commonly used to calculate the primordial density perturbation on large scales and its higher-order correlations, including the use of Feynman-type diagrams and how the δN formalism extends to the description of primordial isocurvature density perturbations. In section 3 I review several examples of models for the origin of structure in the very early universe and the local non-Gaussianity they give rise to. I conclude in section 4 with brief review of current observational constraints.

The δN expansion

A powerful technique for calculating the non-linear primordial density perturbation in many cosmological models is the δN formalism [8,9,5,10], which identifies the primordial density perturbation with the perturbed logarthmic expansion, N = Hdt.

In Friedmann-Robertson-Walker (FRW) cosmology there is a preferred foliation of spatial hypersurfaces which are maximally symmetric, and on which the matter density and pressure are also homogeneous and isotropic. In an inhomogeneous universe we can use the difference between uniform-expansion hypersurfaces and uniform-matter hypersurfaces as a measure of inhomogeneity. At first order we define [11,12]

ζ 1 ≡ -ψ 1 - H ρ δ 1 ρ . ( 5
)
where ψ 1 and δ 1 ρ are the gauge-dependent spatial metric and density perturbations respectively [7,12]. This can be interpreted either as a curvature perturbation on uniform-density hypersurfaces (where δρ = 0) or a dimensionless density perturbation on uniform-curvature hypersurfaces (where ψ = 0). Local energy conservation equation [5] ensures that ζ 1 remains constant for adiabatic density perturbations in the long-wavelength limit where (k/aH) 2 Φ 1 → 0 and ζ 1 approaches the comoving curvature perturbation, -R. Thus during inflation it is convenient to evaluate the comoving curvature perturbation on large scales, k/aH → 0, in order to determine the primordial density perturbation from single-field inflation

ζ 1 → -R 1 = -ψ 1 - H φ δ 1 ϕ . (6)
Because the non-adiabatic decaying mode rapidly decays outside the Hubble scale [13] it is sufficient to evaluate the primordial perturbation due to scalar field perturbations shortly after Hubble exit, k = aH. However in multi-field inflation, additional light fields can lead to non-adiabatic pressure perturbations on super-Hubble scales and ζ 1 becomes time-dependent in general.

Beyond linear order, we can identify the non-linear metric perturbation which coincides with ζ 1 at first order and on large scales [14] (see also [15,16] and the article by Langlois and Vernizzi elsewhere in this issue [17])

ζ = δN + 1 3 ρ ρ dρ ρ + P . ( 7
)
where ρ(t) is the homogenous background density, ρ(t, x) denotes the local inhomogeneous density and the local expansion is given by N = Hdt, where H(t, x) = ∇ µ u µ /3 is the local Hubble expansion rate along comoving worldlines, u µ = dx µ /dτ . A local form for the distribution of ζ naturally arises from the evolution of matter fields on scales much larger than the Hubble scale in the very early universe. After averaging on some scale, L H -1 , such that spatial gradients and anisotropy can be neglected, then the local expansion is well-described by the Friedmann equation for a homogeneous universe with the corresponding local density and pressure. This is known as the "separate universe" approach [5], and can be derived from the full inhomogeneous equations of motion as a long-wavelength limit in a gradient expansion [4,15,16]. In this approach the local evolution, along a given worldline, is determined by the initial values of the matter fields on an initial spatial hypersurface.

Therefore one can evaluate ζ(t f , x), defined by Eq. ( 7), as δN , the perturbed expansion on a uniform-density hypersurface (ρ(t f , x) = ρ(t f ) for all x) at some final time t f after inflation has ended by evaluating the integrated expansion from some initial spatially flat hypersurface, δN (t i , x) = 0 for all x [8, 9, 10]. In the separate universe framework [5] the inhomogeneous universe is modelled as a patchwork of locally homogeneous regions which due to causality evolve independently. In particular one can determine the non-linear local expansion N using the Friedmann equation for the local expansion as a function of the initial local fields, ϕ I (t i , x) = φ(t i ) + δϕ(t i , x). Assuming N (ϕ I ) is an analytic function of the initial field values, we have the Taylor series expansion [10]

ζ = N (ϕ I ) -N = N I δϕ I + 1 2 N IJ δϕ I δϕ J + 1 6 N IJK δϕ I δϕ J δϕ K + . . . , (8)
where N I = ∂N/∂ϕ I , etc, and summation is implied over repeated indices. If the field perturbations themselves are constructed as an expansion in a small perturbation parameter

ϕ I = φI + δ 1 ϕ I + 1 2 δ 2 ϕ I + 1 6 δ 3 ϕ I + . . . , (9)
then we can write, order by order in a perturbative expansion

ζ = ζ 1 + ζ 2 /2 + ζ 3 /6 . . ., where
ζ 1 = N I δ 1 ϕ I , (10)
ζ 2 = N IJ δ 1 ϕ I δ 1 ϕ J + N I δ 2 ϕ I , (11)
ζ 3 = N IJK δ 1 ϕ I δ 1 ϕ J δ 1 ϕ K + 3N IJ δ 1 ϕ I δ 2 ϕ J + N I δ 3 ϕ I , (12)
To calculate the statistics of the primordial perturbation, ζ, we therefore need to know the dependence of the large scale expansion upon the initial field values, N (ϕ I ), and the statistical distribution of the initial field values. The distribution of the fields is determined by the quantum fluctuations of the vacuum in any given inflationary model. We take the first-order perturbations to describe the vacuum fluctuations of the non-interacting free fields, and build up the higher-order terms from the interaction Hamiltonian, following Maldacena [2] (see also the article by Koyama elsewhere in this issue [18]).

In canonical slow-roll inflation the linear field perturbations defined on unperturbed spatially flat hypersurfaces [19,20] are given by the flat spacetime vacuum on sub-Hubble scales and are then approximately constant on super-Hubble scales with power spectra

δ 1 ϕ I k 1 δ 1 ϕ J k 2 (2π) 3 P (k 1 )δ IJ δ 3 (k 1 + k 2 ) , (13)
where P (k 1 ) ≈ H 2 * /2k 3 1 and H * is the Hubble scale at Hubble-exit, k = a * H * . Note that the dimensionless power spectrum, related to the variance in real space, is multiplied by a volume factor in k-space to give

P(k) = 4πk 3 (2π) 3 P (k) H * 2π 2 , ( 14
)
and this is approximately scale invariant, with spectral tilt

d ln P d ln k 1 . (15)
Scalar field interactions are suppressed by slow-roll parameters and so field perturbations on scales close to the Hubble scale are approximately Gaussian. In the following we will assume that all connected higher-order moments of the scalar field perturbations vanish at some fixed time during inflation, shortly after all relevant scales have crossed out side the Hubble scale, k < aH. Hence N is well-described by a local function of Gaussian random fields soon after the smallest relevant modes have left the Hubble scale ‡.

We will assume that δ 2 ϕ I , δ 3 ϕ I , etc, vanish sufficiently close to Hubble exit and thus the higher-order moments are given by the first terms on the right-hand-side of Eqs. ( 11) and (12). In particular the primordial power spectrum is given, at leading order, by

ζ k 1 ζ k 2 = N I N J δϕ I k 1 δϕ J k 2 = (2π) 3 P ζ (k 1 )δ 3 (k 1 + k 2 ) , (16)
where

P ζ (k 1 ) = N I N J δ IJ P (k 1 ) , (17)
In the next subsection we will discuss higher-order contributions to the primordial power spectrum coming from higher-order terms in the delta-N expansion. These can be identified as "loop" corrections in a diagrammatic approach [24], where the leading order result ( 16) is the "tree-level" term.

Similarly we can construct the leading order terms in the bispectrum and trispectrum

ζ k 1 ζ k 2 ζ k 3 = (2π) 3 B ζ (k 1 , k 2 , k 3 )δ 3 (k 1 + k 2 + k 3 ) , (18)
ζ k 1 ζ k 2 ζ k 3 ζ k 4 = (2π) 3 T ζ (k 1 , k 2 , k 3 )δ 3 (k 1 + k 2 + k 3 + k 4 ) , (19)
where

B ζ (k 1 , k 2 , k 3 ) = N I N J N IJ [P (k 1 )P (k 2 ) + 2 perms] , (20)
T ζ (k 1 , k 2 , k 3 ) = N IJK N I N J N K [P (k 2 )P (k 3 )P (k 4 ) + 3 perms] + N IJ N J K N I N K [P (k 3 )P (k 4 )P (k 13 ) + 11 perms] , (21)
‡ Alternatives to slow-roll models of inflation based on non-canonical scalar field Lagrangians with nonlinear kinetic terms, such as k-inflation [21] or DBI inflation [22], can lead to significant non-Gaussianity of the field perturbations at Hubble exit and hence produce different types of non-Gaussianity, different from the local form.

Diagrammatic approach

The perturbative expression of the non-linear expansion as a function of the field values during inflation ( 8) is conveniently expressed in terms of Feynman-type diagrams [24] (see also [23]). Similar diagrams have been used in the analysis of large-scale structure in cosmology [25,26]. Every term in the n-point function can be identified with a diagram with n external lines. To perturbative order r one should draw all diagrams with r internal propagators. Tree-level diagrams correspond to r = n -1 while diagrams with r ≥ n are loop corrections. If the expansion is a local function of Gaussian fields then the internal propagators have no self-interactions. Examples are shown in Figure 1. A comprehensive prescription for drawing diagrams and constructing the corresponding terms to any given order are given by Byrnes et al [24].

Here we will consider only diagrams for Gaussian fields, ϕ A . The power spectrum up to one-loop is then shown in figure 1, corresponding to

P ζ (k) = N A N B δ AB P (k) + 1 (2π) 3 d 3 q 1 2 N AB N CD δ 3 (k + k -q) + N A N BCD δ 3 (k -q) δ AC δ BD P (k)P (k ) . ( 22
)
Similarly the bispectrum and higher order correlations can be evaluated including an arbitrary number of loop corrections.

Because the loop corrections include integrations over all internal momenta, and the scalar field perturbations are expected to have almost scale invariant power spectra during inflation, the loop corrections can give corrections that, although only slowly growing on large scales, formally diverge in the infra-red limit. This issue is discussed by Seery elsewhere in this issue [27]. Typically the corrections remain small in a region which is not much larger than the observable universe, but in some cases, for instance if the tree-level terms are absent [28], then loop corrections may play an important role. Some loop corrections, such as the second one-loop term in Eq. ( 22), can be identified as dressing the vertices. Byrnes et al [24] noted that these can be interpreted as renormalising the derivatives of N in terms of the average local derivatives:

ÑA ≡ d 3 qN A + 1 (2π) 3 d 3 qN ABC δ BC P (q) + . . . (23)
Renormalising the vertices in this way simplifies the expressions for the loop corrections, to yield only one new term at each new order. In real space this is sufficient to resum all divergences, but in Fourier space the dependence on IR wavenumbers on the convolution remains. For instance the power spectrum to one loop becomes

P ζ (k) = ÑA ÑB δ AB P (k) + 1 (2π) 3 d 3 q 1 2 ÑAB ÑCD δ AC δ BD P (k)P (k -q) . (24)

Single-field vs multi-variate local models

If the primordial density perturbation ( 8) is a local function of a single scalar field

ζ = N δϕ + 1 2 N δϕ 2 + 1 6 N δϕ 3 + . . . , (25)
then the expressions for the higher order moments of the primordial distribution, e.g., (20) and ( 21), simplify considerably and we can write

B ζ (k 1 , k 2 , k 3 ) = f NL [P ζ (k 1 )P ζ (k 2 ) + 2 perms] , (26)
T ζ (k 1 , k 2 , k 3 ) = 54 25 g NL [P ζ (k 2 )P ζ (k 3 )P ζ (k 4 ) + 3 perms] + τ NL [P ζ (k 3 )P ζ (k 4 )P ζ (k 13 ) + 11 perms] , (27)
where at leading order (tree-level) the non-linearity parameters are given by [10,29]

f NL = 5 6 N N 2 , τ NL = 36 25 f 2 NL , g NL = 25 54 N N 3 . (28)
The δN expression for the primordial perturbation (8) goes beyond the simplest local model (4) by considering the case where the primordial perturbation is a function of more than one Gaussian random field. Including terms up to second order we have

Φ(x) = I Φ I (x) + I,J f IJ Φ I (x)Φ J (x) -Φ I Φ J + . . . , (29)
where we have defined Φ I = N I δϕ I and f IJ = N IJ /N I N J (no sums). The non-linearity parameter f NL defined by Eq. ( 3) is then k-dependent and given by [30]

f NL = I,J N IJ [P I (k 1 )P J (k 2 ) + P I (k 2 )P J (k 3 ) + P I (k 3 )P J (k 1 )] I,J N I N J [P I (k 1 )P J (k 2 ) + P I (k 2 )P J (k 3 ) + P I (k 3 )P J (k 1 )] . ( 30
)
This is a constant if the scale-dependence of all the fields which contribute to the primordial perturbation is the same, but in general it is scale and shape dependent. k-dependence of f NL defined by Eq. ( 3) also arises due to interactions and hence non-Gaussianity of fields on super Hubble scales [30], but this goes beyond the strict definition of local models considered here. If the fields are all approximately massless during inflation then their scale-dependence and interactions are small and the kdependence of f NL is expected to be weak (but see [31]).

Isocurvature non-Gaussianity

Having identified the non-linear primordial density perturbation, ζ, with the perturbed expansion, δN , up to a uniform density hypersurface, it is straightforward to extend this to n primordial density perturbations in an n-component system, where we define

ζ α ≡ δN + 1 3 ρα ρα dρ α ρ α + P α . ( 31
)
Relative density perturbations between the different components which leave the total energy density unperturbed are know as isocurvature perturbations. In particular a relative perturbation between photons and baryons around the time of last scattering of the CMB photons corresponds to a perturbation of the local entropy (determined by the photon number density) per baryon and hence such perturbations are also known as entropy perturbations. At linear order we have

S B = δn B n B - δn γ n γ . ( 32
)
We can identify this with the difference between the two primordial density perturbations ζ B and ζ γ , up to a factor of 3. Hence beyond linear order we define [32,33]

S B = 3 (ζ γ -ζ B ) . (33)
A similar definition is used for the cold dark matter isocurvature perturbation, S CDM , and the neutrino isocurvature perturbation, S ν , relative to the photon number density. More generally for an n component system we can decompose an arbitrary density perturbation into one adiabatic density perturbation, ζ = ζ α for all compoenents, α, and n -1 independent isocurvature modes

S αβ ≡ 3(ζ α -ζ β ) . (34)
One can define non-linearity parameters for the isocurvature bispectrum at leading order analogous to the adiabatic bispectrum (3)

f S (k 1 , k 2 , k 3 ) = B S (k 1 , k 2 , k 3 ) 2 [P S (k 1 )P S (k 2 ) + P S (k 2 )P S (k 3 ) + P S (k 3 )P S (k 1 )] . ( 35
)
If the isocurvature modes are uncorrelated with the adiabatic density perturbation, and we use the linear relation between primordial matter isocurvature perturbations and the Newtonian potential, Φ = S CDM /5, we obtain [32] f NL = 5f S 162

P S (k 1 )P S (k 2 ) + P S (k 2 )P S (k 3 ) + P S (k 3 )P S (k 1 ) P Φ (k 1 )P Φ (k 2 ) + P Φ (k 2 )P Φ (k 3 ) + P Φ (k 3 )P Φ (k 1 ) . ( 36
)
However one should be wary of interpreting bounds on the adiabatic f NL as any constraint on isocurvature non-Gaussianity which leads to a distinctive bispectrum on the CMB sky [32,34]. Moreover the isocurvature modes are in general correlated with adiabatic modes [35] and hence there may exist non-vanishing cross-correlations, Sζζ and SSζ [33].

Models of non-Gaussianity

Single inflaton

A homogeneous scalar field cosmology evolves like that of a fluid with time dependent density and pressure, but an inhomogeneous scalar field is not in general a barotropic fluid [36]. However the non-adiabatic part of the field's pressure perturbation is proportional to the field's comoving density perturbation [13]. The Einstein equations energy and momentum constraints require that the comoving density perturbation vanishes in the large scale limit where the Newtonian potential stays finite [12] and hence if the local energy density and pressure during inflation is dominated by a single scalar field then perturbations must be adiabatic in this large scale limit. The field perturbations enter a squeezed state in phase space and the local time derivative of the field is no longer independent of the local field value. This makes it straightforward to predict the primordial density perturbation and hence the local non-Gaussianity for inflation models with a single inflaton field. As all perturbations are adiabatic then the local density and pressure follow the same phase-space trajectory in phase-space as the background cosmology with δp/δρ = ṗ/ ρ. This ensures that the perturbation ζ is non-linearly conserved in this large scale limit [5,37,15,14,16] and we can calculate the primordial density perturbation, long after inflation, in terms of quantities during inflation. Using the δN -formalism we just need to calculate the effect of field perturbations on the local expansion, N , during inflation.

To first order we have

ζ 1 = N δϕ = - H φ δϕ . (37)
This is time-independent on large scales, but in practice it is convenient to evaluate it soon after Hubble exit (k = aH) where the amplitude of the field perturbations are given by P (k) H 2 * /2k 3 . We then obtain the simple result for the primordial power spectrum at leading order

P ζ (k) = H φ 2 * P (k) . (38)
At second order, assuming a Gaussian field δϕ, we have

ζ 2 = N δϕ 2 = - Ḣ φ + H φ φ2 δϕ 2 , (39)
In terms of the usual slow-roll parameters = -Ḣ/H 2 and η = -( φ/H φ) -( Ḣ/H 2 ) we have

ζ 2 = (2 -η) ζ 2 1 . (40)
Thus we see that the second order primordial density perturbation is suppressed with respect to the square of the first-order perturbations and hence |f NL | 1 [38,39]. In fact at this order in the slow-roll approximaton we can no longer neglect the intrinsic non-Gaussianity of the fields at Hubble exit [2], and a simple argument where we consider long-wavelength perturbations to set the local Hubble expansion as shorter wavelengths leave the Hubble scale [40] shows that in the squeezed limit, where k 2 1 k 2 2 + k 2 3 , we find

f NL (k 1 , k 2 , k 3 ) = 5 6 (3 -η) . (41)
More generally, for any single field model producing adiabatic perturbations during inflation the bispectrum can be related to the scale-dependence of the power spectrum in the squeezed limit and we have [41] f

NL (k 1 , k 2 , k 3 ) = 5 12 (1 -n ζ ) . ( 42
)
where n ζ -1 = d ln P ζ /d ln k. Observational bounds on the spectral tilt, |n ζ -1| 1, thus ensure that local-type non-Gaussianity is very small in single-inflaton field models.

Curvaton

The curvaton scenario [42,43,44,45,46] provides a simple model for the origin of the primordial perturbations which could exhibit significant non-Gaussianity of the local type [47].

The curvaton is a weakly-coupled scalar field which is light during inflation, m H, but it has a negligible energy density at that time. Quantum fluctuations of the field during inflation generate an almost scale invariant spectrum of field perturbations on super-Hubble scales. We assume that the weakly-coupled field remains decoupled from the inflaton and its decay products at the end of inflation. The field begins to oscillate about the minimum of its potential when the Hubble rate drops below the mass of the field some time after inflation. An oscillating massive field, m H, has a pressureless equation of state (averaged over several oscillation times) and hence the curvaton energy density decays as a -3 , but grows relative to the energy density of radiation, ∝ a -4 . This is the Polonyi or moduli problem of weakly coupled scalar fields which can come to dominate the energy density of the early universe, disrupting the conventional hot big bang. This is not a problem so long as the moduli decay and their decay products thermalise before the epoch of primordial nucleosynthesis. The inhomogeneous density of the curvaton is transfered to the radiation when the curvaton decays leading to a primordial density perturbation on super-Hubble scales.

Note that, in contrast to inflaton perturbations, the curvaton field fluctuations are isocurvature field perturbations during inflation and thus give rise to non-adiabatic pressure perturbations after inflation which leads to a change in the value of the perturbation, ζ, on super-Hubble scales. Thus the bispectrum in the squeezed limit is not constrained by the relation (42) for perturbations generated by the adiabatic mode during inflation.

In the simplest example of a curvaton field, χ, with potential V (χ) = m 2 χ 2 /2, the curvaton field is a free field during inflation with no self-interactions and hence is described by a Gaussian field. The quadratic potential naturally leads to a primordial density field that is a quadratic local function of the Gaussian curvaton field and hence the simplest curvaton model is well described by the simplest local model of non-Gaussianity given in Eq. ( 4).

The initial curvaton density perturbation on spatially flat hypersurfaces is given by

ζ χ = 1 3 ρ χ -ρχ ρχ = 2 χδχ + δχ 2 -χ 2 χ2 . ( 43
)
In the case where the curvaton density remains sub-dominant throughout we can assume a linear transfer to the primordial density perturbation, with transfer efficiency r ∼ Ω χ,decay = ρ χ /ρ tot | decay , and we can write

ζ = rζ χ = ζ 1 + 5 6 f NL ζ 2 1 -ζ 2 1 . ( 44
)
where we identify

ζ 1 = 2r 3 δχ χ . ( 45
)
and the non-linearity parameter [47] f

NL = 5 . (46)
For r 1 we can have large, positive f NL §. In this case current observational bounds on f NL [61] place a lower bound on r > 0.01.

For r ∼ 1 we need to include the gravitational effect of the curvaton density when it decays. If we assume an instantaneous decay hypersurface given by the local Hubble rate H = Γ, we obtain the non-linear relation between the curvaton perturbation ζ χ and the primordial perturbation ζ [49] (1 -Ω χ,decay )e -4ζ + Ω χ,decay e 3(ζχ-ζ) = 1 ,

from which we recover Eq. ( 45) at linear order, where r = 3Ω χ,decay (4 -Ω χ,decay ), and at second and third order we obtain the non-linearity parameters as a function of r [50,10,49]

f NL = 5 4r 1 - 4r 3 - 2r 2 3 , (48)
g NL = - 25 6r 1 - r 18 - 10r 2 9 - r 3 3 . ( 49
)
A measurement of both f NL and g NL would be an important consistency test of the simplest curvaton model as both are a function of a single parameter, r. For a subdominant curvaton, r 1, we recover Eq. ( 46) for f NL and g NL -10f NL /3 [49]. In § There has been some confusion in the literature over the "correct" sign for f NL in the curvaton scenario. This is due to different sign conventions for the Newtonian potential Φ and sometimes due to error. Note that different authors also use slightly different definitions for ζ. In my own papers, the original result for f NL in Ref. [47] had the "correct" sign (that used here, which coincides with that used for observational constraints [6]) due to a combination of the "wrong" sign convention and a mistake in the arithmetic. The mistake in the arithmetic was discovered and corrected, for instance in Ref. [48], but then the sign was "wrong" because of the sign convention used.

the opposite limit, as r → 1, where the curvaton dominates the energy density of the universe before it decays, we have f NL → -5/4 and g NL → 25/12.

Other possibilities that have been studied include a kinetic-energy dominated era (rather than radiation dominated) before curvaton decay [51] and multiple curvaton-type fields [52]. Even in the case of multiple curvaton fields contributing to the primordial density perturbation we recover the same lower bound f NL ≥ -5/4 as in the single field case [53,54]. A much wider range of local non-Gaussianity becomes possible if we consider the effect of self-interaction terms in the curvaton potential, such that V = m 2 χ 2 /2 + λχ n + . . . with n ≥ 3 which can lead to non-linear evolution of the field before it decays [55,56,57,58]. In this case one may find large negative values of f NL for specific model parameters and initial conditions.

Another distinctive aspect of the curvaton model is the possibility to leave residual isocurvature perturbations after the curvaton has decayed [42,43,45]. Non-adiabatic perturbations on super-Hubble scales during inflation are necessary, but not sufficient, condition for the existence of primordial isocurvature perturbations after inflation. Whether or not residual isocurvature modes survive after the curvaton decays depends upon the reheating history [59]. If all the curvaton decay products thermalise with vanishing chemical potential then the primordial perturbation must be adiabatic [60]. But if a baryon or lepton asymmetry has already been created, or the curvaton decay itself produces the asymmetry, or the dark matter has already decoupled, then a residual isocurvature perturbation, S X , may be left, and it will be completely correlated (or anticorrelated) with the total density perturbation, ζ.

For example, if the curvaton decay is the out-of-equilibrium process which breaks time-reversal invariance and violates the baryon number then the baryon number inherits the same density perturbation as the curvaton, ζ B = ζ χ , to all orders [33]. In this case bounds on the amplitude of the linear primordial isocurvature perturbation [61] requires that the curvaton to dominate the energy density when it decays, r 1, so that ζ ζ B and, from Eq. ( 33), S B ζ B . In this case the intrinsic non-linearity parameter of the isocurvature perturbation is of order unity, and the contribution to the CMB bispectrum is suppressed relative to that from the total density perturbation.

A mixed curvaton-inflaton scenario yields a much richer phenomenology of adiabatic, non-adiabatic and non-Gaussianity of primordial perturbations [62,63,64,65,33,66].

Multiple-field inflation

The curvaton is just one example of how fields other than a single inflaton driving inflation, could play a significant role in determining the primordial density perturbation after the end of inflation. The presence of non-adiabatic perturbations on super-Hubble scales allows in principle for the evolution of the large-scale density perturbation after Hubble-exit [13] and hence local non-Gaussianity in the primordial density perturbation some time after inflation.

One of the best studied models is double inflation with two massive but noninteracting fields [8,35] where the non-linearity parameter though not necessarily slowroll suppressed is not expected to be much larger than unity [67,68,69]. Primordial isocurvature perturbations have also been studied in this model where one of the massive fields is identified as a dark matter candidate [70,35,33].

In general it seems surprisingly difficult to produce large non-linearity parameters during slow-roll inflation, but examples can be constructed [71]. For more discussion see the recent review by Byrnes and Choi [72].

In inflationary models where more than one field remains light until the end of inflation it is necessary to consider possible evolution of the density perturbation on large scales at or after the end of inflation. The end of slow-roll inflation is an epoch at which the large-scale density perturbation may have a non-linear dependence upon non-adiabatic modes, especially it there is an abrupt change in the equation of state [73,74,75,76].

It is very natural to realise this possibility in hybrid inflation models [77] where inflation is ended by a tachyonic instability triggered in a waterfall field, leading to a rapid phase transition and decay of the false vacuum. In string theory models of inflation this may describe the collision of branes in a higher-dimensional space [78]. The instability corresponds to a surface in field space, and if the inflaton trajectory is not orthogonal to this surface then non-adiabatic field perturbations, orthogonal to the trajectory [13], are converted into density perturbations at the phase transition. In the simplest case, taking the extreme slow-roll limit and assuming instantaneous reheating, the energy density will be almost uniform on the spatial hypersurface corresponding to the phase transition, and thus the perturbed expansion to this surface, δN , can be identified with the primordial density perturbation, ζ in Eq.( 7).

Modulated decay

A specific example where the primordial density perturbations may be produced just after the end of inflation is when the decay rate of the inflaton is a function of one or more moduli fields [79,80]. Because this occurs some time after Hubble exit during inflation, this produces local type non-Gaussianity [81,82,83].

If we approximate the inflaton reheating by a sudden decay, as we did earlier for the curvaton decay, we find an analytic estimate of the non-linear density perturbation. In the case of modulated reheating, the decay occurs on a spatial hypersurface with variable local decay rate and hence local Hubble rate H d = Γ(χ). Before inflaton decay, the oscillating field has a pressureless equation of state and there is no density perturbation perturbation. Setting ζ = 0 in Eq. ( 7) thus gives the perturbed expansion on the decay hypersurface

δN d = - 1 3 ln ρ d ρd . ( 50
)
After the decay the we have radiation with equation of state p = ρ/3 and hence from Eq. ( 7) a density perturbation

ζ = δN d + 1 4 ln ρ d ρd . (51)
Eliminating δN d and using the local Friedmann equation, ρ ∝ H 2 , to determine the local density in terms of the local decay rate, H d = Γ(χ), we have

ζ = - 1 6 ln Γ(χ) Γ . (52)
At first order we recover the linear relation [79,82]

ζ 1 = - 1 6 Γ Γ δχ . (53)
At higher order we obtain [83]

ζ 2 = 1 6   Γ Γ 2 - Γ Γ   δχ 2 , (54)
and hence

f NL = 5 1 - Γ Γ Γ 2 , (56)
g NL = 50 3 2 -3 ΓΓ Γ 2 + Γ Γ 3 , (57)
and so on to higher order. Thus for an approximately linear modulation function Γ(χ) we have f NL 5 and g NL 100/3 while for Γ ∝ χ 2 we obtain f NL 5/2 and g NL 25/3. Similar results are found if the mass of the inflaton decay products is modulated, socalled inhomogeneous mass domination [84,82].

Resonant decay or preheating may be sensitive to non-adiabatic modes [85,86,87,88,89,83]. In these models observational constraints on primordial non-Gaussianity impose significant constraints on the allowed parameter values and initial conditions [90]. For a review of inflaton dynamics and reheating see Bassett et al [48].

Ekpyrotic model

There have been numerous attempts to construct alternatives to inflation as a model for the origin of primordial perturbations. Many of these exploit the similarities between an inflationary expansion and a collapse phase where the comoving Hubble scale decreases and thus quantum vacuum fluctuations evolve into the super-Hubble regime [91,92]. As the Hubble rate grows during collapse, these models generally require some form of instability for the perturbations to grow on super-Hubble scales and maintain a scaleinvariant spectrum. For example, a simple collapse model driven by a scalar field with pressureless equation of state generates a scale-invariant spectrum for ζ on super-Hubble scales [92,93]. The non-Gaussianity in this has recently been calculated [94] and shown to yield f NL ∼ 1, though not of the local form in Eq. ( 4). Ultimately the four-dimensional low-energy effective theory must break down as the collapse rate approaches the Planck scale and this leaves some uncertainty about how the perturbations are transfered to the expanding hot big bang, but at least within the collapse phase it is possible to study the growth of perturbation ζ and its non-Gaussianity.

Recent attention has focussed on the ekpyrotic model [95,96]. In the 4D effective theory this corresponds to a cosmology driven by scalar fields with steep exponential potentials, V I (ϕ I ) ∝ exp(-c I ϕ I ) (no sum) leading to an ultra-stiff equation of state p/ρ 1. The separate universe picture is an excellent approximation on super-Hubble scales during an ekpyrotic phase [106]. Thus, for fields with canonical kinetic Lagrangians, the local values of the fields perturbations at Hubble-exit have an approximately Gaussian distribution and set the initial conditions the subsequent local expansion history. As a result we expect the primordial non-Gaussianity to have a local form, and due the the strong self-interaction terms in steep potentials, we expect the non-linearities to become large [98].

The adiabatic mode during an ekpyrotic collapse leads to a steep blue spectrum for ζ [99] but in the presence of two or more fields with steep exponential potentials it is possible to produce an almost scale-invariant spectrum of isocurvature perturbations [100,101,97,102,98]. Consider the simple case of two fields with potential

V = -V 1 exp -c 1 ϕ (1) -V 2 exp -c 2 ϕ (2) , (58)
Performing a rotation in field-space this can be re-written as [103] V

= V 0 exp(-cσ) -1 - c 2 2 s 2 - K 3 c 3 12 √ 2 s 3 - K 4 c 4 96 s 4 . . . , (59)
where the fast-roll parameter c -2 = c -2 1 + c -2 2 and we identify s as the tachyonic field direction which acquires an almost scale-invariant spectrum of perturbations about s = 0 on super-Hubble scales, with spectral tilt

d ln P d ln k = 4 c 2 . ( 60
)
For an almost scale-invariant spectrum we require c 2 1. Note that in the δN formula the scale dependence of the primordial power spectrum (at leading order) follows directly from the scale-dependence of the initial field perturbations, n ζ -1 = 4/c 2 , but the amplitude depends on the subsequent expansion history. In the ekpyrotic scenario different models have been proposed for the conversion of the isocurvature field perturbations to density perturbation ζ. The tachyonic instability itself will lead to a phase transition to an ekpyrotic phase dominated by just one of the fields, ϕ I [104]. In this case N (ϕ (1) , ϕ (2) ) can be calculated analytically for the simple model (58) and we find [105,106].

f NL - 5 12 c 2 I , g NL 25 108 c 4 I . (61)
Note that c 2 I > c 2 1 and therefore the non-linearity parameters are expected to be large. Tight observational bounds on a negative value for f NL > -10 [61] require c 2 I < 24 in this simple model and hence from Eq. ( 60) an unacceptably blue primordial spectral tilt n ζ -1 > 0.16. An acceptable tilt and non-Gaussianity would require modifications to the simple potential Eq. ( 60) which itself could trigger a transition and a isocurvaturecurvature conversion. The amplitude and sign of the non-linearity parameters then becomes model-dependent but one generally finds f 2 NL ∼ g NL ∼ c 2 [107]. Alternative conversion mechanisms would produce a different amplitude of primordial perturbations and hence non-Gaussianity. In models where the conversion happens at a reflection in field space during a kinetic dominated phase give estimates for the non-linearity parameters [106].

f NL ∼ - 3 2 √ 2 K 3 c + 5 , g NL -20c 2 1 - K 2 3 32 - K 4 24 . (62)
K 3 and K 4 are expected to be of order unity. Again the non-linearity parameters are expected to be large with |g NL | ∼ f 2 NL . In this case we see that even if K 3 is small, such that |f NL | ∼ 1 [corresponding to c 2 1 -c 2 2 1/c in Eq. ( 58)], then we would expect |g NL | to be large, unless K 4 is also unexpectedly small. We note the further possibility that isocurvature field fluctuations are converted to density perturbations only after the bounce [108], as in curvaton or modulated decay models.

Conclusions

The distribution of primordial density perturbations contains more information than just the power-spectrum. Among a plethora of different possible forms on non-Gaussianity the local models provide a clearly defined class which arise naturally from the evolution of the density perturbation on super-Hubble scales of initially Gaussian vacuum fluctuations during inflation. In particular the (2 + n)-point function of a non-Gaussian distribution generated by a local function of a single Gaussian field can be described at leading order by the two-point function and n non-linearity parameters, f NL , g NL , etc.

Non-Gaussianities have the potential to reveal the physical interactions at work in the very early Universe. For adiabatic perturbations during inflation the non-linearity parameters are related to the scale-dependence of the power spectrum [2,41] and hence must be small. Thus any detection of local non-Gaussianity of the primordial density perturbation would be evidence of non-adiabatic perturbations on super-Hubble scales, e.g., the presence of multiple light fields during slow-roll inflation.

Current observational limits on f NL in the single-field local model come from combining WMAP cosmic microwave background bounds (-10 < f NL < 74 [61]) with large-scale structure (-29 < f NL < 70 [109]), yielding -5 < f NL < 70 at 95% confidence limit [61]. Recent work also places a bound -3.80 < g NL /10 6 < 3.88 from the WMAP 5-year data [110]. Care should be taken when applying these bounds to non-local or even multi-variate local models. CMB bounds are obtained using matched filtering techniques to construct an optimal estimator for f NL defined by Eq. ( 4). It estimates the amplitude of a second-order primordial perturbation which has a specified correlation with the first-order perturbation. On the other hand bounds from large scale structure arise from the form of the (general relativistic) Poisson equation [111] which implies that a local form for the primordial Newtonian potential requires a non-local form for the comoving density contrast, leading to a distinctive scale-dependent bias [112].

There is great potential for future discovery coming from future CMB missions such as ESA's Planck satellite which has the potential to bound |f NL | < 5 and largescale structure surveys, including 21cm line radio surveys [113]. There is also plenty of scope for further theoretical developments given the data that already exists to test non-Gaussian models. New theoretical templates will need to be derived for optimised constraints to be placed on alternative theoretical models of non-Gaussianity. At some point primordial non-Gaussianity will be discovered as it inevitably arises from the nonlinear evolution of density perturbations and the last-scattering of the CMB photons [114], and it will then be a question of using all the available data to disentangle the different contributions.

Figure 1 .

 1 Figure 1. Power spectrum diagrams for δN expansion with Gaussian fields (a) at tree level, and (b) one loop, corresponding to each term in Eq. (22).

ζ 3 = 1 6   -2 Γ Γ 3 + 3 Γ Γ Γ 2 -Γ Γ   δχ 3 ,(55)

Acknowledgements

I am grateful to my many collaborators for the work on which this highly subjective review is based, and especially Chris Byrnes, David Langlois, David Lyth, Karim Malik, Misao Sasaki, Jussi Valiviita and Filippo Vernizzi. DW is supported by the STFC and is grateful to the Yukawa Institute of Theoretical Physics, Kyoto, for their hospitality.


J B Griffiths
email: j.b.griffiths@lboro.ac.uk

N O Santos
email: n.o.santos@qmul.ac.uk

A rotating cylinder in an asymptotically locally anti-de Sitter background

A rotating cylinder in an asymptotically locally anti-de

Sitter background

Introduction

Bonnor, Santos and MacCallum [1] have identified a family of compound space-times for which the interior of an infinite cylinder of finite radius is described by the Gödel solution [2]. They showed that this can be matched to a vacuum exterior in a form obtained by Santos [3]. The resulting space-time is cylindrically symmetric and stationary. It claims to describe the interior and exterior fields of a rigidly rotating cylinder of dust in a vacuum background with a negative cosmological constant. However, the exterior space-time was not expressed explicitly and some properties were only outlined. It is the purpose of this paper to construct this solution explicitly in a form that is suitable for interpretation. In this way, the properties of the space-time are identified for all values of the characterising parameters.

After identifying an appropriate form of the Gödel metric that is to be used as an internal solution, we present a modified form of metric for the family of solutions that are considered for the external region. We then apply the appropriate junction conditions and identify expressions for the parameters for which it is matched to a Gödel interior. Some properties of the space-time are then described in detail.

The Gödel solution

The perfect-fluid solution obtained by Gödel [2] can be expressed in comoving coordinates in the cylindrical form

ds 2 = -dt 2 + dρ 2 + dz 2 - 2 √ 2 ω
sinh 2 ωρ dφ dt + 1 ω 2 sinh 2 ωρsinh 4 ωρ dφ 2 , (1) where ω is an arbitrary constant, ρ ∈ [0, ∞), φ ∈ [0, 2π) and φ = 2π is identified with φ = 0. The coordinate ρ may be interpreted as the proper distance from a regular axis at ρ = 0, and the rotational symmetry of the metric (1) about the axis can be clearly seen. The fluid four-velocity is u = ∂ t , and the constant density µ (the pressure p = 0) and cosmological constant Λ are given by

8πµ = 4ω 2 , Λ = -2ω 2 .
This implies that the fluid density is positive and the cosmological constant is negative. The Weyl tensor is of type D and there are no curvature singularities. This form of the metric has been preferred here since it explicitly reduces to a vacuum Minkowski space in the slowly rotating limit as ω → 0. This solution, which is clearly stationary, is characterised by the fact that it is spatially homogeneous and the fluid has no expansion, acceleration or shear. The only non-zero kinematic quantity is the rotation (vorticity), which has constant magnitude ω that is aligned with the axis. Its properties are well known (see e.g. [2] and [START_REF] Hawking | The large scale structure of space-time[END_REF]- [START_REF] Griffiths | Exact space-times in Einstein's general relativity[END_REF]).

This solution may be interpreted as describing a homogeneous space-time with a fluid source that is rigidly rotating. In the Newtonian analogue of such a situation, relative to any point, there will exist a cylinder of finite radius around the axis of rotation on which the particles of the fluid would move at the speed of light. For all points outside this cylinder, the particles of the classical fluid will be moving faster than light. Such a situation is not possible in a relativistic theory but this feature is reflected in the appearance of closed timelike curves.

In fact, it can be seen from the metric (1) that curves on which t, ρ and z are constant are either spacelike or timelike according to whether sinh 2 ωρ is, respectively, less than or greater than 1. They are null when ωρ = log(1

+ √ 2) (or sinh(2ωρ) = 2 √ 2)
. Moreover, since φ = 2π is identified with φ = 0, these are closed timelike curves whenever ρ is larger than this value, but they are not geodesics. In the present paper, we are considering an infinite cylinder described by this metric and so it is assumed that, in the interior region, ωρ < log(1 + √ 2).

Possible exterior solutions

We now consider a family of compound space-times which contain an infinite rigidlyrotating cylinder of matter with proper radius ρ 1 , such that ωρ 1 < log(1 + √ 2). The region inside this cylinder is taken to be described by the Gödel metric (1) with 0 < ρ < ρ 1 . The exterior region is taken to be vacuum with, of course, a negative cosmological constant. Our purpose is both to find the exterior solution which matches to (1) appropriately on the boundary ρ = ρ 1 , and to determine its properties.

The exterior metric may be assumed to take the general stationary form

ds 2 = -A dt 2 -2K dt dφ + dρ 2 + C dφ 2 + D dz 2 , (2)
where ρ remains as a proper radial distance and A, C, D and K are functions of ρ only. It is assumed here that t ∈ (-∞, ∞), ρ ∈ (0, ∞), z ∈ (-∞, ∞) and φ ∈ [0, 2π). With φ = 2π identified with φ = 0, it may be seen that the space-time contains closed timelike curves whenever the metric function C is negative. In this case, φ is a timelike coordinate and curves with constant t, ρ and z are closed timelike curves, though they are not necessarily geodesics.

It is also convenient to introduce a function ∆(ρ) such that

∆ 2 = AC + K 2 , ( 3
)
and it is reasonable to assume that ∆ > 0. With this, the metric (2) could be written in the alternative form

ds 2 = -A dt + K A dφ 2 + dρ 2 + ∆ 2 A dφ 2 + D dz 2 .

The BSM exterior solution

In [1], Bonnor, Santos and MacCallum have considered the exterior solution to be a special case of the family of vacuum solutions given by Santos [3]. Using a slightly different notation involving the above constant ω = -Λ/2, this can be expressed in terms of two functions G(ρ) and F (ρ), where

G = √ -g = ∆ √ D.
In particular, they have taken G in the form

G = c 1 cosh √ 6 ωρ + c 2 sinh √ 6 ωρ , (4)
where c 1 and c 2 are constants, and have required F (ρ) to satisfy the equation

F = γ G -1 , (5)
where γ is a constant. With these, the metric function D takes the form

D = ∆ e δF ,
where and δ are real constants with > 0. The function ∆ is then given by

∆ = -1/3 G 2/3 e -δF/3 , (6)
and the remaining metric functions were taken to be

A = ∆ 2β cosh αF - α 2 + β 2 α sinh αF , K = ∆ cosh αF - β α sinh αF , (7)
C = ∆ 1 α sinh αF,
which contain the additional constants α and β, of which α could be imaginary. The parameters of this solution satisfy the constraint

24 ω 2 (c 2 2 -c 1 2 ) = (3α 2 + δ 2 )γ 2 . ( 8
)
On application of the boundary conditions, however, it is found that this form of the exterior solution does not easily reduce to Minkowski space in the slowly rotating limit as ω → 0. A different approach is therefore preferred here.

A general exterior solution

In order to construct an exterior solution that more transparently contains the slowly rotating limit, we take the same solution as given by ( 4)-( 6) and ( 8) but apply a general transformation to the t and φ coordinates. In this way, in place of [START_REF] Linet | [END_REF], the remaining metric functions can be expressed in the more general form

A = ∆ 2κµ cosh αF -(κ 2 + µ 2 ) sinh αF , K = ∆ [(κν -λµ) cosh αF + (κλ -µν) sinh αF ] , (9)
C = ∆ 2λν cosh αF + (λ 2 + ν 2 ) sinh αF ,
where κ, λ, µ and ν are arbitrary constants which satisfy the constraint

κν + λµ = 1, (10)
and are such that κ and µ have dimension L -1/2 and λ and ν have dimension L 1/2 . Even with the condition (10), however, this parametrisation contains more freedom than is required. A further constraint may be imposed. However, this will be left until after the application of junction conditions and it will then be used in order to simplify the expressions obtained.

Our second point of departure from [1], is to note that it is possible to determine the exterior solution explicitly by first rewriting the constants c 1 and c 2 as

c 1 = - c √ 6 ω sinh( √ 6 ωρ 0 ), c 2 = c √ 6 ω cosh( √ 6 ωρ 0 ),
where c and ρ 0 are constants. This re-expresses the function (4) in the form

G = c √ 6 ω sinh √ 6 ω(ρ -ρ 0 ) . (11)
In this case, the equation ( 5) can be integrated to give F (ρ) explicitly in the form

F = γ c log 2k √ 6 ω tanh √ 6 ω 2 (ρ -ρ 0 ) , ( 12
)
where k is an arbitrary constant. Recalling that ω = -Λ/2, it is now convenient to introduce the functions

P (ρ) = 2 tanh √ -3Λ(ρ -ρ 0 )/2 √ -3Λ , Q(ρ) = sinh √ -3Λ(ρ -ρ 0 ) √ -3Λ , (13)
so that G = c Q and e F = (kP ) γ/c . This metric could have a singularity when ρ = ρ 0 . However, when taken to represent the exterior of a cylinder of proper radius ρ 1 , this does not occur for ρ ∈ [ρ 1 , ∞).

In terms of the above functions,

∆ = -1/3 (c Q) 2/3 (kP ) -γδ/3c , (14)
and the remaining metric functions are given explicitly by

A = 1 2 ∆ -(κ -µ) 2 (kP ) αγ/c + (κ + µ) 2 (kP ) -αγ/c , (15)
K = 1 2 ∆ (κ -µ)(λ + ν) (kP ) αγ/c -(κ + µ)(λ -ν) (kP ) -αγ/c , (16)
C = 1 2 ∆ (λ + ν) 2 (kP ) αγ/c -(λ -ν) 2 (kP ) -αγ/c , (17)
D = ∆ (kP ) γδ/c , (18)
in which the constants satisfy the constraint

(3α 2 + δ 2 )γ 2 = 4c 2 , (19)
replacing (8). The above expressions remain valid when α and c are either real or imaginary. However, when c is imaginary, ρ 0 is complex.

Matching conditions

It is now appropriate to consider the compound space-time in which an infinite cylinder of finite proper radius ρ 1 , whose interior is described by the Gödel metric, is matched to an exterior described by the metric in the family described above.

The interior Gödel solution in the form (1) is characterised by a single parameter ω, which determines both the density of the dust source and the value of the cosmological constant. A cylinder of such a source is also determined by its proper radius ρ 1 . A rigidly rotating cylinder described in this way is thus characterised by these two parameters.

Such a source should now be matched to a vacuum exterior solution described by the metric (2) in its general form with (15)-( 18) and ( 13) and ( 14). It may first be noted that the metrics (1) and ( 2) for each region are expressed in terms of the same proper radial coordinate ρ. Then, according to the Lichnerowicz conditions, the metric functions and their derivatives must be continuous across the junction at ρ = ρ 1 . This provides eight explicit constraints on the parameters in the exterior metric. Given that the parameters ω (and hence Λ) and ρ 1 are determined by the source, the eight junction conditions determine the values of the quantities c, ρ 0 , , γδ, αγ, κ, λ, µ, ν and k, in which κ, λ, µ and ν are subject to (10) and some additional constraint that is yet to be identified.

It can immediately be seen that the continuity of G = √ -g and its derivative on the junction ρ = ρ 1 implies the two conditions √ 2 c sinh

√ 6 ω(ρ 1 -ρ 0 ) = √ 3 sinh(2ωρ 1 ), c cosh √ 6 ω(ρ 1 -ρ 0 ) = cosh(2ωρ 1 ).
Thus c is given by

c 2 = 1 -2 sinh 2 ωρ 1 -2 sinh 4 ωρ 1 , (20)
and ρ 0 is defined by the equation

√ 2 tanh √ 6 ω(ρ 1 -ρ 0 ) = √ 3 tanh (2ωρ 1 ) . (21)
Thus Q 1 ≡ Q(ρ 1 ), and hence P 1 ≡ P (ρ 1 ), are now determined explicitly as

c Q 1 = sinh(2ωρ 1 ) 2ω , P 1 = 2 cosh(2ωρ 1 ) -c 3ω sinh(2ωρ 1 )
.

It may also be noted that equations ( 21) and (20), which define ρ 0 and c, only have real solutions when

0 < ωρ 1 < sinh -1 √ 3 -1 2 (< 0.573108),
or when 0 < sinh(2ωρ 1 ) < √ 2. For values of ωρ 1 outside this range, c is imaginary and ρ 0 complex. Now consider the (assumed positive) quantity ∆ defined by (3). The continuity of this and its derivative across ρ = ρ 1 leads to the two further constraints = 2ω sinh(2ωρ 1 ) (kP 1 ) -δγ/c , γδ =cosh(2ωρ 1 ).

These two conditions are equivalent to the required continuity of D and its derivative. Now consider the continuity of A/∆, K/∆, C/∆ and their derivatives. These six (not independent) conditions can be expressed as

-(κ -µ) 2 (kP 1 ) αγ/c + (κ + µ) 2 (kP 1 ) -αγ/c = 2ω sinh ωρ 1 cosh ωρ 1 , ( 23
) (κ -µ)(λ + ν) (kP 1 ) αγ/c -(κ + µ)(λ -ν) (kP 1 ) -αγ/c = 2 √ 2 sinh ωρ 1 cosh ωρ 1 , (24)
(λ + ν) 2 (kP 1 ) αγ/c -(λ -ν) 2 (kP 1 ) -αγ/c = 2 sinh ωρ 1 (1 -sinh 2 ωρ 1 ) ω cosh ωρ 1 , (25)
-(κ -µ) 2 (kP 1 ) αγ/c -(κ + µ) 2 (kP 1 ) -αγ/c = - 2ω αγ (1 + 2 sinh 2 ωρ 1 ) sinh ωρ 1 cosh ωρ 1 , ( 26
) (κ -µ)(λ + ν) (kP 1 ) αγ/c + (κ + µ)(λ -ν) (kP 1 ) -αγ/c = 2 √ 2 αγ sinh ωρ 1 cosh ωρ 1 , (27)
(λ + ν) 2 (kP 1 ) αγ/c + (λ -ν) 2 (kP 1 ) -αγ/c = 2 sinh ωρ 1 (1 -3 sinh 2 ωρ 1 -2 sinh 4 ωρ 1 ) αγω cosh ωρ 1 . ( 28
)
The combinations (25)( 27

αγ(λ 2 -ν 2 ) = 2 √ 2 ω sinh 4 ωρ 1 , αγ(κλ + µν) = 1 -2 sinh 2 ωρ 1 , ( 29
) αγ(κ 2 -µ 2 ) = 2 √ 2 ω.
These equations can be solved by putting

κ = A cosh ψ, λ = B cosh χ, µ = A sinh ψ, ν = B sinh χ,
where, according to (10), the constants A, B, ψ and χ are subject to the constraint

AB sinh(ψ + χ) = 1.
The conditions (29) then become

A 2 = 2 √ 2 ω αγ , B 2 = 2 √ 2 αγω sinh 4 ωρ 1 , ABcosh(ψ + χ) = 1 -2 sinh 2 ωρ 1 αγ .
Together, these imply that

α 2 γ 2 = 1 -4 sinh 2 ωρ 1 -4 sinh 4 ωρ 1 . ( 30
)
It follows that α is only real for

0 < ωρ 1 < sinh -1 √ 2 -1 2 (< 0.440687),
i.e. for 0 < sinh(2ωρ 1 ) < 1. Significantly, it can now be seen that the expressions (20), ( 22) and (30) satisfy the required condition (19).

At this point, it can be seen to be convenient to use the remaining freedom in the choice of the parameters λ, κ, µ and ν, to put χ = ψ (so that κν = λµ = 1 2 ). With this additional assumption it can then be seen that

sinh 2 ψ = 1 -2(1 + √ 2) sinh 2 ωρ 1 4 √ 2 sinh 2 ωρ 1 , cosh 2 ψ = 1 + 2( √ 2 -1) sinh 2 ωρ 1 4 √ 2 sinh 2 ωρ 1 ,
and thus

(κ + µ) 2 = ω(1 -2 sinh 2 ωρ 1 + αγ) αγ sinh 2 ωρ 1 , (λ + ν) 2 = sinh 2 ωρ 1 (1 -2 sinh 2 ωρ 1 + αγ) αγω , (κ -µ) 2 = ω(1 -2 sinh 2 ωρ 1 -αγ) αγ sinh 2 ωρ 1 , (λ -ν) 2 = sinh 2 ωρ 1 (1 -2 sinh 2 ωρ 1 -αγ) αγω ,
So far, seven independent junction conditions and two constraints have been used to obtain explicit expressions for the quantities c, ρ 0 , k γδ/c , γδ, αγ, κ, λ, µ and ν. One more condition is required to separately identify the parameters and k. Any of the conditions (23)-(28) or combinations of them is sufficient for this. In particular, subtracting (26) from (23) gives

(kP 1 ) αγ/c = cosh ωρ 1 (1 + αγ -2 sinh 2 ωρ 1 ) sinh ωρ 1 (1 + αγ + 2 sinh 2 ωρ 1 ) .
This determines k, and is then obtained from (22).

The slowly rotating limit

It is now appropriate to investigate the limit in which ω → 0 while ρ 1 is kept constant. In this limit, in which the cosmological constant vanishes, the metric (1) for the interior region approaches that of a vacuum Minkowski space in cylindrical coordinates. It therefore needs to be checked that the exterior metric approaches the same limit.

Expanding expressions to the least order in ω that is required, it can be seen that

c → 1, ρ 0 → 0, γδ → -1, → (ρ 1 kP 1 ) -1 , c Q 1 → ρ 1 , and hence ∆ → ρ 1 Q Q 1 2/3 P P 1 1/3 → ρ.
It can then be seen that

αγ → 1 -2ωρ 1 -14 3 (ωρ 1 ) 2 + . . . Hence (κ+µ) 2 → 2ω (ωρ 1 ) 2 , (κ-µ) 2 → 4ω(ωρ 1 ) 2 , (λ+ν) 2 → 2 ω (ωρ 1 ) 2 , (λ-ν) 2 → 4 ω (ωρ 1 ) 6 .
Since kP 1 → (ωρ 1 ) -1 and P → ρ, it follows that k → ω -1 ρ -2 1 . It can then be seen that the remaining metric functions approach the following limits:

A → 1, K → 0, C → ρ 2 .
It is thus confirmed that the exterior metric reduces exactly to the required form of Minkowski space in this limit.

Asymptotic properties

It is also appropriate to investigate the asymptotic properties of this family of solutions as ρ → ∞. It may first be noted from (11) that, in this limit,

G = c Q → c 2 √ 6 ω e √ 6 ω(ρ-ρ 0 ) .
Moreover, P approaches the finite quantity

√ 2/( √ 3
ω). It can then be seen from ( 14) that ∆ approaches a constant multiplied by e 2 √ 6 ωρ/3 . Thus, A, K, C and D all behave asymptotically like e 2 √ 6 ωρ/3 and it is possible to rotate coordinates to set K → 0. Then, after appropriately rescaling the coordinates, the metric asymptotically approaches a form that is equivalent to

ds 2 = exp 2 - Λ 3 ρ (-d t2 + dz 2 + d φ2 ) + dρ 2 , (31)
which is one of the standard forms for the metric of anti-de Sitter space. This confirms that this solution is also asymptotic to the known static cylindrically symmetric solution with a negative cosmological constant [START_REF] Linet | [END_REF], [8]. However, it should be noted that the coordinate φ in (31) is periodic. The exterior region is therefore not strictly asymptotic to anti-de Sitter space in any global sense.

The exterior region

It can now be shown that the coordinate transformation

t = 1 √ 2 (λ + ν) t -(λ -ν) φ , φ = 1 √ 2 (κ -µ) t -(κ + µ) φ , (32)
takes the vacuum exterior metric (2), with components given by ( 15)-( 18), to the diagonal form

ds 2 = ∆ -(kP ) -αγ/c d t 2 + (kP ) γδ/c dz 2 + (kP ) αγ/c d φ2 + dρ 2 ,
where ∆ is given by (14). It is now possible to rescale the coordinates t, φ and z to absorb the constants , c and k, so that the resulting metric becomes

ds 2 = Q 2/3 -P -(3α+δ)γ/3c d t2 + P 2δγ/3c dz 2 + P (3α-δ)γ/3c d φ2 + dρ 2 . ( 33
)
This may now be seen to be identical to the general static metric with a cosmological constant obtained by Linet [START_REF] Linet | [END_REF] and Tian [8], which may be expressed in the form

ds 2 = Q 2/3 -P -2(1-8σ+4σ 2 )/3Σ dt 2 + P -2(1+4σ-8σ 2 )/3Σ dz 2 +P 4(1-2σ-2σ 2 )/3Σ dϕ 2 + dρ 2 , (34)
where Σ = 1 -2σ + 4σ 2 , and in which Q and P are given exactly as in (13) for this case with Λ < 0. (The periods of φ and ϕ and the existence of related conicity parameters are ignored in both metrics.) Notice that the sum of the different powers of P in (34) vanishes, while the sum of the squares of the powers of P is equal to 8 3 . For the metric (33), the first result is satisfied trivially and the second is precisely the condition (19).

The Linet-Tian metric (34) is the static generalisation of the Levi-Civita solution which includes a cosmological constant. The parameter σ is usually interpreted as the mass per unit length of the source as it has this interpretation in the limit as Λ → 0. By comparing the indices in the metrics (33) and (34), it can be seen that the parameter σ is given by

σ = 1 4 ∓ √ 3 4 1 + γδ 2c 1 -γδ 2c
.

Thus σ = 0 corresponds to γδ = -c which, as expected, occurs when ωρ 1 = 0 and the source vanishes.

The results given here indicate that the exterior field in locally static. However, it is not globally static. In addition, it only has this property when the expressions in the transformation (32) are real, and this only occurs when α is real. The situation here is therfore qualitatively the same as that described by the van Stockum solution [9], which describes a rigidly rotating cylinder of dust when Λ = 0 and in which, provided ωρ 1 is not too large, the vacuum exterior region is locally isomorphic to the Levi-Civita solution.

Conclusion

The solution described above represents the gravitational field a rotating infinite cylinder of finite radius. The interior region, given by the Gödel metric (1), contains a rigidly rotating dust source. This has been matched to a vacuum exterior. Both regions have the same negative cosmological constant. The complete space-time is determined by two parameters -the vorticity ω of the fluid interior, which determines both the cosmological constant and the density of the fluid in the interior region, and ρ 1 , which is the proper radius of the cylinder.

When 0 < sinh(2ωρ 1 ) < 1 (i.e. when 0 ≤ ωρ 1 < 0.441 approximately), α is real and the exterior solution is locally isomorphic to the static Linet-Tian solution. When 1 < sinh(2ωρ 1 ) < √ 2, α is imaginary and c and ρ 0 are real. For √ 2 < sinh(2ωρ 1 ) (i.e. when 0.573 < ωρ 1 roughly), α and c are both imaginary and ρ 0 is complex. The heavy line illustrates the metric function g φφ of the Gödel solution as a function of ωρ. This is considered for ωρ ≤ ωρ 1 . Extensions to the exterior solution are also illustrated for three particular values of ωρ 1 . Closed timelike curves occur when g φφ is negative. For 0 < ωρ 1 < 0.646, the extension remains positive for all ρ. For 0.646 < ωρ 1 < 0.881, the extension becomes negative after some finite distance and, for 0.881 < ωρ 1 , the metric function is already negative in the interior region.

The space-time contains closed timelike lines whenever the metric function C becomes negative. This function is plotted in figure 1 for extensions at three different values of ωρ 1 . It is found numerically that, if 0 < ωρ 1 < 0.6456, the exterior region does not contain closed timelike curves. For 0.6456 < ωρ 1 < 0.881, the exterior region extends after a finite distance into a region which contains closed timelike curves. For 2 √ 2 < sinh(2ωρ 1 ), closed timelike curves occur in both the interior and exterior regions.

  )-(24)(28), (25)(26)-(23)(28) and (24)(26)-(23)(27) with the constraint (10) give, respectively,

Figure 1 :

 1 Figure 1: The heavy line illustrates the metric function g φφ of the Gödel solution as a func-

Acknowledgements

The authors are most grateful to Dr Fátima da Silva for hospitality at the Departamentode Física Teórica, Universidade do Estado do Rio de Janeiro, Brazil, where this work was started. They are also grateful for financial support from CNPq, CAPES, Coordenação de Pós Graduação em Física da UERJ and the Santander Fund.


Dr Frédérique Battin-Leclerc
email: frederique.battin-leclerc@ensic.inpl-nancy.fr

Dr Olivier Herbinet

Pierre-Alexandre Glaude

René Fournet

Dr Zhongyue Y Zhou

Dr Liulin L Deng

Dr Huijun J Guo

Dr Mingfeng F Xie

Fei Qi

Dr P.-A Glaude

Prof R Fournet

Prof. F Qi

Experimental Confirmation of the Low-Temperature Oxidation Scheme of Alkanes

Keywords: hydroperoxides, mass spectrometry, molecular beams, oxidation, photoionization

The control of auto-ignition can be used to increase the efficiency of internal combustion engines which has clear potential positive implications for the problem of global warming [1]. The design of internal combustion engines [2], as well as improved safety in oxidation processes [3], rely on a good understanding of the kinetic mechanism of the autoignition of organic compounds. Here we experimentally demonstrate a key assumption of this mechanism that has been accepted for more than 20 years but never proven [START_REF] Walker | Basic Chemistry of Combustion[END_REF][START_REF] Cox | [END_REF][6]. A detailed speciation of the hydroperoxides responsible for the gas-phase auto-ignition of organic compounds has been achieved for the first time, thanks to the development of a new system coupling a jet-stirred reactor to a molecular-beam mass spectrometer combined with tunable synchrotron vacuum ultraviolet (SVUV) photoionization. The formation of alkylhydroperoxides (ROOH) and carbonyl compounds having a hydroperoxide function (ketohydroperoxide) has been observed under conditions close to those actually observed before the auto-ignition. This result gives the experimental confirmation of an assumption made in all the detailed kinetic mechanisms developed to model autoignition phenomena. A good understanding of the kinetic mechanism governing the oxidation and the auto-ignition of organic compounds is important in two significant applications: the design of combustion engines and the safety of oxidation processes in the chemical industry (e.g. petrochemistry). Auto-ignition governs many features of the combustion in internal combustion engines (e.g. knock and related phenomena, octane and cetane rating) [2]. The potential explosive properties of mixtures of organic compounds with oxygen seldom have catastrophic consequences, but it very often imposes stringent constraints on the operating conditions of partial-oxidation processes [3]. In addition to their propensity to auto-ignite, mixtures of organic compounds with oxygen possess the following specific reactive features. Single or multiple small temperature pulsations (a few tenths of K) accompanied by weak emission of blue light due to excited formaldehyde, so-called "cool flames" [START_REF] Pollard | Hydrocarbons[END_REF], are often observed under conditions preceding those of auto-ignition (usually from 550 K). Closely linked with "cool flames", a zone of temperature (usually around 650 K) where the reactivity decreases with temperature, commonly called "negative temperature coefficient" (NTC) zone, is a second well-known characteristic of these systems [2,[START_REF] Pollard | Hydrocarbons[END_REF]. These intriguing features have made the gas-phase low-temperature oxidation of organic compounds a fascinating field of investigation for kineticists since the end of the 19th century. It was proven early on that the autoignition of hydrocarbons cannot be explained by solely thermal theory, [START_REF] Frank-Kamenetskii | Diffusion and Heat Exchange in Chemical Kinetics[END_REF] but is mainly a result of free-radical chain reactions [START_REF] Semenov | Some Problems in Chemical Kinetics and Reactivity[END_REF]. Following Semenov [START_REF] Semenov | Some Problems in Chemical Kinetics and Reactivity[END_REF], who proposed the concept of degenerate branched-chain reactions (production of free radicals from products formed in a chain reaction) to explain explosive reactions, much work has been devoted to elucidate the species responsible for degenerate branched-chain reactions during the oxidation of organic compounds [START_REF] Pollard | Hydrocarbons[END_REF]. Scheme 1 presents the current commonly accepted mechanism for the oxidation of hydrocarbons at low temperatures (below 1000 K). [START_REF] Walker | Basic Chemistry of Combustion[END_REF] After a short initiation period, a hydrocarbon (RH) mainly reacts with a hydroxyl radical (COH) to give an alkyl radical (RC) through reaction (1), followed by the formation of alkylperoxy radicals (ROOC) after a barrierless reaction with an oxygen molecule (reaction (2)). The existence of the NTC zone is mainly due to the enhanced reversibility of this reaction as the temperature increases. From ambient temperature up to about 550 K, the only prevalent reaction of alkylperoxy radicals is with CHO 2 radicals (reaction (3)) or H abstraction from an organic molecule to yield alkylhydroperoxides (ROOH). The fragility of the RO-OH bond of hydroperoxides (the bond dissociation energy is around 43 kcal.mol -1 ), which can easily break when the temperature increases, has made them very good candidates as species responsible for the degenerate branched-chain reactions explaining auto-ignition and the occurrence of cool flames [START_REF] Walker | Basic Chemistry of Combustion[END_REF][START_REF] Pollard | Hydrocarbons[END_REF]. Scheme 1. Simplified scheme of the primary mechanism of oxidation of alkanes at low temperature.

Above 550 K (in the zone of cool flames), a second consumption pathway becomes faster for alkylperoxy radicals: their isomerization (reaction (4)) by internal transfer of a hydrogen atom to give hydroperoxyalkyl radicals (CQOOH). These radicals can easily decompose to yield oxygenated products, amongst which cyclic ethers are the most abundant species (reaction ( 5)). As cyclic ethers are stable compounds easily analyzed by gas chromatography, the first steps of the oxidation of organic compounds through reactions ( 1)-( 4) (in Scheme 1) are relatively well known [START_REF] Pollard | Hydrocarbons[END_REF]. A second important fate of CQOOH radicals is the second addition to oxygen (reaction ( 6)). The formation and the reactions of the resulting COOQOOH radicals have not yet been directly experimentally investigated. However, following Fish [START_REF] Fish | [END_REF], all the detailed kinetic mechanisms developed nowadays to model the low-temperature ignition of organic compounds consider decompositions of COOQOOH radicals yielding three radicals, amongst them two COH radicals. Cox and Cole [START_REF] Cox | [END_REF] have proposed this global step to occur through the formation of alkylhydroperoxides including a carbonyl group (reaction (8)), mainly ketohydroperoxides, which in turn can easily decompose to give two radicals (reaction ( 9)) and lead to another possible chain-branching reaction. Many modeling studies have suggested that the direct formation of hydroperoxides by reactions of peroxy radicals was not fast enough and that the inclusion of a chain-branching step deriving from the second addition to oxygen was necessary to satisfactorily simulate the experimental observations. While several detailed models can now reproduce most features of the oxidation and auto-ignition of alkanes representative of those present in automotive fuels [6], experimental evidence supporting this commonly used reaction scheme is extremely scarce [11]. The purpose of this study is to give experimental evidence for the formation of hydroperoxides under conditions similar to those observed prior to the auto-ignition of alkanes and using a method that makes it possible to distinguish between alkylhydroperoxides and ketohydroperoxides. n-Butane was studied as it is the smallest alkane with oxidation behavior close to that of the species present in gasoline and diesel fuel. Minimizing the potential reactions of hydroperoxides between their formation and their detection is possible by creating a molecular beam from a high-pressure reactive zone toward a detection device maintained under vacuum; this results in the "freezing" of the chemical composition of the sampled gas. Since hydroperoxides are expected to be formed in very low amounts along with a large number of other oxidation products, a highly sensitive mass spectrometer with low photon energy to avoid product fragmentation has been used, that is, a time-of-flight mass spectrometer combined with tunable SVUV photoionization. This analytical method has already been used successfully to investigate many reactive systems [12] such as laminar premixed flames [13,14]. Since we must accumulate several scans to obtain good sensitivity, this analysis device must be coupled with a reactive system working under steady conditions. Our purpose was then not to really observe unsteady phenomena, such as cool flames or auto-ignition, but to mimic the chemistry leading to these phenomena. We used a heated jet-stirred reactor. This type of reactor coupled with gas chromatography analyses has already been used many times for studying the low-temperature oxidation of organic compounds [15,16]. In these experiments the presence of a NTC zone was shown and the formation of the same type of products was proved as those actually analyzed prior auto-ignition or during cool flames [17]. Figure 1 presents a simplified scheme of the jet-stirred reactor associating a molecular-beam sampling system to a reflectron time-of-flight mass spectrometer combined with SVUV photoionization. The study of the oxidation of n-butane was performed at temperatures between 560 and 720 K, with a mean residence time of 6 s and for an n-butane/oxygen/argon mixture composition of 4:26:70 (in mol%), which corresponds to a stoichiometric mixture for the complete combustion reaction. Figure 2a presents the evolution of the mole fraction of n-butane with temperature. The mole fraction of n-butane (m/z 58) was derived directly from the normalized ion signal at m/z 58 (obtained with a photon energy of 11.0 eV (1 eV = 96.5 kJ.mol -1 )) assuming that no reaction occurred below 590 K. Raw signals of every compound were normalized by the ion signal obtained for argon (m/z 40) with a photon energy of 16.2 eV which acted as an internal standard. Many products were obtained; the major C 4 products were butenes at m/z 56 and C 4 H 8 O products at m/z 72. Figure 2b also displays the normalized ion signal at m/z 56 and 72 obtained with a photon energy of 10.0 eV to avoid ion fragmentation. The ionization energy (IE) of butenes is between 9.11 and 9.55 eV [START_REF]NIST Standard Reference Database 69[END_REF]. The signal at m/z 72 corresponds to the C 4 H 8 O oxygenated products which have an IE below 10 eV, for example, butanal, butanone, and all cyclic ethers (except 2-ethyloxirane which has an IE greater than 10 eV). Note the NTC zone obtained above 630 K (see Figure 2a), which makes the evolution with temperature of the normalized signal of all the products have the same shape as those at m/z 72 with a kind of "plateau" between 620 and 680 K. This plateau is not observed in the case of species corresponding to four specific masses: 48, 62, 90, and 104 (Figure 2b). For these species, a sharp peak is obtained around 590 K meaning that they are very reactive molecules that decompose very quickly when the temperature increases. In a system containing only carbon, hydrogen, and oxygen atoms, m/z 48 can only be related to methylhydroperoxide (CH 3 OOH). For m/z 62, 90, and 104, several possibilities exist. However the mass of ethylhydroperoxide (C 2 H 5 OOH) is 62, that of butylhydroperoxides (C 4 H 9 OOH) is 90, and that of C 4 ketohydroperoxides (C 4 H 8 O 3 ) is 104. Figure 3 shows a mass spectrum with signals at m/z 48, 62, 90, and 104. A substantiation of this identification is given because these four compounds are the only ones having the same particular evolution with temperature, with a sharp peak, characteristic of compounds rapidly decomposing when temperature increases.

Figure 3. Typical mass spectrum obtained for the oxidation of n-butane. The temperature in the reactor was 590 K and the photon energy was 10.0 eV.

Additional supporting evidence for this identification relies on IEs from measurements of photoionization efficiency spectra. When such values were not available in the literature [START_REF]NIST Standard Reference Database 69[END_REF], zero-point energy corrected adiabatic IEs have been calculated from the CBS-QB3 method [START_REF] Montgomery | [END_REF] using Gaussian03 [START_REF] Gaussian ; Revisionb | Gaussian[END_REF]. An experimental sweeping of photon energies from 8.5 to 11.5 eV has been made. The obtained experimental IEs were 9.91, 9.69, 9.37, and 9.29±0.05 eV for m/z 48, 62, 90, and 104, respectively (see the Supporting Information). The calculated values for methylhydroperoxide, ethylhydroperoxide, butylhydroperoxides, and C 4 ketohydroperoxides were 9.83, 9.61, 9.33-9.36, and 9.34-9.39 eV, respectively. For the last two compounds, a range of energy is given because the calculations were made for all the expected isomers (see Table 1). Note that 1,2-dihydroxyethane, another possibility for m/z 62, has an ionization energy of 10.16 eV [START_REF]NIST Standard Reference Database 69[END_REF]. Finally, even further substantiation of the proposed assignments is given by simulations performed with a detailed kinetic mechanism of the oxidation of n-butane previously proposed [START_REF] Buda | [END_REF] (see the Supporting Information). Figure 2a andc show that, while small differences are encountered for the start of the reactivity and the extent of the NTC zone, the model reproduces well the variations of the mole fraction of n-butane with temperature, as well as the "plateau" obtained below 680 K for the major products, such as butenes and C 4 H 8 O compounds. As shown in Figure 2c, simulations reproduce well the particular mole fraction profiles (with a sharp peak) obtained for the four hydroperoxides detected in the experiments compared to those of the major products.

Table 1: Ionization energies of most expected isomers of C 4 hydroperoxides deriving from the oxidation of n-butane.

Mass

Formula Structure IE (eV) [a] 90 C4H9OOH

C H 3 CH 3 O O H 9.33 C H 3 O OH 9
.36 [c] 104 [b] C3H7COOOH

C H 3 O O H O 9.39 C H 3 O OH O 9
.34 [a] When no reference is given the value has been calculated. The mean absolute error of CBS-QB3 for the G2 test is less than 0.05 eV. For hydroperoxides, which can involve hydrogen bonds, lowest energy conformers were searched for a systematic manner. [b] For kinetic reasons, the probability of having two neighboring C atoms each bonded to an oxygen atom is lower. [c] Ref. [START_REF]NIST Standard Reference Database 69[END_REF].

This work gives the first speciation of the hydroperoxides formed during the low-temperature oxidation of an organic compound. Further investigations are necessary to connect the structure of the reactant to the type of hydroperoxides obtained. This should help to increase the accuracy of detailed kinetic models of combustion in order to make them really predictive [22] and able to reproduce the formation of minor pollutants, which have a deleterious impact on environment and human health [23].

Experimental Section

The utilized spherical quartz jet-stirred reactor (diameter around 5 cm), in which a gas mixture is continuously flowing, was operated at a constant temperature and pressure [16,24]. This type of reactor, which can be heated up to 1200 K, is well suited for kinetic studies: the gas phase inside the reactor is well stirred, meaning the concentrations and temperature are homogenous [25]. Quartz wall effects can usually be considered as negligible. This study has been performed under quasiatmospheric pressure (1.05 atm). As the type of diluent gas has negligible chemical significance, argon was used instead of nitrogen, for the ease of the mass spectrometric analysis. The coupling between the quartz reactor and the mass spectrometer was made through a quartz conelike nozzle with a 75 mm orifice (the tip of which has been inserted inside the reactor) generating a molecular beam, which was skimmed before the entrance in the spectrometer chamber. In situ detection of intermediates was made possible by combining tunable VUV photoionization and the molecularbeam-sampling mass spectrometer. The produced ions were detected by a reflectron time-of-flight mass spectrometer which can distinguish between different masses [26].

Figure 1 .

 1 Figure 1. Simplified illustration of the experimental device.

Figure 2 .

 2 Figure 2. Plots of the experimental and simulated mole fractions or signals of the oxidation of n-butane, the main products and hydroperoxides with temperature: a) the experimental (black circles) and simulated (line) mole fractions of n-butane; b) the experimental ion signals (in arbitrary units) at m/z 48 (green circles), 56 (white squares, the signal is divided by 7.5), 62 (purple circles), 72 (black squares), 90 (blue circles, the signal is divided by 4) and 104 (red circles); and c) the computed mole fractions (in ppm) of CH 3 OOH (green line, mole fraction divided by 5), C 2 H 5 OOH (purple line), C 4 H 9 OOH (blue line), the C 4 ketohydroperoxides (red line), butenes (dash line, mole fraction divided by 5), and C 4 H 8 O products (black line).

Acknowledgments:

This work was supported by the European Commission through the "Clean ICE" Advanced Research Grant of the European Research Council and by a grant from Région Lorraine (Soutien Jeune Chercheur, Olivier Herbinet). F.Q. acknowledges funding from the Chinese Academy of Sciences, Natural Science Foundation of China (50925623 and 20533040), and the Ministry of Science and Technology of China (2007CB815204 and 2007DFA61310). We thank Prof. J. F. Pauwels and D. Leray for their help in designing the coupling with the quartz reactor and S. Bax, H. Legall, and P. Aury for technical assistance in France.