ERA OF PRECISION COSMOLOGY 1. THE ERA OF PRECISION COSMOLOGY

"The Cosmos is all that is or was or ever will be. Our feeblest contemplations of the Cosmos stir us -there is a tingling in the spine, a catch in the voice, a faint sensation, as if a distant memory, of falling from a height. We know we are approaching the greatest of mysteries."

Carl Sagan, Cosmos Agradecimientos Tout d'abord j'aimerais remercier à Julien Lavalle et Vivian Poulin, mes directeurs de thèse. Julien, je te remercie énormément pour m'avoir accueilli dans ton équipe, pour ton soutien et pour les conseils très précieux que tu m'as donné pendant ces dernières années. Vivian, un grand merci pour ton implication constante, ton suivi et ton aide dans le travail. Je suis aussi très reconnaissant de l'opportunité que tu m'as donné de voyager en Europe et aux États-Unis pour présenter mon travail (quand la pandémie l'a permis!) et de participer à des projets hyper-interessantes avec des collaborateurs internationaux. Cela m'a permis de m'intégrer du mieux possible à la communauté et de contribuer activement à un domaine de recherche qui est vraiment ma passion. Sans oublier la bonne humeur et tous les bons moments passés en dehors du laboratoire. Je te garderai non seulement comme un guide exceptionnel et un modèle à suivre, mais aussi comme un bon ami. J'espère qu'on gardera le contact et on continuera à collaborer ensemble dans le futur.

I would like to thank all the wonderful people that I had the chance to meet during my stay at the LUPM: Mihael, a.k.a. the bicycle man; Rodrigo, official founder of the latino sub-group; Riki and Marco, my dearest sardinian (although not necessarily italian) friends; the ever smiling Casey (&Jabba); Théo, Alice, Hugo and Chadi, the most tenacious Wordle players that could possibly exist, and Eva, la meva compatriota catalana. Without all the karaoke nights, the barbecues or the cycling trips to the beach, these three years would have been substantially less fun. A special thanks to Gaétan (monsieur le président), my office mate for more than two years. I will never forget our intense physics discussions and all the patience you had for answering my numerous questions. I will also keep a fond memory of all the PhD students that I met during the unforgettable Les Houches summer school 2021, including Adriana (Menki), Gonzalo (=Rafa), JoaQUIM, Arman, Nudzein, Maxl, Dror... I hope our paths will cross again in the future. I would also like to address my thanks to my colleagues in the USA for the various discussions and lively meetings we had in the last years: Zackaria Chacko, Peizhi Du, Yuhsin Tsai and Abhish Dev. For the same reasons, I am very thankful to all the people at Aachen and Amsterdam with whom I had the pleasure to collaborate in the "H 0 Olympics": Nils Schöneberg, Andrea Pérez, Sam Witte and Julien Lesgourgues. Let me thank warmly the members of my thesis committee, for accepting to judge my work. Special thanks to the rapporteurs for accepting to review the manuscript.

Me gustaría pasar a agradecer a todas las personas que han estado a mi lado desde mucho antes de que empezase mis andaduras en el mundo de la investigación. Para empezar, quiero agradecer a mis amigos murcianos de toda la vida: Edu y Salva (a.k.a los pollaringas), Carlos linkin, Gubensiko, Dory, Marta Takian y Guille (no-lennon). Muchas gracias por los momentos de risas que he podido compartir con vosotros (a pesar de la distancia Montpellier-Murcia) y que me han permitido mantener una buena salud mental durante las épocas más duras de mi doctorado. Como no agradecer a mis amigos de la carrera, en particular a todos los integrantes de El bussón de Higgs (Chino, Edu-cuñao, Pablezno, Josexo...), con los que tantísimas buenas experiencias he tenido. Especial mención a Chema la señu nostre y Castelo/Pincelo, por contagiarme vuestra enorme pasión por la física, y por haberme hecho alguna visitilla durante mi estancia en Montpellier.

Por supuesto, quiero dar las gracias a MJ, mi pareja en la vida desde hace ya más de 7 años. Estas pocas líneas no me dan para expresar la enorme gratitud que siento por todo el enorme apoyo que me has dado, y tu inestimable compañía y ayuda en los momentos más difíciles de mi tesis. Si he llegado hasta aquí ha sido en parte gracias a ti. Estoy convencido de que en los próximos años podremos seguir compaginando nuestra relación con nuestra vocación de investigadores, porque como dice la canción," te quiero libre, y libre te quiero conmigo".

Por último, pero no menos importante, me gustaría agradecer a todos los miembros tan maravillosos de mi familia. En particular, quiero dar las gracias a mis primos Felipe y Antonio, que me han dado consejos muy valiosos sobre diversos aspectos de la carrera investigadora. Y como no, infinitas gracias mamá y papá por haber creído siempre en mí, por haberme educado tan bien, y por haberme apoyado incondicionalmente en todas las decisiones que he tomado en mi vida. Si he llegado hasta aquí también es en parte gracias a vosotros, y me llena de felicidad saber que estaréis orgullosos de mí. Gracias también a Ferni y Poté por ser los mejores hermanos que uno podría desear.

Cosmology is the study of the evolution and the properties of the universe as a whole. This scientific discipline addresses questions which are as old as humanity itself (i.e. "has the universe as we know existed forever?" "what's the ultimate fate of the cosmos?"). However, it was not until the beginning of the 20 th century that it became a predictive science, when the theory of General Relativity (GR) was established as a solid theoretical framework to describe the universe. Since then, the field of cosmology has been constantly evolving, thanks to the rapid increase in the amount and precision of observations. In the last couple of decades, the so-called Λ Cold Dark Matter (ΛCDM ) model of cosmology has emerged as a successful paradigm to explain several independent probes with an astonishing degree of accuracy. This model assumes that the universe is well described by a flat Friedman-Lemaître-Robertson-Walker metric, and is primarily composed of around 5% ordinary matter, 26 % cold dark matter, and 69 % dark energy in the form of a cosmological constant Λ. Unfortunately, the nature of its main constituents, dark matter and dark energy, still remains a mystery. In addition, the increase in precision in cosmological data has recently led to the appearance of several experimental discrepancies between probes of the early and late universe. In particular, there is a 5σ tension in the determination of the current expansion rate of the universe (the Hubble constant), and a 2 -3σ tension in the determination of the clustering strength in the universe (described through the S 8 parameter). For these reasons, in recent years there has been a growing interest in exploring different extensions of the ΛCDM model, which could shed some light on the very mysterious dark components, and possibly offer an explanation for the aforementioned discrepancies. On the other hand, the discovery of neutrino oscillations provides firm evidence of the existence of tiny but non zero neutrino masses, a phenomenon that cannot be explained within the Standard Model of particle physics. Cosmology is currently the most powerful probe of neutrino masses, and it can offer very valuable clues about its properties.

In this thesis we will investigate in detail the cosmological implications of several extensions of the ΛCDM paradigm. In the first part we will study two models, early dark energy (EDE) and decaying dark matter (DDM), which are directly targeted at explaining the Hubble and S 8 tensions, respectively. In the second part, we will analyze another exotic scenario motivated by the neutrino mass puzzle, decaying neutrinos, whose formalism is very similar to that of the DDM. We will argue how the very precise measurements of the cosmic microwave background (CMB) anisotropies always play a central role in setting new and robust constraints for each of these models. To make this work as much self-contained as possible, in this introductory chapter we outline the most relevant mathematical tools and observables in cosmology. We start with a short historical introduction in Sect. 1.1 and a sketch of the history of the universe in Sect. 1.2. Then we move on to recapitulate the formalism that is required in order to understand cosmological observables such as the CMB anisotropy and matter power spectra. We will start with the description of the smooth background in Sect. 1.3, and say a few words about inflation in Sect. 1.4. Later we proceed to study the departures from equilibrium by introducing the Boltzmann treatment in Sect. 1.5. We continue with a presentation of the theory of linear perturbations in Sect. 1.6. We end with Sect. 1.7, where we discuss the most important challenges to the ΛCDM model: the nature of the dark sector and the cosmological tensions.

Historical introduction

In this section we describe the historical context of the principal milestones that made cosmology be the very rich field that we know today. These milestones are the metric expansion of the cosmos, the cosmic microwave background, and the discoveries of dark matter and dark energy. Here the purpose is just to give a global picture of the recent history of cosmology, for more details we refer the reader to HELGE (2013).

Evidences of an expanding universe

In 1915, Albert Einstein publishes his theory of General Relativity (GR), which provides the most precise description of gravity until the present date. In 1917, Einstein studied the cosmological implications of his theory and proposed his own model of the universe (EINSTEIN 1917), which he described as being static and closed. Einstein noticed that, in order to keep the mean matter density ρ m constant in time, he was forced to introduce a new constant in his field equations (today known as the cosmological constant Λ) and had to adjust it to a very concrete value, Λ = 4πGρ m /c 2 , where G is Newton's gravitational constant and c is the speed of light. It is important to note that, at that time, the general consensus in the scientific community was that the universe was static and eternal.

On the other side, some indications about the cosmic expansion had already started to appear. In particular, the American astronomer Vesto Slipher reported in 1915 the first observations of shifts in the spectral lines of 15 galaxies 1 [START_REF] Slipher | Spectrographic Observations of Nebulae[END_REF]. Out of the 15 nebulae that Slipher observed, 11 were actually receding from us, since they were redshifted to longer wavelengths. This phenomenon is typically quantified through the redshift factor z, defined as z ≡ λ obs -λ emit λ emit .

(1.1)

However, the majority of scientists at the epoch didn't agree on the physical origin of these redshifts. In 1927, the Belgian physicist Georges Lemaître proposed an explanation. Using GR as a main tool, he theorized the idea of an expanding universe, and used this fact to explain the extra-galactic redshifts observed by Slipher (LEMAÎTRE 1927). In reality, the metric expansion had already been found in 1922 by the Russian mathematician Alexander Friedmann, as one of the possible cosmological solutions of Einstein's field equations [START_REF] Friedmann | Über die Krümmung des Raumes[END_REF]. However, Friedmann regarded this merely as a mathematical curiosity, and didn't try to make a connection with observations, as Lemaître did. It is known that Einstein read Lemaître's article, but refused roundly the idea of an expanding universe, even saying to him "Your calculations are correct, but your physic is abominable".

A big discovery was provided in 1929 by the American astronomer Edwin Hubble. At the Mount Wilson Observatory (which at the time had the most powerful telescope in the world), Hubble measured the redshifts (and thus, the recessional velocities v cz), as well as the distances of 24 galaxies [START_REF] Hubble | A relation between distance and radial velocity among extra-galactic nebulae[END_REF]. By placing the velocities and distances on a diagram, he obtained what is shown in Fig. 1.1. What this plot suggested is that there exists a linear positive correlation between those quantities, i.e. the further away the galaxies are, the faster they recede from us. This can be mathematically formulated in the following way

v = H 0 d, (1.2)
where the factor of proportionality H 0 is known as the Hubble's constant. This constant is typically expressed in units of km/s/Mpc, since velocities and distances to distant objects are commonly measured in units of km/s and Megaparsecs, respectively.

The first measurements by Hubble yielded a value of around H 0 500 km/s/Mpc, which is significantly higher than the values accepted at present (H 0 65 -75 km/s/Mpc, see Sect. 1.7.2), due to errors in the distance calibrations. The work of Hubble provided the first solid evidence of the expanding universe, although it is important to note that Hubble himself never associated his empirical law with the cosmic expansion. In 1930, the British astrophysicist Arthur Eddington reexamined the static universe model proposed by Einstein and proved that it was unstable, i.e. any tiny perturbation would lead to either an exponential contraction or expansion [START_REF] Eddington | On the Instability of Einstein's Spherical World[END_REF]. After that, Einstein definitely abandoned his idea of an static universe and a cosmological constant, which he regarded as "the biggest blunder of his life."

From the Big Bang theory to the first hints of dark matter

The following decades after Hubble's big discovery were very fruitful in cosmology. In the 30s, scientists realized that the observed abundances of light elements such as hydrogen or helium could not have a stellar origin. In 1931, Lemaître suggested the existence of a very hot phase at the beginning of the universe, that he called the primaeval atom. During the 40s, the Russian physicist George Gamow expanded on the idea of Lemaître, and realized that densities and temperatures in the primordial universe were high enough to allow nucleosynthesis. The first computation of the relic abundances of light elements was carried out in 1948 by Alpher, Bethe and Gamow ALPHER et al. (1948). In the same year, Alpher and Hermann improved upon this calculation to account for the universe's expansion, and showed that at initial times the universe was dominated by radiation, rather than matter. They also predicted the existence of a blackbody radiation as a remnant of this hot early phase, which today should have a temperature of a few K.

In those days the theory of the expanding universe still had many detractors. This is especially the case for the British astronomer Fred Hoyle, who coined the name Big Bang on BBC radio in 1949 to denigrate the model of the evolving universe, which had the serious issue of having a singularity at the origin. Another problem of this model was the age paradox. The age of the universe can in general be written as t 0 = F/H 0 , where F is a dimensionless numerical factor depending on the cosmological model. In 1932, Albert Einstein and Willem de Sitter had proposed a model of an evolving flat universe, full of non-relativistic matter and with no cosmological constant, for which F = 2/3 [START_REF] Einstein | On the Relation between the Expansion and the Mean Density of the Universe[END_REF]. With the estimations of H 0 made during the 50s, this formula was giving an age of the universe shorter than the age of the Earth. The reality is that, even with the most recent estimates of H 0 , this model leads to an age t 0 9.3 Gyrs, still shorter than the age of the oldest stars that have been observed (as we will discuss later, this problem is automatically solved when considering dark energy).

In 1965, the American physicists Arno Penzias and Robert Wilson discovered by accident a persistent radio noise coming from all directions, which could not be easily explained with terrestrial or known astronomical sources. It was soon realized that they had just detected the cosmic microwave background (CMB), a relic radiation coming from the primordial universe, for which they were awarded the Nobel prize in 1978. This discovery gave strong support to the Big Bang cosmological model. The spectrum of this radiation has been measured thanks to the FIRAS interferometer (onboard the COBE satellite) in 1996, revealing an almost perfect blackbody spectrum, with a mean temperature of T 0 = 2.7255 ± 0.0006 K and tiny anisotropic deviations at the level of 10 -5 (the latter measured by the COBE DMR instrument). For this breakthrough, the American physicists George F. Smoot and John C. Mather were awarded the Nobel prize in 2006. The anisotropies in the CMB radiation carry a wealth of information about the early universe, since they constitute the primordial seeds for the formation of galaxies. After COBE, these anisotropies have been subsequently measured with increasing angular resolution by the WMAP satellite in 2003 [START_REF] Spergel | First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters[END_REF]) and the Planck satellite in 2013 [START_REF] Ade | Planck2013 results. XVI. Cosmological parameters[END_REF]).

As the 20 th century advanced, many observations started to suggest that around 85 % of the total matter in the universe was in the form of a mysterious component called dark matter, whose effects have only been probed through gravity. The first convincing hint of dark matter came in 1933 by the Swiss physicist Fritz Zwicky. He observed a mismatch between the velocity dispersion measured in the Coma Cluster, and the one that was theoretically predicted from the luminous matter using the virial theorem [START_REF] Zwicky | Die Rotverschiebung von extragalaktischen Nebeln[END_REF]. This led him to suggest the existence of large amounts of invisible matter in the Coma Cluster, that he called dunkle Materie (dark matter). However, Zwicky's proposal was not taken seriously by the scientific community until 1970, year in which the American astronomers Vera Rubin and Kent Ford carried out a pioneering work on galactic rotation curves [START_REF] Rubin | Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions[END_REF]. They observed that the velocities in spiral galaxies at large distances were significantly larger than the ones predicted from Newtonian gravity, considering only the luminous components (i.e. the stars and the neutral hydrogen gas). It was later understood that this excess velocity could be due to the presence of large spherical halos of dark matter, in which galaxies were immersed. Over the following years, many other probes have es- The galaxies and dark matter in the cluster act as a lens that warps the light of the background galaxies, leading to a plethora of arcs around the lens, as predicted by GR. This allows to reconstruct the mass distribution of dark matter, which is represented in purple. Taken from https://esahubble.org/images/heic1014a/ tablished the presence of dark matter on both galactic and cosmological scales, such as gravitational lensing (see Fig. 1.2), the CMB anisotropy spectra, or the distribution of large scale structure (for a review of the history of dark matter, see BERTONE et al. 2018). All these observations tell us that dark matter should be mostly cold (i.e. non-relativistic), collisionless, stable, and posses no (or very weak) non-gravitational interactions with the Standard Model (SM) particles 2 . However, its precise nature remains unknown until today.

The universe is accelerating

The last milestone in cosmology was set in 1998. That year, two independent groups (the Supernova Cosmology Project led by Saul Perlmutter, and the Supernova Search Team led by Adam Riess and Brian Schmidt) carried out the measurements of redshifts and distances to supernovae of type Ia (SNIa) [START_REF] Riess | Observational evidence from supernovae for an accelerating universe and a cosmological constant[END_REF][START_REF] Perlmutter | Measurements of Ω and Λ from 42 high redshift supernovae[END_REF]). These objects have several advantages: they are believed to be good standard candles for measuring distances in the universe, their brightness is very intense and they are ubiquitous in the cosmos. This allowed the two teams to extend the Hubble diagram to much larger distances, where the distance-redshift relation becomes dependent on the cosmological model. Their analysis showed that the current cosmic expansion was accelerating, and that SNIa data favored a universe where vacuum energy (parameterized through the cosmological constant Λ) dominated over matter 3 (see Fig. 1.3). This was reflected in the fact that SNIa appeared to be dimmer (due to the longer light-travel time) than what would have 2 Current observations also allow to test small deviations from these hypotheses. Just to give an example, in Chapter 3 we put bounds on the lifetime of dark matter for a decay scenario happening entirely in the dark sector. 3 It might appear contradictory that the cosmological constant that Einstein introduced to get a static universe has now been used to account for an accelerated expansion. However, one should bear in mind that Einstein had assumed a closed universe, and in addition he had cherry-picked the value of Λ to counteract the dynamics encoded in his theory. been expected in a decelerating universe full of matter. For this revolutionary finding, later corroborated by CMB and LSS data, Perlmutter, Riess and Schmidt received the Nobel prize in 2011.

The cosmic acceleration can more generally be interpreted as a result of a dynamical substance with a negative pressure which dominates the energy content of the universe. Any component with these characteristics is typically referred to as dark energy. Another advantage of dark energy is that it allowed to solve the age paradox [START_REF] Krauss | Age Estimates of Globular Clusters in the Milky Way: Constraints on Cosmology[END_REF]. For an universe composed by around 31 % matter and 69 % dark energy in the form of a cosmological constant (as is now suggested by many different observations apart from SNIa), the age of the universe is predicted to be t 0 13.8 Gyrs (see end of Sect. 1.3.2 for a derivation), which is consistent with the ages of all astronomical objects detected so far.

Timeline of the cosmos

Thanks to all the progress that was achieved in cosmology during the 20 th century and in the recent decades, we have nowadays a clear picture of all the key events in the history of the universe, ranging from the complex interactions in the primordial soup to the formation of large scale structures at later times. In this section we give a brief summary of the key events in the history of the universe, with the aim of framing the relevant epochs for all the models and observables considered in this work. These events are listed in Tab. (1.1).

If the cosmic expansion is extrapolated back in time, one quickly realizes that at earlier times the universe must have been denser and also hotter. At some critical point, all the physical quantities (such as densities or temperatures) become infinite, signaling a breakdown of our theory. This is the so-called Big Bang (BB) singularity, which is conventionally defined as the origin of cosmic time.We remark that this is a fictitious instant, which in principle cannot be described without a complete theory of quantum gravity.

It is now commonly accepted that, at very early times, there was a phase of rapid accelerated expansion named inflation, which could have been sourced by a scalar field called the inflaton. After inflation ended, there was a period of reheating, during which the inflaton decayed into all the known SM elementary particles. This led to the formation of a hot dense plasma where all particles were kept in thermal equilibrium due to the constant interactions between them. The densities of all these particles inherited the inhomogeneities that were created during inflation, when the quantum fluctuations in the inflaton were stretched to cosmic distances. Roughly 20 ps after the BB, when the temperature of the plasma decreased below the scale of electroweak (EW) symmetry breaking, T 100 GeV, the Higgs got trapped at the bottom of its potential, making all SM particles to acquire a mass. Later, when the temperature decreased below 150 MeV (approximately 20 µs after the BB), the strong force between free quarks and gluons became significant, so that they could assemble to form the first hadrons. Only 1 s after the BB, the weak interactions between the neutrinos and the rest of particles became inefficient, and they decoupled from the thermal bath. Afterwards, at 6 s, electrons and positrons annihilated through the process e + + e -→ γ + γ, leaving just a residual amount of non-relativistic electrons 4 . Then, a few minutes after the BB, the first nuclei of light elements, such as helium or lithium, were formed during a period called Big Bang Nucleosynthesis (BBN). Eventually, the energy density of non-relativistic matter started to dominate over the radiation one, point at which the inhomogeneities in the DM started to grow at a significant rate. When the universe was 380 kyr old, the bath temperature had decreased enough to allow the formation of the first hydrogen atoms at the recombination era, according to the process e -+ p + → H + γ. This led to a sharp reduction in the free electron fraction, and the decoupling of photons from matter. The universe became transparent and photons started free-streaming through the universe. These photons constitute the CMB radiation that we observe today.

After recombination, the dark ages started, a period during which the only light in the universe came from the CMB photons and the 21 cm signal of neutral hydrogen. The first DM halos are believed to have formed during this era, due to the gravitational collapse of the large DM density fluctuations. The dark ages extended over a few hundred million years, until the first stars and galaxies were born and started emitting radiation that gradually reionized the hydrogen atoms in the intergalactic medium (IGM). The details of this epoch of reionization (EoR) are very poorly understood, but thanks to the measurements of Lyman-α absorption lines in quasar spectra, today we know the universe was nearly totally reionized around z 6 (i.e.

1 Gyr after the BB). Approximately 5 Gyr ago, dark energy began to dominate the energy budget of the universe, producing an accelerated expansion and impeding the formation of large scale structures to develop further.

The smooth universe

In this section we recapitulate the main tools required to describe the homogeneous and isotropic universe, starting with a discussion of the Friedmann equations and then moving to give several definitions of distances in cosmology. We use the (-+ ++) metric convention, and work in units where c = k B = 1. As usual, latin indices indicate spatial components, e.g. i = 1, 2, 3, while greek indices are reserved for spacetime components, e.g. µ = 0, 1, 2, 3. Einstein summation convention is assumed for greek indices. For this section and two subsequent ones we mainly follow For producing some of the plots shown in these sections, we use the public Boltzmann solver CLASS 5 (BLAS et al. 2011) .

Geometry of the expanding universe

Modern cosmology is built upon the idea that the universe is homogeneous and isotropic on sufficiently large scales (bigger than 100 Mpc). This simple but transcendental assumption is what we normally call the cosmological principle. There is plenty of observational data that confirm this hypothesis, such as the temperature map of the CMB (that we show in Fig. 1.4), exhibiting anisotropies only at the level of 10 -5 .

Homogeneity and isotropy, meaning 'no preferred point in space' and 'no preferred direction', impose a unique spacetime geometry of the universe. This geometry is described by the so-called Friedmann-Lemaître-Robertson-Walker (FLRW) metric g µν , that we can read from the line element: 2 dr 2 1 -kr 2 + r 2 dθ 2 + sin 2 (θ)dϕ 2 .

ds 2 = g µν dx µ dx ν = -dt 2 + a(t)
(1.3)

In the previous expression, we have introduced the cosmic time t, which is the time measured by observers who move with the cosmic expansion, and the comoving coordinates r, θ and φ, which label the points in constant-time slices. The metric contains a single function of time, the scale factor a(t), and a constant, the curvature parameter k. The scale factor relates physical distances d phys with comoving distances d com (that is, distances that do not change with the cosmic expansion), d phys = a(t)d com .

The parameter k determines the three possibilities for the constant curvature of spatial slices: k = 0 (euclidean), k = +1 (spherical) and k = -1 (hyperbolic).

We note that in Eq. (1.3) the scale factor a(t) has the dimension of a length, while r is dimensionless. Sometimes it is more convenient to make the scale factor dimen-sionless and set it to unity today, a 0 ≡ a(t 0 ) = 1. This can be achieved by making use of the following rescaling symmetry: a → a/a 0 , r → ra 0 , k → k/a 2 0 .

(1.4)

In this case, r now carries dimensions of length, and the curvature k is not normalized in general, |k| = 1. For convenience, we will often switch from cosmic time to conformal time τ , defined by the relation dτ = dt/a(t) 6 .

One of the ways to infer the cosmic expansion is by using the light we receive from distant objects. However, one must take into account that photons loose energy (or, equivalently, their wavelengths are stretched) due to the expansion. In order to understand this, we need to discuss first about the motion of particles. In a given spacetime, when no other forces apart from gravity are present, particles move along geodesics, which correspond to the shortest paths x µ (σ) in a curved geometry (here σ is an affine parameter that increases monotonically along the path of the particle) . They satisfy the geodesic equation, which can be written in terms of their 4-momentum vector P µ ≡ dx µ dσ = (P 0 , P i ) as 7dP µ dσ + Γ µ αβ P α P β = 0, (

where Γ µ αβ are the Christoffel symbols:

Γ µ αβ = 1 2 g µλ (∂ α g βλ + ∂ β g αλ -∂ λ g αβ ) .
(

The only non-zero components of the FLRW metric in a spatially flat universe k = 0 are:

Γ j 0i = ȧ a δ j i , Γ 0 ij = ȧ a g ij .
(1.7)

Using the µ = 0 component of the geodesic equation in the FLRW geometry, one can show an important result: the amplitude p of the physical three-momentum, defined by p ≡ ij g ij P i P j , is inversely proportional to the scale factor

p ∝ 1 a , ( 1.8)
where p = E and p = √ E 2 -m 2 for massless and massive particles, respectively. Since the wavelength of photons is inversely proportional to their momentum λ = h/p, then according to Eq. (1.8) the wavelength scales as a(t). Photons emitted at time t with wavelength will arrive at t 0 > t with wavelength λ 0 = a 0 a(t) λ. This confirms that the wavelength of light is increased, λ 0 > λ, due to the expansion of spacetime, a 0 > a(t). In terms of the redshift parameter introduced in Eq. (1.1), this fact implies the following relation

1 + z = a 0 a(t) .
(1.9)

Cosmo-dynamics

All the discussion until now was purely kinematic, but in order to get the dynamical evolution of a(t), we need to resort to Einstein field equations of GR. At the heart of this theory lie the field equations, which indicate how the matter and energy content (encoded in the stress-energy tensor T µν ) curve space-time (encoded in the Einstein tensor G µν )

G µν ≡ R µν - 1 2 g µν R = 8πGT µν -Λg µν . (1.10)
Here G is Newton constant and Λ is the cosmological constant originally introduced by Einstein in 1917 (see Sect. 1.1). The quantities R µν and R denote respectively the Ricci tensor and Ricci scalar, and they are constructed from the first and second derivatives of the metric

R µν = ∂Γ σ µν ∂x σ - ∂Γ σ µσ ∂x ν + Γ σ ρσ Γ ρ µν -Γ σ ρν Γ ρ µσ , R = R µν g µν . (1.11)
The symmetries of the FLRW metric enforce T µ ν to take the form of the stress-energy tensor corresponding to a perfect fluid

T µ ν = (ρ + P )U µ U ν + P δ µ ν , (1.12)
where U µ is the four-velocity of the fluid, which takes the simple form U µ = δ µ 0 if the fluid is at rest in the comoving frame. The fluid can be completely characterised by two functions, the mean density ρ and mean pressure P , which are the sum over the density and pressure of each single fluid species ρ = I ρ I and P = I P I . Any extra degrees of freedom, such as energy fluxes, momentum densities or anisotropic stresses, should be discarded as they would break the hypothesis of isotropy and homogeneity. Interestingly, the cosmological constant (typically assumed to be positive) can be absorbed as a contribution to the stress-energy tensor, T Λ µν = -ρ Λ g µν , with ρ Λ ≡ Λ/8πG. By comparing with Eq. (1.12), we see that this corresponds to a fluid with constant density and a negative pressure given by P Λ = -ρ Λ .

Because of the Bianchi identities, ∇ µ G µν = 0, the stress-energy tensor is conserved under the covariant derivative

∇ µ T µν ≡ ∂T µν ∂x µ + Γ ν µρ T ρµ + Γ µ µρ T ρν = 0. (1.13)
The previous equation is valid for the sum of all species present in the cosmic fluid, and also for each species separately if those are non-interacting. In the FLRW spacetime, the 0 component of Eq. (1.13) leads to the following continuity equation

ρ + 3H(ρ + P ) = 0, (1.14)
where H ≡ ȧ a is the so-called Hubble parameter (in a similar fashion, the conformal Hubble parameter is defined as H ≡ a a = aH). The term with H accounts for the expansion of the universe, and adds friction to the conservation equations. Because of the cosmological principle, only two Einstein equations are independent (00 and ij trace components), and they lead to the Friedmann equations for the evolution of the scale factor

H 2 = 8πG 3 ρ - k a 2 , ( 1.15)
ä a = -4πG 3 (ρ + 3P ).

(1.16) By combining the time derivative of Eq. (1.15) with Eq. (1.16), one can arrive again at Eq. (1.14). This shows that the continuity equation is not adding any new information with respect to the Friedmann eqs., which is to be expected since the stress-energy conservation is a direct consequence of the Bianchi identities. However, Eq. (1.14) is useful to directly solve for the mean densities, once an equation of state is specified. In many cases of interest, fluids can be simply described with a parameter that relate pressure and density in a linear way, w i = P i /ρ i . In the case of a constant w i , the solution of Eq. (1.14) for each non-interacting fluid species reads

ρ i (a) = ρ i (a 0 ) a a 0 -3(1+w i )
.

(1.17)

It is useful to classify the different species that populate the universe according to their Equation of State (EoS) parameter w i :

• Non-relativistic matter or dust. This refers to any component for which the pressure is much smaller than the density, w m 0. Baryons8 and cold dark matter fall into this category. As we will show in Sect. 1.5, for a non-relativistic fluid the EoS is indeed of order w m ∼ p 2 /m 2 1, where p 2 denotes an average momentum squared. In this case, Eq. (1.17) gives ρ m ∝ a -3 . This dilution of energy density is simply reflecting the fact that the number of particles is conserved in a growing volume V ∝ a 3 .

• Relativistic matter or radiation. This corresponds to any substance for which pressure makes a significant contribution, of about one third of the density, w r 1/3. This is the case of CMB photons or massless neutrinos. According to Eq. (1.17), this yields ρ r ∝ a -4 , because the dilution is now including the effect of an increasing wavelength E ∝ λ -1 ∝ a -1 .

• Dark energy. This alludes to some kind of exotic substance responsible for the recent accelerated expansion of the universe. From Eq. (1. [START_REF] Jeffreys | Theory of Probability[END_REF]), we see that any dominating species satisfying w < -1/3 can lead to cosmic acceleration, ä > 0.

The cosmological constant is the simplest model, corresponding to w Λ = -1.

• Massive neutrinos. As we will discuss in more detail in Sect. 1.5.3, massive neutrinos have a more complicated evolution, that cannot be captured by a constant EoS. In particular, they transition from being relativistic to non-relativistic, meaning that their EoS evolves from w ν = 1/3 in the early universe to w ν = 0 at late times. The transition time is fixed by the value of their mass. However, once the detailed evolution of w ν (a) is known, it is always possible to split their energy density into a relativistic contribution, ρ ν,r = 3w ν ρ ν , and a non-relativistic one, ρ ν,m = (1 -3w ν )ρ ν .

Considering that we know the scalings in a of each of the components in the total energy density ρ = ρ m + ρ r + ρ Λ , we can express Eq. (1.15) in the more convenient form 3 + Ω r (1 + z) 4 + Ω k (1 + z) 2 + Ω Λ .

H 2 (z) = H 2 0 E 2 (z) = H 2 0 Ω m (1 + z)
(1.18)

We have defined for each fluid component i the dimensionless density parameters

Ω i ≡ ρ i,0 ρ c , (1.19)
where ρ c is the so called critical density today ρ c ≡ 3H 2 0 /8πG. It is sometimes more convenient to express the physical energy densities as ω i ≡ Ω i h 2 , where h denotes the Hubble constant in units of 100 km/s/Mpc. The curvature density parameter has a different definition, Ω k ≡ -k/(a 0 H 0 ) 2 , and it describes a flat (Ω k = 0), spherical (Ω k < 0) or hyperbolic (Ω k > 0) geometry depending on whether the total energy density today ρ(a 0 ) is respectively equal, larger or smaller than the critical density ρ c . In Fig. 1.5 we show the time-dependent fractional densities Ω i (z) = 8πGρ i (z)/3H 2 (z) for various species in the cosmic inventory. According to the best-fit ΛCDM cosmology from Planck18 Ω r 9.1 × 10 -5 , Ω b 0.05, Ω cdm 0.26, Ω Λ 0.69, |Ω k | < 0.003.

(1.20)

In Sect. 1.6.5 we will describe in detail how these parameters are determined from the CMB temperature anisotropy spectra. Given these values of Ω i , one could now integrate directly Eq. (1.18) to obtain the evolution of the scale factor a(t). However, the very different scalings of matter (a -3 ), radiation (a -4 ) and dark energy (a 0 ) mean that for most of its history, the universe has been dominated by a single component (first radiation, then matter, then dark energy). We can thus consider only one single component i in Eq. (1.18) and integrate to get a(t) in each era. Restricting to a spatially flat universe, we can write down the solutions for the scale factor and the Hubble parameter in each era, expressed in terms of both cosmic and conformal time.

• Radiation era:

a ∝ t 1/2 ∝ τ, H = 1 2t , H = 1 τ .
(1.21)

• Matter era:

a ∝ t 2/3 ∝ τ 2 , H = 2 3t , H = 2 τ . ( 1.22)
• Λ era: a ∝ e Ht ∝ (C -τ ) -1 , H = const., H ∝ 1 C -τ , (1.23) where C is an integration constant determined by the initial conditions of the Λ era. These solutions assume that in each era we are far from the transition times, which are given by RM equality : ρ r (a eq ) = ρ m (a eq ) =⇒ a eq = Ω r Ω m =⇒ z eq 3400,

MΛ equality :

ρ m (a Λ ) = ρ Λ (a Λ ) =⇒ a Λ = Ω m Ω Λ 1/3
=⇒ z Λ 0.3.

(1. [START_REF] Hawking | Gravitationally collapsed objects of very low mass[END_REF] where we used the values from Eq. (1.20) 9 . Close to the transition redshifts given in Eq. (1.24) and Eq. (1.25), one should take into account the effects of the two components. For a universe full of matter and radiation , one can get an analytical expression for a in terms of conformal time τ , but not in terms of cosmic time t. Nevertheless, it is possible to get the inverse relation t(a):

t(a) = 4a 2 eq 3H 0 √ Ω r   1 -1 - a 2a eq 1 + a a eq 1/2   . (1.26)
For a a eq (a a eq ), one recovers the solutions for the radiation (matter) eras given in Eq. (1.21) and Eq. (1.22). In the case of matter and Λ, one can derive an analytical expression for a(t):

a(t) = Ω m Ω Λ 1/3 sinh 2/3 3 2 Ω Λ H 0 t . (1.27)
For H 0 t 1 (H 0 t 1) we arrive at the solutions for matter (Λ) eras shown in Eq. (1.22) and Eq. (1.23).

Let us end this section by giving the expression of a very relevant quantity in cosmology, the age of the universe t 0 . This can be obtained thanks to the useful relation dt = da/ ȧ = da/(aH(a)) = -dz/((1 + z)H(z)). Therefore, we get .28) where in the last step we assumed a universe full of dust and Λ (the contribution from radiation is negligible since this was dominating during only a tiny fraction of the total age). Using Ω Λ 0.69, Ω m 0.31, H 0 67.6 km/s/Mpc, and bearing in mind the conversion 1km/s/Mpc = 1.02 × 10 -3 Gyr -1 , we get t 0 13.8 Gyr, which is the value we quoted at the end of Sect. 1.1.

t 0 = ∞ 0 dz (1 + z)H(z) 2 3H 0 √ Ω Λ ln 1 + √ Ω Λ √ Ω m , ( 1

Distances in cosmology

As a consequence of the cosmological expansion, the notion of distance is not uniquely defined for very distant objects, and different operational procedures will yield in general different results. In this section we will introduce the concepts of 1. THE ERA OF PRECISION COSMOLOGY particle horizon, Hubble radius, luminosity distance and angular diameter distance.

Horizons

Let us start by rewriting the FLRW metric of Eq. (1.3) with the function r m (χ) given by

r m (χ) =        sinh(χ) k = -1, χ k = 0, sin(χ) k = +1.
(1.30)

Now, photons travel along null geodesics (ds 2 = 0) and, because of the isotropy of the FLRW metric, one can always assume that they follow a radial path (dθ 2 = dϕ 2 = 0). Then, according to Eq. (1.29), the comoving distance χ travelled by a photon emitted at a given time t em and observed at a time t obs is given by

χ = τ obs -τ em = t obs tem dt a(t) = 1 a 0 zem z obs dz H(z ) . ( 1.31)
For a photon emitted at a very early time t i (i.e. z em → ∞) and observed at a later time t (i.e. z obs = z), the previous expression describes what is known as particle horizon χ p (z). This corresponds to the maximum comoving distance from which an observer can receive information at a certain time t, and it is equal to the elapsed conformal time between t i and t (in units of c = 1). In other words, two observers separated by a distance λ > χ p (t) could have never been in causal contact. If a light signal is received today on Earth (z obs = 0), then the distance given by a 0 χ p (a 0 ) can be interpreted as the radius of the observable universe (although we will see that in practice the oldest light we have been able to observe comes from the surface of last scattering, located at z em 1100). For the parameter values given in Eq. (1.20), the radius of the observable universe is approximately equal to a 0 χ p (a 0 ) 1.41 × 10 4 Mpc 45 Glyr, which is about three times larger than the naive ∼ 13.8 Glyr that one would estimate by neglecting the cosmic expansion.

A related quantity is the so called Hubble radius, .32) which corresponds to the comoving distance that light can travel in the course of a Hubble time, t H = H -1 . That is, two observers separated by a distance λ > R H (t) can't be in causal contact at a given time t. For a dominant fluid component satisfying 1 + 3w > 0 (such as radiation or dust), the particle horizon and the Hubble radius are of the same order, and grow with time, χ p (t) ∼ R H (t) ∼ a(t) 1 2 (1+3w) . This is the reason why one typically refers to both the particle horizon and the Hubble radius simply as the horizon. However, for a dominant component satisfying 1 + 3w < 0, χ p (t) R H (t), and the Hubble radius actually decreases with time, a case which is very relevant for inflation (see Sect. 1.4) as well as dark energy.

R H ≡ 1 aH = dχ p dlna , ( 1

Luminosity distance

Even if the concepts of Hubble radius and particle horizon are extremely important in cosmology (as we shall see later), these distances are not directly observable. One way to make a link with observations is to consider a pointlike source, of which we know the absolute luminosity L (units of emitted energy per second), i.e. a standard candle. Then, the observed flux F (units of emitted energy per second per receiving area) can be used to infer the luminosity distance of the source. In a static Euclidean space, the flux is given by

F = L 4πr 2 ,
(1. [START_REF] Weinberg | A New Light Boson?[END_REF] where r represents the distance between the source and the observer. In an expanding universe, the previous formula needs to be modified in three different ways:

1. Emitted photons are spread over a sphere of area 4π(a 0 r m (χ)) 2 , where r m (χ) is given by Eq. (1.30) and χ denotes now the comoving distance between the source (that emitted light at z em = z) and us (z obs = 0).

2. The rate of arrival of photons is lower than the one at the moment of emission by a factor (1/∆t obs )/(1/∆t em ) = a/a 0 = (1 + z) -1 .

3. The energy of photons has decreased by the same factor ∆E(t em )/∆E(t obs ) = (1 + z) -1 since the time of emission.

Taking these aspects into account, the observed flux can be written as

F = 1 (1 + z) 2 L 4πa 2 0 r 2 m (χ) ≡ L 4πd 2 L .
(1.34)

In the previous expression, the luminosity distance d L has been introduced in order to keep the same relation between F and L as the one given by Eq. (1.33). Hence, we can write d L as .35) where in the last equality we have used the definition of Ω k in order to eliminate a 0 10 . A well-known example of standard candles are Supernovae Ia. As opposed to Supernovae II or Ib (which arise from the core-collapse of very massive stars), Supernovae Ia are thought to be nuclear explosions in binary systems, which occur when a white dwarf accretes mass from its companion star above the Chandrasekhar limit of ∼ 1.4M . An interesting characteristic of SNIa is that they all exhibit qualitatively similar light curves, i.e. they first brighten to a peak and then fade away after a period of few weeks. One is typically interested in the apparent magnitude at this peak. The apparent magnitude m x in some spectral band x is simply a logarithmic measurement of the flux, m x ≡ -2.5log 10 (F x /F x,0 ), where F x,0 is the flux of some reference object. In a similar way, one can define the absolute magnitude M x as the apparent magnitude an observer would perceive at a distance of 10 pc to the object. By subtracting these quantities we can get rid of F x,0 , µ = m x -M x = -2.5 log 10 F(d L ) F (10 pc) .

d L (z) = (1 + z)a 0 r m (χ) = (1 + z) H 0 √ Ω k sinh Ω k z 0 dz E(z ) , ( 1
(1.36)

where dL is the luminosity distance measured in units of Mpc. By comparing observations of m x,i vs. z i in a SNIa sample against the predicted values of d L (z i ), we can infer the underlying cosmology. Regarding the absolute magnitude, two strategies exist: 1) fit M x to the data together with the rest of cosmological parameters (in which case it becomes totally degenerate with H 0 ), or 2) get a direct measurement of M x using a cosmic distance ladder method (see Sect. 2.2). As discussed at the end of Sect. 1.1, it is thanks to the observations of SNIa light curves up to z ∼ 1 that dark energy was discovered in 1998.

Angular diameter distance

Another way of measuring distances is to consider an extended object of known physical transverse size L, i.e. a standard ruler. If this object emitted light at z em = z and is at a comoving distance χ from us, then according to Eq. (1.29) its transverse size is given by L = a(t em )r m (χ)θ, (1.38) where θ is the angular scale subtended by this object on the sky. The angular diameter distance d A is defined in analogy with the formula in static Euclidean space, L/d A ≡ 2 tan(θ/2) θ, where in the last equality we assumed θ 1 (which is true for all cosmological objects). Combining this with Eq. (1.38), we get

d A = a(t em )r m (χ) = a 0 1 + z r m (χ).
(1.39) 11 In practice, SNIa are not truly standard candles, since their peak luminosity vary from one SNIa to another. However, the peak luminosity is tightly correlated with other parameters, such as the width x1 or color c of the light curve, meaning they are standardisable. Hence, it is possible to correct the absolute magnitude M = M + αx1 + βc (where α, β are some empirical parameters to be fitted to data) such that M is common to all SNIa [START_REF] Mandel | The Type Ia Supernova Color-Magnitude Relation and Host Galaxy Dust: A Simple Hierarchical Bayesian Model[END_REF]).

The angular diameter distance measures the distance between us and the object at the moment of the emission (see Fig. 1.6). A widely used standard ruler is the so called comoving sound horizon at recombination, r s (z), where z is of the order of 10 3 . As we will discuss in Sect. 1.6.4, the sound horizon corresponds to the maximum distance acoustic waves could propagate in the primordial plasma up to a certain redshift z. Depending on the observable, this redshift can correspond to the moment of photon decoupling, z dec 1090, or the time at which baryons were released from photons (typically called the baryon drag epoch), which takes place slightly after photon decoupling, z drag 1060 < z dec . The sound horizon is mainly determined by the relative abundances of photons, baryons and dark matter, and it is of the order of 100 Mpc. This scale is imprinted in the map of CMB anisotropies as well as in the large scale distribution of galaxies, and can be seen as a characteristic angle that we can express in general as

θ X = r X D A , ( 1.40)
where D A = a -1 d A denotes the comoving angular diameter distance to the redshift at which observations are made, z obs . In the case of the CMB measurements, one observes the photons (r s = r s (z dec )) at the surface of last scattering, z obs = z dec , and the angle in Eq. (1.40) defines the position of the peaks in the angular correlation function of photon temperature fluctuations. In the case of LSS data, one observes the baryonic matter (r d = r s (z drag )) at the redshifts accessible by galaxy surveys, 0 z obs 3, and the angle in Eq. (1.40) results from a feature in the galaxy correlation function perpendicular to the line of sight. The LSS determinations of this angular scale are commonly referred as Baryon Acoustic Oscillations (BAO), and when complemented with a knowledge on r s (z drag ), they probe the distance-redshift relation at small z 12 . Hence, BAO data provide an alternative to SNIa to constrain the late-time expansion history, and allow to break parameter degeneracies when combined with other probes. In particular, the combination of CMB anisotropy with BAO or SNIa data indicate that our universe is flat to a high degree of accuracy. For these reasons and for simplicity, we will assume Ω k = 0 in the rest of the text.

Let us end this part by noting that the angular diameter and luminosity distances are not independent. Indeed, by comparing Eq. (1.39) with Eq. (1.35), we observe that

d L = (1 + z) 2 d A , (1.41)
which is also known as the Etherington distance duality relation. At very small redshifts, z 0.1, all distances essentially reduce to the Hubble law

d L d A a 0 χ z H 0 v H 0 , ( 1.42)
where we used the fact that the recessional velocity is approximately given by v z for nearby objects. Notice that the Hubble law also breaks down at even shorter distances, for which the peculiar velocities v pec ≡ a χ become comparable to the Hubble flow. Thus, in full generality one writes v = ḋ = d(aχ)/dt = Hd + v pec .

A few words on inflation

Cosmic inflation is a paradigm which was proposed in the '80s while investigating the problem of why no magnetic monopoles are seen today. It was quickly realized that inflation would resolve many other long-standing problems in Standard FLRW Cosmology (GUTH 1981):

• The horizon problem. Spacetime regions that could not have been in causal contact appear to have the same properties. This is reflected in the fact that spots in the CMB with non-overlapping past light-cones have an almost identical temperature. We can estimate the size of these spots by computing the angle subtended by the particle horizon at the moment of photon decoupling,

θ H = a dec χ p (a dec ) d A (a dec ) = ∞ z dec dz/H(z) z dec 0 dz/H(z) = (1 + z dec ) -1/2 1 -(1 + z dec ) -1/2 0.03 1.7 • , (1.43)
where we assumed matter domination at all times in order to simplify the calculations. This corresponds to 4π/0.03 2 ∼ 10 4 patches in the CMB sky which are causally disconnected. How can we explain the homogeneity of regions that have not had sufficient time to communicate?

• The curvature problem. Using the Friedmann equation Eq. (1.18), it is easy to see that the evolution of the time-dependent curvature density parameter is approximately given by |Ω k (a)| ≡ |k|(aH(a)) -2 ∼ |k|a 1+3w , where w is the EoS of the dominating energy component in the universe. During the radiation and matter eras, 1 + 3w > 0, implying that the curvature parameter increases with time. Hence, in order to explain why we observe the present curvature parameter |Ω k (a 0 )| to be very close to zero, we would need an "incredibly" small value of curvature in the early universe, which appears very unnatural.

• Initial conditions. Standard cosmology is able to tell how perturbations have evolved during the history of the universe, until they grew enough to collapse and form the rich structure that we see. However, it does not provide a physical origin for these perturbations.

Inflation solves these problems by considering that, at very early times, the comoving Hubble radius was actually decreasing with time, rather than increasing. In this way, two points that appear not to be causally connected at the moment of decoupling (meaning they were separated by a comoving distance λ > (a dec H(a dec )) -1 ), were actually in causal contact early on, i.e. λ < (a ini H(a ini )) -1 . Put differently, the decreasing Hubble sphere can lead to a physical size particle horizon at the moment of decoupling, a dec χ p (a dec ) which can now encompass all the scales observed in the CMB. The condition of a decreasing Hubble radius, ṘH < 0, is equivalent to consider a period of early rapid accelerated expansion, ä > 0, which in turn is equivalent to consider a dominant energy source satisfying 1 + 3w < 0. To be more precise, inflation solves the horizon problem by requiring a growth of spacetime between the beginning and the end of inflation of at least a end /a ini ∼ e 60 . The curvature problem is also solved, since the curvature parameter |Ω k (a)| now rapidly decreases with time, and becomes very tiny by the end of inflation. In this way, the curvature parameter needs not to be fine-tuned to an incredibly small value at initial times. Many inflation models consider a scalar field, the inflaton φ(t, x i ), as the mechanism responsible for driving the accelerated expansion. The stress-energy of a scalar field is given by

T µ ν = ∂ µ φ∂ ν φ -δ µ ν g αβ 2 ∂ α φ∂ β φ + V (φ) , (1.44)
where V (φ) is the scalar field potential. Assuming a FLRW background, the homogeneous part of the scalar field satisfies ∂ i φ = 0, and the stress energy tensor in the previous equation has the same form as the one of a perfect fluid (see Eq. (1.12)). Hence, we can identify the energy density and pressure as

ρ φ = 1 2 φ2 + V (φ), P φ = 1 2 φ2 -V (φ). (1.45)
Applying the continuity equation of Eq. (1.14) to ρ φ and P φ , we find the Klein-Gordon equation that dictates the dynamical evolution of the scalar field,

φ + 3H φ + dV dφ = 0, where H 2 = 8πG 3 ρ φ . (1.46)
By looking at Eq. (1.45), we see that the condition for cosmic acceleration, 1 + 3w φ < 0, can be satisfied as long as the potential dominates over the kinetic energy. In addition, this period of cosmic acceleration should last enough time to solve the horizon problem. This amounts to saying that the scalar field should be initially at a high part of the potential and then roll slowly towards its minimum. In order to quantify these effects, it is useful to introduce the so called slow-roll parameters,

ε ≡ - Ḣ H 2 , η ≡ ε Hε . ( 1
ε M 2 pl 2 V ,φ V 2 , η 2M 2 pl V ,φ V 2 - V ,φφ V , ( 1.48)
where

M pl = 1/ √ 8πG 2.4 × 10 18
GeV is the reduced Planck mass. Inflation ends when ε = 1. Then, if V (φ) is of the form V (φ) ∝ φ n (with n 2) close to its minimum, the inflaton starts to oscillate rapidly, and releases all its energy to the Standard Model particles through decay processes. This is the reheating era that preceded the standard hot Big-Bang epoch, which we will describe in the following section.

The global picture that emerges is the following: at the beginning of inflation, all comoving scales λ that we observe today on the CMB were sub-horizon, i.e. they were inside the Hubble radius. In terms of the corresponding Fourier mode, k = 2π/λ, this condition can be expressed as k H. As H -1 decreased during inflation, these scales exited the Hubble radius and became super-horizon (k H). In this process, the microscopic quantum fluctuations of the inflaton got stretched to cosmic sizes and became classical. After inflation ended and H -1 started to grow, the scales k -1 gradually re-entered the Hubble radius. This is depicted in Fig. 1.7. The small fluctuations of the inflaton provide the initial conditions for the evolution of perturbations of the different components in the universe.

In and out of equilibrium

Until now, we have taken a macroscopic view of the energy content of the universe. That is, we have always expressed the stress-energy tensor in terms of thermodynamic quantities, as the density and the pressure (related by the EoS parameter). However, in many cases of interest, it is possible to switch to a microscopic point of view, where all thermodynamic quantities can be derived from a distribution function of the particles in phase space, f (x µ , P µ ). This function gives the number of particles at a given time t and at a certain point in the phase-space (x i , P 0 , P i ) (in practice P 0 is not an independent variable because of the mass-shell condition, g µν P µ P ν = -m 2 ). In terms of the phase-space distribution (PSD), one can define the particle current density and the stress energy tensor as

n µ (x i , t) = g d 3 P (2π) 3 (-det(g)) -1/2 P µ P 0 f (x i , P i , t), (1.49) T µ ν (x i , t) = g d 3 P (2π) 3 (-det(g)) -1/2 P µ P ν P 0 f (x i , P i , t), (1.50)
where g represents the internal degrees of freedom of the particle, and the integration is over covariant 3-momenta P i . Since we are assuming a FLRW background, we can make some simplifications. First, we notice that det(g) = -a 6 and P i = j g ij P j = j a 2 δ ij P j , so that d 3 P (-det(g)) -1/2 = d 3 p, which means that the integration is now over physical 3-momenta p i ≡ aP i . By construction, this vector has a norm i (p i ) 2 = ij a 2 δ ij P i P j = ij g ij P i P j = p, so the integral measure in spherical coordinates reads d 3 p = dΩ p 2 dp. Second, homogeneity and isotropy imply that f (x i , P i , t) = f (p, t). Hence, by virtue of the integrals dΩP i = 0 and dΩP i P j = 0 (for i = j), we see that the components n i , T 0 i and T i j (for i = j) necessarily vanish. The remaining non-zero components correspond to the particle number density, the energy density and the pressure

n(t) = n 0 = g 2π 2 dp p 2 f (p, t), (1.51) ρ(t) = -T 0 0 = g 2π 2 dp p 2 Ef (p, t), (1.52) P (t) ≡ 1 3 i T i i = g 2π 2 dp p 2 p 2 3E f (p, t). (1.53)
We can alternatively express them in terms of the comoving momentum modulus q = ap and comoving energy E = aE:

n(t) = 4π a 3 dq q 2 f (q, t), (1.54) ρ(t) = 4π a 4 dq q 2 Ef (q, t), ( 1.55
)

P (t) = 4π a 4 dq q 2 q 2 3E f (q, t), ( 1.56)
where we have absorbed a factor g/(2π) 3 inside f (q, t). From the previous equations we see that, regardless of the shape of the PSD function, relativistic particles (E = p) satisfy P/ρ = 1/3, while for non-relativistic particles (E m) we have ρ mn and P/ρ ∼ p 2 /m 2 1.

The Boltzmann formalism

What about the shape of the PSD function? The almost perfect blackbody spectrum of the CMB gives a firm proof that the early universe was well characterized by local thermodynamic equilibrium. Indeed, the enormous temperatures after reheating favored the collisions between particles, making them exchange energy and momentum very efficiently. In this case, particles were distributed in the phase space according to the Bose-Einstein or Fermi-Dirac distributions. As the universe cooled, interactions became less frequent and particles progressively exited from equilibrium. The key to quantify this effect is to compare the interaction rate, Γ, to the expansion rate of the universe, H. For two-to-two interactions of the form 1 + 2 ↔ 3 + 4, the interaction rate for the species 1 is given by

Γ 1 = n 2 σv , ( 1.57)
where σv denotes the thermally averaged cross-section (a formal definition will be given below). Equilibrium is maintained if Γ 1 H. When Γ 1 H, particles decouple from the thermal bath and start free-streaming through the universe. In order to accurately describe the departures from equilibrium, we need to use the Boltzmann formalism. For a species 1 interacting with i = 2, ..., N species, the Boltzmann equation reads

L[f 1 ] ≡ df 1 (x µ (σ), P µ (σ)) dσ = C[f 1 ; f 2 , ..., f N ], (1.58)
where L[f ] is the so-called Liouville operator, and C[f 1 ; f i ] is the collision integral, whose expression depends on the particular type of interactions considered. Let us focus first on the l.h.s. of Eq. (1.58). In the absence of interactions, the collisionless Boltzmann equation, L[f 1 ] = 0, can be expressed in the following way

L[f 1 ] = ∂f 1 ∂x µ dx µ dσ + ∂f 1 ∂P µ dP µ dσ = ∂f 1 ∂x µ P µ - ∂f 1 ∂P µ Γ µ αβ P α P β = 0, (1.59)
simply reflecting the fact that f 1 is conserved as we follow a geodesic in phase space.

In the FLRW background, we have f 1 = f 1 (t, p), and we can use the expression for the Christoffels in Eq. (1.7), to arrive at

L[f 1 ] = E ∂f 1 ∂t -Hp df 1 dp = 0. (1.60)
We will sometimes find convenient to switch to the modulus of the comoving momentum, q ≡ ap, which doesn't evolve with time. In terms of this new variable,

∂f 1 (t, q) ∂t = ∂f 1 ∂t + ∂f 1 ∂q dq dp ṗ = ∂f 1 ∂t -Hp ∂f 1 ∂p = 0, (1.61)
which means that any PSD which depends exclusively on comoving momentum, f = f (q), is automatically a solution of the collisionless Boltzmann equation. We can easily express Eq. (1.60) in terms of the number density n 1 by integrating over (g/2π 2 ) dp 1 p 2 1 . Doing integration by parts on the second term and assuming that f 1 → 0 for p → ∞, then we get

ṅ1 + 3Hn 1 = 0, (1.62)
meaning that, in the absence of interactions, the number of particles is conserved on a comoving volume, n 1 ∝ a -3 .

Let us now move to the r.h.s. of of Eq. (1.58). For two-to-two interactions, the collision integral is written as

C[f 1 ; f 2 , f 3 , f 4 ] = d 3 p 2 (2π) 3 2E 2 d 3 p 3 (2π) 3 2E 3 d 3 p 4 (2π) 3 2E 4 δ 4 (P 1 + P 2 -P 3 -P 4 )(2π) 4 × |M 34,12 | 2 f 3 f 4 (1 ± f 1 )(1 ± f 2 ) -|M 12,34 | 2 f 1 f 2 (1 ± f 3 )(1 ± f 4 ) . (1.63)
We have introduced the Lorentz-invariant amplitudes M 34,12 and M 12,34 for the processes 3 + 4 → 1 + 2 and 1 + 2 → 3 + 4 (which include the sum over all internal degrees of freedom). These amplitudes are equal in modulus for interactions with a time reversal symmetry, which is often the case. The Dirac delta is inserted just to guarantee the energy-momentum conservation. From Eq. (1.63), we see that the collision integral is simply giving the difference between the rate of production of species 1 (proportional to f 3 f 4 ) and the rate of destruction of species 1 (proportional to f 1 f 2 ). Finally, the factors (1 ± f i ) account for the stimulated emission for bosons (+) and Pauli blocking effects for fermions (-). Similar equations as Eq. (1.63) can be written for species 2, 3 and 4. In the most general case, one should therefore solve a complicated system of coupled integro-differential equations.

In a static universe, equilibrium distributions can be obtained by demanding that the collision term vanishes, which is sometimes called the detailed balance condition. We remark that this doesn't imply at all that collisions aren't happening. Rather, it is reflecting the idea that forward and backward collision rates should be equal in equilibrium. Using the unitarity of M ij,kl and taking C = 0 in Eq. (1.63) yields

a=1,2 1 ± f a f a = b=3,4 1 ± f b f b . (1.64)
By taking the logarithm of Eq. (1.64), we see that the quantity S ≡ a ln[(1 ± f a )/f a ] remains unchanged during collisions. This can be achieved if S is a linear combination of conserved quantities in the collisions, such as 4-momentum P µ a or any other conserved charge Q a (like baryon number or electric charge). Thus, we can write

ln 1 ± f a f a = βE a + ξQ a , ( 1.65)
where we took P i a = 0 since we assume a fluid whose center of mass is not moving. The coefficients β and ξ characterize the different equilibrium states, so in order to make a connection with thermodynamic quantities we can set β = 1/T and ξQ a = µ a /T , where T is the temperature and µ a the chemical potential associated with the conserved charge Q a . Solving for f a yields the Fermi-Dirac and Bose-Einstein equilibrium distributions,

f a = 1 e (Ea-µa)/T ∓ 1 , (1.66)
where the signstands for bosons and the sign + for fermions. Notice that the distribution of Eq. (1.66) satisfies C = 0 for all kind of two-to-two interactions as long as all particles share the same temperature T (kinetic equilibrium condition) and the chemical potentials fulfill the chemical equilibrium condition,

µ 1 + µ 2 = µ 3 + µ 4 . (1.67)
Since particles and antiparticles carry opposite charges, their chemical potentials must be opposite in sign, µ a = -µ ā. Furthermore, photons carry no additive charges (i.e. they are equal to their antiparticles), so they can't have a chemical potential, µ γ = 0. For a non-degenerate fluid satisfying E a -µ a T (which is relevant for many applications in cosmology), the distribution in Eq. (1.66) reduces to the classical Maxwell-Boltzmann distribution f a e -(Ea-µa)/T .

(1.68)

Now, to see if the PSD in Eq. (1.66) is also an equilibrium solution in an expanding universe, we should check if they are also in the kernel of the Liouville operator, L[f a ] = 0. Inserting Eq. (1.66) in Eq. (1.60) and allowing the independent variables T and µ a to vary with time, yields the conditions

μa µ a = Ṫ T = - p 2 E 2 a H. (1.69)
Specializing to the case of massless particles, E a = p, we find that the temperature and chemical potential evolve as µ a ∝ T ∝ a -1 . For the case of massive particles, there is no general solution to Eq. (1.69). The reason for this is that, strictly speaking, there are no equilibrium solutions for massive particles in a FLRW geometry. However, one can show that the PSD function of massive particles remains always very close to the equilibrium form of Eq. (1.66) as long as interactions are very efficient,

Γ 1 /H 1 (BERNSTEIN 1988).
An important conclusion that can be extracted from the previous considerations is that any species that decouples while being relativistic (i.e. a hot relic), will approximately retain an equilibrium distribution with a decreasing temperature also at later times. To see this, let's consider that a hot species decouples instantaneously at some temperature T D (that is, we neglect the fact that the decoupling process actually takes some time). Before decoupling, the collision term is zero since forward and backward rates exactly compensate, meaning that the PSD of the species will be either a Bose-Einstein or Fermi-Dirac until T D :

f (q) = 1 e (q-a D µ D )/a D T D ∓ 1 . (1.70)
After decoupling, the collision term is still zero because the species don't interact anymore with the rest of particles, and it will keep its equilibrium form with a decreasing temperature and chemical potential. Indeed, the distribution on Eq. (1.70) depends only on comoving momentum q provided that aT = a D T D = const. and aµ = a D µ D = const., so it is automatically a solution of the collisionless Boltzmann equation by virtue of Eq. (1.61). This is the reason why CMB photons exhibit an almost perfect black body spectrum, even if they decoupled at z D ∼ 10 3 . Another example of hot relics are neutrinos, which decoupled from the plasma much earlier, at z D ∼ 10 9 . Even if neutrinos have nonzero masses of the order m ν O(0.1) eV, they decoupled at a temperature of the order T D ∼ 1 MeV m ν , so they were clearly relativistic at that epoch. However, we will see that the fact the decoupling process wasn't exactly instantaneous induces some small distortions in their Fermi-Dirac distribution.

Relativistic degrees of freedom and entropy conservation

Let us now compute the different thermodynamic quantities for particles in equilibrium. Inserting the Fermi-Dirac or Bose-Einstein distribution in Eq. (1.51), Eq. (1.52) and Eq. (1.53), we get

n eq = g 2π 2 dp p 2 exp[( p 2 + m 2 -µ)/T] ∓ 1 , ( 1.71
)

ρ eq = g 2π 2 dp p 2 p 2 + m 2 exp[( p 2 + m 2 -µ)/T] ∓ 1 , ( 1.72
)

P eq = g 2π 2 dp p 4 3 p 2 + m 2 (exp[( p 2 + m 2 -µ)/T] ∓ 1)
.

(1.73)

In general, the previous integrals have to be computed numerically. However, it is possible to get analytical expressions in some limiting cases. Let us start considering the relativistic limit, T m. In this regime, the chemical potentials of all particles are typically so small that we can safely neglect them 13 . Hence, we get

n eq = ζ(3) π 2 gT 3 1 bosons 3 4 fermions
(1.75) 13 The chemical potential of electrons and protons can't be exactly zero, since a non-zero value of µ is required to account for the particle-antiparticle asymmetry that we observe. As an example, consider temperatures mp T me, such that the universe is full of non-relativistic protons p + as well as relativistic electrons e -and positrons e + . One can then compute the ratio r ≡ (n e --n e + )/nγ using the relativistic limit of Eq. (1.71) for fermions and µ e -= -µ e + = 0. This yields r = ge gγ π 2 6ζ( 3)

µ e - T + 1 π 2 µ 3 e - T 3 . (1.74)
This ratio stays constant after the e ± annihilation stage, and can be related to the baryon-to-photon ratio η ≡ n b /nγ np/nγ = r, where the last equality follows from the neutrality of the universe. Observations ρ eq = 3P eq = π 3 30 gT 4 1 bosons ρ eq m n eq , P eq = n eq T ρ eq .

(1.78)

From Eq. (1.77) we see that, for null chemical potentials, the number density of nonrelativistic components is exponentially suppressed as the temperature decreases. This is a result of particle-antiparticle annihilations, which aren't prevented from any conservation law (since µ = 0) and can't be balanced by particle-antiparticle pair production (which are inefficient for T m) 14 .

Thanks to Eq. (1.76), we can easily compute the total energy density associated to radiation

ρ r = i ρ i = π 2 30 g (T )T 4 , (1.79)
where g (T ) is the effective number of relativistic degrees of freedom at a temperature T . Since we are summing over relativistic species with a negligible chemical potential, we can write g (T ) as

g (T ) = i,boson g i T i T 4 + 7 8 i,fermion g i T i T 4 , ( 1.80)
where we have allowed the possibility that particles might have different temperatures from photons, T i = T . When the temperature drops below the mass m i of a certain species, its contribution is removed from Eq. (1.80), consequently decreasing g (T ).

Let us consider now a very important quantity to describe the early universe, the entropy density, s = S/V , where V ∝ a 3 . Here S is the total entropy, which stays constant in equilibrium. From the thermodynamic identity s = ∂P ∂T | µ , and using Eq. (1.66), one can show that the entropy density is given by

s = ρ + P -µn T . (1.81)
For relativistic particles, since ρ ∝ P ∝ µn ∝ T 4 , the entropy density s scales as a -3 , so that S is conserved. In a similar way as we did for the energy density, we of CMB and BBN indicate that η 5.5 × 10 -10 (ω b /0.02). Comparing this with the previous formula for r, we obtain µ e -/T ∼ 10 -9 , which confirms that the contribution of µ e -to the number and energy density is actually very small. 14 The fact that annihilations can't be balanced by pair production doesn't contradict the idea that forward and backward rates are equal in chemical equilibrium. Indeed, when the number density is exponentially supressed, the pair production cross-section is reduced accordingly, such that annihilation and pair production rates remain equal. can compute the total entropy associated to radiation. For relativistic particles with vanishing chemical potentials,

s r = i 4 3 ρ i T i = 2π 2 45 g ,S (T )T 3 , (1.82)
where g ,S (T ) is the effective number of degrees of freedom in entropy,

g ,S (T ) = i,boson g i T i T 3 + 7 8 i,fermion g i T i T 3 . (1.83)
We note that g ,S (T ) = g (T ) as long as all relativistic particles share the same temperature of the photon bath, T i = T . As we will see, this is the case until the e ± annihilation stage. In Fig. 1.8 we show the evolution of g (T ) as well of g ,S (T ), assuming the particle content of the Standard Model. Applying entropy conservation S r = const. to Eq. (1.82), we can readily obtain the evolution of the photon temperature in equilibrium,

a 3 g ,S (T )T 3 = const., =⇒ T ∝ g -1/3 ,S a -1 . (1.84)
We see that, away from mass thresholds, the temperature evolves as T ∝ a -1 . When a particle becomes non-relativistic, the factor g

-1/3 ,S
makes T to decrease more slowly, which is simply reflecting the fact that annihilations are transferring entropy to the other species in the thermal bath.

Neutrino decoupling and evolution

An interesting application of entropy conservation is the calculation of neutrino temperature. At temperatures T > 1 MeV, neutrinos were in equilibrium with the rest of the plasma through processes like ν e + νe ↔ e -+ e + and e -+ ν e ↔ e -+ ν e .

The cross-section for theses process are of the order σv ∼ G 2 F T 2 (where G F 1.17 × 10 -5 GeV -2 is Fermi's constant) meaning that Γ ν ∼ G 2 F T 5 . The decoupling temperature can thus be estimated by comparing this interaction rate with the expansion rate, which by virtue of Eq. (1.79) is given by

H = ρ r 3M 2 pl 1/2 π 3 g 10 1/2 T 2 M pl . (1.85)
Setting Γ ν = H and using the useful conversion 1 eV = 11605 K yields a decoupling temperature of around T D ∼ 1 MeV. As we already discussed, after decoupling the neutrinos inherit a Fermi-Dirac distribution with a temperature T ν ∝ a -1 . This means that their entropy is also separately conserved, so we can just consider the entropy of the only relativistic species that were present in the plasma for T < 1 MeV: electrons, positrons and photons. When electrons and positrons annihilate at T = m e ∼ 0.5 MeV, they inject entropy that reheats the photons. The relation between the temperature of photons before and after e ± annihilation can be derived using entropy conservation of the e ± γ fluid:

T before T after = g ,after g ,before 1/3 . (1.86)
The number of relativistic degrees of freedom before and after e ± annihilation is g ,before = 2 + 7 8 (2 + 2) = 11/2 and g ,after = 2, respectively. Thus, the photon temperature increases by a factor T before /T after = (4/11) 1/3 . Neutrinos don't feel this raise of temperature, since they already decoupled from the plasma by the time electrons and positrons were annihilating, meaning that T ν = T before . We conclude that the relation between neutrino and photon temperature at T γ = T after < m e is the following

T ν T γ = 4 11
1/3 0.71.

(1.87)

After e ± annihilation, photons and neutrinos constituted the sole relativistic species in the universe. In this context, we can use Eq. (1.87) to write the total density in radiation as

ρ r = ρ γ 1 + N eff 7 8 4 11 4/3 , (1.88)
where N eff is the effective number of neutrino species. This parameter would be equal to the number of neutrino flavors, N eff = 3, only if the assumptions we have made until now were valid; namely, if neutrinos decoupled instantaneously and they were described by Fermi-Dirac distributions with null chemical potentials. In practice, a N eff = 3 is used to capture deviations from these hypothesis. For instance, neutrinos with different momenta decoupled at slightly different times, and some of the energy released in e ± annihilations did leak to the more energetic neutrinos. The full calculation of neutrino decoupling yields a value of N eff 3.044 (J. J. BENNETT et al. 2020). It is customary to also use N eff to parameterize the presence of any possible massless invisible species (dark radiation) in the early universe, which would be constrained by BBN and CMB data. To be more precise, the energy density of this dark radiation, ρ dr , is re-absorbed as change in the effective number of neutrinos, given by ∆N eff ≡ N eff -3.044 = ρ dr /ρ 1ν , where ρ 1ν is the density of a single relativistic neutrino.

THE ERA OF PRECISION COSMOLOGY

At late times, the mass of neutrinos becomes important compared to their energy, so Eq. (1.88) is not valid anymore. Neutrinos become non-relativistic when the average momentum per neutrino species i, p = dp p 3 (exp(p/T ν ) + 1) -1 dp p 2 (exp(p/T ν ) + 1) -1 = 7π

Ω ν = i m i n i,0 ρ c 3 11 n γ i m i ρ c i m i 94.1h 2 eV . ( 1.91)
This means that neutrinos can only constitute at present a very small fraction of the total energy budget of the universe.

WIMP freeze-out

We have now two alternative ways to compute the evolution of thermodynamic quantities in equilibrium. On the one hand, we can use the integrated collisionless Boltzmann equation, which yields n ∝ a -3 , as we saw in Eq. (1.62). On the other hand, we know that the temperature of the bath evolves as T ∝ a -1 (away from mass thresholds), so plugging this in the relativistic equilibrium functions shown in Eq. (1.75), we recover n ∝ a -3 . However, if we plug T ∝ a -1 in the non-relativistic equilibrium function of Eq. (1.77) with µ = 0, then we don't recover n ∝ a -3 . This means that something must have happened to non-relativistic particles that prevented the exponential suppression exp(-m/T ) to develop further. There are two possibilities:

• Massive particles in chemical equilibrium have a non-zero chemical potential µ associated with a charge conserved by all interactions in the thermal bath. This is the case of baryons (protons and neutrons) as well as electrons. The chemical potential adjusts itself as a function of temperature, µ(T ), to ensure that the abundance of baryon matches the value that we observe, n b = ηn γ ∼ 5.5×10 -10 n γ . Stated otherwise, if there's an initial excess of baryons over antibaryons, and all interactions conserve a non-zero baryonic charge, B = n B -n B s = 0, then particles acquire a positive chemical potential (such that µ/T grows as T decreases) that cancels the effect of the exponential factor. Conversely, antiparticles are rapidly depleted because their chemical potential is negative, -µ/T .

• Massive species with no particle-antiparticle asymmetry (µ = 0) might exit from chemical equilibrium after they become non-relativistic, when their interaction rate becomes smaller than the Hubble rate. In other words, when the expansion timescale starts to dominate over the interaction timescale, particles and antiparticles cannot find each other to annihilate, so their comoving number density a 3 n freezes out at a certain value . This is the case of weakly interacting massive particle (WIMP) scenario for CDM, that we proceed to discuss now.

In order to describe the departure from equilibrium for the WIMP dark matter, we are going to derive a simplified version of the Boltzmann equation in Eq. (1.58) for two-to-two interactions of the form 1 + 2 ↔ 3 + 4. The assumptions we will make are sufficiently general so that the resulting equations can be applied to many situations of interest. First, we will assume that the particles are in kinetic equilibrium with other particles in the plasma thanks to fast scattering processes. Second, we will consider that particles are either non-relativistic, or relativistic with negligible or vanishing chemical potential (as in the case of photons), which means that all the effects due to quantum statistics can be neglected. The distribution functions can then be written in general as

f i = e (-E i +µ i )/T
1. Taking M 12,34 = M 34,12 = M, the integrand of the collision operator in Eq. (1.63) simplifies to

f 3 f 4 (1 ± f 1 )(1 ± f 2 ) -f 1 f 2 (1 ± f 3 )(1 ± f 4 ) → e -(E 1 +E 2 )/T e (µ 3 +µ 4 )/T -e (µ 1 +µ 2 )/T , (1.92)
where we used energy conservation

E 1 + E 2 = E 3 + E 4 .
In this context, the Boltzmann equation describes how the chemical potentials evolve from a chemical equilibrium configuration where µ 1 + µ 2 = µ 3 + µ 4 to more general values that will fix the number densities of particles after they have decoupled from each other. Nevertheless, we can choose to evolve the number densities instead of the chemical potentials, since they keep a simple relation,

n i = e µ i /T n (0) i , where n (0) i ≡ g i 2π 2 dp p 2 e -E i /T .
(1.93)

We introduced the variable n (0) i to simplify notation, but we remark that it is not necessarily equal to the equilibrium distribution, since it is lacking the factor with µ i . Integrating the Boltzmann equation Eq. (1.58) over (g/2π 2 ) dp 1 p 2 1 , we arrive at the following differential equation for the number density of species 1,

a -3 d(n 1 a 3 ) dt = n (0) 1 n (0) 2 σv n 3 n 4 n (0) 3 n (0) 4 - n 1 n 2 n (0) 1 n (0) 2 , (1.94)
where σv is the thermally averaged cross section

σv ≡ 1 n (0) 1 n (0) 2 dΠ 1 dΠ 2 dΠ 3 dΠ 4 δ 4 (P 1 + P 2 -P 3 -P 4 )(2π) 4 e -(E 1 +E 2 )/T |M| 2 , (1.95)
with dΠ i ≡ d 3 p i (2π) 3 2E i . Eq. (1.94) can be intepreted straightforwardly. The l.h.s. of the equation is of the order n 1 H, while the r.h.s. is of the order n 1 n 2 σv . Therefore, if the interaction rate Γ 1 = n 2 σv is much bigger than the Hubble rate H, the term in parenthesis in Eq. (1.94) needs to vanish,

n 1 n 2 n 3 n 4 eq = n (0) 1 n (0) 2 n (0) 3 n (0) 4
.

(1.96) This is an alternative way of expressing chemical equilibrium, and it is often referred to as the Saha equation. When Γ 1 drops below H, the r.h.s. of Eq. (1.94) goes to zero, and the comoving number density n 1 a 3 approaches a constant value.

Let us now apply Eq. (1.94) to the case of WIMP dark matter. The standard scenario considers that the dark matter particle χ and its antiparticle χ are initially in chemical equilibrium with light SM fermions f and antifermions f through annihilation processes χ + χ ←→ f + f.

(1.97)

We assume there's no particle-antiparticle asymmetry for the dark matter particles (i.e. µ χ = 0 initially), so that n χ = n χ and n

(0) χ = n (0)
χ . In addition, we assume that the fermions are in chemical and kinetic equilibrium with the rest of the plasma (i.e.

µ f = -µ f is always satisfied), meaning that n f n f = n (0) f n (0) f . Then, the Boltzmann equation for n χ reads a -3 d(n χ a 3 ) dt = -σv n 2 χ -(n (0) χ ) 2 (1.98)
It is convenient to express the previous equation in terms of a comoving density N χ ≡ n χ /T 3 and the time variable x ≡ m χ /T ,

dN χ dx = - λ x 2 N 2 χ -(N (0) χ ) 2 , where λ ≡ m 3 χ σv H(m χ )
.

(1.99)

We will take λ as a constant, since many theories predict the so-called s-wave annihilation channels, where σv is independent of temperature. In general, Eq. (1.99) needs to be solved numerically. In Fig. 1.9 we show the qualitative aspect of the solution. We see that the Boltzmann equation correctly predicts the transition from the exponentially suppressed N (0) 15 . This happens soon after the freeze-out temperature x f defined by n χ (x f ) σv = H(x f ), which one can show is of the order x f ∼ 10. Assuming that

χ to the relic constant value of N ∞ χ ≡ N χ (x f → ∞)
N χ (x > x f ) N (0)
χ and that N ∞ χ N χ (x f ), we can analytically solve Eq. (1.99) to estimate the value of the N ∞ χ ,

N ∞ χ x f λ . (1.100)
The relic density is inversely proportional to the cross section σv , which makes sense intuitively, since the larger the interaction rate is, the longer the particles stay in the exponentially suppressed regime. Using this result, it is possible to compute the present CDM abundance, and one notices that in order to reproduce the observed value Ω cdm h 2 ∼ 0.1, a cross-section of roughly σv 2.6 × 10 -9 GeV -2 3 × 10 -26 cm 3 s -1 is required (see [START_REF] Dodelson | Modern Cosmology[END_REF] for details). This is precisely the value we expect from a new particle in the 100 GeV mass range that interacts via the weak force. The fact that such a simple production mechanism leads to the right DM abundance, and the required energy scale corresponds to the one that can be currently probed at colliders, is what is known as the WIMP miracle.

As we already said, the Boltzmann equation in the form of Eq. (1.94) has many interesting applications beyond WIMP freeze-out, such as BBN or hydrogen recombination, that we are going to introduce in the following subsections.

Big-Bang Nucleosynthesis

We briefly discuss here the formation of the light elements during the first few minutes in the history of the universe. This is one of the pillars of the standard Big-Bang model, and the earliest cosmological probe to which we have access. In general, BBN has to be described through a complicated system of coupled Boltzmann equations, in order to track all nuclear abundances (WAGONER et al. 1967; M. S. [START_REF] Smith | Experimental, computational, and observational analysis of primordial nucleosynthesis[END_REF]). However, with very simple tools it is possible to get an analytical estimate of the He 4 fraction, which is the most abundant nucleus that is formed during this epoch. That's the goal of this subsection.

The starting point to describe BBN is to consider temperatures slightly above T ∼ 1 MeV, when the primordial soup was full of relativistic electrons, positrons and neutrinos, as well as small traces of non-relativistic baryons (including free protons and neutrons). Firstly, we determine the relative abundances of protons and neutrons. At those temperatures, protons and neutrons were coupled by weak interactions of the type

p + + νe ←→ n + e + ,
(1.101)

p + + e -←→ n + ν e . (1.102)
We can compute the neutron to proton ratio n n /n p using the non-relativistic profile in Eq. (1.77). Notice that protons and neutrons share the same baryon number (B = +1) so the associated chemical potential is also equal, µ n = µ p = µ B . Therefore,

n n n p eq = n (0) n n (0) p = m n m p 3/2
e -∆m/T e -∆m/T , (1.103) 15 In terms of chemical potentials, the PSD of the dark matter has evolved from fχ = e -Eχ/T initially to fχ = e µ ∞ χ /T e -Eχ/T after chemical decoupling, where

µ ∞ χ (T ) satisfies e µ ∞ χ /T = n ∞ χ /n (0) χ .
where ∆m ≡ m n -m p 1.3 MeV. It is useful to express the previous result in terms of the free neutron fraction, X n ≡ n n /n b , where the baryon abundance at that time was roughly n b n n + n p . Then,

X eq n =
e -∆m/T 1 + e -∆m/T .

(1.104)

Initially T ∆m, so there are as many neutrons as protons (X n = 1/2). Later, when T ∆m, the neutron abundance becomes exponentially smaller. However, we saw already in Sect. 1.5.3 that weak interactions switch off at around T D ∼ 1 MeV, although a more accurate computation gives a value of T D 0.8 MeV. We can take the equilibrium neutron fraction at this temperature T D as a proxy for the neutron fraction at freeze-out, X ∞ n ∼ X eq n (T D ) (the precise calculation of X ∞ n would require solving the Boltzmann equation). In reality, the neutron fraction keeps evolving even after freezeout, because neutrons decay to protons through the weak decay n → p + + e -+ νe . The neutron lifetime (τ n 890 s) is comparable with the age of the universe at that time, so this should be taken into account. Putting all together, the neutron fraction for T < T D reads

X n (t) X eq n (T D )e -t/τn . (1.105)
At this point, He 4 still cannot form since reactions involving three or more incoming particles are very rare. The first nucleus that can form is deuterium through

n + p + ←→ D + γ. (1.106)
Once deuterium is produced, two-particle reactions allow He Let us focus on deuterium production. We can apply the Saha condition in Eq. (1.96) to the process in Eq. (1.106), and use that photons satisfy n γ = n (0) γ , to get the following ratio at chemical equilibrium

n D n p n n eq = n (0) D n (0) p n (0) n = 3 4 4π m p T e B D /T , ( 1.109)
where

B D ≡ m n + m p -m D = 2.
22 MeV is the deuterium binding energy, and we used m D 2m n in the prefactor, as well as

g D = 3, g p = g n = 2.
To get an order of magnitude, we can approximate the neutron abundance as n (0) n ∼ n b = ηn γ (since X n was not much smaller than 1 at the time) and apply Eq. (1.75) to compute n γ . This yields

n (0) D n (0) p η T m p 3/2 e B D /T . (1.110)
Because of the smallness of η, temperature needs to decrease well bellow B D in order for deuterium to form in significant amounts. Thus, heavier elements cannot be formed until deuterium is produced, even if they have larger binding energies. This is the so called deuterium bottleneck. The temperature of formation of these elements can hence be estimated as the temperature T BBN at which the ratio in Eq. (1.110)

n tot H 1 2 n n (t BBN ) n p = 1 2 X n (t BBN ) 1 -X n (t BBN ) ∼ 1 16 , ( 1
Y p 4f He 1 + 4f He ∼ 0.2. (1.114)
The result of the full Boltzmann calculation yields Y p 0.24, which is not too far from the very rough estimate of Eq. (1.114). This prediction is in excellent agreement with current observations, as can be seen in Fig. 1.10. This constitutes one of the major triumphs of the standard cosmological model. Notice that Fig. 1.10 shows the predicted element abundances in terms of the baryon-to-photon ratio η (or equivalently, Ω b ), which in the standard case constitutes the only free parameter of BBN. More generally, BBN physics are also very sensible to the extra number neutrino species ∆N eff and many other deviations from the standard cosmological model, such as exotic electromagnetic energy injection or variations of fundamental constants. It is thus a very powerful probe to constrain new physics (POSPELOV et al.

2010).

As we will discuss in the following subsection, the value of Y p has a non-negligible impact on recombination and thus on the shape of the CMB spectra. Additionally, the value of Ω b that is inferred from CMB analysis affects the prediction of Y p . Therefore, in order to determine the value of Y p self-consistently, the CLASS code reads a table produced with the BBN code PArthENoPE [START_REF] Pisanti | PArthENoPE: Public Algorithm Evaluating the Nucleosynthesis of Primordial Elements[END_REF], that gives Y p as a function of Ω b and N eff . 16 To convert BBN temperature to BBN time, it suffices to integrate Eq. (1.85),

T 1 MeV 1.5g -1/4 1 s t 1/2
, and apply g = 3.38 (see Fig. 1.8). 17 We use the superscript "tot" to indicate the total H and He 4 in the universe, which initially are only in ionized forms (n tot H = np and n tot He = n ++ He ), but will be gradually converted into neutral atoms during the recombination era, that we discuss in the following section.

Hydrogen recombination

We focus here on the recombination 18 between electrons and protons to form neutral hydrogen. As we will see later, a precise knowledge of the recombination history is essential to understand the shape of the CMB anisotropy spectra.

Saha equilibrium

At temperatures above 1 eV, the universe consisted in a plasma of free electrons, free protons and light nuclei such as helium. Photons were strongly coupled to electrons through Compton scattering e -+ γ ←→ e -+ γ, whereas electrons and protons were coupled via Coulomb scattering e -+ p + ←→ e -+ p + . There was very little hydrogen at that epoch: temperature needs to drop well below the binding energy of hydrogen H until recombination can happen. To see why, let's consider the following process in chemical equilibrium e -+ p + ←→ H(1s) + γ.

(1.115)

We labelled the ground state 1s for hydrogen, since we will discuss excited states later.

It is possible to get an estimate of the time of recombination thanks to the Saha condition Eq. (1.96). We take into account that protons, electrons and hydrogen have a non-relativistic profile, given that m H , m p , m e T b (here T b is the temperature of the baryon fluid, which is equal to the photon temperature if photons and baryons are in kinetic equilibrium, T b = T ). We can write n H,1s n e n p eq = n

(0) H,1s n (0) e n (0) p = g H,1s g e g p m H m e m p 2π T 3/2 e (mp+me-m H )/T . (1.116)
In the prefactor of Eq. (1.116) we can approximate m H m p , but in the exponential the difference between m p +m e and m H is crucial: it is the binding energy of hydrogen, ε 0 = m p + m e -m H = 13.6 eV. The number of internal degrees of freedom are g H,1s = 4 and g p = g e = 2. At this stage, it is useful to introduce the free electron fraction,

X e ≡ n e n tot H = n e n p + n H,1s , (1.117)
i.e., it is the ratio of free electrons to total hydrogen, both in ionized and neutral form. As a consequence of the neutrality of the universe, we have n e n p (we are neglecting some small amount of ionized helium in the r.h.s. of this equation). Using the definition of the He 4 mass fraction (c.f. Eq. (1.113)), we can write the total hydrogen density in terms of baryon density as

n tot H = n b -4n tot He = n b (1 -4m H n tot He /(m H n b )) n b (1 -Y p )
. Bearing all of this in mind, we can recast Eq. (1.116)

in terms of X e , X 2 e 1 -X e eq = 1 η(1 -Y p ) m e T 3/2 √ π 2 5/2 ζ(3) e -ε 0 /T , ( 1.118)
where we used n b = ηn γ , and applied Eq. (1.75) to compute n γ . We see that, when T ∼ ε 0 , the r.h.s. of Eq. (1.118) is of order 10 9 (m e /T ) 3/2 ∼ 10 15 . Thus, the denominator at the l.h.s. needs to be very close to zero, i.e. almost all hydrogen is still ionized, X e 1. This is because the number of photons compared to baryons is so high, that even for T < ε 0 the high-energy tail of the photon distribution still contains many ionizing photons with E > ε 0 . In fact, we can use Eq. (1.118) to estimate the temperature T rec at which X e = 0.1 (this could serves as a definition of recombination temperature), to get

T rec 0.3 eV 3600 K. (1.119)
Using T rec = T 0 (1 + z rec ), this leads to a recombination redshift of around z rec ∼ 10 3 .

One could now compute the time at which photons decoupled from the plasma, or in other words, the time of last scattering. For this, it would suffice to compare the photon scattering rate

Γ γ ∼ σ T n b X e (where σ T 2 × 10 -3 MeV -2 is the Thomson cross-section) with the Hubble rate, H = H 0 √ Ω m (T /T 0 ) 3/2
. This procedure yields a value of the order z dec ∼ 10 3 , indicating that photon decoupling took place around recombination. In Sect. 1.6.5, we will give a more precise definition of z dec as the maximum of the visibility function.

Even if the Saha equation gives a correct estimate of the time of recombination, it fails at predicting the residual amount of free electrons at later times. To track accurately the evolution of X e , we need to apply the Boltzmann equation. Using Eq. (1.94) for the electron density n 1 = n e gives

a -3 d(n e a 3 ) dt = -σv   n 2 e - n (0) e n (0) p n (0) H,1s n H,1s   , (1.120)
It is more convenient to rewrite Eq. (1.120) in terms of the free electron fraction X e . Using Eq. (1.116) and Eq. (1.117), we obtain

dX e dt = (1 -X e )β -αn tot H X 2 e , (1.121)
where β and α denote the photoionization and recombination rates,

β ≡ α m e T b 2π 3/2 e -ε 0 /T b , α ≡ σv (1.122)
However, Eq. (1.121) is still an incomplete description of the recombination process, as we shall see. The first compelling model of recombination was developed by ZELDOVICH et al. (1968) and PEEBLES (1968), that we briefly discuss in the following.

The effective three-level model

This recombination model assumes that hydrogen can be in three different states 1. Hydrogen in the ground state 1s, (n H,1s ).

2. Hydrogen in the excited states, mostly 2p or 2s (n H,2l ).

Ionized hydrogen (n p n e ).

This is schematically depicted in the left panel of Fig. The first important thing to note is that recombination to the ground state (n = 1) is very inefficient, since the emitted photon has enough energy to reionize another neutral hydrogen atom, leaving X e unaffected. Indeed, the cross section for absorption of photons just above 13.6 eV is σ pi 6 × 10 -18 cm 2 (C. HIRATA 2006), while the total density of hydrogen is around

n tot H = (1 -Y p )ηn γ ∼ 200[(1 + z)
/1000] 3 cm -3 (we temporarily restore I.S. units). The typical time before a photon is reabsorbed is thus of order

t reabs ∼ 1 n tot H x 1s σ pi c ∼ 10 4 x 1s 1000 1 + z 3 s, (1.124)
where x 1s ≡ n H,1s /n tot H . We see that around recombination, and as long as x 1s > 10 -9 , this reabsorption time is much smaller than the typical expansion time,

t H = 1/H = Ω -1/2 m H -1 0 (1 + z) -3/2 ∼ 10 13 s.
Hence, efficient recombination only happens when electrons initially get captured into an excited state n ≥ 2. In this case, the population of the n = 2 states are suppressed with respect to n = 1 states by a factor e -3ε 0 /4T ∼ 10 -17 (according to Eq. (1.123)), so there is no reabsorption problem. Therefore, the α factor in Eq. (1.121) has to be replaced by the so called case B recombination rate,

α B = ∞ n=2 n-1 l=0 σ [p+e→H(nl)+γ] v .
(1.125) This process is then followed by a fast radiative cascade to n = 2. The resulting excited hydrogen atom could subsequently reach the ground state in a much slower decay, through one of the following processes:

H(2s) ←→ H(1s) + γ + γ, (1.126)
H(2p) ←→ H(1s) + γ Lyα . (1.127)
The two-photon decay process has a rate Λ 2γ 8.2 s -1 , and neither of the emitted photons has enough energy to excite another hydrogen atom. On the contrary, the emitted Lyman-α photons are usually re-absorbed by other hydrogen atoms, since the optical depth for these photons during recombination is very large. This effectively delays recombination. However, expansion eventually redshifts the Lyman-α photons off the resonance, so that they can no longer be re-absorbed by other atoms . This is characterized by the quantity Λ α ≡ (where ω Lyα = 3ε 0 /4 is the Lyman-α energy), which gives the net 2p → 1s transition rate accounting for re-absorptions. To properly include all those effects, the r.h.s. of Eq. (1.121) gets an additional factor (see C. HIRATA (2006) for a proof),

Hω 3 Lyα π 2 n tot H (1-Xe)
C = Λ α + Λ 2γ Λ α + Λ 2γ + 4β B , ( 1.128)
where

β B = α B meT b 2π 3/2
e -ε 0 /4T b is the photoionization rate from n = 2. We see that C gives the probability for an electron in the excited n = 2 state to reach the ground state before being ionized. The equation describing the evolution of X e then reads .129) Given that the recombination rate per free electron Cα B n tot H X e is always much smaller than H, this model predicts that recombination happens much slower than in Saha equilibrium. In addition, it gives a non-zero freeze-out electron fraction of X ∞ e ∼ 10 -4 .

dX e dz = - C (1 + z)H (1 -X e )α B m e T b 2π 3/2 e -ε 0 /T b -α B n tot H X 2 e . ( 1

Evolution of the baryon temperature

So as to complete the description of recombination, we need to track not only the free electron ratio X e , but also the baryon temperature T b , which enters in several places in Eq. (1.129). The assumption T b = T is valid in the early universe, but it ceases to be true at later times. The evolution of matter temperature receives two main contributions (C. HIRATA 2006)

dT b dz = 2T b 1 + z adiabatic + 1 (1 + z)H 8σ T a r T 4 3m e c X e 1 + f He + X e (T b -T ) Compton scattering = 1 1 + z [2T b + γ(T b -T )] . (1.130)
Here we introduced the dimensionless parameter

γ ≡ X e 1 + f He + X e 8σ T a r T 4 3Hm e , ( 1.131)
where a r = 4σ B /c = π 2 k 4 B /(15 3 c 3 ) is the radiation constant. The first term in Eq. (1.130) simply describes the effect of the cosmic expansion, while the second one describes the Compton scattering between CMB photons and the gas. Initially γ 1, so the second term dominates and Compton interactions couple baryons to radiation, so that T b = T ∝ a -1 . Later on, the γ factor decreases until becoming very suppressed (at a redshift of around z ∼ 150), time after which baryons start to cool down adiabatically, T b ∝ a -2 . The gas temperature will eventually increase again in the reionization era, due to the ionizing radiation emitted by the first stars and galaxies.

Modern recombination codes

It was realized by the end of the 90s that the upcoming high-precision CMB data from the Planck satellite was going to require to compute the recombination history with a ∼ 0.1% precision. Thus, many improvements have been added to the threelevel model in the last years, such as:

• Following not only n = 2, but a virtually infinite number of excited levels.

• Computing accurately the radiative transfer in the Lyman-α line.

• Including Helium recombination.

These corrections have been implemented approximately in the code RECFAST [START_REF] Seager | A new calculation of the recombination epoch[END_REF]), which adds some fudge factors to Eq. (1.129) to match a complicated multilevel calculation, and exactly in the more sophisticated codes HYREC (ALI-HAIMOUD et al. 2011) and COSMOREC [START_REF] Chluba | Towards a complete treatment of the cosmological recombination problem[END_REF]). The three recombination codes are currently implemented in CLASS . In the right panel of Fig. 1.11, we show the evolution of the free electron fraction using both the Saha equilibrium condition in Eq. (1.118) and the more advanced calculation with RECFAST .

As a side note, RECFAST also includes the first and second Helium recombinations, He ++ → He + and He + → He, whose binding energies are respectively given by ε He ++ = 4ε 0 = 54.4 eV and ε He + = 24.6 eV. Since these energies are higher than the H binding energy ε 0 , He recombines first. The first He recombination (happening between 5000 ≤ z ≤ 8000) can be safely described by Saha equilibrium, while the second He recombination requires a more careful treatment. Therefore, RECFAST actually solves for three variables (coupled between each other): the proton fraction X p ≡ n p /n tot H , the singly-ionized He fraction X He + ≡ n He + /n tot He (where n tot He = n He + n He + + n He ++ ), and the baryon temperature T b . From the neutrality condition n e = n p + n He + + 2n He ++ , we see that the free electron fraction is given by

X e = X p + f He (X He + + 2X He ++ ), (1.132)
where f He Y p /(4(1 -Y p )) (c.f. Eq. (1.114)). Eq. (1.132) tells us that, even when hydrogen still hasn't recombined (X p = 1), we can still have X e > 1 due to He recombinations not being completed yet (X He + = 0 and X He ++ = 0). This explains the small bumps that can be seen in X e at high z in the right panel of Fig. 1.11.

Reionization

A few hundred million years after recombination, the first stars and galaxies were born. Around these objects, bubbles of ionized plasma formed due to the emitted ultraviolet radiation. Over time, these bubbles grew and eventually overlapped, leaving the IGM fully ionized again (except for some small traces of neutral hydrogen).

As opposed to what happens with recombination, the details of this epoch of reionization are very poorly understood, both at the theoretical and observational level. The best probe that we have nowadays comes from the the optical spectra of very distant quasars, such as the one we show in Fig. 1.12. The Lyman-α absorption lines in these spectra are a very good indicator of the presence of neutral hydrogen. If a quasar from the epoch of reionization shines photons with λ < λ Lyα = 121.57 nm, then these photons get stretched due to the cosmic expansion, until reaching λ = λ Lyα and being absorbed by the neutral hydrogen in the surrounding medium. This leads to the so-called Gunn-Peterson (GP) trough, which was predicted in 1965 [START_REF] Gunn | On the Density of Neutral Hydrogen in Intergalactic Space[END_REF]). The GP trough has been observed only in the spectra of quasars located at z 6 (BECKER et al. 2015), indicating that the universe had to be completely reionized below this redshift. Notice that photons reaching λ = λ Lyα after the end of reionization can still be absorbed, due to encounters with individual clouds of neutral hydrogen along their path. This produces a jagged region in the quasar spectra that is known as the Lyman-α forest.

Observations of the CMB can also probe the EoR. The increase in X e (z) at low redshifts enhances the scattering rate between photons and electrons, which leads to peculiar patterns in the temperature and polarization spectra (as we will discuss in Sect. 1.6.5). The CMB spectra are primarily sensitive to the column density of electrons along the line of sight, and therefore to the optical depth at reionisation, defined by

τ reio ≡ z reio,start 0 X e (z)n tot H (z)σ T dt dz dz, ( 1.133)
where z reio,start is the redshift at the beginning of reionization. For reionization histories centered around z ∼ 6 -7 (as expected from GP measurements), the predicted τ reio is rather small, in agreement with the values recently reported by Planck18. Because of the large theoretical uncertainties associated with the EoR and the fact that the CMB is not very sensitive to the details of reionization, Boltzmann codes usually adopt parametrizations, that interpolate a low X e at high-z to a X e ∼ 1 at low-z. The most common parametrization and the one used in CLASS by default is

X e (z) = 1 + f He 2 1 + tanh y(z reio ) -y(z) ∆y , ( 1.134)
where y = (1+z) 3/2 and ∆y = 3 2 (1+z reio ) 1/2 ∆z, with ∆z = 0.5. With this parametrisation, there is a one-to-one relation between the mid-point redshift of reionisation z reio and τ reio . For other parametrisations or approaches to model reionization sources, we refer to POULIN (2017). There is a current effort in planning and building instruments to observe the 21-cm hyperfine line of neutral hydrogen, such as HERA (ABDURASHIDOVA et al.

The linearly perturbed universe

Until now, we have been considering the universe as perfectly isotropic and homogeneous. However, to understand the pattern of temperature and polarization fluctuations of the CMB, or how the primordial fluctuations eventually formed all the large-scale structures, we need to introduce anisotropies and inhomogeneities. We can treat them in linear perturbation theory as long as these perturbations remain small. This provides an excellent description of the CMB physics, which are mostly determined by processes in the early universe, when over-densities were of the order O(10 -5 ). For small scales at small redshift, non-linear effects become important, and perturbation theory needs to be replaced by more complicated descriptions, such as N-body simulations.

The basic idea of cosmological perturbation theory is to decompose all relevant quantities into spatial averages (from now on, denoted by an overline) plus a linear perturbation,

g µν (τ, x) = ḡµν (τ ) + δg µν (τ, x), T µ ν (τ, x) = T µ ν (τ ) + δT µ ν (τ, x). (1.135)
Here ḡµν (τ ) is the FLRW metric, which for a flat spacetime in cartesian coordinates reads

ds 2 = ḡµν dx µ dx ν = a(τ ) 2 -dτ 2 + δ ij dx i dx j , ( 1.136)
while T µ ν (τ ) is the stress energy tensor of a perfect fluid that we introduced in Eq. (1.12). We will often switch from real to Fourier space when expressing equations for the perturbations. This is very advantageous, because it will allow to reduce a partial differential equation to several (decoupled) ordinary differential equations, one per mode k. Put another way, in linear theory each mode evolves independently. We will adopt the common practice (not suitable to mathematicians) of calling the variables in real space in the same way as in Fourier space. For instance, the Fourier transform of the density contrast δ(τ, x) ≡ ρ(τ, x)/ρ(τ ) -1 will read

δ(τ, x) = d 3 k (2π) 3 δ(τ, k)e i k• x -→ δ(τ, k) = d 3 x δ(τ, x)e -i k• x . (1.137)
It is customary to express the perturbed metric components in the following way

δg 00 = -2a 2 (τ )A, δg 0i = a 2 (τ )B i , δg ij = a 2 (τ )H ij , (1.138)
where A, B i and H ij are generic functions of space and time. On the other hand, perturbing the stress-energy tensor in Eq. (1.12) gives δT µ ν = (δρ + δP ) Ū µ Ūν + (ρ + P )(δU µ Ūν + Ū µ δU ν ) + δP δ µ ν + S µ ν .

(1.139)

The perturbed stress-energy tensor can now receive a contribution from the anisotropic stress tensor S µ ν (whose non-zero elements are the off-diagonal components of δT i j ). This can be relevant for some fluids such as neutrinos, as we shall see later. In addition, perturbations can induce non-vanishing energy flux T 0 j , as well as momentum density T i 0 . So as to give an expression for each of the δT µ ν components in Eq. (1.139), we first need to know the expression for the four velocity U µ = Ū µ + δU µ . As we show in App. A.1, it can be written as

U µ = 1 a [1 -A, v i ], (1.140)
where v j is the coordinate velocity. With this, the covariant four-velocity is easily obtained from

U µ = g µν U ν , U µ = a[-(1 + A), v i + B i ]. (1.141)
Inserting Eq. (1.140) and Eq. (1.141) in Eq. (1.139), we arrive at

δT 0 0 = -δρ, δT i 0 = -(ρ + P )v i , δT 0 j = (ρ + P )(v j + B j ), δT i j = δP δ i j + S i j . (1.142)

Scalar-vector-tensor decomposition

Both tensors g µν and T µν are symmetric, so in the most general case each one contains 10 degrees of freedom (d.o.f). It was shown in BARDEEN (1980) that these d.o.f. can be decomposed in scalar, vector and tensors (SVT). Here the notion of scalar, vector or tensor is referred to their transformation properties under the group of spatial rotations SO [START_REF] Einstein | Cosmological Considerations in the General Theory of Relativity[END_REF], which is a symmetry of the FLRW metric. This SVT decomposition is extremely useful, because Einstein equations for scalars, vectors and tensors don't mix at linear order. This powerful result is known as the decomposition theorem, and a proof can be found in KNOBEL (2012). For B i the decomposition is simply

B i = ∂ i B + Bi , ( 1.143)
with ∂ i Bi = 0 (divergenceless quantities are denoted with hats). For the tensor H ij the decomposition reads (STEWART 1990):

H ij = 2Cδ ij + 2 ∂ i ∂ j - δ ij 3 ∇ 2 E + ∂ i Êj + ∂ j Êi with ∂ i Êi = 0 + 2 Êij with ∂ i Êij = 0 and Êi i = 0. (1.144)
Thus we have split the 10 d.o.f. of the metric into:

• 4 scalar d.o.f. corresponding to A, B, C, E.

• 4 vector d.o.f. corresponding to the independent components of Bi and Êi

• 2 tensor d.o.f. corresponding to the independent components of Êij

In the presence of matter sources, scalars represent the response of the metric to irrotational velocity fields (that is, with vanishing vorticity). Vectors represent the response of the metric to vorticity, and they are usually irrelevant in cosmology, since they decay with the cosmic expansion. Tensor perturbations are the only ones that can propagate in vacuum, and their d.o.f. describe the two polarizations of gravitational waves. In this work we will exclusively concentrate on scalars (the only modes with a newtonian analog among the three) and their associated matter perturbations, although we'll say a few words about tensors in Sect. 1.6.5.

It is also possible to perform a SVT decomposition for the stress-energy tensor. For the coordinate velocity v i we may write,

v i = ∂ i v + vi , (1.145)
while for the anisotropic tensor S ij ,

S ij = ∂ i ∂ j - δ ij 3 ∇ 2 S + 1 2 ∂ i Ŝj + ∂ j Ŝi + Ŝij . (1.146)
In practice, we will be only interested in v and S, since we just focus on the scalar sector. It is traditional to trade the scalar components of the stress-energy tensor (δρ, v, δP, S) for the more common variables (δ, θ, Π, σ), which are defined by: ρδ ≡ -δT 0 0 , (1.147)

(ρ + P )θ ≡ i ∂ i δT 0 i , (1.148) ρΠ ≡ 1 3 i T i i , (1.149)
(ρ + P )∇ 2 σ ≡ - i,j ∂ i ∂ j - δ ij 3 ∇ 2 δT i j . ( 1
δ = δρ/ρ, Π = δP/ρ, θ = i ∂ i v i = ∇ 2 v, (ρ + P )∇ 2 σ = - 2 3 ∇ 2 ∇ 2 S.
(1.151)

The total stress-energy tensor is simply obtained by summing over the stress-energy tensor of each species. To give an example, the total perturbed density is δρ tot = I ρI δ I , from which we infer that δ tot = I ρI δ I /ρ tot .

The gauge problem

In the FLRW universe, there is only one possible slicing of constant-time hypersurfaces compatible with the assumption of homogeneity. On the contrary, in the perturbed FLRW universe, there are infinite possibilities of time slicings compatible with perturbation theory (i.e. such that on each constant-time hypersurface, all quantities remain close to the average value). For example, the density perturbation is defined by δρ(τ, x) ≡ ρ(τ, x) -ρ(t). While ρ(τ, x) is an unambiguously defined quantity, ρ(t) depends on the particular time slicing passing through (τ, x) over which we have decided to perform the average. Hence, δρ(τ, x) also depends on this choice, as illustrated in Fig. 1

.13.

A gauge is simply a choice of time slicing, with a specific choice of spatial coordinates on these constant-time slices. Gauge transformations are thus induced by coordinate transformations,

x µ -→ xµ = x µ + ξ µ (τ, x), ( 1.152)
where ξ 0 ≡ T and ξ i ≡ L i = ∂ i L + Li . These coordinate transformations map the points of once slicing to those of another slicing. We remark that not all coordinate transformations give rise to valid gauge transformations. The parameters T and L i need to be small so that perturbed quantities remain at linear order after the transformation.

To see how metric elements transform under this gauge transformation, it suffices to use the transformation law for tensors,

g µν (x) = ∂ xα ∂x µ ∂ xβ ∂x ν gαβ (x).
(1.153)

A -→ Ã = A -T -HT, (1.154)
B i -→ Bi = B i + ∂ i T -L i , (1.155)
H ij -→ Hij = H ij -(∂ i L j + ∂ j L i ) -2HT δ ij . (1.156)
We give a proof of Eq. (1.155) in App. A.2 as an example. It will be convenient to express Eq. (1.154), Eq. (1.155) and Eq. (1.156) in terms of the SVT decomposition:

A -→ A -T -HT, (1.157)
B -→ B + T -L , Bi -→ Bi -L i , (1.158) C -→ C -HT - 1 3 ∇ 2 L, (1.159) E -→ E -L, Êi -→ Êi -Li , Êij -→ Êij . (1.160)
In a similar vein, one can obtain the gauge transformation rules for the stressenergy tensor using,

T µ ν (x) = ∂x µ ∂ xα ∂ xβ ∂x ν T α β (x). ( 1
v i -→ ṽi = v i + L i , (1.164) S ij -→ Sij = S ij . (1.165)
We have proved Eq. (1.162) in App. A.2 to show the spirit of the calculation. We have seen that cosmological perturbations are subject to gauge freedom, and also how the metric and the stress-energy tensor components change under these gauge transformations. However, this freedom can be problematic, since it can easily produce fake perturbations with no physical meaning. There are two strategies to deal with this issue: either work with gauge invariant quantities, or fix a gauge. We describe these two strategies in the following.

Gauge invariant variables

By looking at the way the metric perturbations change under gauge transformations, one can define linear combinations of them that are left invariant. These are the so-called Bardeen variables:

ψ ≡ A + H(B -E ) + (B -E ) , (1.166) φ ≡ -C -H(B -E ) + 1 3 ∇ 2 E, (1.167) φi ≡ Ê i -Bi , Êij . ( 1
δρ GI ≡ δρ -ρ (E -B), (1.169
)

δP GI ≡ δP -P (E -B), (1.170)
v GI i ≡ v i + ∂ i (E -B).
(1.171)

One could study the dynamical equations of perturbations in terms of these gauge invariant variables. In practice, this is not what it is usually done, since these equations can be quite tricky.

Gauge fixing

This just consists in introducing a condition such that the time slicing is unique. In the scalar sector, this is typically achieved by using the freedom of the transformation variables T and L in order to set two of the four scalar metric perturbations to zero. There is no problem with this method, since observable quantities do not depend on the gauges, only the computations do. Different gauges have their advantages. We are going to discuss the two gauge choices that are more commonly used in the literature.

• The Newtonian gauge. This is defined by the choice B = Ẽ = 0, which can be achieved by setting L = E and T = E -B. This prescription fixes a unique time slicing. A big advantage of this gauge is that the metric is diagonal and that the perturbed quantities can be trivially linked to the Bardeen variables. Namely, A = ψ and C = -φ. Thus, the metric reads

ds 2 = a 2 (τ ) -(1 + 2ψ)dτ 2 + (1 -2φ)δ ij dx i dx j . (1.172)
The name of the gauge stems from the fact that Eq. (1.172) looks very similar to the weak-field limit of GR about the Minkowski metric. In fact, ψ plays the role of the Newtonian gravitational potential on sub-horizon scales. Furthermore, the term a 2 (τ )(1 -2φ) represents a scale factor with a local distortion.

• The synchronous gauge. This gauge, originally introduced by LIFSHITZ (1946), is obtained by the choice à = B = 0, which can be achieved by picking L and T such that T + HT = A and L = T + B. Thus, scalar perturbations only enter in the spatial part of the metric:

ds 2 = a 2 (τ ) -dτ 2 + (δ ij + H ij )dx i dx j , ( 1.173)
where

H ij = 2Cδ ij + 2(∂ i ∂ j - δ ij 3 ∇ 2 )E.
It is traditional to trade C and E for the scalar variables h an η. The first is defined as the trace of H ij , h ≡ i H i i = 6C, while the second is defined through the relation 2∇ 2 E = 6η + h. Hence, we can write in Fourier space:

-2k 2 E = 6η + h, ( 1.174
)

H ij = δ ij 3 h -2 k i k j - δ ij 3 k 2 E = ki kj h + ki kj - δ ij 3 6η, (1.175)
where ki ≡ k i /k. Unfortunately, this still doesn't fix the gauge. Indeed, Ã = 0 is obtained for any T satisfying T + HT = A, i.e. aT = dτ aA + c 1 , so the time slicings haven't yet been fixed due to this integration constant c 1 . It is always possible to use the synchronous gauge as long as one uses some extra condition that eliminates the residual gauge freedom. One common approach is to set the initial velocity divergence of the CDM to zero, θ ini cdm = 0 (synchronous gauge comoving with the CDM).

It is practical to have equations that relate both gauges. For the metric perturbations, this can be straightforwardly achieved using the Bardeen variables in Eq. (1.166) and Eq. (1.167). Setting B = E = 0 for the newtonian gauge and A = B = 0 for the synchronous gauge, we get (in Fourier space):

ψ = Hα + α , (1.176) φ = η -Hα, (1.177)
where α ≡ -E = (6η + h) /2k 2 . In a very similar manner, we can relate elements of the perturbed stress-energy tensor using the gauge invariant variables that we defined in Eq. (1.169)-Eq. (1.171). We obtain:

δρ new = δρ syn + ρ α, ( 1.178
)

δP new = δP syn + P α, (1.179) v new i = v syn i -∂ i α. (1.180)
The last equation can equivalently be expressed as θ new = θ syn + k 2 α (after taking the divergence and going to Fourier space). We observe that all the terms with α at the r.h.s. of Eq. (1.178)-Eq. (1.180) are of the order ∼ H 2 /k 2 or ∼ H/k, meaning that they are very suppressed on sub-Hubble scales (k H). Indeed, well inside the Hubble radius we expect to recover a Newtonian perturbation theory, where the gauge ambiguity (which is a purely relativistic effect) is absent.

The synchronous gauge is the one that is commonly adopted in Boltzmann codes, owing to its numerical stability. For this reason, it is also the one we have adopted in the Chapter 2, Chapter 3 and Chapter 4. However, physics is more intuitive using the Newtonian gauge, so we will restrict to this gauge for the rest of this Chapter.

Perturbation equations

Now that we have defined the perturbation d.o.f and fixed a gauge, it is time to concentrate on the dynamical evolution of these perturbed variables. More precisely, we need to know:

1. How the evolution of matter fluctuations is influenced by gravitational potentials and by the interactions among the different species.

2. How gravitational potentials respond to changes in the matter fluctuations.

The latter is dictated by the perturbed version of Einstein equations (see Eq. (1.10)).

For the former, there are a priori several strategies. We have seen that there are four scalar variables in the perturbed stress-energy tensor, that can be taken to be the density contrast δ, the perturbed pressure δP , the velocity divergence θ and the anisotropic stress σ. The stress-energy conservation ∇ µ T µν = 0 (see Eq. (1.13)) leads to two dynamical equations in the scalar sector, the continuity and the Euler equation. Therefore, two more relations are needed to close the system. For ideal fluids, adiabaticity allows to relate the sound speed c 2 s ≡ δP/ρδ with the EoS w, whereas isotropy imposes σ = 0. In the most general case, for fluids which are non-ideal and/or have non-negligible interactions with other species, the stress-energy conservation provides an incomplete description. The most general description is obtained thanks to the perturbed version of the Boltzmann equation Eq. (1.58), that we proceed to discuss.

Perturbed Boltzmann equations

Let us consider a particle with mass m and energy E = m 2 + p 2 , where we remind that the amplitude of the physical three-momentum is defined as p ≡ ij g ij P i P j . As we show in App. A.3 , the elements of the 4-momentum vector P µ = dx µ dσ = (P 0 , P i ) are given by

P 0 = E a (1 -ψ), P i = pn i a (1 + φ). (1.181)
In terms of these variables, the 0 component of the geodesic equation leads to (see App. A.3 for the derivation)

dq dτ = qφ -E n • ∇ψ, (1.182)
where q = ap is the comoving momentum and E = aE is the comoving energy. This equation describes the change in comoving momentum for a particle following geodesic motion in a perturbed FLRW universe. The first term at the r.h.s of Eq. (1.182) accounts for a local energy loss due to dilation, i.e. the fact that (locally) the expansion is a bit ahead or behind the average. The second term describes the energy increase for photons falling in gravitational wells, or the energy decrease of those leaving the gravitational wells. In absence of inhomogeneities, φ = ψ = 0, we recover the familiar momentum redshift, q = const., i.e. p ∝ a -1 . A crucial observation is that for massless particles, we can use E = q and rewrite Eq. (1.182) as

1 q dq dτ = φ -n • ∇ψ. (1.183)
That is, for massless particles, the relative change in momentum doesn't depend on momentum itself. This means that any massless particle that was in equilibrium in the early universe, will retain the Fermi-Dirac or Bose-Einstein form, even at the level of perturbations. The only way for this massless particle to acquire non-thermal distortions is through non-gravitational interactions. On the contrary, gravitational interactions can induce nonthermal distortions for massive particles, since the momentum dependence can't be eliminated from the r.h.s. of Eq. (1.182). This will be important when discussing the evolution of perturbations for massive neutrinos.

We are ready to derive the Boltzmann equation for the PSD f in a inhomogeneous universe. According to Eq. (1.58) we have

df dσ = P 0 df dτ = C[f ]. (1.184)
Using the chain rule, we can expand the derivative df /dτ , taking account that the PSD has in this case a general dependence f = f (τ, x i , q, ni ). Hence,

df
+ qn i E ∂f ∂x i + q ∂f ∂q φ - E q ni ∂ψ ∂x i = C[f ] P 0 . ( 1.186)
At this stage, it is useful to switch to Fourier space, which means that the PSD is now a function of the type f = f (τ, q, k, µ), where µ ≡ k • n. Notice that the unit vector n enters only through µ, i.e. through the cosine of the angle between the direction of propagation and the wavenumber. This indicates a rotational symmetry around n, which arises just due to the isotropy of the FLRW background. It is customary to split the PSD between a background and a linearly perturbed contribution f (τ, q, k, µ) = f (τ, q)(1 + Ψ(τ, q, k, µ)).

(1.187)

Plugging the decomposition of Eq. (1.187) into Eq. (1.186), we get two equations, one at zeroth-order,

∂ f ∂τ = a 2 E C (0) ! = 0, (1.188)
and another at first-order 1) , (1.189) where we have split the collision term in a zeroth-order and first-order contribution, C = C (0) + C (1) . In Eq. (1.188), we set C (0) = 0 since we assume that the relevant interactions are such that they maintain an equilibrium distribution for f . This is the case for the Compton scattering between baryons and photons, that we shall discuss later. Eq. (1.189) is the master equation governing the evolution of perturbations of all relevant species in cosmology: photons, neutrinos, dark matter and baryons.

∂Ψ ∂τ + iµ kq E Ψ + dln f dlnq φ -iµ kE q ψ = a 2 f E C (
In the following, we will particularize Eq. (1.189) for each species, by specifying the energy-momentum relation and the collision term.

Before doing this task, let us detail how to obtain the elements of the perturbed stress-energy tensor from Ψ. For this, it is convenient to eliminate the dependence of Ψ on µ by doing an expansion over Legendre multipoles

Ψ(τ, q, k, µ) = (-i) (2 + 1)Ψ (τ, q, k)P (µ) (1.190)
We already showed in Eq. (1.50) how the stress-tensor components are obtained from moments of the PSD. The linearly perturbed version of this equation is .191) As we show in App. A.4, the elements of δT µ ν can be obtained from the multipoles Ψ in the following manner: .192) .195) From Eq. (1.194) and Eq. (1.195), we see that the perturbed pressure and anisotropic stress of non-relativistic particles is suppressed by a factor q/am 1. Let us now describe the dynamics of each cosmological species.

δT µ ν (τ, k) = g d 3 P (2π) 3 (-det(g)) -1/2 P µ P ν P 0 f (τ, q)Ψ(τ, q, k, µ). ( 1
ρδ = 4π a 4 ∞ 0 dqq 2 E f Ψ 0 , ( 1
(ρ + P )θ = 4πk a 4 ∞ 0 dqq 2 q f Ψ 1 , (1.193) ρΠ = 4π 3a 4 ∞ 0 dqq 2 q 2 E f Ψ 0 , (1.194)
(ρ + P )σ = 8π 3a 4 ∞ 0 dqq 2 q 2 E f Ψ 2 . ( 1

Photons

Before decoupling, photons were in kinetic equilibrium with electrons thanks to the frequent Compton interactions. We have seen that after decoupling the photon distribution maintains the Bose-Einstein form both at the background (c.f. Eq. (1.70)) and the linearly perturbed level (c.f. Eq. (1.183)). Hence, the only way the photon PSD can acquire anisotropies and inhomogeneities is through the temperature. Thus, we can write the photon PSD as .196) where Θ ≡ δT / T stands for the relative temperature shift. Expanding at first order, .197) We have used the fact that the background PSD fγ = exp(q/a T ) -1

f γ (τ, q, k, µ) = exp q a T (1 + Θ(τ, k, µ)) -1 -1 , ( 1
f γ = fγ + ∂ fγ ∂ T T Θ = fγ -q ∂ fγ ∂q Θ. ( 1
-1 satisfies ∂ fγ ∂ T = -q
T ∂ fγ ∂q . This implies a very interesting relation

Ψ γ (τ, q, k, µ) = - dln f dlnq Θ(τ, k, µ) (1.198)
That is, the perturbed PSD Ψ is a separable function on q and k . Inserting Eq. (1.198) in Eq. (1.189), and using E = q, we get

∂Θ ∂τ + ikµΘ -φ + ikµψ = - a 2 q 2 ∂ f ∂q C (1) γ = - a p 2 ∂ f ∂p C (1)
γ (1.199) We have carried out the full derivation of the collision integral for photons, C

γ , in App. A.5. The final result reads

∂Θ ∂τ + ikµΘ -φ + ikµψ = an e σ T Θ 0 -Θ + µv b - P 2 (µ) 2 Θ 2 (1.200)
where

v b ≡ | v b | = θ b /
ik is the common bulk velocity of baryons and electrons. The terms Θ 0 ≡ 1 4π dΩΘ and Θ 2 ≡ 1 4π dΩP 2 (µ)Θ denote the monopole and quadrupole perturbation, respectively. In Eq. (1.200) we have neglected for simplicity some extra terms associated with photon polarization, which don't affect the qualitative behavior of Θ. We will say a few words about polarization in Sect. 1.6.5. The interaction term at the r.h.s. of Eq. (1.200) can be understood intuitively, by considering the so called tightly coupled limit (Γ γ = an e σ T H). In this limit, Compton scattering enforces Θ 2 → 0 (as we will see more clearly later), and the temperature perturbation becomes equal to Θ = Θ 0 + µv b , i.e. it is isotropic up to a dipole term accounting for the relative motion between the photon-baryon fluid and the observer. This just reflects the idea that, when interactions are strong, photons and electrons behave as a single fluid. In this context, one typically introduces the optical depth, giving the opacity of the universe at a given time τ seen from today

κ γ (τ ) ≡ τ 0 τ
dτ an e σ T , -→ κ γ = -an e σ T .

(1.201)

The optical depth tends to infinity at very early times, drops below one at recombination, and then decreases like an e ∼ a -2 (except during reionization, when it slightly increases). Notice that the optical depth at reionization shown in Eq. (1.133) is computed in the same way as in Eq. (1.201), but in a restricted time interval.

It is useful to perform a Legendre decomposition for Θ, similar to the one we did for Ψ:

Θ(τ, k, µ) = (-i) (2 + 1)Θ (τ, k)P (µ) (1.202)
We can relate the temperature multipoles Θ with elements of the perturbed stresstensor by inserting Ψ γ, = -dln f dlnq Θ in Eq. (1.192)-Eq. (1.195). After doing integration by parts, q 4 d f dq = d dq ( f q 4 ) -4 f q 3 , and discarding the total derivative term (since we assume f decreases sufficiently rapid for q → ∞), we end up with

δ γ = 4Θ 0 , θ γ = 3kΘ 1 , σ γ = 2Θ 2 . (1.203)
The factor 4 in the δ γ -Θ 0 relation could have been anticipated by noting that the photon density scales with temperature as ρ γ ∝ T 4 . We can use the multipole expansion in Eq. (1.202) in order to eliminate the µ dependence from the Boltzmann equation. The inverse relation of Eq. (1.202) is

Θ = (-i) - +1 -1 dµ 2 P (µ)Θ(µ), (1.204)
so the idea is to perform the integral (-i) -+1 -1 dµ 2 P (µ) on each side of Eq. (1.200) 20 . This leads to the following hierarchy of equations for the photon multipoles:

δ γ = - 4 3 θ γ + 4φ , (1.206) θ γ = k 2 δ γ 4 -σ γ + ψ + κ γ (θ γ -θ b ), (1.207
)

σ γ = 4 15 θ γ - 6 5 kΘ 3 + 9 10 κ γ σ γ , (1.208) Θ = k 2 + 1 [ Θ -1 -( + 1)Θ +1 ] + κ γ Θ , ∀ > 2.
(1.209)

From the previous equations, we can understand why the anisotropic stress σ γ and higher moments are very suppressed in the tightly coupled limit, κ γ 1. The l.h.s. of Eq. (1.208) is of order σ γ /τ , which is much smaller than the third term at the r.h.s., of the order κ γ σ γ /τ . Neglecting also the term with Θ 3 , we see that σ γ ∼ τ θ γ /κ γ , so σ γ is very suppressed with respect to θ γ . In general, one shows that in the tightly coupled regime, Θ ∼ kτ Θ -1 /κ γ (which is the reason why we neglected the Θ 3 term before).

Obviously, the infinite hierarchy of Eq. (1.206) -Eq. (1.207) needs to be truncated at some multipole max . The truncation scheme that has been recently adopted by all Boltzmann codes is the one proposed by MA et al. (1995)

Θ max kΘ max-1 - max + 1 τ Θ max + κ γ Θ max . ( 1.210)
The first two terms at the r.h.s. are inspired by the recurrence relation for spherical Bessel functions. Indeed, in the absence of time-varying gravitational potentials or interaction terms, one can show that the exact solution of the Boltzmann hierarchy gives Θ (τ, k) ∝ j (kτ ) for > 0. As a side note, the Boltzmann hierarchy for massless neutrinos is identical to the one showed in Eq. (1.206) -Eq. (1.208), except that all collision terms proportional to κ γ are set to zero.

Massive neutrinos

Neutrinos decoupled from the thermal plasma in the very early universe, as we already saw in Sect. 1.5.3, so their collision term in Eq. (1.189) can be safely set to zero, C

(1) ν = 0. Even if their background PSD is well described by a Fermi-Dirac because they decoupled when they were relativistic, gravitational interactions induce non-thermal distortions at the linearly perturbed level, due to the late-time effects of their non-zero mass (c.f. Eq. (1.182)). Hence, the momentum d.o.f. cannot be integrated out from the Boltzmann equation as was done for photons, and we have to follow the full evolution in phase space.

Following the same logic that we applied to photons, we can eliminate the µ dependence from the Boltzmann equation using the expansion in Legendre multipoles. The inverse relation of Eq. (1.190) is Ψ = (-i) -+1 -1 dµ 2 P (µ)Ψ(µ), so performing the 20 Doing this procedure, one encounters the integral +1 -1 dµµP (µ)Θ(µ). In order to handle this term, we use the recurrence relation .205) integral (-i) -+1 -1 dµ 2 P (µ) in Eq. (1.189), we arrive at the following hierarchy of equations for the neutrino multipoles21

µP (µ) = 1 2 + 1 [( + 1)P +1 (µ) + P -1 (µ)]. ( 1
Ψ ν,0 = - kq E Ψ ν,1 - dln f dlnq φ , (1.211) Ψ ν,1 = kq 3E [Ψ ν,0 -2Ψ ν,2 ] - Ek 3q dln f dlnq ψ, (1.212) Ψ ν, = kq (2 + 1)E [ Ψ ν, -1 -( + 1)Ψ ν, +1 ], ∀ > 1.
(1.213)

A similar truncation scheme as the one formerly discussed has been proposed for massive neutrinos (MA et al. 1995)

Ψ max qk E Ψ max-1 - max + 1 τ Ψ max . ( 1
δ ν = -3H(c 2 s -w ν )δ ν -(1 + w ν ) θ ν -3φ , (1.215
)

θ ν = -H(1 -3c 2 g )θ ν + k 2 c 2 s 1 + w ν δ ν + ψ -σ ν , (1.216)
where c 2 s ≡ δP ν /δρ ν is the sound speed of the fluid and c 2 g ≡ P ν /ρ ν is the adiabatic sound speed. In general, c 2 s is a spatially-and gauge-dependent quantity, as opposed to c 2 g . Eq. (1.215) and Eq. (1.216) could have been alternatively obtained using the covariant conservation of the stress-energy tensor, ∇ µ T µν = 0 (c.f. Eq. (1.13)). However, these conservation equations don't provide a closed evolution of the system, unless one specifies some recipe to compute c 2 s and σ ν . The full description of the system has to be obtained by solving for the multipoles Ψ (τ, q, k) in the Boltzmann hierarchy Eq. (1.211)-Eq. (1.213), and then using the momentum integrals in Eq. (1.192)-Eq. (1.195) to get the elements of δT µ ν . As this is very time-consuming, several approximate descriptions for the neutrino dynamics have been proposed. These approaches typically describe neutrinos as a imperfect fluid, where the sound speed is set to c 2 2011) ). We will make use of such approximation to describe the warm decay products of dark matter in Chapter 3.

Cold Dark Matter

Dark matter is also assumed to have decoupled very early from the primordial bath, through some process such as the freeze-out mechanism that we sketched in Sect. 1.5.4. Thus, its collision term in Eq. (1.189) can be set to zero, C

(1) cdm = 0. CMB and LSS observations indicate that all or most of the dark matter needs to be cold, i.e. non-relativistic. Therefore, the description of CDM is easily achieved by taking the non-relativistic limit of Eq. (1.215)

θ cdm = -Hθ cdm + k 2 ψ. (1.218)
Let us express previous equations also in the synchronous gauge, since this will be relevant for our discussion about decaying dark matter in Chapter 3. This can be done straightforwardly with the help of the relations between the two gauges that we wrote in Eq. (1.176)-Eq. (1.177) and Eq. (1.178)-Eq. (1.180). Applying these transformations to the continuity equation, we get (δ new cdm ) = (δ syn cdm -3Hα) = -θ new cdm + 3φ = -θ syn cdm -k 2 α + 3η -3(Hα) . We see that the terms -3(Hα) cancel out and that -k 2 α + 3η = -h 2 . We do the same game with the Euler equation:

(θ new cdm ) = (θ syn cdm + k 2 α) = -Hθ new cdm + k 2 ψ = -Hθ syn cdm -Hk 2 α + Hk 2 α + k 2 α .
In this case, both terms Hk 2 α and k 2 α cancel out. In summary, the CDM equations in the synchronous gauge read

δ cdm = - h 2 , ( 1.219
)

θ cdm = -Hθ cdm . (1.220)
As we already mentioned at the end of Sect. 1.6.1, to fully fix the synchronous gauge, one typically imposes the initial condition θ ini cdm = 0. Since θ cdm is just proportional to θ cdm according to Eq. (1.220), we see that in this case the velocity divergence of the CDM remains zero at all times, θ cdm = 0. This is the reason why in the synchronous gauge comoving with the CDM, the only dynamical variable to describe CDM at the perturbed level is δ cdm .

Baryons

In the context of CMB physics, the only fully non-relativistic particles of the "visible" sector with a non-negligible cosmic abundance are hydrogen and helium (both in neutral and ionized forms) as well as electrons. At all epochs of interest, electrons and protons remain tightly coupled thanks to frequent Coulomb scatterings. In addition, neutral atoms keep a high scattering rate with protons, even after recombination (when the free proton ratio as small as X p ∼ 10 -4 ) (HANNESTAD 2001). Hence, baryons and electrons can be generally described by a single fluid with δ e = δ b and θ e = θ b . Since the energy density is dominated by baryons (m p m e ), cosmologists often refer to the baryon-electron fluid simply as baryons. Electrons and photons remain coupled until recombination due to Compton interactions, as we already mentioned. The Compton scattering between protons and photons can be neglected, because the Thomson cross-section scales as σ T ∝ 1/m 2 , and thus σ p T /σ e T ∼ 10 -6 .

The dynamical equations equations of such baryon fluid can be derived by considering a separate Boltzmann equation like Eq. (1.189) for each species (electrons, protons and neutral atoms), adding them together, and integrating over the phase space. Given that baryons and electrons are non-relativistic, the non-collisional part of such equations looks like the continuity and Euler equations for CDM (c.f. Eq. (1.217)-Eq. (1.218)), except for the inclusion of a pressure term, k 2 c 2 s δ b (that we will motivate later). Regarding the collision term, the contributions from electronbaryon interactions cancel out (as a consequence of particle number and momentum conservation), and we are only left with the Compton electron-photon interaction term. This can only enter through the Euler equation, given that Compton scattering conserves the number of particles.

The collision term in the θ b equation can easily be calculated using the covariant conservation of energy flux in the photon-baryon fluid, ∇ µ (δT µ i | b + δT µ i | γ ) = 0. This means that the collision terms in the equations for 22 . However, we are writing equations for θ i (with i = γ, b), so the associated collision term in each case is actually C i ≡ C i (ρ i + Pi ) . In Eq. (1.207) we already showed the collision term appearing in the θ γ equation:

[(ρ γ + ρ γ )θ γ ] and [(ρ b + ρ b )θ b ] have to cancel out, C γ = -C b
C γ = κ γ (θ γ -θ b ).
Therefore, for baryons we have

C b = C b (ρ b + Pb ) = - Cγ (ρ b + Pb ) = - (ργ + Pγ ) (ρ b + Pb ) C γ .
In summary, the equations for the baryon fluid read

δ b = -θ b + 3φ , ( 1.221
)

θ b = -Hθ b + k 2 c 2 s δ b + k 2 ψ + R -1 κ γ (θ b -θ γ ), (1.222)
where we defined R ≡ 3ρ b /4ρ γ . We see that for ρb → ∞, the collision term vanishes, reflecting the fact that photons cannot change the electron velocity if the average mass of the baryon-electron fluid is infinite. It is possible to define the drag depth for baryons, similar to the optical depth of photons, but taking into account the extra factor R -1

κ dr (τ ) = τ 0 τ dτ R -1 an e σ T . (1.223)
This also goes from infinity in the early tightly-coupled regime to zero after decoupling. The time of baryon drag is defined by the condition κ dr (τ dr ) = 1, giving the moment at which baryons stop dragging the photons towards gravitational wells. Because recombination happened already during the matter era, R = 3ρ b /4ρ γ < 1, so baryon drag took place slightly after photon decoupling, z drag < z dec .

What about the pressure term k 2 c 2 s δ b ? The motivation for including this term is that it can become important for scales smaller than the so called Jeans length. To see this more clearly, let's suppose we have some non-relativistic decoupled species I which dominates the energy content of the universe. It can thus be described by equations like Eq. (1.221)-Eq. (1.222), but without the collision term. As we will see in the next subsection, it is often a good approximation to neglect the time variation of gravitational potentials, φ 0. Furthermore, on sub-Hubble scales we can use the Poisson equation (c.f. Eq. (1.229)) to infer that k 2 ψ 4πGa 2 ρtot δ I = 3 2 H 2 δ I . Then, combining the time derivative of the continuity equation with the Euler equation, we arrive at a closed equation for the evolution of δ i :

δ I + Hδ I + (k 2 -k 2 J )c 2 s δ I = 0, (1.224
) 22 The fact that Cγ = -C b is by itself a consequence of momentum conservation. Indeed, since these collision terms appear in the equation for (ρ + P )θ (c.f. Eq. (1.193)), they are given by

Cγ = dΠ2p2Cγ = dΠ1dΠ2dΠ3dΠ4|M| 2 δ 4 (P1 + P2 -P3 -P4)(2π) 4 [fe(p3)fγ(p4) -fe(p1)fγ(p2)] p1, C b = dΠ1p1C b = dΠ1dΠ2dΠ3dΠ4|M| 2 δ 4 (P1 + P2 -P3 -P4)(2π) 4 [fe(p3)fγ(p4) -fe(p1)fγ(p2)] p2,
where dΠi ≡ d 3 p i (2π) 3 2E i . The total collision term in the photon-baryon fluid is thus C γb = Cγ + C b . Applying momentum conservation in the integrand of C γb , p1 + p2 = p3 + p4, and relabelling p1 ↔ p3 and p2 ↔ p4, one quickly notices that

C γb = -C γb → C γb = 0.
where k J ≡ Neglecting the time variation of µ, and assuming ρ b -3H ρb , the baryon sound speed can then be computed as:

c 2 s = P b ρ b = w b 1 + w b w b ρb ρ b = Tb µ 1 - 1 3
dln Tb dlna , (1.226) where the evolution of Tb is given by Eq. (1.130).

Perturbed Einstein equations

To close the system of Boltzmann equations presented before, we still need to determine the gravitational potentials ψ and φ. This can be achieved thanks to Einstein equations, which relate ψ and φ with the total perturbed stress-energy tensor. In the scalar sector, there are four independent Einstein equations, two of which are redundant. This is just a consequence of Bianchi identities: Einstein equations can be combined to arrive at the continuity and Euler equation for the total stress-energy tensor. We have given a derivation of the four linearly perturbed Einstein equation in App. A.7. The traceless longitudinal part of the (ij) components gives

k 2 (φ -ψ) = 12πGa 2 I (ρ I + PI )σ I , ( 1.227)
where I runs over all cosmic species. This equation tells us that in a universe full of perfect fluids (with σ I = 0), the two metric perturbations are the same. In practice, neutrinos and photons can have non-negligible anisotropic stress. For photons, we already saw that σ γ remains very suppressed until recombination, due to the tight coupling with baryons. After recombination, σ γ can be sizeable, but ργ is already very small because photons decoupled during the matter era. This makes the overall contribution of photons to the r.h.s of Eq. (1.227) still very minor. On the contrary, neutrinos are free-streaming from the first instants of the universe (i.e. for T 1 MeV), and behave relativistically until late times (after recombination). Hence, they are the only components capable of producing a significant departure of φ -ψ from zero. This happens mainly during the radiation era, when the contribution from both ρν and σ ν is important. Afterwards, neutrinos transition to the non-relativistic regime, and the metric perturbations become equal, φ = ψ.

We still need one more Einstein equation to close the system. We can use the (00) component, which gives wherek 2 a 2 is the Fourier transform of the Laplacian in physical coordinates. We note that it is only the total perturbed density δρ tot and not ρtot what sources the gravitational potential. Indeed, in a FLRW universe without over-densities or under-densities, there is no possibility for gravitational forces to appear.

k 2 φ + 3H(φ + Hψ) = -4πGa
Under some simplifying assumptions, it is possible to derive a closed equation for the evolution of the gravitational potential φ, from which we can extract some qualitative aspects of its evolution. This is important in order to understand some of the key features in the CMB spectra that we will discuss in Sect. 1.6.4. First, let us neglect the effects of the anisotropic stress from neutrinos, such that φ = ψ. In this case, the trace of the (ij) Einstein equation (written in Eq. (A.89)), simplifies to

φ + 3Hφ + (2H + H 2 )φ = 4πGa 2 I δP I .
(1.230)

As we will explain in Sect. 1.6.3, if the universe is dominated by a single component with a constant EoS w, and in addition we assume its fluctuations evolve adiabatically, then we have δP tot wδρ tot . This allows to relate Eq. (1.230) with Eq. (1.228). The resulting equation can be simplified by noting that 2H + H 2 (1 + 3w) = 0, as a consequence of Friedmann equations. We end up with the following closed equation for the evolution of φ φ + 3Hφ (1 + w) + wk 2 φ = 0.

(1.231)

One could now move to solve Eq. (1.231) for matter, radiation and Λ eras, by looking at the corresponding H and w in each case. On super-Hubble scales, the growing mode solution is always φ = const. The fact that φ is frozen reflects the natural expectation that no causal physics should influence perturbations outside the horizon (although this result is in reality only valid in the Newtonian gauge). However, φ can still change of value on super-Hubble scales during the transition between eras. For instance, it is a famous result of linear theory that the super-Hubble values of φ during matter dominated and radiation dominated eras are related by: 23

φ MD = 9 10 φ RD . (1.233)
It is interesting to find how density perturbations behave on super-Hubble scales. To do that, let us express the generalized Poisson equation Eq. (1.228) in the following manner

δ tot = δρ tot ρtot = - 2 3 k 2 H 2 φ - 2 H φ -2φ.
(1.234) 23 This can be shown by using the expression for the comoving curvature perturbation R in Eq. (1.254). As we discuss later, this quantity is conserved on super-Hubble scales. In this case, we can use that φ 0 and H 2 = 8πG 3 a 2 ρtot, in order to rewrite R as

R = 5 + 3w 3 + 3w φ. (1.232)
Evaluating R in the matter (w = 0) and radiation eras (w = 1/3), we find R = 5 3 φ MD = 3 2 φ RD and hence φ MD = 9 10 φ RD . (blue) and k = 10 -3 Mpc -1 (green), which crossed the horizon before, during and after matterradiation equality, respectively.

On super-Hubble scales, φ 0 and k/H 1, so we find that the total overdensity is also frozen and its value is given by δ tot = -2φ.

(1.235)

Regarding the sub-Hubble limit of Eq. (1.231), the behavior is different depending on which epoch we consider. During radiation domination, the metric potential oscillates with an amplitude that decreases as a -2 , while it stays constant during matter domination. In summary, the gravitational potential only evolves when the EoS of the universe changes (on all scales), and during the radiation era on sub-Hubble scales. All those aspects are illustrated in Fig. 1.14, where we show the numerical resolution of φ and ψ for different wavenumbers

Initial conditions

Before moving with our discussion about linear perturbations, it is essential to introduce two important concepts related with the study of initial conditions: adiabatic/isocurvature modes and the primordial power spectrum.

Adiabatic and isocurvature modes

Any mode of interest (those that can be observed on the CMB and linear matter spectra) was outside the Hubble radius R H if we look far enough into the past (i.e. for z 10 5 ). Inflation establishes the initial conditions for these super-Hubble modes. Let us suppose that the universe was initially filled with N decoupled perfect fluids. This means that at the linearly perturbed level, each fluid is completely specified by δ I and θ I , so there are 2N first-order equations, requiring 2N independent initial conditions 24 . One can show that half of them seed decaying solutions and are thus irrelevant for most applications. Therefore, the task is to find which combinations of the remaining N initial conditions get excited. There is one combination that is particularly simple and well-motivated.

In scenarios of single-field inflation, perturbations are generated due to a common, local shift in time δτ ( x) of the background quantities:

ρ I (τ, x) = ρI (τ + δτ ( x)) ρI (τ ) + ρ I δτ ( x),
(1.236)

P I (τ, x) = PI (τ + δτ ( x)) PI (τ ) + P I δτ ( x), (1.237)
meaning that δρ = ρ I δτ and δP = P I δτ . Using the fact that δτ ( x) is the same for all species and Eq. (1.14), we can write

δτ = δρ I ρ I = δρ J ρ J -→ δ I 1 + w I = δ J 1 + w J , ( 1.238)
for all species I and J. This is a great simplification, implying that in the early universe there is a single inhomogeneous degree of freedom, from which the rest of initial conditions can be derived. For example, if the universe is initially filled with photons (w γ = 1/3), baryons and cold dark matter (w b w cdm 0), we have

δ cdm = δ b = 3 4 δ γ . ( 1
I ρ I = c 2 g,I , δS I = 0, (1.241)
i.e. the sound speed of each component is adiabatic and the entropy perturbations vanish. For this reason, such kind of initial condition is often referred as adiabatic or isentropic mode. If in addition the EoS is constant for every species,

w I = const., then c 2 s,I = c 2 g,I = w I .
• The total pressure perturbation is also described by a sound speed c 2 s which is a weighted sum of the sound speeds for each species

δP tot δρ tot = I c 2 g,I δρ I /δτ I δρ I /δτ = I c 2 g,I (ρ I + PI ) I (ρ I + PI ) ≡ c 2 s . (1.242)
From the previous equation we see that, even if c 2 g,I = w I for every component, in general we will have c 2 s = w tot in the presence of several components.

Remark that adiabatic modes don't necessarily have to remain adiabatic at later times.

In fact, without assuming Eq. (1.236)-Eq. (1.237), the relations Eq. (1.239), Eq. (1.241) and Eq. (1.242) are not even true at initial times. In general, the total pressure perturbation can be arranged as a sum over N independent functions

δP tot = c 2 s δρ tot + I =J Υ IJ δρ I ρI + PI - δρ J ρJ + PJ . (1.243)
The term in brackets corresponds to the entropy perturbation δS IJ of the fluid I with respect to a reference fluid J (typically the photons), while Υ IJ = ∂P tot /∂S IJ | ρtot . The initial entropy perturbations δS IJ are also commonly referred as isocurvature modes. To understand the origin for the name, let us rewrite the defining condition for adiabatic modes in terms of the number densities of each species, n

I . From n I (τ, x) = nI (τ + δτ ( x)) = nI (τ ) + n I δτ ( x), we observe δτ = δn I n I = δn J n J -→ δn I nI = δn J nJ -→ δ n I n J = 0. (1.244)
That is, for the adiabatic mode, the relative ratios in number densities are unperturbed. It is associated with a global perturbation in the matter content, which is translated into a fluctuation of the spatial curvature via Einstein's equations (see Eq. (1.252)). The isocurvature modes δS IJ = δn I nI -δn J nJ are thus the complement of adiabatic modes, since they are characterized by variations in the particle number ratios, but with a vanishing curvature perturbation [START_REF] Langlois | Isocurvature cosmological perturbations and the CMB[END_REF]. In summary, the most general initial condition can be expanded in a basis formed by 1 adiabatic mode, N -1 isocurvature modes, and N decaying modes which are unobservable. From the previous discussion, it is clear that non-adiabatic modes are generated when there are more than one inhomogeneous degrees of freedom in the early universe (for example, this could naturally arise in scenarios of multi-field inflation). The presence of isocurvature modes is tightly constrained by Planck18 data (N. LEE et al. 2021), so in the following we will restrict to adiabatic initial conditions.

Primordial power spectrum

The theory of cosmological perturbations is stochastic by nature. Its main goal is to determine how the probability distribution of perturbed quantities evolves in time. The simplest hypothesis that is agreement with observations is that primordial perturbations follow gaussian statistics. As long as perturbations are linear, it is guaranteed that the gaussian shape of the distributions is preserved, so that all observable properties can be encoded in two-point correlation functions. For some arbitrary perturbed quantity A(τ, x), the two-point correlation function in real space is given by The last equality in Eq. (1.245) follows as a consequence of statistical isotropy and homogeneity of the perturbed FLRW universe. The two-point correlation function is often written in Fourier space. Following our Fourier convention in Eq. (1.137) and using that A * (τ, k) = A(τ, -k) for a real quantity,

A(τ, x), A(τ, x ) = ξ A (τ, x, x ) = ξ A (τ, r), ( 1
A(τ, k), A * (τ, k ) = d 3 xd 3 x ξ A (τ, | x -x|)e -i k• x+i k • x . (1.246)
Now, changing variables from x to r = x -x, and using the identity

d 3 xe i( k -k)• x = (2π) 3 δ 3 ( k -k), we get A(τ, k), A * (τ, k ) = δ 3 ( k -k) d 3 rξ A (τ, r)e i k • r = δ 3 ( k -k)4π ∞ 0 drr 2 ξ A (τ, r) sin (kr) kr ,
where we have absorbed the factor (2π) 3 inside ξ A . If we define the power spectrum of the quantity A as

P A (τ, k) ≡ 4π ∞ 0 drr 2 ξ A (τ, r) sin (kr) kr , ( 1.247)
we thus arrive at the very important relation

A(τ, k), A * (τ, k ) = P A (τ, k)δ 3 ( k -k). (1.248)
The Dirac delta indicates that different wavenumbers are uncorrelated, and arises as a consequence of statistical homogeneity. On the other hand, the fact that P A only depends on k = | k| is a consequence of statistical isotropy. It is common to introduce the dimensionless power spectrum, defined as

P A (k, τ ) ≡ k 3 2π 2 P A (τ, k). (1.249)
The motivation for the definition of P A is that it frequently appears convoluted with some window function W (k) when computing the average of a certain quantity in real space

d 3 k (2π) 3 P A (k)W (k) = 4π dkk 2 (2π) 3 P A (k)W (k) = dlnkP A (k)W (k). (1.250)
The notion of "scale-invariant power spectrum" refers to P A being independent of k, i.e.

P A ∝ k -3 .
It is useful to present the primordial power spectrum in terms of some variable that is conserved on super-Hubble scales. Let us consider the spatial part of the metric in some arbitrary gauge,

g ij = a 2 (1 + 2C)δ ij + 2 ∂ i ∂ j - δ ij 3 ∇ 2 E . One can show that the three-dimensional Ricci scalar associated with g ij is given by a 2 R (3) = 4∇ 2 -C + 1 3 ∇ 2 E (BAUMANN 2021).
The term is parenthesis is the so called comoving curvature perturbation, and it is defined in the comoving gauge (where B = 0 and v tot = 0):

R = -C + 1 3 ∇ 2 E B=vtot=0 . (1.251)
It is more convenient to have a gauge-invariant expression for R. Since B and v tot vanish by definition in the comoving gauge, it is always possible to add linear combination of these to R so as to make R gauge invariant. By looking at the gauge transformations Eq. (1.157)-Eq. (1.160) and Eq. (1.164), one quickly realizes that the following combination is gauge-invariant 25

R ≡ -C + 1 3 ∇ 2 E -H(B + v tot ).
(1.253) 25 There is an alternative way to make -C + 1 3 ∇ 2 E gauge-invariant, which results in a similar quantity We can now evaluate R in any gauge, such as the Newtonian one. In this gauge, B = E = 0 and C = -φ. In addition, we can express v tot = -θ tot /k 2 in terms of the metric potentials using the 0i Einstein equation (see Eq. (A.86)). This yields,

ζ ≡ -C + 1 3 ∇ 2 E + H δρtot ρ tot . ( 1
R = φ + H(φ + Hψ) 4πGa 2 (ρ tot + Ptot ) . (1.254)
In App. A.8 we show that R is conserved on super-Hubble scales and for adiabatic initial conditions. This is a big advantage, since it allows to relate predictions made at horizon exit (given by some inflationary theory) to observables at horizon re-entry. The power spectrum in Eq. (1.248) can be decomposed into the primordial spectrum P R (k) and the square of the so called transfer function A(τ, k):

A(τ, k), A * (τ, k ) = δ 3 ( k -k)A(τ, k) 2 P R (k). (1.255)
The transfer function accounts for the linear evolution of A(τ, k), irrespective of its initial conditions. It is given by

A(τ, k) = A(τ, k)/R( k),
where it is common to normalize initially R( k) = 1. Note that we have written the transfer function with the same name as the original variable. The only difference is that the transfer function doesn't depend on k, only on the modulus k = | k|. This just reflects the idea that isotropy is respected by the linear evolution of perturbations in a FLRW background.

The dimensionless primordial spectrum P R is usually parameterized as a powerlaw near some reference scale (often taken to be k * = 0.05 Mpc -1 ):

P R = A s k k * ns-1 . (1.256)
Here A s is the scalar amplitude and n s is the scalar spectral index. Inflation predicts small deviations from a scale invariant primordial spectrum (n s = 1) as a consequence of a nearly constant energy density during the inflationary period. In fact, one can relate the spectral index with the slow-roll parameters that we introduced in Sect. 1.4 as (BAUMANN 2021):

n s -1 = -2ε -η. (1.257)
The last measurements from Planck18 indicate strong evidence for a nearly scaleinvariant spectrum with n s = 0.9665±0.0038. This constitutes one of the great triumphs of the inflationary paradigm. In Chapter 2, we will comment on the consequences of the Hubble tension for this measurement.

Acoustic oscillations

The goal of Boltzmann codes such as CLASS is to solve numerically the set of perturbed Einstein and Boltzmann equations that we presented in Sect. 1.6.2. However, one can adopt several approximations in order to gain a semi-analytic understanding of the evolution of photon and baryon perturbations. This is essential in order to understand the features in the CMB anisotropy spectra that we will describe in Sect. 1.6.5.

As we discussed earlier, photons and baryons form a tightly coupled fluid until recombination. Photons exert a strong pressure on this fluid that counteracts gravitational forces. As a consequence, inhomogeneities in the photon-baryon fluid don't experience a continuous growth, but rather propagate in the form of acoustic waves. The sound speed at which waves propagates can be computed as

c 2 s = δP γ + δP b δρ γ + δρ b 3δρ γ δρ γ + δρ b 1 3(1 + R) , ( 1.258)
where we have used that the baryon pressure is negligible compared to that of the photons, δP b δP γ = 3δρ γ , and that tight-coupling imposes δ b = 3δ γ /4. Since R = 3ρ b /(4ρ γ ) ∝ a, we see that the sound speed is close to the speed of light during the radiation era, c s 1/ √ 3, and then slowly drops down to zero during the matter era. With this, we can define a quantity which is extremely relevant in the context of CMB physics (and for the Hubble tension as we shall see): the sound horizon. At a fixed time t, this corresponds to the comoving distance travelled by a wavefront since some initial time t i , and it is given by

r s (τ ) = τ τ ini c s (τ )dτ = t t ini c s (t)dt a(t) = z ini z c s (z )dz H(z ) . (1.259)
This quantity is independent on the choice of τ ini as long as τ ini τ . During the radiation era, the sound speed is constant and thus the sound horizon is of the order of the particle horizon, r s c s τ χ p / √ 3.

We can get more insight about the details of these acoustic oscillations by building an equation that dictates the evolution of Θ 0 = δ γ /4 at leading order in the tightcoupling limit. When the optical depth κ γ is very large, we have already seen that the photon hierarchy Eq. (1.206)-Eq. (1.208) implies σ γ → 0 and θ γ → θ b . Combining Eq. (1.207) for θ γ with Eq. (1.222) for θ b , we can get rid of the interaction term (notice the baryon pressure term is totally negligible in this context). Then, we can combine with Eq. (1.206) for δ γ in order to eliminate θ γ . Using RH = R and 3(1 + R) = c -2 s in the remaining equation, we end up with

Θ 0 + R 1 + R Θ 0 + k 2 c 2 s Θ 0 = - k 2 3 ψ + R 1 + R φ + φ . (1.260)
This corresponds to the equation of an harmonic oscillator with a time-dependent frequency k 2 c 2 s , a baryon-induced friction term R 1+R Θ and a gravitational forcing function on the r.h.s. If we were to consider a constant R and ignore the gravitational terms, Eq. (1.260) becomes the equation of a perfect harmonic oscillator, whose solution is

Θ 0 (τ, k) = Θ ini cos (kc s τ + ϕ), ( 1.261)
where Θ ini and ϕ stand for the two integration constants. On the super-Hubble limit, kτ 1, the value of Θ 0 should be frozen (for instance, Eq. (1.235) tells us that during radiation era, δ tot δ γ = -2φ = const.). This fixes the phase to ϕ = 0. On the sub-Hubble limit, we expect oscillations to appear. However, from Eq. (1.261) we see that the threshold between constant and oscillatory behavior is set by kc s τ and not by kτ . In particular, the oscillatory condition kc s τ 1 is equivalent to say that a comoving wavelength λ = 2π/k has entered the sound horizon, r s c s τ λ. Modes with smaller λ crossed the sound horizon earlier and therefore experienced more oscillations. Since the phase kc s τ of the cosine roughly stands for 2πr s /λ, the number of oscillations is given by the ratio r s /λ.

There is another important effect that cannot be described with Eq. (1.260). This equation assumes that the scattering rate is infinite, i.e. that the distance between scatters is virtually zero. Close to recombination, this approximation breaks down, and inhomogeneities are washed out by random scatterings on small scales, below the diffusion length (SILK 1968). One can get an estimate of this length by treating photon diffusion as a random walk. The mean-free path of photons is given by r 2 mfp = 1/Γ γ , where Γ γ = an e σ T is the comoving photon scattering rate. Then, at a given time τ , the comoving distance travelled by a photon since some initial time τ i is

r 2 d (τ ) τ τ ini dτ Γ γ r 2 mfp = τ τ ini dτ an e σ T . (1.262)
This equation is essentially telling us that the diffusion length is of the order r d ∼ r mfp √ N , where N ∼ τ Γ γ is the total number of scatters. This distance is close to zero in the tight-coupling limit, and abruptly increases when approaching the time of photon decoupling. A more accurate estimate for r d can be obtained by considering a nonzero quadrupole Θ 2 in the Boltzmann hierarchy Eq. (1.206)-Eq. (1.208). The idea is to combine this set of equations with Eq. (1.222) for θ b , restrict to small scales (where the metric potentials can be neglected), and try solutions of the form e i wdτ . The imaginary part of w in the resulting dispersion relation then accounts for damping. The result is (see [START_REF] Dodelson | Modern Cosmology[END_REF] for details):

r 2 d = (2π) 2 τ τ ini dτ an e σ T R 2 + 8 9 (1 + R) 6(1 + R) 2 . ( 1.263)
Had we included polarization terms in the photon hierarchy, the numerical factor 8/9 would be replaced by 16/15 [START_REF] Hu | The Damping tail of CMB anisotropies[END_REF]).

We proceed now to discuss the behavior of Θ 0 in the different parts of the (k, τ ) space. In Fig. 1.15, we have labelled four different regions, in order to highlight for which times and scales the different effects of baryon damping, gravitational driving and diffusion damping can play a role. Fig. 1.15 also depicts three different lines corresponding to Hubble crossing (k = H), sound horizon crossing (k = 1/r s ) and diffusion damping crossing (k = 1/r d ). Let us discuss each of the regions separately:

• On super-Hubble scales, we've already seen that the perturbation Θ 0 is frozen.

However, its constant value is different in the matter and radiation domination epochs. Using Eq. (1.235) and considering adiabatic initial conditions, one quickly realizes that

Θ 0 = - ψ 2 = - φ 2 (Radiation era), (1.264)
Θ 0 = - 2ψ 3 = - 2φ 3 (Matter era). (1.265)
• The region 1 corresponds to modes that are crossing the sound horizon before photon decoupling. In this region, gravitational source terms are important. A key consequence is that the zero point of oscillations is shifted. An approximation to the new zero-point of oscillations is achieved by setting to zero all time derivatives in Eq. (1.260)

Θ eq 0 = - 1 3c 2 s ψ = -(1 + R)ψ.
(1.266) This only lasts a small time, since ψ rapidly decays on sub-sound horizon scales during the radiation era. Remark that the shifting effect crucially depends on R and thus on the baryon fraction.

• The region 2 refers to scales that are inside the sound horizon during the radiation era. In this region, the metric potentials have decayed, the friction terms are negligible since R 1, and c s 1/ √ 3. Thus, this is the case of the perfect harmonic oscillator solution for Θ 0 that we showed in Eq. (1.261).

• The region 3 is located between matter-radiation equality and decoupling, on scales well below r s . Here the metric potentials have decayed, but the effects of baryons are important (R cannot be neglected). Hence the oscillator equation has a friction term (increasing with time) and a time-dependent pressure term (decreasing with time). This results in damped oscillations. The damping effect is controlled by the duration of the stage between equality and decoupling.

• The region 4 corresponds to scales inside the diffusion length r d , that we wrote in Eq. (1.262). This scale increases rapidly close to decoupling, encompassing a big portion of the sub-sound horizon scales. Fluctuations become strongly damped, as diffusion tends to erase any small-scale photon perturbation.

In addition, Fig. 1.15 shows in which parts of the (k, τ ) space the gravitational potentials decay. This will be important when discussing the integrated Sachs-Wolfe effect in Sect. 1.6.5.

To get a confirmation of the qualitative description we just gave, we show in Fig. 1.16 a numerical resolution of the photon transfer function Θ 0 (τ, k), evaluated at the times of radiation-matter equality and photon decoupling. We also show the zero-point of oscillations, Θ eq 0 = -(1 + R)ψ, at the decoupling time. We observe that Θ 0 is frozen on super-Hubble scales, and starts oscillating when kr s (τ ) > 1. The negative values of Θ 0 and -ψ in the long-wavelength limit come from the fact that transfer functions are all normalized with respect to the curvature perturbation, R = 1. Notice Moving to the sub-sound-horizon limit in Fig. 1.16, we clearly see differences between the shape of the oscillations at τ eq and τ rec . At equality, Θ 0 exhibits oscillations with constant amplitude, roughly symmetric around Θ eq 0 = -ψ 0. At decoupling, Θ 0 is affected by the three important effects we discussed previously: a suppression in the global amplitude due to baryon friction, a shift in the zero point of oscillations by an amount -Rψ, and the exponential damping at k larger than the diffusion wavenumber k d = 2π/r d . Importantly, the shift of the zero-point leads to an enhancement of the absolute value of the odd extrema in Θ 0 with respect to the even extrema.

The CMB power spectra

We would like to understand the pattern of anisotropies appearing the famous temperature map of the CMB (shown in Fig. 1.4). Note that although temperature fluctuations depend on space and time, Θ(τ, x, n), we only measure them on Earth (that for simplicity we can take at the origin x = 0) and at present time (τ = τ 0 ). All the information that we can extract comes from the variations in temperature for each direction of the incoming photons, n. Then, one can expand the temperature map in spherical harmonics

δT (n) T = Θ(τ 0 , 0, -n) = ∞ =0 m= m=- a m Y m (n), (1.267)
(since when observing at n, we see photons traveling in the opposite direction). In an all sky experiment, an observer can collect the coefficients a m 's, given by the inverse of the previous relation

a m = (-1) dΩ nY * m (n)Θ(τ 0 , 0, n), (1.268)
where we did a change of variables n → -n and used the properties dΩ -n = dΩ n as well as Y m (-n) = (-1) Y m (n). We can relate the coefficients a m 's with theoretical quantities such as the photon multipoles Θ (τ, k). To do so, we can expand Θ(τ 0 , 0, n) in Fourier space and in Legendre multipoles, and then make use of the following identity

P ( k • n) = 4π 2 + 1 m =+ m =- Y * m ( k)Y m (n). (1.269)
Proceeding in this way,

a m = (-1) (2π) 3 dΩ nY * m (n) d 3 k ∞ =0 (-i) (2 + 1)P ( k • n)Θ (τ 0 , k) = (-1) 2π 2 d 3 k m (-i) dΩ nY * m (n)Y m (n) δ mm δ Y * m ( k)Θ (τ 0 , k) = i 2π 2 d 3 k Y * m ( k)Θ (τ 0 , k), (1.270)
where to reach the last equality we used the orthogonality of the spherical harmonics. We see that the a m 's and Θ (τ, k) linearly related. Hence, for gaussian linear perturbations, the statistics of the a m 's is fully encoded in the two-point correlation function a * m a m . Using again the orthogonality of the spherical harmonics and

Θ ( k)Θ ( k ) = Θ (k)Θ (k) 2π 2 k 3 P R (k)δ 3 ( k -k ), (1.271)
it is straightforward to get the two-point correlation function for the a m 's

a * m a m = δ mm δ dlnk 2π 2 Θ 2 (τ 0 , k)P R (k) ≡ δ mm δ C TT . (1.272)
The quantity C TT is referred as the power spectrum of temperature anisotropies (in harmonic space). The fact that a * m a m is proportional to δ mm δ indicates that different multipoles are uncorrelated, while the fact that C TT is independent of m is a consequence of statistical isotropy.

An important caveat is that we can never really observe the true harmonic spectrum. This is because C TT is obtained by carrying out an average of the a m 's over many realizations of the theory, i.e. over many universes obeying the same cosmological model. However, we only observe the CMB sky of our universe. This implies there will necessarily be some scattering around C TT . We can reduce this scattering by taking advantage of the fact that, for a given -mode, the distribution of the observed a * m a m is independent of m. Thus, for an ideal full sky observation, the best estimator of the true C TT (= C , to shorten notation) is the average of the coefficients a * m a m over all m-modes

Ĉ ≡ 1 2 + 1 m= m=- a * m a m . (1.273)
We notice that the mean of Ĉ is given by

Ĉ = 1 2 + 1 m= m=- a * m a m C = C , (1.274)
which means that it is an unbiased estimator of the true C . However, there is a non-zero variance associated to Ĉ :

σ 2 C = ( Ĉ -C )( Ĉ -C ) C 2 = Ĉ Ĉ -2C Ĉ + C 2 C 2 = Ĉ Ĉ C 2 -1, (1.275)
where

Ĉ Ĉ = (2 + 1) -2 m,m a * m a m a * m a m .
To compute this quantity, we can make use of the Wick's probability theorem, which says that for gaussian distributions, any even N -point correlation function is a sum over all possible pairings of the products of 2-point functions. Hence

a * m a m a * m a m = a * m a m a * m a m + a * m a * m a m a m + a * m a m a m a * m .
Now, from the requirement that Θ(n) has to be real, one infers the property a * m = (-1) m a -m , and thus

a m a m = a * m a * m = δ m-m (-1) m C .
In short, we can write the variance σ 2 C as

σ 2 C = 1 (2 + 1) 2 C 2 m,m (C 2 + δ 2 m-m C 2 + δ 2 mm C 2 ) -1 = 2 2 + 1 . (1.276)
We see that σ 2 C decreases for larger , which is to be expected, because at high-there are more m-modes available to sample the distribution. This fundamental uncertainty, which can be seen as theoretical error, is known as cosmic variance.

Let us discuss how we compute in practice the temperature power spectrum that we wrote before,

C TT = dlnk 2π 2 Θ 2 (τ 0 , k)P R (k).
(1.277)

In the previous expression, we didn't specify the integration limits of the k-integral.

For k min , a reasonable choice is the inverse of the present Hubble radius,

k min ∼ H 0 ∼ τ -1 0 , since R H = H -1 0
gives roughly the largest scale we can observe. What about k max ? To find the best choice, we have to remember that the approximate solution for Θ (τ, k) (i.e. when neglecting the gravitational source terms) is given in terms of the spherical Bessel function as Θ (τ, k) ∝ j (kτ ). The functions j (x) peak near x = + 1 2 , and present damped oscillations for x > . Therefore, at present time τ 0 and as a function of , Θ (τ 0 , k) peaks around ∼ kτ 0 , while multipoles with kτ 0 are very suppressed. This can be understood from a geometrical point of view. The multipoles index is given by = π/θ, where θ is the angle subtending a comoving scale λ/2 = θD A (τ rec ) on the last scattering surface, with D A (τ rec ) the comoving angular distance at decoupling26 (see Fig. 1.17). This implies the following relation between and k:

λ 2 = π k = θD A (τ rec ) -→ k = D A (τ dec ) . (1.278)
In a flat universe, the comoving angular distance at decoupling is simply given by where (τ 0 -τ rec ) τ 0 . Hence, if the CMB experiment is observing multipoles up to max , the maximum wavenumber should be chosen of the order k max ∼ max /τ 0 . This guarantees that the multipoles of the photon transfer function, Θ (η 0 , k), are very suppressed for > max , so it is safe to compute them just until max . Remark that the one-to-one relation between and k in Eq. (1.279) is only valid for small angles, or equivalently, for large values of .

D A (τ rec ) = (τ 0 -τ dec ), meaning k = (τ 0 -τ rec ) , ( 1.279)
A brute-force approach for computing C TT would consist in truncating the photon Boltzmann hierarchy Eq. (1.206)-Eq. (1.208) at max , i.e. to solve as many equations as observed multipoles. The problem is that this task is extremely time-consuming, since for a Planck-like experiment the maximum multipole is of the order max ∼ O( 103 ). In the next subsection, we introduce the alternative line-of-sight formalism, which allows to get Θ (η 0 , k) up to max without having to solve thousands of differential equations. In addition, it makes the underlying physics much more transparent.

The line-of-sight formalism

The main idea of the line-of-sight formalism, originally introduced by ZALDARRIAGA et al. (1995), is to integrate the Boltzmann equation over the photon trajectory and relate the temperature anisotropies observed today to that at the last scattering surface. In doing this, it is assumed that photons travel along a straight line. Strictly speaking, this is not true, since matter fluctuations produce deviations in the travel direction of photons. However, this effect can be neglected when restricting to first-order perturbations. The starting point is to re-express the photon Boltzmann equation Eq. (1.200) in the following manner (we will neglect the quadrupole Θ 2 for simplicity): .280) We integrate this equation between some initial time τ ini and today τ 0 , and use that κ γ (τ 0 ) = 0, to get

e -ikµτ +κγ d dτ Θe ikµτ -κγ = φ -ikµψ -κ γ [Θ 0 + µv b ] ≡ ST (k, µ, τ ). ( 1
Θ(τ 0 , k, µ) = τ 0 τ ini dτ ST (k, µ, τ )e ikµ(τ -τ 0 )-κγ (τ ) . (1.281)
We have also used the fact that τ ini is early enough so that e -κγ (τ ini ) 1. Now, we would like to remove the dependence of ST on µ, so that the only factor that depends on µ in the integrand is the exponential. This can be achieved by noting that µe ikµ(τ -τ 0 ) = 1 ik d dτ e ikµ(τ -τ 0 ) . After performing integration by parts for the two terms with µ in S T and using v b = θ b /ik, we get

Θ(τ 0 , k, µ) = τ 0 τ ini dτ e ikµ(τ -τ 0 ) e -κγ (φ -κ γ Θ 0 ) + d dτ e -κγ ψ - θ b κ γ k 2 . (1.282)
We have dropped all surface terms, including one that yields ψ(τ 0 , k). In real space, this corresponds to an isotropic term ψ(τ 0 , 0) that cannot be distinguished from a shift in the average photon temperature. This justifies that we discarded ψ(τ 0 , k) in the previous expression. At this point, it is useful to introduce the so-called visibility function

g(τ ) = -κ γ e -κγ . (1.283)
This gives the probability that the photons that we see today last scattered at the time τ (notice it is correctly normalized, τ 0 0 g(τ )dτ = 1). The visibility function is sharply peaked around the time of photon decoupling. Indeed, at early times (when κ γ 1), the exponential e -κγ is very small, while later (after τ dec ), the term κ γ = -an e σ T is also very tiny. Thus, the time of photon decoupling can be defined as the peak of g(τ ). The terms of Eq. (1.282) inside brackets can now be expressed in terms of this function. Then we can multiply each side by i P (µ)/2 and integrate over µ. Using the identity,

i +1 -1 dµ 2 P (µ)e ikµ(τ -τ 0 ) = j (k(τ 0 -τ )), (1.284)
we arrive at

Θ (τ 0 , k) = τ 0 τ ini dτ g(Θ 0 + ψ) + (gk -2 θ b ) + e -κγ (φ + ψ ) j (k(τ 0 -τ )). (1.285)
This is what we were looking for. This equation allows to obtain any Θ (τ 0 , k) in terms of only a few transfer functions: Θ 0 , θ b , ψ.... The truncation scheme should be designed to compute with accuracy only these quantities. To give an example, truncating the Boltzmann hierarchy in Eq. (1.206)-Eq. (1.208) at max,γ ∼ O( 10) is enough for computing Θ (τ 0 , k) (and hence C TT ) up to max ∼ O(10 3 ). Notice that, even if we have significantly reduced the number of differential equations to solve, it is max and not max,γ that should be used to compute the maximum wavenumber, k max ∼ max /τ 0 .

In order to gain further insight from Eq. (1.285), we will make the instantaneous decoupling approximation. That is, we consider that the transition from an opaque to a transparent universe was very abrupt. This means that we can approximate the visibility function as a Dirac delta, g(τ )

δ(τ -τ dec ), and the exponential as a step function, e -κγ Θ H (τ -τ dec ). By integrating the θ b term by parts, we get

Θ (τ 0 , k) [Θ 0 (τ dec , k) + ψ(τ dec , k)] j (k(τ 0 -τ dec )) ←-SW + k -2 θ b (τ dec , k)j (k(τ 0 -τ dec )) ←-Doppler + τ 0 τ dec dτ (φ + ψ )j (k(τ 0 -τ )) ←-ISW (1.286)
Here the prime in the Bessel function indicates derivative with respect to its argument. As we see, there are three different contributions to Θ (τ 0 , k). Let us describe each of them separately

• The first contribution is called the Sachs-Wolfe (SW) term, in honor of the first scientists that studied the large-scale anisotropies [START_REF] Sachs | Perturbations of a cosmological model and angular variations of the microwave background[END_REF]. It corresponds to the quantity Θ 0 + ψ evaluated at the last-scattering surface, conveying the general idea that the CMB map is "the picture of the universe when it was only 380 000 years old". The presence of ψ simply indicates that photons have to climb out of their potential wells to reach us today. Its interpretation becomes more clear when moving to real space. To do this, we can first go back to angle space using Eq. (1.204) and Eq. (1.284), and then integrate over

d 3 k (2π) 3 , to get: Θ SW (τ 0 , 0, n) Θ 0 (τ dec , x dec , n) + ψ(τ dec , x dec ), (1.287)
where x dec = (τ dec -τ 0 )n is the comoving coordinate of the photon arriving from a direction -n. For large (super-Hubble) scales and during the matter era, we know from Eq. (1.265) that there is a relation Θ 0 = -2ψ/3. Hence, the SW contribution to the CMB anisotropies smoothed over small scales is written as

Θ SW,smoothed (τ 0 , 0, n) 1 3 ψ(τ dec , x dec ) - 1 2 Θ 0 (τ dec , x dec , n). (1.288)
Surprisingly, hot regions in the observed CMB map corresponds to cold regions on the last scattering surface, and vice-versa. This just arises because the gravitational redshift effect of photons climbing our of the potential well wins against their intrinsic temperature monopole.

• The second contribution is the standard Doppler term. Photons are emitted from a tightly coupled baryon-electron fluid with a different peculiar velocity at different points on the last scattering surface. The projection along the line of sight induces a Doppler shift in the photon wavelength.

• The third contribution is called the Integrated Sachs-Wolfe (ISW) term, and appears in the presence of time variations of the metric fluctuations. Physically, the integral over ψ describes a cumulative temperature shift acquired by photons in the presence of a non-conservative gravitational potential. That is, for non-static potentials, it is not true that the integral along the line-of-sight of the gradient, τ 0 τ dec dτ n • ∇ψ, is simply equal to ψ(τ 0 ) -ψ(τ dec ). In a similar manner, the integral over φ accounts for the cumulative effect of local time dilation.

In this formalism, the temperature spectrum C TT is obtained by computing Θ with the line-of-sight integral of Eq. (1.285) and then taking its square and convoluting with the primordial spectrum as demanded by Eq. (1.277). Therefore, C TT receives contributions from the three terms we just discussed: the SW, Doppler and ISW contributions. These are shown in Fig. 1.18.

We can understand the qualitative shape of the different contributions to C TT by making two simplifying assumptions. First, we assume again that decoupling happened instantaneously. Second, we restrict to high-(i.e. small scales), for which the functions j (x) and j (x) are very peaked in x . This means that the k-integrals mainly receive contributions from k(τ 0 -τ dec ), which has a simple geometrical interpretation as we discussed around Eq. (1.279). Hence, in these two limits, the SW, Doppler and ISW contributions are roughly given by27

C SW ∼ |Θ 0 + ψ| 2 (τ,k) (τ dec , /(τ 0 -τ dec )) , (1.290) C Doppler ∼ |θ b | 2 (τ,k) (τ dec , /(τ 0 -τ dec )) , (1.291)
C ISW ∼ τ 0 τ dec dτ (τ 0 -τ ) |φ + ψ | 2 (τ,k) (τ, /(τ 0 -τ )) . (1.292)
Note that C TT is not exactly given by the sum of these three contributions because of the different correlations among terms. Let us start discussing the SW part. From Eq. (1.290), it is clear that the shape of the SW contribution should be qualitatively similar to the square of the transfer function Θ 0 + ψ at the time of decoupling, as can be confirmed by comparing with Fig. 1.16. At low-, corresponding to super-Hubble scales at decoupling, we see a plateau since perturbations are frozen. On larger , we see a series of acoustic peaks modulated by the effects we discussed in Sect. 1.6.4. Notoriously, we observe an enhancement of the odd peaks with respect to the even peaks (due to the shift in the zero point of oscillations), as well as a an exponential diffusion damping for multipoles above d ∼ k d (τ 0 -τ dec ). Moving to the Doppler term, we see that its contribution is negligible at low-, whereas at high-it exhibits oscillations with a phase shifted by π/2 with respect to those of the SW term. The former can be again explained by the fact that super-horizon perturbations are frozen. The latter can be understood by noting that in the tight-coupling limit, and neglecting variations of the metric potential, the baryon velocity is of the order θ b ∼ θ γ ∼ Θ 0 ∼ sin (kc s τ ) (see Eq. (1.206)). Finally, the ISW contribution is only important

Eq. (1.232) and Eq. (1.288), we see that the SW transfer function on large scales is simply Θ0 +ψ = ψ 3 = 1

5

(remember that we set R( k) = 1). Hence,

C SW,smoothed = As 2π 2 ∞ 0 dk k (Θ0 + ψ) 2 j 2 (k(τ0 -τ dec )) = As 2π 2 1 25 ∞ 0 dxj 2 (x) = As 100π 2 1 ( + 1) . (1.289)
This is the reason why it is common to plot D = ( + 1)C , in order to roughly get a plateau at small .

THE ERA OF PRECISION COSMOLOGY

when φ = 0 and ψ = 0. In Fig. 1.15, we had indicated in which regions of the k -τ plane the gravitational potentials decay:

• Since the moment of decoupling happened shortly after the beginning matter domination , the metric fluctuations (which have decayed inside the Hubble radius during the radiation era) didn't completely stabilize. This produces a residual variation in ψ and φ, which is known as the Early Integrated Sachs Wolfe (EISW) effect.

• At late times, when dark energy begins to dominate the energy budget of the universe, the metric fluctuations start decaying again. This produces the so called Late Integrated Sachs Wolfe (LISW) effect.

It can be shown that both the EISW and LISW decrease as a function of . Since the EISW effect is only relevant inside the sound horizon after decoupling, its contribution is peaked around ∼ 200 (as can be seen in Fig. 1.18), enhancing the first acoustic peak. The LISW effect happens at all scales, so it peaks at the smallest , slightly tilting the Sachs-Wolfe plateau.

Parameter dependence of C

We have now everything we need to understand the specific effects of the different cosmological parameters on the shape of C TT . For simplicity, we consider a spatially flat universe. We also ignore the effects of free-streamings neutrinos, which are discussed in great details in LESGOURGUES et al. ( 2018). Nevertheless, we will sketch the dominant effects of a non-zero neutrino mass when discussing decaying neutrinos in Chapter 4.

There are many parameter basis that can be chosen. For convenience, we will take the six free parameters to be

{A s , n s , ω b , ω cdm , Ω Λ , τ reio }, (1.293)
where we remind that ω i ≡ Ω i h 2 . Remark that the dimensionless Hubble parameter h is not included since it can be obtained from the condition ω m h -2 = 1 -Ω Λ . The radiation density parameter is not included either, as it can be obtained from the very precise measurement of the present CMB temperature (T 0 = 2.7255 ± 0.0006 K), and the theoretical prediction of the effective number of neutrino species, N eff 3.044.

It is important to remember that the scale factors at matter-radiation and matter-Λ equality are controlled by the ratios ω r /ω m and ((1

-Ω Λ )/Ω Λ ) 1/3
, respectively (c.f. Eq. (1.24)-Eq. (1.25)). Since we assume ω r to be fixed, in practice the scale factor at equality only depends on ω m . The degrees of freedom that sculpt the shape of the temperature spectrum are:

1. The position of the peaks. All the peaks of the CMB spectrum in multipole space, C TT , correspond to the harmonics of one single correlation length of the CMB spectrum in real space, C TT (θ). This characteristic length is precisely the sound horizon, r s (τ dec ), giving the comoving distance travelled by a wavefront by the time of the decoupling. In multipole space, the peak location is given by π/ peak = θ s = r s (z dec )/D A (z dec ). The sound horizon depends on the pre-decoupling expansion history and on the sound speed, and thus is affected by ω m and ω b . The comoving angular distance D A (z dec ) depends on the post-decoupling expansion history and thus it is affected by Ω Λ and ω m .

2. The ratio of odd/even peaks. As we have already discussed, the asymmetry between odd and even peaks arises from squaring acoustic oscillations that are shifted with respect to 0 by an amount -(1 + R)ψ. This effects depends on R and therefore on ω b .

3. The overall amplitude of the peaks.

Between the time of radiation-matter equality and decoupling, the amplitude of acoustic oscillations decreases due to baryon friction (see region 3 in Fig. 1.15). If radiation-matter equality happened later, there would be fewer time for oscillations to decrease, and thus all peaks in the spectrum would move upwards. The first peak would be additionally increased by a larger EISW, since metric fluctuations would be less stabilized at decoupling. Those effects are controlled by ω m .

The damping envelope. The diffusion processes induce an exponential cutoff

shaped like exp(-

( / d ) 2 ), being d = π/θ d with θ d = r d (z dec )/D A (z dec ).
The diffusion length at decoupling, r d (z dec ) depends on ω m (governing the conformal time at equality) and ω b (governing the evolution of the free electron fraction ). In addition, D A (z dec ) depends on Ω Λ and ω m , as we said earlier.

5. The overall amplitude of the spectrum. The global amplitude of the spectrum depends on A s , since this just enters as a multiplicative factor in the expression Eq. (1.277) for C TT .

6. The overall tilt of the spectrum. The global slope of the spectrum depends on the one of the primordial spectrum, P R (k), controlled by n s .

7. Tilting of the plateau. The plateau at small is tilted further as a result of the LISW. This comes from the fact that ψ and φ start decaying again after matter-Λ equality. Consequently, this effect is enhanced by a larger value of Ω Λ .

8. Effect of reionization. A small fraction of the CMB photons that we see today last-scattered not at the time of recombination, but at the time of reionization. These rescattered photons come in random directions that average to zero, so they tend to erase the CMB anisotropies. This produces a steplike suppression in the overall amplitude of the peaks starting at ∼ 40 (which corresponds to modes crossing the Hubble radius around z ∼ 10). This effect depends on the optical depth at reionization, τ reio .

We have assumed for simplicity that the redshift of recombination is fixed, but in reality it has a small dependence on the baryon density ω b and the helium mass fraction Y p . Notice also that τ reio and Ω Λ are poorly constrained by the temperature spectrum, since they are inferred from measurements of the lowest multipoles, dominated by cosmic variance. Fortunately, the combination with polarization anisotropy data and low-redshift probes breaks parameter degeneracies that allow to constrain τ reio and Ω Λ (respectively) with better precision.

A few words on CMB Polarization

At each point in space, photons are described by their temperature, but also by a set of Stokes parameters specifying their polarization state. The invariant amplitude for Thomson scattering depends on photon polarization, more precisely on the product |ˆ • ˆ | 2 , where ˆ and ˆ denote the polarizations of the incoming and outgoing photons. However, during the tight-coupling regime, isotropy implies that photons cannot acquire a net polarization in the scatterings with electrons, so they remain unpolarized on average. The situation changes close to the epoch of photon decoupling. Photons experienced their last interactions in regions with increasing anisotropies. The quadrupole Θ 2 (τ, x) associated to these anisotropies produces a net linear polarization of scattered photons. As a consequence, polarization patterns appear on the last scattering surface, which are strongly correlated with the temperature patterns.

The polarization map is described by a vector field on a sphere, as opposed to the temperature map, which is a scalar field. Therefore, it can be decomposed into two scalar maps, a gradient and a curl field, or an E-polarization and B-polarization component, by analogy with electromagnetism. It is possible to define harmonic power spectra for these polarization modes. It can be shown that scalar perturbations can produce both temperature and E-polarization anisotropies. If the primordial universe features only gaussian scalar fluctuations, all the information of the CMB maps is encapsulated in the temperature power spectrum, C TT , the E-type polarization power spectrum C EE , and also the cross-correlation between both, C TE . The latter really adds new information, since polarization anisotropies are only partially correlated with temperature anisotropies.

At this stage, it is convenient to say a few words about tensor modes. As we mentioned at the beginning of Sect. 1.6, tensor modes are associated to the traceless divergenceless spatial components of the metric and the stress-energy tensor, and their d.o.f. represent the polarization states of gravitational waves (GW). Tensor perturbations can interact with any species having non-negligible tensor components in their perturbed stress tensor δT µν (such as photons after recombination), and generate temperature and polarization anisotropies, including B-type polarization. But how are these tensor modes produced? Decoupled neutrinos and photons possess non-zero anisotropic stress that can seed tensor perturbations, but these are far too small to generate any detectable effect. The most famous generation mechanism is inflation. STAROBINSKY (1979) showed that, during inflation, quantum fluctuations of the metric can generate a significant amount of primordial GWs, which are described by a nearly scale-invariant power spectrum,

P h = A t k k nt
. The amplitude of this spectrum is directly related to the square of the Hubble parameter when observable wavelengths crossed the Hubble radius during inflation. Hence, these primordial GWs directly probe the energy scale of inflation. In this scale is large enough, they can contribute to C TT , but only at 100, because GWs quickly decay inside the Hubble radius. They also generate E-type and B-type polarization (at all scales). However, temperature and E-polarization anisotropies are predominantly dominated by scalars. A better way to detect primordial GWs by measuring the B-type polarization spectrum, C BB , since in this case there is no scalar contamination. In practice, weak gravitational lensing of the CMB produce a leak from E-type to B-type polarization, so disentangling the tensor contribution to C BB is challenging. Tensors have not been observed yet, so current experiments set an upper limit on the tensor amplitude normalized to the scalar one, i.e. on the so called tensor-to-scalar ratio r ≡ A t /A s . The last measurements from Planck18 indicate r < 0.10 (at 95 % CL). Notice that, even in the presence of tensors, the cross-correlation spectra C TB and C EB should vanish. Indeed, T and E modes remain unchanged under parity transformations, while B changes sign, so in the absence of parity violating physics we expect C TB = C EB = 0.

The calculation of C EE and C TE in a universe featuring just scalar perturbations can be performed along the same line as for C TT . The photon polarization is described by a new degree of freedom, Θ P (τ, k, µ), whose evolution is described by a new Boltzmann equation coupled to the Boltzmann equation for the photon temperature perturbation Θ(τ, k, µ). We neglected this effect when writing Eq. (1.200), but it turns out that the impact of polarization on photon temperature anisotropies is quite small, so our previous qualitative discussion on C TT remains valid. One can express the photon polarization as the line-of-sight integral of an appropriate source function, and then expand in Legendre multipoles (just as for temperature). For brevity, we just quote the final result without providing any explicit calculation (we refer to DURRER (2020) for more details on this rich subject). Under the assumption of instantaneous decoupling, and restricting to small scales (kτ 0 1), one can show that the photon polarization multipoles today are roughly given by

E (τ 0 , k) = Θ P, (τ 0 , k) 15 8 Θ 2 (k, τ dec ) 2 j (kτ 0 ) (kτ 0 ) 2 5k 6κ γ (τ dec ) Θ 1 (k, τ dec ) 2 j (kτ 0 ) (kτ 0 ) 2 , (1.294)
where the last equality follows from the tight-coupling condition, Θ 2 -4kΘ 1 /(9κ γ ). There are three important aspects to discuss. First, the polarization spectrum is seen to be smaller than the temperature spectrum by a factor of order k/κ γ at recombination. This arises as a combinaition of the facts that polarization can only be generated by a quadrupole, and the quadrupole is suppressed in the early universe due to the frequent Compton interactions. Second, we see that, E ∝ Θ 1 , so we expect the polarization spectrum to exhibit oscillations which are out of phase with the temperature spectrum. Finally, there is no analogue to the ISW effect, since polarization is not induced by photons movings through time-varying metric potentials.

Measuring the EE and the T E power spectrum is interesting, since it contains additional information that allows to break some degeneracies among parameters. For instance, reionization imprints a very characteristic effect on C EE , as we illustrate in Fig. 1.19. A small fraction of photons rescattered during reionization acquire additional polarization that manifests as a bump at 10 in C EE . This effect is so large that, despite cosmic variance, τ reio can be well determined using polarization data. In Chapter 4, we will see the implications of this measurement for a model of decaying neutrinos.

A few words on CMB lensing

The anisotropies measured in the CMB can be classified as primary or secondary, depending on whether they are affected by processes around the time of recombination or at much later times, respectively. We have already discussed two examples of secondary anisotropies: the LISW effect and reionization. There is still a very important secondary anisotropy that we haven't described: CMB lensing. In general, gravitational lensing occurs when matter deflects the path of light as it travels from a source to the observer. There are three types of gravitational lensing. Strong lensing occurs when the deflection is large and multiple images are formed. Microlensing occurs when the angular separation of images is too small to be resolved, and instead the source is temporarily brightened. Finally, weak lensing occurs when the deflection is small, and no multiple images are formed. The latter is the one which is most relevant in cosmology. The main idea of weak CMB lensing is that the trajectories of photons coming from the last scattering surface are deflected by matter fluctuations localized at redshifts z 3. At leading order in perturbations, it's possible to describe CMB lensing in terms of a two-dimensional deflection field d(n), giving the difference between the direction n at which photons are emitted, and the observed direction n + d(n). Using the spatial component µ = i of the perturbed geodesic equation, one can show that the deflection field is given by the gradient of a lensing potential ϕ, related to the Newtonian metric potentials φ and ψ through a convolution along the line of sight (see [START_REF] Dodelson | Modern Cosmology[END_REF] for details)

ϕ(n) = - τ 0 τrec dτ χ(τ rec ) -χ(τ ) χ(τ rec )χ(τ ) (φ + ψ) (τ, x=r(τ )n) , (1.295)
where χ(τ ) represents the comoving distance introduced in Eq. (1.31). During the matter era and inside the Hubble radius, we can relate ψ = φ to the total matter density fluctuations δ m using the Poisson equation Eq. (1.229). Hence, the harmonic power spectrum C ϕϕ of a given map ϕ(n) can be obtained from the matter power P m (z, k) (which we define in Eq. (1.296)) convolved with an appropriate kernel. Using C ϕϕ , one can compute the two-point statistics associated to the lensed temperature map

Θ(n) = Θ(n + d(n)) from that of the unlensed map Θ(n) (LEWIS et al. 2006).
Lensing smoothes out the peaks in the temperature power spectrum, and adds power on very small scales, as we illustrate in Fig. 1.20. The lensing effects are so small that they are only evident in the fractional differences. However, these effects are already larger than the error bars in current CMB experiments, so lensing needs to be taken into account. Inferring the lensing spectrum C ϕϕ from observations is not a trivial task, since we only observe CMB anisotropies after lensing effects have taken place. Luckily, lensing produces non-zero correlations between off-diagonal modes of the temperature maps, i.e. Θ( ) Θ( ) = f T T ( , )ϕ( -) = 0 for = (here Θ( ) denotes the flat-sky 2D Fourier transform of Θ(n)). Thanks to this effect, one can build estimators that allow to reconstruct the deflection potential ϕ(n) directly from the anisotropy maps Θ(n) [START_REF] Okamoto | CMB lensing reconstruction on the full sky[END_REF]).

As we will see in Chapter 3 and Chapter 4, secondary CMB anisotropies (and specially, CMB lensing) are essential for constraining unstable relics, such as decaying dark matter or decaying neutrinos, since these scenarios typically produce deviations from ΛCDM only after recombination.

The matter power spectrum

Many Large Scale Structure (LSS) observables can be inferred from the matter power spectrum at different scales and redshifts, P m (z, k), which is defined as

δ m (z, k)δ * m (z, k ) = P m (z, k)δ (3) ( k -k ). (1.296)
Here δ m is the total density contrast of non-relativistic matter components.

In the ΛCDM model, it receives contributions from CDM, baryons and nonrelativistic neutrinos

δ m = δρ m ρm = ρcdm δ cdm + ρb δ b + ρν δ ν ρcdm + ρb + ρν . (1.297)
In the following, we will discuss the shape of the matter power spectrum in the linear regime (although we will briefly comment the importance of non-linearities later on). The assumptions of linearity and gaussian initial conditions imply that, at any time, the matter power spectrum can be expressed as the product of the primordial spectrum and the square of the transfer function δ m (z, k):

P m (z, k) = 2π 2 k 3 A s k k * ns-1 δ 2 m (z, k). (1.298)
Hence, one can get some insight about the shape of P m (z, k) by studying the qualitative evolution of δ m (z, k), as we proceed to do now.

Theoretical understanding

For simplicity, let us first consider a neutrinoless universe with a negligible amount of baryons Ω b Ω cdm . This means that δ m δ cdm , so all we need to know is the evolution of CDM perturbations. By combining the first derivative of the continuity equation Eq. (1.217) with the Euler equation Eq. (1.218), we easily arrive at a master equation for the evolution of δ cdm

δ cdm + Hδ cdm = -k 2 ψ + 3φ + 3Hφ . (1.299)
This equation tells us that the evolution of δ cdm is governed by a competition between friction due to the cosmic expansion (Hδ cdm ), gravitational forces (-k 2 ψ) and dilation effects (3φ + 3Hφ ).

As we have already discussed, δ cdm is frozen on super-Hubble scales, and only varies when the EoS of the universe changes (assuming the Newtonian gauge). Let us focus on the evolution at sub-Hubble scales. In this regime, we saw earlier that the dilation terms can be safely ignored. What about the gravitational force term k 2 ψ? The Poisson equation tells us that k 2 ψ -4πGa 2 δρ tot -4πGa 2 (ρ cdm δ cdm + ργ δ γ ). During matter and Λ domination, the contribution from photons is negligible, so we can simply replace k 2 ψ by -4πGa 2 ρcdm δ cdm . During radiation domination, the situation is more subtle, since both δ cdm and δ γ can be important. As pointed out by [START_REF] Weinberg | Cosmological fluctuations of short wavelength[END_REF], in this case the full solution of the system can be decomposed into fast modes (oscillating with a pulsation of the order k or kc s ) and slow modes (evolving over a Hubble time scale). It turns out the gravitational back reaction between the two modes (through the Poisson equation) can be neglected. Since we are only looking at CDM, we just need to consider slow modes and thus make the same replacement as for the matter era, k 2 ψ = -4πGa 2 ρcdm δ cdm . In short, inside the Hubble radius we can rewrite Eq. (1.299) as

δ cdm + Hδ cdm - 3 2 H 2 Ω cdm (τ )δ cdm = 0, (1.300)
where It is possible to get analytical solutions at all times, which are scale independent since the Mészáros equation doesn't involve k. However, the integration constants can generally depend on k, and their expression is obtained by matching to the solutions at horizon crossing. For simplicity, we don't detail this procedure here. In fact, for understanding the shape of the matter spectrum, it will be enough to just look at the asymptotic solutions:

Ω cdm (τ ) = ρcdm (τ )/ρ c (τ ) = 8πGρ cdm (τ )/3H
• During the radiation era (Ω cdm (a) 1) the growing solution has a sub-Hubble limit δ cdm ∝ log (ka) ∝ log (kτ ). The full solution (valid at all scales) involves the cosine integral, and its expression can be found in Eq. ( 22) of GREEN et al. (2005).

• During the matter era (Ω cdm (τ ) 1), the growing sub-Hubble solution evolves linearly with the scale factor δ cdm ∝ a ∝ τ 2 . This is a famous result from linear theory; it means that it is necessary to wait until the universe becomes matter dominates in order for CDM perturbations to grow appreciably.

• During the Λ era, the growth of structures is slowed down, since Λ enhances the expansion rate and consequently the friction term. It is common to parametrize the sub- We have everything we need to understand the global shape of the matter spectrum as a function of time. We will have to consider three different wavenumbers, corresponding to modes crossing the Hubble radius at the time of matter-radiation equality, matter-Λ equality and today: k eq = a eq H eq , k Λ = a Λ H Λ , and k 0 = a 0 H 0 . Let us now describe the evolution of P m (k) with respect to time.

1. At initial times, all modes of interest were super-horizon, so P m just follows the shape of the primordial spectrum P R (k). Assuming a scale invariant spectrum for simplicity (n s = 1), then P m (k) ∝ k -3 .

2. During the radiation era, sub-Hubble modes evolve like log (kτ ). Thus, at matter-radiation equality, super-Hubble modes (k k eq ) are shaped like initially (P m (k) ∝ k -3 ), whereas the sub-Hubble modes (k k eq ) have been amplified by a k-dependent factor [δ cdm (τ eq , k)/δ cdm (τ ini , k)] 2 [log (kτ eq )] 2 (meaning that

P m (k) ∝ k -3 [log (k)] 2 ).
3. At the time of matter-Λ equality, super-Hubble modes (k k Λ ) keep being shaped like initially. Modes with k k eq have been enhanced during the matter era by a k-independent factor [δ cdm (τ Λ , k)/δ cdm (τ eq , k)] 2 (τ Λ /τ eq ) 4 . Finally, intermediate modes crossing the Hubble radius during matter domination (k Λ k k eq ) have been amplified by a k-dependent factor (τ Λ /τ * ) 4 , where τ * is their time of Hubble crossing, τ * 1/k. In summary, the spectrum has three branches, which scale like

• P m (k) ∝ k -3 , for k < k Λ , • P m (k) ∝ k -3 k 4 = k, for k Λ < k < k eq , • P m (k) ∝ k -3 (log k) 2 , for k > k eq . 4.
During the Λ era, δ cdm grows more slowly than τ 2 , but this suppression of growth is the same for all scales. Hence, the shape of the matter spectrum is unaffected by this stage, and the three branches are the same as during the matter era. The only difference is that the primordial spectrum P m ∝ k -3 is now recovered for scales k < k 0 , which in practice are never directly observed. It still remains to see the effects associated to baryons, since in reality Ω b is not negligible. As we already discussed in Sect. 1.6.2, baryons remain coupled to photons until the time of baryon drag, which took place slightly after photon decoupling, τ dr > τ dec . After that time, the collision term vanishes, and the evolution of baryons is dictated by an identical equation as that for CDM, i.e. Eq. (1.299) (notice that will always consider scales bigger than the Jean's length, which means that we can discard the baryon pressure term). Let us discuss separately the evolution of modes that crossed the Hubble radius before and after baryon drag.

For modes that crossed the Hubble radius after baryon drag, the evolution of baryons is trivial, since it matches that of CDM, δ b = δ cdm . Indeed, baryon and CDM density perturbations are equal on super-Hubble scales due to the adiabatic initial conditions. After Hubble crossing, they are subject to the same evolution equation of the form Eq. (1.299), so they remain equal at all times.

The situation is a bit more complex for modes that crossed the Hubble radius before baryon drag. In this case, baryons develop the same oscillatory behavior as photons, since the tight-coupling between these species enforces δ b = 3 4 δ γ . This means that baryons experience all the effects discussed in Sect. 1.6.4, such as decreasing oscillations during the matter era, diffusion damping on the small scales, etc. As long as baryons remain coupled to photons, they don't cluster, but still contribute to enhance the expansion rate. This is translated into a reduced growth of CDM. For instance, one can consider sub-Hubble scales during the matter era, and solve a Mészáros equation identical to Eq. (1.300) but including the contribution from baryons in H. This gives a solution δ cdm ∝ a 1- linear growth of the CDM perturbations produces a suppression in the matter power spectrum for wavenumbers k > k eq . After baryon drag, baryons and CDM are collisionless species feeling the same gravitational forces, so their perturbations are described the following set of coupled equations at sub-Hubble scales

δ cdm + Hδ cdm = 4πGa 2 (ρ cdm δ cdm + ρb δ b ), (1.301
)

δ b + Hδ b = 4πGa 2 (ρ cdm δ cdm + ρb δ b ). (1.302)
By solving this system of equations, one notices that δ b (which at baryon drag was much more suppressed than δ cdm ) rapidly approaches δ cdm . In other words, baryons fall into the potential gravitational wells created by the CDM. The time needed to reach δ b = δ cdm is greater for smaller scales, because in the large-k limit the ratio |δ b /δ cdm | is smaller at τ dr due to Silk damping. This phenomenon, essential to understand the formation of large-scale structures, is illustrated in Fig. 1.21, where we show the numerical resolution of δ γ , δ b and δ cdm for a mode that entered the Hubble radius deep in the radiation era (k = 0.5 Mpc -1 ). Hence, in order to get an approximate description of the matter power spectrum, it is enough to perform a matching of the solution δ c = δ b for τ > τ dr with the individual solutions δ c and δ b during τ < τ dr (as described in detail in EISENSTEIN et al. (1998)). The important outcome is that baryons imprint small oscillations as a function of k in the matter power spectrum, which are known as Baryon Acoustic Oscillations (BAO).

In Fig. 1.22 we show a numerical calculation of the matter power spectrum at z = 0, with and without taking baryons into account 28 . The shape of P m (k) agrees very well with our qualitative discussion. It shows a peak at k = k eq , with the two branches at k < k eq and k > k eq going roughly as P m ∝ k and P m ∝ k -3 [log (k)] 2 , respectively. In addition, baryons lead to an overall suppression for k > k eq and imprint the BAO feature, which has already been detected in galaxy surveys [START_REF] Eisenstein | Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies[END_REF].

Parameter dependence of P m (k)

We can now discuss the effects of the different cosmological parameters on the shape of the matter power spectrum, similarly as we did in Sect. 1.6.5 with the CMB temperature spectrum. In this case, P m (k) is impacted by only 5 parameters, since the optical depth at reionization only affects the scattering rate of photons. Taking as a parameter basis,

{A s , n s , ω b , ω cdm , Ω Λ }, (1.303)
we can enumerate each of the different effects.

1. The time of equality fixes the location of the peak, k eq . In units of h/Mpc, this scale is entirely determined by z eq and Ω m , or equivalently, by ω m and

Ω Λ = 1-Ω m .
2. The abundance of baryons relative to CDM controls the branch at k > k eq . A higher fraction ω b /ω cdm implies more suppression at these scales and BAOs more pronounced.

3. The phase of the BAOs is dictated by the comoving sound horizon at baryon drag, r d = r s (τ dr ). Similarly to the scale of the acoustic oscillations in the CMB spectra, r d depends on ω b and ω m . However, r d depends much more strongly on ω b , because the time of baryon drag is affected by R = 3ρ b /4ρ γ .

4. The global amplitude depends on the amplitude of the primordial spectrum, A s , but also on Ω Λ , since we have seen that the Λ produces a scale-independent suppression of growth during the Λ era.

5. The global tilt depends directly on the tilt of the primordial spectrum, i.e. on n s .

For simplicity we have not included a description of the effects of neutrinos, which are nicely explained in LESGOURGUES et al. (2018). Let us just mention that massive neutrinos have a very strong impact on the matter spectrum. In particular, they induce a smooth steplike suppression on wavenumbers larger than the free-streaming wavenumber at the non-relativistic transition, k nr = k fs (z nr ) (where the function k fs (z) exhibits a minimum). Indeed, on scales k > k nr , neutrinos do not contribute to clustering but still enhance the Hubble friction term, producing a reduced growth of CDM perturbations (akin to the suppression due to baryons). We will encounter a very similar behavior when discussing the dynamics of warm particles arising from DM decays in Chapter 3.

Probes of structure formation

The theory of linear cosmological perturbations cannot provide a description of matter fluctuations at small scales and small redshifts, where the approximation k/aH. Thus, it is convenient to plot the matter power spectrum as a function of k/(a0H0), such that any other choice of H0 would leave the overall shape of the spectrum invariant. However, H0 = h/3000 Mpc -1 and it is typically assumed a0 = 1, so in practice it is equivalent to simply write k in units of h/Mpc.

δ( x, z)

1 breaks down. LSS observations are sensitive to such range of z and k, so it is important to be able to compute the power spectra on mildly or even strongly non-linear scales, in order to extract as much information as possible from these observables 29 . This is typically done with the help of N -body simulations. In this approach, CDM (and eventually other species like baryons or neutrinos) is represented as a set of N -body particles, with masses of the order of million solar masses (not to be confused with real particles!). Initially, these particles are distributed inside a box such that the power spectrum of the smoothed distribution is given by the linear spectrum P m (z ini , k) for a given cosmology. Afterwards, particles are evolved using equations from Newtonian gravity, but including the effects of the expansion through a Hubble friction term. These simulations are performed in boxes in comoving space with periodic boundary conditions. ). We show in Fig. 1.22 the impact of the non-linear matter power spectrum, which starts to deviate from the linear one at k ∼ 0.1h/Mpc. In Chapter 2, we will use the HALOFIT code to compute the non-linear matter spectrum associated to a model of Early Dark Energy, in order to test this model against LSS data.

Many LSS observables probe the σ 8 parameter, measuring the present strength of clustering on a certain range of scales. Given the importance that this parameter has for the so-called "S 8 tension" and for this work, it is worth pausing for a moment to define this parameter properly. Often, one would like to smooth the matter density field δ m ( x, z) over some "window" in space. In fact, the actual density field is grainy due to the presence of structures (stars, galaxies, etc) and the theory can only predict the statistics of a smoothed density field. There are several choices for the smoothing function, a popular one is the top-hat window function in real space:

W R (r) = 1 (4π/3)R 3 Θ H (R -r), (1.304)
where r = | x -x| and Θ H is the Heaviside step function. The smoothed density field is obtained by a convolution with the window function

δ m,R ( x, z) = d 3 x W R (| x -x|)δ m ( x , z). (1.305)
In Fourier space, this corresponds to a simple multiplication

δ m,R ( k, z) = W R (k)δ m ( k, z), where W R (k) is the Fourier transform of the top-hat window function W R (k) = 3 sin (kR) -kR cos (kR) (kR) 3 . (1.306)
Hence, the smoothed matter power spectrum reads

P m,R (k, z) = |W R (k)| 2 P m (k, z
). Now, from Eq. (1.247) we know how to obtain the matter power spectrum P m (k, z) from the 2-point correlation function in real space, ξ m (r, z). One can perform the inverse Fourier-transform and evaluate the zero-lag correlation function ξ m (r = 0, z),

ξ m (r = 0, z) = 1 2π 2 ∞ 0 dkk 2 P m (k, z) lim r→0 sin (kr) kr = ∞ 0 dlnkP m (k, z), (1.307) with P m (k, z) = k 3 P m (k, z)/2π 2 .
We can now consider the autocorrelation function associated to the smoothed matter power spectrum (often referred as

σ 2 (R, z) = ξ m,R (r = 0, z) in the literature), σ 2 (R, z) = ∞ 0 dlnk|W R (k)| 2 P m (k, z). (1.308)
This gives the r.m.s. amplitude of mass fluctuations smoothed over a scale R. Historically, cosmologists have first studied clustering on scales about the size of a galaxy cluster (5 -10 h -1 Mpc), so an important quantity to choose was

σ 8 ≡ σ(R = 8 h -1 Mpc, z = 0). (1.309)
The σ 8 parameter also provides an alternative way to normalize the matter power spectrum, since it is proportionally related to the primordial amplitude A s (but note this relation is model-dependent). In the ΛCDM framework, σ(R, z) increases smoothly with decreasing R, corresponding to a "bottom-up" scenario where the smallest, least massive objects form first (such as stars and galaxies), and the larger, more massive objects (like galaxy clusters) form later. Notice that the matter power spectrum entering in the definition of σ 8 is the linear one. In fact, one can estimate the redshift at which a given scale R enters in the non-linear regime from the condition σ(R, z nl ) ∼ 1.

Let us briefly comment on one technique used to probe the large-scale structures, called galaxy weak lensing, which is very relevant in the context of the S 8 tension. The images of distant galaxies are slightly distorted due to the gravitational lensing of the intervening large scale structures. The two main areas of galaxy weak lensing are cosmic shear, the study of correlations between the shapes of pairs of galaxies, and galaxy-galaxy lensing, the study of correlations between the position of foreground galaxies and background galaxy shapes. Cosmic shear allows to probe the LSS more directly than galaxy-galaxy lensing, where one needs to makes assumptions about the relation between galaxies and the underlying matter distribution.

The basic observable in weak lensing studies is the ellipticity of a galaxy. The measured ellipticity receives contributions from the intrinsic galaxy ellipticity and the shear effect due to gravitational lensing (i.e. the squeezing of the image in one direction, and its stretching on the orthogonal direction). In the weak lensing regime, the intrinsic ellipticity largely dominates over the small shear, so it is necessary to perform an average over many galaxies. The main idea behind this approach is that galaxies are intrinsically randomly oriented on the sky, so the intrinsic ellipticities average to zero, leaving just the shear. Like for CMB lensing, galaxy lensing is described by a deflection potential, which is related to the Newtonian metric potentials φ and ψ through

ϕ(n) = - τ 0 0 dτ s g(τ s ) τ 0 τs dτ χ(τ s ) -χ(τ ) χ(τ s )χ(τ ) (φ + ψ) (τ, x=r(τ )n) , (1.310)
This expression is very similar to the one we wrote in Eq. (1.295) when presenting the CMB lensing, although there are two important differences. First, the source is now located at an arbitrary comoving distance χ(τ s ), rather than on the last-scattering surface. Second, since we are measuring the distortions for a ensemble of galaxies, we need to consider a distribution of galaxy redshifts, g(τ s ), normalized such that

τ 0 0 dτ s g(τ s ) = 1.
As for the CMB lensing spectrum, the associated harmonic spectrum C ϕϕ is obtained by a convolution of the matter power spectrum P m (k, z) with an appropriate kernel. However, as opposed to the CMB lensing, it is strictly necessary to consider the non linear matter power spectrum, since galaxy lensing covers much smaller scales. Often, it is possible to split the catalog of galaxies in several redshift bins i, described by selection functions g i (τ ) such that g(τ ) = i g i (τ ), and to measure the lensing spectrum C ϕ i ϕ j at each redshift bin. This tomographic approach is particularly useful for measuring neutrino masses as well as dark energy properties.

Observational constraints are often reduced to the parameter combination to which weak lensing surveys are mostly sensitive. In particular, constraints are typically quoted in terms of S 8 ≡ σ 8 Ω m /0.3, which modulates the amplitude of weak lensing measurements. We remark that the S 8 parameter is not directly measured by the collaborations, but inferred from Eq. (1.308) by adjusting the ΛCDM model to their weak-lensing data (HEYMANS et al. 2021).

Let us just mention that other LSS probes can be similarly obtained from the matter power spectrum. For instance, galaxy clustering probes measure the galaxy power spectrum, which in the simplest formulation is assumed to be

P g (z, k, µ) = (b g (z) + f µ 2 ) 2 P m (z, k),

Successes and challenges of the ΛCDM paradigm

The ΛCDM model provides outstanding explanation for a wide variety of early universe data, such as Cosmic Microwave Background (CMB) and Big Bang Nucleosynthesis (BBN), as well as late universe observations of Large Scale Structure (LSS) including the Baryon Acoustic Oscillation (BAO), and uncalibrated luminosity distance to SuperNovae of type Ia (SNIa). In Fig. 1.23 we show a schematic representation of the approximate redshift z and wavenumber k corresponding to different cosmological observables. The fact that the ΛCDM model agrees well with such a disparate compilation of observations (spanning many orders of magnitude in time and scale) is an impressive proof of its success. Nevertheless, the ΛCDM model does not teach us about the intrinsic nature of its dark sector, made up of both cold dark matter and dark energy. In addition, as the accuracy of cosmological observations has improved, the concordance cosmological model starts showing several experimental discrepancies. In the following, we proceed to discuss each of this problems.

The nature of the dark sector

During the last couple of decades, a myriad of models have been proposed to describe the dark components of the universe. Unfortunately, their nature remains a mystery, due to the absence of a clear signal independent from the gravitational ones. Below we sketch some of the main candidates for dark matter and dark energy.

Dark matter candidates

• WIMPs. These remain amongst the most popular models, due to their simple production mechanism in the primordial universe (discussed in Sect. • Axions. This corresponds to a very different class of DM candidates, which were originally introduced to solve the CP (charge-parity) problem of the SM. This is a fine-tuning problem that deals with the question of why there is no experimental sign of CP-violating physics in strong interactions, even if the QCD Lagrangian requires in full generality a CP-violating term. In a nutshell, the idea is to introduce a new global chiral symmetry U (1) PQ that is spontaneously broken at scales much higher than the QCD confinement scale (f a Λ QCD ∼ 150 MeV). This generates the "axion" field as a pseudo-scalar Goldstone boson, whose dynamics are such that it can naturally cancel the CP-violating term in QCD [START_REF] Peccei Y | CP Conservation in the Presence of Instantons[END_REF][START_REF] Weinberg | A New Light Boson?[END_REF][START_REF] Wilczek | Problem of Strong P and T Invariance in the Presence of Instantons[END_REF]). The axion field, which is initially massless, acquires a non-zero mass when temperature of the bath drops below T ∼ Λ QCD . At this point, the field starts rolling down to the minimum of its potential, oscillating and diluting in the same way as a CDM fluid, ρa ∝ a -3 . One can show that the axion mass scales as m a ∝ 1/f a , and in order to produce the right DM abundance, it should fall in the range 10 µeV < m a < 0.1 eV. Many axion models predict a coupling to photons, g aγ , which is the source of Primakoff effect: the conversion of axions into detectable photons in a strong magnetic field. With the years, QCD axions have paved the way to the study of more generic light bosons, such as axion-like particles (generalizing the QCD axion but not necessarily connected to the strong CP-problem) or ultralight axions (with masses as low as m a ∼ 10

Dark energy candidates

• Cosmological constant. The simplest possible explanation for the cosmic acceleration which is in agreement with current observations is that dark energy is in the form of a cosmological constant Λ, whose energy is independent of of space and time. However, this triggers a number of questions. First, the stressenergy tensor associated to Λ has the same form of that associated to the ground state energy of the vacuum, T Λ µν = -ρ Λ g µν , with ρ Λ = Λ/8πG. Unfortunately, the predicted size of the vacuum energy ρ vac from the zero-point energy suggested by quantum field theory is completely off, ρ vac /ρ Λ ∼ 10 120 . This is known as the cosmological constant (CC) problem. Another issue is related to the fact that the present abundance of dark energy Ω Λ is extremely similar to that of matter Ω m , despite the very different scalings of each one (ρ Λ ∝ a 0 and ρ m ∝ a -3 ). This is the so-called coincidence problem. Faced with these questions, theorists have developed extensions of GR, which are pursued to understand gravity at a more fundamental level. But how should one modify gravity? In fact, GR turns out to be a very unique theory. According to Lovelock's theorem [START_REF] Lovelock | The Einstein tensor and its generalizations[END_REF], in a four-dimensional space-time, the only second order equations derived from an action that depends only on the metric, S = d 4 xL(g µν ), are precisely Einstein field equations (c.f. Eq. (1.10)). Hence, new theories of gravity can be built by breaking some of the assumptions of this theorem, e.g. by considering new fields, extra dimensions, higher-order equations etc. It should be stressed that, even if most of these models address the coincidence problem, there is still no widely accepted solution for the CC problem.

• Quintessence. This is one of the simplest extensions, which consists in adding a scalar field φ (playing the role of dark energy) minimally coupled to gravity and described by some generic potential V (φ) [START_REF] Copeland | Dynamics of dark energy[END_REF]. A wellknown example is the Ratra-Peebles potential

V (φ) = k/φ α (RATRA et al. 1988).
In quintessence models, the EoS is simply given by

w φ = φ2 /2-V (φ)
φ2 /2+V (φ) , so the scalar field can produce cosmic acceleration (w φ < -1/3) as long as the field is in slowroll (V (φ) φ2 /2), similarly to inflationary models. However, the EoS can never become strictly smaller than w φ = -1. The regime w < -1 actually defines a different class of models, dubbed phantom dark energy. These scenarios produce such a rapid acceleration that a big rip can take place, i.e., a → ∞ is reached in a finite amount of time, meaning that all the matter in the universe is eventually torn apart (R. • Other theories of modified gravity. There are many other proposals of modified gravity outside the scalar-tensor framework. For instance, theories of massive gravity propose that the elementary spin-2 excitations of the gravitational field (the gravitons) can have a non-zero mass. In order to satisfy current constraints, this mass must be tiny (less than 10 -23 eV), but can already have effects on the very large scales [START_REF]Massive Gravity[END_REF]. Another important class of models are higher-dimension theories. They are based on the idea that our universe contains extra hidden dimensions only visible by gravity, while the rest of SM particles are confined in a four-dimensional brane. A well-known example is the Dvali-Gabadadze-Porrati (DGP) model [START_REF] Dvali | 4-D gravity on a brane in 5-D Minkowski space[END_REF]. These models share many features with massive gravity and scalar-tensor theories, but the behavior at large distances is different.

Cosmic discordance

The H 0 tension

The so-called "Hubble tension" refers to the inconsistency between local measurements of the current expansion rate of the Universe, i.e. the Hubble constant H 0 , and the value inferred from early-Universe data using the flat ΛCDM model. This tension is predominantly driven by the Planck collaboration's observation of the cosmic microwave background (CMB), which predicts a value in ΛCDM of H 0 = (67. [START_REF] Carr | Black holes in the early Universe[END_REF] 30 . These probes, which invoke at least one measurement from high redshifts, are often dubbed "early-Universe calibrations", and all result in H 0 values below 70 km/s/Mpc (typically in strong agreement with the value inferred by Planck using ΛCDM ). Similarly, several alternative methods for directly measuring the local expansion rate have been proposed in the literature. A large number of these techniques offer alternative methods for calibrating the cosmic distance ladder, removing any bias introduced from Cepheid observations. One example is the recent determination of H 0 obtained by the Chicago-Carnegie Hubble program (CCHP), which calibrates SNIa using the tip of the red giant branch (TRGB); this observation yielded a value of H 0 = (69.8 ± 0.6 (stat) ± 1.6 (sys)) km/s/Mpc [START_REF] Freedman | The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch[END_REF][START_REF] Freedman | Measurements of the Hubble Constant: Tensions in Perspective[END_REF], in between the Planck CMB prediction and the SH0ES calibration measurement.

However, alternative analyses using similar techniques have yielded values significantly closer to the value obtained by SH0ES, in particular the latest calibration of the TRGB using the parallax measurement of ω-Centauri from GAIA DR3 leads to H 0 = ( 72 we show an updated compilation of some of the direct and indirect determinations of H 0 we just discussed. While not all measurements are in tension with Planck, these direct probes tend to yield values of H 0 systematically larger than the value inferred by Planck 31 . Depending on how one chooses to combine the various measurements, the tension may be elevated to as much as 6σ (DI VALENTINO et al. 2021b).

Intense experimental efforts are underway to establish whether this discrepancy can be caused by yet unknown systematic effects (appearing in either the early or late Universe measurements, or both). This includes (but is not limited to) issues in SNIa dust extinction modeling and intrinsic variations (MORTSELL et al. ). Yet, the appearance of this discrepancy across a wide array of probes seems to suggest that a single systematic effect may not be sufficient to resolve this discrepancy. The alternative possibility is that the Hubble tension reflects a breakdown of the ΛCDM model; properly accounting for new physics operating either in the earlyor late-Universe could change the inference of H 0 from the early-Universe probes to be in agreement with the direct measurements. In Chapter 2 we will explain that any viable ΛCDM extension likely requires reducing the sound horizon at recombination.

The S 8 tension

The "S 8 tension" is a longstanding discrepancy between the strength of matter clustering measured by large-scale structure probes, and the value inferred from early-Universe data. In particular, the value of the clustering strength estimated by the primary anisotropies of the CMB (as measured by Planck) is typically 2 -3σ higher than that obtained from lower redshift probes, such as weak gravitational lensing and galaxy clustering. As mentioned in Sect. 1.6.6, this is often quantified through the S 8 ≡ σ 8 Ω m /0.3 parameter, where σ 8 is the root mean square of matter fluctuations on a 8h -1 Mpc scale, and Ω m is the total matter abundance. This parameter combination is chosen to encapsulate the main degeneracy direction of weak lensing measurements. The S 8 parameter is intimately related to f σ 8 (z = 0) measured by redshift space distortions (RSD) (GIL-MARIN et al. 2017), where f = [Ω m (z)] 0.55 approximates the growth rate in GR. Notice that measuring S 8 is model dependent, and all current experimental determinations assume the flat ΛCDM model.

The CMB anisotropy data from Planck18 TT,TE,EE+lowE yields a constraint S 8 = 0.834 ± 0.016, which can be tightened when adding the CMB lensing reconstruction, S 8 = 0.832 ± 0.013. As we mention later, this high S 8 estimate could be related to an excess of lensing inferred from the smoothing of the acoustic peaks within Planck data. However, the combination of high-data from ACT (S. ) show a similar trend with a better precision. The S 8 parameter can also be measured from redshift-space galaxy clustering; in particular, with the BOSS full-shape galaxy power spectrum and bispectrum: S 8 = 0.751 ± 0.039 (PHILCOX et al. 2022). The precision of these measurements can be improved in a joint analysis with weak lensing data. The combination of cosmic shear, galaxy-galaxy lensing and galaxy clustering at the level of 2-point statistic is known as "3 × 2pt" analysis. For instance, the recent 3 × 2pt analysis by KIDS1000+BOSS+2dfLenS has yielded a constraint S 8 = 0.766 +0.020 -0.014 , which has the overall same precision as Planck but it is ∼ 3σ smaller. This kind of combined analysis between galaxy clustering with weak lensing allows to break the degeneracy between σ 8 and Ω m , revealing that the tension is driven by σ 8 , rather than Ω m . Another possibility to measure S 8 is to use the number density of DM halos, which is very sensitive to the growth of structure through the cosmological halo mass function. The Planck mission has provided measurements of the number counts of galaxy clusters detected through the Sunyaev-Zel'dovich (SZ) effect, reporting a deficit with respect to the value expected from the fiducial Planck TT,TE,EE+lowE ΛCDM model (ADE et al. 2016b). This can be translated into a 2 -3σ tension in S 8 , which has been confirmed by other cluster count experiments (PRATT et al. 2019). In Fig. 1.25 we show an updated compilation of some of the high-z and low-z determinations of S 8 we just discussed.

There is a large number of known systematic effects that can affect the low-redshift measurements of large-scale structures. Weak-lensing systematics include: noise bias affecting the galaxy shape measurements (MELCHIOR et al. Still, the appearance of this mismatch across such a variety of probes suggest that a single systematic might not be enough to explain the discrepancy, similarly to what happens with the H 0 parameter. Hence, the S 8 tension could also be a signal of a failure of the ΛCDM model. In particular, it seems to point towards a suppression of power in the matter spectrum P m (k) at the scales selected by the window function k 2 |W 8 (k)| 2 , i.e. for k ∼ (0.1 -1)h/Mpc. In Chapter 3, we will study a model that can achieve precisely this type of suppression.

The A lens anomaly and other discrepancies

Another well studied 'curiosity' in the recent literature consists in the anomalous amount of lensing estimated from the smoothing of the acoustic peaks at high-'s within Planck data, as quantified by the 'A lens ' parameter (CALABRESE et which shows no tension with the LSS measurement of S 8 , a weaker H 0 tension, and no lensing anomaly. Pin-pointing the source of such lensing anomaly (perhaps a simple statistical fluke, although quantifying its likelihood and how to treat it is complicated) is therefore of utmost importance to understand whether the S 8 tension derives from it. In Chapter 2 and Chapter 3 we will study the implications of the A lens anomaly for models of Early Dark Energy and Decaying Dark Matter, respectively.

There are many other intriguing anomalies that we haven't addressed in this thesis, such as those related to the small-scale crisis of ΛCDM (BULLOCK et al.

Outline

After this general introduction, the rest of the document is divided into three other chapters. In Chapter 2, we reassess the viability of a pre-recombination Early Dark Energy component as a solution to the Hubble tension, by confronting the model against a host of high-and low-redshift measurements, including LSS observations from recent weak lensing surveys, as well as CMB, BAO and SNIa data. We also perform a systematic comparison of seventeen different models which have been proposed to resolve the H 0 tension, and quantify the relative success of each using different metric and a variety of datasets. In Chapter 3 we perform the first thorough cosmological analysis of a scenario in which dark matter can decay into massless and massive invisible particles, showing that the model can successfully explain the low-S 8 measurements reported by weak-lensing surveys. Finally, in Chapter 4 we update the bounds on the mass and lifetime of neutrinos decaying into dark radiation after becoming non-relativistic, thanks to the latest Planck data release and an improved Boltzmann formalism.

Complementary information can be found in the numerous appendices. In App. A we detail some important calculations concerning linear perturbation theory. In App. B we say a few words about statistical analysis and parameter inference. In App. C, App. D and App. E we provide tables, plots and calculations that are intended to complement the content of Chapter

Best-fit EDE cosmology solving the H 0 discrepancy

Early dark energy (EDE) refers to a model in which a scalar field φ is frozen-in at times prior to recombination, thus behaving during this epoch like a dark energy component. EDE representing ∼ 10% of the total energy density of the universe around matter-radiation equality and diluting faster than radiation afterwards has been shown to provide a very good resolution to the H 0 tension. To be more precise, the reconstructed parameters from the Planck15+BAO+SNIa+SH0ES analysis of T. L. Furthermore, in order to preserve a good fit to CMB data while solving the H 0 tension, EDE predicts values of some of the cosmological parameters which are somewhat different from those of baseline ΛCDM:

• A higher spectral index n s . For instance, a Planck18+BAO+SNIa+SH0ES analysis of the EDE model indicates n s = 0.988 ± 0.007 (see Tab. (2.1)), i.e. it is compatible with a scale-invariant primordial spectrum at less than 2σ. If this trend turns out to be confirmed with future data, it might require going beyond the simplest slow-roll scenarios for inflation (see F. TAKAHASHI et al. (2021) for a recent discussion on inflationary models that can achieve n s = 1).

• A smaller age of the universe t 0 . To give an example, the age of the universe derived from a Planck18+BAO+SNIa+SH0ES analysis of the EDE gives t 0 = 13.24 ± 0.17 Gyrs. This feature is actually generic to any early-time solution, since these models increase H 0 while leaving Ω m and Ω Λ unaffected. As a consequence, the age of the universe, which scales as t 0 ∝ 1/H 0 (c.f. Eq. (1.28)), decreases roughly by the same amount as H 0 increases. This can therefore be tested against measurements of the age of old objects such as globular clusters of stars, which was recently determined as t 0 = 13. ) algorithms, which have not been calibrated for EDE cosmologies. We thus check the predictions of these algorithms against the results of a set of dedicated cosmological N -body simulations, confirming that the qualitative departures from ΛCDM arising in the EDE cosmology are small enough to make use of these standard algorithms. We perform a series of Monte Carlo Markov Chain (MCMC) analyses with various combination of the latest Planck, BAO, growth factor and SNIa Luminosity distance measurements, the SH0ES measurement of H 0 , and KiDS/Viking/DES data, in order to assess whether current observations exclude the EDE resolution to the Hubble tension.

We find that, even if the S 8 prediction from the best fit EDE cosmology is indeed ∼ 2.5σ higher than the weak-lensing determinations, KiDS data currently provide very little constraining power on the EDE parameters. Yet, it has been found in ASGARI et al. (2020) and JOUDAKI et al. (2020) that a combination of KiDS and DES-Y1 (after re-calibration of the DES photo-metric redshifts) yields S 8 = 0.755 +0.019 -0.021 , a result that is in 3.2σ tension with Planck ΛCDM prediction 3 . At such a level of discrepancy, one should be cautious when interpreting results obtained from combining Planck and WL data, even within ΛCDM. Indeed, we show that despite the inclusion of a Gaussian S 8 likelihood, the resulting cosmological model yields a very bad fit to the S 8 data, while providing very strong constraints on any parameter correlated with S 8 (e.g. ω cdm , A s , f EDE ). It is easily conceivable that the resolution to the S 8 tension lies elsewhere (whether new physics related -or not -to the EDE, or systematic effects), such that any constraints derived from these combined data are artificial.

As already mentioned in Sect. 1.7.2, It is also possible that the S 8 tension is related to the 'A lens ' anomaly, i.e. the anomalous amount of lensing estimated from the smoothing of the acoustic peaks at high-'s within Planck data. Motivated by this fact, we perform an analysis of the ΛCDM and EDE cosmology against Planck and a prior on S 8 from the joint DES-Y1 and KiDS results, while marginalizing over the lensing information. We find that both the unlensed ΛCDM and EDE cosmology spectrum agrees better with LSS data, and that the presence of EDE does not affect the amount of anomalous lensing. This means that the anomalous lensing is not due to the EDE, and also that the success of EDE is not due to opening up a new degeneracy direction with some exotic lensing parameters.

In the last part, we make a systematic comparison of seventeen different models which have been proposed to resolve the H 0 tension (spanning both early-and late-Universe solutions), and quantify the relative success of each model using a series of metrics and a vast array of data combinations. Among these models, we include the EDE proposal as well as the decaying DM scenario discussed in Chapter 3. We note that the ranking that we've established is based on a full reanalysis of all models, and not just on a bibliographic compilation. This analysis establishes a foundation of tests which will allow the success of novel proposals to be meaningfully "benchmarked". In particular, it shows that the EDE scenario currently stands as one of the most promising mechanisms that could be responsible for the observed discrepancy.

The rest of this chapter is structured as follows. In Sect. 2.2 we discuss the general characteristics that any model attempting to solve the Hubble tension might require. In Sect. 2.3 we present the basic equations of the EDE scenario. In Sect. 2.4 we present the results of our different data analyses for the EDE model. In Sect. 2.5 we discuss the different aspects of the "H 0 olympics": the proposed models, the tests to quantify model success and the results of the contest. Finally, in Sect. 2.6 we draw our conclusions.

Guidelines to solve the H 0 tension

The inference of H 0 from "early-Universe" probes (like the CMB) heavily depends on the assumed cosmological model, as opposed to local determinations (such as the cosmic distance ladder). For this reason, it is commonly stated that the H 0 tension necessarily signals a breakdown of the ΛCDM model, if not originated by unaccounted systematics. Let us explore this statement in more details, focusing on the CMB and cosmic-distance ladder determinations of H 0 .

Most local determinations aim to obtain H 0 from the slope of the Hubble diagram, which gives the luminosity distance d L (z) to some standard candle (typically, SNIa) in terms of redshift. The redshift range should be chosen to reduce as much as possible the dependence on peculiar velocities at low-z and the dependence on cosmology at high-z. As already mentioned in Chapter 1, a SNIa can be standardized using light curve color c and shape x corrections, such that the peak absolute magnitude in the B-band, M b , is common to all SNIa. Hence, the peak apparent magnitude m b at redshift z can be written as:

m b (z) = M b + 25 + log 10 d L (z) = -5a b + 5log 10 dL (z), (2.2)
where

5a b ≡ -(M b + 25 -5log 10 H 0 ) is the intercept and dL (z) ≡ H 0 d L (z)
is the uncalibrated luminosity distance. For a spatially flat universe, the low-z expansion of dL (z) gives

dL (z) = z 1 + (1 -q 0 ) z 2 - 1 6 (1 -q 0 -3q 2 0 + j 0 )z 2 , ( 2.3)
where q 0 ≡ -äa ȧ2 0

and j 0 ≡ ... aa 2 ȧ3 0 are the deceleration and jerk parameters, respectively.

Therefore, in order to get H 0 from the value of a b , a knowledge of M b is required, i.e., the luminosity distances of SNIa need to be calibrated. The goal of the cosmic distance ladder is precisely to calibrate the distances to SNIa by using a succession of overlapping methods, where each rung of the "ladder" is needed to determine greater distances at the next rung. Getting direct measurements of distance is only possible for nearby galaxies (anchors), located at a few ∼ kpc from us. The anchor galaxies considered by SH0ES include the megamaser host NGC 4258, the Milky Way where distances are obtained using parallaxes, and the Large Magellanic Cloud (LMC) where distances are measured by detached eclipsing binaries. Those galaxies host a class of pulsating stars known as Cepheids. These were the first standard candles to be identified, when Henrietta Leavitt discovered that they exhibit a strong correlation between their period and their intrinsic luminosity [START_REF] Leavitt | Periods of 25 Variable Stars in the Small Magellanic Cloud[END_REF]). The Wesenheit apparent magnitude 4 of a jth Cepheid in the ith galaxy can generally be written as

m W H,i,j = µ i + M W H + b W [P ] i,j + Z W [O/H] i,j . (2.5)
Here µ i is the distance modulus to the galaxy, M W H is the zeropoint Cepheid absolute magnitude corresponding to a period P = 10 days, and b W , Z W denote the slope parameters that capture the dependence on period, [P ] ≡ log 10 P -1 (with P in units of days), as well as metallicity, [O/H] ≡ log 10 (O/H) -log 10 (O/H) (with O and H the number of oxygen and hydrogen atoms per unit of volume, respectively). Thus, the first step of the ladder consists in measuring distances to the aforementioned anchors (hosting 980 Cepheids according to the latest SH0ES data relase), in such a way to 4 The dereddended apparent magnitude m W H is connected to the Wesenheit dust extiction parameter RW as (PERIVOLAROPOULOS et al. 2022)

m W H ≡ mH -RW (V -I), (2.4)
where mH is the apparent magnitude in the near-infrared H (F160W) band, V (F555W) and I (F814W) are optical apparent magnitudes in the corresponding bands.

determine M W H , b W and Z W . Once the calibration of the period-luminosity-metallicity relation is achieved, one can measure distances to much farther Cepheids. In the second step of the ladder, a few thousands of Cepheids are observed in galaxies hosting SNIa (at distances between ∼ kpc to ∼ Mpc), hence permitting a calibration of SNIa. In particular, the latest SH0ES analysis considers 37 galaxies with redshifts 0.0015 z 0.011 hosting 2150 Cepheids and 42 SNIa. This procedure yields (see In this procedure, the deceleration and jerks parameters are set to q 0 = -0.55 and j 0 = 1 (corresponding to the standard cosmological model). From this, we see that the SH0ES determination of H 0 is not fully cosmology-independent. However, this procedure is expected to be very robust under changes in the cosmological model, since H 0 is extrapolated from measurements at very low z, as opposed to the early universe probes, which extrapolate H 0 from observations at z ∼ O( 103 ).

RIESS
Let us move to describe the "early-Universe" determination of the Hubble constant. The key scales at play in the ΛCDM prediction are the observed angular scale of sound horizon at recombination θ s , measured at O(0.1%) precision in CMB data, and the related angular scale of sound horizon at baryon drag θ d , measured at O(1%) precision within the latest BOSS data. As discussed in Chapter 1, these angular scales are defined as θ X ≡ r X /D A (z obs ), where the numerator is either the sound horizon at recombination r s or sound horizon at baryon drag r d , while the denominator is the comoving angular diameter distance D A (z obs ) to recombination, z obs z rec , or to the redshift of the survey, z obs 0.1 -3. Given that the relationship between r s and r d is fixed within the ΛCDM cosmology, it is common to say that one can calibrate BAO using the CMB sound horizon measurement. The sound horizon is calculated as

r s = ∞ zrec c s dz H(z) , ( 2.7)
while the comoving angular diameter distance in flat space is given by :

D A (z obs ) = z obs 0 dz H(z) . (2.8)
There are different ways to approach the question of how the angles θ X depend on the Hubble parameter. Let us take the view in which the physical densities ρI contribute to the Hubble parameter H(z) via the Friedmann law (in other words, we parametrize densities with ω i , rather than Ω i ). One recognizes that a change of the Hubble parameter today (H 0 ≡ H(z = 0)) can only be produced by a change in the total physical density at present. Therefore, this change in H 0 yields an impact on the angular diameter distance D A (affected by densities at z ∈ [0, z obs ]), but not on the sound horizon r s (affected by densities at z ∈ [z rec , ∞]). This means that there are two ways to increase the inferred value of H 0 while keeping the angles θ X fixed • Early time solutions. This class of solutions aim to decrease the sound horizon r s , by changing the physics before or around recombination, z z rec . In this way, the shift in θ X is compensanted by an increase of H 0 , which rescales the angular diameter distance.

• Late time solutions. Those solutions change physics well after recombination (z z rec ) in such a way to attain a higher energy density today and thus a higher H 0 . However, given that r s is unchanged, one has to be careful to keep the angular diameter distance fixed in order not to change the measured θ X . This can be accomplished by requiring the energy density to be smaller in the past (but still at z < z rec ), in such a way so as to compensate for the higher energy density today. This is known as the "geometrical degeneracy" within CMB data

The problem for late time solutions, however, arises from the multitude of low redshift measurements (such as supernovae, BAO, cosmic chronometers), which tightly constrain H(z)/H 0 in a redshift range of around z 2. This becomes particularly clear when combining BAO and supernovae data. Notice that, because of the distance duality relation in Eq. (1.41), those two sets of observations are actually providing a measurement of the same type of distance. This is illustrated in Fig. 2.1, where we show the difference in luminosity distances with respect to the prediction from the best-fit Planck18 ΛCDM model (Pl18-ΛCDM, to shorten notation), for two different datasets: 1) the Pantheon SNIa calibrated using the SH0ES value of M b , and 2) the BAO data calibrated using the Pl18-ΛCDM value of r d and converted from angular diameter distance with 5 d L (z) = (1 + z)D A (z). We see a clear discrepancy between the BAO and SNIa data. This makes it hard for late-time modifications to resolve the tension, as it is not possible to fit simultaneously BAO and SNIa data without changing r d . This problem can be reformulated in several equivalent ways: We conclude that in order to quantify the ability of a certain model to solve the tension with SH0ES, it is very important to use a prior on the intrinsic SNIa magnitude, M b , rather than a prior on H 0 , for two main reasons: inverse distance ladder calibration of SNIa. In order to correctly penalize this kind of models, the use of a prior on M b is essential.

From the above discussion, it becomes clear that a minimum ingredient that any model attempting to solve the tension should have (although likely not the only one) is the ability to decrease the sound horizon at recombination. In the next section we proceed to describe one of such models.

Formalism of Early dark energy

The idea that an anomalous era of expansion arising from EDE at early times might resolve the Hubble tension was first suggested in KARWAL et al. ( 2016

Basic equations

In

V n (Θ) = m 2 f 2 [1 -cos(Θ)] n , ( 2.9)
where m represents the axion mass, f the axion decay constant, and Θ ≡ φ/f is a re-scaled dimensionless field variable defined such that -π ≤ Θ ≤ π. The dynamics of EDE are dictated by the Klein-Gordon equations at the background level

φ + 2Hφ + a 2 V n,φ = 0, (2.10)
and at the linearly perturbed level

δφ + 2Hδφ + k 2 + a 2 V n,φφ δφ = - h 2 φ , ( 2.11)
where h denotes the scalar metric potential in the synchronous gauge. The elements of the scalar field stress-energy tensor are given by 7

ρφ = a -2 φ 2 2 + V n , Pφ = a -2 φ 2 2 -V n , ( 2.13
) The background dynamics of EDE is easy to understand: at early times the scalar field is frozen-in due to Hubble friction and contributes as dark energy to the expansion; once the Hubble rate drops below its mass, Hubble friction is removed and the field begins to perform damped oscillations about the minimum of the potential; the rate of energy dilution is related to the period-averaged equation of state, roughly given by w(n) = (n -1)/(n + 1). In Fig. 2.2 we show the redshift evolution for the fractional energy density and the equation of state in the EDE. We see that EDE provides exactly the required behavior to explain a high value of H 0 : thanks to the very peaked shape of its energy injection, it can boost the expansion rate just prior to recombination, thereby reducing the value of the sound horizon.

δρ φ = a -2 φ δφ + V n,φ δφ, δP φ = a -2 φ δφ -V n,φ δφ, ( 2.14)
(ρ φ + Pφ )θ φ = k 2 a -2 φ δφ, σ φ = 0. ( 2
To perform our analyses, we use the modified version of CLASS presented in T. L. SMITH et al. (2020). The code is publicly available at https: //github.com/PoulinV/AxiCLASS. The main goal of this code is to solve the Klein-Gordon equations Eq. (2.10) and Eq. (2.11), and to perform a shooting method in order to map a choice of {z c , f EDE } into the theory parameters {m, f }. 7 The expressions for the overdensity, perturbed pressure, energy flux and shear can easily be obtained by considering linear perturbations in the general expression for the stress energy tensor of a scalar field in Eq. (1.44),

δT µ ν = ∂ µ φ∂ν δφ + ∂ µ δφ∂ν φ -δ µ ν ḡαβ 2 (∂αφ∂ β δφ + ∂αδφ∂ β φ) + V n,φ δφ = -δ µ 0 a -2 φ ∂ν δφ + δ 0 ν φ ḡαµ ∂αδφ + δ µ ν a -2 φ δφ -V n,φ δφ , ( 2.12)
and then using the definitions of δρ, δP , θ and σ in Eq. (A.30). In the following section, we detail the results of our different data analyses for the EDE model. We start by performing an 'anatomy' of the EDE resolution to the Hubble tension, to understand how each data set reacts to the presence of EDE. We additionally introduce the new baseline 1-parameter EDE model that is favored by Planck18 2018 data. Then, we present our N -Body simulations validating for the first time the HMcode prediction and confront the 1-parameter EDE model against WL data. We end by discussing the H 0 and S 8 tension in light of the lensing-marginalized CMB spectrum.

Cosmological constraints

In this section, we test the EDE scenario with various combinations of data-sets, in order to extract the cosmology that would resolve the Hubble tension, and compare with results from past literature making use of Planck 2015 data.

Baseline analysis: anatomy of the 3-parameter EDE model

Our baseline cosmology consists in the following combination of the six ΛCDM parameters {ω b , ω cdm , H 0 , n s , A s , τ reio }, plus three parameters describing the EDE sector, namely {f EDE (a c ), Log 10 (a c ), Θ i }8 . We use wide flat priors on all these parameters. We follow the Planck convention and assume two massless neutrinos and one massive with m ν = 0.06 eV. We perform our MCMC analyses using MontePython-v3 9(AUDREN et al. 2013; BRINCKMANN et al. 2019a), and consider chains to be converged with the Gelman-Rubin criterion10 R -1 < 0.1 [START_REF] Gelman | Inference from Iterative Simulation Using Multiple Sequences[END_REF]). To extract best-fit parameters, we make use of the MINUIT algorithm [START_REF] James | Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations[END_REF]) through the iMinuit python package 11 . Starting from Planck only, we now discuss the impact of adding data-sets on the reconstructed EDE parameters. We compare the evolution of the χ 2 in the EDE cosmology as we add data-sets, to that of the ΛCDM model in the same combined fit, with and without SH0ES data. The results are presented in Tab. (2.1) and we show the 1D and 2D posterior distributions of {H 0 , f EDE (z c ), Θ i , Log 10 (z c ), ω cdm , n s , S 8 } in Fig. 2 Table 2.1 -The mean (best-fit) ±1σ error of the cosmological parameters reconstructed from the combined analysis of various data sets (from left to right, each column adds a set of data to the previous one). We also report the ∆χ 2 min with respect to a ΛCDM fit to the same data-sets, with and without a prior on H 0 from SH0ES.

the EDE model 13 . One can already note a curiosity: the best fit value of f EDE (z c ) is very close to the 2σ bound that we obtain. This, as we will discuss later, is due to the choice of flat, uninformative prior on Log 10 (z c ) and Θ i .

Planck TT,TE,EE+SH0ES: Once a prior on H 0 given by SH0ES is included in the analysis, the sampler explores more easily a part of parameter space with higher H 0 values, and the EDE is now well detected: {f EDE (z c ) 0.11 (z c ) are in perfect agreement with that obtained Planck18 only. This is highly non-trivial, and seem to indicate that Planck18 does favor the region of the {Log 10 (z c ),Θ i }-space that resolves the Hubble tension. However, the best fit fraction reaches 15%, a value that one would naively consider to be strongly excluded by the Planck18 only analysis. In fact, that is not the case, as the the fit to Planck18 data is barely affected by the additional H 0 prior, while one can get an excellent fit of SH0ES data. Concretely, the χ 2 min (EDE) when fitting Planck18+SH0ES increases by ∼ 3, such that even in this combined fit, the χ 2 of Planck18 data is smaller than that of ΛCDM fitted on Planck18 data only. This indicates that the limit on f EDE (z c ) derived in a Planck18 only analysis is not robust, as it is entirely driven by our choice of flat priors. This was also discussed in NIEDERMANN et al. (2020) and T. L. SMITH et al. (2020), and the reason for that is clear 14 : there exists a strong χ 2 degeneracy in Planck18 data between ΛCDM and the EDE cosmology, that, given our choice of uninformative priors on Θ i and Log 10 (z c ), leads to an artificially strong bound on f EDE (z c ). Indeed, once f EDE (z c ) drops below 4% (as seen from the 2D posterior), its impacts on the power spectrum is not detectable given current measurement accuracy. As a result, the quantity Log 10 (z c ) and Θ i have no impact on observables, such that any choice of Log 10 (z c ) and Θ i leads to a cosmology indistinguishable from ΛCDM. Therefore, the sampler spends much more time exploring this degeneracy direction, rather than efficiently sampling the narrow degeneracy between f EDE (z c ) and H 0 , which requires a specific choice of Log 10 (z c ) and Θ i to appear 15 . Following NIEDERMANN et al. (2020), we will discuss a natural way to alleviate this issue in Sect. 2.4.2.

Planck TT,TE,EE+PP+BAO+SNIa+SH0ES:

We now add to our analysis the lensing reconstruction PLANCK φφ , the PANTHEON SNIa data-set, and the BAO data from BOSS. Strikingly, the addition of these three data-sets has Table 2.2 -The mean (best-fit) ±1σ error of the cosmological parameters reconstructed from the combined analysis of various data-sets (from left to right, each column adds a sets of data to the previous one). We also report the ∆χ 2 min with respect to a ΛCDM fit to the same data-sets. In the last row, we also report the ∆χ 2 with respect to ΛCDM fit to the combined data without SH0ES in parenthesis.

almost no impact on the reconstructed posteriors, nor on the best fit. This is far from a trivial test to pass, as many of the suggested resolutions to the Hubble tension are strongly constrained by the addition of these datasets (BERNAL et al. ), this opens up the possibility of constraining the EDE resolution using LSS data, and in particular the S 8 measurement from weak gravitational lensing surveys. However, combining KiDS+VIKING/HSC data with Planck18 to constrain extension to ΛCDM can be problematic as: i) they require the ability to predict the non-linear power spectrum at relatively small scales in models beyond ΛCDM; ii) the ΛCDM best fit model from Planck18 is not a good fit to these data.

All Data: As a starter, we add the 'consensus' f σ 8 BOSS likelihood to the analysis, which is consistent with the ΛCDM model from Planck18; we find a mild ∼ 0.4σ decrease in the reconstructed mean, now being H 0 71 ± 1 km/s/Mpc and f EDE 0.1 ± 0.03. This is consistent with the fact that the f σ 8 measurements are sensibly lower than the ΛCDM prediction, while the EDE cosmology leads to slightly larger values. Still, the χ 2 of the FS data is barely affected; in fact, as before, ΛCDM provides a slightly worse fit to the joint data-set, even when the SH0ES prior is not included in the analysis. Before including weak lensing measurements to the analysis, we therefore conclude that the 3-parameter EDE model under study performs very well in resolving the Hubble tension, but future measurement of f σ 8 will certainly put the model under crucial tests.

Towards a 1-parameter resolution to the H 0 tension

Before turning to the inclusion of WL data, we show that the apparently tight bounds obtained when the SH0ES prior on H 0 is not included is due to our choice of uninformative priors for Θ i & Log 10 (z c ). In fact, one can strongly weaken the bound on f EDE by reducing the EDE parameter space to a suitable choice of Log 10 (z c ) and Θ i . This might sound counter-intuitive: in principle, one expects to relax constraints on a given parameter by enlarging the parameter space such as to introduce a new degeneracy. Here however, it is the poor prior choice which leads to a strong bound on f EDE (z c ) independently of the data combination. Fixing Log 10 (z c ) and Θ i to some fiducial values surely rises the question of what values should one choose. In a realistic scenario, one might know these values a priori; one example is the scenario discussed in SAKSTEIN et al. (2020) in which a scalar field experiences a phase-transition around the redshift at which neutrinos become non-relativistic 16 , such that the critical redshift is specified by the value of the neutrino mass, while 2020) for different EDE models with fewer free parameters). Here however, we have been considering a phenomenological model whose primary characteristics is to have enough freedom to extract information from the data to resolve the tension -we will therefore make use of that information and fix Θ i & Log 10 (z c ) to their best fit value from Planck18 data only -which, we recall, are close-to-identical to that obtained in the combined fit.

We report in Tab. (2.2) the reconstructed cosmological parameters from Planck18 only and from the combined fit of all data, with and without including SH0ES. We show the reconstructed 2D posteriors of {f EDE (z c ), H 0 , S 8 } in Fig. 2.4. Notice how the degeneracy direction f EDE (z c )-H 0 clearly opens up. Furthermore, the mild ∆χ 2 preference in favor of the EDE cosmology now leads to reconstructing f EDE (z c )= 0.082 ± 0.037, i.e., a ∼ 2σ preference for non-zero EDE from Planck18 data only.

The inferred H 0 = 70.1 ± 1.4km/s/Mpc is now in agreement with the SH0ES determination at better than 17 2σ. The addition of BAO, FS and PANTHEON measurements has little impact; the reconstructed EDE fraction shifts downward by ∼ 0.3σ, slightly degrading the success of the resolution to the Hubble tension, while the 2σ preference for non-zero EDE is still present. These results are in excellent agreement with these presented in NIEDERMANN et al. (2020) for a different EDE model.

Finally, the inclusion of a prior from SH0ES pulls up the fraction of EDE to f EDE (z c )= 0.118 ± 0.029 and the value of H 0 = 71.7 ± 1 km/s/Mpc, at the cost of a small degradation in χ 2 min (∆χ 2 ∼ +6). Yet, as before, the χ 2 min of the combined fit Planck18+BAO+FS+PANTHEON+SH0ES in the 1-parameter EDE cosmology is slightly lower than a ΛCDM fit to Planck18+BAO+FS+PANTHEON (no SH0ES). This attests that, despite this small degradation in χ 2 min , the goodness of fit is still excellent. However, as discussed previously, the values of S 8 are in significant tension with weak lensing measurements, and one might expect that it is possible to strongly constrain the EDE model by including LSS data. We study this possibility in detail in Sect.

Confronting EDE to weak-lensing data

In order to make use of weak gravitational lensing data to perform LSS analyses, one needs to accurately model the matter power spectrum in the late-time non-linear regime. To this purpose, one can adopt the HALOFIT semi-analytical prescription R. E. 2015), its main advantage being its flexibility to account for the effects of accounting for the coupling between EDE and the neutrino sector, since adding non-standard neutrino interaction typically shifts neutrino mass constraints (see Chapter 4). 17 For all the EDE analysis carried out in this section, we quote 'tension' and 'agreement' assuming Gaussian posteriors for simplicity. While this is surely a crude approximation, we believe this is justified because the posterior of interest (H0, S8) are close to Gaussian. In Sect. First, we confront the non-linear matter power spectra produced by using HALOFIT /HMCODE in the EDE framework, against the outputs of dedicated cosmological N -Body simulations, to explicitly demonstrate the accuracy of our LSS data analyses. Afterwards, we discuss the results of our MCMC analysis against weak lensing data.

Non-linear matter power spectrum: a comparison with N -Body simulations

The goal of this Section is to show that the impact on the non-linear matter power spectrum due to the presence of EDE is mainly due to changes in the standard ΛCDM free parameters with respect to their reference values, and therefore such impact can be safely studied without further modifying or re-calibrating HALOFIT /HMCODE .

To this end, we perform two sets of N -Body DM-only simulations (one set for the EDE and one for the ΛCDM scenarios), as reported in Tab. To bind together the matter power spectra extracted from simulations with different resolutions we adopt a splicing technique described in detail in App. C.2. Our results are summarized in Fig. 2.5, and we refer to App. C.2 for a deeper technical discussion.

In the top panel of Fig. 2.5 we compare the matter power spectra extracted from our simulations, with the ones computed with HALOFIT /HMCODE , at redshift z = 0. The blue curves refer to the ΛCDM scenario -dubbed hereafter as ΛCDM "equivalent" .5 -In the top left panel we show the matter power spectra extracted from our simulations, and the ones computed with HALOFIT /HMCODE . The blue curves refer to the ΛCDM scenario, while the red ones refer to the EDE best fit model. We also report the best fit ΛCDM case from Planck 2018.The spliced power spectra are reported as thick dot-dashed lines. Symbols stand for the outputs of the LB and HR simulations. The solid/dotted lines are the non-linear power spectra from HALOFIT /HMCODE , whereas the dashed lines are the corresponding linear power spectra used to produce the initial conditions for the simulations. The cyan shaded band roughly corresponds to the scales probed by DES-Y1. In the top right panel, we show the ratio between the non-linear matter power spectra from our simulations and the ones computed with HALOFIT /HMCODE , for both the ΛCDM "equivalent" and the EDE best fit models, adopting the same linestyle-code and color-code. In the bottom right panel we compare departures from the ΛCDM model in terms of ratios of non-linear matter power spectra, adopting the same linestyle-code and color-code.

-while the red ones refer to the EDE best fit model. As a reference, we also report the best fit ΛCDM case from Planck 2018. The spliced power spectra are denoted by thick dot-dashed lines. Symbols stand for the output power spectra of the "non-spliced" LB and HR simulations. The solid/dotted lines are the non-linear power spectra from HALOFIT /HMCODE , while the dashed lines are the corresponding linear power spectra used to set the initial conditions for the simulations. In the right panel, we adopt the same linestyle-code and color-code to show the ratio between the non-linear power spectra produced by HALOFIT /HMCODE with respect to the ones extracted from our simulations. The thick horizontal lines highlight ±5% deviations.

In App. C.2 we extend the analysis to three additional redshift binsz = 0.5, 1.5, 2 -obtaining analogous results. We can thus conclude that the differences between HALOFIT /HMCODE predictions with respect to the outputs of our N -Body simulations are below 5% level, for scales 10 -2 k 10 h/Mpc, at redshifts 0 ≤ z ≤ 2, for both ΛCDM and EDE models. Whereas this is a very well established result for the ΛCDM paradigm, this is not often the case for alternative cosmological scenarios, such as the one considered in this work. Let us note that the exponential increase in the difference between the outputs by simulations and HALOFIT at k ∼ 10 h/Mpc is absent when one compares the outputs from simulations with the predictions by HMCODE . As expected, the latter method is more accurate than HALOFIT in modeling the very small-scale and very low-z regime. In this work, we therefore make use of HMCODE to model the non-linear evolution of perturbations, following the approach adopted by the KiDS collaboration.

We also present our results in terms of ratios between the matter power spectra in the EDE and the ΛCDM models in Fig. 2.5 bottom panel. The comparison between the EDE best fit and the ΛCDM equivalent confirms that it is not the intrinsic presence of EDE that enhances the matter power spectrum on small scales, exacerbating the S 8 tension. Rather, the EDE reduces the growth of perturbations at fixed ω cdm . As already pointed out, such power enhancement is instead due to variations in the standard ΛCDM parameters -mostly an increase in ω cdm -induced to balance the EDE impact on the CMB. This suggests that the limitations of the EDE are not intrinsic to its presence, but rather to an accidental degeneracy that could be alleviated in an extended model (i.e. by adding non-standard interactions to the dark matter sector, see Chapter 3).

In view of these considerations, it is straightforward to conclude that LSS surveys constitute an ideal counterpart to CMB data, given the complementarity between the regimes that they probe. However, in section Sect. 2.4.3 we will show that currently available weak lensing data are not sensitive enough to unequivocally capture the signature of EDE. This will clearly not be the case when more precise data (e.g. from Euclid (AMENDOLA et al. 2018)) will become available. As also our results suggest, it will soon be necessary to go beyond the HALOFIT /HMCODE prescription for modeling the non-linear power spectrum (see e. We leave these tasks for future work.

MCMC analysis against weak lensing data

In the following, we will focus on the 1-parameter EDE cosmology, fixing Θ i and Log 10 (z c ) to their best fit values from Planck18 only. Firstly, we test the model against the KIDS-VIKING cosmic shear measurements. We follow the prescription described in HILDEBRANDT et al. (2020) and make use of the HMCODE algorithm (MEAD et al. 2015) (with 9 nuisance parameters) to model the non-linear matter power spectrum. Secondly, we perform an analysis trading KIDS-VIKING data for a split-normal

Results for ΛCDM :

Starting with the ΛCDM cosmology, we find that combining Planck18 with KIDS-VIKING data leads to a mild degradation of the χ 2 min of the combined fit: While one might naively expect that the χ 2 min of the global fit should be roughly the sum of the χ 2 min of individual fits, we find that the global χ 2 min is degraded by ∼ +6.5. Similarly, the inclusion of a tight Gaussian likelihood on S 8 as measured by KIDS-VIKING+DES leads to a degradation in the combined χ 2 ∼ +15.5, while one expects ∼ +1 for a good fit. In Fig. 2.6, we show the reconstructed 2D posteriors of {H 0 , S 8 , 10 -9 A s , ω cdm , Ω m } in the ΛCDM model. One can see that the degradation in χ 2 min is accompanied by shifts in the mean of any parameter correlated with S 8 , in 18 We stress that S8 is a model-dependent quantity, and it is particularly sensitive to the treatment of the neutrino mass. We therefore make use of the value that was derived following our convention, i.e. at fixed mν = 0.06 eV. particular A s , ω cdm and H 0 , without succeeding in getting a good fit to the WL data. We therefore stress that any of the combined results should be taken with a grain of salt, even in the ΛCDM framework. This joint analysis serves mostly to demonstrate that the EDE cosmology does not sensibly degrade the fit to the S 8 measurement as compared to ΛCDM, and that currently available WL measurements do not strongly constrain the EDE resolution to the Hubble tension.

Results for EDE against Planck+KiDS-VIKING:

In Fig. 2.7 we show the reconstructed 2D posteriors of {f EDE (z c ), H 0 , S 8 , Ω m } in the 1-parameter EDE realization for various data combinations. We start by performing an analysis of EDE against KIDS-VIKING data only; as expected we find that the KIDS-VIKING data have no constraining power on the fraction of EDE. However, the reconstructed S 8 = 0.738 KIDS-VIKING to Planck18+BAO+PANTHEON+FS, a non-zero EDE contribution is still favored at ∼ 1.5σ, but the reconstructed mean fraction has moved downwards by ∼ 0.7σ. This was expected, given the positive correlation between f EDE (z c ) and S 8 . In this cosmology, Planck18 data are still slightly better fitted (∆χ 2 min ∼ -6) than in ΛCDM , while the fit to KIDS-VIKING data is degraded by ∼ +2. Once a prior on H 0 from SH0ES is added, we find again f EDE (z c ) ∼ 10 ± 3%, at the cost of increasing the total χ 2 min ∼ +5.5. The increase in χ 2 is partly due to the inclusion of SH0ES (χ 2 ∼ 1.62, a reasonably good fit), and also to a mild degradation in the fit to Planck18 (∼ +3) and BAO (∼ +1.6). The reason is that the inclusion of KIDS-VIKING data reduces the degeneracy between f EDE (z c ) and the ΛCDM parameters, in particular the one with ω cdm . Note that the goodness of Planck18 fit is not sensibly degraded as compared to ΛCDM , since the χ 2 stays better than that from ΛCDM fitted on Planck18 only. In fact, when compared to ΛCDM, the combined χ 2 is improved by ∼ -13 (for one extra parameter), indicating a significant preference for EDE despite the presence of KIDS-VIKING data. Looking at the individual χ 2 min , we find indeed that the quality of the fit to KIDS-VIKING data in the EDE cosmology that resolves the Hubble tension is barely changed (∆χ

Results for EDE against Planck+KIDS+DES:

We now trade KIDS-VIKING data for a split-normal likelihood on S 8 as inferred from the joint KIDS-VIKING+DES data. We note that the tension between the value of this joint S 8 measurement and that predicted by our fiducial EDE model (obtained from the global fit of Planck18+BAO+FS+ PANTHEON+SH0ES) is at the ∼ 3.8σ level (slightly increased from 3.2σ tension in ΛCDM). It would be interesting to quantify the level of tension between these data sets in the ΛCDM framework using more robust statistical tools than the 'difference in the mean' used here, as done for instance in HANDLEY et al. ( 2019) and RAVERI et al. (2019). However, we note that authors from RAVERI et al. (2019) found that the less precise KIDS-VIKING data available at that time were already in significant statistical disagreement with the prediction from ΛCDM . We anticipate that this more robust approach would strengthen the case for a statistically significant discrepancy, even in ΛCDM, and therefore the need to apply caution when drawing conclusions from the combined analyses. We show the reconstructed 2D posteriors of {f EDE (z c ), H 0 , S 8 , Ω m } in the 1-parameter EDE model in Fig. 2 space. We have simply adopted a different 'prior' choice on Θ i and Log 10 (z c ) (i.e. here we fix them), demonstrating that the current constraints from WL -besides being derived from statistically inconsistent data set 19 -are not robust. Looking at χ 2 min , we find that the resulting best fit cosmology degrades the fit to Planck by ∼ +6 while providing a poor fit to the S 8 likelihood (χ 2 = 8.3 for a single data point). Still, the best-fit is marginally better than that of ΛCDM adjusted on the same sets of data (∆χ 2 min ∼ -1.5). Once we include the SH0ES prior, we find again f EDE (z c ) to be nonzero at more than 3σ, f EDE (z c ) 9 ± 3%, with a global ∆χ 2 min -11.6. Looking at individual χ 2 min , we find that the fit to PLANCK TTTEEE , PLANCK φφ , BAO and FS data is somewhat degraded compared to the best fit EDE cosmology obtained without S 8 prior, as a consequence of the breaking of the f EDE (z c )-ω cdm degeneracy. However, as expected, we note that the S 8 likelihood has a χ 2 9, which is not particularly worst that the one obtained in the ΛCDM case without SH0ES (χ 2 8.3). This indicates that any constraint on the EDE derived from this combined analysis should be regarded with caution, as the cosmology reconstructed from the analysis does not provide a good fit to the S 8 data. This naturally impacts the reconstructed H 0 , which is ∼ 0.6σ lower than without the S 8 likelihood, although the fit to SH0ES is still reasonably good (χ 2 2.4). We therefore conclude that current S 8 measurements do not exclude the EDE resolution to the Hubble tension; however, they do call for new physics beyond EDE -or unknown systematics -to explain the intriguingly low measured S 8 values.

EDE and the S 8 tension in light of the A lens anomaly

It has been noted that there exists a number of 'curiosities' in Planck18 that can potentially shed light on cosmological tensions. In particular, there is a residual oscillatory feature in the Planck18 TT data at 1100 2000 compared to the best fit ΛCDM prediction [START_REF]Planck intermediate results. LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters[END_REF](AGHANIM et al. , 2020b)). This feature can be captured by an extra source of smoothing of the acoustic peaks, as modelled by the 'A φφ lens ' 19 Since Pl18-ΛCDM provides an even worse fit to SH0ES data, one could argue that Planck and SH0ES are also statistically inconsistent. However, we remind that the notion of "statistically inconsistent datasets" is generally model-dependent (specially when including CMB or other early-universe data), and in fact Pl18 -EDE gives a very good fit to SH0ES. This latter point is only evident when looking at the long tails of the EDE posteriors, so to properly quantify the agreement with SH0ES one should consider metrics that generalize the standard Gaussian Tension estimators, as we do in Sect. 2.5.2. parameter which is used to re-scale the amplitude of the lensing potential power spectrum C φφ → A φφ lens C φφ, at every point in parameter space. However, the amplitude of the lensing potential power spectrum can also be estimated directly from the lensing-reconstruction and is compatible with the ΛCDM expectation, such that while this extra smoothing looks like lensing, it cannot be attributed to actual gravitational lensing. A thorough investigation of the lensing-like tensions in the Planck legacy release was performed in AGHANIM et al. (2017) and MOTLOCH et al. (2018,2020). It has been noted in particular that, once marginalizing over the lensing information, the 'unlensed' CMB temperature and polarization power spectra favor a cosmology with a lower A s and Ω cdm h 2 . Indeed, these parameters are strongly correlated with the amplitude of the lensing power spectrum, such that the lensing-like anomaly pulls these parameters up. Additionally, since the acoustic feature of the CMB tightly constraints the parameter combination Ω m h 3 , a lower Ω cdm h 2 is compensated by a higher H 0 . As a consequence, the unlensed ΛCDM cosmology shows no S 8 tension, and a milder (although still > 3.5σ significant) H 0 tension. It was also pointed out that this 'unlensed' cosmology is in good agreement with the ΛCDM cosmology reconstructed from the SPTPol data (HENNING et al. 2018; CHUDAYKIN et al. 2020a).

It is therefore reasonable to ask what is the impact of such anomalies on extensions to ΛCDM like the EDE under study. To that end, we introduce two additional parameters A lens and A φφ lens whose goal is to marginalize over the lensing information in Planck 20 . The latter parameter re-scales the amplitude of the theory lensing potential power spectrum, while the former only re-scales the amplitude of the acoustic peak smoothing. In practice, the amplitude of the acoustic peak smoothing is then determined by the product A TTTEEE lens ≡ A lens × A φφ lens . We first perform MCMC analysis of the ΛCDM and EDE cosmologies against Planck18 data. In a second step, we perform a global analysis combining all the data considered in this work. As before, the joint KIDS-VIKING+DES results is modeled via a split-normal likelihood distribution on S 8 . The results of these analysis are reported in Tab. (2.6) and shown in Fig. 2.9 and Fig. 2.10 .

Results for ΛCDM:

We start by analyzing the ΛCDM cosmology in light of the 'unlensed' Planck spectra. We confirm the results of AGHANIM et al. (2020b) and MOTLOCH et al. (2020): we find that the amount of lensing determined from the peak smoothing A TTTEEE lens is ∼ 2.8σ higher than the expectation from the ΛCDM model deduced from the 'unlensed' CMB power spectrum. Moreover, the difference between the reconstructed A φφ lens 1.07 ± 0.04 and A TTTEEE lens 1.2 ± 0.07 illustrates the fact that this extra smoothing component cannot be due to actual gravitational lensing. Still, this ΛCDM 'unlensed' cosmology is now in much better agreement with the S 8 measurements from KiDS and DES, as can be seen in Fig. 2.9. This is due to the fact that the reconstructed A s and ω cdm are lower than in the analysis including lensing information. We then perform a global analysis, including all data sets considered in this work. We find that the 'unlensed' ΛCDM cosmology can indeed accommodate a low S 8 , however this is at the cost of worsening somewhat the fit to BAO+FS data (∆χ 2 +3.5), when compared to the 'concordance' ΛCDM model obtained from a fit to the full Planck data, BAO and FS (without S 8 priors). Additionally, we note that accommodating such a low S 8 requires a somewhat smaller ω cdm and A s (by a little less than 1σ), which are compensated for by pulling up the A φφ lens and A TTTEEE lens by a similar amount. The fit to SH0ES on the other hand is still very poor, χ 2 min 10, suggesting that the global unlensed cosmology is still in strong tension with SH0ES.

Results for EDE:

Turning now to the 1-parameter EDE model, we wish to check whether the EDE cosmology deduced from 'unlensed' Planck18 spectra only is in better agreement with both S 8 and H 0 direct measurements 21 . As one can see from Fig. 2.10, the lensing-marginalized CMB data does favor non-zero f EDE (z c ) at ∼ 2σ (∆χ 2 min -4 with respect to ΛCDM) and predicts H 0 71.2 ± 1.5 and S 8 0.81 ± 0.02. Compared to the EDE cosmology reconstructed from the full Planck data, the 'tension' with H 0 and S 8 has therefore decreased by ∼ 1σ due to a shift in the mean of the reconstructed posterior in the unlensed cosmology. It is now in 1.4σ agreement with SH0ES but stays in mild (∼ 2.7σ) tension with the combined S 8 measurement. The S 8 prediction is however in very good agreement with the KIDS-VIKING and DES measurements when considered individually (an important note since the combined low S 8 value relies on a re-calibration of DES'photo-metric redshift by the KIDS-VIKING team). Additionally, the A φφ lens and A TTTEEE lens parameters are unchanged with respect to that reconstructed in the ΛCDM cosmology. Therefore, while the anomalous amount of lensing in Planck18 data is not an effect due to the presence of the EDE, these parameters do not correlate with a non-zero f EDE (z c ), i.e., they do not take values different from the ΛCDM ones to 'hide' the effect of the EDE.

Once all data sets are included in the analysis, a non-zero fraction of EDE is favored at ∼ 3.5σ. Interestingly, most of the reconstructed parameters do not shift by more than ∼ 0.5σ; rather, the uncertainty on the reconstructed parameters tighten significantly, as one would expect from making use of additional data. However, 21 A similar study was performed in MOTLOCH et al. (2020) for N eff . There, it was found that polarization and BAO data exclude N eff as a resolution to the Hubble tension, even after marginalizing over the lensing anomaly in Planck18.

similarly to what happens in the ΛCDM cosmology, the inclusion of the tight-andlow S 8 value does force a slightly (∼ 1σ) smaller A s , that is compensated by slightly higher A φφ lens and A TTTEEE lens parameters. The fit to SH0ES is good (χ 2 ∼ 1.9) and stable when compared to that obtained including Planck lensing information. On the other hand, as expected, the fit to the joint KIDS-VIKING+DES S 8 is better than in the 'lensed' cosmology (∆χ 2 ∼ -4.3), but its value is still somewhat poor (χ 2 ∼ 4). We emphasize again that the fit to individual S 8 measurements, on the other hand, is excellent. If future S 8 measurements stay low while becoming more precise, they will be in tension even with the 'unlensed' cosmology (whether ΛCDM or EDE), confirming the need for new physics beyond EDE (or an unknown systematic effect).

EDE in the context of other proposed solutions

We have seen that EDE provides an excellent fit to Planck and SH0ES data, even when LSS data is considered in the analysis. However, there are hundreds of solutions beyond the ΛCDM framework that have been proposed in the recent years, so it is fair to ask whether these models perform worse, better or equally good than EDE in explaining the H 0 discrepancy. Hence, in the second part of this Chapter, we seek to ascertain the relative success of various cosmological models proposed to solve the H 0 tension. We do this by systematically confronting each of the considered models to data from the early and late Universe, assessing at each point the extent to which the tension between the Planck and local measurements remains. This allows a direct comparing between EDE and other proposed solutions, but the broader intent is really to better understand the successes and drawbacks of each approach, and to generate a meaningful set of benchmarks for future proposals.

Brief overview of competitors

We have considered seventeen different models which are intended to be representative of the wide variety of models presented in the literature, such as those listed in DI VALENTINO et al. (2021b). For clarity, we split the models in three different categories: early Universe models that invoke extra relativistic dark relics (in addition to other ingredients), alternative early Universe models that do not involve dark radiation, and finally, models modifying the cosmological expansion at late times (i.e.well after recombination). Below we list all the models that we have considered. 2020a)) that ∆N eff fails at resolving the Hubble tension. Thus, rather than a real competitor, we will treat it as a useful benchmark model, in order to assess the extent to which the additional complexity introduced by other models really helps for relieving the Hubble tension. Remark also that the SIDR model is very similar to the ∆N ur model, except that the anisotropic stress vanishes, σ dr = 0, in order to reflect the fact that the dark radiation behaves as a perfect fluid due to the strong self-interactions. We note that, in order to decrease the sound horizon at recombination r s (z rec ), the first three models in this category increase the recombination redshift z rec , while the last three models change the expansion rate H(z)/H 0 near recombination. The combination varying m e +Ω k might appear surprising (since curvature has an entirely disjoint cosmological effect to that of a varying electron mass), but it is mainly included because it produces a strong shift in H 0 . The reason for this degeneracy is further explained in SEKIGUCHI et al. (2021). We have verified explicitly that this is a peculiar degeneracy that uniquely appears in this model -allowing curvature to float in other models seems to have no significant impact. 2020a). Therefore, rather than a real competitor, we will consider the CPL parameterization of the Dark Energy equation of state as a historical benchmark model, useful to gauge the extent to which more complex late-universe models truly performs. In addition, notice that the PEDE model considers a parametrization of the DE energy density with no new free parameters (i.e. it has not ΛCDM limit), while GPEDE has an additional parameter such that ΛCDM can be recovered for a particular value of this parameter.

Dark

Other early

Late Universe solutions

Statistical tests to quantify model success

For each model M and compilation of data sets D, we discuss three ways to quantify the tension between the cosmological inferred value and the SH0ES experiment 22 , each one related to a different question:

• Criterion 1: When considering a dataset D not including SH0ES, what is the residual level of tension between the posterior on M b inferred using D and the SH0ES measurement?

The tension on x = M b can be quantified through the Gaussian Tension (GT), defined as

xD -xSH0ES (σ 2 D + σ 2 SH0ES ) 1/2 , ( 2.16)
where x i and σ i are the mean and standard deviation of observation i. A problem of this metric is that it is only strictly valid if the parameter's posteriors are Gaussian, and it could disfavor models with long tails in the posterior. This can happen for instance if the data set D cannot disentangle between ΛCDM and a more complex model which has parameters that become irrelevant when others are close to their ΛCDM limit. As a result, the posterior is necessarily dominated by the Gaussian ΛCDM limit, and the easing power of the model can only show up in the aforementioned tails of the probability distribution (this is precisely what happens with EDE cosmologies, as we saw in Sect. 2.4.2). In addition, this criterion does not quantify how good (or bad) the χ 2 of the new model is. As a result, a model which does not contain the ΛCDM best fit (like the PEDE model) can appear arbitrarily good. In order to avoid such problems, we instead use the two additional tests listed below.

• Criterion 2: How does the addition of the SH0ES measurement to the data set D impact the fit within a particular model M?

We compute the change in the effective best-fit chi-square χ 2 = -2 ln L between the combined data set and the dataset D as

∆χ 2 = χ 2 min,D+SH0ES -χ 2 min,D .
(2.17)

In the ΛCDM framework, the χ 2 of the combined fit to D+SH0ES is notably worse than the sum of the separate best-fitting χ 2 to D and to SH0ES, reflecting the fact that the data sets are in tension. Since we are comparing the χ 2 values within a given model, there is no change in the number of model parameters, and the tension can simply be expressed as ∆χ 2 in units of σ, which is identical to the Q DMAP metric discussed in RAVERI et al. ( 2019). This criterion naturally generalizes the commonly used criterion discussed in point 1 to the case of non-Gaussian posteriors. Indeed, for any Gaussian posterior, it is equivalent to that criterion. However, it is sensitive to the effect of over-fitting (i.e. a model with arbitrarily large number of parameters could fit any features better than ΛCDM), which usually requires Bayesian methods to compute Occam's razor factors. For this reason, we also consider a third criterion, which attempts to penalize overly complex models, and to quantify the intrinsic success of a model.

• Criterion 3: When the dataset D includes the SH0ES likelihood, does the fit within a particular model M significantly improve upon that of ΛCDM?

We compute the Akaike Information Criterium (AIC) of the extended model M relative to that of ΛCDM , defined as

∆AIC = χ 2 min,M -χ 2 min,ΛCDM + 2(N M -N ΛCDM ), (2.18)
where N M stands for the number of free parameters of the model. This metric attempts to penalize models which introduce new parameters that do not subsequently improve the fit. Thus, the ability of a model M to resolve the tension at a significant level despite having more parameters can be assessed through In the case of Tests 1 and 2 , we require models to reduce the tension below the 3σ level. This may not seem like a very stringent threshold for success since, at the end of the day, 3σ may still be considered as a significant tension. However, we will show that only a limited number of models are capable of reducing the tension to this rather meager level. In addition, it is important to bear in mind that some of the current data likelihoods might underestimate systematic errors. In the future, as long as systematic errors are revised slightly but not drastically, there is a good chance that the models that do not pass our 3σ criteria will remain excluded, while the models not passing some possible 2σ criterion could be rescued. For Test 3, we demand that the preference for the extended cosmology M over ΛCDM is larger than a "weak preference" on Jeffrey's scale (JEFFREYS 1961; NESSERIS et al. 2013), that is, p = 10 1.5 . Using exp (∆AIC/2) = p in the AIC formalism, this leads to the criterion ∆AIC ≤ -6.91.

Since the questions addressed by criteria 1 and 2 are very similar 23 , and since criterion 2 does not assume Gaussian posteriors, we consider that criterion 2 supersedes criterion 1. On the other hand, criteria 2 and 3 address significantly different questions. Each of them has its pros and cons and they complement each other. Thus, as long as the ∆χ 2 computed for the AIC test is negative (i.e. as long as a model is not giving a worse combined fit than ΛCDM), we will conservatively consider that it is successful when one of criterion 2 or 3 is fulfilled. To summarize the success of each suggested solution we attribute "medals" to models passing our tests: A model passing either criterion 2 (obtaining a good combined fit) or 3 (strongly improving the fit over ΛCDM) receives a bronze medal. A model passing both criteria receives a silver medal. We reserve the gold medal for models that additionally pass criterion 1, that is, whose posterior distributions allow for high values of H 0 (or M B ) independently of the inclusion of a local distance ladder prior. (2.7) and represented graphically in Fig. 2.11. For the sake of completeness, we also present the results obtained using criterion 1 (Gaussian Tension on M b ), both in Tab. (2.7) and in the discussion. First and foremost, no model is perfect -in fact, none of the models studied here are capable of reducing the tension below the ∼ 1.6σ level. A number of models, however, are capable of passing the criteria identified above (with varying levels of success). We enumerate the results for each of the test criteria below:

Results of the contest

• Adopting the GT estimator, only four models can reduce the tension to the 3σ level, with the best model (varying m e +Ω k ) showing a residual 2.0σ tension. From best to worse, they are: varying m e +Ω k , PEDE, varying m e in a flat universe, and the Majoron.

• Making use instead of the more robust Q DMAP criterion, which compares χ 2 of models with and without the inclusion of the SH0ES determination of M b , we find that models with non-Gaussian tails perform significantly better. This most strongly impacts the two models of EDE and the EMG model, reducing their level of tension from roughly 3.1 -3.7σ to 1.6 -2.3σ. From best to worse, models that pass criterion 2 are: EDE, varying m e +Ω k , NEDE, EMG, PEDE, varying m e , and the Majoron.

• Adopting the ∆AIC criterion, which attempts at quantifying the role of enlarged model complexity in the improvement of the fit to D baseline +SH0ES, we find that Before declaring the official finalists, let us briefly comment on models that do not make it to the final, starting with late-universe models. The CPL parameterization, our "late-universe defending champion", only reduces the tension to 3.7σ, inducing a minor improvement to the global fit. The PEDE model noticeably degrades the χ 2 of BAO and Pantheon data, leading to an overall worse fit than ΛCDM. Thus, according to our general rules, we must exclude PEDE from the final.The GPEDE model, which generalises PEDE to include ΛCDM as a limiting case, does not pass any of the tests. This shows the danger of using only criterion 1 or 2 for models that do not include ΛCDM as a limit. Ideally, one should always perform a test equivalent to the ∆AIC or consider models in which ΛCDM is nested. As emphasized in Sect. 2.2, for late-time modifications of ΛCDM, it is also important to treat the SH0ES observation as a model-independent measurement of M b , rather than a model-dependent measurement of H 0 . We checked explicitly that using a SH0ES likelihood on H 0 rather than M b incorrectly yields more favorable results for these late-time models, a result consistent with the claims of (BENEVENTO et al. 2020; CAMARENA et al. 2021; EFSTATHIOU 2021). Finally, the models of decaying dark matter studied here are only capable of reducing the tension from 4.4σ to 4.2σ, despite only introducing two new parameters. Consequently, the ∆AIC criteria disfavors both DDM models. We thus conclude that the late-time DE and dark matter decay models We additionally display the thresholds that have to be reached as purple dashed lines, and the regions of successful models as a purple region.

considered in this work cannot resolve the Hubble tension.

Secondly, the class of models invoking extra relativistic degrees of freedom perform significantly better than late-universe models, but a majority are not successful enough to pass our pre-determined thresholds. Self-Interacting Dark Radiation [SIDR], Self-interacting Dark Radiation scattering on Dark Matter [DR-DM], and Free-streaming plus self-interacting Dark Radiation [mixed DR], all improve upon the "early-universe defending champion", that is, free-streaming DR (for all three criteria). However, none of them reduces the tension below the ∼ 3.2σ level. Perhaps the most surprising case is that of Self-interacting neutrinos plus free-streaming Dark Radiation [SIν+DR], which has long been claimed as a promising solution to the Hubble tension, but performs worse on ∆AIC and Q DMAP than the benchmark of free-streaming DR. It may also sound surprising that the DR-DM model does not perform significantly better than the SIDR model (the latter model passing the ∆AIC criterion). We emphasize that in several previous papers, the success of this model was boosted by a lower prior on the amount of DR that excluded ΛCDM as a limit, a situation comparable to that of PEDE. The only model which successfully passes both criteria is that of the Majoron, which reduces the tension to the level of ∼ 2.9σ and shows a significant improvement to the fit. It is perhaps interesting to point out that this is the only model in this categoriza- tion which invokes a non-trivial evolution of H(z). It is thus in some ways more similar to Early Dark Energy than to the other Dark Radiation models presented here.

In summary, the models that pass at least one of criterion 2 or 3 without leading to a worse global fit, ranked from the best to worst ∆AIC, are the following: These models constitute our "finalist" sample. In Fig. 2.12 we highlight the predicted distributions of S 8 for each of the finalist models 25 . As we can see, none of these models are able to relieve the S 8 tension, however neither do any of the models dramatically increase the tension. As discussed extensively in Sect. There has been attempts in the literature at resolving both tensions simultaneously. In particular, models of DM-DR interactions seemed particularly promising given that they predict both a higher H 0 and a smaller S 8 (BUEN-ABAD et al. 2015; CHACKO et al. 2016; BUEN-ABAD et al. 2018). However, none passed the tests we considered in this work. Similarly, the primordial magnetic field model was advertised as promising to resolve both tensions (JEDAMZIK et al. 2020). While it does pass our ∆AIC criterion, it only reduces the H 0 tension at the 3.5σ level, and the S 8 tension is barely affected (S 8 is even slightly larger than in ΛCDM). Nevertheless, it is possible that the resolutions of the S 8 and H 0 tension lie in different sectors, or require more involved modifications than the ones discussed here. In fact, as we discussed in Sect. 1.7.2, it is becoming clear that the S 8 tension lies in the amplitude of fluctuation σ 8 rather than Ω m , in particular due to constraints on Ω m from uncalibrated SNIa data (SCOLNIC et al. 2018), and from the cross-correlation of weak lensing and galaxy surveys that breaks the Ω m -σ 8 degeneracy (HEYMANS et al. 2021). Therefore, one might expect that the σ 8 tension hints at new perturbation properties, while H 0 , which is a measure of the total energy density, hints at a new background contribution. Many models discussed so far are mainly motivated by their impact at the background level, and it is perhaps not surprising that they fail at simultaneously reducing σ 8 due to the small freedom they have at the level of perturbations. For instance, it was recently noted that extending the (N)EDE sector with an additional ultra-light axion of mass m ∼ 10

1. EDE,

Summary and outline

In the first part of this work, we have reassessed the viability of the EDE against a host of high-and low-redshift measurements, by combining LSS observations from recent weak lensing surveys KIDS-VIKING and DES with Planck18 2018 CMB data, BOSS-DR12 BAO and growth function measurements, and the PANTHEON compilation of luminosity distance to SNIa. Our results can be summarized as follows:

1. Within a phenomenological 3-parameters EDE model (3pEDE), we confirm that Planck18+ BAO+FS+PANTHEON+SH0ES favor f EDE (z c ) 0.1 ± 0.03, z c 4000 +1400 -500

and Θ i = 2.6 +0.4 -0.03 , with a ∆χ 2 = -18.7 compared to ΛCDM fitted on the same data set. The inclusion of the latest Planck18 data (and in particular the more precise polarization measurements) does not spoil the success of the EDE resolution to the Hubble tension. When compared to the 'concordance' ΛCDM model (i.e. obtained from analysis without SH0ES data), the EDE cosmology fits Planck18+BAO+FS+ PANTHEON equally well, but can additionally accommodate the high local H 0 values.

Following the approach of NIEDERMANN et al. (2020), we have then shown that

reducing the parameter space to a 1-parameter EDE (1pEDE) model by fixing Log 10 (z c ) and Θ i to their best fit values as obtained from a Planck18 data only analysis -which strikingly coincide with those from the combined analysis with SH0ES -leads to ∼ 2σ preference for non-zero EDE, namely f EDE (z c ) 0.08±0.04 from Planck18 CMB data alone. In this cosmology, the inferred H 0 70 ± 1.5 km/s/Mpc is in agreement at better than 2σ with its local measurement from SH0ES. The addition of BAO, FS and PANTHEON data has no significant impact on the result. Including a prior on H 0 from SH0ES pulls up the reconstructed fraction to the ∼ 10% level, with H 0 71.7 ± 1, while the fit to Planck18 is slightly better than in the concordance ΛCDM cosmology (∆χ 2 ∼ -5).

3.

To justify the inclusion of LSS data in our analyses, we have confronted the EDE non-linear matter power spectrum as predicted by standard semi-analytical algorithms against a dedicated set of N -body simulations. We have then tested the 1pEDE cosmology against WL data, finding that it does not significantly worsen the fit to the S 8 measurements as compared to ΛCDM, and that current WL observations do not exclude the EDE resolution to the Hubble tension.

4. We also caution against the interpretation of constraints obtained from combining Planck18 with KIDS-VIKING+DES. As we showed, the 'compromise' cosmology that is obtained is a poor fit to KIDS-VIKING+DES and degrades the fit to Planck18 data, even in ΛCDM. This illustrates that these data sets are statistically inconsistent in a ΛCDM framework, and it is easily conceivable that the resolution of this tension lies elsewhere (whether systematic effect or new physics).

5. In light of the CMB lensing anomaly, we have shown that the lensingmarginalized CMB data favor non-zero EDE at ∼ 2σ, predicts H 0 in 1.4σ agreement with SH0ES and S 8 in 1.5σ and 0.8σ agreement with KIDS-VIKING and DES, respectively. There still exists however a ∼ 2.5σ tension with the joint results from KIDS-VIKING and DES. Moreover, the presence of EDE does not affect the amount of anomalous lensing. This suggests that the anomalous lensing is not due to the presence of EDE, but also that the success of EDE is not due to opening up a new degeneracy direction with some exotic lensing parameters.

Interestingly, recent analysis of the 3pEDE model using data from the Atacama Cosmology Telescope's fourth data release (ACT DR4) (S. K. CHOI et al.

(2022).

There is no evidence for 3pEDE over ΛCDM using SPT-3G alone or when combined with the Planck temperature power spectrum restricted to the WMAP multipole range. Combining ACT and/or SPT-3G with the full Planck CMB power spectra returns an upper limit on f EDE (z c ), albeit less restrictive than for Planck alone. In HILL et al. (2021) and POULIN et al. (2021) it was argued that the ACT preference for 3pEDE is mainly driven by a feature in the ACT EE power spectrum around ∼ 500 when ACT is considered alone, with an additional broadly-distributed contribution from the TE spectrum when in combination with restricted Planck TT data ( < 650 or < 1060). Building on these previous studies, in T. L. SMITH et al. (2022) we derived constraints on 3pEDE using ACT DR4, SPT-3G 2018, Planck polarization, and restricted Planck temperature data, finding a 3.3σ preference for EDE over ΛCDM. This is the first time that a moderate preference for EDE has been reported for these combined CMB data sets including Planck polarization. However, with the inclusion of Planck TT data at > 1300, the constraint on f EDE (z c ) becomes compatible with ΛCDM at 1σ. In addition, we explored whether systematic errors in the Planck polarization data could affect our conclusions and found that changing the TE polarization efficiencies 26 significantly reduces the Planck preference for EDE. More work will be necessary to establish whether these hints for EDE within CMB data alone are the sole results of systematic errors or an opening to new physics.

In the second part of this work, we have used a common analysis pipeline to compare and contrast the relative success of seventeen models (including EDE) proposed to ease the Hubble tension; this approach is thus intended as a fair comparison between proposed solutions, and provides a useful benchmark for those wishing to put forth novel ideals. We have broken down the various models into three generic categories: those that modify the sound horizon by including a component of Dark Radiation (DR) impacting the early expansion history, solutions which modify the sound horizon through some other mechanism (such as a delay of recombination or some pre-recombination contribution to the expansion), and solutions that attempt to modify the late-time expansion history.

For each model and data-set, we quantify the residual tension using a series of metrics, each of which has both advantages and shortcomings, and attempting to answer slightly different questions, namely: given a model, (i) to what extent does confrontation with data (other than SH0ES) generate posteriors compatible with high values of H 0 , (ii) to what extent can one obtain a good combined fit to all data, and (iii) to what extent is that model favored over ΛCDM? The summary of our findings is that no model does perfectly well in all of our tests -all models are left with a residual tension, with the most promising reducing the tension to the ∼ 1.6σ level, and with very few models actually reducing the tension below 3σ. Similarly, only a small subset of the models are capable of improving the fit sufficiently to the pass the AIC test. Six models, EDE, NEDE, EMG, varying m e (with and without curvature), and the Majoron, are able to simultaneously satisfy the Q DMAP and ∆AIC criteria. Furthermore, only the varying m e (with and without curvature) passes the Gaussian criterion and allows for high H 0 (or M B ) without a SH0ES prior, receiving the only "gold medal" of our tournament.

Additionally we note that, besides EDE, none of the models of interest alleviate (nor exacerbate) the S 8 tension. Some models which had previously shown some success in reducing the S 8 tension, namely DM-DR and the strongly interacting neutrinos, are now disfavored by the data. Finding a common resolutions to both tensions would certainly reinforce the degree of belief into the new concordance cosmology; however, we reiterate that the resolution of these tensions could arise from independent sectors -either from new physics or systematics. We conclude that some of the models presented in this work can provide a good combined fit to all data considered (including SH0ES), demonstrating that there at least exist potential solutions to the H 0 26 Polarization efficiencies are calibration factors multiplying polarization spectra. In principle, the polarization efficiencies found by fitting the TE spectra should be consistent with those obtained from EE. However, in Planck, small differences (at the level of 2σ) are found between the two estimates at 143 GHz. There are two possible choices: the 'map-based' approach, which adopts the estimates from EE (which are about a factor of 2 more precise than TE) for both the TE and EE spectra; or the 'spectrumbased' approach, which applies independent estimates from TE and EE.

tension, but there are still many difficulties to be overcome in the model building and in explaining the growing S 8 -tension. Further work must be done to establish whether these remaining theoretical and observational issues can be overcome in a new concordant cosmology, one that may either build upon the models studied here, or perhaps lie in a yet unexplored direction.

III

Unstable dark matter and cosmic tensions

From the point of view of particle physics model building, the stability over cosmological timescales is one of the most peculiar property of the dark matter particle, reviewed e.g. in HAMBYE (2011). Often, an additional symmetry (typically a discrete Z 2 symmetry) has to be assumed to make the DM candidate stable. Nevertheless, DM decays at late-times are known signatures of many models in the literature such as (for instance) models with R-parity violation [START_REF] Berezinsky | Cosmological signatures of supersymmetry with spontaneously broken R-parity[END_REF] 2019), they conclude that a ΛDDM scenario does not predict higher H 0 values. This is in good agreement with model-independent analyses existing in the literature in which it has been established that a combination of BAO and uncalibrated SNIa data strongly constrain any late-time modification as a resolution to the Hubble tension (see our discussion in Sect. 2.2). A similar conclusion is also reached when CMB data are considered (CLARK et al. 2021a). Yet, these recent analyses were limited to the study of the effects of ΛDDM on the background evolution of the universe.

In this work, we reassess the phenomenology of this 2-body Decaying Cold Dark Matter (DCDM) scenario, where the decay products are one massive Warm DM (WDM) particle and one (massless) DR component, interacting only through gravitation with the standard model particles. We will refer to the full model as ΛDDM. We perform the first thorough analysis of the ΛDDM model including a realistic treatment of linear cosmological perturbations. To that end, we introduce a new approximation scheme that allows to accurately and quickly compute the dynamics of the WDM linear perturbations by treating the WDM species as a viscous fluid. Thanks to this new fluid approximation, we show that the ΛDDM model, while unable to ease the Hubble tension, can fully explain the low-S 8 measurement from recent weak lensing surveys. We then investigate the implications for the 'S 8 tension' against a number of changes in the analysis: different S 8 priors, marginalization over the lensing information in Planck data, trading Planck high-polarization data for those from the SPTpol and ACTPol surveys, and the inclusion of the recent results from the Xenon1T collaboration. We conclude that the preference for decaying DM, apparent only when the S 8 value determined from weak lensing data is added to the analysis, does not sensibly degrade the fit to any of the cosmological data-sets considered, and that the model could potentially explain the anomalous electron recoil excess reported by the Xenon1T collaboration. Furthermore, we explicitly show that while current CMB data alone are not sensitive enough to distinguish between standard CDM and decaying DM, next-generation CMB observations (CMB-S4) can unambiguously detect its signature.

The rest of this chapter is structured as follows. In Sect. 3.2 we introduce the formalism of ΛDDM (both at the background and linearly perturbed level) and the novel approximation scheme for the WDM species. In Sect. 3.3 we discuss in detail the impact of the ΛDDM model on the CMB and linear matter power spectra. In Sect. 3.4 we discuss the results of our numerous data analyses. Finally, in Sect. 3.5 we draw our conclusions.

Formalism of the two-body Dark Matter decay

Hereafter, we adopt the Boltzmann formalism by AOYAMA et al. (2014), where the time-evolution of the PSD for both the mother and the daughter particles was derived. However, in Sect. 3.2.1 we explicitly show that, at the background level, such formalism is equivalent to the one by BLACKADDER et al. (2014).

While the (cold) mother particle can be safely described as a perfect fluid, computing the density perturbation evolution for the daughter particles requires a more sophisticated treatment. The central role in the game is played by max , i.e. the highest multipole to consider when drawing up the hierarchy of equations describing the PSD of the daughter particles. In the massless case, the degrees of freedom associated to momentum can be removed after the PSD multipole decomposition and integration over q [START_REF] Audren | Strongest model-independent bound on the lifetime of Dark Matter[END_REF]. Due to its non-trivial energy-momentum relation, this approach is not possible for the warm daughter particle. One has to study the full evolution in phase space, which would be computationally prohibitive when performing MCMC analyses. For this reason, in Sect. 3.2.3 we provide a detailed description of a novel approximation scheme, based on describing the WDM component as a viscous fluid on sub-Hubble scales. This allows us to integrate out the momentum degrees of freedom and the hierarchy of equations to be truncated at max = 1. We will show that the new, computationally faster scheme is accurate enough to be used for cosmological analyses, allowing to establish accurate and robust CMB limits on this class of models.

Our framework is characterized by two additional free parameters with respect to ΛCDM: the DCDM lifetime, Γ -1 , and the fraction of DCDM rest mass energy converted into DR, given by (BLACKADDER et al. 2014):

ε = 1 2 1 - m 2 wdm m 2 dcdm , ( 3.1)
where 0 ≤ ε ≤ 1/2. The lower limit corresponds to the standard CDM case, so that Ω cdm = Ω dcdm + Ω wdm , whereas ε = 1/2 corresponds to DM decaying solely into DR. In general, small ε values (i.e. heavy massive daughters) and small Γ values (i.e. lifetimes much longer than the age of the universe) induce little departures from ΛCDM .

We choose to work in the synchronous gauge co-moving with the mother particle (that we introduced in Sect. 1.6.1), where the scalar metric potentials are referred as h and η. This gauge choice, in conjunction with the assumption that the mother particles are fully cold, imply the PSD of mother particles should be proportional to the Dirac delta function of q (AOYAMA et al. 2014):

f dcdm (k, q, τ ) = N dcdm (k, τ )δ (3) ( q) = N dcdm (k, τ ) δ(q) 4πq 2 , ( 3.2)
where

N dcdm (k, τ ) ≡ a 3 ρ dcdm (k, τ ) m dcdm = a 3 ρdcdm (τ ) m dcdm (1 + δ dcdm (τ, k)) (3.3)
denotes the comoving number density of the DCDM. Then, the Boltzmann equations describing the evolution of the I-th dark component (I = {dcdm, dr, wdm}) reads 2

∂f I ∂τ + iµ kq E I f I + ∂f I ∂q dq dτ = ∂f I ∂τ C , (3.4)
where the evolution of the comoving momentum is related to the metric potentials as

1 q dq dτ = η - h + 6η 2 µ 2 , ( 3
.5) 2 The second term at the l.h.s. of Eq. (3.4) is assumed to be a first order perturbation, see App. D.1.

and the collision terms of each species are

∂f dcdm ∂τ C = -aΓf dcdm , (3.6) ∂f dr ∂τ C = ∂f wdm ∂τ C = aΓN dcdm 4πq 2 δ(q -ap max ). (3.7)
In the previous expression p max = p wdm = p dr = m wdm ε/ √ 1 -2ε3 denotes the modulus of the initial physical momentum of the daughter particles. We provide a derivation of Eq. (3.4)-Eq. (3.7) in App. D.1.

Let us decompose the PSD function of the I-th dark component into a background contribution fI plus a linear perturbation ∆f I as f I (k, q, µ, τ ) = fI (q, τ ) + ∆f I (k, q, µ, τ ).

(3.8)

Notice that this time we are considering an absolute perturbation to the background PSD, instead of the relative perturbation we considered in Eq. (1.187). As a matter of fact, the zeroth order collision term for decaying dark matter is not zero (as opposed to the standard species we discussed in Sect. 1.6.2), so the Boltzmann equations for the daughter particles are simpler when expressed in terms of an absolute perturbation.

As usual, the linear perturbation term ∆f I is expanded over Legendre polynomials:

∆f I (k, q, µ, τ ) = ∞ =0 (-i) (2 + 1)∆f I, (k, q, τ )P (µ). (3.9)
We remind that the mean energy density and pressure are obtained by

ρI = 4π a 4 ∞ 0 dq q 2 E I fI , (3.10) PI = 4π 3a 4 ∞ 0 dq q 2 q 2 E I fI , (3.11)
while perturbed energy density, pressure, energy flux and shear stress are given by

δρ I = ρI δ I = 4π a 4 ∞ 0 dqq 2 E I ∆f I,0 , (3.12
)

δP I = ρI Π I = 4π 3a 4 ∞ 0 dqq 2 q 2 E I ∆f I,0 , (3.13) ( ρI + PI )θ I = 4πk a 4 ∞ 0 dqq 2 q∆f I,1 , (3.14)
( ρI + PI )σ I = 8π 3a 4 ∞ 0 dqq 2 q 2 E I ∆f I,2 . (3.15)
Throughout the rest of this Chapter and unless stated otherwise, we compare ΛDDM models at fixed ω ini dcdm ≡ Ω ini dcdm h 2 with ΛCDM models having the same ω cdm . All other cosmological parameters are fixed to {H 0 = 67.7 km/s/Mpc, ω cdm = ω ini dcdm = 0.1194, ω b = 0.0224, n s = 0.9673, ln(10 10 A s ) = 3.052, τ reio = 0.0582}, which constitutes our baseline ΛCDM model. These values correspond to the best-fit from the combined analysis (including S 8 data from weak lensing) that we present in Sect. 3.4.2 (see fourth column of Tab. (3.1)), which are however very similar to the ΛCDM best-fit parameters from Planck18. We implement the equations describing the ΛDDM model in a modified version of CLASS [START_REF] Blas | The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes[END_REF][START_REF] Lesgourgues | The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics[END_REF]). Our code is available at https://github.com/PoulinV/class_majoron.

Background equations

Gathering the zeroth order terms of the Boltzmann equations Eq. (3.4), we arrive at the background evolution of the mother and daughter particles

f dcdm = -aΓ fdcdm , ( 3.16
)

f dr = f wdm =
aΓ Ndcdm 4πq 2 δ(q -ap max ).

(3.17)

Here Ndcdm = a 3 ρdcdm /m dcdm denotes the mean value of the DCDM comoving number density. From Eq. (3.16), one sees that it evolves as Ndcdm = (Ω ini dcdm ρ c /m dcdm )e -Γt4 . By combining Eq. (3.10) and Eq. (3.11) with Eq. (3.16) and Eq. (3.17) we obtain:

ρ dcdm = -3H ρdcdm -aΓρ dcdm , ( 3.18
)

ρ dr = -4H ρdr + εaΓρ dcdm , ( 3.19
)

ρ wdm = -3(1 + w)H ρwdm + (1 -ε)aΓρ dcdm . (3.20)
Here w(τ ) ≡ Pwdm /ρ wdm refers to the WDM Equation of State5 . We give a proof of Eq. (3.20) in App. D.2.

Eq. (3.20) will be useful to analytically derive the fluid equations that we present in Section Sect. 3.2.3, but for a numerical resolution it is much simpler to use an integral formula for ρwdm , as it was done in BLACKADDER et al. (2014). This formula can be obtained by integrating Eq. (3.17) firstly with respect to τ , and then with respect to q. The first integration requires using the relation δ(q -ap max ) = δ(τ -τ q )/qH, where τ q represents the conformal time when daughter particles with co-moving momentum q are born, q = a(τ q )p max [START_REF] Aoyama | Evolution of perturbations and cosmological constraints in decaying dark matter models with arbitrary decay mass products[END_REF]). This yields

fwdm (τ, q) = a q Γ Ndcdm (τ q ) 4πq 3 H q Θ(τ -τ q ). (3.21)
where a q = a(τ q ) and H q ≡ H(a q ). Now, we can obtain an expression for ρwdm by taking the integral 4πa -4 ∞ 0 dqq 2 E wdm at each side of the equation. Then, changing the integration variable from q to a q , and applying the Heaviside function Θ(a-a q ) leads to

ρwdm (a) = ρ c Ω ini dcdm Γ a 4 a 0 da q e -Γtq H q ε 2 a 2 q + (1 -2ε)a 2 , (3.22)
where t q ≡ t(a q ). We note the equivalence between Eq. the mean energy density of the massless DR species, ρdr , we simply take the limit ε → 1/2 of Eq. (3.22). In both cases, the background evolution Eq. (3.22) needs to be solved iteratively, as the Hubble parameter H depends on ρwdm and ρdr through the Friedmann equation. For a flat universe, this equation reads Here ργ , ρν , ρb and ρΛ denote the mean densities of photons, neutrinos, baryons and dark energy, respectively.

H 2 (a) = 8πGa 2 3 i ρi (a), (3.23
In Fig. 3.2, we show the evolution of the Hubble parameter for ΛCDM and several ΛDDM models. In Fig. 3.1, we also show the evolution of the fractional densities Ω dcdm (z), Ω wdm (z) and Ω dr (z) for a particular ΛDDM model (Γ -1 = 30 Gyrs and ε = 0.1) as well as Ω cdm (z) with the same initial amount of dark matter, namely Ω 0 cdm = Ω ini dcdm . The two new parameters affect H(a) as follows: at fixed ε, a shorter lifetime Γ -1 implies a lower Hubble parameter. This is clearly visible at z 1. The behavior below z ∼ 1 is due to our assumption of a flat universe and the requirement of fixing H 0 : to fulfill the budget equation, a lower DM abundance requires a larger Ω Λ , meaning that the period of accelerated expansion begins earlier with respect to ΛCDM. At fixed lifetime Γ -1 , a larger ε also induces a lower H(a), because more energy is converted into radiation, which dilutes faster. We thus anticipate a negative correlation between Γ -1 and ε at the background level. The degeneracy can be captured by the matter contribution from the WDM species, Ω m wdm ≡ Ω 0 wdm (1 -3w), which is well constrained by data. This quantity is roughly approximated by6

Ω m wdm Ω ini dcdm (1 -e -Γt 0 ) √ 1 -2ε, (3.25)
For small Γ and small ε, Ω m wdm ∝ Γ(1 -ε) and we expect data to constrain a parameter combination of the form ε ∝ Γ -1 .

Linear perturbation equations

The continuity and the Euler equations for the DCDM can be obtained by integrating over momenta the first two multipoles (∆f dcdm,0 and ∆f dcdm,1 ) of the Boltzmann equation in Eq. (3.4). The calculation is analogous to one we have done in App. A.6 to arrive at the conservation equations for massive neutrinos. The main differences is that we have to take into account the fact that DCDM is pressureless and has a non-zero collision term associated to the decay. When rewriting the equations for δρ dcdm and ρdcdm θ dcdm as equations for δ dcdm and θ dcdm (respectively), the decay terms coming from the background and perturbed equations cancel out and we are left with the same equations as for CDM (c.f. Eq. (1.219)-Eq. (1.220)):

δ dcdm = - h 2 , ( 3.26
)

θ dcdm = -Hθ dcdm . (3.27)
Given our gauge choice, θ ini dcdm = 0, so we can set θ dcdm = 0 at all times. Let us recall that, strictly speaking, the synchronous gauge in CLASS [START_REF] Blas | The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes[END_REF]) is coded with respect to the CDM. Nevertheless, for adiabatic initial conditions, one can choose θ ini,dcdm = θ ini,cdm = 0, such that the gauge co-moving with CDM is also co-moving with DCDM at all times [START_REF] Audren | Strongest model-independent bound on the lifetime of Dark Matter[END_REF]). Hence, hereinafter we neglect this irrelevant complication.

To obtain the dynamical equations for the daughter particles, we plug the Legendre decomposition of Eq. (3.9) in the Boltzmann equation of Eq. (3.4), so that we have the following hierarchy of equations

∂ (∆f I,0 ) ∂τ = - qk E I ∆f I,1 + q ∂ fI ∂q h 6 + aΓ Ndcdm 4πq 2 δ(q -ap max )δ dcdm , (3.28) ∂ (∆f I,1 ) ∂τ = qk 3E I [∆f I,0 -2∆f I,2 ] , (3.29) ∂ (∆f I,2 ) ∂τ = qk 5E I [2∆f I,1 -3∆f I,3 ] -q ∂ fI ∂q (h + 6η ) 15 , (3.30) ∂ (∆f I, ) ∂τ = qk (2 + 1)E I [ ∆f I, -1 -( + 1)∆f I, +1 ] ( ≥ 3), (3.31)
where I = {dr, wdm}.

F dr, = k (2 + 1) [ F dr, -1 -( + 1)F dr, +1 ] ( ≥ 3). (3.36)
We have adopted the convention r dr ≡ a 4 ρdr /ρ c , as in POULIN et al. (2016), which in the 2-body decay scenario under study leads to: r dr = εaΓ(ρ dcdm /ρ dr )r dr .

(3.37)

The first three multipoles are given by:

F dr,0 =

Fluid approximation for the warm component

In order to compute the WDM dynamics one cannot integrate over the momentum degrees of freedom, as we did in the DR case. Indeed, when taking the moments of the hierarchy of Eq. (3.28)-Eq. (3.31), higher velocity-weight integrals appearing at = 2 cannot be computed from the system of equations itself. Therefore, one has to follow the evolution of the full time-dependent PSD to obtain the elements of the perturbed stress-energy tensor δ wdm , θ wdm and σ wdm through Eq. (3.12)-Eq. (3.15). A typical set-up for CMB analyses requires roughly ∼ 500 wavenumbers, ∼ 20 multipoles and ∼ 10 4 momentum bins, i.e. O(10 8 ) linear differential equations to be computed. On a single processor, this leads to runs with a CPU time of 1 -2 days per each parameter choice, making a systematic scan of the parameter space computationally prohibitive.

To overcome the problem, we make use of a new fluid approximation for the WDM species,where the momentum dependence is removed, and one only needs to track the evolution of the first two multipoles. The number of linear differential equations to be solved is now reduced to O( 103 ), with a CPU time per single run ∼ 30 -40 s. The novel approximation scheme is based on the treatment of massive neutrinos as a viscous fluid by LESGOURGUES et al. ( 2011), and it is only valid at scales deeply inside the Hubble radius, where high-and low-modes are effectively decoupled. In App. App. D.4 we explicitly demonstrate the accuracy of this approximation.

Similarly to the DR case, the fluid equations can be derived by multiplying both sides of Eq. (3.28) and Eq. (3.29) by 4πq 2 E wdm a -4 and 4πq 3 ka -4 , respectively, and integrating over q. Then, by using Eq. (3.12)-Eq. (3.15) and Eq. (3.20), one can write down the continuity equation,

δ wdm = -3H(c 2 s -w)δ wdm -(1 + w) θ wdm + h 2 + (1 -ε)aΓ ρdcdm ρwdm (δ dcdm -δ wdm ), (3.41)
and the Euler equation,

θ wdm = -H(1 -3c 2 g )θ wdm + c 2 s 1 + w k 2 δ wdm -k 2 σ wdm -(1 -ε)aΓ 1 + c 2 g 1 + w ρdcdm ρwdm θ wdm . (3.42)
Notice that we have introduced the WDM sound speed in the synchronous gauge, c 2 s ≡ δP wdm /δρ wdm , and the WDM adiabatic sound speed, c 2 g ≡ P wdm /ρ wdm . The latter can be written as

c 2 g = w P wdm Pwdm ρ wdm ρwdm -1 , (3.43)
and it can be computed as follows

c 2 g = w 5 - p wdm Pwdm - ρdcdm ρwdm aΓ 3wH ε 2 1 -ε 3(1 + w) - ρdcdm ρwdm aΓ H (1 -ε) -1 . (3.44)
Here p wdm denotes the so-called pseudo-pressure, a higher momenta integral of fwdm which reduces to the standard pressure in the relativistic limit (LESGOURGUES et al. Obtaining an analytical expression for c 2 s is less straightforward, since we do not have a dynamical equation for the pressure perturbation δP wdm . In LESGOURGUES et al. ( 2011) it is assumed that c 2 s is scale-independent and approximately equal to c 2 g . For the WDM species, we have found that this assumption leads to accurate results for the CMB power spectrum, but not for the matter power spectrum. In fact, calculations using the full Boltzmann hierarchy of Eq. (3.28)-Eq. (3.31) reveal that c 2 s exhibits a particular k-dependence that cannot be captured by a background quantity such as c 2 g . In particular, c 2 s gets slightly enhanced on scales k > k fs , where k fs is the free-streaming scale of the WDM species, defined as (c.f. Eq. (1.224))

k fs (τ ) ≡ 3 2
H(τ ) c g (τ ) . (3.45)
It is possible to gain a semi-analytic understanding of this behavior by building a formal equation for the evolution of c 2 s , as detailed in App. D.3.5. To account for such an enhancement, we adopt the following prescription

c 2 s (k, τ ) = c 2 g (τ ) [1 + (1 -2ε)T (k/k fs )] , (3.46)
where the function T (x) = 0.2 √ x has been fitted to the sound speed obtained using the full Boltzmann hierarchy, for the parameter values ε = 0.5, 0.1, 0.01, 0.001 and Γ/H 0 = 0.1, 1, 10. The factor (1 -2ε) is inserted to make the k-dependent correction vanishingly small close to the relativistic limit, where c 2 s c 2 g 1/3.

In order to trace the evolution of the shear σ wdm one could follow a similar approach to that of LESGOURGUES et al. (2011), where the authors obtained a dynamical equation for the neutrino shear by means of an improved truncation scheme at max = 2. We tested the implementation of a generalization of that equation suitable to the decaying case (which can be found in App. D.3), but we found it to be only relevant close to the relativistic case ε 1/2, when it reduces to the DR shear equation from [START_REF] Enqvist | Decaying dark matter and the tension in σ8[END_REF]. In this regime, the dynamics of the daughter particles do not significantly impact the CMB and matter power spectra, we thus decided to not include any dynamical equation for the shear of the WDM species, when switching to the fluid approximation. In practice, we simply set σ wdm to a constant value, obtained via integration of the second PSD multipole in the Boltzmann hierarchy (see Eq. (3.15)). We explicitly checked that this approach yields better results rather than simply setting σ wdm = 0 in the fluid equations, or than using the DR shear equation of ENQVIST et al. ( 2015) when ε = 1/2.

Dynamics of perturbations

Before discussing the signatures of varying the parameters Γ and ε on the relevant cosmological observables, it is worth having a look at the linear perturbations of the mother and daughter particles. In Fig. 3.3 we show the evolution of the linear density perturbations for the DCDM, WDM and DR species, corresponding to a mode that enters the horizon very early (i.e. k = 1 Mpc -1 ). To clarify the impact of the two extra free parameters, the perturbations are shown for DCDM lifetimes similar and smaller than the age of the universe (Γ -1 = H -1 0 ∼ 14.5 Gyrs and Γ -1 = (10H 0 ) -1 ∼ 1.5 Gyrs) , and for massive daughters behaving either as warm or cold particles (ε = 10 -2 and One can see that initially, the perturbations of the daughter species always track those of the mother, because the coupling term dominates the dynamics (i.e. the ratios ρ dcdm /ρ wdm and ρ dcdm /ρ dr are large). When a mode crosses the free-streaming scale, the pressure support of the daughter particles becomes important and the perturbations develop oscillatory features. For the DR species, the free-streaming scale simply corresponds to the horizon (k fs ∼ H,) while for the WDM it corresponds to a larger value, given by k fs ∼ H/c g (see Eq. (3.45)). This time-and scale-dependent power suppression, together with the different background dynamics, lead to key signatures on the CMB and matter power spectra, as we discuss later.

By comparing upper and lower panels of Fig. 3.3, it is clear that the value of the WDM free-streaming scale is essentially determined by the value of ε, as expected. On the other hand, by comparing left and right panels, one notices that the intensity of the oscillations due to the pressure support can be compensated by the coupling to the DCDM if the lifetime is long enough, as the daughter particles keep being produced.

Interestingly, the decoupling time of the daughter perturbations from the mother perturbations is always set by the free-streaming crossing time, and not by the characteristic decay time. In order to illustrate that, in Fig. 3.4 we show perturbations corresponding to a smaller wave-number, k = 10 -2 Mpc -1 , that enters the horizon much later. For this mode, the WDM species does not have time to cross the freestreaming length (the crossing will occur in the future), so that δ dcdm and δ wdm remain equal, even if the lifetime is smaller than the age of the universe. This can be understood from the fluid description in Eq. (3.41)-Eq. (3.42): when the decay term that includes Γ is relevant, δ wdm is driven by δ dcdm . Therefore, the WDM density perturbation δ wdm will continue to track the behaviour of δ dcdm , as long as the freestreaming scale is not crossed, i.e., as long as the pressure term, containing c 2 s k 2 δ wdm , is small compared to δ wdm ∼ δ wdm H 2 .

Observable effects of the Dark Matter decay

Impact on the matter power spectrum

Firstly, we focus on describing the effects of the 2-body decay on the linear matter power spectrum, since this will allow to better understand some of the effects on the CMB spectra. In this section, we use the same ΛCDM parameters as in Sect. 3.2.1, except for fixing 100θ s = 1.04217 instead of H 0 , to better connect with CMB observations, that accurately pin down θ s . In Fig. 3.5, we compare the residual differences in the linear power spectra (at redshifts z = 0 and z = 2) with respect to our baseline ΛCDM. The left panel shows several lifetimes and a fixed DR energy fraction ε = 0.1, while the right panel shows a fixed lifetime Γ -1 = 30 Gyrs and several values of the DR energy fraction ε.

One important feature of the C+WDM scenarios, such as the one considered in this work, is that they are expected to produce a suppression in the linear matter power spectrum at scales smaller than a 'cutoff scale', with a non-trivial shape On scales k > k fs , pressure becomes important and WDM particles cannot stay confined in gravitational potential wells, which inhibits structure formation. Fig. 3.5 clearly illustrates that: while the parameter ε fixes the value of the cut-off k fs8 , the lifetime Γ -1 essentially determines the depth of the suppression at very small scales. This is to be expected, since the amount of power suppression grows with the WDM abundance, which increase for smaller lifetimes. In a similar way, the effects of decay become less important when considering the matter spectrum at a higher redshift z = 2, since the abundance of WDM was smaller in the past.

In general, for late-time decay scenarios (well after recombination) as the one studied in this work, it is possible to distinguish three different regimes depending on the value of ε, as it is shown in the right panel of Fig. 3.5:

• Non-relativistic decay: if ε 0.001 (black curve), the WDM leaves the expansion rate unaffected, since its contribution to the matter density, ρwdm (1 -3ω) ρwdm , compensates the reduction in the DCDM density, ρdcdm . In addition, the WDM free-streaming length is very small, inducing a power suppression at k 1 hMpc -1 . Such scales are beyond the range of scales probed by the observables considered in this work, so in this regime the WDM is almost degenerate with standard CDM.

• Relativistic decay: if ε 0.5 (red curve), the WDM component acts as DR, which can appreciably reduce the expansion rate 9 . Moreover, the free-streaming length k fs gets as large as the horizon, so that the WDM does not cluster at all. The reduction in the Hubble friction is balanced by a reduction in the clustering density of the daughter particles, δρ wdm δρ dr 0, inducing a very little overall suppression of the growth of fluctuations. However, there is an impact coming from another background effect: the significant decrease in the co-moving matter density, a 3 ρ m , leading in turn to an increase of the angular diameter distance, thereby a reduction of θ s . This effect can be compensated by increasing H 0 , which shifts the location of the peak, k eq /(a 0 H 0 ), towards smaller k in the matter power spectrum -since we are keeping the matter-radiation equality era fixed. The net effect on the residuals is twofold, a large-scale enhancement and smallscale suppression of power.

• Warm decay: For intermediate values of ε, namely 0.001 ε 0.5 (green and blue curves), the WDM component partially contributes to the matter energy density, leaving to an expansion rate almost unchanged. However, the values of k fs that determine the cut-off scale in the matter power spectrum are not as small as in the case of non-relativistic decay, leading to δρ wdm 0 for k > k fs . Hence, on scales k > k fs the Hubble friction gets enhanced with respect to the clustering density δρ tot , slowing down the growth of DCDM perturbations. The net impact on the matter power spectrum is thus a suppression on intermediate scales, somewhat similar to that induced by massive active neutrinos [START_REF] Lesgourgues | Massive neutrinos and cosmology[END_REF]

Impact on the CMB anisotropy spectra

We now discuss the impact of the 2-body decay scenario on the CMB anisotropy temperature and polarization angular power spectra, as well as on the lensing potential power spectrum reconstructed from the CMB 4-point correlation function. In the left panel of Fig. 3.6 we report the residuals of the (lensed) TT, EE and lensing potential power spectra with respect to our baseline ΛCDM, for different lifetimes Γ -1 and a fixed DR energy fraction ε = 0.1. The effects, more and more pronounced as the lifetime decreases, can be understood as follows:

• At the background level, the decay decreases the value of Ω m with respect to ΛCDM. This is compensated by an increase in Ω Λ (earlier beginning of Λ-domination) and thus an enhancement in the Late Integrated Sachs-Wolfe (LISW) effect, leaving a signature in the low-TT power spectrum. Furthermore, a modified background history alters quantities integrated along z, such as τ reio , which impacts the multipoles ∼ 10 in the EE power spectrum.

• At the perturbation level, the late-time reduction of a 3 ρdcdm implies a reduction of the quantity a 2 δρ dcdm , which acts as a source of gravity through the Poisson equation. This induces a damping in the metric fluctuations, and hence yields a further enhancement of the LISW effect. Furthermore, the suppression in the matter power spectrum and in Ω m lowers the amplitude of the lensing potential power spectrum, consequently reducing the smoothing of the peaks in the highpart of both the TT and EE spectra, as one can see from the 'wiggles' in the corresponding plots.

In the right panel of Fig. 3.6 we show the CMB residuals for a fixed Γ -1 and various values of ε. The effects can be readily understood:

• At the background level, smaller values of ε weaken the effects previously discussed, because the decay product dilute in a way similar to dark matter. Namely, the decrease in Ω m is less prominent due to the significant WDM contribution (i.e. the increase in Ω Λ is shallower), and the impact on τ reio is smaller. Therefore, the signatures in the low-part of the TT and EE spectra become less visible.

• At the level of perturbations, ε leads to some interesting signatures on the LISW effect and on the lensing potential. Since the LISW effect is only relevant for small multipoles (i.e. very large scales), one just needs to look at wavenumbers such that k < k fs . On these scales, the growth suppression does not play any role, and the decrease in a 2 δρ dcdm due to the decay gets partially compensated by the increase in a 2 δρ wdm , which is more significant for smaller values of ε. Thus, the damping in the metric fluctuations is less relevant for smaller ε, reducing the LISW enhancement.

• Regarding the effects on the lensing potential, one can see that the suppression in the corresponding power spectrum monotonically decreases for smaller ε. Naively, one might expect the opposite, since we have argued that the matter power suppression increases for small ε. This can be understood by looking at the CMB lensing kernel W (z), given by the following expression (MANZOTTI 2018)

W (z) = 3Ω m 2 H 2 0 H(z) (1 + z)χ(z) χ * -χ(z) χ * , (3.47)
where χ * is the co-moving distance to the last-scattering surface. Firstly, the CMB lensing kernel peaks at z ∼ 2, where the suppression is less important (see bottom panels of Fig. 3.5). Secondly, it gets highly suppressed for higher values of ε, and this effect can dominate over the effect on the matter power spectrum. This suppression happens mainly due to the smaller Ω m , as we have verified by computing W (z) for several ΛDDM models, with and without including the factor Ω m . Note that coincidentally, the effects on the lensing power spectrum at high are very similar for ε = 0.5 and ε = 0.1: this is because in the former case Ω m is highly reduced and the small scales power spectrum is almost unaffected, while the opposite occurs in the latter case.

Let us finally remark that, even if the effects of varying either ε or Γ on the observables are different, one can easily exploit the degeneracy mentioned at the background level to get different couples of values (large Γ and small ε or vice-versa) with a similar cosmological signature, especially on the CMB. We do indeed expect the Monte Carlo analysis to show a negative correlation in the reconstructed 2D posteriors for ε and Γ.

Cosmological constraints

We now confront the ΛDDM cosmology to a host of recent cosmological observations. Our goal is to set constraints on the lifetime of DM and the mass-ratio of the mother and daughter particles. We also wish to check to what extent the ΛDDM model can play a role in resolving cosmological tensions. To that end, we perform comprehensive MCMC analyses with the MontePython-v3 (AUDREN et We start by performing two distinct sets of studies in order to illustrate the importance of taking CMB data into account when studying the ΛDDM scenarios, even in the long-lived regime (i.e.when the DCDM decays after photon decoupling):

(1) A background-only analysis against BAO 12 and Pantheon SNIa data.

(2) A full analyses including linear perturbations, where we combine the data-set used in (i) with CMB TT, TE, EE + lensing data, with and without including the aforementioned informative priors on S 8 and H 0 .

In the case of (1), the parameter space is fully characterized by the following free parameters:

Ω ini dcdm , H 0 , Γ, ε ,
whereas in (2) the whole parameter space is described by:

Ω b h 2 , ln 10 10 A s , n s , τ reio , Ω ini dcdm , H 0 , Γ, ε .
For both ( 1) and ( 2) we adopt logarithmic priors on ε and Γ 13 , namely,

-4 ≤ Log 10 ε ≤ Log 10 (0.5), -4 ≤ Log 10 (Γ/Gyrs -1 ) ≤ 1,
and a flat prior on the initial DCDM abundance:

0 ≤ Ω ini dcdm ≤ 1.
Secondly, we explore the possibility of resolving the infamous Hubble and S 8 tensions, and the 'A lens ' anomaly that exists within Planck data. We also briefly discuss the viability of the 2-body decay scenario as solution for the Xenon1T anomaly (APRILE et al. 2020). We then test the robustness of our results to various changes in the pipeline, and in particular to trading the high-Planck CMB data for those from the SPT collaboration [START_REF] Henning | Measurements of the Temperature and E-Mode Polarization of the CMB from 500 Square Degrees of SPTpol Data[END_REF], which are known to be less in tension with local S 8 measurements, as well as ACTPol data (AIOLA et al. 2020), which shows a level of tension with S 8 measurements similar to Planck. We end by showing that next-generation CMB observations (CMB-S4) can detect the ΛDDM model.

We adopt flat priors on the rest of parameters, and we set two massless and one massive neutrino species with m ν = 0.06 eV, following Planck18 conventions. We assume our chains to be converged when the Gelman-Rubin criterion R -1 < 0.02 [START_REF] Gelman | Inference from Iterative Simulation Using Multiple Sequences[END_REF]). To extract the best-fit parameters, we make use of the MINUIT algorithm [START_REF] James | Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations[END_REF]) through the iMinuit python package. In App. D.5 we report all individual χ 2 's per each of the analyses performed.

General constraints: background vs. linear perturbations

In Fig. 3.7 we show the 2σ posterior distributions of the parameters characterizing the ΛDDM model, Log 10 (ε) and Log 10 (Γ/Gyrs -1 ). Red contours refer to the 12 As discussed in Sect. 3.1, we calibrate BAO data by imposing a Gaussian prior on the sound horizon at recombination rs(zrec) = 144.7 ± 0.5 Mpc, to not to spoil CMB data. 13 For comparisons with previous works, an useful conversion is the following Log 10 (Γ/Gyrs -1 ) Log 10 (Γ/km s -1 Mpc -1 ) -2.991. We also note that the lower limits for the Γ and ε priors are simply chosen because we verified that the ΛCDM limit is fully recovered for Γ < 10 -4 Gyrs -1 and ε < 10 -4 (at least for the observables considered in this work). Hereafter, unless otherwise stated, the green shaded bands refer to the joint S 8 measurement from KiDS-1000+BOSS+2dFLens, while the gray bands stand for the H 0 measurement by the SH0ES collaboration. Note that the BAO+SNIa analysis is based on background evolution only, whereas the BAO+SNIa+CMB analysis also includes linear perturbations (see the main text for further details).

background-only analysis (1), while blue contours refer to the full analysis [START_REF] Slipher | Spectrographic Observations of Nebulae[END_REF], at the perturbation level, when CMB data are also added 14 . From BAO+SNIa and Planck data, the DDM is not detected. We confirm our expectation from Sect. 3.2.1 and Sect. 3.3 that there is a negative correlation between Log 10 (ε) and Log 10 (Γ/Gyrs -1 ), apparent within all data sets: as ε decreases, the 95% C.L. upper limit on Γ relaxes.

In practice, we find that for decay rates Γ ∼ 10 -3 -10 -1 Gyrs -1 , our 2σ exclusion curve is roughly described by ε 1.6 × 10 -4 (Γ/Gyrs -1 ) -1.1 .

For large Γ and small ε, the factor '1 -e -Γt 0 ' in Eq. (3.25) reaches 1 faster than ' √ 1 -2ε' (assuming t 0 13.8 Gyrs), explaining why our constraint on the DR energy fraction becomes flat even for very large decay rates. In particular, for small decay rates Log 10 (Γ/Gyrs -1 ) -3 and very massive daughters Log 10 (ε) -2.7 the DCDM behaves like CDM, leading to departures indistinguishable from ΛCDM. Of utmost importance, we find that constraints on the ΛDDM free parameters become much stronger when CMB data are included, increasing by more than one order of magnitude over the whole parameter space, contrarily to the naive expectations that CMB data do not weigh-in on late-time decays.

Interestingly, we see that the background-only analysis predicts a value of H 0 slightly higher than the one inferred assumed ΛCDM, though it can be noticed that the standard ΛCDM value for H 0 is still perfectly compatible, due to larger uncertainties with respect to the full analysis, which anyhow pulls H 0 back to its standard value. This suggests that this kind of models is not suitable for relieving the 2021a), we observe a significant decrease in the S 8 contours for ε ∼ 0.01 and Γ -1 ∼ 10 2 Gyrs. We attribute this disagreement to the fact that CLARK et al. (2021a) does not include a treatment of WDM perturbations, which are responsible for the suppression in the matter power spectrum. Hence, the 2-body decay presented here could potentially reconcile the inferred value of S 8 with its direct measurements from LSS observations. We present an explicit comparison of our constraints with those from CLARK et al. (2021a) in App. D.6.

Finally, in order to compare our constraints on Γ with previous literature, we have carried out a MCMC analysis including BAO + SNIa + Planck data, but fixing15 ε = 0.5 (i.e. in the limit in which the daughter particle behaves as dark radiation). We find a 2σ upper limit on the DCDM decay rate of log 10 (Γ/[Gyr -1 ]) -2.67, corresponding to Γ -1 468 Gyrs. Our constraints on DM decays to DR are three times tighter than those found in works using older Planck data [START_REF] Audren | Strongest model-independent bound on the lifetime of Dark Matter[END_REF][START_REF] Poulin | A fresh look at linear cosmological constraints on a decaying dark matter component[END_REF]) but also ∼ 40% tighter than NYGAARD et al. (2021) due to the use of a logarithmic prior on Γ as opposed to linear (as we argue in App. D.7, a logarithmic prior is more agnostic because it doesn't carry any scale).

Implications for the S 8 tension

So as to quantify the ability of the model to solve the S 8 tension, we have re-conducted the BAO + SNIa + Planck analysis for the ΛDDM scenario, but including the S 8 prior as measured by KiDS-1000+BOSS+2dFLens. Note that we neglect the potential co-variance between S 8 and BOSS BAO/FS data for simplicity, and we checked that removing the BAO/FS data from the analysis does not affect the result. To gauge the importance of the late-time decay in the success of the solution, we compare the ΛDDM model with another cosmological scenario that features a power suppression at small scales, namely massive neutrinos (νΛCDM). We model these as three degenerate states and vary the total mass M ν , on top of the standard ΛCDM parameters. We assess the remaining level of tension by computing the Q DMAP (for "difference in the maximum a posteriori") tension metric introduced in RAVERI et al. (2019), which essentially (for flat priors) makes use of the difference in χ 2 between the fit of a given model with and without including the S 8 data point. The tension is then estimated as ∆χ 2 in unit of σ. Finally, we also compute the Bayesian evidence with the sampler MULTINEST (FEROZ et al. 2009), taking 1000 live points and a tolerance condition on the evidence for stopping the sampling equal to 0.1. We perform model comparison by calculating ∆logB = logB(ΛDDM) -logB(νΛCDM).

Our results are reported in Tab. (3.1) and summarized in Fig. 3.8: In the ΛDDM scenario (red contours) we find that the best-fit (when including the S 8 prior) has ε 0.7% and Γ -1 55 Gyrs, yielding S 8 0.767 and Ω m 0.31, in excellent agreement with the KiDS1000+BOSS+2dfLenS measurement. Moreover, the decrease in S 8 is driven by a smaller σ 8 , while Ω m is not affected, which is also what is favored by the data. We find a strong negative correlation between ε and Γ, which approximately scales like Γ -1 55 (ε/0.007) 1.4 Gyrs. On the other hand, the νΛCDM model can only achieve S 8 0.81, with M ν < 0.1614 (95% C.L.). Remarkably, we find a ∆χ 2 min = χ 2 min (ΛDDM) -χ 2 min (νΛCDM) -5.5 in favor of the ΛDDM model. The negative ∆χ 2 is driven entirely by the low S 8 value (the χ 2 min per experiment is reported in App. D.5). The fit to other data set is barely affected by the inclusion of a S 8 prior in the ΛDDM model (but degrades in the νΛCDM case), such that without the S 8 prior, there is no preference for DDM and S 8 seems unchanged. Looking at the reconstructed S 8 = 0.821 might naively expect S 8 0.767 to be largely excluded. In fact, the combined χ 2 with the prior on S 8 only increases by ∼ +1.6 (as opposed to ∼ +7.1 in the νΛCDM). This is because S 8 has a non-Gaussian posterior with a tail extending to low values due the degeneracy between Γ and ε. This degeneracy becomes clear when incorporating the S 8 prior. As a result, the Q DMAP estimator indicates that the tension evolves from 2.7σ within ΛCDM to 1.3σ within ΛDDM.

Nevertheless, the model comparison is slightly in favor of νΛCDM, ∆logB = -0.81, although based on the modified Jeffrey's scales (JEFFREYS 1961; TROTTA 2008; NESSERIS et al. 2013) the preference is 'weak' or 'inconclusive'. We thus conclude that, while the tension between our baseline data set and S 8 is resolved in the ΛDDM, current data do not favor the model in a Bayesian sense 16 .

As we show later in Sect. 3.4.2, the ΛCDM model with two massless neutrinos and one massive with M ν = 0.06 eV yields results very similar to the νΛCDM model. Similarly, letting the neutrino masses free to vary in the ΛDDM model does not affect the results (see App. D.8). We also note that making use of linear priors on ε and Γ does not affect the reconstructed S 8 value, but the scale chosen for the prior (i.e. sampling over the original prior range or in a more restricted range where ε ∼ O(10 -2 )) affects the reconstructed 2D posteriors of ε and Γ. We discuss these issues in more details in App. D.7.

Best-fit cosmology solving the S 8 tension

To better understand the ΛDDM success in resolving the S 8 tension compared to the case of massive neutrinos, in Fig. 3.9 we shown the residual differences in These scenarios feature two key differences: i) a distinct redshift evolution for the power suppression. In the ΛDDM scenario, it is less significant at higher redshifts, since the abundance of the WDM daughter is smaller; ii) a time-evolving cut-off scale; in the ΛDDM model, k fs = 3/2H(a)/c g (a), while in the νΛCDM it is obtained by evaluating k fs at the redshift at which neutrinos become non-relativistic [START_REF] Lesgourgues | Massive neutrinos and cosmology[END_REF]). As a consequence, the CMB power spectra, well constrained by Planck, are vastly different. This is illustrated in Fig. 3.10, for both the best-fit ΛDDM scenario and the νΛCDM model which yields the same S 8 value. The νΛCDM predicts different early-integrated Sachs-Wolfe effects, as well as different amount of lensing, because of a significant power suppression at z ∼ 2 -3, where the CMB lensing kernel peaks MANZOTTI 2018. On the other hand, the differences between ΛCDM and ΛDDM until z ∼ 2 are very small, explaining why Planck cannot disentangle between the two scenarios.

The role of S 8 priors

Instead of using a split-normal likelihood on S 8 , a more accurate approach would have been to include the full galaxy shear and clustering power spectra. Making use of the full likelihood would however require the ability to compute the matter power spectrum on non-linear scales in a ΛDDM universe, a task that is beyond the scope of this work. Let us note that it has been established in various cases that the reconstructed S 8 value only mildly vary from one model to another. In particular, the KiDS collaboration has established that the reconstructed value of S 8 is insensitive to the neutrino mass (HILDEBRANDT et al. 2020) -a model that has physical effects very similar to the ΛDDM model 18 . This provides confidence in making use of a prior on S 8 derived in the ΛCDM context. Nevertheless, to highlight the impact of a different S 8 measurement, we conduct two extra BAO + SNIa + Planck + S 8 runs for ΛDDM, but replacing the S 8 prior from KiDS-1000+BOSS+2dFLens by the one determined in the combined analysis KiDS+Viking+DES-Y1 19 -0.02 . We also performed three analogous ΛCDM runs (one per each S 8 prior), in order to compute ∆χ 2 min = χ 2 min (ΛDDM)χ 2 min (ΛCDM) in each case. From the reconstructed parameters and the ∆χ 2 min : one can see that the level of preference is higher for the KiDS+Viking+DESY1 case while it is lower in the DES-only case. This explicitly demonstrates that the statistical significance of the DDM "detection" is strongly driven by the level of tension of the S 8 value used in the analysis. If the S 8 tension increases in the future, the preference for ΛDDM over ΛCDM would likely increase. On the other hand, if the S 8 tension disappears, cosmological data would not favour the ΛDDM scenario compared to the standard ΛCDM scenario.

Log 10 (Γ/[Gyr -1 ]) = -1.89 +0.82

Implications for other cosmic tensions and the Xenon1T anomaly

The H 0 tension

In order to test the implications of the 2-body decay for the Hubble tension, we conduct a run that includes the local measurement of H 0 from SH0ES (RIESS et al. 2019), CMB, BAO and SNIa data. For the sake of brevity we do not report the results of the runs here. We find that the shape of the posterior probabilities is almost unchanged, except for a tiny shift in H 0 to a higher value, H 0 = 68.21 ± 0.4 km/s/Mpc. We thus confirm the inability of this model to resolve the Hubble tension. In fact, any late-time solution that does not modify the sound horizon at recombination is expected to fail when combining BAO with SNIa data, as we discussed in Sect. 2.2. Although this is not of material importance in the context of the present work, let us note that a more correct way of combining Pantheon and SH0ES is through a prior on the intrinsic magnitude of SN1a (see Sect. 2.2). Yet, this does not affect our conclusions, as it was explicitly shown in Sect. 2.5.

The A lens anomaly in Planck data

We focus now on the 'A lens ' anomaly, i.e. the anomalous amount of lensing estimated from the smoothing of the acoustic peaks at high-'s within Planck data, as quantified by the 'A lens ' parameter. We wish to check whether this anomaly could impact constraints on the ΛDDM model, and conversely if the ΛDDM model could help explaining the existence of such anomalies, since it has already been noted that this anomaly could be related to the S 8 tension (see our discussion in Sect. 1.7.2). To do so, we follow the approach of the SPTpol collaboration and implement two new parameters in CLASS that allows to (roughly) marginalize over the lensing information in Planck. The parameter A TTTEEE lens re-scales the amplitude of the lensing power spectrum entering in the high-part of the CMB TT,TE,EE spectra, while the parameter A φφ lens re-scales the amplitude of the lensing power spectrum reconstruction. We present the result of a MCMC analysis including data from Planck One can see that the S 8 parameter reconstructed once marginalizing over the 'A lens ' anomaly is lower by ∼ 0.5σ than in the baseline analysis. However, the preference for ΛDDM decreases, with Log 10 (ε) now unconstrained. Note also that the A TTTEEE lens is still more than 2σ away than the fiducial value A TTTEEE lens = 1. We can therefore conclude that the ΛDDM model cannot explain this anomaly and that the preference for ΛDDM would likely disappear if the S 8 tension turns out to be explained by a systematic in Planck data leading to the anomalous value of the A lens parameters.

The Xenon1T excess

Let us explore the implications of the ΛDDM model for the excess of events in the electronic recoils recently reported by the Xenon1T Collaboration (APRILE et al. ). Interestingly, the WDM daughter species in the ΛDDM scenario could in principle play the role of such a fast component, since our results indicate that the 1-σ range for the speed of the daughter particle extends up to v/c ε 0.05.

In order to test this hypothesis further, we perform another MCMC analysis including Planck high-TT,TE,EE+lensing+BAO+SNIa+S 8 (from KiDS-1000+BOSS+2dFLens), with the DR energy fraction now fixed to ε = 0.05. This serves as a proxy for taking into account Xenon1T measurement (alternatively, one could enforce ε > 0.05). The results are summarized in Fig. 3.12 and Tab. (3.3). We find best-fit values Log 10 (Γ/Gyrs -1 ) -2.4 and S 8 0.784, at the cost of a mild degradation in the fit to Planck data (∆χ 2 +1.7), indicating that the 2-body decaying scenario has indeed the potential to provide a common resolution to the S 8 and Xenon1T anomalies. We leave the construction of a realistic model and study of the subsequent cosmological implications beyond the effect the decay to another study (see (G. CHOI et al.

Detecting DDM in the CMB: impact of current and future data

In this section, we confront the DDM model to additional CMB data from current ground based surveys and perform forecast for future surveys. In addition to Planck, we consider first, the high-CMB EE and TE (

2057.6

Table 3.3 -The mean (best-fit) ±1σ errors of the cosmological parameters from our Planck high-TT,TE,EE+lensing+BAO+SNIa+S 8 analysis, when fixing the DR energy fraction to ε = 0.05. We also report the best-fit χ 2 .

Confronting ΛDDM to SPTpol data

It is interesting to test the robustness of the DDM "detection" to a change of CMB data sets, especially given the impact of marginalizing over the 'A lens ' anomaly as discussed in the previous section. We thus confront the ΛDDM scenario under study against a set of CMB data constituted by low-temperature and polarization as well as high-temperature data from Planck, in combination with high-polarization data from SPTpol (see Sect. 3.4 for further details and references). It has been shown indeed that such a joint analysis predicts an amount of CMB lensing consistent with the ΛCDM expectation (CHUDAYKIN et al. 2020a), i.e. no 'A lens ' anomaly, and no S 8 tension. This is manifest in Fig. 3.13, where we compare predictions from the ΛCDM and the ΛDDM models, the latter both with and without including information on S 8 from KIDS1000+BOSS+2dFLens. As one can easily see, both cosmological models predict a S 8 value in excellent agreement with the KIDS1000+BOSS+2dFLens measurement, displayed as a green horizontal band. Our results regarding the ΛDDM model are also reported in Tab. (3.4).

In Fig. 3.14, instead, we report a comparison between our baseline ΛDDM analysis and the Planck+SPT one. First and foremost, SPTpol appears in very good agreement with the ΛDDM model resolution of the S 8 tension required by Planck. However, the ΛDDM parameters are largely unconstrained in that case and no deviations from ΛCDM are visible, which further establishes that if the S 8 tension turns out to be explained by a systematic in Planck high-polarization data, the preference for ΛDDM is likely to vanish.

Confronting ΛDDM to ACTpol data

Second, we confront the ΛDDM model to the combination of Planck and ACTPol data, to test whether more accurate measurements at high-can further constrain the model. Within ΛCDM, it has been found that ACTPol data (when combined with WMAP) also favor relatively high S 8 , in 2.1σ disagreement with KiDS1000+BOSS+2dFLens (AIOLA et al. 2020). To limit double counting of information, we follow the procedure of the ACT collaboration and truncate multipoles < 1800 in the ACT TT data. The results of this analysis are presented in Fig. 3.15 and Tab. (3.5). Interestingly, the ΛDDM parameter ε is more precisely measured with the inclusion of the ACTPol data, while the mean value is barely affected. Compared with ΛCDM, the ∆χ 2 in favor of ΛDDM is now -6.7, and the level of -0.12 -6.7

Table 3.5 -The mean (best-fit) ±1σ errors of the cosmological parameters from our BAO + SNIa + Planck + ACTPol analysis, with and without imposing a split-normal likelihood on S 8 (from KiDS+BOSS+2dFLens). For each data-set, we also report the best-fit χ 2 and the ∆χ 2 (ΛCDM) ≡ χ 2 min (ΛDDM) -χ 2 min (ΛCDM).

tension between Planck+ACT+BAO+SN1a and S 8 from KIDS1000+BOSS+2dFLens is 1.3σ. We conclude that ACT data are in very good agreement with the ΛDDM model, slightly increasing its preference over ΛCDM.

Towards detecting the ΛDDM model with CMB-S4

As we have extensively discussed, current CMB data are not sensitive enough to detect DDM, so that the preference for non-standard values for ε and Γ is fully driven by the inclusion of the S 8 measurement from weak lensing data in the analysis. To further stress this aspect we have performed additional analyses fitting a set of mock CMB data generated starting from our reference best-fit ΛDDM model (i.e. that from the Planck+BAO+SNIa+KiDS+BOSS+2dFLens run, reported in the fourth column of Tab. (3.1)). The resulting contour plots are shown in Fig. 3.16, where we compare the constraints that Planck would obtain if the "true" cosmological model actually contained DDM, with those that a future generation CMB survey (CMB-S4) would get. Concretely, this task was pursued by using the 'Planck-fake-realistic' and 'CMB-S4' likelihoods available in MontePython-v3 . The former allowed us to generate synthetic Planck data, whereas the latter includes multipoles from 30 to 3000, assuming a sky coverage of 40%, uncorrelated Gaussian error on each a m 's, uncorrelated temperature and polarization noise, and perfect foreground cleaning up to max 20 . All details about the likelihood can be found in Tab. 1 of BRINCKMANN et al. (2019b).

As expected, from Fig. 3.16 it is manifest that Planck alone could not detect DDM even if its signature was truly imprinted in CMB data. Note indeed that the red contours barely features an overlap at 2σ in S 8 , and only upper limits on Log 10 Γ, Log 10 ε. This, as we explained earlier, is a consequence of the degeneracy that exists within ΛDDM and leads to a bias in the Bayesian analysis. Therefore, the information that matters in quantifying the success of the resolution is rather contained in the χ 2 values: just like in the analysis of real data, we find that, when testing the ΛCDM model against the mock data that contains the ΛDDM signal, the χ 2 in the ΛCDM model is identical to that of the ΛDDM model. This clearly shows that Planck cannot disentangle between ΛCDM and ΛDDM, while the S 8 measurements favors ΛDDM (in terms of χ 2 ). This is not the case for CMB-S4, which contours are over-plotted in blue: if the real Universe contains DDM, CMB-S4 would unequivocally detect its signature, finding ε = 0 at 2 σ level. The ∆χ 2 in favor of the ΛDDM model from (mock) CMB-S4 data alone is in fact +8.

Summary and outline

In this work, we have performed a comprehensive cosmological study of the CDM 2-body decay scenario dubbed 'ΛDDM', whereby decays are characterized both by the decay rate Γ and energy fraction converted to radiation ε, including for the first time a fully consistent treatment of the linear perturbations of the WDM daughter component. To that end, we have made use of a new approximation scheme, that allows to accurately and quickly compute the dynamics of the WDM linear perturbations by treating the WDM species as a viscous fluid. Close to the best-fit values, our approximation scheme is accurate at the O(0.1%) level in the CMB power spectra and O(1%) level in the linear matter power spectrum (see App. D.4). We have then discussed in detail the dynamics of linear density perturbations of the mother and daughter particles, as well as the physical effects of the ΛDDM model on the CMB and matter power spectra.

In a second part, we have performed a set of MCMC analyses of the ΛDDM model against a suite of up-to-date low-and high-redshift data-sets. We have compared the constraints obtained from BAO and SNIa data, thereby solely based on background effects, to those obtained from the full Planck data-set, that requires instead an accurate description of the WDM linear perturbations. We find that Planck CMB data constrain the ΛDDM model ∼ 1 order of magnitude better than current BAO+SNIa data. However, we also show that despite these constraints, the ΛDDM model provides a promising possibility to resolve the S 8 tension.

We have then tested the robustness of the ΛDDM resolution to the S 8 tension to a number of change in the analysis. We show that the mild preference for the ΛDDM model over ΛCDM is tied to the S 8 value chosen in the analysis. Concretely, the S 8 value from the KiDS+Viking+DES analysis, which has a higher level of tension with the Planck ΛCDM prediction than the baseline KiDS+BOSS+2dFLens value, leads to a stronger preference in favor of the ΛDDM model. However, the DES-only result, which is in reasonable agreement with Planck, leads to a weaker preference of the ΛDDM model. Similarly, once marginalizing over the lensing information in Planck through the A lens parameter (we used two extra parameters in practice describing the normalization of the lensing power spectrum and the normalization of the lensing smoothing effect in the high-TT,TE,EE power spectra), or when trading the Planck high-TE,EE power spectra for the SPTpol ones, the preference for the ΛDDM model decreases. This is because in these two cases the inferred ΛCDM model has a smaller S 8 value, showing less of a tension with the weak lensing surveys. This indicates that if the S 8 tension ends up coming from an unknown systematic within weak lensing surveys or within Planck data, the preference for the ΛDDM model would likely disappear. On the other hand, when combining Planck with ACTPol the mild preference for ΛDDM increases, and the remaining 'tension' with S 8 is now only ∼ 1.3σ.

We have also tested the intriguing possibility that the recent Xenon1T excess is due to the ΛDDM model. To that end, we have performed an additional MCMC analysis fixing ε = 0.05 as required by Xenon1T. We find that it is easy to resolve the S 8 tension in that case, pointing to a DCDM lifetime of Log 10 (Γ/[Gyr -1 ]) = -2.72 +0. 61 -0.21 . Interestingly, this comes at the cost of a very minor degradation in Planck fit (∆χ 2 +1.7), indicating that Planck, BAO and SNIa data are in good agreement with this model. Finally, by performing an analysis where we artificially introduce a DDM signal in a set of mock CMB data, we explicitly demonstrate that while current CMB data alone are not sensitive enough to distinguish between standard CDM and DDM, next-generation CMB experiments (CMB-S4) can unambiguously detect its signature.

It will be very interesting to go beyond the linear aspects discussed in this work and study the non-linear evolution of density perturbations, in order to be able to make use of the full power of the KiDS and DES likelihoods. This task would likely require using N-body simulations, as in WANG et al. (2012,2014). This will be even more crucial with upcoming surveys such as Euclid (AMENDOLA et al. 2018), LSST (MANDELBAUM et al. 2018), and DESI (AGHAMOUSA et al. 2016). For instance, these surveys will be able to measure the differences between the growth rate f σ 8 in the ΛDDM scenario and ΛCDM at 0 z 1, which at present remain below the sensitivity of current experiments (see Fig. In SIMON et al. (2022), we have already extended the analysis to the mildly non-linear scales using the Effective Field Theory of LSS (EFTofLSS) (CHUDAYKIN et al. 2020b; D' AMICO et al. 2021a). This allowed us tot test the ΛDDM model against the full shape galaxy power spectrum as measured by BOSS, significantly improving the constraints at 68 % C.L. on the CDM lifetime. We also showed that, in order to fit EFTofBOSS data while lowering S 8 to match the values measured by weak-lensing surveys, the best-fit model has a longer lifetime Γ - 1 120 Gyr and a larger kick velocity v kick /c ε 1.2% than without EFTofBOSS.

To finish, let us point out that the DES collaboration recently derived new constraints on the ΛDDM scenario using observations of Milky Way satellites (MAU et al. 2022). These constraints exclude Log 10 (Γ/Gyr -1 ) -1.5 for Log 10 (v kick /c) Log 10 (ε) -4. The best-fit model of our baseline BAO+SNIa+Planck+S 8 analysis, and a large fraction of the 68 % C.L., lie well within the allowed region, but these observations certainly provide a crucial test of the DCDM cosmology, as a deficit of satellites compared to ΛCDM is expected in this model.

Constraints on neutrino mass and lifetime from cosmological data

Even though neutrinos were first detected more than six decades ago, they remain among the most mysterious particles in nature, with many of their fundamental properties still to be determined. In particular, although oscillation experiments have provided convincing evidence that neutrinos have non-vanishing masses, these measurements are only sensitive to the mass-squared splittings and consequently the spectrum of neutrino masses remains unknown. The lifetimes of the neutrinos are also poorly constrained, especially in comparison to the other particles in the Standard Model (SM). The determination of the masses and the lifetimes of these mysterious particles remain some of the most important open problems in fundamental physics.

The fact that cosmic neutrinos are among the most abundant particles in the universe, contributing significantly to the total energy density at early times, provides an opportunity to measure their properties. In particular, the evolution of the cosmological density fluctuations depends on m ν , the sum of neutrino masses. This translates into characteristic effects on the CMB and LSS (BOND et al. 2018)), that are large enough to allow the sum of neutrino masses to be determined in the near future. This determination is based on the observation that massive neutrinos contribute differently to cosmological observables than either massless neutrinos or CDM. At early times, while still relativistic, massive neutrinos contribute to the energy density in radiation, just as in the case of massless neutrinos. However, after neutrinos become non-relativistic, their energy density redshifts as matter and therefore contributes more to the expansion rate than massless neutrinos, which would continue to redshift as radiation. As a result, over a given redshift span, the higher expansion rate reduces the time available for the growth of matter density perturbations. However, since massive neutrinos retain pressure until late times, their contribution to the density perturbations on scales below their free streaming lengths is too small to compensate for the shorter structure formation time. Therefore, if neutrinos become non-relativistic after recombination, the net effect of non-vanishing neutrino masses is a suppression of the matter power spectrum and the CMB lensing potential. Based on this, current observations are able to place a bound on the sum of neutrino masses, m ν 0.12 eV (AGHANIM et al. 2020b). It is important to note that this result assumes that neutrinos are stable on timescales of order the age of the universe. In scenarios in which the neutrinos decay [START_REF] Serpico | Cosmological Neutrino Mass Detection: The Best Probe of Neutrino Lifetime[END_REF][START_REF] Serpico | Neutrinos and cosmology: a lifetime relationship[END_REF], or annihilate away into lighter species [START_REF] Beacom | Neutrinoless universe[END_REF][START_REF] Farzan | Neutrinos secretly converting to lighter particles to please both KATRIN and the cosmos[END_REF]) on timescales shorter than the age of the universe, this bound is no longer valid and must be reconsidered.

Cosmological observations can also be used to place limits on the neutrino lifetime. In the case of neutrinos that decay to final states containing photons, the bounds on spectral distortions in the cosmic microwave background (CMB) can be translated into limits on the neutrino lifetime, τ ν 10 19 s for the larger mass splitting and τ ν 4 × 10 21 s for the smaller one (AALBERTS et al. 2018). In the case of decays to invisible final states, the limits are much weaker. For neutrinos that decay while still relativistic, the decay and inverse decay processes can prevent neutrinos from free streaming. Measurements of the CMB power spectra sets a lower bound on the neutrino lifetime, τ ν ≥ 4 × 10 6 s (m ν /0.05eV) 5 In the case of non-relativistic neutrino decays into dark radiation, the energy density of the decay products redshifts faster than that of stable massive neutrinos. Unstable neutrinos therefore have less of an effect on structure formation than stable neutrinos of the same mass. Consequently, cosmological observables depend both on the masses of the neutrinos and their lifetimes, and heavier values of m ν may still be allowed by the data provided the neutrino lifetime is short enough. In CHACKO et al. 2020, Planck15 and LSS data were used to place constraints on the neutrino mass as a function of the lifetime, and found that values of m ν as large as 0.9 eV were still allowed by the data. Future LSS measurements at higher redshifts may be able to break the degeneracy between the neutrino mass and lifetime and measure these parameters independently (CHACKO et al. 2021).

It is worth noting that there are also bounds on the neutrino lifetime from Supernova 1987A [START_REF] Frieman | Neutrino Mixing, Decays and Supernova Sn1987a[END_REF] ). However, these constraints are in general much weaker than the limits from cosmology.

We revisit the scenario in which neutrinos decay into dark radiation after becoming non-relativistic, and obtain updated limits based on the newer data from Planck18. In order to take advantage of the greater precision of the new data, the analysis we perform is also more accurate. We find that, under the assumption that neutrinos decay after becoming non-relativistic, the neutrino mass bound from Planck18 data (in combination with BOSS BAO data and Pantheon SN1a data) is relaxed to m ν 0.42 eV (95% C.L.) 2 . While this represents a remarkable relaxation of the constraints as compared to the case of stable neutrinos, we note that it is much stronger than the limit derived from Planck15 data for the same decaying neutrino scenario, m ν 0.9 eV at (95% C.L.). We show that the improvement of the bound arises primarily from the more precise low-polarization data from Planck18, which allows an improved determination of the optical depth to reionization τ reio , thereby breaking the correlation with m ν that appears (for relatively high neutrino masses) through the impact of neutrinos on the overall height of the acoustic peaks (i.e. the "early integrated Sachs-Wolfe effect") (LESGOURGUES et al. 2018).

Besides using up-to-date cosmological data, we also improve the analysis from CHACKO et al. 2020 -hereafter C19 -by incorporating higher order corrections due to neutrino decays into the Boltzmann equations that describe the evolution of Universe's energy and metric fluctuations. Recently, BARENBOIM et al. 2021 -hereafter B20 -provided a complete set of Boltzmann equations for the neutrino decay, but did not conduct MCMC runs necessary to update the bounds on non-relativistic neutrino decays. In this work, we derive Boltzmann equations exactly valid in the absence of 'inverse-decays' and quantum statistics (i.e. Fermi blocking and Bose enhancement terms). For the numerical implementation, we follow a consistent T dec /m ν expansion, where T dec is the temperature at the time of the decay, so that the analysis is under control when neutrinos decay after become non-relativistic.

The rest of this chapter is organized as follows. In Sect. 4.2 we review the neutrino mass conundrum and its interplay with cosmology. In Sect. 4.3, we present a summary of constraints on the parameter space of decaying neutrinos. In Sect. 4.4, we derive the set of Boltzmann equations to describe neutrino decay that are valid in the non-relativistic regime and compare our improved analysis to past work. In Sect. 4.5, we present a MCMC analysis of the decaying neutrino scenario against up-to-date cosmological data. Finally, we conclude in Sect. 4.6.

The neutrino mass conundrum

Neutrinos are neutral fermions with spin 1/2 which were postulated by W. Pauli in 1930 in order to explain the continuous spectrum observed in the β-decay of nuclei. They come in three flavors which are paired to leptons within SU (2) L doublets, and are charged only under the weak interaction. Because of this, neutrinos typically pass through matter unimpeded and therefore are very hard to detect. Electron neutrinos (ν e ) were discovered by Reines and Cowan in 1956, muon neutrinos (ν µ ) in 1962 at Brookhaven and tau neutrinos (ν τ ) in 2000 at Fermilab, by the DONUT experiment. Neutrinos were long thought to be massless, but this started to change with the observation of the solar neutrino flux. The sun is a potent source of ν e in the energy range E ∼ O(MeV), produced in thermonuclear fusion reactions inside the core. After the technology required to detect the solar neutrino flux was developed in the 70s, the observations showed that the number of detected solar ν e was only about one third of the number predicted by the standard solar model. This is the famous solar neutrino problem. The breakthrough came in 2001, thanks to the data from the SuperKamiokande experiment and the Sudbury Neutrino Observatory. Those combined observations allow to compare the total number of solar neutrinos of all types, to the number of just ν e : the first was in good agreement with theoretical expectations, while the second still showed the same deficit, pointing at conversion of neutrino flavor during their propagation. The most plausible explanation invoked neutrino oscillations, which were confirmed in subsequent experiments.

Neutrino oscillations had been theorized in the 50s by Bruno Pontecorvo [START_REF] Pontecorvo | Inverse beta processes and nonconservation of lepton charge[END_REF]. They arise from a quantum mechanical phenomenon linked to the fact that neutrino possess non-zero masses. The principle of superposition allows the three flavor states to be (orthogonal) combinations of three neutrino states with a definite mass. Introducing the Pontecorvo-Maki-Nakgawa-Sakata (PMNS) mixing matrix U , a 3 × 3 unitary matrix, a certain "ket" state can generally be written as

|ν α = 3 i=1 U * αi |ν i with α = e, µ, τ, (4.1)
where greek (latin) indices label flavor (mass) states, and it is assumed the normalization ν α |ν β = δ αβ and ν i |ν j = δ ij . Considering neutrinos propagating in vacuum at semi-relativistic speeds, one can compute the survival probability P να→να as a function of the distance of the detector from the neutrino source (GIUNTI et al. 2007)

P να→να (L) = 1 -4 i>j |U αi | 2 |U αj | 2 sin 2 2π L L osc ij , ( 4.2)
where L osc ij is the characteristic oscillation length

L osc ij ≡ 4πE ∆m 2 ij 2.47 E/MeV ∆m 2 ij /eV 2 m, (4.3)
and ∆m 2 ij ≡ m 2 i -m 2 j denote the (squared) mass splittings. Therefore, flavor transitions can take place only if neutrinos are massive, their masses are non-degenerate, and the corresponding PMNS matrix elements are non-zero. From the argument of the sinusoidal function, we see that for a given distance L and source energy E, experiments are mainly sensitive to mass splittings ∆m 2 ij such that L L osc ij . In fact, the sun is not the single source of neutrinos available. Nuclear reactors typically produce neutrinos with energy of the order E ∼ O(MeV), and are classified between short (L ∼ 10 m) (SBL), long (L ∼ 1 km) (LBL) and very long (L ∼ 10 2 km) (VLBL) baseline experiments, sensitive to ∆m 2 ij 0.1, 10 -3 and 10 -5 eV 2 , respectively. An analogous classification can be made for neutrino beams produced at accelerators in the energy range E ∼ O(GeV) due to the pion, kaon and muon decays. In this case, SBL are characterized by L ∼ 1 km and LBL by L ∼ 10 3 km, sensitive to ∆m 2 ij 0.1 and 10 -3 eV 2 , respectively. Finally, atmospheric neutrinos are produced when cosmic rays, consisting mainly of protons, interact with atoms in our atmosphere. These interactions create a lot of pions which subsequently decay into muons and neutrinos. In this case, the energy and distance cover a very wide range, but neutrinos are typically detected with energies centered at E ∼ O(GeV) and distances up to L ∼ 10 4 km (for neutrinos produced on the other side of the Earth), which corresponds to a sensitivity ∆m 2 ij 10 -4 eV 2 .

The PMNS matrix is unitary, so it contains a priori nine real d.o.f. However, five of these real parameters can be absorbed as phases of the lepton fields, and thus the PMNS matrix is fully described by four parameters (assuming Dirac neutrinos), and can be parametrized in the following form

U =    1 0 0 0 c 23 s 23 0 -s 23 c 23       c 13 0 s 13 e -iδ CP 0 1 0 -s 13 e iδ CP 0 c 13       c 12 s 12 0 -s 12 c 12 0 0 0 1    , (4.4)
where c ij = cos (θ ij ) and s ij = sin (θ ij ), with θ ij the mixing angle of mass states i and j, and δ CP is a Dirac CP violating phase. Hence, neutrino oscillations are described in general by six parameters, that can taken to be {θ 13 , θ 23 ,θ 12 , δ CP , ∆m 2 21 , ∆m 2 31 }. However,

(NH), if m 3 m 2 > m 1 , or inverted hierarchy (IH), if m 2 > m 1 m 3 .
The absolute scale of neutrino mass is unknown as well. The sum of neutrino masses m ν can be related to the mass splittings as follows

m NH ν = m 1 + m 2 1 + ∆m 2 21 + m 2 1 + ∆m 2 31 , (4.5
)

m IH ν = m 3 + m 2 3 + |∆m 2 31 | + m 2 3 + |∆m 2 31 | + ∆m 2 21 , (4.6)
Hence, the minimum value for the total neutrino mass depends on the mass ordering. For the normal hierarchy, setting m 1 = 0 gives m NH ν 0.06 eV, while for the inverted hierarchy, setting m 3 = 0 gives m IH ν 0.10 eV. In Fig. 4.1 we show the sum of neutrino masses (= , to shorten notation) as a function of the lowest neutrino mass (m L ), together with some current and projected upper bounds on . Currently, cosmology provides the strongest constraints on the sum of neutrino masses. As mentioned previously, the latest constraints from Planck18 TT,EE,TE+lowE+lensing+BAO give cosmo < 0.12 eV (AGHANIM et al. 2020b). This bound strengthens to cosmo < 0.102 eV when combining with redshift-space distortions (RSD) from eBOSS (ALAM et al. 2021) and to cosmo < 0.089 eV when combining with the Lyman-α forest 1D flux power spectrum (PALANQUE-DELABROUILLE et al. 2020).

It is also possible to measure the absolute neutrino mass scale with laboratory experiments on Earth. These experiments typically look at the energy spectrum of electrons emitted from the tritium beta decay

H 3 -→ He 3 + e -+ νe , ( 4.7)
which have

Q β = m H 3 -m He 3
-m e 18.58 keV. The main idea is that the endpoint of the spectrum (or maximum electron energy) is shifted with respect to Q β due to the presence of neutrino masses. Assuming mass splittings much smaller than the experimental energy resolution, one expects to measure a shift Q β -m νe , where m νe is the effective electron mass

m 2 νe ≡ i |U ei | 2 m 2 i . ( 4.8)
The combination of the first and second campaign of the KArlsruhe TRItium Neutrino (KATRIN) β-experiment have yielded a constraint on the effective neutrino mass of m νe < 0.8 eV (AKER et al. 2022), corresponding to KATRIN < 2.4 eV 3 (at 90 % C.L.). The expected final sensitivity of KATRIN is KATRIN < 0.6 -0.8 eV. One important benefit of KATRIN is that the neutrino mass determination is fully model independent (even independent of the Dirac or Majorana nature of neutrinos), as opposed to the CMB/LSS determinations, which always rely on the cosmological model.

The origin of neutrino masses cannot be successfully explained in the framework of the SM of particle physics, so they currently provide the only certain evidence of physics beyond the SM [START_REF] Giunti | Fundamentals of Neutrino Physics and Astrophysics[END_REF]). Besides the absolute mass scale and hierarchy, other open questions in the neutrino sector include: their Majorana/Dirac nature, the existence of new sterile neutrinos (inducing a small non-unitarity in the mixing matrix) or the possibility of CP violation in the leptonic sector (possibly related to the matter-antimatter asymmetry). As we discussed in the previous section, massive neutrinos influence both the cosmic expansion as well as the growth of structure in the universe, so cosmological observations provide very valuable clues about their properties. Future LSS surveys such as Euclid (AMENDOLA et al. 2018) aim at a detection of the absolute neutrino mass scale. Notice also from the above discussion that a possible detection of by KATRIN would immediately enter in conflict with CMB/LSS determinations. In the rest of this chapter, we study the cosmological implications of a model of decaying neutrinos that could potentially reconcile those measurements.

Parameter space of decaying neutrinos

Here we outline the constraints on the mass and lifetime of neutrinos decaying into dark radiation. As explained in Sect. 4.1, current cosmological observables 3 For the range of masses probed by KATRIN, neutrino mass states are quasi-degenerate, mi m, so the effective electron mass is roughly given by m 2 νe m 2 i |Uei| 2 = m 2 , where the last equality follows from unitarity. Hence the sum of neutrino masses is indeed given by ∼ 3m ∼ 3mνe.

Figure 4.2 -

The plot shows the current constraints on decaying neutrinos in the m ν -Γ ν parameter space. The colored regions are excluded by current data while the white region is allowed. The orange dashed line represents Γ ν = H(a nr ). Our study focuses on the region below this line, meaning decay happens after neutrinos have become non-relativistic. The light grey region shows current constraints on neutrino mass and lifetime coming from the requirement that neutrinos are free streaming close to recombination (B20). The shaded region indicates that this bound may not be applicable when neutrino mass is larger than the temperature of recombination: m ν > T * ∼ 0.2 eV. Our analysis excludes the red (blue) region labelled "Planck 2015" ("Planck 2018") based on the data (Planck+BAO+Pantheon). The vertical brown line shows the projected KATRIN sensitivity.

only place limits on a combination of the sum of neutrino masses and their lifetime. Therefore, in this study we will map out the constraints in the two-dimensional parameter space spanned by the sum of neutrino masses ( m ν ) and the neutrino decay width (Γ ν ), as shown in Fig. 4.2.

In our analysis we assume that all three neutrinos are degenerate in mass. This is a good approximation because the current bounds on m ν are larger than the observed mass splittings (see Fig. 4.2). We further assume that all three neutrinos have the same decay width Γ ν . Since the mixing angles in the neutrino sector are large, this is a good approximation in many simple models of decaying neutrinos if the spectrum of neutrinos is quasi-degenerate. While this is a simple parameterization of neutrino decays, our bounds can easily be applied to specific models, as done in great details in ESCUDERO et al. (2020b).

The CMB can be used to constrain the masses and decay widths of neutrinos that decay prior to recombination. When neutrinos decay while still relativistic, decay and inverse decay can prevent neutrinos from free-streaming. If this happens before recombination, it can alter the well-known 'neutrino drag' effect that manifests as a phase-shift at high-'s in the CMB power spectrum [START_REF] Bashinsky | Neutrino perturbations in CMB anisotropy and matter clustering[END_REF][START_REF] Audren | Robustness of cosmic neutrino background detection in the cosmic microwave background[END_REF][START_REF] Follin | First Detection of the Acoustic Oscillation Phase Shift Expected from the Cosmic Neutrino Background[END_REF][START_REF] Baumann | Phases of New Physics in the CMB[END_REF]. Therefore, CMB data can place a constraint on the decay width of neutrinos. The resulting bound depends on neutrino masses, and was recently updated in B20, τ ν ≥ 4 × 10 6 s (m ν /0.05 eV) 5 . This bound excludes the grey region at the top of Fig. 4.2.

In addition, based on the analysis in this paper, part of the 'late-decay' parameter space can also be excluded based on the gravitational impacts of massive neutrinos on the CMB and LSS. Through the Monte Carlo study presented in Sect. 4.5, the blue (red) shaded region in Fig. 4.2 is excluded by the data combination Planck18 (Planck15) + BAO + Pantheon. 4 . The orange dashed line in the figure (Γ ν = H(a nr )) separates the region where neutrinos decay when non-relativistic from the region where they decay while still relativistic. Here a nr corresponds to the approximate scale factor at the time that neutrinos transition to non-relativistic, and is defined as 3T ν (a nr ) = m ν . This simple definition is based on the fact that for relativistic neutrinos at temperature T ν , the average energy per neutrino is approximately 3T ν . The Hubble scale at a nr is given by,

H(a nr ) = H 0 Ω m m ν 9T ν,0 3/2 (4.9) 7.5 × 10 5 km/s/Mpc H 0 68km/s/Mpc Ω m 0.3 1/2 m ν 1eV 3/2 1.5 × 10 -4 eV T ν,0 3/2
, where T ν,0 is the present neutrino temperature. Since our study focuses on the decay of neutrinos after they become non-relativistic, we only present constraints below the orange dashed line. Our analysis shows that m ν as large as 0.42 eV is still allowed by the data.

Our results have important implications for current and future laboratory experiments designed to detect neutrino masses. As discussed in Sect. 4.2, next generation tritium decay experiments such as KATRIN (ANGRIK et al. 2005) are expected to be sensitive to values of m νe as low as 0.2 eV, corresponding to m ν of order 0.6 eV. Naively, a signal in these experiments would conflict with the current cosmological bound for stable neutrinos, m ν < 0.12 eV. However, since the unstable neutrino paradigm greatly expands the range of neutrino masses allowed by current cosmological data, it is interesting to explore whether this scenario can accommodate a potential signal at KATRIN. In Fig. 4.2, we display a brown vertical line m ν = 0.6 eV that corresponds to the expected KATRIN sensitivity. We see that this value of m ν is too large to be accommodated in the non-relativistic decay regime, where our analysis is valid. However, our result, in combination with those from the 'relativistic decay' scenario studied in B20, leaves open the interesting possibility that neutrinos decaying with a decay width between Log 10 Γν km/s/Mpc ∼ 5.5 -9 could reconcile cosmological observations with a potential detection at KATRIN, thereby opening a large discovery potential for laboratory experiments. To confirm this conjecture, more work needs to be done to cover the 'intermediate' decay regime (i.e. where neutrinos are neither fully relativistic nor fully non-relativistic). We leave this for future work.

In recent years, a number of studies have attempted to constrain the neutrino mass ordering, showing that under the assumption of stable neutrinos, the inverted 4 Note that in our analysis we scanned the region between 0 ≤ Log 10 Γν km/s/Mpc ≤ 6. In Fig. 4 ). However, these arguments are centered on the fact that these analyse leads to a constraint on m ν at odds with the lower bound on the sum of neutrino masses in the case of inverted ordering, m ν 0.1 eV. Our result suggests that these constraints are strongly dependent on the assumption of neutrino stability over cosmological timescales, and therefore that the inverted ordering is not robustly excluded. It would be very interesting to extend our analysis to the inclusion of Ly-α data to confirm this conclusion.

Formalism of invisible non-relativistic neutrino decay

In this section, we revisit the set of Boltzmann equations describing the evolution of the phase space distribution (PSD) of massive particles decaying into daughter radiation. In our analysis, we assume the decay happens after the neutrinos have become non-relativistic so that the contribution from inverse decay processes and quantum statistics can be safely neglected.

Derivation of Boltzmann equations

We denote the phase space distribution of each species as f I (k, q, µ, τ ) (in Fourier space), where I = {ν, dr1, dr2}, since we are assuming a two-body decay to massless daughters. As in previous chapters, we work in the synchronous gauge co-moving with the CDM (introduced in Sect. 1.6.1), where the scalar metric potentials are referred as h and η. In this gauge, and neglecting all terms that are at least second order in perturbations, the Boltzmann equations controlling the evolution of f I is written as (c.f. Eq. (3.4)) :

∂f I ∂τ + iµ kq E I f I + df I dlnq η - h + 6η 2 µ 2 = ∂f I ∂τ C . ( 4.10)
The collision terms ∂f I ∂τ C

describe the decay process

ν(Q) -→ DR(Q 1 ) + DR(Q 2 ), (4.11)
where we labelled the comoving four-momentum Q = (E, q) of each species, and are given by (C19) :

∂f ν ∂τ C = - a 2 2E ν d 3 q 1 (2π) 3 2E 1 d 3 q 2 (2π) 3 2E 2 |M| 2 (2π) 4 δ (4) (Q -Q 1 -Q 2 )f ν (q), (4.12
)

∂f dr1 ∂τ C = + a 2 2E 1 d 3 q (2π) 3 2E ν d 3 q 2 (2π) 3 2E 2 |M| 2 (2π) 4 δ (4) (Q -Q 1 -Q 2 )f ν (q), (4.13
)

∂f dr2 ∂τ C = + a 2 2E 2 d 3 q (2π) 3 2E ν d 3 q 1 (2π) 3 2E 1 |M| 2 (2π) 4 δ (4) (Q -Q 1 -Q 2 )f ν (q). (4.14)
To separate the Boltzmann equations into zeroth and first order contributions, we will eventually perform the same decomposition as in Eq. (3.8) f I (k, q, µ, τ ) = fI (q, τ ) + ∆f I (k, q, µ, τ ). (4.15)

Let us start with the derivation of the zeroth order equations. As we show in App. E.1, the collision term in Eq. (4.12) for the decaying neutrinos simplifies to

∂f ν ∂τ C = - a 2 Γ ν m ν E ν f ν = - aΓ ν γ f ν , ( 4.16)
where γ ≡ E ν /(am ν ) denotes the Lorentz boost factor. Hence, the background equation for the neutrino mother particles can be simply written as

∂ fν ∂τ = - aΓ ν γ fν , ( 4.17)
Multiplying each side of the former equation by 4πa -4 q 2 E ν , and integrating over q, we find the continuity equation for ρν :

ρ ν + 3H ρν (1 + w ν ) = -aΓ ν m ν nν , ( 4.18)
where nν = 4πa -3 dqq 2 fν (q) is the neutrino number density. On the other hand, the continuity equation for the total background density of daughter radiation, ρdr = 4πa -4 i=1,2 dq i q 3 i fdri , is given by ρ dr + 4H ρdr = aΓ ν m ν nν , (

where the term at the r.h.s. follows from Eq. (4.18) and conservation of energy. In practice, we never use Eq. (4.18) to solve for the evolution of the neutrino background density. Rather, we directly compute the PSD fν at each time step, and then integrate over momenta to find ρν and nν . The formal solution fν (q, τ ) to the differential equation Eq. (4.17) is fν (q, τ ) = fν,i (q)e -Γν where τ i denotes the initial conformal time and fν,i represents the initial momentum distribution, which we take to be of the Fermi-Dirac form fν,i = 1/(e q/T ν,0 + 1).

We now turn to the Boltzmann equations describing the perturbations of the PSD of decaying neutrinos and their decay products. From Eq. (4.10) and Eq. (4.16), the Boltzmann equation for ∆f ν reads

∂∆f ν ∂τ + iµ kq E ν ∆f ν + d fν dlnq η - h + 6η 2 µ 2 = - a 2 Γ ν m ν E ν ∆f ν . (4.21)
As usual, we will expand ∆f I in Legendre multipoles

∆f I (k, q, µ, τ ) = ∞ =0 (-i) (2 + 1)∆f I, (k, q, τ )P (µ). (4.22)
With this, we can turn Eq. (4.21) into the following hierarchy of equations

∆f ν,0 = - qk E ν ∆f ν,1 + q ∂ fν ∂q h 6 - a 2 Γ ν m ν E ν ∆f ν,0 , (4.23) ∆f ν,1 = qk 3E ν [∆f ν,0 -2∆f ν,2 ] - a 2 Γ ν m ν E ν ∆f ν,1 , (4.24) ∆f ν,2 = qk 5E ν [2∆f ν,1 -3∆f ν,3 ] -q ∂ fν ∂q (h + 6η ) 15 - a 2 Γ ν m ν E ν ∆f ν,2 , (4.25) ∆f ν, >2 = qk (2 + 1)E ν [ ∆f ν, -1 -( + 1)∆f ν, +1 ] - a 2 Γ ν m ν E ν ∆f ν, . (4.26)
We have derived fluid equations for the decaying neutrinos in order to integrate out the dependency on momenta, similarly as we did for the warm daughter species in Chapter 3. However, we found that it is not so trivial to relate the collision terms in the Boltzmann hierarchy for the DR daughter (see Eq. (4.34)-Eq. (4.37)) with the elements of the neutrino stress-energy tensor: δ ν , θ ν and σ ν . In addition, the gain in speed provided by these fluid equations is not a high as for the decaying dark matter scenario of Chapter 3. This is mainly because the CLASS code implements an optimal sampling in momentum space 5 that makes the resolution of the neutrino Boltzmann hierarchy very manageable. For these reasons, we don't use any fluid equations for our analysis, but we show them in App. E.3 for the sake of completeness.

To study the perturbations of the daughter radiation, we can take moments of the perturbed PSD, following the same convention of Chapter 3,

F dr, ≡ 4π ρ c i=1,2 dq i q 3 i ∆f dri, , (4.27)
In terms of F dr, , the Boltzmann hierarchy for the radiation daughter reads

F dr,0 = -kF dr,1 - 2 3 h r dr + C 0 , (4.28) F dr,1 = k 3 F dr,0 - 2k 3 F dr,2 + C 1 , (4.29
C = 2i dΩ k 4π P (µ) 4π ρ c dq 1 q 3 1 ∂f dr ∂τ C [q 1 , µ] , (4.32)
with µ = k • q 1 . The overall factor of two in the equation above arises because we are adding the collision integrals of the two massless daughters, which are of the same form,

∂f dr1 ∂τ C = ∂f dr2 ∂τ C = ∂f dr ∂τ C
. In App. E.2, we show that C can be written as

C = 8πm ν Γ ν a 2 ρ c dq E ν q∆f ν, q + 1 q - 1 dq 1 q 1 P 2E ν q 1 -a 2 m 2 ν 2qq 1 , ( 4.33)
with q ± 1 = 1 2 (E ν ± q). Eq. (4.33) may also be obtained by taking the appropriate limit of the more general expression in B20 6 . Performing the integral over q 1 , we can obtain the following expressions for the first few C 's,

C 0 = 4πa 2 Γ ν m ν ρ c dqq 2 ∆f ν,0 , (4.34
)

C 1 = 4πa 2 Γ ν m ν ρ c dqq 2 q E ν ∆f ν,1 , (4.35
)

C 2 = 4πa 2 Γ ν m ν ρ c dqq 2 g 2 (q, E ν )∆f ν,2 , (4.36
)

C 3 = 4πa 2 Γ ν m ν ρ c dqq 2 g 3 (q, E ν )∆f ν,3 . (4.37)
Here the functions g 2 (q, ε ν ) and g 3 (q, ε ν ) are given by,

g 2 (q, E ν ) ≡ 5 2 - 3 2
E 2 ν q 2 + 3 4 (E 2 ν -q 2 ) 2 E ν q 3 ln E ν + q E ν -q , ( 4.38
)

g 3 (q, E ν ) ≡ 25 2 E ν q - 4q E ν - 15 2
E 3 ν q 3 + 15 4 (E 2 ν -q 2 ) 2 q 4 ln E ν + q E ν -q . (4.39)
Given the complicated integrals in Eq. (4.33), it is technically challenging to keep track of all the collision terms in the Boltzmann hierarchy. Instead, we choose to keep just the first few C 's for ≤ max . The idea behind this approach is that C is of order O((T dec /m ν ) ) around the time of decay. Therefore, for non-relativistic decay (T dec /m ν 1), it is self-consistent to set C > max = 0 because those terms only have negligible effect on physical observables. To understand the scaling of C , we first note that the integral over q in Eq. (4.34)-Eq. (4.37) receives most of its support from the region around q ∼ T ν,0 because ∆f ν, inherits features of the Fermi-Dirac distribution from fν,i = 1/(e q/T ν,0 + 1). Deep in the non-relativistic region, q E ν and T ν0 am ν . In this regime, we can employ a Taylor expansion for the functions g 2 and g 3 in powers of q/E ν to obtain,

g 2 (q, E ν ) ≈ 4 5
q 2 E 2 ν , g 3 (q, E ν )≈ 4 7

q 3 E 3 ν , for (q E ν ). (4.40)
Inserting Eq. (4.40) above into Eq. (4.34)-Eq. (4.37), it is straightforward to see that C ∝ (T ν0 /am ν ) . Moreover, if we assume decay happens deep in the non-relativistic region, we will get C ∝ (T dec /m ν ) when decay happens, where T dec = T ν0 /a dec . Therefore, C is suppressed by powers of T dec /m ν 1 for higher . To further justify this argument, we show in Sect. 4.4.3 that setting max = 2 or max = 3 makes negligible difference to cosmological observables (see Fig. 4.5). Therefore, we only keep C ≤3 and set C >3 = 0 in our numerical study for simplicity.

Physically, the expansion in the small parameter T dec /m ν corresponds to perturbing about the ultra-nonrelativistic limit in which the momentum of the mother particle has completely redshifted away, so that it has come to rest in the cosmic frame. Energy and momentum conservation is respected order by order in this expansion. The earlier work C19 approximated the Boltzmann hierarchy for daughter radiation (Eq. (4.28)-Eq. (4.31)) by just keeping C 0 and setting all the C ≥1 = 0. It is clear from the above discussion that this is a consistent approximation to zeroth order in an expansion in the small parameter T dec /m ν . The authors in B20 argued that the Boltzmann hierarchy for daughter radiation in C19 does not reproduce the standard decaying CDM scenario and does not respect momentum conservation. Both criticisms can be addressed by considering the term C 1 . Since C 1 begins at O(T dec /m ν ), we see that the Boltzmann hierarchy in C19 does in fact reproduce the decaying CDM scenario and respects momentum conservation up to O(T dec /m ν ) corrections, consistent with the approximation. In this limit, the momenta of the daughter particles arise entirely from the rest mass of the mother. In practice, since the contributions of neutrinos to the density perturbations are small, we will see that the higher order terms do not significantly affect the constraints derived in C19 with Planck15 data.

Impact of the non-relativistic neutrino decay on the CMB spectra

We briefly summarize the impact of the non-relativistic invisible neutrino decays on the CMB spectra, following the discussion in C19. In Fig. 4.3, we display the residuals in the CMB (lensed) TT, EE and lensing power spectra, for the sum of neutrino masses m ν = 0.6 eV and several decay widths Log 10 (Γ ν /km/s/Mpc) = 0, 2, 4, 6. In all cases, the ΛCDM parameters are set to their best-fit values from Planck18, that is, {100θ s = 1.04089, ω cdm = 0.1198, ω b = 0.02233, n s = 0.9652, ln(10 10 A s ) = 3.043, τ reio = 0.0540}. Our reference ΛCDM model makes use of the same parameters and assumes standard massless neutrinos.

For the value of the mass considered ( m ν = 0.6 eV) and at fixed angular size of the sound horizon θ s , neutrino masses primarily impact the lensing spectrum. Indeed, as they reduce power below the free-streaming scale, they produce a significant matter power suppression at small scales, which leads to a ∼ 20% reduction in the C φφ at large (blue curve in Fig. 4.3). Consequently, this power suppression decreases the smoothing in the high-part of the TT and EE spectra, which can be seen as 'wiggles' in the corresponding plots.

In addition, stable neutrinos dilute like non-relativistic matter at late times (ρ ν ∼ a -3 ), which increases the value of Ω m . As we impose the closure relation Ω m + Ω Λ = 1 at late-times, this is compensated for by a decrease in Ω Λ (later beginning of Λ-domination), and thus a reduction in the Late Integrated Sachs-Wolfe effect (LISW), leaving a signature in the low-TT spectrum. Furthermore, the modified expansion history H(z) changes quantities integrated along z, such as τ reio , which affects the multipoles at ∼ 10 in the EE spectrum.

When a non-negligible Γ ν is considered (orange, green and red curves in Fig. 4.3), one can see that the aforementioned effects typically become less prominent for earlier decays. This is particularly true for the high-part of the lensing spectrum (and consequently the smoothing at high-in TT and EE) since decay of neutrinos reduce their impact on structure formation. The reduction of the effect in the low-part of the TT and EE spectra is not entirely monotonic, as intermediate values of Γ ν can induce additional time variation in the gravitational potentials (thereby affecting the LISW effect), as well as time variations in H(z) (thereby affecting τ reio ). As a result, the ΛCDM limit is reached not only for small values of m ν , but also for high values of Γ ν . This will be reflected in the MCMC analysis in Sect. 4.5, which shows a large positive correlation between both parameters. It is precisely this degeneracy which relaxes the neutrino mass bounds.

Consistency of the implementation of the Boltzmann equations

We begin by comparing the approximation used in C19 for the background energy density of decaying massive neutrinos to the more accurate results obtained by evaluating the integral in Eq. (4.20) numerically. In C19, the phase space distribution of neutrinos in Eq. (4.20) is approximated through the following analytic formula, fν (q, τ ) = fν,i (q)e -Γν t/γ . (4.41)

As argued in C19, this approximation is valid under the assumption that the decay happens deep in the non-relativistic regime. To see the difference between the approximation and the full result, we plot the ratio r ≡ (ρ ν + ρdr )/ρ ur in Fig. 4.4, for several values of the decay width Γ ν and a fixed value of the total neutrino mass m ν = 0.06 eV. Here ρur denotes the energy density of stable massless neutrinos. If neutrinos decay while relativistic, this ratio always gives r 1. However, if the decay happens when the neutrinos are already non-relativistic (ρ ν ∼ a -3 ), then the ratio evolves from r 1 to r ∼ a, and will eventually reach a plateau once all the neutrinos have decayed. From Fig. 4.4, we can see that the approximate formula in Eq. (4.41) gradually improves as we go to smaller decay widths (that is, going deeper into the regime of non-relativistic decays), as expected. The error in the case of neutrinos decaying right around the time of the non-relativistic transition (Log 10 (Γ ν /[km/s/Mpc]) 4 for 0.06 eV) is around 25%. Nevertheless, as we argue below, the impact on observables is much smaller given that neutrinos only contribute a small fraction of the total energy density for masses considered in this work. Not surprisingly, the approximate formula fails in the relativistic regime, leading to r < 1 at late-times. Therefore future work focusing on this regime should We consider a very small value of the neutrino mass sum, m ν = 0.06 eV, and several values for the decay width, Log 10 (Γ ν /km/s/Mpc). "approx. PSD" refers to the approximated phase space distribution in Eq. (4.41) while "Full PSD" refers to the exact solutions of Eq. (4.20). make use of the exact formula.

In Fig. 4.5 and Fig. 4.6, we show the effects of various approximations in dealing with decaying neutrinos (at the background and perturbation level) on the CMB TT, EE and lensing spectra. We compare the impact of using either the approximated or the exact PSD of neutrinos discussed above, as well as the impact of only keeping C ≤ max in the Boltzmann hierarchy of daughter particles in Eq. (4.28)-Eq. (4.31), where we vary max from zero to three. We show the residuals of these approximations with respect to the 'optimal' case (i.e. including all terms up to max = 3 and the exact background PSD) for a fixed value of the neutrino mass ( m ν = 0. For decays close to the non-relativistic transition T dec /m ν 0.3 shown in Fig. 4.5, we find that the biggest improvement in the CMB TT spectrum occurs when including C ≤1 (i.e., the contribution from the decaying neutrino bulk velocity) in the Boltzmann hierarchy of daughter radiation, which impacts the integrated Sachs-Wolfe (ISW) effect at multipoles 100. On the other hand, the approximate background distribution of neutrinos does not have a significant effect. For the CMB EE spectrum shown in the same figure, which is not sourced by the ISW effect, the impact of the approximate background distribution of neutrinos is comparable to the effect of the approximate perturbed hierarchy. Nevertheless, one can see that for max ≥ 2, additional contributions to the daughter hierarchy have negligible impacts, which justifies our choice of cutting the collision term C contribution at max = 3. Finally for the CMB lensing spectrum, the effects due to the approximate treatment of the background PSD dominate over the ones due to including higher order terms in the Boltzmann hierarchy of the dark radiation. This is expected given that the matter power spectrum suppression scales approximately with ρν /ρ m [START_REF] Hu | Weighing neutrinos with galaxy surveys[END_REF][START_REF] Lesgourgues | Neutrino Cosmology[END_REF] where ρm is the total matter density, while neutrino perturbations are very small well below the free-streaming scale, so that their detailed dynamics is not as important as on larger scales.

The impact of the various approximations in the case of decays deep in the non-relativistic regime T dec /m ν 0.03, displayed in Fig. 4.6, is much less visible. In that case, one can therefore safely neglect C >0 and consider the approximate PSD, as done in C19.

Cosmological constraints 4.5.1 Details of the analysis

In this section we perform a numerical scan over the parameter space to obtain updated limits on the neutrino mass and lifetime. We perform comprehensive MCMC analyses with the MontePython-v3 [START_REF] Audren | Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code[END_REF]BRINCKMANN et al. 2019a) code interfaced with our modified version of CLASS . We fit the decaying neutrino model to a combination of the following data-sets:

• The Planck18 high-TT, TE, EE + low-data TT, EE + lensing data. We will also compare these results with the use of Planck15 data to disentangle the effects of our improved formalism and that of the new data.

• The BAO measurements from 6dFGS at z = 0. We adopt flat priors on the following six ΛCDM parameters: {ω b , ω cdm , H 0 , n s , A s , τ reio }. We assume three degenerate neutrinos decaying into massless radiation and consider flat priors on m ν /eV and Log 10 (Γ ν /[km/s/Mpc]). In order to accelerate convergence, we split the parameter space between Log 10 (Γ ν /[km/s/Mpc]) ∈ [0.1, 2.5] and Log 10 (Γ ν /[km/s/Mpc]) ∈ [2.5, 6.5]. In both cases we take wide priors on m ν ∈ [0.06, 1.5] eV. We assume our MCMC chains to be converged when the Gelman-Rubin criterion R -1 < 0.05 [START_REF] Gelman | Inference from Iterative Simulation Using Multiple Sequences[END_REF]). In our baseline analysis, we do not apply any specific cut to the parameter space, even if neutrinos decay in the (mildly-)relativistic regime (this occurs for low m ν and high Γ ν ). In App. E.4, we investigate the impact of excluding the parameter space corresponding to relativistic decay from our analysis and show that the limit at 95% on m ν agrees within a few percent.

Updated limits on the neutrino mass and lifetime

The results of our analyses are presented in Fig. 4.7. For very late decays, corresponding to Log 10 (Γ ν /[km/s/Mpc]) 2.5, no relaxation of the constraints on m ν /eV is visible, in agreement with what was found in C19. The impact of the new Planck data is visible as a significantly improved bound on the sum of neutrino mass, namely we find m ν < 0.127 eV (95%C.L.), an improvement of about ∼ 35% over 2015 data, in good agreement with AGHANIM et al. (2020b). For Log 10 (Γ ν /[km/s/Mpc]) 2.5, one can see that the bound relaxes as expected, although not as much with Planck18 data as for Planck15 data. Taking the intersect of the non-relativistic decay line with our 2σ limit, we find that Planck18 excludes neutrino decaying non-relativistically with masses m ν > 0.42 eV (95% C.L.). This is significantly stronger than the limits from Planck15 data, for which m ν ∼ 0.9 eV was still allowed in the non-relativistic decay scenario.

Our result also has implications for laboratory searches. For m ν = 0.6 eV, the smallest mass scale that the KATRIN experiment is designed to probe, Planck18 data requires decay rate Γ ν 10 5.5 km/s/Mpc, a constraint roughly one order of magnitude stronger than from Planck15 data. However, this value of the decay rate is now slightly beyond the regime of validity of our work 7 , indicating that, in the event of a neutrino mass discovery at KATRIN, a more involved analysis including inversedecays would be necessary to confirm that the decay scenario can reconcile laboratory and cosmological measurements.

Comparison with former results and impact of Planck 2018 data

Comparing with the constraints presented in C19 for Planck15, we find that, while the impact of our improved treatment is clearly visible in the CMB power spectra (and will be relevant for future experiments), it has only a marginal impact on the constraints, and our bounds are in very good agreement with those derived in C19, which only included the leading order term in the daughter radiation hierarchy 8 . The bulk of the improvement is due to the newest Planck18 data and can be understood as follows.

As shown in Fig. 4.3, for the masses we consider, the main effect is an almost scale independent suppression of CMB lensing spectrum. This suppression can be compensated for by increasing the primordial amplitude A s or by adjusting the matter density ω cdm (see [START_REF] Archidiacono | Physical effects involved in the measurements of neutrino masses with future cosmological data[END_REF] for a discussion of the correlation between { m ν , A s , τ reio , ω cdm }). Due to the well-known degeneracy between A s and e -2τ reio , Planck15 data, which was limited in polarization, were unable to place a tight constraint on A s , and thus the constraining power on the sum of neutrino mass and lifetime was limited. The precise measurements of low-polarization from Planck18 leads to constraints on τ reio that are tighter by a factor of two than those from Planck15. As a result, parameters degenerate with τ reio such as A s are now much better constrained. Consequently, the constraints on the sum of neutrino mass and lifetime have significantly improved with Planck18 data.

To confirm this simple argument, we perform another run with Planck15 data and a tight gaussian prior on τ reio = 0.0540 ± 0.0074, chosen to match the optical depth to reionization reconstructed from Planck18. Given that the constraints on m ν are independent of Γ ν below Γ ν 10 3 , and the scaling above Γ ν 10 5.5 is monotonic, we focus on the parameter space Log 10 (Γ ν /[km/s/Mpc]) ∈ [3, 5.5] to accelerate convergence. Our 7 For mν = 0.6 eV and assuming degenerate neutrino masses, the non-relativistic condition requires Γν < 10 5.3 km/s/Mpc. 8 Let us note that the implementation of the BAO/f σ8 DR12 likelihood used in C19 within the Mon-tePython code had an issue that led to constraints on mν that were somewhat milder than the true bounds. MontePython has since then been corrected, leading to an improvement on the constraints on the stable/long-lived (Γν < 10 3 km/s/Mpc) case by about 20%. However, we have verified that this bug had no impact in the short-lived case (Γν > 10 2.5 km/s/Mpc). results are presented in Fig. 4.8, where one can see that this simple prescription leads to constraints that are very similar to those from the full Planck18 data. We attribute the remaining differences to the additional constraining power of Planck18 data on the parameters ω cdm and ω b , which are mildly correlated with m ν (see Fig. 4.7, top panel). Note that our constraints are a factor of two weaker than those advocated in LORENZ et al. (2021), which performed a 'model-independent' reconstruction of the neutrino mass as a function of redshift, but neglects the decay products. As we show here, including details about the daughter radiation is necessary to accurately compute the effect of neutrino decays even in the non-relativistic regime.

Summary and outline

Cosmological observations are known to set the strongest constraints on the sum of neutrino masses. Yet, the existing mass bound from CMB and LSS measurements, which assumes that neutrinos are stable, is significantly weakened if neutrinos decay. In this work, we provide up-to-date limits on the lifetime of massive neutrinos that decay into dark radiation after becoming non-relativistic, from a combination of CMB, BAO, growth factor measurements, and Pantheon SN1a data.

Compared to the earlier analysis in C19, we have incorporated higher-order corrections up to O((T dec /m ν ) 3 ) when solving the dark radiation perturbations, and also performed the full calculation of the background energy density of the decaying neutrino using Eq. (4.20). The more precise treatment of the Boltzmann equations and the background energy evolution in our MCMC study improves the coverage of the case when the neutrinos decay early so that their average momenta are close to their masses. As shown in Fig. 4.6, if neutrinos decay when having T ν m ν /3, the inclusion of higher moment perturbations C ≥2 gives a negligible change to the power spectra as compared to the experimental uncertainties. However, the complete calculation of the neutrino energy does improve the prediction for the power spectrum significantly from the approximate result using Eq. (4.41) when the decays happen semi-relativistically. Nevertheless, we have found that constraints from Planck15, given their limited precision, are unaffected by these considerations. However, we anticipate that these effects will be relevant for future experiments (as well as an essential contribution in the relativistic case, to be considered in the future).

In fact, we have shown that the bulk of the improvement in the constraining power compared to C19 comes from the use of Planck18 data. Indeed, we have demonstrated that the improved τ reio measurement from the low-polarization data helps breaking the degeneracy in the CMB power spectrum amplitude and strengthens the bound on the neutrino mass and lifetime. As a result, we have found that neutrinos with m ν > 0.42 eV (2σ) cannot be made consistent with cosmological data if they decay while non-relativistic, a significant improvement from Planck15 data for which masses as high as m ν ∼ 0.9 eV were consistent with the non-relativistic decay scenario (C19).

We have argued that one notable application of this result is that, if the KATRIN experiment sees an electron neutrino with m ν ≈ 0.2 eV (the advocated sensitivity), our result would constrain Γ ν 10 5.5 km/s/Mpc, i.e. the neutrinos would need to decay between z ≈ 2 × 10 2 -4 × 10 3 , while they are still relativistic, so that our bounds and the bounds studied in B20 would not apply. In case of a neutrino mass discovery at KATRIN, a more involved analysis including inverse-decays would be necessary to firmly confirm that the decay scenario can reconcile laboratory and cosmological measurements. Additionally, our results show that the tentative exclusion of the inverted mass ordering [START_REF] Simpson | Strong Bayesian Evidence for the Normal Neutrino Hierarchy[END_REF] Finally, let us mention that even though current exclusion bounds in Fig. 4.7 do not set independent constraints on the neutrino mass and lifetime, next generation measurements of the matter power spectrum at different redshifts can help break that degeneracy [START_REF] Chacko | Determining the Neutrino Lifetime from Cosmology[END_REF]). It will be interesting to revisit the forecast on the sensitivity of future cosmological data to the sum of neutrino masses and their lifetime in light of our improved formalism.

V

Conclusions

The main goal of this work was to constrain or search for signs of exotic physics using current cosmological data. An important focus has been the theoretical interpretation of two longstanding experimental discrepancies in cosmology: the H 0 and S 8 tensions. While unknown systematic effects at the origin of these discrepancies are not excluded, the existence of several independent observations disfavoring the CMB predictions could point to a major failure of the ΛCDM scenario. I have put new and robust constraints on several extensions of ΛCDM that are targeted at explaining these tensions, such as Decaying Dark Matter and Early Dark Energy. Additionally, I have used cosmic data to better characterize the properties (mass and lifetime) of one of the most elusive particles in the universe, the neutrino. In all my projects, a special attention has been given to the CMB anisotropies, which provides a powerful tool to test deviations from the standard model of cosmology and the standard model of particle physics. My approach has always been very phenomenological, so that the derived constraints can apply to many different theoretical high-energy physics models.

In Chapter 1, I started with a thorough review of the standard tools needed to compute the most relevant cosmological observables, including the CMB and matter power spectra. At the end of this Chapter, I briefly discussed the challenges that the ΛCDM paradigm has to face: the nature of the dark sector and the cosmological tensions.

In Chapter 2, I have analyzed a model of Early Dark Energy, which has been shown to provide a simple resolution to the Hubble tension if the EDE component contributes a fraction f EDE 10% of the energy density of the universe at z c 3500, and dilutes as or faster than radiation afterwards. However, some authors have pointed out that the best-fit EDE cosmology requires a larger value of S 8 , so that including LSS data in the analysis highly restricts the resolution to the tension. I have reassessed this claim by confronting an EDE model whose only parameter is f EDE (z c ) against a wide variety of datasets. This analysis shows that EDE is detected at ∼ 2σ from Planck CMB data alone, and that the addition of BAO, SNIa and weak-lensing data does not significantly worsen the fit to the S 8 measurements with respect to ΛCDM. The main conclusion is that current weak-lensing data is not constraining enough to rule out the resolution to the H 0 tension provided by EDE.

In addition, I have participated in a review of different models that have proposed to resolve the Hubble tension, spanning early-time solutions (such as Early Dark Energy) and late-time solutions (such as Decaying Dark Matter). The main goal was to go beyond a simple bibliographic compilation, and quantify for the first time the relative success of several models in explaining the discrepancy. To achieve that, we confronted seventeen different models proposed in the literature against a variety of cosmological data, and quantified the relative success of each model using a series of metrics. This work provides a series of benchmarks for future model-builders wishing to assess the level of success of a given model/scenario. In Chapter 3, I have studied the cosmological impact of a scenario where CDM is allowed to decay into massless (dark radiation) and massive (warm DM) species. I have performed the first thorough analysis of this 2-body DM decay scenario including a realistic treatment of linear perturbations. In this way, I have obtained the strongest constraints up-to-date for these kind of models (using CMB, BAO and SNIa data), and showed that the S 8 tension can be resolved if DM decays with a lifetime of Γ -1

55 Gyr and transfers around 0.7% of its rest-mass energy to the massless component. This analysis was possible thanks to a new and accurate fluid approximation scheme that I developed for the warm species, and which allowed to reduce the CPU time in the calculations from ∼ 1 day to only ∼ 1 minute. I also considered alternative CMB data-sets (like those from ACT-DR4 and SPTPol), discussed a potential connexion with the recent XENON1T excess, and showed that, while current data are not sensitive enough to distinguish between standard CDM and decaying DM, future CMB observations can unambiguously detect its signature.

In Chapter 4, I have updated the constraints on the mass and lifetime of active SM neutrinos. Currently, the strongest limit of the sum of neutrino mass comes from cosmology, m ν < 0.12 eV, but these determinations assume that neutrinos are stable on cosmic timescales. The idea of neutrino decay is particularly interesting in the eventuality that laboratory experiments (like KATRIN) will measure a value of the neutrino mass bigger than the one inferred from cosmological observations. It has been shown that, for neutrinos decaying to dark radiation while being non-relativistic, the stringent CMB constraints on the sum of neutrino masses can be significantly relaxed (up to m ν ∼ 0.9 eV). I have gone beyond previous work by improving the Boltzmann treatment of neutrino decays both at the background and linear perturbation level. This has allowed us to show that, given the new Planck 2018 CMB measurements, non-relativistic neutrino decays can relax the bound of neutrino masses only up to m ν ∼ 0.4 eV.

The work presented here is based on the following publications • G. F. Abellán, R. Murgia Expanding the derivatives and always keeping linear order:

g 0i (x) = a 2 (τ )B i = ∂ xα ∂x 0 ∂ xβ ∂x i gαβ (x) = ∂ x0 ∂x 0 ∂ x0 ∂x i g00 + ∂ x0 ∂x 0 ∂ xj ∂x i g0j + ¨¨¨¨∂ xj ∂x 0 ∂ x0 ∂x i gj0 + ∂ xl ∂x 0 ∂ xs ∂x i gls = a 2 (τ + T ) -∂ i T (1 + T )(1 + 2 Ã) + (1 + T )(δ ij + ∂ i L j ) Bj + L l (δ is + ∂ i L s )(δ ls + Hls ) = a 2 (τ )(1 + 2HT ) -∂ i T + Bi + L i = a 2 (τ ) -∂ i T + Bi + L i . (A.5)
Let us now prove the gauge transformation for the overdensity

δρ -→ δ ρ = δρ -T ρ , (A.6)
Expanding the derivatives in Eq. (1.161) and keeping linear order:

T 0 0 (x) = -(ρ(τ ) + δρ) = ∂x 0 ∂ xα ∂ xβ ∂x 0 T α β (x) = ∂x 0 ∂ x0 ∂ x0 ∂x 0 T 0 0 + ¨¨¨¨¨∂ x 0 ∂ x0 ∂ xi ∂x 0 T 0 i + ¨¨¨¨¨∂ x 0 ∂ xi ∂ x0 ∂x 0 T i 0 + ¨¨¨¨¨∂ x 0 ∂ xi ∂ xj ∂x 0 T i j = -(1 -T )(1 + T ) (ρ(τ + T ) + δ ρ) = -ρ(τ ) + T ρ + δ ρ . (A.7)

A.3 The 0 component of the geodesic equation

The goal is to compute the 0 component of the geodesic equation. But for this, we first need to compute the Christoffel symbols in a perturbed FLRW universe, which will also be useful for other calculations. From now on, we work in conformal time and assume the Newtonian gauge. We will simply have to plug the elements of the metric

g 00 = -a 2 (1 + 2ψ), g ij = a 2 (1 -2φ)δ ij , (A.8)
and the inverse metric

g 00 = -a -2 (1 -2ψ), g ij = a -2 (1 + 2φ)δ ij , (A.9)
in the definition for the Christoffel symbols:

Γ µ αβ = 1 2 g µλ (∂ α g βλ + ∂ β g αλ -∂ λ g αβ ) . (A.10)
We start with Γ 0 00 , systematically discarding all terms beyond linear order

Γ 0 00 = g 00 2 (2∂ 0 g 00 -∂ 0 g 00 ) = g 00 2 ∂ 0 g 00 = (1 -2ψ) 2a 2 ∂ 0 a 2 (1 + 2ψ) = H + ψ . (A.11)
We compute in the same way the rest of Christoffels, having in mind that g 0i = g0i = 0.

Γ 0 0i : Γ 0 0i = 1 2 g 0λ (∂ 0 g λi + ∂ i g 0λ -∂ λ g 0i ) = g 00 2 ∂ i g 00 = (1 -2ψ) 2a 2 2a 2 ∂ i ψ = ∂ i ψ.
(A.12)

Γ i 00 : Γ i 00 = 1 2 g iλ (2∂ 0 g 0λ -∂ λ g 00 ) = 1 2 g ij (2∂ 0 g j0 -∂ j g 00 ) = - 1 2 g ij ∂ j g 00 = δ ij 2a 2 (1 + 2φ)2a 2 ∂ j ψ = δ ij ∂ j ψ. (A.13) Γ 0 ij : Γ 0 ij = 1 2 g 0λ (∂ i g jλ + ∂ j g iλ -∂ λ g ij ) = - 1 2 g 00 ∂ 0 g ij = δ ij 2a 2 (1 -2ψ)(a 2 -2a 2 φ) = (1 -2ψ)δ ij H -φ -2φH = δ ij H -φ -2H(φ + ψ) .
(A.14)

Γ i j0 : Γ i j0 = 1 2 g iλ (∂ j g 0λ + ∂ 0 g jλ -∂ λ g j0 ) = 1 2 g ik ∂ 0 g jk = δ ik δ jk (1 + 2φ) 2a 2 a 2 -2φa 2 = (1 + 2φ)δ i j (H -φ -2φH) = δ i j (H -φ ). (A.15) Γ i jk : Γ i jk = 1 2 g iλ (∂ j g kλ + ∂ k g jλ -∂ λ g jk ) = 1 2 g il (∂ j g lk + ∂ k g lj -∂ l g jk ) = a 2 δ il (1 + 2φ) 2a 2 (-2δ lk ∂ j φ -2δ lj ∂ k φ + 2δ jk ∂ l φ) = -2δ i (j ∂ k) φ + δ jk δ il ∂ l φ. (A.16)
The remaining ingredients needed to compute the geodesic equation are the elements of the 4-momentum vector P µ = dx µ dσ = (P 0 , P i ). For computing P 0 , we can expand the closure relation -m 2 = g µν P µ P ν = -a 2 (1 + 2ψ)(P 0 ) 2 + p 2 .

(A.17)

From this, we get that P 0 = E a √ 1+2ψ . Now, to determine P i , we write P i = C ni , where ni is a unit vector and C is a constant to be determined. From the definition of p, we have

p 2 = C 2 ij g ij pi pj = a 2 (1 -2φ)C 2 ij δ ij ni nj = a 2 (1 -2φ)C 2 , (A.18) which means C = p a √ 1-2φ
. In short, using that (1 + 2ψ) -1/2 1 -ψ and (1 -2φ) -1/2 1 + φ, we can write:

P 0 = E a (1 -ψ), P i = pn i a (1 + φ). (A.19)
We are finally ready to compute the 0-component of the geodesic equation. Setting µ = 0 in Eq. (1.5):

dP 0 dσ = dτ dσ dP 0 dτ = P 0 dP 0 dτ = -Γ 0 αβ P α P β . (A.20)
This can also be written as

d dτ E a (1 -ψ) = -Γ 0 αβ P α P β E a(1 + ψ), (A.21)
where we used (P 0 ) -1 a E (1 + ψ). By expanding the derivative and then multiplying by a(1 + ψ) we get (after dropping all quadratic terms): Using the expressions for the Christoffels that we derived in Eq. (A.11), Eq. (A.12) and Eq. (A.14), we can work out the factor Γ 0 αβ P α P β at the r.h.s of Eq. (A.23):

dE dτ = EH + E dψ dτ -Γ 0 αβ P α P β E a 2 (1 + 2ψ
Γ 0 αβ P α P β = Γ 0 00 (P 0 ) 2 + 2Γ 0 0i P 0 P i + Γ 0 ij P i P j = E 2 a 2 (H + ψ )(1 -2ψ) + 2 pE a 2 ni ∂ i ψ + p 2 a 2 H -φ -2H(φ + ψ) (1 + 2φ) = E 2 a 2 (H(1 -2ψ) + ψ ) + 2 pE a 2 ni ∂ i ψ + p 2 a 2 H(1 -2ψ) -φ (A.24)
We plug this in Eq. (A.23). After using (1 -2ψ)(1 + 2ψ) 1, many terms simplify and we are left with

dp dτ = -pH + pφ -E ni ∂ i ψ. (A.25)
In terms of comoving momentum q = ap and comoving energy E = aE:

dq dτ = qφ -E ni ∂ i ψ. (A.26)

A.4 Elements of perturbed stress-energy tensor

We want to derive the relations between the elements of the perturbed stressenergy tensor and the Legendre multipoles of the perturbed PSD Ψ. We remind that

δT µ ν (τ, k) = g d 3 P (2π) 3 (-det(g)) -1/2 P µ P ν P 0 f (τ, q)Ψ(τ, q, k, µ). (A.27)
For the newtonian gauge, the determinant gives (-det(g)) -1/2 = a -4 (1 -ψ -3φ), but the ψ and φ terms are irrelevant since the integrand in Eq. (A.27) contains Ψ, which is already first-order. We remind that the integral in Eq. (A.27) is over covariant three-momenta, which is given by

P i = g ij P j = a 2 (1 -2φ)(1 + φ)pn i /a = (1 -φ)qn i . (A.28)
This means that the integral measure can be expressed as

d 3 P = 2π 0 dϕ 1 -1 dµ ∞ 0 dqq 2 (1 -3φ), (A.29)
where the term with φ is again irrelevant for being first-order. Now, the µ integrals will select some particular multipole Ψ for each of the components of δT µ ν 1 , which can easily be seen by plugging the expansion of Eq. (1.190) in Eq. (A.27) , and using the orthogonality property of the Legendre polynomials

1 -1 dµP (µ)P m (µ) = 2δ m 2 + 1 . (A.31)
The overdensity ρδ picks the = 0 multipole (P 0 = 1), since in this case the integrand contains no µ terms, just the factor P 0 = g 00 P 0 = -E(1 + ψ). The same arises for the pressure perturbation δP , whose integrand has i

P i P i P 0 = q 2 E (1 + ψ) i ni nj δ ij , (A.32) with i ni nj δ ij = 1.
The velocity divergence selects the = 1 multipole (P 1 = µ) , given that the integrand includes i ki P i = (1 -φ)qµ. Lastly, the anisotropic stress σ picks the = 2 multipole (P 2 = 1 2 (3µ 2 -1)), as the integrand contains i,j

(-ki kj + δ ij 3 )

P i P j P 0 = q 2 E (1 + ψ) i,j (-ki kj + δ ij 3 )n i nj -µ 2 + 1 3 . (A.33)
In short,

ρδ = 4π a 4 ∞ 0 dqq 2 E f Ψ 0 , (A.34) (ρ + P )θ = 4πk a 4 ∞ 0 dqq 2 q f Ψ 1 , (A.35) ρΠ = 4π 3a 4 ∞ 0 dqq 2 q 2 E f Ψ 0 , (A.36) (ρ + P )σ = 8π 3a 4 ∞ 0 dqq 2 q 2 E f Ψ 2 , (A.37)
where we have absorbed a factor g/(2π) 3 inside f .

A.5 Collision term for photons

The task is to determine the collision integral C γ appearing in the Boltzmann equation for the photon perturbations (c.f. Eq. (1.199)). We'll start assuming totally generic PSD for the particles involved, and perform the decomposition into a background and a linear contribution at the end. In practice, we'll see that the collision integral only contains first-order terms C γ = C

(1) γ . The process we want to describe is the Compton scattering off electrons e -( q) + γ( p) ←→ e -( q ) + γ( p ), (A.38) 1 In Fourier space, the elements of the perturbed stress-energy tensor are computed as ρδ = -δT 0 0,

(ρ + P )θ = ik i kiδT 0 i, ρΠ = 1 3 i T i i, (ρ + P )σ = i,j -ki kj + δij 3 δT i j . (A.30)
where we have already labelled the momenta of each particle (beware of notation, we're working with physical and not comoving momenta, even if we use the letter q). According to the general expression that we wrote in Eq. (1.63), the collision term reads2

C γ [f γ (p)] = d 3 q (2π) 3 2E e (q) d 3 q (2π) 3 2E e (q ) d 3 p (2π) 3 2E γ (p ) |M| 2 (2π) 4 δ 3 p + q -p -q × δ E γ (p) + E e (q) -E γ (p ) -E e (q ) f e (q )f γ (p ) -f e (q)f γ (p) , (A.39)
where M = M eγ,eγ denotes the invariant amplitude for Compton scattering. In the previous expression, we have neglected (1 ± f ) factors related to stimulated emission and Pauli blocking, since they play a negligible role in this context. Energies are of the order of relativistic limit for photons and non-relativistic for electrons. Namely, E γ (p) = p and E e (q) = m e + q 2 2me . During the recombination epoch, the kinetic energy of electrons is very small compared to their mass, so we can set E e m e in the denominator of Eq. (A.39). Performing the d 3 q integral with the 3-dimensional delta distribution yields

C γ = π 4m 2 e d 3 q (2π) 3 d 3 p (2π) 3 p δ p + q 2 2m e -p - | q + p -p | 2 2m e × |M| 2 f e (q + p -p )f γ (p ) -f e (q)f γ (p) . (A.40)
We can simplify the scary-looking aspect of the Dirac delta by noticing that, in the non-relativistic Compton scattering process, very few energy is transferred

E e (q) -E e (q + p -p ) = q 2 2m e - ( q + p -p ) 2 2m e ( p -p) • q m e , (A .41)
since q p, p . That is, we can restrict ourselves to the Thomson limit of Compton scattering. Now, given that p p , the rhs of Eq. (A.41) is roughly pq/m e , where q/m e is of the order of the baryon velocity q/m e ∼ v b 1. Therefore, we can consider the quantity E e (q) -E e (q + p -p ) to be a small perturbation around (p -p ) inside the delta distribution, and formally perform an expansion up to first order. With this and using that f e (q + p -p ) f e (q), we write Eq. (A.40) as

C γ = π 4m 2 e d 3 q (2π) 3 f e (q) d 3 p (2π) 3 p |M| 2 δ(p -p ) + ( p -p ) • q m e ∂δ(p -p ) ∂p f γ (p ) -f γ (p) .
(A.42) The expansion of the Dirac delta is obviously very ill-defined, but it will soon find justification when doing integration by parts. What about |M| 2 ? This amplitude can be computed with the help of the Feynman rules for quantum electrodynamics, as explained for example in [START_REF] Peskin | An Introduction to quantum field theory[END_REF]. It can be written as

|M| 2 = 6πσ T m 2 e (1 + (p • p ) 2 ) (A.43)
where we have introduced the Thomson cross-section σ T = 8πα 2 3m 2 e , with α = e 2 /4π the fine-structure constant. In reality, the invariant amplitude also depends on polarization (∝ |ˆ • ˆ | 2 , where ˆ and ˆ are the polarizations of the incoming and outgoing photons), which has been implicitly summed over in Eq. (A.43). This means that Compton scattering induces some level of polarization on the CMB. Since the description of the polarization photon field is beyond the scope of this work, and its impact on the evolution of the photon temperature field is very small, we will neglect this effect in the following (however see Sect. 1.6.5 for a short discussion). For the moment, let us consider the angle average of Eq. (A.43)

|M iso | 2 = 8πσ T m 2 e (A.44)
Using Eq. (A.44), Eq. (A.42) can be expressed as

C γ = 2π 2 ne σ T d 3 p (2π) 3 p δ(p -p ) + ( p -p ) • v b ∂δ(p -p ) ∂p f γ (p ) -f γ (p) . (A.45)
The factors with ne and ne v b have appeared as a result of two integrals in q:

d 3 q (2π) 3 f e = d 3 q (2π) 3 fe + ¨¨¨¨¨ d 3 q (2π) 3 δf e = ne , (A.46) d 3 q (2π) 3 f e q = ¨¨¨¨¨ d 3 q (2π) 3 fe q + d 3 q (2π) 3 δf e q = (ρ e + Pe ) v b m e ne v b . (A.47)
In Eq. (A.46), we discard the integral over δf e since we neglect the impact of perturbed recombination on the CMB. In Eq. (A.47), the integral over fe q vanishes by symmetry, and we use Pe 0 and ρe m e ne , since electrons were non-relativistic by this time. The next step is to plug in Eq. (A.45) the expansion for the photon PSD that we showed already in Eq. (1.197)

f γ = fγ -p ∂ fγ ∂p Θ. (A.48)
This yields:

C γ = C (1) γ = 2π 2 ne σ T d 3 p (2π) 3 p δ(p -p ) + ( p -p ) • v b ∂δ(p -p ) ∂p × f (p ) -f (p) -p ∂ f ∂p Θ(p ) + p ∂ f ∂p Θ(p) = ne σ T 4π ∞ 0 dp p dΩ δ(p -p ) -p ∂ f ∂p Θ(p ) + p ∂ f ∂p Θ(p) + ( p -p ) • v b ∂δ(p -p ) ∂p f (p ) -f (p) (A.49)
In Eq. (A.49) there are only two terms that depend on p and thus must be accounted in the solid angle integral: Θ(p ) and p • v b . The first one will just give rise to the monopole perturbation Θ 0 = 1 4π dΩ Θ(p ). The second term will vanish, since we are integrating p • v b = v b µ over dΩ = -dϕ dµ (notice that v b = v b k since we only consider the scalar/ irrotational component of the velocity). Therefore,

C (1) γ = ne σ T ∞ 0 dp p δ(p -p ) -p ∂ f ∂p Θ 0 + p ∂ f ∂p Θ(p) + pµv b ∂δ(p -p ) ∂p f (p ) -f (p) . (A.50)
It just remains to use integration by parts on the second term inside the integrand, and then apply the Dirac deltas. This leads to

C (1) γ = -p 2 ∂ f ∂p ne σ T [Θ 0 -Θ + µv b ] . (A.51)
This is not yet the final expression, since we totally neglected the angular dependence in Eq. (A.43). Taking this dependence into account, leads an extra term in the collision integral which is proportional to the quadrupole Θ 2 . To see this, let us extract the anisotropic part of the invariant amplitude

δ|M| 2 = |M| 2 -|M iso | 2 = 2πσ T m 2 e [3(p • p ) 2 -1]. (A.52)
The term in square brackets is equal to 2P 2 (p • p ). It will be convenient to factorize the dependence on p and p by means on the useful identity

P (p • p ) = 4π 2 + 1 m=+ m=- Y m (p)Y * m (p ). (A.53)
Hence,

δ|M| 2 = σ T m 2 e 16π 5 m=+2 m=-2 Y 2m (p)Y * 2m (p ) ! = σ T m 2 e 16π 5 Y 20 (p)Y 20 (p ). (A.54)
In the last equality, we have taken only the m = 0 part, since the other Y 2m contain factors e ±iϕ , e ±2iϕ , which would vanish under the 2π 0 dϕ integral. The spherical harmonic Y 20 is directly related to the second Legendre polynomial, Y 20 (p) = -5 4π P 2 (µ), with µ = p • k. Thus, the extra anisotropic contribution to the collision term is

δC (1) γ = ne σ T 8π P 2 (µ) ∞ 0 dp p dΩ P 2 (µ ) δ(p -p ) -p ∂ f ∂p Θ(µ ) + p ∂ f ∂p Θ(µ) + ( p -p ) • v b ∂δ(p -p ) ∂p f (p ) -f (p) . (A.55)
In the previous expression, the only term that survives after performing the solid angle integral is the one proportional to Θ(µ ), since +1 -1 dµ P 2 (µ ) = 0 and +1 -1 dµ P 2 (µ )µ = 0. Thus,

δC (1) γ = - ne σ T 2 P 2 (µ) ∞ 0 dp p 2 δ(p -p ) ∂ f ∂p +1 -1 dµ 2 P 2 (µ )Θ(µ ). (A.56)
The µ integral in the previous equation gives rise to -Θ 2 .

After applying the Dirac delta, δC (1)

γ = p 2 ∂ f ∂p ne σ T 2 P 2 (µ)Θ 2 , (A.57)
so the full collision term reads

C (1) γ = -p 2 ∂ f ∂p ne σ T Θ 0 -Θ + µv b - P 2 (µ) 2 Θ 2 .
(A.58)

A.6 Conservation equations from the collisionless Boltzmann equation

Here we derive the two conservation equations, by integrating the collisionless Boltzmann hierarchy over the phase space. This calculation is relevant for massive neutrinos but also for any warm species that decoupled in the very early universe. We'll be using repeatedly the integral expressions for δ, θ, δP and σ shown in Eq. (1.192)-Eq. (1.195), as well as the integral expressions for ρ and P shown in Eq. (1.55) and Eq. (1.56). We start with the perturbed continuity equation. This can be obtained from the equation for the zeroth multipole (c.f. Eq. (1.211))

∂Ψ ν,0 ∂τ = - kq E Ψ ν,1 - dln f dlnq φ , (A.59)
The idea is to integrate previous equation over 4πa -4 dqq 2 E f , in order to get a dynamical equation for δρ ν . This yields

4π a 4 dqq 2 E ∂( f Ψ ν,0 ) ∂τ I = - 4πk a 4 dqq 3 f Ψ ν,1 II - φ 4π a 4 dqq 3 E d f dq III . (A.60)
Let us focus on each term separately. The term II is the easiest one, since it directly gives (ρ ν + Pν )θ ν by definition. For the term III, we need to perform integration by parts. Using ∂E ∂q = q E , we get

q 3 E d f dq = d dq q 3 E f -3 f q 2 E + f q 4 3E . (A.61)
When this is inserted in the integral for the term III, the total derivative term vanishes, and we are left with -3(ρ ν + Pν )φ . Moving to the term I, the goal is to take the time derivative out of the integral, which will require to know the time derivative of the energy, E . Since E = q 2 + a 2 m 2 ν , this gives

E = m 2 ν a 2 E H = E 2 -q 2 E H = EH - q 2 E H. (A.62)
We can now proceed to take the time derivative out of the term I in order to make δρ ν appear, and evaluate the remaining terms:

I = 4π a 4 ∂ ∂τ dqq 2 E f Ψ ν,0 - 4π a 4 dqq 2 f Ψ ν,0 E . = (δρ ν ) + 4Hδρ ν -H 4π a 4 dqq 2 E f Ψ ν,0 - 4π a 4 dqq 2 q 2 E f Ψ ν,0 = (δρ ν ) + 4Hδρ ν -H(δρ ν -3δP ν ) = (δρ ν ) + 4Hδρ ν -Hδρ ν (1 -3c 2 s ) = (δρ ν ) + 3Hδρ ν (1 + c 2 s ). (A.63)
Putting all together,

(δρ ν ) = δ ν ρν + δ ν ρ ν = -3Hδρ ν (1 + c 2 s ) -ρν (1 + w ν )θ ν + 3ρ ν (1 + w ν )φ . (A.64)
Dividing by ρν , it can be equivalently written as:

δ ν = - ρ ν ρν δ ν -3Hδ ν (1 + c 2 s ) -(1 + w ν ) θ ν -3φ . (A.65)
Using ρ ν ρν = -3H(1 + w ν ) 3 , we arrive at the perturbed continuity equation:

δ ν = -3H(c 2 s -w ν )δ ν -(1 + w ν ) θ ν -3φ . (A.66)
Let us now consider the Euler equation. It can be derived from the equation for the first multipole (c.f. Eq. (1.212))

∂Ψ ν,1 ∂τ = kq 3E [Ψ ν,0 -2Ψ ν,2 ] - Ek 3q dln f dlnq ψ, (A.67)
To get a dynamical equation for θ ν , we integrate the former equation over 4πka -4 dqq 3 f , yielding

4π a 4 dqq 3 ∂( f Ψ ν,1 ) ∂τ I = 4πk 2 3a 4 dq q 4 E f Ψ ν,0 II - 8πk 2 3a 4 dq q 4 E f Ψ ν,2 III - ψ4πk 2 3a 4 dqq 3 E d f dq IV . (A.68)
The terms II and III give by definition k 2 δP ν and k 2 (ρ ν + Pν )σ ν , respectively. For the term IV we need to integrate by parts on q 3 E d f dq , but this is precisely what we already did in Eq. (A.61). Thus, the term IV just gives -(ρ ν + Pν )k 2 ψ. Regarding the term I, the goal is again to take the time derivative out of the integral to make θ ν appear:

I = 4π a 4 ∂ ∂τ dqq 3 f Ψ ν,1 = ∂ ∂τ (ρ ν + Pν )θ ν + 4H(ρ ν + Pν )θ ν = ρ ν (1 + c 2 g )θ ν + ρν (1 + w ν )θ ν + 4H ρν (1 + w ν )θ ν . (A.69)
Putting all together,

ρν (1+w ν )θ ν = -ρ ν (1+c 2 g )θ ν -4H ρν (1+w ν )θ ν +k 2 δP ν -k 2 ρ ν (1+w ν )σ ν +k 2 ρ ν (1+w ν )ψ. (A.70)
The expression simplifies after diving by ρν (1 + w ν ),

θ ν = - ρ ν ρν (1 + c 2 g ) (1 + w ν ) θ ν -4Hθ ν + k 2 c 2 s δρ ν ρν (1 + w ν ) -k 2 σ ν + k 2 ψ. (A.71)
Finally, using ρ ν ρν = -3H(1 + w ν ), we arrive at the Euler equation

θ ν = -H(1 -3c 2 g )θ ν + k 2 c 2 s 1 + w δ ν + ψ -σ ν . (A.72)

A.7 Perturbed Einstein equations

The goal is to determine Einstein equations in the perturbed FLRW universe, focusing only in the scalar sector. We'll have to compute the Ricci tensor R µν and Ricci scalar R appearing in the Einstein tensor at the l.h.s. of Einstein equations:

G µν ≡ R µν - 1 2 g µν R = 8πGT µν . (A.73)
In order to compute the components of the perturbed Ricci tensor, we just need to plug the Christoffels we computed in Eq. (A.11)-Eq. (A.16) into the definition for R µν :

R µν ≡ ∂ λ Γ λ µν -∂ ν Γ λ µλ + Γ λ λρ Γ ρ µν -Γ ρ µλ Γ λ νρ . (A.74) R 00 : R 00 = ∂ λ Γ λ 00 -∂ 0 Γ λ 0λ + Γ λ λρ Γ ρ 00 -Γ ρ 0λ Γ λ 0ρ = ¨¨∂ 0 Γ 0 00 + ∂ i Γ i 00 - ¨¨∂ 0 Γ 0 00 -∂ 0 Γ i 0i + $ $ $ $ $ Γ 0 00 Γ 0 00 + $ $ $ $ Γ 0 0i Γ i 00 + Γ i i0 Γ 0 00 + Γ i ij Γ j 00 O(2) -$ $ $ $ $ Γ 0 00 Γ 0 00 -$ $ $ $ Γ i 00 Γ 0 0i -Γ 0 0i Γ i 00 O(2) -Γ i 0j Γ j 0i = ∇ 2 ψ -3∂ 0 H -φ + 3(H + ψ )(H -φ ) -(H -φ ) 2 δ j i δ i j =3 = -3H + ∇ 2 ψ + 3H(φ + ψ ) + 3φ . (A.75) R 0i : R 0i = ∂ λ Γ λ 0i -∂ i Γ λ 0λ + Γ λ λρ Γ ρ 0i -Γ ρ 0λ Γ λ iρ = ∂ 0 Γ 0 0i + ∂ j Γ j 0i -∂ i Γ 0 00 -∂ i Γ j 0j + $ $ $ $ Γ 0 00 Γ 0 0i + Γ j j0 Γ 0 0i + ¨¨¨Γ 0 0j Γ j 0i + Γ k kj Γ j 0i -$ $ $ $ Γ 0 00 Γ 0 0i - ¨¨¨Γ 0 0j Γ j 0i -Γ j 00 Γ 0 ij -Γ k 0j Γ j ik = ¨∂ i ψ -∂ i φ - ¨∂ i ψ + 3∂ i φ + 3(H -φ )∂ i ψ -3δ j i (H -φ )∂ j φ -3H∂ i φ -δ jk δ ij δ k i ∂ k ψ{H -φ -2H(φ + ψ)} -(H -φ )δ k j {-2δ j (i ∂ k) φ + δ ik δ jl ∂ l φ} +3H∂ i φ = 2∂ i φ + 3H∂ i ψ -H∂ i ψ = 2∂ i φ + 2H∂ i ψ. (A.76)
We have of course neglected all higher order terms and used that

∂ i H = 0. R ij : R ij = ∂ λ Γ λ ij -∂ j Γ λ iλ + Γ λ λρ Γ ρ ij -Γ ρ iλ Γ λ jρ = ∂ 0 Γ 0 ij + ∂ k Γ k ij -∂ j Γ 0 i0 -∂ j Γ k ik + Γ 0 00 Γ 0 ij + Γ k k0 Γ 0 ij + Γ 0 0k Γ k ij O(2) + Γ k kl Γ l ij O(2) -Γ 0 i0 Γ 0 j0 O(2) -Γ k i0 Γ 0 jk -Γ 0 ik Γ k j0 -Γ k il Γ l jk O(2) = H -φ -2(φ + ψ) δ ij + ∂ k -2δ k (i ∂ j) φ + δ ij δ kl ∂ l φ -∂ j ∂ i ψ + 3∂ j ∂ i φ + (H + ψ ) H -φ -2H(φ + ψ) δ ij + (H -φ ) H -φ -2(φ + ψ) (3δ ij -δ k i δ kj -δ k j δ ki ) = H + 2H 2 -φ + ∇ 2 φ -2(H + 2H 2 )(φ + ψ) -Hψ -5Hφ δ ij + ∂ i ∂ j (φ -ψ). (A.77)
Let us now compute the Ricci scalar, R ≡ g µν R µν . Its expression gets simplified because of the fact that g 0i = 0, meaning that R = g 00 R 00 + g ij R ij . By collecting the previous results for the Ricci tensor and using δ ij ∂ i ∂ j = ∇ 2 and δ ij δ ij = 3, we have

a 2 R = -(1 -2ψ) -3H + ∇ 2 ψ + 3H(φ + ψ ) + 3φ + (1 + 2φ)∇ 2 (φ -ψ) + 3(1 + 2φ) H + 2H 2 -φ + ∇ 2 φ -2(H + 2H 2 )(φ + ψ) -Hψ -5Hφ = 6(H + H 2 ) -2∇ 2 ψ + 4∇ 2 φ -12(H + H 2 )ψ -6φ -6H(ψ + 3φ ).
(A.78)

We are now ready to compute each component of G µν .

G 00 :

G 00 = R 00 - 1 2 g 00 R = -3H + ∇ 2 ψ + 3H(φ + ψ ) + 3φ + 3(1 + 2ψ)(H + H 2 ) + 1 2 -2∇ 2 ψ + 4∇ 2 φ -12(H + H 2 )ψ -6φ -6H(ψ + 3φ ) = 3H 2 + 2∇ 2 φ -6Hφ . (A.79) G 0i : G 0i = R 0i = 2∂ i (φ + Hψ). (A.80) G ij : G ij = R ij - 1 2 g ij R = H + 2H 2 -φ + ∇ 2 φ -2(H + 2H 2 )(φ + ψ) -Hψ -5Hφ δ ij + ∂ i ∂ j (φ -ψ) -3(1 -2φ)(H + H 2 )δ ij - 1 2 -2∇ 2 ψ + 4∇ 2 φ -12(H + H 2 )ψ -6φ -6H(ψ + 3φ ) δ ij = -(2H + H 2 )δ ij + ∇ 2 (ψ -φ) + 2φ + 2(2H + H 2 )(φ + ψ) + 2Hψ + 4Hφ δ ij + ∂ i ∂ j (φ -ψ). (A.81)
It is time to relate the components of the perturbed Einstein tensor we just obtained, to the components of the perturbed stress-energy tensor in Eq. (1.147)-Eq. (1.150), as demanded by Einstein equations. We switch at this point to Fourier space, meaning that we can make the replacements ∂ i → ik i and ∇ 2 → -k 2 .

• We start with the trace-free projection of the ij equation. That is, we consider G = 8πG T , where G ≡ i,j ki kj -

δ ij 3 G ij and T ≡ i,j ki kj - δ ij 3 T ij , with T ij = g ik T k j .
According to Eq. (A.81), G = -2 3 k 2 (φ -ψ), whereas T gives at linear order -a 2 I (ρ I + PI )σ I , by definition of the anisotropic stress. Hence, we have

k 2 (φ -ψ) = 12πGa 2 I (ρ I + PI )σ I (A.82)
The index I runs over all cosmological species: photons, neutrinos, cold dark matter and baryons.

• We move to the 00 equation. From Eq. (A.79) we get

3H 2 -2k 2 φ -6Hφ = 8πGg 0µ T µ 0 = 8πGg 00 T 0 0 = 8πGa 2 (1 + 2ψ) I (ρ I + δρ I ). (A.83)
The zeroth-order part gives the first Friedmann equation (in conformal time)

H 2 = 8πG 3 a 2 I ρI , (A.84)
while the first-order part gives -k 2 φ -3Hφ = 4πGa 2 ( I δρ I + 2ψ I ρI ). By using Eq. (A.84) this last equation can be more conveniently expressed as

k 2 φ + 3H(φ + Hψ) = -4πGa 2 I δρ I . (A.85)
• Next, we consider the 0i equation, G 0i = 8πGT 0i . By virtue of Eq. (A.80), we find it to be ik i (φ +Hψ) = 4πGg 00 T 0 i = -4πGa • At last, we compute the trace of the ij equation, i G i i = 8πG i T i i . For the r.h.s. we have i T i i = 3 I ( PI + δP I ), whereas the l.h.s. gives:

i G i i = i,k g ik G ki = 3a -2 -(2H + H 2 ) + 2k 2 3 (φ -ψ) + 2φ + 2ψ(2H + H 2 ) + 2H(ψ + 2φ ) (A.87)
At zeroth order we get the second Friedmann equation

2H + H 2 = -8πGa 2 I PI . (A.88)
At first order, we obtain

φ + H(ψ + 2φ ) + (2H + H 2 )ψ + k 2 3 (φ -ψ) = 4πGa 2 I δP I . (A.89)

A.8 Conservation of the comoving curvature perturbation

We want to show that the comoving curvature perturbation, whose expression in the Newtonian gauge is given by

R = φ + H(φ + Hψ) 4πGa 2 (ρ tot + Ptot ) , (A.90)
is indeed constant for super-Hubble scales and for adiabatic initial conditions. Differentiating Eq. (A.90) and multiplying by 4πGa 2 (ρ tot + Ptot ), we get:

4πGa 2 (ρ tot + Ptot )R = 4πGa 2 (ρ tot + Ptot )φ + H (φ + Hψ) + H(φ + H ψ + Hψ ) + H 2 (φ + Hψ) + 3H 2 P tot ρ tot (φ + Hψ), (

B

Complements on parameter inference

In this appendix, we explain some basic tools about bayesian statistics and parameter extraction, that are routinely used to test cosmological models (such as the ones considered in this work) against actual data. We mainly follow VERDE (2010), HEAVENS (2009) and TROTTA (2008).

B.1 Parameter extraction

Statistics has become an essential tool for cosmologists. A minimum knowledge of some sophisticated statistical techniques is required in order to properly extract cosmological information from the current observables, allowing us to test deviations from the standard ΛCDM model, or to better determine the values of a specific set of parameters.

There are two different ways of dealing with statistics: the frequentist and the Bayesian approach. As the name suggests, the first one is based on the classical thought that probability can be seen as a frequency, that is, the number of times some event occurs over the total number of trials. That is, for a frequentist, data are repeatable but the model is fixed. For several reasons, the frequentist approach is not the most preferred in cosmology. In particular, we cannot conduct the same procedure for measuring a particular quantity (let's say, H 0 ), since we only observe one realization of the universe. The predominant method in cosmology is the Bayesian approach, in which probability is seen as a measurement of the degree of belief about a proposition. With this notion, one can consider the probability of any event, without necessarily having to repeat many experiments. Thus, for a Bayesian, what is repeatable is the model, while data are fixed. Another consequence of this definition is that, unlike in the frequentist case, hypothesis can also be considered as events with a certain probability distribution. The prescription for dealing with states of belief is given by Bayes theorem, that we proceed to describe below.

B.1.1 Bayes Theorem

The Bayes Theorem is a trivial consequence of the axioms of probability. Let us consider some propositions A and B. The probabilities that these propositions have a certain outcome, assuming some information I is true, are denoted by P (A|I) and P (B|I). These probabilities can be discrete or continuous. The multiplication rule says that the joint probability of A and B equals the probability of B, times the conditional probability of A given that B has ocured (both conditional on I), i.e. P (A, B|I) = P (A|B, I)P (B|I).

( • The posterior distribution. This is what we aim to determine, it gives the probability of observing the parameters of the model, given some data. One is in general interested in the 1-dimensional posterior distribution for the ith parameter, P (θ i |d, M). If the parameter space is N -dimensional, then P (θ i |d, M) is obtained by integrating the posterior over the remaining N -1 parameters (this is known as marginalization). For example, if our model aims to describe the data with only the parameters Ω M and H 0 , we can get the marginal 1D posterior of H 0 by

P (H 0 |d, M) = dΩ M P ({H 0 , Ω M }|d, M). (B.4)
Once the 1D distribution P (X|d, M) of a certain parameter X is determined, one can compute the corresponding moments, such as the mean

µ = X = X = XP (X|d, M)dX, (B.5)
or the variance (giving the width of the distribution)

σ 2 = (X -µ) 2 = (X -µ) 2 P (X|d, M)dX. (B.6)
For the uncertainty, it's customary to quote the 68.3 % confidence intervals +b -a . This can be obtained with the following prescription: 1) find the value that maximizes the posterior, X max , 2) go to smaller and larger values than X max until the 68.3 % of the total probability is enclosed, i.e. find a and b such that P (X = a|d, M) = P (X = b|d, M) and In a similar way, one can find the 95.4 % confidence intervals1 .

• The likelihood. This represents the probability of observing the data, given the parameters of the model. For a fixed data and model, it is typically written as L(θ) = P (d|θ, M), so it describes a surface over the parameter space. One can find the most likely model (or best-fit), by maximizing L(θ). For normally distributed data consisting in N data points d i (for example, the distance luminosity curves to several SNIa), and a model y i (θ) for these data, the likelihood is expressed as a multivariate gaussian 2

L(θ) = 1 (2π) N /2 (det[C ij ]) 1/2 e -χ 2 /2 (B.8)
where χ 2 is the the chi-squared statistics, given by

χ 2 = N i,j=1 (d i -y i (θ))C -1 ij (d j -y j (θ)), (B.9)
and C ij ≡ (d i -y i (θ))(d j -y j (θ)) is the covariance matrix. The variances are given by the diagonal elements of the correlation matrix, σ 2 i = C ii . The ubiquity of gaussian likelihoods is motivated by the central limit theorem. This theorem tells us that the joint distribution of a collection of data points (independent and identically distributed, but not necessarily in a gaussian way), approaches that of a gaussian as the sample size increases. For gaussian likelihoods, we see that maximizing the likelihood is equivalent to minimizing the χ 2 . In many cases, the distribution of χ 2 around its minimum follows a χ 2 distribution with n = N -N degrees of freedom, where N is the number of independent data points and N is the number of parameters. The χ 2 distribution is formally defined as the distribution for the sum of the squares of gaussian variables, Y = X 2 1 + X 2 2 + ... + X 2 n (where X i are gaussian variables with zero mean and variance of one), and is given by

P (Y ) = 1 2 n/2 Γ(n/2) Y n/2-1 e -Y /2 (B.10)
This distribution has a mean given by n and a variance of 2n. An alternative way to derive confidence intervals by looking at regions of the N -dimensional parameter space with constant χ 2 around the minimum (best-fit) χ 2 min . This is a common approach in frequentist analysis, since one doesn't need to integrate probabilities over prior regions in order to get confidence limits. Using the properties of the χ 2 distribution, one can find the intervals of ∆χ Apart from finding the best-fit parameter values (as well and the confidence limits), one might wonder whether the model, evaluated at the best-fit, provides a good fit to the data. A simple rule of thumb to estimate the goodness-of-fit is the compute the χ 2 statistics and compare its minimum value to the number of degrees of freedom n. If the model provides a good fit to the data, we expect the data to scatter symmetrically around the model by 1-σ (on average), which is translated into the requirement χ 2 min n = N dof .

• The Bayesian evidence. This plays the role of a normalization constant, that can be computed by means of an integral over the parameter space

P (d|M) = d N θ P (d|θ, M)P (θ|M). (B. 11
)
This quantity is important in model selection, and we shall discuss it in more detail later Thus, given some data sets (whose likelihood is for example gaussianly distributed) and some choice of priors, we can obtain the 1D posterior distributions of the parameters in which we are interested, by means of the previous formulas. Unfortunately, when the parameter space is big, these analytical recipes are very time consuming. For example, let us suppose we take a model with N = 10 parameters, a typical amount in cosmology. In order to compute the integrals appearing in Eq. (B.4) and Eq. (B.11), we decide to grid each parameter in M = 20 discrete points. In this case, one is forced to carry out M N = 20 10 likelihood evaluations, a huge number that makes the data analysis completely unachievable. In order to overcome these complications, one typically resort to Monte Carlo methods.

B.2 Monte Carlo Markov chains

Monte Carlo Markov chains (MCMC) are a class of algorithms which allow to sample the posterior distribution in an extremely efficient way. Instead of going exponentially with the number of parameters, the MCMC calculation goes linearly with N . The name "Monte Carlo" indicates that they are based on the generation of random numbers, while the name "Markov process" indicates that we consider processes (or chains) in which the future state depends only on the present state, chain converges in a reasonable amount of time. A sign that the chain is efficient is that the acceptance ratio is ∼ 0.3, that is to say, that new points θ (new) are accepted about 1/3 of the times. To speed up the MCMC analysis, it is also common to decompose the covariance matrix into fast and slow parameters, using the so-called Cholesky decomposition (LEWIS 2013). This allows to handle separately the part of the likelihood associated with nuisance parameters, that do not require to re-run the full Boltzmann code.

Another way of decreasing the runtime is to initialize several different chains and then analyze them together. Since the Markov chains often take same time to converge, normally the first ∼ 1000 steps of each chain need to be discarded, this is the so called burn-in stage. Finally, it is necessary to use some convergence diagnostic to determine if the chains have converged. One of the most used is the Gelman-Rubin criterium. It is based on the calculation of a ratio R (for a given parameter of the model), which compares the variance of means within each chain with the mean of variances among different chains. In practice, one calculates first the mean xi and the variance σ 2 i from each chain i = 1, .., n c , as well as the total mean from all chains, x = i xi /(n c -1). Afterwards, one calculates the variance of the chain means σ 2 (x) = i (x 2 i -x2 )/(n c -1) and the mean of the chain variances σ = i σ i /(n c -1). The ratio R ≡ σ(x)/σ should be close to one if the chains are converged. The threshold for convergence is a matter of taste, but a common criterion to stop the chains is when we have R -1 < 0.01 for all the parameters of the model.

B.3 Model selection

We introduced previously the χ 2 test to quantify the goodness-of-the-fit. However, this method is not adequate to compare the goodness-of-the-fit among different models, since it does not include Occam's razor factors to penalize models with a larger number of free parameters. A tool that is widely used to perform model comparison in the Bayesian framework is the Bayes factor.

Until now we have considered that all probabilities were conditional on some model M, that we assume to be true. However, the Bayesian framework also allows to do comparison between models. For example, given CMB data, ¿is there preference for dark energy as a cosmological constant (ΛCDM) or as fluid with a generic equation of state (wCDM)? For that purpose, let us apply Bayes theorem again, but on a model level P (M|d) = P (d|M)P (M) P (d) .

(B.12)

Here P (M) is the model prior (irrespective on the data), while P (d) is the data prior (irrespective of the model). Let us suppose we have two different models M 0 and M 1 that we wish to compare. In that case, we can take the ratio between the posteriors on the models, which according to eq. (B. where B 01 is the Bayes factor, given by the ratio of the Bayesian evidences in each model,

B 01 ≡ P (d|M 0 ) P (d|M 1 ) . (B.14)
Given the observed data d, and assuming the same prior probability for both models P (M 0 ) = P (M 1 ), a value of the Bayes factor that is bigger (smaller) than 1, increases (decreases) the support in favor of model M 1 with respect to model M 2 . Bayes factor are usually interpreted against the empirical "Jeffreys' scale" for the strength of evidence (JEFFREYS 1961), that we report in Tab. (B.2).

From the definition of Bayes factor, it's clear that we need to integrate the likelihood over the prior volume for each model. In general, this is something not easy to achieve with MCMC, so one needs to use Nested sampling algorithms such as MultiNest [START_REF] Feroz | MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics[END_REF]). For brevity we don't describe this algorithm in details, but the main idea is the following: one samples from the prior volume, and gradually concentrates more points near the maximum of the likelihood, by repeatedly replacing the point with the lowest target density by one drawn from the prior volume with higher target density. Nested sampling methods are also more adequate for handling complicated multi-modal posteriors, for which the MH algorithm could easily get stuck in local minima.

Bayes factors correctly account for the increased complexity of a model, but they are very sensitive to the choice of priors, which can be problematic when a parameter is not well constrained. There are frequentist tools for model comparison that do not suffer from this problem, such as the Akaike Information Criterium (AIC) that we introduced in Chapter 2. In this work, we have made use of the MontePython-v3 code [START_REF] Audren | Conservative Constraints on Early Cosmology: an illustration of the Monte Python cosmological parameter inference code[END_REF]BRINCKMANN et al. 2019a), which incorporates both the Metropolis-Hastings and the MultiNest algorithms. Figure C.1 -Here we compare the matter power spectra extracted from our simulations, with the ones computed with HALOFIT /HMCODE , in three different redshift bins from z = 1.5 to z = 0.5. The blue curves refer to the ΛCDM scenario, whereas the red ones refer to the EDE best fit model. As a reference, we also report the best fit ΛCDM case from Planck 2018. The spliced power spectra are denoted by thick dot-dashed lines. Symbols stand for the output power spectra of the "non-spliced" LB and HR simulations. The solid/dotted lines are the non-linear power spectra from HALOFIT /HMCODE , while the dashed lines are the corresponding linear power spectra used to set the initial conditions for the simulations. The cyan shaded band approximately corresponds to the scales probed by DES-Y1.

where k 250 MIN is the minimum k-mode in our small-box simulations (HR and LR), while k LB Nyq is the Nyquist wave-number of the LB one.

Besides the aforementioned systematical uncertainties, numerical simulations are also affected by two primary sources of statistical errors: the cosmic variance, affecting the large-scale part of the spectra, and the shot noise due to the discreteness of the DM particles, thereby affecting the smallest scales.

Concerning the shot noise term, its contribution to the power spectrum is simply given by P SN = (L/N ) 3 . It is straightforward to see that it is largely subdominant at the scales and redshifts considered in this work, from ratio between the power spectrum predicted by HALOFIT /HMCODE and that extracted from the numerical simulation in order to explicitly demonstrate that the differences are below 5% level, for scales 10 -2 k 10 h/Mpc, at redshifts 0.5 ≤ z ≤ 2, for both ΛCDM and EDE models. This extends the z = 0 result presented in the main text to cover the full redshift range from KIDS-VIKING.

It is also informative to compare the prediction from algorithms with N -body at larger scales than that depicted in Indeed, these are affected by higher statistical noise, due to cosmic variance, as one might already guess from the lower-k part of both figures. To beat down cosmic variance, one should run several statistical realizations of the same simulation, by producing initial conditions starting from different random seeds. To circumvent this issue and save computational time, we adopted the simple solution to run the two sets of simulations (EDE and ΛCDM) with identical random seeds for the realization of their initial conditions, and to present our results in terms of ratios in the matter power spectra between the EDE and the ΛCDM models, in lead to a decrease in power. However, the increase in ω cdm , leads the EDE best-fit model to predict O(20%) increase in power, when compared to the best-fit ΛCDM model. Note how the differences become even more manifest at higher redshift. This illustrates that high-z LSS measurements have the potential to put EDE under crucial tests (KLYPIN et al. 2021).

Another way of presenting our results is in terms of the accuracy at which HALOFIT /HMCODE can predict deviations in the non-linear power spectrum of EDE models with respect to the ΛCDM 'equivalent' case (as opposed to predicting the absolute power spectrum). This is what we show in Fig. C.4, where we now compare the ratio between the EDE and ΛCDM power spectra from HALOFIT /HMCODE against the same ratio extracted from simulations. The thick horizontal lines highlight ±5% deviations. In light of all of this, we conclude that, in the EDE framework, HALOFIT /HMCODE predictions on ΛCDM departures are reliable at ≤ 5% level with respect to the outputs of N -Body simulations, for scales 10 -2 k 10 h/Mpc, at redshifts 0 ≤ z ≤ 2.

D

Complements on decaying dark matter

In this appendix we detail some important calculations concerning the dynamics of decaying dark matter with warm decay products, that were too long to be included in the main text of Chapter 3. We also describe our numerical implementation, and include some extra tables and plots.

D.1 Boltzmann equation for decaying dark matter in the synchronous gauge

The goal is to derive the generic Boltmann equation for decaying dark matter models with arbitrary decay mass products. We'll work with the synchronous gauge comoving with the mother particles, following AOYAMA et al. (2014). The metric is thus written as

ds 2 = a 2 (τ ) -dτ 2 + (δ ij + H ij )dx i dx j , (D.1)
where H ij reads in Fourier space

H ij = ki kj h + ki kj - δ ij 3 6η. (D.2)
Let us denote the PSD distribution of the I-th dark component as f I (k, q, µ, τ ), where I = {dcdm, dr, wdm}. The starting point is the same as in Eq. (1.185)

∂f I ∂τ + ∂f I ∂x i dx i dτ + ∂f I ∂q dq dτ + ∂f I ∂ ni dn i dτ = ∂f I ∂τ C = C[f I ] P 0 . (D.3)
We focus first on the l.h.s. of previous equation. The last term at the l.h.s. can be neglected as usual because it's second order in perturbations. For the second term, we note that

dx i dτ = dx i dσ dσ dτ = P i P 0 . (D.4)
Let us obtain P 0 and P i1 . To get P 0 , we simply use the closure relation g µν P µ P ν = g 00 (P 0 ) 2 + p 2 = -m 2 and the fact that g 00 = -a 2 (τ ). To get P i , we write P i = C ni (where ni is some unit vector), and obtain the constant C from the definition of p

p 2 = g ij P i P j = a 2 1 + H ij ni nj C 2 -→ C p a 1 - 1 2 H ij ni nj . (D.5)
Therefore, the elements of the 4-momentum in synchronous gauge are

P 0 = E a 2 , P i = qn j a 2 δ ij - 1 2 H ij , (D.6) so that dx i dτ = qn j E δ ij - 1 2 H ij . (D.7)

D.1.1 Geodesic equation

For the factor dq dτ in Eq. (D.3), we need to use the 0 component of the geodesic equation:

P 0 dP 0 dτ = -Γ 0 αβ P α P β . (D.8)
The calculation is similar to the one we did in App. A.3, except that now we're working in a different gauge. Firstly, we need to know Γ 0 00 , Γ 0 i0 and Γ 0 ij . Using the definition of the Christoffel symbols given in Eq. (A.10), we get Γ 0 00 :

Γ 0 00 = g 0λ 2 (2∂ 0 g λ0 -∂ λ g 00 ) = g 00 2 ∂ 0 g 00 = H. (D.9) Γ 0 i0 : Γ 0 i0 = g 0λ 2 (∂ 0 g λi + ∂ i g λ0 -∂ λ g i0 ) = g 00 2 ∂ i g 00 = 0. (D.10) Γ 0 ij : Γ 0 ij = g 0λ 2 (∂ i g λj + ∂ j g λi -∂ λ g ij ) = - g 00 2 ∂ 0 g ij = 1 2 H ij + H(δ ij + H ij ). (D.11)
Now, we want to express the l.h.s. of Eq. (D.8) in terms of q. This can be straightforwardly achieved by differentiating the relation a 4 (P 0 ) 2 = q 2 + a 2 m 2 . After a little bit of algebra, this yields

P 0 dP 0 dτ = - 2 a 4 Hq 2 - m 2 a 2 H + q a 4 dq dτ . (D.12)
Using the expression for the Christoffels that we computed in Eq. (D.9)-Eq. (D.11), we can compute the r.h.s. of Eq. (D.8)

-Γ 0 αβ P α P β = -Γ 0 00 (P 0 ) 2 -Γ 0 ij P i P j = - H a 4 q 2 + m 2 a 2 - H ij 2 + H(δ ij + Hij) q k a 2 δ ik - H ik 2 q l a 2 δ jl - H jl 2 = - H a 4 q 2 + m 2 a 2 - 1 a 4 H ij 2 + H(δ ij + Hij) q i q j - q j q k 2 H ik - q i q l 2 H jl = - 2H a 4 q 2 - H a 4 m 2 a 2 - 1 2a 4 H ij q i q j . (D.13)
Equating Eq. (D.12) with Eq. (D.13), as demanded by the geodesic equation, some terms cancel out and we are left with

1 q dq dτ = - 1 2 H ij ni nj . = - 1 2 ki kj h + ki kj - δ ij 3 6η ni nj = η - h + 6η 2 µ 2 . (D.14)
In summary, we write the Boltzmann equation in Fourier space as

∂f I ∂τ + iµ kq E I f I + df I dlnq η - h + 6η 2 µ 2 = ∂f I ∂τ C , (D.15)
where it is understood that the second term at the l.h.s is first order.

D.1.2 Collision integrals

Let us now move to describe the collision term for the decay process

DCDM (Q 1 ) -→ WDM (Q 2 ) + DR (Q 3 ), (D.16)
where we labelled already the comoving four-momentum of each species, Q I = (E I , q I ).

Notice that E 1 = am dcdm and E 3 = q 3 . Neglecting quantum statistics and inverse decays, the collision integrals for the mother and daughter particles can be written as

∂f dcdm ∂τ C = - a 2 2E 1 d 3 q 3 (2π) 3 2E 3 d 3 q 2 (2π) 3 2E 2 |M| 2 (2π) 4 δ (4) (Q 1 -Q 2 -Q 3 )f dcdm (q 1 ), (D.17) ∂f wdm ∂τ C = + a 2 2E 2 d 3 q 1 (2π) 3 2E 1 d 3 q 3 (2π) 3 2E 3 |M| 2 (2π) 4 δ (4) (Q 1 -Q 2 -Q 3 )f dcdm (q 1 ), (D.18) ∂f dr ∂τ C = + a 2 2E 3 d 3 q 1 (2π) 3 2E 1 d 3 q 2 (2π) 3 2E 2 |M| 2 (2π) 4 δ (4) (Q 1 -Q 2 -Q 3 )f dcdm (q 1 ). (D.19)
These integrals are performed over comoving three-momenta. As we already discussed in Sect. 3.2, because of our gauge choice and the assumption that mother particles are fully non-relativistic, we can take the following ansatz for the PSD of the DCDM species:

f dcdm (k, q, τ ) = N dcdm (k, τ ) δ(q) 4πq 2 , (D.20)
with N dcdm the comoving number density of the DCDM. This will allow to simplify the collision integrals written above. In the context of two-body decays to massive and massless particles, the invariant amplitude squared |M| 2 is simply related to the restframe decay width of the DCDM as |M| 2 = 8πΓm dcdm /ε, with ε = 0.5(1 -m 2 wdm /m 2 dcdm ) the fraction of DCDM rest mass energy converted into DR [START_REF] Barnett | Review of particle physics. Particle Data Group[END_REF]. In addition, we will be using two useful mathematical relations associated with the Dirac delta function:

d 3 q I 2E I = d 4 Q I δ(E 2 I -q 2 I -a 2 m 2 I )Θ(E I ), (D.21) δ(f (x)) = i δ(x -x 0 ) |f (x 0 )| , with x 0 such that f(x 0 ) = 0. (D.22)
Let us start with Eq. (D.17):

∂f dcdm ∂τ C = - a|M| 2 f dcdm (q 1 ) 2m dcdm (2π) 2 d 3 q 3 2E 3 δ (E 1 -E 3 ) 2 -| q 1 -q 3 | 2 -a 2 m 2 wdm Θ(E 1 -E 3 ) = -aΓf dcdm (q 1 ) 2 ε dq 3 q 3 δ 2am dcdm q 3 -a 2 (m 2 dcdm -m 2 wdm ) Θ(am dcdm -q 3 ) = -aΓf dcdm (q 1 ) 2 ε dq 3 q 3 δ(q 3 -εam dcdm ) 2am dcdm Θ(am dcdm -q 3 ) = -aΓf dcdm (q 1 ). (D.23)
We proceed in a similar fashion for Eq. (D.18):

∂f wdm ∂τ C = a 2 |M| 2 2E 2 (2π) 2 d 3 q 1 2E 1 δ (E 1 -E 2 ) 2 -| q 1 -q 2 | 2 Θ(E 1 -E 2 )f dcdm (q 1 ) = a 2 Γm dcdm E 2 2πε d 3 q 1 E 1 δ 2E 1 E 2 + 2 q 1 • q 2 -a 2 (m 2 dcdm + m 2 wdm ) Θ(E 1 -E 2 )f dcdm (q 1 ) = aΓN dcdm E 2 2πε δ 2am dcdm q 2 2 + a 2 m 2 wdm -a 2 (m 2 dcdm + m 2 wdm ) = aΓN dcdm E 2 2πε δ(q 2 -am dcdm ε)E 2 2am dcdm q 2 = aΓN dcdm 4πq 2 2 δ(q 2 -ap max ). (D .24)
A similar procedure could be carried out for Eq. (D. [START_REF] Wagoner | On the Synthesis of elements at very high temperatures[END_REF]) and the result would be the same as in Eq. (D.24).

D.2 Background continuity equation for the WDM species

We seek to obtain the background continuity equation for the WDM species, by integrating the corresponding Boltzmann equation over the phase space. The starting point is (c.f. Eq. (3.17))

∂ fwdm ∂τ = aΓ Ndcdm (τ ) 4πq 2 δ(q -ap max ), (D.25) with p max = m dcdm ε = m wdm ε/ √ 1 -2ε.
We integrate the previous equation over 4πa -4 dqq 2 E 2 :

4π a 4 dqq 2 E 2 (q) ∂ fwdm (q) ∂τ = Γ Ndcdm (τ ) a 3 dqE 2 (q)δ(q -ap max ). (D.26)
Let us focus first on the l.h.s. of the equation. To get an equation for ρwdm , we have to move the time derivative out of the integral. This is easily achieved using that E 2 = E 2 H -q 2 E 2 H. Hence:

4π a 4 dqq 2 E 2 (q) ∂ fwdm (q) ∂τ = 4π a 4 ∂ ∂τ dqq 2 E 2 fwdm - 4π a 4 dqq 2 fwdm E 2 . = ρ wdm + 4H ρwdm -H 4π a 4 dqq 2 E 2 fwdm - 4π a 4 dqq 2 q 2 E 2 fwdm = ρ wdm + 4H ρwdm -H(ρ wdm -3 Pwdm ) = ρ wdm + 4H ρwdm -H ρwdm (1 -3w) = ρ wdm + 3H ρwdm (1 + w). (D.27)
We move now to compute the r.h.s. of Eq. (D.26). It's just a matter of applying the Dirac delta and simplify

Γ Ndcdm a 3 dqE 2 (q)δ(q -ap max ) = Γ Ndcdm a 3 a 2 m 2 wdm + a 2 p 2 max = aΓ Ndcdm m dcdm a 3 1 -2ε + ε 2 = aΓρ dcdm (1 -ε). (D.28)
In summary, the continuity equation for the WDM reads:

ρ wdm = -3(1 + w)H ρwdm + (1 -ε)aΓρ dcdm . (D .29)

D.3 Linear fluid equations for the WDM species

In this section we proceed to derive the fluid equations for the WDM daughter particles that we wrote in Eq. (3.41)-Eq. (3.44). We start considering the Boltzmann hierarchy for the WDM linear multipoles (c.f. Eq. (3.28)-Eq. (3.31)):

∂ (∆f wdm,0 ) ∂τ = - qk E 2 ∆f wdm,1 + q ∂ fwdm ∂q h 6 + aΓ Ndcdm 4πq 2 δ(q -ap max )δ dcdm , (D.30) ∂ (∆f wdm,1 ) ∂τ = qk 3E 2 [∆f wdm,0 -2∆f wdm,2 ] , (D.31) ∂ (∆f wdm,2 ) ∂τ = qk 5E 2 [2∆f wdm,1 -3∆f wdm,3 ] -q ∂ fwdm ∂q (h + 6η ) 15 , (D.32) ∂ (∆f wdm, ) ∂τ = qk (2 + 1)E 2 [ ∆f wdm, -1 -( + 1)∆f wdm, +1 ] ( ≥ 3). (D.33)
The continuity and Euler equations are obtained by multiplying both sides of Eq. (D.30) and Eq. (D.31) by 4πq 2 E 2 a -4 and 4πq 3 ka -4 , respectively, and integrating over q. We already did most of the job in App. A.6, when writing the conservation equations for massive neutrinos, so there is no need to repeat all the steps. However, we have to take into account three important differences. Firstly, these equations were written in the Newtonian gauge, and not the synchronous one. This simply amounts to replacing -3φ by h /2 in the continuity equation, and removing the ψ term in the Euler equation. Secondly, the background density has a more complicated evolution, ρ wdm /ρ wdm = -3H(1 + w). And thirdly, there is an extra term associated to the decay at the l.h.s. of Eq. (D.30).

D.3.1 Continuity equation

Having all of this in mind, after integrating 4πq 2 E 2 a -4 ×Eq. (D.30) over momenta and dividing by ρwdm , we get:

δ wdm = - ρ wdm ρwdm δ wdm -3Hδ wdm (1+c

D.3.2 Euler equation

In a similar way, we can integrate 4πq 3 ka -4 ×Eq. (D.31) over momenta and divide by ρwdm (1 + w), to get:

θ wdm = - ρ wdm ρwdm (1 + c 2 g ) (1 + w) θ wdm -4Hθ wdm + k 2 c 2 s δ wdm (1 + w) -k 2 σ wdm , (D.37)
where c 2 g = P wdm /ρ wdm . Using again that ρ wdm /ρ wdm = -3H(1 + w) + (1 -ε)aΓρ dcdm /ρ wdm , we arrive at the Euler equation

θ wdm = -H(1 -3c 2 g )θ wdm + c 2 s 1 + w k 2 δ wdm -k 2 σ wdm -(1 -ε)aΓ 1 + c 2

D.3.3 Adiabatic sound speed

We still need to compute the adiabatic sound speed c 2 g . As we mentioned in Sect. The factor (ρ wdm /ρ wdm ) -1 is just given by the background continuity equation.

To get P wdm , we have to take the time derivative of the general expression for the background pressure:

Pwdm = 4π 3a 4 ∞ 0 dq q 4 E 2 fwdm (D.40)
Thus,

P wdm = -4H Pwdm - 4π 3a 4 ∞ 0 dq q 4 E 2 2 E 2 fwdm + 4π 3a 4 ∞ 0 dq q 4 E 2 f wdm = -4H Pwdm -H Pwdm + H 4π 3a 4 ∞ 0 dq q 6 E 3 2 fwdm + 4π 3a 4 ∞ 0 dq q 4 E 2 f wdm , (D.41)
where we used once more time that E 2 = E 2 H -q 2 E 2 H. We see that P wdm involves a new higher momenta integral of fwdm , called the pseudo-pressure

p wdm ≡ 4π 3a 4 ∞ 0 dq q 6 E 3 2 fwdm . (D.42)
For the last term in Eq. (D.41), we just have to insert the background Boltzmann equation Eq. (D.25) and again apply the Dirac delta:

4π 3a 4 ∞ 0 dq q 4 E 2 f wdm = Γ Ndcdm 3a 3 ∞ 0 dq q 2 E 2 δ(q -ap max ) = Γ Ndcdm 3a 3 a 2 p 2 max a 2 m 2 wdm + a 2 p 2 max = Γ Ndcdm 3a 3 am dcdm ε 2 √ 1 -2ε + ε 2 = aΓρ dcdm 3 ε 2 1 -ε . (D.43)
To summarize, the time derivative of the pressure is

P wdm = H(p wdm -5 Pwdm ) + aΓρ dcdm 3 ε 2 1 -ε (D.44)
so that the adiabatic sound speed reads

c 2 g = w 5 - p wdm Pwdm - ρdcdm ρwdm aΓ 3wH ε 2 1 -ε 3(1 + w) - ρdcdm ρwdm aΓ H (1 -ε) -1 . (D.45)
When Γ → 0, this expression reduces to the adiabatic sound speed for massive neutrinos (LESGOURGUES et al. 2011). In addition, it yields c 2 g → 1/3 in the relativistic limit, as it should.

D.3.4 Anisotropic stress

To close the system of equations Eq. (D.36) and Eq. (D.38), we need to select an appropriate truncation formula for ∆f wdm,2 . One might naively apply Eq. (3.40) with max = 2. However, Eq. (3.40) relies on the fact that the formal solution for the PSD multipoles are approximately proportional the spherical Bessel functions, j (kτ ), and thus they inherit the same recurrence relation [START_REF] Ma | Cosmological perturbation theory in the synchronous and conformal Newtonian gauges[END_REF]. This is only true when the source term S (which in the collisionless DM case only depends on h and η ) can be neglected. If S = 0, the the PSD multipole solution contains an additional non-trivial integral (convoluted with a Bessel function), involving derivatives of terms in S (BLAS et al. 2011). To circumvent this problem, publicly available Boltzmann codes generally consider max 1.

Alternatively, it is possible to limit the analysis to sub-Hubble scales (kτ 1), truncate at a lower max (= 2 for a viscous fluid), and obtain an approximated analytical result for the aforementioned convolution integral. This is what is done in BLAS et al. (2011) in order to derive fluid equations for the massless neutrinos, and is further generalized in LESGOURGUES et al. (2011) to the case of massive neutrinos. We apply the same philosophy of those two works to the WDM case, where the source function S now also includes a term related to the decay. Inspired by those works, we write the truncation formula as

∆f wdm,2 = qk E 2 ∆f wdm,1 - 3 τ ∆f wdm,2 -q ∂ fwdm ∂q h 6 -f wdm δ dcdm . (D.46)
Now, in a similar way as we did before for the δ wdm and θ wdm equations, we multiply at each side of Eq. (D.46) by (8π/3)[ρ wdm (1 + w)a 4 ] -1 (q 4 /E 2 ), and then integrate over q. After some algebra, we find

σ wdm = -3 1 τ + H 2 3 -c 2 g - 1 3 Σ wdm σ wdm + aΓ ρdcdm ρwdm (1 -ε) 1 + c 2 g 3(1 + w) σ wdm + 2 3 Θ wdm + h 2 w (1 + w) 5 - p wdm Pwdm -aΓ ρdcdm ρwdm ε 2 (1 -ε) δ dcdm (1 + w) . (D.47)
We have introduced two higher momenta variables, defined as where w σ and w θ can be any function of time going from 1/3 in the relativistic limit to 0 in the non-relativistic one. We set w σ = p wdm /3 Pwdm and w θ = 4wc 2 g /(1 + w), which were found to be good choices in [START_REF] Lesgourgues | The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics[END_REF]. With this in mind, Eq. (D.47) can be rewritten as When integrating this relation to find Θ wdm , we thus get a term proportional to 4π a 4 dq q 5 E2 d fwdm dq = -w ρwdm 5 -p wdm Pwdm .

(
However, it was found in BLAS et al. (2011) that removing the η term leads to better results for the matter power spectrum, as can be justified by an analytic approximation of the full solution. That's why we simply replace αk 2 by h /2.

where ξ ≡ 2/3 -c 2 g -p wdm /3 Pwdm is a function that vanishes in the relativistic limit. Let us now consider the relativistic limit of Eq. (D.52), and multiply the decay terms by a factor of 2. Such an expression, corresponding to a CDM decay into 2 massless components, reads σ dr = -

D.3.5 Semi-analytic understanding of the WDM sound speed

Here we obtain a formal equation that dictates the evolution of the WDM sound speed in the synchronous gauge, c 2 s . The first natural step is to write a dynamical equation for the normalized pressure perturbation, Π wdm = δP wdm /ρ wdm . This can be achieved by multiplying Eq. (D.30) by 4πq 2 q 2 3E 2 a -4 , integrating over q and then using Eq. (D.29). By doing so, two higher velocity-weight integrals appear. One of them is Θ wdm , that we already wrote in Eq. (D.48), and the other is

δP wdm ≡ 4π 3a 4
∞ 0 dq q 6 E 3 wdm ∆f wdm,0 .

(D. [START_REF] Barnett | Review of particle physics. Particle Data Group[END_REF] Similarly to what happens with Θ wdm , we see that δP wdm → δP wdm in the relativistic limit, while δP wdm /δP wdm 1 in the non-relativistic one. This means that one can write δP wdm = δP wdm 3ω p and Θ wdm = θ wdm 3ω θ 4 , where ω p and ω θ are arbitrary functions, going from 1/3 in the relativistic limit, to 0 in the non-relativistic case. In terms of these functions, the equation for Π wdm becomes Π wdm = -3HΠ wdm We remark that the previous equation is highly non-linear in the perturbed quantities, so it can easily give rise to numerical instabilities. In addition, there is no closed expression for computing ω p and ω θ . If these functions were scale independent, one possible approximation would be to trade them for some background functions, 3 In particular, in their derivation they used that the value of the integral such as w or c 2 g . However, calculations using the full hierarchy show that ω p and ω θ exhibit a k-dependence similar to that of c 2 s . For these reasons, we do not implement Eq. (D.56) in our code.

Nonetheless, by making some simplifying assumptions, Eq. (D.56) allows to qualitatively understand why there is a particular k-dependence of c 2 s in the decaying scenario, that is not present in the case of massive neutrinos. Let us consider the non-relativistic limit of Eq. (D.56), since data favors in general very small DR energy fractions, ε 1. This also implies that w p , c 2 s , w 1. Let us further assume that ω θ and c 2 s behave similarly, so that the difference ω θ -c 2 s can be neglected. Finally, let us also restrict to sub-Hubble scales, for which the term h /δ wdm is very small and can be also neglected. In this case, Eq. (D.56) reduces to In absence of the decay term, we see that the sound speed dilutes as c 2 s ∝ a -2 , which is a well-known result for massive neutrinos. This dilution can be compensated by the presence of the decay term, leading to a c 2 s ∼ cte, as long as the ratio δ dcdm /δ wdm doesn't change. In practice, for scales and times such that k < k fs (τ ), we have δ dcdm /δ wdm = 1. In this regime, the sound speed c 2 s is well approximated by the adiabatic sound speed c 2 g . However, when k > k fs (τ ), δ wdm oscillates and starts to become suppressed with One can see that this k-dependent effect appears only because of the coupling term in Eq. (D.57), which is not present for massive neutrinos. This also justifies why the sound speed c 2 s is well approximated by a background function such as c 2 g in the case of massive neutrinos. In the case of the WDM daughter species, the approximation c 2 s c 2 g will only work when k < k fs (τ ). This motivates the use of the fitting formula introduced in Eq. (3.46), that accounts for the small enhancement at scales smaller than the free-streaming scale. While this simple fitting formula is not able to capture the oscillatory features described previously, it leads to results that are accurate enough for all the observables analyzed in this work.

D.4 Numerical implementation and accuracy of the fluid approximation

In order to solve the cosmic evolution equations discussed in Sect. 3.2, we modified the publicly available numerical Boltzmann solver CLASS [START_REF] Blas | The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes[END_REF][START_REF] Lesgourgues | The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics[END_REF]. We now briefly illustrate our implementation of the ΛDDM model. First let us notice that, when solving the background equations for all cosmological species, the dark energy abundance is iteratively derived through the budget equation, Ω Λ = 1i Ω i , where the sum includes the current abundance of all other components, which are not known a priori. We thus applied a shooting method for the aforementioned parameter, i.e. we guess an initial Ω Λ , we solve the system of background equations to obtain i Ω i , and re-compute Ω Λ5 . The procedure is iterated until convergence is achieved. The WDM density is computed by solving Eq. (3.22) in 2800 momentum bins, approximately as many as the time-steps used to describe its background evolution.

At the linear perturbation level, we truncate the hierarchy of the PSD multipole equations for both the daughter particles at a max = 17 (see Eq. (3.39) and Eq. (3.40)). We set the initial conditions for the WDM species following the same procedure of AOYAMA et al. (2014). On conformal times τ < τ q , we set all ∆f wdm, = 0, since no daughter particle with comoving momentum q could have been produced. On the crossing time τ = τ q , one should be more careful, as the terms with fwdm in Eq. (D.30) and Eq. (D.32) contain a Dirac delta and, when integrated, a Heaviside function. Thus, the corresponding initial conditions for ∆f wdm,0 (τ q ) and ∆f wdm,2 (τ q ) are not-vanishing. We set them according to the analytical formulas (A.5) and (A. ∆f wdm,0 (q, τ q ) = -h (τ q ) 6H q + δ dcdm (τ q ) a q Γ Ndcdm (τ q ) 4πq 3 H q , (D.58) ∆f wdm,2 (q, τ q ) = 1 H q h (τ q ) 15 + 2η (τ q ) 5 a q Γ Ndcdm (τ q ) 4πq 3 H q . (D.59)

Finally, on times τ > τ q , we treat the WDM component as a massive neutrino species, and we solve the corresponding hierarchy of equations in 300 momentum-bins. This number of bins is chosen simply because it gives a good compromise between speed and accuracy: it is large enough to accurately describe the super-Hubble and Hubble-crossing scales, where the dynamics is relatively simple, and small enough to not become too computationally expensive 6 . On sub-Hubble scales, when kτ is larger than a threshold value (kτ ) fluid , we switch-on the fluid approximation described in section Sect. The predictions for the linear matter power spectrum P (k) are less accurate than for the anisotropy spectra, because the former is more sensitive to the dynamics of we find that the constraints on the ΛDDM models are up to (roughly) one order of magnitude stronger when our improved treatment is considered.

D.7 Results with a linear prior on Γ and ε

In our baseline analysis we have made use of log-prior on ε and Γ, to ease comparison with earlier works [START_REF] Vattis | Dark matter decaying in the late Universe can relieve the H0 tension[END_REF]CLARK et al. 2021a) who adopted the same choice. Here we present results using linear priors on the DDM parameters. Let us however stress that the use of a linear prior is less informative than adopting a logarithmic one. That is because a linear prior carries a scale (due to the large error bars used in the proposal distribution of the MCMC sampler), so that it is hard for the sampler to detect fine structure over 4 orders of magnitude by using a linear scale, in particular at very small values. In other words, given the difference between the scale of the upper limit on Γ/Gyrs -1 an ε (∼ 10 -1 ) and that of the lower limit (∼ 10 -3 )), it is very difficult to accurately reconstruct the parameter space with a linear prior. Such a difficulty is illustrated in Fig. D.6, where we provide the results of two linear-prior analyses: the upper panel corresponding to the original prior range, the lower panel corresponding to a more restricted range: ε ∈ [0.0001, 0.015] and Γ/Gyrs -1 ∈ [0.0001, 0.05]. While in the latter case ε and Γ are detected at the 2σ level -similarly to the log-prior results -the former case weighs in favor of larger ε values, so that one would incorrectly deduce an upper limit only.

D.8 Results with the neutrino mass free in the ΛDDM cosmology

We show in Fig. D.7 the 2D posteriors of {S 8 , Ω m , Log 10 (Γ/Gyrs -1 ), Log 10 (ε), m ν /eV} with the (individual) neutrino mass fixed to 0.06 eV (red) or let free to vary (blue). When considering the neutrino mass a free parameter, we model neutrinos as three degenerate state, while when the neutrino mass is fixed to 0.06 eV we consider one massive, two massless neutrinos. One can see that the results are unaffected by our choice of keeping the neutrino mass fixed to 0.06 eV in our fiducial analysis.

E

Complements on decaying neutrinos

In this appendix we detail some important calculations concerning the dynamics of decaying neutrinos to dark radiation, that were too long to be included in the main text of Chapter 4. We also include an extra run to show the effects of imposing the non-relativistic limit as a prior on the MCMC.

E.1 Collision term for decaying neutrinos

The goal is to simplify as much as possible the general expression for the decaying neutrino collision term (c.f. Eq. (4.12)):

∂f ν ∂τ C = - a 2 f ν (q) 2E ν (2π) 2 d 3 q 1 2E 1 d 3 q 2 2E 2 |M| 2 δ (4) (Q -Q 1 -Q 2 ). (E.1)
In the context of two-body decays to massless particles, the invariant amplitude squared |M| 2 is simply related to the rest-frame decay width of the neutrinos as |M| 2 = 16πΓ ν m ν [START_REF] Barnett | Review of particle physics. Particle Data Group[END_REF]. As in App. D.1, we will be using the following relations associated with the Dirac delta function: The calculation is very similar to the one we performed for decaying dark matter in App. D.1. However, there are two important differences we have to take into account: 1) the momentum of mother particles is not negligible anymore, q ν = q = 0, and 2) both daughter particles are massless, E 1 = q 1 , E 2 = q 2 . The former implies that now we'll have to take care of the angular integral over q • q1 = cos θ. Hence:

d 3 q I 2E I = d 4 Q I δ(E 2 I -q 2 I -a 2
∂f ν ∂τ C = - 2a 2 Γ ν m ν f ν πE ν d 3 q 1 2E 1 δ (E ν -E 1 ) 2 -| q -q 1 | 2 Θ(E ν -E 1 ) = - 2a 2 Γ ν m ν f ν E ν ∞ 0 dq 1 q 1 +1 -1 d cos θ δ 2E ν q 1 -2qq 1 cos θ -a 2 m 2 ν Θ(E ν -q 1 ). = - 2a 2 Γ ν m ν f ν E ν ∞ 0 dq 1 q 1 +1 -1
d cos θ δ (cos θ -cos θ * ) 2qq 1 Θ(E ν -q 1 ), (E.4)

where cos θ * is given by

cos θ * = 2E ν q 1 + a 2 m 2 ν 2qq 1 . (E.5)
The integral over cos θ is performed trivially thanks to the Dirac delta. However, this will introduce a step function Θ(1 -cos 2 θ * ), in order to enforce the condition cos θ * ∈ [-1, 1]. This will in turn restrict the integration limits for the remaining momentum integral to q ± 1 = 1 2 (E ν ± q), which are nothing but the roots of 1-cos 2 θ * . Thus, we arrive at

∂f ν ∂τ C = - a 2 Γ ν m ν f ν E ν q q + 1 q - 1 dq 1 = - a 2 Γ ν m ν f ν E ν . (E.6)

E.2 Collision term for dark radiation daughters

The starting point is the collision term appearing in the equations for the perturbed PSD multipoles of the dark radiation, F dr, . It is given by (c.f. Eq. (4.32))

C = 2i dΩ k 4π P ( k • q 1 ) 4π ρ c dq 1 q 3 1 ∂f dr ∂τ C [q 1 , k • q 1 ] = i 8a 2 Γ ν m ν πρ c dΩ k P ( k • q 1 ) dq 1 2E 1 q 3 1 d 3 q 2 2E 2 d 3 q 2E ν ∆f ν (q, k • q)δ (4) (Q -Q 1 -Q 2 ). (E.7)
where in the second line we inserted the expression for ∂f dr ∂τ C in Eq. (4.13) and used that |M| 2 = 16πΓ ν m ν . In this expression dΩ k represents the differential solid angle along the direction k, while q 1,2 are the momenta of daughter particles. The d 3 q 2 integral can be easily evaluated using the delta function corresponding to momentum conservation. In order to perform the integral over dΩ k , we notice that the direction of k enters only via P ( k • q1 ) and ∆f ν (q, k • q). Now, using the Legendre expansion of ∆f ν (q, k • q) in Eq. (4.22) and employing the identity

dΩ k P ( k • q)P ( k • q1 ) = 4π 2 + 1 P (q • q1 )δ , (E.8)
we can evaluate the dΩ k integral to obtain

C = 32m ν Γ ν a 2 ρ c d 3 qdq 1 8E ν E 1 E 2 q 3
1 P (q 1 • q)∆f ν, (q)δ(E ν -E 1 -E 2 ). (E.9)

Now, notice that the direction of the neutrino momentum only enters the integrand via the angle between the neutrino momentum q and the daughter momentum q 1 , cos θ ≡ q • q1 . The energy conserving delta function can be expressed in terms of this angle as

δ(E ν -E 1 -E 2 ) = E 2 qq 1 δ (cos θ -cos θ * ) , (E.10)
where cos θ * is given by Eq. (E.5). As before, The energy conservation restricts the daughter momentum to a range of values (q + 1 , q - 1 ). The edges of this range occur when the extreme values, cos θ * 1 = ±1, are reached. These values correspond to q ± 1 = 1 2 (E ν ± q). After integrating over the delta function corresponding to energy conservation, this reduces to

C = 8πm ν Γ ν a 2 ρ c dq E ν q∆f ν, q + 1 q - 1 dq 1 q 1 P 2E ν q 1 -a 2 m 2 ν 2qq 1 .
(E.11)

E.3 Fluid equations for decaying neutrinos

We seek to derive viscous fluid equations for decaying neutrinos, in order to simplify the dynamics dictated by the following Boltzmann hierarchy of equations: .

∆f ν,0 = - qk E ν ∆f ν,1 + q ∂ fν ∂q h 6 - a 2 Γ ν m ν E ν ∆f ν,
(E.20)

When setting Γ ν to 0, Eq. (E.16)-Eq. (E.20) reduce to the fluid equations for massive stable neutrinos considered in LESGOURGUES et al. (2011). In Eq. (E.20), the quantities p ν and n ν denote the pseudo-pressure and pseudo-number density, respectively. These are higher momenta integrals of the background p.s.d., fν (q, τ ), that reduce to the standard pressure Pν and number density nν in the relativistic limit. They can be computed as: .

p ν = 4π 3a
(E.24)

The variables Θ ν and Σ ν were already discussed in App. D.3 when writing fluid equations for the warm daughter particles. In order to close the system of Eq. (E.16)-Eq. (E.18) we need some prescription for computing the quantities ω δ , ω θ and ω σ without having to solve the full Boltzmann hierarchy for each momentum bin. By having a look at Eq. (E.22)-Eq. (E.24), we realize that m ν δn ν /δρ ν → 1, Θ ν /θ ν → 1 and Σ ν /σ ν → 1 in the non-relativistic limit, while m ν δn ν /δρ ν → 0, Θ ν /θ ν → 0 and Σ ν /σ ν → 0 in the relativistic limit (by a factor am ν /q 1). Therefore, we could approximate ω δ , ω θ and ω σ by some background functions that go from 0 in the non-relativistic limit to 1/3 in the relativistic limit. An obvious guess would be the following w δ w θ w σ w ν , (E. [START_REF] Hawking | Gravitationally collapsed objects of very low mass[END_REF] where w ν ≡ Pν /ρ ν is the EoS of the neutrino species. However, there are many other possible choices, such as c 2 g , 3w ν c 2 g , 3w ν (1 + w ν )/4, p ν /3 Pν , n ν /3n ν , etc. In order to get the best candidates, one should compare these background functions with the exact formulas Eq. (E.22)-Eq. (E.24) computed with the full hierarchy, to see which one reproduces better the transition regime. For all the choices we tried, we verified that the decay terms in Eq. (E.16)-Eq. (E.18) play a very minor role, i.e. they give almost the same results as the fluid equations for stable massive neutrinos. This is not so surprising, given that these decay terms vanish both in the relativistic limit (for which w i w ν c 2 g 1/3 and nν m ν ρν ) and in the non-relativistic one (for which w i w ν c 2 g 0 and nν m ν ρν ).

We would still need to relate the collision terms up to max = 2 in the Boltzmann hierarchy for the dark radiation daughter with δ ν , θ ν and σ ν , and then set C > max = 0.

The first three collision terms are (c.f. Eq. (4.34)-Eq. (4.37)) C 0 = 4πa 2 Γ ν m ν ρ c dqq 2 ∆f ν,0 , (E.26)

C 1 = 4πa 2 Γ ν m ν ρ c dqq 2 q E ν ∆f ν,1 , (E.27) C 2 = 4πa 2 Γ ν m ν ρ c dqq 2 5 2 - 3 2
E 2 ν q 2 + 3 4
(E 2 ν -q 2 ) 2 E ν q 3 ln E ν + q E ν -q ∆f ν,2 .

(E.28)

By looking at the asymptotic limits of Eq. (E.26) and Eq. (E.27), it's easy to see that they can be interpolated in the following manner:

C 0 = a 5 Γ ν ρ c (1 -3w δ )ρ ν δ ν , (E.29) C 1 = a 5 Γ ν ρ c (1 -3w θ )ρ ν (1 + w ν ) θ ν k . (E.30)
Unfortunately, it's not so easy to relate C 2 with the neutrino shear σ ν , given the complicated integrand appearing in Eq. (E.28). In addition, the fluid equations do not provide such a high gain in speed as compared to the decaying dark matter scenario, because CLASS implements an optimal momentum sampling that makes the resolution of the neutrino Boltzmann hierarchy in Eq. (E.12)-Eq. (E.13) very manageable. For these reasons, we don't use the fluid equations in our analysis. Nevertheless, Eq. (E.16)-Eq. (E.18) provide a first step to achieve a fluid description of the relativistic neutrino decay, for which the inverse decay terms cannot be neglected and would likely make the resolution of the full Boltzmann hierarchy computationally prohibitive.

E.4 Excluding the relativistic decay regime from the MCMC analysis

In our baseline analysis, we have extrapolated our scans to the (mildly-)relativistic decay regime, despite the fact that the equations do not include inverse decays. We have then interpreted the bound on the sum of neutrino masses when considering non-relativistic decays as the intersect between the non-relativistic decay condition Γ ν > H(T ν = m ν /3) and the 2σ limit derived from our analysis.

In this appendix, we investigate how excluding the relativistic decay regime of parameter space from the scan can affect the bounds on m ν /eV and Log 10 (Γ ν /[km/s/Mpc]). As we are interested in (semi-)relativistic decays, we focus on the parameter space Log 10 (Γ ν /[km/s/Mpc]) ∈ [3, 6.5]. Our results are presented in Fig. E.1. In the 2D plane {Log 10 Γ ν , m ν } and below the non-relativistic line Γ ν = H(T ν = m ν /3), we find that imposing the condition directly within the MCMC prior relaxes the bound by ∼ 10 -20%. Nevertheless, after marginalizing over Log 10 (Γ ν ), we find that the 'naive' bound coming from the intersect between the non-relativistic line ( Γ ν > H(T ν = m ν /3)) and the 2σ limit without priors is in excellent agreement with that coming from imposing this condition as a prior in the analysis, both yielding m ν < 0.42 eV.

F

Résumé détaillé en français F.1 L'ère de la cosmologie de précision

La cosmologie est l'étude de l'évolution et des propriétés de l'univers dans son ensemble. Cette discipline scientifique aborde des questions aussi anciennes que l'humanité elle-même (c'est-à-dire "l'univers tel que nous le connaissons a-t-il existé depuis toujours ?" "quel est le destin ultime du cosmos ?"). Cependant, ce n'est qu'au début du 20 e siècle qu'elle est devenue une science prédictive, lorsque la théorie de la relativité générale (RG) s'est imposée comme un cadre théorique solide pour décrire l'univers. Depuis lors, le domaine de la cosmologie n'a cessé d'évoluer, grâce à l'augmentation rapide de la quantité et de la précision des observations. Au cours des deux dernières décennies, le modèle de cosmologie appelé Λ Cold Dark Matter (ΛCDM ) est devenu un paradigme réussi pour expliquer plusieurs sondes indépendantes avec un degré de précision étonnant. Ce modèle suppose que l'univers est bien décrit par une métrique plate de Friedman-Lemaître-Robertson-Walker, et qu'il est principalement composé d'environ 5 % de matière ordinaire, 26 % de matière noire froide et 69 % d'énergie noire sous forme de une constante cosmologique Λ. Malheureusement, la nature de ses principaux constituants, la matière noire et l'énergie noire, reste encore un mystère. De plus, l'augmentation de la précision des données cosmologiques a récemment conduit à l'apparition de plusieurs divergences expérimentales entre les sondes de l'univers primordial et tardif. En particulier, il existe une tension de 5σ dans la détermination du taux d'expansion actuel de l'univers (la constante de Hubble), et une tension de 2 -3σ dans la détermination de l'amplitude d'agglutination dans l'univers (décrite par le paramètre S 8 ). Pour ces raisons, ces dernières années, il y a eu un intérêt croissant pour l'exploration de différentes extensions du modèle ΛCDM , qui pourraient donner des indices sur les composants sombres très mystérieux, et éventuellement offrir une explication aux divergences expérimentales susmentionnés. D'autre part, la découverte des oscillations des neutrinos fournit des preuves solides de l'existence de masses de neutrinos minuscules mais non nulles, un phénomène qui ne peut pas être expliqué dans le cadre du modèle standard de la physique des particules. La cosmologie est actuellement la sonde la plus puissante des masses de neutrinos, et elle peut offrir des indices très précieux sur ses propriétés.

Dans cette thèse, nous avons étudié en détail les implications cosmologiques de plusieurs extensions du paradigme ΛCDM . Dans la première partie, nous avons étudié deux modèles, l'énergie sombre précoce (EDE) et la matière noire instable (DDM), qui sont directement ciblés à expliquer les tensions de Hubble et S 8 , respectivement. Dans la deuxième partie, nous avons analysé un autre scénario exotique motivé par le mystère de la masse des neutrinos, la désintégration des neutrinos, dont le formalisme est très similaire à celui de la matière noire instable.

Nous avons expliqué comment les mesures très précises des anisotropies du fond diffus cosmologique (CMB) jouent toujours un rôle central dans la définition de nouvelles contraintes robustes pour chacun de ces modèles. Pour rendre ce travail aussi autonome que possible, tout d'abord nous avons décrit les outils mathématiques et les observables les plus pertinents en cosmologie. Nous avons commencé par une courte introduction historique de la cosmologie moderne et une esquisse de les événements majeurs de l'histoire de l'univers. Ensuite, nous avons passé à la récapitulation du formalisme nécessaire pour comprendre les observables cosmologiques telles que l'anisotropie du fond diffus cosmologique et le spectre de puissance de la matière. Nous avons commencé par la description de l'univers homogène, et avons expliqué quelques concepts essentiels sur l'inflation. Plus tard, nous avons procédé à l'étude des écarts à l'équilibre en introduisant le formalisme de Boltzmann. Nous avons continué avec une présentation très détaillé de la théorie des perturbations linéaires. Dans la partie finale, nous avons discuté des défis les plus importants du modèle ΛCDM: la nature du secteur sombre et les tensions cosmologiques.

F.2 Energie noire précoce et la tension H 0

Une composante constante d'énergie sombre précoce (EDE) contribuant à une fraction f EDE (z c ) ∼ 10% de la densité d'énergie de l'univers autour de z c 3500 et se diluant au même rythme ou plus rapidement que le rayonnement par la suite, peut fournir une résolution simple à la tension de Hubble. Cependant, il a été souligné que l'inclusion de données de structure à grande échelle (LSS), qui sont en tension ∼ 3σ avec les cosmologies ΛCDM et EDE, pourrait briser une certaine dégénérescence des paramètres et modifier ces conclusions. Nous réévaluons la viabilité de l'EDE en combinant les observations LSS des récentes enquêtes à faible lentille (WL) avec les données CMB, BAO, fonction de croissance (FS) et SNIa. Nos résultats peuvent être résumés comme suit :

1. Dans un modèle EDE phénoménologique à 3 paramètres (3pEDE), nous confirmons que Planck18+ BAO+FS+PANTHEON+SH0ES favorisent f EDE (z c ) 0.1 ± 0.03, z c 4000 +1400 -500 et Θ i = 2, 6 +0,4 -0,03 , avec un ∆χ 2 = -18, 7 par rapport à ΛCDM ajusté sur le même ensemble de données. L'inclusion des dernières données Planck18 (et en particulier les mesures de polarisation plus précises) ne gâche pas le succès de la résolution EDE à la tension de Hubble. Dans la deuxième partie de ce travail, nous avons utilisé une méthodologie commune pour comparer et contraster le succès relatif de dix-sept modèles (y compris EDE) proposés pour atténuer la tension de Hubble; cette approche se veut donc une juste comparaison entre les solutions proposées, et fournit un repère utile pour ceux qui souhaitent proposer des idéaux nouveaux. Nous avons réparti les différents modèles en trois catégories génériques: ceux qui modifient l'horizon sonore en incluant une composante de rayonnement sombre (DR) impactant l'histoire de l'expansion précoce, les solutions qui modifient l'horizon sonore par un autre mécanisme (tel qu'un retard de recombinaison ou une certaine contribution à l'expansion avant la recombinaison), et des solutions qui tentent de modifier l'histoire de l'expansion tardive.

Nous avons alors montré que

Pour chaque modèle et ensemble de données, nous quantifions la tension résiduelle à l'aide d'une série de mesures, chacune présentant à la fois des avantages et des inconvénients, et en essayant de répondre à des questions légèrement différentes, à savoir : étant donné un modèle, (i) dans quelle mesure la confrontation avec des données (autres que SH0ES) génèrent des distributions a posteriori compatibles avec des valeurs élevées de H 0 , (ii) dans quelle mesure peut-on obtenir un bon ajustement combiné à toutes les données, et (iii) dans quelle mesure ce modèle est-il préféré à ΛCDM ? Le résumé de nos conclusions est qu'aucun modèle ne fonctionne parfaitement bien dans tous nos tests -tous les modèles se retrouvent avec une tension résiduelle, le plus prometteur réduisant la tension au niveau ∼ 1, 6σ, et avec très peu modèles réduisant réellement la tension en dessous de 3σ. De même, seul un petit sous-ensemble de modèles est capable d'améliorer suffisamment l'ajustement pour réussir le test AIC. Six modèles, EDE, NEDE, EMG, varying m e (avec et sans courbure), et le Majoron, sont capables de satisfaire simultanément les critères Q DMAP et ∆AIC. De plus, seul le varying m e (avec et sans courbure) passe le critère gaussien et permet un haut H 0 (ou M B ) sans SH0ES préalable, recevant la seule "médaille d'or" de notre tournoi.

De plus, nous notons qu'en dehors de l'EDE, aucun des modèles d'intérêt n'atténue (ni n'exacerbe) la tension S 8 . Certains modèles qui avaient précédemment montré un certain succès dans la réduction de la tension S 8 , à savoir DM-DR et les neutrinos à forte interaction, sont maintenant défavorisés par les données. Trouver une résolution commune aux deux tensions renforcerait certainement le degré de croyance dans la nouvelle cosmologie de la concordance; cependant, nous réitérons que la résolution de ces tensions pourrait provenir de secteurs indépendants -soit de la nouvelle physique, soit des erreurs systématiques. Nous concluons que certains des modèles présentés dans ce travail peuvent fournir un bon ajustement combiné à toutes les données considérées (y compris SH0ES), démontrant qu'il existe au moins des solutions potentielles à la tension H 0 , mais il reste encore de nombreuses difficultés à surmonter dans la construction du modèle et dans l'explication de la tension S 8 croissante. Des travaux supplémentaires doivent être effectués pour établir si ces problèmes théoriques et observationnels restants peuvent être surmontés dans une nouvelle cosmologie concordante, qui peut soit s'appuyer sur les modèles étudiés ici, soit peut-être se situer dans une direction encore inexplorée.

F.3 Matière noire instable et la tension S 8

Dans ce travail, nous avons réalisé une étude cosmologique complète du scénario de désintégration de la matière noire froide (CDM) à 2 corps appelé 'ΛDDM', dans lequel les désintégrations sont caractérisées à la fois par le taux de désintégration Γ et la fraction d'énergie convertie en rayonnement ε, incluant pour la première fois un traitement totalement cohérent des perturbations linéaires de la composante fille tiède (WDM). À cette fin, nous avons utilisé un nouveau schéma d'approximation, qui permet de calculer avec précision et rapidité la dynamique des perturbations linéaires de la WDM en traitant l'espèce WDM comme un fluide visqueux. Proche des valeurs les mieux ajustées, notre schéma d'approximation est précis au niveau O(0, 1%) dans les spectres de puissance de CMB et O(1%) dans le spectre linéaire de puissance de la matière. Nous avons ensuite discuté en détail la dynamique des perturbations linéaires des particules mères et filles, ainsi que les effets physiques du modèle ΛDDM sur les spectres de puissance du CMB et de la matière.

Dans une deuxième partie, nous avons effectué un ensemble d'analyses MCMC du modèle ΛDDM par rapport à une suite d'ensembles de données à jour à faible et à haut décalage vers le rouge. Nous avons comparé les contraintes obtenues à partir des données BAO et SNIa, donc uniquement basées sur les effets de fond, à celles obtenues à partir du jeu de données complet Planck, qui nécessite plutôt une description précise des perturbations linéaires WDM. Nous constatons que les données CMB Planck contraignent le modèle ΛDDM ∼ 1 ordre de grandeur mieux que les données BAO+SNIa actuelles. Cependant, nous montrons également que malgré ces contraintes, le modèle ΛDDM offre une possibilité prometteuse pour résoudre la tension S 8 .

Nous avons ensuite testé la robustesse de la résolution de ΛDDM à la tension S 8 à un certain nombre de changements dans l'analyse. Nous montrons que la légère préférence pour le modèle ΛDDM par rapport à ΛCDM est liée à la valeur S 8 choisie dans l'analyse. Concrètement, la valeur S 8 de l'analyse KiDS+Viking+DES, qui a un niveau de tension plus élevé avec la prédiction Planck ΛCDM que la valeur de référence KiDS+BOSS+2dFLens, conduit à une préférence plus forte en faveur du modèle ΛDDM. Cependant, le résultat DES uniquement, qui est raisonnablement en accord avec Planck, conduit à une préférence plus faible pour le modèle ΛDDM. De même, une fois marginalisé sur les informations de lentille dans Planck via le paramètre A lens , ou lors de l'échange des spectres de puissance TE,EE élevés Planck contre les spectres SPTpol, la préférence pour les Le modèle ΛDDM diminue. En effet, dans ces deux cas, le modèle inféré ΛCDM a une valeur S 8 plus petite, montrant moins de tension avec les enquêtes à faible lentille. Cela indique que si la tension S 8 finit par provenir d'une systématique inconnue dans les enquêtes à faible lentille ou dans les données Planck, la préférence pour le modèle ΛDDM disparaîtra probablement. D'un autre côté, en combinant Planck avec ACTPol, la légère préférence pour ΛDDM augmente, et la 'tension' restante avec S 8 n'est plus que ∼ 1.3σ.

Nous avons également testé la possibilité intrigante que le récent excès de Xe-non1T soit dû au modèle ΛDDM. À cette fin, nous avons effectué une analyse MCMC supplémentaire fixant ε = 0.05 comme requis par Xenon1T. Nous constatons qu'il est facile de résoudre la tension S 8 dans ce cas, indiquant une durée de vie DCDM de Log 10 (Γ/[Gyr -1 ]) = -2.72 +0. 61 -0.21 . Fait intéressant, cela se fait au prix d'une dégradation très mineure de l'ajustement Planck (∆χ 2 +1.7), indiquant que les données Planck, BAO et SNIa sont en bon accord avec ce modèle. Enfin, en effectuant une analyse où nous introduisons artificiellement un signal DDM dans un ensemble de données CMB fictives, nous démontrons explicitement que, même si les données CMB actuelles ne sont pas suffisamment sensibles pour faire la distinction entre CDM standard et DDM, les expériences CMB de nouvelle génération (CMB-S4) pourront détecter sans ambiguïté sa signature.

F.4 Nouvelles contraintes sur la masse et la durée de vie des neutrinos

Les observations cosmologiques sont connues pour imposer les contraintes les plus fortes sur la somme des masses des neutrinos. Pourtant, les contraintes liées aux mesures CMB et LSS, qui supposent que les neutrinos sont stables, est considérablement affaiblie si les neutrinos se désintègrent. Dans ce travail, nous fournissons des limites à jour sur la durée de vie des neutrinos massifs qui se désintègrent en rayonnement sombre après être devenus non-relativistes, à partir d'une combinaison de mesures de CMB, BAO, de facteurs de croissance et de données Pantheon SN1a.

Par rapport à l'analyse précédente dans C19, nous avons incorporé des corrections d'ordre supérieur jusqu'à O((T dec /m ν ) 3 ) lors de la résolution des perturbations de rayonnement sombre, et a également effectué le calcul complet de la densité d'énergie moyenne du neutrino instable. Le traitement plus précis des équations de Boltzmann et de l'évolution de l'énergie moyenne dans notre étude MCMC améliore la couverture du cas où les neutrinos se désintègrent tôt de sorte que leurs impulsions moyennes sont proches de leurs masses. Si les neutrinos se désintègrent lorsqu'ils ont T ν m ν /3, l'inclusion de perturbations de moment plus élevées C ≥2 donne un changement négligeable au spectre de puissance par rapport aux incertitudes expérimentales. Cependant, le calcul complet de l'énergie du neutrino améliore considérablement la prédiction du spectre de puissance à partir du résultat approximatif lorsque les désintégrations se produisent de manière semi-relativiste. Néanmoins, nous avons trouvé que les contraintes de Planck15, étant donné leur précision limitée, ne sont pas affectées par ces considérations. Cependant, nous prévoyons que ces effets seront pertinents pour les expériences futures (ainsi qu'une contribution essentielle dans le cas relativiste, à considérer dans le futur).

En fait, nous avons montré que l'essentiel de l'amélioration du pouvoir contraignant par rapport à C19 provient de l'utilisation des données de Planck18 . En effet, nous avons démontré que la mesure améliorée de τ reio à partir des données de polarisation à faible aide à briser la dégénérescence de l'amplitude du spectre de puissance CMB et renforce la limite sur la masse et la durée de vie des neutrinos. En conséquence, nous avons constaté que les neutrinos avec m ν > 0, 42 eV (2σ) ne peuvent pas être rendus cohérents avec les données cosmologiques s'ils se désintègrent alors qu'ils ne sont pas relativistes, une amélioration significative par rapport AUX données de Planck15 pour lesquelles des masses aussi élevées que m ν ∼ 0, 9 eV étaient cohérentes avec le scénario de désintégration non-relativiste (C19).

Nous avons soutenu qu'une application notable de ce résultat est que, si l'expérience KATRIN detecte un neutrino électronique avec m ν 0, 2 eV (la sensibilité prédite), notre résultat contraindra Γ ν 10 5.5 km/s/Mpc, i.e. les neutrinos auraient besoin de se désintégrer entre z ≈ 2 × 10 2 -4 × 10 3 , alors qu'ils sont encore relativistes, de sorte que nos bornes et les bornes étudiées dans B20 ne s'appliqueraient pas. Dans le cas d'une découverte massive de neutrinos à KATRIN, une analyse plus complexe comprenant des désintégrations inverses serait nécessaire pour confirmer fermement que le scénario de désintégration peut concilier les mesures de laboratoire et cosmologiques.

Abstract

With the advent of large astronomical surveys coming from ground-and space-based telescopes, the field of cosmology has undergone a scientific revolution in the last decades. It is said that we have entered in the era of precision cosmology: not only do we have plenty of measurements from the early and late universe, but also we can use them to test theoretical models with astonishing accuracy. One of such models, describing the universe with a flat Friedman-Lemaître metric and containing around 5 % baryons, 25 % Cold Dark Matter (CDM) and 70 % dark energy in the form of a cosmological constant (Λ), has shown to provide a remarkable fit to a wide variety of observables. Unfortunately, the nature of its main constituents -dark matter and dark energy -lacks identification. In addition, several experimental discrepancies have emerged in recent years, possibly shedding light on the properties of these mysterious components. On the other hand, the discovery of neutrino oscillations has provided convincing evidence that neutrinos possess non-zero masses, a phenomenon that cannot be explained within the Standard Model of Particle Physics. In my work, I have put new and robust constraints on many different extensions of the ΛCDM paradigm, aiming at explaining some of the present anomalies in cosmology, or to better characterize the properties of the elusive neutrinos. I start confronting a 1-parameter model of Early Dark Energy against a wide variety of data sets, showing that it can resolve the H0 tension, and that the addition of weak-lensing data doesn't affect this conclusion. In addition, I have made a systematic comparison of seventeen different models which have been proposed to resolve the H0 tension (including Early Dark Energy), and quantified the relative success of each model using a series of metrics and a wide array of data combinations. Secondly, I have performed a detailed cosmological analysis of a scenario in which dark matter can decay into dark radiation and warm dark matter species, including for the first time a full treatment of perturbations. This study has shown that this model, while unable to ease the Hubble tension, can fully explain the low-S8 measurement from recent weak lensing surveys. I end discussing updated cosmic constraints on a scenario in which neutrinos can decay into dark radiation while being non-relativistic. I show that this allows to significantly relax the bounds on the neutrino masses, making a potential detection in the laboratory compatible with cosmological constraints.

Résumé

Avec l'avènement de grands relevés astronomiques provenant de télescopes terrestres et spatiaux, le domaine de la cosmologie a connu une révolution scientifique au cours des dernières décennies. On dit que nous sommes entrés dans l'ère de la cosmologie de précision : non seulement nous disposons de nombreuses mesures de l'univers ancien et tardif, mais nous pouvons également les utiliser pour tester des modèles théoriques avec une très grande précision. L'un de ces modèles, décrivant l'univers avec une métrique de Friedman-Lemaître plate et contenant environ 5 % baryons, 25 % Matière Noire Froide (acronyme anglais CDM) et 70 % d'énergie noire sous la forme d'une constante cosmologique (Λ), s'est avérée fournir un ajustement remarquable à une grande variété d'observables. Malheureusement, la nature de ses principaux constituants -la matière noire et l'énergie noire -n'est pas encore identifiée. De plus, plusieurs divergences expérimentales sont apparues ces dernières années, qui pourraient révéler les propriétés de ces composants mystérieux. D'autre part, la découverte des oscillations des neutrinos a fourni des preuves convaincantes que les neutrinos possèdent des masses non nulles, un phénomène qui ne peut pas être expliqué dans le modèle standard de la physique des particules. Dans mon travail, j'ai mis de nouvelles contraintes robustes sur de nombreuses extensions différentes du paradigme ΛCDM, visant à expliquer certaines des anomalies actuelles en cosmologie, ou à mieux caractériser les propriétés des neutrinos insaisissables. Je commence par confronter un modèle à 1 paramètre supplémentaire d'énergie noire précoce à une grande variété d'ensembles de données, montrant qu'il peut résoudre la tension H0, et que l'ajout de données du lentillage gravitationnelle faible des galaxies n'affecte pas cette conclusion. De plus, j'ai fait une comparaison systématique de dix-sept modèles différents qui ont été proposés pour résoudre la tension H0 (y compris l'énergie noire précoce), et quantifié le succès relatif de chaque modèle en utilisant une série de métriques et un large éventail de combinaisons des données. Deuxièmement, j'ai effectué une analyse cosmologique détaillée d'un scénario dans lequel la matière noire peut se désintégrer en un rayonnement sombre et une particule de matière noire "chaude", incluant pour la première fois un traitement complet des perturbations. Cette étude a montré que ce modèle, bien qu'incapable d'atténuer la tension de Hubble, peut pleinement expliquer la mesure de S8 par des récentes enquêtes à lentillage gravitationnelle faible des galaxies. Je termine en discutant des nouvelles contraintes cosmologiques sur un scénario dans lequel les neutrinos massifs non-relativistes se désintégrent en rayonnement noir. Je montre que cela permet de réduire significativement les limites de masse des neutrinos, rendant compatible une possible détection en laboratoire avec les contraintes cosmologiques.

Figure 1 . 1 -

 11 Figure 1.1 -Velocity-Distance relation for the different extra-galactic nebulae observed by Hubble in 1929. The units on the X-and Y-axis are parsecs and km/s, respectively. Taken from HUBBLE (1929)

Figure 1 . 2 -

 12 Figure 1.2 -Image of the galaxy cluster Abell 1689, taken with the Hubble Space Telescope.The galaxies and dark matter in the cluster act as a lens that warps the light of the background galaxies, leading to a plethora of arcs around the lens, as predicted by GR. This allows to reconstruct the mass distribution of dark matter, which is represented in purple. Taken from https://esahubble.org/images/heic1014a/

Figure 1 . 3 -

 13 Figure 1.3 -Hubble diagram for SNIa, showing the distance modulus (defined in Eq. (1.36)) vs. redshift. Data points are taken from Tabs. 5 and 10 in RIESS et al. (1998). The SNIa data at high-z clearly select a universe dominated by a cosmological constant, Ω Λ > Ω m . The small red region in the lower left marks the maximum span of Hubble's original diagram from 1929.

Figure 1 . 4 -

 14 Figure 1.4 -The CMB map of the temperature anisotropies, as seen by AGHANIM et al. 2020b -hereafter refereed to as Planck18. Colors label variations at the level of 10 -5 around a mean temperature of T 0 = 2.725 K

Figure 1 . 5 -

 15 Figure 1.5 -Evolution of the fractional energy densities of photons, baryons, three neutrino species (one massless and two massive, 0.06 and 0.01 eV) and a cosmological constant Λ.

Figure 1 . 6 - 2 L= 5

 1625 Figure 1.6 -Geometries corresponding to the notions of luminosity distance to SNIa (left) and angular diameter distance to the BAO feature in the galaxy distribution (right).

Figure 1 . 7 -

 17 Figure 1.7 -Evolution of the comoving Hubble radius during and after inflation. Taken from BAUMANN 2021.

76 ) 3 / 2 e

 7632 In the non-relativistic limit T m, and for chemical potentials µ a m, we can use the Maxwell Boltzmann distribution of Eq. (1.68), to obtain n eq = g mT 2π (-m+µ)/T , (1.77)

Figure 1 . 8 -

 18 Figure 1.8 -Temperature evolution of the effective number of relativistic degrees of freedom g (T ), assuming the particle content of the Standard Model. The dotted line indicates the effective number of degrees of freedom in entropy, g ,S (T ). Taken from BAUMANN 2021.

Figure 1 . 9 -

 19 Figure 1.9 -Schematic view of dark matter freeze-out. A comoving density is plotted against m χ /T . The red region corresponds to a time when annihilations and pair productions compensate. The yellow region, a time when pair productions become inefficient. The purple region, a time when annihilations stop due to the cosmic expansion. Adapted from D. HOOPER (2008)

4 3 + D ←→ He 4 +

 34 and other heavier nuclei to be sequentially built from lighter ones,p + + D ←→ He 3 p + .(1.108)

Figure 1 . 10 - 4 ,

 1104 Figure 1.10 -Predicted primordial abundances of He 4 , D, He

  of η or Ω b . The colored bands indicate theoretical 95 % CL uncertainty. Yellow boxes indicate observed element abundances. The narrow vertical band indicates the CMB measurement on Ω b from ADE et al. 2016a -hereafter refereed to as Planck15. Taken from TANABASHI et al. (2018)

Figure 1 . 11 -

 111 Figure 1.11 -Left panel: Schematic view of the three-level model of recombination. Taken from https://cosmo.nyu.edu/yacine/teaching/cmb_fudan/lecture_2_yah.pdf. Right panel: Evolution of the free electron fraction, computed using either the Saha equation in Eq. (1.118) or the RECFAST code. We note that RECFAST also takes into account the two Helium recombinations. We have assumed Y p = 0.24 and Ω b = 0.049.

Figure 1 . 12 -

 112 Figure 1.12 -Optical spectra of quasar ULAS J1319+0959 at z = 6.13, accompanied by a pictorial description of the reionization process. The image has been adapted from Figure 1 in BECKER et al. (2015).

  2022) or SKA (BARRY et al. 2021) radio interferometers. These observations promise to open a new window onto the dark ages and the beginning of reionization.

Figure 1 . 13 -

 113 Figure 1.13 -The gauge ambiguity. Darker (brighter) regions have larger (smaller) density, diluting with the cosmic expansion. Left panel: A homogeneous universe, with the only valid time slicing shown in red. Right panel: A universe with perturbations. In this case there are many possible time slicings, two of them are shown in blue and yellow. In both slicings, density remains homogeneous on average

s c 2 g

 2 , and the anisotropic stress σ ν is expressed in terms of other thermodynamic quantities (see for example SHOJI et al. (2010) or LESGOURGUES et al. (

Figure 1 . 14 -

 114 Figure 1.14 -Evolution of the two scalar metric perturbations in the Newtonian gauge, φ and ψ. This is shown for three different wavenumbers, k = 1 Mpc -1 (purple), k = 10 -2 Mpc -1(blue) and k = 10 -3 Mpc -1 (green), which crossed the horizon before, during and after matterradiation equality, respectively.

. 239 )

 239 Eq. (1.236)-Eq. (1.237) have other interesting consequences: • The pressure P I of each species can be generally expressed as a function of the density ρ I and the entropy S I . Performing a Taylor expansion around the background quantities, we get P I (ρ I , S I ) P I (ρ I , SI ) + c 2 s,I δρ I + Υ I δS I , (1.240) where c 2 s,I ≡ ∂P I /∂ρ I | S I is the sound speed and Υ I ≡ ∂P I /∂S I | ρ I . From Eq. (1.236) and Eq. (1.237) we have δP I = ( P I /ρ I )δρ I , meaning that c 2 s,I = P

. 245 )

 245 with r = | x -x|. The symbol ... indicates an average over many different realizations.

. 252 )

 252 This is called the constant density curvature perturbation, and it corresponds to the comoving curvature defined in the uniform density gauge (where B = 0 and δρtot = 0). Both R and ζ are widely used, and it can be shown that on super-Hubble scales, R ζ (FERGUSSON 2015).

Figure 1 . 15 -

 115 Figure 1.15 -Different portions of the (k, τ ) space, leading to specific behaviors of Θ 0 . The shaded areas indicate regions in which the gravitational potentials decay. Taken from POULIN (2017).

Figure 1 .

 1 Figure 1.16 -A snapshot of the photon transfer function Θ 0 and the zero-point of oscillations, Θ eq 0 = -(1 + R)ψ, at the time of photon decoupling.The former is also shown at the time of matter-radiation equality, in order to highlight the differences in the shape of the oscillations. In the x axis, we have rescaled the wavenumbers by the corresponding value of the sound horizon, to indicate that the phase of the oscillations is given by kr s (τ ).

  also that the super-Hubble values of Θ 0 are not exactly the same at τ eq and τ rec . In fact, combining Eq. (1.233) with Eq. (1.264)-Eq. (1.265), one finds that the values of Θ 0 during radiation and matter domination are related by Θ MD 0

Figure 1 . 17 -

 117 Figure 1.17 -Diagram to illustrate the geometric relation between the multipole and the Fourier scale λ observed on the last scattering surface. Adapted from LESGOURGUES (2013).

Figure 1 . 18 -

 118 Figure 1.18 -Different contributions to the temperature anisotropy spectrum C TT .

Figure 1 . 19 -

 119 Figure 1.19 -Type-E polarization spectrum, with and without including the effects of the cosmic reionization at low redshift.

Figure 1 . 20 -

 120 Figure 1.20 -Top panel: Temperature power spectrum before and after lensing corrections are applied. Bottom panel: Fractional difference between these two spectra.

Figure 1 . 21 -

 121 Figure 1.21 -Linear evolution of the density perturbations for photons, baryons and CDM, corresponding to a mode (k = 0.5 Mpc -1 ) that crossed the Hubble radius during radiation domination.

  3 5 f b , with f b ≡ ρb ρb +ρ cdm . This reduction of

Figure 1 . 22 -

 122 Figure 1.22 -The matter power spectrum at z = 0 computed with only CDM, adding baryons (ω b = 0.022), and including non-linear corrections with HALOFIT (R. E. SMITH et al. 2003; R. TAKAHASHI et al. 2012). In all cases, the present matter abundance is set to ω m = 0.142.

  Their associated spatial resolution simply scales like the volume of the box divided by the number of particles N . State-of-the-art codes such as GADGET -2 (SPRINGEL et al. 2001; SPRINGEL 2005) are based on the Tree Particle-Mesh (Tree-PM) approach, which is an intelligent algorithm that avoids having to compute N (N -1) forces between pairs of particles, as would be required in a brute-force calculation. The Boltzmann solver CLASS makes use of an accurate fitting formula based on the results of the GADGET -2 code, called HALOFIT (R. E. SMITH et al. 2003; R. TAKAHASHI et al. 2012), improved to account for the effects of massive neutrinos (BIRD et al. 2012

  where the first term includes the light-to-mass bias function b g (z), and the second term proportional to the growth rate f ≡ dln(D(a))/dlna (with D(a) = ag(a)/g(a 0 )) is accounting for Redshift Space Distortions (KAISER 1987). On the other hand, probes of the Lyman-α forest measure the flux power spectrum P F (k), linearly related to the one-dimensional matter spectrum P m,1D (k) = 1 2π ∞ k dk k P m (k ) through another scale-dependent bias function, P F (k) = b 2 (k)P m,1D (MURGIA et al. 2017).

Figure 1 . 23 -

 123 Figure 1.23 -Schematic view of the coverage of current and future cosmological probes across various epochs and scales. The ranges of each individual probes are approximate. Similar figures can be found in BODDY et al. (2022) and SABTI et al. (2022).

  .1 ± 2.0) km/s/Mpc (YUAN et al. 2019; SOLTIS et al. 2021). Additional methods intended to calibrate SNIa at large distances include: surface brightness fluctuations of galaxies (KHETAN et al. 2021), MIRAS (HUANG et al. 2019), or the Baryonic Tully Fisher relation (SCHOMBERT et al. 2020). There also exists a variety of observations which do not rely on observations of SNIa -these include e.g. time-delay of strongly lensed quasars (BIRRER et al. 2020; K. C. WONG et al. 2020), maser distances (PESCE et al. 2020), or gravitational waves as "standard sirens" (B. P. ABBOTT et al. 2021). In Fig. 1.24

Figure 1 . 24 -

 124 Figure 1.24 -Whisker plot showing 68% CL determinations of the Hubble constant H 0 through a series of direct and indirect measurements. The yellow vertical band denotes the H 0 value from Planck18 (AGHANIM et al. 2020b), whereas the blue vertical band denotes the H 0 value reported by SH0ES (RIESS et al. 2021a). For clarity, we restricted to a subset of constraints with error bars smaller than σ H0 3 km/s/Mpc, a full compilation of measurements can be found in DI VALENTINO et al. (2021b).

Figure 1 . 25 -

 125 Figure 1.25 -Whisker plot showing 68% CL determinations of the S 8 parameter by various CMB experiments and other low-redshift cosmological probes. The yellow vertical band denotes the S 8 value from Planck18 (AGHANIM et al. 2020b), whereas the blue vertical band denotes the S 8 value reported by the joint analysis of KIDS1000+BOSS+2dfLenS (HEYMANS et al. 2021). For clarity, we restricted to a subset of constraints with error bars smaller than σ S8 0.03, a full compilation of measurements can be found in ABDALLA et al. (2022).

  2012), errors in photometrically determined redshift distributions (HUTERER et al. 2006), intrinsic alignment of galaxies (BRIDLE et al. 2007), uncertainties in the modeling of baryon feedback (MEAD et al. 2020) and small-angle approximations (KITCHING et al. 2017). Regarding the galaxy clustering systematics, there can be issues with stellar contamination, atmospheric extinction and blurring (ROSS et al. 2012), fiber collisions (HAHN et al. 2017) and selection bias (C. M. HIRATA 2009). Finally, the dominant source of systematic uncertainty in cluster count observations is the hydrostatic bias parameter relating the mass observed through the hot gas and the true mass of the host halo, 1 -b = M obs /M true (BLANCHARD et al. 2021).

  2017), the Lithium problem (MATHEWS et al. 2020), the cosmic dipole tension (SECREST et al. 2021; DALANG et al. 2022) or the Ly-α tension in n s (PALANQUE-DELABROUILLE et al. 2020; D. C. HOOPER et al. 2022) (a thorough review of cosmological tensions can be found in ABDALLA et al. (2022)).

  SMITH et al. (2020) indicate 1 f EDE = 0.10 ± 0.03, Log 10 (z c ) = 3.57 +0.05 -0.14 , H 0 = 71.4 ± 1.2 km/s/Mpc. (2.1)However, taken at face value, this model triggers a number of questions. On the theoretical side, it suffers from a strong coincidence problem as the fluid needs to become dynamical around a key era of the universe, matter-radiation equality z c 3500. However, this coincidence might be the sign of a very specific dynamics to be uncovered; in fact there exist models in which the field becomes dynamical precisely around matter-radiation equality, either because of a phase-transition triggered by some other process (SAKSTEIN et al. 2020) or because of a non-minimal coupling to the Ricci curvature (BRAGLIA et al. 2020a). Other models set on a stronger theoretical ground have been proposed in the literature, see for example ALEXANDER et al. (2019), KALOPER (2019), BERGHAUS et al. (2020), BRAGLIA et al. (2020b), GONZALEZ et al. (2020), NIEDERMANN et al. (2020), ALLALI et al. (2021), FREESE et al. (2021), GOGOI et al. (2021), MCDONOUGH et al. (2021), NIEDERMANN et al. (2021a), KARWAL et al. (2022), and SABLA et al. (2022)).

  models as a resolution to the Hubble tension (HILL et al. 2020). A similar conclusion was reached in IVANOV et al. (2020) and D'AMICO et al. (2021b) with the inclusion of BOSS data in the effective field theory (EFT) of LSS framework. In the first part of this chapter, we analyze the EDE cosmology resolving the Hubble tension in light of the latest Planck18 data (and the more precise polarization measurement) and confront it to the KIDS-VIKING measurement of the cosmic shear power spectrum (ASGARI et al. 2020) and the joint measurement of S 8 from KIDS-VIKING+DES 2 . The KIDS-VIKING+DES measurements however rely on modeling the non-linear matter power spectrum on relatively small scales. This is done within numerical Einstein-Boltzmann solvers such as CAMB (LEWIS et al. 2000) or CLASS (LESGOURGUES 2011), through the HALOFIT (R. E. SMITH et al. 2003; R. TAKAHASHI et al. 2012) or HMCODE (MEAD et al. 2015

  et al. 2021b for details) M b = 19.2435 ± 0.0373 (2.6) In the third and last step of the ladder, the Hubble constant is determined from the observations of SNIa in the Hubble flow (at distances of the order of ∼ Gpc). More precisely, the SH0ES team converts the intrinsic magnitude M b into a value of H 0 via the magnitude-redshift relation (see Eq. (2.2)) of 277 SNIa from the Pantheon catalog (SCOLNIC et al. 2018) in the redshift range 0.023 < z < 0.15.

Figure 2 . 1 -

 21 Figure 2.1 -Differences in the luminosity distances, with respect to the prediction from the best-fit ΛCDM model from Planck18. These distances are obtained from two sets of data: measurements of SNIa apparent magnitudes calibrated using M SH0ES b = 19.2435 ± 0.0373 (RIESS et al. 2021b), and BAO measurements calibrated with r Pl18-ΛCDM d = 147.18±0.29 Mpc (AGHANIM et al. 2020b) and converted into luminosity distance with d L (z) = (1 + z)D A (z). The blue dots correspond to SNIa data from the Pantheon catalog (SCOLNIC et al. 2018), while the red dots correspond to BAO data from BOSS DR12 at z = 0.38, 0.51 and 0.61 (ALAM et al. 2017), eBOSS DR16 quasars at z = 1.48 (HOU et al. 2020), and the joint constraints from the eBOSS DR14 Ly-α cross-correlation at z = 2.35 (BLOMQVIST et al. 2019).

•

  The "inverse distance ladder" determination of M b , obtained from the combination of d L (z) = (1 + z)D A (z) as extracted from the Pl18-ΛCDM-calibrated BAO data and the apparent magnitudes from the Pantheon survey, cannot be made compatible with the direct determination from SH0ES. This has been shown for example in CAMARENA et al. (2021) and EFSTATHIOU (2021).• The value of the sound horizon r s obtained from a combination ofD A (z) = d L (z)/(1 +z) as extracted from the SH0ES-calibrated SNIa data and the BAO data, is in disagreement with that inferred from Planck18 under ΛCDM . This was highlighted in BERNAL et al. (2016), KNOX et al. (2020) and AYLOR et al. (2019). • The reconstructed expansion histories H(z) inferred from Pl18-ΛCDM-calibrated BAO data and SH0ES-calibrated SNIa data are inconsistent with each other. This was the approach of LEMOS et al. (2019) and POULIN et al. (2018b).

  ), where computation only at the level of the background was shown to partially alleviate the tension. However, it is the work of POULIN et al. (2019) that showed through a fluid approximation the key role played by perturbations in the scalar field to allow for a resolution of the Hubble tension. In particular, it has been shown that Planck data not only provide a detection of the background dynamics of the EDE component, but also severely restricts the dynamics of perturbations, favoring either a non-canonical kinetic term, whereby the equation of state w is approximately equal to the effective sound speed c 2 s (M.-X. LIN et al. 2019), or a potential that flattens close to the initial field value (T. L. SMITH et al. 2020).

  this work, we study the modified axion potential introduced in KAMIONKOWSKI et al. (2014), KARWAL et al. (2016), POULIN et al. (2018a, 2019), and T. L. SMITH et al. (2020),

. 15 )

 15 We assume that the field always starts in slow-roll, Θ i = 0 (as enforced by the very high value of the Hubble rate at early times), and without loss of generality we restrict 0 ≤ Θ i ≤ π. The model at this point has four free parameters: {m, f, n, Θ i }. Given that the dynamics are relatively insensitive to changes of 2 n 4.5 (AGRAWAL et al. 2019; T. L. SMITH et al. 2020), we further restrict the parameter space by taking n = 3. In order to make this model more physically accessible, instead of parameterizing the dynamics using the mass m and decay constant f , we use the critical redshift when the field becomes dynamical, z c , and the the fractional energy density at this redshift, f EDE (z c ) ≡ ρφ (z c )/ρ tot (z c ), which is roughly the maximum energy contribution induced by EDE. The final degree of freedom is encoded in the dynamics of the linear perturbations, which is fully characterized via the effective sound speed c 2 s , and physically related to the curvature of the potential close to the initial field valuenote that after fixing all other phenomenological parameters, this is fully described by Θ i (POULIN et al. 2018a; T. L. SMITH et al. 2020).

Figure 2 . 2 -

 22 Figure 2.2 -Fractional energy density (upper) and equation of state (lower) in the EDE as a function of redshift. Cosmological parameters are set to the best-fit values from the Planck18+BAO+SNIa+SH0ES analysis (see third column of Tab. (2.1)).

Figure 2 . 3 -

 23 Figure 2.3 -Reconstructed 2D posterior distributions of a subset of parameters for various data set combinations (see legend) in the 3-parameters EDE cosmology.

  2016; DI VALENTINO et al. 2017a,b; ZHAO et al. 2017; ADDISON et al. 2018; POULIN et al. 2018b; AYLOR et al. 2019; RAVERI 2020). However, as noted in POULIN et al. (2019), NIEDERMANN et al. (2020), and T. L. SMITH et al. (2020), we find that the reconstructed ω cdm and n s in the EDE cosmology are somewhat higher than in ΛCDM, such that the S 8 tension is slightly increased. As suggested in past literature (HILL et al. 2020; IVANOV et al. 2020; D'AMICO et al. 2021b

Θ

  i is set by the dynamics of the phase-transition (see also BERGHAUS et al. (2020), BRAGLIA et al. (2020a), GONZALEZ et al. (2020), and NIEDERMANN et al. (

  2.4.3, following HILL et al. (2020), IVANOV et al. (2020), and D'AMICO et al. (2021b).

  SMITH et al. (2003), as revised by R. TAKAHASHI et al. (2012), which has been shown to be accurate at 5% level in reproducing the non-linear power spectra of ΛCDM models up to wavenumbers k ≤ 10 h/Mpc. However, the version developed by authors of R. TAKAHASHI et al. (2012) does not consider the impact of baryon feedback. A further improvement, dubbed as HMCODE , has been developed in MEAD et al. (

  (2.3), by using the N -body code GADGET -3, a modified version of the publicly available numerical code GADGET -2 (SPRINGEL et al. 2001; SPRINGEL 2005). The initial conditions have been produced by displacing the DM particles from a cubic Cartesian grid according to second-order Lagrangian Perturbation Theory, with the 2LPTIC public code (CROCCE et al. 2006), at redshift z = 99. The corresponding input linear matter power spectra, for both the EDE and ΛCDM cases, were computed with AXICLASS, the aforementioned modified version (T. L. SMITH et al. 2020) of the publicly available code CLASS (BLAS et al. 2011). For all of the simulations, we kept the cosmological parameters fixed to their EDE best fit values from T. L. SMITH et al. (2020) (very close to ours), namely H 0 = 72.81, Ω m = 0.2915, A s = 2.191 • 10 -9 , n s = 0.986 for both cosmological scenarios; plus the additional parameters log 10 (z c ) = 3.53, f EDE (z c ) = 0.132, Θ i = 2.72, n = 2.6 for the EDE model.

Figure 2

 2 Figure2.5 -In the top left panel we show the matter power spectra extracted from our simulations, and the ones computed with HALOFIT /HMCODE . The blue curves refer to the ΛCDM scenario, while the red ones refer to the EDE best fit model. We also report the best fit ΛCDM case from Planck 2018.The spliced power spectra are reported as thick dot-dashed lines. Symbols stand for the outputs of the LB and HR simulations. The solid/dotted lines are the non-linear power spectra from HALOFIT /HMCODE , whereas the dashed lines are the corresponding linear power spectra used to produce the initial conditions for the simulations. The cyan shaded band roughly corresponds to the scales probed by DES-Y1. In the top right panel, we show the ratio between the non-linear matter power spectra from our simulations and the ones computed with HALOFIT /HMCODE , for both the ΛCDM "equivalent" and the EDE best fit models, adopting the same linestyle-code and color-code. In the bottom right panel we compare departures from the ΛCDM model in terms of ratios of non-linear matter power spectra, adopting the same linestyle-code and color-code.

  g. (CHUDAYKIN et al. 2020b; IVANOV et al. 2020; D'AMICO et al. 2021b; KLYPIN et al. 2021)). Furthermore, it might be already possible to test O(20%) deviations in the small-scale power, as the ones shown in the bottom panel of Fig. 2.5, with current Lyman-α forest flux power spectrum data (MURGIA et al. 2017, 2018; ARCHIDIACONO et al. 2019; MILLER et al. 2019; BALDES et al. 2020) and the EFT of LSS data analysis of BOSS data (see e.g. IVANOV et al. (2020) and D'AMICO et al. (2021b) for a recent analysis in the 3-parameter model).

Figure 2 . 7 -

 27 Figure 2.7 -Reconstructed 2D posterior distributions of a subset of parameters for various data set combinations (see legend) in the 1-parameter EDE cosmology. "All data" stands for the combination Planck+BAO/FS+Pantheon+SH0ES+KiDS/Viking.

Figure 2 . 10 -

 210 Figure 2.10 -Reconstructed 1-and 2D-posterior of a subset of parameters in the 1-parameter EDE cosmology for various data sets (see legend), once marginalizing over A lens and A φφ lens .

  Radiation solutions • Free-streaming Dark Radiation [∆N ur ]. • Self-Interacting Dark Radiation [SIDR]. • Free-streaming plus self-interacting Dark Radiation [mixed DR], (BRUST et al. 2017; BLINOV et al. 2020b). • Self-interacting Dark Radiation scattering on Dark Matter [DR-DM], (LESGOURGUES et al. 2016; BUEN-ABAD et al. 2018; ARCHIDIACONO et al. 2019). • Self-interacting neutrinos plus free-streaming Dark Radiation [SIν+DR], (PARK et al. 2019; KREISCH et al. 2020). • Interacting neutrinos with a Majoron [Majoron], (ESCUDERO et al. 2020a, 2021). Let us emphasize that it has been shown in many different studies (e.g. BERNAL et al. (2016), POULIN et al. (2019), and DI VALENTINO et al. (

  Universe solutions • primordial magnetic field [primordial B], (JEDAMZIK et al. 2011, 2020). • varying effective electron mass [varying m e ], (HART et al. 2018, 2020). • varying effective electron mass in a curved universe [varying m e +Ω k ], (SEKIGUCHI et al. 2021; SOLOMON et al. 2022). • Early Dark Energy [EDE]. This is the model introduced in Sect. 2.3 of this Chapter. • New Early Dark Energy [NEDE], (NIEDERMANN et al. 2020, 2021a). • Early Modified Gravity [EMG], (BRAGLIA et al. 2021).

For

  our baseline dataset, D baseline , where D baseline = Planck18 (including TTTEEE and lensing) + BAO (including BOSS DR12 (ALAM et al. 2017) + MGS (ROSS et al. 2015) + 6dFGS (BEUTLER et al. 2011)) + Pantheon (SCOLNIC et al. 2018) , the result of our main tests (Q DMAP for M b and Akaike Information Criterion) are summarized in Tab.

Table 2 . 7 -

 27 Test of the models based on dataset D baseline , using the direct measurement of M b by SH0ES as the GT for the quantification of the tension (4th column), the Q DMAP criterion (5th column), or the computation of the ∆AIC (7th column).eight models are capable of significantly improving over ΛCDM24 . They are, in decreasing level of success: EDE, varying m e +Ω k , NEDE, EMG, varying m e , the Majoron model, a primordial magnetic field, and SIDR.

Figure 2 . 11 -

 211 Figure 2.11 -∆χ 2 (Test 2) and ∆AIC (Test 3) of the various models considered in this work.We additionally display the thresholds that have to be reached as purple dashed lines, and the regions of successful models as a purple region.

Figure 2 . 12 -

 212 Figure 2.12 -Contours of {H 0 , M b , S 8 } obtained when considering the D extended dataset for the finalists models compared to ΛCDM. The purple band represents the S 8 measurement from KiDS-1000 (ASGARI et al. 2021).

2 .

 2 varying m e +Ω k ,

  2.4 and in other works (HILL et al. 2020; IVANOV et al. 2020; D'AMICO et al. 2021b; NIEDERMANN et al. 2021b; T. L. SMITH et al. 2021), the inclusion of LSS data doesn't spoil the resolution to the H 0 tension provided by the EDE and NEDE models, even if they slightly increase the mean value of S 8 . Nevertheless, this figure clearly illustrates that resolving the both H 0 and S 8 tensions will likely require multiple extensions.

  2020) alone have shown a slight (∼ 2.2σ) preference for the presence of an EDE component with a fraction f EDE (z c ) ∼ 0.15 and H 0 ∼ 74 km/s/Mpc (HILL et al. 2021; POULIN et al. 2021). These studies showed that the inclusion of large-scale CMB temperature measurements by the Wilkinson Microwave Anisotropy Probe (WMAP) (C. L. BENNETT et al. 2013) or Planck18 restricted to the WMAP multipole range increases the preference to ∼ 3σ. A similar analysis using the third generation South Pole Telescope 2018 (SPT-3G) data (DUTCHER et al. 2021) was presented in LA POSTA et al.

( 3 .

 3 22) and the analogous expression derived in BLACKADDER et al. (2014) with a different formalism. Concerning

Figure 3 . 1 -

 31 Figure 3.1 -Redshift evolution of the abundances of the DCDM, WDM and DR species, assuming Γ -1 = 30 Gyrs and ε = 0.1. We also show the abundance for a standard CDM species with Ω 0 cdm = Ω ini dcdm .

Figure 3 . 2 -

 32 Figure 3.2 -Redshift evolution of the Hubble parameter for the ΛCDM and four different ΛDDM models. The Hubble parameter today is fixed to H 0 = 67.7 km/s/Mpc.

  ) = ρdcdm (a) + ρdr (a) + ρwdm (a) + ργ (a) + ρν (a) + ρb (a) + ρΛ . (3.24)

  2011). Taking the relativistic limit of Eq. (3.41) and Eq. (3.42) and multiplying the decay terms by 2 (to account for the decay into two massless particles), we recover the conservation equations for DR written in POULIN et al. (2016). We give a complete proof of Eq. (3.41)-Eq. (3.44) in App. D.3.

Figure 3 . 3 -

 33 Figure 3.3 -Time evolution of the linear density perturbations of the DCDM, WDM and DR species, corresponding to a wavenumber k = 1 Mpc -1 . Each panel displays the perturbations for a different combination of the parameters Γ and ε. The black dashed and solid lines indicate the times of horizon crossing (at H(τ ) = k) and WDM free-streaming scale crossing (at k fs (τ ) = k), respectively. The purple dotted line indicates the characteristic decay time, given by t(τ ) = Γ -1 .

Figure 3 . 4 -

 34 Figure 3.4 -Same as in Fig. 3.3, but corresponding to a wavenumber k = 10 -2 Mpc -1 . In this case, the WDM perturbations never cross the free-streaming scale.

Figure 3 . 5 -

 35 Figure 3.5 -Left panel: Residuals of the linear matter power spectrum at z = 0 (upper) and z = 2 (lower) for several values of the lifetime Γ -1 = 10, 30, 100, 300 Gyrs and a fixed DR energy fraction ε = 0.1. Residuals are taken with respect our baseline ΛCDM model. Right panel: Same as in the left, but for several DR energy fractions ε = 0.5, 0.1, 0.01, 0.001 and a fixed lifetime Γ -1 = 30 Gyrs.

Figure 3 . 6 -

 36 Figure 3.6 -Left panel: Residuals (with respect our baseline ΛCDM model) of the CMB lensed TT (upper), EE (middle) and lensing potential (lower) power spectra for several values of the lifetime Γ -1 = 10, 30, 100, 300 Gyrs and a fixed DR energy fraction ε = 0.1. Right panel: Same as in the left, but for several values of the DR energy fraction ε = 0.5, 0.1, 0.01, 0.001 and a fixed lifetime Γ -1 = 30 Gyrs.

- 3 - 2 - 1 0Figure 3 . 7 -

 32137 Figure3.7 -1 and 2D marginalized posterior distributions for the cosmological parameters relevant for our analysis. Hereafter, unless otherwise stated, the green shaded bands refer to the joint S 8 measurement from KiDS-1000+BOSS+2dFLens, while the gray bands stand for the H 0 measurement by the SH0ES collaboration. Note that the BAO+SNIa analysis is based on background evolution only, whereas the BAO+SNIa+CMB analysis also includes linear perturbations (see the main text for further details).

  Hubble tension, contrarily to earlier claims (VATTIS et al. 2019), and in agreement with HARIDASU et al. (2020) and CLARK et al. (2021a). However, in contrast to CLARK et al. (

- 3 - 2 - 1 0Figure 3 . 8 -

 32138 Figure 3.8 -Reconstructed 2D posterior distributions of a subset of parameters in the ΛDDM and in the νΛCDM models when confronted to BAO + SNIa + Planck data and a prior on S 8 from KiDS-1000+BOSS+2dFLens.

Figure 3 . 9 -

 39 Figure 3.9 -Residuals in the linear matter power spectrum P (k) at redshifts z = 0, 3, with respect to our baseline νΛCDM model, for the best-fit ΛDDM model (red lines) and a νΛCDM scenario yielding the same S 8 (blue lines). The gray band indicates the approximate range of comoving wavenumbers contributing to σ 8 .

Figure 3 . 10 -

 310 Figure 3.10 -Same as in Fig. 3.9, for the (lensed) CMB TT, EE and lensing power spectrum. In this case, the gray bands show Planck 1σ errors.

2 .

 2 A νΛCDM model with three degenerate massive neutrinos of total mass M ν = 0.27 eV 17 , which yields σ 8 0.75 and Ω m 0.31, in agreement with weak lensing data (HEYMANS et al. 2021).

  (JOUDAKI et al. 2020), S 8 = 0.755 +0.019 -0.021 and with the DES-Y1 data only (T. M. C. ABBOTT et al. 2018), S 8 = 0.773 +0.026

7 , 81 - 1 48 -0. 59 S 8 = 6 , 41 - 1 . 8 DES 38 - 1 . 5 S 8 =

 7811485986411838158 Log 10 (Γ/[Gyr -1 ]) = -1.62 +0.KiDS+ Viking + DES -Y1 Log 10 (ε) = -2.23 +0.Log 10 (Γ/[Gyr -1 ]) = -2.15 +0.-only Log 10 (ε) = -2.52 +0.

Figure 3 . 11 -

 311 Figure 3.11 -2D posterior distribution of a subset of parameters from our BAO + SNIa + Planck + S 8 (from KiDS+BOSS+2dFLens) analysis, with and without including the extra A TTTEEE lens

  2020b; XU et al. 2021) for examples).

  50 ≤ ≤ 8000) (HENNING et al. 2018) measurements and the reconstructed gravitational lensing potential (100 ≤ ≤ 8000) (BIANCHINI et al. 2020) from the 500deg SPTpol survey (CHUDAYKIN et al. 2020a). We then include the high-CMB TT, EE and TE (350 ≤ ≤ 4125) data from the DR4 of the ACTPol survey (AIOLA et al. 2020; S. K. CHOI et al. 2020). Finally, we demonstrate that an experiment like CMB-S4 can unambiguously detect the DDM model.

8 Figure 3 . 12 -

 8312 Figure 3.12 -2D posterior distribution of a subset of parameters in our BAO + SNIa + Planck + S 8 (from KiDS+BOSS+2dFLens) analysis when fixing the DR energy fraction to a value motivated by the recent Xenon1T anomaly (ε = 0.05).

Figure 3 . 16 -

 316 Figure 3.16 -2D posterior distribution of a subset of parameters reconstructed from a fit to simulated Planck and CMB-S4 data. The fiducial model has Log 10 (ε) = -2.16 and Log 10 (Γ/[Gyr -1 ]) = -1.74, as denoted by the gray dotted lines.

Figure 3 . 17 -

 317 Figure 3.17 -Growth rate of matter fluctuations for our baseline νΛCDM model (solid line), compared to the best-fit ΛDDM model (dashed line) and to the νΛCDM scenario yielding the same σ 8 and Ω m (dotted line). The observational constraints are taken from AGHANIM et al. (2020b) and references therein.

  3.17). It might also be possible to test the ΛDDM model with current Lyman-α forest flux power spectrum data (WANG et al. 2013; MURGIA et al. 2017; ARCHIDIACONO et al. 2019; ENZI et al. 2021).

  , in the case of decay into dark radiation (BARENBOIM et al. 2021) 1 (for earlier work, see HANNESTAD et al. (2005), BASBOLL et al. (2009), ARCHIDIACONO et al. (2014), and ESCUDERO et al. (2019)).

Figure 4 . 1 - 21 7. 6 ×

 41216 Figure 4.1 -Sum of neutrino masses as a function of the lowest neutrino mass, together with the most recent upper limits (at 95 % C.L.) from cosmology and the estimated KATRIN sensitivity limit. The grey bands represent 5σ uncertainty on the oscillation measurements. Taken from JIMENEZ et al. (2022).

Figure 4 . 3 -

 43 Figure 4.3 -Residuals in the the CMB lensed TT (upper), EE (middle) and lensing (lower) spectrum for a fixed value of the neutrino mass and several decay widths. The residuals are taken with respect to the ΛCDM best-fit parameters from Planck 2018. The ΛCDM parameters are kept fixed in all cases.

Figure 4 . 4 -

 44 Figure 4.4 -Redshift evolution of the quantity (ρ ν + ρdr )/ρ ur (where ρur denotes the energy density of stable massless neutrinos), which should be equal to 1 in the limit of relativistic decays.We consider a very small value of the neutrino mass sum, m ν = 0.06 eV, and several values for the decay width, Log 10 (Γ ν /km/s/Mpc). "approx. PSD" refers to the approximated phase space distribution in Eq. (4.41) while "Full PSD" refers to the exact solutions of Eq. (4.20).

  6 eV) and two different decay widths (Log 10 (Γ ν /[km/s/Mpc]) = 5.5 in Fig. 4.5 and Log 10 (Γ ν /[km/s/Mpc]) = 4 in Fig. 4.6). Fig. 4.5 corresponds to decays happening around the time of the non-relativistic transition, T dec /m ν 0.3, where the effects of the approximations are expected to be largest. Fig. 4.6 on the other hand refers to decays happening deep in the non-relativistic regime, T dec /m ν 0.03. We also show the Planck18 1-σ error bars, as well as the (binned) cosmic variance.

Figure 4 . 5 -

 45 Figure 4.5 -Fractional change in the CMB TT (upper), EE (middle) and lensing (lower) spectrum, when imposing different prescriptions for the background energy density distribution and Boltzmann hierarchies. "approx. PSD" refers to the approximate phase space distribution in Eq. (4.41) while "Full PSD" refers to the exact solution of Eq. (4.20). "C " in the plot means we only keep those collision terms in Eq. (4.28)-Eq. (4.31). The chosen values of the neutrino mass ( m ν = 0.6 eV) and decay width (Log 10 (Γ ν /[km/s/Mpc]) = 5.5) correspond to the case when neutrinos decay close to non-relativistic transition (T dec /m ν 0.3). The gray shaded region indicates Planck18 1-σ uncertainties, while the pink boxes indicate the (binned) cosmic variance.

Figure 4 . 6 -

 46 Figure 4.6 -Same as in Fig. 4.5 , but with a smaller decay width (Log 10 (Γ ν /[km/s/Mpc]) = 4), corresponding to a neutrino decay happening deep in the non-relativistic limit (T dec /m ν 0.03).

Figure 4 . 7 -

 47 Figure 4.7 -2D posterior distribution of the decaying neutrino model reconstructed from the analysis of BAO + FS + Pantheon together with either Planck15 or Planck18 data. In the top panel, we show the correlation with other cosmological parameters.

  /f σ 8 + Pan P15 + BAO/f σ 8 + Pan + τ P18 P18 + BAO/f σ 8 + Pan

Figure 4 . 8 -

 48 Figure 4.8 -Posterior distribution of m ν and Log 10 (Γ ν /[km/s/Mpc]) with Planck18 and Planck15 + a prior on τ reio from Planck18.

  Finally, as discussed in ARCHIDIACONO et al. (2017) and CHACKO et al. (2021), a combination of CMB data with future tomographic measurements of the power spectrum by DESI (FONT-RIBERA et al. 2014) or Euclid (AMENDOLA et al. 2018), and an improved determination of the optical depth to reionization by 21-cm observations with SKA (MAARTENS et al. 2015; LIU et al. 2016), could greatly increase the sensitivity of cosmological probes to neutrino masses and lifetimes.

  ; VAGNOZZI et al. 2017; PALANQUE-DELABROUILLE et al. 2020; DI VALENTINO et al. 2021a), based solely on the fact that the inverted ordering predicts m ν > 0.1 eV, is highly dependent on the hypothesis that neutrinos are stable on cosmological time-scales. Non-relativistic decays can still easily reconcile the inverted ordering with cosmological data.

  2 = χ 2 -χ 2 min enclosing regions of 68.3 %, 95.4 % and 99.5% probability, as a function of the number of parameters in the joint distribution. Some of these ∆χ 2 values are shown in Tab. (B.1).

Table B. 2 -

 2 [START_REF] Lifshitz | Republication of: On the gravitational stability of the expanding universe[END_REF] gives P (M 0 |d) P (M 1 |d) = P (d|M 0 )P (M 0 ) P (d|M 1 )P (M 1 ) ≡ B 01 P (M 0 ) P (M 1 ) , (B.13) Empirical Jeffreys' scale for evaluating the strength of evidence when comparing two models M 0 and M 1 . The probability column refers to the posterior probability of the favored model assuming P (M 0 ) = P (M 1 ) and that the two models cover entirely the model space P (M 0 |d) + P (M 1 |d) = 1.

  Fig. C.1, where we compare the matter power spectra extracted from our simulations with the ones computed with HALOFIT /HMCODE , in three different redshift bins from z = 1.5 to z = 0.5 -given that we have already discussed the z = 0 case in Sect. 2.4.3. In Fig. C.2 we plot the

Figure C. 2 -

 2 Figure C.2 -Here we show the ratio between the non-linear matter power spectra from our simulations and the ones computed with HALOFIT /HMCODE , for both the ΛCDM equivalent and the EDE best-fit models. We have adopted the same linestyle-code and color-code of Fig. C.1.

  Fig. C.1 and Fig. C.2.

130 PFigure C. 3 -

 1303 Figure C.3 -Here we compare departures from the ΛCDM "equivalent" model in terms of ratios of non-linear matter power spectra. The EDE best-fit case is shown in red, while the gray lines refer to the ΛCDM best-fit model. Solid and dotted lines stand for the non-linear power spectra from HALOFIT and HMCODE , respectively. Dot-dashed lines refer to the outputs of our simulations. The cyan shaded band approximately corresponds to the scales probed by DES-Y1.

Figure C. 4 -

 4 Figure C.4 -Here we compare the outputs of our simulations with the HALOFIT /HMCODE predictions, in terms of deviations in the ratios of the EDE best-fit power spectra over the ΛCDM "equivalent" ones, in four redshift bins from z = 2 to z = 0. Solid and dotted lines stand for HALOFIT and HMCODE , respectively.

  3.2.3, we can write c 2 g as

2

  The extra terms proportional to h in Eq. (D.51) arise due to the gauge dependence of θ, θ new = θ syn + αk 2 , with α = (6η + h )/2k 2 . At the level of the phase space variable ∆f wdm,1 , this corresponds to a

  al.(2015), the authors derived an equation for the shear of the dark radiation daughters. It turns out that their equation is formally incorrect3 , but if one places all numerical factors fairly, their equation is identical to Eq. (D.53).

2 3 - 3 -

 33 ω p -ω -ω θ (1 + ω)θ wdm -(1 -ε)Π wdm (D.55)One can convert this into an equation for the sound speed by noting that Π wdm = c 2 s δ wdm , and using the background continuity equation Eq. (D.29). The final result reads:

∞ 0 j 2

 02 (x) x is -1/3, while the correct value is 1/3.

Figure D. 1 -

 1 Figure D.1 -Left panel: The sound speed of the WDM species in the synchronous gauge, as a function of conformal time and wavenumber. Right panel: The ratio between the perturbed densities of the DCDM and the WDM components, as a function of conformal time and wavenumber. The black solid and dashed lines indicate the horizon and free-streaming crossing scales, respectively. We have set ε = 0.007 and Γ -1 = 55 Gyrs.

Figure D. 2 -

 2 Figure D.2 -Schematic view of the implementation of the ΛDDM model in the public Boltzmann solver CLASS . The grey boxes denote the standard modules of CLASS , while the pink boxes indicate which parts of these modules have been modified in order to describe the new species.

  7) from AOYAMA et al. (2014):

3 . 2 . 3 .

 323 The WDM dynamics is now described by Eq. (3.41)-Eq.(3.44) and Eq.(3.46). After trying several values of (kτ ) fluid , we found (kτ ) fluid = 25 to provide the speed yet accurate enough for the purposes of the current analyses. Our implementation is summarized in Fig. D.2. In Fig. D.3 and Fig. D.4 we explicitly compare the novel approximation scheme with the results of the "exact" computation for the WDM species. For the latter, we solve the full Boltzmann hierarchy using 10 4 momentum-bins and max = 17. In Fig. D.3 we show the residuals of the lensed CMB TT and EE power spectra in the WDM fluid approximation, with respect to the full computation, for a grid of parameter values given by Γ/H 0 = 0.1, 1, 10 and ε = 0.5, 0.1, 0.01, 0.001. These values span most of the parameter space in the ΛDDM framework, and for none of them the residuals exceed the Planck 1σ uncertainties, which are indicated by the gray shaded regions, nor the error bars for a cosmic-variance-limited experiment (close to CMB-S4 errors), indicated by the pink shaded regions.

Figure D. 3 -

 3 Figure D.3 -Residuals of the lensed CMB TT power spectrum (upper) and EE power spectrum (lower) in the WDM fluid approximation, with respect to the full hierarchy calculation, for a grid of values covering most of the parameter space: Γ/H 0 = 0.1, 1, 10 and ε = 0.5, 0.1, 0.01, 0.001. The gray shaded regions indicate Planck 1σ errors, while the pink shaded areas indicate cosmic variance up to = 3000.

Figure D. 4 -

 4 Figure D.4 -Residuals of the linear matter power spectrum (at z = 0), with respect to the baseline ΛCDM model, for the same grid of parameter values considered in Fig. D.3, both from the full hierarchy calculation (solid lines) and the WDM fluid approximation(dashed lines).

Figure D. 5 -

 5 Figure D.5 -Comparison between the 2σ exclusion bounds (solid lines) from the Planck analysis of CLARK et al. (2021a) and our Planck+BAO+SNIa analysis. In each case, the dashed line indicates a fit that roughly describes the 2σ limit in the range Γ ∼ 10 -3 -10 -1 Gyrs -1 .

Figure D. 6 -

 6 Figure D.6 -Reconstructed 2D posteriors of a BAO + SNIa + Planck + S 8 (from KiDS+BOSS+2dFLens) analysis, with linear priors and sampling either with the original prior range (upper panel) or within a restricted prior range (lower panel).

8 ,

 8 Figure D.7 -Reconstructed 2D posterior distributions of {S 8 , Ω m , Log 10 (Γ/Gyrs -1 ), Log 10 (ε), m ν /eV} with the neutrino mass fixed to 0.06 eV (blue) or let free to vary (red).

m 2 I

 2 )Θ(E I ), (E.2) δ(f (x)) = i δ(x -x 0 ) |f (x 0 )| , with x 0 such that f(x 0 ) = 0. (E.3)

Figure E. 1 -

 1 Figure E.1 -Posterior distribution of m ν and Log 10 (Γ ν /[km/s/Mpc]) when confronted to Planck 2018 + BAO + FS + Pantheon for two different choices of priors on Γ ν (see legend).

Table 1 . 1 -

 11 List of key events in the history of the universe.

	Event	Pictorial description	t	z	T CMB
	Inflation		?	?	?
	EW transition		20 ps	10 15	100 GeV
	QCD transition		20 µs	10 12	150 MeV
	Neutrino
	decoupling		1 s	6 × 10 9	1 MeV
	electron-positron
	annihilation		6 s	2 × 10 9	500 keV
	Nucleosynthesis		3 min	4 × 10 8	100 keV
	Recombination		380 kyr	1100	0.27 eV
			380 kyr -		7 meV -
	Dark ages		400 Myr	30 -1100	0.27 eV
			400 Myr -		1.7 meV -
	Reionization		1 Gyr	6 -30	7 meV

  in terms of conformal time and a new radial coordinate dχ = dr/ √ 1 -kr 2 , such that ds 2 = a(τ ) 2 -dτ 2 + dχ 2 + r 2 m (χ) dθ 2 + sin 2 (θ)dϕ 2 , (1.29)

  As we will discuss in Sect. 4.2, oscillation experiments tell us that at least two neutrinos need to have mass, whose minimum values correspond to m 2,min 0.01 eV and m 3,min 0.05 eV in the case of normal mass ordering, and to m 2,min ∼ m 1,min 0.05 eV in the case of an inverted mass ordering. Combining this information with Eq. (1.90), we see that at least two neutrinos are necessarily non-relativistic today. In addition, current CMB data places a stringent upper limit on the sum of neutrino masses of neutrinos became non-relativistic after decoupling, z nr,i < z dec . The present contribution of neutrinos to the total energy density can easily be computed by summing over all neutrinos that are non-relativistic today

			4
		180ζ(3)	T ν 3.15 T ν ,	(1.89)
	drops below its mass m ν,i (SHOJI et al. 2010). Solving 3.15 T ν,0 (1 + z nr ) = m ν,i leads to
	1 + z nr,i 1890	m ν,i 1 eV	.		(1.90)

i m i 0.12 eV (AGHANIM et al. 2020b), which by virtue of Eq. (1.90) implies that

  He = 7.1 MeV per nucleon) is much larger than the one of deuterium. We can thus assume that effectively all remaining neutrons at t ∼ t BBN are incorporated into He

			4	. Since
	each He 4	nucleus contains 2 neutrons, n tot He = 1 2 n n (t BBN ) 17 , or equivalently,
		f He ≡	n tot He

T BBN 0.06 MeV -→ t BBN 340 s. (1.111) After t BBN , He 4 is produced almost immediately, given that its binding energy (B

  .[START_REF] Trotta | Bayes in the sky: Bayesian inference and model selection in cosmology[END_REF] where we used Eq. (1.105) to compute X n (t BBN ). Traditionally, this result is expressed as the mass fraction of He

	4	, which is formally defined as
		Y p ≡	ρ tot He ρ b	.	(1.113)
	We can take ρ b ρ tot He + ρ tot H (notice that the abundances of the rest of nuclei produced
	during BBN, like D, He

3 or Li 7 , are too small to have any cosmological impact). In addition, ρ tot H m H n tot H and ρ tot He m He n tot He , where m He 4m H . Hence,

  Both factors ∂f /∂ ni and ∂ ni /∂τ are first order in perturbations, since particles follow straight lines in a homogeneous universe. Thus, we can can discard the last term in Eq. (1.185). For the factor dx i /dτ , we notice that dx i /dτ = (dx i /dσ)(dσ/dτ ) = P i /P 0 . Finally, for dq/dτ , we insert the result of the geodesic equation Eq.(1.182). We get

	dτ	=	∂f ∂τ	+	∂f ∂x i	dx i dτ	+	∂f ∂q	dq dτ	+	∂f ∂ ni	dn i dτ	.	(1.185)
	∂f
	∂τ

  and Eq. (1.216): c 2

		s	c 2 g	0, w	0
	and σ	0. This yields
		δ cdm = -θ cdm + 3φ ,		(1.217)

3 2 H

 2 cs is the Jeans scale. When k > k J , the effective mass term (k 2 -k 2 J )c 2 s becomes positive, and δ I develops oscillations. Hence, we see that the Jeans length is the scale below which pressure prevents gravitational collapse of the fluid. The same scale can be defined for neutrinos, although in that context it is typically referred as free-streaming scale k fs , and c s is referred as the velocity dispersion.

	The baryon sound speed is typically identified with the adiabatic sound speed,
	c 2 s = c 2 g . Since the baryon pressure follows the ideal gas law (c.f. Eq. (1.78)), Pb = nb Tb ,
	the baryon EoS reads w b = Tb µ , where µ is the mean molecular weight
	µ ≡	ρb nb	ρtot H + ρtot He ntot H + ntot He + ne	m H	1 + 4f He 1 + f He + X e	.	(1.225)

  Hubble solution as δ cdm ∝ ag(a, Ω m ), where g(a, Ω m ) is a function capturing the deviations from the linear evolution with a: it is equal to 1 deep in the matter era, and monotonically decreases afterwards. g(a, Ω m ) can be written in terms of a hypergeometric function, but simple fitting formulas in terms of Ω m = 1 -Ω Λ can be found in the literature (e.g. see Eq. (A4) in EISENSTEIN et al.

(1998)

).

  Indeed, they are neutral, stable, have masses in the GeV-TeV range, and interact weakly with the SM. Specific WIMP models arise for example in Supersymmetry (SUSY). This is an extension of the gauge theories that was introduced in the 70s in order to fill some of the gaps of the SM, such as the electro-weak hierarchy problem. It predicts that each particle should have a super-partner with a spin that differs by a half-integer, i.e. all fermions have bosonic super-partners and vice-versa. The lightest of the neutralinos (which are the mass eigenstates of four neutral super-partners, namely, two Higgsinos, the bino and the wino) has all the required properties to be a WIMP candidate. Another good candidate for WIMP DM is the gravitino, the super-partner of the graviton (see[START_REF] Jungman | Supersymmetric dark matter[END_REF] for a review on SUSY DM). Outside the SUSY framework, WIMPs also arise in mod-) and the Universal Extra Dimensions (UED) model[START_REF] Appelquist | Bounds on universal extra dimensions[END_REF]. In this context, particles that are allowed to propagate through all dimensions exhibit excited, more massive states, denominated Kaluza-Klein (KK) states.In UED, all SM particles can propagate in all dimensions, and the lightest KK particle (LKP), if stable and neutral, becomes a perfect WIMP candidate. During the recent years, WIMPs have lost their quasi-monopoly due to the null searches in colliders and direct/indirect detection experiments, and the community is starting to focus on other candidates.

1.5.4), and the fact that they are excellent CDM candidates (B. W. LEE et al. 1977). els of extra-dimensions. These models are based on the seminal work by KALUZA (1921) and KLEIN (1926), who pursued to unify electromagnetism and gravity by postulating the existence of a compact fifth dimension. Modern examples of extra-dimension theories include the Randall-Sundrum model (RANDALL et al. 1999

  -22 eV), making what is known as fuzzy DM (see MARSH (2016) for a review on axion cosmology). These particles arise in the context of the neutrino mass puzzle. Even if active neutrinos are assumed to be massless in the SM, oscillation experiments have provided convincing evidence that at least two of them must posses tiny non-zero masses (see Sect. 4.2). One simple construction to explain the smallness of neutrino masses is given by the type-I seesaw mechanism (DREWES et al. 2017). The main idea is to add to the SM content a right-handed, sterile neutrino, i.e. a fermion which is a singlet for all interactions. Then, one can construct the most general mass matrix that is allowed by gauge invariance, and from its eigenvalues one finds that the mass of the active neutrino is inversely proportional to the mass of the heavy sterile neutrino. Notice that, a priori, the mass of sterile neutrinos can cover a wide range of values. Those with masses of the order M N R ∼ O(keV) have all the necessary properties to constitute cold DM or warm DM, i.e. they are neutral, stable on cosmological scales, and posses DM models don't necessarily have to introduce new particles. Primordial black holes (PBHs) are a class of non-baryonic MACHOs (standing for Massive Astrophysical Compact Halo Object) that are naturally good DM candidates. PBHs are believed to have formed in the very early universe, from rare and extremely large density fluctuations that collapsed into BHs right after entering the Hubble radius (ZEL'DOVICH et al. 1967;[START_REF] Hawking | Gravitationally collapsed objects of very low mass[END_REF][START_REF] Carr | Black holes in the early Universe[END_REF]). The abundance of PBHs is controlled by the amplitude of the primordial spectrum, and should be vanishingly small if the latter were at the level constrained by CMB and LSS observations. Nonetheless, this amplitude remains very unconstrained at small scales, so it could still be possible that PBHs represent a sizable fraction of the DM. There has been recently a renewed interest in PBHs thanks to the latest observations of black hole binary mergers with the LIGO/Virgo gravitational wave detectors (B. P. ABBOTT et al. 2016; BIRD et al. 2016). However, they are subject to many observational constraints, such as those coming from γ-rays, microlensing, the CMB or the Ly-α forest, to name a few examples (see GREEN et al. (2021) for a recent compilation of constraints).

• Sterile neutrinos. very weak interactions with other particles (if the mixing angles are small). Interestingly, sterile neutrinos can decay into an active neutrino and a photon, providing a clean astrophysical signature than can be searched with X-ray observations (see BOYARSKY et al. (2019) for a recent review).

• Primordial black holes.

  R. CALDWELL 2002; R. R.[START_REF] Caldwell | Phantom energy and cosmic doomsday[END_REF]). These theories postulate the existence of one extra scalar field, with a possible non-minimal coupling to gravity. This leads to a vast array possibilities, from the simplest forms of Brans-Dicke theories[START_REF] Brans | Mach's principle and a relativistic theory of gravitation[END_REF] to the most recent higher-derivative theories, where the scalar and the metric dynamics are strongly intertwined. Scalar-tensor theories are constrained by local tests of GR and the equivalence principle, and by cosmological probes (since they typically modify the growth of large-scale structures). Moreover, the recent precise measurement of the gravitational wave speed from the detection of GW170817 has already put under pressure many scalar-tensor theories de-

• Scalar-tensor theories. signed to account for the cosmic acceleration (CREMINELLI et al. 2017). It was recently discovered that scalar-tensor theories can generally be grouped in the Horndeski action, which was first written down in HORNDESKI (1974). This action gives the most general scalar-tensor theory described by second order equations of motions and universally coupled to matter. The Horndeski action covers quintessence, Brans-Dicke, f (R), covariant Galileons as well as many other exotic modifications (see CLIFTON et al. (2012) for a review).

  ± 0.60)km/s/Mpc (AGHANIM et al. 2020b), and the value measured by the SH0ES collaboration using the Cepheid-calibrated cosmic distance ladder, whose latest measurement yields H 0 = (73.04 ± 1.04)km/s/Mpc (RIESS et al. 2021a). Taken at face value, these observations alone result in a 5σ tension. The problem, however, is more severe than the naïve comparison between Planck and SH0ES may suggest. Today, there exist a variety of different techniques for calibrating ΛCDM , and subsequently inferring the value of H 0 , which do not involve Planck data -for example, one can use alternative CMB data sets such as WMAP (C. L. BENNETT et al. 2013), ACT (S. K. CHOI et al. 2020), or SPT (DUTCHER et al. 2021), or one can remove observations of the CMB altogether and combine measurements of Big Bang nucleosynthesis (BBN) with data from baryonic acoustic oscillations (BAO) (AUBOURG et al. 2015; BLOMQVIST et al. 2019; CUCEU et al. 2019; SCHÖNEBERG et al. 2019) or with supernovae constraints (VERDE et al. 2017; BERNAL et al. 2021)

  K. CHOI et al. 2020) with the low-measurements from WMAP (C. L. BENNETT et al. 2013) also yields a high value, S 8 = 0.834 ± 0.016, despite not exhibiting any lensing anomaly. On the side of the low-z probes, there is a host of galaxy weak lensing surveys that have provided accurate measurements of S 8 . The Canada France Hawaii Lensing Survey (CFHTLenS) (HEYMANS et al. 2013) was the first to report a ∼ 2σ tension in S 8 with the initial Planck data release (ADE et al. 2014), which has persisted with subsequent analysis of CFHTLenS data (JOUDAKI et al. 2017a). This motivated the weak lensing community to carry out blinding procedures in their analysis. The first blinded analysis of this type was performed by the Kilo Degree Survey on their first 450 square degrees of data (KiDS-450, HILDEBRANDT et al. 2017), which found a 2.3σ lower value of S 8 compared to Planck 2015 (ADE et al. 2016a). All subsequent published results from weak-lensing measurements have obtained S 8 values which are systematically lower than the high-z estimates. The first year of data from the Dark Energy Survey (DES-Y1, TROXEL et al. 2018) and Hyper Suprime Cam (HSC, HIKAGE et al. 2019) found slightly higher but consistent values with the results of KiDS-450. The improved analysis of KiDS-450 with photometric data from VIKING (KV450) consistently found low S 8 results (HILDEBRANDT et al. 2020). A combination of KV450 and DES-Y1 was performed in ASGARI et al. (2020), finding stronger constraints in good agreement with former results. The latest analysis of KiDS (KiDS-1000, ASGARI et al. 2021) and DES (DES-Y3, SECCO et al. 2022

  al. 2008; AGHANIM et al. 2017; EFSTATHIOU et al. 2019; AGHANIM et al. 2020b). However, this anomalous 'lensing' is not supported by the lensing power spectrum reconstruction, such that it is now commonly admitted that this tension (oscillating between the 2 -3σ statistical level) cannot originate from a true lensing effect. On the other hand, it has been understood that this anomaly can be easily resolved if the universe is closed (DI VALENTINO et al. 2019; EFSTATHIOU et al. 2020; HANDLEY 2021), in certain modified gravity theories (MOSHAFI et al. 2021), or in early-universe scenarios inducing a pattern of primordial oscillatory features (X. CHEN 2010; CHLUBA et al. 2015; SLOSAR et al. 2019; DOMÈNECH et al. 2020). In fact, it has already been noted that this anomaly could be related to the S 8 tension: indeed, once including A lens as an extra free-parameter in the analysis, it has been shown that the reconstructed cosmology has a smaller A s and ω cdm (as well as a higher H 0 ), showing no S 8 tension, but a remnant ∼ 3.5σ Hubble tension (DI VALENTINO et al. 2018a; MOTLOCH et al. 2018, 2020). Interestingly, the cosmology deduced once marginalizing over the lensing information is in better agreement with the recent results from the SPTPol and SPT-3G (HENNING et al. 2018; CHUDAYKIN et al. 2020a; DUTCHER et al. 2021),

  2 , Chapter 3 and Chapter 4, respectively.

	II
	Early dark energy and the H 0
	tension
	"Sometimes science is more art than science, Morty. Lot of people don't get that."
	Rick character in Rick and Morty
	This Chapter is based on:
	R. Murgia, G. F. Abellán, V. Poulin "Early dark energy resolution to the H0 tension in light of weak lensing surveys and lensing anomalies"
	PRD 103 063502 (2021); arXiv:2009.10733
	N. Schöneberg, G. F. Abellán, A. P. Sánchez, S. Witte, V. Poulin, J. Lesgourgues "The H0 Olympics: A fair ranking of proposed models"
	Phys. Rept. 984 1-55 (2022); arXiv:2107.10291

Contents 2.1 Best-fit EDE cosmology solving the H 0 discrepancy . . . . . . . . . . . 2.2 Guidelines to solve the H 0 tension . . . . . . . . . . . . . . . . . . . . . 2.3 Formalism of Early dark energy . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cosmological constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Baseline analysis: anatomy of the 3-parameter EDE model . . 2.4.2 Towards a 1-parameter resolution to the H 0 tension . . . . . . 2.4.3 Confronting EDE to weak-lensing data . . . . . . . . . . . . . . 2.4.4 EDE and the S 8 tension in light of the A lens anomaly . . . . . . 2.5 EDE in the context of other proposed solutions . . . . . . . . . . . . . 2.5.1 Brief overview of competitors . . . . . . . . . . . . . . . . . . . . 2.5.2 Statistical tests to quantify model success . . . . . . . . . . . . 2.5.3 Results of the contest . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  5 ± 0.027 Gyrs (VALCIN et al. 2020, 2021). This issue was discussed in great details in the context of the EDE cosmology, but current data are not accurate enough to play a decisive role in arbitrating the tension (BERNAL et al. 2021; BOYLAN-KOLCHIN et al. 2021; VAGNOZZI et al. 2021b). • Higher values of ω cdm and S 8 . Namely, the Planck18+BAO+SNIa+SH0ES analysis indicates ω cdm = 0.130 ± 0.004 and S 8 = 0.838 ± 0.013. This is related to some "unexpected" effects within CMB power spectra, introduced by the increase in H(z) at early-times. More precisely, the increase in ω cdm needed for EDE is required to match the observed height of the first CMB peak (POULIN et al. 2019; VAGNOZZI 2021). Additionally, it was noted that a similar increase in ω cdm is required to simultaneously preserve the BAO angular scale, because of the different r s -H 0 degeneracy lines for BAO and CMB data (JEDAMZIK et al. 2021). As a consequence of the larger ω cdm , the peak of the matter power spectrum is shifted, and the growth rate of perturbation at late times increases. This leads to a larger value of σ 8 and hence a larger value of S 8 . The last point is the focus of this work. As explained in Sect. 1.7.2, a number of cosmic shear surveys (CFHTLenS (HEYMANS et al. 2013), KiDS/Viking (HILDEBRANDT et al. 2020), DES (T. M. C. ABBOTT et al. 2018), HSC (HIKAGE et al. 2019)) have provided accurate measurements of the cosmological parameter S 8 which are systematically lower than the Pl18-ΛCDM prediction. Since EDE predicts a higher value of S 8 than ΛCDM, taken at face values these experiments pose a challenge to EDE cosmologies, and could exclude these

  This potential is a phenomenological generalization of the well motivated axion-like potential (which can be recovered by setting n = 1) that arise generically in string theory (SVRCEK et al. 2006; DOUGLAS et al. 2007; ARVANITAKI et al. 2010; MARSH 2016). Such a potential may be generated by higher-order instanton corrections (KAPPL et al. 2016), but taken at face values would suffer from strong fine-tuning issues necessary to the cancelling of the lowest orders instantons (if n > 1). Therefore, it should not be interpreted beyond a phenomenological description.

  • PLANCK TTTEEE+φφ : the high-TT,TE,EE, low-TT and EE data from Planck 2018 through the baseline PLIK, COMMANDER and SIMALL likelihoods (AGHANIM et al. 2020a,c), alone and combined with the lensing amplitude reconstruction (SIMCA BAO: the measurements from 6dFGS at z = 0.106 (BEUTLER et al. 2011), SDSS MGS at z = 0.15 (ROSS et al. 2015), and BOSS DR12 data at z = 0.38, 0.51 and 0.61 (ALAM et al. 2017). • FS: the measurements of the growth function f σ 8 (z) (FS) from the CMASS and LOWZ galaxy samples of BOSS DR12 at z = 0.38, 0.51, and 0.61 (ALAM et al. 2017). In practice, we make use of the "consensus" BAO and FS result that combines both in a single likelihood • PANTHEON: the Pantheon SNIa catalogue, spanning redshifts 0.01 < z < 2.3; we marginalize over the nuisance parameter M describing the SNIa calibration (SCOLNIC et al. 2018). • SH0ES: the SH0ES result, modeled with a Gaussian likelihood centered on H 0 = 74.03 ± 1.42 km/s/Mpc (RIESS et al. 2019). We could have alternatively considered a prior on M b as advocated in Sect. 2.2, but this is not expected to affect the results, given that EDE only changes physics at very early times.

likelihood); we make use of a Cholesky decomposition to handle the large number of nuisance parameters (LEWIS 2013). Following the procedure of Planck 2018, we don't consider the low-TE part of the spectra, given its lower signal-to-noise ratio (AGHANIM et al. 2020a).

•

TE,EE only: with

  .3. All relevant χ 2 information is given in App. C.1. PLANCK TTTEEE data only and three free parameters, the EDE model under study is not detected. In agreement with HILL et al. (2020) and NIEDERMANN et al. (2020), we find that the fraction of EDE at z c is limited to 12 f EDE (z c ) < 0.088, while Log 10 (z c ) and Θ i are unconstrained. Interestingly, we also find that the best fit within Planck18 data only has {f EDE (z c ) ∼ 8.5%, Log 10 (z c ) ∼ 3.56, Θ i ∼ 2.8, H 0 ∼ 70.5 km/s/Mpc} and a ∆χ 2 min ≡ χ 2 min (ΛCDM) -χ 2 (EDE) -5 in favor of

	Parameter Planck TT,3-parameter EDE cosmology PLANCK TTTEEE +SH0ES +PLANCK φφ +BAO+PANTHEON H 0 [km/s/Mpc] 68.29(70.49) +0.75 -1.3 71.49(73.05) ± 1.2 71.34(72.41) +1 -1.1 100 ω b 2.252(2.270) +0.019 -0.023 2.284(2.281) +0.022 -0.024 2.282(2.292) +0.021 -0.022 ω cdm 0.1232(0.1278) +0.0019 -0.004 0.13(0.135) +0.0042 -0.004 0.1297(0.1327) +0.0036 -0.0039 10 9 A s 2.116(2.124) +0.035 -0.041 2.153(2.160) +0.036 -0.042 2.152(2.183) +0.031 -0.035 n s 0.9706(0.9829) +0.0058 -0.0087 0.9889(0.9966) +0.0076 -0.0075 0.9878(0.9963) +0.0066 -0.007 τ reio 0.0552(0.0524) +0.0076 -0.0086 0.0586(0.0558) +0.0077 -0.0091 0.0585(0.0633) +0.007 -0.008 f EDE (z c ) < 0.088(0.085) 0.108(0.152) +0.035 -0.028 0.106(0.133) +0.031 -0.028 Log 10 (z c ) 3.705(3.569) +0.37 -0.22 3.612(3.569) +0.13 -0.049 3.615(3.602) +0.11 -0.029 Θ i unconstrained (2.775) 2.604(2.756) +0.33 0.0087 2.722(2.759) +0.17 -0.092 100 θ s 1.04165(1.04371) +0.00039 -0.00034 1.04131(1.04070) +0.00039 -0.0004 1.04143(1.04122) +0.00036 -0.00039 r s (z rec ) 142.8(140.1) +1.9 -0.72 138.8(136.4) +1.7 -1.9 139(137.5) +1.7 -1.7 S 8 0.839(0.834) +0.018 -0.019 0.838(0.842) +0.018 0.838(0.846) ± 0.013 -0.019 Ω m 0.314(0.304) +0.0088 -0.0091 0.3004(0.2969) +0.0079 -0.0084 0.301(0.2980) +0.0051 -0.0055 ∆χ 2 --20.8 -19.1 min (ΛCDM w/ SH0ES) ∆χ 2 min (ΛCDM w/o SH0ES) -4.9 -1.5 -0.02	+FS 71.01(71.96) +1.1 -1 2.28(2.285) +0.021 -0.022 0.1289(0.1323) ± 0.0039 2.144(2.135) +0.032 -0.033 0.9859(0.9895) +0.007 -0.0071 0.0574(0.0528) +0.007 -0.0079 0.097(0.126) +0.035 -0.029 3.61(3.572) +0.13 -0.054 2.557(2.705) +0.37 0.025 1.04145(1.04098) +0.00038 -0.00039 139.4(137.8) +1.7 -1.9 0.837(0.838) ± 0.013 0.3022(0.3009) +0.0053 -0.0054 -18.7 -0.6

  +0.036 -0.031 , Log 10 (z c )= 3.6 +0.14 -0.039 , Θ i = 2.569 +0.36 -0.032 }, with H 0 = 71.4 ± 1 km/s/Mpc. This is in excellent agreement with results from past literature (M.-X. LIN et al. 2019; POULIN et al. 2019; NIEDERMANN et al. 2020; T. L. SMITH et al. 2020). Remarkably, the best fit values of both Θ i and Log 10

Table 2 . 3 -

 23 2.5 we use a different estimator of tension, following RAVERI et al.(2019), that allows to better capture the non-gaussian shape of the posteriors. Mpc 7.5 •10 10 h -1 M LB ΛCDM/EDE 256 3 250 h -1 Mpc 7.5 •10 10 h -1 M LR Summary of the properties of the cosmological simulations used in this work. Notice that the Figures shown in thisSection have been obtained by splicing together (for each redshift and model) the non-linear matter power spectra extracted from the first two simulations listed here, by using the third one to correct for finite-volume and resolution effects (see App. C.2 for details). The labels listed in the last column stand for High Resolution, Large Box, and Low Resolution, respectively. baryon physics on the small-scale clustering of matter, particularly important at very low redshifts. Both HALOFIT and HMCODE have been shown to be suitable to describe the ΛCDM scenario, as well as some common extensions beyond it, such as models with varying DE EoS or massive neutrinos(JOUDAKI et al. 2017b).

	Model	Particles (N )	Box size (L)	Mass resolution Label
	ΛCDM/EDE	1024 3	250 h -1 Mpc	1.2 • 10 9 h -1 M	HR
	ΛCDM/EDE	1024 3	1000 h -1

  We report results of MCMC analysis of ΛCDM and EDE against the KIDS-VIKING data and the joint KIDS-VIKING+DES data in table Tab. (2.4) and Tab. (2.5).

							Planck+BAO/FS+Pantheon
							+KiDS/Viking
							+DES (COSEBIs)
			0.84
	S8	0.80
			0.76
			2.25
	10 -9 As	2.15 2.20
			2.10
			2.05
		0.122
	ωcdm	0.120
		0.118
		0.116
			0.32
	Ωm	0.31
			0.30
			0.29
			67	68	69	0.80	0.85	2.1	2.2	0.117	0.120	0.30	0.32
				H0		S8		10 -9 As		ωcdm	Ωm
	Figure 2.6 -Reconstructed 2D posterior distributions of a subset of parameters for various
	data set combinations (see legend) in the ΛCDM cosmology.
	likelihood on S 8 as inferred from the joint KIDS-VIKING+DES data using Complete
	Orthogonal Sets of E/B-Integrals (COSEBIs), namely 18 S 8 = 0.755 +0.019 -0.021 (ASGARI et al.
	2020).

Table 2 . 4 -

 24 +0.041 -0.038 is ∼ 2.4σ discrepant with that obtained from previous analyses, suggesting a potential discordance between the cosmologies. For comparison, the prediction for S 8 in the ΛCDM model obtained from Planck18 data is 2.3σ discrepant with that from KIDS-VIKING data (HILDEBRANDT et al. 2020). Therefore, although the mean value has increased, the level of the S 8 tension in the EDE cosmology is similar to that in ΛCDM because of larger error bars. Combining The mean (best-fit) ±1σ error of the cosmological parameters reconstructed from the combined analysis of KIDS/Viking with other data. The 'Base' dataset refers to Planck18+BAO/FS+PANTHEON. We also report the χ 2 min for each model and data set combination.

	Model	ΛCDM	EDE
	Parameter	Base+KIDS/VIKING	+SH0ES	Base+KIDS/VIKING		+SH0ES
	H 0 [km/s/Mpc] 100 ω b ω cdm 10 9 A s n s τ reio f EDE (z c ) 100 θ s S 8 Ω m χ 2 min	67.97 ± 0.38 2.248(2.248) ± 0.013 0.1187(0.1188) ± 0.0009 2.10(2.08) ± 0.028 0.9685(0.9667) +0.0038 -0.0036 0.0556(0.0520) +0.0069 -0.0066 -1.04198(1.04165) ± 0.00028 1.04207(1.04210) ± 0.00028 1.04165(1.04181) +0.00035 68.4 ± 0.38 69.75(68.95) +0.99 -1.1 2.257(2.256) ± 0.013 2.261(2.253) +0.014 -0.015 0.1179(0.1180) ± 0.0009 0.1245(0.1235) +0.0028 -0.0039 2.11(2.12) +0.03 -0.032 2.117(2.116) +0.03 -0.033 0.9708(0.9691) +0.0039 -0.0035 0.9778(0.9740) +0.0061 -0.0068 0.0585(0.0589) +0.0074 -0.0075 0.0547(0.0559) +0.0067 -0.0074 -0.058(0.042) +0.028 -0.034 -0.00032 0.8172(0.8137) +0.009 -0.0096 0.8092(0.8094) +0.0091 -0.0098 0.826(0.831) ± 0.011 0.307(0.309) ± 0.005 0.302(0.302) ± 0.005 0.3037(0.3085) +0.0054 -0.0055 3996.82 4011.16 3992.11	71.58(72.22) +1 -0.97 2.277(2.282) +0.013 -0.015 0.1291(0.1310) ± 0.0034 2.136(2.130) +0.03 -0.033 0.9872(0.9907) +0.0064 -0.0055 0.0552(0.0536) +0.0068 -0.0073 0.104(0.122) +0.029 -0.025 1.04146(1.04130) +0.00031 -0.00034 0.829(0.828) +0.012 -0.011 0.2976(0.2962) +0.005 -0.0051 3997.67
	Model	ΛCDM		EDE
	Parameter	Base+KIDS/VIKING/DES	+SH0ES	Base+KIDS/VIKING/DES	+SH0ES
	H 0 [km/s/Mpc] 100 ω b ω cdm 10 9 A s n s τ reio f EDE (z c ) 100 θ s S 8 Ω m χ 2 min	68.16(68.15) ± 0.38 2.251(2.253) ± 0.013 0.1183(0.1183) +0.00084 -0.00082 2.094(2.091) +0.029 -0.03 0.9691(0.9705) ± 0.0037 0.0546(0.0538) +0.0069 -0.0073 -1.04198(1.04195) +0.00028 -0.00029 0.8043(0.8102) +0.0055 -0.0057 0.3045(0.3046) +0.0048 -0.005 3821.93	68.56(68.69) +0.38 -0.39 2.26(2.263) +0.013 -0.014 0.1175(0.1172) +0.00085 -0.00083 2.104(2.115) +0.029 -0.032 0.9712(0.9731) ± 0.0037 0.0576(0.0602) +0.0069 -0.0077 -1.04207(1.04209) +0.00028 -0.00029 0.8039(0.8023) +0.0056 -0.0058 0.2994(0.2978) ± 0.0049 3837.98	69.56(69.55) +0.72 -1.2 2.262(2.270) +0.014 -0.015 0.1223(0.1199) +0.002 -0.0036 3.046(2.107) +0.014 -0.015 0.9765(0.9782) +0.0051 -0.0065 0.05339(0.0559) +0.0071 -0.0072 < 0.094(0.029) 1.04178(1.04190) +0.00032 -0.00031 0.8145(0.8036) +0.0098 -0.01 0.3008(0.2961) +0.0054 -0.0053 3820.46		71.29(71.81) +0.94 -0.94 2.278(2.288) ± 0.014 0.1264(0.1270) +0.003 -0.0032 2.121(2.117) ± 0.031 0.9854(0.9892) +0.0055 -0.0057 0.05441(0.05254) +0.007 -0.0072 0.087(0.097) +0.029 -0.024 1.04157(1.04149) +0.00033 -0.00032 0.817(0.812) +0.01 -0.011 0.2949(0.2919) +0.0047 -0.005 3826.35

Table 2 .

 2

5 -

The mean (best-fit) ±1σ error of the cosmological parameters reconstructed from the combined analysis of the KIDS/Viking/DES data with other data discussed in the text, with and without a prior on H 0 from SH0ES. The 'Base' dataset refers to Planck18+BAO/FS+PANTHEON. We also report the χ 2 min for each model and data set combination.

  2 ∼ +1.6 for 195 data points (HILDEBRANDT et al. 2020)) compared to the ΛCDM fit to the same data set.

					Planck+BAO/FS+Pantheon+KiDS/Viking/DES
					Planck+BAO/FS+Pantheon+SH0ES
					All Data
		74
		72
	H0
		70
		68
		0.88
	S8	0.84
		0.80
		0.31
	Ωm	0.30
		0.29
		0.28
		0.03	0.10	0.17	68 70 72 74	0.80	0.84	0.88	0.285 0.300 0.315
			fEDE(zc)		H0		S8		Ωm
	Figure 2.8 -Reconstructed 2D posterior distributions of a subset of parameters for various
	data set combinations (see legend) in the 1-parameter EDE cosmology. "All data" stands for
	the combination Planck+BAO/FS+Pantheon+SH0ES+KiDS/Viking/DES.

  .8.Without the SH0ES prior, f EDE (z c ) is compatible with 0 at 1σ, and we find an upper limit on f EDE (z c ) < 0.094 at 95% C.L. This constraint is significantly weaker than that derived in HILL et al. (2020), despite the fact that we have reduced the parameter

	Model	ΛCDM		EDE
	Parameter	PLANCK TTTEEE +PLANCK φφ	All Data	PLANCK TTTEEE +PLANCK φφ	All Data
	H 0 [km/s/Mpc]	68.44(68.66) +0.74 -0.72	69.16(69.37) ± 0.41	71.17(72.18) +1.4 -1.6	71.64(72.07) +0.94 -1
	100 ω b	2.262(2.269) ± 0.018	2.277(2.284) ± 0.014	2.284(2.292) ± 0.02	2.292(2.294) ± 0.015
	ω cdm 10 9 A s n s τ reio A φφ lens A TTTEEE lens f EDE (z c ) 100 θ s S 8 Ω m χ 2 min (ΛCDM)	0.1179(0.1174) ± 0.0016 2.069(2.071) +0.038 -0.035 0.9718(0.9730) ± 0.005 0.0494(0.0506) +0.0089 -0.0079 1.071(1.075) +0.04 -0.043 1.195(1.208) +0.066 -0.07 -1.04205(1.04207) ± 0.00031 1.04215(1.04214) ± 0.00029 0.1164(0.1159) ± 0.00087 2.048(2.053) +0.039 -0.032 0.9755(0.9786) ± 0.0037 0.0464(0.0480) +0.0093 -0.0074 1.104(1.110) +0.034 -0.038 1.247(1.266) +0.06 -0.066 -0.800(0.795) +0.019 0.780(0.776) ± 0.011 -0.02 0.302(0.297) +0.009 -0.01 0.2924(0.2883) +0.0049 -0.0051 2765.98 3816.23	0.1253(0.1286) +0.0037 -0.0045 2.101(2.122) ± 0.041 0.9862(0.9925) +0.008 -0.0088 0.0507(0.0529) +0.0087 -0.008 1.064(1.056) +0.04 -0.043 1.187(1.188) +0.065 -0.07 0.078(0.108) +0.035 -0.038 1.04165(1.04343) +0.00036 -0.00035 0.801(0.812) ± 0.02 0.2938(0.2870) +0.0095 -0.01 2761.98	0.1248(0.1255) +0.003 -0.0033 2.064(2.066) +0.047 -0.033 0.9884(0.9925) +0.0059 -0.0056 0.0429(0.0431) +0.012 -0.0071 1.093(1.099) +0.035 -0.039 1.222(1.238) +0.061 -0.067 0.082(0.092) ± 0.027 1.04165(1.04164) ± 0.00034 0.794(0.793) ± 0.013 0.2891(0.2870) ± 0.0052 3808.40
	Table 2.6 -The mean (best-fit) ±1σ error of the cosmological parameters recon-
	structed from the lensing-marginalized Planck18 data only and in combination with
	BAO/FS+PANTHEON+KIDS-VIKING-DES. We also report the χ 2 min for each model and data
	set combination.

  Similarly, to the case of ∆N eff , it is already well known that the CPL model does not resolve the Hubble tension (POULIN et al. 2019; DI VALENTINO et al.

	• Generalized Phenomenological Emergent Dark Energy [GPEDE],
	(YANG et al. 2021b).
	• fraction of CDM decaying into DR [DM → DR],
	(AUDREN et al. 2014; POULIN et al. 2016; NYGAARD et al. 2021).
	• CDM decaying into DR and WDM [DM → DR+WDM].
	This model will be discussed in detail in Chapter 3.
	• Late Dark Energy with Chevallier-Linder-Polarski parametrization [CPL],
	(CHEVALLIER et al. 2001; LINDER 2003).
	• Phenomenological Emergent Dark Energy [PEDE],
	(LI et al. 2019; PAN et al. 2020; YANG et al. 2021a).

  -26 eV representing ∼ 5% of the DM would resolve both tensions (ALLALI et al. 2021). Alternatively, while less pleasing from the Occam's razor point of view, new properties of DM, such as decays (see Chapter 3), cannibalism (HEIMERSHEIM et al. 2020), interactions with DR (BUEN-ABAD et al. 2015; CHACKO et al. 2016), with neutrinos (DI VALENTINO et al. 2018b; MOSBECH et al. 2021) or with DE (DI VALENTINO et al. 2020b; LUCCA 2021), as well as new neutrino properties (KREISCH et al. 2020; DAS et al. 2022) could independently resolve the σ 8 tension, and leave unaffected the (relative) success of the models studied here at resolving the H 0 tension. In this context, it was recently shown that a naïve combination of the EDE model with the DM → DR+WDM decay scenario can indeed address both tensions successfully (CLARK et al. 2021b).

  Observable effects of the Dark Matter decay . . . . . . . . . . . . . . . 3.3.1 Impact on the matter power spectrum . . . . . . . . . . . . . . . 3.3.2 Impact on the CMB anisotropy spectra . . . . . . . . . . . . . . 3.4 Cosmological constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 General constraints: background vs. linear perturbations . . . 3.4.2 Implications for the S 8 tension . . . . . . . . . . . . . . . . . . . 3.4.3 Implications for other cosmic tensions and the Xenon1T anomaly 3.4.4 Detecting DDM in the CMB: impact of current and future data 3.5 Summary and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Decaying dark matter and the S 8
	tension
	"Caminante, son tus huellas el camino y nada más; Caminante, no hay camino, se hace camino al andar. Al andar se hace el camino, y al volver la vista atrás se ve la senda que nunca se ha de volver a pisar. Caminante no hay camino sino estelas en la mar. ."
	Antonio Machado, Proverbios y cantares
	This Chapter is based on:
	G. F. Abellán, R. Murgia, V. Poulin, J. Lavalle "Implications of the S8 tension for decaying dark matter with warm decay products"
	PRD 105 063525 (2022); arXiv:2008.09615
	G. F. Abellán, R. Murgia, V. Poulin "Linear cosmological constraints on two-body decaying dark matter scenarios and the S8 tension"
	PRD 104 123533 (2021); arXiv:2102.12498

Contents 3.1 Unstable dark matter and cosmic tensions . . . . . . . . . . . . . . . . 3.2 Formalism of the two-body Dark Matter decay . . . . . . . . . . . . . . 3.2.1 Background equations . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Linear perturbation equations . . . . . . . . . . . . . . . . . . . 3.2.3 Fluid approximation for the warm component . . . . . . . . . . 3.2.4 Dynamics of perturbations . . . . . . . . . . . . . . . . . . . . . 3.3

  ; KIM et al. 2002), super Weakly Interacting Massive particles (super WIMPs) (COVI et al. 1999; FENG et al. 2003; ALLAHVERDI et al. 2015), sterile neutrinos (ABAZAJIAN et al. 2012) or models with an additional U (1) gauge symmetry (C.-R. CHEN et al. 2009; G. CHOI et al. 2020a,c) 1 . Decays to electromagnetically charged particles are tightly constrained by Planck data (POULIN et al. 2017; SLATYER et al. 2017), γ-ray (CIRELLI et al. 2012; ESSIG et al. 2013) and cosmic-ray searches (JIN et al. 2013; GIESEN et al. 2015), typically requiring Γ -1 > O(10 26 s), with some level of dependence on the decay channel. Still, a purely gravitational constraint, although weaker, is very interesting in the spirit of being 'model-independent', while applying to models with decay to a dark sector, or to (non-interacting) neutrinos. The canonical example is perhaps that of the keV majoron (BEREZINSKY et al. 1993; LATTANZI 2008) decaying into relativistic neutrinos. Models of CDM decays with massive daughters have also been invoked as a potential solution to the observational discrepancies with CDM on small (sub-galactic) scales after structure formation (W. B. LIN et al. 2001; CEMBRANOS et al. 2005; KAPLINGHAT 2005; PETER et al. 2010a,b). Even more recently, decaying dark matter models were proposed (G. CHOI et al. 2020b; XU et al. 2021) as a way to explain the excess of events in the electronic recoils reported by the Xenon1T collaboration (APRILE et al. 2020). In the literature, most studies restricted themselves to massless daughter particles (AUDREN et al. 2014; POULIN et al. 2016; NYGAARD et al. 2021), with the benefit of greatly simplifying the cosmological analysis, but limiting the true 'model-independence' of the bound, and therefore its robustness. Nevertheless, some studies have attempted at including the effect of massive daughters in a cosmological context but either neglected cosmological perturbations of the daughter particles (VATTIS et al. 2019; HARIDASU et al. 2020; CLARK et al. 2021a) or were limited by computational power to perform a complete analysis against a host of cosmological data (AOYAMA et al. 2011, 2014). It has originally been suggested that DM decaying into massless daughters could help with cosmological tensions (BEREZHIANI et al. 2015; ENQVIST et al. 2015), but careful analysis of this scenario in light of Planck 2015 data has excluded this possibility (CHUDAYKIN et al. 2016; POULIN et al. 2016) (although see BRINGMANN et al. (2018) for a different conclusion if the decay rate is not constant). Attempting to go beyond these studies, the authors of VATTIS et al. (2019) suggested that considering a non-zero mass for (at least one of) the decay product would affect the phenomenology and allow for a resolution of the Hubble tension. However, a recent study based on a combination of both BAO and uncalibrated SNIa data-sets has been carried out in HARIDASU et al. (2020). As opposed to VATTIS et al. (

  The system of Eq. (3.28)-Eq.(3.31) is in the same form as the one for massive neutrinos[START_REF] Ma | Cosmological perturbation theory in the synchronous and conformal Newtonian gauges[END_REF], except for the last term in Eq. (3.28) and the fact that the partial derivative ∂ fI /∂q in Eq. (3.28) and Eq. (3.30) is now time-dependent.Given that the DR species satisfies the condition q/E dr = 1, the hierarchy of equations can be simplified by taking the moments7

				F dr, ≡	1 ρ c	0	∞	dq 4πq 2 q∆f dr,	(3.32)
	F dr,1 =	k 3	F dr,0 -	2k 3	F dr,2 ,	(3.34)
	F dr,2 =	2k 5	F dr,1 -	3k 5	F dr,3 +	4 15	r dr (h + 6η ),	(3.35)

of Eq. (3.28)-Eq.

(3.31)

, and integrating over all the momentum degrees of freedom, so that

F dr,0 = -kF dr,1 -2 3 r dr h + r dr δ dcdm ,

(3.33)

  We choose the maximum multipole max to truncate the hierarchies of equations according to the scheme proposed in MA et al.(1995) for both massless and massive neutrinos, i.e. F dr, max = kF dr, max-1 -

	r dr δ dr ,	F dr,1 =	4r dr 3k	θ dr ,		F dr,2 = 2σ dr r dr .	(3.38)
			max + 1 τ	F dr, max ,	(3.39)
	∆f wdm, max =	qk∆f wdm, max-1 E wdm	-	max + 1 τ	∆f wdm, max .	(3.40)

  ; GIUSARMA et al. 2016; PARIMBELLI et al. 2019).

  al. 2013; BRINCKMANN et al. 2019a) code interfaced with our modified version of CLASS , considering various combinations of the following data-sets:

	• The Pantheon SNIa catalogue, spanning redshifts 0.01 < z < 2.3 (SCOLNIC et al.
	2018).
	• The low-CMB TT, EE, and the high-TT, TE, EE data 10 + the gravitational
	lensing potential reconstruction from Planck18.
	• The KIDS-1000+BOSS+2dFLens (HEYMANS et al. 2021), DES-Y1 (T. M. C. ABBOTT
	et al. 2018) and KIDS-1000+Viking+DES-Y1 (JOUDAKI et al. 2020) weak lens-
	ing data, compressed as a a split-normal likelihood, i.e., S 8 = 0.766 +0.02 -0.014 , S 8 = 0.773 +0.026 -0.02 , S 8 = 0.755 +0.019 -0.021 , respectively.
	• The local measurement of the Hubble constant from SH0ES 11 , modelled with a
	Gaussian likelihood centered on H 0 = 74.03 ± 1.42 km/s/Mpc (RIESS et al. 2019).

• The BAO measurements from 6dFGS at z = 0.106 (BEUTLER et al. 2011), SDSS DR7 at z = 0.15 (ROSS et al. 2015), BOSS DR12 at z = 0.38, 0.51 and 0.61 (ALAM et al. 2017), and the joint constraints from eBOSS DR14 Ly-α auto-correlation at z = 2.34 (SAINTE AGATHE et al. 2019) and cross-correlation at z = 2.35 (BLOMQVIST et al. 2019). • The measurements of the growth function f σ 8 (z) (FS) from the CMASS and LOWZ galaxy samples of BOSS DR12 at z = 0.38, 0.51, and 0.61 (ALAM et al. 2017).

  +0.017 -0.011 in the ΛDDM model without the prior information, one

	Model	νΛCDM	ΛDDM
	Parameter	w/o S 8	w/ S 8	w/o S 8	w/ S 8
	100 ω b	2.245(2.242) ± 0.013 2.251(2.253) ± 0.013	2.243(2.244) +0.014 -0.013	2.246(2.241) ± 0.013
	ω cdm or ω ini dcdm	0.1193(0.1194) ± 0.0009 0.1182(0.1184) +0.0009 -0.0008 0.1195(0.1195) ± 0.00095 0.1191(0.1194) +0.0009 -0.001
	H 0 /[km/s/Mpc]	67.55(67.76) +0.46 -0.44	67.85(68.08) +0.47 -0.44	67.71(67.71) ± 0.42	67.92(67.70) +0.43 -0.42
	Ln(10 10 A s )	3.052(3.045) +0.014 -0.016	3.047(3.043) +0.014 -0.015	3.051(3.052) +0.014 -0.015	3.048(3.052) +0.014 -0.016
	n s	0.9676(0.9663) ± 0.0037 0.9697(0.9683) +0.0037 -0.0036 0.9674(0.9672) ± 0.0038 0.9682(0.9673) ± 0.0037
	τ reio	0.058(0.055) +0.007 -0.008	0.0569(0.0549) +0.007 -0.008	0.0576(0.0582) +0.0071 -0.0076	0.0570(0.0582) +0.0071 -0.0077
	M ν /eV	< 0.1395	< 0.1611	-	-
	Log 10 (ε)	-	-	-2.69(-2.97) +0.32 -1.3	-2.28(-2.16) +0.8 -0.78
	Log 10 (Γ/[Gyr -1 ])	-	-	unconstrained (-3.86)	-1.89(-1.74) +0.82 -1.5
	Ω m	0.3127(0.3104) +0.0057 -0.0061	0.3083(0.3061) +0.0056 -0.006	0.3102(0.3109) +0.0056 -0.0058	0.3071(0.3099) +0.0053 -0.0058
	S 8	0.824(0.824) ± 0.011	0.81(0.816) ± 0.01	0.821(0.828) +0.017 -0.011	0.795(0.767) +0.024 -0.016
	χ 2 min	2053.4	2060.5	2053.4	2055.0
	∆logB	0	0	-1.4	-0.81
	Q DMAP tension	2.7σ	1.3σ

Table 3 . 1 -

 31 The mean (best-fit) ±1σ errors of the cosmological parameters from the analysis of Planck, BAO/FS, SN1a data, with and without a split-normal likelihood on S 8 from KiDS-1000+BOSS+2dFLens. For each model and data-set, we also report the best-fit χ 2 , the level of tension estimated through the Q DMAP metric (RAVERI et al. 2019) and the Bayesian evidence.

Table 3 . 2 -

 32 The mean (best-fit) ±1σ errors of the cosmological parameters from our BAO + SNIa + Planck + S 8 (from KiDS+BOSS+2dFLens) analysis performed by marginalizing over the amplitude of the lensing potential A lens . We also report the best-fit χ 2 . 2020). It has been pointed out that this excess could potentially be explained by the elastic interactions of electrons with a fast DM component of mass m

		Parameter	w/ A lens
		100 ω b Ω ini dcdm H 0 /[km/s/Mpc] Ln(10 10 A s ) n s τ reio Log 10 (Γ/[Gyr -1 ])	2.262(2.260) +0.016 -0.015 0.2506(0.2526) +0.0049 -0.0059 68.56(68.38) +0.5 -0.45 3.025(3.032) +0.02 -0.018 0.9725(0.9718) +0.0042 -0.0039 0.0474(0.0506) +0.0098 -0.008 -2.10(-1.49) +0.39 -1.9
		Log 10 (ε) Ω m S 8 A TTTEEE lens A φφ lens χ 2 min	unconstrained (-2.47) 0.2991(0.3009) +0.0053 -0.0066 0.784(0.768) +0.018 -0.014 1.208(1.192) +0.066 -0.064 1.086(1.072) +0.035 -0.041 2043.2
				0.1 MeV
	and velocities 0.05	v/c	0.13 (KANNIKE et al. 2020

Table 3 . 4 -

 34 Figure3.13 -2D posterior distribution of a subset of parameters in the joint BAO + SNIa + Planck + SPTpol analysis, with and without imposing a prior on S 8 from KiDS+BOSS+2dFLens, compared to the ΛCDM scenario. The mean (best-fit) ±1σ errors of the cosmological parameters from our BAO + SNIa + PlanckTT + SPTpol analysis, with and without imposing a split-normal likelihood on S 8 (from KiDS+BOSS+2dFLens). For each data-set, we also report the best-fit χ 2 .

	ΛDDM w/o S 8
	ΛCDM w/o S 8
	ΛDDM w/ S 8
	.75 0.80 0.85	0.30	0.32
	S8	Ωm

  1980; HU et al. 1998) (for reviews see Y. Y. Y. WONG (2011), LATTANZI et al. (2018), and LESGOURGUES et al. (

  , solar neutrinos (BEACOM et al. 2002; JOSHIPURA et al. 2002; BANDYOPADHYAY et al. 2003; BERRYMAN et al. 2015), astrophysical neutrinos measured at IceCube (BAERWALD et al. 2012; PAGLIAROLI et al. 2015; BUSTAMANTE et al. 2017; DENTON et al. 2018; ABDULLAHI et al. 2020; BUSTAMANTE 2020), atmospheric neutrinos and long baseline experiments (GONZALEZ-GARCIA et al. 2008; GOMES et al. 2015; CHOUBEY et al. 2018; AHARMIM et al. 2019

  is now disfavored by a compilation of cosmological data (A. CALDWELL et al. 2017; GERBINO et al. 2017; SIMPSON et al. 2017; VAGNOZZI et al. 2017; DI VALENTINO et al. 2021a; JIMENEZ et al. 2022) (see also SCHWETZ et al. (2017), GARIAZZO et al. (2018), HERGT et al. (2021), and GARIAZZO et al. (2022) for a different take) as well as from Ly-α observations (PALANQUE-DELABROUILLE et al. 2020

					.2,
	we have extrapolated the bound at Log 10	Γν km/s/Mpc	= 0 to Γν = 0, because the constraint on	mν is
	independent of Γν when Γν	H0.

ordering

  )

	F dr,2 =	2k 5	F dr,1 -	3k 5	F dr,3 +	4(h + 6η ) 15	r dr + C 2 ,	(4.30)
	F dr, >2 =	k (2 + 1)	[ F dr, -1 -( + 1)F dr, +1 ] + C ,	(4.31)
	where r dr ≡ a 4 ρdr /ρ c . The terms C appearing in Eq. (4.28)-Eq. (4.31) arise from
	the integrated daughter collision terms in Eq. (4.13)-Eq. (4.14) projected in Leg-
	endre polynomials:

  The measurements of the growth function f σ 8 (z) (FS) from the CMASS and LOWZ galaxy samples of BOSS DR12 at z = 0.38, 0.51, and 0.61 (ALAM et al. 2017).

					Planck 2015+BAO/f σ 8 +Pantheon		Planck 2018+BAO/f σ 8 +Pantheon
	ν /eV	1.0
	m	0.6
	0.2 )
	10 (Γ ν /kms -1 Mpc -1	2 4
	Log	2.20	2.25	0.117 0.120	0.96 0.97		67 68 69	2.0 2.1 2.2 2.3	0.05	0.10
			10 -2 ω b	ω cdm	n s		H 0	10 -9 A s	τ reio
				2.5
							6
			Log 10 (Γ ν /kms -1 Mpc -1 )	1.0 1.5 2.0		Log 10 (Γ ) ν /kms -1 Mpc -1	4 5	Γ ν	=	H	( a n r )
				0.5			3			Planck 2015 + BAO/f σ8 + Pantheon
										Planck 2018 + BAO/f σ8 + Pantheon
				0.1	0.2	0.3		0.2	0.4	0.6	0.8	1.0	1.2
					m ν /eV					m ν /eV
								106 (BEUTLER et al. 2011), SDSS
	DR7 at z = 0.15 (ROSS et al. 2015), BOSS DR12 at z = 0.38, 0.51 and 0.61 (ALAM
	et al. 2017), and the joint constraints from eBOSS DR14 Ly-α auto-correlation at
	z = 2.34 (SAINTE AGATHE et al. 2019) and cross-correlation at z = 2.35 (BLOMQVIST
	et al. 2019).
	• The Pantheon SNIa catalogue, spanning redshifts 0.01 < z < 2.3 (SCOLNIC et al.
	2018).

•

  2 δT 0 i . Contracting each side of the equation with i i k i , we get k 2 (φ + Hψ) = 4πGa 2

	(ρ I + PI )θ I .	(A.86)
	I

  A.91) where we used ρ tot = -3H(ρ tot + Ptot ). Let us try to simplify the previous expression. Using the two Friedmann equations Eq. (A.84) and Eq. (A.88) we can replace 4πGa 2 (ρ tot + Ptot ) by H 2 -H . In the last term on the r.h.s., we can use the Poisson equation Eq. (A.85) in order to replace 3H(φ + Hψ) by -4πGa 2 δρ tot -k 2 φ. Hence:We can now add and substract 4πGa 2 HδP tot + k 2 H 3 (ψ -φ) on the r.h.s.This quantity vanishes for adiabatic modes. In addition, we notice that the first term in brackets at the r.h.s. of Eq. (A.93) vanishes by virtue of the ii Einstein equation Eq. (A.89). We are thus left with

							and
	simplify, to find
	(H 2 -H )R = H φ + H(ψ + 2φ ) + (2H + H 2 )ψ +	k 2 3	(φ -ψ) -4πGa 2 δP tot
	+ 4πGa 2 HδP nad -Hk 2 φ	P tot ρ tot	-	(φ -ψ) 3	,	(A.93)
	where δP nad is the non-adiabatic pressure perturbation
	δP nad ≡ δP tot -	P tot ρ tot	δρ tot .	(A.94)
	(H 2 -H )R = -Hk 2 φ	P tot ρ tot	-	(φ -ψ) 3	.	(A.95)
	This equation tells us that the variation of R is of the order dR dlna ∼ k H	2	φ, meaning
	that the comoving curvature perturbation doesn't evolve on super-Hubble scales.

(H 2 -H )R = (H 2 -H )φ + H (φ + Hψ) + H(φ + H ψ + Hψ ) + H 2 (φ + Hψ) -H P tot ρ tot (k 2 φ + 4πGa 2 δρ tot ). (A

.92)

  What is less obvious is the way we interpret previous formula in order to extract parameter information. Let us suppose that I is a model M we assume to be true (for example, the flat ΛCDM model with gravity described by General Relativity). We now replace B by some hypothesis θ we want to test (here θ = {θ 1 , ..., θ N } can represent the set of N parameters of the underlying model M) and A by the observed data d. In this manner, we can rewrite the Bayes theorem in a more informative way

				B.1)
	Since we have P (A, B|I) = P (B, A|I), one easily arrives at the Bayes theorem
	P (B|A, I) =	P (A|B, I)P (B|I) P (A|I)	.	(B.2)

P (θ|d, M) = P (d|θ, M)P (θ|M) P (d|M) . (B.3)

Here P (θ|M) is the prior, P (θ|d, M) is the posterior, P (d|θ, M) is the likelihood, and P (d|M) is the Bayesian evidence. Let us describe each of these quantities:

• The prior. This represents our state of knowledge before observing the data. It is a key quantity in Bayesian inference, since we can update a posterior probability by using it as a prior on a new application of the Bayes theorem. For example, a prior of the Hubble parameter P (H 0 |ΛCDM) can be just a top-hat function on the interval

[START_REF] Kass | Bayes Factors[END_REF][START_REF] Lesgourgues | Massive neutrinos and cosmology[END_REF]

km s -1 Mpc -1 .

Table B .

 B 1 -∆χ 2 values for the conventionals 1, 2 and 3-σ as a function of the number of parameters in the joint distribution.

  ρ wdm + Pwdm )Θ wdm ≡ As we see from above definitions, it's clear that Θ wdm → θ wdm and Σ wdm → σ wdm in the relativistic limit, and that Θ wdm /θ wdm 1 and Σ wdm /σ wdm 1 in the non-relativist regime. Thus, one way of closing the system of fluid equations is to replace Θ wdm and Σ wdm by the usual θ wdm and σ wdm multiplied by functions depending only on background quantities. Following LESGOURGUES et al. (2011), we write 2 Σ wdm = 3w σ σ wdm , (D.50)

					4πk a 4	0	∞	dqq 2 q 3 2 E 2	∆f wdm,1 ,	(D.48)
	(ρ wdm + Pwdm )Σ wdm ≡	8π 3a 4	0	∞	dqq 2 q 4 2 E 3	∆f wdm,2 .	(D.49)
	Θ wdm +	h 2	w (1 + w)	5 -	p wdm Pwdm			= 3w θ θ wdm +	h 2	,	(D.51)

Table D . 2 -

 D2 Planck high-TT,TE,EE 586.67 587.57 584.82 585.74 Planck low-EE 396.06 396.05 396.92 396.92 Best-fit χ 2 per experiment (and total) in νΛCDM and ΛDDM, with and without a split-normal likelihood on S 8 from KiDS-1000+BOSS+2dFLens.

	Model	νΛCDM	ΛDDM
	Experiment	w/o S 8	w/ S 8	w/o S 8	w/ S 8
	Planck low-TT	23.18	22.66	23.12	23.09
	Planck lensing	8.93	9.60	8.78	9.07
	Pantheon	1026.93 1026.73 1026.94 1026.93
	BAO BOSS low-z	1.23	1.62	1.20	1.21
	BAO FS BOSS DR12	6.51	5.88	6.63	6.95
	eBOSS DR14 Ly-α	4.93	4.68	4.94	4.91
	KIDS1000+BOSS+2dfLenS	-	5.64	-	0.15
	total	2053.4 2060.5 2053.4 2055.0

  The fluid viscous equations are obtained by truncating Eq. (E.12)-Eq. (E.15) at = 2 and integrating over q as discussed in BLAS et al. (2011) and LESGOURGUES et al.(2011). As already mentioned in Chapter 3, the fluid description is only valid at scales deeply inside the Hubble radius, where high-and low-modes are effectively decoupled. The calculations are totally analogous to the ones carried out for the warm dark daughter in App. D.3, except that the decay terms are slightly different. The continuity and Euler equation are:δ ν = -3H(c 2 syn -w ν )δ ν -(1 + w ν ) θ ν + -c 2 g -p ν /(3 Pν ).As usual, we have introduced the adiabatic sound speed, c 2 g ≡ P ν /ρ ν , and the sound speed in the synchronous gauge, c 2 syn ≡ δP ν /δρ ν . For the latter, one can follow the same approximation as for stable massive neutrinos[START_REF] Lesgourgues | The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics[END_REF]

												0 ,	(E.12)
		∆f ν,1 =	qk 3E ν	[∆f ν,0 -2∆f ν,2 ] -	a 2 Γ ν m ν E ν	∆f ν,1 ,	(E.13)
		∆f ν,2 =	qk 5E ν	[2∆f ν,1 -3∆f ν,3 ] -q	∂ fν ∂q	(h + 6η ) 15	-	a 2 Γ ν m ν E ν	∆f ν,2 ,	(E.14)
		∆f ν, >2 =	qk (2 + 1)E h 2	-aΓ ν δ ν 1 -3w δ -	nν m ν ρν	,	(E.16)
	θ ν = -H(1 -3c 2 g )θ ν +	c 2 syn 1 + w ν	k 2 δ g 1 + w ν	nν m ν ρν	,	(E.17)
	while the dynamical equation for the shear stress is written as
	σ ν = -3	1 τ	+ Hξ σ ν +	8w ν c 2 g 1 + w ν	θ ν +	h 2	-aΓ ν σ ν 1 -3ω σ -	1 + c 2 g 1 + w ν	nν m ν ρν	,	(E.18)
	where ξ ≡ 2/3 c 2 syn		c 2 g .	(E.19)
	The adiabatic sound speed can straightforwardly be computed using the fol-
	lowing expression
			c 2 g = w ν 5 -	p ν Pν	+	aΓ ν 3w ν H	n ν m ν ρν	3(1 + w ν ) +	H aΓ ν	ρν nν m ν	-1

ν [ ∆f ν, -1 -( + 1)∆f ν, +1 ] -a 2 Γ ν m ν E ν ∆f ν, . (E.15) ν -k 2 σ ν -aΓ ν θ ν 1 -3w θ -1 + c 2

  In Eq. (E.16)-Eq. (E.18), the quantities w i (for i = δ, θ, σ) are determined through the following expressions:

	4 dq	q 6 E 3 ν	fν ,				n ν =	4π a 3 dq	q 4 ν E 2	fν .	(E.21)
	1 -3ω δ ≡	m ν δn ν δρ ν	=	4π a 4 4π a 4	dqq 2 am ν ∆f ν,0 dqq 2 E ν ∆f ν,0	,	(E.22)
	1 -3ω θ ≡	(ρ ν + Pν )Θ ν (ρ ν + Pν )θ ν	=	4πk a 4 4πk a 4	dqq 3 amν Eν ∆f ν,1 dqq 3 ∆f ν,1	,	(E.23)
	1 -3ω σ ≡	(ρ ν + Pν )Σ ν (ρ ν + Pν )σ ν	=		8π 3a 4 8π 3a 4	dqq 4 amν E 2 ν dq q 4 Eν ∆f ν,2 ∆f ν,2

  réduisant l'espace des paramètres à un modèle EDE à 1 paramètre (1pEDE) en fixant Log 10 (z c ) et Θ i à leurs meilleures valeurs d'ajustement tel qu'obtenu à partir d'une analyse de Planck18 données uniquement -qui coïncide de manière frappante avec celles de l'analyse combinée avec SH0ES -conduit à une préférence ∼ 2σ pour l'EDE non nul, à savoir f EDE (z c ) 0.08 ± 0.04 à partir des données Planck18 CMB seules. Dans cette cosmologie, le H 0 70 ± 1.5 km/s/Mpc inféré est en accord à mieux que 2σ avec sa mesure locale de SH0ES. L'ajout des données BAO, FS et PANTHEON n'a pas d'impact significatif sur le résultat. 3. Pour justifier l'inclusion des données LSS dans nos analyses, nous avons confronté le spectre de puissance de la matière non linéaire EDE tel que prédit par des algorithmes semi-analytiques standard à un ensemble dédié de simulations N -corps. Nous avons ensuite testé la cosmologie 1pEDE par rapport aux données WL, constatant qu'elle n'aggrave pas de manière significative l'ajustement aux mesures S 8 par rapport à ΛCDM, et que les observations WL actuelles n'excluent pas la résolution EDE à la tension de Hubble. 4. Nous mettons également en garde contre l'interprétation des contraintes obtenues en combinant Planck18 avec KIDS-VIKING+DES. Comme nous l'avons montré, la cosmologie de « compromis » obtenue est un mauvais ajustement à KIDS-VIKING+DES et dégrade l'ajustement aux données de Planck18 , même en ΛCDM. Cela illustre que ces ensembles de données sont statistiquement incohérents dans un cadre ΛCDM, et il est facilement concevable que la résolution de cette tension se situe ailleurs (qu'il s'agisse d'un effet systématique ou d'une nouvelle physique). 5. À la lumière de l'anomalie A lens du CMB, nous avons montré que les données CMB marginalisées par A lens favorisent l'EDE non nul à ∼ 2σ, prédisent H 0 en accord 1, 4σ avec SH0ES et S 8 en 1.5σ et 0.8σ concordent respectivement avec KIDS-VIKING et DES . Il existe cependant toujours une tension ∼ 2.5σ avec les résultats conjoints de KIDS-VIKING et DES. De plus, la présence d'EDE n'affecte pas la quantité de lentilles anormales.

The existence of other galaxies beyond the Milky Way had not yet been established in 1914, so they were actually known by the name of extra-galactic nebulae.

This residual amount of electrons corresponds to an initial excess of matter over anti-matter, whose origin is still not understood. Models of baryogenesis try to predict the matter-antimatter asymmetry, and rely on physics beyond the SM.

http://class-code.net/.

We will use a dot . for derivatives with respect to cosmic time t, and a prime for derivatives with respect to conformal time τ .

This definition is consistent with the 4-momentum that is typically introduced for particles with non-zero mass m, P µ ≡ m dx µ dσ , since one can always make a parameter redefinition σ = σ /m.

After hydrogen recombination, the only non-relativistic SM particles with a non-negligible cosmic abundance are neutral hydrogen, neutral helium, and some residual amount of free electrons and protons. They form a single fluid, whose energy density is dominated by the baryons. For this reason, cosmologists often refer to the electron-baryon fluid simply as baryons.

The acceleration of the universe actually started a bit before dark energy domination. Indeed, from Eq. (1.16) we see that ä = 0 implies zacc = (2Ω Λ /Ωm) 1/3 -1 0.6.

Notice that Eq.(1.35) includes in a compact way also the flat and spherical geometries, as can be seen by using the mathematical identities limx→0 sin(ax)/x = a and sinh(ix) = i sin(x), respectively.

BAO observations also measure a feature in the galaxy correlation parallel to the line of sight, in which case they constrain the combination ∆z = rs(z drag )H(z obs ), see Fig.1.6.

The term recombination is misleading: this is the first time electrons and nuclei combined.

We take B = 0 since we can always use the gauge freedom to do so, as we will see promptly.

This set of equations can also be applied to describe any kind of collisionless warm dark matter component.

This is still true for imperfect fluids that were initially tightly coupled to other species, such as photons. Indeed, we already saw that in the tightly coupled limit, all photon multipoles above = 0 and = 1 are negligible. The initial conditions for neutrinos are a bit more subtle and we don't discuss them here, we refer to LESGOURGUES et al. (2018) for details about this topic.

The reason for dividing λ by 2 is that, in harmonic space, θ is the angle between a maximum and a minimum, while in Fourier space the distance between a maximum and a minimum is λ/2.

The SW contribution to C TT can actually be computed analytically assuming large (super-Hubble) scales instead of small ones, and considering also a scale invariant primordial spectrum, ns = 1. From

Let us briefly motivate the choice of displaying k in units of h/Mpc. The evolution of sub-Hubble perturbations depends primarily on how deep inside the Hubble radius a given mode is, i.e. on the ratio

Note however, that the amount of information that can be extracted tends to saturate at large kmax, because, deeply in the non-linear regime, the non-linear matter spectrum gradually loses memory of initial conditions, i.e. of early linear perturbations.

The robustness of such probes have been investigated for example in BERNAL et al. (2020) and CAR TER et al. (2020).

However, there is some debate about the robustness and independence of these additional measurements, see e.g. FREEDMAN (2021).

As we'll explain later, these phenomenological parameters can be mapped into the theory parameters entering into the scalar field potential. They correspond roughly to an ultra-light axion with a mass m ∼ 10 -27 eV and a decay constant f ∼ M pl . The latter value is in stark contrast with the expectation from the Weak Gravity Conjecture (HILL et al.

2020), but we remark that the EDE potential considered here is just a way to capture the phenomenology that is required to explain the H0 tension.

The re-analysis of BOSS data in the EFT of LSS has been performed in T. L. SMITH et al. (2021).

The joint analysis of KIDS1000+BOSS+2dfLenS has determined S8 = 0.766 +0.020 -0.014[START_REF] Heymans | KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints[END_REF]) in 3σ tension with Planck. Making use of these data would not affect our conclusions.

It is a fully cosmology-independent approach. Indeed, we saw earlier that what the SH0ES team directly measures is M b , and not H 0 . Focusing on M b allows to derive H 0 from the supernovae self-consistently within the given exotic expansion history. This point is effectively irrelevant for early-Universe solutions[START_REF] Klein | Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English)[END_REF] , but might be important for late-Universe solutions. Nevertheless, DHAWAN et al. (2020) explored several parametric forms of the expansion history at late times, and concluded that the distance ladder values of H 0 inferred from these expansion histories agreed to within about 0.5 km/s/Mpc. Hence, a SH0ES-like determination of H 0 is expected to be quite robust under late-time changes in the model. Still, using a M b prior is very important for the reason given below.

2.It highlights the inconsistency between the SH0ES-calibrated Pantheon data and the ΛCDMcalibrated BAO data. In fact, one could further split the late-Universe solutions into two different categories. The first type considers smooth modifications of the expansion history at late times. This is the case of the decaying dark matter scenario that we will discuss in

Chapter 3. For this kind of solutions, the inconsistency between SNIa and BAO data shown in Fig.2.1 already prevents them from predicting a high value of H 0 , even when a SH0ES prior is imposed. The second type of late-time solutions generically consider sharp transitions in the dark energy equation of state at very low z (BENEVENTO et al. 2020). Such kind of solutions can in principle predict a high value of H 0 , even when both BAO and SNIa data are considered, giving the impression that they have resolved the tension after imposing a SH0ES H 0 prior. However, they are not resolving the true origin of the tension, namely, the inconsistency between the distance ladder and[START_REF] Friedmann | Über die Krümmung des Raumes[END_REF] This is sometimes referred as the the "inverse distance ladder".[START_REF] Klein | Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English)[END_REF] In fact, all our EDE analysis presented in Sect. 2.4 consider a SH0ES prior on H0.

One could have alternatively considered a logarithmic prior on f EDE , but this would have given more weight to very small values of f EDE , which are uninteresting in the context of finding a solution to the Hubble tension.

https://github.com/brinckmann/montepython_public

Most chains are in fact converged at the R -1 ∼ 0.01 level, this somewhat 'loose' but reasonable criterion was only used once including KIDS-VIKING data, which are much longer to converge.

https://iminuit.readthedocs.io/

Hereinafter, we quote 1-sided constraints at 95%C.L., and two-sided ones at 68%C.L.

To guide the reader, we mention that a 1σ shift in the quality of the fit to Planck18 data roughly corresponds to a ∆χ 2 of ∼ 6 (see the distribution of Planck18's χ 2 in the tables available at this link).

Here, let us mention that IVANOV et al. (2020) make the comment that such degeneracy does not exist. This is of course only true because they include the 3σ discrepant S8 data to their analysis. The degeneracy is very clear within Planck18 data.

Let us mention that HEROLD et al. (2022) recently derived new constraints on EDE with the profile likelihood method, which does not suffer from volume effects. Using Planck CMB and BOSS full-shape data, they obtain f EDE = 0.072 ± 0.036, which is extremely similar to the results of the 1-parameter EDE analysis of T. L. SMITH et al. (2021) using the same data-sets.

Current constraints on the sum of neutrino masses set mν < 0.12 eV (Planck18+lensing+BAO), meaning that neutrinos become non-relativistic after recombination. Taken at face value, this would exclude the proposal of SAKSTEIN et al. (2020), but to confirm this one should perform a full analysis

EARLY DARK ENERGY AND THE H 0 TENSION

An alternative, more thorough, way is to use CMB lensing principal components as introduced in MOTLOCH et al. (2018, 2020). As we will show shortly, our reconstructed 'unlensed' cosmologies are in good agreement. Our approach follows that introduced in SIMARD et al. (2018) and WU et al. (2019).

We treat SH0ES not as a model-dependent measurement of H0, but as a direct measurement of the intrinsic magnitude of supernovae Ia (SNIa), M b = -19.2435 ± 0.0373 (RIESS et al. 2021b). As discussed in Sect. 2.2, this allows to take into account the correlation between the information contained in SH0ES and Pantheon catalog of remote SNIa.

The most notable difference is that one is based on average properties of the posterior distribution, as opposed to the χ 2 of a single point, and the ability to capture the effect of long tails in non-Gaussian posterior distributions.

None of our models show a significant preference in AIC over ΛCDM when only considering the baseline D baseline

We remark that this analysis corresponds to an extended dataset D extended = D baseline + Redshift-Space-Distortions (from BOSS DR12 (ALAM et al. 2017), 6dFGS (BEUTLER et al. 2011), MGS (ROSS et al. 2015) and eBOSS DR 14 quasars (ZARROUK et al. 2018)) + Cosmic Chronometers (VAGNOZZI et al. 2021a) + Lyman-α based high-z BAO (BLOMQVIST et al. 2019; SAINTE AGATHE et al. 2019).

Recently, the authors of G. CHOI et al. (2022) engineered a model in the context of supergravity that explicitely reproduces the kind of late

2-body decays considered here, and also provides a natural explanation for the small mass splitting that seems to be favoured by cosmic data.

This expression is easily derived using energy-momentum conservation. In the rest frame of the DCDM, we have E dcdm = m dcdm = E wdm + E dr , with E wdm = p 2 wdm + m 2 dcdm and E dr = p dr = p wdm . Solving for p wdm yields p wdm = (m dcdm /2)(1 -m 2 wdm /m 2 dcdm ) = m dcdm ε = m wdm ε/ √ 1 -2ε.

The initial abundance of DCDM, Ω ini dcdm , is related to the present DCDM abundance by Ω 0 dcdm ≡ Ω ini dcdm e -Γt 0 , with t0 the age of the universe.

This expression does not coincide with the EoS used in BLACKADDER et al. (2014), which regarded it as an average squared velocity.

One can arrive at this expression simply by taking the limit ε 1 in Eq.(3.22) and then performing the integral analytically noticing that daq/Hq = dtq.

Even if the momentum degrees of freedom are integrated out, we can not write equations in terms of temperature multipoles as we did for photons in Sect. 1.6.2, since the DR species are never in thermal equilibrium.

By looking at Eq. (3.44) and Eq. (3.45), we see that for small values of ε, the cut-off approximately satisfies the scalings k fs ∝ c -1 g ∝ w -1/2 ∝ ε -1 .

Note that since now we are fixing

100θs instead of H0, the Hubble rate H(z) can increase with respect to ΛCDM at z 1, once dark energy starts to dominate. However, this effect is small for long lifetimes, and at early times the Hubble rate is still smaller than in ΛCDM.

In our main analysis, we use the 'lite' version of the CLIK likelihood. We have verified that this leads to negligible differences with respect to the full likelihood.

A new version of the SH0ES measurement (RIESS et al. 2021a) was published during completion of this work. We do not expect it to have any impact on our conclusions.

The results of the full BAO+SNIa+Planck analysis are reported in the third column of Tab.(3.1)

This is not equivalent to directly reading the constraints on Γ at ε = 0.5 from the Γ vs. ε contours, since the 95% C.L. derived from a χ 2 distribution with different degrees of freedom correspond to different ∆χ 2 .

Let us point out that, assuming the ∆χ 2 is χ 2 -distributed with 2 degrees of freedom, the ΛDDM model is favored at 93.3% (∼ 2σ) over ΛCDM (in the combined analysis). This indicates that part of the 'inconclusive' evidence is driven by our choice of wide priors and that different choices can affect the Bayesian evidence.

We adjust ω cdm = 0.1154 whereas all other parameters are fixed to the baseline νΛCDM model

The differences between the ΛDDM and νΛCDM models that we have highlighted in Fig.3.9 concern the shape of the power suppression with respect to a reference model adjusted to Planck data. This doesn't mean that similar differences will appear between these models when looking directly at the corresponding matter power spectra adjusted to KiDS data.

Note that this combined analysis includes a photo-metric redshift correction applied to DES result, slightly lowering the S8 value compared to what is advocated by the DES collaboration (T. M. C. ABBOTT et al.

2018).

To overcome the lack of low-data in the CMB-S4 analysis, we have imposed a Gaussian prior on the optical depth to reionization, centered on its best-fit value from our reference Planck+BAO+SNIa+KiDS+BOSS+2dFLens analysis, namely τreio = 0.0582 ± 0.008.

More concretely, this work considered a scenario in which a heavier neutrino decays relativistically into a lighter neutrino and a massless scalar, νH → ν l + φ, and derived bounds assuming mν l = 0. Recently, J. Z. CHEN et al. (2022) updated those bounds by accounting for the experimentally determined mass splittings between the parent and the daughter neutrino, ∆m

ν . They showed that, for a parent neutrino with mν H 0.1 eV, the constraints on its lifetime weaken by a factor 50 and 10 5 , if ∆m 2 ν corresponds to the atmospheric and solar mass gap, respectively.[START_REF] Slipher | Spectrographic Observations of Nebulae[END_REF] This is a factor of two weaker than the constraints advocated in LORENZ et al.(2021), which used a model-independent approach to constrain the neutrino mass as a function of redshift, but neglected the effect of the daughter particles.

The optimal momentum sampling of CLASS typically involves a Gauss-Laguerre quadrature, which is expected to give good results for PSD which are close to a Fermi-Dirac, like the one considered in Eq. (4.20). On the contrary, the PSD of the warm daughter particles considered in Chapter 3 has a very different shape from a Fermi-Dirac, so that's why the optimal sampling was not used in that case.

We note that the same Boltzmann hierarchy has been derived in the context of warm dark matter decaying into dark radiation (BLINOV et al. 2020a).

To shorten notation, we will omit the dependence of the PSD fi on µ, τ, k, and write only the dependency on momenta.

The background continuity equation, ρ ν = -3H(1 + wν )ρν , could be derived in a very similar way by integration of the background Boltzmann equation, ∂ f ∂τ = 0, over

4πa -4 dqq 2 E.

The 68.3 % and 95.4 % confidence interval are often referred as the 1-σ and

2-σ values, because the nomenclature agrees for gaussian distributions. However, this is an abuse of notation, since in most cases the posterior distribution do not have a gaussian shape.[START_REF] Slipher | Spectrographic Observations of Nebulae[END_REF] It's important to note that the likelihood is gaussian in the data, but in general it is not gaussian in the parameters, due to the non-linear relations between the observable and the parameters.

Typically the covariance matrix is scaled up by a a certain factor, whose optimal value has been shown to be approximately 2.4.

We omit the label I in P 0 and P i to shorten notation.

We forget about the issue of the gauge dependence of ∆f wdm,0 and ∆f wdm,1 , since we will be restricting to sub-Hubble scales later on.

This procedure corresponds to the case in which Ω ini dcdm is passed as an input parameter. Alternatively, the user can pass as input the present abundance of DCDM, WDM and DR, Ω dcdm+dr+wdm , in which case a shooting method is also performed in order to find the corresponding Ω ini dcdm .

[START_REF] Klein | Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English)[END_REF] Note that the number of bins used at both the background and perturbation level is much larger than the one typically used in standard CLASS analysis for massive active neutrinos, given that the time-dependence of the background PSD of the WDM requires a finer momentum resolution. Regarding the momentum spacing, we have considered a logarithmic Simpson quadrature instead of the Gauss-Laguerre quadrature typically used in standard CLASS analysis.

IV

New cosmological limits on the neutrino mass and lifetime "When I heard the learn'd astronomer, When the proofs, the figures, were ranged in columns before me, When I was shown the charts and diagrams, to add, divide, and measure them, When I sitting heard the astronomer where he lectured with much applause in the lecture-room, How soon unaccountable I became tired and sick, Till rising and gliding out I wander'd off by myself, In the mystical moist night-air, and from time to time, Look'd up in perfect silence at the stars."

Walt Whitman, Leaves of Grass

Contents

• N. Schöneberg, G. F. Abellán, A. P. Sánchez, S. Witte, V. Poulin, J. Lesgourgues "The H 0 Olympics: A fair ranking of proposed models" Phys. Rept. 984 1-55 (2022); arXiv:2107.10291.

• G. F. Abellán, Z. Chacko, A. Dev, P. Du, V. Poulin, Y. Tsai "Improved cosmological constraints on the neutrino mass and lifetime" JHEP 08 076 (2022); arXiv:2112.13862.

Besides the aforementioned articles, I have also been working on a few other projects, which are not included in this thesis, but have ended up in the following publications:

• T. Smith In conclusion, current cosmological data carry a wealth of information that is still far from being fully exploited. It is an exciting time for cosmology, since the upcoming arrival of new and more precise data will allow to test models suggested to resolve cosmic tensions (such as the ones studied in this work), and to possibly establish a new concordance model. I hope that my research work has been useful in that direction.

Appendices

A

Complements on linear perturbation theory

In this appendix we provide some important calculations regarding the theory of linear cosmological perturbations, that were a bit too long to be included in Chapter 1. We always assume a spatially flat background metric.

A.1 The four velocity in a generic gauge

The goal is to compute the four-velocity U µ = Ū µ + δU µ at linear order in a generic gauge. We remind that metric perturbations are generically written as

For a comoving observer, Ū µ = a -1 δ µ 0 (the a -1 factor appears because we are working in conformal time), which means Ūµ = ḡµν Ū ν = -aδ 0 µ . Let us now obtain δU µ and δU µ . Using that ḡµν Ū µ Ū ν = g µν U µ U ν = -1, we get at linear order

from which we can infer that δU 0 = -a -1 A. Writing δU i = a -1 v i , where v i is the coordinate velocity, we arrive at

A.2 Gauge transformations

We want to prove the following gauge transformation for the 0i metric perturbation

We simply have to substitute the coordinate transformation Eq. (1.152) and the perturbed metric elements Eq. (A.1) in the transformation law for tensors Eq. (1.153).

and not on the past ones.

The main idea is to explore the parameter space in some intelligent manner, that is, to cover only the regions which are close enough to the best-fit values. For doing so, the MCMC starts from sone initial point θ (t) in the N -dimensional parameter space, and evaluates its likelihood, L(θ (t) ). Then a new point θ (new) is proposed in a random way (but using a specific probability distribution), and the likelihood is computed again. Depending on the value of L(θ (new) ), a specific algorithm must decide whether this new point should be accepted or rejected.

B.2.1 Metropolis-Hastings algorithm

A commonly used MCMC algorithm is the Metropolis-Hastings (MH) algorithm, defined by the following iterative steps:

• We propose a random point in the parameter space around the last point in a chain, drawn from a proposal probability distribution Q(θ (new) |θ (t) ).

• We compute the likelihood at this point, L(θ (new) ), and the ratio r = L(θ (new) )/L(θ (t) ).

• If r ≥ 1, we accept and move to the new point in the chain, θ (t+1) = θ (new) . Otherwise, we select a random number α from a uniform probability distribution U [0, 1]:

-If α ≥ r, we reject the new point and stay where we were: θ (t+1) = θ (t) .

-If α < r, we accept and move to the new point, θ (t+1) = θ (new) .

Therefore, the MH algorithm does not accept new points only when the likelihood is higher, in which case it would reach a maximum too rapidly. The algorithm also accepts points with smaller likelihood, at a smaller rate, so that it can explore the parameter space and trace the underlying posterior distribution. To be more precise, we end generating the weights of each point in the parameter space, i.e., the number of times we waited and did not move. If the number of steps N steps performed in the chain is big enough (the order of 10 4 -10 6 ), then these weights are proportional to the posterior distribution that we are looking for. Obtaining marginal 1D distribution of some parameter θ i is also very easy, we simply have to plot the histogram with the values of θ i versus the corresponding weights. From these histograms, quantities such as the mean, best-fit and 1-σ values (the first two do not necessarily agree for asymmetrical distributions) are straightforward to compute. Notice that the MH algorithm does not provide information about the bayesian evidence in Eq. (B.11).

The idea of the algorithm is pretty simple, but there are questions that remain. What is the best starting point for the chain? And which is a good choice for the proposal distribution? A good strategy is the following: for the starting point, use an estimate of the best-fit value, and for a proposal distribution, use a multi-variate gaussian (centered in the current point), with an estimate of the covariance matrix 3 . These estimates can be pre-computed with a short initial run. Using good initial values and a good proposal distribution is important in order to make sure the C

Complements on Early Dark Energy

In this appendix we provide some complements on Chapter 2, concerning the χ 2 information and the details about our N -body simulations.

C.1 χ 2 Tables

We report all χ

C.2 A closer look to N -body simulations

The main systematical uncertainties in numerical simulations come from their limited box size and resolution, as it has been thoroughly discussed in past literature (see, e.g. 2019)). In order to minimize the missing large-scale modes, potentially affecting small-box simulations, and to overcome the impossibility of capturing the very non-linear scales in our large-box simulations, we adopted a splicing technique to bind together the matter power spectra extracted from simulations with different resolutions, for each redshift and model, as in MCDONALD ( 2003

the daughter particles. Close to the best-fit parameter values, and in general close to ΛCDM, the residual errors between the full and the fluid calculations in P (k) are O(1%), but they can become higher far away from the best-fit. In particular, we have verified that inside the parameter region delimited by Log 10 ε ∈ [-2.3, -0.7] and Log 10 (Γ/Gyrs -1 ) ∈ [-1.3, 1], the residual errors are typically larger than 10%, so the fluid approximation should be used with caution in this region. However, this portion of the parameter space is deeply inside the 2σ exclusion region, as one can check by looking at Fig. 3.7. In addition, given that current data are mostly sensitive to integrals over P (k) (e.g. S 8 , CMB lensing), we are mainly interested in getting accurate predictions for the departures from ΛCDM (rather than the exact shape of the matter spectrum itself). To illustrate that, we have computed the residuals of the linear matter power spectrum (at z = 0) with respect to our baseline ΛCDM model, for both the fluid and the full hierarchy calculations.