


Hybrid Inflation: Multifield Dynamics and Cosmological Constraints 
Sébastien Clesse
Thèse présentée en vue de l’obtention du titre de
Docteur en Sciences
Juin 2011
Promoteurs:  Professeur C. Ringeval (UCL, Louvain) 
Professeur M. Tytgat (ULB, Bruxelles)  
Jury:  Professeur A.C. Davis (Cambridge University) 
Professeur T. Hambye (ULB, Bruxelles)  
Professeur J. Martin (IAP, Paris)  
Professeur S. Van Eck (ULB, Bruxelles) 



Hybrid Inflation: Multifield Dynamics and Cosmological Constraints 
Sébastien Clesse
Thèse présentée en vue de l’obtention du titre de
Docteur en Sciences
Juin 2011
Promoteurs:  Professeur C. Ringeval (UCL, Louvain) 
Professeur M. Tytgat (ULB, Bruxelles)  
Jury:  Professeur A.C. Davis (Cambridge University) 
Professeur T. Hambye (ULB, Bruxelles)  
Professeur J. Martin (IAP, Paris)  
Professeur S. Van Eck (ULB, Bruxelles) 
Abstract: Hybrid models of inflation are particularly interesting and well motivated, since easily embedded in various high energy frameworks like supersymmetry/supergravity, Grand Unified Theories or extradimensional theories. If the original hybrid model is often considered as disfavored, because it generically predicts a blue spectrum of scalar perturbations, realistic hybrid models can be in agreement with CMB observations. The dynamics of hybrid models is usually approximated by the evolution of a scalar field slowly rolling along a nearly flat valley. Inflation ends with a waterfall phase, due to a tachyonic instability. This final phase is usually assumed to be nearly instantaneous.
In this thesis, we go beyond these approximations and analyze the exact 2field nonlinear dynamics of hybrid models. Several non trivial effects are put in evidence: 1) the possible violation of the slowroll conditions along the valley induce the non existence of inflation at small field values. Provided superplanckian fields, the scalar spectrum of the original model is red, in agreement with CMB observations, independently of the position of the critical instability point. 2) Contrary to what was thought, the initial field values leading to inflation are not finetuned along the valley but also occupy a considerable part of the field space exterior to it. They form a complex connected structure with fractal boundaries that is the basin of attraction of the valley. Using bayesian methods, their distribution in the whole parameter space, including initial velocities, is studied. Natural bounds on the potential parameters are derived. 3) For the original model, after the field evolution along the valley, inflation continues for more than 60 efolds along the waterfall trajectories in some part of the parameter space. Observable predictions are modified, and the scalar power spectrum of adiabatic perturbations is generically red, possibly in agreement with CMB observations. Moreover, topological defects are conveniently stretched outside the observable Universe. 4) The analysis of the initial conditions is extended to the case of a closed Universe, in which the initial singularity is replaced by a classical bounce. Contrary to some other scenarios, due to the attractor nature of the valley, the field values in the contracting phase do not need to be extremely finetuned to generate a bounce followed by a phase of hybrid inflation.
In the third part of the thesis, we study how the present CMB constraints on the cosmological parameters could be ameliorated with the observation of the 21cm cosmic background from the dark ages and the reionization, by the future generation of giant radiotelescope. Assuming ideal foreground removals, forecasts on the cosmological parameters are determined for a characteristic Fast Fourier Transform Telescope experiment, by using both Fisher matrix and MCMC methods.
Une page ne serait pas suffisante si je devais nommer tous ceux qui, de près ou de loin, ont contribué à la réussite de ces quatre années de doctorat. Je citerai tout de même mes collaborateurs directs, avec qui j’ai eu l’honneur et le très grand plaisir de travailler: Dr. Jonathan Rocher, Dr. Marc Lilley, Dr. Larissa Lorenz, Dr. Laura H. Lopez et Dr. H. Tashiro. Mes remerciements les plus sincères vont aussi naturellement à tous les membres du Service de Physique Théorique de l’ULB et du Centre de Cosmologie, de Physique des Particules et de Phénoménologie de l’UCL, en particulier les doctorants et postdoctorants, pour leur accueil, pour les échanges d’idées, et pour l’ambiance toujours chaleureuse et amicale.
D’autre part, j’ai été particulièrement touché par la grande disponibilité de personnalités de renom pour répondre à mes questions parfois naïves. Je pense en particulier aux discussions fructueuses entretenues avec Juan GarciaBellido, David Lyth, Patrick Peter, Jerôme Martin et André Fuzfa.
J’ai l’immense privilège de compter comme membres du jury AnneChristine Davis, Sophie Van Eck, Jerôme martin et Thomas Hambye. Je les remercie pour leur présence et pour l’intérêt qu’ils ont porté à mes travaux.
Ces quatre années de doctorat ont été financées par le Fond pour la Recherche dans l’Industrie et l’Agriculture (FRIA). Je remercie cet organisme et particulièrement les membres des commissions scientifiques, pour la confiance accordée à mon projet de recherche en Cosmologie.
Enfin, je suis extrêmement reconnaissant du soutien et de la confiance que mes promoteurs, Michel Tytgat et Christophe Ringeval, m’ont accordés. Toujours disponibles, ouverts à de nouvelles idées, ils sont sans conteste les éléments décisifs qui ont conduit à l’accomplissement de mes travaux et à la rédaction de cette thèse. Plus particulièrement, je dois à Christophe l’apprentissage de la rigueur scientifique, mes compétences numériques et la découverte des indices de style romain. Je profite aussi de ces quelques lignes pour lui adresser mes plus plates excuses pour avoir été si souvent un “imbitable goret”. Michel et sa patience inestimable m’ont appris l’intuition physique et les formalismes liés à la cosmologie 21cm. Tous deux ont réussi a tirer le meilleur de moi, tout en me laissant la liberté nécessaire à mon épanouissmeent. Pour toutes ces raisons, et pour bien plus encore, je vous adresse, Michel et Christophe, un immense et chaleureux Merci!
Mais le plus important à mes yeux est le soutien inconditionnel d’Alice et de ma famille. Je remercie plus particulièrement Alice pour avoir supporté mon stress et mes sautes d’humeur durant la phase de rédaction de cette thèse.
Merci à tous ceux que j’oublie, qui se reconnaîtront…
“L’essentiel est invisible”,
Antoine de SaintExupéry
“Ce n’est pas parce qu’on fait des choses sérieuses
qu’il faut le faire en tirant une gueule d’enterrement!”
Alain Moussiaux
Table of contents
 Abbreviation List
 Introduction and motivations

I General context

1 Standard cosmological model
 1.1 Introduction
 1.2 The homogeneous FLRW model
 1.3 Matter content of the Universe
 1.4 Thermodynamics in an expanding spacetime
 1.5 Thermal history of the Universe
 1.6 Precision observational cosmology

1.7 Unresolved problems
 1.7.1 Nature of dark matter
 1.7.2 Nature of dark energy
 1.7.3 Horizon problem
 1.7.4 Flatness problem
 1.7.5 Problem of topological defects
 1.7.6 Why is the primordial power spectrum scaleinvariant?
 1.7.7 Contribution of isocurvature modes
 1.7.8 Why are the perturbations Gaussian?
 1.7.9 Initial singularity
 2 The inflationary paradigm

1 Standard cosmological model

II Multifield dynamics of hybrid inflationary models
 3 Hybrid models of inflation
 4 Slowroll violations in hybrid inflation

5 Initial field and natural parameter values in hybrid inflation
 5.1 Introduction
 5.2 Exact twofield dynamics and initial conditions
 5.3 Initial conditions for extended models of hybrid inflation
 5.4 Fractal properties of subplanckian initial field values
 5.5 Probability distributions in hybrid inflation
 5.6 Probability distributions in FSUGRA inflation
 5.7 Conclusion and discussion
 6 The waterfall phase
 7 Hybrid inflation in a classical bounce scenario
 III 21cm Forecasts
 A Fast Fourier Transform Telescope
 B Fisher matrix formalism
 C MCMC Bayesian methods
Abbreviation List
2DF  Two degree field 
ACBAR  Arcminute Cosmology Bolometer Array Receiver 
BAO  Baryonic Acoustic Oscillations 
BBN  Big Bang Nucleosynthesis 
CDM  Cold Dark Matter 
CDMS  Cryogenic Dark Matter Search 
CMB  Cosmic Microwave Background 
DM  Dark Matter 
dof  degree of freedom 
e.o.m.  equation of motion 
FFTT  Fast Fourier Transform Telescope 
FL  FriedmannLemaître 
FLRW  FriedmannLemaîtreRobertsonWalker 
GR  General Relativity 
HI  neutral hydrogen 
i.c.  Initial Condition 
IGM  Inter Galactic Medium 
KG  KleinGordon 
LHC  Large Hadron Collider 
LOFAR  LOw Frequency ARray 
LSS  Large Scale Structures 
MCMC  MonteCarloMarkovChain 
MWA  Murchison Widefield Array 
PNGB  Pseudo Nambu Goldstone Boson 
QUaD  Q and U Extragalactic Submm Telescope at DASI 
SDSS  Sloan Digital Sky Survey 
SM  Standard Model 
SKA  Square Kilometre Array 
SUGRA  Supergravity 
SUSY  Supersymmetry 
vev  Vacuum Expectation Value 
WMAP  Wilkinson Microwave Anisotropy Probe 
WIMP  Weakly Interacting Massive Particle 
Introduction and motivations
Since the 1990’s and the COBE experiment, the cosmology has entered into an era of high precision. Measurements of the anisotropies in the Cosmic Microwave Background (CMB) have become increasingly accurate. Combined with the observations of the large scale structures and the typeIa supernovae, they have permitted to measure and constrain with accuracy the parameters of the standard cosmological model. The main contributions to the energy density of the Universe today are the dark energy (71%) and the dark matter (23%). But their nature and origin still have to be understood.
Moreover, for the cosmological model to be in accord with observations, the inhomogeneities at the origin of the galaxies are required to follow precise statistical properties in the early Universe. These can be obtained in a natural way if a phase of quasiexponentially accelerated expansion is assumed to occur in the early stages of the Universe’s evolution. Such a phase of inflation can be realized if the Universe is filled with one or more scalar fields slowly rolling along their potential. However, such scalar fields cannot be integrated in the standard model (SM) of particle physics. A major challenge is thus to identify these fields in a theory beyond the standard model.
Among the zoo of inflationary models, the hybrid class is particularly interesting and motivated since such models are easily embedded in various high energy frameworks like supersymmetry/supergravity (SUSY/SUGRA) [Halyo:1996pp, Binetruy:1996xj, Dvali:1994ms, Kallosh:2003ux], Grand Unified Theories (GUT) [Jeannerot:1997is, Jeannerot:2003qv] or extradimensional theories (see e.g. Refs. [Dvali:1998pa, Brax:2007xq, Koyama:2003yc, Fukuyama:2008dv, Linde:2005dd, ArkaniHamed:2003wu]). It is therefore important to determine correctly and accurately their dynamics and their observable signatures.
In hybrid models, inflation takes place in the false vacuum along a nearly flat valley of the scalar field potential. The accelerated expansion ends due to a Higgstype tachyonic instability that forces the field trajectories to deviate and to reach one of the global minima of the potential during the socalled waterfall phase. A phase of tachyonic preheating [Felder:2000hj, Felder:2001kt] is triggered during the waterfall when the mass of the transverse field becomes larger than the Hubble expansion rate. Topological defects like cosmic strings can be formed when the initial symmetry is broken. They can affect the primordial power spectrum of curvature perturbations.
In the usual way to study hybrid models, it is assumed that inflation is realized when the fields are slowly evolving in the valley along an effective 1field potential. It is also assumed that inflation ends instantaneously once the trajectories have reached the critical instability point. In that 1field slowroll description, the observable quantities like the amplitude of the primordial power spectrum of curvature perturbations, its spectral index, and the ratio between tensor and scalar metric perturbations can be calculated after a Taylor expansion of the power spectra about a pivot length scale.
For the original hybrid model [Linde:1993cn, Copeland:1994vg], the predictions on the spectral index are such that it is disfavored by CMB experiments. For its supersymmetric (SUSY) version, the Fterm hybrid model [Dvali:1994ms], the flat valley along which inflation takes place is lifted up by radiative and supergravity corrections [Coleman:1973jx, Senoguz:2003zw, Jeannerot:2005mc, Jeannerot:2006jj]. The spectral index can be in accord with observations, but the model is in tension with the data [Battye:2010hg]. Many other hybridtype models have been proposed, from various high energy frameworks, and their 1field slowroll predictions have been determined. They are more or less in agreement with observations.
In the first part of this thesis, the standard cosmological model, the inflationary paradigm and the present status of observations are explained. In the second part, we analyze the exact 2field classical dynamics of hybrid models and put in evidence several non trivial effects.

Slowroll violations (chapter 4): For the original hybrid model, the slowroll conditions can be violated during the field evolution along the valley. Determining the effects of these slowroll violations require the integration of the exact field dynamics. We show that such violations induce the non existence of inflation at small field values. This modifies the primordial power spectrum of curvature perturbations that can be in agreement with CMB observations, provided superplanckian initial field values. These results are independent of the position of the critical instability point.

Set of initial field values (chapter 5): In a flat Universe, for generating more than 60 efolds of accelerated expansion, the 2field trajectories were usually required to be initially finetuned in a very narrow band along the inflationary valley or in some subdominant isolated points outside it. From a more precise investigation of the dynamics, we have shown with C. Ringeval and J. Rocher [Clesse:2008pf, Clesse:2009ur, Clesse:2009zd, Clesse:2010ht] that original hybrid inflation does not suffer from any finetuning problem, even when the fields are restricted to be subplanckian. Because of the attractor nature of the inflationary valley, a nonnegligible part of the field trajectories initially exterior to the valley reach the slowroll regime along it, after some oscillations around the bottom of the potential. We show that the set of successful initial field values is connected, of dimension two and possesses a fractal boundary of infinite length exploring the whole field space.
The relative area covered by successful initial field values depends on the potential parameters. Therefore, a MonteCarloMarkovChain (MCMC) bayesian analysis is performed on the whole parameter space consisting of the initial field values, field velocities and potential parameters. For each of these parameters, we give the marginalized posterior probability distributions such that inflation is long enough to solve the standard cosmological problems. It is found that inflation is realized more probably by field trajectories starting outside the valley. Natural bounds on potential parameters are also deduced.
Finally, the genericity of our results are confirmed for 5 other hybrid models from various framework, namely the SUSY/SUGRA Fterm, smooth and shifted hybrid models, as well as the radion assisted gauge inflation model.

Inflation along waterfall trajectories (chapter 6): For the original hybrid model, the exact integration of the classical 2field trajectories reveals that inflation can continue for more than 60 efolds after crossing the critical instability point[Clesse:2010iz]. We first check that the classical dynamics is not spoiled by quantum backreactions of the fields. Then, by performing a MCMC analysis of the parameter space, we show that inflation along the waterfall trajectories lasts for more than efolds in a large part of this space. When this occurs, the predictions on the spectral index are modified, and the primordial power spectrum of curvature perturbations is possibly in agreement with the CMB constraints. Moreover, the topological defects formed when the initial symmetry is broken at the critical instability point are diluted by the subsequent phase of inflation along the waterfall trajectories. They become therefore nonobservable.

Classical bounce plus hybrid inflation (chapter 7): With M. Lilley and L. Lorenz, we have extended the analysis of the chapter 5 to the case of a closed Universe, for which the initial singularity is replaced by a classical bounce. Contrary to previously proposed scenarios, we show that the initial conditions in the contracting phase do not need to be extremely finetuned for hybrid inflation to be triggered after the bounce, provided that spatial curvature was initially sufficiently large.
Currently the best constraints on inflationary models come from the observations of the CMB temperature anisotropies. In the future, a major challenge will consist in improving CMB measurements and in observing new cosmological signals.
In the third part of the thesis, we are interested in one of these promising signals: the 21cm cosmic background. This could be used to probe the dark ages and the reionization epochs. The 21cm cosmic background is induced by the transitions between the hyperfine ground states of the neutral hydrogen (HI) atoms. The signal corresponds to a stimulated emission or an absorption of 21cm CMB photons. Compared to CMB, the 21cm signal is in principle observable over a wide range redshifts (). The observation of its anisotropies is expected to improve in the future the accuracy of the cosmological parameter measurements.
With L. H. Lopez, C. Ringeval, H. Tashiro and M. Tytgat, we focus on a concept of 21cm dedicated giant radiotelescope, the Fast Fourier Transform Telescope (FFTT), and analyze its ability to put significant constraints on the cosmological parameters. Our first motivation was to determine forecasts directly on the parameters of some inflation models, including hybrid ones, as well as on the reheating energy scale. This objective is on his way and we give here a particular attention to the forecasts on the spectral index of the primordial power spectrum of curvature perturbations. More specifically, we compare the interest of observing a 21cm signal from the dark ages and from the reionization. We show that the observation of the 21cm signal from the dark ages should only contribute to put significant constraints on the spectral index for idealistic configurations of the FFTT experiment. For the signal from the reionization, we obtain forecasts similar to those of Ref. [Mao:2008ug].
The thesis is organized as follows: In the first part, the general context is introduced and explained. In chapter 1, we introduce the standard cosmological model, its observational confirmations and the current bounds on its parameters, as well as several problems and unanswered questions rising in this context. In chapter 2, we explain how some of these problems can be solved naturally if one assumes a phase of inflation in the early Universe’s evolution. The homogeneous dynamics of 1field inflationary models is described and the slowroll approximation is introduced. By using the linear theory of cosmological perturbations, observable predictions are derived in the slowroll approximation. Then, the dynamics of multifield inflation models is described and we explain how to calculate the exact primordial power spectrum of scalar and tensor perturbations in this context. Finally, the theory of the reheating after inflation is introduced.
In the second part, we study the exact multifield dynamics of some hybrid models. In chapter 3, these models are introduced and motivated. In chapter 4, the effects of slowroll violations during the field evolution along the valley are determined and discussed for the original hybrid model. In chapter 5 we study, for a flat Universe, the set of the initial conditions leading to a sufficient amount of inflation, for all the hybrid models we have considered. Chapter 6 is dedicated to the end of inflation in the original hybrid model. In particular, we show that in a large part of the parameter space, inflation only ends after more than 60 efolds of expansion are realized along the waterfall trajectories. In chapter 7, we study the case of a closed Universe, in which the initial singularity is replaced by a classical bounce.
The third part of the thesis is dedicated to 21cm forecasts. In chapter 8, the theory of the 21cm cosmic background from the dark ages and the reionization is introduced. In chapter 9, we analyze the ability of a typical FFTT radiotelescope to detect the 21cm power spectrum. For two configurations of the experiment and ideal foreground removal, we calculate the forecasts on the cosmological parameters.
The perspectives to this work are discussed in the conclusion. In annexes, the bayesian MCMC methods and the Fisher matrix formalism are described. The concepts of the FFTT radiotelescope are given and its advantages over the standard interferometers are explained.
Part I General context
Chapter 1 Standard cosmological model
1.1 Introduction
The standard cosmological scenario accurately describes the evolution and the structure of the Universe. It relies on three major hypothesis:

The gravitational interaction obeys to the General Relativity (GR) theory.

At very large scales, the Universe can be considered as isotropic and homogeneous

The Universe is of trivial topology
A dynamical cosmological model based on these assumptions was first proposed independently by Alexander Friedmann [Friedmann1922, Friedmann1924] and Georges Lemaître [Lemaitre1927], respectively in 1922 and 1927. In 1929, Hubble first measured the expansion rate of the Universe [Hubble1929], by interpreting the observation of redshifted spectral lines for the nearest galaxies [Slipher1915]. Related to the expansion, the idea that the Universe was born in a BigBang, from an extremely dense initial state, has emerged and is today a cornerstone of the standard cosmological model. This scenario is today confirmed by the observation of the Cosmic Microwave Background (CMB), relic of the period when free electrons recombined to atomic nuclei. It relies also on the measurements of light element abundances. These have been formed during a phase of primordial nucleosynthesis in the early Universe.
In the next section the equations governing the spacetime expansion are given. In section 1.3 we will apply these equations to various types of fluid filling the Universe. We will find the corresponding expansion laws and energy density evolutions. A brief description of the thermal history of the Universe will be given in section 1.5. Section 1.6 is dedicated to the observations of astrophysical and cosmological signals that have permitted to measure precisely the various cosmological parameters. Some unresolved questions rising from this cosmological scenario will be developed in section 1.7.
1.2 The homogeneous FLRW model
In comoving spherical coordinates , imposing the hypothesis of isotropy and homogeneity leads to the FriedmannLemaîtreRobertsonWalker (FLRW) metric^{1}^{1}1The metric signature is used and we work in the natural system of units . Greek indices go from 0 to 3. Latin indices go from 1 to 3. The Einstein convention of summing repeated indices is used.
(1.1) 
where is the cosmic time, is the scale factor and is the spatial curvature normalized to unity, such that if the Universe is flat, if it is closed and if it is open. The cosmological dynamics is given by the Einstein equations^{2}^{2}2 is the Ricci tensor, is the Riemann tensor and is the Ricci scalar. is the stressenergy tensor. In the comoving frame of an isotropic and homogeneous Universe, it is diagonal. For a perfect fluid, the (0,0) component is the energy density and (i,i) components are the pressure . is the Planck mass. The reduced Planck mass will be denoted . is a possible cosmological constant.
(1.2) 
applied to the FLRW metric. This gives the FriedmannLemaître (FL) equations
(1.3)  
(1.4) 
in which a dot denotes the derivative with respect to the cosmic time and where the Hubble expansion rate has been introduced. Furthermore, the conservation of the energymomentum tensor () leads to
(1.5) 
which is not independent since it can be derived also from Eq. (1.3) and Eq. (1.4).
1.3 Matter content of the Universe
The expansion dynamics depends on the characteristics of the fluid(s) filling the Universe. One can define the equation of state parameter as
(1.6) 
From the energymomentum tensor conservation equation Eq. (1.5) one concludes that the energy density of a perfect fluid characterized by a constant behaves like
(1.7) 
Combined with Eq. (1.3), if and , it is straightforward to show that the scale factor evolves like
(1.8) 
It is also useful to define the redshift, corresponding to the spectral shift of photon wavelengths due to the expansion,
(1.9) 
The dynamics and the evolution of the energy density for some typical fluids are given below:

Cosmological constant, : the cosmological constant acts like a perfect fluid whose energy density is constant, that is . If the Universe is filled with such a fluid, the Hubble expansion term is constant and the scale factor grows exponentially, .

Curvaturelike fluid, : if , the curvature term in the FL equations is equivalent to a perfect fluid with . The energy density goes like and the scale factor grows linearly with the cosmic time, .

Pressureless matter dominated Universe, : the energy density decreases like , because of the volume growth of any comoving region. The scale factor grows like . A nonrelativistic baryonic fluid belongs to this class.

Radiation dominated Universe, : the energy density decreases like . Compared to the nonrelativistic matter case, the additional factor can be viewed as the decrease of a photon energy whose wavelength increases due to the expansion. The scale factor evolves like . Relativistic species (like relativistic massive neutrinos), belong to this class.
As long as and , the scale factor reaches in a finite past while the energy density tends to infinity [PeterUzan]. All known kinds of matter verify this bound on the equation of state parameter. If the Universe is closed and if the curvature term was dominant in the past, the singularity is replaced by a bounce [Starobinsky:1980]. A model belonging to this specific case will be described and discussed in chapter 7.
In the standard cosmological scenario, the socalled CDM model, the Universe has been successively dominated by radiation, pressureless matter and finally by cosmological constant, or identically a fluid with . It is usual to introduce the notion of critical energy density , corresponding to the energy density of a flat Universe,
(1.10) 
For each fluid filling the Universe, one can define the ratio of its energy density to the critical density today,
(1.11) 
Let us define also and . Then the adimensional first FL equation reads
(1.12) 
In the CDM model, the species contributing to the energy density today are the cosmological constant (), a pressureless matter component (), the socalled cold dark matter (CDM), the nonrelativistic baryonic matter (), the photons () and the relativistic neutrinos (). One may also consider a possible curvature term (). The Hubble expansion rate therefore evolves like
(1.13) 
where and are respectively the value of the Hubble parameter and the scale factor today. The six cosmological parameters describing completely the homogeneous evolution of the CDM model are therefore^{3}^{3}3Instead of , it is usual to consider as a cosmological parameter the number of relativistic species , , , and or today, as well as . These parameters have been measured by observations, as explained later in section 1.6.
In a realistic scenario, the universe is only nearly isotropic and homogeneous. The theory of cosmological perturbations [Mukhanov:1990me] permits to describe how density and metric perturbations are growing at the linear level. Therefore, some additional cosmological parameters are required to provide initial conditions for the perturbative quantities.
1.4 Thermodynamics in an expanding spacetime
Let us assume that the Universe is filled with several cosmological fluids. The FermiDirac or BoseEinstein distribution function for the species in kinetic equilibrium can be used to define the species temperature ,
(1.14) 
where is the momentum, is the degeneracy factor, is the chemical potential and where is the particle mass. Other macroscopic quantities such as the particle number density, the energy density and the pressure are defined from this distribution function,
(1.15)  
(1.16)  
(1.17) 
Let us consider several fluids in thermal equilibrium. On one hand, the pressure variation with respect to the temperature reads
(1.18) 
On the other hand, the energymomentum tensor conservation can be rewritten
(1.19) 
Then, let us define a quantity as
(1.20) 
By combining the last two equations, one obtains that satisfies to
(1.21) 
For a constant number of particles in each species, one sees that is conserved. For a relativistic fluid, one has and . One recognizes in the entropy.
In the standard CDM model, the early Universe was dominated by interacting relativistic species. Because their interaction rates were much larger than the expansion rate , each species had the same temperature. As the Universe expands, the interaction rates can become lower than the expansion rate. In such a case, the corresponding species decouples from the other fluids and becomes a relic.
1.5 Thermal history of the Universe
1.5.1 Early Universe
At temperatures above 10 MeV ()^{4}^{4}4In this section, the ratios corresponding to the given temperatures are evaluated for the best fit values of the CDM parameters. These are given in section 1.6.6., the standard model of particle physics predicts that the universe was filled with a mixture of photons, neutrinos, relativistic electrons and positrons, as well as nonrelativistic protons and neutrons. In the CDM model, an additional nonrelativistic component is assumed, the cold dark matter, that can be considered as interacting only gravitationally with the others species. Due to the weak and electromagnetic interactions,
(1.22)  
(1.23)  
(1.24)  
(1.25) 
these are in thermal equilibrium.
Below 1 MeV (), neutrinos decouple and stop to interact with the other species. A cosmic background of neutrinos is therefore expected.
Below 511 keV (), electronpositron pairs annihilate into photons and thus the photon fluid is reheated compared to the neutrino fluid.
When the temperature decreases below 0.1 MeV (), the photon energy becomes insufficient to photodissociate the eventually formed atomic nuclei. Therefore light elements (deuterium, tritium, helium and lithium) can be formed [Alpher:1948, Gamow:1948, Alpher:1948b, Peebles:1966]. This phase is called the primordial nucleosynthesis, or BigBang nucleosynthesis (BBN). The resulting abundances in light elements depend on the baryon to photon ratio and on the number of relativistic species (for a recent review, see [Mukhanov:2003xs]).
The present light element abundances in the Universe can be evaluated with astrophysical observations such that strong constraints have been established on the state of the Universe at the primordial nucleosynthesis epoch. This will be explained in more details in the next section on observations.
1.5.2 Matter dominated era
Since radiation and relativistic matter energy densities decrease more slowly than the nonrelativistic matter energy density, the Universe undergoes a transition from a radiation dominated era to a matter dominated era. From Eq. (1.13), if we neglect the cosmological constant, this happens when
(1.26) 
The time of matterradiation equality thus depends on the energy density of neutrinos, itself depending on the effective number of neutrino species . For the best fit of the CDM parameters, the matterradiation equality occurs at a redshift [Komatsu:2010fb].
1.5.3 Recombination and cosmic microwave background
When the photon energy goes below the binding energy of hydrogen atoms, eV, free electrons can start to bind with protons and helium nuclei without being ionized anymore. The mean free path of photons increases suddenly and becomes so large that they can propagate until today. These photons can be observed as a black body, whose temperature was actually well below K. Indeed, since the number density of photons was much larger than the number density of electrons, (this ratio can be determined with the BBN theory and the observation of light element abundances), the recombination occurs only when the number density of photon with an energy is smaller than . The corresponding temperature of the photon black body distribution is about . Due to the expansion by a factor , the photon black body spectrum is observed today with a temperature . For an observer on Earth, it corresponds in the sky to an isotropic microwave radiation. This was discovered accidently by Penzias and Wilson in 1964 [Penzias:1965]. This is the socalled Cosmic Microwave Background (CMB).
Let us study more in details the recombination process. The neutrality of the Universe imposes that . The free electron fraction is defined as . Neglecting the Helium fraction, one has . As long as the interaction
(1.27) 
permits to maintain the equilibrium, one has from Eq. (1.15) (in the limit )
(1.28) 
and the free electron fraction is given by the Saha equation
(1.29) 
where eV. Because , when , the right hand side is of the order of and the free electron fraction remains . The recombination therefore only occurs at .
At late time, the equilibrium is not maintained anymore and the Saha equation is not accurate. To determine the free electron fraction, one needs to solve the Boltzmann equation for . A good approximation^{5}^{5}5For more accurate results, the recombination can calculated numerically by using the RECFAST code [website:recfast], taking account for additional effects like Helium recombination and 3level atom. is given in Refs. [Seager:1999bc, Seager:1999km, Dodelson],
(1.30) 
where
(1.31) 
is the ionization rate, and where
(1.32) 
is the recombination rate. The superscript indicates that recombination at the ground state is not relevant. Indeed, this process leads to the production of a photon that ionizes immediately another neutral atom and thus there is no net effect. The free electron fraction evolution as a function of the redshift, calculated with the Saha equation Eq. (1.29) for , and with Eq. (1.30) for , is represented in Fig. 1.2.
1.5.4 The baryon decoupling of photons  the dark ages
Between and , the first luminous objects are not yet formed. Because photons only interact weakly with the remaining small free electron fraction, no astrophysical or cosmological signal has been observed from this era. This period is called the dark ages. Nevertheless, because of the collisions between atoms, spinflip transitions between the first hyperfine states of the neutral hydrogen atoms are possible. This results in an absorption of 21cm CMB photons, a signal in principle observable. The interest of this signal and its ability to constrain cosmology will be studied and detailed in the last chapters of this thesis.
Between and , even if photons interact weakly, the remaining free electron fraction, coupled to the baryons through Coulomb interaction, is sufficient for the gas temperature to be driven to the photon temperature. The energy transfer between photons and electrons is due to the Compton interaction. The rate of energy transfer per unit of comoving volume between photons and free electrons is given by (see e.g. [Seager:1999bc, Seager:1999km])
(1.33) 
where is the Thomson cross section, is the scattering rate, is the photon temperature and is the gas temperature. This energy transfer influences the baryon gas temperature. After using Eq. (1.16), it results that the gas temperature evolves according to
(1.34)  
(1.35) 
where is the Helium fraction. The first term on the right hand side is due to the volume expansion. The second term accounts for the energy injection due to the Compton scattering between CMB photons and the residual free electrons. Its last factor accounts for its distribution over the ionized fraction. When the photon energy density and the ionized fraction are sufficiently large, the Compton heating drives such that the gas and the CMB photons have the same temperature. With the expansion the photon energy density decreases, together with the ionized fraction. At a redshift , the gas temperature decouples from the radiation. Past this point it cools like , as expected for an adiabatic nonrelativistic gas in expansion.
1.5.5 Reionization
Around the first luminous astrophysical objects are formed. These inject a large amount of radiation in the intergalactic medium (IGM) such that all the Universe is reionized.
The reionization process is until now weakly known. During reionization, free electrons can diffuse CMB photons and thus affect the optical depth of the signal. Our current knowledge about the reionization history relies on one hand on the measurement of the optical depth of CMB photons that can be used to determine the reionization redshift, for a given reionization model. For instantaneous reionization, one has [Komatsu:2010fb].
On the other hand, Gunn and Peterson [1965ApJ...142.1633G] have predicted in 1965 that the highredshift quasar spectra must be suppressed at wavelengths less than that of the Lyman line at the redshift of emission, due to absorption by the neutral hydrogen in the IGM. A GunnPeterson trough has been observed in the spectrum of quasars at [Becker:2001ee]. This method is used to fix a lower bound () on the reionization redshift.
The reionization process itself can be investigated using complex numerical and seminumerical methods (see e.g. [Santos:2009zk, Thomas:2010mz]), simulating the growth of structures and the energy transfer to the IGM.
A 21cm signal in absorption/emission against CMB from reionization is also in principle observable. However, because collision rates are much lower than during the dark ages, the physical process generating hyperfine transitions is different. Spinflip transitions are induced by absorption and reemission of Lyman photons emitted by the first stars. Here again, details about the reionization 21cm signal and its ability to probe cosmology will be given in the last chapters of this thesis.
1.6 Precision observational cosmology
With the measurements of temperature anisotropies in the CMB, the cosmology has entered into an era of high precision. Combined with light element abundances, the observations of large scale structures and type Ia distant supernovae, they have permitted to determine with accuracy the standard cosmological parameter values. The combination of several signals is important, for breaking the degeneracies between parameters that can affect a given observable in the same way.
The aim of this section is to give a brief review of these observations and to describe qualitatively how they can be used to constraint the cosmological parameters of the CDM model. At the end of this section, we give the current bounds on these parameter values.
1.6.1 Hubble diagram
The Hubble expansion rate today can be determined by measuring the relative velocity of a large number of astrophysical objects as a function of their distance. Velocities are calculated by measuring the spectral shift of the distant objects. The original Hubble diagram [Hubble1929] (see Fig. 1.4) put in evidence for the first time the expansion rate of the Universe by determining the velocity and the distance of 18 near galaxies. Hubble found that . This value is far from the present measurement, [Komatsu:2010fb] (see Fig. 1.5). The difference between the original and the present values is due to inadequate methods for determining how distant the galaxies are [Webb:1999].
Today, distance measurements are much more accurate and several methods have permitted to calculate how distant extremely far objects are. The present relative errors on the Hubble parameter are less than 5%. The methods for measuring distances include:

cepheids: variable stars whose luminosity period has been empirically shown to be linked to their intrinsic luminosity [Udalski:1999pc].

The TullyFisher relation: this technique uses the correlation between the total luminosity of spiral galaxies with their maximal rotation velocity [Tully:1977fu]. An analogous technique can be used for elliptic galaxies [PeterUzan].

Type Ia supernovae: they correspond to the explosion of white dwarf stars. They are so luminous that they can be observed at a few hundreds of Mpc. Their distance can be measured due to the correlation between their characteristic evolution time and their maximal luminosity [1979ApJ...232..404C]. This technique can be used to estimate the Hubble expansion rate as a function of redshift.
In 1998, the present acceleration of the Universe’s expansion has been detected using type Ia supernova observations [Riess:1998cb]. Combined with other signals, they have permitted to measure the value of and to put constraints on the equation of state parameter of the dark energy fluid.

Other techniques [Freedman:2000cf], like type II supernovae (by measuring their angular size and their spectral Doppler shifts), or the fluctuations of the surface brightness of galaxies on the pixels of a CCD camera, that depend on the distance of the galaxy.
1.6.2 Abundances of light elements
During the history of the Universe, the primordial abundances in light elements have been modified by various nuclear processes, like nuclear interactions in the stars. Nevertheless, these can be measured in sufficiently primitive astrophysical environments to be connected to the relative abundances at the end of the primordial nucleosynthesis. From these measurements, and by using the theory of the primordial nucleosynthesis in an expanding spacetime [Alpher:1948, Gamow:1948, Alpher:1948b, Peebles:1966, Mukhanov:2003xs], it is possible to determine a range of acceptable values of the parameter at the time of BBN.
Fig. 1.6 illustrates the observational status for the primordial abundances Helium4 [Izotov:1999wa, Luridiana:2003jy, Izotov:2003xn], Deuterium [Kirkman:2003uv] and Lithium [Ryan:2000zz]. One can see that observations are all compatible with a parameter . This value corresponds today to a fractional energy density for baryons . Before CMB anisotropy observations, the light element abundances have been for a long time the only indirect measurement of the baryon energy density. Today, observations also help to constraint other parameters, like the number of neutrino species [Shimon:2010ug]. They also constrain a possible variation of the fundamental constants [Dent:2007zz].
Finally, it must be noticed that the value of the parameter can be determined independently by CMB observations (see next point). This results in a stronger constraint on the parameter [Peiris:2005dt, Larson:2010gs], as illustrated in Fig. 1.6.
1.6.3 The CMB anisotropies
Before recombination, photons were tightly coupled to electrons and protons via Compton scattering. The small inhomogeneities were prevented to collapse due to the pressure of photons. Therefore, instead of growing like they do after recombination, the density perturbations have performed acoustic oscillations. These left imprints in the cosmic microwave background, on the form of temperature anisotropies of the order of K. These anisotropies have been put in evidence for the first time by the COBE (COsmic Background Explorer) satellite [1992ApJ...396L...1S] in 1992. The statistical properties of these temperature anisotropies have proved to be the an efficient tool to constrain the cosmological parameters.
The angular power spectrum
For gaussian temperature fluctuations, the statistical properties of the CMB sky map are encoded in the socalled angular power spectrum.
Let us define , the temperature fluctuation compared to the average sky temperature in a specific sky direction . Then let us decompose these fluctuations in spherical harmonics with coefficients ,
(1.36) 
The statistical properties of the sky map are encoded in the twopoint correlation function
(1.37) 
where we have used the isotropy hypothesis and the property of the Legendre polynomials,
(1.38) 
Thus the coefficients are related to the through the relation
(1.39) 
The next step is to compare the to the theoretical . The theoretical temperature field is an homogeneous and isotropic random field. The theoretical coefficients are also independent stochastic fields, with vanishing mean value,
(1.40) 
The theoretical are not directly observable, but an estimator can be obtained by summing over ,
(1.41) 
If initial fluctuations follow a gaussian statistic, the probability distribution function is also gaussian and reads
(1.42) 
It results that the follow a distribution with dof. One sees that the can be used to estimate the theoretical . This estimation can not be perfect, because it is obtained by averaging over the finite () set of the of the unique CMB sky map, whereas we want to estimate an ensemble average. The intrinsic variance of the estimators reads
(1.43) 
As expected, the error is smaller if one has a larger number of for the estimation. Actually, it is possible to show that the are the best estimators, and that the resulting variance is the smallest possible [Grishchuk:1997pk]. This is called the cosmic variance.
The present best measurements of the CMB angular power spectrum are shown in Fig. 1.7. As already mentioned in Sec. 1.6.4, prior to recombination the inhomogeneities in the tightly coupled baryon and photon fluid are prevented to collapse and perform acoustic oscillations. These oscillations result from the competition between gravitational attraction and photon pressure. At recombination, the density perturbations reaching for the first time a maximal amplitude lead to a maximum of temperature fluctuation for the emerging CMB photons, and induce a first peak in the CMB angular power spectrum. The following peaks can be seen as its harmonics. For instance, the second peak corresponds to perturbations having performed one complete oscillation at time of recombination. These peaks are damped at high multipoles . This socalled Silk damping [1968ApJ...151..459S] occurs because the acoustic waves can not propagate for perturbation modes whose wavelength is smaller than the mean free path of photons. Large angular scale temperature fluctuations () are sourced at recombination by perturbations larger than the Hubble radius . In the superHubble regime, perturbations remain constant in time, and thus they conserve at recombination their initial amplitude.
The complete evolution of the perturbations before the recombination can be determined in the context of the theory of cosmological perturbations [Mukhanov:1990me]. This is done by solving both the perturbed Einstein equations and the firstorder Boltzmann equations for all the species. Since in the next chapters of the thesis we will focus mainly on models of inflation and on the postrecombination evolution, this calculation has not been reproduced here. A detailed description of the evolution of perturbations prior to recombination and their effect on the angular power spectrum of CMB temperature anisotropies can be found in most textbooks on modern cosmology (see e.g. [Dodelson, PeterUzan]).
Dependance on cosmological parameters
In this section, we give a qualitative description of the effects of the CDM cosmological parameters on the shape of the CMB angular power spectrum, and more particularly on the positions and magnitudes of the acoustic peaks. In the section 1.6.6, the best fits of these parameters are given.
The density of baryons :
The fractional energy density of baryons at fixed total matter density modifies the shape of the angular power spectrum in three ways:

It fixes the sound velocity and thus the frequency of oscillations in the primordial baryonphoton plasma. An increase of leads to a reduction of the sound velocity, and thus a reduction of the oscillation frequency. The acoustic peaks are therefore induced by perturbations of smaller wavelengths, entering earlier inside the Hubble radius, and they are thus shifted to higher multipoles .

It fixes the relative amplitude of odd and even peaks. At constant total matter energy density, a reduction of the baryon energy density means that more dark matter can accumulate and dig deeper gravitational wells. This induces a reduction of the relative magnitudes of the odd peaks, because the amplitude of the baryon perturbations are reduced each time they climb the more steep gravitational wells.

The mean free path of photons due to the Compton scattering depends on the electron number density, and thus is affected by the baryon density (since the Universe is neutral, ). The Silk damping is therefore affected by . For an augmentation of , the diffusion length is reduced and the damping is less efficient, inducing a higher magnitude for the peaks at high multipoles in the angular power spectrum.
The total matter density :
The total matter energy density, for a fixed ratio , has two main effects on the angular power spectrum.

When CMB photons emerge from an overdensity, their wavelength is affected by the gravitational Doppler effect. Increasing affects the Doppler effect on the CMB photons and change the contrast between maxima and minima in the angular power spectrum.

Increasing also shifts acoustic peaks to higher multipoles, because it affects the Hubble expansion rate. At fixed value of^{6}^{6}6 denotes the fractional energy density for the radiation and , a higher value of reduces the time of matter/radiation equality and the moment of the last scattering.
The cosmological constant and the curvature :
At fixed values of , and , fixing a value of is equivalent to fix . Their respective effect on the CMB angular power spectrum can thus be considered simultaneously.

The Hubble expansion rate, and thus the relation between angular distances in the sky and the corresponding distance at a given redshift is modified with (as well as ).

Spatial curvature induces geodesic deviations for CMB photons. If the Universe is open, the positions of the acoustic peaks is shifted to higher multipoles, if it is closed, to lower multipoles.
The neutrino density :
The neutrino energy density, characterized by the effective number of relativistic neutrinos , affects the total radiation energy density, and thus the time of the radiation/matter equality as well as the recombination time [riazuello]. Thus the position of the acoustic peaks also depends on .
1.6.4 The matter power spectrum
With the recent surveys of galaxies and quasars (e.g. 2dF [Percival:2001hw, website:2dF], SDSS [website:SDSS]), the large scale distribution of structures has been probed. Galaxies are observed to be arranged in a complex structure of "walls" and "filaments" (see Fig. 1.8). The statistical properties of the matter distribution in the Universe are encoded in the matter power spectrum. If the mean density of galaxies is denoted , the fractional inhomogeneity can be expanded in Fourier modes . The power spectrum is defined via^{7}^{7}7The adimensional form of the power spectrum is also often used.
(1.44) 
where is the 3dimensional Fourier transform of , and where the brackets denote an average over the whole distribution. The measurements of the matter power spectrum today are plotted in Fig. 1.9.
Some oscillations have been detected in the matter power spectrum [Eisenstein:2005su], for perturbation wavelengths of a few Mpc (see Fig. 1.10). These have been identified as the relic of the Baryon Acoustic Oscillations (BAO) that took place in the early Universe, the same that are observed in the CMB.
The shape of the matter power spectrum and the BAO are sensitive to the cosmological parameter values. For instance, the largest possible wavelength for perturbation modes to oscillate is referred as the sound horizon. It can be measured at recombination with CMB observations and compared to its present value measured with the matter power spectrum. The ratio is sensitive to the expansion history, and thus to the cosmological parameters [see Eq. (1.13)]. This method can be used to determine the latetime acceleration of the Universe’s expansion and to put a bound on the dark energy equation of state, independently of the type Ia supernova measurements.
1.6.5 Other signals
To break the degeneracy between the cosmological parameters, it is necessary to combine data from several cosmological and astrophysical signals. Besides the main signals described above, one could also mention: Gravitational weak lensing [Bartelmann:1999yn], Galaxy clusters [Voit:2004ah] , Ly forest [Viel:2006yh] and rotation curves of galaxies [Battaner:2000ef].
1.6.6 Current bounds
The best fits for the cosmological parameter values for the CDM model are given in the table below [Komatsu:2010fb]. The mean values of the probability distributions of these parameters and the corresponding 1 errors are also given.
Parameter  Best fit  Mean value and 1 errors 

1.7 Unresolved problems
The hot BigBang standard cosmological model has raised several questions and problems that remain today unresolved. Some of them are described in this section.
1.7.1 Nature of dark matter
The nature of the cold dark matter component remains unknown. Since the SM does not contain any dark matter candidate that is in agreement with all the observations, dark matter is a strong indication for new physics beyond the standard model. A large number of models and dark matter candidates in agreement with cosmological and astrophysical observations have been proposed (for a review, see [Bertone:2004pz]).
In the next few years, a major challenge will consist in identifying the nature of dark matter. Dark matter particles could be produced and detected directly in particle accelerators, e.g. in the Large Hadron Collider (LHC) at CERN [DiCiaccio:2011zz]. Other direct detection experiments in laboratories attempt to measure dark matter interactions with nuclei, e.g. in cryogenic detectors (CDMS [website:CDMS], CRESST [Angloher:2008jj], Edelweiss [website:Edelweiss],…). Indirect detection experiments attempt to measure the decay/annihilation products of dark matter particles, that may lead to positron, antiproton, neutrino or gamma excesses.
Recently, an excess of positrons has been reported by the PAMELA experiment [Adriani:2008zr]. But this could be due to astrophysical sources [Delahaye:2008ua]. The DAMA experiment has measured an annual modulation [Bernabei:2000qi] that could be due to weakly interacting massive particles (WIMP’s). These results are subject to intense discussions in the community [Angle:2007uj].
1.7.2 Nature of dark energy
The dark energy component is today the main contribution to the energy density of the Universe, representing about 71% of the total energy density. Its energy density is thus comparable to the total matter energy density, and the epoch from which the Universe became dominated by the dark energy coincides approximatively with the epoch of structure formation. In the CDM model, dark energy is identified with a cosmological constant.
But the dark energy could be also a dynamical quantity, due to an unknown fluid or a modification of gravity at cosmological scales. A large number of models have been proposed in this context (for a recent review, see [Sapone:2010iz]). But because dark energy is not expected to be related to the matter content of the Universe, several model are said to suffer from a socalled coincidence problem. Recently, it has been proposed that the current cosmic acceleration can be due to an almost massless scalar field experiencing quantum fluctuations during a phase of cosmic inflation close to the electroweak energy scale [Ringeval:2010hf].
A possible contribution to the cosmological constant could be the vacuum fluctuations. However, when it is estimated using quantum field theories, it is found to be larger than the energy of the electroweak breaking scale . But the measured value of the cosmological constant is . Its energy density is therefore at least 60 orders of magnitude smaller than expected [Weinberg:2000yb].
1.7.3 Horizon problem
It is convenient to define the conformal time
(1.45) 
that is the maximal comoving distance covered by the light between an initial hypersurface at time and the hypersurface at time . Two points separated by a comoving distance larger than the conformal time do not have a causal link if one consider that the Universe’s evolution begins at . Usually, the initial hypersurface is identified with the Plancktime, and points separated by a comoving distance larger than are said to be causally disconnected. For an observer in at a time (see Fig. 1.11), is the comoving radius of the sphere centered in separating particles causally connected to the observer of particles causally disconnected. is called the comoving horizon or the particle horizon. It is important to distinguish between the particle horizon and the event horizon, which is, for the observer, the hypersurface separating the universe in two parts, the first one containing events that have been, are or will be observable, the second part containing events that will be forever unobservable. Mathematically, the event horizon exists only if the integral
(1.46) 
converges. Finally, it is useful to define the comoving Hubble radius, . It is smaller than the conformal time, that is the logarithmic integral of the Hubble radius.
The horizon problem is linked to the isotropy of the CMB. Indeed, how to explain that regions in the sky have the same temperature whereas their angular size is too large to correspond to causally connected patches at the time of last scattering, if the CDM model alone is assumed to describe the whole Universe’s expansion?
In the standard cosmological model, the early Universe is dominated by the radiation and the chemical potentials can be neglected most of the time. One has therefore , and in a comoving coordinate system, any physical distance growths like
(1.47) 
The temperature of CMB photons today is .
On the other hand, assuming that the expansion rate is dictated by the CDM model at every time, the radius of the observable universe, that is the radius of the spherical volume in principle observable today by an observer at the center of the sphere, is . At the time corresponding to the last scattering surface , the radius of the observable universe was
(1.48) 
Under the same assumption, at recombination, the maximal distance between two causally connected points would roughly be
(1.49) 
At last scattering, our observable Universe would therefore have been constituted of about causally disconnected regions. But CMB photons emerging from these regions are observed to have all the same temperature, to a accuracy. At the Planck time, the number of causally disconnected patches would have been much larger, about .
1.7.4 Flatness problem
From the FL equations (1.3), one can write the equation for the evolution of the curvature. If we neglect the cosmological constant^{8}^{8}8This is a good approximation because dominates the energy density only at late times., we have
(1.50) 
This equation is easily integrated when is constant. One has
(1.51) 
where is the curvature today. Since it is constrained by observations ( [Komatsu:2010fb]) one has roughly at radiationmatter equality
(1.52) 
and at the Planck time,
(1.53) 
If the Universe is not strictly flat, the CDM model does not explain why the spatial curvature is so small.
1.7.5 Problem of topological defects
In Grand Unified Theories (GUT), the standard model of particle physics results from several phase transitions induced by the spontaneous breaking of symmetries. Such symmetry breakings are triggered during the early Universe’s evolution due to its expansion and cooling, and they can lead to the formation of topological defects like domain walls, cosmic strings and monopoles. These defects correspond to configurations localized in space for which the initial symmetry remains apparent (see Fig. 1.12).
Let us consider the symmetry breaking of a group resulting to an invariance under the subgroup : . The vacuum manifold is isomorphic to the quotient group [Nakahara:1990th]. Domain walls are formed when the 0thorder homotopy group of is not trivial. They can be due to the breaking of a symmetry, or if the resulting vacuum contains several distinct elements. Cosmic strings are formed when the first homotopy group of is not trivial, for instance for the breaking scheme . Monopoles are formed when the second homotopy group of the vacuum manifold is not trivial. This is the case for the breaking of a symmetry into . For higher homotopy groups, the resulting topological defects are called textures.
Groups involved in GUT are such that the first and second homotopy groups are trivial, . In the SM, there remains a U(1) invariance corresponding to electromagnetism. The first homotopy group of U(1) is . Therefore, by using the property of homotopy groups [PeterUzan]
(1.54) 
one obtains that the second homotopy group of the vacuum manifold corresponding to the breaking of a GUT group is not trivial. That induces necessarily the formation of monopoles [Langacker:1980kd].
However, monopole annihilation has been found to be very slow [Zeldovich:1978wj, PhysRevLett.43.1365]. As a consequence, their energy density today should be 15 orders of magnitude larger than the current energy density of the universe. Domain walls can also lead to catastrophic scenarios, but they can be avoided in the schemes of symmetry breaking in GUT. Cosmic strings are observationally allowed, but their contribution to the CMB angular power spectrum [Kaiser:1984iv] is constrained [Battye:2010xz].
1.7.6 Why is the primordial power spectrum scaleinvariant?
The density perturbations at the origin of the CMB temperature fluctuations start to oscillate when their size becomes smaller than the Hubble radius. On the contrary, the perturbations whose wavelength is much larger at recombination have remained constant and thus conserve their initial amplitude. In the CMB angular power spectrum, these superHubble perturbations correspond to temperature fluctuations at low multipoles (). The CMB temperature fluctuations at large angular scales therefore directly probe the initial state of those density perturbations.
With CMB observations, it has been established that the primordial power spectrum of density perturbations is (nearly) scale invariant. The present measurements of the shape of the primordial power spectrum will be given in details in section 2.2. This constrains the possible physical processes at the origin of the initial density perturbations.
1.7.7 Contribution of isocurvature modes
There are two different kinds of primeval fluctuations: the curvature (or adiabatic) and isocurvature (or entropic).
The adiabatic density fluctuations are characterized as fluctuations in the local value of the spatial curvature (hence the name of curvature perturbations). By the equivalence principle, all the species contribute to the density perturbation and one has for any fluid ,
(1.55) 
where is the entropy density. Furthermore, one can write
(1.56) 
That means that the fluctuation in the local number density of any species relative to the entropy density vanished.
The entropic fluctuations are perturbations for which and therefore they are not characterized by fluctuations in the local curvature (hence the name isocurvature). They correspond to fluctuations in the equation of state.
CMB observations have been used to determine that the temperature fluctuations are sourced by curvature perturbations, and the contribution of isocurvature perturbation modes is constrained [Komatsu:2010fb]. The mechanism leading to initial inhomogeneities therefore needs to generate (at least mostly) curvature perturbations.
1.7.8 Why are the perturbations Gaussian?
The statistical properties of the CMB anisotropies are encoded in the power spectrum of the temperature fluctuations, that is the twopoint correlation function in the Fourier space. Within a general framework, those are also encoded in the threepoint, fourpoint, and higher order correlation functions. But if the fluctuations follow a Gaussian statistic, these are all vanishing.
The point is that the observations of the CMB have not detected a nonzero value neither for the threepoint neither for higherorder correlation functions. Since the temperature fluctuations in the CMB are induced by density perturbations, the mechanism generating the primordial density perturbations needs to be such that their statistics is Gaussian.
1.7.9 Initial singularity
As already mentioned in section 1.3, if the Universe has not been dominated by a positive spatial curvature, an initial singularity is generic for all known types of fluids. In the CDM model, the gravitation is assumed to be described correctly by GR at every time. However, some theories predict that GR is not valid anymore at the Planck energy scale. Let us mention String Theories, Loop Quantum Gravity [Rovelli:1995ac] and HoravaLifshitz theory [PhysRevD.79.084008]. In some of these frameworks, the initial singularity is avoided and replaced by a bounce.
It is nevertheless important to remark that all these theories are still highly hypothetic and not at all confirmed by observation.
In the next chapter, the concept of inflation, that is an hypothetic phase of accelerated expansion in the early Universe, is introduced. Models of inflation solve in an unified way several problems mentioned above. They can provide Gaussian adiabatic primordial perturbations whose power spectrum is nearly scale invariant. They solve also naturally the monopole, the horizon and the flatness problems.
Chapter 2 The inflationary paradigm
2.1 Motivations for an inflationary era
Inflation is a phase of quasiexponentially accelerated expansion of the Universe. By combining the F.L. equations (1.3) and (1.4), and assuming , one obtains a necessary condition for inflation to take place,
(2.1) 
The amount of expansion during inflation is measured in term of the number of efolds, defined as
(2.2) 
where is the scale factor at the onset of inflation.
The inflationary paradigm is motivated since it provides a solution to several problems of the standard cosmological model.

The horizon problem: Inflation solves naturally this paradox if the number of efolds of expansion is sufficiently large. Indeed, isothermal regions in the CMB sky, appearing as causally disconnected at recombination if the CDM model alone is assumed, can actually be causally connected because of a primordial phase of inflation. If the Universe’s expansion was exponential during the inflationary era,
(2.3) (it will be shown later that this condition is nearly satisfied) one can evaluate the number of efolds required to solve the horizon problem. At the end of inflation, the size of the current observable Universe must have been smaller than the size of a causal region at the onset of inflation ,
(2.4) where is the scale factor at the end of inflation. If inflation ends at the Grand Unification scale ( GeV), one needs
(2.5) where is the photon temperature today, and for which we have assumed , where and are respectively the Planck length and the Planck temperature. If this condition is satisfied, the entire observable Universe can thus emerge out of the same causal region before the onset of inflation.

The flatness problem: During inflation, the Universe can be extremely flattened. Indeed, if we assume to be almost constant during inflation, one has (see section 1.3)
(2.6) With and a curvature of the order of unity at the Planck scale, the flatness problem discussed in section 1.7 is naturally solved.

Topological defects: During inflation, topological defects are diluted due to the volume expansion and can have been "pushed" outside the observable Universe.

The primordial power spectrum: Models of inflation generically predict a nearly scale invariant power spectrum of curvature perturbations, and thus can provide good initial conditions for the perturbations in the radiation era. It will be explained later in this chapter how this power spectrum can be determined for a large class of models of inflation (single and multifield models).

Gaussian perturbations: Inflation models predict that the classical perturbations leading to the formation of structures in the Universe are due to quantum metric and scalar field fluctuations. As the Universe grows exponentially, the quantumsize fluctuations become classical, are stretched outside the Hubble radius, and source the CMB temperature fluctuations. All the preinflationary classical fluctuations are conveniently stretched outside the Hubble radius today and can be safely ignored. The Gaussian statistic of the perturbations therefore takes its origin in the Gaussian nature of the quantum field fluctuations.

Isocurvature modes: Most models of inflation source only curvature perturbations. Nevertheless, for some models (like multifield models), the isocurvature mode contribution can be potentially important and eventually observable (e.g. in Ref. [Langlois:1999dw]). In multifield models, these are induced by field fluctuations orthogonal to the field trajectory, as explained more in details in section 2.4.2.
2.2 Observables
The CMB angular power spectrum is sensible to the initial conditions of the density and curvature fluctuations. A model of inflation provide these initial conditions and it can therefore be confronted to CMB observations.
Observations have permitted to measure and constrain the amplitude of the power spectrum of curvature perturbations at the end of inflation, its spectral tilt, as well as the ratio between curvature and tensor metric perturbations.
2.2.1 Power spectrum of primordial curvature perturbations
The primordial power spectrum of the curvature perturbation is defined from,
(2.7) 
where is the 3dimensional Fourier transform of . The spectral index of this power spectrum is defined as
(2.8) 
where is a pivot scale in the observable range, e.g. . A power spectrum showing an excess at large angular scales () is called redtilted while for an excess at small scales, it is called bluetilted. Deviation from a scale invariant primordial power spectrum have been detected by recent CMB experiments. The power spectrum is observed to be redtilted, and the case is disfavored. The present 1 bound on the spectral index is [Komatsu:2010fb] , as illustrated in Fig. 2.1.
On the other hand, measurements of the CMB temperature monopole and quadrupole have permitted to fix the amplitude of the curvature power spectrum [Komatsu:2010fb],
(2.9) 
2.2.2 Tensortoscalar ratio
The tensor metric perturbations, characterized by a power spectrum at the end of inflation, can also affect the CMB angular power spectrum (for details, see e.g. [PeterUzan]). CMB observations have permitted to put a significant limit on the primordial power spectrum of gravitational waves. It is convenient to express this limit as an upper bound on the ratio between tensor and curvature power spectra. At 2, one has [Komatsu:2010fb],
(2.10) 
The 1 and 2 bounds in the plane are shown in Fig. 2.1, as well as the predictions for some models of inflation.
2.2.3 Other observables
Other observable quantities are potentially interesting to improve the constraints on inflation models. The observational bounds on these quantities are not yet sufficient to provide significant constraints on inflation. Nevertheless, the future data from the Planck satellite could change this perspective. Some of these observables are briefly described below:

The running spectral index : it is defined as
(2.11) Present data are compatible with .
Inflationary predictions can be compared to the constraints on , , and . However, it is more efficient to constrain directly the parameter space of a given model of inflation by confronting it directly to the ’s measurements, and by using Bayesian analysis to obtain the posterior probability density distributions of its parameters marginalized over all the cosmological parameters.

The parameter: this parameter characterizes the amplitude of the socalled local form bispectrum of ,