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Why in ation? The puzzles of the standard Hot Big Bang scenario

The Hot Big Bang scenario can be understood as a logical implication of the observed expansion of space. It states that the current inhomogeneous Universe made of stars, galaxies and large-scale structures, is the deterministic result of the forward evolution of a dense and homogeneous initial state, under the combined eects of Standard Model forces and gravity. Pushing the reasoning, there must be a time before which the Universe was so dense that it must have been opaque. And indeed, before photon decoupling, the Thomson scattering of photons o electrons was so ecient that light could only travel on very short distances, and an observer in the primordial Universe could only see its immediate surroundings, in a very analog way to what happens in fog. When the Universe cooled enough, photons were able to free-stream, forming the Cosmic Microwave Background (CMB) that one can observe nowadays and which carries an impressive amount of information about the photon{baryon{dark-matter primordial plasma. A few interesting properties of this relic radiation are listed here:
• The CMB is almost isotropic over the whole sky, and its spectrum corresponds to a black-body radiation (see, e.g., the historical though accidental discovery [8] and subsequent interpretation in the Big Bang scenario [9], the COBE measurement by the instrument FIRAS [157] and latest calculations [137] of the current CMB temperature yielding T = (2:72548 0:00057) K). See Fig. 3.1 for the CMB blackbody spectrum as measured by COBE-FIRAS.
• Although their amplitude is small, the CMB contains temperature anisotropies that are distributed according to Gaussian statistics, and that have been more and more precisely measured on smaller and smaller scales, and over the whole sky, rst by the COBE (see [10] for the rst detection), then WMAP (see [14] for the conclusions after the 9-year release) and eventually Planck (see [15] for a summary of the Planck legacy) satellites. See upper panel of Fig. 3.2 for the CMB temperature anisotropies as measured by Planck.
• CMB photons are polarized in E- and B-modes, but currently dectected B-modes are understood as produced only by secondary eects after the emission of CMB, such as weak lensing (represented by the local potential ). The temperature and E-modes are cross-correlated. See the lower panels of Fig. 3.2 for the power spectra of E- and -modes, and the middle panel for the cross-power spectrum of temperature and E-modes. Given perturbative quantities X(^n); Y (^n), etc. dened on the celestial sphere, where ^n is a unit vector denoting the direction in the sky, the rotationally invariant angular power spectra are CXY ` = X` m=􀀀` aX `maY `m=(2` + 1) .

Dynamics of single-eld in ation

Although the exact dynamics of in ation are still to be determined, a large class of minimalist models now constitute our paradigm for the very early universe. Historically denoted as \chaotic in ation » models (see Ref. [158] for a historical review of early works in in ationary cosmology), they feature a single scalar eld in the matter sector, evolving under the force derived from its potential V () and the expansion of space as described by general relativity. Importantly, the initial conditions can be quite generic (hence the name \chaotic »), and the scalar potential can be in principle of any type, as long as it is suciently at as one shall see soon. In the remaining of this section, the dynamics of these single-eld, slow-roll, models of in ation at the level of the background and linear perturbations, are shortly reviewed.

CMB observations and constraints on single-eld models

Observations of the CMB not only constrain the background dynamics of in ation, Ntot & 60, but also the statistics of linear uctuations. Indeed, taking the power spectra at the end of in ation as initial conditions for the radiation-dominated and matterdominated eras, Boltzmann codes like CMBFast [166], CAMB [167] and CLASS [168] can be used to evolve the probability density functions of the positions and momenta of particles in the early universe, and to predict the acoustic peaks in the temperature and polarisation spectra of the CMB. The physics of the CMB is extremely rich and I could not give enough credits to the physicists that helped understanding it, however in the spirit of keeping this thesis reasonably long, it shall not be further explained.
The attitude in the following is rather to take the information on the inferred values of, or constrains on, the primordial parameters (ns; s; r; : : :), as granted and explore their consequences in terms of model-building. Indeed, it is both a blessing and a curse that many dierent models of in ation succeed to provide the minimal amount of efolds of expansion to solve both the horizon and the atness problems, and even to predict accurately the values of ns; r, etc. Therefore, it is of great importance to study in detail the constraints that cosmological observations put on these models that can be well-motivated or not in terms of high-energy physics. To put it in a nutshell, studying in ation amounts to studying physics beyond the Standard Model, at energy scales much larger than those accessible in particle accelerators on Earth, by taking full advantage of the great cosmic accelerator. This approach is extended in Chapter 5 to the CMB constraints on primordial non-Gaussianities and how they can help us to favour or rule out models of the very early universe. Rather than long sentences, one is invited to observe Fig. 3.4.
A few comments can be made, rst about the gure, and then more generally about constraints on single-eld, slow-roll in ation:
• A perfectly scale-invariant scalar power spectrum, ns = 1, is ruled out at more than 7, and the deviation from 1 is very well measured: ns = 0:9649 0:0042 at 68% condence (1). It is a huge achievement of the simple single-eld, slow-roll in ationary paradigm to predict such small but non-vanishing deviations from 1 (remember the typical example of quadratic in ation for which ns 0:967), and precise measurements of ns enable one to distinguish amongst various models, e.g. \Low scale SB SUSY » models are excluded.
• Primordial gravitational waves have not been observed yet, as one is only given upper bounds on the tensor-to-scalar ratio: r < 0:056 at 95% condence. This fact can be thought of as disappointing, since such observation would provide a smoking-gun evidence for a quantum generation of primordial perturbations, both of the scalar and tensor kinds, as predicted almost solely by cosmic in ation. But the constraints are so strong that they enable one to rule out a huge number of in ationary models, in particular so-called \large-eld models » in which the scalar elds take super-Planckian values, > MPl and the rst slow-roll parameter, , is not extremely small (such as the famous and simple quadratic in ation model: 10MPl, 1 10􀀀2). Actually, all monomial potentials V / p where p 2 R+ are strongly disfavored by the combined (ns; r) constraints. \Small-eld models », for which < MPl or \plateau models » that have a very at potential, and models with concave rather than convex potentials, are generally favored by these constraints since they feature smaller values of and thus a smaller tensorto- scalar ratio.
• Although not represented in this plot, the running of the spectral index, s is measured to be consistent with zero, an observation that is again in good agreement with the single-eld, slow-roll picture where this quantity is a second-order one in the slow-roll expansion. The running of the running, @s= (@ln k) is also constrained around a vanishing value.
• The amplitude of the scalar power spectrum, As = PR(kpivot) is also well measured, one has As = 2:10 10􀀀9, which xes the value of H2 = for single-eld, slowroll in ation. Since P / (H=MPl)2, the absence of detection of gravitational waves (P < 0:056 2:10 10􀀀9) provides us with an upper bound on the value of the Hubble parameter at the time of horizon exit for the pivot scale: H < 7:6 10􀀀5MPl.
For completeness, the few in ationary models that are represented on Fig. 3.4, are spelled out here, but no comment is made on their motivations:
• Natural in ation: V () = 4 [1 + cos (=f)], where is a mass scale that can be inferred from the measured value of As, and f is another mass scale often called the axion decay constant, that when varied together with the pivot-scale, horizon-exit time N, gives rise to the purple contour on Fig 3.4.
• Hilltop models: Vp() = 4 [1 􀀀 (=)p], with p 2 R and where and are again mass scales. The hilltop quartic model simply corresponds to V4 and varying both and N gives the green contour on Fig. 3.4 (hilltop quadratic, V2, is less favored by the Planck analysis).
• Power-law in ation has a dierent status, since it is rather dened as in ation happening with a scale factor growing as tq with q 2 R+. Exponential potentials like Vq() = 4exp 􀀀 p 2=q =MPl can induce such background dynamics but overall these models are strongly disfavored because they predict a large tensor-toscalar ratio except if ns 1, indeed one has the following relation: ns 􀀀1 = 􀀀r=8.
• -attractors correspond to a class of models where the potential is stretched exponentially when going away from = 0. E-models are a subset of these and feature of potential V() = 4 h 1 􀀀 exp p 2=(3) =MPl i2 . Interestingly, they provide a continuous description between the simplest quadratic in ation model for 1.

The eective eld theory of in ation: a model-independent theory of uctuations

Having explained and elucidated the degeneracy amongst the various possible single- eld models, it would appear appealing to have a model-independent description of in ationary uctuations. This is exactly the aim of the so-called Eective Field Theory of In ation (EFToI) [175,176], which is brie y sketched in this Section following closely the spirit and sometimes the notations of this original paper.
As any Eective Field Theory (EFT), the EFToI relies on the symmetries of the system at hand to describe it in the most generic way. An important feature of the EFToI is that it describes only the in ationary uctuations, and assumes that the background quantities are given functions of time. In particular, its aim is not to provide a mechanism for solving the horizon and atness problems (any serious model of single-eld slow-roll in ation does that), and it simply assumes a quasi-exponential expansion of the FLRW background spacetime, with a(t) / eHt and H(t) a given function of time that is slowly varying: 􀀀 _H H2. This is both a strength and a weakness: it enables one to describe cosmological perturbations in a model-independent way, but that might not be realised in concrete models where a Lagrangian is specied and dictates both the background dynamics and the one of linear uctuations in a consistent way.

Indirect motivations: limitations of the single-eld picture

First, indirect motivations for the study of multield in ation come from the limitations of the slow-roll, single-eld picture, that one shall here investigate upon. In order to do this, one must rst remember the strong theoretical constraints on such models of the early universe based on CMB observations, and more particularly on the derivatives of the scalar potential for the in aton, V (). Indeed, remember that for standard slow-roll in ation to proceed, one needs to meet the two conditions V ; V 1, meaning that both the slope and the curvature of the scalar potential must be small in Planck mass units. In particular, those constraints put into question the naturalness of slow-roll, single-eld in ation, as it seems that two dimensionless parameters of the model must be ne-tuned to small values. This problem of naturalness can be exemplied with the so called -problem that is summarised in the following. Moreover, recent years have seen the quick growth of a discipline aiming at eliminating low-energy (at the scale of in ation) eective eld theories that do not possess a UV completion at higher energy scales where a quantum theory of gravity must apply: the so-called swampland program that is also in tension with the simplest single-eld paradigm and is shortly mentioned here too.

Table of contents :

Introduction
I The cosmological paradigm 
1 The homogeneous and isotropic Universe 
1.1 Special Relativity and General Relativity: a description of spacetime
1.1.1 Interferometer experiments and the constancy of the speed of light
1.1.2 Special Relativity: a consistent framework for particles, challenged by gravity
1.1.3 General relativity and the bending of spacetime
1.2 Cosmology: the Universe has a history
1.2.1 Expansion of the Universe, from philosophy to modern science .
1.2.2 Modern approach to cosmology
1.2.3 A brief reverse history of 13.8 billion years
2 Cosmological perturbation theory 
2.1 Perturbing spacetime
2.2 Perturbing the matter content
2.2.1 Perturbations of perfect uids
2.2.2 Covariant perturbations of scalar elds
2.3 Gauge freedom and gauge-invariant quantities
2.3.1 Generalities
2.3.2 Spacetime
2.3.3 Matter content
2.3.4 Popular gauges and relations with gauge-invariant variables
2.4 ADM formalism and constraint equations
II Canonical ination: a classical background and quantum pertur- bations 
3 Single-eld ination, the minimal working example 
3.1 Why ination? The puzzles of the standard Hot Big Bang scenario
3.2 Dynamics of single-eld ination
3.2.1 Background
3.2.2 Linear perturbations
3.3 CMB observations and constraints on single-eld models
3.4 The eective eld theory of ination: a model-independent theory of uctuations
3.4.1 Action in the unitary gauge
3.4.2 Action for the pseudo-Goldstone
4 Multield ination: exploring implications from high-energy physics 
4.1 Motivations
4.1.1 Indirect motivations: limitations of the single-eld picture
4.1.2 Direct motivations for multi-scalar ination
4.2 Dynamics of multield ination
4.2.1 Background
4.2.2 Linear perturbations
4.3 The adiabatic-entropic decomposition
4.3.1 A new parameterisation
4.3.2 Super-Hubble evolution and power spectra at the end of ination .
4.3.3 Possible multield instabilities
5 Primordial non-Gaussianities as a probe of extra particle content 
5.1 Formalism and generalities
5.1.1 Correlation functions as a parameterisation of Non-Gaussianities .
5.1.2 Primordial non-Gaussianities in single-eld ination
5.1.3 Primordial non-Gaussianities in multield ination
5.1.4 Observational constraints
5.2 Transient tachyonic instability and enhanced non-Gaussianities inattened congurations (article)
5.3 Revisiting non-Gaussianity in multield ination with curved eld space .
5.3.1 Two-eld case (article)
5.3.2 General case with Neld scalars (article)
III Stochastic ination: a non-perturbative treatment of large-scal uctuations 
6 Single-eld, slow-roll stochastic ination 
6.1 Coarse-graining and the emergence of stochasticity
6.1.1 A separation of scales
6.1.2 Langevin equation: a stochastic dierential equation
6.2 Correlation functions
6.2.1 Fokker-Planck equation and test scalar elds in de Sitter
6.2.2 Cosmological observables in stochastic ination
7 Multield stochastic ination: a path to the discretisation ambiguity and its resolution 
7.1 Inationary stochastic anomalies, or the discretisation ambiguity (article)
7.2 A manifestly covariant, anomaly-free theory of multield stochastic ination (article)
IV Cosmological reheating: the transition 
8 Single-eld (p)reheating and the growth of small-scale perturbations 
8.1 Generalities and growth of small-scale perturbations during (p)reheating (article)
8.2 Metric preheating and formation of Primordial Black Holes
8.3 Temperature of reheating
9 Multield { multi-uids reheating and the evolution of isocurvature perturbations 
9.1 General formalism and application to double ination (article)
9.2 Going further: a few possibilities
Conclusion and prospects
Compte-rendu en francais
Bibliography

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