Aspects of Inflationary Cosmology

The inflationary community has undoubtedly been on a rollercoaster ride during the 2014. At first, given the detection of a high level of $B$-modes polarization around the scale of the recombination bump, there was a strong indication that the simplest potential $V=\frac12 m^2\phi^2$ could represent the benchmark model for inflation. This suggested to look for observables that could parametrize deviations from this benchmark. For a quadratic potential, the quantity $(n_s-1)+r/4+11 (n_s-1)^2/24$ vanishes (up to corrections which are cubic in slow roll) and can be used to parametrize small deviations from the minimal scenario independently on the reheating process. Constraints on this quantity would be able to distinguish a quadratic potential from a pseudo-Nambu-Goldstone boson with $f \lesssim 30 \Mpl$ and set limits on the deviation from unity of the speed of sound $| c_s-1| \lesssim 3\times 10^{-2}$ (corresponding to an energy scale $\Lambda\gtrsim 2\times 10^{16}\, \mathrm{GeV}$), and on the contribution of a second field to perturbations ($\lesssim 6 \times 10^{-2}$). Furthermore, for the quadratic potential, one can provide predictions accurate up to $1\%$ for the spectral index $n_s$ and the tensor-to-scalar ratio $r$ assuming instantaneous reheating and a standard thermal history: $n_s = 0.9668\pm0.0003$ and $r=0.131\pm 0.001$. This represents the simplest and most informative point in the $(n_s,r)$ plane. The result is independent of the details of reheating (or preheating) provided the conversion to radiation is sufficiently fast. A slower reheating or a modified post-inflationary evolution (with an equation of state parameter $w\leq 1/3$) push towards smaller $n_s$ (and larger $r$), so that our prediction corresponds to the maximum $n_s$ (and minimum $r$) for the quadratic potential. The relations and the predictions so far considered can be derived for a general $V \propto \phi^p$ potential, however this typically requires some additional assumption. Eventually, the presence of $B$-modes polarization was mainly due to Galactic dust. The part due to primordial signal is still unknown. Is there any theoretical prior to guess what is the size of the primordial tensor modes? In this respect, we investigated the possible implications of the measured value of the scalar tilt $n_s$ for the tensor-to-scalar ratio $r$ in slow-roll, single-field inflationary models. The measured value of the tilt satisfies $n_s -1\sim 1/N_*$, where $N_* \sim 60$ is the number of $e$-folds for observationally relevant scales. If this is not a coincidence and the scaling holds for different values of $N$, it strongly suggests that either $r$ is as big as $10^{-1}$, or smaller than $10^{-2}$ and exponentially dependent on $n_s$. A large region of the ($n_s$,$r$)-plane is not compatible with this scaling. Given the small value for $r$ we expect, we update the forecasts for various ground-based experiments (AdvACT, CLASS, Keck/BICEP3, Simons Array, SPT-3G), balloons (EBEX 10k and Spider) and satellites (CMBPol, COrE and LiteBIRD), taking into account the recent Planck data on polarized dust and using a component separation method. The forecasts do not change significantly with respect to previous estimates when at least three frequencies are available, provided foregrounds can be accurately described by few parameters. We then argue that a theoretically motivated goal for future experiments, $r\sim2\times10^{-3}$, is achievable if the noise is reduced to $\sim1\,\mu$K-arcmin and lensing is reduced to $10\%$ in power. Of course, the constraints on inflationary cosmology do not lonely come from 2-points statistics. Given the tight bound on the local shape of non-Gaussianities and the room still available for non slow-roll models, we show that in the Effective Field Theory (EFT) of inflation an ISO(4,1) symmetry (like the one in DBI inflation) uniquely fixes, at lowest order in derivatives, all correlation functions in terms of the speed of sound $c_s$. In the limit $c_s\rightarrow1$, the ISO(4,1) symmetry reduces to the Galilean symmetry. On the other hand, we point out that the non-linear realization of SO(4,2), the isometry group of 5D AdS space, does not fix the cubic action in terms of $c_s$. Last, we go beyond the conformal consistency condition for the scalar three-point function. In single-field models the effect of a long mode with momentum $q$ reduces to a diffeomorphism at zeroth and first order in $q$. This gives the well-known consistency relations for the $n$-point functions. At order $q^2$ the long mode has a physical effect on the short ones, since it induces curvature, and we expect that this effect is the same as being in a curved FRW universe. We verify this intuition in various examples of the three-point function, whose behavior at order $q^2$ can be written in terms of the power spectrum in a curved universe. This gives a simple alternative understanding of the level of non-Gaussianity in single-field models. The non-Gaussianity is always parametrically enhanced when modes freeze at a physical scale $k_{\rm ph,\,f}$ shorter than~$H$:~$f_{\rm NL} \sim (k_{\rm ph,\,f}/H)^2$. The outline of this thesis is rather simple. We will introduce the basic concepts that we need in the Introduction. Then, each chapter that follow will treat a particular aspect of the inflationary observables, often related to broad classes of models. Given the diversity of topics contained in this thesis, we preferred to have the conclusions at the end of each chapter.