Inflation, Cosmic Perturbations and Non-Gaussianities

TL;DR

$P_\zeta(k)$ is given by $P_\zeta(k)=\frac{H^2}{8\pi^2\epsilon M_p^2}$ in the slow-roll regime, and the tensor-to-scalar ratio obeys $r=16\epsilon$, with the tensor tilt $n_T=-\frac{r}{8}$. The review synthesizes linear and nonlinear perturbation theory via the in-in formalism and the $\delta N$ formalism, showing how a wide class of inflationary models imprint characteristic non-Gaussian signatures through local, equilateral, and quasi-local shapes, including the curvaton, modulated-reheating, and quasi-single-field scenarios. It highlights the role of the EFT of perturbations and Horndeski-type theories in extending the landscape of viable inflationary Lagrangians, and discusses observational constraints on $n_s$, $r$, and $f_{NL}$ as discriminants among models. The work emphasizes the predictive power of a conserved comoving curvature perturbation on super-Hubble scales and the importance of soft limits and unitarity in shaping higher-point correlation functions, linking early-universe dynamics to present-day cosmological data. The practical impact lies in guiding model-building and data analysis for current and upcoming CMB and large-scale structure surveys to constrain inflationary physics with non-Gaussian observables and gravitational waves.

Abstract

We review the theory of inflationary perturbations. Perturbations at both linear and nonlinear orders are reviewed. We also review a variety of inflation models, emphasizing their signatures on cosmic perturbations.

Figures (28)

• Figure 6: Left panel: a rolling scalar field is a modest deformation of pure vacuum energy. Right panel: the inflationary potential. The shape of the inflationary potential is largely unconstrained, as long as it is flat enough. Inflation ends at where the potential is no longer flat and a process of reheating follows to reheat the universe.
• Figure 10: The ADM decomposition of the metric. The green arrows are the components of the metric, namely, lapse of time $Ndt$, a shift of hypersurface due to the time flow $N^idt$ and the spatial distance $dx^i$. The distance $N^idt+dx^i$ is first measured by 3-metric $h_{ij}$, and then together with $Ndt$, makes $ds$. If $dx^i=0$, the time part themselves sums up to $dt^\mu$, along the flow direction of time.
• Figure 11: The meaning of $\zeta$, $E$ and $h_{ij}$. $\zeta$ is an overall rescaling. $E$ rescale the points with respect to its original radius to the origin. $h_{ij}$ rescale the points with respect to its own direction.
• Figure 12: Time difference between local FRW universes. After reheating, the universe become radiation dominated and the energy density dilutes quickly with the expansion of the universe. The change of color illustrate the drop of energy density.
• Figure 13: The left panel is the contour of integration in the in-in formalism. In the right panel, the contour is deformed to a vertical line running top-down. The deformed contour is used numerically in Chen:2009zpChen:2012ge and more generally in Behbahani:2012beGreen:2013rd.