Evolution of Second-Order Cosmological Perturbations and Non-Gaussianity

TL;DR

This work develops a second-order gauge-invariant perturbation formalism for a Friedmann-Robertson-Walker universe with multiple interacting fluids and applies it to standard single-field inflation, curvaton, and inhomogeneous reheating scenarios. It provides exact expressions for second-order large-scale temperature anisotropies and clarifies the precise definition of the non-linearity parameter $f_{\rm NL}$ as it enters the CMB bispectrum, deriving scenario-dependent predictions. The results show that in the standard scenario $|f_{\rm NL}|$ tends to be small, while the curvaton and inhomogeneous reheating mechanisms can yield larger non-Gaussianities depending on parameters like the transfer coefficient $r$ or fluctuations in the decay rate $\delta\Gamma$. These findings offer a concrete, testable link between early-universe physics and CMB observations from Planck and future missions, enabling discrimination among perturbation-generation mechanisms through non-Gaussianity measurements.

Abstract

We present a second-order gauge-invariant formalism to study the evolution of curvature perturbations in a Friedmann-Robertson-Walker universe filled by multiple interacting fluids. We apply such a general formalism to describe the evolution of the second-order curvature perturbations in the standard one-single field inflation, in the curvaton and in the inhomogeneous reheating scenarios for the generation of the cosmological perturbations. Moreover, we provide the exact expression for the second-order temperature anisotropies on large scales, including second-order gravitational effects and extend the well-known formula for the Sachs-Wolfe effect at linear order. Our findings clarify what is the exact non-linearity parameter f_NL entering in the determination of higher-order statistics such as the bispectrum of Cosmic Microwave Background temperature anisotropies. Finally, we compute the level of non-Gaussianity in each scenario for the creation of cosmological perturbations.