Second Order Gauge-Invariant Perturbations during Inflation

TL;DR

This paper investigates second-order gauge-invariant scalar perturbations during inflation, analyzing their spectra and the conditions under which second-order effects become significant compared to first-order perturbations.

Contribution

It provides a detailed evaluation of second-order gauge-invariant curvature and scalar field fluctuations during slow-roll inflation, including mode-mode coupling effects and their impact on spectra.

Findings

Second-order spectra are nearly scale-invariant with logarithmic corrections.

The amplitude depends on the total number of e-folds.

An upper limit exists for e-folds where second-order effects become comparable to first-order.

Abstract

The evolution of gauge invariant second-order scalar perturbations in a general single field inflationary scenario are presented. Different second order gauge invariant expressions for the curvature are considered. We evaluate perturbatively one of these second order curvature fluctuations and a second order gauge invariant scalar field fluctuation during the slow-roll stage of a massive chaotic inflationary scenario, taking into account the deviation from a pure de Sitter evolution and considering only the contribution of super-Hubble perturbations in mode-mode coupling. The spectra resulting from their contribution to the second order quantum correlation function are nearly scale-invariant, with additional logarithmic corrections to the first order spectrum. For all scales of interest the amplitude of these spectra depend on the total number of e-folds. We find, on comparing first and second order perturbation results, an upper limit to the total number of e-folds beyond which the two orders are comparable.