Second-order cosmological perturbations produced by scalar-scalar coupling during inflation stage

TL;DR

This paper analytically derives second-order cosmological perturbations during inflation driven by a scalar field, revealing the coupling between scalar modes and the behavior of tensor and vector perturbations.

Contribution

It provides the first comprehensive analytical solutions for second-order scalar, vector, and tensor perturbations during inflation, including scalar-scalar coupling effects.

Findings

Second-order scalar and tensor perturbations are coupled waves.

Vector perturbations at second order are also wave-like.

Solutions include homogeneous and inhomogeneous parts with effective sources.

Abstract

We study the perturbations up to the 2nd-order for a power-law inflation driven by a scalar field in synchronous coordinates. We present the 1st-order solutions, and analytically solve the 2nd-order perturbed Einstein equation and scalar field equation, give the 2nd-order solutions for all the scalar, vector, and tensor metric perturbations, as well as the perturbed scalar field. During inflation, the 1st-order tensor perturbation is a wave and is decoupled from other perturbations, the scalar metric perturbation and the perturbed scalar field are coupled waves, propagating at the speed of light, differing from those in the dust and relativistic fluid models. The 1st-order vector perturbation is not wave and just decreases during inflation. The 2nd-order perturbed Einstein equation is similar in structure to the 1st-order one, but various products of the 1st-order perturbations occur as the effective source, among which the scalar-scalar coupling is considered in this paper. The solutions of all the 2nd-order perturbations consist of a homogeneous part similar to the 1st-order solutions, and an inhomogeneous part in a form of integrations of the effective source. The 2nd-order vector perturbation is also a wave since the effective source is composed of the 1st-order waves. We perform the residual gauge transformations between synchronous coordinates up to the 2nd-order, and identify the 1st-order and 2nd-order gauge modes.