Consistency of Equations for the Single Scalar Field Case in Second-order Gauge-invariant Cosmological Perturbation Theory

TL;DR

This paper derives and confirms the self-consistency of second-order gauge-invariant perturbation equations for a single scalar field in cosmology, ensuring their correctness and discussing their physical implications.

Contribution

It provides a comprehensive derivation and verification of second-order perturbation equations in a gauge-invariant framework for cosmological scalar fields.

Findings

All second-order Einstein and Klein-Gordon equations are consistent and correct.

The equations are self-consistent and physically meaningful.

Discussion of the physical implications of these equations.

Abstract

We derived the second-order perturbations of the Einstein equations and the Klein-Gordon equation for a generic situation in terms of gauge-invariant variables. The consistency of all the equations is confirmed. This confirmation implies that all the derived equations of the second order are self-consistent and these equations are correct in this sense. We also discuss the physical implication of these equations.