Inclusion of the first-order vector- and tensor-modes in the second-order gauge-invariant cosmological perturbation theory

TL;DR

This paper develops a gauge-invariant framework for second-order cosmological perturbations, including vector and tensor modes, revealing mode-coupling effects in Einstein's equations without gauge fixing.

Contribution

It introduces a comprehensive gauge-invariant formulation that incorporates first-order vector and tensor modes into second-order cosmological perturbation theory.

Findings

Derived Einstein equations showing mode-coupling at second order

Formulated gauge-invariant treatment without gauge fixing

Included all modes in the perturbation analysis

Abstract

Gauge-invariant treatments of the second-order cosmological perturbation in a four dimensional homogeneous isotropic universe are formulated without any gauge fixing. We have derived the Einstein equations in the case of the single perfect fluid without ignoring any modes. These equations imply that any types of mode-coupling arise due to the second-order effects of the Einstein equations.