Consistency relations between the source terms in the second-order Einstein equations for cosmological perturbations

TL;DR

This paper derives and confirms the self-consistency of second-order Einstein equations and matter equations in cosmological perturbation theory, ensuring all perturbation modes are correctly included and consistent.

Contribution

It provides a gauge-invariant derivation of second-order perturbations of Einstein and matter equations, including all modes, and verifies their mutual consistency.

Findings

All second-order Einstein and matter equations are mutually consistent.

The derivations are gauge-invariant and include all perturbation modes.

The set of equations is confirmed to be self-consistent.

Abstract

In addition to the second-order Einstein equations on four-dimensional homogeneous isotropic background universe filled with the single perfect fluid, we also derived the second-order perturbations of the continuity equation and the Euler equation for a perfect fluid in gauge-invariant manner without ignoring any mode of perturbations. The consistency of all equations of the second-order Einstein equation and the equations of motion for matter fields is confirmed. Due to this consistency check, we may say that the set of all equations of the second-order are self-consistent and they are correct in this sense.