Consistency of Equations in the Second-order Gauge-invariant Cosmological Perturbation Theory

TL;DR

This paper rigorously re-derives second-order Einstein equations in gauge-invariant cosmological perturbation theory, confirming their internal consistency for universes dominated by perfect fluids or scalar fields.

Contribution

It provides a comprehensive, mode-inclusive derivation of second-order Einstein equations in a gauge-invariant framework, confirming their consistency and correctness.

Findings

All second-order Einstein equations are self-consistent.

The equations are valid for universes with perfect fluid or scalar field dominance.

The derivation confirms the correctness of previously proposed equations.

Abstract

Along the general framework of the gauge-invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we re-derive the second-order Einstein equations on four-dimensional homogeneous isotropic background universe in gauge-invariant manner without ignoring any mode of perturbations. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We also confirmed the consistency of all equations of the second-order Einstein equation and the equations of motion for matter fields which are derived in the paper [K. Nakamura, arXiv:0804.3840 [gr-qc]]. This confirmation implies that the all derived equations of the second order are self-consistent and these equations are correct in this sense.