Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --

TL;DR

The paper develops a complete second-order gauge invariant cosmological perturbation theory on a Friedmann–Robertson–Walker background for two matter models: a single perfect fluid and a single scalar field. Building on a general gauge invariant perturbation framework, it defines gauge invariant variables for both metric and matter at first and second order and derives the corresponding Einstein equations purely in terms of these variables, without gauge fixing. A key result is that second-order vector and tensor modes can be generated by mode coupling of first-order scalar perturbations, with explicit master equations for the scalar sector and structured source terms Γ that encode quadratic nonlinearities. The formalism provides a clean, gauge independent route to study non-Gaussianity and nonlinear effects in early universe cosmology and offers a bridge to compare with and extend previous approaches. Overall, the work yields a self-contained, first-principles framework for evolving second-order cosmological perturbations in a rigorous gauge invariant manner, applicable to inflationary and post-inflationary epochs.

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 formulate the second order gauge invariant cosmological perturbation theory in a four dimensional homogeneous isotropic universe. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We derive the all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. All equations are derived in terms of gauge invariant variables without any gauge fixing. These equations imply that the second order vector and tensor modes may be generated due to the mode-mode coupling of the linear order scalar perturbations. We also briefly discuss the main progress of this work by the comparison with some literatures.