Cosmological Perturbation Theory Revisited

TL;DR

This paper simplifies the formulation of linear cosmological perturbation equations using gauge invariants, facilitating the extension to nonlinear perturbations and comparing different approaches for scalar perturbations.

Contribution

It introduces a concise formulation of linear perturbation equations using intrinsic gauge invariants, aiding future nonlinear analysis.

Findings

Formulation of governing equations in a simple, concise form

Identification of two complementary approaches for scalar perturbations

Establishment of fundamental operators for nonlinear perturbation equations

Abstract

Increasingly accurate observations are driving theoretical cosmology toward the use of more sophisticated descriptions of matter and the study of nonlinear perturbations of FL cosmologies, whose governing equations are notoriously complicated. Our goal in this paper is to formulate the governing equations for linear perturbation theory in a particularly simple and concise form in order to facilitate the extension to nonlinear perturbations. Our approach has several novel features. We show that the use of so-called intrinsic gauge invariants has two advantages. It naturally leads to: (i) a physically motivated choice of a gauge invariant associated with the matter density, and (ii) two distinct and complementary ways of formulating the evolution equations for scalar perturbations, associated with the work of Bardeen and of Kodama and Sasaki. In the first case the perturbed Einstein tensor gives rise to a second order (in time) linear differential operator, and in the second case to a pair of coupled first order (in time) linear differential operators. These operators are of fundamental importance in cosmological perturbation theory, since they provide the leading order terms in the governing equations for nonlinear perturbations.