Cosmological perturbations: a new gauge-invariant approach

TL;DR

This paper introduces a new gauge-invariant method for analyzing cosmological perturbations by splitting the stress-energy tensor, simplifying the Einstein equations, and providing explicit solutions in a flat universe model.

Contribution

It develops a novel gauge-invariant framework based on stress-energy tensor splitting, enabling decoupled and solvable perturbation equations in cosmology.

Findings

Explicit gauge-invariant Einstein equations for perturbations

Decoupled equations for metric components in flat universe

Solutions expressed as retarded integrals

Abstract

A new gauge-invariant approach for describing cosmological perturbations is developed. It is based on a physically motivated splitting of the stress-energy tensor of the perturbation into two parts - the bare perturbation and the complementary perturbation associated with stresses in the background gravitational field induced by the introduction of the bare perturbation. The complementary perturbation of the stress-energy tensor is explicitly singled out and taken to the left side of the perturbed Einstein equations so that the bare stress-energy tensor is the sole source for the perturbation of the metric tensor and both sides of these equations are gauge invariant with respect to infinitesimal coordinate transformations. For simplicity we analyze the perturbations of the spatially-flat Friedmann-Lemaitre-Robertson-Walker dust model. A cosmological gauge can be chosen such that the equations for the perturbations of the metric tensor are completely decoupled for the h_{00}, h_{0i}, and h_{ij} metric components and explicitly solvable in terms of retarded integrals.