Scalar Cosmological Perturbations

TL;DR

This paper reviews and unifies methods for analyzing scalar perturbations in Friedmann-Lemaitre cosmologies using gauge-invariant variables, aiming to simplify equations and connect different approaches, with a view toward extending to second order.

Contribution

It introduces an efficient construction of gauge invariants, unifies various formulations of the perturbation equations, and clarifies the connection between metric-based and 1+3 approaches.

Findings

Developed a method for constructing dimensionless gauge invariants.

Unified different formulations of the governing equations.

Clarified the link between metric-based and 1+3 approaches.

Abstract

Scalar perturbations of Friedmann-Lemaitre cosmologies can be analyzed in a variety of ways using Einstein's field equations, the Ricci and Bianchi identities, or the conservation equations for the stress-energy tensor, and possibly introducing a timelike reference congruence. The common ground is the use of gauge invariants derived from the metric tensor, the stress-energy tensor, or from vectors associated with a reference congruence, as basic variables. Although there is a complication in that there is no unique choice of gauge invariants, we will show that this can be used to advantage. With this in mind our first goal is to present an efficient way of constructing dimensionless gauge invariants associated with the tensors that are involved, and of determining their inter-relationships. Our second goal is to give a unified treatment of the various ways of writing the governing equations in dimensionless form using gauge-invariant variables, showing how simplicity can be achieved by a suitable choice of variables and normalization factors. Our third goal is to elucidate the connection between the metric-based approach and the so-called 1+3 gauge-invariant approach to cosmological perturbations. We restrict our considerations to linear perturbations, but our intent is to set the stage for the extension to second order perturbations.