Cosmological perturbations and the physical meaning of gauge-invariant variables

Abstract

This paper concerns gauge-invariant perturbations of Robertson-Walker spacetimes, with the aim of (1) giving a complete set of perturbation equations and (2) comparing the coordinate-based method of Bardeen with the covariant approach of Ellis & Bruni. To this end, we first consider covariantly defined quantities which are gauge-invariant in a perturbed Robertson-Walker universe; for these variables we derive a complete set of covariant linearized equations as they follow from the Bianchi and Ricci identities, and we show various possible ways of obtaining a second-order linear equation for the density perturbation variables. Then we systematically expand the covariant and gauge-invariant variables, recovering Bardeen's variables as first-order terms in this expansion: thus the two sets of variables are equivalent to first order. Through this comparison Bardeen's variables are shown to have a natural physical and geometrical meaning, which can be determined without the need of a gauge specification, and Bardeen's equations follow directly. All equations are devised using the hydrodynamic approximation for a fluid with the energy momentum tensor including viscous terms.