Cosmological perturbations in the early universe

Abstract

We elucidate and somewhat extend Bardeen's gauge-invariant formalism for calculating the growth of linear gravitational perturbations in a Friedmann-Robertson-Walker cosmological background. We show that the formalism can be derived from the usual gravitational Lagrangian, by variation with respect to a restricted set of metric perturbation functions. This approach produces a natural decomposition of an arbitrary matter field (whose constitutive equations need not resemble the usual cosmological perfect fluid) into a spatially homogeneous piece, which couples to the background metric, plus a spatially inhomogeneous piece, which is not necessarily small and which is the source term in a second-order differential equation which evolves the gauge-invariant metric perturbation potential. We show how the complete perturbed metric can be reconstructed in arbitrary gauge from the single gauge-independent metric potential, so that the evolution of the matter fields can be concurrently calculated in the usual manner (i.e., in a perturbed coordinate frame). The approach of this paper is designed to be particularly suited to the study of fluctuations generated by classical scalar or gauge fields in "inflationary" cosmological models.