Perturbation theory based on the Einstein–Boltzmann system. I. Illustration of the theory for a Robertson–Walker geometry

Abstract

This article is a discussion of how it is possible to do perturbation theory for the Einstein–Boltzmann system about a dust solution. Explicitly expressed, the background Robertson–Walker universe model and one-parameter families of exact solutions are applied to the Einstein–Boltzmann system to obtain a closed-form solution of the equations governing linearized perturbations at late times, when the nonzero pressure of the gas of massive particles may be regarded as being significantly smaller than the energy density. After splitting the distribution function into two structurally different parts, the analysis given here provides a means of deriving the equations of linear hydrodynamics. In fact, due to the specific properties of the background chosen, one can prove that the evolution of suitably defined hydrodynamic variables is exactly decoupled from the evolution of the phase-space function satisfying the linearized Boltzmann equation. For simplicity, the workings of the method are illustrated by assuming that the perturbed metric is also of the Robertson–Walker form. A detailed treatment of the effect of inhomogeneities in an almost-Robertson–Walker universe model will be the subject of the last article in this series.