Perturbations of a Universe Filled with Dust and Radiation

TL;DR

This paper develops a first-order perturbation method for flat Friedmann cosmologies filled with dust and radiation, deriving coupled equations and solutions that align with known dust solutions at large times.

Contribution

It introduces a novel perturbation approach in comoving coordinates for dust and radiation-filled universes, providing coupled equations and power series solutions.

Findings

Coupled equations for metric trace and velocity potential derived

Power series solutions match dust solutions at large times

Neglects radiation vorticity for simplification

Abstract

A first-order perturbation approach to $k=0$ Friedmann cosmologies filled with dust and radiation is developed. Adopting the coordinate gauge comoving with the perturbed matter, and neglecting the vorticity of the radiation, a pair of coupled equations is obtained for the trace $h$ of the metric perturbations and for the velocity potential $v$. A power series solution with upwards cutoff exists such that the leading terms for large values of the dimensionless time $\xi$ agree with the relatively growing terms of the dust solution of Sachs and Wolfe.