1+3 Covariant Cosmic Microwave Background Anisotropies II: The Almost-Friedmann–Lemaı̂tre Model

Abstract

This is the second of a series of papers extending the 1+3 covariant and gauge-invariant treatment of kinetic theory to an examination of cosmic microwave background temperature anisotropies arising from inhomogeneities in the early universe. The first paper (Paper I) dealt with algebraic issues, representing anisotropies in a covariant and gauge-invariant way by means of projected symmetric and trace-free tensors. Here we derive the mode form of the integrated Boltzmann equations, first, giving a covariant version of the standard derivation using the mode recursion relations, second, demonstrating the link to the the multipole divergence equations and finally various analytic ways of solving the resulting equations are discussed. A general integral form of solution is obtained for the equations with Thomson scattering. The covariant Friedmann–Lemaı̂tre multipole form of the transport equations are found near tight-coupling using the covariant and gauge-invariant generalization of the Peebles and Yu expansion in Thompson scattering time. The dispersion relations and damping scale are then obtained from the covariant approach. The equations are integrated to give the covariant and gauge-invariant equivalent of the canonical scalar sourced anisotropies in the K=0 (flat background) case. We carry out a simple treatment of the matter dominated free-streaming projection, slow-decoupling, and tight-coupling cases in covariant and gauge-invariant theory, with the aim of both giving a unified transparent derivation of this range of results and clarifying the formal connection between the usual approaches (for example, works by Hu and Sugiyama) and the covariant and gauge-invariant like treatments for scalar perturbations (for example, works by Challinor and Lasenby).