1+3 covariant cosmic microwave background anisotropies II: The almost-Friedmann-Lemaitre model

lThis 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 mi crowave background temperature anisotropies arising from inhomogeneities in the early universe. The first paper (Paper Ij 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 o f the standard derivation using the mode recursion relations, second, demon strating the link to the the multipole divergence equations and finally var ious analytic ways of solving the resulting equations are discussed. A gene ral integral Form of solution is obtained for the equations with Thomson sc attering. The covariant Friedmann-Lemaitre multipole form of the transport equations are found near tight-coupling using the covariant and gauge-invar iant 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 m atter dominated free-streaming projection, slow-decoupling, and tight-coupl ing cases in covariant and gauge-invariant theory, with the aim of both giv ing a unified transparent derivation of this range of results and clarifyin g 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). (C) 20 00 Academic Press.