1+3 covariant cosmic microwave background anisotropies I: Algebraic relations for mode and multipole expansions

摘要

This is the first of a series of papers systematically extending a 1 + 3 covariant and gauge-invariant treatment of kinetic theory in curved space-times to a treatment of cosmic microwave background temperature anisotropies arising from inhomogeneities in the early universe. The present paper deals with algebraic issues, both generically and in the context of models linearised about Robertson-Walker geometries. The approach represents radiation anisotropies by projected symmetric and trace-free tensors. The angular correlation Functions for the mode coefficients are found in terms of these quantities, following the Wilson-Silk approach, but derived and dealt with in 1 + 3 covariant and gauge-invariant form. The covariant multipole and mode-expanded angular correlation functions are related to the usual treatments in the literature. The 1 + 3 covariant and gauge-invariant mode expansion is related to the coordinate approach by linking the Legendre functions to the projected symmetric trace-free representation, using a covariant addition theorem for the tensors to generate the Legendre polynomial recursion relation. This paper lays the foundation for further papers in the series, which use this formalism in a covariant and gauge-invariant approach to developing solutions of the Boltzmann and Liouville equations for the cosmic microwave background before and after decoupling, thus providing a unified covariant and gauge-invariant derivation of the variety of approaches to cosmic microwave background anisotropies in the current literature, as well as a basis for extension of the theory to include nonlinearities. (C) 2000 Academic Press. [References: 40]