1+3 Covariant Cosmic Microwave Background anisotropies I: Algebraic relations for mode and multipole representations

Abstract

This is the first of a series of papers extending a covariant and gauge invariant (CGI) treatment of kinetic theory in curved space-times to a treatment of Cosmic Background Radiation (CBR) temperature anisotropies arising from inhomogeneities in the early universe. This paper deals with algebraic issues, both generically and in the context of models linearised about RW geometries. The approach represents radiation anisotropies by PSTF 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 CGI form. The covariant multipole and mode-expanded angular correlation functions are related to the usual treatments in the literature. The CGI mode expansion is related to the coordinate approach by linking the Legendre functions to the PSTF 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 CGI approach to develop solutions of the Boltzmann and Liouville equations for the CBR before and after decoupling, thus providing a unified CGI derivation of the variety of approaches to CBR anisotropies in the current literature.