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

This is the first of a series of papers systematically extending a 1 + 3 co variant and gauge-invariant treatment of kinetic theory in curved space-tim es to a treatment of cosmic microwave background temperature anisotropies a rising from inhomogeneities in the early universe. The present paper deals with algebraic issues, both generically and in the context of models linear ised about Robertson-Walker geometries. The approach represents radiation a nisotropies by projected symmetric and trace-free tensors. The angular corr elation Functions for the mode coefficients are found in terms of these qua ntities, 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 i s related to the coordinate approach by linking the Legendre functions to t he projected symmetric trace-free representation, using a covariant additio n theorem for the tensors to generate the Legendre polynomial recursion rel ation. This paper lays the foundation for further papers in the series, whi ch use this formalism in a covariant and gauge-invariant approach to develo ping solutions of the Boltzmann and Liouville equations for the cosmic micr owave background before and after decoupling, thus providing a unified cova riant 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 Acade mic Press.