Microwave background polarization in cosmological models

Abstract

We introduce a new multipole formalism for polarized radiative transfer in general spacetime geometries. The polarization tensor is expanded in terms of coordinate-independent, projected symmetric trace-free (PSTF) tensor-valued multipoles. The PSTF representation allows us to discuss easily the observer dependence of the multipoles of the polarization, and to formulate the exact dynamics of the radiation in convenient 1+3 covariant form. For the case of an almost-Friedmann-Robertson-Walker (FRW) cosmological model we recast the Boltzmann equation for the polarization in to a hierarchy of multipole equations. This allows us to give a rigorous treatment of the generation and propagation of the polarization of the cosmic microwave background in almost-FRW models (with open, closed or flat geometries) without recourse to any harmonic decomposition of the perturbations. We also show how expanding the intensity and polarization multipoles in derivatives of harmonic functions gives a streamlined derivation of the mode-expanded multipole hierarchies. Integral solutions to these hierarchies are provided, and the relation of our formalism to others in the literature is discussed.