Integral Solution for the Microwave Background Anisotropies in Nonflat Universes

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Abstract
We present an efficient method of computing cosmic microwave background (CMB) anisotropies in nonflat universes. First, we derive the Boltzmann equation for CMB temperature and polarization fluctuations produced by scalar perturbations in a general Robertson-Walker universe. We then apply the integral method in order to solve this equation, writing temperature and polarization anisotropies as a time integral over a geometrical term and a source term. The geometrical terms can be written using ultraspherical Bessel functions, which depend on curvature. These cannot be precomputed, as for flat space. Instead, we solve their differential equation directly for selected values of the multipoles. The resulting computational time is comparable to the flat-space case, improving over previous methods by 2-3 orders of magnitude. This allows one to compute highly accurate CMB temperature and polarization spectra, matter transfer functions, and their CMB normalizations for any cosmological model, thereby avoiding the need to use the various inexact fitting formulae that exist in the literature.