Integral Solution for the Microwave Background Anisotropies in Non-fl at Universes

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Abstract
We present an efficient method to compute CMB anisotropies in non-flat universes. First we derive the Boltzmann equation for cosmic microwave background temperature and polarization fluctuations produced by scalar perturbations in a general Robertson-Walker universe. We then apply the integral method 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 ultra-spherical Bessel functions, which depend on curvature. These cannot be precomputed in advance as in flat space. Instead we solve directly their differential equation for selected values of the multipoles. The resulting computational time is comparable to the flat space case and improves 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 various approximate fitting formulae that exist in the literature.

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