INTEGRAL SOLUTION FOR THE MICROWAVE BACKGROUND ANISOTROPIES IN NONFLAT UNIVERSES

We present an efficient method of computing cosmic microwave backgroun d (CMB) anisotropies in nonflat universes. First, we derive the Boltzm ann equation for CMB temperature and polarization fluctuations produce d by scalar perturbations in a general Robertson-Walker universe. We t hen apply the integral method in order to solve this equation, writing temperature and polarization anisotropies as a time integral over a g eometrical term and a source term. The geometrical terms can be writte n using ultraspherical Bessel functions, which depend on curvature. Th ese 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 allo ws one to compute highly accurate CMB temperature and polarization spe ctra, matter transfer functions, and their CMB normalizations for any cosmological model, thereby avoiding the need to use the various inexa ct fitting formulae that exist in the literature.