Fast and accurate CMB computations in non-flat FLRW universes

We present a new method for calculating CMB anisotropies in a non-flat Friedmann universe, relying on a very stable algorithm for the calculation of hyperspherical Bessel functions, that can be pushed to arbitrary precision levels. We also introduce a new approximation scheme which gradually takes over in the flat space limit, and significant speeds-up calculations. Our method is implemented in the Boltzmann code CLASS. It can be used to benchmark the accuracy of the CAMB code in curved space, which is found to match expectations. For default precision settings, corresponding to 0.1% for scalar temperature spectra and 0.2% for scalar polarisation spectra, our code is two to three times faster, depending on curvature. We also simplify the temperature and polarisation source terms significantly, so the different contributions to the $C_\ell$'s are easy to identify inside the code.

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