Green's function of the cosmological thermalization problem

Energy release in the early Universe leads to spectral distortions of the cosmic microwave background (CMB) which in the future might allow probing different physical processes in the pre-recombination (z>~10³) epoch. Depending on the energy injection history, the associated distortion partially thermalizes due to the combined action of Compton scattering, double Compton scattering and Bremsstrahlung emission, a problem that in general is hard to solve. Various analytic approximations describing the resulting distortion exist, however, for small distortions and fixed background cosmology the Green's function of the problem can be pre-computed numerically. Here we show that this approach gives very accurate results for a wide range of thermal histories, allowing fast and quasi-exact computation of the spectral distortion given the energy release rate. Our method is thus useful for forecasts of possible constraints on early-universe physics obtained from future measurements of the CMB spectrum.