Thermalization and spectral distortions of the cosmic background radiation

Thermalization under double Compton scattering, bremsstrahlung, and elastic Compton scattering of an early energy injection of photons to the cosmic background radiation is examined both analytically and numerically. We find that spectral distortions in the Wien region can include negative as well as positive chemical potential distortions. The former must be evaluated numerically, whereas for the latter we compare analytic and numerical solutions. Spectral distortions in the Rayleigh-Jeans region are fixed by the Wien distortions and by the baryon density ${\ensuremath{\Omega}}_{B}{h}^{2}$. Thus, the Rayleigh-Jeans region can potentially provide constraints to ${\ensuremath{\Omega}}_{B}{h}^{2}$ should spectral distortions be seen in the cosmic background radiation. From an analysis of the time evolution of chemical potential distortions and the observational constraints on such distortions, significant energy injection at ${z}_{h}\ensuremath{\lesssim}5\ifmmode\times\else\texttimes\fi{}{10}^{5}{({\ensuremath{\Omega}}_{B}{h}^{2})}^{\ensuremath{-}\frac{2}{5}}$ is ruled out in all but a very small class of injection scenarios.