Reionization and cosmic microwave background distortions: A complete treatment of second-order Compton scattering

The ionization history of the Universe provides a major source of ambiguity in constraining cosmological models using small angular scale microwave background anisotropies. To clarify these issues, we consider a complete treatment of Compton scattering to second order, an approach which may be applicable to other astrophysical situations. We find that only the $O\left(v\right)\mathrm{}$ Doppler effect and the $O\left(v\delta \right)$ Vishniac effect are important for recent last scattering epochs and realistic power spectra. The $O\left(v^2\right)$ Doppler effect is not significant on any angular scale, and other higher-order effects are completely negligible. However the $O\left(v^2\right)$ effect does lead to Compton-$y$ distortions, which, although generally below current constraints, set an unavoidable minimum level in reionization models. We consider the small-angle approximation for the Vishniac effect in detail, and show several improvements over previous treatments, particularly for low ${\Omega }_0$. For standard cold dark matter models, the effect of reionization is to redistribute the anisotropies to arcminute scales; late reionization leads to partially erased primary fluctuations and a secondary contribution of comparable magnitude. Using recent anisotropy limits from the ATCA experiment, we set new constraints on baryonic dark matter models. Stronger constraints are imposed (in second order) upon models with higher Hubble constant, steeper $n$, and higher density. These limits depend on the specific ionization history assumed, but the factor gained by lowering the ionization fraction is generally small, and may be tested by currently planned experiments on arcminute scales.