The reionization history at high redshifts. II. Estimating the optical depth to Thomson scattering from cosmic microwave background polarization

In light of the recent inference of a high optical depth tau to Thomson scattering from the Wilkinson Microwave Anisotropy Probe (WMAP) data, we investigate the effects of extended periods of partial ionization and ask if the value of tau inferred by assuming a single sharp transition is an unbiased estimate. We construct and consider several representative ionization models and evaluate their signatures in the cosmic microwave background (CMB). If tau is estimated with a single sharp transition, we show that there can be a significant bias in the derived value ( and, therefore, a bias in sigma(8) as well). For WMAP noise levels, the bias in tau is smaller than the statistical uncertainty, but for Planck or a cosmic variance limited experiment the tau bias could be much larger than the statistical uncertainties. This bias can be reduced in the ionization models we consider by fitting a slightly more complicated ionization history, such as a two-step ionization process. Assuming this two-step process, we find that the Planck satellite can simultaneously determine the initial red-shift of reionization to +/- 2 and tau to +/- 0.01. Uncertainty about the ionization history appears to provide a limit of similar to 0.005 on how well tau can be estimated from CMB polarization data, much better than expected from WMAP but significantly worse than expected from cosmic variance limits.