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 τ 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 τ 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 τ 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 σ8 as well). For WMAP noise levels, the bias in τ is smaller than the statistical uncertainty, but for Planck or a cosmic variance limited experiment the τ 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 redshift of reionization to ±2 and τ to ±0.01. Uncertainty about the ionization history appears to provide a limit of ~0.005 on how well τ can be estimated from CMB polarization data, much better than expected from WMAP but significantly worse than expected from cosmic variance limits.