The Reionization History at High Redshifts II: Estimating the Optical Depth to Thomson Scattering from CMB Polarization

In light of the recent inference of a high optical depth to Thomson scattering, tau, from the 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 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 sigma8 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 the Planck satellite can simultaneously determine the initial redshift of reionization to +-2 and tau to +-0.01 Uncertainty about the ionization history appears to provide a limit of about 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.