Reconstructing the Thomson Optical Depth due to Patchy Reionization with 21-cm Fluctuation Maps

Large fluctuations in the electron column density can occur during the reionization process. We investigate the possibility of deriving the electron density fluctuations through detailed mapping of the redshifted 21-cm emission from the neutral medium during reionization. We find that the electron-scattering optical depth and 21-cm differential brightness temperature are strongly anti-correlated, allowing optical depth estimates based entirely on redshifted 21-cm measurements. This should help isolate the CMB polarization fluctuations that are due to reionization, allowing both cleaning of the patchy reionization polarization signal as a contaminating source of confusion to other signals and a measurement of the primordial quadrupole that would be measured at various locations in the universe at the epoch of reionization. This latter application in principle allows mapping of the primordial density field at z~1100 over a large fraction of the Hubble volume.