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 anticorrelated, allowing optical depth estimates based entirely on redshifted 21 cm measurements. This should help us to isolate the cosmic microwave background (CMB) polarization fluctuations due to reionization, allowing for the removal of the patchy reionization polarization signal from the other polarization signals and for a measurement of the primordial CMB quadrupole at various locations in the universe at the epoch of reionization. This latter application, in principle, allows for three-dimensional mapping of the primordial density field at z~1100 over a large fraction of the Hubble volume.