Probing patchy reionization through tau-21cm correlation statistics

We consider the cross-correlation between free electrons and neutral hydrogen during the epoch of reionization. The free electrons are traced by the optical depth to reionization tau, while the neutral hydrogen can be observed through 21 cm photon emission. As expected, this correlation is sensitive to the detailed physics of reionization. Foremost, if reionization occurs through the merger of relatively large halos hosting an ionizing source, the free electrons and neutral hydrogen are anti-correlated for most of the reionization history. A positive contribution to the correlation can occur when the halos that can form an ionizing source are small. A measurement of this sign change in the cross-correlation would directly measure the bubble bias and as, such, the halo mass. We estimate the signal-to-noise of the cross-correlation using the estimator for inhomogeneous reionization tau_{lm} proposed by Dvorkin and Smith (2009). We find that with upcoming radio interferometers and CMB experiments, the cross-correlation is measurable going up to multipoles l ~ 1000. We also derive parameter constraints and conclude that, despite the foregrounds, the cross-correlation proofs a complementary measurement of the EoR parameters to the 21 cm and CMB polarization auto-correlations expected to be observed in the coming decade.