Reconstructing patchy reionization from the cosmic microwave background

We introduce a new statistical technique for extracting the inhomogeneous reionization signal from future high-sensitivity measurements of the cosmic microwave background (CMB) temperature and polarization fields. If reionization is inhomogeneous, then the optical depth to recombination will be a function $\ensuremath{\tau}(\stackrel{^}{\mathbf{n}})$ of position on the sky. Anisotropies in $\ensuremath{\tau}(\stackrel{^}{\mathbf{n}})$ alter the statistics of the observed CMB via several physical mechanisms: screening of the surface of last scattering, generation of new polarization via Thomson scattering from reionization bubbles, and the kinetic Sunyaev-Zel'dovich effect. We construct a quadratic estimator ${\stackrel{^}{\ensuremath{\tau}}}_{\ensuremath{\ell}m}$ for the modes of the $\ensuremath{\tau}$ field. This estimator separates the patchy reionization signal from the CMB in the form of a noisy map, which can be cross correlated with other probes of reionization or used as a standalone probe. A future satellite experiment with sufficient sensitivity and resolution to measure the lensed $B$ modes on most of the sky can constrain key parameters of patchy reionization, such as the duration of the patchy epoch or the mean bubble radius, at the $\ensuremath{\sim}10%$ level.