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 tau(n boolean AND) of position on the sky. Anisotropies in tau(n boolean AND) 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 tau boolean AND(center dot m) for the modes of the 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 similar to 10% level.