Fast estimation of polarization power spectra using correlation functions

We present a fast method for estimating the cosmic microwave background polarization power spectra using unbiased estimates of heuristically weighted correlation functions. This extends the O(N-pix(3/2)) method of Szapudi et al. to polarized data. If the sky coverage allows the correlation functions to be estimated over the full range of angular separations, they can be inverted directly with integral transforms and clean separation of the electric (E) and magnetic (B) modes of polarization is obtained exactly in the mean. We assess the level of E-B mixing that arises from apodized integral transforms when the correlation function can only be estimated for a subset of angular scales, and show that it is significant for small-area observations. We introduce new estimators to deal with this case on the spherical sky that preserve E-B separation; their construction requires an additional integration of the correlation functions but the computational cost is negligible. We illustrate our methods with application to a large-area survey with parameters similar to Planck, and the small-area Background Imaging of Cosmic Extragalactic Polarization experiment. In both cases we show that the errors on the recovered power spectra are close to theoretical expectations.