Accurate estimation of angular power spectra for maps with correlated masks

The widely used MASTER approach for angular power spectrum estimation was developed as a fast $C_{\ell}$ estimator on limited regions of the sky. This method expresses the power spectrum of a masked map ("pseudo-$C_\ell$") in terms of the power spectrum of the unmasked map (the true $C_\ell$) and that of the mask or weight map. However, it is often the case that the map and mask are correlated in some way, such as point source masks used in cosmic microwave background (CMB) analyses, which have nonzero correlation with CMB secondary anisotropy fields and other mm-wave sky signals. In such situations, the MASTER approach gives biased results, as it assumes that the unmasked map and mask have zero correlation. While such effects have been discussed before with regard to specific physical models, here we derive a completely general formalism for any case where the map and mask are correlated. We show that our result ("reMASTERed") reconstructs ensemble-averaged pseudo-$C_\ell$ to effectively exact precision, with significant improvements over traditional estimators for cases where the map and mask are correlated. In particular, we obtain an improvement in the mean absolute percent error from 30% with the MASTER result to essentially no error with the reMASTERed result for an integrated Sachs-Wolfe (ISW) field map with a mask built from the thresholded ISW field, and 10% to effectively zero for a Compton-$y$ map combined with an infrared source mask (the latter being directly relevant to actual data analysis). An important consequence of our result is that for maps with correlated masks it is no longer possible to invert a simple equation to obtain the true $C_\ell$ from the pseudo-$C_\ell$. Instead, our result necessitates the use of forward modeling from theory space into the observable domain of the pseudo-$C_\ell$. Our code is publicly available at https://github.com/kmsurrao/reMASTERed.