General solution to theE−Bmixing problem

Abstract

We derive a general ansatz for optimizing pseudo-${C}_{\ensuremath{\ell}}$ estimators used to measure cosmic microwave background anisotropy power spectra, and apply it to the recently proposed pure pseudo-${C}_{\ensuremath{\ell}}$ formalism, to obtain an estimator which achieves near-optimal $B$-mode power spectrum errors for any specified noise distribution while minimizing leakage from ambiguous modes. Our technique should be relevant for upcoming cosmic microwave background polarization experiments searching for $B$-mode polarization. We compare our technique both to the theoretical limits based on a full Fisher matrix calculation and to the standard pseudo-${C}_{\ensuremath{\ell}}$ technique. We demonstrate it by applying it to a fiducial survey with realistic inhomogeneous noise, complex boundaries, point source masking, and a noise level comparable to what is expected for next-generation experiments ($\ensuremath{\sim}5.75\text{ }\text{ }\ensuremath{\mu}\mathrm{K}\mathrm{\text{\ensuremath{-}}}\mathrm{arcmin}$). For such an experiment our technique could improve the constraints on the amplitude of a gravity wave background by over a factor of 10 compared to what could be obtained using ordinary pseudo-${C}_{\ensuremath{\ell}}$, coming quite close to saturating the theoretical limit. Constraints on the amplitude of the lensing $B$-modes are improved by about a factor of 3.