De-Rotation of the Cosmic Microwave Background Polarization: Full-Sky Formalism

Mechanisms have been proposed that might rotate the linear polarization of the cosmic microwave background (CMB) as it propagates from the surface of last scatter. In the simplest scenario, the rotation will be uniform across the sky, but the rotation angle may also vary across the sky. We develop in detail the complete set of full-sky quadratic estimators for the rotation of the CMB polarization that can be constructed from the CMB temperature and polarization. We derive the variance with which these estimators can be measured and show that these variances reduce to the simpler flat-sky expressions in the appropriate limit. We evaluate the variances numerically. While the flat-sky formalism may be suitable if the rotation angle arises as a realization of a random field, the full-sky formalism will be required to search for rotations that vary slowly across the sky as well as for models in which the angular power spectrum for the rotation angle peaks at large angles.