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.
