Constraining a spatially dependent rotation of the Cosmic Microwave
  Background Polarization
  Following Kamionkowski (2008), a quadratic estimator of the rotation of the
plane of polarization of the CMB is constructed. This statistic can estimate a
spatially varying rotation angle. We use this estimator to quantify the
prospects of detecting such a rotation field with forthcoming experiments. For
PLANCK and CMBPol we find that the estimator containing the product of the E
and B components of the polarization field is the most sensitive. The variance
of this EB estimator, N(L) is roughly independent of the multipole L, and is
only weakly dependent on the instrumental beam. For FWHM of the beam size ~
5'-50', and instrument noise $\Delta_p ~ 5-50 uK-arcmin, the scaling of
variance N(L) can be fitted by a power law N(L)=3.3 x 10^{-7} \Delta^2_p
(FWHM)^{1.3} sq-deg. For small instrumental noise \Delta_p \leq 5 uK-arcmin,
the lensing B-modes become important, saturating the variance to ~10^{-6}
sq-deg even for an ideal experiment. Upcoming experiments like PLANCK will be
able to detect a power spectrum of the rotation angle, C^{\alpha \alpha}(L), as
small as 0.01 sq-deg, while futuristic experiment like CMBPol will be able to
detect rotation angle power spectrum as small as 2.5 x 10^{-5} sq-deg. We
discuss the implications of such constraints, both for the various physical
effects that can rotate the polarization as photons travel from the last
scattering surface as well as for constraints on instrumental systematics that
can also lead to a spurious rotation signal. Rotation of the CMB polarization
generates B-modes which will act as contamination for the primordial B-modes
detection. We discuss an application of our estimator to de-rotate the CMB to
increase the sensitivity for the primordial B-modes.
