Beating cosmic variance with CMB polarization

Part of the CMB polarization signal in the direction of galaxy clusters is produced by Thomson scattering of the CMB temperature quadrupole. In principle this allows measurement of the CMB power spectrum harmonic C_2(z) with higher accuracy (at z>0) than the cosmic variance limit imposed by sample variance on one CMB sky. However the observed signals are correlated if the comoving separation between the clusters is small enough. Thus one cannot reduce the sample variance by more than roughly the number of separate regions available which produce uncorrelated signals. The correlation of the polarization signals is obtained by considering the statistics of the spherical harmonic expansion coefficients of the temperature anisotropy. We consider only the case of scalar anisotropies in a flat universe. We find statistical estimators of the angular power spectrum harmonic C_2 at a given redshift in terms of a set of Stokes parameter measurements in arbitrary directions on the sky at that redshift, and show that these estimators beat the cosmic variance limit. With clusters out to z=2 we find that it is possible to reduce the cosmic variance in C_2 from approximately 60% of the ensemble average to 30%. We discuss the implications for reconstruction of the primordial potential on large scales.