Analysis of the Kamionkowski-Loeb method of reducing 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 statistically 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, as first pointed out by Kamionkowski and Loeb. In this paper we analyze in detail the procedure outlined by Kamionkowski and Loeb, computing the correlation of the polarization signals by considering the variation of the spherical harmonic expansion coefficients of the temperature anisotropy on our past light cone. We find that in fact the cosmic variance is not reducible below the single sky CMB value using the cluster method. Thus this method is not likely to be of use for reconstruction of the primordial power spectrum. However the method does yield a measurement of $C_2$ as a function of redshift with increasing accuracy at higher redshift, and thus potentially a probe of the mechanism which may have suppressed the quadrupole. (abridged)