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. Given a hypothetical set of Stokes parameter measurements of the CMB polarization in the directions of galaxy clusters, distributed at random on a given redshift shell, we show how to construct an estimator of the angular power spectrum harmonic ${C}_{2}$ at that redshift. We then compare the variance of this estimator with the cosmic variance of the CMB multipole on our sky which probes the same scale. 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. We also examine to what extent the redshift dependence of ${C}_{2}$ can be used to probe the time changing potential anisotropy as the universe evolves into the vacuum dominated phase (the late-time integrated Sachs-Wolfe effect). We find that this effect is not observable in the time dependence of ${C}_{2}$ since it is swamped by cosmic variance, but there is an observable signature in the correlation functions of the Stokes parameters.