Probing the Universe on gigaparsec scales with remote cosmic microwave background quadrupole measurements

Scattering of cosmic microwave background (CMB) radiation in galaxy clusters induces a polarization signal proportional to the CMB quadrupole anisotropy at the cluster's location and lookback time. A survey of such remote quadrupole measurements provides information about large-scale cosmological perturbations. This paper presents a formalism for calculating the correlation function of remote quadrupole measurements in spherical harmonic space. The number of independent modes probed by both single-redshift and volume-limited surveys is presented, along with the length scales probed by these modes. In a remote quadrupole survey sparsely covering a large area of sky, the largest-scale modes probe the same-length scales as the quadrupole but with much narrower Fourier-space window functions. The largest-scale modes are significantly correlated with the local CMB, but even when this correlation is projected out the largest remaining modes probe gigaparsec scales (comparable to the CMB at $l=2\mathrm{\text{\ensuremath{-}}}10$) with narrow window functions. These modes may provide insight into the possible anomalies in the large-scale CMB anisotropy. At fixed redshift, the data from such a survey form an $E$-type spin-2 field on the sphere to a good approximation; the near absence of $B$ modes will provide a valuable check on systematic errors. A survey of only a few low-redshift clusters allows an independent reconstruction of the five coefficients of the local CMB quadrupole, providing a test for contamination in the Wilkinson Microwave Anisotropy Probe quadrupole. The formalism presented here is also useful for analyzing smaller-scale surveys to probe the late integrated Sachs-Wolfe effect and hence the properties of dark energy.