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 look-back 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. A remote quadrupole survey sparsely covering a large area of sky can be used to measure modes that are independent of the local CMB and that probe scales about as large as the local CMB quadrupole. Furthermore, the window functions of these modes can be much narrower than those of the local CMB. These modes should 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; the 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 WMAP 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