New geometric representations of the CMB two-point correlation function

Abstract

When searching for deviations of statistical isotropy in CMB, a popular strategy is to write the two-point correlation function (2pcf) as the most general function of four spherical angles (i.e., two unit vectors) in the celestial sphere. Then, using a basis of bipolar spherical harmonics, statistical anisotropy will show up if and only if any coefficient of the expansion with non-trivial bipolar momentum is detected -- although this detection will not in general elucidate the origin of the anisotropy. In this work we show that two new sets of four angles and basis functions exist which completely specifies the 2pcf, while, at the same time, offering a clearer geometrical interpretation of the mechanisms generating the signal. Since the coefficients of these expansions are zero if and only if isotropy holds, they act as a simple and geometrically motivated null test of statistical isotropy, with the advantage of allowing cosmic variance to be controlled in a systematic way. We report the results of the application of these null tests to the latest temperature data released by the Planck collaboration.