A new probe of Gaussianity and isotropy with application to cosmic microwave background maps

Abstract

We introduce a new mathematical tool (a direction-dependent probe) to analyze the randomness of purported isotropic Gaussian random fields on the sphere. If the field is isotropic and Gaussian then the probe coefficients for a given direction should be realizations of uncorrelated scalar Gaussian random variables. To study the randomness of a field, we use the autocorrelation of the sequence of probe coefficients (which are just the Fourier coefficients [Formula: see text] if the [Formula: see text]-axis is taken in the probe direction). We introduce a particular function on the sphere (called the AC discrepancy) that accentuates the departure from Gaussianity and isotropy. We apply the probe to assess the full-sky cosmic microwave background (CMB) temperature maps produced by the Planck collaboration (PR2 2015 and PR3 2018), with special attention to the inpainted maps. We find that for some of the maps, there are many directions for which the departures are significant, especially near the galactic plane. We also look briefly at the noninpainted Planck maps, for which the computed AC discrepancy maps have a very different character, with features that are global rather than local.