A New Probe of Gaussianity and Isotropy applied to the CMB Maps

We introduce a new mathematical tool (a direction-dependent probe) to analyse the randomness of purported isotropic Gaussian random fields on the sphere. We apply the probe to assess the full-sky cosmic microwave background (CMB) temperature maps produced by the {it Planck} collaboration (PR2 2015 and PR3 2018), with special attention to the inpainted maps. To study the randomness of the fields represented by each map we use the autocorrelation of the sequence of probe coefficients (which are just the full-sky Fourier coefficients $a_{ell,0}$ if the $z$ axis is taken in the probe direction). If the field is {isotropic and Gaussian} then the probe coefficients for a given direction should be realisations of uncorrelated scalar Gaussian random variables. We introduce a particular function on the sphere (called the emph{AC discrepancy}) that accentuates the departure from Gaussianity and isotropy. 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 study the effect of varying the highest multipole used to calculate the AC discrepancy from the initial value of $1500$ to $2500$. In the case of Commander 2015, the AC discrepancy now exhibits antipodal blobs well away from the galactic plane. Finally, we look briefly at the non-inpainted Planck maps, for which the computed AC discrepancy maps have a very different character, with features that are global rather than local. For the particular case of the non-inpainted 2018 texttt{SEVEM} map (which has visible equatorial pollution), we model with partial success the observed behaviour by an isotropic Gaussian random field added to a non-random needlet-like structure located near the galactic centre.