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.
