Probing CMB polarization Gaussianity with the statistics of unpolarized points: Non-Gaussianity of Planck data and prospects for future <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>B</mml:mi> </mml:math> -mode me

We present a Gaussianity test of the cosmic microwave background polarization by analyzing the statistics of unpolarized points in the sky, classified into three distinct types: saddles, comets, and beaks. This classification of singular points where both Stokes parameters $Q$ and $U$ vanish stems from the fact that linear polarization is described by a second-rank tensor. By varying the number of spherical harmonics included in the polarization maps, one can probe the statistics of these singularities across a range of angular scales. Applying this method to Planck data we find clear evidence of non-Gaussianity in both E and B modes of polarization. This approach may be especially useful for processing data from current and future experiments such as the Simons Observatory. In particular, it can help to assess the Gaussianity of a potentially detected B mode signal, thereby determining whether it arises from primordial tensor perturbations---as predicted by inflation---or from alternative sources such as polarized foregrounds (e.g., thermal dust), E-to-B mode leakage, systematics, photon noise, or gravitational lensing. We have made publicly available software that finds unpolarized points of all three types on any polarization map in Hierarchical Equal Area isoLatitude Pixelation format with full or incomplete sky coverage to enable testing of the observed signal for Gaussianity.