Probing CMB Polarization Gaussianity with the Statistics of Unpolarized Points: Non-Gaussianity of Planck Data and Prospects for Future B-Mode Measurements

TL;DR

This work introduces a Gaussianity test for CMB polarization based on the statistics of unpolarized points (where $Q=U=0$) classified as saddles, comets, and beaks. By decomposing polarization into the $E$ and $B$ scalar/pseudoscalar fields and varying the maximum spherical harmonic degree $\ell_{max}$, the authors probe scale-dependent Gaussianity and quantify the expected distributions for Gaussian fields. Applying the method to Planck SMICA maps reveals significant non-Gaussian components in both $E$ and $B$ modes, driven by foregrounds, noise inhomogeneities, and possible leakage between $E$ and $B$, reinforcing the need for robust Gaussianity checks in current and future B-mode analyses. The approach offers a powerful diagnostic for assessing whether a detected B-mode signal arises from primordial tensor perturbations or from non-primordial contaminants, and the authors provide publicly available software to facilitate such tests for incomplete or full-sky polarization maps.

Abstract

We present a Gaussianity test of the cosmic microwave background (CMB) 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 (SO). 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 (HEALPix) format with full or incomplete sky coverage to enable testing of the observed signal for Gaussianity.

Figures (6)

• Figure 1: Singularities in vector and tensor fields. Left panel: two-dimensional velocity vector field ${\bf V}$. Continuous lines show the orientation of vectors in the vicinities of points where ${\bf V}=0$ (vector directions are not indicated). Foci cannot exist in a gradient field $\nabla\times{\bf V}=0$, and nodes do not exist in a vortex field $\nabla\cdot{\bf V}=0$. Right panel: two-dimensional tensor field. Lines show the orientation of polarization in the vicinities of points with zero polarization: $Q=U=0$. All three types of points can exist in both E and B modes.
• Figure 2: Polarization map of a small $15'\times 15'$ part of the sky with unpolarized points. Left panel: the orientation of each segment corresponds to the orientation of linear polarization, and the lengths of the segments are proportional to the polarization level $P$. Singularity points: saddles are marked in red, comets in green, and beaks in blue, respectively. Right panel: the same part of the sky. Areas satisfying the conditions for saddles, comets and beaks are colored red, green, and blue, respectively. The same singularity points as in the left panel are marked in black.
• Figure 3: Expected ratios $\frac{N_{type}}{N_{tot}}$ for saddles, comets, and beaks ( left panel) and expected fractions of area corresponding to saddle, comet, and beak conditions ( right panel) for a random Gaussian field. The results are obtained for the CMB E polarization spectrum according to SMICA 2018 Planck data. Solid lines show the mean expected values of the $\frac{N_{type}}{N_{tot}}(\ell_{max})$, $\frac{A_{type}}{A_{tot}}(\ell_{max})$ functions. Dashed lines correspond to the asymptotic values in a flat limit. The shaded areas in both panels show the standard $\sigma_{N_{type}}^\pm$ and $\sigma_{A_{type}}^\pm$ deviations for the all-sky statistics.
• Figure 4: Statistics of singularities and area fractions for the E polarization of the Earth. Upper left panel: deviations of the fractions of points of three different types from the average expected values in a Gaussian field as a function of $\ell_{max}$: $\Delta\tilde{n}_{type}(\ell_{max})$. The shaded areas show $1\cdot\sigma$ deviations which correspond to $\langle(\Delta\tilde{n}_{type}^G)^2\rangle^{\frac{1}{2}}$. Upper right panel: the same as the left panel, but for the area fractions $\Delta\tilde{a}_{type}(\ell_{max})$ (the shaded areas are very small). The bottom three rows from top to bottom in descending order: Maps showing the relative concentrations of each singularity point type $\frac{N_{type}}{N_{tot}}$, their area fractions $\frac{A_{type}}{A_{tot}}$ and their spatial density $\Delta\left(\frac{N_{type}}{A_{type}}\right)$. For comparison, the right column shows such maps for saddles in the case of a Gaussian field with the Earth spectrum. The smoothing angle is $\theta_0=0.5^\circ$, which corresponds to $\sim$ 35 miles on the Earth's surface. The white part of the surface does not contain singularities.
• Figure 5: Statistics of the three singularity types (saddles, comets, beaks) and their corresponding area fractions in the SMICA CMB E mode polarization. The first three rows (from top to bottom): Deviations from the Gaussian mean for the relative number of singularities $\Delta\tilde{n}_{type}(\ell_{max})$, their area fractions $\Delta\tilde{a}_{type}(\ell_{max})$, and their spatial density $\Delta\tilde{\rho}_{type}(\ell_{max})$ in units of $\langle(\Delta\tilde{n}_{type}^G)^2\rangle^{\frac{1}{2}}$, $\langle(\Delta\tilde{a}_{type}^G)^2\rangle^{\frac{1}{2}}$ and $\langle(\Delta\tilde{\rho}_{type}^G)^2\rangle^{\frac{1}{2}}$, respectively. The shaded areas mark the $1\sigma$ confidence region. The last three rows (from top to bottom): Sky maps ($\ell_{max}=2048$, smoothed at $\theta_0 = 0.5^\circ$) showing the relative concentrations of each singularity point type $\frac{N_{type}}{N_{tot}}$, the area fraction $\frac{A_{type}}{A_{tot}}$, and their spatial density $\Delta\left(\frac{N_{type}}{A_{type}}\right)$. For comparison, the right column shows the same analysis performed on a typical Gaussian realization with the CMB SMICA E mode spectrum for saddles.