Entropy methods for CMB analysis of anisotropy and non-Gaussianity

Abstract

We propose several pseudoentropy measures that agree well with the Wehrl entropy, but are significantly faster to compute. All of them are rotationally invariant measures of entanglement very sensitive to non-Gaussianity, anisotropy, and statistical dependence of spherical harmonic coefficients. We provide a simple proof that the projection pseudoentropy converges to the Wehrl entropy with increasing dimensionality of the ancilla projection space. Furthermore, for $l=2$, we show that both the Wehrl entropy and the angular pseudoentropy can be expressed as functions of the squared chordal distance of multipole vectors. We also show that the angular pseudoentropy can distinguish between Gaussian and non-Gaussian temperature fluctuations at large multipoles and henceforth provides a non-brute-force method for identifying non-Gaussianities. This allows us to study possible hints of statistical anisotropy and non-Gaussianity in the CMB up to multipole $l=1000$ using Planck 2015/2018, and WMAP 7-yr data. We find that $l=5$ and $28$ have a large entropy at $2$--$3\sigma$ significance and a slight hint towards a connection of this with the cosmic dipole. On a wider range of large angular scales we do not find indications of isotropy/Gaussianity violation. We also find a small-scale range, $l\in[895,905]$, that is incompatible with the assumptions at about $3\sigma$ level, although how much this significance can be reduced by taking into account the selection effect is left as an open question. We find overall similar results in our analysis of the 2015 and the 2018 data. Finally, we also demonstrate how a range of angular momenta can be studied with the range angular pseudoentropy. Our main purpose is to introduce the methods, analyze their mathematical background, and demonstrate their usage for providing researchers in this field with an additional tool.