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Entropy Methods for CMB Analysis of Anisotropy and Non-Gaussianity

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Entropy Methods for CMB Analysis of Anisotropy and Non-Gaussianity Entropy methods for CMB analysis of anisotropy and non-Gaussianity Momchil Minkov,1, ∗ Marvin Pinkwart,2, y and Peter Schupp2, z 1Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, CA 94305-9505, USA 2Department of Physics and Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany (Dated: June 8, 2021) In recent years, high-resolution cosmic microwave background (CMB) measurements have opened up the possibility to explore statistical features of the temperature fluctuations down to very small angular scales. One method that has been used is the Wehrl entropy, which is, however, extremely costly in terms of computational time. Here, we propose several different pseudoentropy measures (projection, angular, and quadratic) that agree well with the Wehrl entropy, but are significantly faster to compute. All of the presented alternatives are rotationally invariant measures of entangle- ment after identifying each multipole l of temperature fluctuations with a spin-l quantum state and are very sensitive to non-Gaussianity, anisotropy, and statistical dependence of spherical harmonic coefficients in the data. 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 one-dimensional functions of the squared chordal distance of multipole vectors, giving a tight connection between the two measures. We also show that the angular pseudoentropy can clearly 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, Planck 2018, and WMAP 7-yr full sky data. We find that l = 5 and l = 28 have a large entropy at 2{3σ 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 violation of isotropy or Gaussianity. We also find a small-scale range, l 2 [895; 905], that is incompatible with the assumptions at about 3σ level, although how much this significance can be reduced by taking into account the selection effect, i.e.,, how likely it is to find ranges of a certain size with the observed features, and inhomogeneous noise is left as an open question. Furthermore, 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, which measures averages and correlations of different multipoles. Our main purpose in this work is to introduce the methods, analyze their mathematical background, and demonstrate their usage for providing researchers in this field with an additional tool. We believe that the formalism developed here can underpin future studies of the Gaussianity and isotropy of the CMB and help to identify deviations, especially at small angular scales. Keywords: CMB { data analysis { coherent states { pseudoentropy I. INTRODUCTION of order 10−3 K, which is assumed to be of pure kine- matic origin. The anisotropy of the CMB temperature Since its discovery in 1964 by Penzias and Wilson [1], has been measured first by the Cosmic Background Ex- the cosmic microwave background (CMB) has served as plorer satellite from 1989 to 1993, followed up by the the main source of information about the current and Wilkinson Microwave Anisotropy Probe (WMAP) mis- past Universe. Originating in the process of recombina- sion from 2001 to 2010. The most recent CMB investiga- tion at about 380 000 years after the Big Bang at red- tion satellite, Planck, was launched in 2009 and shutdown shift of about 1100, the CMB temperature distribution in 2013. Its 2015 results from the second data release pro- arXiv:1809.08779v2 [astro-ph.CO] 3 May 2019 on the celestial sphere displays the energy density distri- vide the most precise values of cosmological parameters bution on the Last Scattering Surface when the Universe [3, 4] measured up to this point. Recently, they have became transparent to electromagnetic radiation. The been refined in the 2018 data release [5, 6]. CMB intensity follows a nearly perfect Planckian distri- It is commonly assumed that the tiny temperature fluc- bution and its average temperature has been measured tuations follow a Gaussian and statistically isotropic dis- to be T0 = 2:72548±0:00057 K[2] with minor fluctuations −5 −4 tribution. This assumption has been confirmed by the of order 10 {10 K and one larger dipole modulation Planck mission to a large extent[7, 8], but nevertheless the search for possible non-Gaussianities[9] and statisti- cal anisotropies [10{19] has been rich and certain anoma- ∗ [email protected] lies have been found, as, for example, unusual (anti- y [email protected] )correlation of the lowest multipoles with the Cosmic z [email protected] Dipole as well as with each other, a sign of parity asym- 2 metry and a lack of large-angle correlation (see e.g. the it to the statistics used before. Eventually, in Sec. V we review[20]). summarize and discuss our findings. Common tools in these analyses are multipole vectors (MPVs) which were introduced for cosmological data analysis in [21] and whose properties have been elab- II. CMB STATISTICS orated in [11, 22{25]. For the most recent results on possible CMB anomalies using multipole vectors and an As a function on S2 the CMB temperature fluctuations overview over the mathematical approaches see [26{28]. ∆T := δT=T0 can be decomposed uniquely according to MPVs are closely related to Bloch coherent states (see irreducible representations of SO(3), i.e.,, into spherical [11]) which were also used in the past to prove special harmonics cases of Lieb's conjecture[29] for the Wehrl entropy. 1 l In this work, we develop and compare several rota- X X tionally invariant measures of randomness on functions ∆T (θ; φ) = almYlm(θ; φ) 2 R; (1) on the two-sphere, namely, the angular, projection, and l=1 m=−l quadratic pseudoentropies. We show that for l = 2 the where the l = 0-summand is omitted because the fluctua- Wehrl and angular entropy can be expressed as a func- tions average to zero, and θ, φ denote the usual spherical tion of the squared chordal distance of MPVs. We find coordinates. The fact that ∆T is real together with the that all these measures except the quadratic one show property Y = (−1)mY ∗ impose the constraints the same features, making the quadratic pseudoentropy l;−m lm the least preferred measure. Because of the shared fea- a = (−1)ma∗ (2) tures, we then restrict ourselves to the numerically fastest l;−m lm method, and use it to analyze Planck 2015 and 2018 full on the spherical harmonic coefficients, leaving for each sky as well as WMAP 7-year Internal Linear Combina- multipole number l exactly 2l+1 real d.o.f. The multipole tion (ILC) maps. The angular pseudoentropy allows for number l corresponds to angular scales of ≈ 180=l deg. comparing the data to many ensembles of Gaussian and The orthonormality of fY g allows to compute the co- isotropic random maps up to l = 1000 in short computing lm efficients from the temperature map via time. With a better theoretical understanding of confi- dence levels, also the Wehrl entropy could be used easily Z ∗ since the computing time for a single map is still rea- alm = dΩ ∆T (Ω)Ylm(Ω): (3) sonable. In general, it is especially nice to have a single S2 number for each multipole even in the case that the data Simple inflationary models together with linear pertur- would not be Gaussian and isotropic. In this case, the bation theory predict nearly Gaussian temperature fluc- CMB would be described by more than one degree of free- tuations and a further common assumption is statistical dom (d.o.f.) per multipole. Non-Gaussian distributions isotropy. The spherical harmonic coefficients inherit both need higher correlation functions and anisotropic distri- properties from the temperature map, meaning that butions yield an m-dependent two-point function. In the tradition of thermodynamics, with these pseudoentropies 1 − 1 ~ayD ~a p(~a ) = e 2 l l l (Gaussianity), (4) one can approximately reduce a possibly large set of data l N again to one number for each multipole. Since all con- sidered types of entropies show a similar behavior the where p denotes the joint probability distribution, ~al = T l −1 ∗ information does not depend on the definition of the en- (al0; : : : ; all) , Cmn = Dl mn = halmalni and N de- tropy. Eventually there exists also an extension of the notes a normalization constant, and angular entropy to ranges and collections of multipoles, 2 which we call range angular entropy. 8R 2 SO(3);~e1; : : : ;~en 2 S ; n 2 N : This paper is organized as follows: In Sec. II, we briefly Gn(fR~eig) = Gn(f~eig) (isotropy), (5) recapitulate the basic ingredients of CMB spherical har- Qn monic statistics. Afterwards, in Sec. III, we introduce where Gn(f~eig) = h i=1 ∆T (~ei)i denotes the n-point our methods mathematically, clarify their properties, and function of temperature fluctuations. The isotropy con- show the connection to multipole vectors. We also pro- dition is equivalent to rotationally invariance of the joint vide a simple proof of the convergence of the projection alm-probability distribution. The averaging h:i is meant entropy to the Wehrl entropy up to a term which is in- to be performed over all possible universes, which of dependent of the input density matrix.
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