Multipole vectors of completely random microwave skies for l≤50
MMultipole vectors of completely random microwave skies for l ≤ Marvin Pinkwart
1, 2, ∗ and Dominik J. Schwarz † Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany Department of Physics and Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany (Dated: November 1, 2018)The statistical cosmological principle states that observables on the celestial sphere are sampledfrom a rotationally invariant distribution. Previously certain large scale anomalies which violatethis principle have been found, for example an alignment of the lowest multipoles with the cosmicdipole direction. In this work we continue the search for possible anomalies using multipole vectorswhich represent a convenient tool for this purpose. In order to study the statistical behavior ofmultipole vectors, we revisit several construction methods.We investigate all four full-sky foreground-cleaned maps from the Planck 2015 release with respectto four meaningful physical directions using computationally cheap statistics that have a simplegeometric interpretation. We find that the full-sky SEVEM map deviates from all the other cleanedmaps, as it shows a strong correlation with the Galactic Pole and Galactic Center. The other threemaps COMMANDER, NILC and SMICA show a consistent behavior. On the largest angular scales, l ≤
5, as well as on intermediate scales, l = 20 , , , ,
24, all of them are unusually correlatedwith the cosmic dipole direction. These scales coincide with the scales on which the angular powerspectrum deviates from the Planck 2015 best-fit ΛCDM model. In the range 2 ≤ l ≤
50 as a wholethere is no unusual behavior visible globally. We do not find abnormal intramultipole correlation,i.e. correlation of multipole vectors inside a given multipole without reference to any outer direction.
PACS numbers: 02.50.Cw, 02.70.Uu, 07.05.Kf, 98.80.EsKeywords: cosmic microwave background – data analysis – multipole vectors – statistical isotropy
I. INTRODUCTION
High-fidelity observations of the cosmic microwavebackground (CMB) at the largest angular scales be-came available with data from the Wilkinson MicrowaveAnisotropy Probe (WMAP) [1]. Previous full-sky analy-ses based on data from the Cosmic Background Explorer(COBE) suffered at the largest angular scales from theirlimited capacity (only three frequency bands) to reliablyseparate the various foreground components from theCMB. Once confidence on foreground separation tech-niques was built, the WMAP data offered the potentialto address the statistical isotropy of the observed temper-ature anisotropies at large angular scales. The propertyof statistical isotropy is a fundamental assumption in theanalysis of the CMB and for the estimation of cosmolog-ical parameters. It was noted that the quadrupole andoctopole seem to be aligned with each other [2, 3] andwith the CMB dipole [3] and an unexpected hemispher-ical asymmetry was revealed [4]. More signs of violationof statistical isotropy have been found in several publica-tions [5–12].On the other hand, already the WMAP data suggestedthat deviations from Gaussianity and from the angularpower spectrum predicted by the ΛCDM model are neg-ligibly small [13, 14]. The analysis of the data from thePlanck satellite confirmed these findings [15–17], but atthe same time confirmed the existence of isotropy anoma- ∗ [email protected] † [email protected] lies [18].The full Planck mission data allowed to construct fourprecise full-sky maps that use different cleaning algo-rithms to remove the influences of the Milky Way [19, 20].That analysis increases our confidence that the aforemen-tioned isotropy anomalies are not due to instrumental ef-fects, mistakes in the analysis pipeline, or unaccountedforegrounds, which should have been revealed by the widefrequency coverage of Planck. A recent review collectsthe up-to-date knowledge about these isotropy anoma-lies [21].Analyses of the cosmic background radiation are con-veniently performed by means of spherical harmonic co-efficients and angular power spectra, or correlation func-tions in angular space. When investigating the CMBwith respect to possible breaking of statistical isotropy,a third tool has become popular, namely multipole vec-tors (MPVs) [6, 22, 23]. While spherical harmonic coef-ficients transform with Wigner’s symbols under a rota-tion of the celestial sphere, MPVs transform like ordinarythree-vectors, i.e. they rotate rigidly with the tempera-ture fluctuations on the sphere which makes them a con-venient choice for isotropy analysis.In this article we review the common constructionmethods for multipole vectors as well as their theoreticalstatistical behavior and use the four foreground cleanedfull-sky maps from the Planck 2015 data analysis to in-vestigate the statistical isotropy of the CMB, especiallyalignments of multipole vectors within a multipole orwith external directions. It should be noted that the mul-tipole vector method can only test for statistical isotropyif Gaussianity of the temperature fluctuations is assumedto hold since multipole vector statistics are only sensitive a r X i v : . [ a s t r o - ph . C O ] O c t to deviations from a completely random distribution.In Sec. II we review the basic definitions and prop-erties of CMB data analysis by means of the angularpower spectrum and we describe our convention of sta-tistical isotropy. Then, in Sec. III we give an overviewover three convenient extraction methods for MPVs. InSec. IV we review the derivation of the probability dis-tribution of MPVs. In Sec. V we shortly describe thePlanck data used in the analysis. Section VI is dedi-cated to the introduction of the statistics that we use forthe analysis. Section VII introduces the four astrophys-ical directions used in the analysis to estimate sourcesof multipole anomalies. In Sec. VIII we present the re-sults before discussing them in Sec. IX and giving a shortconclusion and outlook in Sec. X. II. ANGULAR POWER SPECTRUM
The relative fluctuations of CMB temperature, whichlive on the celestial sphere, are conveniently decomposedaccording to the irreducible representations of the groupof three-dimensional spatial rotations SO (3), namely theorthonormal set of spherical harmonic functions, δTT ( e ) = ∞ (cid:88) l =1 l (cid:88) m = − l a lm Y lm ( e ) , (1)with the radial unit vector e =(cos( φ ) sin( θ ) , sin( φ ) sin( θ ) , cos( θ )) pointing towardsthe direction of observation. The contribution from agiven integer number l , f l ( e ) = (cid:88) m a lm Y lm ( e ) , (2)is called a multipole of order l , which describes features attypical angular scales of about α l = π/l . The coefficients a lm are called spherical harmonic coefficients. Thanks tothe orthonormality of { Y lm } , i.e., (cid:82) Y lm Y ∗ l (cid:48) m (cid:48) = δ ll (cid:48) δ mm (cid:48) ,the a lm can be calculated from δT /T via integration a lm = (cid:90) d e δT ( e ) T Y ∗ lm ( e ) . (3)Since temperature fluctuations are real, and Y ∗ lm =( − m Y l, − m , the spherical harmonic coefficients obey a ∗ lm = ( − m a l, − m . (4)The particular pattern of the CMB temperature fluctu-ations cannot be predicted. Instead, temperature fluctu-ations are modeled as a real, random field on the sphere,or equivalently we model the measured spherical har-monic coefficients as realizations of an ensemble of ran-dom variables, subject to condition (4). A fundamental assumption regarding the temperaturefluctuations is statistical isotropy, that means ∀ R ∈ SO (3) ∀ e , . . . , e n ∈ S ∀ n ∈ N : (cid:42) n (cid:89) i =1 δTT ( R e i ) (cid:43) = (cid:42) n (cid:89) i =1 δTT ( e i ) (cid:43) , (5)where (cid:104) . (cid:105) denotes the expectation value of random fields,respectively the ensemble average over all “possible uni-verses”. Correlation functions of temperature fluctua-tions at different directions should only depend on theangle between them, respectively on the scalar products e i · e j .Here, we take the point of view of an observer in three-dimensional space that acts with an element of SO (3) onthe sky. There exists also the other convention that thethree degrees of freedom of three-dimensional rotationsare split into a translation on S – and invariance undersuch a transformation would then be called homogeneity– and a rotation around a point on S . The invarianceof the latter would then be associated with isotropy. Inour work all three symmetry operations are viewed asrotations and thus we only speak about isotropy.One usually argues that the smallness of the CMB tem-perature fluctuations provides empirical evidence for sta-tistical isotropy of the Universe and cosmological infla-tion provides an argument on why the observed patch ofthe Universe should be isotropic. But there are a priori no other reasons and a detailed study of the observeddeviations from isotropy in δT /T might reveal that sta-tistical isotropy could be violated, for example due toprimordial anisotropies.A trivial consequence of the definition of δT is thevanishing of the one-point function, (cid:104) δT ( e ) /T (cid:105) = 0 or (cid:104) a lm (cid:105) = 0. Thus the first nontrivial and most interestingobject is the angular two-point correlation or the angu-lar power spectrum C l , which for an isotropic ensembleis given by (cid:104) a ∗ lm a l (cid:48) m (cid:48) (cid:105) = C l δ ll (cid:48) δ mm (cid:48) .As a result of the initial Gaussianity after inflationand the following linear evolution, this random field isassumed to be Gaussian in the standard theory. Thatmeans that higher correlations cannot carry independentinformation. Thus in the standard model of cosmologyall cosmological information is encoded in the angularpower spectrum C l . III. MULTIPOLE VECTORS
MPVs represent a tool for investigating CMBanisotropies in a very natural manner. They behave likeordinary three-vectors under rotation and they do notdistinguish a certain reference frame, unlike the spher-ical harmonics which incorporate a reference to a cho-sen z -axis in their definition. In the following, we re-view different mathematical approaches to the descrip-tion of functions on the sphere via MPVs. This is nec-essary to provide the appropriate tools for an analyticstudy of the statistical distribution of MPVs on com-pletely random skies. While the algebraic and tensorialapproaches yield recursive relations for direct calculationof the MPVs from a given spherical harmonic decompo-sition, the coherent state approach gives the MPVs asroots of a complex polynomial. With the help of thelatter one can calculate analytically the joint probabilitydensity given a fixed multipole. A. Origin and Sylvester’s theorem
MPVs date back to Maxwell who introduced them in[24], in the study of interactions between monopoles. Amonopole creates an electric potential proportional to1 /r . Maxwell argued that the potential of two opposite-signed monopoles can be written as a directional deriva-tive of the monopole potential D v (1 /r ), where v de-notes the linking vector between the point charges. Hecontinued to the case of 3 , , . . . interacting monopolesand received a potential of the form D v . . . D v l (1 /r ) if l monopoles are involved. Later on it has been noticedthat any real, harmonic and homogeneous polynomial on R can be represented in that form.Let f : R → R be a real, harmonic and homoge-neous polynomial of degree l in the variables x, y, z . Thatmeans ∆ R f = 0, implying ∆ S f = − l ( l + 1) f , and f ( λx, λy, λz ) = λ l f ( x, y, z ). Any such polynomial de-fines a polynomial (cid:101) f = f | S : S → R on the sphere andvice versa. Maxwell’s MPV representation states thatthere exist l unique unit directions v , . . . , v l , such that f takes on the following form f ( x, y, z ) = ( v · ∇ ) . . . ( v l · ∇ ) 1 r ( x, y, z )with r ( x, y, z ) = (cid:112) x + y + z . (6)This statement is known as Sylvester’s theorem [25].The expression (6) is equivalent to the in practice moreuseful expression f ( θ, φ ) = C ( e ( θ, φ ) · v ) . . . ( e ( θ, φ ) · v l ) + r F ( θ, φ ) , (7)where θ and φ describe the sphere in spherical coordi-nates and e ( θ, φ ) = ( x ( θ, φ ) , y ( θ, φ ) , z ( θ, φ )) /r ( x, y, z ),and F is a homogeneous polynomial in the variables x, y, z of degree ≤ l −
2. Due to the fact that spher-ical harmonics provide a basis for harmonic functions,each multipole of a spherical harmonic decomposition ofCMB fluctuations on the sky can be identified uniquelywith a set of MPVs.
B. Extraction of multipole vectors
There exist several approaches to MPVs and their cal-culation from a spherical harmonic decomposition, threeof which we will review briefly in the following. While the approach via coherent states appears to be best suited forthe investigation of statistical properties, in this work thetensorial approach has been used to calculate the MPVsnumerically [26].
1. Tensorial construction
Copi et al. first applied the long-forgotten method ofMPVs to the analysis of CMB data in [22].Let the fragments f l be as in Eq. (2). They are har-monic and homogeneous polynomials of degree l in x, y, z and thus f l ( x, y, z ) = F i ··· i l e i . . . e i l . In order to guar-antee the uniqueness of this expression, it is inevitableto impose further restrictions on the coefficients F i ··· i l ,which can be regarded as coefficients of a tensor F , andon the product e i . . . e i l . Both factors have to be trace-free and symmetric: f l ( e ) = F ( l ) i ··· i l [ e i · · · e i l ] =: A ( l ) (cid:104) v ( l, i · · · v ( l,l ) i l (cid:105) [ e i · · · e i l ] . (8)The brackets denote the symmetric trace-free part of theinterior. Equation (8) defines the MPVs, which can becalculated uniquely, up to rescaling, from the sphericalharmonic data. One recovers F from f l via integration F ( l ) i ··· i l = (2 l + 1)(2 l )!(4 π )2 l ( l !) (cid:90) S d e f l ( e ) (cid:2) e i · · · e i l (cid:3) , (9)and afterwards peels off the first MPV by writing F ( l ) i ··· i l = (cid:104) v ( l, i a ( l, i ··· i l (cid:105) , (10)where a ( l, is a rank l − a ( l, leaving arank l − l MPVs. By performing a moredetailed mathematical calculation one can write down asystem of equations that relates the a lm and the v ( l,j ) ; formore details see [22]. Copi’s MPV calculation program[26], which was used by the authors, evaluates this systemand returns the MPVs.Finally note that no information is lost in the tran-sition from spherical harmonics to MPVs. For each l there are 2 l + 1 real degrees of freedom in the sphericalharmonic decomposition, namely the real and imaginaryparts of all a lm with m ≥
0. On the other hand, l unitvectors and an amplitude constitute as well 2 l + 1 realdegrees of freedom since due to the normalization condi-tion a single unit vector in R has 2 degrees of freedom,and the amplitude is just a scalar which contributes onefurther degree of freedom.
2. Algebraic construction
Katz and Weeks [23] applied B´ezout’s theorem fromalgebraic geometry to proof Sylvester’s theorem. Theadvantage of this approach is its mathematically sophis-ticated nature. Furthermore, like the tensorial approach,it yields an iterative prescription calculating the MPVs,and even an explicit expression for the residual polyno-mial F can be obtained. This approach has a long historydating back to Hilbert and Courant, see [27].A homogeneous polynomial P of degree l on R maybe written uniquely up to reordering and rescaling as P ( x, y, z ) = λ ( a x + b y + c z ) · · ·· · · ( a l x + b l y + c l z )+ ( x + y + z ) R, (11)where R denotes a residual polynomial which is homoge-neous of degree l −
2. If l < R ≡
0, and for l = 2the zero can be replaced by a nonvanishing constant.Let now f be an arbitrary, especially not necessarilyhomogeneous, polynomial of degree l restricted to thetwo-sphere. It can be written as a sum of homogeneouspolynomials of degree i , f i , via f = (cid:80) li =0 f i . Accord-ing to (11), up to reordering and rescaling f i can be de-composed into linear factors and a residual polynomial f i = λ i (cid:81) ij =1 ( v ( l,j ) x x + v ( l,j ) y y + v ( l,j ) z z ) + R i − ( x, y, z ).Since R i − is homogeneous of degree i −
2, the sum f (cid:48) i − := f i − + R i − is again homogeneous of degree i − f ( x, y, z ) = l (cid:88) i =0 λ i i (cid:89) j =1 ( v ( i,j ) x x + v ( i,j ) y y + v ( i,j ) z z ) on S . (12)The scalar product of MPVs with the unit vector in the( θ, φ )-direction is given by the i = l -term while the restof the sum constitutes the residual polynomial F . Formore details we refer to [23].
3. Construction via Bloch coherent states
Dennis used a very physical approach to proofSylvester’s theorem and associate MPVs to spherical har-monics [28]. A complex spin-state in nonrelativistic one-particle quantum mechanics with spin 1 / θ and φ with a normalized spin state gives, after stereographic projection, the Ma-jorana polynomial below. Using the Bloch states one candefine an extended version of the von Neumann entropy,called Wehrl entropy, which measures quantum random-ness.The formalism below has already been used by Schuppin 1999 in the proof of some special cases of Lieb’s con-jecture for the Wehrl entropy of Bloch coherent states;see [29].Let | Ψ (cid:105) denote a quantum mechanical state with defi-nite integer spin l , i.e., an eigenstate of the total angularmomentum operator ˆ L . It corresponds to a harmonicfunction in the language of the previous subsections. Theeigenstate property allows us to expand the state in termsof eigenstates of the z -component of the angular momen-tum operator ˆ L z | Ψ (cid:105) = l (cid:88) m = − l Ψ m | m, l (cid:105) , Ψ m ∈ C , (13)which in position space is nothing other than the spher-ical harmonic decomposition. Let ˆ R z ( φ ) denote the op-erator which executes a rotation by the angle φ aroundthe z -axis and ˆ R y ( θ ) the rotation by θ around the y -axisand define | m, l ; θ, φ (cid:105) := ˆ R z ( φ ) ˆ R y ( θ ) | m, l (cid:105) . (14)This is an eigenstate of the e ( θ, φ )-parallel component ofˆ L . The spherical harmonics are then given by Y lm ( θ, φ ) = (cid:114) l + 14 π (cid:104) , l ; θ, φ | m, l (cid:105) . (15)The state | Ψ (cid:105) can now be expressed via spherical har-monics by projecting on the rotated m = 0 stateΨ( θ, φ ) := (cid:104) , l ; θ, φ | Ψ (cid:105) = (cid:114) π l + 1 l (cid:88) m = − l Ψ m Y lm ( θ, φ ) . (16)After stereographic projection from the south pole ζ ( θ, φ ) = tan( θ/
2) exp( iφ ) , (17)and using some group theory, the spin spate | Ψ (cid:105) can bedecomposed according to the SL (2 , C ) basis functions µ k − l ζ k with k ∈ N f Ψ ( ζ ) := (cid:104)− l, l ; ζ | Ψ (cid:105) = exp( − i l arg( ζ ))(1 + | ζ | ) l l (cid:88) m = − l Ψ m µ m ζ l + m , (18)with the numerical factor µ m = ( − l + m (cid:113)(cid:0) ll + m (cid:1) . Therepresentation (18) of the state is called the Majoranafunction. It is a product of a ζ -dependent factor anda polynomial of degree 2 l in ζ which contains all theinformation about the original state. This polynomialis called the Majorana polynomial and it determines theroots of the Majorana function. Since it is a polynomialof degree 2 l in the complex variable ζ , it possesses 2 l complex roots according to the fundamental theorem ofalgebra, and therefore it can be factorized f Ψ ( ζ ) = exp( − i l arg( ζ ))(1 + | ζ | ) l ( − l Ψ m = l l (cid:89) n =1 ( ζ − ζ n ) . (19)The 2 l roots can be backprojected onto the Riemanniansphere S ˜= ˆ C ˜= C ∪ {∞} . These backprojected roots v ( ζ n ) are called Majorana vectors. In the case of a realΨ( θ, φ ) they are identical to the MPVs v ( l,j ) ≡ v ( ζ j ) (20)for a given l .The Majorana function of the rotated state obeys f ˆ R v,θ Ψ ( ζ ) = f Ψ ( M T ( ζ )), where MT denotes a uni-tary M¨obius transformation. Consequently its zeros alsotransform under a unitary M¨obius transformation. Af-ter backprojection this transformation corresponds to arotation through SO (3). Majorana vectors rotate rigidlylike ordinary three-vectors.A further property of Majorana vectors is their appear-ance in antipodal pairs if the original state is real f Ψ ( − /ζ ∗ ) = f Ψ ( ζ ) ∗ . (21)Hence, ζ is a root of the Majorana function if and onlyif − /ζ ∗ is a root, but − /ζ ∗ is the image under thestereographic projection of the antipode of the Majoranavector determined by ζ . This property does not hold ifthe original state is complex. Complex functions on thesphere cannot be represented by l MPVs.There have been several further approaches to MPVs,for example by investigating their topological implica-tions in [30].
IV. STATISTICAL PROPERTIES OFMULTIPOLE VECTORS
The spherical harmonic coefficients a lm of the CMBtemperature fluctuations are attached with a notion ofrandomness implied by inflationary fluctuations. Stan-dard inflationary scenarios lead to Gaussianity of thesecoefficients. Whether they are really Gaussian or not,they definitely constitute a set of random variables. TheMPVs, which depend only on these coefficients, inheritthe randomness from these coefficients. One may now askwhat kind of probability distribution they obey exactly.Dennis and Land attended to this question first in [31],followed up by [32]. For this purpose the coherent stateapproach turns out to be especially useful because MPVsare the roots of a complex polynomial whose coefficientsare the a lm times some numerical factor. Therefore wehave to deal with the probability density of roots of ran-dom polynomials which is currently a much studied fieldof statistical mathematics. In Appendix B we present some first ideas on how toapply results from random matrix theory and the theoryof Gaussian analytic functions to the problem of the jointprobability distribution of MPVs. Future advances inthis direction could allow for determining p-values witharbitrary precision in short computing time.This section is intended to provide a review of thederivation of the MPV joint probability distribution.The essential properties are the statistical decoupling ofMPVs at different angular scales and the nontrivial cor-relation between MPVs at a given angular scale π/l . Fur-thermore, it is important to note that even if the under-lying temperature fluctuation field is Gaussian, the MPVdistribution is not and hence it is not enough to consideronly one- and two-point functions, but one needs the fullset of all n -point functions, where n = 1 , . . . , l for a given l . Although an explicit expression for the probability dis-tribution of the MPVs has been found before in [31], itturns out to be of limited use for practical purposes, ex-cept for the lowest multipoles l = 1 , ,
3. In this workwe use Monte Carlo methods, which appear to yield re-sults faster than numerical integration of the analyticexpression. Nevertheless, the following review yields asolid understanding of what kind of behavior one shouldexpect.
A. Isotropy and Gaussianity
Let us first focus on the description of isotropy andGaussianity in the CMB data and the difference betweenboth.The temperature fluctuation field is Gaussian if for all n ∈ N and all e i ∈ S with i = 1 , . . . , n the probabilitydistribution of δT /T follows p ( δT /T ) = 1 N exp − (cid:88) ij (cid:18) δTT (cid:19) i ( D − ) ij (cid:18) δTT (cid:19) j , (22)with ( δT /T ) i = δT ( e i ) /T and some normalizationconstant N . The matrix D is the correlation matrix D ij = (cid:104) ( δT /T ) i ( δT /T ) j (cid:105) . The Gaussianity of δT im-plies Gaussianity of the a lm that obey p ( { a lm } ) = 1 N (cid:48) exp − (cid:88) l,l (cid:48) ,m,m (cid:48) a ∗ lm ( C − ) lml (cid:48) m (cid:48) a l (cid:48) m (cid:48) , (23)with C lml (cid:48) m (cid:48) = (cid:104) a ∗ lm a l (cid:48) m (cid:48) (cid:105) and N (cid:48) some normalizationconstant which is in general different from N . A Gaus-sian field is fully characterized by its correlation matrixand if we demand isotropy additionally, then C lml (cid:48) m (cid:48) = δ mm (cid:48) δ ll (cid:48) C l . In this case we have p ( { a lm } ) = (cid:89) lm exp( −| a lm | / (2 C l )) √ πC l , (24)i.e., the a lm are identically and independently distributedcomplex Gaussian random variables with variance C l , orFIG. 1: Visualization of the relation betweenGaussianity, statistical isotropy and completerandomness. Dotted lines denote infinite extension.alternatively all real and imaginary parts (cid:60) a lm , (cid:61) a lm aswell as a l are identically and independently distributedreal Gaussian random variables with variance C l .Isotropy and Gaussianity do not necessarily imply eachother. Consider for example a distribution which isgained by an isotropic and Gaussian distribution via in-troducing a cutoff for large values of | a lm | . By this wemean p ( a lm ) = 0 if | a lm | > κ ∈ R for all l and m . Thisdistribution is not fully Gaussian any longer but does notlose its isotropy. On the other hand a general Gaussiandistribution does not need to be isotropic. B. Probability distribution
It turns out that the joint probability density for theMPVs of fixed angular momentum l is the same for allso-called completely random sets of a lm . This meansthat the probability density of the coefficients, p ( a lm ),depends only on the sum (cid:80) m | a lm | , respectively on thepower spectrum estimator ˆ C l , see [31] and [32]. Anisotropic and Gaussian distribution is of course includedin the set of completely random distributions, see Fig. 1.Note that complete randomness does not require the a lm to be statistically independent. Rather, if they are sta-tistically independent and completely random, they au-tomatically have to be Gaussian. Gaussianity itself incombination with complete randomness implies statisti-cal isotropy, but not vice versa, and therefore we shall insist on complete randomness for the rest of this publica-tion and treat it as a basic assumption which incorporatesstatistical isotropy if Gaussianity is given. The most im-portant class of distributions for cosmology is given bythe intersection of completely random and Gaussian a lm ,which we call “standard cosmology” in Fig. 1. Since nosizable deviations from Gaussianity have been observedso far, see e.g. [13] or [17], assuming that Gaussianityholds true allows for investigating isotropy solely.For fixed l the set of Gaussian distributions is de-scribed by a finite number of degrees of freedom, sincedue to Wick’s theorem the expectation values (cid:104) a lm (cid:105) and the two-point functions (cid:104) a lm a ∗ lm (cid:48) (cid:105) uniquely deter-mine the distribution. Respecting the reality condition a ∗ lm = ( − m a l, − m yields 2 l +1+(2 l +1) = (2 l +1)(2 l +2)real degrees of freedom. The sets of statistically isotropicas well as completely random distributions are a priori not bounded in their degrees of freedom. Any n -pointfunction, with n ∈ N , contributes to the knowledge ofthe distribution. Hence both distributions have at leasta countably infinite set of degrees of freedom. The inter-section of the completely random case and the Gaussiancase coincides with the intersection of the isotropic andGaussian case. Nevertheless, there exist completely ran-dom a lm which are not Gaussian, as for example a deltadistribution δ ( ˆ C l ). In principle there can also exist sta-tistically isotropic, non-Gaussian distributions which arenot completely random. For this one just forms somerotationally invariant quantity Q –which shall not be afunction of the C l – out of the a lm and considers its distri-bution p ( Q ). That completely random distributions forma subset of statistically isotropic ones can be seen as fol-lows: since a rotation ˆ R acts on C l +1 as a special unitarytransformation, its determinant vanishes and therefore (cid:104) (cid:89) i O i ( ˆ R ( e i )) (cid:105) = (cid:90) (cid:89) m d a lm (cid:89) i O i ( { a lm } ) p ( { ˆ R − a lm } ) . (25)The property of statistical isotropy reduces to the rota-tional invariance of the joint probability. In the com-pletely random case we have p ( { a lm } ) = p ( (cid:80) m | a lm | ).Let the unitary operator corresponding to ˆ R − acting on C l +1 be denoted by D (which is related to Wigner’s D-matrix; see [33]) and write the set of a lm for fixed l as avector a l ∈ C l +1 ; thenˆ R (cid:32)(cid:88) m | a lm | (cid:33) = (cid:88) m | ˆ R ( a lm ) | = ( D a l ) † · ( D a l )= (cid:88) m | a lm | , (26)due to the unitary representation of SO (3) as SU (2).Hence, completely random sets of a lm always obey sta-tistical isotropy.For CMB analysis one needs the joint probability dis-tribution of MPVs because inside one multipole they arenot independent of each other. This stems directly fromthe behavior of random roots which tend to repel eachother. Contrary to this, the MPVs from different multi-poles are perfectly independent.The first calculation of the joint probability densitiesof random spin- l states was performed in 1995 by Han-nay, see [34]. He notes that the Majorana function equalsexactly the Bargmann function of the spin state in theSegal-Bargmann space [35]. This representation of quan-tum states can be seen as a third leg of standard quan-tum mechanics accompanying Heisenberg’s matrix- andSchr¨odinger’s wave function quantum mechanics.It turns out that in the completely random case the n -point density can be written as a normalized permanent p ln ( ζ , . . . , ζ n , ζ l +1 = − /ζ ∗ , . . . , ζ l + n = − /ζ ∗ n )= 1 π n per( C − B † A − B )det( A ) , (27)with f i := f Ψ ( ζ i ) Majorana function evaluated at the root A ij = (cid:104) f i f ∗ j (cid:105) isotropy = l (cid:88) m,m (cid:48) = − l ( − m + m (cid:48) (cid:20)(cid:18) ll + m (cid:19)(cid:18) ll + m (cid:48) (cid:19)(cid:21) / · (cid:104) a lm a ∗ lm (cid:48) (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) = C l δ mm (cid:48) ζ l + mi ( ζ ∗ j ) l + m (cid:48) = C l (1 + ζ i ζ ∗ j ) l (28) B ij = (cid:104) f i f (cid:48)∗ j (cid:105) isotropy = C l lζ i (1 + ζ i ζ ∗ j ) l − (29) C ij = (cid:104) f (cid:48) i f (cid:48)∗ j (cid:105) isotropy = C l l (1 + 2 lζ i ζ ∗ j )(1 + ζ i ζ ∗ j ) l − , (30)where the second equalities only hold in the isotropiccase; see also [32]. The matrices A, B, C are (2 n × n )-matrices. Calculating A − and inserting the explicit for-mulas from (28) to (30) yields the joint probability den-sity of 1 ≤ n ≤ l MPVs. Note that here the function f can in principle be complex. Isotropy and Gaussian-ity enter the game when the precise expressions (28)-(30)are inserted. Even though the function f can be complex,one should remember that the representation of the func-tion by MPVs is possible only for real functions.An alternative derivation of the full joint density ( n = l ), made by Dennis in [32], uses the fact that the coeffi-cients of any polynomial can be expressed by a symmetricpolynomial of its roots leading to the full joint density p ll ( { ζ i } ) = (2 l − (cid:81) lj =1 j !(2 π ) l l ! (cid:81) lj =1 | ζ j | · (cid:81) lj,k =1 ,j 2. Both probability densitiesare normalized in a way such that (cid:82) π/ dΘ p (Θ) = 1. InFig. 2 the two densities are plotted together. One can seethat the interaction of MPVs leads to repulsion. Biggerangles are more probable in the case of real MPVs thanin the case of uniformity and independence. Such a be-havior is characteristic for roots of random polynomials.The results from (27) are a priori complicated expres-sions, even for the case n = 3, whose integration does notallow for a faster numerical computation of confidencelevels than a full Monte Carlo simulation. V. CMB DATA For our analysis we make use of the Planck 2015 full-sky CMB intensity maps (see [36]) with N side = 2048and treat it within HEALPy, which is a HEALPix [37]implementation for python , see [38].The Planck full-sky maps are provided in nine differ-ent frequency bands. These maps have been foregroundcleaned to produce best estimates of intensity and po-larization of the measured CMB signal from the sky af-ter instrumental and known systematical effects, like thedipole and quadrupole (DQ) induced by the motion ofthe Sun and Earth with respect to the cosmic frame; thelight from the zodiac cloud; and most important galac-tic foregrounds (synchrotron radiation, free-free emission,thermal dust, CO lines, anomalous microwave radiation)FIG. 2: Comparison of the two-point density of MPVsdrawn from a Gaussian ensemble of a lm and identicallyuniformly distributed pseudo-MPVs for l = 2 l COMMANDER NILC SEVEM SMICA2 (6.9, 21.2) (12.8, 20.3) (13.3, 26.2) (5.7, 23.7)(119.1, 18.2) (117.5, 20.1) (83.6, 12.3) (121.5, 21.9)3 (25.6, 8.8) (23.2, 9.3) (33.0, 5.4) (22.1, 8.8)(86.3, 38.4) (86.8, 37.7) (61.3, 35.0) (88.1, 38.8)(317.7, 5.0) (315.3, 7.9) (140.5, 0.6) (314.8, 10.5)4 (69.5, 3.2) (69.9, 4.5) (71.1, 19.2) (68.8, 2.2)(207.5, 72.6) (203.9, 70.5) (189,4, 73.0) (207.8, 38.5)(211.2, 36.7) (212.6, 40.0) (201.7, 38.0) (214.8, 69.9)(333.4, 29.1) (333.5, 27.7) (336.5, 27.3) (335.9, 26.1)5 (43.3, 33.3) (44.2, 35.8) (51.5, 28.2) (44.3, 36.7)(98.7, 35.7) (96.8, 36.1) (79.8, 35.3) (98.3, 36.2)(174.0, 3.9) (175.2, 3.4) (174.2, 3.8) (175.7, 3.3)(232.1, 54.2) (232.4, 55.3) (232.1, 58.2) (234.9, 55.3)(287.4, 31.6) (286.1, 31.6) (290.1, 18.9) (285.2, 32.8) TABLE I: MPVs from l = 2 to l = 5 in galacticcoordinates ( l, b ) in deg with the precision of onedecimal. All MPVs have been taken to lie in thenorthern hemisphere and for a given multipole theywere ordered according to their value of the galacticlongitude. MPVs in the same line cannot necessarily beidentified with each other since they are not a priori ordered inside of a given multipole.have been removed. Using four different cleaning algo-rithms, foreground cleaned full-sky maps of CMB tem-perature intensity are constructed; for the details of thecomponent separation process we refer to [19]. Thesedifferent cleaning algorithms, are called COMMANDER,NILC, SEVEM and SMICA.The four maps have been used to extract the MPVsup to l = 50 by using a tensorial algorithm [26], seealso Sec. III B 1. A list of the MPVs for the multipoles l = 2 , , , l are provided at https://github.com/MPinkwart/MPV-files-Pinkwart-Schwarz . VI. STATISTICS AND SIMULATIONSA. Statistics One needs statistics that deliver information aboutboth intramultipole alignments and alignments of mul-tipoles with some given astrophysical direction, which inthe following will be referred to as the outer direction.Furthermore, possible statistics are not allowed to de-pend on the ordering of MPVs for a fixed l since thisordering is completely arbitrary and contains no infor-mation. Additionally, the statistics may not depend onthe hemisphere, since MPVs are lines rather than vec-tors. Eventually, they may not be sensitive to the equa-tor since the gluing mechanism at the equator shouldbe hidden. In the following, intramultipole statistics aresometimes also referred to as inner statistics and statis-tics that investigate correlations with outer directions asouter statistics.We used the following two outer statistics S || D ( l ) := 1 l l (cid:88) i =1 | v ( l,i ) · D | (34) S v D ( l ) := 2 l ( l − (cid:88) ≤ i 13 + 23 (cid:18) − √ (cid:19) π (cid:19) ≈ . . (40)If the MPVs are not correlated but are all independent, S || D would follow a slightly modified Irwin-Hall distribu-tion, see [39, 40], p S || D ( l ) ( s ) = l l − l (cid:88) k =0 ( − k (cid:18) lk (cid:19) ( ls − k ) l − sgn( ls − k ) , (41)which would result in a variance of S || D (2) of about 0 . a lm and compare their MPV statistics withthe one from the cleaned Planck maps. From Fig. 6a onededuces that the theoretical result (40) for the varianceis compatible with the numerical result for the 1 σ -regionbecause √ . ≈ . S i,l be the data point of statistic S i ( l ) received by one of the four Planck maps, where i ∈ { , , , } runs through the four statistics. Definethe p-value of this data point as P ( S i,l ) := (cid:90) S i,l d s p S i ( l ) ( s ) , (42)i.e. small ( (cid:28) 1) as well as large ( ≈ 1) p-values indi-cate unusual behavior. Now let S Mi,l denote a data pointas above received from map M (COMMANDER, NILC,SEVEM or SMICA). We define the outer likelihood L outer l, D ( M ) := 4 (cid:89) outer P (cid:0) S Ml,i (cid:1) (cid:0) − P (cid:0) S Ml,i (cid:1)(cid:1) , (43)as well as the inner likelihood L inner l ( M ) := 4 (cid:89) inner P (cid:0) S Ml,i (cid:1) (cid:0) − P (cid:0) S Ml,i (cid:1)(cid:1) , (44) S || D S || S || D S v S || D S vD S || S v S || S vD S v S vD FIG. 3: Linear correlation coefficients of statistics basedon a Monte Carlo simulation with 1000 ensembles of a lm . The combination S v − S v D is the one mainly usedin previous studies by means of correlations betweenarea vectors. It shows only slight correlation at thelargest angular scales.and the alignment likelihood L || l, D ( M ) := 4 (cid:89) || P (cid:0) S Ml,i (cid:1) (cid:0) − P (cid:0) S Ml,i (cid:1)(cid:1) . (45)The inner likelihood measures anomalies inside a givenmultipole, while the outer likelihood measures theanomalies with respect to some outer direction. Even-tually, the alignment likelihood measures the combinedeffect of alignment with an outer direction and intramul-tipole alignment.Before studying the Planck maps we need to under-stand the correlation between statistics. For low l Fig. 3shows that the outer as well as the inner statistics arehighly (anti)correlated. For higher l the linear correlationof inner statistics vanishes, the outer statistics keep theircorrelation on a wide range of scales. Apart from sta-tistical fluctuations the alignment statistics show nearlyno correlation at all. Naively one would expect the innerand outer correlations to effect the behavior of the like-lihoods, but the distribution of likelihoods in the range2 ≤ l ≤ 50 is nearly the same for all three consideredtypes of likelihoods. B. Simulations We generate a fixed set of 1000 ensembles of Gaussianand isotropic random a lm on the range 2 ≤ l ≤ 50 whileassuming Planck 2015 best-fit cosmological data to befixed. Then we use the MPV calculation program [26] toextract MPVs for each of the 1000 ensembles and for thefour full-sky foreground cleaned Planck 2015 CMB maps.0From the MPVs we calculate the statistics described be-fore. Then, using the fixed set of 1000 random ensembleswe calculate mean values, confidence levels, p-values andlikelihoods and use them for analysis of all four Planckmaps. VII. TEST DIRECTIONS We use the following four physical directions, whosepossible influences should have different and independentphysical reasons: • The cosmic dipole ( l, b ) = (264 . ◦ , . ◦ ) (withan amplitude ( δT /T ) dip = (3 . ± . × − ),taken from [15]. The CMB dipole is assumed to bedue to the peculiar motion of the Solar System withrespect to the cosmic comoving frame [42]. A cor-relation with this direction could imply that thenature of the kinematic dipole is not fully under-stood yet, that it has not been removed from thedata properly, that the CMB contains an intrinsicdipole for itself, or that the calibration pipeline isodd. • The Galactic Pole ( l, b ) = (0 ◦ , ◦ ). Galactic fore-grounds which are aligned with the disk of theMilky Way could give rise to an alignment withthe Galactic Pole. • The Galactic Center ( l, b ) = (0 ◦ , ◦ ). The fore-ground pollution due to the inner part of the MilkyWay could still be present in the cleaned maps. Acorrelation with this direction would indicate thatthese residuals still play an important role in dataanalysis. • The ecliptic pole ( l, b ) = (96 . ◦ , . ◦ ) (trans-formed from ecliptic to Galactic coordinates withthe NASA conversion tool [43]). The lowest mul-tipoles are known to correlate unusually with theecliptic. Foreground pollution from the Solar Sys-tem could cause such a correlation.In Fig. 4 we plot the MPVs for all pipelines at l = 2 to-gether with the four outer directions and the intersectionof the plane orthogonal to the cosmic dipole with the ce-lestial sphere in stereographic projection from the southpole. One should note that the stereographic projectiondoes not preserve distances. Arcs close to the south poleget stretched with respect to arcs close to the north pole.But since we only consider one hemisphere, distances ofpoints on the sphere are approximately conserved. De-spite this disadvantage the stereographic projection waschosen because it allows for a simple and straightforwardinterpretation and has a nice geometrical meaning. Notethat due to the identification of antipodal MPVs, oppo-site points on the unit circle in stereographic projectionhave to be identified. For l = 2 the plot already showsone feature that we will encounter in Sec. VIII, namely FIG. 4: MPVs for l = 2 and physical directions instereographic projection. The violet curve shows theplane orthogonal to the cosmic dipole.that the MPVs nearly lie in the plane orthogonal to thecosmic dipole.In Appendix C further stereographic projection plotsfor l = 3 , l = 48 , 49 (Fig. 19) can befound. l = 2 , , l =48 , 49 have been chosen because they depict two highermultipoles which are orthogonal in the sense that one isespecially unlikely and one is especially normal. VIII. RESULTS The four statistics mentioned in Sec. VI were cal-culated for all four full-sky maps using all test di-rections and compared to the statistics of one thou-sand Monte Carlo ensembles of Gaussian and isotropic a lm . We provide a qualitative description of the re-sults in Secs. VIII B 1-VIII B 5 before before summariz-ing the findings in more precise statistical statements inSec. VIII B 6. A. Reproduction of known large scale anomaliesand investigation of intermediate scales with outervertical statistic It has been observed in previous studies that on thelargest angular scales, i.e. the quadrupole and octupole,the MPVs correlate with the cosmic dipole. Fig. 5ashows that this correlation is due to a strong orthogo-nality of the MPVs and the dipole direction. The areavectors of the quadrupole and octupole in COMMAN-DER, NILC and SMICA show an alignment with thedipole at 2 σ level. By visualizing the quadrupole asa plane, this means that the cosmic dipole direction is1nearly perfectly orthogonal to this plane which cannotbe achieved in about 96% of random ensembles of Gaus-sian and isotropic temperature fluctuation fields. Thequadrupole value of SEVEM seems less anomalous, butas we will argue later, we find hints that SEVEM stillshows residual foreground effects via a correlation of theMPVs with the Galactic Center and especially with theGalactic Pole. That the MPVs of the quadrupole arealmost normal to the dipole direction is also shown inFig. 4.Fig. 5b shows the vertical outer statistic for smallerangular scales. The region 20 ≤ l ≤ 24 sticks out justlike the largest angular scales. At these scales data pointsoutside the 1 σ regions cluster. Both scale ranges showa similar behavior; first the MPVs are too close to theplane orthogonal to the cosmic dipole, and then theyare too far away from this plane. It should be notedthat the two suspicious scales (large and intermediate)coincide with the scales at which the measured angularpower spectrum deviates from the best-fit ΛCDM modelof the Planck 2015 analysis [15]. This hints towards aconnection between the power spectrum deviation andthe peculiar motion of the Solar System with respect tothe cosmic frame. B. Comparison of directions and pipelines usingaligned statistics In Figs. 6–9 we plot the outer statistic S || D for each ofthe five directions including the 1 σ to 3 σ regions fromthe Monte Carlo simulations in the range of large an-gular scales 2 ≤ l ≤ 11 and in the range of smallerangular scales 12 ≤ l ≤ 50, comparing in each plot allfour pipelines. Figure 10 shows the same for the innerstatistic S || . Figure 17 shows the outer likelihood as afunction of l for the cosmic dipole. In Table II we presentmultinomial probabilities and respective p-values. 1. Cosmic dipole One observes (see Fig. 6a) that the large scale anticor-relation of the quadrupole and octupole with the cosmicdipole – see for example the review [21] – is still presentin the second release data. While for SEVEM the an-tialignment is more pronounced at l = 3 than at l = 2,both multipole data points are equally unusual in theother three pipelines (both nearly 2 σ ). It turns out thatfor l = 4 an even less expected alignment of the COM-MANDER, NILC and SMICA data with the dipole ispresent. Except for SEVEM each of the lowest multi-poles l = 2 , , , ≤ l ≤ 50 SEVEM is less l S ( l ) a) Large angular scales 2 ≤ l ≤ 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Smaller angular scales 12 ≤ l ≤ FIG. 5: Comparison of pipelines for S v D with D thecosmic Dipole. The expectation value and 1 , , σ regions from Monte Carlo simulations are included.aligned with the cosmic dipole than the other cleanedmaps and it admits more unlikely data points.Concerning COMMANDER, SMICA and NILC, all inall 17 out 49 multipoles are outside of the 1 σ region.Despite the large angular scale 2 ≤ l ≥ l S ( l ) a) Large angular scales/small multipoles 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Small angular scales/large multipoles FIG. 6: Comparison of pipelines for S || D with D thecosmic dipole. The expectation value and 1 , , σ regions from Monte Carlo simulations are included.different masks the strong coincidence surprises. 2. Galactic Pole The large scale behavior with respect to the Galac-tic Pole is as expected (even in the SEVEM map); seeFig. 7a. When referring to larger multipoles (see Fig. 7b)SEVEM shows even stronger deviations from the othermaps than in the other directions. From l = 12 onSEVEM is tremendously aligned with the Galactic Pole.Eight out of 39 data points lie even outside of the 3 σ re-gion. Again, we state that this behavior might be a hinttowards Milky Way residuals in the SEVEM map.When comparing Fig. 7 to Fig. 20a from Appendix D,where the statistic S || D is plotted for the Galactic Pole for l S ( l ) a) Large angular scales/small multipoles 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Small angular scales/large multipoles FIG. 7: Comparison of pipelines for S || D with D theGalactic Pole. The expectation value and 1 , , σ regions from Monte Carlo simulations are included.all four pipelines but with the SEVEM mask applied tothe map, it becomes obvious that the strong alignmentof SEVEM with the Galactic Pole is solely due to themasked region. When the mask is applied, all four mapsshow a similar behavior and the deviation of SEVEMfrom the others vanishes nearly completely, especially inthe high l regime. This is not surprising since it is as-sumed by the Planck Collaboration itself that SEVEMcarries residual effects of the Galactic Plane.Hence, one concludes that one should be careful whenusing SEVEM for full-sky analyses. A more detailed in-vestigation of the precise cleaning algorithm needs to betaken into account.Finding such a strong deviation of the SEVEM mapfrom complete randomness is a confirmation of the powerof MPV to identify alignment effects.Concerning the other maps neither alignment nor an-3 l S ( l ) a) Large angular scales/small multipoles 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Small angular scales/large multipoles FIG. 8: Comparison of pipelines for S || D with D theGalactic Center. The xpectation value and 1 , , σ regions from Monte Carlo simulations are included.tialignment is preferred.Altogether, on the whole range 2 ≤ l ≤ 50 the GalacticPole incorporates more low probability multipoles thanthe Galactic Center or the ecliptic pole but approxi-mately as many as the cosmic dipole. 3. Galactic Center The behavior with respect to the Galactic Center tendsto be less unexpected than the cosmic dipole on large an-gular scales; see Fig. 8a. Only l = 2 , σ . Again SEVEM deviates clearly from the othermaps on smaller angular scales (see Fig. 8b), this timeshowing a stronger alignment with the Galactic Center,especially on the midrange scales which correspond ap-proximately to the angular size of the Galactic core. l S ( l ) a) Large angular scales/small multipoles 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Small angular scales/large multipoles FIG. 9: Comparison of pipelines for S || D with D theecliptic pole. The expectation value and 1 , , σ regionsfrom Monte Carlo simulations are included.There are three multipoles far away from the expecta-tion in the non-SEVEM maps. The multipoles l = 16 , σ regions and l = 16 is nearly at 3 σ . 4. Ecliptic pole On large angular scales (see Fig. 9a), the data show aneven more expected behavior with respect to the eclipticpole than with respect to the cosmic dipole.On smaller angular scales (see Fig. 9b) SEVEM againclearly deviates from the other maps, showing more an-tialignment with the ecliptic pole. For the other maps,the ecliptic seems to show less correlation with the CMBdata than the dipole.4 l S ( l ) a) Large angular scales/small multipoles 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 l S ( l ) b) Small angular scales/large multipoles FIG. 10: Comparison of pipelines for S || . Theexpectation value and 1 , , σ regions from Monte Carlosimulations are included. 5. Inner alignment The inner statistic S || comes equipped with a muchsmaller variance than the outer statistic S || D , as can beseen in Fig. 10. Hence, the inner statistic is more sus-ceptible to computational errors or minor fluctuationsthan the outer statistic, but nevertheless the plots painta distinct picture: the inner statistic of COMMANDER,NILC and SMICA lies inside 1 σ for most of the multi-poles and does not leave 2 σ one single time. No strongoutliers are present, either in the low l or in the large l regime. If anomalies are present in the CMB, they seemto be mainly caused by correlation with outer directions,while no remarkable intramultipole correlation of MPVscan be observed. Note that other methods than ourscould reveal hidden intramultipole correlations that can- not be observed with our simple method. 6. Multinomial p-values In Table II we gather the number of multipoles in therange 2 ≤ l ≤ 50 lying within 1 σ ( n ), between 1 σ and2 σ ( n ) as well as outside of 2 σ ( n ) and the probabilityof that based on a multinomial distribution p ( n , n , n ) = (cid:18) n , n , n ) (cid:19) (0 . n (0 . n (0 . n (46)for the statistics S || D and S || . We also define a correspond-ing multinomial p-value, which is the probability to findat least n and n multipoles at 1 σ to 2 σ and above 2 σ deviation from expectation, i.e.multinomial p-value = n (cid:88) i =0 i + n (cid:88) k =0 p ( n − i, n + i − k, n + k ) . (47)The smaller the p-value the more the data deviate fromthe expectation concerning the number of outliers. Analternative definition of the p-value as the sum over allmultinomial probabilities that are smaller than the givenprobability would result in higher p-values, but wouldbe inadequate for our purposes since it would involveconfigurations which are unlikely normal, as for example( n , n , n ) = (49 , , 0) as well.Inspecting Table II, the first thing to note is thatSEVEM is strongly (anti)correlated with the GalacticPole, as it shows a p-value of 10 − %. The correla-tion with the Galactic Center is also significant with ap-value of 0 . 14 %. All non-SEVEM maps behave as ex-pected with respect to the Galactic Pole and the eclipticpole. Regarding the inner alignment a slight deviation ofCOMMANDER from the other maps can be observed.While NILC and SMICA do not possess any data pointoutside of 2 σ , COMMANDER contains some data pointsshifted from 1 σ to beyond 2 σ . Concerning the GalacticCenter NILC seems more normal than COMMANDERand especially SMICA. However, the deviations are notextremely large and taking into account the other two di-rections, the similarity of all non-SEVEM maps emergesagain. The cosmic dipole is the only considered directionfor which all maps show p-values below 10%. Next, weinvestigate from which ranges this slight correlation ofdata and dipole stems.The correlation of MPV with the cosmic dipole is stud-ied in more detail in Fig. 11. It shows the dependence ofthe multinomial p-value on l max as one enlarges the con-sidered multipole range from [2 , 2] to [2 , l max ]. Here weadditionally take into account statistic S v D , which mea-sures orthogonality of a multipole to the given direction,and only focus on the NILC map (it is most normal withrespect to all the other tests considered here). We findthat the correlation with the cosmic dipole is due to thelargest angular scales and the region around l = 20. The5 Direction Map < σ σ > σ p multi (%) Multinomial p-value (%)Cosmic dipole COMMANDER 31 14 4 1.06 6.41NILC 32 13 4 1.12 8.11SMICA 28 19 2 0.84 5.64SEVEM 32 12 5 0.42 3.16Galactic Pole COMMANDER 37 11 1 2.33 79.30NILC 36 12 1 2.96 73.90SMICA 33 16 0 1.42 51.46SEVEM 10 11 28 10 − − Galactic Center COMMANDER 32 14 3 2.25 16.77NILC 34 13 2 3.48 41.87SMICA 29 17 3 1.15 6.16SEVEM 26 17 6 0.04 0.14Ecliptic pole COMMANDER 32 15 2 3.15 27.59NILC 36 10 3 1.33 28.80SMICA 34 12 3 2.15 24.10SEVEM 29 17 3 1.15 6.16Inner alignment COMMANDER 30 16 3 1.58 9.19NILC 32 17 0 1.13 39.43SMICA 32 17 0 1.13 39.43SEVEM 32 15 2 3.15 27.59 TABLE II: Number of multipoles lying inside the 1 σ region, between the 1 σ and 2 σ boundaries and outside of the2 σ region for all maps and directions for statistic S || D as well as for statistic S || . In the last two columns we give themultinomial probability for these distributions of multipoles amongst the σ regions in percent up to two digits andthe respective p-values.FIG. 11: Multinomial p-values for outer statistics S || D and S v D with D the cosmic dipole. The p-value iscalculated from l = 2 to l = l max using the NILC map.p-value of the aligned statistics drops to 0 . 4% for therange [2 , , l = 20 the p-value curveshows a dip with a minimum at l = 22 and p-value of0 . 7% for the aligned statistic and a drop in the verti-cal statistic with a minimum at l = 24. A third re-gion where the p-value of both statistics clearly drops isaround l = 42, but with higher multinomial p-value thanat l around 20. It is remarkable, that these three regions are exactly those regions where the power spectrum de-viates from the best-fit Planck value [15].We conclude that there are three main features foundin our investigation: the SEVEM map is affected by theGalactic Pole and Galactic Center directions and whenused for full-sky analyses a careful treatment of its pro-cessing algorithm should be taken into account. Theother three maps agree remarkably well, except with re-spect to their alignment towards the Galactic Center.The cosmic dipole is the only considered physical direc-tion, for which we are able to identify an effect on allfull-sky maps. The alignments are localized in multipolespace and stem from three ranges l ∈ [2 , l around 20and l around 42. C. Comparison of directions using likelihoodhistograms By plotting histograms for the likelihoods introducedin Eqs. (43) and (44) on logarithmic intervals [0 , , 10) and [10 , ≤ l ≤ 50. The resultsconfirm the findings of the last section. Furthermore,combining two statistics into one likelihood compressesthe information content.6 Likelihood05101520253035404550 N u m b e r o f m u l t i p o l e s FIG. 12: Inner likelihood histogram for SEVEM.Gaussian, isotropic expectation and 1 / / σ regions(black, gray and lightgray, respectively) are included. Likelihood05101520253035404550 N u m b e r o f m u l t i p o l e s Cosmic DipoleGalactic PoleEcliptic PoleGalactic Center FIG. 13: Outer likelihood histogram for SEVEM.Gaussian, isotropic expectation and 1 / / σ regions(black, gray and lightgray, respectively) are included. 1. SEVEM The large deviations from the expectation, which havebeen observed by investigating the alignment statisticsalone, can be seen more easily from the likelihood his-tograms of the inner likelihood (see Fig. 12) and the outerlikelihood (see Fig. 13).The bin [0 , 1) in Fig. 12 contains too many multi-poles at the 2 σ level, while the bin of largest multipoles[10 , σ level. This shows that likelihoods are shifted from thelargest to the smallest values.SEVEM’s strange behavior becomes even more pro-nounced when considering the outer likelihoods, seeFig. 13. While the ecliptic pole and the cosmic dipole Likelihood05101520253035404550 N u m b e r o f m u l t i p o l e s FIG. 14: Inner likelihood histogram for NILC.Gaussian, isotropic expectation and 1 / / σ regions(black, gray and lightgray, respectively) are included.show anomalous behavior at the 2 σ level, the GalacticCenter (3 σ ) and especially the Galactic Pole ( (cid:29) σ ) arefar off from the expectation. Altogether, 21 out of 49values for the Galactic Pole have a likelihood which issmaller than 1%, which is far beyond 3 σ .Hence, we can conclude that SEVEM shows a slightanomaly with respect to intramultipole correlations,while it shows an enormous anomaly with respect toouter correlations with the Galactic Center and moststrongly with the Galactic Pole, whose statistics corre-spond to measures of the influence of the Galactic Plane.The combination of the Galactic Pole and Center anoma-lies evokes the conjecture, that SEVEM is influenced bythe Milky Way when no masking procedure is considered. 2. NILC Now, we exclude SEVEM and only consider the otherthree maps: COMMANDER, NILC and SMICA. Itturns out, as already conjectured in the investigationof the pure statistics, that all three maps deviate onlymarginally. While COMMANDER tends to be the mapwith slightly larger likelihoods than the other two maps,NILC is equipped with the smallest confidence mask andtherefore we choose to present only the NILC results andmention that COMMANDER seems to be closer to theexpectation while SMICA is slightly further away thanNILC. Here again the striking similarity of all three maps,despite their very different cleaning procedures, is quiteremarkable. For (nonlogarithmic) likelihood histogramsof the other maps we refer to [44].The overall structure of inner likelihoods (see Fig. 14)is remarkably normal except for the fact that the numberof multipoles in the likelihood bin [1 , 10) is too low atthe 1 σ level while the single multipole in the lowest bin7 Likelihood05101520253035404550 N u m b e r o f m u l t i p o l e s Cosmic DipoleGalactic PoleEcliptic PoleGalactic Center FIG. 15: Aligned likelihood histogram for NILC.Gaussian, isotropic expectation and 1 / / σ regions(black, gray and lightgray, respectively) are included. Likelihood05101520253035404550 N u m b e r o f m u l t i p o l e s Cosmic DipoleGalactic PoleEcliptic PoleGalactic Center FIG. 16: Outer likelihood histogram for NILC.Gaussian, isotropic expectation and 1 / / σ regions(black, gray and lightgray, respectively) are included.equals the expectation value. It seems that some artificialintramultipole isotropy could have been induced in thecourse of data processing, resulting in a lack of variancein intramultipole correlations. Apart from this, the innerlikelihoods do not show any further noticeable feature.The aligned likelihood, see Fig. 15, shows expectedbehavior for all directions but the cosmic dipole. Thenumber of multipoles with aligned likelihoods in the bin[1 , 10) for the cosmic dipole as outer direction is higherthan one would expect from a Gaussian and isotropic setof a lm at approx. 1 . σ .The outer likelihood (see Fig. 16) shows the same be-havior as the aligned likelihood but with an additionalexcess of likelihoods in the [0 , σ level for thecosmic dipole, the Galactic Pole and the ecliptic pole, and l L ( l )[ % ] NILCCOMMANDERSMICA FIG. 17: Comparison of outer likelihoods in dependenceof l for the range 2 ≤ l ≤ 50 and the cosmic dipole asthe outer direction. SEVEM has been excluded.at 2 σ level for the Galactic Center. Considering both binsof small likelihoods [0 , 1) and [1 , 10) together, the cosmicdipole sticks out most again. The similar behavior ofthe outer and aligned likelihoods confirms the robustnessof the likelihood definition against correlations of the in-cluded statistics.Figure 17 shows the outer likelihoods for the cos-mic dipole. It is clearly shown that there is a range,25 ≤ l ≤ 34, that does not include any unlikely data pointregarding the cosmic dipole. Hence, using the methodapplied here, we cannot identify any statistically signif-icant effect of the cosmic dipole on the data on angularscales of about 5 . . ≤ l ≤ ≤ l ≤ 24 low likeli-hoods cluster. Comparing the two statistics S v D and S || D one sees that the contribution to low likelihoods in thisrange mainly stems from the vertical statistic. We con-clude that the slight excess of low likelihoods regardingthe cosmic dipole in the range 2 ≤ l ≤ 50 mainly stemsfrom the two regions 2 ≤ l ≤ ≤ l ≤ IX. DISCUSSION We find that SEVEM strongly deviates from COM-MANDER, NILC and SMICA in every regard. It isstrongly aligned with the Galactic Center and especiallythe Galactic Pole and antialigned with the dipole andthe ecliptic pole in the S || D statistic. The alignment withthe Galactic Center is most prominent on midrange mul-tipoles, indicating a residual effect of the Galactic Corethat has not been removed in the cleaning process. Fur-thermore, the deviation of SEVEM from the other mapsis most present at l ≥ 12, which indicates that the centralpart of the Milky Way is the dominating source of dis-traction in SEVEM. Nevertheless, the correlation with8the cosmic dipole, that is present in the other maps,could also be seen in SEVEM. It is just overshadowedby the sizable galactic residuals. Since none of the aboveobservations surprised, SEVEM serves us as a controlmap. The fact that we were able to identify the expectedresidual foreground features of SEVEM with our methodshows that our method yields geometrically easily inter-pretable results and that the heuristic geometric intuitionof the used statistics is correct. Hence, we propose thatthe outer statistics truly measure the influence of thegiven directions that are included.COMMANDER, NILC and SMICA show a similar be-havior, deviating only marginally on the observed range.While with respect to the Galactic Center and Pole andthe ecliptic pole the data do not show abnormal statis-tical behavior, a correlation with the cosmic dipole isvisible, concentrating mainly on largest angular scales2 ≤ l ≤ l = 20 , , , , l = 2 , , l = 4), but also the range 35 ≤ l ≤ ≤ l ≤ 34 issurprisingly normal with the absence of small likelihoods.The large scale (anti)alignments might imply that we donot yet fully understand the true nature of the dipole,i.e. the relative motion of the Solar System with respectto the cosmic frame. Since these anomalies are presentin all of the maps such a physical origin could be morelikely than data processing reasons.It should be noted that the SMICA algorithm as-sumes isotropy and Gaussianity from the beginning andthus it is biased. Furthermore some assumptions on thespectrum on synchrotron and free-free emission in thephysics-based cleaning process of COMMANDER as wellas its noise model might induce a bias on isotropy. In aweak sense also NILC might be biased. SEVEM is theonly map where an a priori bias on isotropy and Gaus-sianity can be excluded. This fact might influence theinterpretation of the results. X. CONCLUSION The purpose of this work was to study the completerandomness of the microwave sky by means of multipolevectors (MPV) in the hope of identifying deficits in ourunderstanding or the data analysis of CMB full-sky maps.We gave an overview over different extraction meth-ods for MPVs and their statistical properties. MPVscan be represented via a symmetric and trace-free ten-sor applied to the symmetric and trace-free product ofunit vectors, which yields an algorithm for extractingMPVs from the spherical harmonic decomposition. Al-ternatively methods from algebraic geometry can be usedto identify MPVs as lines in C P . A third approach uses the extension of the Bloch sphere to higher spin and thestereographic projection. The resulting polynomial canbe understood as the scalar product of a spin state withBloch coherent states. The latter approach can be usedto assign joint probability densities to MPVs. It turnsout that the explicit expression for the joint probabilitydensity is the same for the set of all distributions of com-pletely random a lm . This set forms a subset of statisticalisotropic distributions and the intersection of completelyrandom and Gaussian distributions yields the regime ofstandard cosmology.Using different simple statistics we observed numeri-cally a correlation of the full-sky cleaned maps with thecosmic dipole on the largest angular scales 2 ≤ l ≤ l = 20 , , , , 24. Fur-thermore around l = 40 low likelihoods cluster and themultinomial p-value drops. These are the same multipolenumbers which also deviate from the theoretical expecta-tion in the angular power spectrum [15]. To the authors’knowledge, this ”conspiracy” of MPV and power spec-trum has not been observed before. Other covariances ofCMB anomalies have recently been investigated in [45].One main conclusion we draw is that the SEVEM mapis still strongly correlated with the Galactic Center andespecially the Galactic Pole in our analysis. The cross-talk between MPVs and masked skies needs to be studiedin more detail before one can use the foreground cleanedmaps with small galactic masks for MPV analysis.In the future, one could also study cross-multipole cor-relations on the observed range of scales and investigate ifthe previously observed large scale correlations continuesdown to smaller scales.Furthermore one needs more insight about possiblephysical reasons for CMB anomalies. One should espe-cially focus on detailed studies of the dipole and reveal itstrue nature. Analyses of the radio sky with galaxy sur-veys hint towards an increased radio dipole amplitude[46–49], which could be caused by an intrinsic, nonkine-matic CMB dipole. ACKNOWLEDGMENTS python .9 Appendix A: Derivation of one-point density For l = 1 we get from (31) p (cid:18) ζ, − ζ ∗ (cid:19) = 1 π | ζ | (cid:12)(cid:12)(cid:12) ζ + ζ ∗ (cid:12)(cid:12)(cid:12) ((1 + | ζ | )(1 + | /ζ ∗ | )) / = 1 π | ζ | ) . Using ζ = tan( θ/ 2) exp( iφ ) we receive (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂ ( ζ, ζ ∗ ) ∂ ( θ, φ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = tan( θ/ ( θ/ ⇒ d Ω = 2 sin( θ/ 2) cos( θ/ θ/ dζdζ ∗ | ζ | = 2 dζdζ ∗ (1 + | ζ | ) , where d Ω = sin( θ ) dθdφ denotes the solid angle elementand (cid:12)(cid:12)(cid:12)(cid:16) ∂ ( ζ,ζ ∗ ) ∂ ( θ,φ ) (cid:17)(cid:12)(cid:12)(cid:12) the Jacobi determinant of the change ofcoordinates. Hence we have p ( θ, φ ) 2 dζdζ ∗ (1 + | ζ | ) = p ( ζ, − /ζ ∗ ) dζdζ ∗ ⇒ p ( θ, φ ) = 12 π . Appendix B: Joint Probability Distribution ofMultipole Vectors: Connection to Gaussian analyticfunctions and random matrix theory The Majorana polynomial in (18) is a special case ofthe wide class of Gaussian analytic functions (GAFs), see[50]. In general, a GAF is defined as a random field on C n such that for each z , . . . , z n the quantity f ( z , . . . , z n ) isa normally distributed random variable.For every L ∈ N the function f ( z ) = L (cid:88) n =0 (cid:115)(cid:18) LL − n (cid:19) a n z n , (B1)with identically and independently distributed zero meanand unit variance complex random variables a n is a GAFwhose zero set is invariant under the action of SO(3).Its covariance kernel is given by Cov( f ( z ) , f ( w )) = K ( z, w ) = (1 + zw ∗ ) L . The Majorana polynomial equalsthis GAF up to a factor ( − l which can be combinedinto Ψ m , and with L = 2 l and substituting n = m + l ,yielding Ψ m = a m + l . The Ψ m do not have unit variance,but variance C l . By rescaling Ψ m a common factor for allΨ m can be pulled out of the sum. This does not changethe behavior of the zeros.The general density (27) holds for every GAF, whilethe one-point density (32) can be expressed as p ( z ) = ∆ log( K ( z, z )) / π, (B2) for a general GAF. This equals – up to a different nor-malization – the one-point density of the Majorana poly-nomial in C which was used in the proof of (32). Theformula above is known as the Edelman-Kostlan formula;see [51].One can show that one-point statistics, which are com-pactly supported, are asymptotically normal regardingrotationally invariant GAFs. Let φ ∈ C c (Λ) and L L ( φ ) := (cid:88) z ∈ f − (0) φ ( z ) , (B3)then the following asymptotic behavior is valid: √ L ( L L ( φ ) − (cid:104)L L ( φ ) (cid:105) ) l →∞ , distribution → N (0 , κ ( φ )) , (B4)where κ ( φ ) denotes some number that depends on thefunction φ . Unfortunately, the above is a priori not truefor functions φ with arbitrary support. Hence, it doesnot apply to the statistics in Sec. VI. Since we are deal-ing with one hemisphere, one could restrict the scalarproducts appearing in those statistics to the unit disc.This cutoff compactifies the statistic but unfortunatelyit destroys any kind of differentiability. Nevertheless theresult above could be used to study local statistics oncertain patches on the sky in the large l limit in futureinvestigations.Remember that the Majorana polynomial has covari-ance kernel K ( z, w ) = (1 + zw ∗ ) l . The following state-ment will show that MPVs as zeros of the isotropic GAFand eigenvectors of Gaussian random matrices are tightlyconnected: let A , B be independent ( n × n ) randommatrices with identically and independently distributedcomplex standard Gaussian entries. Then the eigenval-ues of A − B form a determinantal point process on C with covariance kernel K ( z, w ) = (1 + zw ∗ ) n − with re-spect to the measure n/ ( π (1 + | z | ) n +1 ) · d m ( z ) and theeigenvectors are distributed as p ( { z i } ) = 1 n ! (cid:16) nπ (cid:17) n n (cid:89) k =1 (cid:81) i 49. We choose to show the small multi-poles l = 3 , a) l = 48b) l = 49 FIG. 19: Multipole vectors (only NILC) and physicaldirections in stereographic projection. The violet curveshows the plane orthogonal to the cosmic dipole.text show that in all maps the quadrupole is unusuallyweakly aligned with the cosmic dipole and l = 4 is un-usually strongly aligned with the cosmic dipole. We alsochoose to plot the stereographic projection for two highervalues of l in Fig. 19, one of which ( l = 48) is close to theexpectation regarding alignment with the cosmic dipoleand one of which ( l = 49) is especially weakly alignedwith the cosmic dipole. 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