CMB statistical isotropy confirmation at all scales using multipole vectors

Renan A. Oliveiraa,b,c, Thiago S. Pereirac, Miguel Quartind,e

aCenter for Computational Astrophysics, Flatiron Institute, 10010-5902, New York, NY, USAbPPGCosmo, CCE, Universidade Federal do Espırito Santo, 29075-910, Vitoria, ES, Brazil

cDepartamento de Fısica, Universidade Estadual de Londrina, 86051-990, Londrina, PR, BrazildInstituto de Fısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil

eObservatorio do Valongo, Universidade Federal do Rio de Janeiro, 20080-090, Rio de Janeiro, RJ, Brazil

Abstract

We present an efficient numerical code and conduct, for the first time, a null and model-independent CMBtest of statistical isotropy using Multipole Vectors (MVs) at all scales. Because MVs are insensitive to theangular power spectrum C`, our results are independent from the assumed cosmological model. We avoid aposteriori choices and use pre-defined ranges of scales ` ∈ [2, 30], ` ∈ [2, 600] and ` ∈ [2, 1500] in our analyses.We find that all four masked Planck maps, from both 2015 and 2018 releases, are in agreement with statisticalisotropy for ` ∈ [2, 30], ` ∈ [2, 600]. For ` ∈ [2, 1500] we detect anisotropies but this is indicative of simply theanisotropy in the noise: there is no anisotropy for ` < 1300 and an increasing level of anisotropy at highermultipoles. Our findings of no large-scale anisotropies seem to be a consequence of avoiding a posterioristatistics. We also find that the degree of anisotropy in the full sky (i.e. unmasked) maps vary enormously(between less than 5 and over 1000 standard deviations) among the different mapmaking procedures and datareleases.

Keywords: observational cosmology, cosmic microwave background, statistical isotropy, multipole vectors

1. Introduction

Cosmic Microwave Background (CMB) maps havebeen the best window to probe the hypotheses thatthe primordial perturbations were Gaussian and sta-tistically homogeneous and isotropic. When these hy-potheses are met, the multipolar coefficients of theCMB temperature map can be treated as randomvariables satisfying

〈a`ma∗`′m′〉 = C`δ``′δmm′ . (1)

where C` is the angular power spectrum and δij is theKronecker delta. CMB experiments have spent thelast decades in pursuit of a precise measurement ofthe C`s. The WMAP mission successfully measuredthis quantity to the cosmic variance limit in the range2 ≤ ` ≤ 600, showing a remarkable accuracy betweentheory and observations (Hinshaw et al., 2013; Ben-nett et al., 2013). The Planck team then extendedthis task to the multipole range 2 ≤ ` ≤ 1800,confirming the predictions of the standard modelwith unprecedented precision (Planck CollaborationI, 2018; Planck Collaboration VI, 2018).

In the standard framework, the C`s are, at each `,the variance of the distribution from which the pri-mordial perturbations were drawn, and at first orderin perturbation theory constitute both the only non-trivial statistical moment of a CMB map and the onlyquantity predicted by theory. Since each multipolehas only 2` + 1 independent components, this im-poses a fundamental lower-bound to the uncertaintyin measuring the C`s, known as cosmic variance. Thismeans that in a typical Planck map, the over 3 mil-lion modes (a`ms) measured in the cosmic variancelimit (apart from the masked regions) are reduced toonly 1800 numbers (C`s): a data reduction of a factorof almost 2000.

Most fundamental extensions of the standard cos-mological model will modify equation (1), either byincluding extra off-diagonal terms and/or by intro-ducing higher order correlations between the a`ms.1

The challenge is that most statistical estimators built

1Some models will also change the C`s while keeping thematrix (1) diagonal. But as we will see, MVs are insensitive tosuch models.

Preprint submitted to Elsevier June 2, 2020

arX

iv:1

812.

0265

4v4

[as

tro-

ph.C

O]

1 J

un 2

020

to extract such extensions will compress the a`ms ina model (Planck Collaboration XXVII, 2014; Amen-dola et al., 2011; Prunet et al., 2005) or geometri-cal (Pullen and Kamionkowski, 2007; Froes et al.,2015; Hajian et al., 2004) dependent way, and thusmeaningful information could still remain undetected.

At the same time, there have been claims of possi-ble anomalies in the CMB data (see Planck Collab-oration XVI, 2016; Schwarz et al., 2016; Muir et al.,2018, for a review). These claims are hard to verifybecause they are mostly related to large-scales whichhave already been measured at the cosmic variancelimit since WMAP. Thus one cannot settle the is-sue with just more observations of the same quan-tities.2 For instance, for the quadrupole–octupolealignment (Tegmark et al., 2003; Bielewicz et al.,2005; Copi et al., 2006; Abramo et al., 2006) mostrecent papers focused either on the study of possi-ble systematics (Francis and Peacock, 2010; Rassatet al., 2014; Notari and Quartin, 2015), or on thesignatures of alignments using galaxy catalogs (Ti-wari and Aluri, 2019). Another issue with the studyof anomalies is how to deal with the so-called lookelsewhere effect: a posteriori selection of a small sub-set of the current 3 million modes of the CMB canartificially lead to low-probability statistics (Bennettet al., 2011).

In this work we circumvent these issues is by mak-ing use of Multipole Vectors (MVs) (Copi et al., 2004;Katz and Weeks, 2004). MVs are a well-known alter-native decomposition for multipole fields (Maxwell,1873; Torres del Castillo and Mendez-Garrido, 2004),and are particularly suited to the analysis of CMBmaps, since they allow for model-independent testsof isotropy. We also introduce a novel set of vectors,dubbed as Frechet Vectors (FVs), which are definedas the mean position of a set of MVs on the sphere.Not only the tools that we use are motivated a priori,but we also choose to work on three well motivatedrange of scales.

The advantage of the FVs is that, besides sharingmost of the nice properties of the MVs, they capturethe correlation of the latter in a model-independentway.

2Although one could expect to see similar effects in the po-larization data.

2. Multipole Vectors and Frechet Vectors

MVs are an alternative representation to themultipole moments of functions on the sphere.For any function X(n), its multipole momentsX`(n) =

∑mX`mY`m(n) can be specified in terms of

a real constant λ` and ` unit and headless (multipole)vectors v` as

X`(n) = λ`∇v1 · · · ∇v`

1

r

∣∣∣∣r=1

, (2)

where r =√x2 + y2 + z2 and ∇v`

= v` · ∇. In thecase of CMB, X could be either the temperature (T )or polarization (E or B-modes) fluctuations. Beingvectors, MVs do not depend on external frames ofreference, but instead rotate rigidly with the data.Moreover, all the information on C` is contained inλ` (see below), so the vectors do not depend on cos-mology in the standard, Gaussian FLRW case. It isthus natural to think about the a`ms as representedby the 2`+1 numbers of the set {C`,v1, · · · ,v`} (Copiet al., 2004).

Previous CMB analysis using MVs have focusedmostly on the low range of scales, 2 ≤ ` . 50, andwere mostly interested in their power to detect largeangle statistical anomalies (Copi et al., 2004; Schwarzet al., 2004; Land and Magueijo, 2005; Bielewiczet al., 2005; Abramo et al., 2006; Bielewicz and Ri-azuelo, 2009; Pinkwart and Schwarz, 2018). Herewe will use these vectors to conduct, for the firsttime, a null test of statistical isotropy in the range2 ≤ ` ≤ 1500. Multipoles ` & 1500 are affected bythe anisotropic instrumental noise (Planck Collabo-ration IX, 2016), so their inclusion is postponed to afuture analysis.

Algorithms for extracting MVs from the a`mswere given in Copi et al. (2004); Weeks (2004) (seePinkwart and Schwarz, 2018, for a recent review onthe existing algorithms). A much more elegant andfaster algorithm was given in (Helling et al., 2006;Dennis, 2004, 2005), and is based in the fact thatMVs can be identified with the roots of a randompolynomial Q` having the a`ms as coefficients:

Q`(z) =∑m=−`

√(2`

`+m

)a`m z

`+m . (3)

For each `, this polynomial has 2` complex roots zi,i = {1, · · · , 2`}. However, only half of these are inde-pendent, since the other half can be obtained by the

2

relation z → −1/z∗.3 Given a root zi, one can obtainthe pair (θi, φi) of coordinates of the vector by meansof a stereographic projection zi = cot(θi/2)eiφi .

We have built a new Python code dubbed polyMV

which uses MPSolve (Bini and Robol, 2014) to findthe roots of Eq. (3) and convert a set of zis into a setof (θi, φi) coordinates. Computational time tests inobtaining all MVs at a given ` show that our codehas computational complexity O(`2), compared toO(`3.5) of the ones in (Copi et al., 2004; Weeks, 2004).See Appendix A for more details. This means thatin a simple 2015 desktop it takes less than 1 sec toextract all MVs at ` = 1000, compared to around 75min and 22 h with the routines of Copi et al. (2004)and Weeks (2004), respectively. Our code comparisonalso served as a cross-check: the absolute differencebetween our MV values and those of Copi et al. (2004)was only ∼ 10−10.

The independence of the MVs on the C` canbe directly seen in (3). By rescaling the a`ms asa`m =

√C`b`m, with 〈b`mb∗`′m′〉 = δ``′δmm′ , one ob-

tains an equivalent class of polynomials R`(z), allhaving the same roots as Q`(z). For this reason theaddition of Gaussian and isotropic noise to a CMBmap will not change the statistics of the MVs, sincethe sum of Gaussian and isotropic variables is stillGaussian and isotropic and the net effect will be justa change on the effective C`s.

If the a`ms are drawn from a Gaussian, isotropicand unmasked random sky, the one-point functionof the roots zi (i.e., the expected values 〈zi〉) followa uniform distribution on the Riemann sphere (Bo-gomolny et al., 1992), so that the normalized4

one-point function of the stereographic angles areP `1(θ, φ) = (sin θdθ)(dφ/2π). In terms of the variables

η ≡ 1− cos θ and ϕ ≡ φ/2π (4)

this reduces to

P `1(η, ϕ) = dηdϕ×

{1 (η, ϕ) ∈ [0, 1] ,

0 otherwise .(5)

Moreover, MVs at different multipoles are uncorre-lated whenever the a`ms are. However, vectors com-ing from the same multipole are correlated, and will

3This reflects the parity invariance of the MVs, which isultimately linked to the reality of the CMB field.

4This normalization uses the fact that MVs are headless, sothat we only consider vectors in the upper hemisphere.

in general have allN -point correlation functions, with1 ≤ N ≤ ` (Dennis, 2004, 2005). Thus, in order toscrutinize CMB data against the null statistical hy-pothesis, it is interesting to test not only Eq. (5), butalso the inner correlations of MVs at a fixed multi-pole. Testing the uniformity of MVs is straightfor-ward as we shall see in the next section. Testingtheir inner correlations, on the other hand, is a muchharder task, given that the number of possible cor-relations grows as `2. This motivates us to look forways of mapping the MV vectors into a smaller setof vectors. While there have been similar approachesin the literature (see e.g. Abramo et al., 2006; Copiet al., 2004), they were designed to test for specificCMB anomalies, which is not our primary goal. Thatis to say, we would like to compress the information inthe MVs without introducing biases towards specificsignatures.

One of the simplest ways is to separate the MVsaccording to multipole, and analyse separately the `MVs in each multipole `. If we think of MVs as pointson the unit sphere, one possibility is to look for thepoint which minimizes the sum of squared distancesbetween itself and all other points. That is, we canlook for the vector umin,` such that

umin,` = arg min Ψ` , (6)

where

Ψ`(u) =2∑i=1

γ2(u,vi,`) , with cos γ = u · vi,` .

(7)The function Ψ` is a generalization of the variance fordata points living on the sphere, and is known as theFrechet variance. For this reason we shall dub umin,`

as the Frechet Vectors (FV).5 Frechet vectors havemany interesting properties that we shall explore ina future publication. For our present purposes, it suf-fices to say that, just like the MVs, they are headlessand their one-point function is uniform under the nullhypothesis; thus, testing for their uniformity is alsostraightforward. As we shall see statistics on FVs canbe more sensitive than similar statistics on the MVs.

The presence of masks will change the statistics ofboth Multipole and Frechet vectors, and we resort tonumerical simulations to estimate their distributions.

5In the mathematics literature it is known as Frechet mean,or sometimes as Riemannian center of mass (Nielsen and Bha-tia, 2013).

3

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

NILC 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SEVEM 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

NILC 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SEVEM 2015

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

NILC 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SEVEM 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

NILC 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SEVEM 2018

Figure 1: Plot of MVs and FVs in galactic coordinates in the range ` ∈ [2, 1500]. Each pair of (antipodal) red dots representsone MV or FV. First row: MVs for the unmasked Planck 2015 maps. Second row: corresponding FVs. Third and fourth rows:same for the unmasked Planck 2018 maps. Note that 3 out of the 8 MV unmasked maps appear clearly anisotropic: SEVEM 2015and 2018, and Commander 2018. Note also that FVs appear anisotropic even for MV sets which, by eye, look isotropic, such asCommander 2015 and NILC 2015 and 2018.

In this work we will focus on the Planck tempera-ture maps in the range 2 ≤ ` ≤ 1500, where eachmode is measured with a high signal-to-noise (S/N)ratio. We will not consider polarization in this firstpaper as both E and B-modes have (individually) lowS/N and Planck has a highly anisotropic noise profile.This would lead to strong noise-induced anisotropiesand we would need to take into account the Plancknoise simulations. The Planck team provided, in eachrelease, 4 different temperature maps: Commander,NILC, SEVEM and SMICA. Each is built using a differ-ent pipeline but all use as input the different intensityfrequency maps and aim at removing all foregroundsas much as possible. Figure 1 shows all MVs andcorresponding FVs for these four full sky (unmasked)2015 and 2018 Planck maps.6

The inclusion of the (apodized) Planck 2018 Com-mon Temperature Mask (henceforth Common Mask,Planck Collaboration IV, 2018) makes the MVs allindistinguishable among themselves and from the

6The polyMV code is available at https://oliveirara.

github.io/polyMV/, and the tables of the MVs and FVs forall the Planck maps considered in this work are availableat https://doi.org/10.5281/zenodo.3866410.

isotropic simulations; this is depicted in Figure 2 forone isotropic simulation and two Planck maps (cho-sen arbitrarily), where we can see that the inclusionof a mask leads to small anisotropies near the poles(see also Figure 3).

Note that the 2018 Commander MVs differs dras-tically from the 2015 ones. Indeed, as remarked bythe Planck team, the use of full-frequency maps (asopposed to single bolometer maps) leads to a sim-pler foreground model employed to the Commander

2018 map, which includes only four different com-ponents in temperature, instead of the seven compo-nents used in the 2015 map (Planck Collaboration IV,2018). Figure 3 shows the effect of foreground maskson the distribution of the multipole and Frechet vec-tors. Because the relation between the a`ms and theMVs is non-linear, the effect of a mask on the latteris hard to predict, as confirmed by the top panel inthis figure, which shows a tendency of the MVs to ac-cumulate on the North-Pole. The FVs, on the otherhand, concentrate around the equator, as can be seenin Figure 2. The FV in all scales exhibit small dif-ferences between data and the isotropic simulations,which we will discuss in Section 4. Notice that the az-

4

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Isotropic Simulation + Common Mask

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2018 + Common Mask

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2018 + Common Mask

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Isotropic Simulation + Common Mask

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Commander 2018 + Common Mask

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

SMICA 2018 + Common Mask

Figure 2: Similar to Figure 1 for the Planck 2018 maps after applying the Common Mask, together with a masked Gaussian andIsotropic map simulation. The inclusion of a mask leads to a small clustering of MVs on the poles (see also Figure 3). The FVswill then concentrate around the equator, since this is the position which minimizes their distance to the MVs.

0.0 0.2 0.5 0.8 1.00.0

0.4

0.8

1.2

1.6

P (η)

0.0 0.2 0.5 0.8 1.00.0

0.4

0.8

1.2

1.6

P (ϕ)

0.0 0.2 0.5 0.8 1.0η

0.0

0.4

0.8

1.2

1.6

0.0 0.2 0.5 0.8 1.0ϕ

0.0

0.4

0.8

1.2

1.6

Figure 3: Top: distribution of MV coordinates for ` ∈ [2, 100] from 100 simulations with the 2018 Common Mask. Vectorsconcentrate on the north-pole (η = 0) and are surrounded by a depression ring of ∼ 15◦ radius. Bottom: distribution of the FVscoordinates from the corresponding MVs of the upper panel.

imuthal distribution of the FVs (bottom-right panelin Figure 3) captures the asymmetries of the Com-mon Mask which are not obvious in the one-pointdistribution of the MVs, but which are nonethelesshidden in their inner correlations.

3. Statistical Tests

Under the null statistical hypothesis (i.e., a Gaus-sian, homogeneous and isotropic universe) both MVsand FVs are uniformly distributed. Thus, in order toconduct a null test of statistical isotropy, we simulate3000 CMB maps with Nside = 1024 (see Appendix

D for more details), which are then masked with theCommon Mask. After extracting the a`ms in therange ` ∈ [2, 1500], we obtain the MVs and their FVsas described above. We verified that, while the ad-dition of Gaussian and isotropic random noise to thea`ms might eventually lead to drastic displacementsof a few individual MVs (due to the ill-conditioning ofthe polynomials at some particular scales), it has noobservable impact on their statistical distributions.This is expected as per the preceding discussion: suchnoise effectively equates to simple re-scaling of theC`s.

The null hypothesis can then be verified by means

5

of a simple chi-square test which, for the multipolevectors, is described as follows: for each simulationand at each `, we obtain normalized histograms forthe variables η and ϕ, which give an estimate of theirone-point functions. From these histograms we com-pute the mean number of events at the i-th angu-lar bin, and the mean covariance among bins, Cij .These quantities allow us to the define the (reduced)chi-square function:

χ2` (x)=

1

Nbins(`)− 1

Nbins−1∑i,j

(xi − xi)(C−1)ij(xj − xj),

(8)where xi stands for the number counts in the i-th binof either η or ϕ. Note that, because

∑Nbinsi xi = `, not

all bins are independent. We thus drop one bin7 inthe computation of each Cij , and use (Nbins−1) as thenumber of independent degrees of freedom. Finally,although one can test the isotropy independently foreach variable, we here focus on just the global quan-tity combining both:

χ2` ≡

1

2

[χ2` (η) + χ2

` (ϕ)]. (9)

This can then be applied to test the uniformity of theMVs of any CMB map.

We used our 3000 simulations to determine Cij .At this point, one would expect Eq. (8) to giveχ2` ≈ 1 when applied to an independent and identi-

cally generated CMB map. This expectation is onlyapproximately met, reflecting the fact that the overallamplitudes of the elements of Cij have poorly con-verged after these 3000 simulations. But we con-firmed that χ2

` → 1 as the number of simulationsincreased (see Appendix B for more details). Eventhough the employed algorithm is efficient, runningmany more simulations at high-` is still very inten-sive, so we chose a more feasible solution by gener-ating control simulations to calibrate Eq. (8). Wehave thus generated 2000 additional (and indepen-dent) CMB maps from which we extracted all MVs,and to which we applied Eqs. (8) and (9). This givesus a mean theoretical χ2

` at each multipole, as well asthe measure of the cosmic variance.

For the number of angular bins, Nbins, a highervalue allows one to better capture the fine effects of amask (see Figure 3) but can lead to numerical insta-bilities in the inversion of the covariance matrix. The

7Chosen, for convenience, to be the last bin.

most straightforward choice would be to use Nbins = `as there are ` multipole vectors at a given multipole `,but this leads to too much numerical noise for higher`s. We tested that for the number of simulations weused for the highest ` (1500) a total of around 600bins gave the optimal trade-off. We thus settled onthe following scheme, which keepsNbins = ` only until`max = 30 (see below on the relevance of this number)and then increases linearly the number of bins untilreaching 600 at ` = 1500:

Nbins(`) =

{` , ` ≤ 30⌈

57147 (`− 30) + 30

⌉, ` > 30

(10)

where d·e denotes the ceiling function.In summary, given a range of multipoles, the above

algorithm will produce a list of chi-square values forthe MVs (one value for each `). We then further con-vert this list into one number describing the overallnull-hypothesis, as follows: given a list of chi-squarevalues in a chosen range ∆` = ` − 2, we computedthe overall fit of the chi-square of the data to the chi-square of the simulations. This quantity, which wedub χ2

MV, is given by

χ2MV ≡`max∑`1,`2=2

[χ2,data`1

− χ2,sim`1

]C−1MV, `1`2

[χ2,data`2

− χ2,sim`2

],

(11)where CMV gives the correlation matrix between theχ2,sim` s at different scales. This correlation is only

non-diagonal if a mask is used, as masks will in gen-eral induce correlation among different multipoles.We however verified numerically that for the Planckmask the non-diagonal terms of CMV are negligible.We illustrate this in Appendix C. Thus, when com-puting Eq. (11), we can treat the multipoles as ap-proximately independent from each other. Note thatwe include the parameter χ2,sim

` in order to account

for the fact that χ2,sim` ≈ 1 but it is not exactly unity,

as discussed above. Due to the negligible correla-tions among `s we can combine the p-values for each` into a global p-value in a straightforward mannerusing Fisher’s method (Fisher, 1992; Brown, 1975).We then tested at different values of ` what was theprobability distribution for our χ2

` (x). It is in generalwell described by a chi-squared distribution, and infact for small scales by a Gaussian distribution.

The uniformity of the Frechet vectors can be testedin exactly the same manner, except that in this case

6

Eqs. (8) and (9) are not computed for each `, butinstead for a single histogram containing all FVs inthe multipolar range of interest. We thus have a asingle chi-square for all the FVs in that range:

χ2FV(x) =

1

Nbins − 1

Nbins−1∑i,j

(xi − xi)(C−1FV)ij(xj − xj).

(12)The FVs angular binning is performed similarly, ex-cept that in this case we replace the multipole ` inEq. (10) by a range of multipoles. Thus, for exam-ple, to test all FVs in the range ` ∈ [2, 600] we useNbins(600− 1) = 251.

While we could compute our p-values using thesedistributions, some of the unmasked maps exhibithigh-levels of anisotropy, and the data points fall farin the tail of the distributions. Due to these extremecases, in order to be very conservative when analysingthe statistics of MVs we relied only on the histogramsthemselves and not on the fitted distributions, andthus put a lower bound of 1/2000 on the resultingprobabilities for each `. Nevertheless, since we havebasically 2000 simulations for ` = [2, 1500], this is stilla very low lower bound. The minimum possible com-bined p-value can be computed from (using Fisher’smethod):

xFisher ≡ −2

`max∑`=2

log(1/2000) . (13)

The corresponding p-valuemin is computed as theprobability of having a value of at least xFisher fora χ2 distribution with 2`max − 2 degrees of freedom.The result is p-valuemin ∼ 10−2980. Since such tinynumbers are not very intuitive, we write all of our p-values in terms of the corresponding Gaussian stan-dard deviations σ:

σ-value ≡√

2 Erf−1(1− p-value

). (14)

This means that our p-valuemin corresponds to a max-imum σ-value of around 117σ. Of course if one doesrely on the fitted distributions on their tails there isno limit.

For the Frechet Vectors, since each is already amean among all MV in a given `, their p-valuemin

is simply 1/2000, which corresponds to a σ-value of3.5σ. Some of our results however show a much higherdiscrepancy than this, so for the FV we also quote thez-score (also referred to as standard score), which is

simply the number of standard deviations betweenthe observed result and the simulated one:

z-score ≡χ2,dataFV − χ2,sim

FV

σχ2

. (15)

4. Results

Aiming to mitigate possible a posteriori effects wechose three well-motivated values of `max even be-fore the analysis began. We chose first `max = 1500,covering all-scales measured by Planck while stilltrying to avoid the effects of the noise anisotropy,which becomes more important at high-`. As weshow below, we do not detect any anisotropies withthe MV statistics, but for the FVs we notice an in-creasing anisotropy for ` ≥ 1300, a strong hint ofanisotropic noise detection. In fact, for ` = 1350in all 4 mapmaking pipelines we have a noise spec-trum N` ' 0.1C` (Planck Collaboration IV, 2018).A 10% contribution in a sufficiently sensitive testwill result in anisotropic results. So for the FVs wealso quote results up to `max = 1200. We also chose`max = 600 which represents the range of scales cov-ered by WMAP; WMAP data resulted in most of theclaimed anomalies still investigated today. Finallywe chose `max = 30 to depict the results on the large-scales only. This value is the one used by the Planckteam in order to separate their low-` and high-` like-lihoods (Planck Collaboration XI, 2016).

4.1. Multipole Vectors

Figure 4 summarizes the result of our analysis ap-plied to the four 2018 masked Planck maps. Westress that these results are totally independent ofexisting measurements of the C`s, and thus of anyof its claimed anomalies, as well as of the additionof isotropic instrumental noise, regardless of the am-plitude of its spectrum. Table 1 gives the globalgoodness-of-fit of the data points in the Figure 4with respect to the theoretical curve. Under this test,all four masked Planck pipelines are consistent withisotropy. In particular, no deviation of isotropy wasdetected at large scales (` ∈ [2, 30]), where most ofCMB anomalies were reported.

The Planck collaboration provides a CommonMask for the community to allow the use of theirdata safe from foreground contamination and advisesone not to use their maps without this mask. How-ever, here testing the isotropy of the full sky Planck

7

500 1000 1500

ℓ

0.0

0.5

1.0

1.5

2.0

Simulations + Common Mask

Commander 2018 + Common Mask

NILC 2018 + Common Mask

SEVEM 2018 + Common Mask

SMICA 2018 + Common Mask

2 10 300.0

0.5

1.0

1.5

2.0

χ2 ℓ

Figure 4: Test of isotropy as a function of ` for the 4 maskedPlanck 2018 maps. The solid (green) curve gives χ2

` averagedover 2000 control simulations, and the green bands show 2σcosmic variance. For clarity, data points are gathered in 49bins of ∆` = 30 in the interval ` ∈ [31, 1500].

Masked MVs (PR3) Commander NILC SEVEM SMICA

Large scalesχ2MV/dof 0.811 0.765 0.579 1.12

σ-value 0.20 0.14 0.02 0.86

WMAP scalesχ2MV/dof 1.06 0.904 1.04 1.04

σ-value 1.2 0.05 1.1 1.3

All scalesχ2MV/dof 0.998 0.977 1.03 0.968

σ-value 0.54 0.24 1.1 0.27

Table 1: Goodness-of-fit of the data points of Figure 4 andtheir associated σ-values (d.o.f. = `max − 1). The analysis wasdivided in three ranges of interest: Large scales (` ∈ [2, 30]),WMAP scales (` ∈ [2, 600]) and “All” scales (` ∈ [2, 1500]). All4 pipelines appear isotropic under this test.

maps is interesting because it provides a good test ofthe power of the method to detect these foregrounds.After all, these regions have allegedly high levels offoreground contamination.

We thus present the results of the same analysisconstructed with full sky simulations and applied tothe four full sky Planck pipelines. In Figure 5 andTable 2 we show the results for the full sky 2015maps, and in Figure 6 and Table 3 the same for2018. Interestingly, all full sky maps (except SEVEM2015, which shows a moderate deviation) are con-sistent with the isotropy hypothesis at large scales(` ∈ [2, 30]), where most of known CMB anomalieswere reported. This is not a counter-proof of exist-ing C` anomalies since, again, our results are inde-pendent of this quantity. We find that 2015 SEVEM

and 2018 SEVEM and Commander maps are in flagrantdisagreement with the isotropy hypothesis, becom-ing highly anisotropic at ` & 300. The differences

500 1000 1500

ℓ

0.0

0.5

1.0

1.5

2.0

Simulations

Commander 2015

NILC 2015

SEVEM 2015

SMICA 2015

2 10 300.0

0.5

1.0

1.5

2.0

χ2 ℓ

Figure 5: Same as Figure 4 but for the unmasked, full skyPlanck 2015 maps. The full sky SEVEM map is in disagreementwith the isotropy hypothesis at over 47σ. See Table 2.

Full sky MVs (PR2) Commander NILC SEVEM SMICA

Large scalesχ2MV/dof 0.933 0.807 1.83 0.808

σ-value 0.58 0.25 2.6 0.20

WMAP scalesχ2MV/dof 1.01 0.907 1.60 0.956

σ-value 0.63 0.10 8.4 0.25

All scalesχ2MV/dof 0.983 1.00 4.26 1.08

σ-value 0.36 0.74 >47 2.2

Table 2: Same as Table 1 but for the unmasked Planck 2015data. Only the SEVEM map shows deviations from isotropy (al-ready at WMAP scales) in this test. See the text for moredetails.

between the two releases of Commander can be at-tributed to the simpler foreground model employedby the Planck team in the 2018 release (Planck Col-laboration IV, 2018). Regarding SEVEM, both fullsky releases show strong traces of residual contam-inations, although the 2018 release shows a slightimprovement (χ2

MV/d.o.f ≈ 3.6) in comparison tothe 2015 release (χ2

MV/d.o.f ≈ 4.3). These findingsare in agreement with the recent results of Pinkwartand Schwarz (2018); Minkov et al. (2019), whereCommander (2018) and SEVEM (2015 and 2018) mapswere also found to contain anisotropic residuals.

Overall, our analysis with full sky maps, whichare known to contain anisotropic residuals, show nodeviations of isotropy in both releases of NILC andSMICA, and the 2015 release of Commander. Thisshould not be seen as a weakness of the MVs to de-tect anisotropies on these maps, but rather as a lim-itation of the simple chi-square test when applied tothe one-point distribution of these vectors, which ig-nores the correlations between the MVs at a given

8

500 1000 1500

ℓ

0.0

0.5

1.0

1.5

2.0

Simulations

Commander 2018

NILC 2018

SEVEM 2018

SMICA 2018

2 10 300.0

0.5

1.0

1.5

2.0

χ2 ℓ

Figure 6: Same as Figure 4 but for unmasked Planck 2018maps. The full sky SEVEM and Commander maps are in disagree-ment with the isotropy hypothesis already at the WMAP scales.At All scales, the discrepancy is over 99σ and 40σ, respectively.See Table 3.

Full sky MVs (PR3) Commander NILC SEVEM SMICA

Large scalesχ2MV/dof 1.35 0.962 0.985 1.27

σ-value 1.4 0.49 0.52 1.3

WMAP scalesχ2MV/dof 12.6 0.973 1.39 1.01

σ-value >42 0.47 5.7 0.66

All scalesχ2MV/dof 215 1.02 3.63 1.01

σ-value >99 1.2 >40 0.74

Table 3: Same as Table 1 but for the unmasked Planck 2018data. Compared to 2015, SEVEM shows a slight improvement,but Commander becomes completely anisotropic.

multipole. As we show below, the same analysis con-ducted with the Frechet vectors clearly pinpoints thepresence of anisotropies in all full sky maps at smallscales. On the other hand, the fact that NILC andSMICA appear isotropic even without masks suggeststhat, for some applications, one may rely on smallermasks than Planck’s Common Mask. We investigatethis possibility in more detail in Appendix E, wherewe apply instead of this Common Mask the muchsmaller inpainting mask, which removes only 2.1% ofthe sky, and show that this is sufficient to make all 4pipelines in agreement with the isotropic hypothesisin our MV statistic.

4.2. Frechet Vectors

We recall that, since we have only one Frechet vec-tor at each ` (as opposed to ` MVs per `), we gatherall vectors from a multipolar range [`min, `max] intoa single histogram. When compared to the theoret-ical (i.e., simulated) distribution of Frechet vectors,this leads to one chi-square value per multipole range

100 300 500 700 900 1100 1300 1500

ℓmax

0.25

0.75

1.25

1.75

2.25

χ2 ℓ

Simulations + Common Mask

Commander 2018 + Common Mask

NILC 2018 + Common Mask

SEVEM 2018 + Common Mask

SMICA 2018 + Common Mask

Figure 7: Test of isotropy for the Frechet vectors of maskedPlanck maps as a function of `max. Green bands stand for 1,2 and 3σ cosmic variance regions. For ` > 1200 anisotropiesstart to significantly increase, likely due to the anisotropy inthe noise which become more important at higher `.

Masked FVs (PR3) Commander NILC SEVEM SMICA

Large scalesχ2FV/dof 1.15 1.16 1.07 1.22

σ-value 1.1 1.1 0.84 1.3

WMAP scalesχ2FV/dof 1.13 1.02 1.10 1.14

σ-value 0.90 0.38 0.72 0.93

` ∈ [2, 1200]χ2FV/dof 1.24 1.20 1.29 1.26

σ-value 0.89 0.69 1.1 0.97

All scalesχ2FV/dof 1.79 1.60 1.72 1.72

σ-value >3.5 2.8 3.5 3.5

Table 4: Same as Table 1 but for the Frechet vectors. The FVare more sensitive to anisotropies, and for scales ` ≥ 1300 itstarts to detect anisotropies in all maps, likely due to the noiseanisotropy. At ` ∼ 1350 the noise N` is ∼ 0.1C`, and it growsquickly with increasing `. We therefore include here also theresults for ` ∈ [2, 1200]. All maps are consistent with isotropy(at < 2σ) at scales ` ≤ 1300.

(instead of one chi-square value per single `, as inFigures 4–6).

As highlighted in the beginning of this Section, theχ2 analysis on the FVs is more sensitive than the oneon the bare MVs. The reason for this is that it takesinto account the inner correlation of the MVs at eachgiven `. Its higher sensitivity picks up an increasinganisotropic signature for ` > 1200 for all 4 mapmak-ing methods. This is illustrated in Figure 7, wherewe also show the mean theoretical values togetherwith their 1, 2 and 3σ bands in our 2000 controlsimulations. At ` ∼ 1500 all maps are inconsistentwith isotropy at around 3σ. A thorough modelling ofanisotropic instrumental noise is needed to confirmthe origin of these anisotropies and to extend our re-

9

Full sky FVs (PR2) Commander NILC SEVEM SMICA

Large scales

χ2FV/dof 1.25 1.08 3.07 0.971

σ-value 1.4 0.85 >3.5 0.54

z-score 1.0 0.3 9.2 -0.2

WMAP scales

χ2FV/dof 1.13 1.22 26.3 1.14

σ-value 1.0 2.1 >3.5 1.2

z-score 0.53 1.8 350 0.71

All scales

χ2FV/dof 1.66 3.15 84.3 1.83

σ-value >3.5 >3.5 >3.5 >3.5

z-score 6.9 33 1400 9.9

Table 5: Same as Table 4 bur for the unmasked 2015 Planckmaps. We also include the z-scores for WMAP scales andAll Scales, which show the huge significance of the detectedanisotropies in some cases. SEVEM is the only method whichexhibits strong anisotropies also in Large scales and WMAPscales. For all scales all maps become anisotropic, but Com-mander and SMICA much less so.

sults to higher `s. This is left for a future publication.Table 4 shows the corresponding significance levels inthe 3 ranges of scales considered here. Our analy-sis confirms the concordance of masked Planck mapswith the null hypothesis at all scales ` . 1300.

The higher sensitivity of the Frechet vector statis-tic can also be seen in the unmasked, full sky maps.In Table 5 and 6 we show the results for the 2015 and2018 Planck releases, respectively. For Planck 2015,all maps are highly anisotropic when considering allscales, but the amount of anisotropy differ signifi-cantly among the different mapmaking procedures.Commander performs the best, with a z-score of 6.9,followed by SMICA and then NILC. SEVEM has a hugez-score of more than a 1000, and in particular showsanisotropies in all multipole ranges considered. Forthe 2018 release, some pipelines perform better, oth-ers worse. As in the MV statistic, Commander showshuge levels of anisotropies at WMAP scales and allscales. SEVEM performs only slightly better than in2015 and NILC slightly worse. SMICA improves sig-nificantly and on all scales shows a relatively smallz-score of 4.8, half the amount than in 2015.

5. Conclusions

Our findings illustrate the usefulness of MV analy-ses. They constitute a great blind tool in the detec-tion of residual anisotropic contaminations in CMBmaps, which are an indication of residual foregrounds.

Full sky FVs (PR3) Commander NILC SEVEM SMICA

Large scales

χ2FV/dof 1.53 1.10 2.18 1.12

σ-value 2.5 0.91 >3.5 0.97

z-score 2.3 0.3 5.2 0.4

WMAP scales

χ2FV/dof 32.4 1.24 18.3 1.19

σ-value >3.5 2.4 >3.5 1.8

z-score 440 2.2 240 1.4

All scales

χ2FV/dof 161 3.55 60.7 1.53

σ-value >3.5 >3.5 >3.5 >3.5

z-score 2700 39 1000 4.8

Table 6: Same as Table 5 but for the unmasked 2018 Planckmaps. Notice that at WMAP scales both SEVEM and Com-mander exhibit strong anisotropies. Including all scales allmaps become anisotropic, but SMICA considerably less so thanthe others.

Therefore it is possible that MV analysis can be ex-plored to help determine which regions to mask. Inorder to achieve this one should ideally find a methodto correlate the positions of the MVs to positions ofanisotropic sources in configuration space. The novelFrechet vectors introduced here partially accomplishthis task. Indeed, Figures 2 show that the Frechetvectors of masked maps fall precisely on the equato-rial line where most of the mask is found.

In any case, a simpler method would be just touse MV and FV isotropy to validate and calibrateother methods of mask determination. The full skymap analyses we performed allow a quick glimpse onwhether the Common Mask is too conservative. Theresults indicate SMICA is the less sensitive procedureto foregrounds at low latitudes as even in the full un-masked sky their FV had a comparatively low amountof anisotropy.

Overall, we confirm the consistency betweenPlanck data and the fundamental hypothesis of thestandard cosmological model. Our results using thebare MV in all masked Planck maps are in agree-ment with a isotropic and Gaussian CMB in all rangeof scales considered. The results using the FVs alsoshow no anisotropy on large or WMAP scales. Theabsence of large-scale anisotropies is likely a conse-quence of our explicit avoidance of a posteriori statis-tics. For ` ∈ [2, 1500] the FV instead show an increas-ing anisotropy for ` ≥ 1300, but this behavior is in-dicative that our statistic is detecting the well-knownanisotropy in the noise. For ` ≤ 1300 even the more

10

sensitive FV statistic is consistent with the isotropicand Gaussian hypothesis. In future work we plan toextend our analysis to higher ` by carefully takinginto account the anisotropy in the noise in each oneof the 4 mapmaking pipelines.

We stress that the main novelty of our resultsare not the detection of anisotropy in full sky maps(which are known to contain residual anisotropies)but the power of the formalism to detect suchanisotropies at high statistical levels. Moreover, sincethey can also be directly applied to polarizationmaps, all the applications discussed here can be ex-tended directly to all primordial CMB maps. Furtherresearch is needed to investigate this possibility in de-tail.

Clearly our results do not rule out all types ofanisotropies. In fact, in this first analysis we areignoring anisotropies that result on correlationsbetween different `s, and perform only 1-point sta-tistical tests on the MVs and FVs (although the FVsthemselves take into account the MV correlationsin a given `). We leave an analysis of 2-point (orhigher) statistics for future work. These correlationsare common to a broad class of models, includingastrophysical sources of anisotropies, aberrationof the CMB and/or non-Gaussianities like lensing.Instead, our main focus was to build a simple andwell-motivated statistic as much as possible free of aposteriori selection effects. There was no guaranteethat the data would pass this test, and the fact thatit did favors the standard isotropic model. It alsoconstitutes a new test that all anisotropic modelsmust pass henceforth.

CRediT authorship contribution statement

Renan A. Oliveira: Software, Formal analysis,Investigation, Visualization. Thiago S. Pereira:Formal analysis, Methodology, Validation, Writing -review & editing. Miguel Quartin: Conceptual-ization, Methodology, Validation, Writing - review &editing.

Acknowledgments

We thank Jeffrey Weeks for providing a copy ofhis code, Alessio Notari for important early dis-cussions, Camila Novaes, Marvin Pinkwart, Omar

Roldan and Dominik Schwarz for useful correspon-dence, and Marcello Oliveira da Costa for the shar-ing of computational resources. RAO thanks Co-ordenacao de Aperfeicoamento de Pessoal de NıvelSuperior (CAPES). TSP thanks Brazilian Fundingagencies CNPq and Fundacao Araucaria. MQ is sup-ported by the Brazilian research agencies CNPq andFAPERJ. This work made use of the CHE cluster,managed and funded by the COSMO/CBPF/MCTI,with financial support from FINEP and FAPERJ,and operating at Javier Magnin Computing Cen-ter/CBPF. The results of this work have been de-rived using HEALPix (Gorski et al., 2005) and Healpypackages.

References

Abramo, L.R., Bernui, A., Ferreira, I.S., Villela, T., Wuen-sche, C.A., 2006. Alignment Tests for low CMB multipoles.Phys. Rev. D74, 063506. doi:10.1103/PhysRevD.74.063506,arXiv:astro-ph/0604346.

Amendola, L., Catena, R., Masina, I., Notari, A., Quartin,M., Quercellini, C., 2011. Measuring our peculiar velocityon the CMB with high-multipole off-diagonal correlations.JCAP 1107, 027. doi:10.1088/1475-7516/2011/07/027,arXiv:1008.1183.

Bennett, C.L., et al., 2011. Seven-Year Wilkinson Mi-crowave Anisotropy Probe (WMAP) Observations: AreThere Cosmic Microwave Background Anomalies? Astro-phys. J. Suppl. 192, 17. doi:10.1088/0067-0049/192/2/17,arXiv:1001.4758.

Bennett, C.L., et al. (WMAP), 2013. Nine-Year WilkinsonMicrowave Anisotropy Probe (WMAP) Observations: FinalMaps and Results. Astrophys. J. Suppl. 208, 20. doi:10.1088/0067-0049/208/2/20, arXiv:1212.5225.

Bielewicz, P., Eriksen, H.K., Banday, A.J., Gorski, K.M., Lilje,P.B., 2005. Multipole vector anomalies in the first-yearWMAP data: A Cut-sky analysis. Astrophys. J. 635, 750–760. doi:10.1086/497263, arXiv:astro-ph/0507186.

Bielewicz, P., Riazuelo, A., 2009. The study of topology of theuniverse using multipole vectors. Mon. Not. Roy. Astron.Soc. 396, 609. doi:10.1111/j.1365-2966.2009.14682.x,arXiv:0804.2437.

Bini, D.A., Robol, L., 2014. Solving secular and polynomialequations: A multiprecision algorithm. Journal of Compu-tational and Applied Mathematics 272, 276–292. doi:10.1016/j.cam.2013.04.037.

Bogomolny, E., Bohigas, O., Leboeuf, P., 1992. Distribution ofroots of random polynomials. Physical Review Letters 68,2726. doi:10.1103/PhysRevLett.68.2726.

Brown, M.B., 1975. 400: A method for combining non-independent, one-sided tests of significance. Biometrics 34,987–992. doi:10.2307/2529826.

Copi, C.J., Huterer, D., Schwarz, D.J., Starkman, G.D.,2006. On the large-angle anomalies of the microwave sky.Mon. Not. Roy. Astron. Soc. 367, 79–102. doi:10.1111/j.1365-2966.2005.09980.x, arXiv:astro-ph/0508047.

11

Copi, C.J., Huterer, D., Starkman, G.D., 2004. Multipole vec-tors - A New representation of the CMB sky and evidencefor statistical anisotropy or non-Gaussianity at 2 ≤ l ≤ 8.Phys. Rev. D70, 043515. doi:10.1103/PhysRevD.70.043515,arXiv:astro-ph/0310511.

Dennis, M., 2004. Canonical representation of spherical func-tions: Sylvester’s theorem, maxwell’s multipoles and ma-jorana’s sphere. Journal of Physics A: Mathematical andGeneral 37, 9487. arXiv:math-ph/0408046.

Dennis, M.R., 2005. Correlations between Maxwell’s mul-tipoles for gaussian random functions on the sphere. J.Phys. A38, 1653–1658. doi:10.1088/0305-4470/38/8/002,arXiv:math-ph/0410004.

Fisher, R.A., 1992. Statistical methods for research workers,in: Breakthroughs in statistics. Springer, pp. 66–70. doi:10.1007/978-1-4612-4380-9_6.

Francis, C.L., Peacock, J.A., 2010. An estimate of the localISW signal, and its impact on CMB anomalies. Mon. Not.Roy. Astron. Soc. 406, 14. doi:10.1111/j.1365-2966.2010.16866.x, arXiv:0909.2495.

Froes, A.L., Pereira, T.S., Bernui, A., Starkman, G.D., 2015.New geometric representations of the CMB two-point cor-relation function. Phys. Rev. D92, 043508. doi:10.1103/PhysRevD.92.043508, arXiv:1506.00705.

Gorski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen,F.K., Reinecke, M., Bartelman, M., 2005. HEALPix - AFramework for high resolution discretization, and fast anal-ysis of data distributed on the sphere. Astrophys. J. 622,759–771. doi:10.1086/427976, arXiv:astro-ph/0409513.

Hajian, A., Souradeep, T., Cornish, N.J., 2004. Statisticalisotropy of the WMAP data: A Bipolar power spectrumanalysis. Astrophys. J. 618, L63–L66. doi:10.1086/427652,arXiv:astro-ph/0406354.

Helling, R.C., Schupp, P., Tesileanu, T., 2006. CMB statisticalanisotropy, multipole vectors and the influence of the dipole.Phys. Rev. D74, 063004. doi:10.1103/PhysRevD.74.063004,arXiv:astro-ph/0603594.

Hinshaw, G., et al. (WMAP), 2013. Nine-Year Wilkinson Mi-crowave Anisotropy Probe (WMAP) Observations: Cosmo-logical Parameter Results. Astrophys. J. Suppl. 208, 19.doi:10.1088/0067-0049/208/2/19, arXiv:1212.5226.

Katz, G., Weeks, J., 2004. Polynomial interpretation of mul-tipole vectors. Phys. Rev. D70, 063527. doi:10.1103/PhysRevD.70.063527, arXiv:astro-ph/0405631.

Land, K., Magueijo, J., 2005. The Axis of evil. Phys. Rev.Lett. 95, 071301. doi:10.1103/PhysRevLett.95.071301,arXiv:astro-ph/0502237.

Maxwell, J.C., 1873. A Treatise on Electricity and Mag-netism. volume 1 of A Treatise on Electricity and Mag-netism. Clarendon Press. URL: https://books.google.

com.br/books?id=92QSAAAAIAAJ.Minkov, M., Pinkwart, M., Schupp, P., 2019. Entropy meth-

ods for CMB analysis of anisotropy and non-Gaussianity.Phys. Rev. D 99, 103501. doi:10.1103/PhysRevD.99.103501,arXiv:1809.08779.

Muir, J., Adhikari, S., Huterer, D., 2018. Covariance of CMBanomalies. Phys. Rev. D98, 023521. doi:10.1103/PhysRevD.98.023521, arXiv:1806.02354.

Nielsen, F., Bhatia, R., 2013. Matrix information geometry.Springer.

Notari, A., Quartin, M., 2015. On the proper kineticquadrupole CMB removal and the quadrupole anomalies.

JCAP 1506, 047. doi:10.1088/1475-7516/2015/06/047,arXiv:1504.02076.

Pinkwart, M., Schwarz, D.J., 2018. Multipole vectors ofcompletely random microwave skies for l ≤ 50. Phys.Rev. D98, 083536. doi:10.1103/PhysRevD.98.083536,arXiv:1803.07473.

Planck Collaboration I (Planck), 2018. Planck 2018 re-sults. I. Overview and the cosmological legacy of PlanckarXiv:1807.06205.

Planck Collaboration IV (Planck), 2018. Planck 2018 results.IV. Diffuse component separation arXiv:1807.06208.

Planck Collaboration IX (Planck), 2016. Planck 2015 results.IX. Diffuse component separation: CMB maps. Astron.Astrophys. 594, A9. doi:10.1051/0004-6361/201525936,arXiv:1502.05956.

Planck Collaboration VI (Planck), 2018. Planck 2018 results.VI. Cosmological parameters arXiv:1807.06209.

Planck Collaboration XI (Planck), 2016. Planck 2015 re-sults. XI. CMB power spectra, likelihoods, and robustnessof parameters. Astron. Astrophys. 594, A11. doi:10.1051/0004-6361/201526926, arXiv:1507.02704.

Planck Collaboration XVI (Planck), 2016. Planck 2015 re-sults. XVI. Isotropy and statistics of the CMB. Astron.Astrophys. 594, A16. doi:10.1051/0004-6361/201526681,arXiv:1506.07135.

Planck Collaboration XXVII (Planck), 2014. Planck 2013results. XXVII. Doppler boosting of the CMB: Eppursi muove. Astron. Astrophys. 571, A27. doi:10.1051/0004-6361/201321556, arXiv:1303.5087.

Prunet, S., Uzan, J.P., Bernardeau, F., Brunier, T., 2005. Con-straints on mode couplings and modulation of the CMBwith WMAP data. Phys. Rev. D71, 083508. doi:10.1103/PhysRevD.71.083508, arXiv:astro-ph/0406364.

Pullen, A.R., Kamionkowski, M., 2007. Cosmic MicrowaveBackground Statistics for a Direction-Dependent PrimordialPower Spectrum. Phys. Rev. D76, 103529. doi:10.1103/PhysRevD.76.103529, arXiv:0709.1144.

Rassat, A., Starck, J.L., Paykari, P., Sureau, F., Bobin, J.,2014. Planck CMB Anomalies: Astrophysical and Cos-mological Secondary Effects and the Curse of Masking.JCAP 1408, 006. doi:10.1088/1475-7516/2014/08/006,arXiv:1405.1844.

Schwarz, D.J., Copi, C.J., Huterer, D., Starkman, G.D.,2016. CMB Anomalies after Planck. Class. Quant.Grav. 33, 184001. doi:10.1088/0264-9381/33/18/184001,arXiv:1510.07929.

Schwarz, D.J., Starkman, G.D., Huterer, D., Copi, C.J., 2004.Is the low-l microwave background cosmic? Phys. Rev.Lett. 93, 221301. doi:10.1103/PhysRevLett.93.221301,arXiv:astro-ph/0403353.

Tegmark, M., de Oliveira-Costa, A., Hamilton, A., 2003. Ahigh resolution foreground cleaned CMB map from WMAP.Phys. Rev. D68, 123523. doi:10.1103/PhysRevD.68.123523,arXiv:astro-ph/0302496.

Tiwari, P., Aluri, P.K., 2019. Large Angular-scale Multipolesat Redshift ∼ 0.8. Astrophys. J. 878, 32. doi:10.3847/1538-4357/ab1d58, arXiv:1812.04739.

Torres del Castillo, G.F., Mendez-Garrido, A., 2004. Differen-tial representation of multipole fields. Journal of PhysicsA Mathematical General 37, 1437–1441. doi:10.1088/0305-4470/37/4/025.

Weeks, J.R., 2004. Maxwell’s multipole vectors and the CMB

12

50 100 500 1000`

0.01

0.10

1.00

10.00

100.00

t[s

]

t ∝ `3 t ∝ `3.5

t ∝ `2

Figure A.8: Computational complexity comparison. Compu-tational time of MVs extraction using polyMV (red) and thealgorithms (Copi et al., 2004) (blue) and (Weeks, 2004) (pur-ple). polyMV has computational complexity O(`2) as comparedto O(`3.5) of both the other available codes. The scatter seenat ` & 400 is a consequence of numerical ill-conditioning of thepolynomials at high multipoles, which demands more CPU timeof MPSolve.

arXiv:astro-ph/0412231.

Appendix A. Computational complexitycomparison

As discussed in the main text, in this work we de-veloped a new code polyMV to efficiently compute allthe MVs of a given map. This code is order of mag-nitudes faster than both existing public algorithmsat high multipoles. In fact, it has computationalcomplexity O(`2) as compared to O(`3.5) of both theother available codes. Figure A.8 depicts the com-putation time of polyMV as a function of ` for ourcode as well as the other two public codes. Startingat ` ∼ 400 some of the polynomial of equation (3)become ill-conditioned due to their very high-order.This is reflected on an order of magnitude longer com-putational time in order to evaluate their roots withthe same numerical accuracy. But even in these casesthe evaluation at, say, ` = 1000 takes less than 1% ofthe time than the best competing code.

Appendix B. Discussion on the total numberof simulations

The mean covariance matrix Cij appearing inEq. (8) is estimated numerically. In practice, wewould expect Cij to have converged to its theoreti-cal value if χ2

` approaches unity when applied to anindependent masked sky. Figure B.9 shows χ2

` as afunction of ` for Cij evaluated with 1000 and 3000

500 1000 1500

ℓ

0.0

0.5

1.0

1.5

2.0

1000 Simulations + Common Mask

3000 Simulations + Common Mask

2 10 300.0

0.5

1.0

1.5

2.0

χ2 ℓ

Figure B.9: Comparison of the simulated Gaussian andisotropic χ2

` using either 1000 and 3000 masked simulationsto compute the covariance matrix Cij of Eq. (8).

500 1000 1500

ℓ

0.2

0.4

0.6

0.8

1000 Simulations + Common Mask

3000 Simulations + Common Mask

2 10 30

0.2

0.4

0.6

0.8σχ2 ℓ

Figure B.10: The standard-deviation of the cases of Fig. B.9.Note that the values start to increase for ` > 500 when we haveonly 1000 simulations, which indicate that numerical noise inthe inversion of Cij starts to dominate. For 3000 simulationsnumerical noise is much suppressed.

simulations. Clearly, a higher number of independentmaps is required to achieve χ2

` → 1 for all `s. As wehave discussed in the main text, this is not a problemsince the theoretical χ2

` can be calibrated with theuse of independent control simulations. However, thenumber of maps used to estimate Cij requires a morecareful investigation since it will also have an impacton the variance of χ2

` , and consequently on the evalua-tion of the inverse correlation (C−1)`1`2 . This is illus-trated explicitly in Figure B.10, but can also be notedin the thickness of the variance bands on the high-`tail of Figure B.9. Note that the blue curve in Fig-ure B.10 has a minimum at ` & 500, which does notcorrespond to the expectation that cosmic varianceshould decrease with increasing `. With 3000 simu-lations this issue does not appear inside the range ofscales we are probing, which indicates that the nu-merical noise is negligible in this case.

13

0 10 200

5

10

15

20

25

0 10 200

5

10

15

20

25

−0.04

−0.02

0.00

0.02

0.04

0.06

Figure C.11: Matrix plot of the residual correlation matrix –Eq. (C.1) – between χ2,sim

`1and χ2,sim

`2. [Left] : full sky maps;

[Right]: masked maps.

Appendix C. Correlation among differentmultipoles

The results of Tables 1–3 are based on the approx-imation M`1`2 ∝ δ`1`2 , where M`1`2 is the covariancematrix whose inverse appears in Eq. (11) due to thepresence of a mask (and possibly due to primordialnon-Gaussianity). While this should be an excellentapproximation at large multipoles, it was not guaran-teed to hold at small `s. Figure C.11 shows a matrixplot of the “residual” correlation matrix, defined asthe correlation matrix minus the identity,

ρ`1`2 =M`1`2

σχ2`1

σχ2`2

− δ`1`2 , (C.1)

for the multipoles in the range ` ∈ [2, 30]. As we cansee, different multipoles are weakly correlated even inthese large scales, where the mask effect is strong. Asa final test we computed the χ2

MV both with and with-out the non-diagonal correlations, and the results hadnegligible differences. We also computed the correla-tion matrix for the full range os scales here consid-ered, and indeed the correlations seem to be negligibleat all scales. This shows that the different multipolescan be treated as independent even in the presenceof Planck’s Common Mask.

Appendix D. Choosing Nside

All MVs in this work were extractedfrom CMB maps constructed at resolutionNside = 1024. Such resolution should enoughto reconstruct maps at a maximum multipole`max = 3Nside − 1 & 3000 (Gorski et al., 2005) —twice the maximum range used in this paper.Nonetheless, because numerical approximations inthe reconstruction of the a`ms is a potential source

2 500 1000 1500`

10−3

10−2

10−1

〈γ〉 `

[′′]

Figure D.12: Mean angular displacement among MVs bychanging resolution from Nside = 1024 to Nside = 2048 asa function of `. Both set of MVs were generated from CMBmaps with the same random seed. The displacement of theMVs induced by a higher Nside is of order of 10−3 arcsec, andthus negligible.

of systematic noise, which will in turn affect theposition of the MVs, it is important to estimate theimpact of choosing a higher Nside in our simulations.Given a set of vectors at a fixed resolution, {vNside

` },we can estimate this effect by evaluating the angle

γj,` = arccos(v2048j,` · v1024

j,` ) , (D.1)

with {v1024` } and {v2048

` } sharing the same randomseed — and averaging it over all values of j. Themean angle 〈γ〉` is shown in Figure D.12 as a functionof `. The impact induced by a choice of a higher Nside

is below 10−3 arcsec ∼ 10−9 rad for the vast majorityof scales we probed. This corresponds to a multi-pole ` = π/〈γ〉` ∼ 108, which is five orders of mag-nitude above the maximum scale we are consideringin this work. Even the highest observed difference of0.1 arcsec corresponds to a multipole ` ∼ 106, whichis safely above our maximum range. Thus, the resolu-tion Nside = 1024 is completely safe for our purposes.Moreover, notice that the number of scales with amean displacement & 10−3 arcsec increase linearly at` & 400 — roughly the same scale and shape abovewhich ill-conditioning of the polynomials is observed,as confirmed by Figure A.8. Thus, such behavior in〈γ〉` seen in Figure D.12 is most-likely due to numer-ical ill-conditioning of the polynomials, and not ourchoice of Nside.

Appendix E. Inpainted maps

Our analysis has shown that the 2018 Commander

and SEVEM full-sky maps are clearly anisotropic in

14

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Inpainted Commander 2018

135° 90° 45° 0° 315° 270° 225°

-45°

0°

45°

Inpainted SEVEM 2018

Figure E.13: Similar to Figure 1 for the MVs of the 2018 Commander and SEVEM maps inpainted inside the small 2.1% mask. MVsfor the inpainted NILC and SMICA maps are visually identical to the ones above, and thus not shown.

the range ` ∈ [2, 1500]. Since these maps containanisotropic residuals which are visible by eye, mostlyvery near the galactic plane. We wanted to test ifthese visible residuals alone could account for thelarge σ-values reported in Table 3. For this taskwe apply a simple inpainting method, which con-sists of masking these maps with the common in-painting mask made available by the Planck team(with fsky = 0.979), and then filling the masked re-gions with a realization of a Gaussian and statisticallyisotropic random sky. We used the same realizationto inpaint all 4 maps. As we can see in Figure E.13,MVs for these inpainted maps are visually indistin-guishable from full-sky Gaussian and isotropy ran-dom maps.

Figure E.14 and Table E.7 show the result of thestatistics (9) applied to these maps, and Table sum-marizes the global fit of these maps when compareto full-sky Gaussian and isotropic simulations. Thisanalysis shows that all anisotropies found in full-skyCommander and SEVEM maps are coming from thissmall 2.1% fraction of the sky. Nevertheless, the factis that the remaining 97.9% of the sky is compatiblewith Gaussianity and isotropy, and this is an inter-esting and non-trivial result which confirms the pre-dictions of the standard model to a great extent.

.

500 1000 1500

ℓ

0.0

0.5

1.0

1.5

2.0

Simulations

Inpainted Commander 2018

Inpainted NILC 2018

Inpainted SEVEM 2018

Inpainted SMICA 2018

2 10 300.0

0.5

1.0

1.5

2.0

χ2 ℓ

Figure E.14: Same as Figure 4 for the four inpainted Planck2018 maps – see the text for details. This small inpainting isenough to remove all anisotropies in our statistic.

Full sky (PR3) Commander NILC SEVEM SMICA

Large scalesχ2MV/dof 1.28 1.07 1.51 0.72

σ-value 1.5 0.79 1.8 0.14

WMAP scalesχ2MV/dof 1.07 1.07 0.93 1.01

σ-value 1.6 1.7 0.17 0.79

All scalesχ2MV/dof 1.02 1.06 0.94 1.00

σ-value 1.0 1.9 0.08 0.70

Table E.7: Same as Table 1 for the data points of Figure E.14.

.

15