Asymmetry of the CMB map: local and global anomalies
PPrepared for submission to JCAP
Asymmetry of the CMB map: localand global anomalies
James Creswell and Pavel Naselsky
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen,DenmarkE-mail: [email protected], [email protected]
Abstract.
We investigate the sources of parity asymmetry in the CMB temperature mapsusing a pixel domain approach. We demonstrate that this anomaly is mainly associated withthe presence of two pairs of high asymmetry regions. The first pair of peaks with Galacticcoordinates ( l, b ) = (212 ◦ , − ◦ ) and (32 ◦ , ◦ ) is associated with the Northern Galactic Spurand the direction of the dipole modulation of the power spectrum of the CMB anisotropy.The other pair ( ( l, b ) = (332 ◦ , − ◦ ) and (152 ◦ , ◦ ) ) is located within the Galactic plane (theGalactic Cold Spot and its antipodal partner). Similar asymmetric peaks, but with smalleramplitudes, belong to the WMAP/Planck Cold Spot and its partner in the Northern GalacticSpur. These local anomalies increase the odd-multipole power to a level consistent withGaussian simulations. In contrast, the deficit of symmetric peaks is accompanied by a deficit inthe even-multipole power and is the source of the parity asymmetry of the CMB temperaturemaps at the level of about 3 sigma. We also evaluate the influence of the quadrupole, whichis another source of the even-multipole deficit. If the low quadrupole is an intrinsic featureof the theoretical model, it will reduce the significance of the parity asymmetry to aroundthe 2 sigma level. We also investigate the relationship between the asymmetry of the powerspectrum and the level of the parity asymmetry in the framework of a model with dipolemodulation of a statistically uniform Gaussian signal. We show that these two anomalies areinnately linked to each other. a r X i v : . [ a s t r o - ph . C O ] F e b ontents Z (n)
34 Statistical properties of Z (n) (cid:96) dominated? 7 The concept of symmetry and its breaking constitute the foundation of modern particle physics.It is clear that it is also important for the study of the statistical properties of CMB [1, 2]. Oneof the most statistically significant anomalies of the CMB temperature in the low multipoledomain ( (cid:96) ≤ ) of the power spectrum C (cid:96) is the hemispherical power asymmetry [3, 4] andthe point parity asymmetry [5].Non-Gaussianity of the CMB at low multipoles has been a longstanding area of interest.In [6], Gaussianity and isotropy for ≤ (cid:96) ≤ was rejected with significance p ≈ . in thepreliminary WMAP data, and in [7] the alignment of the (cid:96) = 2 and (cid:96) = 3 modes was detectedwith significance p ≈ . . These and other low- (cid:96) anomalies were widely investigated in thecontext of the WMAP data, and several theoretical and systematic explanations were discussed(see e.g. [8–10]). The hemispherical power asymmetry was further explored and confirmed withhigh significance using different techniques in [11–14]. Parity asymmetry, the apparent excessof power in the odd- (cid:96) multipoles compared to the even- (cid:96) multipoles was revisited in [15–19]using the WMAP 5-year and 7-year data releases. In these works, which focused on estimatorsof the power spectrum in the multipole range ≤ (cid:96) ≤ , the odd-parity excess was detectedwith significance p = 0 . , and in [19] this result was linked with the correlation function onlarge angular scales. More recently, the problem of the alignment of the quadrupole, octopole,and other odd- (cid:96) multipoles has been confirmed in the Planck data with significance of 2 to 3sigma using various estimators [20, 21], and parity asymmetry and other low- (cid:96) anomalies havealso been detected and investigated using the Planck data [2, 22–24].The goal of our paper is to perform a new pixel-domain analysis and argue that:a) the hemispherical power asymmetry and the parity asymmetry have a common origin;b) they are associated with asymmetric ditribution of high amplitude peaks;c) the parity asymmetry clearly indicates the deficit of negative peaks;d) the sources of asymmetry are only partially associated with the Galactic plane.– 1 –nlike standard analysis of asymmetries in the C (cid:96) domain, we perform our investigationpurely in the pixel domain. This allows us to identify zones of the sky that make the maincontribution to anomalies. Smoothing maps with a Gaussian filter effectively removes some ofthe multipoles from the analysis: the smoothing angle Θ , which is the FWHM of the Gaussianfilter, is connected with the cerresponding multipole (cid:96) via the approximate relationship Θ (cid:39) ◦ /(cid:96) .The comparison between pixel and multipole approaches cannot be done without im-plementation of different masks. Unless otherwise stated, our standard mask in use is a ringmasking Galactic latitudes − ◦ < b < ◦ . Such a mask is quite similar to the Planckconfidence mask, however, it has the important property that a pixel n is masked if and onlyif the corresponding opposite pixel − n is masked, which simplifies the calculation of the parityasymmetry pixel domain estimators below.The outline of our paper is the following: first, in Section 2, we review the decompositionof the sky into symmetric and asymmetric components, and define the pixel-domain Z ( n ) asymmetry operator in use for this work. In Sections 3 and 4 we study the morphology andthe statistics of this estimator, which reveals the sources of the parity asymmetry. In Section 5we compare to Gaussian simulations, especially focusing on the contribution of the quadrupoledeficit to the overall parity asymmetry, and the effect of changing cosmological parameters. InSection 6 we show how this is related to the power asymmetry and a brief conclusion is inSection 7. In the CMB temperature map T ( n ) , for each pixel n , one can denote the symmetric S ( n ) andasymmetric A ( n ) parts: S ( n ) = T ( n ) + T ( − n )2 , (2.1a) A ( n ) = T ( n ) − T ( − n )2 , (2.1b)where we have T ( n ) = S ( n ) + A ( n ) . In the multipole domain, these components can bedecomposed through the spherical harmonics with the corresponding projection operators γ s = cos (cid:0) π(cid:96) (cid:1) and γ a = sin (cid:0) π(cid:96) (cid:1) as follows: S ( n ) = (cid:96) max (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m Y (cid:96)m ( n ) γ s , (2.2a) A ( n ) = (cid:96) max (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m Y (cid:96)m ( n ) γ a , (2.2b) T ( n ) = (cid:96) max (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m Y (cid:96)m ( n ) , (2.2c)where a (cid:96)m are the coefficients of decomposition.For estimation of the degree of asymmetry of the CMB map, we will use the followingfunction: Z ( n ) = T ( n ) T ( − n ) = S ( n ) − A ( n ) . (2.3)– 2 – a1b2b 1a − − − K Figure 1 . Morphology of Z ( n ) map of Planck 2018 SMICA with ◦ smoothing. Note that the peaks1a and 1b correspond to inversion n → − n , as for 2a and 2b as well. If Z ( n ) > for the point n we have dominance of the symmetric part, and vice versa, if Z ( n ) < the asymmetric part is greater than the symmetric part.Averaging over all pixels of the map leads to the following result: (cid:104) Z ( n ) (cid:105) = (cid:88) (cid:96) (2 l + 1) C (cid:96) cos( π(cid:96) ) , (2.4)where C (cid:96) corresponds to the actual realization of the theoretical power spectrum. Due tothe properties of spherical harmonics, any even- (cid:96) mode has Z ( n ) ≥ , and any odd- (cid:96) modehas Z ( n ) ≤ . Consequently, the full sky mean of Z ( n ) is not expected to be . Instead,it has some positive bias (in average over statistical ensemble of realizations) because theleading-order term is the quadrupole, which always has Z ( n ) > . Z (n) By definition Z ( n ) is a local estimator, and we can plot its map on the sky. We show Z ( n ) without masking for the Planck 2018 SMICA map [25] with 5 degrees smoothing in figure 1.Note that by construction this map is symmetric, Z ( n ) = Z ( − n ) . The most interestingobservation from figure 1 is the two pairs of very strong high negative peaks of Z ( n ) (labelled1a/1b and 2a/2b) and about twenty negative peaks with smaller amplitudes, mainly localizedwithin the area | b | ≤ ◦ in Galactic coordinates. The strongest peaks have the followingcoordinates ( l, b ) :
1a : (212 ◦ , − ◦ ) ,
1b : (32 ◦ , ◦ )2a : (332 ◦ , − ◦ ) ,
2b : (152 ◦ , ◦ ) – 3 – . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y ρ = 0 ρ = − . ρ = − . SMICA
Figure 2 . Theoretical Z ( n ) distribution functions according to the Gaussian model, for ρ = 0 (black), ρ = − . (blue), and ρ = − . (green). This is compared to the actual calculated distributionfunction of SMICA, shown in red. There are about nine pairs of high amplitude positive peaks, as it is seen from figure 1, localizedin the same | b | ≤ ◦ belt, and about six highest peaks distant from this area near the poles.It is noteworthy that the coordinates of the 1a peak are practically the same as thedirection found for the power asymmetry in the model of dipole modulation, e.g. [14] reports ( l, b ) ≈ (224 ◦ , − ◦ ) ± ◦ , and [2] adopts ( l, b ) = (221 ◦ , − ◦ ) for the dipole modulationdirection. The peak 1b is nothing but the symmetric partner of the peak 1a. The peak 2acoincides with the position of the Galactic Cold Spot, and 2b is its antipodal partner. Thesedirections will be excluded from the analysis of the power asymmetry due to proximity to theGalactic plane where they are masked out. In the forthcoming sections we will show thatthese anomalies, the parity and power asymmetries, could have the same origin. Z (n) Although we will work with simulated sky maps for estimation of the significance level of theanomalies, it is also valuable to have a basic understanding of the statistical properties offunction Z ( n ) . Suppose T ( n ) is a realisation of statistically isotropic Gaussian field. Then, T ( − n ) is also Gaussian-distributed. However, the two quantities T ( n ) and T ( − n ) are notindependent random variables due to correlations in the pixel domain. The precise details ofthis correlation are determined by the power spectrum and the smoothing angle Θ .For our purposes, it is sufficient to consider the Pearson cross-correlation coefficient of T ( n ) and T ( − n ) : ρ = Corr( T ( n ) , T ( − n )) = (cid:90) T ( n ) T ( − n ) d n (cid:90) T ( n ) d n = (cid:80) (cid:96) ( − (cid:96) (cid:80) m | a (cid:96)m | (cid:80) (cid:96) (cid:80) m | a (cid:96)m | . (4.1)– 4 – . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y SimulationsSMICACommanderSEVEMNILC − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y SimulationsSMICA
Figure 3 . Left panel: Z ( n ) distribution functions from CAMB simulations (blue) and actualdistribution function from SMICA (red) and the other Planck maps. The smoothing angle is Θ = 5 ◦ .The 68%, 95%, and 99.7% uncertainty regions of the simulations are shown in the three shades of blue.On the right side of the distribution, we have a departure of the distribution function at around orgreater than the σ level. Right panel: the same for SMICA with smoothing angle Θ = 2 . ◦ . We have taken that the mean is subtracted, (cid:82) T ( n ) d n = 0 . The integrals are calculated assums over all available pixels.For given ρ , the distribution function of Z ( n ) has a form [26, 27]: P ( Z (cid:48) ) = 1 π (cid:112) − ρ exp (cid:18) ρZ (cid:48) − ρ (cid:19) K (cid:18) | Z (cid:48) | − ρ (cid:19) , (4.2)where K is the 0-th order modified Bessel function of the second kind and Z (cid:48) = Z/ var( T ) is the rescaled Z to unit variance. In figure 2, function P ( Z ) from equation (4.2) is plottedfor three sample values of ρ . When ρ = 0 , the distribution is symmetric. When ρ < , thedistribution skews to the left and corresponds to an excess of odd- (cid:96) power. The value of ρ estimated from the SMICA data is ρ SMICA = − . . Therefore, the simple cross-correlationestimator indicates an excess of odd- (cid:96) power, or a deficit of even- (cid:96) power.However, as seen in figure 2, the SMICA distribution function deviates quite strongly fromthe model with ρ SMICA = − . : on the left, there is a strong bump around Z ∼ − × − ,and on the right above Z > × − , there is a correspondingly smaller observed density inthe histogram. These features will be investigated more carefully using Λ CDM simulations.
Figure 1 reveals one of the advantages of the local estimator: we can directly identity theparts of the sky that are the main sources of asymmetry. For example, the purple and blackzones mark pixels n in which T ( n ) and T ( − n ) have simultaneously large amplitudes anddifferent signs; the red and white zones are pixels in which T ( n ) and T ( − n ) both have largeamplitudes and they also have the same signs. To characterize the occurrence of such featuresin a general way, we investigate to the distribution function of Z ( n ) , which in practice meanscalculating its histogram over the observed sky.To test the significance of departures from the Gaussian model, we run simulations basedon a power spectrum from CAMB [28] with Planck 2018 best fit cosmological parameters from[29, 30]. This is shown in figure 3. The main result is a ∼ σ deficit on the right (symmetric,– 5 – K aCS CS -0.00018 0.00018 K -0.00018 0.00018 K Figure 4 . Top left: The zones of the sky that are located in the left tail of the distribution functionare shown in blue (regions of asymmetric excess), and the zones of the sky from the right tail are shownin red (symmetric excess). Top right: the zones of the SMICA map with Z density D > K − .Bottom left: the zones of SMICA map with density D < K − . Bottom right: the same as left, butfor even (cid:96) -excess. The labels CS and aCS mark the WMAP/Planck Cold Spot. even- (cid:96) ) side of the distribution function. Also, in figure 4, we show the parts of the skycontributing to the tails of the distribution function with density D ≤ K − . To show themorphology of asymmetry, we use the SMICA map as a background and masked out all zoneswhich contributed to symmetric and asymmetric modes (see the four maps in figure 4). Theupper right map corresponds to SMICA map with Θ = 5 ◦ with the symmetric and asymetriczones with D < K − masked out. The bottom maps correspond to the SMICA signal forthe asymmetric tail (on the left) and the symmetric tail (on the right).It is important to note that all known local anomalies of the CMB sky contribute to theasymmetric part. The peaks 1a, 1b, 2a, and 2b, discussed above, the Galactic Cold Spot, theWMAP/Planck Cold Spot and its counterpart (see figure 4) are presented there. At the sametime, from figure 3, the odd- (cid:96) asymmetric modes lie within the range of Gaussian simulationsand show no Z -distribution anomalies when Θ = 5 ◦ . In contract, the even- (cid:96) modes revealsignificant departure from Gaussian simulation (greater then . ), and they are free fromthe contribution of the local anomalies listed above. This result is in agreement with [5, 19],where it was pointed out that odd- (cid:96) tail of the WMAP power spectrum perfectly follows tothe best fit Λ CDM model, while the even- (cid:96) tail significantly departs from it.The analysis presented above of the parity asymmetry for Planck component separationproducts has a strong dependence on the multipole domain under consideration. The maximumof asymmetry corresponds to multipoles ≤ (cid:96) ≤ . In our pixel domain approach thatmeans that the significance of asymmetry should depend on the smoothing angle Θ . Thisstatement is illustrated in the right panel of figure 3, where the smoothing angle is Θ = 2 . ◦ – 6 – . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y SimulationsQuadrupoleadjustedsimulationsSMICA − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y SimulationsQuadrupoleadjustedsimulationsSMICA
Figure 5 . Two different methods of simulating quadrupole-adjusted maps. In the left panel, the (cid:96) = 2 coefficient in the best-fit power spectrum is adjusted to that of SMICA, and simulations are generatedfrom this modified power spectrum. In the right panel, simulations are generated from the originalpower spectrum, and each map is manually adjusted to the SMICA quadrupole. The black shadedregions show the 68%, 95%, and 99.7% uncertainty of the density estimation. Comparison of the twofigure shows that the two methods of simulation give almost the same results. However, the variancefor the power spectrum adjustment (left panel) is slightly higher than for the map adjustment (rightpanel). The power spectrum adjustment is the standard method used in this work. and the range of the corresponding multipoles is extended up to – . Here the asymmetrictail has no anomalies, while the symmetric tail is still marginally peculiar for high positive Z . From figure 4 (the bottom left and right panels) one can see the reason for that: in thesymmetric component we have a deficit of the negative peaks in the comparison with theasymmetric component.We have focused the analysis on the tails of the Z ( n ) distribution function, correspondingto the highly symmetric and asymmetric peaks. Although it is not visible in figure 3 due to thelogarithmic axes scale in use, there is also a slight excess of low- Z ( n ) pixels, complementaryto the general deficits found in the tails, as constrained by the histogram estimator. (cid:96) dominated? One of the features of the best fit Λ CDM cosmological model based on the Planck 2018 powerspectrum is the even- (cid:96) parity dominance (on average), at least for the multipole domain (cid:96) ≤ .This fact attracted less attention in the literature devoted to investigation of parity anomaly,but it is very important for understanding its origin. Actually, the even- (cid:96) dominance can bequantitatively seen right from the CMB power spectrum, which is close to C (cid:96) ∝ (cid:96) ( (cid:96) + 1) for ≤ (cid:96) ≤ [5]. For a power spectrum of this form, the highest amplitude corresponds to thequadrupole and it will determine the even- (cid:96) asymmetry of Z ( n ) . For the actual Planck 2018power spectrum we confirmed that result by simulations presented in figure 3. Here one cansee that the mean value of Z ( n ) over all realisations and the sample variance (blue lines andregion) are skewed towards the symmetric components, while the actual SMICA data revealsthe odd- (cid:96) asymmetry.One of the key reason for dominance of the asymmetric modes is related to the abnormallylow quadrupole. The analysis of the quadrupole anomaly and its influence on the distributionof Z ( n ) is presented below in two directions.– 7 –irst, we generate a power spectrum from CAMB using the best-fit cosmological param-eters from Planck, and calculate realizations of Z ( n ) from simulated Gaussian maps fromthis power spectrum. The average histogram of Z ( n ) and the variation in the counts ofthe histograms was shown in figure 3 by the blue shaded regions. Alternatively, the secondmethod of simulations is based on a new power spectrum in which the (cid:96) = 2 amplitude hasbeen reduced to the SMICA quadrupole power. This is a very large reduction, because theSMICA quadrupole is abnormally small. From this new quadrupole-adjusted power spectrum,simulations are generated and the histograms are calculated like before. The results are shownby the black solid line and black shaded regions in figure 5 (left panel). As it is seen from thatfigure, the significance level of abnormality for the even- (cid:96) multipoles drops from about σ to σ . One can treat this result as a formal solution of the problem of parity asymmetry, but theprice of it is very high. Formally, we move out from the best-fit Λ CDM cosmological modelwith statisticaly abnormal quadrupole to the models with normal low quadrupole, but morecomplicated physics of the CMB anisotropy in the Sachs-Wolfe domain. The third method ofsimulations is based on the Planck best fit cosmological model, as in figure 3, but for eachrealisation of the CMB map the quadrupole component is set equal to the SMICA quadrupole.With this brute-force approach, we completely remove the effect of sample variance for thequadrupole, which should be viewed as a kind of artificial model of systematic effects. Thedifference is slight, but as one can see from the right panel of figure 5, the significance of parityanomaly of symmetric modes is reduced almost to the confidence level.
Discussion of the role of the quadrupole in the framework of the modified Planck 2018cosmological model raises the more general question of how much the statistical significanceof anomalies depends on cosmological parameters. In answering this question, we looked attwo artificial models, changing the cosmological parameters manually. The first modificationconsists in replacing the Hubble constant . to . km/s/Mpc without changing the otherparameters. The second model was discussed in [19] in relation to fitting the power spectrumof even multipoles, in which the main factor is the use of the Harrison-Zeldovich spectral index.Hereafter we will call that model as the “even model”. In the left column of table 1 is shownthe values of the cosmological parameters under consideration, as derived from the Planck2018 combined analysis [29] with the corresponding error bars at 68 % confidence level. Thesimulations used so far in this work are consistent with these parameters. The right column oftable 1 shows alternative parameters for the even model.Parameter Planck 2018 [29] Even model [19] H . ± .
54 71 . b h . ± . . c h . ± . . τ . ± . . n s . ± . . Table 1 . Cosmological parameters derived in the Planck 2018 analysis [29] compared to even model. In this article we will use Gaussian simulations from HEALPIX synfast program. The comparison ofthem and Planck FFP simulations can be found in [31]. Note that at the level of σ we still have symmetric modes anomaly for Z ( n ) after modification of thequadrupole. – 8 – . − . . . . . T ( n ) , T ( − n ))3 . . . . . . . s t d ( T ( n )) × − . % % % . % % % H = 67 . H = 71 Figure 6 . Contours of
Corr( T ( n ) , T ( − n )) and std( T ( n )) within Λ CDM models having H = 67 . (blue) and H = 71 . (red). − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y H = 67 . − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y H = 71 − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . T ( n ) T ( − n ) ( K ) × − D e n s i t y H = 71 (even model) Figure 7 . From left to right: normal model, normal model with H adjusted to , and even-model(see table 1). The most important observation is the significance of the bump on the left side of thedistribution function, which contains the peaks 1a/1b/2a/2b and other high amplitude asymmetricpeaks discussed in Section 3 and shown in figure 1. In the Gaussian theory, the parameters ρ = Corr( T ( n ) , T ( − n )) (see equation (4.1)) and std( T ( n )) determine the shape of the distribution function of Z ( n ) . Both parameters showdependence on the Hubble parameter. Using the 2-sample Kolmogorov-Smirnov test, thedifference in ρ is detectable with p-value < − between the simulations from H = 67 . andthe simulations from H = 71 . std( T ( n )) is also detectable with p-value < − using thesame test between the models for H = 67 . and H = 71 . The contours for these parametersare shown in figure 6.In figure 7, the range of simulations is shown for three models: default model with thebest-fit Planck parameters (this is effectively the same as the left panel of figure 3), the secondmodel derived by increasing the Hubble parameter from . to . while leaving all otherparameters fixed, and third the even-model from the right column of table 1. The generaltrend is a shift of density out of the tails. As we advance through these models, the significanceof the even- (cid:96) deficit of SMICA weakens from around σ to around σ ; correspondingly, thepeak odd- (cid:96) excess becomes a little bit more significant, nearing a maximum departure of σ for the even-model. – 9 – K Figure 8 . G ( n ) in zones where the density of Z ( n ) is ≤ K − for the SMICA map. The power symmetry is one of the most intriguing CMB anomalies. It leads to weak couplingbetween neighbouring multipoles and an anisotropic distribution of the power spectrum. Forthe model of power asymmetry based on dipole modulations, the observed CMB temperature T ( n ) is related with statistically isotropic Gaussian component T g ( n ) as: T ( n ) = (1 + ( m · n ) D ) T g ( n ) (6.1)where m is unit vector in the direction of dipole, m · n denotes the dot product, and D isthe amplitude of the dipole modulation. The dipole modulation of the statistically isotropicGaussian random signal T g ( n ) means that in the direction of the maximum of the dot product m · n we have amplification of the local amplitude of the signal, while in opposite direction, itis effectively decreasing.For characterisation of the asymmetry induced by the dipole modulation of the isotropicGaussian field T g ( n ) , we use the following estimator: G ( n ) = T ( n ) − T ( − n ) = 4 S ( n ) A ( n ) . (6.2)Using equation (6.1), this estimator can be presented in the following form: G ( n ) = (cid:2) − R ( n ) + 2 D cos( ξ )(2 + R ( n )) − R ( n ) D cos ( ξ ) (cid:3) T g ( n ) (6.3)where ξ is the angle between m and n , and R ( n ) + 1 = T g ( − n ) T g ( n ) is the asymmetry parameter.If R ( n ) (cid:28) D we get: G ( n ) ∝ S ( n ) A ( n ) (cid:39) D cos( ξ ) T g ( n ) . (6.4)Thus, from equation (6.4) one can see that combination S ( n ) A ( n ) of the symmetric andasymmetric components of the observable signal can be used as an estimator of the dipolemodulation from equation (6.1). For illustration of this statement in figure 8 we show the map– 10 – ipole of G ( n ) Power asymmetry dipole
Figure 9 . The G ( n ) dipole for the SMICA map and dipole of modulation from equation (6.1). Threshold . . . . . . C o rr e l a t i o n c o e ffi c i e n t Figure 10 . Correlation coefficient between dipole maps as a function of the threshold of the Z ( n ) density below which the correlation is computed. S ( n ) A ( n ) from SMICA signal in zones with Z -density ≤ K − in order to get the parityasymmetry and the dipole modulation simultaneously.We decompose this map into spherical harmonics and get a dipole component shown infigure 9. For comparison, in this figure, we constructed a dipole power modulation model (seeequation (6.1)) and estimated the pixel-to-pixel Pearson cross-correlation coefficient, which is C = 0 . . This coefficient is a function of the threshold amplitude Z ( n ) . In figure 10 we plotthe dependency of the dipole-dipole cross-correlation versus the amplitude of the threshold.The cross-correlation coefficient is only computed within the zones where the estimated densityof Z ( n ) is below the threshold, and then we repeat this for different thresholds. The highthreshold corresponds to using more of the sky for the C : then we get 0.87. Lower thresholdscorrespond to calculating C only in areas with the most asymmetric pixels: then we getconvergence to C = 0 . . Asymptotically, it will be associated only with the peaks 1a/1band 2a/2b discussed in Section 3. In the pixel domain, we investigated the sources of asymmetry hovering in the CMB temperaturemaps. We have demonstrated that this type of anomaly is mainly associated with the presenceof four high amplitude positive and negative peaks (1a, 1b, 2a, 2b), symmetrically spacedfrom each other relative to the Galactic center. Peaks 1a and 1b are associated with the zonesbelonging to the Northern Galactic Spur and the direction of the dipole modulation of the– 11 –ower spectrum CMB anisotropy. The influence of the Northen Galactic Spur on the CMBsignal has been pointed out in [32]. The two other peaks (2a, 2b) are associated with theGalactic plane (the Galactic Cold Spot and its partner asymmetric in amplitude). The sametype of asymmetric peaks, but with a smaller amplitude, belongs to WMAP/Planck Cold Spotand its partner in the Northern Galactic Spur. The most striking result of our analysis is thefact that these local anomalies of the CMB map increase the level of the asymmetric part thedistribution of the estimator Z ( n ) to a level consistent with predictions based on simulationsof a random Gaussian process. In contrast, the deficit of peaks symmetric in amplitude andposition is accompanied by a decrease in the symmetric part Z ( n ) of the σ level, which isthe source of the parity asymmetry of the CMB temperature maps.Another source of the deficit of the symmetric part of the Z ( n ) function is the lowamplitude of the SMICA quadrupole. We investigated the influence of this factor usingGaussian simulations, in which, instead of the theoretical value in the power spectrum, weused the amplitude of the SMICA quadrupole. Moreover, this modified power spectrumstill corresponds to the Planck 2018 best fit cosmological model, with the exception of thequadrupole. We have shown that such a modification of the power spectrum leads to aweakening of the confidence level of Z ( n ) for symmetric modes in simulations and reduces thesignificance of the parity asymmetry to the σ level.The cosmological parameters are also reflected in the distribution function of Z ( n ) andtheir effect on the significance of anomalies. We have considered alternative models in whichthe Hubble parameter is increased to H = 71 . from the Planck best-fit value of around . . The model with higher H has a slightly more symmetric distribution function. Furtheradjustment of the cosmological parameters to the “even model” derived from [19] has the sameoverall effect on the Z ( n ) function.We investigated the relationship between the asymmetry of the power spectrum andthe level of the parity asymmetry in the framework of a model with dipole modulation of astatistically uniform Gaussian signal. We have shown that the dipole modulation amplitude isdirectly related to the level of asymmetry of the G ( n ) = S ( n ) A ( n ) estimator, which is theproduct of the symmetric and asymmetric components of SMICA in the zones of maximumasymmetry of the function Z ( n ) . The dipole the component of the G ( n ) estimator in the zoneswith different levels of parity asymmetry practically coincides with the dipole modulation ofthe power spectrum: the corresponding Pearson cross-correlation coefficient C is localised inthe domain . − . . Again, the maximum of C = 0 . achieved from the peaks 1a, 1b, 2a,2b, where the parity asymmetry has a point of maxima.In conclusion, we would like to note the peculiarity of our analysis of various anomaliesin the pixel domain. At first glance, it may seem trivial that various anomalies turned outto be interdependent due to their joint contribution to each pixel. However, one must notforget that the transition from pixel to multipole analysis of some anomalies is associatedwith integration over the map, in which a significant role belongs to different types of maskswith different sky coverage. Thus, the zones identified by us in the map must be taken intoaccount (masked out) in analysis of anomalies in a multipole domain. Acknowledgments
The HEALPix pixelization [33] scheme was heavily used in this work, and we thank them fortheir contributions to the field. This research was partially funded by Villum Fonden throughthe Deep Space project. – 12 – eferences [1]
Planck collaboration,
Planck 2015 results. XVI. Isotropy and statistics of the CMB , Astron.Astrophys. (2016) A16 [ ].[2]
Planck collaboration,
Planck 2018 results. VII. Isotropy and Statistics of the CMB , Astron.Astrophys. (2020) A7 [ ].[3] H.K. Eriksen, F.K. Hansen, A.J. Banday, K.M. Gorski and P.B. Lilje,
Asymmetries in the CosmicMicrowave Background anisotropy field , Astrophys. J. (2004) 14 [ astro-ph/0307507 ].[4] F.K. Hansen, A.J. Banday, K.M. Gorski, H.K. Eriksen and P.B. Lilje,
Power Asymmetry inCosmic Microwave Background Fluctuations from Full Sky to Sub-degree Scales: Is the UniverseIsotropic? , Astrophys. J. (2009) 1448 [ ].[5] J. Kim and P. Naselsky,
Anomalous parity asymmetry of the Wilkinson Microwave AnisotropyProbe power spectrum data at low multipoles , Astrophys. J. Lett. (2010) L265 [ ].[6] C.J. Copi, D. Huterer and G.D. Starkman,
Multipole vectors - A New representation of the CMBsky and evidence for statistical anisotropy or non-Gaussianity at ≤ (cid:96) ≤ , Phys. Rev. D (2004) 043515 [ astro-ph/0310511 ].[7] D.J. Schwarz, G.D. Starkman, D. Huterer and C.J. Copi, Is the low- (cid:96) microwave backgroundcosmic? , Phys. Rev. Lett. (2004) 221301 [ astro-ph/0403353 ].[8] A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga and A. Hamilton, The Significance of thelargest scale CMB fluctuations in WMAP , Phys. Rev. D (2004) 063516 [ astro-ph/0307282 ].[9] K. Land and J. Magueijo, Is the Universe odd? , Phys. Rev. D (2005) 101302[ astro-ph/0507289 ].[10] K. Land and J. Magueijo, Examination of evidence for a preferred axis in the cosmic radiationanisotropy , Phys. Rev. Lett. (2005) 071301.[11] H.K. Eriksen, D.I. Novikov, P.B. Lilje, A.J. Banday and K.M. Gorski, Testing fornon-Gaussianity in the WMAP data: Minkowski functionals and the length of the skeleton , Astrophys. J. (2004) 64 [ astro-ph/0401276 ].[12] F.K. Hansen, A.J. Banday and K.M. Gorski,
Testing the cosmological principle of isotropy: Localpower spectrum estimates of the WMAP data , Mon. Not. Roy. Astron. Soc. (2004) 641[ astro-ph/0404206 ].[13] H.K. Eriksen, A.J. Banday, K.M. Gorski, F.K. Hansen and P.B. Lilje,
Hemispherical powerasymmetry in the three-year Wilkinson Microwave Anisotropy Probe sky maps , Astrophys. J. Lett. (2007) L81 [ astro-ph/0701089 ].[14] J. Hoftuft, H.K. Eriksen, A.J. Banday, K.M. Gorski, F.K. Hansen and P.B. Lilje,
Increasingevidence for hemispherical power asymmetry in the five-year WMAP data , Astrophys. J. (2009) 985 [ ].[15] P.K. Aluri and P. Jain,
Parity Asymmetry in the CMBR Temperature Power Spectrum , Mon.Not. Roy. Astron. Soc. (2012) 3378 [ ].[16] M. Hansen, A.M. Frejsel, J. Kim, P. Naselsky and F. Nesti,
Pearson’s random walk in the spaceof the CMB phases: Evidence for parity asymmetry , Phys. Rev. D (2011) 103508 [ ].[17] J. Kim and P. Naselsky, Anomalous parity asymmetry of WMAP power spectrum data at lowmultpoles: is it cosmological or systematics? , Phys. Rev. D (2010) 063002 [ ].[18] A. Gruppuso, F. Finelli, P. Natoli, F. Paci, P. Cabella, A. De Rosa et al., New constraints onParity Symmetry from a re-analysis of the WMAP-7 low resolution power spectra , Mon. Not.Roy. Astron. Soc. (2011) 1445 [ ]. – 13 –
19] J. Kim, P. Naselsky and M. Hansen,
Symmetry and anti-symmetry of the CMB anisotropypattern , Adv. Astron. (2012) 960509 [ ].[20] C.J. Copi, D. Huterer, D.J. Schwarz and G.D. Starkman,
Large-scale alignments from WMAPand Planck , Mon. Not. Roy. Astron. Soc. (2015) 3458 [ ].[21] P.K. Aluri, J.P. Ralston and A. Weltman,
Alignments of parity even/odd-only multipoles in cmb , Monthly Notices of the Royal Astronomical Society (2017) 2410–2421 [ ].[22] Y. Akrami, Y. Fantaye, A. Shafieloo, H.K. Eriksen, F.K. Hansen, A.J. Banday et al.,
Powerasymmetry in WMAP and Planck temperature sky maps as measured by a local varianceestimator , Astrophys. J. Lett. (2014) L42 [ ].[23] A. Gruppuso, N. Kitazawa, M. Lattanzi, N. Mandolesi, P. Natoli and A. Sagnotti,
The Evensand Odds of CMB Anomalies , Phys. Dark Univ. (2018) 49 [ ].[24] S. Shaikh, S. Mukherjee, S. Das, B.D. Wandelt and T. Souradeep, Joint Bayesian Analysis ofLarge Angular Scale CMB Temperature Anomalies , JCAP (2019) 007 [ ].[25] Planck collaboration,
Planck 2018 results. IV. Diffuse component separation , Astron.Astrophys. (2020) A4 [ ].[26] S. Nadarajah and T.K. Pogány,
On the distribution of the product of correlated normal randomvariables , Comptes Rendus Mathematique (2016) 201.[27] R.E. Gaunt,
A note on the distribution of the product of zero-mean correlated normal randomvariables , Statistica Neerlandica (2019) 176 [ ].[28] A. Lewis, A. Challinor and A. Lasenby, Efficient computation of CMB anisotropies in closedFRW models , Astrophys. J. (2000) 473 [ astro-ph/9911177 ].[29]
Planck collaboration,
Planck 2018 results. VI. Cosmological parameters , Astron. Astrophys. (2020) A6 [ ].[30]
Planck collaboration,
Planck 2018 results. V. CMB power spectra and likelihoods , Astron.Astrophys. (2020) A5 [ ].[31] J. Muir, S. Adhikari and D. Huterer,
Covariance of CMB anomalies , Phys. Rev. D (2018)023521 [ ].[32] H. Liu, P. Mertsch and S. Sarkar, Fingerprints of Galactic Loop I on the Cosmic MicrowaveBackground , Astrophys. J. Lett. (2014) L29 [ ].[33] K.M. Gorski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke et al.,
HEALPix:A framework for high-resolution discretization and fast analysis of data distributed on the sphere , The Astrophysical Journal (2005) 759.(2005) 759.