Anomalous spacings of the CMB temperature angular power spectrum
AAnomalous separations of the CMB temperature angular powerspectrum
Md Ishaque Khan * , Rajib Saha † , Department of Physics, Indian Institute of Science Education and Research (IISER), Bhopal 462066, India
Abstract
In this article, we propose a novel technique to test for anomalous features in the CMB. We analyse separations ofthe observed CMB angular power spectrum ( C (cid:96) ) using temperature anisotropy data from WMAP 9 year ILC and 2018Planck maps of Commander, NILC and SMICA. We estimate the minimum, maximum, average separations and ratios ofthe maximum to minimum separations between consecutive multipoles of the weighted spectrum, f ( (cid:96) ) C (cid:96) . We see thatsuch f ( (cid:96) ) ’s with higher multipole powers mitigate the parity asymmetry anomaly. For anomalous separations, we find thatdata exhibits anomalous ranges of multipoles defined by different (cid:96) max and (cid:96) min values, specifically for the entire rangeof multipoles from − of this work. Without parity based distinctions, most significantly, the maximum separationof the range ≤ (cid:96) ≤ is seen to be anomalously low at the . confidence level for f ( (cid:96) ) = (cid:96) (WMAP), (cid:96) ( (cid:96) +1)2 π (Planck NILC), the latter indicating a strong deviation from the Sachs-Wolfe plateau for maximum separations among lowmultipoles. The analysis is repeated for odd and even multipoles taken separately, in the same multipole ranges. Mostnoticeably, the even multipoles are seen to have anomalously low maximum and average separations relative to their oddcounterparts, the most outstanding among which is the anomalously low maximum separation for even multipoles in therange ≤ (cid:96) ≤ for f ( (cid:96) ) = (cid:96) (WMAP), at the . confidence level. For separation ratios, the multipole ranges aresimilar to those which turn up as anomalous when only separations are considered. Keywords— cosmic background radiation – inflation – methods: data analysis – cosmology: miscellaneous
Cosmological data analysis in the past unveiled manyanomalies in the observed Cosmic Microwave Background(CMB) patterns of WMAP [12] and Planck [37] satellitemissions. It has hence become imperative to study anoma-lies in both WMAP and Planck maps to make discrepan-cies more discernible if any such exist. Such anomaliesinclude the north-south power asymmetry [5], the anoma-lous cold spot ( [40], [14], [13]), high degree of octupole-quadrupole alignment ( [39], [17], [38]), quadrupole powerdeficit ( [4], [24]) and planarity of the octupole [16], andthe like. Some of the latter ones concerning the quadrupoleand octupole have been somewhat mitigated using a hypo-thetical foreground reduction [1].Anomalies, specifically those of the power excess forlower odd multipoles [30], have been shown to be a partof a more general trend in the observed maps ( [29], [3],[42]) referred to as a ‘parity asymmetry’ of the power dis-tributed among low odd and even parity multipoles. Studiesof the Integrated Sachs-Wolfe (ISW) effect for low multi-poles [15] help explain the power deficit in this range ofmultipoles. [22] show how an ISW foreground subtractionhelps obscure the anomalous effects of the quadrupole andits alignment with the octupole. Besides, modifications tothe scale of the primordial power ( [26], [8], [23]) mayserve as a starting point for explaining the ‘parity asymme-try’ and other power deficit anomalies. An excellent propo- * email: [email protected] † email: [email protected] sition of a Finsler spacetime [9] also helps naturally explainparity asymmetry.Probes of anomalies till now have delved into low mul-tipole ranges in the observational data. Large scale or lowmultipole studies of CMB data have become imperative forvarious reasons as explained by [12]. Our analyses pre-sented here also show that such ranges are distinctivelyanomalous relative to the theoretical Λ CDM temperaturepower spectrum.We propose a novel analysis of separations or absolutedifferences between consecutive CMB temperature angu-lar power spectrum (APS) measures ( C (cid:96) ’s). We investigateanomalous maximum, minimum and average separations,for low multipoles, by using a weighted APS ( f ( (cid:96) ) C (cid:96) ). Weanalyse not just all multipoles, but also even and odd multi-poles taken distinctly. This helps understand the behaviourof parity preference of the APS. We also look for anoma-lous ratios of maximum to minimum separations.Our paper is organised as follows : Section 2 states thebasic framework of the analysis presented thereafter. Sec-tion 3 describes our way of testing consecutive multipoleseparations with simulations. We report anomalies in sec-tion 4, for all multipole and for even/odd multipole sepa-rations, respectively. In section 5, we summarise the mostsignificant results for anomalous separations. In section 6,we present anomalous ratios of the maximum to minimumseparations. In section 7, we present our tests of parityasymmetry with six f ( (cid:96) ) ’s and use the results to furthersubstantiate our argument for the choices of these f ( (cid:96) ) ’swhen testing for anomalous separations. And, in section 8,1 a r X i v : . [ a s t r o - ph . C O ] J a n e deduce any general trends in anomalies based on ourresults and opine on apparent discrepancies. The Cosmic Microwave Background (CMB) provides ameasure of temperature anisotropies relative to the uni-form mean background temperature of nearly . K [21].These anisotropies are denoted by ∆ T (ˆ n ) , where ˆ n definesa direction in the CMB sky specifically given in terms of ( θ, φ ) in spherical polar coordinates with the radius of thelast scattering surface. The unbiased angular power spec-trum estimator is given by C (cid:96) ’s as [27]: C (cid:96) = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) | a (cid:96)m | , (1)where, the a (cid:96)m ’s are coefficients of the expansion: ∆ T (ˆ n ) = ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) a (cid:96)m Y (cid:96)m (ˆ n ) . (2)Hence one may ascribe units of ( µK ) to the C (cid:96) ’s.In this article, we present a new idea of investigatingCMB temperature angular power spectrum anomalies. Wepropose a novel set of estimators for this purpose, basedon the CMB angular power spectrum of the observed CMBmaps. More specifically, we seek to realise nearest neigh-bour spacings in the context of CMB angular power spec-tral measures ( C (cid:96) ’s). Such estimators have not been spo-ken of in existing literature but can be understood in termsof the absolute differences or separations between any twoconsecutive C (cid:96) ’s.For a given range of multipoles [ (cid:96) min , (cid:96) max + 1] ,the separations between consecutive C (cid:48) (cid:96) s were estimated,weighted by f ( (cid:96) ) ’s. Say, such consecutive multipole sep-arations of f ( (cid:96) ) C (cid:96) ’s are [∆ , ∆ , ... ] , then four estimatorsto detect anomalies are as follows: max i = max [∆ i , ∆ i , ... ] ,min i = min [∆ i , ∆ i , ... ] ,avg i = ∆ i + ∆ i + ...N ,r i = max i min i , (3)which stand for the maximum, minimum, average of a setof separations and the ratio of the maximum to minimumseparations, respectively. Here, i = a, o, e , where a, o, e stand for all, odd and even multipoles, and N = number ofseparations for a given range of (cid:96) ’s.These estimators have been proposed here as exhaus-tive probes of anomalous separations, because the lowerand upper limits ( (cid:96) min , (cid:96) max respectively) of a multipolerange have also been varied appropriately for all the anal-yses. Being extreme statistical measures, these estimators(excluding the average measure) represent rare occurrencesof certain values of the nearest neighbour separations, yet along with the average or mean separation and the ratioof the maximum to minimum of a multipole range, these,will characterise anomalous behaviours of separations in aneasily quantifiable way.We present anomalies for all, odd and even multipoleseparations for four simple cases of f ( (cid:96) ) = 1 , (cid:96) ( (cid:96) +1)2 π , (cid:96), (cid:96) .For reference purposes we have included the plots of thefour f ( (cid:96) ) C (cid:96) ’s with multipoles in figure 1. Fractional pow-ers of multipoles in f ( (cid:96) ) are also possible choices, but suchchoices and arguments in favour of those will be postponedfor future work. f ( ℓ ) = 1 f ( ℓ ) = ℓ ( ℓ +1)2 π f ( ℓ ) = ℓ f ( ℓ ) = ℓ Figure 1: f ( (cid:96) ) C (cid:96) versus (cid:96) , the weighted angular powerspectrum : The plot compares the theoretical best fit toPlanck (purple) with COMM (green), NILC (light blue),SMICA (ochre) and WMAP (yellow) maps used here, allequivalently with zero smoothing but with a pixel windowapplication appropriate for n side = 16 We have estimated discrepancies relative to the theoreticalAPS (Planck 2018 best fit) with the help of the HEALPix[25] package, by considering simulations of the theo-retical temperature angular power spectrum to account forstatistical fluctuations while studying the minimum, max-imum and average of the separations between consecutive f ( (cid:96) ) C (cid:96) ’s.Observed data maps used here for probing anomalousseparations are: WMAP 9 year ILC and 2018 release fullmission Planck Commander (COMM), NILC and SMICAmaps from latest sources: [33] and [19]. These have beendowngraded with the help of HEALPix [25] software fa-cilities to a HEALPix n side = 16 , n(cid:96) max = 32 (hence,an appropriate pixel window) with no beam smoothing(fwhm arcmin = 0 . d ) and the simulations are obtainedusing the same resolution. With all data maps, the the-oretical power spectrum, and simulations thereof on anequal footing, we have proceeded with the analysis. We2ave excluded multipoles (cid:96) = 0 , as these correspond re-spectively to the monopole of uniform CMB temperature( ≈ . K ) [21], and the dipole which arises due to ourpeculiar motion relative to the CMB rest frame [7].We have taken minimum, maximum and average of theseparations of f ( (cid:96) ) C (cid:96) ’s for consecutive multipoles firstlywithout any distinction between odd or even ones and laterseparately for odd/even multipoles to study anomalies thatarise upon such parity based distinctions. Here, a separa-tion means the absolute value of the difference between twoconsecutive f ( (cid:96) ) C (cid:96) ’s. For example, for the multipolesin the range [2 , , we have separations with no paritybased distinction namely, | f (2) C − f (3) C | , | f (3) C − f (4) C | , ..., | f (6) C − f (7) C | . Whereas for even andodd multipoles taken distinctly, in the same range, we have separations each for even and odd multipoles namely, | f (2) C − f (4) C | , | f (4) C − f (6) C | and | f (3) C − f (5) C | , | f (5) C − f (7) C | respectively.For characterising the extent to which a separation maybe anomalous in a quantitative way, we define the followingprobabilities:s-value : P t ( x ) , p-value : P b ( x ) = 1 − P t ( x ) , where, x = ( min i , max i , avg i , r i ) ,i = a, o, e, (4)and a, o, e as before, stand for all (no parity based distinc-tion), odd and even multipoles respectively. P t ( x ) (hereafter referred to as the ‘s-value’) is the prob-ability of the theoretical spectrum having a value greaterthan a data map for the four entities, that are, the mini-mum, maximum, average separation and the ratio of themaximum to minimum separations, respectively, denotedby x . This is calculated by counting the number of errantsimulations having these entities greater than the observedmap and dividing the number by the total number of simu-lations, that being throughout this paper. P b ( x ) (or the familiar ‘p-value’) is similarly defined asthe probability of simulations having a value of the quan-tity x less than that from observational data. Probabilityplots for P b are in log-scale along the vertical axis. Plotsfor P t are not in log-scale. All probability plots men-tion . , . (in dotted red lines) corresponding to the – confidence range of hypothesis testing method-ology ( [6], [20], [35], [34]). Therefore all probabilitieswhich feature below or above have been reportedhere. A considerable discourse against usage of such confi-dence intervals is given by [32] or that against the often as-sumed complimentarity of the s- and p-values is presentedin the work of [41].As may be intuitively obvious from figure 1, the max-imum and minimum separations for f ( (cid:96) ) C (cid:96) ’s may lie inany of the lower or higher multipole ranges and hence itbecomes neccessary to scan the multipole ranges by chang-ing not only the upper bound of the multipole range ( (cid:96) max ,whilst keeping the lower bound fixed), but also the lowerbound ( (cid:96) min , whilst keeping the upper bound fixed). Alsoconsidering not just the minimum and maximum separa- tions, but also computing the average value of a set of sep-arations for a given (cid:96) max or (cid:96) min for the same reason, isneeded.We have, therefore, calculated the maximum, minimumand average separations by varying (cid:96) max ’s and (cid:96) min ’s. So,for example, multipoles in the range [2 , will correspondto an (cid:96) max = 6 where the (cid:96) min is fixed at (cid:96) min = 2 . Asimilar scheme holds for the (cid:96) min ’s which have been var-ied keeping (cid:96) max fixed at (cid:96) max = 30 . (cid:96) = 0 , have alreadybeen excluded for reasons aforementioned. Also having amaximum upper limit of (cid:96) max = 30 , implies that we arelooking for anomalies in the range [2 , . This invariablymeans that multipole (cid:96) = 32 is excluded from our anal-ysis. This is because, although including (cid:96) = 32 wouldnot have affected our results for all separations (no paritydistinctions), but, when we scan for anomalies after takingodd and even multipole separations distinctly, we shouldhave an equal number of odd and even separations for anychosen range of multipoles.Hence with this forethought, (cid:96) = 32 is excluded notonly for the analysis with parity based distinction but alsowhen all multipoles are treated without distinction, so thatcomparisons of results between the two analyses later onseem legitimate. This is ensured further by choosing onlyeven numbers for (cid:96) min ’s and (cid:96) max ’s. Thus, (cid:96) max ’s havebeen varied from → (by fixing (cid:96) min = 2 ) in stepsof ∆ (cid:96) = 2 for all separations and (cid:96) min ’s have been var-ied from → (having fixed (cid:96) max = 30 ) in steps of ∆ (cid:96) = 2 for all separations. Similar variations of (cid:96) min and (cid:96) max have been made for odd and even multipole separa-tions when those are considered individually, but those arevaried in steps of ∆ (cid:96) = 4 for, that being the most sensi-ble choice, keeps at least two separations of odd and evenmultipoles for comparison when maximum and minimumamong the set of separations for a given (cid:96) max or (cid:96) min arecomputed. We present the (cid:96) min or (cid:96) max ranges of the four CMBtemperature anisotropy data sources which are anomalousalong with their respective probabilities P t and/or P b . For the entire set of multipoles in [2 , , the anomalousseparations are presented here. f ( (cid:96) ) = 1 When considering all separations, anomalies arise when (cid:96) min is varied for all four maps. For a variation of (cid:96) max however, only WMAP shows anomalies (See figure 2).The most anomalous case for COMM is for (cid:96) min = 8 where the p-value of the average separation is only about . in , and for the maximum separation, it is for (cid:96) min = 16 , and the p-value is . and . in re-spectively. For NILC, the most significant is (cid:96) min = 8 with an s-value of . for the maximum separation;others are (cid:96) min = 12 , , with lesser significant values.3or average separation (NILC) the most anomalous is again (cid:96) min = 8 with a p-value of . in parts, while the oth-ers are (cid:96) min = 12 , , , with lesser significant values.For SMICA, the most anomalous is (cid:96) min = 8 for both av-erage and maximum separations, with the s-value higherthan SMICA being . for maximum, and, . for average separations. Other anomalous multipole sep-arations with lesser significance are for (cid:96) min = 12 , forthe maximum separation, and (cid:96) min = 12 , , for theaverage separation, respectively, for SMICA. Similarly forWMAP we have for (cid:96) min = 24 for minimum separationan s-value of . , and for both maximum and averageseparations, it is for (cid:96) min = 8 with s-values of . and . respectively. Also, WMAP shows maximallyanomalous (cid:96) max = 24 for minimum separation with an s-value of . , whilst other maps do not show anomaliesfor (cid:96) max ’s. Other anomalous ones for WMAP for mini-mum separation are (cid:96) max = 26 , . f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π In this case too, anomalies arise mainly for variations of (cid:96) min for all four maps, save NILC, which also showsanomalies for certain (cid:96) max ’s (See figure 3). For COMM,there is an entire range of values of (cid:96) min = 6 → which is anomalous for the maximum and average sepa-rations. The most significant is (cid:96) min = 8 again, with p-values of . in parts for the maximum and . in for the average separations. NILC has an anomalousrange of (cid:96) max = 14 → for the minimum separation,with (cid:96) max = 14 being most significant with an s-value of . . (cid:96) min = 8 → is anomalous for the maximumseparation with highest s-value of . for (cid:96) min = 8 for NILC. (cid:96) min = 6 → , → are anomalous foraverage separations in NILC with maximum s-value for (cid:96) min = 8 , which is . . For SMICA the anomalousranges are (cid:96) min = 8 → for maximum separation and (cid:96) min = 6 → for average separation. The most sig-nificant is (cid:96) min = 8 again for both kinds of separations,with p-values of . and in for maximum and av-erage separations respectively. For WMAP, again the mostsignificant is (cid:96) min = 8 for both maximum and average sep-arations. The s-values of WMAP for these are . and . respectively. f ( (cid:96) ) = (cid:96) For this case similarly, mostly for (cid:96) min variations, anoma-lous ranges are found for all four maps. Only WMAPshows an anomaly for (cid:96) max = 6 (See figure 4). COMMwith (cid:96) min = 12 for maximum separations has highest s-value of . whereas others of lesser significance are (cid:96) min = 8 , , ; (cid:96) min = 8 for average separation hasthe highest s-value of . , while others with lower s-values are (cid:96) min = 6 , , . NILC with (cid:96) min = 8 for bothmaximum and average separations is most anomalous withp-values of . and . in parts respectively. Otheranomalous ranges for NILC are (cid:96) min = 10 , , , forthe maximum separation and (cid:96) min = 10 , , , , for the average separation. SMICA similarly reveals that (cid:96) min = 8 is most anomalous for both maximum and aver- age separations with p-values of . and . in respec-tively. Other lesser significant anomalous ones for SMICAare (cid:96) min = 10 , , , for the maximum separation and (cid:96) min = 10 , , . Also for WMAP, the most anomalous is (cid:96) min = 8 for maximum and average separations with con-siderable p-values of in , and . in . WMAPalso shows an anomalous (cid:96) max = 6 with a p-value of . in for the minimum separation. f ( (cid:96) ) = (cid:96) In this case, COMM and SMICA show some nearly in-significant anomalous (cid:96) max ’s but as usual all four mapsshow anomalous ranges of (cid:96) min ’s. (See figure 5). Startingwith COMM again, (cid:96) max = 22 for the minimum separa-tion has an s-value for COMM of . , whereas againthe highest s-values are for (cid:96) min = 8 for the maximum andaverage separations of . and . respectively.Other anomalous ranges for COMM are (cid:96) min = 16 , , for the minimum separation, (cid:96) min = 6 → for the max-imum separation and (cid:96) min = 6 → , for the aver-age separation. Again for NILC too, (cid:96) min = 8 is maxi-mally anomalous with a considerable p-value of in parts and an s-value of . for the maximum and av-erage separations. Other anomalous ranges for NILC are (cid:96) min = 10 → for the maximum separation and (cid:96) min =6 → for the average separation. For (cid:96) max = 10 SMICAis anomalous with an s-value of . for the minimumseparation. (cid:96) min = 8 is again maximally anomalous forSMICA with s-values of . and . for the maxi-mum and average separations, while other anomalous mul-tipoles are given by (cid:96) min = 10 , for the maximum sep-aration and (cid:96) min = 6 → for the average separation.Similarly the highest s-values are . and . forthe maximum and average separations for WMAP respec-tively for (cid:96) min = 8 , while the other anomalous rangesare (cid:96) min = 10 , , for the maximum separation and (cid:96) min = 10 , , for the average separation. On treating f ( (cid:96) ) C (cid:96) ’s differently based on parity of the mul-tipoles, some novel anomalies come up. For example,the minimum separation for a certain (cid:96) max or (cid:96) min showsanomalous behaviour in some ranges with a much greatersignificance relative to the case when all separations re-gardless of parity were being treated without distinction.Also many a time, f ( (cid:96) ) C (cid:96) ’s of a certain parity may onlyshow up in anomalous ranges, whereas those of a differentparity remain quiescent until the power to which the multi-pole in f ( (cid:96) ) is raised is changed. f ( (cid:96) ) = 1 Mostly the maximum and average separations in even mul-tipoles (i.e., max e and avg e ) show anomalous behaviourfor all four maps, but at times, the minimum separationamong odd multipoles ( min o ) also shows some anomalies(see figure 6).COMM shows an anomaly for (cid:96) max = 10 , for the min-imum separation among odd multipoles ( min o ), with an4 ( ℓ min ) ( ℓ min ) ( ℓ min ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 2: Probabilities P t for min a , max a , avg a (in purple, green and light blue, respectively) for f ( (cid:96) ) = 1 versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Red dotted-dashed lines indicate (doubledots) and (single dots). ( ℓ min ) ( ℓ min ) ( ℓ min ) ( ℓ min )
10 15 20 25 30NILC ( ℓ max ) Figure 3: Probabilities P t for min a , max a , avg a (in purple, green and light blue, respectively) for f ( (cid:96) ) = (cid:96) ( (cid:96) +1))2 π versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Red dotted-dashed lines indicate (double dots) and (single dots). ( ℓ min ) ( ℓ min ) ( ℓ min ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 4: Probabilities P t for min a , max a , avg a (in purple, green and light blue, respectively) for f ( (cid:96) ) = (cid:96) versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Red dotted-dashed lines indicate (doubledots) and (single dots).s-value of . . For the maximum and average sepa-rations among even multipoles, the entire range of (cid:96) max ’sfrom (cid:96) max = 6 → is anomalous with highest s-valuesof . for (cid:96) max = 10 → for max e and . for (cid:96) max = 26 , for avg e . For variations of (cid:96) min , (cid:96) min = 2 is the only anomalous one for both max e and avg e forCOMM, with s-values of . and . .For NILC, the entire range of (cid:96) max = 6 → isanomalous for maximum and average separations amongthe even multipoles reaching the maximum s-values of . for (cid:96) max = 10 → for max e , and . for avg e for (cid:96) max = 26 , . Again, only for (cid:96) min = 2 , max e is anomalous for NILC, with probability of . . (cid:96) min = 2 , , , are anomalous for avg e for NILC, but (cid:96) min = 2 is most anomalous for avg e with a probability of . .Similarly for SMICA, the entire range of (cid:96) max = 6 → is anomalous for both max e and avg e , with maximalprobability of . for max e for (cid:96) max = 10 → ,and that of . for (cid:96) max = 26 , for avg e . (cid:96) max =26 , are also anomalous for min o for SMICA, with the maximum s-value of . for (cid:96) max = 26 . The range (cid:96) min = 2 → is anomalous for min o with a maximalprobability of . for (cid:96) min = 22 for SMICA. (cid:96) min =2 is anomalous for SMICA for max e with a probability of . . Also (cid:96) min = 2 , , are anomalous for avg e with a maximal probability of . for (cid:96) min = 2 forSMICA.For WMAP, (cid:96) max = 14 , are anomalous for min o with (cid:96) max = 14 being most anomalous with a probabil-ity of . . Again, (cid:96) max = 6 → are all anomalousfor max e and avg e for WMAP, with a maximal probabil-ity of . for (cid:96) max = 10 → for max e , and that of . for (cid:96) max = 26 for avg e . For (cid:96) min variations forWMAP, only (cid:96) min = 2 is anomalous with a probability of . for max e and . for avg e . f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π For COMM, there are no anomalies except one for (cid:96) min =6 for the maximum separation of even multipoles which isalmost insignificant with an s-value of . , hence prob-5 .20.40.60.81 COMM ( ℓ min ) COMM ( ℓ max ) ( ℓ min ) SMICA ( ℓ min ) ( ℓ max ) ( ℓ min ) Figure 5: Probabilities P t for min a , max a , avg a (in pur-ple, green and light blue, respectively) for f ( (cid:96) ) = (cid:96) ver-sus (cid:96) max or (cid:96) min as mentioned in parentheses () beside thename of the data map. Red dotted-dashed lines indicate (double dots) and (single dots).ability plots for COMM for (cid:96) max are not being presentedhere.For the other plots, see figure 7. For (cid:96) max variations,only average separations of even multipoles reveal anoma-lous ranges, which are (cid:96) max = 22 → for NILC, (cid:96) max = 18 → for SMICA, and (cid:96) max = 14 → for WMAP. Their maximum s-values are . , . and . for (cid:96) max = 26 for the three maps.For (cid:96) min variations, (cid:96) min = 6 , , and (cid:96) min =2 , , are anomalous for max e and avg e for NILC, withmaximum s-values of . for (cid:96) min = 10 and . for (cid:96) min = 6 , respectively. For SMICA, (cid:96) min = 10 shows an anomaly with an s-value of . for max o .For max e , (cid:96) min = 6 , are anomalous with maximals-value of . for (cid:96) min = 6 , whereas for avg e , forSMICA, (cid:96) min = 2 , are anomalous with maximal s-valueof . for (cid:96) min = 2 . For WMAP, (cid:96) min = 6 for max e is anomalous with an s-value of . . Also, (cid:96) min = 2 , , are anomalous for avg e for WMAP, with amaximal s-value of . for (cid:96) min = 6 . f ( (cid:96) ) = (cid:96)(cid:96) max = 22 → for avg e are anomalous for COMM,with the maximal s-value being . for (cid:96) max = 26 . (cid:96) min = 6 for max e is anomalous with an s-value of . , whereas (cid:96) min = 2 is anomalous at an s-valueof . for avg e for COMM. There are no anomalies for NILC for variations in (cid:96) max . For (cid:96) min = 6 , there isa . s-value for max e and . for avg e respec-tively for NILC. For SMICA, (cid:96) max = 22 → are anoma-lous with maximal s-value being . for (cid:96) max = 26 . (cid:96) min = 6 with . s-value for max e shows anomalousbehaviour for SMICA, while (cid:96) min = 2 , for avg e with (cid:96) min = 6 of maximal s-value of . show anoma-lous behaviour. For WMAP, there is a change, becauseit shows that (cid:96) max = 6 , for min o are anomalous withmaximal s-value of . for (cid:96) max = 6 . (cid:96) max = 10 , are anomalous for min e with maximal s-value of . for (cid:96) max = 10 . Also for WMAP, (cid:96) max = 10 → areanomalous for avg e with maximal s-value of . for (cid:96) max = 26 . (cid:96) min = 6 , are anomalous for max e forWMAP, but (cid:96) min = 6 is most anomalous with an s-valueof . . Again, (cid:96) min = 2 , are anomalous for avg e but (cid:96) min = 6 is most anomalous with an s-value of . .[See figure 8]. f ( (cid:96) ) = (cid:96) In this case, COMM does not have anomalies for (cid:96) min vari-ations. (cid:96) max = 6 → are anomalous for min e forCOMM, being maximally anomalous for (cid:96) max = 6 withan s-value of . . For NILC, (cid:96) max = 22 → areanomalous for avg e with (cid:96) max = 26 being maximallyanomalous with an s-value of . . (cid:96) min = 6 → are anomalous for max e for NILC, with a maximal s-valueof . for (cid:96) min = 10 . (cid:96) min = 2 → are alsoanomalous for avg e with a maximal s-value of . for (cid:96) min = 6 for NILC. For SMICA, (cid:96) max = 18 → areanomalous with maximal s-value of . for (cid:96) max = 26 for avg e . Also, (cid:96) min = 10 is anomalous for max o withs-value of . for SMICA. (cid:96) min = 6 , are anoma-lous for max e with (cid:96) min = 6 being of maximal s-vlaueof . , whereas for avg e , (cid:96) min = 2 , are anoma-lous with maximal s-value of . for (cid:96) min = 2 . ForWMAP, again, (cid:96) max = 18 → are all anomalous for min o , but (cid:96) max = 18 is most anomalous with an s-valueof . . Also, for WMAP, (cid:96) max = 14 → are allanomalous for avg e , but (cid:96) max = 26 is most anomalouswith an s-value of . . (cid:96) min = 2 → are anomalousfor min o with a maximal s-value of . for (cid:96) min = 14 for WMAP, whereas, for the even separations of WMAP’s (cid:96) C (cid:96) ’s, (cid:96) min = 6 with a an s-value of . is anoma-lous for max e , while, (cid:96) min = 2 → are all anomalousfor avg e , but (cid:96) min = 2 is most anomalous with an s-valueof . . [See figure 9]. We realise that on restricting the anomalous ranges as thosecorresponding to or beyond . as the s-value of a sim-ulation being errant relative to observational data, whichimplies a nearly equal to or greater than σ confidencefor an anomaly (see table 1 for all (cid:38) σ anomalies), wecan sieve out the maximally anomalous results. The mostsignificantly anomalous (cid:96) min = 8 has a maximal proba-6 .20.40.60.81 COMM ( ℓ min ) COMM ( ℓ max ) NILC ( ℓ min ) NILC ( ℓ max ) ( ℓ min )
10 15 20 25SMICA ( ℓ max ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 6: Probabilities P t for min o , max o , avg o , min e , max e , avg e (in purple, green, light blue, ochre, yellow anddark blue) for f ( (cid:96) ) = 1 versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Reddotted-dashed lines indicate (double dots) and (single dots). ( ℓ min ) NILC ( ℓ min ) NILC ( ℓ max ) ( ℓ min ) ( ℓ max ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 7: Probabilities P t for min o , max o , avg o , min e , max e , avg e (in purple, green, light blue, ochre, yellow anddark blue) for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Reddotted-dashed lines indicate (double dots) and (single dots).bility for NILC (for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π ) of . and alsofor WMAP (for f ( (cid:96) ) = (cid:96) ) of . , both for the maxi-mum separations of the respective f ( (cid:96) ) C (cid:96) ’s. These corre-spond to ≈ . σ deviant anomalies. For the average sepa-rations, the maximal probabilities are shown by NILC for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π , (cid:96) min = 8 with value of . , and alsoshown by WMAP for f ( (cid:96) ) = (cid:96) , (cid:96) min = 8 , with value . . Again for (cid:96) max = 26 , the most significantlyanomalous are NILC and SMICA, both for maximum sep-aration with probabilities of . and . respec-tively. So mainly, the maximum separation turns out to be anomalous with highest s-values (lowest p-values) for (cid:96) min = 8 and (cid:96) max = 26 . These anomalies at a signifi-cance ≈ . σ are highlighted in table 1.We further present the plots of p-values (in log-scalealong the vertical axis), which are those of the probabilities P b , and P b = 1 − P t . These plots give a better insight onthe extent to which the observational data are deviant fromthe theoretical predictions, indicated by the lowermost dipsin the plots of figure 10.7 .20.40.60.81 COMM ( ℓ min ) COMM ( ℓ max ) NILC ( ℓ min ) ( ℓ min ) ( ℓ max ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 8: Probabilities P t for min o , max o , avg o , min e , max e , avg e (in purple, green, light blue, ochre, yellow and darkblue) for f ( (cid:96) ) = (cid:96) versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Red dotted-dashedlines indicate (double dots) and (single dots). ( ℓ max ) NILC ( ℓ min ) NILC ( ℓ max ) ( ℓ min ) ( ℓ max ) ( ℓ min )
10 15 20 25 30WMAP ( ℓ max ) Figure 9: Probabilities P t for min o , max o , avg o , min e , max e , avg e (in purple, green, light blue, ochre, yellow anddark blue) for f ( (cid:96) ) = (cid:96) versus (cid:96) max or (cid:96) min as mentioned in parentheses () beside the name of the data map. Reddotted-dashed lines indicate (double dots) and (single dots). On glancing at the most essentially anomalous ranges, werealise that there is only one (cid:38) σ anomaly (shown byWMAP) : . for (cid:96) min = 6 for max e (when f ( (cid:96) ) = (cid:96) ). We present the most significant anomalies for (cid:38) . σ range (corresponding to a confidence range (cid:38) . ) inthe tables in 2 that follow and wherein the (cid:38) σ anomalyhas been highlighted.The general trend (of (cid:38) . σ anomalies) however, seemsto hold for that of avg e being anomalous for all maps for f ( (cid:96) ) = 1 , with high (above ) probability for (cid:96) min = 2 and (cid:96) max = 26 and/or . Further, for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π thereis an isolated anomaly again for avg e with (cid:96) max = 26 for WMAP. WMAP dominates these ‘highly significant’anomalies for f ( (cid:96) ) = (cid:96) , for (cid:96) max = 6 for min o (whichdoesn’t appear for other f ( (cid:96) ) ’s, at this level of signifi-cance), and (cid:96) min = 6 for max e and avg e . For f ( (cid:96) ) = (cid:96) ,COMM has an anomaly for (cid:96) max = 6 for min e , whileWMAP shows that for (cid:96) max = 22 , for avg e .An overall impression from the results is that the evenseparations show up as anomalies for most of the maps; theodd separations are seldom anomalous in as much as their8able 1: (cid:38) σ anomalies for all separations (with (cid:38) . σ results highlighted) f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π Data Map (cid:96) min or (cid:96) max Type s-value ( P t )COMM (cid:96) min = 8 M ax a . % NILC (cid:96) min = 8
M ax a . % NILC (cid:96) min = 10
M ax a . NILC (cid:96) min = 8
Avg a . f ( (cid:96) ) = (cid:96) Data Map (cid:96) min or (cid:96) max Type s-value ( P t )NILC (cid:96) min = 8 M ax a . SMICA (cid:96) min = 8
M ax a . SMICA (cid:96) min = 8
Avg a . WMAP (cid:96) min = 8
M ax a . % WMAP (cid:96) min = 8
Avg a . f ( (cid:96) ) = (cid:96) Data Map (cid:96) min or (cid:96) max Type s-value ( P t )COMM (cid:96) min = 8 M ax a . NILC (cid:96) min = 8
M ax a . % NILC (cid:96) min = 10
M ax a . maximum, minimum and average value are concerned. Be-sides, in the most significant results for anomalies as pre-sented in the table 2, it is evident that the average valueof separations among odd f ( (cid:96) ) C (cid:96) ’s, i.e., avg o is neveranomalous. For the odd separations, min o is anomalousfor (cid:96) max = 6 with a . probability only for WMAPand only for f ( (cid:96) ) = (cid:96) . The average separation amongeven multipoles ( avg e ), is always anomalous for all f ( (cid:96) ) ’s,but shows up selectively in the various data maps for therespective f ( (cid:96) ) ’s. Among the (cid:38) . σ significant results, max e also shows up only once for f ( (cid:96) ) = (cid:96) . For this case,where parity based differentiation is done before the sepa-rations are analysed, there are multiple anomalous rangesfor the four maps and for all the four f ( (cid:96) ) ’s because a widerconfidence interval is taken ( (cid:38) . σ ). The p-value plotsfor the same are presented in figure 11. In addition to checking solely for anomalous separations,we also checked if the ratio ( r i ) of the maximum to theminimum separation (as defined in (3)) of a given rangeof multipoles is anomalous. The s-value plots of the mostsignificantly anomalous results are presented in figure 12,which show that SMICA is not very problematic for sep- − − − f ( ℓ ) = ℓ ( ℓ +1)2 π ( ℓ min ) f ( ℓ ) = ℓ ( ℓ min ) − − − − f ( ℓ ) = ℓ ( ℓ min ) Figure 10: Plots of p-values ( P b ) versus (cid:96) min or (cid:96) max as in-dicated in parentheses (), for (cid:38) σ anomalies for all separa-tions. Top left: COMM ( max a ) in purple, NILC ( max a ) ingreen, NILC ( avg a ) in light blue. Top right: NILC ( max a )in purple, SMICA ( max a ) in green, SMICA ( avg a ) in lightblue, WMAP ( max a ) in ochre, WMAP ( avg a ) in yellow.Bottom left: COMM ( max a ) in purple, NILC ( max a )in green. Dotted-dashed red lines represent: ∼ σ ) (above) and . ∼ σ ) (below).aration ratios sans parity distinction. Although the otherthree maps show most anomalous ratios, yet COMM andWMAP most recurrently show anomalies in as much as all,odd and even multipole separation ratios are concerned.The anomalous ratios of highest significance in each ofthe four subplots of figure 12 are (clockwise from top) :(1) COMM for r a , f ( (cid:96) ) = 1 with an s-value of . at (cid:96) min = 22 , (2) WMAP for r a , f ( (cid:96) ) = 1 with an s-value of . at (cid:96) max = 22 , (3) SMICA for r o , f ( (cid:96) ) = 1 witha p-value of . at (cid:96) min = 6 and, (4) COMM for r e , f ( (cid:96) ) = (cid:96) with a p-value of . at (cid:96) max = 6 . The (cid:96) min and (cid:96) max ranges are similar to those whichturned up as anomalous when only separations were con-sidered . Besides, r e is anomalous at the highest signifi-cance, which agrees with most of the previous analyses thatshowed how even multipole separations are at a dissonancewith theoretical simulations. In the review article [28], the authors using the conceptof constructing symmetric and antisymmetric functions forthe temperature field (that have even and odd parity respec-tively), emphasise the notion of how a significant powerasymmetry between even and odd multipoles may be in-terpreted as a preference for a particular parity of theanisotropy pattern, referred to as ‘parity asymmetry’. Theyuse a ratio of the following two constructed measures to9 − − − f ( ℓ ) = 1( ℓ min )
10 15 20 25 f ( ℓ ) = 1( ℓ max )
10 15 20 25 f ( ℓ ) = ℓ ( ℓ +1)2 π ( ℓ max ) f ( ℓ ) = ℓ ( ℓ min )
10 15 20 25 30 f ( ℓ ) = ℓ ( ℓ max ) Figure 11: Plots for p-values ( P b ) versus (cid:96) min or (cid:96) max as indicated in parentheses (), for (cid:38) . σ anomalies for evenand odd multipole separations. Top left: COMM ( avg e ) in purple, SMICA ( avg e ) in green. Top right: COMM ( avg e ) inpurple, SMICA ( avg e ) in green. Middle left: WMAP ( avg e ) in purple. Middle right: WMAP ( max e ) in purple. Bottomleft: WMAP ( avg e ) in purple. Dotted-dashed red lines represent: ∼ σ ) (above) and . ∼ σ ) (below).Table 2: (cid:38) . σ anomalies for odd/even separations(with (cid:38) σ results highlighted) f ( (cid:96) ) = 1 Data Map (cid:96) min or (cid:96) max Type s-value ( P t )COMM (cid:96) max = 26 , Avg e . COMM (cid:96) min = 2
Avg e . SMICA (cid:96) max = 26 , Avg e . SMICA (cid:96) min = 2
Avg e . f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π Data Map (cid:96) min or (cid:96) max Type s-value ( P t )WMAP (cid:96) max = 22 Avg e . WMAP (cid:96) max = 26
Avg e . f ( (cid:96) ) = (cid:96) Data Map (cid:96) min or (cid:96) max Type s-value ( P t )WMAP (cid:96) min = 6 M ax e . % f ( (cid:96) ) = (cid:96) Data Map (cid:96) min or (cid:96) max Type s-value ( P t )WMAP (cid:96) max = 22 Avg e . WMAP (cid:96) max = 26
Avg e . test this ‘parity asymmetry’: P + = (cid:96) max (cid:88) (cid:96) = (cid:96) min cos (cid:18) (cid:96)π (cid:19) f ( (cid:96) ) C (cid:96) P − = (cid:96) max (cid:88) (cid:96) = (cid:96) min sin (cid:18) (cid:96)π (cid:19) f ( (cid:96) ) C (cid:96) (5)with f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π only and test the same only for WMAPdata for the 3, 5 and 7 year releases and fixed (cid:96) min = 2 .We test the same for six f ( (cid:96) ) ’s, i.e, f ( (cid:96) ) = P t ( r a , ℓ min ) P t ( r a , ℓ max ) P t ( r o , r e , ℓ min )
10 15 20 25 30 P t ( r o , r e , ℓ max ) Figure 12: Anomalous ratios of maximum to minimumseparations: Probabilities P t ( r i ) , i = a, o, e versus (cid:96) max or (cid:96) min as mentioned in parentheses (). Top left: COMM,WMAP for f ( (cid:96) ) = 1 in purple and green, respectively.Top right: COMM, NILC, WMAP for f ( (cid:96) ) = 1 in purple,green and light blue, respectively; NILC for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π in ochre; NILC, WMAP for f ( (cid:96) ) = (cid:96) in yellow and darkblue, respectively. Bottom left: COMM ( r e ), SMICA ( r o ),WMAP ( r e ) for f ( (cid:96) ) = 1 in purple, green and light blue,respectively; COMM ( r e ) for f ( (cid:96) ) = (cid:96) in ochre; WMAP( r o ) for f ( (cid:96) ) = (cid:96) in yellow. Bottom right: COMM( r o ), SMICA ( r o ), WMAP ( r o ), COMM ( r e ), NILC ( r e ),SMICA ( r e ), WMAP ( r e ) for f ( (cid:96) ) = 1 in purple ( + ), green( × ), light blue, ochre, yellow, dark blue and red, respec-tively; WMAP ( r o ) for f ( (cid:96) ) = (cid:96) in black; COMM ( r e ),WMAP ( r o ) for f ( (cid:96) ) = (cid:96) in purple ( (cid:78) ) and green ( (cid:79) ),respectively. Red dotted-dashed lines indicate (doubledots) and (single dots). , (cid:96) ( (cid:96) +1)2 π , (cid:96), (cid:96) , (cid:96) , (cid:96) and for four maps [Planck 2018 mapsof COMM, NILC, SMICA and WMAP 9 year ILC (1)].We also vary (cid:96) min from → and (cid:96) max from → ,both in steps of ∆ (cid:96) = 2 . And also our definition of (cid:96) max is slightly different: we denote (cid:96) max = 6 for (cid:96) (cid:15) [2 , ,etc. When (cid:96) min is varied, (cid:96) max is fixed at , i.e, the range10 min ≤ (cid:96) ≤ is considered. Whereas, when (cid:96) max isvaried, (cid:96) min is fixed at .Comparing the ratio P + P − from these observed data mapswith the same evaluated from simulations generatedfrom the theoretical best fit to Planck, we perform our fa-miliar p-value analysis as discussed in (3). To reiterate,p-value = Number of simulations having P + P − lesser thanthat from data. A low P + P − indicates an odd-parity prefer-ence, which has been shown by several authors ( [10], [42],[29], [3]). This translates to a low p-value, and the lowerthe p-value, the more significant is the anomaly. Worksof [10], [42], [29], [3], show that f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π is stillproblematic for the low multpoles < .Our analysis reveals that as we approach higher powersof (cid:96) in f ( (cid:96) ) , the p-values rise much above the ∼ σ ) and . ∼ σ ) red lines, as can be seen for f ( (cid:96) ) = (cid:96) , (cid:96) for both (cid:96) min and (cid:96) max variations; although for (cid:96) min varia-tions, the anomalies completely disappear for these f ( (cid:96) ) ’s,which is why we have not tested f ( (cid:96) ) = (cid:96) , (cid:96) in our anal-ysis of separations (3, 4.1, 4.2). All the p-value plots are infigure 13. These plots indicate that the most significant p-value inboth cases ( (cid:96) min and (cid:96) max variations) is that of WMAP for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π , given by (cid:96) min = 2 with p-value = 0 . and (cid:96) max = 24 with p-value = 0 . , respectively . Ourlowest p-values are different from [28], because we havetaken equal number of even and odd multipoles for eachcomputation of P + P − , by ensuring ∆ (cid:96) = 2 , whereas [28] haveapparently taken ∆ (cid:96) = 1 . p-values of lower significancefor WMAP are (cid:96) max = 24 , (for f ( (cid:96) ) = 1 ), (cid:96) max =22 , (for f ( (cid:96) ) = (cid:96) ), (cid:96) max = 24 (for f ( (cid:96) ) = (cid:96) ), (cid:96) max =6 (for f ( (cid:96) ) = (cid:96) , (cid:96) ) and (cid:96) min = 2 (for f ( (cid:96) ) = 1 , (cid:96), (cid:96) ).Planck maps also show a highly recurrent anomalous (cid:96) max = 6 (for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π , (cid:96) , (cid:96) , (cid:96) ), (cid:96) max = 8 (for f ( (cid:96) ) = 1 , (cid:96) ) and (cid:96) min = 2 (for f ( (cid:96) ) = 1 for COMM, f ( (cid:96) ) = 1 , (cid:96) ( (cid:96) +1)2 π , (cid:96) for NILC and SMICA, and f ( (cid:96) ) = (cid:96) for NILC). COMM, however does not show anomalies for (cid:96) max variations for f ( (cid:96) ) = (cid:96) . NILC reveals an anomalous (cid:96) max = 24 for f ( (cid:96) ) = (cid:96) .As we approach higher powers of (cid:96) as aforementioned,the p-values tend to recede back into our – confi-dence range and the anomalies are obscured. (cid:96) min = 2 ismost recurrently anomalous, which indicates that as soonas we change the lower limit to exclude the quadrupole (oruse higher powers of the multipole: f ( (cid:96) ) = (cid:96) , (cid:96) ), theanomalies cease to exist in the case of (cid:96) min variations forall four data maps, and this is congruent with earlier stud-ies that have pointed out how anomalously low is the powerassociated with the quadrupole ( [37], [36], [4], [24], [39]).However, as opposed to Planck maps which show this ef-fect with low significance, WMAP shows this anomaly athigh significance.This analysis gives us a cue to use f ( (cid:96) ) with higher pow-ers of the multipole to alleviate the problems of the anoma-lous parity asymmetry. This is also the reason why we haveused a set of such four simple f ( (cid:96) ) ’s while trying to findanomalous separations in f ( (cid:96) ) C (cid:96) ’s as our ‘tested powerspectra’ because in a similar vein, [12] describes how usinga function as a multiplicative factor to the temperature fieldcan be used to explain some large-scale anomalies. We have proposed a new technique for investigatinganomalies in the CMB temperature angular power spec-trum for its nearest neighbour separations and found thatsuch anomalous separations arise in various multipoleranges, across all four data maps, with a variation of thelower or upper limit of the multipole range or a change inan f ( (cid:96) ) . In this section, we will try to deliberate on someobvious trends.Reasonably, since all four maps, namely, COMM, NILC,SMICA, and WMAP, have been obtained by using differ-ent foreground removal methods, the levels of foregroundresiduals in these maps are quite different ( [31], [2], [18]).These systematic differences are clearly visible if we sub-tract any two of these maps at low resolution. Therefore,since each cleaned map is obtained by using different fore-ground filters, the residual foreground in each of them isexpected to be different. This can be a possible cause of dif-ferent results obtained from differently cleaned maps [36].The p-value plots in figure 10, show that for all separa-tions, for (cid:96) min = 8 and (cid:96) max = 26 , the maximum sep-aration from theoretical predictions is most significantlygreater than that from observations for all f ( (cid:96) ) ’s except f ( (cid:96) ) = 1 . For odd and even parity based distinctions,p-value plots in figure 11, reveal how (cid:96) min = 6 and (cid:96) max = 22 are the most anomalous because in these (cid:96) ranges, the maximum separation among f ( (cid:96) ) C (cid:96) ’s for even (cid:96) ’s from theoretical simulations is mostly larger than ob-served data. P t = 99 . is the highest probability amongall the p-values in all our analyses, and is exhibited by (cid:96) min = 8 for all (no parity distinction) multipole separa-tions for f ( (cid:96) ) = (cid:96) ( (cid:96) +1)2 π for NILC and f ( (cid:96) ) = (cid:96) for WMAP .Initially, a low quadrupole anisotropy was reported ([37], [11]), and shown to be a part of the general trend ofparity asymmetry [28]. We too, observe that lower (cid:96) min ’sseem to be more anomalous and after a certain (cid:96) min valueis crossed, the anomalies disappear, most conspicuouslyfor our analysis of parity asymmetry in section 7 .Parity based distinction reveals that the even multipoleshave a more significant tendency to keep their maximumseparations much smaller than what is theoretically pre-dicted . Also a noticeable feature of the Planck maps(COMM, NILC, SMICA) is that without any parity baseddistinction, the minimum separation does not usually showup as an anomaly. However, when the odd and even multi-poles are considered separately, the minimum separationfor odd multipoles begins to show up as being anoma-lously lower than theoretical simulations (especially for f ( (cid:96) ) = 1 ), whereas, the maximum of even separationsis almost always anomalously low and at relatively muchhigher significance. This anomaly disappears for higherpowers of (cid:96) in f ( (cid:96) ) (except for f ( (cid:96) ) = (cid:96), (cid:96) in WMAP).This implies that the very low maximum separations ofeven multipoles override the effect of possibly low oddmultipole separations in f ( (cid:96) ) C (cid:96) ’s when all multipoles arestudied without any parity based distinctions.Our analyses here clearly spell out that for all multipoleseparations taken without parity based distinction, f ( (cid:96) ) =1 is not excluded at as high a significance as the others. Only for NILC, and WMAP, f ( (cid:96) ) = (cid:96), (cid:96) ( (cid:96) +1)2 π are excluded − − f ( ℓ ) = 1( ℓ min ) f ( ℓ ) = ℓ ( ℓ +1)2 π ( ℓ min ) f ( ℓ ) = ℓ ( ℓ min ) f ( ℓ ) = ℓ ( ℓ min ) f ( ℓ ) = ℓ ( ℓ min ) f ( ℓ ) = ℓ ( ℓ min ) − − f ( ℓ ) = 1( ℓ max ) f ( ℓ ) = ℓ ( ℓ +1)2 π ( ℓ max ) f ( ℓ ) = ℓ ( ℓ max ) f ( ℓ ) = ℓ ( ℓ max ) f ( ℓ ) = ℓ ( ℓ max ) f ( ℓ ) = ℓ ( ℓ max ) Figure 13: p-value versus (cid:96) min or (cid:96) max as indicated in parentheses () for mitigating parity asymmetry with f ( (cid:96) ) (section7). Coloured lines are representative of the four maps : COMM (purple), NILC (green), SMICA (light blue), WMAP(dark yellow). Dotted red lines represent: ∼ σ ) (above) and . ∼ σ ) (below). Higher powers of (cid:96) in f ( (cid:96) ) tendto mitigate the anomalies better. at the . σ significance level for max a in the multipolerange ≤ (cid:96) ≤ , which indicate strong deviations fromthe traditional Sachs-Wolfe plateau and from f ( (cid:96) ) = (cid:96) , re-spectively. For odd and even multipole separations treateddistinctly, WMAP excludes f ( (cid:96) ) = (cid:96) with . σ signifi-cance in the range ≤ (cid:96) ≤ for the maximum evenmultipole spectrum separation . Thus, f ( (cid:96) ) = (cid:96), (cid:96) , (cid:96) ( (cid:96) +1)2 π are most significantly excluded for all separations, while f ( (cid:96) ) = 1 is excluded at relatively lower significance; butfor the even multipole spectrum, f ( (cid:96) ) = (cid:96) is excluded at ahigher significance relative to f ( (cid:96) ) = 1 , (cid:96) , (cid:96) ( (cid:96) +1)2 π .The anomalous behaviour of specific multipole rangesand the even multipole separations is consistent with theanalysis of separation ratios in section (6). We must notehowever, that all these results are specific to certain maps asaforementioned and all the four maps show sundry anoma-lies with high significance. Hence, this precludes the pos-sibility of such a clear conclusion about the rejection of f ( (cid:96) ) ’s for separations in the temperature power spectrum.Besides, we need a more succinct characterisation of whyhigher order f ( (cid:96) ) ’s should work for mitigating the parityasymmetry anomaly (7). A study of the physical origin ofthese anomalies may constitute a work in the future. Acknowledgements
We acknowledge some initial use of the HEALPix [25]software package which is publicly available ( http://healpix.sourceforge.io ). Our analyses are basedon observations obtained with Planck ( ), an ESA science mission with instru-ments and contributions directly funded by ESA MemberStates, NASA, and Canada. We acknowledge the use of theLegacy Archive for Microwave Background Data Analysis(LAMBDA), part of the High Energy Astrophysics ScienceArchive Center (HEASARC). HEASARC/LAMBDA is aservice of the Astrophysics Science Division at the NASAGoddard Space Flight Center. MIK would like to thank Uj-jal Purkayastha for immense help in guiding him throughthe basics of HEALPix relevant to this project.
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