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PHYSICAL REVIEW D 70 043515 2004 or non Gaussianity at 2 S6 S8 Craig J Copi and Dragan Huterer Department of Physics Case Western Reserve University Cleveland Ohio 44106 7079 USA Glenn D Starkman Theory Division CERN Geneva Switzerland and Department of Physics Case Western Reserve University Cleveland Ohio 441067079 USA Received 21 October 200339 published 17 August 2004 We propose a novel representation of cosmic microwave anisotropy maps where each multipole order 6 is represented by 6 unit vectors pointing in directions on the sky and an overall magnitude These multipole vectors and scalars transform as vectors under rotations Like the usual spherical harmonics multipole vectors form an irreducible representation of the proper rotation group SO3 However they are related to the familiar spherical harmonic coefficients am in a nonlinear way and are therefore sensitive to different aspects of the cosmic microwave background CMB anisotropy Nevertheless it is straightforward to determine the multipole vectors for a given CMB map and we present an algorithm to compute them A code implementing this algorithm is available at httpwwwphyscwrueduprojectsmpvectors Using the Wilkinson Microwave Anisotropy Probe WMAP fullsky maps we perform several tests of the hypothesis that the CMB anisotropy is statistically isotropic and Gaussian random We nd that the result from comparing the oriented area of planes de ned by these vectors between multipole pairs 2S61 6ZS8 is inconsistent with the isotropic Gaussian hypothesis at the 994 level for the internal linear combination map and at 989 level for the cleaned map ofTegmark et al Aparticular correlation is suggested between the 6 3 and 6 8 multipoles as well as several other pairs This effect is entirely different from the now familiar planarity and alignment of the quadrupole and octupole while the aforementioned is fairly unlikely the multipole vectors indicate correla tions not expected in Gaussian random skies that make them unusually likely The result persists a er account ing for pixel noise and a er assuming a residual 10 dust contamination in the cleaned WMAP map While the de nitive analysis of these results will require more work we hope that multipole vectors will become a Multipole vectors A new representation of the CMB sky and evidence for statistical anisotropy valuable tool for various cosmological tests in particular those of cosmic isotropy DOI 101103PhysRevD70043515 I INTRODUCTION A great deal of attention is currently being devoted to examining the power spectrum of the cosmic microwave CMB 39 39 ATQT ex tracted from the Wilkinson Mcrowave Anisotropy Probe WMAP 174 and other CMB data 5711 Decomposing the temperature in spherical harmonics AT 0 f 2 ammo 1 6m and deducing the angular power spectrum 6 2 2 1mlalml 2 Ct as a fmetion of 6 allows cosmologists to t the parameters of cosmological models to merecedented accuracy possibly even probing the physics of the in ationary epoch Similarly the power spectrum of t e p izatiun cross correlation fmetion is teaching us about the physics of the reionization of the universe presumably by the rst genera tion of stars and the power spectrum of the temperature galaxy crosscorrelation fmetion is teaching us about the 155079982004704043515132250 70 0435151 PACS numbers 9880Es 0230Px 9575Pq statistical distribution of matter in the universe But does the only physics lie in the angular power spectra Is the sky statistically isotropic so that any variation in the values of the individual multipole moments am with xed 6 repre sents only statistical uctuations or could there be subtle correlations between the am If the sky is statistically iso tropic is it Gaussianiare the am of each xed 6 drawn from a Gaussian distribution of variance that is only 6 de pendent Are there other interesting deviations from the sim plest picture In the standard in ationary cosmology the answer to the question just posed is that in the linear regime ie at low 6 the am are realizations of Gaussian random variables of zero mean with variances that depend only on 6 statistical isot ropy This paradigm is so strongly believed because of both considerable observational evidence and considerable theo retical prejudice that relatively little though some eg 12 attention has been paid to searches for deviations from statistical isotropy In is aper we set out to search for one particular deviationispecial directions on the sky We do this by rst con uucu39n from each multipole moment 6 win ammo lt3 2004 The American Physical Society COPI HUTERER AND STARKMAN of the CMB sky a set of 6 unit vectors 136quoti1 6 and a scalar A that completely characterize that multipole We then examine the correlations between pairs of such sets of vectors dww and 6Z39IZ comparing them with Monte Carlo simulations of CMB skies with statistically iso tropic Gaussian random am If the sky is statistically iso tropic with Gaussian random am atsz m Cg gg 6mm Since the 16 quot9 depend only on the 11am 13M1 quot0 and 662quotZ should be uncorrelated for 61 62 We constructed these vectors for 2lt6lt8 for a set of fullsky maps including the WMAP internal linear combina tion ILC 1 and the WMAP map as cleaned by Tegmark et al 13 We applied four statistical tests to the set of vec tors from these fullsky maps We nd that one of the tests is inconsistent with the hypothesis of statistical isotropy and Gaussianity at the 99 con dence level This work comple ments recent work by Eriksen el al 14 looking at north south asymmetries in N point functions work by Park 15 looking at genus curves work by Hajian el al 16 using the socalled K4 test 17 which nds violations of statistical isotropy for 15 6 45 and work by Vielva el al 18 using the spherical mexican hat wavelet technique where they found a strong signal for nonGaussianity II MOTIVATION FOR ANEW TEST OF STATISTICAL ISOTROPY AND GAUSSIANITY Tests of nonGaussianity as opposed to statistical isot ropy have a long and rich history Motivation for those tests originally came from the realization that nonGaussianity is a signature of structure formation by topological defects 19 while in ation predicts Gaussian CMB anisotropies Subse quently it has been realized that even if in ation seeded the structure in the universe CMB nonGaussianity may be present as a signature of features in the in ationary model 20724 Finally latetime processes in the universe will in duce nonGaussianity on small scales 25730 The tests of nonGaussianity include studies of the bispectrum and skew ness 31737 trispectrum 3839 Minkowski functionals and the genus statistic 1540744 spherical wavelets 184546 a combination of these 47751 and many other methods 5275 Of particular interest was the claim for nonGaussianit in the Cosmic Background Explorer COBE fouryear data 56 but this was shown to be an artifact of a particular known systematic 57 Nevertheless efforts to test the Gaussianity of the CMB continue and most though not all eg 1558 have so far given results entirely in agreement with the Gaussian hypothesis As these previous studies have shown it is a challenge to test such fundamental assumptions as statistical isotropy and Gaussianity without theoretical direction on what deviations to expectithey can be violated in a very large number of ways each of which could easily be hidden from the test that was actually performed An instructive example is the effects on the CMB of any nontrivial topology of the universe for example a universe which is a threetorus If the length scale of cosmic topology for example the length of the smallest nontrivial closed curve is suf ciently short this will mani PHYSICAL REVIEW D 70 043515 2004 fest itself in various twopoint temperaturetemperature cor relations such as the socalled circlesinthesky signature 59 if there are closed paths shorter than the diameter of the last scattering surface then the last scattering surface will selfintersect along circles These circles can be viewed by an observer from both sidesifrom one side in one direction on the sky and from the other side in some other direction The temperature as a function of location around the circle as seen from the two sides will be very strongly correlated One can therefore search for such pairs of circles A de nitive direct search is currently being conducted by the original proponents of the signature However for us this example serves to show precisely why it is so dif cult to perform a comprehensive test of statistical isotropy and Gaussianity The topologyinduced temperature correlations are strong only on or very near the matched circle pairs thus the mul tipole coef cients am and other statistics that are weighted averages over the entire sky would be poor tools for search ing for such circles Thus testing these phenomena is in part a matter of continually searching for preferably physically motivated ways in which they manifest The CMB data itself may provide motivation for search ing for deviations from the standard in ationary predictions of statistical isotropy and Gaussianity especially at large angular scales An absence of large angular scale correlations in the CMB sky relative to the in ationary prediction was rst noted by COBE Differential Microwave Radiometer DMR in their rst year data 60 which showed what was reported as an anomalously low quadrupole C2 Because the cosmic variance in the quadrupole is quite large it was widely dismissed as a statistical anomaly The result per sisted and was strengthened by the COBE DMR fouryear data 61 The recent WMAP analysis shows a marked ab sence of power on scales extending from 600 to 1800 to an extent that cannot be explained solely by a low quadrupole 4 Note that the estimator used by the WMAP team has been shown to be nonoptimal when applied to incomplete maps of the sky When an alternative estimator is applied 1362 or a fullsky map is analyzed 1 the discrepancy becomes less signi cant An absence of power on large scales is expected in some topologically nontrivial universes In a compact universe there is a spectral cutoff of long wavelength modes leading to a suppression of power near this cutoff One method of looking for such a cutoff is the circlesinthesky signature as noted above In general such compact topologies would lead to special directions in the universe To search for special directions we need a method of de ning our direc tions Our de nition as discussed in the next section is to decompose the 6th multipole into 6 unit vectors these vec tors are then studied to search for peculiar alignments These vectors contain the full information of the am but encode it in a different way that allows one to more easily look for special directions In particular the components of these vec tors are nonlinear combinations of the am for a xed 6 III DEFINING SPECIAL DIRECTIONS One attempt to look at the statistical isotropy of the CMB on large scales was the analysis of the quadrupole and octu 0435152 MULTIPOLE VECTORS A NEW REPRESENTATION OF pole moments of the WMAP sky by de Oliveira7Costa et al 63 They found that the quadrupole was unusually small and that the octupole was unusually planar and unusually aligned with the quadrupole They identi ed an axis with each multipole by nding for each 6 the axis n4 around which the angular momentum dispersion AT A ltTn6 is maximized Here Imam are the spherical harmonic co ef cients of the CMB map in a coordinate system with its 2 6 L2 AT A 6 A Tn6gtm millimm lz 4 axis in the n direction They found that the n2 and the 13 directions 12 70114550526508424 13 70257850420708698 5 are unusually aligneditheir dot product is 09838 This has only a l in 62 chance of happening if n2 and 13 are uncor related and the dot product is uniformly distributed on the sky De Oliveira7Costa el al 63 point out that these values of n2 and 63 could be explained by a universe which has a compact direction parallel to n2 and 13 and of length ap proximately equal to the horizon radius but this is ruled out by other tests including the absence of matched circles in these directions It has been noticed that the quadrupole and the octupole in the cleaned WMAP skies remain dominated by a hot and a cold spot in the Galactic planewne in the general direction of the Galactic center and the other in the general direction of the molecular cloud in Taurus This raises the possibility that the observed correlation is dominated by foreground contamination One would like therefore to examine in more detail the correlations between the am corresponding to pos sible preferred directions or correlations in directions be tween the various multipo es The question then is how best to associate directions with the CMB multipoles De Oliveira7Costa el al 63 associ ated only one direction with each multipole corresponding to two real degrees of freedom whereas the am of a given 6 have 26 1 real degrees of freedom The 6 th multipole f in the multipole expansion of a function fQ on a sphere f03 mng ammo 6 can be fully represented by a symmetric traceless rank6 tensor F117 M ikl23 Such a tensor can readily be constructed from the outer product of 6 unit vectors 136quot and a single scalar A Strictly speaking these are headless vectors ie points on the projective twosphere The sign of each vector can always be absorbed by the scalar The sign of the scalar takes on physical signi cance when we de ne a PHYSICAL REVIEW D 70 043515 2004 convention for the multipole vectors such as that all of them point into the northern hemisphere A Vector decomposition The correspondence between these vectors and the usual multipole coef cients can readily be seen for a dipole A dipole de nes a direction in spaceithe line along which the dipole lies The standard correspondence is A l A A Ylyoaz Ylytlai xiiy 7 v3 Thus 1 2 11quot Y1m 0 m1 A1z 1391651391131 1sin 6 cos lt15 sin 6sin cos 6 EA111391 8 where is the radial unit vector in spherical coordinates For a real valued function the vectors components are found to be 0171 aie1gt 051391 airfigt Ull39la10gt 9 and A1z1391 which can then be used to construct the unit vector 13 To extend this to the 6th multipole we want to write heuristically that 6 2iaimYmQ Amli39l quot lw39m l lt10 m for each of the 6 directions given by This cannot be quite right since the product of 6 vectors would contain compo nents not only of angular momentum 6 but also of angular momenta 67 2 6 74 etc However a simple power count ing shows that once the reality conditions have been im posed on am 6 unit vectors and a scalar contain the same number of degrees of freedom as does am namely 26 1 real degrees of freedom We therefore expect that the components of lower angular momentum found in the right hand side of Eq 10 are not independent We shall see this explicitly in Eqs 13 and 15 below and more elegantly in Sec III B For now let us treat Eq 10 as motivation and proceed Instead of solving Eq 10 directly for all 12AMquot we peel off one vector at a time nding rst a vector 0 with components riff and a rank 671 symmetric traceless tensor am We can think of i1 as running over x y and z or more conveniently over 1 0 and 1 Similarly we can write a 39 as a 3gtlt3gtlt A X3 671 terms matrix affilvlll39 however this hides its traceless symmetric nature and makes it appear that 1a has far more independent degrees of freedom than it actually does It is therefore more instructive to write the 2671 independent components as 15631 withm 7671 671 0435153 COPI HUTERER AND STARKMAN We repeat this procedure recursively on the remaining symmetric traceless tensor from the previous step Thus we next peel from aw a vector 6a and a rank 2 sym metric traceless tensor aw and repeat until we have found the full set of vectors u gtili 1 The scalarl is found in the last step when the second to last vector NH is peeled off In this case the remaining symmetric traceless tensor adj is rank 1 and is just the product of the nal unit vector 1 and the scalarl To apply the recursive procedure outlined above we use the following recursion relation to peel off one vector YLJ39Yg Lm jC mY mD mY 2m 11 for 101 where Clt gtmgtxi W 0 477 2171211 Genoai W 1 877 2171211 Mambai W 0 477 2173x2171 Damni 1 877 2173x2171 For a given we peel off a vector using 671 1 A a quot1271 WMYM 120571 121 v 1a lrY1JY lr 672 i E bgfilwhzm39 13 m 7 72 and that ll 11 14 The second term on the righthand side of Eq 13 is re quired to guarantee that this rank 7 1 tensor is traceless In other words it subtracts off the trace We can see this is necessary from the Y 52quot term in the recursion relation 1 l The presence of this 2 term is as anticipated in the dis cussion following Eq 10 Plugging in the recursion relation 11 yields 4 1 coupled quadratic equations that must be solved for the 4 l unknowns 135 33 and 17 From them we nd 71 PHYSICAL REVIEW D 70 043515 2004 l iii l T FIG 1 Color online An image of the sky as decomposed into the 278 multipole moments based on the rst year WMAP results 1 as cleaned by Tegmark et a 13 Shown are the Emanggm and the vectors calculated for these multipoles The vec tors are drawn as sticks since they de ned only up to a sign thus they are headless vectors See httpwwwphyscwruedu projectsmpvectorsfor a full sized color picture 1 am 21 cg mhjyhggl m 21l equations 1 1 1 2 1 bm TFZI DJ M Di will 7 21 3 equations 15 WWI 1 1 equation These equations are easily solved numerically though see Appendix A Notice that again as anticipated below Eq 10 the com ponents bi are not relevant for further calculations they are functions of the 651 and ag l m l and are not indepen dent This means that the vectors v m we calculate are indeed unique as claimed The general procedure now follows from the given am we construct 6a and a359 We continue using a fffm to nd 13 M and 52553 This is repeated until we nd agj z which gives the nal two vectors ax and 131 The result of this procedure for the CMB is shown in Fig l B Alternative derivation of vectors In practice we have implemented the procedure described above Appendix A and solved the set of equations 15 for the analysis we have performed A mathematically more so phisticated decomposition procedure that leads to the same set of vectors without the need to calculate the bm begins by recognizing that fg0gtFEf il i2o o i JEF f 0 16 Z A l1fgt1ljlgt2 gt i1 i2 ie Here 5 is again a radial unit vector there is an implicit sum over repeated indices i1 ig and the square brackets represent the symmetric tracefree part of the outer product For a general symmetric tensor SOT 15Squot1 W 0435154 MULTIPOLE VECTORS A NEW REPRESENTATION OF I 2 W 272k I SH 12E 2 71V 26 ko k k i2kl x 51112 gzkrrlzkslzk 39quot1F1quot39sz xgprpz39ngpzkrrpzk 17 where there is an implicit sum over the repeated indices p1 1 and denotes symmetrization of the enclosed indices 1 T 1quotquotE 2 T wlt1gtquotquotwltgt 18 6 my U is the group of permutations of the numbers I The particular combination 1 011E 11 12ee 19 simpli es considerably because a b 6amp1 l 2 1221 7 6 272k O Tango le 6 1 Z5 2k7112k 12k16140 20 More importantly it is easily calculable because of the recur sion relation g11glg1gt 11 lz gigmu 21 Similarly the individual 13 quot can be peeled off one by one by recursion 7 1 1 F53vg a5 7 1 2 2 71v hrvf hafgwii Alt vf vf vf 1 22 assuming that the can be calculated But these are easily represented as integrals over the sky just as are the am I 11quotquot 212 AT0 7 11 F 4w2 2 fSZ T 0 mo 23 Alternatively there is an explicit analytic relation between the Ym and the 01quot so that F11 can be expressed in terms of am Thus given F g calculated from the sky Eq 23 is a sequence of 26 l coupled quadratic equations for the 6 quot and1W analogous to Eq 13 in which the 17 quot have been eliminated PHYSICAL REVIEW D 70 043515 2004 IV SOURCES OF ERROR AND ACCURACY 1N DETERMlNlNG THE MULTIPOLE VECTORS With the actual fullsky CMB maps such as those that we use the main source of error is pixel noise which accounts for the imperfections in measured temperature on the sky Pixel noise depends on a variety of factors one of which is the number of times a given patch has been observed Pixel noise for WMAP is reported as Tm 00 N obs where To is noise per observation and Nobs is the number of observations per given pixel 64 For the WMAP Vband map the re ported noise per pixel is To 3ll mK and the variation in the number of observations of each pixel is moderately small Nevertheless it is important to account for the inho mogeneous distribution of pixel noise as we do in our full analysis in Secs V and VI where we nd that the inhomo geneity of the pixel noise does not signi cantly change the main results For the purposes of illustrating the accuracy in determining the multipole vectors however we assume a homogeneous noise with the measured mean value of N obs 490 observations per pixel The calculation of the pixel noise is straightforward For equalarea pixels as in HEALPIX 65 subtending a solid angle Opix and assuming a fullsky map we have am I dmzmomm dpiXE momma lt24 0 so that using 02 EATOZconst we have mangt3 opimeE minim039 Q 0 gtltATQATQ 20303544 5mm 25 Using the Vband map parameters the pixel noise for NSIDE512 HEALer resolution is Q ixa39pix27 X 10 4 mK We adopt this quantity as an estimate of pixel noise for all maps we use A Accuracy in determining the multipole vectors We would like to nd how accurately the multipole vec tors are determined To do this we add a Gaussian distributed noise with standard deviation rum to each am aim aimtAO iml 26 0435155 COPI HUTERER AND STARKMAN 15 2 A0 or A4 degrees A0 or A degrees I 15 20 01 1 10 100 12 Galm QPix 6 pix FIG 2 The accuracy in 6 lled circles and qS empty squares of a chosen multipole vector as a function of noise added to the am Both the mean value and scatter in the shift of the angles are shown The xaxis value of unity corresponds to the scatter in the am due to WMAP V band pixel noise where Nu02 denotes a Gaussian random variate with mean u and variance 02 We repeat this many times in order to determine the distribution in the noiseadded multipole vectors and their spherical coordinates 6 15 The mean and standard deviations of 6 and 15 as a func tion of 0am which is in units of WMAP V band pixel noise Qpixapix are shown in Fig 2 The two panels show the effects on the 6 2 and 6 8 vectors Note that both bias and scatter in 6 and 15 can be read off this gure It is clear that the vectors are not extremely sensitive to the accuracy in the am and are determined to about i 10 for noise which is of the order of the pixel noise If the noise is much larger however atng 10 Qpixapix the accuracy in multipole vectors deteriorates to the point that they are probably not useful as a representation of the CMB anisotropy Accurate determination of 6 and 15 is expected to be especially impor tant for higher multipoles where the number of vectors is large V TESTS OF NONGAUSSIANITY WITH MULTIPOLE VECTORS Now that we have developed a formalism to compute the multipole vectors we would like to test the WMAP map for any unusual features Clearly there does not exist a test that PHYSICAL REVIEW D 70 043515 2004 simply checks for the weirdness of any particular repre sentation of the map in this case the multipole vectors We can test only for features that we specify in advance Here our general goal is to test the statistical isotropy of the map and search for any preferred directions Our motivation to devise and apply tests of statistical isot ropy comes from ndings that the quadrupole and octupole moments of WMAP maps lie in the plane of the Galaxy as discussed previously One could in principle extend this and ask whether any higher multipoles lie preferentially in this or any other plane and devise a statistic to test for this alignment Clearly the number of tests one can devise is very large Here we would like to be as general as possible and choose tests that suit our multipole vector representation We choose to consider the dot product of multipole vectors be tween two different multipoles as described below A Vector product statistics A dot product of two unit vectors is a natural measure of their closeness One test we consider is dot products between all vectors from the multipole 61 with those from the multi pole 6 2 Since our vectors are really sticks ie each vec tor is determined only up to a ip 13 gt 6 we always use the absolute value of a dot product Furthermore motivated by the fact that the WMAP quadrupole and octupole are located in the same plane and that their axes of symmetry are only about 10 from each other 63 we also consider using the cross products of multipole vectors of any given multi pole if the vectors of 61 and 6 2 lie in a preferred plane their respective cross products are oriented near a common axis perpendicular to the plane Dot products of these two cross products would then be near unity We generalize this argument and make the following four choices for our statistic which we shall call S For any two multipoles 61 and 6 2 we consider the following 1 Dot products of multipole vectors I13 1 i13 2 jl where 619E62 z 16 is the ith vector from the 61 multi pole and z 26 is the jth vector from the 6 2 multipole We call this statistic vectorvector For a given 61 and 6 2 there are clearly M 616 2 distinct products This statistic tests the orientation of vectors 2 Dot products 391 516103 2 jgtltl MMIr 26 X13 26 where z 16 comes from the 61 multipole and z W 139 and z 26 from the 6 2 multipole We call this sta tistic vectorcross For a given 61 and 6 2 and j k there are M 6 16262 12 distinct products This statistic tests the orientation of a vector with a plane 3 Dot products Iz who 140113 26 X1 Hymn 1 50 1139l 526 52gtm where z 16 and z 16 come from the 61 multipole and z 26 and 13 MW come from the 62 multipole We call this statistic crosscross For a given 61 and 6 2 and ti j and kv m there are M61611626214 distinct prod ucts This statistic tests the orientation of planes 4 Dot products Iz WW1 14011 mm 2 ml where z 16 and z 16 come from the 61 multipole and 0435156 MULTIPOLE VECTORS A NEW REPRESENTATION OF Pick statistic S a Calculate the M values for the data 8 Rank order LWAP in MO to nd the these MC to get Run distribution for L e d For each 1112 Calculate LWMAP c With all 2112 MC Q distribution for nal probability VI FIG 3 A owchart of the algorithm we apply in order to extract the likelihood of the statistic S The lower case letters such as a refer to the itemized points in Sec V B where more information can be found about the step Also V C and refer to the sections in the paper where the details of these boxes can be found 13 2 10 and u 52 come from the Q multipole We call this statistic oriented area For a given 1 and Q and ti j and and kv m there are M 1 11 2 2714 distinct products This statistic tests the orientation of areas Notice that it is similar to the previous test but the cross products are unnorrnalized B Rankorder statistic Having computed the statistic in question we would like to know the likelihood of this statistic given the hypothesis that the am are statistically isotropic The most straightfor ward way and possibly the only reliable way to do this is by comparing to Monte Carlo MC realizations of the statisti cally isotropic and Gaussian random am To be explicit we provide the algorithm for computing the rankordered likeli hood and also illustrate it in the owchart Fig 3 a For 1 and Q xed a statistic S will produce M num bers dot products We will use S to test the hypothesis that the multipole vectors come from a map that exhibits statisti cal isotropy and Gaussianity Calculate the M numbers for this statistic for the WMAP map b To determine the expected distributions for these M numbers we begin by generating 100 000 Monte Carlo Gaussian isotropic maps in other words we draw the coef cients am by assuming agmafmC56H6mml We are assuming statistical isotropy and Gaussiarlity and this is the hypothesis that we are testing We add a realization of inho mogeneous pixel noise consistent with WMAP s Vband noise to each MC map For each MC realization we then compute the multipole vectors for multipoles 1 and Q and M dot products of the statistic S Because the vectors of any particular realization do not have an identity e g we do not know which one is the fth vector of the 8 multipole and neither do the dot products we rankorder the M dot products At the end we have M histograms of the products each having 100 000 elements 0 We would like to know the likelihood of the M prod ucts computed from a WMAP map To compute it we use a likelihood ratio test which in the case of a single dot product PHYSICAL REVIEW D 70 043515 2004 of vectors would be the height of the histogram for the value of the statistic S corresponding to WMAP relative to the maximum height Since we have M histograms the likeli hood trivially generalizes to M i N jWMAP wmpi H N j 1 jmax 27 where N jme is the ordinate of the jth histogram corre sponding to WMAP s j th rankordered product N Lmax is the maximum value of the j th histogram and the product runs over all M histograms 1 Now that we have the WMAP likelihood we need to compare it to typical likelihoods produced by Monte Carlo realizations of the map An alternative approach computing the expected distribution of the likelihood from rst principles would be much more dif cult since one would have to explicitly take into account the correlations betweenM products To do this we generate another 50 000 Gaussian random realizations of coef cients a 121m and a152 and compute the multipole vectors and the statistic S for each realization e We rankorder the likelihood WMAP among the 50 000 likelihoods from MC maps to obtain its rank R1112 f Finally we go to step a and repeat the whole proce dure for all pairs of multipoles 1 2 that we wish to test Only when we have the complete set do we assign a prob ability The rank R1112 gives the probability that the statistic SWMAP is consistent with the test hypothesis For example if the likelihood of SWMAP is rank 900 out of 1000 then there is a 10 probability that a Monte Carlo Gaussian random re alization of the CMB sky will give higher likelihood and 90 probability for a lower MC likelihood We say that the rank order of this particular likelihood is 09 If our CMB sky is indeed random Gaussian we would expect the rank orders of our statistics to be distributed between 0 and 1 being neither too small nor too large Conversely if we computed the rank orderings for three different multipoles for a particu lar test and obtained 098 099 and 095 or 001 005 and 002 say we would suspect that this particular test is not consistent with the Gaussian random hypothesis C How unusual are the ranks We now consider how to quantify the con dence level for rejecting the hypothesis of statistical isotropy of the am Let us assume that we have computed the rank orders for N different pairs of multipoles and obtained the rank orderings of x for the ith one where 0leS1 We consider the fol lowing parametric test The test anticipates that the ranks will be unusually high Let us rst order the ranks x in descending order so that x1 is the largest and xN the smallest We calculate the following statistic yNil 1 y1 Qx1 xNNfdJ1f dyz f dyN x1 x2 xN 28 0435157 COPI HUTERER AND STARKMAN PHYSICAL REVIEW D 70 043515 2004 TABLE I Ranks of the vectorvector vectorcross crosscross and oriented area OA statistics for multipoles 2 S61 52S8 for the four tests we consider as applied to the Tegmark et a 13 cleaned map Also listed are the oriented area ranks for the ILC map Mis the number of products for a given statistic for each 4 3 1 4 32 pair In Sec VI B we perform a parametric test to compute the likelihood of oriented area ranks being this high Ranks of Products of Multipole Vectors Vectorvector Vectorcross Crosscross Oriented area OA ILC map 1672 M Rank M 1672 rank 2671 rank M Rank M Rank M Rank 2 3 6 057714 6 3 003176 004814 3 001316 3 000126 3 000886 2 4 8 039167 12 4 051983 020369 6 056747 6 096042 6 062279 2 5 10 0 66656 20 5 085252 010536 10 022285 10 071820 10 077484 2 6 12 053649 30 6 050367 067791 15 087882 15 076320 15 093862 2 7 14 044925 42 7 074254 052205 21 091890 21 086496 21 0 63763 2 8 l6 0 21683 56 8 020861 073486 28 091338 28 068520 28 099986 3 4 l2 0 18093 l8 12 075272 030611 18 017059 18 0 34475 18 0 12562 3 5 l5 0 21511 30 15 036963 078578 30 037187 30 0 67870 30 076284 3 6 l8 0 31507 45 18 026683 054146 45 075052 45 0 93546 45 052839 3 7 21 0 98772 63 21 0 85874 0 57072 63 055147 63 0 73650 63 0 83324 3 8 24 0 76120 84 24 098578 060408 84 099988 84 0 99656 84 097766 4 5 20 0 41209 40 30 028221 0 84716 60 054035 60 0 52936 60 065965 4 6 24 0 68840 60 36 0 58372 0 86140 90 074826 90 0 62266 90 073762 4 7 28 0 85008 84 42 0 51404 0 95584 126 0 53715 126 0 88992 126 032153 4 8 32 048723 112 48 056328 0 85462 168 0 84374 168 0 99006 168 095266 5 6 30 082148 75 60 042361 0 88662 150 0 66327 150 0 82760 150 0 86242 5 7 35 0 86884 105 70 056542 0 96116 210 0 59483 210 0 68920 210 0 98440 5 8 40 0 83380 140 80 096812 0 34287 280 0 30403 280 0 24449 280 033959 6 7 42 0 03742 126 105 013831 0 02203 315 0 20221 315 0 97286 315 078414 6 8 48 092760 168 120 091468 078058 420 072850 420 062894 420 066873 7 8 56 003238 196 168 004440 002060 588 009552 588 025367 588 024871 If the ranks R41 viz were expected to be Lmiformly distributed in 01 Q would be the probability that the highest rank is greater than x1 and the second biggest rank greater than x2 and the smallest rank greater than xN However we do not expect that the ranks from Gaussian random maps are Lmiformly distributed and we treat Q merely as a statistic We then ask that given the possibly very small value of QWMAP what fraction of Gaussian random maps would give an even smaller Q That number is our probability and is computed in the next section Note that although apparently dif cult or impossible to evaluate analytically the righthand side of Eq 28 can trivially be computed using a recursion relation as shown in Appen ix B VI RESULTS Table I shows the nal ranks for the vectorvector vector cross crosscross and oriented area tests for pairs 61 t z These results correspond to the fullsky cleaned WMAP map from Tegmark el al 13 results for their Wiener ltered map from the same reference are essentially identical The ranks from the WMAP ILC fullsky map are similar and to make the presentation concise we show the ILC map ori ented area ranks in Table I but otherwise quote only nal probabilities for the ILC ranks As discussed earlier a de tailed analysis of cutsky maps will be presented in an up coming publication For computational convenience we considered only the multipoles 2lt 1 2lt8 we discuss the upper multipole limit in Sec VI C Note that for the vectorcross test 61 t z and 72 61 products are distinct and both need to be considered A Vectorvector vectorcross and crosscross ranks Table I shows that the vectorvector ranks are distributed roughly as expected nearly Lmiformly in the interval 01 The vectorcross ranks however are starting to shows hints of an interesting feature that will be more pronounced later in the crosscross and oriented area tests ranks that are m usually highiseven ranks out of 42 are greater than 09 The probability of this happening however is not statistically signi cant and it may be purely accidental The rst big surprise comes from the crosscross ranks the 61 362 8 rank is 0999 88 This means that only six out of 50000 MC generated maps had a higher likelihood than the WMAP map In other words the 84 crosscross dot products computed for this pair from WMAP lie Lmusually near the peaks of their respective histograms which recall are built out of 100 000 products from MC map realizations The violation afsla s cal isotropy and0r Gaussianily there 0435158 MULTIPOLE VECTORS A NEW REPRESENTATION OF 11 11 Il 1m ll Illlu L2 3 4 5 6 I L I IIII I 3456784567 IIIII III 567867878 00 239 L2 FIG 4 Ranks of the oriented area statistics for the Tegmark fullsky map The mean values correspond to the actual extracted a gm while the error bars were obtained by adding the pixel noise the error bars are not necessarily symmetric around the noerror values Note that an unusually large fraction of the ranks are high Also note the low value of the A 2 23 rank which is due to the alignment of the quadrupole and octupole which was noted earlier 13 fore manifests itself by a particular correlation between the vectors which makes the statistic SWMAP unusually usual Since we checked that the distribution of Monte Carlo gen erated 1 312 8 crosscross ranks is uniform in the in terval 01 it is easy to see that the probability of this rank being this high or higher is 650 000 or 0012 Admit tedly we checked 21 such crosscross ranks which raises the probability of nding such a result to 026 If we include all vectorvector vectorcross crosscross and oriented area ranks the probability rises to a still rather low 13 Note that this effect is very different from the now famil iar orientation of the quadrupole and octupole axes the quadrupoleoctupole alignment is quite unlikely and results in multipole vector products which preferentially fall on the tails of their respective histograms This is con rmed by the actual ranks for 23 and 32 multipole vectorcross prod ucts and also the 23 crosscross and oriented area prod ucts all of which are fairly low 005 This is easy to understand in the crosscross product case for example the fact that the multipole vectors lie mostly in the Galaxy plane implies that their cross products are roughly perpendicular to this plane The dot products of those are then by absolute value very large and hence unusual What we are seeing here is that the 1 3 2 8 crosscross products from the WMAP fullsky map are unusually usual B Oriented area ranks As mentioned above one of the crosscross ranks was extremely high A much bigger surprise is found when we examine the oriented area ranks see also Fig 4 two out of 21 are greater than 099 a total of ve are greater than 09 and a total of eight are greater than 08 These ranks are clearly not distributed in the same way as those from a typi cal MC map To examine the probability of the ranks being this high we perform the parametric test described in Sec VB and in Appendix B we compute the statistic Q which for a distri PHYSICAL REVIEW D 70 043515 2004 700 an 500 E m E E 300 200 100 171 10 8 6 4 2 0 10g10Q FIG 5 The statistic Q for the oriented area statistic computed from the Tegmark et al cleaned WMAP map is shown by vertical line The shaded region around it corresponds to the uncertainty due to pixel noise while the histogram shows the distribution of the statistic for MC generated Gaussian maps Only 107 of MC val ues of Q are smaller than the noerror value of the WMAP Q The same fraction for the ILC map not shown here is 038 bution of ranks expected to be uniform in 01 would be the probability of the largest one being at least as large as the largest actual WMAP rank and the second largest being at least as large as the secondlargest actual rank etc This sta tistic applied to ranks from the Tegmark et al cleaned map is Q 1046125 29 where the error around the mean value is estimated by re peatedly adding the pixel noise of the map to the pure ex tracted am as in Eq 26 and estimating the effect on the products of multipole vectors and their ranks However we have to be cautious not to overinterpret Q as the nal probability it has to be compared to Monte Carlo probabilities computed under the same conditions to deter mine its distribution To this end we generate 10 000 addi tional MC Gaussian random maps and compute Q for each It turns out that only 107 of them produce Q lower than the WMAP value in Eq 29 This is further illustrated in Fig 5 which also shows the error bars on the WMAP Q The prob ability of WMAP oriented area ranks being this high accord ing to the Q test is 107 which corresponds to a 260 or 9893 evidence for the violation of statistical isotropy andor Gaussianity The connection of this nearly 3 039 de viation to the nearly 30 deviation represented by the 1 3 2 8 crosscross rank remains somewhat unclear C Further tests We have performed a few tests to explore the stability of the oriented area result First we have varied the multipole range from the ducial 2S S8 the results are shown in Table II Increasing the lower limit mm leads to the nal probability of 1 and 56 for mm3 and 4 respectively Therefore evidence for the violation of statistical isotropy weakens but does so relatively slowly This shows that our main result does not completely hinge on the quadrupole and 0435159 COPI HUTERER AND STARKMAN TABLE II Final probabilities of the WMAP oriented area sta tistic as a function of multipole coverage mm l lmw The ducial case is 2S S8 and we have shown how the results change if the lower and upper bounds are changed fQMcltQWMAP is the frac tion of MC random Gaussian maps that give a value of Q smaller than the WMAP value Varying the multipole coverage 3mm QW MAP 2 761x10 7 10710000 3 313x10 6 10510000 4 312x10 4 56510000 4gmax QW MAP 8 761x10 7 10710000 7 372x10 5 39410000 6 362gtlt10 3 207910000 octupole Same can be said for the upper multipole limit which gives strongest results for the violation of statistical isotropy with 6mm 8 but with decreasing mx the result does not immediately go away Finally we have checked that increasing mx to higher values up to 12 does not produce new ranks that are unusually high The correlations are there fore most apparent in the multipole range 2lt 1 l z lt8 We next check if the correlations can be explained by any remaining dust contamination We use WMAP s Vband map of the identi ed thermal dust39 this map was created by tting to the template model from Ref 66 We assume for a mo ment that 10 of the identi ed contamination by dust had not been accounted for and we add it to the cleaned CMB map ie Tm TCMB01Tdust Although we are adding a signi cant contamination the remaining dust is expected to contribute no more than a few percent to the rms CMB tem perature 1 the high oriented area ranks do not change much they actually slightly increase and the Gaussian iso tropic hypothesis is still ruled out at the 993 level Clearly dust contamination does not explain our results Further to mimic remaining foregrounds due to an imperfect cleaning of the map we tested adding a synthetic random Gaussian map which contributed 10 of the rms temperature We nd that the oriented area statistic still disagrees with the Gauss ian isotropic hypothesis at the 994 level Finally we have repeated the analysis with several other available fullsky CMB maps As mentioned earlier both maps analyzed by Tegmark el all give the same probability for the oriented area statistic For the WMAP ILC map we nd similarly high ranks giving an even smaller value for our statistic QEC244gtlt10 7 and only 62 MC maps out of 10 000 have a smaller value of Q39 therefore the high ranks in the ILC map are unlikely at the 9938 level correspond ing to 2717 VII DISCUSSION Does the apparent violation of statistical isotropy or Gaus sianity that we detected have a cosmological origin or is it due to foregrounds or measurement error The results pre PHYSICAL REVIEW D 70 043515 2004 sented in this paper refer to fullsky maps and it is known that there are two large cold and two hot spots in the Galaxy plane and that any result that depends on structure in this plane is suspect Nevertheless it is far from obvious that the result is caused by the contamination in the map for the following reasons 1 As Tegmark el al 13 argue their cleaned map agrees very well with the ILC map on large scales although the two were computed using different methods The results we pre sented indicate a violation of isotropy andor Gaussianity at 2989 con dence using either map Furthermore instru mental noise and beam uncertainties are completely sub dominant on these scales 2 The results come from an effect different from the quadrupole and octupole alignment the latter is fairly un likely as discussed in Ref 63 while we see a particular correlation between the vectors that makes our statistic S unusually likely 3 Perhaps most importantly our results are mostly al though not completely independent of the quadrupole and octupole multipoles that might be suspect For example the secondhighest oriented area rank is 6 1 462 8 Further more Table 11 shows that if we use only the multipoles 4 lt l 2lt8 the oriented area statistic still rules out the Gaussian random hypothesis at the 944 level At this time it is impossible to ascertain the origin of the additional correlation between the multipole vectors that we are seeing One obvious way to nd out more about their origin is to Monte Carlo generate maps that are non Gaussian or violate the statistical isotropy according to a chosen prescription and see whether our statistics SWMAP agree with the S computed from MC maps Of course there are many different ways in which Gaussianity andor isot ropy can be broken and there is no guarantee that we can nd one that explains our results Another possibility is to cut the galaxy or other possible contam inations from the map prior to performing the vector decomposition There are two approaches we can take 1 use the cutsky am to compute the multipole vectors and the statistics S and compare those to S computed from cutsky Gaussian random maps or 2 reconstruct the true fullsky am and compare with fullsky Gaussian random maps The latter procedure is preferred as one would like to work with the true multipole vectors of our universe but reconstructing the fullsky map from the cutsky information is a subtle problem that will introduce an additional source of error Nevertheless the total error with a N10 cut may still be small enough to allow using the vectors as a potent tool for nding any preferred directions in the universe We are ac tively pursuing these approaches at the present time Finally Park 15 recently tested the fullsky WMAP maps using the genus statistic and found evidence for the violation of Gaussianity at the 27317 level depending on the smoothing scale and the chosen aspect of the statistic Furthermore Eriksen el al 14 nd that the WMAP multi poles with 6 lt35 have signi cantly less power in the north ern hemisphere than in the southern hemisphere It is pos sible that these results and the effects discussed in this paper 04351510 MULTIPOLE VECTORS A NEW REPRESENTATION OF have the same underlying cause but this cannot be con rmed without further tests VIII CONCLUSIONS The traditional Y m expansion of the sky has many advan tages For one each set of Y4quot of xed 6 form an irreducible representation of the rotation group in three dimensions 03 for another more than two centuries of effort have lead to a rich mathematical literature on the Y m their prop erties and how to ef ciently calculate them The coef cients am 0 a Y m expansion of a function on the sphere are readily calculated as integrals over the sphere of the function times the Y 2 In this paper we have considered a different basis of an equivalent irreducible representations of the proper rotation group O 39l ifor each 6 the traceless symmetric product of 6 copies of the unit vector of coordi nates Q These are merely linear combinations of the Y m and so share many of their properties albeit with a more sparse mathematical literature explicitly dedicated to their properties In particular the coef cients Ff 1 of an OW expansion of a function on the sky are like the spheri cal harmonic coef cients calculable as integrals over the sphere of the function times 09 We have expressed the F516 as symmetric traceless products of 6 headless unit vectors 1 am and a scalar A The 1 MO are highly nonlinear functions of the am Thus while in principle they encode the exact same infor mation they may make certain features of the data more selfevident In particular we claim that these multipole vectors are natural sets of directions to associate with each multipole of the sky A code to calculate multiple vectors from CMB skies is available on our website at http wwwphyscwrueduprojectsmpvectors We have obtained the multipole vectors of the CMB sky as measured by WMAP as well as the oriented areas de ned by all pairs of such vectors within a particular multipole We have examined the hypothesis that the vectors of multi pole 6 are uncorrelatedwith the vectors of multipole 6 for 6 and 6 up to 8 We have done this by comparing in turn the dot products of the vectors from 6 with those from 6 the dot products of the vectors with the unit normals to the planes the dot products of the unit normals to the planes with each other and the dot products of the normals to the planes with each other We found that while there is nothing unusual about the distribution of dot products of the vectors with each other the dot products of the normals to the planes with each other and to a lesser extent the dot products of the unit normals to the planes with each other are inconsis tent with the standard assumptions of statistical isotropy and Gaussianity of the am To quantify this inconsistency we compared the distribution of these dot products with those from 50 000 Monte Carlo simulations and found that they are inconsistent at the level of 107 parts in 10000 for the Teg mark el al cleaned fullsky map and 62 parts in 10 000 for the ILC fullsky map These results are robust to the inclu sion of appropriate Poisson noise The sensitivity to a Galac PHYSICAL REVIEW D 70 043515 2004 tic cut will be explored in a future publication but prelimi nary results suggest that the results persist within the error bars but eventually decline in statistical signi cance as the uncertainties increase with increasing cuts ACI IOWLEDGMENTS We would like to thank Tom Crawford Vanja Dukic Doug Finkbeiner Gary Hinshaw Eric Hivon Arthur Lue Dominik Schwarz David Spergel JeanPhillipe Uzan Tan may Vachaspati and Ben Wandelt for helpful discussions We have bene ted from using the publicly available HEALPIX package 65 The work of the particle astrophysics theory group at CWRU is supported by the DOE APPENDIX A VECTOR DECOMPOSITION EQUATIONS The vector decomposition equations we derived 15 can be recast in a numerically more convenient form The equa tions as written involve complex valued coef cients Here we rewrite these equations in terms of their purely real com ponents To begin we note that the spherical harmonics sat isfy ngmQ 7 1 quotY m0 Thus the decomposition coef cients of a real valued function such as ATQ T satisfy 1er 71 am This shows that all the informa tion about the function is encoded in the real part of ago the imaginary part is identically zero and the real and imaginary parts of am for 1 ltmlt6 These are the 26 1 independent components we use in the vector decomposition We thus need to solve Eq 15 only for 0ltmlt6 For notational convenience we drop the 6 superscript on awlym j bm and 1 It should be understood that these quantities are associated with a particular multipole and step in the recursive decomposition procedure as outlined in Sec III A The correspondence between the dipole and Cartesian coordinate directions 7 allows us to identify 1 131 60 61 with standard coordinate axes via A 1 A vil 2Uxlvygt 606Z A1 A 1 A A 121 7 02711 Finally the real and imaginary parts of the am are re 1 im 1 m amam and am am am A2 Applying Eqs A1 and A2 to the multipole vector decom position equations 15 gives the following equations 04351511 COPI HUTERER AND STARKMAN 1 re 1m re A Lm re A aim Co quot14771quotszr C71 ai1m1vr 3 im A 1 Cimx re A 7a 1m1v3975 1 114771quotqu alimilmilvygt im C1m im A JFLCMJM im A aim 0 firmvz 71 firm vx 3 re A 1 Am im A a rrm1vy cr aiiLmilvr Taffeirwir yl l A i A 39 A 470 lib Ci malfe pvz ci 390alierr1vx 7 rrvyk A3 1 re 7 ml re A m re A bra Di a 1mvz D71 a 71m1vx xf im A 1Dim re A 7ai1mlvy7 1 aiiLmilvr al mrrmirvyx l 1m 7 1m 1m m 1m quot bra Di atirrmrszr D21 aiilm rlvx rE re A itiw im A ai1m1vy 1 firmrrvx Taffeirwir yl b0 bireDil390alieirolz DiLOa err1lxia m yx M r Note that the equations for bm are identical to those for am with Damquot inserted in place of CL quot Here 1ltmlt and l m 6 72 These equations involve only real quantities and can thus be easily coded and solved These are the equa tions we have implemented to nd the multipole vectors APPENDIX B PROBABILITY OF RANK ORDERINGS Consider N numbers x1 where 0 xl S l and order them in descending order so that x1 is the largest and x N the smallest Let us then consider a set of variates y uniformly distributed in the interval 01 and also order them in de scending order so that yl is the largest one and y N the small est We ask what is the probability that y1 is greater than x1 and that y2 is greater than x2 and that yN is greater than x N PHYSICAL REVIEW D 70 043515 2004 The probability that y1 is in the interval x1 x1dx1 731x1dx1 is N 731x1dx1 N71 1 x1 dx1gt B1 and the probability that yl is larger than x1 P1x1 is ob viously 1 13136C 731y1dy1 B2 Given that y1 is greater than x1 the probability that y2 is in the interval x2 x2 dxz is 71 1 x2 N72 dx2 x1 N 73972lexrdxz B3 x1 and the probability that the largest y is greater than x1 and the secondlargest greater than x2 is 1 y P2x1gtx2f 731y1dy1f 1732012th112 1 1 1C2 1 yr 7 NN1l dyrf yiv 2dlyz 1C1 1C2 B4 We can continue this argument for all other y and x in descending order in x1 The nal probability the joint prob ability of the 1th largest y being greater than x1 for all 139 is given by 1 yr J39Nrr PNx1x2xNN dy1 dy2 dyN 1 1 I2 IN71 B5 We would like to evaluate this integral Even though the result will obviously be a polynomial in x1 there is a total of 2 terms and it is dif cult to do the bookkeeping However there is a simple recursion formula for this integral Assume more generally that we want to compute a7 1 yr Hz a IN dyr dy2 dyNyN 1 1 I2 IN71 One can then perform the innermost integral and this leads to the recursion relation 7 l 7al B6 1 Ifl xllz livrrl B7 We are left with two N 7 ltuple integrals Therefore starting from the N dimensional integral we can recursively bring it down all the way to N l at which point it is an easy onedimensional integral 1 31 for the required B Using the recursion relation B7 to gether with B8 we numerically compute the probability in B5 1 1 dyryf 1 xlm B8 1C1 04351512
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