PRL 96, 151303 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

Global Universe Anisotropy Probed by the Alignment of Structuresin the Cosmic Microwave Background

Y. Wiaux,1,* P. Vielva,2,† E. Martınez-Gonzalez,2 and P. Vandergheynst11Signal Processing Institute, Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

2Instituto de Fısica de Cantabria (CSIC-UC), E-39005 Santander, Spain(Received 23 December 2005; revised manuscript received 14 February 2006; published 21 April 2006)

0031-9007=

We question the global universe isotropy by probing the alignment of local structures in the cosmicmicrowave background (CMB) radiation. The original method proposed relies on a steerable waveletdecomposition of the CMB signal on the sphere. The analysis of the first-year Wilkinson MicrowaveAnisotropy Probe data identifies a mean preferred plane with a normal direction close to the CMB dipoleaxis, and a mean preferred direction in this plane, very close to the ecliptic poles axis. Previous statisticalanisotropy results are thereby synthesized, but further analyses are still required to establish their origin.

DOI: 10.1103/PhysRevLett.96.151303 PACS numbers: 98.70.Vc, 98.80.Es

Introduction.—Last years’ experiments in cosmologyhave resulted in the definition of a consistent picture ofthe structure and evolution of the Universe. The recent dataof the cosmic microwave background (CMB) radiation,together with other cosmological observations, have al-lowed us to determine precise values for the main cosmo-logical parameters [1]. However, the correspondingconcordance cosmological model is based on strong hy-potheses which need to be questioned. They extend fromthe nature of the gravitational interaction underpinning thecosmological evolution of the Universe, to the physicsgoverning the early inflationary era, or also to the cosmo-logical principle for the global homogeneity and isotropyof the Universe.

This Letter defines an original method to test the globaluniverse isotropy through the analysis of the CMB data.The observed CMB anisotropies on the celestial sphere canbe interpreted as a realization of a statistical process orig-inating in the inflationary era. The cosmological principleimplies the isotropy of the corresponding statistical prop-erties. This statistical isotropy has been challenged throughmultiple methods applied to the first-year full-sky CMBdata of the Wilkinson Microwave Anisotropy Probe(WMAP) experiment. The detections quoted in the follow-ing suggest statistical anisotropy with confidence levelshigher than 99%. Analyses based on N-point correlationfunctions [2,3], local curvature [4], local power spectra [5–8], and bispectra [9], suggest a north-south asymmetrymaximized in a coordinate system with the north pole at��;’� � �80�; 57�� in Galactic colatitude � and longitude’, close to the north ecliptic pole lying at ��;’� ��60�; 96��. Analyses of multipole vectors, angular momen-tum dispersion, as well as azimuthal phases correlationsfind an anomalous alignment between the low l multipolesof the CMB, suggesting a preferred direction around��;’� � �30�; 260��, near the ecliptic plane and close tothe axis of the dipole lying at ��; ’� � �42�; 264�� [10–17]. Galactic north-south asymmetries are also found in the

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analysis of the kurtosis and the area of the wavelet coef-ficients of the CMB data. These are mainly due to a verycold spot in the southern hemisphere [18,19]. First resultswith the angular pair separation method, which probes thestatistical isotropy both in real and multipole space, alsoseem to support those results [20]. On the contrary, bipolarpower spectra analyses are consistent with no violation ofthe statistical isotropy of the Universe [21–23]. Finally,theoretical models for an anisotropic universe are beingstudied to account for the observed effects [24].

Our alternative method probes the statistical isotropy ofthe CMB by the analysis of the alignment of structures inthe signal. Preferred directions in the Universe are definedas the directions towards which local features of the CMBare mostly oriented. The level of preference of each direc-tion may be established from simulations, and is repre-sented as a signal on the sphere. The approach is thereforepowerful as it also a priori allows the study of the corre-sponding angular power spectrum in order to probe themultipole distribution of the anisotropy. The analysis de-fined relies on a steerable wavelet decomposition of thefirst-year full-sky WMAP data.

Data and simulations.—The experimental temperaturemap used for the analysis is obtained from the first-yearWMAP data following the procedure originally proposedin [25] for Gaussianity tests. First, a best estimation ofdiffuse Galactic foregrounds is removed from each of theeight frequency maps of the WMAP receivers at the Q, V,and W bands, as prescribed by the specific template fitsmethod in [26]. The eight foreground cleaned maps arethen combined through a noise-weighted linear combina-tion in order to enhance the CMB signal-to-noise ratio. Theso-called Kp0 mask of [26] is applied to account for theremaining strong foreground contamination by diffuseGalactic emissions around the Galactic plane and by brightpoint sources. Finally, the residual monopole and dipoleare subtracted outside the mask. All the WMAP data arereleased in the HEALPIX pixelization on the sphere [27], at a

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FIG. 1 (color online). Mollweide projections of the first (left)and second (right) Gaussian derivative wavelets at position!0 ���=2; 0�, orientation � � 0, and scale a � 0:19. Red and blueregions, respectively, correspond to positive and negative values.

PRL 96, 151303 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

resolution identified by the parameter Nside � 512, corre-sponding to maps with several million pixels with a spatialresolution of 6.9 arcminutes. The temperature map ob-tained from the above preprocessing is downgraded to alower resolution withNside � 32 for the specific purpose ofour analysis. This provides a map with Npix � 12 288pixels with a spatial resolution of 1.8 degrees.

Ten thousand temperature map simulations at the reso-lution Nside � 32 are used in order to define confidencelevels for the analysis results. These simulations specifi-cally assume the Gaussianity and statistical isotropy of theCMB. They are generated following the scheme also pro-posed in [25]. A Gaussian CMB simulation is obtained inspherical harmonics space from the angular power spec-trum determined by the cosmological parameters of theWMAP best-fit cosmological model. The observation ateach receiver is simulated by convolving that map with thecorresponding WMAP beam window function. After trans-forming each map to pixel space, a Gaussian noise real-ization is also added with the proper dispersion per pixel.Following the same procedure as for the data, each one ofthe simulated frequency maps of the eight receivers at theQ, V, andW bands are finally combined through the noise-weighted linear combination which enhances the CMBsignal-to-noise ratio, the Kp0 mask is applied, and themonopole and dipole are subtracted.

Wavelet filtering.—We recall the recently introducedsteerable wavelet decomposition of a signal on the sphereupon which we base our analysis.

Let the function F�!� describe a signal on the sphere,with the point ! � ��; ’� identified by its colatitude � 2�0; �� and longitude ’ 2 �0; 2��. A wavelet on the sphereis a localized function � that can be dilated at any scalea 2 R�� associated with a given angular opening on thesphere, rotated on itself by any angle � 2 �0; 2��, andfinally translated at any position !0 � ��0; ’0�. The wave-let filtering of F by � results from the scalar product(correlation) of the signal with all the dilated, rotated,and translated versions of the wavelet: �!0;�;a. Hence,the wavelet coefficients WF

��!0; �; a� � h�!0;�;ajFi char-acterize the signal locally at each scale a, in each orienta-tion �, and at each point !0 [28,29]. Individualizing theproperties of the signal independently at each scale is amajor advantage of the wavelet scale-space filtering asphysical phenomena may generally be scale dependent.Determining the local orientations of the signal structureat each scale also provides essential information for ourparticular analysis.

Notice that the computation of all wavelet coeffi-cients for a large number of scales, and with the samehigh resolution in local orientation and position is a nu-merically complex calculation. A corresponding fast di-rectional correlation algorithm on the sphere was re-cently defined which strongly relies on the use of so-calledsteerable wavelet filters [30]. Our analysis of the WMAP

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data is performed independently with the first and secondGaussian derivatives. These steerable filters, represented inMollweide projection in Fig. 1, are analytically defined in[29,30].

Analysis method.—Here, we consider the analysis lead-ing to the identification of preferred directions from thefirst-year full-sky WMAP data at the HEALPIX resolutionNside � 32. The complementary study of the correspond-ing angular power spectrum is postponed to a future work.

For a given scale a, the angular size of our wavelets onthe sphere is defined as twice the dispersion of the corre-sponding Gaussian. Twelve scales are selected correspond-ing to angular sizes of the first and second Gaussianderivatives lying between 5 and 30 degrees. At each scale,an extended exclusion maskMa is defined in order to avoidconsidering pixels for which the correlation between theWMAP signal and the wavelet is contaminated by the Kp0mask. The pixels added to the initial mask are those forwhich the wavelet coefficients of a constant signal in theregion of the mask are nonzero, up to a given threshold.

For each pixel !0 of the signal outside the extendedexclusion mask, the direction �0�!0� for which the waveletcoefficient is maximum in absolute value is selected, andthe corresponding absolute value is retained. This selectsthe local wavelet orientation which best matches the ori-entation of the local structure of the signal at each point.The great circle is defined which passes through that pointand admits the corresponding local orientation as a tangent.The directions in the sky lying on that great circle areconsidered to be highlighted by the local structure identi-fied, and are weighted by the absolute value of the waveletcoefficient jWF

��!0; �0�!0�; a�j.Consider each direction ! on the celestial sphere in the

set S of the Npix � 12 288 pixels at Nside � 32. The totalweight Da�!� at scale a is the sum of the Ncros�!� weightsoriginating from all pixels !�c�0 in the original signal, with1 c Ncros�!�, for which the great circle definedcrosses the direction considered:

Da�!� �1

A

XNcros�!�

c�1

jWF��!

�c�0 ; �0�!

�c�0 �; a�j: (1)

The factor A � LN1pix

P!0=2Ma

jWF��!0; �0�!0�; a�j of nor-

malization defines a mean total weight in each directionequal to unity for isotropic CMB simulations without

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PRL 96, 151303 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending21 APRIL 2006

mask: N1pix

P!2SDa�!� � 1. The quantity L � 4Nside

stands for the number of points on a great circle on aHEALPIX grid. Preferred directions in the Universe aretherefore identified as the most weighted ones. Noticethat the procedure obviously assigns identical total weightsto opposite directions. In other words, our directions areheadless vectors.

Results.—In this section, the results of the analysis ofthe first-year WMAP data proposed above are exposed.The filtering of the CMB data by the first Gaussian deriva-tive does not lead to any significant detection. This sug-gests that there is no clear alignment of the local CMBstructures with the morphology captured by that specificfilter. The corresponding analysis with the second Gaussianderivative leads to a strong detection, detailed in thefollowing.

At each scale a and at each pixel !, the total weightobtained can be quantified by the number of standarddeviations �a�!� through which it deviates from themean �a�!�, as estimated from the simulations.Preferred directions, or positive total weights, correspondto Da�!�>�a�!�. Nonpreferred directions, or negativetotal weights, are distinguished as Da�!�<�a�!�. Weconsider, in particular, the scale a3 associated with anangular size of the filter of 8.3 degrees on the sky. Thecorresponding map of total weights clearly depicts thedistribution of anisotropy resulting from our analysis(Fig. 2, top panel).

–3.48 +7.40

+7.40+4.40

FIG. 2 (color online). Mollweide projection of the Galacticcoordinates maps of total weights (top) and directions anomalousat 99.99% (bottom) at the scale a3 of the second Gaussianderivative wavelet, and in �a3

units.

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A rare detection is observed at that specific scale. Itidentifies 20 directions (pairs of opposite points), qualifiedas anomalous at 99.99%, with an associated positive totalweight higher than in any of the ten thousand simulationsconsidered. The 20 anomalous directions obtained at scalea3, with positive total weights lying in the interval�4:4; 7:4� in �a3

�!� units, are mainly concentrated in twoclusters around the ecliptic poles (Fig. 2, bottom panel).The most prominent of these concentrations comprises 16of the 20 anomalous directions. Its mean position is iden-tified by a northern end at ��;’� � �71�; 91�� in Galacticcolatitude � and longitude ’, when each direction isweighted by its corresponding total weight. This definesa mean preferred direction in the sky with a northern endvery close to the north ecliptic pole at ��; ’� � �60�; 96��.The presence of the two elongated clusters also suggeststhat the 20 anomalous directions lie on a great circle,defining a preferred plane in the sky. The normal directionto the mean preferred plane, given by the cross product ofthe mean directions of each of the two clusters, has anorthern end position at ��;’� � �34�; 331��. Alterna-tively, the great circle which best fits the 20 anomalousdirections identifies exactly the same plane. This normaldirection lies close to the northern end of the CMB dipoleaxis at ��; ’� � �42�; 264��.

Let us emphasize that 11 directions anomalous at99.99% are also detected at the neighbor scale a4 corre-sponding to an angular size of the filter of 10 degrees. Theybelong to two clusters with very similar mean positions tothose identified at scale a3, and therefore single out thesame structure of anisotropy.

Discussion and conclusion.—The present originalanalysis of structures alignment in the CMB from thefirst-year WMAP data clearly identifies a mean preferredplane in the Universe with a normal direction close to theCMB dipole axis, and a mean preferred direction in thisplane, very close to the ecliptic poles axis. This result isbased on the observation of 20 directions anomalous at99.99%. Our original method thus singles out the samedirections as those highlighted by previous statistical iso-tropy studies. The wavelet approach also identifies theangular size of the anomalously aligned structures around8.3 degrees on the celestial sphere, corresponding to amultipole range roughly between l � 11 and l � 27.

This new insight into the anisotropy structure might helpus to understand its still unclear origin. First, the angularsize identified for the aligned structures is compatible withthe size of primary CMB anisotropies due to topologicaldefects such as texture fields [31] or secondary anisotropiesdue to the Rees-Sciama effect [32]. Alignment mecha-nisms [33,34] were recently proposed which might begeneralized to such structures. Second, the identified an-gular size is also compatible with the one above whichdiffuse Galactic emissions dominate the CMB emission.Corresponding residual foregrounds can be probed by a

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direct analysis of the WMAPQ, V, andW bands frequencymaps. Finally, the coincidence of the preferred directionsdetected with the ecliptic poles and dipole axes naturallysuggests possible unknown systematic effects [33,35]. Inthat respect, notice that the angular size of the mesh of theWMAP scan pattern defined by the combination of the spinand precession of the satellite is, again, of the order ofseveral degrees [36].

In conclusion, our analysis provides a first synthesis ofprevious statistical anisotropy results. Further analyses ofthe complete total weights distributions at various waveletscales and of the corresponding angular power spectra canallow a deeper probe of the anisotropy structure. But theorigin of the present detection already needs to be thor-oughly investigated. Various hypotheses can be suggestedin terms of cosmological or foreground structures, or sys-tematics. But nothing at present allows us to discard thepossibility of a global universe anisotropy, simple violationof the cosmological principle hypothesis.

The authors acknowledge the use of the LAMBDAarchive, and of the HEALPIX and CMBFAST softwares.Y. W. was supported by the Swiss and Belgian NationalScience Foundations. P. V. and E. M.-G. were supported bythe Spanish MEC Project No. ESP2004-07067-C03-01.

*Electronic address: yves.wiaux@epfl.ch†Electronic address: vielva@ifca.unican.es

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