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Mon.Not.R.Astron.Soc.000,1–10(2007) Printed1February2008 (MNLATEXstylefilev2.2) Non-Gaussianity analysis on local morphological measures of WMAP data Y. Wiaux1, P. Vielva2, R. B. Barreiro2, E. Mart´ınez-González2, P. Vandergheynst1 8 1Signal Processing Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 0 E-mails : yves.wiaux@epfl.ch, pierre.vandergheynst@epfl.ch 0 2Instituto de F´ısica de Cantabria (CSIC - UC), 39005 Santander, Spain 2 E-mails : [email protected], [email protected], [email protected] n a J 1February2008 7 ] h ABSTRACT p The decomposition of a signal on the sphere with the steerable wavelet constructed - from the second Gaussian derivative gives access to the orientation, signed-intensity, o andelongationofthe signal’slocalfeatures.In the presentwork,the non-Gaussianity r t of the WMAP temperature data of the cosmic microwave background (CMB) is an- s alyzed in terms of the first four moments of the statistically isotropic random fields a [ associated with these local morphological measures, at wavelet scales corresponding to angular sizes between 27.5′ and 30◦ on the celestial sphere. While no detection is 2 made neither in the orientation analysis nor in the elongation analysis, a strong de- v tection is made in the excess kurtosis of the signed-intensity of the WMAP data.The 6 non-Gaussianity is observed with a significance level below 0.5% at a wavelet scale 4 ◦ correspondingto an angularsize around10 , andconfirmedat neighbour scales.This 3 2 supports a previous detection of an excess of kurtosis in the wavelet coefficient of the . WMAP data with the axisymmetricMexicanhatwavelet(Vielva et al.2004). Instru- 6 mental noise and foreground emissions are not likely to be at the origin of the excess 0 of kurtosis. Large-scale modulations of the CMB related to some unknown system- 7 atics are rejected as possible origins of the detection. The observed non-Gaussianity 0 : may therefore probably be imputed to the CMB itself, thereby questioning the basic v inflationary scenario upon which the present concordance cosmological model relies. i X Taking the CMB temperature angularpowerspectrum of the concordancecosmologi- calmodelatfacevalue,furtheranalysisalsosuggeststhatthis non-Gaussianityisnot r a confined to the directions on the celestial sphere with an anomalous signed-intensity. Key words: methods: data analysis, techniques: image processing, cosmology: observations, cosmic microwave background 1 INTRODUCTION ground fluctuations are also assumed to arise from Gaus- sian quantum energy density perturbations around the ho- Among other cosmological observations, the recent data mogeneousandisotropicbackgroundduringtheinflationary of the cosmic microwave background, in particular those phase.Thecosmological parametersoftheconcordancecos- released by the Wilkinson Microwave Anisotropy Probe mologicalmodelaredeterminedwithaprecisionoftheorder (WMAP) satellite experiment, have played a central role of several percent, but the hypotheses on which the model in defining a concordance model highlighting a flat ΛCDM relies need to bethoroughly challenged. Universewithaprimordialphaseofinflation(Bennett et al. Inthat context,theCMB constitutes arealization of a 2003;Spergel et al.2003;Hinshawet al.2007;Spergel et al. random field on thesphere. The cosmological principle and 2007). In that framework, the flat Universe is filled in with the basic inflationary scenario respectively imply the sta- cold dark matter (CDM) and dark energy in the form of tistical isotropy and the Gaussianity of this random field. a cosmological constant (Λ), in addition to the standard These basic hypotheses of the concordance cosmological baryonicandelectromagneticcomponents.Thecosmological model may consequently be questioned through the CMB principle is assumed, which postulates global homogeneity analysis. and isotropy. In the basic inflationary scenario considered, the large scale structure and the cosmic microwave back- The statistical isotropy of the CMB tempera- 2 Wiaux et al. ture field has already been largely questioned in Section3,wepresenttheresultsoftheWMAPdataanalysis. the analysis of the WMAP data. Firstly, a North- InSection4,westudysystematiceffectsasapossibleorigin South asymmetry in ecliptic coordinates has been de- of the detections. In Section 5, we discuss the origin of our tected (Eriksen et al. 2004a,b, 2005; Hansen et al. 2004a,b; detectionanditsdetailedinterpretation.Wefinallyconclude Donoghue& Donoghue 2005; Land & Magueijo 2005a; in Section 6. Bernui et al. 2006, 2007; Spergel et al. 2007; Eriksen et al. 2007; Monteser´ın et al. 2007). Secondly, an anomalous alignment of the lowest multipoles of the data was ob- 2 METHODOLOGY served (de Oliveira-Costa et al. 2004; Schwarzet al. 2004; In this section, we firstly recall the formalism for the anal- Copi et al. 2004; Katz& Weeks 2004; Bielewicz et al. ysisofsignals onthespherewith steerablewavelets,aswell 2005; Land & Magueijo 2005b, 2007; Freeman et al. 2006; as the local morphological measures of orientation, signed- Abramo et al. 2006).Finally, wavelet analyses havealso re- intensity,andelongation,definedfromthesteerablewavelet ported statistical isotropy anomalies related to the signed- constructed from the second Gaussian derivative. Secondly, intensity of local CMB features, as well as anomalies re- we explicitly describe the statistics for the non-Gaussianity lated to the alignment of local CMB features toward spe- analysisontherandomfieldsassociatedwiththelocalmor- cific directions on the celestial sphere (Wiaux et al. 2006a; phological measures of the CMB temperature field. These Vielva et al. 2006, 2007). statistics are simply the first four moments of the random The Gaussianity of the CMB temperature field has fieldsconsidered. also been largely questioned in the analysis of the WMAP data. Firstly, departures from Gaussianity were detected using statistics of extrema (Larson & Wandelt 2004, 2005; 2.1 Steerable wavelets and morphological Tojeiro et al. 2006), bispectra (Land & Magueijo 2005a), measures phase correlations (Chiang et al. 2003; Chiang & Naselsky 2006; Coles et al. 2004; Naselsky et al. 2005), Minkowski Weconsiderthethree-dimensionalCartesiancoordinatesys- functionals(Park2004;Eriksen et al.2004b),andlocalcur- tem(o,oxˆ,oyˆ,ozˆ)centeredontheunitsphere,andwherethe vature (Hansen et al. 2004a). Secondly, wavelet analyses direction ozˆ identifies the North pole. Any point ω on the have also reported non-Gaussian deviations. An excess of sphere is identified by its corresponding spherical coordi- kurtosis in the wavelet coefficient of the WMAP tempera- nates (θ,ϕ), where θ∈[0,π] stands for the co-latitude, and ture data with the axisymmetric Mexican hat wavelet on ϕ∈[0,2π) for thelongitude. thespherewasfoundatwaveletscalescorrespondingtoan- Firstly,webrieflysummarizetheformalismofsteerable gular sizes on thecelestial spherearound 10◦,andlocalized wavelets on the sphere S2 (Wiaux et al. 2005). Any filter in the southern galactic hemisphere (Vielva et al. 2004). A invariant under rotation around itself is said to be axisym- cold spot (i.e. with negative wavelet coefficients) was iden- metric.Bydefinition,anynon-axisymmetric,ordirectional, tifiedat(θ,ϕ)=(147◦,209◦),withθ∈[0,π]andϕ∈[0,2π) filterΨis steerable if arotation byχ∈[0,2π) around itself respectively standing for the co-latitude and longitude in may be expressed in terms of a finite linear combination of galactic spherical coordinates, and considered to be a good M non-rotated basis filters Ψm: candidate to explain the observed deviation. The confirma- M tionthatthecoldspotisanomalouswasprovided,stillwith Ψχ(ω)= km(χ)Ψm(ω), (1) the axisymmetric Mexican hat wavelet, in terms of its area m=1 X (Cruz et al.2005).Thedetectionwasfurtherconfirmedwith where the weights k (χ), with 1 6 m 6 M, are called m various wavelets and various statistics (Mukherjee & Wang interpolation functions. The analysis of a signal F with a 2004; Cayón et al. 2005; McEwen et al. 2005; Cruz et al. given wavelet Ψ simply defines a set of wavelet coefficients 2006, 2007; McEwen et al. 2006). Notice that the cold spot WF(ω ,χ,a), which result from the directional correlation Ψ 0 identifiedalsocertainlyrepresentsadeparturefromstatisti- between F and the wavelet dilated at any scale a, Ψ . In a calisotropy,intermsofaNorth-Southasymmetryingalac- other words these wavelet coefficients are defined by the tic coordinates. scalar product between the signal and the wavelet dilated By essence, wavelet analyses present theparticular ad- atscale a,rotatedarounditself byχ,andtranslatedat any vantageof probing, not only thescale but also thelocaliza- point ω on thesphere, also denoted Ψ : 0 ω0,χ,a tion of the features constituting the CMB on the celestial sphere (Wiaux et al. 2005). Steerable wavelets also provide WF(ω ,χ,a)=hΨ |Fi= dΩΨ∗ (ω)F(ω). (2) Ψ 0 ω0,χ,a ω0,χ,a morphological measures of thelocal features, such as orien- ZS2 ∗ tation,signed-intensity,orelongation(McEwen et al.2007), The denotescomplexconjugation.Thewaveletcoefficients atalowcomputationalcost(Wiaux et al.2006b).Theywere of a signal therefore characterize the signal at each scale a, used to probe the statistical isotropy of the WMAP CMB orientation χ, and position ω0. temperature data in the previously quoted signed-intensity In the present work, we consider the second Gaussian and alignment analyses. They were also used to probe the derivative wavelet (2GD), Ψ∂x2ˆ(gau), which is obtained by a Integrated Sachs-Wolfe effect through the correlation of stereographicprojectionofthesecondderivativeindirection WMAPdataand large scale structuredata(McEwen et al. xôfaGaussianinthetangentplaneattheNorthpole.The 2007). In the present work, a further insight into the CMB filter obtained is a steerable wavelet on the sphere which temperature non-Gaussianity is provided through a steer- may be rotated in terms of three basis filters (M =3): the able wavelet analysis of the WMAP data. secondderivativeinthedirectionxîtself,thesecondderiva- In Section 2, we present the methodology adopted. In tiveinthedirection yˆ,andthecross-derivative.Noticethat Non-Gaussianity of WMAP data 3 the value of the scale a identifies with the dispersion of the WMAPdataarealsocontaminatedbynoiseandforeground Gaussianinunitsof2tan(θ/2).Theangularsizeofthe2GD emissions. The statistical analysis is performed by compar- is defined as twice the half-width of the wavelet, where the ison of the data with simulations. The noise present in the half-widthisdefinedbyθ =2arctan(a/2),whichisclosely data is simulated and the regions of the sky in which the hw approximated bya at small scales. data are too much contaminated by foreground emissions Secondly,werecallthatthe2GDgivesaccesstothreelo- are masked, and excluded from the analysis. Typically, a cal morphological measures of orientation, signed-intensity, non-Gaussianity analysis is performed through the evalua- and elongation (McEwen et al. 2007). By linearity, the re- tionofglobalestimatorscomputedassimpleaveragesonthe lation of steerability (1) is automatically transferred on the wholepartofthecelestialspherewherethedataareconsid- wavelet coefficients of F. Consequently, at each scale a and ered to bevalid, explicitly assuming the statistical isotropy ateachpositionω ,theorientationχ (ω ,a)thatmaximizes in thecorresponding part of thesky.Anyanomaly between 0 0 0 the absolute value of the wavelet coefficient, can easily be thedata and thesimulations is consequently interpreted as computed, with an infinite theoretical precision. It corre- a departureof the data from Gaussianity. sponds to the local orientation at which the wavelet best We consider the statistically isotropic real-valued ran- matches the local feature of the signal. As the 2GD is in- dom fields on the sphere associated with the local morpho- variant under rotation around itself by π, orientations may logical measures of orientation, signed-intensity, and elon- arbitrarily beconstrained inarangeoflengthπ,andasthe gation of the CMB temperature field T at each wavelet 2GDoscillatesinthetangentdirectionxˆ,itactuallydetects scale a: X(ω ,a), with X = {DT,IT,ET}. The statis- 0 featuresalignedalongthetangentdirectionyˆ.Thelocalori- tics estimated for the subsequent non-Gaussianity analysis entationofthefeatureitself,DF(ω ,a),isthereforedefined are simply moments of the first four orders. The first two 0 in termsof χ0 =χ0(ω0,a) as: are the mean µX(a) = hX(ω0,a)i, and variance σX(a) = h[X(ω ,a) − µX(a)]2i1/2. The third-order moment is the π π 3π 0 2 6 DF (ω0,a)≡χ0+ 2 < 2 . (3) skewness SX(a) = h[X(ω0,a) − µX(a)]3i/[σX(a)]3. The skewness measures the asymmetry of the probability den- The wavelet coefficient itself at scale a, position ω , and in 0 sity function, and hence a deviation relative to a Gaussian directionχ ,definestoso-calledsigned-intensityofthelocal 0 distribution. Positive and negative skewnesses are respec- feature: tivelyassociated withlargerrightandleftdistributiontails. IF(ω0,a)≡WΨF∂x2ˆ (ω0,χ0,a). (4) TKhXe(afo)u=rthh-[Xor(dωer,ma)o−meµnXt(cao)n]4sii/d[eσrXed(ai)s]4th−e3e.xTcehses kkuurrttoossiiss 0 The elongation of local features is explicitly defined by measuresthepeakednessoftheprobabilitydensityfunction relativetoaGaussiandistribution.Positiveandnegativeex- 06 EF (ω ,a)≡1− WΨF∂x2ˆ ω0,χ0+ π2,a 61. (5) cess kurtoses are respectively associated with distributions 0 ˛˛˛ WΨF∂`x2ˆ (ω0,χ0,a) ´˛˛˛ mfouorremaonmdelnetsssapreeaikneddeptehnadnenatGofatuhsesipanoindtisωt0ribbuectaiouns.eTofhtehsee Numerical tests perform˛˛ed on elliptical Gaus˛˛sian-profile statisticalisotropy.Thecorrespondingestimatorscomputed features show that this elongation measure increases byaverages over thesphereare monotonously in the range [0,1] with the intrinsic eccen- tricitye∈[0,1]ofthefeatures.Whileitispossibletodefine 1 Na µX(a) = X ω(i),a alternative elongation measures, these numerical tests also N 0 a indicate that the chosen definition is not an arbitrary mea- Xi=1 “ ” sureofthenon-axisymmetryoflocalfeatures,butrepresents b 1 Na 2 1/2 aroughestimateoftheeccentricityofaGaussian-profilelo- σX(a) = (Na X ω0(i),a −µX(a) ) cal feature. Xi=1h “ ” i waveIlnetssiusminmtearreys,tinthgeinasnevaelyrasilsreosfpescitgsn.aFlisrstwlyit,hthestweearvaeblelet SbX(a) = 1 Na X ω0(i),a −µbX(a) 3 N 2 “ σX”(a) 3 decomposition enables one to identify the scales a of the a Xi=1 b physical processes which define the local feature of the sig- b 4 54 nalateachpointω0.Secondly,thesteerabilitytheoretically KX(a) = 1 Na X ω0(i)b,a −µX(a) −3. (6) gives access to local morphological measures. For the 2GD, Na 2 “ σX”(a) 3 theorientation,signed-intensityandelongation oflocal fea- Xi=1 b b 4 5 tures are defined. Finally, from the computational point of At each wavelet scale a, Na bstands for the total number view,thecalculationofadirectionalcorrelationateachanal- of valid pixels outside a given exclusion mask M which, a ysis scale is an extremely demanding task. The relation of by definition, identifies the pixels to be excluded from the steerability is essential to reduce the complexity of calcula- analysis (see Subsection 3.1). The values ω(i) identify the 0 tion of the wavelet coefficients when local orientations are centerof thesevalid pixels. considered (Wiaux et al. 2006b). Let us emphasize here that, even under the assump- tion of Gaussianity of the CMB temperature field, postu- lated for the simulations, none of the random fields associ- 2.2 Statistics for non-Gaussianity ated with these local morphological measures at each scale In the context of the concordance cosmological model, the is intrinsically Gaussian. This is simply due to the non- CMB temperature represents a realization of a statistically linearityofthedefinitions(3)fortheorientation,(4)forthe isotropic and Gaussian random field on the sphere. The signed-intensity, and (5) for the elongation. In particular, 4 Wiaux et al. bystatisticalisotropy,themeasureoforientationDF(ω ,a) fromtheNASALAMBDAarchive1.Thesemapsaremasked 0 shouldbeuniformlydistributedateachpointω andateach with the Kp0 mask (Spergel et al. 2007) that cuts the re- 0 wavelet scale a. gionsofbrightestgalacticemissionaroundthegalacticplane For each local morphological measure, statistics, and (≈ 20% of the sky), as well as the brightest galactic and wavelet scale, the value obtained for the data can be com- extragalactic point sources (≈ 5% of the sky). Zero values pared to the corresponding values for the simulations. The are assigned to the corresponding pixels. The instrumental percentiles corresponding to specific cumulative probabili- beam associated with the WMAP radiometers is described tiespinthesimulationsconsideredarecalculated forafirst byanisotropicwindowfunction,andtheinstrumentalnoise comparison with the value of the data. The percentile as- is to first order Gaussian, statistically anisotropic, and un- sociated with p = 50% defines the median value. Cumula- correlated. A map with better signal-to-noise ratio can be tive probabilities p = {15.865%,84.135%} are considered, obtainedbyanoptimalcombinationoftheeightforeground which formally correspond to the percentiles at one stan- cleaned and masked maps. At each pixel, this combination dard deviation (1σ) from the mean in a Gaussian distribu- isobtainedbyweightingeachmapbythecorrespondingin- tion. They define a first, innermost, region for the distri- versenoisevariance. Inorderto minimizeany errorcoming bution of percentiles around the median value. The exact from the cosmological dipole subtraction, the dipole out- valuesconsidered reflect themaximumprecision allowed by side themask is removed (Komatsu et al. 2003).This over- our sample of ten thousand simulations. Cumulative prob- allpre-processing proceduredefinestheso-called three-year abilities p = {2.275%,97.725%} are also considered, which WMAPco-addedCMBmap(Hinshawet al.2007),whichis formallycorrespondtothepercentilesattwostandarddevi- used in thesubsequentanalysis. ations(2σ)fromthemeaninaGaussian distribution.They Secondly, ten thousand simulations of the three-year define a second, middle, region for the distribution of per- WMAP co-added CMB map are considered to compare centiles around the median value. Again, the exact values the results of the analysis of the data to what is expected considered reflect the maximum precision allowed by our from the concordance model. Each simulation is produced sample of ten thousand simulations. Cumulative probabil- as follows. Sphericalharmonics coefficients of astatistically ities p = {0.5%,99.5%} are finally considered, defining a isotropic and Gaussian CMB realization are obtained from third, outermost, region for the distribution of percentiles the angular power spectrum determined by the cosmolog- aroundthemedianvalue.Ifthevalueofthedataforastatis- ical parameters of the three-year WMAP best-fit model ticsatagiven scaleishigher(lower) thanthemedian value (Spergel et al. 2007) with CAMB (Codefor Anisotropies in obtainedfromthesimulations,thesignificancelevelofade- theMicrowaveBackground2).Theobservationateachofthe tection is simply definedas the fraction of simulations with eightWMAPradiometersoftheQ,V,andWfrequenciesis a higher (lower) value than the data. The lower the sig- simulated by convolving that realization in harmonic space nificance level, the stronger the detection. Typically, in the withthecorrespondingisotropicwindowfunction.Eachmap following, a significance level below 0.5%, corresponding to isthentransformedtopixelspaceattheappropriateresolu- values outside the outermost region for the distribution of tion,and aGaussian, statistically anisotropic, and uncorre- percentilesaroundthemedianvalue,willbeassociatedwith latednoiserealizationisaddedwiththepropervarianceper a strong detection. pixel.ThisprovidessimulationsoftheCMB,asseenbythe eight radiometers at the different WMAP frequencies considered.Thesameprescriptionsasthosedescribedabovefor the data are then applied to produce a three-year WMAP co-added CMB map. 3 WMAP ANALYSIS Notice that the WMAP co-added CMB maps for the dataandsimulationsareinitiallyproducedinHEALPixpix- Inthis section, we firstly describethepre-processing proce- elization3 (Górski et al.2005)attheresolutionN =512, dure applied to the three-year WMAP CMB temperature side corresponding to maps with more than three million equal- data, as well as to the corresponding simulations produced area pixels with a spatial resolution of 6.87′. For the sake from theconcordance cosmological model. Secondly,we ex- of our analysis, which is applied at 17 scales of the 2GD pose the results of the application of the non-Gaussianity ′ wavelet, corresponding to angular sizes between 27.5 and analysis defined in the previous section on the three-year 30◦,themapsaredowngradedtotheresolutionN =256. WMAP co-added CMB data, notably highlighting a strong side This provides maps with a bit less than one million equal- detection in the excess kurtosis of the signed-intensity. ′ area pixels with a spatial resolution of 13.7. Also notice that, in pixels close to masked regions, the result of the directional correlation of a signal with a steerable wavelet is inevitably affected by the zero values of the Kp0 mask. 3.1 Data and simulations An exclusion mask M is therefore defined at each wavelet a Firstly, thefollowing pre-processing procedure is applied to scale a, identically on the data and simulations, in order to the three-year WMAP CMB temperature data before the exclude the affected pixels from the analysis (Vielva et al. non-Gaussianity analysis. The original maps of the eight WMAPradiometersattheQ,V,andWfrequencies(Q1and Q2at41GHz,V1andV2at61GHz,andW1,W2,W3,and W4 at 94 GHz) are corrected for foreground emissions con- 1 http://lambda.gsfc.nasa.gov/ tamination by a template fitting technique (Spergel et al. 2 http://camb.info/ 2007). The resulting foreground cleaned maps are available 3 http://healpix.jpl.nasa.gov/ Non-Gaussianity of WMAP data 5 0.2 0.03 0 9997..57 %% ppeerrcceennttiilleess 9997..57 %% ppeerrcceennttiilleess −0.2 84.1 % percentiles 0.025 84.1 % percentiles Excess kurtosis of signed−intensity−−−−1000−....12864 5120W05..35M . 9 A %%%%P cppppoeeee−rrrrccccaeeeednnnndtttteiiiilllleeeedssss data Variance of signed−intensity000.0..001125 5120W05..35M . 9 A %%%%P cppppoeeee−rrrrccccaeeeednnnndtttteiiiilllleeeedssss data −1.4 0.005 −1.6 −1.8 0 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 2GD wavelet half−width in arcminutes 2GD wavelet half−width in arcminutes Figure1.Excesskurtosisofthesigned-intensityofthethree-year Figure 2. Variance of the signed-intensity of the three-year WMAP co-added CMB data as a function of the 2GD wavelet WMAP co-added CMB data as a function of the 2GD wavelet half-widthinarangecorrespondingtoangularsizesbetween27.5′ half-widthinarangecorrespondingtoangularsizesbetween27.5′ and 30◦ on the celestial sphere. Data (black rhombi) are com- and 30◦ on the celestial sphere. Data (black rhombi) are com- paredwithpercentilesestablishedfromtenthousandstatistically paredwithpercentilesestablishedfromtenthousandstatistically isotropic and Gaussian simulations produced from the concor- isotropic and Gaussian simulations produced from the concordance cosmological model. Significance levels lie roughly below dance cosmological model. Significance levels lie roughly above 1.4%atthefourwaveletscalesa8,a9,a10,anda11,respectively 5% between the wavelet scales a2 and a17, corresponding to an- corresponding to angular sizes of 8.33◦, 10◦, 11.7◦, and 13.3◦. gularsizesbetween50′ and30◦.Atthewaveletscalea1 though, Thesignificancelevelreachesaminimumvalueof0.49%atscale corresponding to an angular size of 27.5′, the significance level a9.Thisidentifiesastrongdetectionofnon-Gaussianity,interms reaches 0%. This identifies a strong, but isolated detection of ofanexcess ofkurtosisinthesigned-intensity. non-Gaussianity, in terms of a too high variance of the signed- intensity. 2004), leaving N valid pixels from which statistics may be a estimated. kurtosis in the signed-intensity (see Figure 1). Also notice significancelevelsofroughly1.6% and1.2%,respectivelyat thewaveletsscalesa anda ,correspondingtoangularsizes 3.2 Non-Gaussianity analysis 4 5 ◦ ◦ of2.5 and3.33 .Theexcesskurtosisinthesigned-intensity Theresultsoftheapplicationofthenon-Gaussianityanaly- ofthedataateachofthesewaveletscalesiswelllowerthan sis of the local orientation, signed-intensity, and elongation themedianvaluedefinedbythesimulations,eventhoughit tothethree-yearWMAPco-addedCMBmapareasfollows. is not considered as anomalous. Let us recall that 17 scales of the 2GD wavelet are probed, Thirdly, the significance levels observed for the vari- corresponding to angular sizes on the celestial sphere be- ance of the signed-intensity are roughly above 5% between tween 27.5′ and 30◦. The complete list of the wavelet half- thewaveletscalesa anda ,correspondingtoangularsizes 2 17 widths θ considered in arcminutes reads: {13.7′, 25′, 50′, between 50′ and 30◦, the minimum value being reached at hw ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ 75, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, scale a . Again, these values lie well inside the middle re- 8 600′, 750′, 900′}. giondefinedaboveforthedistributionofpercentilesaround Firstly, the significance levels observed for statistics of the median value. We therefore conclude that no detection thelocal orientation and ofthelocal elongation reach mini- is obtained at those scales. At the wavelet scale a though, 1 ′ mumvaluesbetween5%and10%,andonlyatsomeisolated corresponding to an angular size of 27.5, the value of the scales. In other words, these values lie well inside the mid- variance is well higher than the median value defined by dle region defined above for the distribution of percentiles the simulations, and the significance level actually reaches aroundthemedianvalue.Wethereforeconcludethatnode- 0%. Formally, this represents a strong, but isolated detec- tection is obtained neither for the local orientation nor for tionofnon-GaussianityintheWMAPco-addedCMBdata, thelocal elongation. in terms of a too high variance of the signed-intensity (see Secondly,thesignificancelevels observedfor theexcess Figure 2). In Section 4, we suggest that it might originate kurtosis of the signed-intensity are roughly below 1.4% at in the presence of residual point sources in the data, and thefourwaveletscalesa ,a ,a ,anda ,respectivelycor- that it should therefore be discarded. Notice that no detec- 8 9 10 11 responding to angular sizes of 8.33◦, 10◦, 11.7◦, and 13.3◦ tion appearsneitherin themean norintheskewnessof the on the celestial sphere. The excess kurtosis in the signed- signed-intensity. intensityofthedataateachofthesewaveletscalesishigher In summary, the2GD wavelet gives access to the mea- than the median value defined by the simulations. The sig- suresoforientation,signed-intensity,andelongationoflocal nificancelevelreachesaminimumvalueof0.49%atscalea . features of the WMAP temperature data. But a strong de- 9 Theseresultsidentifyastrongdetectionofnon-Gaussianity tection of non-Gaussianity is only observed in the excess intheWMAPco-addedCMBdata,intermsofanexcessof kurtosis of the signed-intensity. This result actually sup- 6 Wiaux et al. ports the previous detection, with the axisymmetric Mex- and the corresponding ten thousand simulations are down- ican hat wavelet, of an excess of kurtosis in the wavelet co- graded to the resolution N = 32. This provides maps side efficientoftheWMAPtemperaturedata(Vielva et al.2004; withabitmorethantwelvethousandequal-areapixelswith Mukherjee & Wang 2004; Cruz et al. 2005). a spatial resolution of 1.83◦. For coherence, the initial non- Gaussianity analysis is reproduced for the signed-intensity, from the data and simulations preliminary downgraded to theresolutionN =32.Allconclusionsrelativetothefirst side 4 SYSTEMATIC EFFECTS four statistical moments remain obviously unchanged. The strong detection in the excess kurtosis is slightly enhanced. In this section, we firstly suggest that the high variance of The corresponding significance levels are roughly below 1% thesigned-intensity oftheWMAPtemperaturedataat the at the four wavelet scales a , a , a , and a , respectively smallest wavelet scales, and the corresponding isolated de- 8 9 10◦ ◦ 11 ◦ ◦ correspondingtoangularsizesof8.33 ,10 ,11.7 ,and13.3 tection, might originate in the presence of residual point onthecelestialsphere.Thesignificancelevelreachesamin- sourcesinthedata.Secondly,wediscardinstrumentalnoise, imum value of 0.26% at scale a . residual foreground emissions, and large-scale modulations 8 oftheCMBtemperaturefieldrelatedtosomeunknownsys- tematics, as possible origins of the excess of kurtosis in the 4.2 Other systematics signed-intensity. Letusrecallthepreviousdetectionofanexcessofkurtosisin thewaveletcoefficientoftheWMAPtemperaturedatawith the axisymmetric Mexican hat wavelet (Vielvaet al. 2004; 4.1 Residual point sources and lower resolution Mukherjee & Wang2004;Cruz et al.2005),aswellasprevi- Thevarianceofthesigned-intensityoftheWMAPco-added ousdetectionsofanomalies obtainedinthesigned-intensity CMB data is lower than the median value defined by the of the WMAP temperature data with the 2GD wavelet simulations between the wavelet scales a and a , corre- (McEwen et al.2007;Vielva et al.2007).Thecorresponding 6 17 spondingtoangularsizesbetween5◦ and30◦.Thisvariance analysesconcludedthatneitherinstrumentalnoisenorresid- isnotconsideredasanomalousthough,assignificancelevels ual foreground emissions are at the origin of the deviations arealwaysroughlyabove5%.Onthecontrary,thevariance observed. These conclusions were drawn from independent observed between thewavelet scales a and a ,correspond- analysesofthedataproducedbytheeightWMAPradiome- 1 5 ′ ◦ ing to angular sizes between 27.5 and 3.33 , is higher than ters at the Q, V, and W frequencies. These results suggest themedianvalue.Again,thevaluesofthevariancebetween that the wavelets and statistics used are rather insensitive the wavelet scales a and a are not considered as anoma- to instrumental noise and residual foreground emissions in 2 5 lous, as significance levels are always roughly above6%. As theWMAPtemperaturedata. Inthiscontext,eventhough already emphasized, at the wavelet scale a , the signifi- nosimilar analysis is performed here, weconcludethatnei- 1 cance level reaches 0%. The high variance at the smallest ther instrumental noise nor residual foreground emissions wavelet scales, in opposition with the behaviour at larger are likely to be at the origin of the excess of kurtosis ob- wavelet scales (see Figure 2), might originate in systematic served in the signed-intensity of the WMAP temperature effects such as the presence of residual point sources. Let data. us recall that the three-yearWMAP best-fit angular power Further investigations are proposed here, considering spectrum is obtained after correction for a non-zero best- somepossibleform ofunknownsystematics. Itwasrecently fit amplitude of a residual point sources power spectrum proposed that the WMAP data are possibly affected by a (Hinshaw et al. 2007). Consistently, residual point sources large-scale modulation. This modulation was primarily put were recently identified in the WMAP co-added CMB data forward asapossibleexplanationoftheNorth-Southasym- (López-Caniego et al. 2007). These residual point sources metry and low multipoles alignment of the WMAP data arehowevernotaccountedforintheWMAPco-addedCMB (Helling et al. 2006; Gordon 2007). In that framework, the dataanalyzedhere,whiletheyareaccountedforinthesim- WMAPdataareoftheformT(ω)×[1+f(ω)],whereT(ω) ulations based on the three-year WMAP best-fit angular stands for the CMB temperature on the sky, and f(ω) is a powerspectrum.Thiscouldindeedexplainthehighvariance modulationfunctioncontainingonlylowmultipolesl.Dipo- observed in the signed-intensity of the data at the smallest lar (l=1) and dipolar-quadrupolar(l={1,2}) modulation wavelet scales. Obviously,thecontributionof theseresidual functions providing the best-fit cosmological models to the point sources is negligible at larger wavelet scales. three-yearWMAP data were proposed. Let us remark that In the absence of a detailed model of the contribution the best-fit dipolar modulation used (Eriksen et al. 2007), of the residual point sources to the variance of the signed- as well as thebest-fit dipolar-quadrupolarmodulation used intensity,thecorrespondingisolateddetectionatthewavelet Spergel et al. (2007, see arXiv:astro-ph/0603449v1), were scale a is discarded, and the analysis is restricted to the notprimarilycomputedforthethree-yearWMAPco-added 1 waveletscalesbetweena anda ,correspondingtoangular CMBmapitself,buttheywereshownnottobesensitiveto 6 17 ◦ ◦ sizes between 5 and 30 . This does not affect theinterpre- the three-year WMAP data set and sky cut. We therefore tation of thedetection relative to theexcess kurtosis in the also considered them to be adequate for correction of the signed-intensity, which appears between the wavelet scales three-yearWMAP co-added CMB map. ◦ a and a , corresponding to angular sizes between 8.33 We have checked the stability of the excess of kurtosis 8 11 ◦ and 13.3 . For the subsequent analyses dedicated to search in the signed-intensity of the three-year WMAP co-added for the origin of the detection in the excess kurtosis of the CMB data relative to these modulations. Firstly, consider- signed-intensity,thethree-yearWMAPco-addedCMBmap ing thebest-fit dipolar-quadrupolar modulation, the strong Non-Gaussianity of WMAP data 7 detection in the excess kurtosis remains unchanged, if it is 0.5 not slightly increased. Just as for the analysis of the non- 99.5 % percentiles 97.7 % percentiles corrected three-year WMAP co-added CMB data, the sig- 0 84.1 % percentiles nsgcuifialalcerasnsaci8ze,eslae9vo,efals810.a3,r3ae◦n,rdo1au01g◦1,h,l1yr1es.b7pe◦el,cotwainved1l%y13cao.3tr◦rtehsopenofntohdueirncgweltaeovsetalineat-l ed−intensity−0.5 5120T05..h35 .r 9 e s%%%%ho ppppldeeeeerrrrccccdeeee WnnnnttttiiiiMlllleeeeAssssP co−added data sphere. The significance levels reach a minimum value of gn 0.22% atscalesa anda .Secondly,consideringthebest-fit of si dipolar modulatio8n, the9detection in the excess kurtosis is urtosis −1 bsleiglohwtly3%deactretahseefdo.uTrwheavseiglentifisccaalnecsea8le,vae9l,saa1r0e,aonndlyar1o1,uwghitlhy Excess k a minimum value of 0.60% at scale a . As already empha- −1.5 8 sized (Vielvaet al. 2007), a more precise definition of the modulation in terms of specific systematic effects would be requiredbeforestrongconclusionscanbedrawnfromtheap- −2 200 300 400 500 600 700 800 900 2GD wavelet half−width in arcminutes plication ofthecorresponding corrections. But,eventaking the results at face value, the proposed dipolar and dipolar- quadrupolarcorrectionsaretoberejectedaspossibleorigins Figure 3. Excess kurtosis of the signed-intensity of the three- year WMAP co-added CMB data as a function of the 2GD of the observed excess of kurtosis in the signed-intensity of wavelet half-widthina range corresponding to angular sizes be- theWMAP temperature data. tween5◦ and30◦ onthecelestial sphere.Data(redsquares)are Insummary,instrumentalnoiseandresidualforeground comparedwithpercentilesestablishedfromtenthousandstatisti- emissions are not likely to be at the origin of the excess callyisotropicandGaussiansimulationsproducedfromthe con- of kurtosis. Large-scale modulations of the CMB related to cordance cosmological model.The directions withananomalous someunknownsystematicsareexplicitlyrejectedaspossible signed-intensityareexcludedfromthestatisticalanalysis,inthe originsofthedetection.Thenon-Gaussianitydetectedinthe data as well as in each of the ten thousand simulations inde- excesskurtosisofthesigned-intensityoftheWMAPdatais pendently. Significance levels are roughly below 1% at the four therefore probably related to the CMB temperature field wavelet scales a7, a8, a9, and a10, respectively corresponding to itself. angularsizesof6.67◦,8.33◦,10◦,and11.7◦.Thesignificancelevel reachesaminimumvalueof0.28%atscalea8.Thisstillidentifies a strong detection of non-Gaussianity, in terms of an excess of kurtosisinthesigned-intensity. 5 CONFINEMENT AND DISCUSSION We here firstly recall the recent detection of an anomalous signed-intensities) close to the southern end of the eclip- distribution on the sky of anomalous signed-intensities in tic poles axis. The detection is confirmed at the neighbour the three-year WMAP co-added CMB data. We secondly wavelet scales a , a , and a , respectively corresponding 9 10 11 test,and tend toreject, thepossible confinementof theob- toangularsizesof10◦,11.7◦,and13.3◦.Instrumentalnoise, servedexcessofkurtosistothedirectionswithananomalous residual foreground emissions, as well as large-scale modu- signed-intensity. We finally discuss the detailed interpreta- lations of the CMB related to some unknown systematics, tion of our detections. are rejected as possible origins of the detection. The localized anomalous distribution of anomalous signed-intensities identified may therefore probably be imputed to the CMB 5.1 Local signed-intensity anomalies temperaturefield itself. Inaveryrecentanalysisofthethree-yearWMAPco-added CMB data with the 2GD wavelet, the distribution on the 5.2 Confinement analysis celestial sphereofdirectionswithasigned-intensityanoma- lousat99.865%(formallycorrespondingtothepercentilesat PostulatingtheGaussianity oftheCMB,onemayinterpret threestandarddeviations(3σ)fromthemeaninaGaussian theanomalous distributionofdirectionswithananomalous distribution) was observed to be anomalous (Vielva et al. signed-intensityasacleardeparturefromstatisticalisotropy 2007). At the wavelet scale a , corresponding to an angu- (Vielvaet al. 2007). But more generally, the anomaly ob- 8 ◦ lar size of 8.33 , the global significance level of that detec- served highlights a deviation of theCMB temperature field tion, defined as the fraction of the ten thousand simula- fromthewholeassumptionofstatisticalisotropyandGaus- tions with a number of anomalous directions higher than sianity. In the context of the present non-Gaussianity de- in the data, is 1.39%. The anomalous directions are essen- tection, the anomalous distribution of anomalous signed- tially distributed in three clusters in the southern galactic intensities previously identified represents a serious candi- hemisphere, identifying three mean preferred directions in datetoexplaintheexcessofkurtosisobservedinthesigned- theskyVielva et al. (2007, Figure 5). Afirst cold spot (i.e. intensityofthethree-yearWMAPco-addedCMBdata.This with negative signed-intensities) identifies with the anoma- ideaissupportedbythefactthatthetwodetectionsareob- ◦ ◦ louscold spot originally detectedat (θ,ϕ)=(147 ,209 )in served at the same wavelet scales. The hypothesis to check galactic spherical coordinates with the axisymmetric Mexi- istoknowifthewholenon-Gaussianityobservedisconfined canhatwavelet(Vielvaet al.2004;Cruz et al.2005).Asec- tothedirections with an anomalous signed-intensity,in the ondcoldspotliesveryclosetothesouthernendoftheCMB idea that the excess in the number of anomalous directions dipole axis. The third spot is a hot spot (i.e. with positive would bear thedeparturefrom Gaussianity. 8 Wiaux et al. The confinement analysis simply consists in reproduc- 5.3 Discussion ingthepreviousanalysisonthefirstfourstatisticalmoments of the signed-intensity of the three-year WMAP co-added Firstly, if the excess of kurtosis observed in the signed- CMB data, from which the directions with an anomalous intensity of the WMAP temperature data is related to signed-intensity are excluded. The only way to account for the CMB temperature field itself (see Section 4), the fact a possible bias introduced by the exclusion of the extremal that this non-Gaussianity is not confined to the direc- values of the data above a given threshold is to apply the tionswith ananomaloussigned-intensityseemsnatural.In- same exclusion process to each simulation independently. deed, independently of the modification of the basic infla- In particular, the coherence of the procedure can only be tionary scenario that might explain the non-Gaussianity achievedifallvaluesabovethethreshold,andnotonlypart of the CMB temperature field (Bartolo et al. 2004), the ofit,areidentifiedandexcludedfromthestatisticalanalysis, cosmological principle still implies its statistical isotropy, in the data and in the simulations4. Notice that, while the at least as a first approximation. Non-Gaussian pertur- anomalous signed-intensities were originally identified at a bations are therefore more naturally widely spread over thresholdof99.865%,weconsiderhereathresholdat99.5%. the whole sky. We also notice that the wavelet scales at Thislowering isperformed in ordertoavoid apossible neg- which the non-Gaussianity is observed are compatible with ative conclusion, relative to the confinement, simply dueto the size of CMB anisotropies due to topological defects the fact that directions in the immediate vicinity of the di- such as cosmic textures (Turok & Spergel 1990), or due to rectionsthresholdedarenotexcludedfromtheanalysis.For the Integrated Sachs-Wolfe effect, which is associated with completeness, eventhough thedetectionsareonly observed the time evolution of the gravitational potential of large atthefourwaveletscalesbetweena anda ,corresponding scale structures (Sachs& Wolfe1967;Rees & Sciama 1968; 8 11 to angular sizes between 8.33◦ and 13.3◦, the confinement Mart´ınez-González & Sanz 1990). Texture models suggest analysis is performed at all wavelet scales between a and thepresenceofanumberoftextureswithangularsizesabove a , corresponding to angular sizes between 5◦ and 306◦. 1◦,whichcaninducehotspotsorcoldspotsofcorresponding 17 The results of the analysis performed are as follows. angular size in the CMB (Turok & Spergel 1990). A recent The strong detection of an excess of kurtosis in the signed- Bayesian analysis (Cruz et al. 2007) showed that the cold intensityoftheWMAPtemperaturedataispreserved.The spot originally detected at (θ,ϕ) = (147◦,209◦) in galactic significance levels areroughly below 1% at thefour wavelet spherical coordinates with the axisymmetric Mexican hat scales a , a , a ,and a , respectively corresponding to an- wavelet, is satisfactorily described by a texture with an angular si7zes8of 69.67◦, 8.1303◦, 10◦, and 11.7◦ on the celestial gularsizeonthecelestialspherearound10◦.Otheranalyses sphere. The significance level reaches a minimum value of alsoshowed thatthetimeevolutionofthegravitationalpo- 0.28%atscalea (seeFigure3).Thevarianceofthesigned- tential of large scale structures such as voids might induce 8 intensity at each wavelet scale was well lower than the me- cold spotsin theCMB with angularsizes of severaldegrees dian value defined by the simulations before the threshold- on the celestial sphere (Mart´ınez-González & Sanz 1990). ◦ ◦ ing. It was not considered as anomalous as the significance The cold spot identified at (θ,ϕ) =(147 ,209 ) in galactic level reached a minimum value of roughly 5% at scale a . sphericalcoordinateswithanangularsizearound10◦ could 8 Thevarianceof thesigned-intensityat eachwavelet scale is actually be explained in terms of a void at a redshift z ≃1 still well lower than the median value defined by the sim- andwithadiameteraround300h−1Mpc(Inoue& Silk2006, ulations after the thresholding. It is still not considered as 2007; Rudnicket al. 2007). anomalous,eventhoughthesignificancelevelreachesamin- Secondly, as already emphasized, an excess of kurtosis imum value of roughly 1% at scale a . No detection ap- in the wavelet coefficient of the WMAP temperature data 8 pearsneitherinthemeannorintheskewnessofthesigned- waspreviouslydetectedwiththeaxisymmetricMexicanhat intensity. wavelet. Thenon-Gaussian deviation observed with theax- Insummary,removingthedirectionswithananomalous isymmetric Mexican hat wavelet is undoubtedly related to signed-intensitydoesnotsolvetheobserveddiscrepancybe- the present detection in the excess kurtosis of the signed- tween the WMAP temperature data and simulations. Con- intensity with the 2GD wavelet, notably because it is ob- sequently,takingtheCMBtemperatureangularpowerspec- served with a similar statistics at the same angular sizes trum of the concordance cosmological model at face value, on the celestial sphere. From this point of view, both de- wecanconcludethatthestrongdetectionintheexcesskur- tections support one another. But the axisymmetric Mexi- tosisofthesigned-intensityoftheWMAPtemperaturedata can hat wavelet also allowed the detection of the cold spot ◦ ◦ is not confined to thedirections with an anomalous signed- at (θ,ϕ) = (147 ,209 ) in galactic spherical coordinates, intensity. which was interpreted to be the exclusive origin of the ex- cessofkurtosisdetected(Cruz et al.2005).Onthecontrary, we have concluded that the detection in the excess kurto- 4 We therefore have to assume that, in the data, all extremal sis of the signed-intensity of the WMAP temperature data valuesofthebackgroundGaussiandistributionabovethethresh- withthe2GDwaveletisnotconfinedtothepreviouslyiden- old are excluded. The only formal reason for which extremal tified directions with an anomalous signed-intensity. Con- values outside the effectively observed directions with a signed- sequently, even though the two detections are similar, they intensity above the threshold might have been missed would be probablysimplydonotidentifythesamenon-Gaussiancon- thatsomenon-Gaussianitycoincidentallycompensatesfortheex- tent in the WMAPtemperature data. tremalvalueinthecorrespondingdirectiononthecelestialsphere. In that trivial case, one readily knows that the non-Gaussianity Finally,let usalso underlinethat thevaluesofthecos- observedintheWMAPdataisnotconfinedtothelocalizeddis- mological parameters are affected by uncertainties associ- tributionofanomalous signed-intensities. atedwiththelimitedprecision ofmeasurementoftheCMB Non-Gaussianity of WMAP data 9 temperature angular power spectrum. These uncertainties by comparison of the data with ten thousand statistically are associated with the cosmic variance, but also with sys- isotropic and Gaussian simulations produced from the con- tematiceffectssuchasinstrumentalnoiseandresidualfore- cordance cosmological model. No detection is made neither ground emissions. The simulations produced for our anal- in theorientation analysis norin theelongation analysis. A ysis are obtained from the angular power spectrum deter- strongdetectionismadeintheexcesskurtosisofthesigned- mined by the cosmological parameters of the concordance intensityoftheWMAPdata,withasignificancelevelbelow model (i.e. thethree-yearWMAP best-fit model). They do 0.5% at a wavelet scale corresponding to an angular size ◦ not account for thequoteduncertainties. Consequently,be- around 10 , and confirmed at neighbour scales. This sup- fore giving full credit to our conclusions, a deep analysis ports a previous detection of an excess of kurtosis in the should be performed to check the stability of the various waveletcoefficient oftheWMAPdatawiththeaxisymmet- detections considered when the WMAP temperature data ricMexican hat wavelet.Anisolated detection isalso made are compared with simulations produced from any possi- inthevarianceofthesigned-intensityatthesmallestwavelet ble angular power spectrum inside the experimental error scale.Systematiceffectssuchasresidualpointsourcesinthe bars. Formally, any of our conclusions might be affected by WMAPco-addedCMB dataaresuggested tooriginate this thisfurtheranalysis,fromthefactthatthenon-Gaussianity anomaly, which is consequently simply discarded. observed in the WMAP temperature data is not confined Instrumental noise and residual foreground emissions to the directions with an anomalous signed-intensity, up to are not likely to be at the origin of the detection in the the mere detection of an excess of kurtosis in the signed- excess kurtosis of the signed-intensity. Large-scale modu- intensity.However,suchananalysis would beveryinvolved lations of the CMB related to some unknown systematics and is not produced here. Onthe onehand,the stability of are explicitly rejected as possible origins of the detection. the detection of an excess of kurtosis in the wavelet coeffi- The observed non-Gaussianity is therefore probably to be cient oftheWMAPtemperaturedatawith theaxisymmet- imputedtotheCMB temperaturefield itself, therebyques- ric Mexican hat wavelet was suggested (Vielva et al. 2004). tioning the basic inflationary scenario upon which the con- The same conclusion probably holds for the present detec- cordance cosmological model relies. In this context, taking tion of an excess of kurtosis in the signed-intensity of the the CMB temperature angular power spectrum of the con- WMAP temperature data with 2GD wavelet. On the other cordance cosmological model at face value, further analy- hand, the confinement analysis itself is based on the pre- sis also naturally suggests that this non-Gaussianity of the vious detection of the distribution of directions with an WMAP temperature data is not confined to the localized anomalous signed-intensity(Vielva et al. 2007). Atpresent, distribution of anomalous signed-intensities. However, this no analysis confirmed the stability of this distribution rel- last result, in particular, might be sensitive to uncertain- ative to the uncertainties on the cosmological parameters. tiesaffectingthecosmological parameters. Furtheranalyses A possible excess of power in the concordance model rela- should be performed before giving it full credit. tive to the WMAP temperature data (Spergel et al. 2003; Hinshaw et al. 2007; Monteser´ın et al. 2007) might imply that a part of the distribution on the celestial sphere of directions with an anomalous signed-intensity was actually ACKNOWLEDGMENTS notdetected.Thiswouldprobablynotquestionthefactthat The work of Y.W. is funded by the Swiss National Science thisdistributionisanomalousatthewaveletscalesbetween ◦ Foundation (SNF) under contract No. 200021-107478/1. a and a , corresponding to angular sizes between 8.33 8 11 ◦ Y.W.isalsopostdoctoralresearcheroftheBelgianNational and 13.3 , but would simply suggest that the global signif- Science Foundation (FNRS). The work of P.V. is funded icance level for the detection was underestimated. However throughanI3PcontractfromtheSpanishNationalResearch insuchacase,theconfinementanalysisitself, whichexplic- Council (CSIC). P.V., R.B.B., and E.M.-G. are also sup- itly requires the exclusion of all the extremal values above portedbytheSpanishMCYTproject ESP2004-07067-C03- a given threshold, both in the data and in the simulations, 01.Theauthorsacknowledge theuseof theLegacy Archive might not be performed anymore. Noconclusion relative to forMicrowaveBackgroundDataAnalysis(LAMBDA).Sup- the possible confinement of the non-Gaussianity observed port for LAMBDA is provided by the NASA Office of to the directions with an anomalous signed-intensity could SpaceScience.Theauthors also acknowledge theuseof the therefore be reached. 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