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Draftversion September19,2012 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 PROBING WMAP COLD SPOT THROUGH LOCAL MINKOWSKI FUNCTIONS Wen Zhao KeyLaboratoryforResearches inGalaxiesandCosmology,UniversityofScienceandTechnology ofChina,Hefei,230026, China NielsBohrInstituteandDISCOVERYCenter,Blegdamsvej17,2100Copenhagen, Ø,Denmark Draft versionSeptember 19, 2012 ABSTRACT 2 Inthispaper,weintroducethelocalMinkowskiFunctionsandapplythemasthestatisticstostudy 1 the local properties of WMAP Cold Spot (CS) at different scales. We find that these local statistics 0 alwaysexcessatthescaleofR 5◦comparingwithotherspotsofWMAPdata,whichclearlypresents 2 ∼ thenon-GaussianityofCSanditscharacteristicscale. Meanwhile,wefindthatthecosmictexturecan p excellently explain all the excesses in these statistics, which supports the cosmic texture explanation e of WMAP CS. S Subject headings: cosmicmicrowavebackgroundradiation—earlyuniverse—methods: dataanalysis — methods: statistical 8 1 1. INTRODUCTION CS is prefer a large-scale non-Gaussian structure to a ] combination of some small structures. In this paper, we O Soon after the release of observations of the NASA shallfocus on the same topic by using the different local Wilkinson MicrowaveAnisotropy Probe (WMAP) satel- C statistics. We introduce the local Minkowski Functions lite on the Cosmic Microwave Background (CMB) tem- . (MFs),andapplythemtotheWMAPdata,inparticular h perature and polarization anisotropies, some anomalies theWMAPCS.Wefindthatcomparingwithotherspots p in CMB field have also been reported (Bennett et al. in WMAP data, these MFs around CS have significant - 2011). Among these, an extremely Cold Spot (CS) cen- ◦ o ◦ ◦ anomalitiesatthescalewithR 5 ,whichclearlyshows tered at the position (l =209 , b= 57 ) and a charac- ∼ r ◦ − the non-Gaussian signal of CS and indicates its charac- t teristic scale about 10 in the Galactic Coordinate was s detectedintheSphericalMexicanHatWavelet(SMHW) teristic scale. However, in the larger or smaller scales, a the MFs of CS are all quite normal. Meanwhile, similar non-Gaussian analysis (Vielva et al. 2004). Comparing [ to (Zhao2012), we find that these localnon-Gaussianity with the distribution derived from the isotropic and of WMAP CS can be excellently explained by a cosmic 1 Gaussian CMB simulations, due to this CS, the SMHW v coefficients of WMAP data have an excess of kurtosis texture. If subtracting this texture from WMAP data, 1 (Cruz et al. 2005). The non-Gaussian CS has also been all these anomilies of MFs disappear. So our local anal- 2 confirmed by using some other statistics (Cruz et al. ysis of the CS in this paper also strongly supports the 0 2007a, 2005; Cayon et al. 2005; Naselsky et al. 2010; cosmic texture explanation. 4 Therestofthepaperisorganizedasfollows: InSection Zhang & Huterer 2010; Vielva 2010). At the same . 2, we introduce the WMAP data, which will be used in 9 time, some analyses on WMAP CS have also been the analysis. In Section 3, we define the local MFs and 0 done, such as the non-Gaussian tests for the different applythemintoWMAPdata. Section4summarizesthe 2 detectors and different frequency channels of WMAP main results of this paper. 1 satellite (Vielva et al. 2004; Cruz et al. 2005), the in- v: vestigation of the NVSS sources (Rudnick et al. 2007; 2. THE WMAP DATA Smith & Huterer 2010), the survey around the CS i X with MegaCam on the Canada-France-HawaiiTelescope In our analysis, we shall use the WMAP data including the ILC7 map and the NILC5 map. The WMAP in- (Granett et al. 2009), the redshift survey using VIMOS r a onVLT towardsCS (Bremer et al. 2010), and the cross- strument is composed of 10 DAs spanning five frequen- cies from 23 to 94 GHz (Bennett et al. 2003). The in- correlationbetweenWMAP andFaradaydepthrotation ternal linear combination (ILC) method has been used map (Hansen et al. 2012). by WMAP team to generate the WMAP ILC maps Since then, various alternative explanations for the (Hinshaw et al. 2007; Gold et al. 2011). The 7-year ILC CS have been proposed, including the possible fore- (writtenas“ILC7”)mapisaweightedcombinationfrom grounds(Cruz et al.2006;Hansen et al.2012),Sunyaev- allfiveoriginalfrequencybands,whicharesmoothedtoa Zeldovich effect (Cruz et al. 2008), the supervoid in the commonresolutionofonedegree. Forthe5-yearWMAP Universe (Inoue & Silk 2006, 2007; Inoue 2012), and the data,in(Delabrouille et al.2009)theauthorshavemade cosmic texture (Cruz et al. 2007b, 2008). Due to the ahigherresolutionCMBILCmap(writtenas“NILC5”), fact that, nearly all the explanations of CS are related animplementationofaconstrainedlinearcombinationof to the local characters of the CMB field, the studies on the channels with minimum errorvarianceon a frame of the local properties of CS are necessary. In the previ- spherical called needlets1. In this paper, we will con- ous work (Zhao 2012), we have studied the local non- sider both these two maps for the analysis. Note that Gaussian properties of WMAP CS by the local mean these WMAP data have the same resolution parameter temperature,variance,skewnessand kurtosis. We found the excesses of the local variance and skewness in the ◦ 1 The similarmapfor7-year WMAP data isrecently gotten in large scales with R > 5 , which implies that WMAP (Basak&Delabrouille2012). 2 N = 512, and the corresponding total pixel number the cold/hot disk-like structure in the CMB. However, side N =3145728. they found these statistics are noise-dominated for the pix WMAPCS.ForasingleWMAPorPLANCKresolution 3. APPLYING LOCAL MINKOWSKI FUNCTIONS map, the method can only detect the highly prominent TO WMAP DATA disk, i.e. extremely cold or hot disk with quite large area. These can be easily understood as follows: since 3.1. Minkowski Functions the MFs are constructed on the global sky, the local MFs characterizethe morphologicalproperties of con- non-Gaussianity of the cold/hot spot in the relatively vex, compact sets in an n-dimensional space. On small scale has been diluted to be too small. the 2-dimensional spherical surface S2, any morpho- logical property can be expanded as a linear combi- 3.2. Local Minkowski Functions nation of three MFs, which represent the area, cir- To study the local properties of WMAP CS, similar cumference and integrated geodesic curvature of an ex- to the previous works (Bernui & Reboucas 2009, 2010, cursion set (Schmalzing & Gorski 1998). For a given 2012; Zhao 2012), in this paper we shall define the local threshold ν, it is convenient to define the excursion MFs as the statistics of CMB field. For a given full- set Q and its boundary ∂Q of a smooth scalar field u as νfollows: Q = x Sν2 u(x)>ν and ∂Q = sky WMAP data with Nside = 512 (ILC7 or NILC5), ν ∈ | ν we smooth them using a Gaussian filter with a smooth- x S2 u(x)=ν . Th(cid:8)en, the MFs v ,(cid:9)v and v can 0 1 2 ing scale of θ . Since MFs are sensitive to the smooth- ∈ | s b(cid:8)e written as (Sch(cid:9)malzing & Gorski 1998), ing scale of a density field and thereby we can obtain a variety of information from density fields by using dif- da dℓ dℓ κ v (ν):= , v := , v := , ferent smoothing levels. In this paper, we focus on both 0 Z 4π 1 Z 16π 2 Z 8π2 ′ Qν ∂Qν ∂Qν ILCfieldssmoothedbysixdifferentsmoothingscales10, (1) 20′, 30′, 40′, 50′, 60′. Then, we can construct the cor- where da and dℓ denote the surface element of S2 and responding full-sky maps: (x ) and (x ) defined in 1 k 2 k the line element along ∂Qν, respectively. And κ is Eq. (3), where u is the corUrespondingUILC temperature the geodesic curvature. Given a pixelated map with anisotropic map. field u(xi), these MFs can be numerically calculated by Now, we can define the local MFs. Let Ω(θj,φj;R) the formulae (Schmalzing & Gorski 1998; Lim & Simon be a spherical cap with aperture of R degree, centered 2012) at (θ ,φ ). The local MFs v (ν) (i = 0,1,2) in this cap j j i can be calculated by using Eq. (2), which are denoted 1 Npix as vj(ν;R) in the rest of this paper. But here the sum v (ν)= (ν,x ), (i=0,1,2), (2) i i Npix XIi k only counts the pixels inside the cap Ω(θj,φj;R). Note k=1 that in our calculation, the binning range of threshold ν where issettobe-3.0to3.0with24equallyspacedbinsofν/σ (σ is the standard deviation of u-field in this cap) per 0(ν,xk):=Θ(u ν), each MF. In order to quantify the same kind of MFs by I − δ(u ν) a singlequantity, following(Hikage et al. 2009), for each 1(ν,xk):= − 1(xk), i (the type of MFs), j, R (the cap) we can define the I 4 U { } χ2 as follows: δ(u ν) (ν,x ):= − (x ), (3) I2 k 2π U2 k χ2 = [vj(ν ;R) vth(ν ;R)]Σ−1 i α − i α αα′ (x ):= u2 +u2 , Xαα′ UU12(xkk):=q2u;θ;θu;φu;θ;φφu−2 u+2;θuu2;φφ−u2;φu;θθ. where α and α′ de×no[vtije(νthα′e;Rbi)n−ninvitgh(nνuαm′;bRe)r],of thres(4h)- ;θ ;φ old values. vth(ν;R) is the theoretical value of vj(ν;R), i i Note that u denotes the covariant differentiation of u whichisindependentofthesuperscriptj. Σisthecorre- ;i with respect to the coordinate i. The delta function in sponding covariance matrix. Although, in principle the theseformulaecanbenumericallyapproximatedthrough theoreticalvaluesvth(ν;R)fortherandomGaussianfield i a discretization of threshold space in bins of width ∆ν can be calculated by the analytical formulae (Tomita by the stepfunction δ (x) = (∆ν)−1[Θ(x + ∆ν/2) 1986; Schmalzing & Gorski 1998), there are systemati- N − Θ(x ∆ν/2)]. TheexpectationvaluesofthreeMFsfora caldifferencesfromthe numericalresultsdue to the bin- − Gaussianrandomfieldarealsoderivedin(Tomita1986), ning of threshold (Lim & Simon 2012). In this paper, which have been explicitly expressed in equations (14) we avoid this problem by replacing the theoretical value and (15) in (Schmalzing & Gorski 1998). vth(ν;R)bythequantity vj(ν;R) ,whichistheaverage These MFs have been applied by cosmologists vailue of all vj(ν;R) withhbi >30i◦. And the covariance to look from derivations from Gaussianity of the i | j| matrix Σ can also be numerically calculated by these perturbations in the CMB (Winitzki & Kosowsky quantities vj(ν;R). Note that the Galactic plane have 1998; Schmalzing & Gorski 1998; Novikov et al. i been excluded to reduce the effect of foreground resid- 1999; Eriksen et al. 2004; Hikage et al. 2006, 2008; Komatsu et al. 2009; Hikage et al. 2009; Matsubara uals. Hereafter, we denote this χ2 quantity as Xij(R). 2010; Hikage & Matsubara 2012). In particular, in Clearly, the values Xj(R) obtained in this way for each i (Lim & Simon 2012) the authors usedthe MFs to probe cap can be viewed as a measure of non-Gaussianity in 3 Figure 1. X¯0(R) maps (upper), X¯1(R) maps (middle), and Figure 2. X¯0(R) maps (upper), X¯1(R) maps (middle), and X¯2(R) maps (lower) for ILC7 data with θs = 60′ smoothing. In X¯2(R) maps (lower) for NILC5 data with R = 2◦. In the left the leftpanels, wehave usedR=2◦, andinrightpanels, R=5◦ panels, the NILC5 map is smoothed by θs = 10′, and in right ischosen. panels,θs=40′ smoothingisapplied. the direction of the center of cap (θ ,φ ). For a given smoothing case, from the left middle and left lower pan- j j aperture R, we scan the celestial sphere with evenly dis- els we find the clear structure of N , i.e. the effective obs tributed spherical caps, and build the X (R)-, X (R)-, observationsofWMAPforeachpixels. Wefindasmaller 0 1 X (R)-maps. In our analysis, we have chosen the loca- N , which follows a smaller pixel-noise variance, corre- 2 obs tions of centroids of spots to be the pixels in Nside =64 spondsalargerX¯j. So,thesetwolocalMFstatisticscan i resolution. BychoosingdifferentRvalues,onecanstudy also be used to search for the morphology of the pixel- the local properties of CMB field at different scales. noisevariance,whichhasbeenhiddeninthetemperature SametotheV(R)-map(orS(R)-,K(R)-maps)defined anisotropy map. We leave this as a future work. Here, in(Zhao2012),herewealsofindthatXj(R)alwaysmax- in order to reduce the effect of them, we should choose i imize at the edge of the circles, rather than the center a larger smoothing parameter θ as in the right panel s of circles. To overcome this problem and localize the in Fig. 2, where we find the morphology of the pixel- non-Gaussian sources, we define the average quantities noise variance disappears. But the non-Gaussianity in the Galactic place is still there, due to the heavy con- 1 Npix tamination caused by foreground residuals. X¯ij(R):= N Xij(R), (i=0,1,2), (5) Choosing a common smoothing parameter θs = 60′, pix Xj=1 in Fig. 3 we plot the X¯i maps for R = 2◦ (left) and ◦ R = 5 (right). By comparing with those in Fig. 1, we where N is again the pixel number in the jth cap. pix find that although several non-Gaussian point sources We apply the method to the ILC7 data by choosing R = 2◦ and θ = 60′. The X¯ maps are presented in and the foreground residuals in Galactic plane are still s i there, NILC5 is much cleaner than ILC7, as claimed in Fig. 1 (left panels), which clearly show that these local (Delabrouille et al. 2009). statistics, in particular third MF v , are very sensitive 2 to the foreground residuals and various point sources. 3.3. Local Properties of WMAP Cold Spot The Galactic plane is clearly presented, which is the non-Gaussian area caused by the foreground residuals We now focus on the local properties of WMAP CS ◦ ◦ in ILC7 map. In addition, two important point sources at (l = 209 , b = 57 ) in Galactic Coordinate. To − at (l = 209.5◦, b = 20.1◦) and (l = 184.9◦, b = 5.98◦), quantifythe propertiesofCS,wedefinethe probabilities as well as several small ones, also clearly found in the p-values as follows, X¯ maps. So we expect these local statistics with small i p (R):=Prob X¯j(R)>X¯0(R) , (i=0,1,2), (6) R values can be used to identify the point sources and i i i (cid:16) (cid:17) foreground residuals, which will be discussed in a sepa- ratepaper. IfwechooseR=5◦,fromtherightpanelsin whereX¯j(R)arethevaluesofspotsintheregionof b > Fig. 1 we find the similar results, except for some non- 30◦ in X¯i(R) map, and X¯0(R) is that for CS. Note|th|at i i Gaussianities of small point sources, which have been theGalacticplanehasbeenexcludedtoremovetheeffect diluted for this larger R case. of foreground residuals. So for any given X¯ (R) map, a i Let us turn to the NILC5 map, which has the much smaller p (R) value indicates a larger non-Gaussianity i higher resolution than ILC7. We firstly study effect of around CS. ◦ different levels of smoothing. By adopting R = 2 , in InFig. 4, we plot the probabilities p (R)as a function i Fig. 2weplotthe X¯ mapsforθ =10′ (leftpanels)and of R for ILC7 map with different levels of smoothing. i s ′ θ =40 (right panels). Interesting enough for the lower We find that p (R) > 0.01 is holden for the cases with s i 4 100 100 10−2 10−2 10−4 10−4 0 5 10 15 0 5 10 15 MFs100 100 or s f10−2 10−2 e u P−val10−40 5 10 15 10−40 5 10 15 100 100 10−2 10−2 10−4 10−4 0 5 10 15 0 5 10 15 rad (deg) rad (deg) Figure 4. Theprobabilitiespi(R)versusR-valuesforILC7based X¯imaps. Fromupperlefttolowerright,thesmoothingparameter θs=10′, 20′,30′, 40′, 50′,60′ areused. In each panel, black line isfori=0,bluelineisfori=1andredlineisfori=2. Figure 3. X¯0(R) maps (upper), X¯1(R) maps (middle), and X¯2(R) maps (lower)forNILC5data withθs=60′ smoothing. In the leftpanels, wehave usedR=2◦, andinrightpanels, R=5◦ ischosen. 100 100 R 4◦ and R 6◦, which shows that CS in these scales 10−2 10−2 ≤ ≥ isquitenormalcomparingwithotherspotsinILC7data. However, when R 5◦, the p (R) values are system- 10−40 5 10 15 10−40 5 10 15 i atically small for al∼l the three MFs, which means that MFs100 100 CofS ca1u0s◦e,swthhiechsiigsniinfidcaenptenndoenn-tGoafutshsieansmityooitnhitnhgelsecvaelles s for 10−2 10−2 o1,f wI∼LhCer7emwaepfi.nTdhCesSeifseaqtuuirteesncoarnmaallsowhbeenfoRun=d2in◦ (Fleigft. P−value10−40 5 10 15 10−40 5 10 15 panels). However, when R = 5◦ (right panels), X¯ij are 100 100 significantly large around CS position for all the three MF maps. These results are consistent with other au- 10−2 10−2 thors’ discovery in the SMHW non-Gaussian analysis 10−4 10−4 (Vielva et al. 2004; Cruz et al. 2005). So we conclude 0 5 10 15 0 5 10 15 rad (deg) rad (deg) that the local MF statistics can clearly present the non- GaussianityofCSanditscharacteristicscale. Moreover, Figure 5. SamewithFig. 4,butforNILC5. ◦ the normality of CS in the smallscales, i.e. R<4 , also impliesthattheWMAPCSisprefertoalarge-scalenon- Gaussian structure, rather than a combination of some small structures. Same results are also obtained from ′ the NILC5 analysis (see Fig. 5): so long as θs > 30 100 100 is satisfied, where the effect of pixel-noise variances has been well suppressed, the CS causes the significant non- 10−2 10−2 ◦ GaInus(sCiarnuiztyeattatl.h2e0s0c7able, 2o0f0R8)∼, b5y.studying the temper- 10−40 5 10 15 10−40 5 10 15 ature and areaof CS, the authors found that the cosmic MFs100 100 texture, rather than the other explanations, provides an or excellentinterpretationfortheWMAPColdSpot. Here, s f10−2 10−2 e u wweorskhaisllcsotnusdisyteifntthweitlohctahleancoosmmailcietsexoftuCreSefoxupnladnaintiothni.s P−val10−40 5 10 15 10−40 5 10 15 The profile for the CMB temperature fluctuation 100 100 caused by a collapsing cosmic texture is given by 10−2 10−2 ε if ϑ ϑ∗ ∆T =  q1+4(ϑϑc)2 ≤ (7) 10−40 5 10 15 10−40 5 10 15 T − εe−2ϑ12c(ϑ2+ϑ2∗) if ϑ>ϑ∗ Figure 6. Same toradF (idge.g)4, but the cosmic texrtaudr (edesgt)ructure has  2 beensubtracted intheoriginalILC7map. where ϑ is the angle from the center. ε is the amplitude parameter,and ϑc is the scale parameter. ϑ∗ =√3/2ϑc. 5 100 100 We appreciate useful discussions with P. Naselsky, J. Kim, M. Hansen and A.M. Frejsel. We acknowledge the 10−2 10−2 use of the Legacy Archive for Microwave Background 10−4 10−4 Data Analysis (LAMBDA). Our data analysis made 0 5 10 15 0 5 10 15 the use of HEALPix (Gorski et al. 2005) and GLESP MFs100 100 (Doroshkevichet al. 2005). This work is supported by or NSFCNo. 11173021,11075141andprojectofKnowledge es f10−2 10−2 Innovation Programof Chinese Academy of Science. u P−val10−40 5 10 15 10−40 5 10 15 REFERENCES 100 100 10−2 10−2 Basak,S.&Delabrouille,J.2012,MNRAS,419,1163 10−4 10−4 Bennett, C.L.etal.,2003,ApJS,148,1 0 5 10 15 0 5 10 15 Bennett, C.L.etal.,2011,ApJS,192,17 rad (deg) rad (deg) Bernui,A.&Reboucas, M.J.2009,Phys.Rev.D,79,063528 Figure 7. SamewithFig. 6,butforNILC5. Bernui,A.&Reboucas, M.J.2010,Phys.Rev.D,81,063533 Bernui,A.&Reboucas, M.J.2012,Phys.Rev.D,85,023522 Bremer,M.N.,Silk,J.,DaviesL.J.M.&Lehnert,M.D.2010, MNRAS,404,L69 Cayon,L.,Lin,J.&Treaster,A.2005,MNRAS,362,826 By the Bayesian analysis, the texture parameters were Cruz,M.,Martinez-Gonzalez, E.,Vielva,P.&Cayon, L.2005, obtained ε=7.3+2.5 10−5 andϑ =4.9+2.8deg at 95% MNRAS,356,29 −3.6× c −2.4 Cruz,M.,Tucci,M.,Martinez-Gonzalez, E.&Vielva,P.2006, confidence (Cruz et al. 2007b). 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