2 0 0 3MNRAS.344. .686C 2 12 1 Anisotropy Probe(WMAP)project(Bennettetal.2003a)haverede- *E-mail: [email protected] phase information.Andwhilethepowerspectrumisaverypower- Accepted 2003June2.ReceivedMay30;inoriginalformMarch4 Department ofAstrophysicalSciences,PrincetonUniversity,PeytonHall,IvyLane,Princeton,NJ08544,USA Wesley N.Colley*andJ.RichardGott,III Mon. Not.R.Astron.Soc.344,686-695(2003) ture ofadensity(ortemperature) distribution inthreedimensions Dickinson (1986)directlytestsfor theGaussianrandom-phasena- terized bysphericalharmoniccoefficientswithGaussiandistributed in theCMB,atresolutionmeasuredbyWMAP,willbecharac- Namely, standardinflation(e.g.Guth1981;Albrecht&Steinhardt information forcharacterizingtheprimordialdensityfluctuations. ai coefficientsthemselves.Thesephases,however,containcritical it does(explicitly)removeanyphaseinformationcontainedinthe ful toolforassessingcosmologicalparameterssuchasCLand£2, t andmintheYexpansionofCMBsky,summedoverval- the powerspectrum,whereonecalculatesproductsofspheri- Hubble Constant,Hq,andthecosmologicalconstant,Q(Spergel The experimenthasresultedindramaticallyimprovedconstraints Mason etal.2003,CBI;Kovac2002,DASI)isseveralhundred. the orderof500intermsresolutionelements,andskycover- the onlypreviousfull-skydataset(COBE,Smootetal.1992)isof The greatlyanticipatedresultsfromtheWilkinsonMicrowave fined thestateofartincosmicmicrowavebackground(CMB) ues toformCf).Thisa¿a*product,however,explicitlyremoves cal harmoniccoefficientsa£andtheircomplexconjugates(foreach et al.2003). de Bernardisetal.2000,Boomerang;Stompor2002,Maxima; CMB atunprecedentedangularresolution.Theimprovementover opportunity todatetestthathypothesis. amplitudes andrandomphases.TheWMAPdataprovideourbest on cosmologicalparameters,suchasthematterdensity,Cl, age improvementoverballoonandground-basedexperiments(e.g. science. TheWMAPprojecthasprovidedfull-skycoverageofthe 1 INTRODUCTION Genus topologyofthecosmicmicrowavebackgroundfromWMAP DepartmentofAstronomy,UniversityVirginia,POBox3818,Charlottesville,VA22903,USA 1982; Linde1982,1983)predictsthatthetemperaturefluctuations m mA im A mím m m The genustopologymethoddeveloped byGott,Melott& The toolofchoiceforassessingthecosmologicalparametersis © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem We haveindependentlymeasuredthegenustopologyoftemperaturefluctuationsin A genusanalysisoftheWMAPdataindicatesconsistencywithGaussianrandom-phaseinitial ABSTRACT Key words:cosmicmicrowavebackground-cosmology:observations. conditions, aspredictedbystandardinflation. cosmic microwavebackgroundseenbytheWilkinsonMicrowaveAnisotropyProbe(WMAP). but alsotocomparethegenusmeasuredfromdatadirectly by thegenusstudy. real dataset.Theyfoundthatthesetdoesnotdepartsig- profiles, tosimulatetheresultsineachfrequency,andapplied map projectionandthestereographic mapprojection. the nuancesofcarryingoutgenus calculationontheHEALPix two dimensions.Furthermore,we provide somedetailsconcerning the theoreticalgenuscurveforstructures onasphere.Ratherthan tion), andfoundconstraintsonGaussianitysimilartothoseprovided nificantly fromthegenusofsimulatedGaussianrandom-phase this, theycarriedoutalargenumberofsimulationstheCMB,in maps (Gottetal.1992;Park,Gott&Choi2001a;Hoyle,Vogeley mensions (Adler1981;Melottetal.1989).Coles(1988)indepen- compare directlywiththetheoretical predictionforthegenusin Gaussianity bytestingotherMinkowskifunctionals(Minkowski data sets,verymuchinagreementwiththeresultspredictedfor data set.Foreachofthesesimulations,theycomputedthegenus KpO (Bennettetal.2003b)mask,justasonewouldwiththereal which thesphericalharmoniccoefficientsweredrawnfromaGaus- 2001b). Watts&Coles(2003),Chiang(2000),amongmany data sets:onredshiftslices(Parketal.1992;Colley1997),sky dimensional casehasbeenstudiedforavarietyofcosmological dently developedanequivalentstatisticintwodimensions.Thetwo- are consistentwiththeGaussianrandom-phasehypothesis.Todo as lookingattheFouriermodesdirectly. others, haveinvestigatedothermethodsformeasuringphases,such Kogut 1993;etal.1996;Colley,Gott&Park & Gott2002);andontheCMB,inparticular(Smootetal.1992; Weinberg 1986;Gott,&Melott1987)orintwodi- simulating Gaussianrandom-phase realizationsoftheCMB,we (as definedbyMelottetal.1989)andcompareditwiththatofthe sian random-phasedistribution.Theythenusedtheirknownbeam sky (Komatsuetal.2003),anddemonstratedthattheWMAPresults (Adler 1981;Gott,Melott&Dickinson1986;Hamilton,Gott WMAP byParketal.(1998).Bennett(2003b)furtherexplored 1903) andthebispectrum(relatedtothree-pointcorrelationfunc- We seek,first,toconfirmthisresultusingourownmethods, The WMAPteamhasrecentlymeasuredthegenusof © 2003RAS 2 0 0 3MNRAS.344. .686C 1 1 2 beamwidths), respectively. measured theanisotropyatfrequenciesof23,33,41,61and94GHz leased its1-yrdatasetinseveraldifferentskymapsmeasuredat The WilkinsonMicrowaveAnisotropyProbeprojectteamhasre- 2 WMAPOBSERVATIONS that hotspotsareredandcoldspotsbluewithatotalrangeintemperature the GalacticCentreatcentre.Thecolourmapisalinearscale,chosensuch Figure 1.TheWMAPdata(internallinearcombination)inthenorth(top) varying frequencyandangularresolution.Namely,theprojecthas of —200to+200|aK.Themeantemperaturecontourisinwhite. and south(bottom)Galactichemispheres,instereographicprojection,with e 2003RAS,MNRAS 344,686-695 at angularresolutionsof0?82,0?62, 0?49,0?33and0?21(FWHM TheWMAPhomepageisathttp://map.gsfc.nasa.gov/ WMAPdatasetsareavailable athttp://lambda.gsfc.nasa.gov/ © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem 1/22 jection. Secondly,wehavechosenaverydifferentcolourscheme purposes. by 27t,isequaltothecontribution thegenus).Aroundeachvertex The propertiesofthegenusarewellknowninthreedimensions(3D) 3 GENUSTOPOLOGYONASPHERE temperatures belowthemean,amountofblueinkperpixelon ink perpixelislinearlyproportionaltothehotnessofahotspot;for is white.Fortemperatureshigherthanthemeanamountofred used bytheWMAPteam.Thecolourschemeispretty,in- that ismorerelevanttogenusstudiesthantheoriginalcolourscheme First, wewillbestudyingthegenusonstereographicprojections tion calledtheHEALPixprojection(Górskietal.2000).Wehave there arefourpixels,thecontribution tothetotalturn(andgenus) map. Thereisacertainturnoccurringateachvertex(which,divided that isaturntotheleftwhenhotregiononyourright.Thus,in truck aroundanisolatedcoldspot,wewouldturnatotalangleof temperature contourdividedby2tt.Ifweweretodriveatruck The genusisalsoequaltotheintegralofcurvaturearound hotspots: trum, andvisaparameterthatmeasurestheareafractionin For aGaussianrandomfield, map isdefinedas(Melottetal.1989) the pageislinearlyproportionaltocoldnessofacoldspot.The formative andimpressive,butisnotsymmetricwithrespecttohot chosen tore-presenttheWMAPresultsinFig.1forafewreasons. contour threshold.Thenthelineisactuallycomposedofa 2?! withtheoppositesign,anegativeturnangledefinedasone 2?! aswecompleteanentirecircuitaroundthehotspot.Drivinga while forv<0(/>0.5)therearemorecoldspotsthanhotspots. where theamplitude,A,isafunctiononlyofpowerspec- gD =numberofhotspots—coldspots.(1) Gott etal.1990). and coldspots,wefindthiscolourmaphighlyinstructiveforour and coldspots.Inourcolourscheme,themeantemperaturecontour as hotifitisabovethecontourthresholdandcoldbelow a temperaturefieldthatisdividedintopixelswemaydefinepixel around anisolatedhotspotwewouldhavetoturnatotalangleof gD =Avexp(-i+2),(2) sian random-phasefieldisallaboutthesymmetrybetweenhotspots (Bennett etal.2003a).Sincethetwo-dimensionalgenusofaGaus- same rangeasinthetemperaturemapsproducedbyWMAPteam scale runsfrom—200pX(solidblue)to+200pKred),the (Sections 6and7),soitisusefultoseethefull-skymapinthatpro- series oflinesegmentswithturnsoccurringatverticesinthepixel some explanationinthetwo-dimensional(2D)case(Melottetal. (Gott etal.1986;HamiltonGott1987),butrequire f =(27t)/exp(—x2)dx.(3) 1989), particularlyinthecaseofsphere(Coles&Plionis1991; 2 2 2 Genus topologyoftheCMBfromWMAP687 The WMAPteamhasreleasedthedatainauniquemapprojec- For 2Dtopologyonaplanethegenusofmicrowavebackground So forv>0(/<0.5)therearemorehotspotsthancoldspots, J V poo 2 0 0 3MNRAS.344. .686C 2 2 background maponthecelestialsphere.Suppose,forexample,we because itisoneisolatedregion.TheGaussiancurvature+(l/r) - bursttheballoonandyouwillhavesolidcurvedleadshapesthat tour2d programasdescribedinMelottetal.(1989). 4 the equatorialregionsandtherewere coldspotsatthenorthernand is identicalsinceonecanbedeformed intotheother,sogenus for acoldspotinthesouthpolarregion.Thegenuswouldstillbe lated region.Supposethehotregioncoversallofsphereexcept northern hemispherewhilethesoutherniscold.The have onehotspotinthenorthernpolarregion,andrestof handles hasagenusof+1onourdefinitionbecauseitconsists has oneholeandconsistsofisolatedregion.Aspherewithtwo is locatedatthevertices.)Adoughnuthasagenusof0,becauseit tegral ofone-eighththatasphereor7i/2.Theedgeswouldbe the Gaussiancurvatureintegralof7t/2ateachvertex,equalto it hasthesametopologyasasphere.Inthiscase,curvatureof nut) holesminusthenumberofisolatedregions.Our3Dgenusis that willbethe2Dgenusaswedefineit. while theareaofsphereis47tr,sointegralGaus- equal totheintegralofGaussiancurvatureovercontour will haveacertain3Dgenus-taketheminusofthisnumberand 2 0 equatorial band(confinedinathin sphericalshell)isadoughnutin dles, whichisalsooneisolatedregion.Thetopologyineachcase genus wouldstillbe+1,becauseahemisphericalbowlisoneiso- cap isoneisolatedregion.Nowimaginethehotregioncovers conical deficitangleatthevertex.(Ifwesandpaperedoffcorners where threesquaresmeetatapointintroducingcontributionto 3D. Useleadpainttothehotregionsonsurfaceofaballoon of thehotregionsconfinedwithinathinsphericalshell,measuredin 3 are hotandcold: at thatvertexdependsonhowmanyofthefoursurroundingpixels 3D, whichhasa3Dgenusof0.Ifthe spherehadNisolatedhotspots one isolatedregionandhastwoholes. and thefaceswouldhavenoGaussiancurvature,soallcurvature of aspherecontainingcontributiontotheGaussiancurvaturein- and edgesofthecube,wewouldproduceateachvertexanoctant + 1,becausethiswouldlooklikeasugarbowlwithoutanyhan- sphere iscold.Thenthegenuswouldbe+1,becausehotspot sanded offtoquartercylinders,whichhavenoGaussiancurvature surface entirelyconsistsofeightdeltafunctionsatthevertices sphere, andthegenusis—1.Acubehasaof1also,because sian curvatureoverthesphereis4?!,independentofsize surface dividedby—47t.Thus,aspherein3Dhasgenusof—1, surface. Herewewillfollowourpreviousderivation(Gottetal. southern polarregions,the2Dgenus wouldbe0becausein3Dthe should bethesameinallthreecases. Ifthehotregionranaround 688 W.N.ColleyandJ.R.Gott,III 1990). The2Dgenusisdefinedtobeequalminusthe3D 1 hot, 0coldgenus=contourlinedoesnotintersect hot, 1coldgenus=—/4rightangleturnarounda hot, 2coldgenus=0noturn,orturnsthataddtozero hot, 3coldgenus=+1/4thecontourlinemakesaright hot, 4coldgenus=0thecontourlinedoesnotintersect Recall thatwehavedefinedthe3Dgenusasnumberof(dough- Now wewishtodefinerigorouslythe2Dgenusonaspherical So usingthesedefinitionswecandefinethegenusofamicrowave This iscarriedoutbyanidenticalalgorithmtothatinthecon- © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem the vertex. the vertex on average angle turn spot boundaries oftheinnerandoutersphericalcaps-contribution pole (theresthot),becausethelatitudeofcontourlinecanbe background (ahotspotcentredonthenorthpole;restcold)is region mustaddtogive27t. return toitsinitialpositionaftercircling,sothetotalofturning the integralofturningaroundcontourboundaryiszero.Since is saddle-shapedandthereforenegative.Theequatorageodesic, to this:thereisacontributionof27tfromtheoutside(spherical) 47T bytheGauss-Bonnettheorem.Thereareseveralcontributions have anintegraloftheGaussiancurvatureonitssurfaceequalto the sametopologicallyasanisolatedcoldspotcentredonsouth the spherehadNisolatedcoldspotsonahotbackgroundgenus nents oftheboundarycontour(around allthehotspots)dividedby turning integralofatruckdriving around alltheindividualcompo- is theturningintegralthatbeingcalculatedfromformulafor have tokeepyoursteeringwheelturnedtheleft,sincethisnota must turnanadditionalangleoftt.Indeed,ifyouwouldliketodrive If weweretoparalleltransportavectoraroundthisboundary, isolated boundary(turningplusdeviationfromparalleltransport)]. the totalGaussianintegral,47t,istwice2ttvalueforcirclingan turning integralforatruckdrivingonthesphericalsurface[since the boundaryextendstolatitude30°N.Thisisasphericalcapwith integral andtheparalleltransportdeflectionduetocirclingacurved the valueweexpectforaclosedcurveinplane.Thevectormust we arecirclinganisolatedhotspot,whyisthetotalturningnot27t vatures ontheinsideandoutsideedgesatequatormarking cap surfaceandacontributionof27tfromtheinside(spherical) genus is+1becausein3Ditoneisolatedregion.Consideredasa ern hemispherecold.Thisisasphericalcapofarea271sr.Its2D we haveahotspotcoveringthenorthernhemispherewithsouth- would be2—A.[Rememberthatoneisolatedhotspotonacold drives straightaheadwithoutturning. Forthecasewherehotspot 2tt; inthiscaseitis0becausethe truck drivingaroundtheequator example, wheretheonehotspotwas theentirenorthernhemisphere, where/ isthefractionofareasphereinhotspots.Inour geodesic, andthetotalofthatturningintegralistt.Now,locallyit counterclockwise aroundthesphereatlatitude30°N,youwould cap; toreturnitsoriginalorientation,arotationof27t,thetruck would findadeflectionof7tbecausethatistheareainspherical edges oftheboundarymustbe27t,whichequaltotwice flat (partofacone),sothecontributionfrominsideandoutside 7t, andontheinsideofsphericalcapisalsoboundary curvature intheinteriorofboundary,whichthiscaseis27t, we paralleltransport(withoutturning)avectoraroundthebound- of theouteredgeispositive,whilecontributioninner giD =Z/\andg2D,eff0.The valueofg2D,effisequaltothe g2D,eff =g2D—2/,(4) a Gaussianrandomfield.Soletusdefinetheeffectivegenusas 3D mustbe4tt.Theintegralontheoutsideofsphericalcapis of theintegralGaussiancurvatureoversphericalcapin area 7t.Itisanisolatedregionsothe3Dgenus—1,andsum ary itwillsufferadeflectionequaltotheintegralofGaussian as itisontheplane?Becausewearecirclingacurvedregion,andif so wecoulddriveatruckaroundtheequatorwithoutturning;hence, spherical), andanetcontributionof0fromthedeltafunctioncur- surface (asphericalshellhasaninsideandoutsideboundaryboth spherical cap(partofathinshell)thisisolatedregionmust simply moveddownwardtotransformoneintotheother.]Suppose (polka dots)onacoldbackground,the2DgenuswouldbeN.If surface connectingtheinnerandoutersphericalcapshellsislocally Now considerahotspotcentredatthenorthpoleandforwhich e 2003RAS,MNRAS 344,686-695 2 0 0 3MNRAS.344. .686C 2 2 is mappedintotwotrianglesthat appearonadjacentfacesofthe Each diamondoftherhombicdodecahedron isorientedwithitsshort label T(top),N(north),E(east),S(south), W(west)andB(bottom). the spheretorhombicdodecahedron,thenfrom by usingthetopologicalinvarianceofgDtoprojection(firstfrom be complicatedtocalculate,soaversionofcontour2dforthe pixels thatareatleast2.5timessmallerthanthesmoothinglength. resolution toshowtheWMAPdata,whichhasanangular four subdiamonds,wehaveatotalof12x4^pixelsequalarea instead ofsquares,butthetopologyissame.Thesubdivisions produce achequerboardof64squarepixelsbydividing by similarlysubdividingeachdiamond.(Inasimilarwaywemay be subdividedintodiamond-shapedpixels.Thisisdonebyfactors TS, TWallmeetatthetop,andsimilarlyTN,NE,NW,BN the equator),SE(southeast,onSW(southwest, TE (topeast),TSsouth),TWwest),NE(northeast,on that meetatthebottomlikesidesofaninvertedpyramid.Ifwe It has12faces,fourdiamondsthatmeetatthetoplike The sphereiseffectivelyprojectedontoarhombicdodecahedron. the sphere,calledHEALPixprojection(Górskietal.2000). The WMAPdataareplottedusinganunusualmapprojectionof 4 GENUSINTHEHEALPIXPROJECTION Thus, incomparingtheWMAPdatawithrandom-phaseformula because theGaussianrandomfieldlocallybehavesonsphere the valueof#20,err=becauseturningintegralinthiscaseis is centredatthenorthpoleandextendstolatitude30°N,/=|,so cube. Forexample,therhombusTN isdividedintotwotriangles, diagonal alonganedgeofthecube, andeachofthe12rhombuses dodecahedron toacube).Thiscubehassixfaces,whichwewill calculate theturningangles.However,wemaysimplifythisgreatly dimensions ofapproximately0.11xdeg,whichisadequate covering thesphere.Forexample,WMAPusesN=9,wherethere first intofourquadrantsquares,dividingeachofthese equator), NW(northwest,ontheBN(bottomnorth),BE directions wecouldlabelthe12facesasfollows:TN(topnorth), were tosetthisrhombicdodecahedronupalignedwiththecardinal crystal lattice,thepolyhedronrepresentingboundaryofset we willuseg2D,errdefinedrigorouslyasdescribedabove. g2D,eff OCvexp(—v/2),(5) ti. NowforaGaussianrandomfieldonthesphere, e 2003RAS,MNRAS 344,686-695 one ofwhichismappedontotheT maponthecubeandotheris of approximately0?3.Tostudythetopologyproperly,oneneeds are 3145728diamond-shapedpixelscoveringthesphere,eachwith of latitude.WithNsuccessivesubdivisionsthediamondsinto of pixelstouchingpointtohavecentresthatlieoncircles are madesothatallthepixelsofequalarea,anddiagonalrows of 4,subdividingeachdiamondintofourdiamonds,andrepeating at apoint(e.g.TN,TE,NEmeetvertex).Eachdiamondmay at avertex).Thereareeightverticeswherethreediamondsmeet are sixverticeswherefourdiamondsmeetatapoint(i.e.TN,TE, of pointsclosertoaparticularatomicnucleusthananyother.) as ontheplanetoproducethiscontributionturningintegral. sphere wouldseemtobecomplexandrequiremanycalculations squares). IntheWMAPcasethesearealldiamonds(rhombuses) subsquares, andeachofthesesubsquaresintofourchequerboard (bottom east),BSsouth)andBWwest).There sides ofapyramid,fourdiamondsthatcircletheequator,and (This semiregularpolyhedronisthecellforface-centredcubic 2 Each ofthesediamondpixelsonthespherehasanglesthatwould © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem N rhombuses oftheoriginaldodecahedronismappedontotworight point arelikeordinaryverticesintheplanewherefoursquarepixels have equivalently,12squaremapsofApixelseach.Thus, these canbeflattenedouttomakeasquare,withsquarepixels.Sowe triangles meetingattheirhypotenusesalonganedgeofthecube; instead offourmeetateachtheseeightvertices.Eachthe12 is concentratedentirelyineightdeltafunctionsatthevertices, have onlybeendistortedinshape,notchangedtopology.The the sameasiftheywerepaintedontosphere,because the surfaceofcube,and3Dgenustheseleadshapesis the fourrhombusesTN,TE,TS,TW,whichmeetattop. meet inthecentre;thesefourtrianglesrepresentupperhalvesof for whichthehypotenusesformsidesofsquare,and that aremappedontotheSandWfacesofcube.TheTface mapped ontotheNfaceofcube.Infact,letterdesignation This isthenaddedtothehotrhombus asthisisthehotspotweare the sphere)sogenuscontribution (turnplusparalleltransport in theeightvertices.Theedgesofthreerhombusesmeetat120° tiny octantsofasphereeachwithanintegratedGaussiancurvature the cornerofcubeinourprojection)isasfollows: The prescriptionforverticeswherethreepixelsmeetatapoint(at the contributiontogenusfromthatvertexmustthenbedivided meet atapoint,sothisishandledwithcontour2d.Thevalueof there isa7i/2angledeficitinourcubicmapandwherethreepixels for thoseatthecomersofrhombuses.Eachrhombushastwo meet alongeachedge.Thataccountsforallpixelverticesexcept the verticesalongtopandright-handedgesofrhombusare with thenextrhombusaresharedaccordingtoaprescriptionwhere we canconsidereachrhombusasasquaremapwithpixels where thereisanangledeficitof7t/2ateach,asthreesquares total integratedGaussiancurvatureoverthecube,whichtotals4tt two regionsof30°curvaturecontributing totheparalleltransport circling. Inthecaseoftwohot,one cold,thereisa—60°turn,plus deviation 90°)shouldcorrespond to +1/4(or7t/2dividedby27t). cold thereisaturnof60°inthecontourpluscurvature30° would turnintoquartercylinders(withzeroGaussiancurvature) 2 hot,1cold 0 hot,3cold corners wherefourrhombusesmeet(suchasTN,TE,TS,TW edges areassignedtoadjacentrhombuses,sincetworhombuses are withintherhombusincluded.Verticesthatalonganedge and wecancalculateg2Dforeachusingcontour2d.Verticesthat of thecubeisthusasquarethatdividedintofourrighttriangles on to.Forexample,theSWrhombusisdividedintotwotriangles of eachrhombustellsuswhichthetwocubicfacesitismapped angles inthesphericaloctant.Socaseofonehotandtwo of 7i/2.Thefacesthecubeareflatsoallcurvatureislocated and theeightvertices(wherethreesquaresmeetatapoint)turninto 3 hot,0cold and sharedequallyamongeachoftherhombusesthatmeetthere. only meetatapoint.Thecornerswherefourrhombuses at thetop)andtwocornerswherethreerhombusesmeetavertex assigned toitwhenlaidoutproperly,andthebottomleft-hand (equal toone-thirdoftheintegrated curvatureovertheoctantof (such asTN,TE,NE)whichoccursatavertexofthecubewhere 1 hot,2cold Genus topologyoftheCMBfromWMAP689 In suchaprojectionwepaintthehotregionswithleadon If weweretosmooththeedgesofcubebysanding, genus =1/12addedtoeachofthethree genus —0 genus =1/4addedtohotrhombus,0 genus =0thecontourdoesnotintersect rhombuses. to others vertex 2 0 0 3MNRAS.344. .686C 2 but itadds7t/2tothecurvatureintegralwithinthatregionandthis nation oftheskymapsatdifferentfrequenciestoremove HEALPix format.ThisILCmapusestheoptimumlinearcombi- linear combination’(ILC)mapoftheCMB,whichisgivenin The mostdirectdataproductfromtheWMAPteamis‘internal BY WMAP the genusineach.Thetotalforsphereg2Discalculatedby must besharedequallywithallthreerhombusestoadd+1/12 for 95.4percentconfidenceinthestudent’s tdistribution. whole sky.Innererrorbarsarecomputed fromthestandarddeviationof the wholeskyintermsofpixelwisetemperature standarddeviationforthe The abscissa,v,isthedeparturefrom thepixelwisetemperaturemeanfor Figure 2.Totalgenus,g2D,eff,fortheoriginalHEALPixprojectionof histogram oftemperaturewerestrictlyGaussian.Fig.3isidenti- to theareafractionmethoddescribedbyequation(3)ifpixel the pixelwisetemperaturemeanandstandarddeviationmeasured the nextsection,butitisworthcheckinggenusonthismap this map(Bennettetal.2003b).Wewilladdresstheseissuesin has aresolution(beamwidth)of0.3xdeg.Thismappresents the primarymapdistributedasbestrenderingofCMBand genus ineachoftheseregionsandcomputedthetotalforwhole directly. which isthattheWMAPteamadvisedagainststatisticalstudiesof Galaxy andsomeotherforegrounds(Bennettetal.2003b).Thisis 5 THEGENUSOFCMB,MEASURED distribution. deviation, makingatotalcontributionof0tothegenus.Incase cal, exceptthatviscomputedbyareafractionineachindividual a fewproblemsfordirectlycomputingthegenus,notleastof and compareitwiththatexpectedfromaGaussianrandom-phase area. Thus,fromtheWMAPdatawecancalculate2Dtopology of thediamond-shapedpixelsthatarehot,sincetheseallequal g2D,eff bysubtracting2/.Wecancalculate/assimplythefraction adding thegenusfromall12rhombuses.FromgDwecancalculate of threehot,zerocold,thevertexliesentirelywithinhotregion over thewholesky(method1).Thismethodwouldbeequivalent sky, whichweplotinFig.2.Inthisfigureviscomputedtermsof set (sky)into12independentregions.Wethereforemeasuredthe 690 W.N.ColleyandJ.R.Gott,III WMAP data(internallinearcombination) fortheentiresky,withnomasking. 12 HEALPixdiamonds;theoutererror barsreflectthesigmavaluescaled 2 The HEALPixprojectionprovidesanaturaldivisionofthedata © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem i/ 2 rhombus separatelyandthusdiscoversmorestructures. plitude applied.Theamplitudes,A,[g2D,eff=Avexp(—v/2)],in fraction methodmeasurestherandom-phasenatureofdistribu- HEALPix rhombus(method2);seeequation(3).Usingthearea Figure3. AsinFig.2,exceptthevvaluesarecomputedbyareafraction has negligibleerrorcomparedwith anyonepointonthecurve.The is givenexactlybyequation(2)and thatourbestfitfortheamplitude mean-square differencefromthemean (asusual).Weexpectthat/x the truestandarddeviationinmeanexpectedforthatnumber the standarderrorwithdistributionitself.Namely,iftrue lifts someregions(rhombuses)tohighertemperature,consolidating to date.Asexpected,thefirstvalueofAislowerthansecond the highestinanygenusmeasurementsofcosmologicalstructure Figs 2and3areA=34323657,respectively.Thesebyfar freedom, whichwewillexplainshortly. the meaningenusforwholesky,estimatedfromthoseof identical results. tion moredirectly,sinceitseparatesthatfromanydepartures for /xisthearithmeticaverageestimated fromthe12rhombuses Gaussian random-phasefields,Gottetal.1990),thenourestimator genus forthewholeskyateachpointoncurvewere¡iand genus curve,accordingtoequation(2),withthebest-fittingam- 2a) confidenceintervalforastudent’stvariablewith12degreesof error bars.Theouterbarsshowthe95.4percent(Gaussian tained fromeach).Thestandardlaerrorbarsareshownastheinner fine vbyareafractionineachrhombus,thustreatingofthe deviation intemperatureoverthewholesky;method2,wede- directly). Inmethod1,weusevdefinedintermsofthestandard (see thetext). averaged), x,andourestimatorfor a isWn—1,wherestheroot- of structureswerea(weexpectthedistributiontobeGaussianfor one, becausethepoweratlargescales(quadrupoleandoctopole) a contrastofdatatreatment,andyet,asweshallsee,giveessentially a Gaussianhistogramofthetemperature(whichcouldbemeasured (the valuesfromeachrhombusare multipliedby12,thentheyare solidating somecoldspotstherein.Method2setsamedianineach some hotspotstherein,butlowersotherregions(rhombuses),con- 12 independentregionsofthesky(addingvaluesg2D,errob- 12 rhombusescompletelyindependently.Thetwomethodsprovide The student’stformulationisnecessarysinceweareestimating The solidcurvesinFigs2and3givetheGaussianrandom-phase The innererrorbarsinFigs2and3arethestandarddeviationof e 2003RAS,MNRAS 344,686-695 v 2 0 0 3MNRAS.344. .686C theoretical formula.Thus,wecan skip astepbygoingdirectlyafter terested inanywaybecauseitis what willbecomparedwiththe by adding2/areap/(47r).]However, itisreallyg2D,effwearein- the relationbetweeng2D,effonasphericalcap,and#20,mapmea- map directly,whichwouldbecomplicated,letusinsteadcalculate that comesfromcirclingthedensitycontouronmap. that occursatverticesismappedproperly.However,geodesics(great - geodesicsegmentsjoinedatverticeswhereturningonthesphere boundary isacircleofradiusr=2tan[(90°—18°)/2].The radius inthemap,r=2tan[(90°-&)/2](seeFig.1).Theoutside projections ofahemisphereareconformalandpreserveangles.A using stereographicmapprojectionsofthecaps.Stereographic hemisphere). the map,usingstereographicprojectionforbothNorthern four equatorialdiamonds,ceasestobeideal.Instead,were-project tic latitudes\b\>18°.Sincethemaskmustremovemuchof point sources.Furthermore,wehaverestrictedourselvestoGalac- KpO (mostconservative)skymask,whichalsoremovessome200 is theGalaxy.Inasecondstudy,wehavethereforeusedtheir plied toavoidforegroundcontaminants,theprincipaloneofwhich results. Theyprovideaveryusefulskymaskthatshouldbeap- be addressed.Thefirstissueisforegroundemission.Bennettetal. In theILCHEALPixdatarelease,thereareafewissuesthatmust MAP PROJECTION random-phase hypothesis. None theless,genusoftemperaturefluctuationsinCMB, - neighbouringgenusvaluesmeasuremanyofthesamestructures). lations meanthatthe21datapointsarenotcompletelyindependent neighbouring valuesofv,asshownbyColley(1997)(thesecorre- by chancefromcorrelationsbetweenthegenusmeasurementsat miss thecurvebymorethaninnererrorbars.Inbothcases, From thepropertiesofstudent’stdistributionwith12d.o.f, the 95.4percentconfidenceerrorbar.Infact,notevenonemisses. g2D,eff- contribution to£20duethestructures inthenorthgalacticcap which turnonthemapbutnotspheresototalgenus#20 circles) aremappedintoarcsofcirclesratherthanstraightlines, contours onthespherecanbeapproximatedassphericalpolygons 6 GENUSONASTEREOGRAPHIC each figure.Thisbetter-than-expectedagreementcaneasilyarise expression (x—fi)/(s/^/n1)isdefinedtobeastudent’stvariate © 2003RAS,MNRAS 344,686-695 of thecap.[Ofcourseoncewehave g2D,effwecancalculatethe on thesphereisnotautomaticallycalculablefromtotalturning occurs. Thestereographicprojectionpreservesangles,sotheturning azimuthal angleinthemapisgalacticlongitudecap.The and SouthernGalacticHemispheres(asinFig.1,whichthe area neartheGalacticequator,HEALPixprojection,withits as observedbyWMAP,iscertainlyconsistentwiththeGaussian agreement isagainbetterthanexpected,withonlythreemissesin one wouldexpecttofindthat33.7percent(~7)ofthe21points of thegenuscurvesinFigs2and3tomisscurveoutside sured bythecontour2dprogramonflatstereographicmap stereographic projectionofthenorthgalacticcapisacircle,inwhich (2003b) discussinquiteabitofdetailtheforegroundsWMAP (Lupton 1993). \b\ >18°cutoccursat72.7percentofthetotalradiusineach ca b We cancalculate#20fromthenorthandsouthgalacticcapsby On average,wewouldexpectonlyoneofthe21pointsineach So insteadofcalculatinggDforthespherefromstereographic 2 © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem _1 _1 2 1 plane -thinkofflatleadsheetslyinginaplane,becauseonecanbe boundary terms.Eachhotspot(thinkofacurvedleadsheet)that boundary istheequatoranditageodesicsonoturningrequired boundary ofthecap.Ifentirewerehotthendriving boundary. Eachofthese2Nturnswillbesomeanglebetween0and boundary thereareNhotsegments,willbe2Nsuchturns, presented above.Theterm(27t)(turnsatcapboundary)isthe turns madedrivingonthespherearoundallcontourswithin The firstterm,(27r)^(turnsonsphericalcap),isequalto isolated capwillhaveavalueofg2D,exciedcap=g2D,excisedmap;thetotal hotspots in3Dwhenthecapisexcisedfromrestofsphere: the boundarythatwillmakeitanisolatedhotspot.Thinkaboutthese intersects theboundaryofcapwillbeclosedoffbyacurvealong it offandseparatingfromtherestofsphere.Thiswilladd not expecttheboundarytobespecial, soonaveragethefraction the boundarythatishot,andsoitthisfractionof have startedsowesufferedatotalrotationof271,theturning When wehavecompletedthecircuit,returnedtowhere the closedboundarycurveisequaltointegralofGaussian to circletheboundary.)Asareaofcapgoeszero, if thiswereahemisphericalcaptheareaofwouldbe27t is theintegratedGaussiancurvaturewithinhotspotsincap The term/area(27T),whereismeasuredinsteradians, ing thesphericalcap.Nowg2D,eff,caocvexp(—v/2),soitiswhat g2D,eff,ca fortheinteriorofsphericalcap.Itistotalnumber g2D,excised cap—(271)E(turnsonsphericalcap) contributing tog20,excisedcap: continuously deformedintotheother.Wecancalculateterms genus oftheisolatedflatstereographicmapsittingin the spherethatisinhotspots.Onaverage then,weexpectf=f. genus derivedbydrivingalongthe hotsegmentsoftheboundary: we drivearound.Dividingthisby 2tt, givesthecontributionto 27t. Thus,turning=2tt—area.Thenumber/isthefractionof we doonthespherecirclingboundaryplusareaofcapis curvature inside,whichinthiscaseisequaltotheareaofcap. general, theangledeflectionsufferedbyparalleltransportaround curvature withinthecapbecomesunimportanttoparalleltransport 271. Thefinalterm,/[1—area(27i)],representsthetotalturning circling thathotspotasanisolatedregion.Ingeneral,ifalongthe corner theretocontinuedrivingalongthecapboundarycomplete divided by2ti,whichcontributestothegenus3Dargument we wanttocalculatecomparewiththetheoreticalgenusformula. any hotspotthathitstheboundarywillbecomeisolated,floatingin of thecapboundarythatwashotshould beequaltothefractionof and thetotalturnrequiredtocircleboundaryapproaches2tt.In and thetotalturncirclingboundarywouldbezero,because around itwouldhaveatotalturnof271-area.(Forexample, on thespherethatoccursdrivingalongNhotsegmentsof as weenterandthenexiteachoftheseNhotsegmentsalongthe and youencounterthecapboundarythenmustturnasharp space, becausebeyondtheboundarythereisnothing.Thisspherical sum oftheturnsthataremadewhenyoudrivingalongacontour spherical cap,andexcludesanyboundarytermsproducedbyexcis- / [1—area(27r)].UsingtheCopernican principle,onewould S cap P P b capb bcap cap bcap Genus topologyoftheCMBfromWMAP691 To begin,letusexcisethesphericalcapfromspherebycutting 1 + /[l-area(27t)].(6) + /area/(27t) + (27t)-E(turnsatcapboundary) bcap cap 2 0 0 3MNRAS.344. .686C _1 tUmS 011Stere0rahÍCma tums _1 boundary occupiedbytheNhotsegmentsisexactlysameas boundary wewouldhavetoturnatotalof27t,becausethemap boundary ofthemap.Ifweuseastereographicprojection,whichis logically equivalenttotheexcisedcap: Now letusmeasurethegenusonexcisedstereographicmap ical cap,sincetermsproportional to theareaofcapcancelout. ignores theboundary)ourresultwill equal,onaverage,g2D,eff,ca, in theinteriorofstereographic mapusingcontour2d(which the boundary.Equatingg2D,exciedcapandgD,excisedmap(because to azimuthinthestereographicmap,sofractionofmap is flatandEuclideangeometryapplies.TheNhotsegmentscover The finalterm,/,isthetotalturntakenonNhotsegments region anddoesnotincludeanyboundaryeffects.Thereisno the quantitymeasuredbycontour2donflatmap. The firstterm,(27t)^(turnsonstereographicmap)=g2D,map,is conformal andpreservesangles,eacheveryturnatthebound- examines onlytheCartesianverticeswithinstereographicmap which iswhatwewerelookingfor. Thisworksforanyradiusspher- equations wefind cap canbedeformedintothemap),andsubstitutingfromabove create isolatedhotspotsintheplaneoutofhotregionsthathit 27t gives/.Aswehaveremarkedabove,iftheboundaryisnot ary inthemapwillbeequaltothatencounteredonsphere,so adds theturnsthataretakenwhencontourhitsoutercircular (?2D,excised map=—^gPP) ■?2D,excised cap—,?2D,eff,cap+/ (?>2D,excised cap—^2D,eff,cap Substituting wefind: So ‘whatyouseeiswhatget’. Ifwemeasurethe2Dtopology g2D,eff,cap —g2D,map- g2D,eff,cap +/^>(atcapboundary) §2D, excisedcap—map on theplane,theseouterboundarysegmentsmustbeincludedto on thesphere).Thus,totalturnis2nf,anddividingthisby a fraction/oftheboundary(galacticlongitudeismappedon of theboundarymapdividedby271.Ifwecircledentire special weexpect/=/.Sincethemapisexcisedandsitsalone side thehotspots.Thenextterm,(27r)^(turnsatmapboundary), (82D,excised map)?recallingthattheexcisedstereographicmapistopo- 692 W.N.ColleyandJ.R.Gott,III ‘/ area’termbecausethemapisflatandtherezerocurvaturein- — y(turnsatmapboundary) P S2 b b h b b UmSmaP bUndar 1 ^—> = g2D,map+^°y)f 271 1 E © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem (turns atcapboundary). z H >(turnsatmapboundary)+/.(9) b Ztt ^ 1 ^—> 1 ^—> 2tt ^^ 1 1 ^^ (turns atcapboundary). f area cai 271 at capboundary) (13) (12) (ID (10) (8) (7) boundary tobetypical,itscoveragewithhotspotsshouldequal projection, computedbymethods1and2(thesameasin per centvariation).Wenormalizethegenustomeansolidangle ment fortheactualsolidangleineachoctantismade,because independent regionsinwhichtomeasurethegenus.Aslightadjust- that correspondtooctantsonthesphere.Theseformeight mask issimplyignoredbycontour2d. Any vertexinthestereographicmapprojectionthatfallswithin larly neartheGalacticCentre,GumNebulaandOrionComplex. include removaloflocalregionsforpointsourceavoidance,and We maynowcomputethe2-Dgenusonastereographicre-projection 7 GENUSTOPOLOGYONSTEREOGRAPHIC to thefraction/ofspherecoveredbyhotspots,andthiscauses in bothcasesandcanceloutaswell.Finally,sinceweexpectthe And sincethemapisconformal,turnsatboundaryareequal the standarddeviationofeightoctants; theoutererrorbarsreflect elwise temperaturemeanforthewhole sky intermsofpixelwisetemperature tion), projectedstereographicallytoatotal smoothingof0.35°,thenmasked Figure 4.Totalgenus,g2D,eff,fortheWMAPdata(internallinearcombina- for eightoctantsinsteadof12diamonds,whichchangestherelative the HEALPixgenuscomputation).Theerrorbarsarenowcomputed have appliedaslightsmoothingtototalFWHMof masked octant. KpO maskchangesthesolidangleofeachoctantveryslightly(afew exclude theequatorregionupto\b\<18°.TheKpOmaskdoes egy oftheWMAPspacecraft. contamination andnoisevariabilityarisingfromthescanningstrat- according totheKpOstandard.Theabscissa, v,isthedeparturefrompix- 0?35. Figs4and5showthegenusformaskedstereographic of theWMAPdata,inwhichweaddressissuesforeground other termstocancelaswell,leavingtheabovesimpleresult. sigma valuescaledfor95.4percentconfidence inthestudent’stdistribution. standard deviationfromthewholesky. Innererrorbarsarecomputedfrom one ofeightindependentmeasurementsthegenusperaverage of themaskedoctants,andproceedtreatingeachoctantas some additionalGalaxymaskingabovethe\b\>18°cut,particu- size ofthe95.4percenterrorbarsanddirectla WMAP RESULTS We proceedbydividingthestereographicprojectionintooctants First, toalleviatecontaminationfromtheGalaxy,wecompletely In carryingoutthere-projectiontostereographicmap,we e 2003RAS,MNRAS 344,686-695 v 2 0 0 3MNRAS.344. .686C reducing thesignificanceofnoise differencesarisingfromthe map. Atthecostofnumberstructures,thesemapssignificantly been pre-smoothedtothesameeffectivebeamwidth(0?82).These the samefivefrequenciesasusedinILCmap,butonesthathave problems, wehavedownloadedfromtheWMAPwebsitemapsin yields manymoresamplesattheeclipticpolesthan to pixel.First,thebeamsateachfrequencyincombinationhave behaviour isverycomplicated,anddifferssubstantiallyfrompixel from small-numberstatisticsineachcase. per centlevel,theexcessiscertainlywithinuncertaintyarising than theaveragenumberofmissesexpectedatbothlaand95.4 using method2.Thoughthe2resultsshowslightlymore method 1,wefindthatthreepointsmissatthismarginandnine reside atthe65.3percentconfidenceinterval,soonaverage,one miss thetheoreticalcurveoutsidetheir95.4percenterrorbar,while the additionalsmoothingappliedinstereographicprojection. less thanthesecond,asexpected).Theamplitudeshavecomedown plotted ineachfigure,withrespectiveamplitudesofA=2057and Figure 5.AsinFig.4,exceptthevvaluesarecomputedbyareafraction enhance thesignal-to-noiseratioofremainingstructures,thus equator (seeBennettetal.2003a,Fig.3).Toalleviatesomeofthese different widths.Secondly,thescanmethodofWMAPsatellite would expectapproximatelysevenofthe21pointsoneachFigs4 with method2(Fig.5),twopointsmissbyasmallmargin.The1er curve isstriking.Quantitatively,wewouldexpectonaveragethat equatorial band(sothetotalgenuscoversonlyafractionofwhole 2186 (aswiththeHEALPixgenuscurves,firstvalueisslightly e 2003RAS,MNRAS 344,686-695 are thenaddedwiththesameweightsasusedinproducingILC and 5tomissthetheoreticalcurveoutsidetheir1ererrorbars.Using outside the95.4percenterrorbar.Withmethod1(Fig.4),nopoints one ofthe21pointssamplingcurvewouldmisstheoreticalfit (see thetext). stereographic projections.Thebest-fitting amplitudeshavedropped smoothed maps.Figs6and7show thegenusmeasuredinthese smoothed to0?9,slightlymorethan the0?82smoothingofpre- scanning strategy.Thisis,therefore ourmostconservativemap. (inner errorbars)fortheeightdegreesoffreedomoctants sphere), theadditionalmaskedpixelsinKpOprescription,and somewhat fromtheHEALPixmapsduetoexclusionof somewhat. Thebest-fittingtheoreticalgenuscurve(equation2)is In projectingthesemapsintostereographic maps,wehave The secondcontaminantintheILCHEALPixmapisthatnoise In eachofthefigures,agreementwiththeoreticalgenus © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem v jected stereographicallytoatotalsmoothingof0?9,thenmaskedaccording bined accordingtotheinternallinearcombinationcoefficients,thenpro- represents ourmostconservativeestimateofthetruegenusavailable temperature meanforthewholeskyintermsofpixelwisestan- to theKpOstandard.Theabscissa,v,isdeparturefrompixelwise Figure 6.Totalgenus,g2D.eff,forthepre-smoothedWMAPdata,com- Figure 7.AsinFig.6,exceptthevvalues arecomputedbyareafraction in theWMAPdistribution,andshowsexcellentagreementwith method 2,againabitbetterthanexpected.Thefinalfigure,Fig.7, to A=772formethod1and8312,duethe value scaledfor95.4percentconfidenceinthestudent’stdistribution. dard deviationfromthewholesky.Innererrorbarsarecomputed Gaussian random-phasehypothesis. we findisthatwithmethod1(Fig.6),nopointsmissthecurve (see thetext). outside their1ererrorbarstwiceformethod1andfourtimes one pointmissesbyasmallamount.Thepointsmissthecurve outside the95.4percenterrorbar,whilewithmethod2(Fig.7), outside ofthe95.4percenterrorbarsonceoneachfigure.What at the21pointsshouldalsobeexpectedtomisstheoreticalcurve standard deviationoftheeightoctants;outererrorbarsreflectsigma smoothing. Genus topologyoftheCMBfromWMAP693 From thesefinalstereographicprojections,thegenusmeasured V 2 0 0 3MNRAS.344. .686C boundary (byafactorof2fromthepoletoequator,asseenin phase. Thestereographicprojectionisconformal,soshapesarepre- phase. Indeed,theoverallvisualimpressionisstrikinglyrandom phase distributionandtherewerenofeaturesthatlookedunusual planet inthelate19thcentury.Wenowknowthattherearenosuch This featureismorevisuallyapparentwhenviewedinlandscape feature stretchingfromapproximatelythe8:30positiontoward the MolleweideprojectionusedbyWMAPteam,whichisthatit map atallthatlooktobenon-randomphaseinanyway. Fig. 1).Wemayaskwhetherthereareanyfeaturesvisibleinthe As wehaveshown,topologically,withintheerrors,CMBmea- to beapproximatelyrandomphase. lated phasesthatcouldproducelocally linearstructures,thoughthe from agreatwallseenedge-on,butinlookingatseveralgalaxy maps assuggestedbyGottetal.(1990). Also, thereisnoobviousevidenceforcosmicstringcliffsanywhere the CMBwasquitegrey,andsofeature. that whiletheforegroundsweredistinctlypinktoorangeyellow, map, usingtheKbandasred,QgreenandWblue,found the gasinGalaxy,andhencebesubtractedaboutaswell Magellanic Streammeasuredat21cm(Brooks2000).Furthermore, to theeye,thatwouldbeunusual.Sothisfeaturemaymerely in portraitorientation.Infact,ifonehadatrueGaussianrandom- more apparentwhenthefigureisviewedinlandscaperatherthan the samethingisgoingonhere,particularlysincefeature the linearfeaturesreportedinnaked-eyeobservationsofred ture. First,thereisthesimplephenomenonof‘canals’onMars, in theWMAPscanninggeometry. ear featuresoredgeswassimplycreatinganillusion.Mostlikely canals, andthatthehighlyevolvedabilityofoureyestodetectlin- 2:30 positioncurvinglikeabowbelowtheSouthGalacticPole. there isonepossiblynon-randomfeature,afairlynarrowred(hot) correlations arepresumablyanartefact ofthecleaningtechnique, with appropriategeometry. catalogues andX-raymaps,wecoulddetectnoobviousfeature else intheWMAPdata,whichwehavetestedforbymakinggradient cosmic stringswithinthehorizoninsuchanalignmentbychance). each other(aninterestingtheoreticalpossibility(Gott1991),butnot Galaxy isintheILCmap.Totestthis,weconstructedathree-colour we wouldexpectthestreamtohavesimilarmicrowave‘colour’ coincidental. along agreatcircle,butisnotcoincidentwithanyparticularlocus orientation. Ithastheappearanceofascanningerrorapproximately 8 DISCUSSION of Tegmark,deOliveira-Costa&Hamilton(2003)exhibitscorre- a likelyone,sinceweareunlikelytofindthetwomostprominent a movingcosmicstringproducescliff,whichishotterononeside stripe wouldrequiretwoparallelcosmicstrings,havingjustpassed (the trailingside)thantheothersideleadingside).Thusahot strings, however,donotamplifytemperaturealongaline.Instead, shows theGalacticpolarregionsverywell.Insouthregion, served locally,thoughthescalegrowsasoneapproachesouter sured byWMAPappearstobeconsistentwithGaussianrandom smoothing isinvolved,changingamplitudes butnotphases),appears since theWeinerfilteredmapofTegmark etal.(2003)(whereonly 694 W.N.ColleyandJ.R.Gott,III Finally, Chiangetal.(2003)haveshownthatthe‘cleaned’map The featurecouldalsobeduetotheintegratedSachs-Wolfeeffect Another possibilitythatcomestomindiscosmicstrings.Cosmic What aretheotherpossibilities?Itisnotcoincidentwith We haveconsideredseveralpossibilitiestoexplainsuchafea- Our choiceofmapprojectionhasamiscellaneousadvantageover © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem 33 perturbations. ing theprimordialstructures.Thegenustopologytest,inparticular, they arestillinthelinearregime,makingWMAPmeasurements mation. IntheCMB,however,weareseeingthesefluctuationswhile the bigbang.Ingalaxyclusteringsurveys,thesefossilshavebecome mains ofrandomquantumfluctuationsoccurringjust10safter initial conditions. lar resolutionandall-skycoverageofthecosmicmicrowaveback- tion isquitespectacular.Theunprecedentedcombinationofangu- tistical significanceofanysuchaweakfeatureisdifficulttoestimate. 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