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Mon.Not.R.Astron.Soc.000,000–000(0000) Printed14January2015 (MNLATEXstylefilev2.2) ff Environment-based selection e ects of Planck clusters R. Kosyra1,2, D. Gruen1,2, S. Seitz1,2, A. Mana1,2, E. Rozo3,4, E. Rykoff3, A. Sanchez2 and R. Bender1,2 (cid:63) 1Universitäts-SternwarteMünchen,Ludwig-Maximilians-UniversitätMünchen,Scheinerstraße1,D-81679München,Germany 2MaxPlanckInstitutfürExtraterrestrischePhysik,Giessenbachstr.,D-85748Garching,Germany 3SLACNationalAcceleratorLaboratory,MenloPark,CA94025,U.S.A. 5 4UniversityofArizona,DepartmentofPhysics,1118E.FourthSt.,Tucson,AZ85721,U.S.A. 1 0 2 14January2015 n a J ABSTRACT 2 We investigate whether the large scale structure environment of galaxy clusters imprints a 1 selectionbiasonSunyaevZel’dovich(SZ)catalogs.Suchaselectioneffectmightbecaused bylineofsight(LoS)structuresthataddtotheSZsignalorcontainpointsourcesthatdisturb ] thesignalextractionintheSZsurvey.WeusethePlanckPSZ1unioncatalog(PlanckCollab- O orationetal.2013a)intheSDSSregionasoursampleofSZselectedclusters.Wecalculate C the angular two-point correlation function (2pcf) for physically correlated, foreground and . backgroundstructureintheRedMaPPerSDSSDR8catalogwithrespecttoeachcluster.We h compareourresultswithanopticallyselectedcomparisonclustersampleandwiththeoretical p predictions.Incontrasttothehypothesisofnoenvironment-basedselection,wefindamean - o 2pcfforbackgroundstructuresof−0.049onscalesof(cid:46) 40(cid:48),significantlynon-zeroat∼4σ, r whichmeansthatPlanckclustersaremorelikelytobedetectedinregionsoflowbackground st density.WehypothesizethiseffectariseseitherfrombackgroundestimationintheSZsurvey a orfromradiosourcesinthebackground.WeestimatethedefectinSZsignalcausedbythis [ effecttobenegligiblysmall,oftheorderof∼10−4ofthesignalofatypicalPlanckdetection. 1 Analogously,therearenoimplicationsonX-raymassmeasurements.However,theenviron- v mentaldependencehasimportantconsequencesforweaklensingfollowupofPlanckgalaxy 0 clusters:wepredictthatprojectioneffectsaccountforhalfofthemasscontainedwithina15’ 4 radius of Planck galaxy clusters. We did not detect a background underdensity of CMASS 8 LRGs,whichalsoleavesaspatiallyvaryingredshiftdependenceofthePlanck SZselection 2 functionasapossiblecauseforourfindings. 0 . Keywords: galaxies:clusters:general–cosmology:observations. 1 0 5 1 : v 1 INTRODUCTION core(stronglensing,forexampleinZitrinetal.2012;Eichneretal. i 2013;Monnaetal.2014).Themostwidespreadmethodforoptical X Clustersofgalaxiesplayamajorroleinastrophysicsandcosmol- clusterdetection,theso-calledredsequencemethod (Gladders& r ogy,astheycanbeusedtoputconstraintsonthedarkmattercon- Yee2005;Koesteretal.2007;Rykoffetal.2014)isbasedonspatial a tent of the universe. Furthermore galaxy clusters are particularly overdensitiesofredgalaxies.Furthermethodsincludetheobserva- sensitivetotheinterplayofdarkmatteranddarkenergy.Theyare tion of the X-ray Bremsstrahlung emission by the hot gas in the cosmologicalprobesthatcouldpotentiallyhelptodistinguishbe- intra-cluster medium (ICM, Piffaretti et al. 2011; Vikhlinin et al. tween dark energy and modified gravity explanations for the ac- 2009;Mantzetal.2010)andtheobservationofinverseCompton celeratingexpansionoftheuniverse(forareview,see Allenetal. scatteringofthecosmicmicrowavebackground(CMB)photonsby 2011;Borgani&Kravtsov2011;Weinbergetal.2013). theICM,whichisknownastheSZeffect(Sunyaev&Zeldovich Avarietyofdifferentmethodsforclusterdetectionandmass 1972). The latter describes the distortion of the CMB spectrum measurementexists.Gravitationallensingprobesthedarkandlu- along the LoS through clusters and groups. The amplitude of the minousmatterdistributionofaclusterbymeasuringthedistortion SZeffectisproportionaltothedimensionlessComptonparameter of background galaxies (weak lensing, for example in Hoekstra y,definedastheintegraloverthethermalelectronpressurealong etal.2001;Gruenetal.2013,2014),orbydetectingmultipleim- theLoS: agesofsinglebackgroundgalaxiesclosetotheLoSofthecluster (cid:90) σ y= T Pdl, (1) m c2 e (cid:63) E-mail:[email protected] whiletheintegraloverasolidangleyieldstheSZobservableY: (cid:13)c 0000RAS 2 R. Kosyraetal. (cid:90) (cid:90) D2Y =D2 ydΩ= σT PdV, (2) massive RLAGNs. In Yates et al. (1989) the clustering effect of A A mec2 RLAGNs at z ≈ 0.2 is compared to the one at z ≈ 0.5, with the whereσ istheThomsoncrosssection,m c2 therestenergy resultthatthelatterobjectsarefoundinenvironmentsthreetimes T e oftheelectronsandD theangulardiameterdistance. denseronaverage.TheyalsostatethatmorepowerfulRLAGNsare A All of these methods may have selection effects induced by found in denser environments than less powerful ones. Based on thesefindingswehypothesizethatahighbackgroundgroupden- structures along the LoS. Lensing, for example can yield biased sityentailsahigherprobabilityofcontainingradiosourcesandthus mass estimates when there are groups along the LoS which con- increasesthenoisealongthelineofsight,potentiallyleadingtoa tributetotheshearsignal(e.g.Spinellietal.2012).TheX-raysig- lowerSZdetectionprobabilityforclustersindensebackgrounden- nalofLoSstructurescanstack,resultinginabiasedmassestimate. The same is true for the SZ effect, however more severely as the vironments. SZsignalisproportionaltothegasdensityρ,whiletheX-rayflux Thispaperisstructuredasfollows.Insection2,wedescribe isproportionaltoρ2,makingtheeffectofLoSstructureonSZsig- the Planck PSZ1 union catalog and the RedMaPPer SDDS DR8 groupcatalog,aswellasourmatchingofthesetwo.Insection3, nalsmuchlargeratlargerangularseparation.Duetothisreasonwe webrieflydiscusstwo-pointcorrelationfunctions.Furthermorewe will investigate whether SZ selected clusters are possibly biased describe our method of generating random points for the Planck bystructuresalongtheLoS,eitherbyphysicallyuncorrelatedfore- catalogandtheprocedureofdefiningtheclustercomparisonsam- groundorbackgroundstructures,orbycorrelatedstructuresatthe pleoutoftheRedMaPPercatalog.Wealsoincludethedescription sameredshiftastheclusteritself. Severaleffectscouldpotentiallycontributetoaselectionbias. ofourtheoreticalpredictionofthe2pcf.Insection4,wepresent ourresults,giveadetaileddescriptionofourerrorestimationand TheblendingoftheSZsignalofthedetectedclusterwithgroups generalized χ2 analysis and we estimate the implications of the alongtheLoScouldbiastheSZestimatehighandcauseclusters measuredeffectonSZandlensinganalysesofPlanckclusters.We alongoverdenselinesofsighttobemorelikelydetected.If,onthe concludeinsection5. contrary,unresolvedgroupsinthevicinityofclustersincreasethe backgroundlevel,thiscouldleadtoalowerdetectionprobability as the signal from the cluster is partly suppressed by the wrong backgroundestimate.Furthermore,ifthebackgroundofacluster 2 DATA iscontaminatedwithradio-loudgalaxies,thiscouldraisethenoise 2.1 ThePlanckPSZ1catalog suchthatclusterswithaweakSZsignalarenotdetected. In this paper we address this question by analyzing the pro- ThePlanck PSZ1unioncatalog isacluster catalog,coveringthe jectedgroupenvironmentofSZ-selectedclustersfromthePlanck whole sky based on SZ detections using the first 15.5 months of PSZ1 union catalog (Planck Collaboration et al. 2013a) and test Plancksurveyobservations.Itcontainsatotalof1227clusters,861 forgroupoverdensitiesorunderdensitiesalongtheLoSinthefore- ofwhichareconfirmedwhiletheremaining366areclustercandi- ground,backgroundandattheredshiftoftheclusters.Thegroup dates(PlanckCollaborationetal.2013a).ThePlancksatellitefea- sampleistakenfromtheRedMaPPer red-sequencecatalogbased turesalowfrequencyandahighfrequencyinstrument,theformer on SDSS DR8 photometry (Rykoff et al. 2014; Rozo & Rykoff coversthebandsat30,44and70GHz(PlanckCollaborationetal. 2014; Rozo et al. 2014). We compute the angular two-point cor- 2013e) while the latter operates at frequencies of 100, 143, 217, relation function (2pcf) of galaxy clusters and groups for differ- 353, 545 and 857 GHz (Planck Collaboration et al. 2013b) with ent subsamples of our catalogs (correlated, foreground and back- angular resolutions between 9.53’ and 4.42’ FWHM, for a total ground structures) to quantify correlated and physically uncorre- ofninedetectionbands.Thechannelmapsofthesixhighestfre- lated group overdensities and underdensities. We compare these quencybands(100to857GHz)wereusedtobuildtheSZ-detection resultstothe2pcfobtainedforanindependentclustersamplewith catalog, in order to avoid problems caused by strong radio point similarredshiftandrichnessdistribution,drawnasasubsampleof sources in cluster centers,which typically have steep spectra and theRedMaPPerSDSSDR8catalog,andtotheoreticallypredicted thusdonotappearinthehighfrequencybands(PlanckCollabora- values. tionetal.2013a). ThegeneralizedNFW(Navarroetal.1997)profilefromAr- naudetal.(2010)wasadoptedfortheclusterdetection. 1.1 Motivation Threedetectionalgorithmswereusedtocreatetheclustercat- We briefly discuss several possible effects that could cause a se- alog, two realizations of the Matched Multi-filter (MMF) method lectionbias.ThefilterfunctionthatisusedforthePlanckcluster (Herranzetal.2002;Melinetal.2006)and(PowellSnakes(PwS), detectionmightestimateatoolargebackgroundvalueifthereare Carvalhoetal.2009,2012). groupssurroundingtheclusterthatcontributetothesignal,which The MMF method detects clusters by using a linear com- could lead to a decreased detection probability in crowded fields bination of maps and a spatial filtering to suppress foregrounds as the subtracted background estimate is too large. On the other andnoise.Thetwoimplementations(MMF1andMMF3)splitthe hand,theclustersaredetectedbycombiningsixfrequencybands wholeskyin640patchesofsize14.66×14.66squaredegreescov- withdifferentfiltersizes,soitisratherunlikelythatthisstillcauses ering3.33timestheareaofthesky(MMF1),andin504patchesof problemswhendetectingclustersbasedonthedifferentialsignal. size10×10squaredegreescovering1.22timestheareaofthesky Another possible origin of a selection effect might be radio- (MMF3).TheMMF3algorithmisrunintwoiterations:thesecond loud galaxies in the background. Donoso et al. (2010) state that is centered on the positions of the candidates from the first one, radio-loud active galactic nuclei (RLAGNs) are predominantly rejecting all candidates that fall below the signal-to-noise (S/N) found in dense environments compared to radio quiet galaxies threshold.Thematchedmulti-frequencyfilteroptimallycombines and regular red luminous galaxies (LRGs) at redshifts 0.4 < z < thesixfrequenciesofeachpatchandtheresultingsub-catalogsfor 0.8.Theyconcludethatthisclusteringeffectisstrongerformore all patches are finally merged together to a single SZ-catalog per (cid:13)c 0000RAS,MNRAS000,000–000 Environment-basedselectioneffectsofPlanckclusters 3 method,selectingthecandidatewiththehighestS/Nratio.Fores- timatingthecandidatesize,thepatchesarefilteredovertherange 105 ofpotentialscales,selectingthescalewiththehighestS/Nofthe current candidate. Finally, the SZ-signal is estimated by running MMFwithfixedclustersizeandposition. 104 PowellSnakesisaBayesianmulti-frequencydetectionalgo- rithm,optimizedtofindcompactobjectsinadiffusebackground. e Afterclusterdetection,PwSmergesallintermediatesub-catalogs. nc Thecross-channelcovariancematrixiscalculateddirectlyfromthe nda103 u b pixeldata,whichisdoneinaniterativewaytominimizethecon- A taminationofthebackgroundbytheSZsignalitself.Ineachiter- ation step, all detections in the same patch with higher S/N than 102 thecurrenttargetaresubtractedfromthedatabeforere-estimating thecovariancematrix.Thisso-called“native”modeofbackground subtractionproducesS/Nvalues20%higherthanthoseoftheMMF 101 method.Inordertoemulatetheestimationofthebackgroundnoise 10-6 10-5 10-4 1p0de-3t 10-2 10-1 100 cross-powerspectrumoftheMMFmethod,PwSisrunin“compat- ibility”mode,skippingthere-estimationstep. Figure2.Log-loghistogramofthedetectionprobability. EachofthethreedetectionalgorithmscreatesacatalogofSZ sources with an S/N ratio (cid:62) 4.5. Obvious false detections are re- movedfromeachofthethreeindividualcatalogs(PlanckCollabo- Outlierrejection rationetal.2013a). Theunioncatalogcontainsallsourcesthathavebeendetected WerejectallmatchesthatareobviousSZ-projections(5cases),as byatleasttwoalgorithmswithS/N(cid:62)4.5withinadistanceof5(cid:48), identifiedbyRozoetal.(2014).Allclustersthathavebeenflagged fixingthepositionoftheMMF3detectionor,incaseofnoMMF3 as3σredshiftoutliersarecross-matchedwiththeRozoetal.(2014) detection,keepingthepositionofthePwSdetection. tableofredshiftoutliers,andinthecaseofanincorrectPlanckred- shiftandacorrectRedMaPPerredshift,weaccepttheclusterusing theRedMaPPerredshiftandviceversa.Furthermorewerejectclus- terswithbadz-matchingwhenavisualinspectionidentifiedthem 2.2 TheRedMaPPerSDSSDR8catalog clearlyasamismatch(onecaseonly),andwerejectclustersthat areoutliersinthemass-Y -plane(accordingtoRozoetal.2014) The Red Sequence Matched-filter Probabilistic Percolation SZ duetoalowRedMaPPerrichness(onecaseonly).Afterrejecting (RedMaPPer)algorithm(Rykoffetal.2014)isared-sequenceclus- alloutliersthefinalmatchedcatalogincludes265clusters. terfinderbasedontheoptimizedrichnessestimatorλ(Rykoffetal. 2012). It has excellent photo-z performance and λ has been de- signedtobealow-scattermassproxy(Rozo&Rykoff2014;Rozo etal.2014).Thealgorithmisdividedintotwostages.Thefirstisa 3 METHODS calibrationstagewheretheredsequencemodelisderiveddirectly fromthedatabyrelyingonspectroscopicgalaxiesingalaxyclus- 3.1 Two-pointcorrelationfunction ters:givenaninitialmodelofthered-sequence,RedMaPPerselects Wemeasurethecrowdingofclustersandgroupswiththeangular clustermembergalaxies,usesthesetoderiveanewred-sequence two-pointcorrelationfunction,whichtracestheamplitudeofclus- model,andtheniteratesthewholeprocedureuntilconvergenceis ter/groupclusteringasafunctionoftheirseparation.Theangular achieved,aswhichpointthered-sequencemodelisadequatelycal- correlationfunctionw(θ)isdefinedastheexcessprobabilityovera ibrated.Thesecondisthecluster-findingstage,whereRedMaPPer random,uncorrelateddistributionoffindingtwoobjectsseparated utilizesthered-sequencemodeltosearchforclustersaroundevery by an angle θ. The probability of finding two objects in two in- galaxyintheSDSS.Thisworkusestheupdatedversion(v5.10)of theoriginalRedMaPPercatalogofRykoffetal.(2014)presented finitesimalsolidangleelementsδΩ1andδΩ2separatedbyangleθ thenreads: inRozoetal.(2014). δP=n n (1+w(θ))δΩ δΩ , (3) 1 2 1 2 withn andn beingthemeancluster/groupdensitiesinbothsam- 2.3 MatchingofthePlanckandRedMaPPercatalogs 1 2 ples. InordertomatchthePlanckPSZ1unioncatalogtotheRedMaPPer A multitude of different estimators exist for calculating the two- SDSSDR8clustercatalogweusedanalgorithmsimilartotheone point correlation function from data catalogs. Kerscher et al. describedinRozoetal.(2014).WefindallmatchesintheRedMaP- (2000), who compared the nine most important of these estima- Per catalog in a radius of 10(cid:48) around each Planck cluster. In the tors in terms of the cumulative probability of returning a value caseofmultiplematcheswedefinethebestmatchastheRedMaP- within a certain tolerance of the real correlation, show that the Per system with the highest richness. We flag all matches with a Landy-Szalayestimator(hereafterLS,Landy&Szalay1993)per- redshiftdifferencebetweenthePlanckandRedMaPPerredshiftof formsbestaccordingtotheircriteria.Henceweadoptthisestima- morethan3σ,whereσcorrespondstotheredshifterrorgivenin tor,whichreads: theRedMaPPercatalog.Thisgivesusatotalof290matchedclus- DD−2DR+RR ters. wˆLS(θ)= RR , (4) (cid:13)c 0000RAS,MNRAS000,000–000 4 R. Kosyraetal. 1.2 1.2 PCloamncpka rsisaomnp sleample PCloamncpka rsisaomnp sleample 1.0 1.0 malized abundance 00..68 malized abundance 00..68 Nor 0.4 Nor 0.4 0.2 0.2 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 300 photo−z richness Figure1.Redshift(left)andrichnessdistribution(right)ofthePlancksample(black)andthecomparisonsample(red).PlanckerrorbarsarePoissonian. Comparisonsampleerrorbarsarenotshownbutareofcomparablesize. or,incaseofacross-correlationbetweentwodifferentsam- bands (Planck Collaboration et al. 2013c) to generate a map dis- ples: playing the weighted average noise of all channels, averaged to D D −D R −D R +R R 3072pixels,tofindthenoiseatthepositionofeachcluster. wˆ (θ)= 1 2 1 2 2 1 1 2, (5) We test for correlation of the density of Planck detections LS R R 1 2 with the noise quantitatively. In this case we assume the number where DD, DRandRRstandforthedata-data,data-random ofPlanckdetectionsperunitarea f tobeapowerlawofthenoise i andrandom-randompaircounts,respectively.Allpaircountsineq. Nperredshiftbiniwithredshiftdependentexponentα: i 5arenormalizedtothetotalnumberofdatapairsintherespective samples. Random points account for geometrical effects like sur- fi(N)= fi(1)·Nαi. (6) veyboundariesandmasksinthesurveyarea.Wedonotwantthe Weperformalikelihoodanalysisovertheparameters f(1)and randompointstocorrectforenvironment-baseddetectioneffects, α,bycalculatingtheexpectednumberofclustersineachsikycell sincethisistheeffectwewanttomeasure,soweareusingrandom viia eq. 6 and computing the Poisson probability with the actual pointswherethetruedetectionshavebeenerased.Thepaircounts numberofdetectionsinthatskycell.Thepowerα scattersaround i havebeencomputedusingthe2d-treecodeAthena(Kilbingeretal. andisconsistentwithzeroforredshiftsz(cid:54)0.5.Abovethisredshift, 2014). wefindα ≈ 0.8.Inconclusion,thenoiselevelhasnoimpacton i detectionsforz<0.5.Wedecidetoremoveallclusterswithz>0.5 from our catalog, bringing our sample size down to 250 clusters. 3.2 GenerationofrandompointsforthePlanckcatalog Based on these findings, we decide to use uniformly distributed randompointsforthePlanckcatalog. TheLSestimator(eq.5)needsarandomcatalogforeachdatacat- alog,inordertocorrectforgeometricaleffectsthatcouldmimica WeusethePlancksurveymask(HealpixNside =2048)tode- finetheregionwheretogeneratethepointsandcutthemafterwards signal. Therearetwoeffectsthatmightimprintaspatialvariationon totheSDSSfootprint.TherandompointsaregeneratedinHealpix coordinatestoensureauniformdistributionoverthesky. thePlanck detectionfunction:thevariationinthenoiseleveland thedistancefromthegalacticdisk.Inthissectionwedescribeour approachtogeneraterandompointsforthePlanckcatalogtaking 3.3 GenerationofrandompointsfortheRedMaPPercatalog intoaccountthevaryingnoiselevel.Thevariationofthedetection probabilityasafunctionofdistancefromthegalacticdiskisinves- TogeneratearandompointcatalogforRedMaPPer,wefirstdraw tigatedinappendixA. arandompositioninthesky,andthenrandomlydrawaRedMaP- Since the noise level of the Planck observations varies over Per cluster. Given the assigned cluster redshift and richness, we theSDSSregion,weneedtotestwhetherthedensityofSZdetec- useourclustermodeltorandomlydrawclustergalaxiestocreate tions has a significant correlation with the noise level that has to a synthetic cluster. We then run RedMaPPer at this location, and be accounted for when generating a random catalog. We use the determinewhetherthesyntheticclusterisdetectedofnot.Thepro- PlanckSMICAmap(whichcomesinHealpix(Górskietal.2005) cedureisrepeated100times,andwecalculatethefractionoftimes coordinateswithN =2048,whichis50331648pixels,resolution wthattheclusterwasdetectedatthatlocation.Thequantitywis side ∼1.7(cid:48)),whichusesanoptimalcombinationoftheninefrequency theweightassignedtothisrandompoint. (cid:13)c 0000RAS,MNRAS000,000–000 Environment-basedselectioneffectsofPlanckclusters 5 thatweobtainforgroupsinthevicinityofPlanckselectedclusters Background toanindependentsampleofopticallyselectedclustersthatresem- blestheselectionfunctionofourmainsampleintermsoftheirred- z shiftandrichnessdistribution.Tothisend,weneedtomodelthe DEC Planckdetectionprobability.Weassumethattheprobabilitythata RedMaPPerclusterisdetectedbyPlancktakestheform: RA 1(cid:34) (cid:32)Λ−Λ (cid:33)(cid:35) 1° Pdet= 2 1+erf √2σdet , (7) Cluster whereerfistheerrorfunction,Λ istherichnessatwhichthe det detectionprobabilityis50%andσthescatterinrichnessatfixed Correlated SZsignal.Pdetstatestheprobabilitythataclusterofgivenrichness ΛisdetectedbythePlancksurvey,ifitwasinsidethesurveyarea. Structure WeusetheRedMaPPerSDDSDR8catalog,calculateP foreach det clusterandassignitasaweighttotheclusteritself. 0.05 WeparameterizetheredshiftevolutionofΛ andσas: det Λdet=αΛ(1+z)βΛ (8) and 0.05 σ=α (1+z)βσ. (9) σ TofindtheoptimumvaluesforαΛ,βΛ,ασandβσ,weperform alikelihoodanalysisinthesefourparameters: (cid:88) (cid:88) ln(L)= ln[P (i)]+ ln[1−P (i)]. (10) det det iPlck inonPlck Here the first sum is over all RedMaPPer clusters that have beendetectedbyPlanckandthesecondsumisoverallRedMaP- Foreground PerclustersthathavenotbeendetectedbyPlanck.Figure1shows thephoto-zdistributionofthePlancksample(black)comparedto thesubsample(red)definedbytheselectionalgorithmbasedonde- tectionprobability.Thedataagreeinmostbinswithin1σ(ofthe Figure3.Sketchoftheselectionmethodforthegroupcatalogs.Thegreen, Poissonianerrors)andinallbinswithin2σ.Tovalidatethequal- blueandredvolumesshowtheselectionforthecorrelated,foregroundand ityofthecomparisonsamplewedrew1000randomsubsamplesof backgroundsamplesrespectively.Thetotalz-depthofthegreenvolumeis 250clustersaccordingtotheirP anddeterminedthelikelihoodof 0.1,withtheclusterinthecenter.Theangularradiusforallvolumesis1◦ det eachsubsample.ComparingtothelikelihoodoftheoriginalPlanck (seecyancircles). sample,weobtainap-valueof0.27,soweconsiderourcompari- sonsampleasreasonable(i.e.,27%ofsubsampleshavelowerlike- lihoodthantheactualPlancksample). Thereisonesubtletyassociatedwiththeaboveprocedure:by Wegeneratearandomcatalogforthecomparisonsampleby random luck, some fraction of our synthetic clusters will overlap withrealRedMaPPerclustersinbothlocationintheskyandred- using the derived values for the four parameters αΛ, βΛ, ασ and β and calculate the detection probability for each entry in the shift.Ifonedidnotremovethegalaxiesassociatedwiththeorigi- σ RedMaPPerrandomcatalog. nalRedMaPPerclusterbeforeplacingthesyntheticclusteratthat location,uponrunningRedMaPPeronewillalwaysfindacluster there(i.e.theoriginalcluster),andonewoulderroneouslyconclude 3.5 Theoreticaltwo-pointcorrelationfunction w=1irrespectiveofthedetailsofthesyntheticcluster.Thus,itis criticallyimportanttoremovetheoriginalRedMaPPergalaxyclus- Our purpose is to compute the cross-correlation between a refer- tersfromthegalaxycatalogpriortodrawingourrandompoints.We enceclusteratgivenredshiftandcorrelatedstructureswithinacer- eraseclustersprobabilistically:givenaclusterofrichnesslambda tainredshiftrange.Notethatthisdiffersfromcomputingtheusual atredshiftz,wecollectallofitsmembergalaxies,andremoveeach angularcorrelationfunctionbetweentwosamples.Inourcase,in galaxyaccordingtotheassignedmembershipprobability,sothata fact,werestrictourcalculationofthecluster-grouptwo-pointcor- galaxythatis90%likelytobeaclustermemberisremovedfrom relationfunctiontoredshiftbinscenteredaroundthereferenceclus- thegalaxycatalogwith90%probability. ter.Thecorrelatedgroupredshiftdistributionisthusdependenton thereferenceclusterredshift.Wethenobtainthetotalcorrelation function by summing up all the redshift-binned contributions ac- 3.4 Definitionofacomparisonsample cordingtotheclusterredshiftdistribution. Thenumericaltoolweuseforcalculatingthetheoreticalcor- WewanttotestwhetherSZselectedclustersaregenerallyfoundin relationfunctioniscambsources1(Lewis&Challinor2007),which adifferentenvironmentthansimilar(intermsofredshiftandrich- ness)clustersthatareselectedfortheiropticalproperties.Thuswe needtocomparethecluster-grouptwo-pointcorrelationfunctions 1 http://camb.info/sources/ (cid:13)c 0000RAS,MNRAS000,000–000 6 R. Kosyraetal. computestheangularpowerspectrumCsofthematterdensityper- We draw 100 sets of 250 Planck random points, assigning l turbations, for given input redshift distributions and for different themthesameredshiftdistributionastheclustersample.Weper- cosmological models. We restrict our calculation to standard flat form the procedure described above on each set, averaging the ΛCDM cosmology (Ω = 0.25, h = 0.7) and the linear regime results. We proceed analogously for the comparison sample, by m only.Therelationbetweenthecross-spectraandtheprojectedtwo- drawing100samplesofunweightedclustersbyselectingrandomly pointcorrelationfunctionisgivenby amongallclustersofthecomparisonsampleaccordingtotheirde- (cid:88)(cid:32)2l+1(cid:33) tection probability. The sample size is on average 247, the same w(θ)= P(cosθ)C , (11) asthesumoveralldetectionprobabilities.Thesameprocedureis 4π l l l(cid:62)0 performed on the random catalog of the comparison sample (see where P are the Legendre polynomials of degree l. We use section3.4). l a maximum l = 3000 and θ ∈ [0.01,300] arcmin. We calculate theexpectedtwo-pointcorrelation(eq.11)for20referencecluster redshiftszcl ∈ [0.05;0.5].Forthereferenceclusterredshiftdistri- 4.1 Errorestimation bution, we assume a Gaussian distribution centered at the cluster redshiftz ,withstandarddeviationequaltothemeanphotometric For estimating errors and covariance matrices, we use three dif- cl redshifterrorassociatedtotheclusterredshiftinthePlanckcata- ferent methods: for the errors of the Planck sample with respect log,i.e.N(z ,0.02).Forthecorrelatedgroupsredshiftdistribution, totheory,weusea“replace-one”implementationoftheJackknife cl weusetheobservedredshiftdistributionoftheRedMaPPergroups resamplingmethod,fortheerrorsofthecomparisonsamplewith withrichnessλ > 5,limitedtoarangeof±0.06,centeredaround respecttotheory(zero)weuseBootstrapresamplingandfortheer- z . This interval is greater than the bin width in the analysis of rorsofthePlancksamplewithrespecttothecomparisonsamplewe thcleobservationaldataof±0.05(seesection4),inordertoaccount usea“delete-one”Jackkniferesamplingbydrawing100different fortheerrorsinphotometricredshift(∼ 0.02).Theobservedcor- (unweighted)representationsoutofthecompletecomparisonsam- relationistheaverageofthew(θ)calculatedineachredshiftbin plerandomlyaccordingtothedetectionprobabilities.Theformer i i, weighted by the cluster and group distributions and respective twowillbeexplainedinmoredetailinthefollowingsubsection. averagebiases,normalizedbythetotalnumberofobjects: (cid:80)20 dNcdNgb¯cb¯gw(θ) w(θ)theory= i=1 (cid:80)2i0 dNicdiNig i . (12) 4.1.1 ErrorsofthePlancksamplewithrespecttotheory i=1 i i WeuseamodificationoftheJackkniferesamplingmethod.Inthe Here dNc and dNg are the counts per redshift bin of clusters and i i standard”delete-one“Jackknifetechnique,thesurveyareaissub- groupsrespectively.Furthermore,b¯c andb¯g aretheaveragebiases i i divided into a number of subsamples and the analysis is done a for clusters and groups within the bin i. We estimate the bias for number of times equal to the number of subsamples, considering eachclusterinthematchedPlanck catalogandeachgroupinthe eachtimeallsamplesexceptone.TheJackknifecovariancereads: RedMaPPercatalogbyusingtheanalyticformulaofTinkeretal. (w2h0i1c0h).wTeheimspalsosyumtheesreasufilxteodfRmyaksosff-riecthanle.s(s20s1c2a)li.(cid:12)nAgnraenlaat(cid:12)lioogn,esftoi-r Cij= mm−1(cid:88)m (cid:0)xi,k−x¯i(cid:1)(cid:16)xj,k−x¯j(cid:17), (13) matefortheforeground/backgroundstructuresat(cid:12)(cid:12)zcl−zgr(cid:12)(cid:12) > 0.05 k=1 yieldsa2pcfconsistentwithzerowithinthestatisticalerrorsofour wheremisthenumberofJackknifesamples, x isthedata i,k analysis. valueinbiniofsamplekand x¯ isthemeanvalueinbini.Since i galaxygroupsareclusteredintrinsically,theerrorsinneighboring bins may be correlated, so we need to take into account the full 4 ANALYSISANDRESULTS covariancematrixinouranalysis. We define our Jackknife samples to be equal to the data- Therelevantquantitiesweareinterestedinarethe2pcfsofclusters cylindersinoursub-catalogs.Weareusing250samplescontaining and groups for correlated structure (groups with similar redshifts exactlyoneclustereachandallgroupsinitsvicinity. as the cluster), foreground structure (groups with lower redshift Sinceourtheoreticalpredictionismadefortheexactredshift thanthecluster)andbackgroundstructure(groupswithhigherred- distributionofPlanck,weneedtofindtheerrorswithrespecttothis shiftthanthecluster).Wecomputetheangularcorrelationfunction, distribution.Adelete-oneJackknifewouldintroduceasystematic whereclusterpairsaresubjectedtooneofthreeconstraints: errorhere,astheredshiftdistributionofthesamplechangeswhen (i) theRedMaPPer-Planckclusterpairisseparatedbylessthan deletingonecluster.Toovercomethisproblem,weuseamodified |∆z|<0.05. Jackknifemethod:ineachJackknifesampleweleaveoutonesub- (ii) theRedMaPPer-Planckclusterpairissuchthatz < z − sample(cluster)andassignaweightoftwotoanothercluster.This rm pl 0.05. clusterischosentobetheclosestinredshifttotheleft-outcluster, (iii) theRedMaPPer-Planckclusterpairissuchthatz >z + in order to minimize the effect on the redshift distribution of the rm pl 0.05. sample.Wehavetomodifyequation13toaccountforthechanged samplesize: The first set of pairs allows us to test for the environmental impactofphysicallycorrelatedstructures,thesecondfortheimpact C = 1(cid:88)m (cid:0)x −x¯(cid:1)(cid:16)x −x¯(cid:17). (14) offoregroundstructuresandthethirdfortheimpactofbackground ij 2 i,k i j,k j structures.Thisselectionmethodisdisplayedgraphicallyinfigure k=1 3,withthegreenvolumebeingthecorrelatedstructure,inbluethe The validity of the formula has been verified in a Monte- foregroundandinredthebackground. Carlo-simulation. (cid:13)c 0000RAS,MNRAS000,000–000 Environment-basedselectioneffectsofPlanckclusters 7 2.0 Correlated structure Planck sample Comparison sample Theory 1.5 1.0 ) θ ( w 0.5 0.0 0 10 20 30 40 50 60 θ [arcmin] 2.0 2.0 Correlated structure high S/N Planck sample Correlated structure low S/N Planck sample Comparison sample Comparison sample 1.5 1.5 1.0 1.0 w()θ w()θ 0.5 0.5 0.0 0.0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 θ [arcmin] θ [arcmin] Figure4.Two-pointcorrelationfunctionforgroupsinthevicinityofPlanckclusters(blue)andgroupsinthevicinityofclustersinourcomparisonsample (red).Inthisplotweshowthe2pcfforgroupswithredshiftequaltotheclusterredshift±0.05(correlatedstructure).Thecyanlinerepresentsthetheoretical prediction.Top:completesample,bottomleft:onlyclusterswithS/N>median,bottomrightonlyclusterswithS/N<median.InthelowS/Ncasethereisa slightunderdensityinthePlancksampleintheregionbetween10(cid:48)and20(cid:48).Forinterpretationsseesections4.2,4.3and5. 4.1.2 Errorsofthecomparisonsamplewithrespecttotheory sincewedidnotaccountforthemodifiedredshiftdistributiondue tothebootstraphere.Toovercomethisproblemweslightlychange In order to perform an error estimation on the comparison sam- theprocedurebybootstrappingsetsof5groupsinsteadofsingle plewitharesamplingmethod,weneedamultitudeofcomparison groups.Thesetsarecreatedbydividingthecataloginto5subsam- samples. We perform a Bootstrap resampling on the RedMaPPer plessplitbyredshift,andselectingonegroupfromeachofthese catalog by drawing 1000 random catalogs with the same number subsamples.Wesortthesubsamplesbyweight,soweensurethat of clusters as in the original catalog. We then count the number eachpackagecontains5groupswithsimilarweightsanddifferent ofpairsinangularbinsaroundeachclusterweightedwiththede- (equallydistributed)redshifts.Inthiswaythesystematicerrordue tectionprobabilityandcomputethecovarianceineachangularbin tothemodifiedredshiftdistributionisminimized. fromthese1000samples.Itturnsoutthattheerrorsestimatedby thismethodtendtobehigherthantheerrorsofthePlancksample, (cid:13)c 0000RAS,MNRAS000,000–000 8 R. Kosyraetal. Foreground groups Planck sample 0.4 Comparison sample 0.2 0.0 ) (θ −0.2 w −0.4 −0.6 −0.8 −1.0 0 10 20 30 40 50 60 θ [arcmin] Foreground groups high S/N Planck sample Foreground groups low S/N Planck sample 0.4 Comparison sample 0.4 Comparison sample 0.2 0.2 0.0 0.0 w()θ−0.2 w()θ−0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1.0 −1.0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 θ [arcmin] θ [arcmin] Figure5.Sameasfigure4,butforgroupswithredshiftzgr <zcl−0.05(foregroundstructure).Thetwodatasetsagreewellinthecompletesampleandthe highS/Ncase,whileforlowS/NaslightoverdensitycanbeobservedinthePlancksamplenearlyoverthecompleteangularregiontested,albeitmostdata pointsstillagreewithintheerrormargins. 4.2 Results usefulapproachherewouldbetosplitthesampleatS/N7,which isthethresholdabovewhichtheclustersareincludedinthePlanck In thissubsection we presentthe results ofthe angular two-point cosmological sample (Planck Collaboration et al. 2013d). Unfor- correlationfunctionofgalaxyclustersandgroupsobtainedasde- tunately, in this case the high S/N sample would contain too few scribed earlier in this section. We analyze w(θ) in 15 equidistant clusterscausingtheerrorlimitstobecometoolarge,sowedecided angularbinswithawidthof4(cid:48).Wecomparetheresultsobtained tosplitthesampleatthemedianS/N5.4,generatingtwoequally forthePlancksample(bluepointsinfigures4,5and6)withthose largesubsampleswith125clusterseach. for our comparison sample (red points) and with our predictions (cyanline).Alikelihoodanalysisispresentedinsubsection4.3. Thetopoffigure4showsw(θ)forgroupsatthesameredshift Weexpectthatapossibleeffectisstrongerforclustersthatare as the cluster redshift ±0.05 (correlated structure). In the two in- just above the detection threshold S/N of 4.5. Due to this reason nermostangularbinsbothsamplesareaffectedbyblendingeffects wealsosplittheclustersintoahighandlowS/Nsample.Themost andhaloexclusion.Thelatteristheeffectoftwonearbystructures (cid:13)c 0000RAS,MNRAS000,000–000 Environment-basedselectioneffectsofPlanckclusters 9 Background groups Planck sample 0.2 Comparison sample 0.0 −0.2 ) θ ( w −0.4 −0.6 −0.8 0 10 20 30 40 50 60 θ [arcmin] Background groups high S/N Planck sample Background groups low S/N Planck sample 0.2 0.2 Comparison sample Comparison sample 0.0 0.0 −0.2 w()θ−0.2 w()θ−0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1.0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 θ [arcmin] θ [arcmin] Figure6.Sameasfigure4,butforgroupswithredshiftzgr >zcl+0.05(backgroundstructure).WeobserveaslightunderdensityinthePlancksamplewith respecttothecomparisonsample,whichismoresevereinthelowS/Nsubsample. merging into one halo, which has not been included in the theo- Figure5showsthe2pcfforgroupswithredshiftz <z −0.05 gr cl reticalprediction.Inmostbinsuptoapproximately40(cid:48)thePlanck (foreground structure). The fact that we also observe blending sampleshowsaslightunderdensitywithrespecttothecomparison here (in the innermost bins), shows that the detection probabil- sample,albeittheindividualdatapointsstillagreewithintheerror ity of RedMaPPer groups also suffers from blending effects, i.e. margins(likelihoodanalysisshowstheunderdensityisnotsignif- RedMaPPerislesslikelytodetectgroupsinthevicinityofarich icant,seetable1).Theexcessinthethirdbinwithrespecttothe foregroundorbackgroundcluster.Besidesthiseffect,onecansee predicted curve is potentially due to non-linear structure growth. a slight overdensity in the Planck sample at angular scales >10(cid:48), Incaseofthesplitsampleweseeabetteragreementbetweenthe buttheerrorbarssuggestthatthisdifferenceisnotsignificant.The twosamplesforthehighS/Nsubsample(bottomleftplot),while effectisagainweakerinthehighS/NandstrongerinthelowS/N theagreementisworseinthelowS/Ncase(bottomright)where subsample. Planckclustersarefoundinevenmoreunderdensebackgrounden- Figure6showsthe2pcfforgroupswithredshiftz >z +0.05 vironments. gr cl (background structure). Here the 2pcf suffers from blending on (cid:13)c 0000RAS,MNRAS000,000–000 10 R. Kosyraetal. smallangularscales,too.Aslightunderdensitycanbeseeninthe Wealsosplitthegroupsampleintwosubsamplesatrichness Plancksamplewithrespecttothecomparisonsampleinnearlyall 12(highλandlowλintable1),butwefoundnosignificantdiffer- angularbins.Theindividualdatapointsare,however,inagreement encesinthesetwosubsamples. withintheerrormargins.Alsoheretheobservedunderdensityap- Table2givesthebest-fittingconstantvaluesforthe2pcfand pearslesssevereinthehighS/NandmoresevereinthelowS/N thecorresponding1σerrors.Thefirstfourangularbinswhichsuf- case. fer from blending have been ignored in this fit. We see that the backgroundcorrelationisnotconsistentwithzeroforPlanckwith morethan4σ,whilethecomparisonsampleisconsistentwithzero 4.3 Likelihoodanalysis within1.25σ,whichcanstillbeduetostatisticalfluctuations.Since wedetectabackgroundunderdensityof-0.049withasignificance Inthissubsectionweinvestigatethesignificancebywhichthe2pcf inthePlancksampledifferfromthecomparisonsampleandfrom of∼4σwithrespecttozerobutthecomparisonsamplealsodiffers the theoretical prediction. We perform a generalized χ2 analysis fromzerowithavalueof-0.02at∼1.25σ,weconcludethatstatis- ticalfluctuationsintheparticularregionsused(cosmicvariance), that takes into account the full covariance matrix, since as men- likelyalsocontributetotheobserveddefectofPlanckbackground tionedbeforeweexpecttheerrorsinneighboringbinstobecorre- latedpositivelyduetotheclusteringofgroups.Thegeneralizedχ2 groups,butarenosufficientexplanationofthefullobservedeffect. Ontheotherhand,onecouldimaginethatRedMaPPerdetections reads: arebiasedinthevicinityofmassiveclustersduetothecorrelated χ2 =δT ·C−1·δ, (15) structurethatsurroundsthemouttolargeradii,whichmightaffect gen ij thedetectionofgroupsduetotheblendingeffect,asdiscussedin whereC−1istheinversecovariancematrixandδistheresidual vector, contaiinjing the difference between measured and expected section4.2. Whenlookingattheforegroundsample,theslightoverdensity values(wheremeasuredvaluescorrespondtothePlanck2pcfand onemightexpectfromfigure5isnotsignificant,withap-valueof expectedvaluescorrespondtoeitherthecomparisonsampleorpre- 0.72. dictedvalues)inangularbins.Fortheforegroundandbackground samplewecomparetheresultswithzero,sincethetheoreticalpre- dictionsinthesecasesareseveralordersofmagnitudelowerthan 4.4 2pcfforPlanckandLRGs ourmeasurementuncertainty. Table1givesthep-valuesforallourthreedifferentdatasam- Wewanttoverifyourresultsbycomparingthemtoanindependent plesforPlanckwithrespecttothecomparisonsample,Planckcom- sampleofbackgroundsources.WereplacetheRedMaPPergroup paredtotheoryandthecomparisonsamplewithrespecttothethe- catalogwiththeCMASS catalogofluminousredgalaxies(LRG) oreticalprediction.Thefourinnermostangularbinshavebeenig- withspectroscopicredshifts(Eisensteinetal.2011;Dawsonetal. noredintheχ2 calculation,sincethedatainthesebinsapparently 2013;Andersonetal.2014).Asclustersandgroupstendtofeature suffer from halo exclusion and blending effects, which have not mostly red galaxies, we expect the LRGs to show a similar clus- beenconsideredinourtheoreticalprediction.Thus,thenumberof teringbehavior.Furthermore,iftheoriginoftheunderdensitywe degreesoffreedomis11. observedistrulythepresenceofradiosources,whichtendtoclus- Thep-valueswithrespecttothecomparisonsamplearetyp- ter at high redshifts, we expect to see the same effect in CMASS ically quite high (the lowest one being 0.28 for the foreground galaxies. sample),sothenull-hypothesis,whichstatesthatthetwosamples Whenlookingatfigure7,wecannotconfirmtheunderdensity aresimilar,cannotberejected.Thep-valuesaregenerallyslightly thatwefoundforbackgroundRedMaPPergroups.Ifthephysical lowerinthelowS/Ncasewhichsupportsourassumptionthatse- effecthasaz-dependence,thefactthattheredshiftdistributionsof lectioneffectsarepredominatelyobservedinthelowS/Nregime. theRedMaPPergroupsandtheCMASSLRGsdifferlargelymight Nevertheless,wecannotconfirmaselectionbiasbasedonourdata beresponsiblefortheobservedeffect.Asweareusinguniformly sample,sincethevaluesofthePlancksampleandthecomparison distributed random points for the Planck clusters, we do not ac- sampleareinagreementeverywhere. countforapotentialpositiondependenceoftheselectionfunction. WhencomparingthePlanckdatawiththetheoreticalpredic- Inparticular,aspatialvariationoftheredshiftdependenceofthe tion,wefindhighp-valuesinthecorrelatedandforegroundsam- Planckdetectionprobabilitycouldpossiblymimicsuchaselection ples, while we find very low values in the background sample, effect.Weinvestigatedthemostlikelyversionofthispossibilityin whichsuggestsaselectioneffectrelatedtolowerbackgroundden- AppendixA,althoughmorecomplexdependenciesmightexist. sity.Tosupportthisassumption,welookatthep-valueofthecom- parison sample vs zero (for the background sample) which sug- 4.5 ImplicationsforSZandlensingmasses gestsmuchbetteragreementthanthevalueofthePlancksample. When looking at the splitted sample with respect to S/N, the p- The potential underdensity in the background of Planck detected valuesforthePlancksamplearehigherthanforthecompletesam- clustersisnearlyconstantinallangularbinsexcepttheonesthatare pleinbothcases,whichcomesfromthelargeruncertaintiesinthe affectedbyblending.Henceitisstraightforwardtomodeltheeffect splittedsample.Thep-valuesforPlanckrelativetothecomparison asthebestfittingconstant2pcfinthesebins.Weusethisvalueof sampleinthehighS/Ncaseareingoodagreement,yetbothonly -0.049forfurtherapproximations.WeestimatethedefectSZsig- marginallyagreewithzero,whichweassumecomesfromcosmic nal caused by this effect in the average beam size of the Planck variance.InthelowS/Nsamplehowever,theagreementofPlanck channelsthatareinvolvedintheclusterdetection.Weusetheme- with zero is significantly worse than for the comparison sample. dian redshift of the 250 Planck clusters in our sample (0.23) and WeconcludethatthebackgroundunderdensityforPlanckclusters calculatethemeanSZsignalofallRedMaPPer groupswithred- isafunctionofS/NandtheeffectbecomesstrongerforlowS/N shift higher than that value +0.05 (as our background selection), detections. usingthescalingrelationfromPlanckCollaborationetal.(2013d). (cid:13)c 0000RAS,MNRAS000,000–000

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