Environment-based selection effects of Planck clusters

0.75 MB·English

by  Ralf Kosyra

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