On the over-concentration problem of strong lensing clusters

0.5 MB·English

by  M. Sereno

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed11January2010 (MNLATEXstylefilev2.2)
On the over-concentration problem of strong lensing clusters
M. Sereno⋆1,2, Ph. Jetzer1 and M. Lubini1
1Institutfu¨rTheoretischePhysik,Universita¨tZu¨rich,Winterthurerstrasse190,8057Zu¨rich,Switzerland
2DipartimentodiFisica,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129Torino,Italia
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Accepted2009December22.Received2009December14;inoriginalform2009September25
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ABSTRACT
n Λ cold dark matter paradigmpredicts that galaxy clusters follow an universalmass density
a
profile and fit a well defined mass-concentration relation, with lensing clusters being pref-
J
erentially triaxial haloes elongated along the line of sight. Oddly, recent strong and weak
1
lensinganalysesofclusterswith alargeEinsteinradiussuggestedthosehaloesto behighly
1
over-concentrated.Here, we investigatewhat intrinsic shape and orientationan halo should

have to account for both theoretical predictions and observations. We considered a sample
]
O of 10 strong lensing clusters. We first measured their elongation assuming a given mass-
concentration relation. Then, for each cluster we found the intrinsic shape and orientation
C
whicharecompatiblewiththeinferredelongationandthemeasuredprojectedellipticity.We
.
h distinguished two groups. The first one (nearly one half) seems to be composed of outliers
p of the mass-concentrationrelation, which they would fit only if they were characterised by
- afilamentarystructureextremelyelongatedalongthelineofsight,thatisnotplausiblecon-
o
sideringstandardscenariosofstructureformations.Thesecondsamplesupportsexpectations
r
t ofN-bodysimulationswhichprefermildlytriaxiallensingclusterswithastrongorientation
s
a bias.
[
Keywords: galaxies:clusters:general– cosmology:observations– gravitationallensing–

1 methods:statistical
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1 INTRODUCTION galaxyclustersarethenacrucialprobeofthemeandensityofthe
1
. universe at relatively late epochs. State-of-the art models of cos-
1 Clusters of galaxies, the most recent bound structures to form
micstructureformationsuggestthatgalaxyclusterconcentrations
0 in a hierarchical cold dark matter model with a cosmological
decreasegraduallywithvirialmass.However,clusterobservations
0 constant (ΛCDM), offer important clues to the assembly pro-
haveyettofirmlyconfirmthiscorrelation.
1 cessofstructureintheuniverse.N-bodysimulationsaresuccess-
: ful in fitting large-scale structure measurements and are able to On the observational side, the situation at present is
v
i make detailed theoretical predictions on dark matter halo proper- still unclear due to the plurality of methods employed
X ties(Navarroetal.1997;Bullocketal.2001;Diemandetal.2004; (Comerford&Natarajan 2007). The observed concentration-
r Duffyetal. 2008) but some disagreement with observations still mass relation for galaxy clusters has a slope consistent with
a persists.OnepossibleconflictbetweenΛCDMandmeasurements theoretical prediction fromsimulations, though thenormalization
isthedetection of extremelylargeEinsteinradii inmassivelens- factor seems to be higher (Comerford&Natarajan 2007). A
ingcluster(Broadhurst&Barkana2008;Sadeh&Rephaeli2008; critical point is that concentrations measured in massive lensing
Oguri&Blandford 2009; Zitrinetal. 2009). The Einstein radius clusters appear to be systematically larger than X-ray concen-
mirrorsthemasscontainedintheinnerregionsanditsmeasurement trations (Comerford&Natarajan 2007). A similar, though less
isquitemodelindependent. EvenifanuniversalNavarro-Freank- pronounced, effect is also found in simulations (Hennawietal.
Whitedensityprofile(Navarroetal.1996,1997,NFW)reproduces 2007),whichshowthatmassivelensingclustersareusuallyelon-
manycharacteristicsofmassivelenses,suchhaloesshouldbeover- gated along the line of sight. Oguri&Blandford (2009) showed
concentratedtofitthedata. thatthelargertheEinsteinradius,thelargertheover-concentration
The concentration parameter measures the halo central den- problem,withclusterslookingmoremassiveandconcentrateddue
sity, which depends on the assembly history and thereby on the totheorientationbias.
time of formation. The halo concentration is then expected to be Theover-concentrationbiasseemstobemuchlargerinobser-
relatedto itsvirialmass, withtheconcentration decreasing grad- vations than in simulations. Broadhurstetal. (2008) inferred sig-
ually with mass (Bullocketal. 2001). Concentrations of massive nificantly high concentrations for four nearly relaxed high-mass
clusters.Suchatrendhasbeenrecentlyexacerbatedwiththeanal-
ysisofthelargestknownEinsteinradiusinMACSJ0717.5+3745
⋆ E-mail:[email protected](MS) (Zitrinetal.2009).Ogurietal.(2009) found that thedatafroma
(cid:13)c 0000RAS
2 M. Sereno et al.
sampleoftenclusterswithstrongandweaklensingfeatureswere versionmethodwhichundersomegivena-priorihypothesesallows
highly inconsistent with the predicted concentration parameters, to infer intrinsic mass, concentration and elongation of a lensing
evenincludinga50%enhancementtoaccountforthelensingbias cluster;themethodisthenappliedtoasampleoftenstronglensing
(Oguri&Blandford2009).Ontheotherhand,Okabeetal.(2009) clusters.InSec.5wecomparetheobserveddistributionsofelonga-
found that the correlation in the c M relation, as measured in tionalongthelineofsightandellipticityintheplaneoftheskyto
−
a sample of 19 clusters with significant weak lensing signal that differenttheoretical predictions. Section6exploitsthepreviously
werewellfittedbyaNFWprofile,wasmarginallycompatiblewith inferredgeometricalparameterstopredicttheintrinsicaxialratios
predictionsforbothslopeandnormalization. andtheorientationoftheclustersinthesample.Finally,Sec.7is
Different definitions of parameters for spherically averaged devotedtoasummaryandtosomefinalconsiderations.
profiles can play a role when comparing observations to predic- Throughout the paper, we assume a flat ΛCDM cosmology
tions (Broadhurst&Barkana 2008). Triaxiality issues were ad- withdensityparametersΩ =0.3,Ω =0.7andanHubblecon-
M Λ
dressed by Corlessetal. (2009), who derived weak lensing con- stantH =100hkms−1Mpc−1,h=0.7.Wequoteuncertainties
0
straintsonthreestronglensingclusterswithoutassumingaspheri- atthe68.3%confidencelevel.
calhalomodel.Thelargeerrorsthataccompanytriaxialparameter
estimates can make observations compatible, even if marginally,
with theoretical predictions. Investigations in the weak lensing 2 THEORETICALPREDICTIONS
regime demonstrated that neglecting halo triaxiality can lead to
over- and under-estimates of up to50% and afactor of 2 inhalo High resolution N-body simulations have shown that the density
mass and concentration, respectively (Corless&King 2007). An profiles of dark matter halos are successfully described as NFW
analysisofAC114usingonlystronglensingdataandaccounting densityprofiles(Navarroetal.1996,1997),whose3Ddistribution
for triaxiality also supported that projection effects play a major follows
role in the estimate of the concentration (Serenoetal. 2009). Fi- ρ = ρs , (1)
nally,analysesofstackedweaklensingclustersoflessermassdoes NFW (r/rs)(1+r/rs)2
not exhibit the high concentration problem (Johnstonetal. 2007;
where ρ is the characteristic density and r is the characteristic
s s
Mandelbaumetal. 2008), in agreement with theoretical findings
length scale. N-body simulations showed as well that haloes are
(Oguri&Blandford2009).
aspherical and that such profiles can be accurately described by
Severaleffectscanplayarole:over-concentratedclustershave concentrictriaxialellipsoidswithalignedaxes(Jing&Suto2002).
a larger lensing cross section (Hennawietal. 2007); strong lens- A NFW equivalent profile whose density is constant on a family
ingclusterspreferentiallysamplethehigh-massendofthecluster of similar, concentric, coaxial ellipsoids is obtained by replacing
massfunction(Comerford&Natarajan2007);whileextremecases the spherical radius r with an ellipsoidal radial variable ζ in the
oftriaxialityarerare,suchhaloscanbemuchmoreefficientlenses intrinsic orthogonal framework centred on the cluster barycentre
thantheirmoresphericalcounterparts(Oguri&Blandford2009); andwhosecoordinates,x ,arealignedwithitsprincipalaxes,
i,int
the strongest lenses in the universe are expected to be a highly
biased population preferentially orientated along the lineof sight 3
ζ2 e2x2 , (2)
(Hennawietal.2007;Oguri&Blandford2009);estimatesoflens- ≡ i i,int
ing concentrations can be also inflateddue tosubstructures close Xi=1
tothelineofsight(Puchwein&Hilbert2009).Ontheotherhand, where e are the intrinsicaxial ratios. Without loss of generality,
i
contamination of weak lensing catalogues can lead to underesti- wecanfixe > e > e = 1.Inthefollowing,wewillalsouse
1 2 3
matetheconcentration(Limousinetal.2007). theinverseratios,0<q =1/e 61.
i i
In order to check the ΛCDM paradigm is then crucial to According to recent N-body simulations (Netoetal. 2007;
account for all possible biases when comparing theoretical re- Maccio` etal.2008;Gaoetal.2008;Duffyetal.2008),thedepen-
lations with lensing observations. Such approach was taken in dence of dark matter halo concentration c on halo mass M and
Broadhurst&Barkana (2008), whoderived theprobability distri- redshiftzcanbeadequatelydescribedbyapowerlaw
bution of Einsteinradii fromconcentration distributions found in
c=A(M/M )B(1+z)C. (3)
N-body simulations. Also after considering that lensing clusters pivot
areintrinsicallyover-concentratedandthattheinherenttriaxiality Since several assumptions were used by competing groups, re-
ofCDMhaloesalongwiththepresenceofsubstructure enhances sults can be somewhat different, in particular as far as the over-
theprojectedmassinsomeorientations,theyfoundthattheoretical all normalization is concerned. Several values for the linear am-
predictions are excluded at a 4σ significance. Sadeh&Rephaeli plitude of mass fluctuations σ were considered. The higher σ ,
8 8
(2008)reachedasimilarconclusion.TheyimpliedtheEinsteinra- theearlier the formation epoch for haloes of a given mass. Here,
diusdistributionfromtheprobabilitydistributionofclusterforma- we follow Duffyetal. (2008), who used the cosmological pa-
tiontimesandfromaformationredshift-concentrationscalingde- rameters from WMAP5 (σ = 0.796) and found A,B,C =
8
{ }
rivedfromN-bodysimulations. However duetovariousinherent 5.71 0.12, 0.084 0.006, 0.47 0.04 forapivotalmass
{ ± − ± − ± }
uncertainties,thestatisticalrangeofthepredicteddistributionmay Mpivot =2 1012M⊙/hintheredshiftrange0 2fortheirfull
× −
besignificantlywiderthancommonlyacknowledged. sampleofclusters.
Here,wecomparemeasurementswiththeoreticalpredictions ByseparatelystudyingthedistributionofNFWprofileparam-
fromsemi-analyticalinvestigationsandN-bodysimulationsavoid- etersbothforthegeneralhalopopulationandforthelensingpop-
ing some possible biases connected to spherical averaging. The ulation(i.e.haloesweightedbytheirstronglensingcross-section)
paper isasfollows. In Section2, wereview thepredictions from Hennawietal.(2007)showedthatthedistributionof3Dconcentra-
either N-body simulations or semi-analytical investigations. Sec- tionsofthelenspopulationisthesameasthatofthegeneralhalo
tion 3 discusses how projected quantities are related to intrinsic populationexceptforashiftupwardsbyafactorof 17%.Inthe
∼
parametersforanellipsoidalcluster.InSec.4,wedevelopourin- following,wewillthenalsoconsideranenhancedc M relation
−
(cid:13)c 0000RAS,MNRAS000,000–000
Arelensingclustersover-concentrated? 3
forlensingclusters,withA 6.68.Notethatsuchincreasedvalue δ = 200 c200 . (11)
ofAcouldbealsoseenasd∼uetoalargervalueofσ8. c 3 ln(1+c200)−c200/(1+c200)
N-bodysimulationsprefermildlytriaxialhalos.Jing&Suto The virial mass, M , is the mass within the ellipsoid of semi-
200
(2002)investigatedtheprobabilitydistributionofintrinsicaxialra- majoraxisr ,M =(800π/3)q q r3 ρ .Suchdefinedc
200 200 1 2 200 cr 200
tiosandproposedanuniversalapproximatingformulaforthedis- and M have small deviations with respect to the parameters
200
tributionofminortomajoraxisratios, computed fittingspherically averaged density profiles, asdone in
(q q /r )2 N-bodysimulations.Theonlycaveatisthatthesphericalmassob-
P(q1)∝exp − 1−2σµ2 q1 (4) tainedinsimulationsissignificantlylessthantheellipsoidalM200
(cid:20) s (cid:21) forextremeaxialratios(Corless&King2007).However,sincethe
whereq =0.54,σ =0.113and dependence of the concentration on the mass is quite weak, see
µ s
Eq.(3),thiswillhavenegligibleeffectsonouranalysis.
rq1 =(Mvir/M∗)0.07ΩM(z)0.7, (5) Threerotationanglesrelatetheintrinsictotheobserver’scoor-
dinatesystem,i.e.thethreeEuler’sangles,θ,ϕandψ.Afteralign-
withM∗thecharacteristicnonlinearmassatredshiftz.Thecondi-
ment of the observer’s coordinate system with the direction con-
tionalprobabilityforq ,theratiooftheintermediatetothemajor
2 nectingtheobservertotheclustercentre,thelineofsighthaspolar
axis-length,goesas
angles θ,ϕ π/2 intheintrinsicsystem.Withathirdrotation,
{ − }
3 2q /q 1 r ψ,wecanproperlyalignthecoordinateaxesintheplaneofthesky.
P(q1/q2|q1)= 2(1 rmin) 1− 1 12−rm−in min (6) Ifnotstatedotherwisewewilllineupsuchaxeswiththeaxesof
− (cid:20) − (cid:21) theprojectedellipses.
forq1/q2 > rmin max[q1,0.5],whereasisnullotherwise.The Whenviewedfromanarbitrarydirection,quantitiesconstant
≡
lensing population has nearly the same triaxialily distribution as onsimilarellipsoidsproject themselves onsimilarellipses(Stark
the total cluster population (Hennawietal. 2007). This could be 1977).Ingeneral,theprojectedmapF ontheplaneofthesky
2D
explained astheresultof twocounter-balancing effects.Whereas and the intrinsic spheroidal volume density F are related by
3D
bothtriaxialityandconcentrationincreasethelensingcrosssection, (Stark1977;Sereno2007),
theshapeof adarkhaloiscorrelatedwithitsconcentration, with
∞
moreconcentratedclustersbeingmorespherical. F (ξ;l ,p )= 2 F (ζ;l ,p ) ζ dζ, (12)
2D P i 3D s i
Forcomparisonwewillalsoconsideraflatdistributionforthe √f ζ2 ξ2
Zξ −
axialratios,suchthat
where ξ is the elliptical radius in the plapne of the sky, ls is the
P(q1)=1 (7) typical length scale of the 3D density, lP is its projection on the
planeofthesky,p aretheotherparametersdescribingtheintrinsic
forthefullrange0<q 61and i
1 densityprofile(slope, ...)andf isafunction oftheclustershape
P(q q )=(1 q )−1 (8) andorientation,
2 1 1
| −
forq2 >q1 andzerootherwise.Theresultingprobabilityforq2 is f =e21sin2θsin2ϕ+e22sin2θcos2ϕ+cos2θ; (13)
then P(q ) = ln(1 q )−1. Such aflat distribution allowsalso
2 2 thesubscriptPdenotesmeasurableprojectedquantities.
forverytriaxialclust−ers(q < q 1),whicharepreferentially
1 ∼ 2 ≪ Letusseeinsomedetailshowtheparametersdescribingthe
excludedbyN-bodysimulations.
projectedmapdependontheintrinsicshapeandorientationofthe
Finally, semi-analitycal (Oguri&Blandford 2009) and nu-
3Ddistribution.Theaxialratioofthemajortotheminoraxisofthe
merical (Hennawietal. 2007) investigations showed a large ten- observedprojectedisophotes,e (>1),canbewrittenas(Binggeli
P
dencyforlensingclusterstobealignedwiththelineofsight.De-
1980),
notingtheanglebetweenthemajoraxisandthelineofsightasθ,
suchconditioncanbeexpressedas(Corlessetal.2009) j+l+ (j l)2+4k2
e = − , (14)
P
(cosθ 1)2 sj+l p(j l)2+4k2
P(cosθ) exp − , (9) − −
∝ − 2σ2
(cid:20) θ (cid:21) wherej,kandlapredefinedas
withσθ =0.115.Forcomparison,wewillalsoconsiderapopula- j = e2e2sin2θ+e2cos2θcos2ϕ+e2cos2θsin2ϕ, (15)
tionofclustersrandomlyoriented,i.e. 1 2 1 2
k = (e2 e2)sinϕcosϕcosθ, (16)
1− 2
P(cosθ)=1 (10) l = e2sin2ϕ+e2cos2ϕ. (17)
1 2
for06cosθ61.
Inthefollowingwewillalsousetheellipticityǫ=1 1/e .
P
−
Theobservedscalelengthl istheprojectionontheplaneof
P
theskyoftheclusterintrinsiclength(Stark1977),
3 PROJECTIONOFTRIAXIALHALOES e 1/2
l l P f1/4. (18)
Dealingwithellipsoidalhalos,weneedgeneralizeddefinitionsfor p ≡ s e1e2
(cid:16) (cid:17)
theintrinsicNFWparameters.WefollowCorless&King(2007), Equation(18)canberewrittenas
whodefinedatriaxialvirialradiusr suchthatthemeandensity
200 l l
containedwithinanellipsoidofsemi-majoraxisr is∆ = 200 s P , (19)
times the critical density at the halo redshift; the20c0orresponding √f ≡ e∆
concentrationisc r /r .Then,thecharacteristicoverden- where the parameter e quantifies the elongation of the triaxial
200 200 s ∆
≡
sityisexpressedintermsofc asforasphericalprofile, ellipsoidalongthelineofsight(Sereno2007),
200
(cid:13)c 0000RAS,MNRAS000,000–000
4 M. Sereno et al.
e∆ = ee1Pe2 1/2f3/4. (20) M200 = 43π ×200ρcr×(c200rsP)3ef3Pg/eo2. (25)
(cid:16) (cid:17)
ThequantitylP/e∆representsthehalf-size(alongthelineofsight) InordertoestimateM200andc200fromtheprojectedNFWparam-
oftheellipsoidasseenfromabove,i.e.perpendicularlytotheline etersdirectlyinferredfromthelensinganalysis,weneedtoknow
of sight. If e∆ < 1, then thecluster ismoreelongated along the the elongation of the cluster. The problem is intrinsically degen-
lineofsightthanwideintheplaneofthesky.Thesmallerthee∆ erate and can not be solved based on lensing information alone,
parameter,thelargertheelongation.Inthefollowing,wewilluse evenintheidealcaseofobservationswithoutnoise.Ifaclusteris
asanelongationparameteralsoageometricalfactor elongatedalongthelineofsight,theconcentrationparameterand
thevirialmassestimatedfromlensingareoverestimated(Gavazzi
(e e )1/2 e1/2
f 1 2 = P . (21) 2005;Ogurietal.2005).Ontheotherhand,therearemoreineffi-
geo ≡ f3/4 e∆ cientlensingorientationsforatriaxialhalothanthereareefficient
Summarising,thesurfacedensitycanbeexpressedintermsof ones(Corlessetal.2009).
projectedquantitiesas
F = lP f (ξ;e ,ψ;l ;p ,...), (22) 4.1 Datasample
2D e 2D P P i
∆
Wecompiledasampleofstronglensingclustersdrawingfrompre-
wheref hasthesamefunctionalformasforasphericallysym-
2D existing lensing analyses. As selection criteria, we retained only
metrichalo. IInorder towriteEq.(22)initsactual form,weex-
clustersthatarewelldefinedbyasingledarkmatterhaloandwhose
ploitedthat theintegral inζ inEq. (12) isproportional tothein-
lensingdatawerefittedwithanellipticalNFWmodel.Table1lists
trinsic scale length l . The dependence on the elongation e is
s ∆ thefinalclustersample,togetherwithcorrespondingNFWparam-
decoupledfromthedependenceontheapparentellipticityandin-
etersandreferencestowherethelensinganalyseswereperformed.
clination.Theotherparameterscharacterisingthe3Dprofileonly
For clusters that do not have published arc/multiple images red-
account for theradial dependence of theprojected density. Then,
shift, we assumed a source redshift z = 2.5. Many input data
s
whenwedeprojectasurfacedensity,thenormalizationofthevol-
were originally presented with asymmetric uncertainties. To ob-
ume density can be known only apart from a geometrical factor.
tain unbiased estimates, we applied correction formulae for the
Notethatinournotation,theellipticalradiusiswrittenasafunc-
mean andstandard deviation asgiven by equations (15) and (16)
tionofthecoordinatesintheplaneoftheskyas
inD’Agostini (2004). Notethat therearedifferent definitionsfor
ξ2 =(x2+e2x2)(l /l )2, (23) the elliptical radius, which affect the numerical value of the pro-
1 P 2 s P
jectedscale-length.Wetookcaretotranslatepublisheddatatothe
so that in order to obtain the elliptical projection from the corre- notationinthepresentpaper.Furthermore,somestudiesexploited
spondingspherical halowehavei)tomultiplytheoverallprofile elliptical NFW potential instead of elliptical mass density. When
by1/√f ,ii)tosubstitutethepolarsphericalradiuswithξ.The necessary,i.e.forthesub-samplefromComerfordetal.(2006),we
intrinsicscale-lengthhasthentobeexpressedintermsofthepro- convertedthepotentialellipticitytoisodensityellipticityaccording
jectedone,seeEq.(19). totherelationinGolse&Kneib(2002).Asafinalprecaution,we
forcederrorsonκ andr tobeatleastof10%andtheerroronǫ
s sP
tobeatleast0.015.Suchuncertaintiesmirrordiscrepanciesamong
differentstudiesofthesamedataset,seetheanalysesofA1703in
4 LENSINGINVERSION
Richardetal.(2009)andOgurietal.(2009)orMS2137.3-2353in
For gravitational lensing studies, the projected map of interest is Comerfordetal.(2006)andGavazzi(2005).
thesurfacemassdensity.WewilldescribetheprojectedNFWden- Usingthefullprobabilitydistributioninsteadoftheestimate
sityintermsofthestrengthofthelensκ ,seeEq.(24),andofthe of mean and error for the ellipticity and the central convergence
s
projectedlengthscaler ,i.e.thetwoparametersdirectlyinferred would be an improvement. However, we limited our method to
sP
byfittingprojectedlensingmaps.Theprojectedsurfacemassden- quiteregularclusters(uni-modalandwellfittedbyaNFWprofile).
sityΣofthesedensityprofilesisexpressedintermsoftheconver- Fromthedetailedanalysescollectedintheliteratureforeachclus-
genceκ,i.e.inunitsofthecriticalsurfacemassdensityforlensing, ter,wefoundnoevidenceforcomplexparameterdistributions,with
Σ = (c2D )/(4πGD D ), where D , D and D are the probability functions that are single-peaked and generally well-
cr s d ds s d ds
source,thelensandthelens-sourceangulardiameterdistances,re- behaved.
spectively.AccordingtoournotationinSec.3,foraNFWprofile,
theintrinsicl and theprojected l lengths havetobe readasr
s P s 4.2 Inferredparameters
andr ,respectively.
sP
The central convergence of a NFW profile estimated from In order to extract the physical information, i.e. to the determine
lensing can be written in terms of c200 and the projected length theparametersc200,M200 ande∆,wehavethentouseadditional
scalemodulusafactorfgeo(Serenoetal.2009), constraints. Sincewewant totesttheoretical predictions, wewill
employthepriorfromthec M relationasgiveninEq.(3).
Σcr×κs = √fgeePoρsrsP, (24) Slouwcshuasntaodddeittieornmailntehitrhdeceolon2ns0t0graa−tiinotn,t2oo0fg0eththeecrluwsittehrEanqds.i(t2s4m,a2s5s),aanld-
where as usual ρ = δ ρ (z) with ρ being the critical den- concentration. The prior isvery strong so the estimated c and
s c cr cr 200
sityoftheuniverseattheclusterredshift.Theconcentrationenters M willfitnicelythetheoreticalprediction.Ontheotherhand,
200
Eq.(24)throughδ ,seeEq.(11).TheestimateofthemassM e isfreetotakewhatevervalueallowstheclustertofitthelens-
c 200 ∆
dependsalsoonthescale-lengthr whichisknownmodulusafac- ingconstraintsandthec M relationatthesametime.Un-
s 200 200
−
tor√f/e ,seeEq.(19).Then physicalvaluesfore (either 1or 1),thatwoulddescribe
∆ ∆
≪ ≫
(cid:13)c 0000RAS,MNRAS000,000–000
Arelensingclustersover-concentrated? 5
Name zd zs κs rsP(kpc/h) ǫ refa
Abell1703 0.28 0.888 0.19±0.04 540±90 0.37±0.035 1
MS2137.3-2353 0.313 1.501 0.67±0.07 112±11 0.226±0.015 2
AC114 0.315 3.347 0.22±0.02 680±70 0.502±0.018 3
ClG2244-02 0.33 2.237 0.18±0.02 300±30 0.242±0.015 4
SDSSJ1531+3414 0.335 1.096 1.2±0.8 210±110 0.47±0.23 1
SDSSJ1446+3032 0.464 - 3.2±2.0 110±50 0.62±0.34 1
MS0451.6-0305 0.55 0.917 0.28±0.03 350±30 0.425±0.015 4
3C220.1 0.62 1.49 0.18±0.02 320±30 0.497±0.015 4
SDSSJ2111-0115 0.637 - 7.1±1.5 57±11 0.46±0.27 1
MS1137.5+6625 0.783 - 0.26±0.03 330±30 0.300±0.015 4
Table 1. The strong lensing cluster data sample. References: 1 stands for Ogurietal. (2009); 2 for Gavazzi (2005); 3 for Serenoetal. (2009); 4 for
Comerfordetal.(2006).Forclusters withmultiple imagesystems,wepicked outonesourceredshift(Col. 2).Thecentral convergence κs fortheNFW
modelreferstosuchredshift;rsPistheprojectedscalelength.
Name Standardc200−M200 Enhancedc200−M200
c200 M200(1014M⊙/h) e∆ c200 M200(1014M⊙/h) e∆
A1703 2.98±0.16 12±4 0.66±0.19 3.46±0.18 13±5 0.91±0.26
MS2137 3.41±0.13 2.0±0.4 0.068±0.011 3.95±0.15 2.3±0.5 0.093±0.014
AC114 2.94±0.13 12±3 1.06±0.17 3.40±0.16 14±3 1.44±0.024
ClG2244-02 3.26±0.13 3.3±0.7 0.69±0.11 3.77±0.15 3.7±0.8 0.95±0.15
SDSS1531 3.1±0.4 7±7 0.04±0.04 3.6±0.4 8±8 0.06±0.05
SDSS1446 3.2±0.4 3±3 0.015±0.014 3.7±0.5 3±3 0.020±0.019
MS0451 2.82±0.13 7.6±1.6 0.29±0.05 3.27±0.15 8.6±1.9 0.40±0.06
3C220.1 3.03±0.13 2.6±0.6 0.79±0.13 3.51±0.15 2.9±0.6 1.09±0.17
SDSS2111 3.0±0.2 2.6±1.6 0.004±0.002 3.5±0.2 2.8±1.8 0.006±0.003
MS1137 2.77±0.13 4.3±0.9 0.71±0.12 3.21±0.15 4.9±1.0 0.97±0.16
Table2.Concentration, massandelongationforeachclusterinferredthroughlensinginversionassumingasaprioreitherastandardoranenhancedmass
concentrationrelation.Massesareinunitsof1014M⊙/h.
morefilamentarystructuresthanvirializedclusters,willpointmore plane of the sky. Lensing parameters of SDSS1531, SDSS1446
tooutlierswithrespecttopredictionsthantoextremelyelongated andSDSS2111hadquitelargeobservational uncertaintieswhich
structures. The most likely explanation for extreme e values is propagateintheestimateoftheintrinsicclusterparameters.How-
∆
thenthatthecorrespondingclustersdonotfollowtherelationim- ever,theestimatedvaluesofe aresosmallthattheordinaryvalue
∆
posedapriori.Thiscanbeviewasasortofproofababsurdo. of 1 can be excluded for such clusters at a high confidence
∼
ResultsarelistedinTable2.Weconsidered boththec level.Evendoublingthenormalizationfactorofthec M relation
200 − −
M as determined in Duffyetal. (2008) and the case of over- (i.e. assuming A 11.4), elongation parameters for two cluster
200 ∼
concetratedclusters(A =6.68).Toaccountformeasurementser- (SDSS2112andSDSS1446)wouldremainsmallerthanonetenth
rors,wedrawlensingparameters(κsandrsP)fromrandomnormal (e∆ ∼0.019and0.070,respectively).
distributionswithmeananddispersiongivenbythereportedcentral Notethatfinalresultsonelongationwouldhavebeenconsis-
locationandscale,seeTable1.Similarly,theoreticaluncertainties tentifwehadchosendifferentmethodsforderivingthestronglens-
onthemass-concentrationrelationwereaccountedforbydrawing ingparameters.TheelongationofA1703calculatedusingthedata
theparametersA,BandC,whichdescribeEq.(3),fromGaussian reportedinRichardetal.(2009)or Limousinetal.(2008), which
distributionswithmeananddispersionvaluesfoundinDuffyetal. both fitted the lensing potential, turns out to be 0.35 0.11 or
±
(2008).Then,foreachsetofparameters(κ ,r ,A,BandC)we 0.77 0.17 respectively, the value of e based on Ogurietal.
s sP ∆
±
solvedthesystemofequationsEqs.(3,24,25),discardingonlyso- (2009),thatdirectlyfittedtheconvergence,beingintermediatebe-
lutionswitheitherc200 > 40orM200 > 1018M⊙/h.Thevalues tween the two, see Table 2. The elongation of MS 2137 using
listedinTable2arethebiweightestimatorsforlocationandscale datainComerfordetal.(2006),thatfittedthelensingpotential,is
ofthefinalinferreddistributions(Beersetal.1990). 0.051 0.008,compatiblewiththeresultbasedondirectconver-
±
Ifweusetheenhancedc M relation,theconcentration gencefittinginGavazzi(2005),seeTable2.Then,independently
200 200
−
ofeachclusterincreasesby 16%,themassby 13%ande of the lensing technique used, results are quite consistent within
∆
∼ ∼
by 27%, i.e.theelongation shrinks. Evenif clusterscome out theerrors,bothformildly(A1703)orveryelongated(MS2137)
∼
less elongated if we assume that the lensing population is intrin- clusters.
sicallyover-concentrated,weseethatsomeoutliersarestillthere. Itisquitereassuringthatwheneveraclusterhasbeenanalyzed
Fouroutof10clustershavee < 0.1,i.e.thesizealongtheline eitherfittingthepotentialortheconvergence,theestimatedelonga-
∆
ofsightshouldbelargerthantentimesthemaximumlengthinthe tiondoesnotchangeinasignificantway.Togetherwiththecentral
(cid:13)c 0000RAS,MNRAS000,000–000
6 M. Sereno et al.
n 4 q1,2 θ All e∆>0.1
nctio ffllaattqq,,bfliaatsΘedΘ flat flat 4.8×10−2 1.5×10−2
Fu 3 NN--bbooddyyqq,,bfliaatsΘedΘ flat biased 2.6×10−2 1.0×10−2
sity N-body flat 7.9×10−1 5.9×10−1
Den 2 N-body biased 8.6×10−3 1.2×10−1
y
bilit Table5.Kolmogorov-Smirnovsignificancelevelthatthemeasuredelliptic-
a 1
ob itiesoftheobservedsamplesofclusters(eitherallofthemorthesixwith
Pr e∆>0.1)aredrawnfromagivenpopulation.
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
apart from the group at e < 0.1. Cumulative distributions are
eD ∆
plottedinFig.2andprobabilitiesatverylowthresholdvaluesare
listedinTable3.Chancesforveryelongatedclustersareverytiny.
Figure1.Probabilitydensityfunctionsfortheelongationsofdifferentclus-
Evenforbiaseddistributionsjustoneoutoffewthousandsclusters
terpopulations,seelegend.Thebarsdenotethemeasuredvaluesforourfull
sampleassumingastandardc200−M200relation. isexpected to have e∆ < 0.1. Even in the most favourable case
ofapopulation of clustersbiased inθ andflatinaxial ratios,the
chancetohavefouroutoftenclusterswithe < 0.1wouldbea
∆
verytiny1.7 10−16.Sowecanconcludethatsuchclustersare
q1,2 θ P(e∆<0.1) P(e∆<0.2) P(e∆<0.3) verylikelyou×tliersofthemass-concentrationrelations.
flat flat <∼10−3% 8×10−3% 2.7×10−3% Further quantitative comparisons can be performed exploit-
flat biased 4.5×10−3% 3.5×10−2% 1.87×10−2% ingtheKolmogorov-Smirnov(KS)test.Whenweconsiderthefull
N-body flat <∼10−3% 1×10−3% 9.5×10−3% sample,noneoftheinvestigatedpopulationsgivesagoodfittothe
N-body biased <∼10−3% 1.5×10−3% 5.7×10−2% data.Thebetterperformer,i.e.apopulationwithN-bodylikeaxial
ratiosandbiasedalignments,giveaKSsignificancelevelof<1%,
∼
bothforthestandardortheenhancedc M relation,seeTa-
Table3.Probability(in%)tohaveanelongationlargerthanagiventhresh- 200− 200
ble4.
oldvaluefordifferentpopulationsofgalaxyclusters.
The significance levels improve very significantly when we
consider the subsample with e > 0.1. The prediction from
∆
convergence, our method needs only anestimateof theprojected N-body simulations reproduce very well the observed distribu-
ellipticity,whichisquitewellmeasuredwithstronglensinganal- tion both for the standard ( 31.1%) and the enhanced relation
∼
yses.Golse&Kneib(2002)discussedindetailshowpotentialand ( 4.7%).Fortheenhancedrelation,alsodistributionsflatinthe
∼
surfacemassellipticitiesarerelated,andtheiranalysisshowedhow axialratiosperformwellforbothpopulationssufferingorientation
themassdensityellipticitycanbeestimatedusingapreviousdeter- bias( 2.1%)orunbiased( 97.2%).
∼ ∼
minationofthepotentialellipticity.Oncetheellipticityofthesur-
facemassdensityisknown,ourmethodreliesonlyongeometrical
5.2 Ellipticity
projectionsandisnotaffectedanymorebylensingnon-linearities.
Infact,wealwaysde-projectthesurfacemassdensity(insteadof Theellipticitydistributionofoursampleisnotnearasinformative
thepotential)toobtaintheintrinsicmassdistribution. astheelongationone.Probabilitydensityfunctionsbothforunbi-
asedorbiasedpopulationshavenotnegligiblevaluesincorrespon-
denceoftheobservedellipticities,seeFig.3.Populationswithflat
5 EXPECTEDVS.OBSERVEDELONGATIONAND axialratiosarepreferentiallyrounder(ǫ >∼ 0,e∆ ∼ 1)sincehigh
ELLIPTICITY valuesofq1arenotpenalized,buttheobservedsampledonothelp
todiscriminate.TheKStestisinconclusivetoo,seeTable5,even
Thechancetoobserveaveryelongatedclustercanbeassessedon
if the biased N-body-like population performs remarkably better
a more firm ground. We derived the probability density function
consideringthee > 0.1subsample.However,theellipticitiesof
∆
(PDF)foragivenelongation,P(e∆),andagivenellipticityP(ǫ) such subsample are nothing special. According to a KS test, the
underdifferentassumptions.AsdiscussedinSec.3,elongationand
ellipticitiesof theoutliers(i.e.clusterswithe < 0.1) might be
∆
ellipticitydependontheintrinsicaxialratios,q1andq2andtheori- drawnfromthefullsamplewithasignificancelevel 98%.
entationanglesθandϕ.Weconsideredfourscenarios.Fortheaxial Since our sample is neither homogeneous or ∼statistical, we
ratios,weconsideredeithertheN-bodypredictionsinEqs.(4,6) are cautious in drawing conclusions, but some indications seem
or a flat distribution, see Eqs. (7, 8). For the alignment we con-
to be quite strong. There is a number of clusters whose over-
sidered either thebiased distributionfor P(θ)inEq. (9) or aflat concentrationproblemcannotbesolvedjustconsideringsomepar-
distribution, Eq. (10). For theazimuth angle ϕwealways used a ticular geometrical configurations. Even strong biases inintrinsic
flatdistribution,P(ϕ)=const. triaxialityandalignmentwouldnotsolvetheproblem.Oncesuch
outliersareexcludedfromtheanalysis,theoreticalpredictionsare
inverygoodagreement withdata.Populationswithanalignment
5.1 Elongation
bias perform much better than randomly oriented clusters. There
ThePDFfortheelongation,P(e ),isplottedinFig.1.Itispretty isalsosomeevidence forintrinsicaxialratiosdistributedaccord-
∆
evidentthatpopulationsofclusterspreferentiallyalignedwiththe ing tothe outputs of N-body simulations, even if,under suitable
lineof sightmakeabetterjobtoreproduce theobserved sample, circumstances,flatpopulationscangivegoodresultstoo.
(cid:13)c 0000RAS,MNRAS000,000–000
Arelensingclustersover-concentrated? 7
Standardc200−M200 Enhancedc200−M200
q1,2 θ All e∆>0.1 All e∆>0.1
flat flat 9×10−8 2.5×10−4 3×10−5 2.1×10−1
flat biased 9×10−6 3.5×10−3 9.3×10−3 9.7×10−1
N-body flat 6×10−7 2.5×10−4 5×10−5 9.1×10−3
N-body biased 7.8×10−3 3.1×10−1 9.6×10−3 4.7×10−2
Table4.Kolmogorov-Smirnovsignificancelevelthattheelongationsoftheobservedsamplesofclusters(eitherallofthemorthesixwithe∆ > 0.1)are
drawnfromagivenpopulation.Weconsiderede∆obtainedfromeitherastandardoranenhancedmass-concentrationrelation.
Standardc -M Enhancedc -M
200 200 200 200
1.0 1.0
n n
o o
cti cti
n 0.8 n 0.8
u u
F F
y y
bilit 0.6 bilit 0.6
a a
b b
o o
Pr 0.4 Pr 0.4
e e
v v
ulati 0.2 ffllaattqq,,bfliaatsΘedΘ ulati 0.2 ffllaattqq,,bfliaatsΘedΘ
m N-bodyq,flatΘ m N-bodyq,flatΘ
Cu N-bodyq,biasedΘ Cu N-bodyq,biasedΘ
0.0 0.0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
e e
D D
Figure2.Predictedcumulativedistributionfunctionsforelongationversusmeasurements.Thefullanddashedstep-linesareforthefullobservedsampleand
fortheclusterswithe∆ >0.1,respectively;thesmoothfunctionsarethepredicteddistributionsunderdifferentassumptions,seelegends.Theleftandright
panelsshowtheobservedelongationscomputedassumingeitherastandardoranenhancedmass-concentrationrelation,respectively.
The assumption that lensing clusters are intrinsically over- (DeFilippisetal.2005;Serenoetal.2006),evenifdiffuseprolate-
concentrated partially reduce the problem, but expected distribu- nesscannotbeexcluded(Plionisetal.1991;deTheijeetal.1995;
tionsandobservationswouldbecompatiblewithaverylowsignif- Basilakosetal. 2000; Cooray 2000; Plionisetal. 2004; Pazetal.
icancelevelof<1%andtheproblemofhavingnearlyhalfofthe 2006).Oncetheelongationofaclusterisknowntogetherwithits
∼
samplewithveryextremeelongationwouldbestillthere.Ingen- projected ellipticity, strong constraints can be put on its intrinsic
eral, the data analysis performed on our limited sample does not shape (Sereno2007).Axial symmetry reduces thenumber of un-
provide evidence for intrinsic over-concentrations, an orientation knownparameterstoacouple:asingleaxialratio,q(61),andthe
bias being enough to account for observations of normal clusters inclinationangleofthesymmetryaxis,i.
(e >0.1). Aprolate-likesolutionisadmissiblewhenthesizealongthe
∆
lineofsightislargerthantheminimumwidthintheplaneofthe
sky,thatis,whene 6e .Theintrinsicparameterscanbewritten
∆ P
6 INTRINSICAXIALRATIOSANDORIENTATION asq1 = q2 = q and θ = i. Interms of themeasured quantities
(Sereno2007),
Knowledge of the sizes of a cluster in the plane of the sky and
e
alongthelineof sightallowsustoputconstraintsonitsintrinsic q = ∆, (26)
e2
geometry(Sereno2007).However,evenexploitingsuchstrongas- P
sumptionsontheshape,inversioncanbenotunique:intrinsically e2 e2
cosi = e P− ∆. (27)
differentellipsoidscancastontheplaneoftheskyinthesameway P e4 e2
(Sereno 2007). In order to infer the properties of the cluster and r P− ∆
Anoblate-likesolutionisadmissiblewhenthesizealongthe
deriveitsorientationandshape,wehavetoexploitsomeexternal
lineofsightissmallerthanthemaximumsizeintheplaneofthe
information.Wewillconsidertwokindsofprior:firstasharpone
whichassumestheclustertobeaxiallysymmetric;thensomeless sky,thatis,whene∆ >1.Accordingtoournotation,foranoblate
informative priors on the distribution of intrinsic axial ratios for ellipsoid,q1 = q,q2 = 1andcosi = sinθsinϕ.Inversiongives
(Sereno2007),
triaxialhaloes.
1
q = , (28)
e e
6.1 Axialsymmetry P ∆
e2 1
As a working hypothesis, let us first consider if the cluster cosi = ∆− . (29)
e2e2 1
shape can be approximated as an ellipsoid of revolution. Previ- r P ∆−
ous studies have shown that clusters seems to be quite triaxial Theprolateandtheoblatesolutionsareadmissibleatthesametime
(cid:13)c 0000RAS,MNRAS000,000–000
8 M. Sereno et al.
1.0
3.0 on
flatq,flatΘ ncti 0.8
2.5 flatq,biasedΘ u
N-bodyq,flatΘ F
LΕ 2.0 N-bodyq,biasedΘ ability 0.6
HP 1.5 ob
Pr 0.4 flatq,flatΘ
e flatq,biasedΘ
1.0 ativ NN--bbooddyyqq,,bfliaatsΘedΘ
ul 0.2
0.5 m
u
C
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Ε Ε
Figure3. Probability functionfortheprojectedellipticity. Leftpanel:probability densityfunctionsfortheellipticity ofdifferentclusterpopulations, see
legend. The bars denote the measured values for our full sample. Right panel: predicted cumulative distribution functions for a given elongation versus
observations.Thefullanddashedstep-linesareforthefullobservedsampleandfortheclusterswithe∆ > 0.1,respectively;thesmoothfunctionsarethe
predicteddistributionsunderdifferentassumptions,seelegend.
Prolate Oblate
Name Comp. q cosi Comp. q cosi
A1703 ∼1 0.26±0.08 0.94±0.04 0.0342 0.59±0.05 0.25±0.11
MS2137 ∼1 0.040±0.007 0.9994±0.0002 <∼10−5 NA NA
AC114 ∼1 0.26±0.05 0.88±0.05 0.622 0.43±0.04 0.26±0.09
ClG2244-02 ∼1 0.40±0.07 0.93±0.03 2.31×10−3 0.74±0.03 0.24±0.12
SDSS1531 ∼1 0.012±0.011 0.999795±0.0002 <∼10−5 NA NA
SDSS1446 ∼1 0.0014±0.0014 0.999989±0.000015 <∼10−5 NA NA
MS0451 ∼1 0.098±0.017 0.9902±0.003 <∼10−5 NA NA
3C-220.1 ∼1 0.20±0.03 0.94±0.02 0.0542 0.48±0.02 0.16±0.07
SDSS2111 ∼1 0.0010±0.0009 0.999998±10−6 <∼10−5 NA NA
MS1137 ∼1 0.35±0.06 0.93±0.03 6.45×10−3 0.68±0.03 0.23±0.11
Table6.Intrinsicparameters(axialratioqandinclinationanglei)assumingeitherprolatenessoroblateness.Thecolumn”Comp.”givesthesignificancelevel
foraclustershapetobecompatiblewithagivensetofdata.Foraverylowcompatibilitywithagivenshapehypothesis,parametervaluesarenotavailable
(NA).
onlywhenthesizealongthelineofsightisintermediate,i.e.1 6 gationandellipticityallowedbythedataiscompatiblewithsuch
e 6e . an hypothesis. Only AC 114, with e 1, has a good chance
∆ P ∆
∼
to be described by an oblate shape ( 60%), otherwise signif-
ResultsoftheinversionarelistedinTable6.Intrinsicparam- icance levels are < 5%. For the few∼clusters for which oblate-
etershavebeenobtainedbymeansofEqs.(26,27)fortheprolate ∼
nessismarginallycompatible,inclinationangleswouldstillbebi-
case and Eqs. (28, 29) for the oblate-case. Input values for elon-
ased,symmetryaxisbeingnearlyperpendiculartothelineofsight
gationandellipticitywererandomlyextractedfromnormaldistri-
(cosi < 0.26), whereas intermediate axial ratios would be pre-
butionscentredinthemeasuredvalueandwithdispersionequalto ∼
ferred(0.43<q<0.74).
theobservational uncertainty. ThevalueslistedinTable6arethe ∼ ∼
bi-weigthestimatorsofthefinaldistributionsoftheinferredparam-
eters.Thesignificancelevelforagivenshapehasbeenobtainedas 6.2 Triaxialclusters
the fractionof drawn e and e for which a given compatibility
P ∆
Inordertoexactlydeterminetheintrinsicshapeofatriaxialclus-
conditions is fulfilled. We considered only elongation values ob-
ter, weshould know both ellipticityand elongation together with
tainedassuming astandard mass-concentration relation.Thepro-
two additional observational constraints. The problem of invert-
late hypothesis is compatible with the full sample but the shapes
shouldbeextremelylongandnarrow (q < 0.35) andnearlyper- ingaprojectedmapisintrinsicallydegenerateandevenaddingX-
∼
fectlyalignedwiththelineofsight(cosi>0.88).Theconclusion rayobservationsormeasurementsoftheSunyaev-Zeldovicheffect
∼
wouldnotmaketheinversionunique(Sereno2007).Analternative
thatclusterswithe < 0.1areoutliersisfurtherstressedbythe
∆
approachistousepriorsontheintrinsicparameters(Corlessetal.
verysmallspacevolumeallowedfortheintrinsicparameters(under
2009).Here,wetrytosolvethesystemofequations
wronghypotheses,uncertaintiesareverylikelytobeverysmall).
Apopulationofoblateclustersdonotprovideagooddescrip- eP = eP(q1,q2;θ,ϕ); (30)
tionofthedata.Onlyatinyregionintheparameterspaceofelon- e = e (q ,q ;θ,ϕ)
∆ ∆ 1 2
(cid:13)c 0000RAS,MNRAS000,000–000
Arelensingclustersover-concentrated? 9
6 7
45 233...505 56 23..50
HLPq123 HLPq2112...050 HLΘPcos234 HLjPcos112...050
1 0.5 1 0.5
0 0.0 0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
q1 q2 cosΘ cosj
Figure4.Posteriorprobability densityfunctionsfortheintrinsicparametersofA1703.PanelsfromthelefttotherightareforthePDFofq1,q2,cosθ
andcosϕ,respectively.FullanddashedthicklineshavebeenobtainedassumingaN-body-likeandaflatpriorontheaxisratios,respectively.Thefulland
dashedthinlineintheleftpanelrepresenttheN-bodyandtheflatpriorforP(q1),respectively;thefullanddashedthinlineintheq2-panelrepresentthe
priordistributionsaccordingtoeitheraN-bodyoraflatprior,respectively;thethinanddashedfulllineinthecosθ-panelrepresentthebiasedandtheflat
distributionsontheorientationangle.SuchpriorsoncosθwerenotusedtoderivethePDFs.Finallytheflatlineinthecosϕ-panelrepresentsanuniform
distribution.
6 3.0 7 8
5 2.5 6
HLPq1234 HLPq2112...050 HLΘPcos2345 HLjPcos46
2
1 0.5 1
0 0.0 0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
q1 q2 cosΘ cosj
Figure5.ThesameasFig.4fortheclusterAC114.
7 5 10
6 4 8 2.0
5
HLPq134 HLPq223 HLΘPcos 46 HLjPcos11..05
2
1 1 2 0.5
0 0 0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
q1 q2 cosΘ cosj
Figure6.ThesameasFig.4fortheclusterClG2244.
8
6 3.0 10
5 2.5 6 8
HLPq1234 HLPq2112...050 HLΘPcos24 HLjPcos 46
1 0.5 2
0 0.0 0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
q1 q2 cosΘ cosj
Figure7.ThesameasFig.4forthecluster3C220.
8 5 10 2.5
6 4 8 2.0
HLPq14 HLPq223 HLΘPcos 46 HLjPcos11..05
2 1 2 0.5
0 0 0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
q1 q2 cosΘ cosj
Figure8.ThesameasFig.4fortheclusterMS1137.
(cid:13)c 0000RAS,MNRAS000,000–000
10 M. Sereno et al.
N-bodyq Flatq
Name Comp. q1 q2 cosθ cosϕ Comp. q1 q2 cosθ cosϕ
A1703 0.094 0.39±0.07 0.55±0.11 0.87±0.10 0.81±0.16 0.025 0.37±0.10 0.61±0.15 0.90±0.09 0.76±0.17
MS2137.3 <∼10−4 NA <∼10−4 NA
AC114 0.104 0.36±0.06 0.52±0.13 0.77±0.11 0.93±0.07 0.054 0.31±0.09 0.61±0.18 0.84±0.12 0.85±0.13
ClG2244 0.149 0.43±0.06 0.57±0.08 0.93±0.04 0.67±0.20 0.026 0.45±0.10 0.60±0.12 0.92±0.05 0.67±0.22
SDSS1531 1.2×10−3 0.44±0.06 0.60±0.10 0.82±0.25 0.70±0.17 4.3×10−4 ∼0.26 ∼0.49 ∼0.83 ∼0.60
SDSS1446 1.7×10−4 ∼0.45 ∼0.70 ∼0.65 ∼0.39 1.7×10−4 ∼0.16 ∼0.76 ∼0.75 ∼0.63
MS0451.6 <∼10−4 NA 3×10−4 ∼0.14 ∼0.27 ∼0.99 ∼0.64
3C220.1 0.037 0.33±0.06 0.50±0.12 0.85±0.07 0.95±0.05 0.025 0.29±0.09 0.56±0.15 0.92±0.06 0.86±0.13
SDSS2111 <∼10−4 NA <∼10−4 NA
MS1137.5 0.174 0.38±0.05 0.52±0.08 0.92±0.04 0.74±0.16 0.028 0.41±0.09 0.59±0.12 0.92±0.06 0.73±0.19
Table7.Intrinsicparametersforatriaxialshape(axialratios,q1andq2,andorientationanglesθandϕ)inferredusingdifferentpriorsfortheintrinsicaxial
ratiosdistributions.Thecolumn”Comp.”givesthesignificancelevelforaclustershapetobecompatiblewithagivensetofdata.
usingacoupleofproxiesfortheaxialratios.Inparticularweex- bothlensingdataandtheoreticalpredictions.Finally,westudiedfor
ploit either a flat distribution or the guess from N-body simula- eachclusterwhichintrinsicshapeandorientationwerecompatible
tions. Operatively, we randomly extract the axial ratios from the withtheinferredelongationandthemeasuredprojectedellipticity.
assumedpriordistributionandthensolveforθandϕinEqs.(30). Ateachstep,wechecked theexploitedhypothesisababsurdo by
Foreachiteration,acoupleofinput valuesfore ande isalso findingany inconsistency between theoretical predictionsand ac-
P ∆
randomlydrawn.Ifthereisasolutiontothesystem,weconsider tualconditionsunderwhichexpectationscanbeinagreementwith
thedrawnq andq andthecorrespondingθandϕtobeasample data.
1 2
fromtheposteriordistribution. We first considered the inferred distribution of elongations,
ResultsarelistedinTable7,whereasusual wereportedbi- comparing that with the probability to really see them observ-
weightestimators.Finalestimatesarequiteinsensitivetothepriors. ingagivenpopulation. Wefound twogroups inour sample. The
Thisassertsthevalidityofourinversionapproach,sinceaBayesian first group is in very good agreement with what expected from
analysisiseffectiveasfarastheeffectofpriorsisasnotinforma- a population of clusters fitting the mass-concentration relation
tiveaspossible.Theparameterspaceforsolutionsisquitenarrow and preferentially oriented along the line of sight, as suggested
forveryelongatedclusters(e < 0.3),andweactuallywereable by theoretical analyses of lensing clusters (Hennawietal. 2007;
∆
tofindveryfewofthem,lessthanoneoutoftenthousandsdraw- Oguri&Blandford2009).Observedellipticitiesandinferredelon-
ings.Assumingthesharpprolatepriorontheshape,wecouldfind gationsarealsoinagreement withintrinsicaxialratiosfollowing
someextremelyelongatedconfigurations,butsuchintrinsicshapes distributionsderivedinN-bodysimulations.Thereisnoevidence
areprettymuchexcludedbyassumingmorerealisticpriorsonthe forsuchlensingclusterstobeintrinsicallyover-concentratedeven
axial distributions as the ones expected for general triaxial con- ifdatacannotexcludethat.
figurations.Thisfurthersuggeststhatoursamplecontainsseveral Thesecondsubsampleinouranalysisismadeofclustersvery
outliersofthemass-concentrationrelation. likelytobeoutliersofthemassconcentrationrelation.Tofulfilthe
Posterior probabilities for A1703, AC114, ClG2244, 3C220 expectedrelation,theyshoulddevelopalongthelineofsightasa
and MS1137 are plotted in Figs. 4, 5, 6, 7, and 8, respectively. filamentary structure with extreme elongation, a clearly poor de-
ThefinaldistributionshavebeensmoothedusingaGaussianker- scriptionformassivehaloes.Evenallowingformoreconcentrated
nelestimatorwithreflectiveboundaryconditions(Vioetal.1994; halosforagivenmassbyenhancingthec Mrelation,elongations
−
Ryden1996).Foreachcluster,whateverthepriorontheaxialra- wouldstillbeextreme.
tios,theposteriorprobabilitiesarequitesimilar.Evenifweassume Even if our sample was not statistical, we took care of se-
a flat distribution for the axial ratios, the posterior probability is lectingquite regular clusters. However, bimodal structures nearly
quitesimilartothepredictionfromN-bodysimulations.Notethat alignedwiththelineofsightwouldseeminglyhavearegularmor-
thealignmentbiasisconfirmedbytheaboveanalysiswithoutany phology intheplaneof thesky. Suchconfigurations wouldboost
priorassumptionontheorientation. theapparentconcentration,buttheyareveryrareanditisproblem-
atictoconsiderallofouroutlierswithinthisscenario.Furthermore,
weusedamass-concentration relationderivedforthefullsample
ofclusters,notjustthevirializedandregularones.
7 DISCUSSION
Thesecondstepinouranalysisfurtherstrengthenssuchview.
Recentobservationalanalyseshavebeenfindingmanylensingclus- Using a series of statistical priors, we found for each one of the
ters with Einstein radii much larger than expected in a standard mildlyelongatedclustersanintrinsicstructureandorientationcom-
ΛCDMmodel(Broadhurstetal.2008;Ogurietal.2009).Weper- patiblewiththeinferredelongation. Prolateshapes make abetter
formed a statisticalanalysis on a sample of 10 clusters that were workinexplainingdatathanoblateclusters,butingeneraldataare
wellfittedbyasingleNFWmodel.Ourmethodwasasfollow.We fullycompatiblewithtriaxialstructures.Whatever thehypothesis
supposed theoreticalexpectationsfromN-bodysimulationstobe exploitedaspriorontheintrinsicaxialratios,inferredintrinsicpa-
true and modelled clusters as NFW haloes fitting standard mass- rameterssuggestsmildlytriaxialclusterswithanalignmentbias,in
concentration relations. Then, we found the elongation along the verygoodagreementwithexpectationsfromN-bodysimulations.
line of sight of the clusters required to satisfy at the same time Analternativeapproachwouldhavebeentoapplythefitting
(cid:13)c 0000RAS,MNRAS000,000–000