DRAFTVERSIONJANUARY10,2012 PreprinttypesetusingLATEXstyleemulateapjv.11/12/01 LENSINGANDX-RAYMASSESTIMATESOFCLUSTERS(SIMULATIONS) E.RASIA1,2,M.MENEGHETTI3,R.MARTINO4,S.BORGANI5,6,7,A.BONAFEDE8,K.DOLAG9,S. ETTORI3,10 D.FABJAN7,11,12,C.GIOCOLI3,P.MAZZOTTA4,13,J.MERTEN14,M.RADOVICH15,L. TORNATORE5,6,7 DraftversionJanuary10,2012 ABSTRACT Wepresentacomparisonbetweenweak-lensingandX-raymassestimatesofasampleofnumericallysimulated clusters. Thesampleconsistsonthe20mostmassiveobjectsatredshiftz=0.25andM >5×1014M h−1. They vir (cid:12) werefoundinacosmologicalsimulationofvolume1h−3 Gpc3,evolvedintheframeworkofaWMAP-7normal- 2 izedcosmology. Eachclusterhasbeenresimulatedathigherresolutionandwithmorecomplexgasphysics. We 1 processeditthoughtSkylensandX-MAStogenerateopticalandX-raymockobservationsalongthreeorthog- 0 2 onal projections. The final sample consists on 60 cluster realizations. The optical simulations include lensing effectsonbackgroundsources. Standardobservationaltoolsandmethodsofanalysisareusedtorecoverthemass n profilesofeachclusterprojectionfromthemockcatalogue. TheresultingmassprofilesfromlensingandX-ray a areindividuallycomparedtotheinputmassdistributions. Giventhesizeofoursample,wecouldalsoinvestigate J the dependence of the results on cluster morphology, environment, temperature inhomogeneity, and mass. We 7 confirmpreviousresultsshowingthatlensingmassesobtainedfromthefitoftheclustertangentialshearprofiles with NFW functionals are biased low by ∼5−10% with a large scatter (∼10−25%). We show that scatter ] O couldbereducedbyoptimallyselectingclusterseitherhavingregularmorphologyorlivinginsubstructure-poor C environment. The X-ray masses are biased low by a large amount (∼25−35%), evidencing the presence of both non-thermal sources of pressure in the ICM and temperature inhomogeneity, but they show a significantly . h lowerscatterthanweak-lensing-derivedmasses. TheX-raymassbiasgrowsfromtheinnertotheouterregions p oftheclusters. Wefindthatbothbiasesareweaklycorrelatedwiththethird-orderpowerratio,whileastronger - correlationexistswiththecentroidshift. Finally,theX-raybiasisstronglyconnectedwithtemperatureinhomo- o r geneities. ComparisonwithpreviousanalysisofsimulationsleadstotheconclusionthatthevaluesofX-raymass t biasfromsimulationsisstilluncertain,showingdependencesontheICMphysicaltreatmentand,possibly,onthe s a hydrodynamicalschemeadopted. [ Subjectheadings: cosmology: miscellaneous–methods: numerical–galaxies: clusters: general–X-ray: 1 hydrodynamics–lensing v 9 1. INTRODUCTION 6 Galaxy clusters are important test sites for cosmology and astrophysics. First, they are ideal laboratories for studying how the 5 dark-matterbehavesindenseenvironmentandevolvesinthenon-linearregime. Second, theirmassfunctionishighlysensitiveto 1 . cosmology,sinceitsevolutiontracesthegrowthofthelineardensityperturbationswithexponentialmagnification(Press&Schechter 1 1974;Sheth&Tormen2002;Jenkinsetal.2001;Warrenetal.2006). Indeed, clustersarethemostmassivegravitationallybound 0 structuresintheuniverseand,intheframeworkofthehierarchicalstructureformationscenario,theyarealsotheyoungestsystems 2 formedtodate. Therefore,clustersareamineofcosmologicalinformation,alargefractionofwhichiscontainedinthemassprofile 1 ofthesestructures. Severalmethodscanbeusedtodeterminethematterdistributioningalaxyclusters. Twowidelyusedapproaches : v arebasedonX-rayandlensingobservations. Xi X-rayobservationsallowtheclustermassprofilestobederivedbyassumingthatthesesystemsaresphericallysymmetricandthat theemittinggasisinhydrostaticequilibrium(e.g.Henriksen&Mushotzky1986;Sarazin1988;Ettorietal.2002). Thismethodhas r a theadvantagethat,sincetheX-rayemissivityisproportionaltothesquareoftheelectrondensity,itisnotverysensitivetoprojection 1DepartmentofAstronomy,UniversityofMichigan,500ChurchSt.,AnnArbor,MI48109-1120,USA,[email protected] 2FellowofMichiganSocietyofFellows 3INAF,OsservatorioAstronomicodiBologna,viaRanzani1,I-40127,Bologna,Italy,[email protected] 4DipartimentodiFisica,UniversitàdiRomaTorVergata,viadellaRicercaScientifica1,I-00133,Roma,Italy 5DipartimentodiFisicadellÕUniversitàdiTrieste,SezionediAstronomia,viaTiepolo11,I-34131Trieste,Italy 6INAF–OsservatorioAstronomicodiTrieste,viaTiepolo11,I-34131Trieste,Italy 7INFN–IstitutoNazionalediFisicaNucleare,Trieste,Italy 8JacobsUniversityBremen,CampusRing1,D-28759Bremen,Germany 9INAF–OsservatorioAstronomicodiPadova,vicolodell’Osservatorio5,I-35122,Padova,Italy 10INFN,SezionediBologna,vialeBertiPichat6/2,I-40127Bologna,Italy 11CenterofExcellenceSPACE-SI,Aškercˇeva12,1000Ljubljana,Slovenia 12FacultyofMathematicsandPhysics,UniversityofLjubljana,Jadranska19,1000Ljubljana,Slovenia 13Harvard-SmithsonianCentreforAstrophysics,60GardenStreet,Cambridge,MA02138,USA 14ITA,Zentrumfu¨rAstronomie,Universita¨tHeidelberg,Germany 15UniversityObservatoryMu¨nchen,Scheinerstr.1,81679,Mu¨nchen,Germany 1 2 Rasiaetal. effectsofmassesalongthelineofsighttotheclusters. However,itisstillnotwellestablishedhowsafelythehydrostaticequilibrium approximationcanbemade(Rasiaetal.2004,2006,hereafterR06). Asthehighestmassconcentrationsintheuniverse,galaxyclustersarethemostefficientgravitationallensesonthesky. Theirmatter distortsbackground-galaxyimageswithanintensitythatincreasesfromtheoutskirtstotheinnerregions. Strongdistortionsoccur in the cores of some massive galaxy clusters, leading to the formation of “gravitational arcs” and/or to the formation of systems of multiple images of the same source. Weak distortions, which can be only measured statistically, are impressed on the shape of distantgalaxiesthatlieontheskyatlargeangulardistancesfromtheclustercenters(e.g.Bartelmann&Schneider2001). Boththese lensingregimescanbeusedtomapthemassdistributioningalaxyclusters. Gravitationallensingcandirectlyprobetheclustertotal masswithoutanystrongassumptionsontheequilibriumstateofthelens. Further,massprofilescanbemeasuredoverawiderange of scales, from (cid:46)100 kpc out to the virial radius. However, lensing measures the projected mass instead of the 3D mass and it is sensitivetoprojectioneffects,suchastriaxialityandadditionalconcentrationsofmassalongthelineofsight. Giventheprosandconsofeachmethod, wecanconcludethatlensingandX-rayarecomplementaryinmanyways. Inparticular, thecomparisonofthesetwomassestimatescangreatlyhelpinimprovingtheaccuracyofthemeasurementsandunderstandingthe systematicerrors. Numerical simulations provide a unique way to investigate the performance of the lensing and X-ray techniques for measuring themassprofilesofgalaxyclusters. Severalstudieswereperformedinthepastwhichmadeuseofrelativelysimpledescriptionsof galaxyclustersandsimulationset-ups,butstillwereabletoassesssomefundamentallimitsofthesetechniquesandpossiblysuggest improvements(seee.g.Metzleretal.2001;Cloweetal.2004;Becker&Kravtsov2011;Rasiaetal.2004;Piffarettietal.2003).Inthe lastyears,usingtheincreasingnumberofobservationalconstraintsandprofitingofthehugeincrementofcomputationalefficiency, thesimulationshavebecomeevenmoresophisticatedandcannowincludealargenumberofrealisticandimportantfeatures. These improvementsregardboththedescriptionofthephysicalprocessesdeterminingtheevolutionofthecosmicstructures(seeBorgani & Kravtsov 2009, for a review) and the interface between simulations and observations. In particular, few pipelines have been developedwhichproducesimulatedobservationsofthenumericallymodeledclustersatdifferentwavelengths(Gardinietal.2004; Nagaietal.2007;Rasiaetal.2008;Meneghettietal.2008;Heinz&Brüggen2009). Thesepipelinescansimulateobservationswith avarietyofexistingandfutureinstrumentsandincludemostobservationalnoiseswhichtypicallyaffectandlimitrealmeasurements. Thus,theyareidealfortestingdatareductionpipelines(Rasiaetal.2006;Mazzottaetal.2004;Nagaietal.2007). InMeneghettietal.(2010)(M10,hereafter)wecombinedouropticalsimulator,SkyLens(Meneghettietal.2008),withourX- rayone,XMAS(Rasiaetal.2008;Gardinietal.2004),tostudythesystematiceffectsinmassmeasurementsencounteredfollowing standardlensingandX-rayanalysis. Inthatwork,weusedthreesimulatedclustersandstudythemalongthreeindependentlines-of- sight. Inthispaper,weextendthatworktoamuchlargersample. Weconsider60mockopticalandX-rayimages(20independent clustersforthreeorthogonallines-of-sight). Throughoutthepaperthequotederrorscorrespondto1σlevel. Thepaperisstructuredasfollow.Section2containsashortreviewoftheresultsobtainedinpreviousnumericalstudies,especially in M10. Section 3 presents a description of the simulated clusters. Section 4 and Section 5 describe the lensing and the X-ray simulationpipelinesandthemethodsofanalyses. WepresenttheresultsinSection6,wherewefirstdiscussthelensingandX-ray massestimatesindividually. Weshowhowthebiasandthescatterofthemassmeasurementsdependontheclustermorphologyand environmentinSection7. Finally,wediscussourresultsinSection8. 2. PREVIOUSSTUDIES 2.1. Stronglensing In M10, we used the parametric code Lenstool to construct mass models from the multiple image systems detected in synthetic Hubble-Space-Telescope(HST)observations. Intheregionwherestronglensingconstraintswerefound(withintheEinsteinradius), themassprofilesrecoveredagreewiththeinputmassdistributionswithanaccuracyofafewpercent. Similarresultswereobtained byJulloetal.(2007)testingtheperformancesofLenstoolwithlensmodelsproducedusingsemi-analyticalmethods. Thestrong- lensingmodelsareconstructedbycombiningamainhalocomponentandadditionalmassiveclumpsassociatedtostarclumps(the galaxiesofthecluster). Fundamental,inthisprocess,isthemodelingofthecentralgalaxy,BCG(Donnarummaetal.2011,2009; Comerfordetal.2006). M10demonstratedthatawrongparametrizationoftheBCGleadstoasevereunder-orover-estimateofthe strong-lensingmassesextrapolatedatlargeradii. Indeed,whenthecentralgalaxywasexcludedduringthecreationandtheanalysis of the synthetic optical images, both the bias and scatter were largely reduced. Discrepancies were seen already at R , a radius 2500 whichistypically2–3timeslargerthantheEinsteinradius. TheparametrizationoftheBCGisalsoimportantforamorerealistic estimate of the lensing cross-section: its presence increases the strong lensing signal up to a twenty percent in cluster size haloes (Giocolietal.2011). The necessity of having an accurate model for the BCG in order to extrapolate the strong lensing mass at large radii makes the comparison between strong lensing and X-ray mass estimates highly uncertain. Indeed, X-ray emission from the central region (∼70−100kpc)isoftenexcludedfromtheX-rayanalysisbecausemoredifficulttomodel(seee.g.Vikhlininetal.2006). 2.2. Weaklensing Theweak-lensinganalysisisbasedonthemeasurementoftheshapeofgalaxiesinthebackgroundoftheclusters,whoseellipticity canbeusedtoestimatetheshearproducedbythelens. Detailsontheweak-lensinganalysiscanbefoundinSection4. InM10,we found that fitting the reduced tangential shear profile with a Navarro-Frenk-White (NFW, Navarro et al. 1997) functional or using theaperturemassdensitometryproducequitesimilarresults. Themeasuredprojectedmassisaccurateatlevelof∼10%forthose clustersthatdonotshowanymassivesubstructuresnearby. Twolensplanespresentedmassiveclumpsjustoutsidethevirialradius ofthecluster. Thisdilutesthesheartangentialtothemainclusterclumpevenatsmallerradii. Asaconsequence,themassprofilesof LensingandX-raymassestimates-Simulations 3 thesetwolensesresultedtobeseverelyunder-estimated. Suchproblemaffectsthemethodswheretheshearismeasuredtangentially. Instead,wetestedthattechniqueswhichcombinestrong-andweak-lensing,suchasthosebyCacciatoetal.(2006)andbyMerten etal.(2009)arenotinfluenced. Toreconstructthelensingpotentialthelattermethodusesanadaptivegrid(seealsoBradacˇ etal. 2005; Diego et al. 2007; Merten et al. 2011) and naturally incorporates the effects of substructure. As result, the scatter in the projectedmassmeasurementisreducedandlimitedto(cid:46)10%. Themaincausesofsubstantialscatterinthedeprojectedmassesaretriaxialityandpresenceofsubstructure. Underthestandard assumption of spherical symmetry, three-dimensional masses are over- or under-estimated, depending on the orientation of the clustermajoraxiswithrespecttotheline-of-sight. Forsystemswhosemajoraxispointstowardtheobserver, massesaretypically over-estimated. The opposite occurs for clusters elongated in the plane of the sky. In M10, the resulting scatter of our sample is of order ∼17%. For clusters with substructure, the unknown location of the substructure along the line of sight also makes the three-dimensionalmassestimatehighlyunsure. More recently, Becker & Kravtsov (2011) used a large number of simulated halos extracted from a large cosmological box to discuss the accuracy of weak lensing masses measured by fitting the cluster shear profiles with NFW models. Their results are consistentwithours. Giventhelargesizeoftheirsample,theysignificantlyprobethatweak-lensingmassesmeasuredbyfittingthe tangential shear profiles are biased low, concluding that the NFW model is actually a poor description of the actual shear profiles of clusters at the radii used in the fitting. At the radius enclosing an overdensity of 500 times the critical density of the universe, R ,theyfoundthatthebiasamountsto∼10%forbothclustersatz=0.25andz=0.5. Theyvariedtheintegrationlengthtosee 500 thedependenceonlarge-scalestructureonthedeflectionfield. Thescatterfoundusingourintegrationlength(i.e. 20h−1 Mpc)is comparabletoours. Withintheintegrationdepthwechose,thelarge-scalestructurecanbeconsideredcorrelated. Iftheintegrationlengthislarger,we willincludealsouncorrelatedstructures. Theireffectsontheweaklensingmassestimateshavebeendiscussedindetailinseveral papers: uncorrelatedstructuresintroduceanoiseinthemassestimatesandtheircontributiontothetotalerrorbudgetiscomparable tothestatisticalerrors(Cen1997;Metzleretal.1999;Hoekstra2001,2003;White&Vale2004;Hoekstraetal.2011). Becker& Kravtsov (2011), specifically, showed that the scatter in the weak-lensing masses of low-mass halos increases more than for high- masshalosasafunctionofline-of-sightintegrationlengthbecausethehigh-masshalosgeneratemoreshearthanthelow-masshalos. Thelargescalestructurehasdifferentimpactdependingontheredshiftofthelensesandonthedepthoftheobservations(Hoekstra 2003). Finally, Becker & Kravtsov (2011) also discussed how the scatter and the bias changes under varying number density of back- groundsources,n . Theyshowthatasthenumberdensityincreases,theshapenoisecontribution(duetotheintrinsicellipticityof g thesources)tothescatterdecreasesandeventuallybecomessubdominantwithrespecttotheintrinsicscatterinweak-lensingmass measurements. Forclustersatz=0.25,thetotalscatteronM changesfrom∼37%forn =10galsarcmin−2to∼25%forn =40 500 g g galsarcmin−2. Asforthebias,theyfoundthatfittingtheNFWfunctionalformwithinR canreducethebiasby∼5%. 500 2.3. X-ray RegardingtheX-rayanalysis,wetestedtwodifferentapproachesinM10thatwedubbedthebackwardandforwardmethods. The backward procedure assumed a priori a functional form for the mass (such as NFW), spherical symmetry and hydrostatic equilibrium(Eq. 26inM10): −Gµm n M (<r)/r2=dP/dr=d(n ×T)/dr (1) p gas tot gas whereGisthegravitationalconstant,µ=0.59ismeanmolecularweightina.m.u.,m istheprotonmass,kistheBoltzmanncon- p stant, and n and T are the gas density and temperature profiles. These are estimated at once by geometrically de-projecting the gas measuredX-raysurfacebrightnessandtemperaturedata. The3DtemperatureiscomputedfollowingtherecipebyMazzottaetal. (2004). Moredetailscanbefoundin(Ettorietal.2002;Morandietal.2007,R06,M10). The forward method, instead, uses a complex parametric formulae to fit the projected surface brightness and temperature profiles. Subsequently,theanalytic2Dexpressionsarede-projectedassumingsphericityandfinallythetotalmassiscomputedthroughEq.1. Thetwomethodsarebasedonthesamebasichypothesis: sphericalsymmetryandhydrostaticequilibrium. Theydifferforthequan- titytheyanalyticallyparametrized. Thefirstmethodimposesafixedmassprofile(usuallyNFW)whilethesecondusesparametric formulasforthesurfacebrightnessandthetemperaturedistributions(seeSec.5.2). Inthisway,theforwardapproachhassmoother radialprofilestobederived,butalsomoreparameters.Thetwoproceduresconsistentlyreconstructboththetotalandthegasmasses, aswedemonstrated. Forthisreason, here, welimitourX-rayanalysistotheforwardmethod(presentedwithmoredetaillateron thepaper). TheX-raymasseswereshowntosystematicallyunderestimatethetruemassofthesimulatedclustersby5-20%withan averagebiasof10%betweenR andR andascatterof6%. Thegasmassesreconstructedwereusually5%higherthanthetrue 2500 200 oneswithintheregionwithsufficientsignal. Thankstothehighexposuretimeused(500ks)andthefieldofviewoftheimages,we comparedthemassprofilesuptoR . 200 M10resultsweresimilartothefindingsofNagaietal.(2007). Inthesamefashion,theauthorscreatedmockX-rayimagesof16 objectssimulatedwithanadaptivemeshcode. Theexposuretimewas1Msecandthefieldofviewselectedextendedwellbeyond R . Processingtheimages,theyfollowedtheforwardmethod. Intheirwholesample,theaveragedifferencebetweenthetotalmass 200 andtheX-rayderivedmasswas16%atR withascatterof9%,reducingto13%±10%forregularsystems. 500 Rasiaetal.(2006)studiedasmallersampleoffiveobjects. Wefollowthebackwardmethodassumingdifferentparametrization forthetotalmass: β modeleitherisothermalorwithpolytropictemperatureprofile, NFWandthemodelpresentedbyRasiaetal. (2004). Undertheconditionofperfectbackgroundsubtraction,wefoundanaveragedbiasof23%atR and20.6%atR . The 500 2500 causesofthebiasweredouble: theneglectcontributionofthegasbulkmotionstothetotalenergybudgetandthetemperaturebias towardslowervaluesoftheX-raytemperatures. ThecontributionofthelastfactorwasconfirmedbyPiffaretti&Valdarnini(2008) 4 Rasiaetal. whoanalyzedmorethan150SPH-simulatedclusters. Bothpapersfoundatemperaturebiasof10-15%(seealsoRasiaetal.2005). Ameglioetal.(2009)pointedoutthedirectcorrelationbetweenthisbiasandtheclustermass(ortemperature): thebiasishigherin mostmassivesystemsbecausetheyhavealargerspreadintemperature. 3. SIMULATIONS Ouranalysisisbasedon20simulatedclustersidentifiedatz=0.25,allhavingvirialmassM >5×1014h−1M atthatredshift, vir (cid:12) andeachobservedalongthreeorthogonalprojectiondirections. Theseclustersbelongtothesetofradiativesimulationspresented byFabjanetal.(2011),whoseinitialconditionshavebeendescribedindetailsbyBonafedeetal.(2011). TheLagrangianregions aroundeachoftheseclustershavebeenidentifiedwithinalow–resolutionN-bodycosmologicalsimulation,thatfollowed10243DM particleswithinaboxhavingacomovingsideof1h−1Gpc.ThecosmologicalmodelassumedisaflatΛCDMone,withΩ =0.24for m thematterdensityparameter,Ω =0.04forthecontributionofbaryons,H =72kms−1Mpc−1 forthepresent-dayHubbleconstant, bar 0 n =0.96fortheprimordialspectralindexandσ =0.8forthenormalizationofthepowerspectrum,thusconsistentwiththeCMB s 8 WMAP7 constrains (Komatsu et al. 2011). Within each Lagrangian region mass resolution is increased following the Zoomed Initial Condition (ZIC) technique (Tormen et al. 1997). Resolution is progressively degraded outside such regions, so as to save computational time, while preserving a correct description of the large–scale tidal field. Within the high–resolution region, it is m =8.47×108M h−1andm =1.53×108M h−1forthemassesoftheDMandgasparticles,respectively. DM (cid:12) DM (cid:12) SimulationshavebeencarriedoutusingtheTreePM/SPHGADGET-3code,anewerandmoreefficientversionoftheGADGET-2 codeoriginallypresentedbySpringel(2005). APlummer–equivalentsofteninglengthforthecomputationofthegravitationalforce inthehigh–resolutionregionwasfixedto(cid:15)=5h−1kpcinphysicalunitsatredshiftz<2,whilebeingkeptfixedincomovingunitsat higherredshift. Asforthecomputationofhydrodynamicforces,weassumetheSPHsmoothinglengthtoreachaminimumallowed valueof0.5(cid:15). Oursimulationsincludemetal–dependentradiativecoolingandcooling/heatingfromaspatiallyuniformandevolving UVbackground,accordingtotheprescriptionpresentedbyWiersmaetal.(2009). Followingthestar-formationmodelbySpringel & Hernquist (2003) gas particles whose density exceeds a given threshold value are treated as multi-phase particles, where a hot ionizedphasecoexistsinpressureequilibriumwithacoldphase,whichisthereservoirforstarformation. Wealsoincludeadetailed descriptionofmetalenrichmentfromdifferentstellarpopulations, usingthemodeloriginallydescribedbyTornatoreetal.(2007). TheeffectofSNfeedbackisincludedthroughtheeffectofgalacticwindshavingavelocityof500kms−1. Theclustersignificantradii(R ,R ,R ,R ,R )16 andthecorrespondingmassesarelistedinTable1. Tocomputethese 2500 1000 500 200 vir quantities we center on the minimum of the potential well as done in Rasia et al. (2011). In the following, we will refer to these numbersasthetrueortheintrinsicvalues. TABLE1 TRUERADIIIN h−1KPCANDMASSESIN h−11014M(cid:12)ATDIFFERENTOVERDENSITY(∆=2500,1000,500,200,vir). cluster R R R R R M M M M M 2500 1000 500 200 vir 2500 1000 500 200 vir CL1 388 669 989 1561 1988 2.089 4.277 6.900 10.852 12.394 CL2 491 823 1161 1731 2241 4.227 7.948 11.170 14.796 17.76 CL3 341 558 790 1181 1515 1.410 2.484 3.510 4.702 5.489 CL4 314 513 747 1204 1615 1.099 1.923 2.974 4.979 6.641 CL5 415 654 925 1495 1962 2.557 3.985 5.637 9.518 11.921 CL6 437 719 1010 1557 2048 2.966 5.296 7.342 10.772 13.543 CL7 396 656 934 1476 1949 2.218 4.021 5.807 9.169 11.698 CL8 404 655 921 1487 1951 2.357 4.003 5.563 9.367 11.719 CL9 372 615 857 1277 1647 1.830 3.315 4.480 5.941 7.046 CL10 393 708 1052 1637 2075 2.163 5.051 8.299 12.514 14.091 CL11 458 739 1019 1528 1943 3.427 5.751 7.546 10.187 11.565 CL12 317 568 836 1343 1763 1.131 2.617 4.171 6.902 8.640 CL13 304 541 868 1405 1827 1.005 2.257 4.655 7.913 9.621 CL14 452 723 998 1503 1930 3.289 5.381 7.079 9.686 11.346 CL15 373 608 902 1467 1965 1.847 3.200 5.238 9.008 11.971 CL16 400 653 911 1392 1822 2.278 3.970 5.392 7.691 9.547 CL17 370 616 892 1459 1891 1.809 3.332 5.052 8.863 10.662 CL18 277 475 700 1147 1504 7.584 1.528 2.444 4.303 5.365 CL19 289 513 780 1249 1585 0.858 1.922 3.380 5.551 6.279 CL20 403 660 920 1410 1858 2.337 4.092 5.544 7.993 10.121 4. WEAKLENSINGSEC:LENSING 4.1. SkyLenssimulations 16R∆andM∆aretheradiusandthemassofthespherewhosedensityis∆timesthecriticaldensityattheclusterredshift. LensingandX-raymassestimates-Simulations 5 Tosimulatetheirlensingeffectsonapopulationofbackgroundsources,weprocessthehalosusingourwelltestedopticalsimula- tionpipelineSkyLens(e.g.Meneghettietal.2008;Meneghettietal.2011,andM10). Here,webrieflysummarizethebasicsteps towardtherealizationofthesimulatedimagesandreferthereadertothosepapersforfurtherdetails. We begin selecting particles falling into a cylinder centered on the cluster and having its width and depth set equal to 10 and 20h−1Mpc, respectively. This ensures to include in the simulation the effects of filaments apart from the cluster and of additional mass clumps that could produce addition shear signal. Since we are focusing on high resolution re-simulated clusters, we do not includetheeffectsofun-correlatedlarge-scale-structures. Asmatteroffact,theimportanceofmatteralongthelineofsightisfairly small for rich clusters at intermediate redshifts, like those in our sample, provided that the bulk of the sources are at high redshift comparedtothecluster(seeSection1). Weprojectthemassdistribution(i.e. theselectedparticles)onalensplaneattheredshiftofthecluster,z =0.25. Foreachcluster L inthesamplewederivethreelensplanes,correspondingtotheprojections(named1,2,and3)alongthethreeaxesofthesimulation box. Thefinalnumberoflensplanesusedinthisstudyis60. Thisisafactorof∼7largerthanthesamplepreviouslyinvestigatedin M10. Thedeflectionfieldofeachclusterisdeterminedbytracingabundleof4096×4096light-raysfromtheobserverpositionthrough thelensplane(seeM10forthedescriptionofthetree-code). Thefinaldeflectionmatrixisusedtofurthertracethelightraystoward thebackgroundsources,allowingustoreconstructtheirdistortedimages. Inshort,thecodeusesasetofrealgalaxiesdecomposed into shapelets (Refregier 2003) to model the source morphologies on a synthetic sky. In the current version of the simulator, the shapelet database contains ∼3000 galaxies in the z-band from the GOODS/ACS archive (Giavalisco et al. 2004) and ∼10000 galaxies in the B,V,i,z bands from the Hubble-Ultra-Deep-Field ( HUDF) archive (Beckwith et al. 2006). Most galaxies have spectralclassificationsandphotometricredshiftsavailable(Benitez2000;Coeetal.2006),whichareusedtogenerateapopulation ofsourceswhoseluminosityandredshiftdistributionsresemblethoseoftheHUDF. SkyLensallowsustomimicobservationswithavarietyoftelescopes,bothfromspaceandfromtheground. Forthiswork,we simulatewidefieldobservations,onwhichwecarryoutaweaklensinganalysis,usingtheSUBARUSuprime-Cam. Allsimulations includerealisticbackgroundandinstrumentalnoise. Thegalaxycolorsarerealisticallyreproducedbyadopting22SEDstomodel thebackgroundgalaxies,followingthespectralclassificationspublishedbyCoeetal.(2006). ComparedtoM10,weusehereadifferentsetup. First,weassumeanexposuretimeof2000sintheI-band,whichisafactorof threeshallowerthaninM10. Thisisaimedattestingtheweaklensinganalysisundermorerealisticconditions. Second,weusereal stars observed with SUBARU to model the PSF. The PSF model is characterized by a FWHM of ∼0.6(cid:48)(cid:48). M10 used an isotropic gaussian PSF instead. For all lens planes, we produce wide-field images covering a region of 2400(cid:48)(cid:48)×2400(cid:48)(cid:48) around the cluster center. Thisallowsustomeasuretheshearsignaluptoadistanceof∼3.5h−1Mpcatz=z , wellbeyondthevirialradiusofany L clusterinthesample. 4.2. Weak-lensinganalysis TheweaklensingmeasurementsaredoneusingthestandardKaiser-Squires-Broadhurst(KSB)method,proposedbyKaiseretal. (1995)andsubsequentlyextendedbyLuppino&Kaiser(1997)andbyHoekstraetal.(1998). Thegalaxyellipticitiesaremeasured fromthequadrupolemomentsoftheirsurfacebrightnessdistributions,correctedforthePSF,andusedtoestimatethereducedshear undertheassumptionthattheexpectationvalueoftheintrinsicsourceellipticityvanishes. In this study we use the publicly available pipeline KSBf90 by C. Heymans17 to process our images and measure the shear fields. Thefinalgalaxycatalogsareconstructedbyselectingonlythegalaxieswithsignal-to-noiseratioS/N>10(asprovidedby SExtractor,Bertin&Arnouts1996),half-lightradiuslargerthan1.15timesthePSFsize,andreducedshear|g|<1. Giventhe above mentioned exposure time and seeing conditions, the effective number density of galaxies in the final shear catalogs is ∼17 arcmin−2. Inobservationsexploitingadepthsimilartooursimulatedimages,thenumberofsourcesavailableforthelensinganalysis maybesmallerbecausethelightemissionfromclustergalaxies(notincludedinthesesimulations)arepotentialcontaminants.These non-lensed galaxies bias low the lensing signal if they are accidentally included in the shear catalogues. Color based techniques (e.g. Medezinski et al. 2007, 2010) allow to separate the foreground and the background galaxy populations efficiently, but, to be conservative,severalsourceswhichmayhaveadubiousclassificationareusuallyexcludedfromthelensingcatalogs. Amongthem severalbackgroundgalaxies. InM10,weconsideredseveralmethodstomeasurethetotalmassusingtheobservedshearfield. Asdiscussedabove,wefound thatthe mostprecisemass measurementsareobtained bycombining weakandstrong lensingnon-parametrically(see e.g.Merten et al. 2009). The disadvantage of this approach is that it is very expensive both in terms of time needed to carry out the analysis andintermsofdatarequirements. Theidentificationofthestrong-lensingfeatures,usedtoconstrainthemodelintheinnerregion, usually requires deep and high-resolution Hubble-Space-Telescope imaging. Moreover, strong-lensing clusters are relatively rare andknowntobeaffectedbymanybiases(Meneghettietal.2010,2011;Hennawietal.2007). Fittingthetangentialshearprofiles with functionals describing the cluster density profiles is a very common and easy alternative to measure the mass (e.g. Clowe & Schneider2002;Hoekstraetal.2000;Jeeetal.2005;Dahle2006;Paulin-Henrikssonetal.2007;Bardeauetal.2007;Ogurietal. 2009;Kuboetal.2007;Okabeetal.2010;Romanoetal.2010;Umetsuetal.2011;Zitrinetal.2011). Further,thismethodcanbe appliedtoclustersdowntorelativelysmallmasslimitsandinabsenceofstronglensingfeatures. Here,weassumethatthedensityprofilesofclustersarewelldescribedbytheNavarro-Frenk-Whiteprofile(Navarroetal.1997), ρ ρ (r)= s (2) NFW r/r (1+r/r )2 s s whereρ andr arethecharacteristicdensityandthescaleradius,respectively. Thecharacteristicdensityisoftenwrittenintermsof s s 17http://www.roe.ac.uk/∼heymans/KSBf90/Home.html 6 Rasiaetal. theconcentrationparameter,c =r /r ,as 200 200 s 200 c3 ρ = ρ 200 . (3) s 3 cr[ln(1+c )−c /(1+c )] 200 200 200 Wederivethemassbyfittingtheone-dimensionalreducedtangentialshearprofilewiththecorrespondingNFWfunctional(Bartel- mann1996;Wright&Brainerd2000;Meneghettietal.2003). Thetangentialshearprofileisderivedfromthedatabyradiallybinningthegalaxiesandaveragingthetangentialcomponentof theirellipticitywithineachbin. Thetangentialandcrosscomponentsare,respectively,definedas (cid:15) =−Re[(cid:15)e−2iφ] and (cid:15) =−Im[(cid:15)e−2iφ]. (4) + × The angle φ specifies the direction from the galaxy centroid towards the center of the cluster, which we identify with most bound particleinthesimulation. Whenaveragingovermanygalaxies,theexpectationvalueoftheintrinsicsourceellipticityvanishes,and thereducedtangentialshearisgivenbyg =(cid:104)(cid:15) (cid:105). Onthecontrary,inabsenceofsystematicstheaveragedcrosscomponentofthe + + ellipticityshouldbezero. 5. X-RAY 5.1. X-MAS simulations BeforeproducingtheX-raysyntheticcatalogue,wehaveappliedthetechniquedescribedinAppendixAtoremoveover-cooled particles. Subsequently, our clusters are processed through X-MAS to obtain Chandra mock images. The characteristics of this software package are described in detail in other works (Gardini et al. 2004; Rasia et al. 2008). To create the photon event file, we assumed the Ancillary Response Function (ARF) and Redistribution Matrix Function (RMF) typical of the ACIS-S3 detector aimpoint. Weconsidertheredshiftasthatofthesimulatedtimeframe(z=0.25)andthemetallicityconstantandequalto0.3solar inrespecttothetablesbyAnders&Grevesse(1989)18. Thefieldofviewofourimageshasasideof16arcmin. Forourcosmology andredshift,thiscorrespondsto2561h−1 kpc. AlltheclustershavetheirR regionswithinthefieldofview(seeTable1),evenif 500 someofthematthatradiusdonotemitasufficientnumberofphotonstoallowaprecisespatialandspectralanalysis(seemoreinthe nextsession). Weaccountfortheemissionbyalltheparticleswithinadepthof10Mpcalongthelineofsightdirectionandcentered onthecluster. Theexposuretimechosenis100ksec. ThissettingdiffersfromwhatadoptedinM10: thereductionoftheexposure timeallowsamorerealisticcomparisonwithobserveddata. Inthefinaleventfiles,weaddacontributionforthegalacticabsorption byaWABSmodelwithN =5×1020cm−2. AsinM10,wedonotincludetheinfluenceofthebackgroundsinceR06provedthatits H neteffectistoenlargetheerroronthemassestimateswithoutintroducinganextrabias. Furthermore,newbackgroundmodelsare capabletopredictthespatialvariationoftheChandrabackgroundwithanaccuracybetterthan1%(Bartaluccietal. inpreparation). We note that tools as X-MAS are not suitable to address calibration problematics since the same response files are used both to create and to analyze the data. In this sense, in our analysis we assume a perfect knowledge of the instrument calibration and the resultsdonotdependontheinstrumentreproduced. Intheanalysisofrealobservationaldata,systematicinstrumentaluncertainties arehighlyimportant,inparticularinsituationofhighstatistics(highnumberofcounts). Totreatthemcorrectlyoneneedstoinclude themintheanalysis. Leeetal.(2011)haverecentlyprovidedabayesianstatisticalmethodtotacklethisproblem. 5.2. X-rayanalysis UsingCIAOtool(Fruscioneetal.2006)weextractsoftbandimagesinthe[0.7−2]keVband. Weapplythewaveletalgorithm ofVikhlininetal.(1998)toidentifyclumps. Theseandanymajorsubstructurehavebeenmaskedandexcludedfromthefollowing analysis. The surface brightness profiles are centered in the X-ray centroid (Rasia et al. 2011) and account 15-30 linearly spaced annuli with at least 100 counts. The innermost annulus is selected outside the central 10% of R , the outermost one is always 500 beyondR andreachesR inthemajorityofthecases(seeTable3). Theradialcoverageiscomparabletorecentobservations, 1000 500 someofwhichextendbeyondR (Neumann2005;Ettori&Balestra2009;Leccardi&Molendi2008;Vikhlininetal.2009). The 500 temperatureprofileiscalculatedin6-10annulispanningoverthesameradialrangeofthesurfacebrightnessprofile. Theminimum numberofphotonpertemperatureannulusis1000. Thespectraaregroupedandfittedbyasingle-temperatureMEKALmodelinthe XSPECpackage(Arnaud1996). Thestatisticsusedisχ2andtheenergybandconsideredis[0.8-7]keV.Inthepipelinethevaluesof galacticabsorption,redshift,hydrogencolumndensityandmetallicityarefixedequaltotheinputones. TocomputethetotalmassfromtheX-rayanalysis,wefollowthe“forward"methodofM10(seealso,Vikhlininetal.2006). The surfacebrightnessandthetemperatureprofilesarefittedbytheanalyticformulae: (r/r )−α 1 (r/r)−a n n =n2 c ; T =T t . (5) p e [1+(r/r )2]3β−α/2[1+(r/r )γ](cid:15)/γ 0[1+(r/r)b]c/b c s t Since we exclude the cluster central part from our analysis, we do not model the cooling core region as done in Vikhlinin et al. (2006).Thisexcisioniscommoninbothsimulations(e.g.M10,Nagaietal.2007)andobservations(e.g.Vikhlininetal.2009;Ettori etal.2010)toavoid,intheformercase,theinfluenceoftheovercooledcentralregionand,inthelattercase,thepresenceofcentral activegalaxy,cool-coreregions,gassloshing. The2Danalyticformulaearedeprojectedandthetotalmassisrecoveredbyassuming hydrostaticequilibrium,thatfromEq. 1itcanbewrittenas (cid:18) (cid:19) kT(r)r dlnρ dlnT M (<r)=− + , (6) X Gµm dlnr dlnr p 18 TheHeliumabundanceusedintheplasmaemissionwasmodifiedfrom9.77e-02to7.72e-2tobeconsistentwiththehydrogenmassfractionusedasinputin GADGET-3code. LensingandX-raymassestimates-Simulations 7 whereT andρarethedeprojected3Danalyticprofiles. Followingthisprocedure,weobtaintheX-raymassthatwecomparewith the true mass of the simulated cluster. The uncertainties in the estimate of this mass, eM , were obtained through Monte Carlo X simulations. IneachMonteCarlorealization,surfacebrightnessandtemperatureprofileswerevariedwithintheirmeasurederrors. Eachtimeanewmasswasthenderived. Theresultinguncertaintyweredefinedasthestandarddeviationcomputedover100such realizations. 6. RESULTS 6.1. Weak-lensingmassestimates Weak-lensing allows us to measure the mass of the cluster projected on the plane of the sky. The NFW analytic formula of the integralalongthelineofsightofthemasscontainedinacylinderisM(R )=4ρ r3h(x),wherex=R /r and 2D s s 2D s  (cid:113) x  √x22−1arctan xx−+11 (x>1) (cid:113) h(x)=ln2+ √2 arctanh 1−x (x<1) . (7)  1−x2 1+1x (x=1) Theprofileparametersr andρ areobtainedfromfittingthetangentialshearprofile,asdiscussedabove. s s Unfortunately, inmostcosmologicalapplicationstheprojectedmassisnottheinterestingquantity. Rather, weneedtomeasure thethree-dimensionalmass. Toderiveitbyde-projectingthetwo-dimensionalmass,oneneedstomakestrongassumptionsabout theshapeofthecluster,whichisusuallyassumedspherical. Inthiscase,theNFWmodelisgivenby (cid:20) (cid:21) y M(r)=4πr2ρ ln(1+y)− , (8) s s 1+y wherey=r/r . Toboththe2Dandthe3Dmassesweassociateerrors,eM andeM ,computedbypropagatingtheerrors s WL,2D WL,3D onr andρ ,asobtainedfromfittingthetangentialshearprofiles. s s In the following, we show how well we measure projected and de-projected mass profiles of the clusters in our sample. In both cases, the quality of the mass measurement is assessed by means of the ratio Q between measured and true mass: Q = WL WL M /M . Theuncertaintyonthisratio,eM /M ,accountsonlyfortheerrorsintheweak-lensingmasssincethetruemassis WL true WL true perfectly known from simulations. The weighted-mean of both the 2D and the 3D weak-lensing bias radial profiles, <Q >, is WL shownbythesolidredlineinFig.1. Itsscatterisquantifiedbythestandarddeviationofitsdistribution,andisrepresentedbythe shadedyellowregion. Informulae: ΣQ (R/R )×eM (cid:20)Σ(Q (R/R )−<Q >)2×eM (cid:21)0.5 <Q >= i WL,i vir,i WL,i and scatter= i WL,i vir,i WL WL,i (9) WL ΣeM ΣeM i WL,i i WL,i TheprofilesareplottedinunitsofR . Theover-imposedcrossesrefertotheweightedaveragedQ computedatthesignificant vir WL radiusofeachobject,withaveragecomputedataradiuscorrespondingtoafixedoverdensity∆,(R /R ),andnotoverthewhole ∆,i vir,i radialprofile(R/R ). Theyarelocatedattherespectiveaveragedover-densityradii. Thecrosshorizontalbarsshowthedispersion vir,i aroundtheradiiinunitsofR . ThequantitiveversionofFig.1isreportedonTable2whereweresumeallourresults. Eachvalue vir ofQ anditscorrespondinguncertaintyislistedinTable B6ofAppendixB. 3D,WL Twoimportantconclusionsemergefromthisanalysis. First,themassmeasuredfittingthereducedtangentialshearprofilewithan NFWfunctionalisbiasedlow. Thebiasamountsto∼7−10%betweenR andR andreaches∼13%forlargerdistancesfrom 2500 500 theclustercenter. Second,thescatterinbothtwo-andthree-dimensionalmassesrangesbetween∼10%atsmallradiiand∼25%at largerradii,being20%atR . 500 These results agree with the findings of M10 and Becker & Kravtsov (2011), confirming that the weak-lensing analysis via the KSBpipelinedoesnotintroducesignificantsystematics 6.2. X-raymassestimates Contrarily to the optical mass measurements, the X-ray mass derivation gives directly the 3D mass profile. Therefore, we can straightlydefinetheratiobetweentheX-raymassandthetrueones:Q =M /M .Similarlytoweaklensinganalysis,wecompute X X true theweightedaverageofQ overthewholesampleandthestandarddeviationofthedistribution(Eq.9). InFig.2,weplotthevalues X onlywithinR ,whichistheradiusreachedbymostofthesurfacebrightnessandtemperatureprofiles(Table3).Thecrossshown 500 atR istheresultoftheextrapolationoftheanalyticformulae. OnthethirdpartofTable2,wereporttheweighted-averagedQ 200 X anditsscatter. EachQ valueanditscorrespondinguncertaintyislistedinTableB7ofAppendixB. X Fig.2confirmssomepreviousfindingsthatwewillsynthesizeherepostponingtoSection8amoreprofounddiscussion. TheaverageX-raymassisconsistentlyunderestimatingthetruemass. TheX-raybiasisaround25%atthecenterand30-35%at R . The decline in the most external regions is expected since the cluster outskirts present a more dramatic lack of hydrostatic 500 equilibrium(Lauetal.2009)andastrongerinfluenceofgasclumpiness(Nagai&Lau2011). Thepresenceofgasclumpsaffects the X-ray mass determination in two ways: it shallows the surface brightness profile and it cools X-ray temperature (see more on thisintheSection8.) Massivesystems,astheonesstudiedinthispaper,areexpectedtobestillaccreatingandthereforefarforan equilibriumstate. Moreover, thetemperaturebinsintheexternalregions, wherethetemperatureprofiledeclinesmoresteeply, are usuallylargercontainingmoretemperaturestructure. Finally,thelargebiasonthemostexternalregionhastobetakenwithcaution sinceitisnottheresultofameasurementbutofanextrapolation. Thedispersionaroundtheaverageisquitesmall. Thestandard deviation is less than 10% at all radii apart from R where it is 12%. These numbers are twice or three times smaller than the 2500 gravitationallensingdispersion. 8 Rasiaetal. FIG.1.—Comparisonbetweenweak-lensingandtruemassesusingallclustersinthesample(60lensplanes).Thesolidlinesintheleftandrightpanelsshowthe averageratiosbetween2D-andthe3D-weak-lensingmassesandthetruemasses,respectively.Theseareplottedasafunctionofthedistancefromtheclustercenter inunitsofthevirialradius. Thecrossesineachpanelmarktheaveragelocationsofvariousover-densityradiiandtheiramplitudeistheweightedaveragebiasat eachsignificantradius.Theyellow-shadedregionmarksthestandarddeviationateachradius. TABLE2 WEIGHTEDAVERAGEMASSBIAS,QWLANDQX,ANDTHEIRSTANDARDDEVIATIONFORTHEWHOLESAMPLEANDTHEDIFFERENT SUB-SAMPLESBASEDONX-RAYANDENVIRONMENTALCLASSIFICATION. THEENVIRONMENTALCLASSIFICATIONISPERFORMEDON THEVISUALINSPECTIONOFBOTHINTRINSICSIMULATEDMAPS,I.POOR,ANDONOPTICALSYNTHETICIMAGES,O.POOR(SEE SECTION7FORDETAILS). (1−Q )×100 2D,WL radius allcluster regular I.Poor O.Poor bias rms bias rms bias rms bias rms R 7.0 11.5 7.9 11.0 4.4 4.3 7.0 9.5 2500 R 7.6 16.5 8.7 15.8 1.7 3.3 6.0 12.6 1000 R 9.7 20.8 10.1 19.5 0.0 5.0 4.9 16.4 500 R 12.7 26.2 12.8 23.1 -4.2 7.0 4.1 22.2 200 (1−Q )×100 3D,WL radius allcluster regular I.Poor O.Poor bias rms bias rms bias rms bias rms R 6.5 16.1 6.9 9.5 3.0 13.3 6.2 15.2 2500 R 6.7 18.5 8.2 12.8 4.5 12.2 5.2 16.4 1000 R 8.4 20.5 8.9 16.9 3.5 11.3 4.8 16.7 500 R 12.8 25.0 13.3 22 0.0 9.8 5.8 20.0 200 (1−Q )×100 X radius allcluster regular I.Poor O.Poor bias rms bias rms bias rms bias rms R 23.9 11.0 19.0 7.6 21.9 4.9 20.8 8.2 2500 R 27.5 7.9 25.6 7.8 22.5 2.2 26.4 6.2 1000 R∗ 33.0 9.4 34.4 10.4 26.1 7.7 33.1 8.8 500 ∗TheX-raymeasuresareextrapolatedforsomeclusters. LensingandX-raymassestimates-Simulations 9 7. CLUSTERCLASSIFICATION We investigate in this Section the efficiency in reducing bias and scatter on both X-ray and gravitational lensing masses of two selecting criteria. We create different sub-samples determined by the morphology of the X-ray images or by the presence of sub- structuresontheirenvironment. 7.1. MassesandX-raymorphology To limit the impact of the non-thermal processes on the X-ray mass estimates, clusters are often selected on the basis of their appearance. Theliteratureisrichofstudieswhereclustershavebeenclassifiedintorelaxed,orregular,andunrelaxed,ordisturbed, because of their X-ray morphology (e.g. Zhang et al. 2008; Vikhlinin et al. 2009). Most of the time, the classification is done “visually”, i.e. simply quantifying the regularity of an object from the X-ray image in the soft band. More objective criteria, proposedinthepast,arethepowerratios,centroid-shift,asymmetryandfluctuationparameters,andhardnessratio. Wetestallof themandpresenthereourresult. Third order power ratio and centroid shift. Buote & Tsai (1995) suggested to decompose the surface brightness distribution in multipoles. The high order multipoles, usually normalized by the monopole and called power ratios, are used to quantify the contributionofdifferentscalecomponents(asymmetriesandsubstructures)tothesurface-brightnesspowerspectrumrelativetothe large-scale smooth cluster emission. Most information in the power spectrum is contained in the first four multipoles. P is the 0 monopole. Thepowerratio P/P measuresthedipoleoftheX-rayemission, whichiszeroifmeasuredwithrespecttotheX-ray 1 0 centroid. The power ratio P/P measures the ellipticity (quadrupole). The third order power ratio P/P can be used to quantify 2 0 3 0 asymmetriesandisthebestindicatorofclusterswithmultimodaldistributions. Substructuresonsmallerscalescontributetohigher ordermultipoles. Another indicator of the dynamical state and of the asymmetry of the X-ray emission is the centroid-shift, i.e. the shift of the surfacebrightnesscentroidinaperturesofincreasingsize. Thisparameterpointsoutthedynamicalstateoftheclusteraswellasthe asymmetry. FollowingPooleetal.(2006)andMaughanetal.(2008),wedefinethecentroid-shiftas (cid:115) (cid:80) 1 (∆ −(cid:104)∆(cid:105))2 w= × i i , (10) R (N−1) max whereR istheradiusofthelargestaperture,and∆ =R(cid:126) −R(cid:126) istheshiftofthecentroidinthei−thaperturewithrespectto max i c,i c,max thecentroidinthelargestaperture,R(cid:126) . (cid:104)∆(cid:105)isthemeanvalueofthevarious∆ andthesumisdoneoveralltheN apertureswith c,max i radiiuptoR . InthisworkweassumedN=17apertureswithradiirangingbetweenR =0.15×R andR =R . max min 500 max 500 Thethird–orderpowerratioandthecentroidshiftwereshowntobeeffectiveinclassifyingclustersbytworecentworksbyCassano et al. (2010) and Böhringer et al. (2010). Clusters are located in a rather well defined region in the P/P −w plane: objects with 3 0 smallcentroidshiftandsmallP/P areclassifiedas“regular”. Themajorityofthemarecoolcoresystems, notverydynamically 3 0 activeandshowingabsenceorverylittleradioemission. Forallthesereasons,often,theseobjectsarereferredas“relaxed”. Inthiswork,wecomputethepowerratio,P/P,andthecentroid-shift,fromthesignaloftheregionwithinR ofthemasked 3 0 500 images. Withthisattention,weaimtoevaluatethe“irregularity”oftheactualportionoftheimagethatweusetoretrievethemass. BoththevaluesandtheiruncertaintiesarederivedfromMonteCarlosimulations. Wecreate100newimageswherethephotonsare re-distribuited accordingly to a Poisson statistics. We evaluate the estimators in each image. Finally, we extract the medians and the16th and84th percentileoftheMonteCarlodistributionstorepresentthefinalvaluesofthemorphologicalestimatorsandtheir uncertainties. FIG.2.—ComparisonbetweenX-rayandtruemassesusingthewholesample.Themeaningoflines,crossesandshadedregionsisthesameofFig.1. 10 Rasiaetal. ThethirdorderpowerratiosandthecentroidshiftsofoursamplewiththeiruncertaintiesareshownintheleftpanelofFig.3. We recognizetheregionwithregularsystemsbyslightlyrelaxingthecriteriaadoptedbyCassanoetal.(2010). Inourwork,wedefine aclustertoberegularwhenw<0.03andP/P <2×10−7. OurchoiceismotivatedbythefactthatourapertureisequaltoR , 3 0 500 thuslargerthanthe500kpcapertureradiusanalyzedbyCassanoetal. (2010). Reducingthisradius,theynaturallymeasuredlower valuesofthemorphologicalestimatorsand,inparticular,ofthepowerratios(Böhringeretal.2010). The 17 regular objects are denoted by asterisks in the figure. In most cases, these are systems with small companions or some minorirregularityinthesurfacebrightnessmap.ThefullclassificationislistedinTable3belowthecolumn“P/P,w”.Twoextreme 3 0 examples are represented in the right panel of Fig. 3. The X-ray images of the most disturbed system of our sample (CL 13) are shownonthetoppanels,whilethebottompanelsrefertoarelaxedcluster(CL9). The uncertainties on our parameters are smaller than those of Böhringer et al. (2010) because of the better spatial resolution of ChandrawithrespecttoXMM-Newton(seeBöhringeretal.2010,foradetaileddiscussionabouttheinfluenceonspatialresolution orpoint-spreadfunction). ThecomparisonbetweentheirworkandthatbyCassanoetal. (2010),basedonChandradata,confirms thisstatement. Thelargeexposuretimeassumedinourmockobservationsensuresahighcountsstatisticsand,therefore,afurther reduction on the uncertainties. If future missions will reproduce the great spatial resolution of Chandra, both power-ratios and centroidshiftswillbeavailablewithsufficientaccuracyforalargenumberofobjects,thusallowinghighydetailedstudiesofcluster morphologies. WenowproceedbycheckingwhetherthelensingandtruemassesbiasesimprovewhenselectingonlytheX-rayregularsystems. Similarly to Fig. 1, we show in Fig. 4, Q as a function of the distance from the cluster center for the sub-samples of regular WL systems. QuantitativeresultsarelistedinTable2includingthoseforQ . X The scatter on lensing bias is reduced by 20%-40%. However, the bias itself worsen with respect to the whole sample. Clearly, a selection based on the X-ray morphology is not optimal for lensing purposes. The reason of this behavior can be explained by comparingTable3andTableB6. AmongtheX-rayregularclusters,therearethreeimages(projection2ofCL1,CL9,andCL20) whose lensing measurements are severely under-estimated. All of them present in the outskirts of the optical images filaments or fallingsubstructures,thatdonothaveanyobviouscounterpartintheX-rayimagesorarelyingoutsidetheChandrafieldofview. Jeltemaetal.(2008)andPiffaretti&Valdarnini(2008)claimedtofindasignificanttrendoftheX-raymassesbiasonthemorpho- logicalestimators. Weinvestigatethisaspectfurtherbyincludingananalysisofthebiasesintheweak-lensingmassreconstruction. To this purpose, we considered the absolute values of |(1−Q )| to evaluate the dependence for any deviation. A linear fit be- 3D,WL tween the mass biases and the morphological estimators has been computed accounting for measurement errors in both variables. TheresultsarereportedinTable4. Thecentroidshiftperformsbetterthatthethirdorderpowerratio. TheslopesoftheQ−wrela- tionsarealwayssignificantlydifferentfromzerowiththecorrectsign(negativefortheX-raybiasandpositivefortheweak-lensing deviations). The best fit values are similar to those found by Jeltema et al. (2008) and Piffaretti & Valdarnini (2008). We further quantifythecorrelationbetweenthemassbiasesandP/P orwbymeansofthePearsoncorrelationcoefficient. Wefindalwaysa 3 0 negativecorrelation,beingthevaluesbetween-0.3and-0.4forwandaround-0.2and-0.3forP/P. 3 0 InFig.5wepresentthebestcombinations: Q −wforR ,and|1−Q |−wforR andR . Thetoppanels,aresimilarto X 2500 3D,WL 1000 500 Fig.3wherethedifferentcolorsandsymbolsrefertodifferentvaluesofthemassbias.TheredtrianglesrefertoclusterswhoseX-ray massbiases,Q ,arewithin20%orwhoseweak-lensing-massesdeviations,|(1−Q )|,arewithin10%. Inallthreetoppanels, X 3D,WL we distinguish no segregation of colors. This implies that a better estimate of the total mass (red triangles) does not necessarily come from regular clusters defined on the basis of P/P and w values. However, for the centroid shift, even if this condition is 3 0 notbenecessary,itissufficientatallradii: theweighted-averagebiasforclusterswhosecentroidshiftislowerthan0.3is15-20% lowerthanthosewithw>0.06(seedifferenceonhorizontallinesinthecentralpanels). Asconfirmation,thethirdorderpowerratio weaklydiscriminatebetweengoodandbadestimates. FIG. 3.—Ontheleft:thedistributionofclustersintheP3/P0−wplane.Theasterisksrepresentclustersclassifiedasregular.Ontheright:softX-rayimagesof adisturbedclusterandaregularoneseenalongthethreeprojections:CL13onthetopandCL9atthebottom. Toemphasizethemorphologyweover-plottedthe iso-fluxcontoursingreen.