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Giles, P. A., Maughan, B. J., Pacaud, F., Lieu, M., Clerc, N., Pierre, M., Adami, C., Chiappetti, L., Démoclés, J., Ettori, S., Le Févre, J. P., Ponman, T., Sadibekova, T., Smith, G. P., Willis, J. P., & Ziparo, F. (2016). The XXL Survey: III. Luminosity-temperature relation of the Bright Cluster Sample. Astronomy and Astrophysics, 592, [A3]. https://doi.org/10.1051/0004-6361/201526886 Publisher's PDF, also known as Version of record License (if available): CC BY-NC Link to published version (if available): 10.1051/0004-6361/201526886 Link to publication record in Explore Bristol Research PDF-document This is the final published version of the article (version of record). It first appeared online via Astronomy & Astrophysics at 10.1051/0004-6361/201526886. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ Astronomy&Astrophysicsmanuscriptno.xxl_lt (cid:13)cESO2015 December2,2015 The XXL Survey(cid:63) III. Luminosity-temperature relation of the Bright Cluster Sample P.A.Giles,1 B.J.Maughan,1 F.Pacaud,2 M.Lieu,3 N.Clerc,4 M.Pierre,5 C.Adami,6 L.Chiappetti,7 J.Démoclés,3 S. Ettori,89 J.P.LeFévre,5 T.Ponman,3 T.Sadibekova,6 G.P.Smith,3 J.P.Willis,10 F.Ziparo3 1 SchoolofPhysics,HHWillsPhysicsLaboratory,TyndallAvenue,Bristol,BS81TL,UK e-mail:[email protected] 2 Argelander-InstitutfurAstronomie,UniversityofBonn,AufdemHugel71,D-53121Bonn,Germany 3 SchoolofPhysicsandAstronomy,UniversityofBirmingham,Edgbaston,Birmingham,B152TT,UK 4 MaxPlanckInstitutfurExtraterrestrischePhysik,Postfach1312,D-85741GarchingbeiMunchen,Germany 5 LaboratoireAIM,CEA/DSM/IRFU/Sap,CEASaclay,91191Gif-sur-Yvette,France 6 LAM(Laboratoired’AstrophysiquedeMarseille)UMR7326mAix-MarseilleUniversite,CNRS,F-13388Marseille,France 7 INAF,IASFMilano,viaBassini15,I-20133Milano,Italy 8 INAF,OsservatorioAstronomicodiBologna,viaRanzani1,1-40127Bologna,Italy 9 INFN,SezionediBologna,vialeBertiPichat612,1-40127Bologna,Italy 10 DepartmentofPhysicsandAstronomy,UniversityofVictoria,3800FinnertyRoad,Victoria,BC,Canada December2,2015 ABSTRACT Context.TheXXLSurveyisthelargesthomogeneoussurveycarriedoutwithXMM-Newton.Coveringanareaof50deg2,thesurvey containsseveralhundredgalaxyclustersouttoaredshiftof≈2aboveanX-rayfluxlimitof∼5×10−15ergcm−2s−1.Thispaperbelongs tothefirstseriesofXXLpapersfocusingonthebrightclustersample. Aims.Weinvestigatetheluminosity-temperature(LT)relationforthebrightestclustersdetectedintheXXLSurvey,takingfullyinto accounttheselectionbiases.WeinvestigatetheformoftheLT relation,placingconstraintsonitsevolution. Methods.Wehaveclassifiedthe100brightestclustersintheXXLSurveybasedontheirmeasuredX-rayflux.These100clusters havebeenanalysedtodeterminetheirluminosityandtemperaturetoevaluatetheLT relation.Weusedthreemethodstofittheform oftheLT relation,withtwoofthesemethodsprovidingaprescriptiontofullytakeintoaccounttheselectioneffectsofthesurvey.We measuretheevolutionoftheLT relationinternallyusingthebroadredshiftrangeofthesample. Results.Takingfullyintoaccountselectioneffects,wefindaslopeofthebolometricLT relationofB =3.08±0.15,steeperthanthe LT self-similarexpectation(B =2).Ourbest-fitresultfortheevolutionfactorisE(z)1.64±0.77,fullyconsistentwith“strongself-similar” LT evolutionwhereclustersscaleself-similarlywithbothmassandredshift.However,thisresultismarginallystrongerthan“weakself- similar”evolution,whereclustersscalewithredshiftalone.Weinvestigatethesensitivityofourresultstotheassumptionsmadein ourfittingmodel,findingthatusinganexternalLT relationasalow-zbaselinecanhaveaprofoundeffectonthemeasuredevolution. However,moreclustersareneededinordertobreakthedegeneracybetweenthechoiceoflikelihoodmodelandmass-temperature relationonthederivedevolution. 1. Introduction times, including pre-heating, supernovae feedback, and heating from active galactic nuclei (AGN) at high redshift. These non- Under the assumption of self-similarity (Kaiser 1986), simple gravitationalprocessesshouldhavethestrongesteffectinlower scalinglawscanbederivedbetweenvariouspropertiesofgalaxy mass systems owing to their shallower potential wells that ex- clusters.Thesescalinglawsareadvantageousastheyprovidea pelgasfromtheinnerregionsandsuppresstheluminosity.Ob- cheapwayofmeasuringthemassesoflargesamplesofclusters, servations have shown further steepening of the LT relation at an important ingredient for cosmological studies using galaxy the low-mass regime (e.g. Osmond & Ponman 2004; Sun et al. clusters (see Allen et al. 2011, for a review). One of the most 2009).However,recentwork,usingmethodstocorrectforsam- explored scaling relations is that between the X-ray luminos- ple selection effects, has suggested that the slope of the LT re- ity (L) and temperature (T), expected to follow a relationship lation on group scales is consistent with massive clusters (e.g. of L ∝ T2.However,amultitudeofstudieshavefoundthatthe Bharadwajetal.2015;Lovisarietal.2015). slope of the LT relation is ∼3 (e.g. Pratt et al. 2009; Eckmiller etal.2011;Takeyetal.2013;Connoretal.2014),steeperthan X-rayfluxlimitedsamplessufferfromtwoformsofselection the self-similar expectation. One of the main explanations for bias,Malmquistbias,wherehigherluminosityclustersarepref- the deviation from this theoretical expectation is energy input erentially selected out to higher redshifts, and Eddington bias, bynon-gravitationalprocessesduringclusterformationatearly whereinthepresenceofintrinsicorstatisticalscatterinluminos- ity for a given mass, objects above a flux limit will have above (cid:63) Based on observations obtained with XMM-Newton, an ESA sci- average luminosities for their mass. This effect is amplified by ence mission with instruments and contributions directly funded by the steep slope of the cluster mass function, which results in a ESAMemberStatesandNASA. net movement of lower mass objects into a flux limited sam- Articlenumber,page1of16page.16 Article published byEDP Sciences,to be cited ashttp://dx.doi.org/10.1051/0004-6361/201526886 A&Aproofs:manuscriptno.xxl_lt ple. The net effect on the LT relation is to bias the normalisa- Inthispaper,partofthefirstreleaseofXXLresults,wein- tionhighandtheslopelow(seeAllenetal.2011,forareview). vestigate the form of the luminosity-temperature relation (LT) Therefore, taking these biases into account is paramount when for a sample of the 100 brightest clusters detected in the XXL modelling cluster scaling relations in order to uncover the true Survey.TheLT relationwillbederivedtakingtheselectionfunc- natureofanynon-gravitationalheatingdrivingdeparturesfrom tionoftheclustersamplefullyintoaccount.Theoutlineofthis self-similar behaviour with mass or redshift. Although scaling paper is as follows. In § 2 we discuss the data preparation and relation studies have had a rich history, only a relatively small sample selection. Section 3 outlines the cluster analysis. In § 4 number of published relations attempt to account for selection wepresentourresultsandderivethesampleandbias-corrected biases (e.g. Stanek et al. 2006; Pacaud et al. 2007; Pratt et al. scaling relations. Our discussion and conclusions are presented 2009;Vikhlininetal.2009;Andreon&Bergé2012;Bharadwaj in § 5 and § 6, respectively. Throughout this paper we assume etal.2015;Lovisarietal.2015),whileMantzetal.(2010a)pro- a WMAP9 cosmology of ΩM=0.282, ΩΛ=0.719, and H0=69.7 videsthemostrobusthandlingofselectioneffectstodate. (Hinshawetal.2013). In the self-similar model, with cluster properties mea- sured within overdensity radii defined relative to the crit- ical density, the evolution of the scaling relations can 2. Dataprocessingandsampleselection be parameterised by the factor E(z) (where E(z) = (cid:112) The data processing and sample selection are fully detailed in ΩM(1+z)3+(1−ΩM−ΩΛ)(1+z)2+ΩΛ). In this frame- XXL paper II, Pacaud et al (2015, submitted, hereafter Paper work, the evolution of the normalisation of the LT relation II ), and is briefly summarised here. The XXL Survey contains goes as E(z)γLT, where γLT ≡ 1 if clusters scale self-similarly 542XMMpointingscovering50.8deg2.Afterlight-curvefilter- with both mass and redshift, or γ ≈ 0.42 if the observed LT ing(followingPratt&Arnaud2002),rejectionofbadpointings massdependenceoftheclusterbaryonfractionisalsoincluded andexclusionofpointingswithhighbackgroundlevels,thetotal (Maughan2014).ThesetworeferencevaluesforγLT assumethe XXLareaspans46.6deg2 (from454pointings).Images,expo- same underlying evolution driven by the changing critical den- sure maps, and detector masks were generated and processed sity of the Universe; the difference arises purely from the alge- usingtheXpipeline(Pacaudetal.2006;Clercetal.2012). braiccombinationofthescalinglawsofgasmass,temperature, SEwasthenruntogenerateaconservativesourcecat- and gas structure with total mass that are used to construct the alogue and source masks, followed by a dedicated XMM maxi- LT relation(Maughan2014). mumlikelihoodfittingproceduretodeterminelikelihoodratios We refer to γ = 1 as “strong self-similar” evolution, re- toassessthedetectionandsourceextentprobabilities.Extended LT flecting the fact that it is based on the assumption that clusters sourcesweredefinedwithanextentlargerthan5(cid:48)(cid:48)andextension scale self-similarly with both mass and redshift, while γ = likelihoodlargerthan15.Theseextendedsourceswerethensep- LT 0.42isreferredtoas“weakself-similar”evolutionandassumes arated into two classes, the C1 class with extension likelihood clusters only scale self-similarly with redshift. The latter is a larger than 33 and detection likelihood larger than 32, and the more realistic prediction of the self-similar evolution against C2classwithanextensionlikelihood15<EXT_LH<32. which to test for departures in the observed evolution of clus- The sample used in this work is the 100 brightest clusters tersintheLT plane. (hereafter XXL-100-GC1) selected from the source list gener- Understanding how scaling relations evolve with redshift is ated above. Count rates were estimated from a growth curve important for two main reasons: (i) scaling relations must be analysis (GCA, following Clerc et al. 2012) within a 60(cid:48)(cid:48) aper- well-calibrated at high redshift to provide mass estimates for ture, and converted into fluxes using an energy conversion fac- cosmologyand(ii)therelationsprovideinsightintothehistory tor (assuming T=3 keV, z=0.3, and Z=0.3) of 9.04×10−13, or ofheatingmechanismsinclusters.Aconsensusontheevolution 1.11×10−13 forclustersfallingonthedamagedmos1chip.The fromobservationshasyettobeachieved;variousstudiesfindan clusterswererankedinorderofdecreasingfluxandthebrightest evolutionconsistentwithself-similar(e.g.Vikhlininetal.2002; 100clusterswereselected.Wenotethatfiveclusterswithinthe Maughanetal.2006;Pacaudetal.2007),whileothersfindde- XXL-100-GC fell on observations with high periods of flaring. parturesfromtheself-similarexpectation(e.g.Ettorietal.2004; The five flared observations were not used and instead the next Kotov&Vikhlinin2005;Reichertetal.2011;Hiltonetal.2012; five brightest objects were included. The clusters XLSSC 113, Clercetal.2014).Again,theimportanceoftakingintoaccount 114,115,550,and551werereplacedwithXLSSC091,506,516, selectioneffectsiscrucialforunderstandingevolutionandrecent 545,and548.Takingthisintoaccount,ourselectioncorresponds workhasshownthatdeparturesfromself-similarevolutioncan toafluxlimitofFX,cut=3×10−14 ergss−1 cm−2.ThefinalXXL- beexplainedbyselectionbiases(Pacaudetal.2007;Mantzetal. 100-GCsampleconsistsof96C1and4C2clusters,respectively, 2010a;Maughanetal.2012).Althoughtheseprocessesneedto with 51 falling within the northern field of the XXL footprint, be understood to explain the differences between the observed and49withinthesouthernfield.Followingarobustredshiftval- andtheoreticalprediction,furtherconsiderationsmustbemade idation(seePaperI),theXXL-100-GCspanaredshiftrangeof totakeintoaccounttheselectioneffects. 0.04≤z≤1.05,with98spectroscopicand2photometricredshifts. ThesampleandthederivedpropertiescanbefoundinTable1. Inordertoaddressthepointsraisedabove,wehavecarried out the largest X-ray survey undertaken by XMM-Newton. The XXL Survey (XXL paper I, Pierre et al 2015, submitted, here- after Paper I), is a 50 deg2 survey with an X-ray sensitivity of 3. Analysis ∼5×10−15 erg s−1 cm−2 (for point-like sources), in the [0.5-2] In this section we describe the cluster analysis process used in keV band. With the aim of detecting several hundred clusters thiswork. outtoaredshiftof≈2,thissurveyprovidesauniqueopportunity to constrain cluster scaling relations that fully accounts for se- 1 A master XXL-100-GC cluster catalogue will also be available in lectioneffects,andinthefuturewillplacerobustconstraintson electronicformathttp://cosmosdb.isaf-milano.inaf.it/XXL/ cosmologicalparameters(seePaperI). andviatheXMMXXLDataBaseathttp://xmm-lss.in2p3.fr Articlenumber,page2of16page.16 P.Gilesetal.:XXLIII:LT RelationoftheBrightClusterSample Theextentoftheclusteremissionwasdefinedastheradius beyondwhichnosignificantclusteremissionisdetectedusinga threshold of 0.5σ above the background level. This is intended 5’’ 0 to provide a conservative estimate of the radius beyond which 0 3 thereisnosignificantclusteremission.Aone-dimensional(1D) -3. surfacebrightnessprofileofeachcamerawasproduced.Aback- ground annulus was defined with an inner radius of 250(cid:48)(cid:48) and 00 4 modelled by a flat (particle) and vignetted (X-ray) component. -3. Thetwomodelcomponentswerethenfittotheradialprofilein n o ttoheprboadcukcgeroautnodtarlebgaiocnkgurosiunngdtmheodχe2l.sTtahtiestriacdaianldbtihnesnwseurmecmoend- clinati -3.500 structed to be 15.4(cid:48)(cid:48) in width, chosen to ensure greater than 20 De counts per bin and with enough bins to perform the fitting pro- 00 6 cedure.Theradialprofileforeachcamerawasthensummedto 3. - produce an overall profile. An initial source extent was deter- mined based on the 0.5σ radius, and the background fitting re- 00 7 peated using the extent as the inner radius for the background 3. - annulus.Thisprocesswasiterateduntilthesourceextentradius changedbylessthan1%. Toaccountforthebackgroundinthespectralanalysis,local 37.100 37.000 36.900 36.800 36.700 36.600 backgroundswereused.However,owingtothesurveydetection Right ascension oftheclusters,theyaredetectedatarangeofoff-axispositions on the XMM cameras. Therefore, if possible, an annulus cen- Fig.1:Exampleofthebackgroundmethodusedinthespectralanalysis tred on the aimpoint of the observation (BG ) with a width (see§3).TheimageoftheclusterXLSSC010withtheregionsshow- aim ingthespectralextractionregion(blackcircle),clusterdetectionradius equaltothediameterofthespectralextractionregionwasused. Thisensuresthatthelocalbackgroundistakenatthesameoff- (dashedredcircle),andthebackgroundannuluscentredontheaimpoint oftheobservation(blueannulus). axispositionasthecluster,whichhelpstoreducethesystematic uncertainties due to the radial dependencies of the background components. To ensure that no cluster emission was included in the background subtraction, a region centred on the cluster rc=0.15r500 andβ=0.667.Theβ-profileparameterswerechosen withradiusequaltotheclusterextent(seeabove)wasexcluded. tomatchthoseusedinPaperII.Wenotethattheuncertaintieson Figure 1shows an exampleof this local backgroundfor one of theluminosityarescaledbythisextrapolation,butdonotinclude our clusters. However, if this method was not possible (owing anyuncertaintyontheβ-profileparameters.Theimpactofthese to close proximity to the aimpoint or a large cluster extent), an assumptions is tested in section 5.3.4. Values for cluster r500 annulus centred on the cluster (BGclust) with inner radius equal arecalculatedusingamass-temperaturerelation(r500,MT,seebe- to the cluster extent and an outer radius of 400(cid:48)(cid:48) was used for low). We denote LXXL as the luminosity in the [0.5-2.0] keV 500,MT the local background. The clusters XLSSC 060 and XLSSC 091 band(clusterrestframe)withinr ,andLbol asthebolo- 500,MT 500,MT had a source extent larger than 400; therefore, an outer radius metric luminosity within r . All the properties quoted in- 500,MT of500(cid:48)(cid:48) and800(cid:48)(cid:48),respectively,wasusedfortheclustercentred cludetheclustercorebecausetheexclusionoftheclustercoreis localbackground. notpossibleforall100clusters.Theresultsofourspectralanal- Cluster spectra were extracted for each of the XMM cam- ysis are given in Table 1. The cluster XLSSC 504 was dropped erasandfitswereperformedinthe0.4-7.0keVbandwithanab- becauseoftheunconstrainederrorsreportedbyXSPEC. sorbedAPEC(Smithetal.2001)model(v2.0.2)withtheabsorb- Amass-temperature(M −T)relationisusedinthiswork ingcolumnfixedattheGalacticvalue(Kalberlaetal.2005).The WL for two purposes: (i) to calculate r for the XXL-100-GC spectra for each camera were fit simultaneously with the tem- 500,MT and (ii) to convert the mass function to a temperature function perature of the APEC components tied together. The fits were (see§4.3)forthebiascorrection.The M −T relationispre- performedusingXSPEC(v12.8.1i)andtheabundancetablefrom WL sented in XXL paper IV, Lieu et al (2015, submitted, hereafter Anders&Grevesse(1989).Becauseofthelownumberofcounts PaperIV),baseduponweaklensingclustermassesandthetem- for many of the clusters, the spectra were fitted using the cstat peratures presented in this work. Briefly, the M −T relation statistic. The background spectra were grouped such that they WL was determined using 37 XXL-N clusters that fell within the containedatleast5countsperbin,andthisgroupingappliedto Canada-France-HawaiiTelescopeLensingSurvey(CFHTLenS) the source spectra. A similar method was employed and justi- footprint, utilising the CFHTLenS shear catalogue (Heymans fiedinWillisetal.(2005)whoanalysedasampleof12galaxy et al. 2012; Erben et al. 2013) for the mass measurements. In groups and clusters in the XMM-LSS. Throughout the spectral order to increase the statistics and the mass range covered, the analysisweassumedafixedmetalabundanceofZ=0.3Z . (cid:12) XXL-100-GC is combined with 10 groups from the COSMOS Theclustertemperaturesarederivedwithin300kpcforeach survey (Kettula et al. 2013), and 50 massive clusters from the cluster,denotedasT .Thisradiusrepresentsthelargestra- 300kpc Canadian Cluster Comparison Project (CCCP, Hoekstra et al. diusforwhichatemperaturecouldbederivedfortheentireclus- 2015). tersample.Thenormalisationsforeachcamerawerefreeinthe spectralfit,withthepncamerausedtocalculatetheluminosity. ThecombinedMWL−T relationisfittedwithapower-lawof We denote the luminosity within 300kpc as LXXL , where the theform(assumingself-similarevolution) 300kpc superscript XXL refers to the [0.5-2.0] keV band (cluster rest frame). Luminosities quoted within r are extrapolated from 500 log (M E(z))= A +B log (T) (1) 300kpc out to r by integrating under a β-profile assuming 10 WL MT MT 10 500 Articlenumber,page3of16page.16 A&Aproofs:manuscriptno.xxl_lt using the Bayesian code linmix_err (Kelly 2007). In Paper IV Wefinda2.8σdifferenceinthenormalisationascompared we find A =13.56+0.10 and B =1.69+0.12. See Paper IV for to the REXCESS clusters, with the XXL normalisation being MT −0.08 MT −0.13 furtherdiscussionsontheresultsfromtheM −T relation. lower. This is due to the presence of strong cool core clusters WL in the REXCESS sample. These clusters are apparent as high- luminosityoutliersinFig.2(right),whichshowstheREXCESS 4. Results clustersplottedontheXXLLT relation.Qualitatively,itisclear Theresultsarepresentedintheformofastudyofthe LT rela- thatintheabsenceofthesestrongcoolcoreclusters,theremain- tion, taking the selection effects fully into account. We present ingREXCESSclustersareconsistentwiththeLXXL−T relation. two implementations to account for the selection effects: (i) an Asdiscussedinsection5.2,theabsenceofstrongcoolcoreclus- updatedmethodofthePacaudetal.(2007)implementation,de- ters from the XXL-100-GC sample is due to a combination of finedastheXXLlikelihood,and(ii)amethodbaseduponMantz surveygeometryandcoolcoreevolution. etal.(2010b),definedastheM10likelihood.Wefirstpresentthe sampleLT relation,notcorrectingfortheselectioneffects.This 4.2. Selectionfunction allows for comparison of the LT relation when taking into ac- countselectioneffects,whatimpactthishasonthederivedscal- Fulldetailsoftheconstructionoftheselectionfunctionaregiven ing relation, and also for the comparison with previously pub- in Paper II, but the key points are summarised here. The se- lishedrelationsthatdonotaccountforselectioneffects. lection function takes into account three aspects of the XXL- For the analysis of the scaling relations, we account for the 100-GC selection, (i) the pipeline detection, (ii) the flux cut of factthatthelikelihoodcurveforameasuredtemperatureisap- the XXL-100-GC, and (iii) the survey sensitivity. The pipeline proximatelyGaussianinlogspace(consistentwiththeasymmet- detection, i.e. the C1+2 classification in pointing p (denoted ric errors usually found in temperature measurements). We use PC1+2,pi(I|CR∞,rc,RA,Dec)), was studied in depth in Pacaud the method of Andreon (2012) to convert the generally asym- et al. (2006) and updated following the methodology of Clerc metric errors reported by XSPEC into a log-normal likelihood. etal.(2012).Wereferthereadertotheseworksforfulldetails. We note that the errors on the temperature given in Table 1 are ThemodellingofthefluxcutassumesthattheerrorsontheGCA thosereportedbyXSPEC. countratecanbemodelledasaGaussiandistribution.Theprob- abilityofatrueaperturecountrate(CR )beinggreaterthatthe 60 count-ratecut(CR )foragivenpointing(p)is cut i 4.1. SampleLT Relation Figure2(leftpanel)showsthe LXXL −T relationfortheXXL- PPi(I|CR∞,rc,RA,Dec)= PC1+2,pi(I|CR∞2,rc,RA,Dec) 100-GC.Forsimplicity,theLXXL−T notationexplicityrefersto (cid:32) (cid:34) (cid:15) (r )CR −CR (cid:35)(cid:33) theL5X0X0L,MT −T300kpcrelation.Afittothedatausingapowerlaw × 1+erf 60 c ∞ cut √ , oftheform σm((cid:15)60(rc)CR∞,RA,Dec) 2 (3) (cid:32)L (cid:33) (cid:32)T (cid:33)BLT LXX0L = E(z)γLTALT T0 (2) where (cid:15)60(rc) = (cid:18)1−(cid:104)1+(60(cid:48)(cid:48)/rc)2(cid:105)1.5−3β(cid:19) and CR∞ is the pipeline count rate extrapolated to infinity. The final ingredient wasperformed,whereA ,B ,andγ representthenormali- LT LT LT isthemodellingofpointingoverlaps,whereweassumethede- sation,slope,andpoweroftheevolutioncorrectionrespectively. We note that the two clusters at low luminosity offset from the tection is independent over the different pointings. For a given position,pointingsaresortedbyincreasingoff-axisdistance,re- L − T are the clusters XLSSC 011 and 527. These clusters arXeXlLowS/Nclustersatlowredshift,0.054and0.076,suchthat producing the overlap cross-matching procedure of the survey. Thecombinedselectionfunctionisthengivenby the 300 kpc region is large on the detector. The temperature mstielalsiunrcelumdeentthiesmliikneltyheafifftectotetdhebyLXtXhLes−eTfacretolartsi.oHn.oTwheevepro,wweer P(I)=(cid:88)N Ppi(I)(cid:89)(cid:16)1−PC1+2,pj(I)(cid:17). (4) lawwasfittothedatausingtheBCESorthogonalregressionin i=1 j<i base ten log space (Akritas & Bershady 1996) assuming self- similarevolution.ThefitisgivenbytheblacksolidlineinFig- ure2,assuming L =3×1043 ergs−1,T =3keV,andγ =1(self- 4.3. Likelihood 0 0 LT similar). We find a normalisation of ALT=0.90±0.06, and slope The observational data used in this study is the distribution of BLT=3.03±0.27.Forcomparison,wefittheLT relationtothe31 the XXL-100-GC clusters in L, T, and z. Our physical model REXCESSclustersstudiedinPrattetal.(2009,hereafterP09), assumes that the cluster population is described by a power- and 52 clusters selected within the 11deg2 XMM-LSS survey law correlation between L and T with log-normal scatter in L (Clerc et al. 2014, hereafter C14). To fit for these LT relations andevolutionin Lby E(z)γLT.TheXXL-100-GCdatarepresent we use the data from Table B1 (Cols. T1 and L[0.5 − 2]1) in a subset of this population selected according to our selection P09,andthedatagiveninTable1(Cols.T andL[0.5−2])inC14. function and with noisy measurements of L and T, denoted Lˆ X 500 The P09 and C14 relations are hereafter denoted as L − T andTˆ.Weneglectanymeasurementerrorsonz. P09 and L −T, respectively, and the fit parameters are given in Thenumberdensityofclustersinthesurveyvolumeistaken C14 Table 2. The fits to the P09 and C14 clusters are given by the intoaccountbyusingamassfunction,assumedtobedescribed red dot-dashed line and green dashed line, respectively, assum- by a Tinker mass function (Tinker et al. 2008). We then trans- ingthesameL ,T ,andγ asfortheXXL-100-GCfit.Wefind formthistoatemperaturefunction,dn/dT,usingthe M −T 0 0 LT WL nosignificantdifferencebetweentheXXL-100-GCL −T re- relation given in Sect. 3. In the present analysis we neglect the XXL lationandtheL −T relation.ThisisunsurprisingastheC14 intrinsic scatter in the M −T relation, and the measurement C14 WL clustersare selectedfrom theXMM-LSSarea, whichhas many errorsontheparametersofthescalingrelation,andassumethat clustersincommonwiththeXXL-100-GC. itsevolutionisself-similar(i.e.M ∝ E(z)TBMT). Articlenumber,page4of16page.16 P.Gilesetal.:XXLIII:LT RelationoftheBrightClusterSample Table1:X-raypropertiesoftheXXL-100-GC.r isestimatedfromtheM −T relationinSect.3.LXXL andLbol areextrapolatedout 500,MT WL 500,MT 500,MT tor byintegratingunderaβ-profile(seeSect.4).†indicatesthatonlyaphotometricredshiftwasavailablefortheclusteranalysis. 500,MT XLSSCNum z E(z) r T LXXL LXXL Lbol 500,MT 300kpc 300kpc 500,MT 500,MT Mpc (keV) 1043(ergss−1) 1043(ergss−1) 1043(ergss−1) XLSSC001 0.614 1.38 0.777 3.8+0.5 7.56±0.54 10.10±0.72 23.48±1.68 −0.4 XLSSC003 0.836 1.57 0.643 3.4+1.0 10.04±1.21 12.32±1.49 26.64±3.22 −0.6 XLSSC006 0.429 1.24 0.982 4.8+0.5 11.44±0.55 17.42±0.83 46.72±2.24 −0.4 XLSSC010 0.330 1.18 0.751 2.7+0.5 1.97±0.16 2.58±0.21 5.58±0.46 −0.3 XLSSC011 0.054 1.02 0.831 2.5+0.5 0.11±0.01 0.15±0.01 0.33±0.02 −0.4 XLSSC022 0.293 1.15 0.671 2.1+0.1 2.45±0.09 3.06±0.11 6.06±0.22 −0.1 XLSSC023 0.328 1.17 0.655 2.1+0.3 1.32±0.14 1.63±0.18 3.21±0.35 −0.2 XLSSC025 0.265 1.14 0.751 2.5+0.2 1.68±0.08 2.21±0.11 4.69±0.23 −0.2 XLSSC027 0.295 1.15 0.768 2.7+0.4 1.11±0.09 1.48±0.11 3.20±0.25 −0.3 XLSSC029 1.050 1.77 0.626 4.1+1.0 16.03±1.38 19.46±1.68 43.67±3.76 −0.6 XLSSC036 0.492 1.29 0.801 3.6+0.5 8.21±0.53 11.14±0.72 25.78±1.68 −0.4 XLSSC041 0.142 1.07 0.670 1.9+0.1 0.96±0.06 1.19±0.07 2.31±0.14 −0.2 XLSSC050 0.140 1.07 0.897 3.1+0.2 1.93±0.05 2.78±0.07 6.68±0.17 −0.2 XLSSC052 0.056 1.02 0.387 0.6+0.0 0.09±0.01 0.09±0.01 0.14±0.01 −0.0 XLSSC054 0.054 1.02 0.723 2.0+0.2 0.21±0.02 0.28±0.02 0.56±0.04 −0.2 XLSSC055 0.232 1.12 0.843 3.0+0.3 1.88±0.11 2.61±0.15 6.02±0.34 −0.4 XLSSC056 0.348 1.19 0.824 3.2+0.5 3.03±0.18 4.16±0.25 9.58±0.58 −0.3 XLSSC057 0.153 1.07 0.734 2.2+0.3 0.91±0.05 1.18±0.07 2.46±0.14 −0.1 XLSSC060 0.139 1.07 1.136 4.8+0.2 3.75±0.05 6.31±0.08 18.57±0.24 −0.2 XLSSC061 0.259 1.13 0.678 2.1+0.5 0.87±0.11 1.09±0.14 2.16±0.27 −0.3 XLSSC062 0.059 1.03 0.422 0.7+0.1 0.10±0.02 0.11±0.02 0.16±0.02 −0.1 XLSSC072 1.002 1.73 0.613 3.7+1.1 12.35±1.49 14.89±1.79 32.61±3.92 −0.6 XLSSC083 0.430 1.24 0.943 4.5+1.1 3.14±0.25 4.67±0.38 12.11±0.98 −0.7 XLSSC084 0.430 1.24 0.945 4.5+1.6 1.38±0.21 2.04±0.31 6.18±0.84 −1.3 XLSSC085 0.428 1.24 0.976 4.8+2.0 2.56±0.26 3.88±0.40 11.46±1.18 −1.0 XLSSC087 0.141 1.07 0.619 1.6+0.1 0.76±0.07 0.92±0.09 1.67±0.16 −0.1 XLSSC088 0.295 1.15 0.726 2.5+0.6 1.22±0.11 1.57±0.15 3.28±0.31 −0.4 XLSSC089 0.609 1.38 0.769 3.7+1.6 4.85±0.66 6.44±0.87 14.84±2.01 −1.2 XLSSC090 0.141 1.07 0.507 1.1+0.1 0.38±0.05 0.43±0.05 0.67±0.08 −0.1 XLSSC091 0.186 1.09 1.149 5.1+0.2 7.73±0.11 13.12±0.19 39.12±0.58 −0.2 XLSSC092 0.432 1.24 0.771 3.1+0.8 2.11±0.24 2.81±0.31 6.24±0.70 −0.6 XLSSC093 0.429 1.24 0.810 3.4+0.6 4.75±0.31 6.47±0.42 14.91±0.96 −0.4 XLSSC094 0.886 1.62 0.742 4.7+1.3 19.85±1.71 25.93±2.24 62.01±5.35 −0.9 XLSSC095 0.138 1.06 0.450 0.9+0.1 0.15±0.03 0.17±0.03 0.24±0.04 −0.1 XLSSC096 0.520 1.31 1.000 5.5+2.0 3.77±0.40 5.80±0.62 16.05±1.71 −1.1 XLSSC097 0.760 1.50 0.794 4.6+1.5 9.91±1.25 13.37±1.69 32.53±4.11 −1.0 XLSSC098 0.297 1.15 0.801 2.9+1.0 1.28±0.16 1.74±0.21 3.88±0.48 −0.6 XLSSC099 0.391 1.22 1.032 5.1+3.1 1.44±0.26 2.26±0.41 6.31±1.13 −1.5 XLSSC100 0.915 1.64 0.694 4.3+1.7 11.12±2.50 14.06±3.17 32.42±7.28 −1.2 XLSSC101 0.756 1.50 0.788 4.6+0.8 12.27±0.96 16.53±1.29 39.95±3.13 −0.8 XLSSC102 0.969 1.69 0.574 3.2+0.8 13.31±1.41 16.07±1.70 33.56±3.56 −0.5 XLSSC103 0.233 1.12 0.913 3.5+1.2 0.90±0.10 1.30±0.14 3.20±0.35 −0.8 XLSSC104 0.294 1.15 1.038 4.7+1.5 0.86±0.10 1.36±0.15 4.36±0.42 −1.0 XLSSC105 0.429 1.24 1.024 5.2+1.1 7.91±0.57 12.39±0.89 34.34±2.47 −0.8 XLSSC106 0.300 1.16 0.856 3.3+0.4 3.16±0.15 4.44±0.21 10.48±0.49 −0.3 XLSSC107 0.436 1.25 0.711 2.7+0.4 3.82±0.32 4.89±0.41 10.33±0.88 −0.4 XLSSC108 0.254 1.13 0.705 2.2+0.3 1.49±0.10 1.90±0.13 3.86±0.26 −0.2 XLSSC109 0.491 1.29 0.787 3.4+1.3 4.71±0.78 6.30±1.04 14.34±2.37 −0.8 XLSSC110 0.445 1.25 0.525 1.6+0.1 1.43±0.22 1.63±0.25 2.82±0.43 −0.1 XLSSC111 0.299 1.16 1.017 4.5+0.6 4.27±0.21 6.65±0.32 18.06±0.87 −0.5 XLSSC112 0.139 1.07 0.653 1.8+0.2 0.49±0.06 0.61±0.08 1.15±0.15 −0.2 XLSSC501 0.333 1.18 0.768 2.8+0.6 1.84±0.22 2.44±0.29 5.34±0.64 −0.4 XLSSC502 0.141 1.07 0.532 1.2+0.0 0.55±0.04 0.63±0.05 1.00±0.08 −0.1 XLSSC503 0.336 1.18 0.642 2.0+0.3 2.01±0.19 2.47±0.24 4.79±0.46 −0.2 Articlenumber,page5of16page.16 A&Aproofs:manuscriptno.xxl_lt Table1:continued.... XLSSCNum z E(z) r T LXXL LXXL Lbol 500,MT 300kpc 300kpc 500,MT 500,MT Mpc (keV) 1043(ergss−1) 1043(ergss−1) 1043(ergss−1) XLSSC504 0.243 1.12 1.953 13.8+−13.8 0.48±0.17 1.35±0.48 6.87±2.46 −5.4 XLSSC505 0.055 1.02 0.661 1.7+0.2 0.38±0.02 0.47±0.03 0.90±0.05 −0.1 XLSSC506 0.717 1.49 0.798 4.5+2.1 6.31±1.25 8.53±1.69 20.69±4.08 −1.5 XLSSC507 0.566 1.34 0.612 2.4+0.6 3.68±0.63 4.43±0.76 8.77±1.50 −0.5 XLSSC508 0.539 1.32 0.742 3.3+0.7 3.48±0.33 4.55±0.43 10.08±0.95 −0.5 XLSSC509 0.633 1.39 0.806 4.2+1.1 6.61±0.64 8.99±0.86 21.49±2.07 −0.8 XLSSC510 0.394 1.22 0.711 2.6+0.4 2.31±0.16 2.96±0.20 6.22±0.43 −0.3 XLSSC511 0.130 1.06 0.545 1.3+0.1 0.25±0.03 0.29±0.04 0.48±0.06 −0.1 XLSSC512 0.402 1.22 0.848 3.6+0.6 2.14±0.14 2.99±0.19 7.11±0.46 −0.4 XLSSC513 0.378 1.21 0.936 4.2+0.8 4.40±0.30 6.50±0.44 16.63±1.12 −0.5 XLSSC514 0.169 1.08 0.582 1.5+0.2 0.40±0.06 0.47±0.07 0.81±0.12 −0.1 XLSSC515 0.101 1.05 0.540 1.2+0.1 0.32±0.03 0.37±0.04 0.59±0.06 −0.1 XLSSC516 0.866 1.60 0.695 4.8+1.0 18.38±1.97 23.31±2.50 55.21±5.91 −0.7 XLSSC517 0.699 1.45 0.698 3.4+1.1 5.77±0.88 7.33±1.12 16.09±2.47 −0.6 XLSSC518 0.177 1.09 0.535 1.3+0.0 0.50±0.05 0.58±0.05 0.93±0.09 −0.0 XLSSC519 0.270 1.14 0.555 1.5+0.2 0.81±0.15 0.94±0.18 1.59±0.30 −0.2 XLSSC520 0.175 1.08 0.805 2.7+0.2 1.72±0.05 2.34±0.07 5.19±0.16 −0.1 XLSSC521 0.807 1.54 0.775 4.7+1.3 12.99±1.46 17.32±1.95 42.03±4.72 −0.8 XLSSC522 0.395 1.22 0.711 2.6+0.4 2.11±0.15 2.71±0.19 5.69±0.40 −0.3 XLSSC523 0.343 1.18 0.779 2.9+0.6 2.17±0.17 2.90±0.23 6.40±0.51 −0.4 XLSSC524 0.270 1.14 0.754 2.6+0.5 0.92±0.09 1.21±0.12 2.59±0.25 −0.4 XLSSC525 0.379 1.21 0.832 3.4+0.3 4.83±0.24 6.68±0.33 15.54±0.76 −0.3 XLSSC526 0.273 1.14 0.794 2.8+0.4 3.91±0.20 5.27±0.27 11.68±0.60 −0.2 XLSSC527 0.076 1.03 0.926 3.1+2.8 0.14±0.03 0.20±0.05 0.50±0.12 −1.0 XLSSC528 0.302 1.16 0.839 3.2+0.8 1.49±0.11 2.07±0.16 4.82±0.36 −0.4 XLSSC529 0.547 1.33 0.769 3.5+0.7 5.20±0.44 6.91±0.58 15.71±1.32 −0.4 XLSSC530 0.182 1.09 0.686 2.0+0.2 0.60±0.05 0.75±0.06 1.49±0.13 −0.2 XLSSC531 0.391 1.22 0.966 4.5+2.2 1.81±0.25 2.73±0.38 7.18±0.99 −1.4 XLSSC532 0.392 1.22 0.772 3.0+0.6 2.58±0.23 3.43±0.31 7.59±0.68 −0.5 XLSSC533 0.107 1.05 0.789 2.4+0.1 1.64±0.04 2.21±0.05 4.80±0.12 −0.1 XLSSC534 0.853 1.58 0.725 4.3+1.7 12.49±1.87 16.14±2.41 37.74±5.66 −1.0 XLSSC535 0.172 1.08 0.756 2.4+0.3 1.83±0.10 2.41±0.13 5.14±0.28 −0.2 XLSSC536 0.170 1.08 0.659 1.8+0.3 0.38±0.06 0.47±0.08 0.91±0.14 −0.2 XLSSC537 0.515 1.30 0.934 4.8+1.2 5.47±0.45 8.07±0.67 21.11±1.76 −0.9 XLSSC538 0.332 1.18 0.804 3.1+0.9 1.35±0.14 1.83±0.19 4.13±0.42 −0.6 XLSSC539 0.184 1.09 0.520 1.2+0.1 0.38±0.07 0.44±0.08 0.69±0.12 −0.2 XLSSC540 0.414 1.23 0.776 3.1+0.4 4.13±0.25 5.52±0.34 12.31±0.75 −0.4 XLSSC541 0.188 1.09 0.805 2.7+0.3 1.05±0.06 1.42±0.09 3.16±0.20 −0.3 XLSSC542 0.402 1.22 1.202 6.8+0.5 28.69±0.60 50.37±1.05 157.70±4.59 −0.3 XLSSC543 0.381 1.21 0.689 2.4+0.5 1.06±0.14 1.33±0.18 2.73±0.36 −0.3 XLSSC544 0.095 1.04 0.788 2.4+0.2 0.57±0.02 0.77±0.03 1.68±0.07 −0.2 XLSSC545 0.353 1.19 0.668 2.2+1.6 1.13±0.33 1.41±0.41 2.82±0.81 −0.6 XLSSC546 0.792 1.53 0.668 3.5+0.7 10.49±1.09 13.08±1.36 28.47±2.97 −0.6 XLSSC547 0.371 1.20 0.920 4.0+1.1 2.80±0.27 4.09±0.40 10.31±1.00 −0.8 XLSSC548 0.321 1.18 0.428 1.0+0.1 0.47±0.13 0.51±0.13 0.74±0.19 −0.1 XLSSC549 0.808 1.54 0.709 4.0+2.4 8.87±1.50 11.34±1.92 25.92±4.40 −0.9 Thetotalnumberofclusters N inthevolumeisthenthein- The number of clusters predicted by our model to be ob- tegraloverthetemperatureandredshiftrangeconsidered,multi- served in the subsample defined by our selection function is pliedbythesolidangleofthesurveyΩ: the integral of the mass function over the volume of the sur- vey, weighted by the probability that a cluster of a given mass would be included in the subsample given the LT relation and (cid:90) (cid:90) dn dVdΩ (cid:104)N(cid:105)=Ω dT dz . (5) dT dz Articlenumber,page6of16page.16 P.Gilesetal.:XXLIII:LT RelationoftheBrightClusterSample 1e+45 1e+45 XXL Clusters XXL Clusters XXL fit, Slope=3.01 XXL fit, Slope=3.01 Clerc et al. (2014) Clerc et al. (2014) Pratt et al. (2009) Pratt et al. (2009) Pratt et al. (2009, CC) 1) 1) Pratt et al. (2009, NCC) -s s 1e+44 -s s 1e+44 g g er er 1( 1( -z) -z) E( E( T 1e+43 T 1e+43 M M XXL500, XXL500, L L 1e+42 1e+42 1 10 1 10 T (keV) T (keV) 300kpc 300kpc Fig.2:(Left)LT relationfortheXXLsample.Theblacklinerepresentsanunbiasedfittothedata(seeSect.4.1).TheLT relationsofPrattetal. (2009)andClercetal.(2014),givenbythereddot-dashedlineandthegreendashedline,respectively,areoverplotted;(Right)Sameastheleft plot,butwiththeREXCESSclustersstudiedinPrattetal.(2009)overplotted,splitbetweencool-core(openredsquares)andnon-cool-core(filled redsquares)clusters.Inbothplots,theerrorsonthetemperaturesfortheXXLclustershavebeentransformedviathemethoddescribedinSect.4. theintrinsicandstatisticalscatterontheluminosity, finallikelihood (cid:104)Ndet(cid:105)=(cid:90) dT(cid:90) dzddTn dVddzΩΩ L(Lˆ,Tˆ,I|z,θ)=(cid:89)Ndiet (cid:82) dTˆ P(cid:82)i(dLˆLˆ,TPˆi,(ILˆ|z,iT,ˆθ,)I|zi,θ). (8) (cid:90) Thisnormalisationpenalisesthemodelforexcessprobabil- × dLP(L|T,z,θ)×P(I|L,T,z), (6) itydensityinregionsoftheparameterspacewherethedatadis- favourtheexistenceofclusters.Forexample,weconsideraset where θ stands for our full set of model parameters ofmodelparametersthatgiveagoodfittothepropertiesofthe (ALT,BLT,γLT,σLT) describing the LT relation. In this expres- observedclusters,butalsogiveahighprobabilityofcool,highly sion,thefirstprobability,P(L|T,z,θ),istheprobabilitythatclus- luminousclusters,whennoneareobserved.Thiswouldleadto teroftemperatureT hassomeintrinsicallyscatteredluminosity. a larger denominator in equation 7, and hence a lower overall Thesecondprobability,P(I|L,T,z),istheselectionfunction,i.e. likelihoodcomparedwithanalternatesetofparametersthatde- the probability that a cluster with a luminosity L and tempera- scribes the properties of the observed clusters equally well, but tureT ataredshiftzwouldbeincludedinthesubsample.Thisis does not predict the unobserved clusters. This likelihood is re- summarised in Sect. 4.2. We note the change in notation as we ferredtoastheXXLlikelihood. arewritingthelikelihoodintermsoftheclusterpropertiesLand Wealsoconsideranalternativeconstructionofthelikelihood T. setoutbyM10.Inthisapproach,thefinallikelihoodforthesam- The (un-normalised) likelihood of a cluster i in our sample ple of clusters and their observed properties is the product of a havingtheobservedproperties(Lˆ ,Tˆ)isgivenby PoissonlikelihoodofN totalclusters(detectedplusundetected) i i giventhemodelprediction(cid:104)N(cid:105),abinomialcoefficientaccount- (cid:90) (cid:90) dndVdΩΩ ing for the number of ways of drawing N detected clusters Pi(Lˆi,Tˆi,I|zi,θ)= dT dL dT(cid:104)Ndz(cid:105) fromthetotalN,thejointprobabilityofthesdeettofobservedclus- ter properties (the product of Eq. 7 over the N clusters), and ×P(L|T,z,θ)P(Lˆ|L)P(Tˆ|T)P(I|L,T,z). theprobabilityofnotdetectingtheremainingNde−t N clusters. det (7) Neglecting terms not dependent on the model parameters, the likelihoodsimplifiesto Theinclusionof(cid:104)N(cid:105)normalisesthetemperaturefunctionto apeprraotubraebTilitaytrdeidstsrhibifuttzi.oTnhfeorpraonbaarbbiiltirtyaroyfcPlu(Lst|eTr,zto)ihsaavsedaefitenmed- L(Lˆ,Tˆ,I|z,θ)∝e−(cid:104)Ndet(cid:105)(cid:89)Ndet (cid:10)n˜det,i(cid:11), (9) above,andtheremainingtermsaretheprobabilityofeachofthe i=1 observablesusingthemeasureduncertaintyforthatobservable. where(cid:10)n˜ (cid:11)= P(Lˆ,Tˆ)(cid:104)N(cid:105)fortheithcluster. det,i The joint probability of the full set of observed cluster proper- The principal difference between Equations 8 and 9 is that ties is the product of P(Lˆ,Tˆ) over all N observed clusters in the latter has a stronger requirement that the model must accu- i det thesample. ratelypredictthenumberofobservedclustersinadditiontotheir Equation 7 is an improper probability because P(I|L,T,z) distributionin(L,T,z).Thishastheadvantagethatitusesaddi- does not integrate to unity, and it does not penalise the model tional information to constrain the model, and is a requirement forpredictingtheexistenceofclustersinpartsoftheL,T,zspace whentheanalysisisbeingusedtoconstraincosmologyinaddi- wheretheyarewithintheselectionfunctionbutarenotobserved. tiontotheformofthescalingrelations,asinM10. Inotherwords,themodelisnotpenalisedforexcessprobability However,whentheaimoftheanalysisistoinfertheformof densityinregionsofthespacewherethelackofdetectionsdis- thescalingrelationsunderanassumedcosmology,Eq.8hascer- favourstheexistenceofclusters.Thisisresolvedbynormalising tainadvantages.Thisimplementationisinsensitivetosystemat- Eq. 7 by the integral over the observed Lˆ,Tˆ space to give the icsaffectingthenumbersofclusters(suchasthenormalisationof Articlenumber,page7of16page.16 A&Aproofs:manuscriptno.xxl_lt themassfunctionandhenceσ8)orthenormalisationoftheMT 1045 relation, and so gives more robust measurements of the scaling l relationparameters.Furthermore,sincethenumberofobserved clustersisnotusedtoconstrainthemodelparametersinthisap- pp4Trr.h4oeed.aciIlcnihktf,ieevtlhreieehcnonhcoueedmckoob.ffemrEooqfd.de7eltpewcaatresadmccoelmutesbtriesnresdcawnibtheupsreiodrassoanpoeastcehrioofr ()g-1 Ez (ergs)LTMT11004434 l llllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll themodelparameterstocomputetheposteriorprobability.The XXL500, l l l priorsusedwereuniformintherange(−∞,∞)forA ,B ,and L 1042 l l LT LT γ and uniform in the range (0.01,2.0) for σ (expressed in LT LT naturallog,sorepresentingfractionalscatter). The posterior distribution was analysed using the Bayesian 1041 inference package Laplace’s Demon2 within the R statistical 0.5 1.0 2.0 5.0 10.0 T (keV) 300kpc computingenvironment(RCoreTeam2014).Theposteriordis- tribution was first explored using a Laplace approximation for computationalefficiencybeforerefiningthefitusinganMCMC Fig.3: LXXL-T relationwiththebest-fittingmodel.Thelightbluecir- algorithm.Weusedthe“AdaptiveMetropolis-within-Gibbs”al- clesshowtheXXL-100-GCclusters;thebest-fittingmodelisshownas thesolidblacklinethe1σuncertaintyrepresentedbythegreyshaded gorithminLaplace’sDemonforthispurpose,andusedfourpar- region. allelchainsof50,000iterationseach,initialisedtorandomised startingvalues(nearthemodeoftheposterioridentifiedbythe Laplaceapproximation).Thestationarypartsofthechainswere compared using the Gelman and Rubin (1992) convergence di- agnostic, and the largest value of the 95% upper bound on the potential scale reduction factor was 1.01, giving a strong indi- cationthatthechainshadconverged.Thestationarypartsofthe 8 chainswerethenconcatenatedgivinganeffectivesamplesizeof atleast500foreachparameter. del l o m 4Tfith.the5teii.nrgLTuXhmnXecLoeL-drTtXeaXlirLn(eb-tlTialeatsiRcokaenrllaeiinstiteohp)nel.onTtthseuedmbinmesaFt-rifiigstuetirdnegb3ypaatlrhoaenmgmetweearitnhvaatlnhudeessbtaeansntd-- XXLL / L500,MT04 llllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllll lllllll lll ll lllllllll lll dard deviation of the posterior chains for each parameter. The values are given in Table 2, and illustrated with the scatterplot matrixinFig.5.WefocusourattentionontheXXLlikelihood 0.00 0.25 0.50 0.75 1.00 method, noting that the results do not change significantly be- z tween the XXL and M10 methods. Using the XXL likelihood method, we find a normalisation and slope of A =0.71±0.11 LT andB =2.63±0.15,respectively.Wefindashallowerslopethan Fig. 4: Evolution of the L -T relation for the XXL-100-GC. The LT XXL thatfoundwhenusingtheBCESregressionfit(whichdidnotac- XXL-100-GC are represented by the light blue circles and the best- count for biases, see Sect 4.1), although the difference in slope fitting model is given by the black solid line; the grey shaded region isonlysignificantatthe1σlevel.TheL -T relationisconsis- highlightsthe1σuncertainty.The“strong”and“weak”self-similarex- XXL pectations are given by the red dashed and blue dashed lines, respec- tentwithrecentresultswhichalsoaccountsforselectionbiases. tively. Bharadwaj et al. (2015) studied the LT relation of 26 groups (spanning the 0.6 < T < 3.6 keV range) and found a slope of B ≈ 2.7, converting their LT relation from bolometric to LT,B14 the soft band (see Sect. 4.7). The comparison is complicated, ters drawn from a single homogeneous survey, fully account- however, owing to the differing implementations of the bias- ing for selection biases. As introduced previously, we expect correction and fitting methods used in Bharadwaj et al. (2015) γLT=1forstrongself-similarevolution,andγLT=0.42forweak andinourwork.Interestingly,theyfindthattheslopeofthesam- self-similar evolution. We find γLT=1.64±0.74, fully consistent pleLT relationisshallowerthanthebias-correctedfit,opposite with both the the strong and weak self-similar evolution mod- to what we find in this work. However, this could be driven by els. Figure 4 plots the evolution of the LT relation as inferred thelargenumberofstrongcoolcore(SCC)systemsintheirsam- from our best-fitting model. The best-fit evolution is given by ple(≈50%);theSCCLT relationshowsachangeinslopefrom the black solid line along with the 1σ uncertainty, the strong 2.56±0.22to3.60±0.22whencorrectingforbiases. andweakself-similarexpectationsaregivenbytheredandblue Nextwefocusontheevolutionofthe L -T relation,γ , dashed lines, respectively. Our best-fit model appears to lie be- XXL LT wheretheevolutionisexpressedasE(z)γLT.Forthefirsttimewe low all the high-redshift clusters, due to the fit being driven by areabletomeasuretheevolutionofthe LT relationusingclus- thelargernumberoflowerredshiftclusters.However,theevolu- tionfavouredbyourbest-fitmodelisintensionwithotherrecent 2 http://www.bayesian-inference.com/software results,andisdiscussedfurtherinSect.5.1. Articlenumber,page8of16page.16 P.Gilesetal.:XXLIII:LT RelationoftheBrightClusterSample 2.2 2.6 3.0 3.4 −2 −1 0 1 2 3 4 2 A 1. LT 0 1. 8 0. 6 0. 4 4 0. 3. B LT 0 3. 0.46 6 2. 2 2. 8 s 0. LT 7 0. 6 0.19 0.50 0. 5 0. 4 0. 3 0. 4 g 3 LT 2 1 0.69 0.59 0.40 0 1 − 2 − 0.4 0.6 0.8 1.0 1.2 0.3 0.4 0.5 0.6 0.7 0.8 Fig. 5: Scatterplotmatrixforthefitofthe L -T relationoftheXXL-100-GCsample.Theposteriordensitiesareshownalongthediagonal; XXL the1σ,2σ,and3σconfidencecontoursforthepairsofparametersareshownintheupperrightpanels.ThelowerleftpanelsshowthePearson’s correlationcoefficientforthecorrespondingpairofparameters(textsizeisproportionaltothecorrelationstrength). 4.6. Posteriorpredictivechecks simulatedandobservedproperties.Inallcasesthevisualagree- mentisgood. While the modelling process described above determines the Thesecondtestwastocomparethenumberofclusterspre- best-fittingvaluesofthemodelparametersforthechosenmodel, dictedtobeobservedbythebest-fittingmodelwiththenumber itdoesnotguaranteethatthemodelisagooddescriptionofthe observed.Themodelpredicted116.7clusters,sotheprobability data.Forthisweusethreetypesofposteriorpredictivecheckto of observing a sample as discrepant as or more discrepant than assesshowwellthefinalmodeldescribesthedata. the99 clustersobservedisthe sumofthePoisson probabilities P(N < 99|116.7)+ P(N > 134|116.7) = 0.11. The number of First,wemakevisualcomparisonsofthepopulationofclus- observedclustersisthusreasonablyconsistentwiththenumber ters predicted by the best-fitting model and those observed. To predicted. We emphasise that this is not trivial, as the model is do this, a large number of model clusters were generated using notrequiredtoreproducethenumberofobservedclustersinour the best-fitting model parameters, and including the selection likelihood. function. Figure 6 shows contours of the simulated population along with the observed clusters in different projections of the Thefinaltestwemakeistomeasuretheposteriorpredictive observedproperties.Figure7,instead,showshistogramsofthe p-value(Meng1994).Thisisdonebymeasuringthediscrepancy Articlenumber,page9of16page.16
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