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SSRvmanuscriptNo. (willbeinsertedbytheeditor) Thermodynamical properties of the ICM from hydrodynamical simulations S.Borgani · A.Diaferio · K.Dolag · S.Schindler 8 0 0 2 n Received:1October2007;Accepted:8November2007 a J 7 Abstract Modern hydrodynamical simulations offer nowadays apowerful means to trace the evolution of the X–ray properties of the intra–cluster medium (ICM) during the cos- ] mological history of the hierarchical build up of galaxy clusters. In this paper we review h the current status of these simulations and how their predictions fare in reproducing the p - mostrecentX–rayobservationsofclusters.Afterbrieflydiscussingtheshortcomingsofthe o self–similar model, based on assuming that gravity only drives the evolution of the ICM, r t we discuss how the processes of gas cooling and non–gravitational heating are expected s tobring model predictions into betteragreement withobservational data.Wethenpresent a [ resultsfromthehydrodynamical simulations,performedbydifferentgroups,andhowthey compare withobservational data.Astermsofcomparison, weuseX–ray scalingrelations 1 betweenmass,luminosity,temperatureandpressure,aswellastheprofilesoftemperature v andentropy.Theresultsofthiscomparisoncanbesummarisedasfollows:(a)simulations, 2 3 whichincludegascooling,starformationandsupernovafeedback,aregenerallysuccessful 0 in reproducing the X–ray properties of the ICM outside the core regions; (b) simulations 1 generally fail in reproducing the observed “cool core” structure, in that they have serious . difficultiesinregulatingovercooling,therebyproducingsteepnegativecentraltemperature 1 0 profiles.Thisdiscrepancycallsfortheneedofintroducingotherphysicalprocesses,suchas 8 energyfeedbackfromactivegalacticnuclei,whichshouldcompensatetheradiativelosses 0 : v S.Borgani i DepartmentofAstronomy,UniversityofTrieste,viaTiepolo11,I-34143Trieste,Italy X E-mail:[email protected] INAF–NationalInstituteforAstrophysics,Trieste,Italy r a INFN–NationalInstituteforNuclearPhysics,SezionediTrieste,Italy A.Diaferio DipartimentodiFisicaGenerale“AmedeoAvogadro”,Universita`degliStudidiTorino,Torino,Italy E-mail:[email protected] INFN–NationalInstituteforNuclearPhysics,SezionediTorino,Italy K.Dolag Max-Planck-Institutfu¨rAstrophysik,Karl-SchwarzschildStrasse1,GarchingbeiMu¨nchen,Germany E-mail:[email protected] S.Schindler Institutfu¨rAstro-undTeilchenphysik,Universita¨tInnsbruck,Technikerstr.25,6020Innsbruck,Austria E-mail:[email protected] 2 ofthegaswithhighdensity,lowentropyandshortcoolingtime,whichisobservedtoreside intheinnermostregionsofgalaxyclusters. Keywords Cosmology: numerical simulations · galaxies: clusters · hydrodynamics · X–ray:galaxies 1 Introduction Clustersofgalaxiesformfromthecollapseofexceptionallyhighdensityperturbationshav- ing typical size of ∼10 Mpc in a comoving frame. As such, they mark the transition be- tweentwodistinctregimesinthestudyoftheformationofcosmicstructures.Theevolution of structures involving larger scales is mainly driven by the action of gravitational insta- bilityofthedarkmatter(DM)densityperturbations and,assuch,itretainsthememoryof the initial conditions. Onthe other hand, galaxy–sized structures, which form from initial fluctuationsonscalesof∼1Mpc,evolveunderthecombinedactionofgravityandofcom- plexgas–dynamicalandastrophysicalprocesses.Onsuchscales,gascooling,starformation andthesubsequentreleaseofenergyandmetalfeedbackfromsupernovae(SN)andactive galacticnuclei(AGN)haveadeepimpactontheobservationalpropertiesofthediffusegas andofthegalaxypopulation. Inthissense,clustersofgalaxiescanbeusedasinvaluablecosmologicaltoolsandastro- physicallaboratories(seeRosatietal.2002andVoit2005forreviews).Thesetwoaspects are clearly interconnected with each other. From the one hand, the evolution of the pop- ulation ofgalaxy clusters and theiroverall baryonic content provide inprinciple powerful constraintsoncosmologicalparameters.Ontheotherhand,forsuchconstraintstoberobust, one has to understand in detail the physical properties of theintra–cluster medium (ICM) anditsinteractionwiththegalaxypopulation. The simplest model to predict the properties of the ICM and their evolution has been proposedbyKaiser(1986).Thismodelisbasedontheassumptionthattheevolutionofthe thermodynamicalpropertiesoftheICMisdeterminedonlybygravity,withgasheatedtothe virialtemperatureofthehostingDMhalosbyaccretionshocks(Bykovetal.2008-Chapter 7,thisvolume). Sincegravitational interactiondoesnotintroduce anypreferredscale,this modelhasbeencalled“self–similar”1.Asweshalldiscuss,thismodelprovidesprecisepre- dictionsontheshapeandevolutionofscalingrelationsbetweenX–rayluminosity,entropy, totalandgasmass,whichhavebeentestedagainstnumericalhydrodynamical simulations (e.g.,Ekeetal.1998;Bryan&Norman1998).Thesepredictionshavebeenrecognisedfor severalyearstobeatvariancewithanumberofobservations.Inparticular,theobservedre- lationbetweenX–rayluminosityandtemperature(e.g.Markevitch1998;Arnaud&Evrard 1999; Osmond&Ponman 2004) is steeper and the measured level of gas entropy higher thanexpected(e.g.,Ponmanetal.2003;Pratt&Arnaud2005),especiallyforpoorclusters and groups. This led to the concept that more complex physical processes, related to the heatingfromastrophysicalsources ofenergyfeedback, andradiativecooling ofthegasin thecentral clusterregions, play akeyroleindetermining theproperties ofthediffuse hot baryons. 1 Strictlyspeaking,self–similarityalsorequiresthatnocharacteristicscalesarepresentintheunderlying cosmologicalmodel.ThismeansthattheUniversemustobeytheEinstein–de-Sitterexpansionlawandthat theshapeofthepowerspectrumofdensityperturbationsisafeaturelesspowerlaw.Inanycase,theviolation ofself–similarityintroducedbythestandardcosmologicalmodelisnegligiblewithrespecttothatrelatedto thenon–gravitationaleffectsactingonthegas. 3 Althoughsemi–analyticalapproaches(e.g.,Tozzi&Norman2001;Voit2005,andref- erences therein) offer invaluable guidelines to this study, it is only with hydrodynamical simulationsthatonecancapturethefullcomplexityoftheproblem,soastostudyindetail theexistinginterplaybetweencosmologicalevolutionandtheastrophysicalprocesses. Inthelastyears,everimproving codeefficiencyandsupercomputing capabilitieshave opened the possibility to perform simulations over fairly large dynamical ranges, thus al- lowing to resolve scales of a few kiloparsecs (kpc), which are relevant for the formation of single galaxies, while capturing theglobal cosmological environment on scales of tens or hundreds of Megaparsecs (Mpc), which are relevant for the evolution of galaxy clusters. Starting from first attempts, in which only simplified heating schemes were studied (e.g.,Navarroetal.1995;Bialeketal.2001;Borganietal.2002),anumberofgroupshave studied the effect of introducing alsocooling (e.g. Katz&White 1993; Lewisetal. 2000; Muanwongetal.2001;Dave´ etal.2002;Tornatoreetal.2003),ofmorerealisticsourcesof energyfeedback(e.g.,Borganietal.2004;Kayetal.2007;Nagaietal.2007a;Sijackietal. 2007),ofthermalconduction(e.g.,Dolagetal.2004),andofnon–thermalpressuresupport frommagneticfields(e.g.,Dolagetal.2001)andcosmicrays(e.g.,Pfrommeretal.2007). Inthispaper,wewillreviewtherecentadvancementperformedinthisfieldofcomputa- tionalcosmologyandcriticallydiscussthecomparisonbetweensimulationpredictionsand observations, byrestrictingthediscussiontothethermal effects.Assuch,thispapercom- plements thereviews by Borganietal. 2008 -Chapter 18, this volume, whichreviews the studyoftheICMchemicalenrichmentandbyDolagetal.2008b-Chapter15,thisvolume, which reviews the study of the non–thermal properties of the ICM from simulations. We refertothereviewsbyDolagetal.2008a-Chapter12,thisvolumeforadescriptionofthe techniquesofnumericalsimulationsandbyKaastraetal.2008-Chapter9,thisvolumefor anoverviewoftheobservedthermalpropertiesoftheICM. The scheme of the presentation is as follows. In Sect. 2 we briefly discuss the self– similarmodeloftheICMandhowtheactionofnon–gravitational heatingandcoolingare expectedtoalterthepredictions ofthismodel.Sect.3and4overviewtheresultsobtained onthecomparisonbetweenobservedandsimulatedscalingrelationsandprofilesofX–ray observablequantities,respectively.InSect.5wesummariseandcriticallydiscusstheresults presented. 2 ModellingtheICM 2.1 Theself–similarscaling The simplest model to predict the observable properties of the ICM is based on the as- sumptionthatgravityonlydeterminesthethermodynamicalpropertiesofthehotdiffusegas (Kaiser1986).Sincegravitydoesnothavepreferredscales,weexpectclustersofdifferent sizestobethescaledversionofeachother.Thisisthereasonwhythismodelhasbeencalled self-similar. If,atredshiftz,wedefineMD c tobethemasscontainedwithintheradiusrD c,encom- passing a mean density D c times the critical density r c(z), then MD c (cid:181) r c(z)D crD3c. The criticaldensityoftheuniversescaleswithredshiftasr (z)=r E2(z),whereE(z)isgiven c c0 by E(z) = (1+z)3W m+(1+z)2W k+W L 1/2, (1) (cid:2) (cid:3) 4 whereW mandW L arethedensityparametersassociatedtothenon–relativisticmatterandto thecosmologicalconstant,respectively,W k=1−W m−W L andweneglectanycontribution fromrelativisticspecies. Therefore, theclustersizerD c scaleswithzand MD c as rD c (cid:181) MD1/c3E−2/3(z),sothat, assuminghydrostaticequilibrium,clustermassscaleswithtemperatureT as MD c (cid:181) T3/2E−1(z). (2) Ifr isthegasdensity,thecorrespondingX–rayluminosityis gas r 2 L = gas L (T)dV, (3) X ZV(cid:18)m mp(cid:19) whereL (T)(cid:181) T1/2 for pure thermal Bremsstrahlung emission. Ifgasaccretes along with DMbygravitationalinstabilityduringtheformationoftheclusterhalo,thenweexpectthat r (r)(cid:181) r (r),sothat gas DM LX (cid:181) MD cr cT1/2 (cid:181) T2E(z). (4) Another useful quantity characterising the thermodynamical properties of the ICM is theentropy(Voit2005)which,inX–raystudiesoftheICM,isusuallydefinedas k T B K = , (5) m m r 2/3 p gas wherek istheBoltzmannconstant,m themeanmolecularweight(≃0.58foraplasmaof B primordialcomposition)andm theprotonmass.Withtheabovedefinition,thequantityK p istheconstantofproportionality intheequationofstateofanadiabaticmono-atomicgas, P=Kr 5/3.Usingthethermodynamicdefinitionofspecificentropy,s=c ln(P/r 5/3)(c : gas V gas V heat capacity at constant volume), one obtains s=k lnK3/2+s , where s is aconstant. B 0 0 Anotherquantity,oftencalled“entropy”intheclusterliterature,whichwewillalsousein thefollowing,is S = k Tn−2/3, (6) B e wheren istheelectronnumberdensity.Accordingtotheself–similarmodel,thisquantity, e computedatafixedoverdensityD ,scaleswithtemperatureandredshiftaccordingto c SD c (cid:181) T(1+z)−2. (7) Asalreadymentioned intheintroduction, anumberofobservational factsfromX–ray data point against the simple self–similar picture. The steeper slope of the L –T relation X (Markevitch1998;Arnaud&Evrard1999;Osmond&Ponman2004),L (cid:181) Ta witha ≃3 X for clustersand possiblylarger forgroups, theexcess entropy inpoor clusters andgroups (Ponmanetal. 2003; Pratt&Arnaud 2005; Piffarettietal. 2005) and the decreasing trend ofthegasmassfractioninpoorersystems(Linetal.2003;Sandersonetal.2003)allpoint towardthepresenceofsomemechanismwhichsignificantlyaffectstheICMthermodynam- ics. 5 Fig.1 Profilesofreducedentropy,S/T,fornon–radiative simulations(fromBorganietal.2001).Theleft panelisforsimulationsincludingonlygravitationalheating,thecentralpanelisforrunsincludingSNfeed- backaspredictedbyasemi–analyticalmodelofgalaxyformationandtherightpaneliswithpre–heatingwith anentropythresholdatredshiftz=3.Solid,short–dashedandlong–dashedcurvesarefora3keVcluster, fora1keVgroupandfora0.5keVgroup,respectively.Thedottedstraightlineintheleftpanelshowsthe analyticalpredictionbyTozzi&Norman(2001)fortheentropyprofileassociatedtogravitationalheating. 2.2 HeatingandcoolingtheICM The first mechanism, that has been introduced to break the ICM self–similarity, is non– gravitationalheating(e.g.Evrard&Henry1991;Kaiser1991;Tozzi&Norman2001).The ideaisthatbyincreasingthegasentropywithagivenextraheatingenergypergasparticle E prevents gasfromsinkingtothecentreofDMhalos,therebyreducing gasdensityand h X–ray emissivity. This effect will be large for small systems, whose virial temperature is k T .E , while leaving rich clusters with k T ≫E almost unaffected. Therefore, we B h B h expectthat theX–rayluminosityandgascontent arerelativelymoresuppressedinpoorer systems,thusleadingtoasteepeningoftheL –T relation. X Thenotionofnon–gravitationalheatinghasbeenfirstimplementedinnon–radiative(i.e. neglectingtheeffectofcooling)hydrodynamicalsimulationsbyeitherinjectingentropyin animpulsivewayatagivenredshift(Navarroetal.1995;Bialeketal.2001),orbyadding energy in a redshift–modulated way, so as to mimic the rate of SN explosions from an external model of galaxy formation (Borganietal. 2002). In Fig. 1 we show the different efficiencythatdifferentheatingmechanismshaveinbreakingtheself–similarbehaviourof theentropyprofilesinobjectsofdifferentmass,rangingfromaVirgo–likeclustertoapoor galaxy group. According to the self–similar model, the profiles of reduced entropy, S/T, shouldbeindependent oftheclustermass.Thisisconfirmed bytheleftpanel,whichalso shows that these profiles have a slope consistent with that predicted by a model in which gas is shock heated by spherical accretion in a DM halo, under the effect of gravity only (Tozzi&Norman2001).Thecentralpanelshowsinsteadtheeffectofaddingenergyfrom SN, whose rate is that predicted by a semi–analytical model of galaxy formation. In this case, which corresponds to a total heating energy of about 0.3 keV/particle, the effect of extra heating starts being visible, but only for thesmaller system. It is only withthe pre– heatingscheme,basedonimposinganentropy floorof50keVcm2,thatself–similarityis clearly broken. While this heating scheme is effective in reproducing the observed L –T X relation,itproduceslargeisentropiccores,apredictionwhichisatvariancewithrespectto observations(e.g.,Donahueetal.2006). 6 Fig.2 Leftpanel:therelationbetweenentropyandtemperatureforgashavingafixedvalueofthecooling time(fromVoit&Bryan2001).Thecrosseswitherrorbarsareobservationaldataontheentropymeasured atone–tenth ofthevirial radiusforclusters andgroupsbyPonmanetal.(1999).Rightpanel:acompari- sonbetweenobservations(errorbarswithcrosses;Ponmanetal.(1999)andsimulations,includingradiative coolingandstarformation(numbers),fortheentropyinthecentralregionsofgalaxygroupsandclusters. Thesolidlineshowsthepredictionoftheself–similarmodel(fromDaveétal.2002). Although it may look like a paradox, radiative cooling has been also suggested as a possible alternative to non–gravitational heating to increase the entropy level of the ICM andsuppressingthegascontentinpoorsystems.AsoriginallysuggestedbyVoit&Bryan (2001),coolingprovidesaselectiveremovaloflow–entropygasfromthehotX–rayemitting phase(seealsoWu&Xue2002).Asaconsequence,whiletheglobalentropyofthebaryons decreases,theentropyoftheX–rayemittinggasincreases.Thisisillustratedintheleftpanel ofFig.2(fromVoit&Bryan2001).Inthisplot,eachofthetwocurvesseparatestheupper portion of the entropy–temperature plane, where the gas has cooling time larger than the ageofthesystem,fromthelowerportion, wheregaswithshortcoolingtimeresides.This impliesthat onlygashaving arelativelyhigh entropy willbeobservedasX–rayemitting, whilethelow–entropygaswillbeselectivelyremovedbyradiativecooling.Thecomparison withobservationaldataofclustersandgroups,alsoreportedinthisplot,suggeststhattheir entropylevelmaywellbetheresultofthisremovaloflow–entropygasoperatedbyradiative cooling.Thisanalyticalpredictionhasbeenindeedconfirmedbyradiativehydrodynamical simulations. The right panel of Fig. 2 shows the results of the simulations by Dave´ etal. (2002)onthetemperaturedependence ofthecentralentropy ofclustersandgroups. Quite apparently,theentropylevelinsimulationsiswellabovethepredictionoftheself–similar model,byarelativeamountwhichincreaseswithdecreasingtemperature,andinreasonable agreementwiththeobservedentropylevelofpoorclustersandgroups. Althoughcoolingmaylooklikeanattractivesolution,itsuffersfromthedrawbackthat atoolargefractionofgasisconvertedintostarsintheabsenceofasourceofheatingenergy whichregulatesthecoolingrunaway.Indeed,whileobservationsindicatethatonlyabout10 per cent of the baryon content of aclusteris inthe stellarphase(e.g., Baloghetal. 2001; Linetal. 2003), radiative simulations, like those shown in Fig. 2, convert into stars up to ∼50percentofthegas. 7 Anotherparadoxical consequenceofcoolingisthatitincreasesthetemperature ofthe hotX–rayemittinggasatthecentreofclusters.ThisisshownintheleftpanelofFig.3(from Tornatoreetal.2003),whichcomparesthetemperatureprofilesforthenon–radiativerunof aVirgo–like clusterwithavariety ofradiative runs, based on different ways of supplying non–gravitational heating. The effect of introducing cooling is clearly that of steepening thetemperatureprofilesinthecoreregions,whileleavingitunchangedatlargerradii.The reasonforthisisthatcoolingcausesalackofcentralpressuresupport. Asaconsequence, gas starts flowing in sub-sonically from more external regions, thereby being heated by adiabaticcompression.AsweshalldiscussinSect.4,thisfeatureofcoolingmakesitquite difficulttoreproducethestructureofthecoolcoresobservedingalaxyclusters. Steepening of the central temperature profiles and overcooling are two aspects of the same problem. In principle, the solution to this problem should be provided by a suitable scheme of gas heating which regulates star formation, while maintaining pressurised gas in the hot phase. The right panel of Fig. 3 compares the temperature–density phase dia- gramsforgasparticleslyinginthecentralregionofanSPH–simulatedcluster,whenusing twodifferentfeedbackefficiencies.Thetwosimulationsincludecooling,starformationand feedbackintheformofgalacticwindspoweredbySNexplosions,followingtheschemein- troducedbySpringel&Hernquist(2003a).Theupper(green)andthelower(red)cloudsof hightemperatureparticlescorrespondtoawindvelocityof500kms−1 andof1000kms−1, respectively. This plot illustrates another paradoxical effect: in the same way that cooling causes anincreaseofthetemperature ofthehot phase, supplying energy withanefficient feedbackcausesadecreaseofthetemperature.Thereasonforthisisthatextraenergycom- pensatesradiativelosses,therebymaintainingthepressuresupportforgaswhichwouldoth- erwisehaveaveryshortcoolingtime,therebyallowingittosurviveonaloweradiabat.Itis alsoworth reminding that cooling efficiencyincreaseswiththenumerical resolution (e.g., Baloghetal.2001;Borganietal.2006).Therefore,forafeedbackmechanismtoworkprop- erly, it should be abletostabilisethecooling efficiency inawaywhich is independent of resolution. In the light of these results, it is clear that the observed lack of self–similarity in the X–rayproperties ofclusters cannot besimplyexplained onthegrounds ofasingleeffect. The emerging picture is that the actionof cooling and offeedback energy, e.g. associated to SN explosions and AGN, should combine in a self–regulated way. As we shall discuss inthefollowingsections,hydrodynamicalsimulationsofgalaxyclustersinacosmological context demonstrate that achieving this heating/cooling balanceisnot easyandrepresents nowadaysoneofthemostchallengingtasksinthenumericalstudyofclusters. Asanexample,weshowinFig.4howthegasdensityofasimulatedclusterchanges, both at z=2 (left panels) and at z=0 (right panels), when cooling and star formation arecombinedwithdifferentformsofnon–gravitational heating(fromBorganietal.2005). Thecomparisonofthetopandcentralpanelsshowstheeffectofincreasingthekineticen- ergy carried by galacticoutflows by afactor of six. Thestronger winds are quiteefficient in stopping star formation in the small halos, which are washed out, and make the larger onesslightlypuffier,whilepreservingthegeneralstructureofthecosmicwebsurrounding theLagrangianclusterregion.Comparingthetopandthebottompanelsshowsinsteadthe effect ofadding togalacticwinds alsotheeffect ofan entropy floor. Although the energy budgetofthefeedbackschemesofthecentralandbottompanelsarequitecomparable,the effectofthegasdistributionisradicallydifferent.Imposing anentropyflooratz=3with animpulsiveheatinggeneratesamuchsmoothergasdensitydistribution,bothatz=2and atz=0.Inthiscase,thefilamentarystructureofthegasdistributioniscompletelyerased, while only the largest halos areable toretain part of their gas content. This demonstrates 8 Fig.3 Leftpanel:temperatureprofilesfromhydrodynamicalsimulationsofa∼3keVgalaxycluster.Inall panelsthedottedandthesolidcurvescorrespondtoanonradiativerunandtoarunincludingcoolingand starformation.Theothercurvesarefordifferentrecipesofgasheating(fromTornatoreetal.2003).Right panel:therelationbetweentemperatureandoverdensityforgasparticleswithin0.1r200forSPHsimulations ofacluster ofmass≃1014h−1M⊙.Upper(green) points are forarunwhich includes feedback through galactic windswithavelocity of500 kms−1,whilethelower(red)pointsareforarunbasedonassum- ingstrongerwinds(sw),withatwiceaslargevelocity.Bothrunsincludeamodelofchemicalenrichment (Borganietal.2008-Chapter18,thisvolume)whichassumesanInitialMassFunctionforstarformationby Arimoto&Yoshii1987(AY).Thepointsinthebottomrightcornerarestar–forminggasparticles. thatafixedamountofenergyfeedbackcanprovidelargelydifferentresultsontheICMther- modynamicalproperties,dependingontheepochandonthedensityatwhichitisreleased. 3 Scalingrelations So far, we have qualitatively discussed how simple models of pre–heating and radiative cooling canreproduce theobserved violation ofself–similarity in theX–ray properties of galaxyclusters.Inthisandinthefollowingsectionswewillfocusthediscussiononamore detailedcomparisonbetweensimulationresultsandobservationaldata,andontheimplica- tions ofthis comparison onour current understanding ofthefeedback mechanisms which regulatestarformationandtheevolutionofthegalaxypopulation.Asastartingpointforthe comparisonbetweenobservedandsimulatedX–rayclusterproperties,wedescribehowob- servablequantitiesarecomputedfromhydrodynamical simulationsandhowtheycompare totheanalogousquantitiesderivedfromobservationaldata. AsfortheX–rayluminosity,itiscomputedbysummingthecontributions totheemis- sivity, e , carried by all the gas elements (particles in a SPH run and cells in an Eulerian i grid–based run), L =(cid:229) e , where the sum extends over all the gas elements within the X i i regionwhereL iscomputed.Thecontributionfromthei-thgaselementisusuallywritten X as e = n n L (T,Z)dV , (8) i e,i H,i i i i wheren andn arethenumberdensitiesofelectronsandofhydrogenatoms,respectively, e,i H,i associated to the i-th gas element of given density r , temperature T and metallicity Z. i i i 9 Fig. 4 The maps of the gas density for simulations of a Virgo–like cluster at z=2 and z=0 (left and rightpanels,respectively),includingcooling,starformationanddifferentformsofnon–gravitationalheating. Upperpanelsareforarunwhichincludes galactic windswithavelocity ofabout340kms−1,thecentral panelsisforgalacticwindswithvelocityofabout830kms−1 andthebottompanelsisforgalacticwinds asinthetoppanels,butalsoaddingapre–heatingwithanentropythresholdof100keVcm2 atz=3.At z=0thesizeoftheboxisof11.7h−1Mpc,whileatz=2iscorrespondsto17.5h−1Mpccomoving(from Borganietal.2005). 10 Furthermore,L (T,Z)isthetemperature–andmetallicity–dependentcoolingfunction(e.g., Sutherland&Dopita1993)computedwithinagivenenergyband,whiledV =m/r isthe i i i volumeofthei-thgaselement,havingmassm. i Asforthetemperature,differentproxiestoitsX–rayobservationaldefinitionhavebeen proposed in the literature, which differ from each other in the expression for the weight assignedtoeachgaselement.Ingeneral,theICMtemperaturecanbewrittenas (cid:229) wT i i T = i , (9) (cid:229) w i i whereT isthetemperatureofthei-gaselement,whichcontributeswiththeweightw.The i i mass–weighted definition of temperature, T , is recovered for w =m (m: mass of the mw i i i i-thgaselement),whichalsocoincideswiththeelectrontemperatureT forafullyionised e plasma. A moreobservation–oriented estimateoftheICMtemperature isprovided bythe emission–weighteddefinition, T ,whichisobtainedforw =e (e.g.,Evrardetal.1996). ew i i Theideaunderlyingthisdefinitionisthateachgaselementshouldcontributetotheoverall spectrumaccordingtoitsemissivity. Mazzottaetal.(2004)pointed outthatthethermalcomplexity oftheICMissuchthat the overall spectrum is given by the superposition of several single–temperature spectra, eachoneassociatedtoonethermalphase.Inprinciple,thesuperpositionofseveralsingle– temperaturespectracannotbedescribedbyasingle–temperaturespectrum.However,when fitting it to a single–temperature model in a typical finite energy band, where X–ray tele- scopes are sensitive, the cooler gas phases are relatively more important in providing the high–energy cut–offofthespectrumand,therefore, indetermining thetemperatureresult- ing from the spectral fit. In order to account for this effect, Mazzottaetal. (2004) introduced a spectroscopic–like temperature, T , which is recovered from Eq. 9 by using the sl weightw =r mTa −3/2.Byusinga =0.75,thisexpressionforT wasshowntoreproduce i i i sl withinfewpercentthetemperatureobtainedfromthespectroscopicfit,atleastforclusters withtemperatureabove2–3keV.Morecomplexfittingexpressionshavebeenprovidedby Vikhlinin(2006),whogeneralisedthespectroscopic–liketemperaturetothecasesoflower temperatureandarbitrarymetallicity. 3.1 Theluminosity–temperaturerelation TheL –T relationrepresentedthefirstobservationalevidenceagainsttheself–similarmodel. X Thisrelationhasbeenshownbyseveralindependentanalysestohaveaslope,L (cid:181) Ta ,with X a ≃3forT &2keV(e.g.,Whiteetal.1997),withindicationsforaflatteningtoa .2.5for themostmassivesystems(Allen&Fabian1998).Thescatterinthisrelationislargelycon- tributedbythecool–coreemission,sothatitsignificantlydecreaseswhenexcisingthecores (Markevitch 1998) or removing cool–core systems (Arnaud&Evrard 1999). A change of behaviour isalsoobserved at thescalesof groups, T .2keV, whichgenerally displays a verylargescatter(Osmond&Ponman2004). HydrodynamicalsimulationsbyBialeketal.(2001)andbyBorganietal.(2002)demon- stratedthatsimplepre–heatingmodels,basedontheinjectionofentropyatrelativelyhigh redshift,canreproducetheobservedslopeoftheL –T relation.Dave´ etal.(2002)showed X thatasimilarresultcanalsobeachievedwithsimulationsincludingcoolingonly,theprice to be paid being a large overcooling. Muanwongetal. (2002) and Tornatoreetal. (2003)