PreprinttypesetusingLATEXstyleemulateapjv.10/10/03 EVOLUTION OF THE CLUSTER X-RAY LUMINOSITY FUNCTION C.R. Mullis1, A. Vikhlinin2,3, J.P. Henry4, W. Forman2, I.M. Gioia5, A. Hornstrup6, C. Jones2, B.R. McNamara7, and H. Quintana8 To appear in the Astrophysical Journal ABSTRACT We report measurements of the cluster X-ray luminosity function out to z = 0.8 based on the final sample of 201 galaxy systems from the 160 Square Degree ROSAT Cluster Survey. There is little evidence for any measurable change in cluster abundance out to z ∼ 0.6 at luminosities less than a few times 1044 h−2 ergs s−1 (0.5–2.0 keV). However,between0.6<z <0.8 andatluminosities above 50 4 1044 h−2 ergs s−1, the observedvolume densities are significantly lowerthan those ofthe present-day 50 0 population. Wequantifythisclusterdeficitusingintegratednumbercountsandamaximum-likelihood 0 analysis of the observed luminosity-redshift distribution fit with a model luminosity function. The 2 negative evolution signal is > 3σ regardless of the adopted local luminosity function or cosmological n framework. Our results and those from several other surveys independently confirm the presence a of evolution. Whereas the bulk of the cluster population does not evolve, the most luminous and J presumably most massivestructures evolveappreciably between z =0.8 andthe present. Interpreted 9 in the context of hierarchicalstructure formation,we are probing sufficiently large mass aggregations 2 atsufficientlyearlytimesincosmologicalhistorywheretheUniversehasyettoassembletheseclusters to present-day volume densities. 1 Subject headings: cosmology: observations — galaxies: clusters: general — X-rays: general v 5 0 1. INTRODUCTION lection functions, it is straight-forward to convert these 6 numbercountsintovolume-normalizedmeasuressuchas 1 Galaxy clusters form via the gravitational amplifica- the luminosity function. Finally, given the strong corre- 0 tionofrare,highpeaksinthecosmicmatterdensityfield. lation between X-ray luminosity and cluster mass, it is 4 The redshift evolution of cluster abundance depends on possible to transform the observed XLF into the cluster 0 thegrowthrateofdensityperturbationswhichis,inturn, mass function which is the fundamental relation in the / sensitive to the mean cosmic matter density (Ω ) and, h M theoretical treatment. to a lesser extent, the dark-energy density (Ω ). Thus p Λ Though the cluster XLF was first measured with observationsof cluster space density with sufficient tem- - an X-ray flux-limited sample over two decades ago o poral sampling can provide powerful cosmological con- r straints(e.g.,Oukbir & Blanchard1992;Eke et al.1998; (Piccinotti et al. 1982), a definitive characterization of t its evolution has proven very difficult. The latter is a s Bahcall et al. 1999). a particularly important test in observational cosmology A directly measurable androbustdiagnostic of cluster : because the variation of the cluster XLF as a function v abundance is the X-ray luminosity function (XLF), that ofredshiftreflectstheevolutionoftheclustermassfunc- i is the volume density of clusters per luminosity interval. X tion. SuchameasureallowsforstrongconstraintsonΩ X-rayselectedgalaxyclustersareparticularlywell-suited M r forthis type ofanalysis. Clusters areefficiently detected even accounting for uncertainty in the luminosity-mass a relation(e.g.,Borgani et al.2001). Earlytheoreticalpre- at X-ray wavelengths to high redshift (currently out to dictions (e.g., Kaiser 1986) postulated dramatic positive z ∼ 1.25) thus providing the leverage for evolutionary evolution in the XLF where the volume density of clus- studies(seereviewbyRosati, Borgani, & Norman2002). ters of a fixed luminosity would increase with redshift. The resulting samples feature high statistical complete- This would be an “observer-friendly” universe since the nesswhichisclearlyimportantforderivingreliablenum- lossinsensitivityathighredshiftinaflux-limitedsurvey ber counts. Since X-ray surveys have well-defined se- would be offset by the growing population of detectable 1 European Southern Observatory, Headquarters, Karl- sources. Contrary to this scenario, observations of the Schwarzschild-Strasse 2, Garching bei Mu¨nchen D-85748, Ger- clusterXLFrangefromzeroevolutionto negativeevolu- many,[email protected] tiondepending onthe redshiftsandluminositiesprobed. 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Thesefindingsareconsistentwithcurrentpredictionsfor Street,Cambridge,MA02138,USA 3 SpaceResearchInstitute,Profsoyuznaya84/32,Moscow,Rus- a low-density universe where mild negative evolution is sia restricted to the most luminous, high-redshift clusters 4InstituteforAstronomy,UniversityofHawai‘i,2680Woodlawn while there is little change in the bulk of the population Drive,Honolulu,HI96822,USA (e.g., Borgani & Guzzo 2001, and references therein). 5 Istituto di Radioastronomia del CNR-INAF,viaGobetti 101, Bologna,I-40129,Italy Taking advantage of the first time an X-ray sur- 6 Danish Space Research Institute, Juliane Maries Vej 30, vey extended into cosmologically interesting redshifts, Copenhagen 0,DK-2100,Denmark Gioia et al.(1990a)andHenry et al.(1992)used67clus- 7 Department of Physics and Astronomy, Ohio University, ters (0.14 < z < 0.60) from the pioneering Einstein Ex- Athens,OH45701, USA 8 DepartamentodeAstronomiayAstrofisica,PontificiaUniver- tended Medium Sensitivity Survey (EMSS; Gioia et al. sidadCatolicadeChile,Casilla306,Santiago, 22,Chile 1990b; Stocke et al. 1991; Maccacaro et al. 1994) to 2 MULLIS ET AL. 1045 1] − s s g r e 2 − 01044 5 h ) [ V e k 0 2. − 5 0. 1043 ( x L 160SD EMSS BCS 1042 0.0 0.2 0.4 0.6 0.8 z Fig. 1.—X-rayluminosityand redshiftdistribution of the 160SD (Mullisetal. 2003), the EMSS(Gioia&Luppino 1994 withupdates fromthe literature, e.g.,Henry 2000), and the BCS(Ebeling etal.1998)cluster samples. The dotted curves (left to right)areindicative fluxlimitsof2.7×10−12,1.5×10−13,and3×10−14 ergscm−2 s−1(0.5–2.0keV). TheEMSSandBCSluminositieswereconverted to thisenergybandassumingaRaymond-Smithplasmaspectrum(Raymond&Smith1977)withametallicityof0.3solarandthereported gastemperature(eitherdirectlymeasuredorestimatedfromtheluminosity-temperaturerelation). make the first detection of evolution in the observed (Burke et al. 2003), and WARPS (Scharf et al. 1997; XLF. Based on a steepening of the high-redshift lumi- Perlman et al. 2002). Complementary surveys at low nosity function, they found a deficit of high-luminosity redshifts (z . 0.3) yielded accurate determinations of clusters at z > 0.3 with a statistical significance of ap- the local luminosity function, thus providing the crucial proximately 3σ. Though most subsequent investigations baselineforthedetectionofredshiftevolution. Theselo- corroborate these findings, some questions have been calsurveysinclude the BCS+eBCS(Ebeling et al. 1998, raised concerning the reliability of the EMSS evolution 2000a), RASS1BS (De Grandi et al. 1999a), and RE- detection (e.g., Nichol et al. 1997; Ebeling et al. 2000b; FLEX(B¨ohringer et al.2001). Fromtheshearnumberof Ellis & Jones2002;Lewis et al.2002). Ofhistoricalnote, projectsitshouldbeclearthatROSAT wasawatershed the only other pre-ROSAT measurement of significant event for X-ray cluster surveys. clusterevolutioncamefromEdge et al.(1990)whofound Critical comparisons of the ROSAT and EMSS lumi- rapid negative evolution in the luminosity function at nosity functions must account for the overlap (or lack z <0.2basedonaHEAO-1sample. Thiswasultimately thereof) of the measurements in redshift and luminos- overruledbyadefinitiveandnon-evolvingmeasureofthe ity. Insufficient attention to this point led to confusion local XLF (Ebeling et al. 1997). in some early analyses and debates. Different combina- SeekinginparttoconfirmorrefutetheEMSS’scontro- tionsofsensitivityandarealcoverageinflux-limitedsur- versial claim of negative cluster evolution, a large num- veys, when convolved with the intrinsic cluster luminos- ber of cluster surveys were initiated in the 1990s based ity function, result in populating different regions of the on ROSAT data (Voges 1992; Tru¨mper 1993). The 160 observedluminosity-redshiftplane(seeindicativeresults SquareDegreeROSAT ClusterSurvey(hereafter160SD, inFigure 1). Inpractice itis difficultto directlytest the Vikhlinin et al.1998a;Mullis et al.2003)isonesuchpro- cluster evolution seen in the EMSS detection because it gram and the subject of this paper. Additional sur- liesattheextremebrightendoftheluminosityfunction. veys probing to high redshift include the NEP (Mullis Thus large search volumes are required to detect ade- 2001; Henry et al. 2001; Gioia et al. 2003), BMW-HRI quate numbers of such rare clusters. The 160SD is one (Moretti et al. 2001; Panzera et al. 2003), BSHARC ofthelargestserendipitousX-raysurveysconductedwith (Romer et al. 2000), MACS (Ebeling, Edge, & Henry ROSAT. With this substantial areal coverage and high 2001), RDCS (Rosati et al. 1995, 1998, 2000), RIXOS sensitivity, the 160SD survey is well-positioned to probe (Castander et al. 1995; Mason et al. 2000), SSHARC cluster evolution. Preliminary analyses of our sample, CLUSTER XLF EVOLUTION 3 shifts. Note that the number of false-detections in the sampleagreesverywellwiththatexpectedfromthecon- 111000000 110000 fusion of point sources as demonstrated in the Monte- Carlo simulations of Vikhlinin et al. (1998a). Two false- detections (Nos. 77 and 141) serendipitously imaged by 22g]g] XMM-Newton andChandra,respectively,areindeedcon- a [dea [de 111000 1100 fused point sources. Finally, the optical survey imaging ee arar was sufficiently deep to demonstrate that none of the urvey urvey false-detections should be galaxy clusters at z .0.9. ss 111 11 Although spatial X-ray extent was our primary selec- tion criterion, detailed comparisons with other surveys demonstrate that no known clusters were missed as un- resolved sources (see §4 of Mullis et al. 2003, and ref- 000...111 00..11 erences therein). Thus the 160SD clusters are in ef- 11 00 −− 11 44 11 00 −− 11 33 11 00 −− 11 22 fflluuxx ((00..55−−22..00 kkeeVV)) [[eerrggss ss−−11 ccmm−−22]] fect drawn from a statistically complete, flux-limited survey with an areal coverage (Ω) and sensitivity char- Fig. 2.—Theselection functionofthe160SD surveyspecifying acterized by the selection function shown in Figure the square degrees of area covered as a function of limitingX-ray flux (0.5–2.0keV). These data are tabulated in Vikhlininetal. 2. A total of 158.5 deg2 were surveyed and the me- (1998a,Table5). Theshadingencodes thisinformationforsubse- dian survey flux (where Ω = Ω /2) is 1.2 × 10−13 total quentuseinFigure3. Theverticallineindicatestheminimumflux ergs cm−2 s−1(0.5–2.0keV). We restrict our analysis to alinmailtys(i4s.×10−14 ergscm−2 s−1, 0.5–2.0keV) used inthe present a minimum flux of 4 × 10−14 ergs cm−2 s−1 to avoid anyuncertaintiesintheskycoverageatveryfaintfluxes. Thereare190clustersinthe160SDsurveysampleabove with at-the-time incomplete optical follow-up, seemed thisfluxlimit. Figure 1showsthepositioninluminosity- to confirm the deficit of high-luminosity, high-redshift redshift space of our sample relative to the BCS, one of clusters first seen by the EMSS (Vikhlinin et al. 1998b, thekeyreferencesamplesatlowredshift,andtheEMSS. 2000). Figure 3 shows the 160SD data in greater detail. Here In this paper we present measurements of the clus- the cluster luminosities are plotted as a linear function ter XLF out to z = 0.8 and describe their implications ofcomovingvolumewhichprovidesamoreuniformview for cluster evolution based on the final 160SD sample of ofthe volumesampling. Thegreyscaleinthefigureindi- 201 clusters. In §2 we outline the basic construction of cates the parameter space probed by our survey — sen- our cluster sample and the associated selection function sitivity is maximal in light regions and minimal in dark used herein. The formalism associated with the XLF is regions. defined in §3 and used to measure cluster abundances In our subsequent derivations of the cluster XLF and at both low and high redshifts. In §4 we characterize tests forevolution,weexclude severalobjects(allatz < the evolution in the cluster population using integrated 0.3) to minimize the biasing of our results. To avoid numbercountsandamaximum-likelihoodanalysisofthe potentially skewing the impartiality of the sampling, we observed luminosity-redshift distribution relative to an rejectnineclusterswhoseredshiftsarewithin∆z =0.015 evolving Schechter function. We discuss our results in of the original target of the ROSAT PSPC observations thecontextofpreviousworksanddrawconclusionsin§5. (Nos. 16, 32, 112, 134, 165, 166, 174, 177, 206). Four Throughoutthisanalysisweusethecosmologicalparam- X-ray-overluminous elliptical galaxies or “fossil groups” eters H =50 h km s−1 Mpc−1, Ω =1, and Ω =0 (Nos. 110, 144, 201, and 211) are discounted because 0 50 M Λ (Einstein–de-Sitter, EdS) for direct comparison to pre- thesespecialstructuresareunlikelytomeettheselection vious work in this field. We also repeat calculations in criteriaofthelocalsamples(Vikhlinin et al.1999). Thus the currently preferred cosmology where Ω = 0.3 and thefinalclustersampleusedhereconsistsof177clusters. M Ω = 0.7. X-ray fluxes and restframe luminosities are Λ quoted in the 0.5–2.0 keV energy band unless otherwise 3. THEX-RAYLUMINOSITYFUNCTION stated. We define the cluster differential luminosity function 2. THE160SDCLUSTERSAMPLE to be The 160SDsample of201galaxyclustersisthe largest high-redshift,X-rayselectedsamplepublishedsofar. For d2N φ(L ,z) = (L ,z) (1) instance, there are 73 objects at z > 0.3 and 22 objects X dVdL X X at z > 0.5. The 160SD sample was constructed via the serendipitousdetection ofextendedX-raysourcesin 647 where N is the number of clusters of luminosity L in X archivalROSAT PSPCobservations. Of223clustercan- a volume V at a redshift z. The standard approach didates, we identified 201 as galaxy clusters, 21 as prob- for deriving a nonparametric representation of the dif- able false-detections due to blends of unresolved point ferential cluster XLF is based on the 1/V technique max sources,andonesourceisunidentifiedduetoitsproxim- first proposed by Schmidt (1968) and generalized by ity to a bright star. We have secured spectroscopic red- Avni & Bahcall (1980). Here the observed luminosity shiftsfor200ofthe201clusters. Vikhlinin et al.(1998a) range is parsed into bins of width ∆L, within each of give a complete description of the survey methodol- which there are N observed clusters. The XLF is es- j ogy. Mullis et al. (2003) detail the optical follow-up and timated by summing the density contributions of each present the final cluster catalog with spectroscopic red- cluster in the considered luminosity/redshift bin, 4 MULLIS ET AL. z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −1−1s]s] 11004444 s s gg rr ee 22 −− 00 55 hh ) [) [ VV ee 0 k0 k 11004433 2.2. −− 55 0.0. (( xx LL 11004422 00 11 22 33 44 ccoommoovviinngg vvoolluummee VV((<<zz)) [[110088 hh −−33 MMppcc33]] 5500 Fig. 3.— X-ray luminosity of the 160SD clusters versus redshift z (top axis) and comoving volume out to redshift z (bottom axis). The volume corresponds to a survey solid angle of 158.5 deg2 in an Einstein–de-Sitter cosmology. The shading, established in Figure2, demonstrates the range of survey flux limits and associated sky coverage in this parameter space. The thick, solid curve indicates the minimum flux limit (4×10−14 ergscm−2 s−1, 0.5–2.0keV) used in the present analysis. The horizontal (vertical) dashed lines are the boundariesoftheluminosity(redshift)binsusedincomputingtheluminosityfunction. transected by the survey flux limit (e.g., faint luminosi- 1 Nj 1 ties). WewillusethePage-Carrera(PC)estimatorinour φ(L ,z) = , (2) subsequentderivations of the cluster XLF. However,the X ∆L V (L ) i=1 max X,i results are essentially identical to the classical 1/Vmax X procedureexceptat the very faintend wherethe XLF is whereV isthetotalcomovingvolumeinwhichaclus- max marginally increased. terofluminosityL couldhavebeendetectedabovethe X,i As for a parametric representation of the cluster flux limits ofthe survey. Overa specific redshift interval XLF, observations are well fit by a Schechter function (z <z <z ), this search volume is defined to be min max (Schechter 1976) of the form zmax dV(z) −α V (L ) = Ω(f (L ,z)) dz. (3) L L dL max X X X dz φ(L ,z)dL = φ⋆ X exp − X X (5) Zzmin X X L⋆ L⋆ L⋆ (cid:18) X(cid:19) (cid:18) X(cid:19) X Here Ω(f ) is the sky area surveyed in steradians as a X where φ⋆ is the normalization (units h3 Mpc−3), α is function of X-ray flux and dVd(zz) is the differential, co- thefaint-endslope,andL⋆ isthecharac5t0eristicluminos- moving volume element per steradian. X itymarkingthe interfacebetweenthepower-lawandthe Page & Carrera (2000) describe a refinement to the exponential regimes. An equivalent expression for the canonicalapproachinwhichtheestimatortakestheform XLF commonly used in the literature is φ(LX,z) = LLXX,,mmianx zzmmianxΩ(fX(NLjX,z)) dVdz(z) dz dL(X4) With L φi(nLXu,nzi)ts=ofA1L0−X4α4ehx−p2(cid:18)−erLLgs⋆XXs(cid:19)−1., the (a6s)- R R X 50 where the boundaries of the luminosity bin are specified sociated normalization, A, has units of h3 Mpc−3 50 by LX,min and LX,max. The effect of bringing the lumi- (1044 h−2 ergs s−1)α−1, and the two normalizations are nosityintervalintothedoubleintegralresultsinabetter related5b0y φ⋆ = A(L⋆)1−α. estimate of the effective ∆L which can be smaller than X thefullbinwidthforregionsofluminosity-redshiftspace CLUSTER XLF EVOLUTION 5 Table1. Best-Fit SchechterParametersfortheLocalCluster XLF (z<0.3) Sample L⋆† α φ⋆ Reference X (1044 h−2 ergss−1) h3 Mpc−3 50 50 BCS 5.70+1.29 1.85+0.09 (7.56+0.82)×10−8 Ebelingetal.(1997) −0.93 −0.09 −0.75 RASS1BS 3.80+0.70 1.52+0.11 (2.53+0.22)×10−7 DeGrandietal.(1999b) −0.55 −0.11 −0.22 REFLEX‡ 4.21+0.37 1.63+0.06 (1.80+0.5 )×10−7 Bo¨hringeretal.(2002) −0.34 −0.06 −0.4 5.18+−00..5560 1.69+−00..004455 1.07 ×10−7 ΩM =0.3,ΩΛ=0.7 Note. —ParametersforanEinstein–de-Sitter universeunlessotherwiseindicated †L⋆ isquotedinthe0.5–2.0keVband X ‡fitincludesacorrectionformissingfluxforconsistency withtheothersurveys(seedetailsinBo¨hringeretal.2002) 3.1. The Local XLF Before considering the cluster population at high red- shift, we first examine the low-redshift diagnostic using Knowledge of the local or near present-day abun- the 160SD cluster sample. Using the procedure pre- dance of clusters is fundamental to evolutionary stud- viously described, we estimate the local XLF between ies because it serves as the no-evolutionbaseline against ∼1042 and 3 × 1044 ergs s−1 using the 110 clusters at which distant cluster samples can be tested. As previ- 0.02 < z < 0.3 in the 160SD survey. Our measurement ously noted, one of the significant achievements of the isplottedinFigure4andistabulatedinTable2. Thelu- ROSAT era is the accurate determination of the lo- minositybinning is uniformin logspace anddata points cal cluster XLF (z . 0.3). Three principal measure- are plotted at the center of the luminosity interval. The ments are based on the BCS, RASS1BS, and REFLEX error bars are the equivalent ±1σ uncertainties based samples which were constructed by surveying large por- on Poissonian errors for the number of clusters per bin tions of the two-thirds of the sky outside the Galactic plane9 at relatively bright fluxes. Ebeling et al. (1997) (Gehrels1986). Notethegoodagreementatlowredshift betweenthe160SDandthethreeprincipallocalsamples. reportedthefirstresultsbasedon199BCSclustersinthe northern hemisphere (f > 2.8 × 10−12 ergs cm−2 s−1, In Figure 5 we add the measurements from four addi- X tional deep surveys (RDCS, EMSS, NEP, and WARPS) 0.1–2.4keV). Part of a pilot program of the larger thus producing a compilation of all of the ROSAT (plus REFLEX survey, De Grandi et al. (1999b) presented a EMSS)localXLFspublishedtodatefromX-rayselected, measurement based on 129 RASS1BS clusters (f > X 3–4 × 10−12 ergs cm−2 s−1, 0.5–2.0keV)fromthe south X-ray flux-limited cluster samples. The shallow all-sky surveysaccuratelymeasurethelocalluminosityfunction Galactic cap. Finally, the XLF determination based on at intermediate to high luminosities (& 1043 ergs s−1) thelargestsampletodatecomesfromtheREFLEXsur- butarerelativelyinsensitivetoverylowluminosityclus- vey. B¨ohringer et al. (2002) made a detailed analysis of ters (.1043 ergs s−1). In a complementary fashion, the 452 clusters extracted from the southern celestial hemi- sphere (f > 3 × 10−12 ergs cm−2 s−1, 0.1–2.4keV). deep surveys better measure the faint end of the local X XLF, provide reasonable precisionat intermediate lumi- Nonparametric determinations of the local XLF (z < nosities,butpoorlyconstraintheverybrightend(&1044 0.3) from the all-sky samples (BCS, RASS1BS, and RE- ergs s−1) due to limited survey volumes at low redshift. FLEX) are shown in Figure 4. We also plot the best- The cluster luminosity-redshift distributions in Figure 1 fitting Schechter functions for these data and list the alsoillustratethisdependenceonfluxlimitandsolidan- associated best-fit parameters in Table 1 for future ref- gle. erence. The results demonstrate that we have accurate knowledgeofthe localclusterluminosity function. Inde- 3.2. The High-Redshift XLF pendentinvestigatorsusingdifferentX-rayselectionpro- ceduresoverdifferentregionsoftheskyagreeonthelocal We measure the distant XLF using the 66 clusters space density of clusters. For example, in the luminos- from the 160SD sample at 0.3 < z < 0.8 and fluxes ity interval 5 × 1043 – 1045 ergs s−1 (0.5–2.0keV), the above 4 × 10−14 ergs cm−2 s−1. The numerical results internalaccuracyofthese XLF measurementsis approx- are shown in Table 3 and plotted in Figure 6. We have imately ±10%–20% (estimated from the ±1σ excursion probed a sufficiently large volume such that we can de- oftheerrorenvelopesplottedinFigure4). Moreover,the rive useful results in two intervals: 0.3<z <0.6 and systematics are also small — the results from the BCS 0.6<z <0.8. In addition to our high-redshift measure- and RASS1BS surveys vary a maximum of about ±25% ments,wealsoshowinFigure6thelocaldeterminations relative to the Schechter fit of REFLEX. of the XLF. These establish the regime where the high- redshiftresultsshouldlieifthespatialdensityofclusters 9 Note that the significant gap in all-sky coverage due to the does not evolve out to the considered redshifts. former zone of avoidance (|b| < 20o) is being redressed by the Our measurement of the XLF at 0.3<z <0.6 CIZAsurvey(Ebeling,Mullis,&Tully2002) probes the luminosity range 2 × 1043 – 7 × 1044 6 MULLIS ET AL. 1100−−33 160SD (0.02 < z < 0.30) BCS (z < 0.3) 1100−−44 RASS1BS (z < 0.3) REFLEX (z < 0.3) 1100−−55 −1−1−1−1s)]s)] s s 1100−−66 gg erer 44440 0 11 −3−3c (c ( 1100−−77 pp MM 55h h 5050 1100−−88 φφ [ [ 1100−−99 1100−−1100 11004422 11004433 11004444 11004455 LL ((00..55−−22..00 kkeeVV)) [[eerrggss ss−−11]] XX Fig. 4.—DeterminationsofthelocalclusterX-rayluminosityfunctionasmeasuredbythe160SDsurveyandthelocalreferencesamples (BCS,RASS1BS,andREFLEX)inanEinstein–de-Sitter universe. The160SDdatavaluesalongwiththenumberofclustersandaverage cluster redshift for each luminosity bin are given in Table 2. Nonparametric data points and Schechter fits for the reference samples are fromEbelingetal.(1997),DeGrandietal.(1999b),andBo¨hringeretal.(2002),respectively. TheBCSdatapointsarebasedonamerged analysis of the BCS+eBCS samples (H. Ebeling 2003, private communication). The shaded region indicates the 1σ uncertainty envelope oftheSchechter fitsassumingtheerrorsonL⋆ andαarecorrelated. Theindicateduncertainties ondatapointsare±1σ. X 111000−−−333 160SD (0.02 < z < 0.30) BCS (z < 0.3) 111000−−−444 RASS1BS (z < 0.3) REFLEX (z < 0.3) 111000−−−555 −1−1−1−1−1−1s)]s)]s)] s s s 111000−−−666 ggg ererer 4444440 0 0 111 −3−3−3c (c (c ( 111000−−−777 ppp MMM 555h h h 505050 111000−−−888 φφφ [ [ [ RDCS (0.04 < z < 0.25) EMSS (0.14 < z < 0.30) 111000−−−999 NEP (0.02 < z < 0.30) WARPS (0.03 < z < 0.30) 111000−−−111000 111000444222 111000444333 111000444444 111000444555 LLL (((000...555−−−222...000 kkkeeeVVV))) [[[eeerrrgggsss sss−−−111]]] XXX Fig. 5.—Compilation of local XLFs as measured byeight X-rayflux-limitedsurveys. RDCS:Rosatietal. (1998), EMSS: Henryetal. (1992),NEP:Gioiaetal.(2001),andWARPS:Jonesetal.(2000b)andthereferencesinFigure4(Einstein–de-Sitter universe). CLUSTER XLF EVOLUTION 7 160SD (0.3 < z < 0.6) 1100−−55 160SD (0.6 < z < 0.8) BCS (z < 0.3) RASS1BS (z < 0.3) 1100−−66 REFLEX (z < 0.3) −1−1−1−1s)]s)] 1100−−77 s s gg erer 44440 0 −3−3c (1c (1 1100−−88 pp MM 55h h 5050 1100−−99 φφ [ [ 1100−−1100 1100−−1111 11004433 11004444 11004455 LL ((00..55−−22..00 kkeeVV)) [[hh −−22 eerrggss ss−−11]] XX 5500 Fig. 6.—Thehigh-redshiftclusterX-rayluminosityfunctionfromthe160SDsample(Einstein–de-Sitter universe). Thedatapointsfor 0.6<z <0.8 have been slightly offset to avoid confusion. The 160SD data values along with the number of clusters and average cluster redshiftforeachluminositybinaregiveninTable3. SchechterfitstothelocalXLFarealsoplottedasdescribedinFigure4. Ifnoclusters aredetected inaluminositybin,weplotthe1σ upperlimitforthePoissonerroronzeroclusters. 116600SSDD ((00..33 << zz << 00..66)) 1100−−55 116600SSDD ((00..66 << zz << 00..88)) BCS (z < 0.3) RASS1BS (z < 0.3) 1100−−66 REFLEX (z < 0.3) −1−1−1−1s)]s)] 1100−−77 s s gg erer 44440 0 −3−3c (1c (1 1100−−88 pp MM 55h h 5050 1100−−99 φφ [ [ 1100−−1100 Ω Ω = 0.3, = 0.7 M Λ 1100−−1111 11004433 11004444 11004455 LL ((00..55−−22..00 kkeeVV)) [[hh −−22 eerrggss ss−−11]] XX 5500 Fig. 7.— The high-redshiftcluster X-rayluminosity function fromthe 160SD sample fora cosmology characterized bythe parameters ΩM = 0.3 and ΩΛ = 0.7. The data points for 0.6 < z < 0.8 have been slightly offset to avoid confusion. The 160SD data values along with the number of clusters and average cluster redshift for each luminosity bin are given in Table4. The REFLEX XLF was explicitly measuredforaΛ-dominatedcosmologybyBo¨hringeretal.(2002). WetransformtheBCSandRASS1BSresultstothiscosmologybased ontheREFLEXresults(seetextfordetails). 8 MULLIS ET AL. Table2. Local 160SDCluster X-rayLuminosity Function (0.02<z< 0.3,EdS) LX(center)† LX(min)† LX(max)† φ(LX) φ(LX)−1σ φ(LX)+1σ <z> Ncl (1044 h−2 ergss−1) ··· ··· (h5 Mpc−3 (1044 ergss−1)−1) ··· ··· 50 50 0.016 0.013 0.019 5.09×10−4 1.78×10−4 1.18×10−3 0.045 2 0.022 0.019 0.026 2.19×10−4 7.65×10−5 5.07×10−4 0.035 2 0.031 0.026 0.037 1.42×10−4 6.45×10−5 2.81×10−4 0.062 3 0.044 0.037 0.053 2.08×10−5 3.59×10−6 6.85×10−5 0.135 1 0.063 0.053 0.075 4.59×10−5 2.61×10−5 7.69×10−5 0.142 5 0.089 0.075 0.106 2.87×10−5 1.80×10−5 4.42×10−5 0.136 7 0.125 0.106 0.149 1.86×10−5 1.28×10−5 2.65×10−5 0.156 10 0.177 0.149 0.211 6.98×10−6 4.55×10−6 1.04×10−5 0.154 8 0.251 0.211 0.298 7.44×10−6 5.65×10−6 9.72×10−6 0.200 17 0.355 0.298 0.422 5.49×10−6 4.35×10−6 6.89×10−6 0.197 23 0.502 0.422 0.597 1.27×10−6 8.55×10−7 1.86×10−6 0.195 9 0.710 0.597 0.844 9.89×10−7 6.94×10−7 1.39×10−6 0.239 11 1.004 0.844 1.194 3.60×10−7 2.16×10−7 5.76×10−7 0.226 6 1.420 1.194 1.688 8.26×10−8 2.89×10−8 1.92×10−7 0.225 2 2.008 1.688 2.388 5.76×10−8 2.01×10−8 1.34×10−7 0.180 2 2.840 2.388 3.377 2.01×10−8 3.47×10−9 6.63×10−8 0.296 1 †LX isquotedinthe0.5–2.0keVband ergs s−1 (Figure 6: filled circles). Except for the data data points of the latter are offset diagonally down and point near 5 × 1043 ergs s−1 which is 1.9σ off the local to the right approximately along the former. This is relation, there is excellent agreement between these the combined effect of the increase in both the cluster results and the no-evolution benchmark at least out to luminosities and the search volumes in a Λ-dominated intermediate luminosities, L ∼ 2 × 1044 ergs s−1. In universe. The actual positioning of the two XLFs rela- X this regionthe 160SDbest matches the normalizationof tive to each other depends on the cluster redshifts and the REFLEX XLF. At higher luminosities the distant the specific luminosity interval. For example in the RE- XLF is lower than the local population but we will FLEX survey (< z >= 0.08), between 1043 and 1045 demonstrate in §4.1 that this effect is only marginally ergs s−1 the ratio of the fitted XLFs (φ /φ ) is less Λ EdS significant with respect to the local XLF with the than unity with a broad minimum of ∼0.8 around 1044 highest normalization (RASS1BS). The median redshift ergs s−1. REFLEX is the only local sample for which for these depressed data points is < z >= 0.50 (see the XLF in a Λ-cosmology has been explicitly measured Table 3). (B¨ohringer et al. 2002). However,giventhe similarity of At the highest redshifts probed by the 160SD in the redshift and luminosity distributions, we have used the present analysis, 0.6<z <0.8, we measure use- the ratio φ /φ from REFLEX to make an approxi- Λ EdS ful constraints over the luminosity interval 1 – 6 × 1044 mate transform of the BCS and RASS1BS to this alter- ergs s−1,andthemedianclusterredshiftis<z >=0.70. nate cosmology. (Figure 6: open squares). The distant cluster volume It is apparent from Figure 7 that the high-redshift densities are below the local level at all measured lu- 160SD XLFs and the local XLFs shift in similar ways; minosities. We will show that this result is significant thus the apparentdeficitofhigh-luminosityclustersper- (> 3σ) in both the Einstein–de-Sitter and Λ-dominated sists in the Λ-dominated cosmology. The only im- models evenwith respecttothe lowest-normalizationlo- portant difference, of course, is that the point where cal XLF (§4.1). This is a deficit of high luminosity, the 160SD data depart significantly from the non- high redshift clusters in a manner similar to that seen evolution baseline is shifted to larger luminosities, and in the EMSS results. Note that although the optical this is anticipated given the increase in luminosity dis- CCD imaging used to identify the 160SD clusters was tance (L & 3 × 1044 ergs s−1 for 0.3<z <0.6, and X sufficientlydeeptorevealmassiveclusterstoz =0.9,we L & 2 × 1044 ergs s−1 for 0.6<z <0.8). X haveconservativelycutoffthemostdistantredshiftshell We will examine the significance and strength of this at z =0.8. This lessens any potential negative evolution apparent cluster evolution in the following section. in the results since the larger redshift boundary would increase the search volume without adding clusters and 4. QUANTIFYINGEVOLUTION hence depress the data points. X-ray luminosity functions like those shown in To assess the impact of changing the cosmological Figures 4–7 are useful for visualizations and qualitative framework from an Einstein–de-Sitter to a Λ-dominated assessments of the cluster population; however,they are universe, we repeat our calculation of the cluster lu- non-optimal for quantitative analyses. For example am- minosity function setting ΩM = 0.3 and ΩΛ = 0.7, biguities exist in the selection of the luminosity binning the results of which are shown in Figure 7 and Table 4. (e.g.,fixedoradaptiveintervals)andthelocioftheplot- The Einstein–de-Sitter XLF (φEdS, Figure 6) and Λ- teddatapointsinluminosityspace(e.g.,atthebincenter dominated XLF (φΛ) appear very similar because the orthe density-weightedmeanluminosity). Furthermore, CLUSTER XLF EVOLUTION 9 Table3. Distant160SDCluster X-rayLuminosity Function (EdS) LX(center)† LX(min)† LX(max)† φ(LX) φ(LX)−1σ φ(LX)+1σ <z> Ncl (1044 h−2 ergss−1) ··· ··· (h5 Mpc−3 (1044 ergss−1)−1) ··· ··· 50 50 0.3<z<0.6 0.251 0.211 0.298 2.86×10−6 4.95×10−7 9.45×10−6 0.329 1 0.355 0.298 0.422 2.01×10−6 9.10×10−7 3.96×10−6 0.334 3 0.502 0.422 0.597 2.80×10−6 2.06×10−6 3.77×10−6 0.376 14 0.710 0.597 0.844 5.83×10−7 3.80×10−7 8.72×10−7 0.420 8 1.004 0.844 1.194 3.17×10−7 2.18×10−7 4.53×10−7 0.461 10 1.420 1.194 1.688 1.61×10−7 1.10×10−7 2.29×10−7 0.418 10 2.008 1.688 2.388 9.18×10−8 6.32×10−8 1.31×10−7 0.517 10 2.840 2.388 3.377 2.88×10−8 1.64×10−8 4.83×10−8 0.501 5 4.016 3.377 4.776 7.67×10−9 2.68×10−9 1.78×10−8 0.485 2 5.679 4.776 6.754 4.85×10−9 ··· 4.85×10−9 ··· 0 0.6<z<0.8 1.004 0.844 1.194 3.65×10−7 ··· 3.65×10−7 ··· 0 1.420 1.194 1.688 5.17×10−8 8.95×10−9 1.71×10−7 0.699 1 2.008 1.688 2.388 1.88×10−8 3.25×10−9 6.20×10−8 0.625 1 2.840 2.388 3.377 1.59×10−8 ··· 1.59×10−8 ··· 0 4.016 3.377 4.776 4.71×10−9 8.15×10−10 1.55×10−8 0.700 1 5.679 4.776 6.754 5.33×10−9 ··· 5.33×10−9 ··· 0 †LX isquotedinthe0.5–2.0keVband Table4. Distant160SDClusterX-rayLuminosityFunction(ΩM =0.3 andΩΛ=0.7) LX(center)† LX(min)† LX(max)† φ(LX) φ(LX)−1σ φ(LX)+1σ <z> Ncl (1044 h−2 ergss−1) ··· ··· (h5 Mpc−3 (1044 ergss−1)−1) ··· ··· 50 50 0.3<z<0.6 0.355 0.298 0.422 9.52×10−7 1.65×10−7 3.14×10−6 0.329 1 0.502 0.422 0.597 9.51×10−7 4.97×10−7 1.70×10−6 0.351 4 0.710 0.597 0.844 1.04×10−6 7.62×10−7 1.40×10−6 0.372 14 1.004 0.844 1.194 1.93×10−7 1.21×10−7 2.97×10−7 0.429 7 1.420 1.194 1.688 1.20×10−7 8.22×10−8 1.71×10−7 0.439 10 2.008 1.688 2.388 5.99×10−8 4.12×10−8 8.56×10−8 0.440 10 2.840 2.388 3.377 3.05×10−8 2.05×10−8 4.45×10−8 0.509 9 4.016 3.377 4.776 1.47×10−8 9.26×10−9 2.27×10−8 0.507 7 5.679 4.776 6.754 1.39×10−9 2.41×10−10 4.59×10−9 0.516 1 8.032 6.754 9.552 1.75×10−9 ··· 1.75×10−9 ··· 0 0.6<z<0.8 1.420 1.194 1.688 1.59×10−7 ··· 1.59×10−7 ··· 0 2.008 1.688 2.388 3.75×10−8 ··· 3.75×10−8 ··· 0 2.840 2.388 3.377 6.76×10−9 1.17×10−9 2.23×10−8 0.699 1 4.016 3.377 4.776 2.90×10−9 5.02×10−10 9.57×10−9 0.625 1 5.679 4.776 6.754 1.51×10−9 2.61×10−10 4.98×10−9 0.700 1 8.032 6.754 9.552 1.65×10−9 ··· 1.65×10−9 ··· 0 †LX isquotedinthe0.5–2.0keVband for the case of negative evolution the effect that we are 4.1. Integrated Number Counts attempting to measure is either a diminishing signal or Foragivenregionofluminosity-redshiftspacewecom- a non-detection. Thus we will apply the alternate ap- pare the number of observed clusters (N ) with the obs proaches of integrated number counts and a maximum- number that are expected (N ) assuming there is no exp likelihood fit of an evolving model XLF to quantify the evolution in the population. The latter is computed by significanceandstrengthoftheapparentevolutioninthe integrating the local luminosity function, φ(L ,z), over X 160SD clusters. luminosityandredshift,andfoldingthisthroughthesur- vey selection function, Ω(f ), using the following equa- X tion, 10 MULLIS ET AL. Table 5. Observed versus Expected Number of Clusters (LX,min < LX<∞) LX,min† Nobs Nexp significance(σ) (1044 h−2 ergss−1) 50 REFLEX BCS RASS1BS REFLEX BCS RASS1BS 0.3<z<0.6,EdS 4.776 0 4.5 3.4 5.2 2.3 1.8 2.6 3.377 2 9.1 6.7 10.8 2.5 1.8 3.0 2.388 7 16.2 11.8 19.2 2.4 1.3 3.0 1.688 17 25.8 18.7 30.4 1.7 0.2 2.5 0.3<z<0.6,ΩM =0.3andΩΛ=0.7 6.754 0 3.5 3.0 4.0 1.9 1.6 2.1 4.776 1 7.8 6.1 9.1 2.7 2.2 3.1 3.377 8 14.8 11.0 17.4 1.7 0.7 2.3 2.388 17 24.4 17.8 28.8 1.4 0.0 2.2 0.6<z<0.8,EdS 4.776 0 4.1 3.2 4.8 2.1 1.7 2.4 3.377 1 7.9 5.8 9.3 2.7 2.0 3.1 2.388 1 12.6 9.1 14.9 3.9 3.1 4.4 1.688 2 17.2 12.5 20.2 4.4 3.4 5.0 1.194 3 20.5 15.0 24.1 4.6 3.5 5.2 0.844 3 22.0 16.1 25.7 4.9 3.8 5.5 0.6<z<0.8,ΩM =0.3andΩΛ=0.7 6.754 0 3.8 3.3 4.4 2.0 1.8 2.2 4.776 1 7.8 6.1 9.0 2.7 2.2 3.0 3.377 2 12.7 9.6 14.9 3.4 2.7 3.9 2.388 3 17.4 12.9 20.5 4.0 3.1 4.6 1.688 3 20.6 15.2 24.2 4.6 3.6 5.2 1.194 3 21.8 16.1 25.6 4.8 3.8 5.5 †ValuesofLX,min (0.5–2.0keV)arebasedonthelowerlimitsoftheluminositybinsusedinthederivationofthenon-parametricXLF XLF predicts 4.5 clusters according to Equation 7. This difference is only 2.3σ significant. The next two bins LX,max zmax dV(z) (L & 3.4,2.4) have N = 2,7 and N = 9.1,16.2 N = φ(L ,z)Ω(f (L ,z)) dzdL . X obs exp exp X X X dz Xwith the associated significances of 2.6σ and 2.4σ. Con- ZLX,min Zzmin tinuing from here to lower luminosity bins decreases the (7) significance. These results are summarized in Table 5. Note that the non-evolving XLF is strictly a function of Examining the run of significance versus minimum lu- luminosity as parameterized by the Schechter fits to lo- minosity without the constraints of the arbitrary bins calclusters(e.g.,Equation 5withthebest-fitparameters of Figure 6 indicates that the significance briefly peaks from Table 1). However, we explicitly indicate the po- above 3σ near 3.4 × 1044 ergs s−1. However with the tentialredshiftdependenceinthisequationtogeneralize BCS XLF as the no-evolution reference, the apparent it for subsequent treatment with an evolving XLF. The cluster deficit is not significant (i.e. always < 3σ). statisticalsignificanceofthedifferencebetweenN and obs Conversely, the RASS1BS XLF indicates a statistically N is computed based on Poisson confidence intervals exp strong signal across much of the probed luminosity. Re- (Gehrels 1986). peating this analysis in the context of a Λ-dominated Of the three local XLFs (BCS, RASS1BS, and RE- universe uniformly reduces the significance of the clus- FLEX), we use the REFLEX measure as the preferred ter deficit such that only relative to the RASS1BS does referenceforthreereasons: 1)theREFLEXlocalXLFis thedeviationseeninthe160SDmeasureat0.3<z <0.6 basedonthe largestsampleofclusters usedtodate (452 appear marginally significant (see Table 5). clusters), 2) the REFLEX normalization lies intermedi- Our highest redshift measure of the XLF, ate to the BCS and RASS1BS, and 3) at low to inter- 0.6<z <0.8, is clearly lower than all the determi- mediate luminosities, our 160SD low-redshift data most nations of the local XLF (Figures 6 & 7). Integrating closely match the REFLEX normalization. Nonetheless, overthe entire luminosity range sampled in this redshift we will also quote our results relative to the BCS and shell, a non-evolving model of the cluster population RASS1BS to demonstrate the full range of possible sig- based on the REFLEX XLF predicts 22.1 clusters. This nificances. strongly conflicts with the actual observed sample of First we consider the redshift interval 0.3<z <0.6 3 clusters, a 4.9σ difference. The situation in the Λ- starting with the highest luminosity bin and then in- cosmologyisessentiallythesame;21.8clustersexpected, tegrating to lower luminosities in an Einstein–de-Sitter a 4.8σ difference. If we instead use the RASS1BS XLF model. In and above the highest bin of the XLF (L & X as our baseline, the predicted count is greater than 26 4.8),weobservezeroclusterswhereastheREFLEXlocal