Observational constraints on the redshift evolution of X-ray scaling relations of galaxy clusters out to z ~ 1.5
aa r X i v : . [ a s t r o - ph . C O ] S e p Astronomy&Astrophysicsmanuscript no. AA-2011-16861 c (cid:13)
ESO 2018August 15, 2018
Observational constraints on the redshift evolution ofX-ray scaling relations of galaxy clusters out to z ∼ . A. Reichert , H. B ¨ohringer , R. Fassbender , M. M¨uhlegger Max-Planck-Institut f¨ur extraterrestrische Physik, D 85748 Garching, Germany, [email protected]
Submitted 09 / / ABSTRACT
Context.
A precise understanding of the relations between observable X-ray properties of galaxy clusters and cluster mass is a vitalpart of the application of X-ray galaxy cluster surveys to test cosmological models. An understanding of how these relations evolvewith redshift is just emerging from a number of observational data sets.
Aims.
The current literature provides a diverse and inhomogeneous picture of scaling relation evolution. We attempt to transformthese results and the data on recently discovered distant clusters into an updated and consistent framework, and provide an overallview of scaling relation evolution from the combined data sets.
Methods.
We study in particular the most important scaling relations connecting X-ray luminosity, temperature, and cluster mass (M–T, L X –T, and M–L X ) combining 14 published data sets supplemented with recently published data of distant clusters and new resultsfrom follow-up observations of the XMM-Newton Distant Cluster Project (XDCP) that adds new leverage to e ffi ciently constrain thescaling relations at high redshift. Results.
We find that the evolution of the mass-temperature relation is consistent with the self-similar evolution prediction, whilethe evolution of X-ray luminosity for a given temperature and mass for a given X-ray luminosity is slower than predicted by simpleself-similar models. Our best fit results for the evolution factor E ( z ) α are α = − . ± .
07 for the M–T relation, α = − . + . − . for theL-T relation, and α = − . + . − . for the M–L X relation. We also explore the influence of selection e ff ects on scaling relations and findthat selection biases are the most likely reason for apparent inconsistencies between di ff erent published data sets. Conclusions.
The new results provide the currently most robust calibration of high-redshift cluster mass estimates based on X-rayluminosity and temperature and help us to improve the prediction of the number of clusters to be found in future galaxy cluster X-raysurveys, such as eROSITA. The comparison of evolution results with hydrodynamical cosmological simulations suggests that earlypreheating of the intracluster medium (ICM) provides the most suitable scenario to explain the observed evolution.
Key words.
X-rays: galaxies: clusters, Galaxies: clusters: Intergalactic medium, Cosmology: observations
1. Introduction
Galaxy clusters have become important probes for the studyof the evolution of the large-scale structure and for thetest of cosmological models (e.g. Borgani & Guzzo (2001),Holder et al. (2001), Rosati et al. (2002), Schuecker et al.(2003), Haiman et al. (2005), Voit (2005), Vikhlinin et al.(2009), Mantz et al. (2009), B¨ohringer et al. (2010)) andprovide an interesting laboratory to study galaxy evolution.Nowadays, the application of galaxy cluster surveys to cosmo-logical studies is limited mainly by a lack of understanding ofgalaxy cluster properties and the precise scaling relations ofobservables with cluster mass, in particular at higher redshifts.To date, X-ray observations provide the most reliable anddetailed characterization of galaxy clusters and X-ray pa-rameters and are most widely used in cosmological galaxycluster studies for several reasons: (i) X-ray luminosity istightly correlated to the cluster mass (Reiprich & B¨ohringer(2002), Pratt et al. (2009)), (ii) bright X-ray emission is onlyobserved for evolved clusters with a deep gravitational po-tential well, and (iii) the X-ray emission is highly peaked,minimizing projection e ff ects. While great progress has beenmade in the characterizing clusters in X-rays at low redshifts(Arnaud et al. (2005), Vikhlinin et al. (2006, 2009), Zhang et al. Send o ff print requests to : H. B¨ohringer, [email protected] (2008), Pratt et al. (2009), Mantz et al. (2009), B¨ohringer et al.(2010), Arnaud et al. (2009)), the understanding of the evolu-tion of the scaling relations towards higher redshifts is less clear.Results are now emerging that provide insight into the redshiftrange beyond z =
1, but the di ff erent available cluster samplesprovide inconsistent scenarios for the redshift evolution of thescaling relations. Moreover, it is not trivial to compare the dif-ferent results since they have been partly derived for di ff erentcosmologies, for di ff erent definitions of the scaling radius, andcompared with di ff erent schemes of the self-similar scaling lawsto quantify the evolutionary trend. Therefore, this work makesan e ff ort to put the di ff erent results into a uniform frameworkand combine the available data to study the evolutionary trendover the widest redshift baseline available.The data sets used from the literature comprise the workof Andersson et al. (2010), Pratt et al. (2009), Mantz et al.(2009), Zhang et al. (2008, 2007), Hicks et al. (2008),Pacaud et al. (2007), Branchesi et al. (2007), O’Hara et al.(2007), Vikhlinin et al. (2006), Maughan et al. (2006),Arnaud et al. (2005), Kotov & Vikhlinin (2005), and Ettori et al.(2004) and we include the data of an additional 15 clustersfrom recent publications and the XMM-Newton Distant ClusterProject (XDCP, Fassbender (2008), Boehringer et al. (2005)).The latter set of clusters significantly improve the statistics inthe redshift range z = . − . Reichert et al.: Evolution of X-ray scaling relations
The derived evolution results are compared to the findingsof hydrodynamical cosmological simulations assuming di ff erentheating and cooling scenarios and therefore allow an investiga-tion of the ICM thermal history.Tighter constraints on the evolution of the X-ray luminosityof clusters for a given mass also allow us to make refined predic-tions about the number of high redshift clusters to be observedin future X-ray surveys. We illustrate this in the context of theeROSITA mission (Predehl et al. 2010).The paper is structured as follows. In Sect. 2, we briefly de-scribe the cluster samples used and the way in which we havetransformed these public results into a unified framework. Wealso outline the theoretical expectations of the scaling relationsand an estimate of the selection bias inherent to our combinedcluster sample. The local scaling relations and the results on theirevolutionary trend with redshift are given in Sect. 3. In Sect. 5.1,these results are compared to the predictions of numerical simu-lation studies and the implications for the ICM heating scenarioare discussed. In Sect. 5.2, we outline the impact of our results onthe number of clusters to be detected with eROSITA and Sect. 6,we provide a summary and our conclusions.Throughout the article, we adopt a Λ CDM cosmology with( Ω Λ , Ω M , H , w ) = (0 . , . ,
70 km s − Mpc − , −
2. Data and data analysis
The evolution of galaxy cluster X-ray scaling relations is in-vestigated by means of a combined cluster sample compiledfrom a number of recent publications. To enable constraints onthe redshift evolution of scaling relations, the clusters were se-lected to cover a wide redshift range from local systems out to z = .
46. We attempted to avoid the complications caused by theexpected deviation of galaxy groups from the scaling laws formore massive clusters by applying an ICM temperature thresh-old of T min = z = . z > . · M ⊙ to 3 · M ⊙ . For the dis-tant cluster sample, cluster masses derived by means of the Y X –M relation were not considered owing to their dependence onthe Y X –M scaling relation and its redshift evolution. The z > . z = . − .
46 (see Appendix B) and by construc-tion, the distant cluster sample shows no strong morphologicalselection bias.
Assuming that the evolution of the ICM during cluster forma-tion is governed solely by gravitational processes, clusters areexpected to be self-similar objects whose X-ray properties andmasses are connected by scaling relations predicted by the self-similar model ( e.g.
Kaiser (1986)). Since galaxy clusters have noclearly defined natural outer boundary, a fiducial radius withinwhich cluster properties are considered has to be chosen. Scalingtheory is used to define this radius in such a way that it de-scribes the same corresponding boundary for clusters of all sizesin the framework of the self-similar cluster structure model.In accordance with the homogeneous spherical collapse modelof Gunn & Gott (1972) and detailed N-body simulations ( e.g.
Navarro et al. (1995)), the fiducial radius is defined to enclose aspherical region with a mean overdensity of ∆ times the criticaldensity of the Universe ρ crit ( z ) r ∆ = M ( r < r ∆ )4 πρ crit ( z ) ∆ . (1)While this first-order self-similar model seems to describe thestructure of dark matter haloes fairly well, additional gas physicsincluding heating and cooling processes are needed to ex-plain the ICM structure and the resulting X-ray properties.Consequently, the local scaling relations have been found todi ff er from the self-similar predictions in some cases (see e.g. Pratt09), e.g. the L X –T relation is steeper than expected.The self-similar model also predicts the evolution of scal-ing relations with redshift or lookback time. However, there aredi ff erent schemes for defining the fiducial radius in order to en-close self-similar regions for clusters at di ff erent redshifts. Oneapproach based on a spherical top-hat collapse model assumesthat a galaxy cluster as it is observed has only recently formedat the given redshift and proposes adopting a redshift-dependentdensity contrast ∆ z when defining fiducial radii, where ∆ z can beexpressed in terms of the density contrast at the virial radius atthe cluster redshift ∆ z = ∆ ( z = ∆ vir ( z ) ∆ vir ( z = . (2)Bryan & Norman (1998) give an expression for ∆ vir ( z ) in a flat Λ CDM-cosmology of ∆ vir ( z ) = π + Ω m ( z ) − − Ω m ( z ) − , (3)where Ω m ( z ) = Ω M (1 + z ) / E ( z ) and E ( z ) = H ( z ) H . The expectationfor the M–T, L X –T, and M–L X relation and their evolution withredshift in a model only taking into account gravitational e ff ectsis then M ∝ T / E ( z ) − ∆ − / z , (4) L X ∝ T E ( z ) ∆ / z , (5) M ∝ L / X E − / ∆ − / z , (6)where L X is the bolometric luminosity integrated out to the scaleradius. This or related definitions of fiducial radii were used in eichert et al.: Evolution of X-ray scaling relations 3 Table 1.
Overview of the publications used to compile the combined cluster sample.
Publication Acronym Survey Instrument Cluster
Andersson et al. (2010) Andersson10 SPT XMM / Chandra 9 0.4-1.1Pratt et al. (2009) Pratt09 REXCESS XMM 26 ≤ . / ROSAT 42 ≤ . ≤ . / XMM 4 0.25-0.46O’Hara et al. (2007) OHara07 archival Chandra 26 0.29-0.82Zhang et al. (2007) Zhang07 pilot LoCuSS XMM 4 0.27-0.3Vikhlinin et al. (2006) Vikhlinin06 archival Chandra / ROSAT ≤ . / Chandra 8 0.6-1Arnaud et al. (2005) Arnaud05 archival XMM 5 ≤ . a number of recent publications on scaling relation evolution( e.g. Ettori et al. (2004) and Maughan et al. (2006)). The sec-ond commonly used definition is to measure cluster propertieswithin regions with a redshift-independent value for the den-sity contrast ∆ , which was applied e.g. by Pacaud et al. (2007)and Kotov & Vikhlinin (2005). The expected evolution of scal-ing relations in this framework is similar to Eqs. 4, 5, and 6, al-beit omitting the ∆ z -factors. We are currently unable to decidebetween the two approaches on the basis of observational dataowing to the lack of su ffi ciently extensive and precise data setsneeded to catch the subtle di ff erences between the two models.In a forthcoming paper (B¨ohringer et al. 2011), we explore thisquestion by means of numerical simulations, finding that therecent formation approximation is imprecise and that the fixedoverdensity approach (not including the ∆ z factors) describes thesimulation results more accurately than the formulae includingthis extra term. For this paper we decided to explore both ap-proaches and find that the di ff erences are negligibly small forour conclusions. A number of corrections had to be applied to the subsamples inorder to correct for the slightly di ff erent methods used by theauthors.Throughout the cluster samples, we used both means ofdefining fiducial radii described in the previous section. Withinthese two schemes, various values of the mean overdensity areused. Most recent studies use a density contrast of 200, 500,or 2 500. The use of r , which approximately corresponds tothe virial radius, has the drawback that in many cases the areainside r is not fully covered by the available X-ray data. Inmany clusters, r typically corresponds to the most relaxedcentral part of the cluster and is also used in some publica-tions. The most common definition of the cluster radius thatis also used in this work is r (corresponding to either ∆ z = · ∆ vir ( z ) / ∆ vir ( z =
0) or a fixed ∆ = ff er-ent definitions and values of density contrast. For this rescalingscheme, the cluster density profiles were assumed to follow theisothermal β -model (Cavaliere & Fusco-Femiano 1976), since for the majority of the clusters included in the combined samplethe shape parameters β and r c are known. In the framework ofthe β -model, the mass enclosed within the radius r is given by M ( r ) = β k B TG µ m p r ( r / r c ) + ( r / r c ) . (7)Using Eqs. 7 and 1, the radius corresponding to a density contrastof ∆ z is given by r ∆ z = s β k B T µ m p H ( z ) ∆ z − r c . (8)For the β -model, the fractional luminosity within r is L X ( < r ) / L tot = π S f tot " − (1 + ( r / r c ) ) / − β β − , (9)where S designates the central surface brightness and f tot the to-tal X-ray flux. Using these equations, a simple correction schemecan be applied to the cluster observables. First x = r / r c , the ra-dius in units of r c for which the cluster properties are given is cal-culated using Equ. 8, where ∆ z is calculated by means of Equ. 2.The radius corresponding to the density contrast to which thecluster properties will be rescaled, x = r / r c , is calculated in thesame way. Using Eqs. 7 and 9 we obtain the expressions M ( x ) / M ( x ) = x + x / x + x (10) L X ( x ) / L X ( x ) = − (1 + x ) / − β − (1 + x ) / − β . (11)The cluster properties are then multiplied by the correction fac-tors obtained by means of Eqs. 10 and 11.Some publications give cluster X-ray luminosities in eitherthe 0.1-2.4 keV or 0.5-2 keV energy band. For the combinedcluster sample, bolometric X-ray luminosities, that is cluster lu-minosities in the 0.01-100 keV-band, are used. The band lumi-nosities were converted to bolometric values by means of theX-ray spectral fitting package XSPEC (Mewe et al. 1985), as-suming a Mekal model with an ICM metallicity of 0 . Z ⊙ .It has been shown ( e.g. Markevitch (1998), Pratt09) thattighter scaling relations involving X-ray luminosity are obtained
Reichert et al.: Evolution of X-ray scaling relations by excising the cluster core region. For the combined sampleselected in this work, however, the cluster observables for thelocal sample are given in a way that allows direct comparisonto distant clusters for which core excision is not always feasi-ble. Therefore, luminosities in the entire r < r aperture areconsidered, the emission from central regions is not excludedor replaced by any extrapolated profile to make the results oflocal samples comparable to the distant cluster studies. In thecase of the ICM temperatures, obtaining a homogeneous datasetis less straightforward because in some studies of local clustersonly core-excluded temperatures are given. Compiling a com-bined sample directly from studies of clusters where the core isincluded and others with core-excluded temperatures can leadto systematic bias. The source of this bias is the diversity ofthe central ICM temperature profiles that a ff ects the di ff erencebetween core-included and core-excised temperatures. In mostcases, non-CC clusters have a flat central temperature profile butCC clusters with a rather pronounced cool core (CC) have a cen-tral temperature is between about one-third and one-half of thesurrounding regions ( e.g. Peterson et al. (2003)). The fraction ofCC clusters therefore determines the magnitude of the inducedbias.Subsamples with both core-included and core-excised tem-peratures available allow us to estimate the errors caused by theinhomogeneous measurement schemes. The sample of Pratt09consists of 31 low redshift clusters and was designed to be mor-phologically unbiased, hence its relative error introduced by us-ing core-excluded instead of core-included temperatures wasfound to be 7%. This value was added to the estimated temper-ature errors for samples with only core-excluded temperaturesavailable (Andersson10, Zhang08, Zhang07, Arnaud05). For thecluster SPT-CL J2106-5844 (Foley et al. 2011), an emission-weighted core-included temperature of T = . −
70% ( e.g.
Hudson et al. (2010)). No consensus has emerged yet on howthis CC fraction evolves with redshift. While Vikhlinin et al.(2009) find a decrease in the CC fraction from ∼
70% locallyto ∼
15% at z > .
5, Santos et al. (2008) find that the fractionof CC clusters at high redshift ( z ∼ . − .
2) is very similar tothe local value (with an absence of very strong CCs). In sum-mary, using inhomogeneous temperature measurement schemesthroughout the combined sample is believed to bias the evolutionresults only marginally.The values of most cluster properties depend on the assumedcosmological model. In all publications used to compile ourcombined cluster sample, the model of choice is the standard Λ CDM scenario. However, the value assumed for the Hubbleconstant H = h km s − Mpc − varies slightly from h = . h = .
73 within the list of source publications considered.The e ff ects of this change slightly a ff ect the values of the ba-sic cluster properties. To obtain comparable cluster subsamples,a common h of 0.7 is chosen and the necessary corrections areapplied to the subsamples with a di ff erent h .Cluster mass depends on h as M ∝ h − , (12)while for the bolometric X-ray luminosities L X ∝ d L ∝ h − . (13) The cluster properties were rescaled accordingly, e.g.M / M = / = .
043 and L / L = (73 / = . The cluster samples obtained in typical X-ray surveys are notstrictly volume-limited but rather, at least approximately, flux-limited since the limited observation time, and detector areaand sensitivity generally only enables the detection of objectsbrighter than a certain flux limit f min . Various selection biasescomplicate the analysis of these flux-limited samples and haveto be taken into account when determining scaling relations andtheir evolution with redshift. For a well-controlled, homoge-neous survey, the cluster selection function, i.e. the probabilityof detecting a cluster with given properties taking into accountthe adopted observation strategy, can be modeled. A realistic se-lection function enables us to correct for selection e ff ects in amore consistent and exact way than a survey for which only anapproximate flux limit is known. An example of the applicationof this correction strategy can be found in Pacaud et al. (2007).However, the sample used in our work is highly heterogeneous,including clusters from numerous di ff erent surveys. For the ma-jority of these surveys, not all information necessary to recon-struct the survey selection function with high accuracy is avail-able. Therefore, a di ff erent strategy has to be applied to obtain atleast a realistic estimate of the influence of selection e ff ects forthis sample.The approach adopted here consists of simulating a clus-ter population with as realistic as possible properties and se-lecting a cluster sample comparable to the observed one fromthis population. In this situation, in contrast to observed clus-ter samples, the characteristics of the underlying cluster popu-lation are known and the properties of the selected sample canbe compared to those of the entire population to estimate se-lection e ff ects. To probe the selection bias in the local scalingand the evolution of the L X –T relation, a temperature function n T ( T , z ) was assumed. Since no su ffi ciently exact measured tem-perature function was available, n T ( T , z ) was deduced from themore tightly constrained luminosity function n L ( L , z ) by multi-plying it with the determinant d L / d T and converting X-ray lumi-nosities to temperatures by means of the L . − . –T relation.This method is only exact for the unrealistic case of no intrinsicscatter about the scaling relation, but provides a su ffi ciently ex-act approximation of the cluster temperature function to obtaina rough estimate of the selection bias. We note that for clusterdiscoveries and the related selection e ff ects the luminosities inthe X-ray observatory’s detection band rather than the bolomet-ric X-ray luminosities used throughout this work are relevant.Since the majority of combined sample clusters originate fromthe ROSAT surveys, ROSAT band luminosities (0.1 - 2.4 keVobserver frame) are used throughout this section. The numberof clusters in a given redshift bin is therefore determined by thetemperature function and the solid angle covered by the survey.Cluster luminosities were calculated assuming a non-evolving L . − . –T relation since this evolution model rep-resents a fair first-order approximation to the observationaldata. The ROSAT band L . − . –T relation given in Pratt09, L . − . = . · T [keV]) . erg s − , was used for thispurpose. The luminosities were then displaced from the meanrelation assuming a log-normal scatter of 0 .
25 dex ( ∼ eichert et al.: Evolution of X-ray scaling relations 5 Table 2. L X –T relations derived from the simulated local sam-ples. Sample 1 f min = · − erg s − cm − , sample 2 f min = · − erg s − cm − , sample 3 f min = · − erg s − cm − . Theinput relation used to model the bolometric X-ray luminositiesof the simulated cluster population is given for comparison. input relation L X = . · ( T [keV]) . erg s − sample L X –T relation L X = (0 . ± . · ( T [keV]) . ± . erg s − L X = (0 . ± . · ( T [keV]) . ± . erg s − L X = (0 . ± . · ( T [keV]) . ± . erg s − tight correlation between mass and temperature compared to thegreater intrinsic scatter in the L X –T relation. As for the observedcluster sample, a temperature threshold of T min = L min ( z ) resulting from the flux limit as L min ( z ) = π d L f min / K ( z , T ) , (14)where d L is the luminosity distance and K ( z , T ) the k-correctionquantifying the relation between observer-frame band lumi-nosity and cluster rest-frame band luminosity, i.e. K ( z , T ) = L obs / L rest .The influence of selection e ff ects on the observed local scal-ing relations was estimated by means of flux-limited samples se-lected from the simulated cluster population with limiting fluxesof f min = · − erg s − cm − (sample 1), 1 · − erg s − cm − (sample 2), and 1 · − erg s − cm − (sample 3), covering theredshift range used to fit the local relations, i.e. < z < . i.e. ∼
100 clusters. The bolometric X-ray luminosities of the clusterpopulation were assumed to follow the input L X –T relation ofEqu. 17. We then fitted L X –T relations to the three samples us-ing the BCES(L | T) method (Akritas & Bershady 1996). Table 2summarizes the derived relations.In summary, the characteristics of the selection e ff ects for lo-cal scaling-relation fits depend on whether the flux limit of thesample cuts away a significant fraction of the luminosity func-tion. For the faintest of the three samples described above (sam-ple 3), this is not the case, and consequently selection bias isnegligible for this sample. Most of the clusters within the z < . f min = · − erg s − cm − and the f min = · − erg s − cm − -sample. As is clearly visible in the L X –T relations fitted to thesesamples, the resulting bias for this range of flux limits is fairlyinsensitive to the exact limiting flux. This situation justifies acommon bias estimate for the entire local sample used in thiswork. According to the simulated samples, the measured slopeof the local L X –T relation is decreased only slightly by selec-tion bias, whereas the normalization is raised by almost 100%.The selection bias in the measured evolution of the L X –T rela-tion with redshift displays similar trends, i.e. a higher normaliza- L ob s / L z = ( T ) z Fig. 1.
Evolution bias of the L X –T relation: Flux-limited sam-ples with f min = · − erg s − cm − (green), f min = · − erg s − cm − (orange), f min = · − erg s − cm − (black),and combined sample (blue curve). The bias for the combinedsample rescaled to remove the bias in the local relation is plottedin red. The self-similar prediction for the evolution is plotted ingrey.tion caused by selection e ff ects, and is analyzed in greater detailthroughout the remainder of this section.The simulated counterpart of the combined cluster sampleused in this work was constructed from an all-sky survey with aflux limit of f min = · − erg s − cm − , representing the clus-ter samples based on the ROSAT All-Sky Survey (RASS), e.g. the REFLEX (B¨ohringer et al. 2004) survey. A second samplewith a flux limit of f min = · − erg s − cm − and an area of400 square degrees was added, representing the clusters fromROSAT PSPC-based surveys such as WARPS (Perlman et al.2002) or the 400sd (Burenin et al. 2007) survey. As a third com-ponent, a sample with f min = · − erg s − cm − , an area of 80square degrees and a minimum redshift of z min = . X –T relation can be estimated. To achievethis, we calculated the logarithmic mean of the bolometriccluster luminosity divided by the luminosity resulting from thebolometric L X –T relation at the cluster temperature in redshiftbins. This number quantifies the selection bias, with a value of1 corresponding to no selection e ff ects. The value of the biascurve in a redshift bin with a width of ∆ z and centered on theredshift z is therefore given by B ( z ) = P clusters log (cid:18) LXA · Tb (cid:19) N clusters , (15)where the subscript ”clusters” designates all clusters within [ z − ∆ z ; z + ∆ z ] included in the flux-limited sample and A and b quantify the normalization and slope of the local L X –T relationand were set according to Equ. 19.The bias curve deduced for the simulated combined sampleis indicated with a blue-dashed line in Fig. 1. However, in theredshift range used to fit ”local” scaling relations, 0 . < z < . ff ects are already non-negligible, visible in a clear de-viation of the bias curve from 1. A value of 1 on the vertical axis, Reichert et al.: Evolution of X-ray scaling relations theoretically corresponding to no bias, therefore already includesthe bias present in the local sample used to fit the L X –T relation.This e ff ect is redshift-independent because the same L X –T rela-tion is used for clusters at all redshifts to compare their measuredluminosity to the expected one. To distinguish the component ofthe selection bias that may mimic evolution of the L X –T rela-tion, the e ff ects of the local bias have to be taken into account.The local scaling relation bias leads to a division of cluster lumi-nosities by a higher expectation value than the unbiased relation.To determine the evolution bias relative to this higher expecta-tion value and not relative to the underlying cluster population,the bias curve has to be divided by the mean bias in the redshiftrange that was used to fit the ”local” scaling relation. The biascurve was therefore rescaled by a factor of 0.8, corresponding tothe red-dashed curve in Fig. 1. To summarize, the rescaled biascurve shows the additional redshift-dependent bias of the loga-rithmic mean of the bolometric cluster luminosity divided by theluminosity resulting from the local L X –T relation relative to thealready bias-a ff ected local sample.The bias curves of Fig. 1 have various characteristics that caneasily be explained in terms of the underlying cluster sample.The non-rescaled bias is generally greater than 1 because forany flux-limited sample in the presence of scatter more clustersbelow the mean relation than above it are too faint to be includedin the sample. With increasing redshift, the fraction of the clusterpopulation that is excluded for being too faint increases, causingan increase in the fraction of clusters above the mean L X –T rela-tion that have no counterpart below the relation. Hence, for a sin-gle flux limit the bias increases with redshift. This trend is visiblein the bias curves for the three flux-limited subsamples, f min = · − erg s − cm − (green), f min = · − erg s − cm − (orange),and f min = · − erg s − cm − (black). Up to a certain red-shift threshold, the influence of selection bias on the subsam-ples is negligible because no significant part of the cluster pop-ulation is excluded from the sample owing to the flux limit.Naturally, the unbiased redshift range increases with the sur-vey sensitivity from about z ∼ .
05 for the sample with f min = · − erg s − cm − to z ∼ . f min = · − erg s − cm − and z ∼ . f min = · − erg s − cm − .Surveys with a high limiting flux display a faster increase in biaswith redshift than deeper surveys.The characteristics of the mean bias curve for the combinedcluster sample (blue curve in Fig. 1) are a ff ected by the domi-nant contribution in terms of cluster detections with increasingredshift shifting from the all-sky f min = · − erg s − cm − -survey to the more sensitive but smaller solid-angle serendipi-tous surveys. The combined bias curve therefore decreases whenthe contribution from the f min = · − erg s − cm − -survey be-comes dominant at about z ∼ .
15. At z ∼ .
8, the curve dropsagain because from there toward higher redshift the contribu-tion of the f min = · − erg s − cm − -surveys dominates. From z = . X relation was not determined indepen-dently but based on the results of the L X –T relation. Clustermasses were set according to the local M–T relation (Equ. 16)and the ICM temperatures of the simulated cluster sample out-lined before. The logarithmic mean of the cluster masses dividedby the masses expected from the M–L X relation (Equ. 18) wasthen calculated for the simulated sample to obtain a bias curveanalogous to the one derived for the L X –T relation. By construc-tion, the bias curves for the M–L X relation show the same fea-tures as those for the L X –T relation. However, the bias curve is σ i n t [ de x ] z Fig. 2.
Redshift-dependent bias in the measured intrinsic scatter.Red-dashed line: true intrinsic scatter of the simulated samples.Black: fitted intrinsic scatter for f min = · − erg s − cm − .Red: fitted intrinsic scatter for f min = · − erg s − cm − .inverted and the selection e ff ects generally lead to an underesti-mation of the mean mass for a given luminosity.The intrinsic scatter in the cluster observables about themean scaling relations is of great importance to the interpreta-tion of results about scaling relation evolution. While the intrin-sic scatter for local clusters is roughly known (see Pratt09), itsredshift-dependence has not been well-constrained. Measuringthe intrinsic scatter at high redshifts is challenging since in addi-tion to the intrinsic scatter, the observed scatter in the evolutionplots of Sect. 3.2 has a number of other causes. First of all, mea-surement errors naturally have an influence on the observed scat-ter. Furthermore, increased scatter can also result from a changein the scaling-relation slope with redshift. Finally, selection ef-fects of flux-limited cluster samples may have an influence onthe observed scatter.To estimate this bias in the L X –T relation, we measured theintrinsic scatter of various simulated cluster samples. The sim-ulated cluster population has an intrinsic scatter in luminosityof 0.25 dex ( ∼ f min = · − erg s − cm − and f min = · − erg s − cm − were selected. The intrinsic scatter of the samples was then fit-ted in redshift bins centered around z = .
1, 0.3, 0.7, 0.9, 1.1, and1.3. The results are shown in Fig 2.For all subsamples, the fitting routine generally underesti-mates the real intrinsic scatter. The fitted scatter for local clus-ter samples is about 0.22 dex and comparable for the two sam-ples. As more and more luminous clusters are excluded by theflux limit at higher redshifts, the fitted scatter shows a decreas-ing trend. As expected, the decrease is more rapid with red-shift for the f min = · − erg s − cm − -sample than the deeper f min = · − erg s − cm − -sample. For the most distant sub-samples at both flux limits ( z ∼ . i.e. the selection bias causes the scat-ter to be underestimated by about 50%. eichert et al.: Evolution of X-ray scaling relations 7
3. Results
The results for scaling relations fitted to observed clus-ter samples may di ff er significantly depending on the fittingscheme used. In this work, we use the BCES fitting method(Akritas & Bershady 1996) that has been widely used in re-cent studies and correctly accounts for intrinsic scatter aboutthe mean relation and inhomogeneous measurement errors.However, several slightly di ff erent variations of this method ex-ist. When choosing one of these alternatives, it is most importantto distinguish between a fundamental independent and depen-dent variable. For the cosmological applications related to the re-sults of this study, cluster mass is the fundamental property, andluminosity and temperature take the role of dependent variables.Therefore for the M–T and M–L X relation, the BCES(T | M) andBCES(L | M) method is used in this work. Owing to the largeintrinsic scatter in the M–L X relation, the ICM temperature dis-plays a tighter correlation with cluster mass than the X-ray lumi-nosity. As a consequence, for the L X –T relation the BCES(L | T)scheme is used.Only clusters with z < . ffi ciently large number of systems and improve the quality ofour statistical analysis but small enough to keep evolution e ff ectsto a negligible level. To determine the influence of evolutionarye ff ects, the fits were repeated with a maximum redshift of z = . z = .
3. The results derived with this lower redshiftthreshold are fully consistent with the sample at z < .
3, that iseven though the expected evolution factor is non-negligible at z = .
3, no obvious evolutionary e ff ects on the scaling fit for theentire local sample are observed. However, the statistical errorsincrease significantly with redshift because of the smaller samplesize. The cluster properties were not rescaled by any assumedevolutionary model before the fit.The scaling relations derived from the combined local clustersample are M = (0 . ± . · ( T [keV]) . ± . M ⊙ , (16) L X = (0 . ± . · ( T [keV]) . ± . erg s − , (17) M = (1 . ± . · ( L X [10 erg s − ]) . ± . M ⊙ , (18)where L X is the bolometric X-ray luminosity and all propertiesare considered to be within r . However, for the analysis ofscaling relation evolution with redshift, the relations derived byPratt09 were considered instead of the fitted scaling relations be-cause the former were derived from a more homogeneous sam-ple with well-known selection criteria, that have smaller relativeerrors. The L X –T relation of Pratt09 L X = (0 . ± . · ( T [keV]) . ± . erg s − (19)is consistent with our result in terms of both slope and normal-ization. No M–T relation is provided in Pratt09, but a fit to theirsample with the BCES(T | M) method yields M = (0 . ± . · ( T [keV]) . ± . M ⊙ , (20)which is consistent with our result. Pratt09 provide an L-M in-stead of an M–L X relation. Fitting the inverse relation to theirsample leads to M = (1 . ± . · ( L X [10 erg s − ]) . ± . M ⊙ , (21) which di ff ers slightly from our result at the < σ level.The local scaling relations derived by means of the di ff er-ent fitting methods and the relations by Pratt09 are shown inAppendix D. The central goal of this work is to obtain a clearer understandingof the redshift evolution of X-ray scaling relations. The figures inthis section help us to visualize these evolutionary trends. Theyshow the redshift-dependent distribution of cluster properties di-vided by the expected value assuming local scaling relations. Forthe L X –T relation, this means that all cluster luminosities for in-stance are divided by L z = ( T ), that is the luminosities inferredfrom the local L X –T relation at the measured cluster tempera-ture, and that we plot the quantity L obs L z = ( T ) as a function of redshift.Plotting the data this way, the properties of the local clustershave an approximately log-normal scatter around 1. A change inthe normalization of the scaling relation with redshift translatesinto similar scatter around a di ff erent mean value in the evolu-tion plot, whereas a change in the slope would result in a largerscatter around the mean value. We note, however, that this is nota very suitable test for changes in slope as a greater intrinsicscatter at earlier times also leads to larger scatter in the evolu-tion plot and as outlined in Sect.2.4, selection biases may leadto an underestimation of the scatter of up to 50%. To investigatechanges in the slope of high-z scaling laws, relations were fittedto the available z > . E ( z ) in thecase of fixed overdensity. To test these predictions, a power-law E ( z ) α was fitted to all cluster data points with z > . i.e. the unknown exponent α in M obs M z = ( T ) = E ( z ) α (22)was constrained by fitting M obs M z = ( T ) versus E ( z ) in log-log-space.In the redshift range 0 . < z < .
6, an estimate of the influenceof selection biases on the evolution results is challenging, since,in contrast to the more distant systems, the properties of thesesystems were obtained from both shallow surveys with a veryhigh influence of selection biases but also deeper recent surveys.Therefore, this redshift range was excluded from the evolutionfits for the L X –T and M–L X relation, for which selection biasesplay a more critical role than for the M–T relation. The excludedredshift range contains 45 sample clusters, while 65 clusters at z > . i.e. to avoid the result being exclusively determined bythe large number of relatively low-redshift clusters with smallerrors, the data points were weighted by the inverse numberof clusters in the corresponding redshift bin ( ∆ z = . Reichert et al.: Evolution of X-ray scaling relations M ob s / M z = ( T ) z Arnaud05Vikhlinin06Pratt09Zhang08Kotov05Mantz09Zhang07Pacaud07 M ob s / M z = ( T ) z Maughan06Ettori04Hicks08z>0.8 sampleself-similar evolution *E(z) -1 best fit evolution *E(z) -1.04 ± Fig. 3.
Redshift evolution of the M–T relation. Black-dashed line: self-similar prediction ( ∝ E ( z ) − ). Continuous red line: best-fitevolution ( ∝ E ( z ) − . ± . ). Table 3.
Evolution results based on the combined cluster sample.First column: Scaling relation. Second column: Observed evolu-tion of scaling relations, bias e ff ects have been accounted for bygreater uncertainties. Third column: Evolution results includinga tentative selection-bias correction. Fourth column: Self-similarexpectations and self-similar predictions relation observed evolution bias-corrected self-similarM–T ∝ E ( z ) − . ± . ∝ E ( z ) − L X –T ∝ E ( z ) − . + . − . E ( z ) − . ± . ∝ E ( z ) + M–L X ∝ E ( z ) − . + . − . E ( z ) − . ± . ∝ E ( z ) − / We first discuss the M–T relation, which is expected amongall relations to most closely follow the self-similar predictions.Figure 3 shows the redshift evolution of the M–T relation for thecombined cluster sample. The best fit to the data correspondsto a redshift dependence of the normalization proportional to E ( z ) − . ± . , which is consistent with the self-similar predictionof E ( z ) − . We note that selection bias is not taken into accountin this plot. However, for the the M–T relation this is not as im-portant as for scaling relations including cluster luminosity. Ananalysis similar to the one shown in Fig. 3 was performed withcluster radii defined with a variable density contrast ∆ z instead of a fixed ∆ at cluster redshift. The results are consistent withinthe errors with those of Fig. 3, i.e. using the redshift-dependentdensity contrast does not significantly influence our results aboutthe evolution of the M–T relation. The M–T relation fitted to the z > . ffi ciently good X-ray data has a slope of M ∝ T . ± . , which is fully consistent with the local result.Figure 4 shows the redshift evolution of the L X –T relation for thecombined cluster sample. The best-fit relation for the evolutionis E ( z ) − . ± . , that is there is a slightly negative evolution. Thisresult is clearly inconsistent with the self-similar prediction thatthe normalization increases with redshift in proportion to E ( z ) + .The evolution result was uncorrected for the estimated selec-tion bias (see Sect.2.4) because this bias estimate relies on a toymodel that is only approximately comparable to the real clustersample. The error budget was instead increased by the estimatedbias, i.e. the confidence region was enlarged by the size of theestimated bias (see Fig. 1) in the direction of the supposed biascorrection. This led to a final evolution result of E ( z ) − . + . − . . Asexpected, applying an approximate bias correction based on therescaled bias curve of Fig. 4 before the fit as a test of the influ-ence of selection biases results in an even more negative evolu-tion result of E ( z ) − . ± . .The slope of the L X –T relation fitted to the z > . ffi ciently accurate X-ray data available is L ∝ T . ± . .This result is slightly steeper but still consistent within the errorswith the local slope derived by Pratt09. Owing to the small clus- eichert et al.: Evolution of X-ray scaling relations 9 L ob s / L z = ( T ) z Pratt09Zhang08Mantz09Zhang07OHara07Kotov05Branchesi07Pacaud07 L ob s / L z = ( T ) z Maughan06Ettori04Hicks08Andersson10z>0.8-sampleself-similar evolution *E(z) best fit evolution *E(z) -0.23 +0.12-0.62 rescaled bias
Fig. 4.
Redshift evolution of the L X –T relation. Black-dashed line: self-similar prediction ( ∝ E ( z )). Continuous red line: best-fitevolution ( ∝ E ( z ) − . + . − . ). Red-dashed line: estimated mean bias for the combined sample rescaled to remove the e ff ects of bias inthe local scaling relation.ter sample and the large errors, this result heavily depends onthe fitting method used and therefore provides no significant ev-idence of a steepening of the high redshift L X –T relation. As forthe M–T relation, the use of a redshift-dependent density con-trast ∆ z instead of a fixed ∆ leads to comparable results withsimilar scatter about the mean relation.Figure 5 shows the redshift evolution of the M–L X rela-tion for the combined cluster sample. The best-fit relation is E ( z ) − . ± . , i.e. negative evolution as predicted by the self-similar model. However, our result is significantly less steepthan the self-similar prediction of E ( z ) − / . As for the M–T relation, the estimated selection bias is taken into accountin the error budget and leads to the evolution being propor-tional to ∝ E ( z ) − . + . − . . This observed evolutionary trend isclose to the one expected after combining the results for theevolution of the L X –T and M–T relations, which would be ∝ E ( z ) − . . Applying an approximate bias correction to test theinfluence of selection biases as for the L X –T relation results ina slightly less negative evolution fit of E ( z ) − . ± . . In Pratt09,a bias-corrected local L X –M relation is provided. Using the in-verted bias-corrected BCES(L | M)-relation ( M = (1 . ± . · ( L X [10 erg s − ]) . ± . ) and correcting for the total estimatedevolution bias (not the curve rescaled to remove the e ff ects ofbias in the local scaling relation) leads to an evolution result of E ( z ) − . + . − . . The estimated errors in this result include statisti- cal errors and an estimate of the systematical error caused by aninexact bias correction.The slope of the fitted high redshift M–L X relation is M ∝ L . ± . , which is consistent with the local result. Using the ∆ z -scheme instead of a redshift-independent density contrast againleads to similar results. The observed scatter in cluster propertiesabout the mean relation for the high redshift clusters is consis-tent with the local scatter in all three relations. The real scatterabout the L X –T and M–L X relation for distant clusters may beup to a factor of two larger than the observed result because ofthe influence of selection biases (see Sect. 2.4). However, owingto conservative error estimates, no constraints on the intrinsicscatter in cluster properties can be placed based on the measuredtotal scatter for the distant cluster sample.
4. Discussion
The results on scaling relation evolution presented in the previ-ous section were obtained by means of a number of input as-sumptions and results of preceding studies that have an influ-ence on the obtained results and may introduce additional errors.Throughout the remainder of this section, we briefly discuss thestability of the results under these assumptions. M ob s / M z = ( L ) z Zhang08Pratt09Kotov05Mantz09Zhang07Pacaud07 M ob s / M z = ( L ) z Maughan06Ettori04Hicks08z>0.8 sampleself-similar evolution *E(z) -7/4 best fit evolution *E(z) -0.93 +0.62-0.12 rescaled bias
Fig. 5.
Redshift evolution of the M–L X relation. Black-dashed line: self-similar prediction ( ∝ E ( z ) − / ). Continuous red line: best-fitevolution ( ∝ E ( z ) − . + . − . ). Red-dashed line: estimated mean bias for the combined sample rescaled to remove the e ff ects of bias inthe local scaling relation.The assumed local scaling relations have a direct influenceon the observed evolution. For our analysis in Sect. 3.2, the red-shift evolution of scaling relations was determined using the lo-cal scaling relations of Pratt09. Using the relations derived forthe entire local combined sample (see Sect.3.1) instead of theseresults does not lead to fundamentally di ff erent findings on theevolution coe ffi cient. For the M–T relation, using the relationderived from the combined sample leads to a best-fit evolutionof E ( z ) − . ± . , which is only slightly di ff erent from the resultfor the Pratt09 relation ( E ( z ) − . ± . ). Using the L X –T relationfitted to our sample implies an evolution result of E ( z ) − . ± . instead of E ( z ) − . + . − . and for the M–L X relation E ( z ) − . ± . instead of E ( z ) − . + . − . , both of which are fully consistent withthe results presented above for both relations.An incomplete or incorrect homogenization scheme appliedto the di ff erent subsamples naturally influences the evolution re-sults. However, the combined cluster sample provides no hintsthat this might be a major problem (see Appendix C). Incorrecthomogenization would be visible as larger scatter about themean behavior in the cluster sample or di ff erent evolutionarytrends for di ff erent subsamples. Taking into account selectionbiases, these significant trends are not observed for the clustersample (see Sect.4.2).Our study again highlights the importance of selection bi-ases when investigating scaling relation evolution and the prob- lems inherent to small cluster samples over a limited redshiftrange. Although the simulated cluster sample used to estimatebias e ff ects in this study is only a rough approximation of thetrue situation, it reveals the apparent evolutionary trends causedby selection e ff ects, which have been taken into account in theestimated errors. Despite the lack of knowledge about the exacte ff ects of selection bias in a highly inhomogeneous combinedcluster sample such as ours, at least a fair estimate of the influ-ence on the evolution results can be given. For both the M–L X and the L X –T relations, a bias correction of the evolution resultswould render the di ff erence to the self-similar predictions evenmore significant. Our finding about the inconsistency with theself-similar model can therefore not be attributed to selection ef-fects.The X-ray properties of the systems within a cluster sam-ple and their evolution with redshift might depend on the clus-ter selection strategy. While the subsamples of clusters selectedby means of the Sunyaev-Zel’dovich e ff ect (SZ) and their opti-cal / infrared (IR) properties are not su ffi ciently extensive to placeindependent constraints on scaling relation evolution, the X-ray,SZ,and optical / IR-selected subsamples display no obvious dif-ferences in their evolutionary trends.In recent studies, we note that di ff erent definitions of the den-sity contrast ∆ and the resulting cluster radii have been used.However, the choice of either a fixed ∆ at the cluster redshift or eichert et al.: Evolution of X-ray scaling relations 11 a redshift-dependent ∆ z has no significant influence on the deter-mined evolution results. We now briefly discuss previously published results on scalingrelation evolution comparing these to our findings:Kotov05: For the M–T relation, their evolution result of ∝ E ( z ) − . ± . is consistent with both our results and the self-similar expectation. For the L X –T relation, Kotov05 find thenormalization to be ∝ (1 + z ) . ± . , i.e. a positive evolu-tion that is even steeper than self-similar. This trend is eas-ily visible for their sample in Fig. 4. The seven clusters ofthe Kotov05 sample cover a redshift range of 0 . < z < . ff ects, in addition to both limited sample size andredshift range. We note that the evolution result derived inKotov05 approximately traces the bias curve for the samplewith f min = · − erg s − cm − visible in Fig. 1. Althoughno uniform flux limit can be assigned to this archival sam-ple, f min = · − erg s − cm − represents an appropriateestimate of the mean flux limit, i.e. the bias estimate sug-gests that after correcting for selection e ff ects the observedevolution should be close to zero.Maughan06: Their result for the L X –T evolution is ∝ (1 + z ) . ± . when using the local relation of Arnaud & Evrard(1999) and ∝ (1 + z ) . ± . for the Markevitch (1998) re-lation, i.e. a slightly positive evolution. Using the scalingrelation derived by Pratt09 instead removes this evolution-ary trend, causing the evolution of the Maughan06 subsam-ple ( ∝ E ( z ) − . ± . ) to be consistent with the results of ourstudy.OHara07: Their result about the evolution of the core-included L X -T relation within r is E ( z ) (1 + z ) − . + . − . , i.e. consistentwithin the errors with our result.Branchesi07: No significant evolution of the L X –T relation inthe redshift range 0 . < z < . ∼ ff erent local scaling relation.Pacaud07: Since a detailed selection function was derived inthis work, this analysis provides an important insight intothe influence of selection e ff ects. Before correcting for those,their result about the evolution of the L X –T relation isroughly similar to that of Kotov05. Afterwards, they obtainan evolution factor of ∝ E ( z ) (1 + z ) − . + . − . , i.e. slightly lessthan self-similar. This result is still marginally inconsistentwith ours. We note, however, that a significant part of theirsample was not included here because the temperatures werebelow 2 keV.Ettori04: Their inferred evolution of the L X –T relation variesdrastically depending on whether clusters below z < . ∝ (1 + z ) . ± . for theentire sample and ∝ (1 + z ) . ± . if only clusters with z > . z > . ∝ E ( z ) − . ± . ). The markeddi ff erence between the clusters at z < . ff ects,that is to the fact that the low-z sample consisting mostly ofarchival clusters detected in relatively shallow observations,as discussed above for the results of Kotov05, in accordancewith which Ettori04 find the M–T evolution to be consistentwith the self-similar prediction.Vikhlinin et al. (2009): When applying X-ray galaxy clusterdata to cosmological tests these authors also provide a newevaluation of the L X –M relation and its evolution with red-shift in their Equ. 22. In contrast to our calculations, this re-lation is determined for luminosities in the 0.5 - 2 keV band.Using the results of Pratt about the di ff erence between thebolometric and 0.5 - 2 keV band scaling relations for com-parison, we find that the results of Vikhlinin et al. and oursare in very good agreement for the zero redshift relation.Analyzing the redshift-dependent term in the relation infer-ring L X from M, we find a term of E ( z ) . ± . for Vikhlininet al. and a term of E ( z ) . + . − . by inverting our Equ. 26.There is again good agreement within the large error bars.Taking the mean trends of both relations, we find a di ff er-ence in the evolution parameter of ∼
5% at redshift z = . ∼
10% at z = .
5. Therefore, the di ff erence between thetwo relations is smaller than can be detected with any currentdata set.Leauthaud et al. (2010): In their study, the M –L X relationfor 206 X-ray-selected galaxy groups from the COSMOSsurvey is determined in the redshift range 0 . < z < . ∝ E ( z ) − . ± . is consistent with our result.In summary, the present study provides a clearer picture ofthe scaling-relation evolution in three aspects. The first is thatthe use of the very early results by Markevitch (1998) andArnaud & Evrard (1999) as a local reference of the scaling rela-tions introduces a positive evolutionary trend as already shownin Branchesi07. Using more recent results with higher qualitystatistics, and in particular the use of the weakly biased (not flux-limited) REXCESS sample results by Pratt09 removes most ofthis trend. The second aspect is that the overview of a larger setof cluster samples with di ff erent biases leads to the identificationof bias e ff ects from high flux-limits. Thirdly, the extension of theredshift range to newly detected high-redshift clusters increasesthe leverage for the evaluation of evolutionary e ff ects.
5. Implications of results
The observed modification of the evolution of scaling relationswith respect to the self-similar model is caused by changes inthe thermodynamical state of the ICM. Therefore, di ff erent evo-lutionary trends are the signature of di ff erent histories of heatingand cooling processes in the clusters. Comparing observationalresults with the predictions made by hydrodynamical cosmolog-ical simulations permits us to assess whether the heating schemeimplemented in the simulations is realistic. A variety of exist-ing simulations taking into account di ff erent sources of heatingand assumptions on the time evolution of non-gravitational heat-ing and cooling allow a constraint of the most realistic heatingscheme.Previous attempts these comparison analyses were compli-cated by the inconsistent observational results about scaling re-lation evolution. Owing to updated local scaling relations basedon morphologically unbiased cluster samples and the availabil- ity of a larger cluster sample over a wider redshift range, evolu-tion constraints derived from the combined cluster sample usedin this work permit a meaningful comparison. The simulationsconsidered on that account are the Millennium Gas Simulations(MGS, Stanek et al. (2010), Short et al. (2010)), a series ofhydrodynamical resimulations of the original dark-matter-onlyMillennium simulations (Springel et al. 2005). The MGS runsincorporate a (500 h − Mpc) -volume with the same initial con-ditions and cosmological model as the original Millennium run,but include an equal number of gas particles in addition to the5 · dark matter particles.The MGS consist of three simulation runs with di ff erent im-plementations of heating schemes:GO run: This simulation does not include any additional non-gravitational heating sources. As a consequence, it mostly re-produces the self-similar model expectations but drasticallyfails to reproduce the observed local galaxy cluster X-rayscaling relations, hence is not considered in the comparisonanalysis.PC run: This run employs a simplistic preheating model, raisingthe entropy of the gas particles to 200 keV cm at z =
4. In ad-dition to preheating, radiative cooling according to the cool-ing function of Sutherland & Dopita (1993) is implemented.FO run: The third simulation run includes no radiative coolingbut incorporates a model of the energy input by supernovaand AGN feedback computed by means of a semi-analyticmodel of galaxy formation, i.e. a gradual injection of energyinto the surrounding ICM gas.The PC and FO runs represent two opposing heatingschemes, one assuming an early preheating of the gas be-fore its accretion onto the cluster, the other incorporating non-gravitational heating as a relatively recent and ongoing process.Both runs are able to reproduce the observed local scaling re-lations, i.e. they predict the correct properties of the ICM forlow-redshift clusters. However, owing to the contrasting heat-ing schemes, the two models make opposing predictions aboutthe thermal evolution of the ICM with redshift. Table 4 gives anoverview of the MGS evolution results fitted in the redshift range0 < z < . Table 4.
Results on the redshift evolution of X-ray scaling rela-tions from the Millennium Gas Simulations (Short et al. 2010).
GO run redshift evolution
M–T ∝ E ( z ) − · (1 + z ) . ± . L X –T ∝ E ( z ) · (1 + z ) . ± . M–L X ∝ E ( z ) − / · (1 + z ) . ± . PC run redshift evolution
M–T ∝ E ( z ) − · (1 + z ) − . ± . L X –T ∝ E ( z ) · (1 + z ) − . ± . M–L X ∝ E ( z ) − / · (1 + z ) . ± . FO run redshift evolution
M–T ∝ E ( z ) − · (1 + z ) . ± . L X –T ∝ E ( z ) · (1 + z ) . ± . M–L X ∝ E ( z ) − / · (1 + z ) − . ± . Depending on the selection criteria of the cluster sample, theadopted fitting scheme, and since the fitted slope and normaliza-tion of scaling relations are not independent of each other, themeasured normalization of local scaling relations di ff ers by upto a factor of two (see e.g. the results of Arnaud et al. (2005) M ob s / M z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( T ) z Fig. 6.
Redshift evolution of the M–T relation. Continuous redline and light grey confidence area: observed evolution. Green-dashed line and dark grey confidence area: MGS FO run. Blue-dashed line and dark grey confidence area: MGS PC run. Black-dashed line: self-similar prediction. L ob s / L z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 L ob s / L z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 L ob s / L z = ( T ) z 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 L ob s / L z = ( T ) z Fig. 7.
Redshift evolution of the L X –T relation. Red-dashed line:Estimated mean bias of the combined sample rescaled to removethe e ff ects of bias in the local scaling relation. Additional linesand confidence areas have the same meaning as in Fig. 6.and Pratt et al. (2009)). Therefore, instead of directly comparingthe redshift-dependent normalization derived from the MGS tothe results of this work, we do not take into account the di ff er-ent local normalizations but only the evolution with respect tothe local values. Fig. 6, 7, and 8 show the redshift evolution ofthe normalization of the M–T, L–T, and M–L X relations with re-spect to its local value for both the MGS PC and FO runs andthe observational results deduced in this work. We note that incontrast to this work, Short et al. (2010) assumed the self-similarevolution model and then fitted the observed di ff erence from thismodel as powers of (1 + z ).It is clearly visible that the FO simulation provides no gooddescription of the observed evolution for all three relations. Incontrast, the PC run shows good agreement with the observa-tions. For the M–T and M–L X relations, the PC results are fullyconsistent with ours within the errors. The prediction for the L X – eichert et al.: Evolution of X-ray scaling relations 13 M ob s / M z = ( L ) z 0.1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( L ) z 0.1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( L ) z 0.1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 M ob s / M z = ( L ) z Fig. 8.
Redshift evolution of the M–L X relation. Lines and con-fidence areas have the same meaning as in Fig. 7.T relation shows slight deviations from our result at the 2 σ levelfor z . .
2. This di ff erence can partly be attributed to the di ff er-ent functional form assumed for the evolution fit in Short et al.(2010).Care has to be taken when interpreting the constraints on theICM thermal history deduced from this finding because none ofthe MGS runs provide a complete model of the necessary ICMheating and cooling processes. As an example, the PC run doesnot include ongoing heating, which is known to be of crucial im-portance to balance the cooling in cluster cores. Such an additionof mild ongoing feedback to the model is likely to bring the pre-dicted evolution for the L X –T relation in even closer agreementwith the observed trend (see Fig. 7). Nevertheless, the observedevolution of X-ray scaling relations provides strong evidence ofthe preheating scenario. A further refinement of the simulations, e.g. the combination of preheating with mild ongoing feedback,is desirable to permit more detailed comparisons between theobserved and the simulated evolutions. eROSITA is the main instrument on the Spektrum-R¨ontgen-Gamma mission scheduled for launch in 2012 and will carry outa new all-sky X-ray survey in the energy range from 0.1 to 10keV (see Predehl et al. (2010) and Predehl et al. (2007)). One ofthe main science goals of the project is to provide a sample of ∼
100 000 X-ray selected galaxy clusters. A cluster catalogue ofthis size is necessary to test cosmological models to higher accu-racy and place reliable constraints on cosmological parameterssuch as the dark energy equation of state (Haiman et al. 2005).The achievable accuracy of the parameter constraints de-rived from the eROSITA all-sky survey depends on the numberof cluster detections and the knowledge of X-ray scaling rela-tions up to high redshift. Whether a cluster can be detected byeROSITA is mainly determined by the cluster’s soft band X-rayluminosity and its distance. Since the mean cluster luminosityat a given redshift depends on the evolution of X-ray scaling re-lations, the results of this work have a direct influence on theexpected number of eROSITA clusters to be detected at highredshift and therefore on the expected cosmological constrain-ing power of the mission. The expected redshift distribution and total number ofeROSITA clusters was recalculated by means of the clustercounter presented in chapter 8 of M¨uhlegger (2010). The clustercounter provides an estimate of the number of eROSITA clus-ter detections based on various simplifying assumptions. Theassumed criterion for a cluster detection is whether the num-ber of photons detected from the observed system exceeds thecount limit c lim . Until now, the eROSITA count limit has not beenknown in detail owing to its dependence on as of now unspeci-fied instrumental parameters such as the eROSITA point spreadfunction, and furthermore its significant dependence on the to-tal X-ray background, which varies with sky position, exposuretime and other conditions. A constant detection count limit of c lim =
100 is currently used as a conservative estimate.The number of clusters in each redshift and luminosity binwas set according to the luminosity function used in Sect. 2.4.This luminosity function is based on a cluster mass function re-sulting from a cosmological model consistent with the seven-year WMAP results (Komatsu et al. 2010) and was convertedinto a luminosity function by means of the L-M relation and theevolution results of Sect. 3.2. Further necessary input data forthe cluster counter are an all-sky exposure map for the specialgeometry of the eROSITA survey and an all-sky map of Galacticneutral hydrogen. The ICM temperature was set according to the(ROSAT band) L X –T relation of Pratt09 and the observed evolu-tion of Sect. 3.2.Thereafter, the eROSITA count rate at each position in thesky, cluster luminosity, and redshift is calculated by means ofXSPEC, assuming an absorbed Mekal model with the param-eters z, L X , T, and n H . The count rate is then converted into anumber of detected photons by multiplying it with the exposurevalue at the respective sky position. If the number of detectedphotons for a given set of parameters exceeds c lim , the number ofclusters determined by the luminosity function for the given red-shift and luminosity is added to the cluster number in the currentredshift bin and sky position. The resulting redshift distributionand total number of detected clusters is presented in the follow-ing section for three L X –T evolution models: The self-similarmodel, i.e. positive evolution, the no-evolution scenario assumedin M¨uhlegger (2010), and the slightly negative evolution foundin this work.Table 5 shows the total number of expected cluster detec-tions with eROSITA under the assumption of a count limit c lim = | b | > ◦ . The total number of achievable cluster detectionsis expected to be closer to this last number, since the high col-umn density of the absorbing Galactic interstellar medium andhigh density of other X-ray and stellar sources in the Galacticplane make cluster detections in this area challenging. For theextragalactic area, the number of expected clusters with z > . , , . , .
4, and 1.6 is also listed.As can be seen in Fig. 9, which shows the number of cluster de-tections with redshift within the redshift interval [ z − . z + . > z , a change in the assumed evolution model hasa significant influence on the number of eROSITA cluster de-tections. However, the total number of clusters that should bedetected is hardly a ff ected because the majority of the eROSITAclusters are expected to be discovered at low redshifts, wherethe scaling relation evolution is of little importance. As an ex-ample, the total number of clusters for the extragalactic areais decreased by ∼
5% when changing from the previously as-
Table 5.
Number of expected cluster detections with eROSITAassuming self-similar, no, or slightly negative evolution of theL X –T relation. The rows labeled ”extragalactic” refer to aGalactic latitude | b | > ◦ . A count limit of c lim =
100 was as-sumed. self-sim. N tot N z > . N z > N z > . N z > . N z > . all sky 132 787extragal. 97 195 5 834 2 062 699 227 69 no evol. N tot N z > . N z > N z > . N z > . N z > . all sky 120 965extragal. 88 238 4 297 1 414 447 135 38 best fit N tot N z > . N z > N z > . N z > . N z > . all sky 114 803extragal. 83 603 3 505 1 083 320 90 22 N C l u s t e r / π s r / ∆ z = . Fig. 9.
Achievable number of cluster detections with redshiftwithin the redshift interval [ z − . z + . | b | > ◦ ).Black: Self-similar L X –T evolution ( ∝ E ( z ) + ). Green: No L X –Tevolution. Red: Best fit L X –T evolutionsumed no-evolution scenario to the negative evolution found inthis work.The situation is clearly di ff erent for distant cluster detec-tions. Owing to the smaller number of luminous high-redshiftclusters in the negative evolution scenario, the number of ex-pected detections is smaller by ∼
18% at z = . ∼
28% at z = . z > .
6. Summary and conclusions
The main goal of this study has been to to investigate the redshiftevolution of galaxy cluster X-ray scaling relations by means ofa combined cluster sample. To this end, a cluster sample wascompiled from both recent publications and newly discoveredclusters provided by the XMM-Newton Distant Cluster Project(XDCP). Our gathered sample of recently discovered distantclusters has allowed tighter constraints to be made on scalingrelation evolution than possible in previous studies. N C l u s t e r w it h r e d s h i f t > z Fig. 10.
Achievable number of cluster detections with redshift > z for eROSITA. The lines have the same meaning as in Fig. 9.The definition of cluster observables slightly di ff ers betweenthe individual publications used to compile the combined clustersample. A homogenization scheme that accounts for these dif-ferent analysis schemes was therefore applied to the subsamples.In detail, the definition of cluster radii, either relying on a fixeddensity contrast ∆ at the cluster redshift or a redshift-dependentaverage overdensity ∆ z and the di ff erent values for the densitycontrast were corrected for. Furthermore, the applied correctionstake into account the slight di ff erences in the assumed cosmolog-ical parameters, in the energy band for which L X is given.On the basis of data for clusters at z < . X –T, and M–L-relations werefitted. The derived relations generally agree well with the pub-lished results of Pratt09 and show deviations from the self-similar model similar to those found in earlier studies, e.g. asteeper L X –T relation.Typical X-ray selected cluster samples can be approximatedto be flux-limited. Various selection biases complicate the anal-ysis of these flux-limited samples and are important to the in-terpretation of results on scaling relations and their evolution.The bias inherent to the sample used in this work was estimatedby means of comparable cluster samples selected from a simu-lated cluster population. In detail, selection biases were found toraise the measured normalization of the local L X –T relation anddecrease the apparent scatter about the mean relation for high-zclusters. For the L X –T relation, the bias appears to generate apositive evolution, whereas for the M–L X relation it generatesthe opposite.The redshift evolution of X-ray scaling relations was investi-gated in the redshift range 0 < z < .
46. Throughout this redshiftrange, no significant variation in the slope of the relations wasfound. The normalization, however, evolves with redshift for allthree examined relations. For the M–T relation, the measuredevolution is ∝ E ( z ) − . ± . , consistent with the self-similar pre-diction. The results for the L X –T ( ∝ E ( z ) − . + . − . ) and M–L X relation ( ∝ E ( z ) − . + . − . ), however, di ff er significantly from themodel predictions. As for the local relations, these deviations in-dicate that the influence of non-gravitational ICM heating andcooling is not negligible. The inconsistent results of recent stud-ies have been found to be caused by limited sample sizes and red-shift ranges in combination with selection biases and to a lesserdegree by the use of di ff erent local relations. eichert et al.: Evolution of X-ray scaling relations 15 On the basis of our results and the local scaling of Pratt09and assuming that h = .
70, the M–T, L–T, and M–L X relationsfor cluster properties within r have the explicit form M = (0 . ± . · ( T [keV]) . ± . · E ( z ) − . ± . M ⊙ , (23) L X = (0 . ± . · ( T [keV]) . ± . · E ( z ) − . + . − . erg s − , (24) M = (1 . ± . · ( L X [10 erg s − ]) . ± . · E ( z ) − . + . − . M ⊙ . (25)This work once again highlights the importance of selection bi-ases to scaling relation studies. For distant cluster mass estimatesbased on the bolometric X-ray luminosity L X , we recommendusing Equ. 26, which is based on the bias-corrected local relationof Pratt09 and includes a correction for the estimated selectionbias on the observed evolution, given by M = (1 . ± . · ( L X [10 erg s − ]) . ± . · E ( z ) − . + . − . M ⊙ (26)To provide tighter constraints on scaling-relation evolution andimprove the mass estimate of Equ. 26 in the future, a more homo-geneous, extensive distant cluster sample with a precisely knownselection function is necessary. Such a sample would allow amore precise bias estimate and correction to be made than pos-sible for the sample used in this study.Comparing the observed evolution with predictions made bythe Millennium Gas Simulations has allowed us to discriminatebetween di ff erent proposed scenarios and attempt a physical in-terpretation of the thermodynamic history of the ICM. The com-parison analysis strongly suggests an early preheating, i.e. an en-tropy increase for the gas particles before the infall of the ICMgas into the cluster potential well.The expected number of cluster detections for the upcom-ing eROSITA survey was recalculated taking into account theresults of this work. In general, the total number of achievabledetections is slightly lower than assumed before, while the num-ber of high redshift clusters to be detected shows a significantdecrease.Future more detailed studies of the redshift evolution of X-ray scaling relations will be important to more tightly constrainthe early thermodynamic history of the ICM and provide cali-brated mass-observable relations for upcoming large cluster sur-veys and their cosmological applications. Acknowledgements.
We acknowledge Ben Maughan for helpful suggestions andthe XDCP collaboration for providing some data prior to publication. We thankthe anonymous referee for helpful comments and suggestions. The work of thispaper was supported by the DfG cluster of excellence Origin and Structure ofthe Universe, by the German DLR under grant no. 50 QR 0802 and by the DfGunder grant no. BO 702 / References
Akritas, M. G. & Bershady, M. A. 1996, Astrophys. J., 470, 706Andersson, K., Benson, B. A., Ade, P. A. R., et al. 2010, arXiv:1006.3068Arnaud, M., Bohringer, H., Jones, C., et al. 2009, arXiv:0902.4890Arnaud, M. & Evrard, A. E. 1999, MNRAS, 305, 631Arnaud, M., Pointecouteau, E., & Pratt, G. W. 2005, Astronomy & Astrophysics,441, 893Boehringer, H., Mullis, C., Rosati, P., et al. 2005, The Messenger, 120, 33B¨ohringer, H., Pratt, G. W., Arnaud, M., et al. 2010, Astronomy & Astrophysics,514, A32 + B¨ohringer, H., Schuecker, P., Guzzo, L., et al. 2004, Astronomy & Astrophysics,425, 367Borgani, S. & Guzzo, L. 2001, Nature, 409, 39Branchesi, M., Gioia, I. M., Fanti, C., & Fanti, R. 2007,Astronomy & Astrophysics, 472, 739 Bremer, M. N., Valtchanov, I., Willis, J., et al. 2006, MNRAS, 371, 1427Brodwin, M., Stern, D., Vikhlinin, A., et al. 2010, arXiv:1012.0581Bryan, G. L. & Norman, M. L. 1998, Astrophys. J., 495, 80Burenin, R. A., Vikhlinin, A., Hornstrup, A., et al. 2007, Astrophys. J., Suppl.Ser., 172, 561Cavaliere, A. & Fusco-Femiano, R. 1976, Astronomy & Astrophysics, 49, 137Ettori, S., Tozzi, P., Borgani, S., & Rosati, P. 2004, Astronomy & Astrophysics,417, 13Fassbender, R. 2008, arXiv:0806.0861Fassbender, R., B¨ohringer, H., Santos, J. S., et al. 2011,Astronomy & Astrophysics, 527, A78 + Foley, R. J., Andersson, K., Bazin, G., et al. 2011, arXiv:1101.1286Gioia, I. M., Wolter, A., Mullis, C. R., et al. 2004, Astronomy & Astrophysics,428, 867Gunn, J. E. & Gott, III, J. R. 1972, Astrophys. J., 176, 1Haiman, Z., Allen, S., Bahcall, N., et al. 2005, arXiv:astro-ph / + Kaiser, N. 1986, MNRAS, 222, 323Komatsu, E., Smith, K. M., Dunkley, J., et al. 2010, arxiv:1001.4538Kotov, O. & Vikhlinin, A. 2005, Astrophys. J., 633, 781Lamer, G., Hoeft, M., Kohnert, J., Schwope, A., & Storm, J. 2008,Astronomy & Astrophysics, 487, L33Leauthaud, A., Finoguenov, A., Kneib, J., et al. 2010, Astrophys. J., 709, 97Lubin, L. M., Mulchaey, J. S., & Postman, M. 2004, Astrophys. J., Lett., 601, L9Mantz, A., Allen, S. W., Ebeling, H., Rapetti, D., & Drlica-Wagner, A. 2009,arxiv:0909.3099Markevitch, M. 1998, Astrophys. J., 504, 27Maughan, B. J., Jones, L. R., Ebeling, H., & Scharf, C. 2006, MNRAS, 365, 509Maughan, B. J., Jones, L. R., Pierre, M., et al. 2008, MNRAS, 387, 998Mewe, R., Gronenschild, E. H. B. M., & van den Oord, G. H. J. 1985,Astronomy & Astrophysics, Suppl. Ser., 62, 197M¨uhlegger, M. 2010, PhD thesis, , Technische Universit¨at Munich, (2010)Navarro, J. F., Frenk, C. S., & White, S. D. M. 1995, MNRAS, 275, 56O’Hara, T. B., Mohr, J. J., & Sanderson, A. J. R. 2007, arXiv:0710.5782Pacaud, F., Pierre, M., Adami, C., et al. 2007, MNRAS, 382, 1289Perlman, E. S., Horner, D. J., Jones, L. R., et al. 2002, VizieR Online DataCatalog, 214, 265Peterson, J. R., Kahn, S. M., Paerels, F. B. S., et al. 2003, Astrophys. J., 590, 207Pierre, M., Valtchanov, I., Altieri, B., et al. 2004, Journal of Cosmology andAstro-Particle Physics, 9, 11Pratt, G. W., Croston, J. H., Arnaud, M., & B¨ohringer, H. 2009,Astronomy & Astrophysics, 498, 361Predehl, P., Andritschke, R., B¨ohringer, H., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7732,Society of Photo-Optical Instrumentation Engineers (SPIE) ConferenceSeriesPredehl, P., Andritschke, R., Bornemann, W., et al. 2007, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6686,Society of Photo-Optical Instrumentation Engineers (SPIE) ConferenceSeriesReiprich, T. H. & B¨ohringer, H. 2002, Astrophys. J., 567, 716Romer, A. K., Viana, P. T. P., Collins, C. A., et al. 2002, in Astronomical Societyof the Pacific Conference Series, Vol. 268, Tracing Cosmic Evolution withGalaxy Clusters, ed. S. Borgani, M. Mezzetti, & R. Valdarnini, 43Rosati, P., Borgani, S., & Norman, C. 2002, Annual Rev. of Astronomy andAstrophysics, 40, 539Rosati, P., Tozzi, P., Gobat, R., et al. 2009, Astronomy & Astrophysics, 508, 583Santos, J. S., Rosati, P., Gobat, R., et al. 2009, Astronomy & Astrophysics, 501,49Santos, J. S., Rosati, P., Tozzi, P., et al. 2008, in Astronomical Society of thePacific Conference Series, Vol. 399, Astronomical Society of the PacificConference Series, ed. T. Kodama, T. Yamada, & K. Aoki, 375Schuecker, P., Caldwell, R. R., B¨ohringer, H., et al. 2003, A&A, 402, 53Schwope, A. D., Lamer, G., de Hoon, A., et al. 2010, Astronomy & Astrophysics,513, L10Short, C. J., Thomas, P. A., Young, O. E., et al. 2010, MNRAS, 408, 2213Siemiginowska, A., Burke, D. J., Aldcroft, T. L., et al. 2010, Astrophys. J., 722,102Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629Stanek, R., Rasia, E., Evrard, A. E., Pearce, F., & Gazzola, L. 2010, Astrophys.J., 715, 1508Sutherland, R. S. & Dopita, M. A. 1993, Astrophys. J., Suppl. Ser., 88, 253
Ulmer, M. P., Adami, C., Lima Neto, G. B., et al. 2009,Astronomy & Astrophysics, 503, 399Vikhlinin, A., Burenin, R. A., Ebeling, H., et al. 2009, Astrophys. J., 692, 1033Vikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, Astrophys. J., 640, 691Voit, G. M. 2005, Reviews of Modern Physics, 77, 207Watson, M. G., Schr¨oder, A. C., Fyfe, D., et al. 2009,Astronomy & Astrophysics, 493, 339Zhang, Y., Finoguenov, A., B¨ohringer, H., et al. 2007,Astronomy & Astrophysics, 467, 437Zhang, Y., Finoguenov, A., B¨ohringer, H., et al. 2008,Astronomy & Astrophysics, 482, 451
Appendix A: Clusters included in the combinedsample eichert et al.: Evolution of X-ray scaling relations 17
Table A.1.
Clusters included in the combined sample and their X-ray properties within r . T indicates the global ICM temperature, L X the bolometric X-ray luminosity and M the (hydrostatic) cluster mass. Cluster names printed in boldface designate the additional z > . Cluster z kT [keV] L X [10 erg s − ] M [10 M ⊙ ] Source publicationXMMXCSJ2215.9-1738 . + . − . . + . − . Hilton et al. (2010)
ISCSJ1438.1 + . + . − . . + . − . Brodwin et al. (2010)
XMMUJ2235.3-2557 . + . − . . ± . . ± + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . XLSSJ022303.0-043622 . + ∞− . . ± . RXJ1053.7 + . ± . . ± . SPT-CLJ2106-5844 . + . − . . ± . + . ± . . ± . . ± . XMMUJ100750.5 + . + ∞− . . + ∞− . Schwope et al. (2010)SPTJ0546-5345 1.0665 7 . ± . . ± . . + . − . . ± . . + . − . Siemiginowska et al. (2010)
XLSSJ022404.1-041330 . ± . . + . − . . + . − . Maughan et al. (2008)XLSSJ022709.2-041800 1.05 3 . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± XMMUJ1229.5 + . + . − . . ± . XMMUJ1230.3 + . + . − . . ± . . ± . + . ± . . ± . . ± . + ± . ± . . ± . CLJ1604 + . + . − . ± . + . ± . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . RXJ1257.2 + . + . − . . + . − . Ulmer et al. (2009)ClJ1559.1 + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . RXJ1821.6 + . + . − . . + . − . Gioia et al. (2004)RX J1716 + . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . continuedonnextpage8 Reichert et al.: Evolution of X-ray scaling relationscontinuedfrompreviouspage Cluster z kT [keV] L X [10 erg s − ] M [10 M ⊙ ] Source publication
ClJ1342.9 + . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± .
147 Hicks08RX J1334.3 + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . + . ± . . ± . + . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
04 0 . ± . + . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . + . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± .
02 0 . ± . . ± . . ± . . ± . . ± . continuedonnextpageeichert et al.: Evolution of X-ray scaling relations 19continuedfrompreviouspage Cluster z kT [keV] L X [10 erg s − ] M [10 M ⊙ ] Source publication
Abell 1300 0.3075 9 . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
04 1 . ± . . ± . . ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
02 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . + . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . continuedonnextpage0 Reichert et al.: Evolution of X-ray scaling relationscontinuedfrompreviouspage Cluster z kT [keV] L X [10 erg s − ] M [10 M ⊙ ] Source publication
Abell 2163 0.203 12 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ± . . ± . . ± .
08 4 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
03 Pratt09RXC J0547.6-3152 0.148 6 . ± . . ± . . ± . . ± . . ± .
03 Pratt09Abell 1413 0.143 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
04 Pratt09RXC J0605.8-3518 0.1392 4 . ± . . ± .
04 Pratt09Abell 1068 0.1386 3 . ± . . ± . . ± . . ± .
03 7 . ± .
02 Pratt09MS J1111.8 0.1306 5 . ± . . ± . . ± . . ± . . ± .
02 Pratt09RXC J1141.4-1216 0.1195 3 . ± .
03 3 . ± .
01 Pratt09RXC J2149.1-3041 0.1184 3 . ± .
04 3 . ± .
02 Pratt09RXC J1516.3 + . ± . . ± .
02 Pratt09RXC J0145.0-5300 0.1168 5 . ± . . ± .
03 Pratt09RXC J0616.8-4748 0.1164 4 . ± . . ± .
02 Pratt09RXC J0006.0-3443 0.1147 5 . ± . . ± . . ± . . ± . . ± . . ± . . ± .
02 Pratt09PKS 0745-191 0.1028 8 . ± . . ± . . ± . . ± .
01 Pratt09Abell 2244 0.0989 5 . ± . . ± . . ± . . ± .
03 2 . ± .
02 Pratt09Abell 3921 0.094 5 . ± . . ± . . ± . + . ± . . ± .
01 Pratt09Abell 2142 0.0904 10 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
01 Pratt09RXC J0821.8 + . ± . . ± .
01 Pratt09Abell 2255 0.0809 6 . ± . . ± . . ± . . ± . . ± .
02 Pratt09RXC J1236.7-3354 0.0796 2 . ± . . ± .
01 Pratt09Abell 2029 0.0779 8 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
01 Pratt09RXC J0345.7-4112 0.0603 2 . ± .
04 0 . ± .
01 Pratt09 continuedonnextpageeichert et al.: Evolution of X-ray scaling relations 21continuedfrompreviouspage
Cluster z kT [keV] L X [10 erg s − ] M [10 M ⊙ ] Source publication
Abell 3266 0.0602 8 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± .
01 Pratt09Abell 133 0.0569 4 . ± . . ± . . ± . . ± .
01 Pratt09Abell 85 0.0557 6 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Appendix B: L-z distribution of the cluster sample L X ( e r g s - ) z Pratt09Zhang08Mantz09Zhang07OHara07Kotov05 L X ( e r g s - ) z Branchesi07Pacaud07Maughan06Ettori04Hicks08Andersson10z>0.8-sample
Fig. B.1.
Bolometric X-ray luminosity L X of the clusters in-cluded in the combined cluster sample. Appendix C: Comparison of cluster properties forsystems included in more than one subsample M ( M O • ) L ( e r g s - ) A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll A be ll Z wC l Z wC l J . - J . - J . - J . - J . - J . - J . - J . + J . - J . - R X J . + R X J . + R X J . + M S . + k T ( k e V ) Mantz09Zhang08Arnaud05Vikhlinin06O‘Hara07Zhang07Pratt09
Fig. C.1.
Comparison of cluster properties for z < . ff erent authors. These measurements of cluster observablescan be compared to each other after correcting for di ff erent anal-ysis schemes and provide a useful tool to test the applied homog-enization scheme. Furthermore, the comparison analysis givesan estimate of whether the error budget assumed in the di ff er-ent publications is realistic and reveals systematic di ff erencesbetween the results derived by the various studies.Fig. C.1 shows the cluster properties of the low redshift( z < .
3) overlap sample. In Fig. C.2, we present the deviations <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ M , N C l u s t e r <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ L X , N C l u s t e r <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ T , N C l u s t e r σ Zhang08Arnaud05Vikhlinin06OHara07 Pratt09Mantz09Zhang07
Fig. C.2.
Distribution of deviations from the mean value for the z < . σ . Top panel: Mass deviations. Middle panel: L X deviations. Bottom panel: ICM temperature deviations. M ( M O • ) L ( e r g s - ) J . - S J . - NR X J + R X J . - R X J . + R X J + C L0024 . + C l J . + C l G J + C l G J - C l G J - C l G J + C l G + C l G J + C l G J + C l G J + C l G J + C l G J + M S . + M S + M S . - M S . - M S . + M S . - C k T ( k e V ) Maughan06Ettori04Kotov05Branchesi07O‘Hara07
Fig. C.3.
Comparison of cluster properties for z > . σ , and the error estimated in the individual publications. Notethat the true values of the cluster observables are unknown. Thecomparison to the mean value therefore does not allow any state-ments about the reliability of the results derived in the di ff er-ent studies. Figure C.2 instead provides an insight into the sys-tematic di ff erences between the results of di ff erent studies andwhether the assumed error estimates are realistic.The derived spectroscopic temperatures agree well for mostof the clusters. As visible in the bottom panel of Fig. C.2, themajority of the measured values deviate less than 1 σ from themean value. In detail, 59% of the measurements lie within 1 σ and 82% within 2 σ of the mean. Only 5% of the results deviateby more than 5 σ . This indicates that the spectral fitting methodgenerally leads to secure and consistent results, that the proba-bility of severe misestimations is low, and that the assumed er-ror budgets are likely to be realistic. Di ff erences may result fromdi ff erent spectral extraction regions or di ff erent treatments of pa-rameters, such as the ICM metallicity or the background subtrac-tion process. However, these di ff erent measurement schemes do eichert et al.: Evolution of X-ray scaling relations 23 <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ M , N C l u s t e r <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ L X , N C l u s t e r <-5 σ -4 -3 -2 -1 0 1 2 3 4 >5 σ T , N C l u s t e r σ Ettori04Maughan06Branchesi07Kotov05OHara07
Fig. C.4.
Distribution of deviations from the mean value for the z > . σ . Top panel: Mass deviations. Middle panel: L X deviations. Bottom panel: ICM temperature deviations.not lead to completely incomparable data sets. The temperaturedi ff erences between the subsamples are rather uncorrelated andreveal no systematic trends between di ff erent studies. We notethat for the samples of Zhang07, Zhang08, and Arnaud05 onlycore-excised temperatures were available. However, comparingthose to the core-included temperatures given in other studies( e.g. Mantz09), the observed di ff erences remain small.For the X-ray luminosity L X , the situation is clearly di ff erent.As visible in the middle panel of Fig. C.1, most of the derived lu-minosities do not agree within the errors. Furthermore, the dif-ferences between the results of some studies clearly show sys-tematic trends. In terms of the deviations from the mean value,only 19% of the values lie within 1 σ and 29% within 2 σ , while39% of the measurements show deviations of more than 5 σ . Thedi ff erent samples exhibit systematic di ff erences when comparedto each other, especially for the Mantz09-Zhang08 overlap butto a lesser degree also for the common clusters of Zhang08 andOHara07. The reason for these deviations remains unclear sinceall known systematic di ff erences, such as the definition of clus-ter radii and the di ff erent energy bands used, were corrected for.These deviations therefore imply that there are additional sys-tematic di ff erences between the samples. However, for the cen-tral goal of this work, constraining the redshift evolution of scal-ing relations, this open question is of negligible importance be-cause systematic di ff erences mostly occur for low-redshift sam-ples and the choice of sample from which multiply analyzedclusters are taken has no significant influence on the evolutionresults. In addition to systematic trends, even for samples thatshow no trends at all, the di ff erences between the results consid-erably exceed the estimated errors. This indicates that the errorestimations made by the di ff erent studies are too optimistic orthat there are additional sources of measurement errors not in-cluded in the error budget.Similar but less significant systematic trends are also visiblewhen comparing the results for cluster masses. As visible in thetop panel of Fig. C.2, 51% of the results deviate by less than 1 σ from the mean value, while 89% lie within 2 σ and no measure-ment shows deviations of more than 5 σ . However, apart fromthese systematic trends the estimated errors for cluster massseem realistic as most measurements deviate by less than 1 σ from the mean. The masses derived in Pratt09 based on the Y X - parameter and the Y X -M relation show no significant systematicdi ff erence from the hydrostatic mass estimates. However, ow-ing to the small overlap sample of five clusters, the comparisonanalysis provides no suitable tool to identify these di ff erences.In Fig. C.3, the derived cluster properties of the z > . σ and 90% by less than 2 σ from the mean value, while no devi-ations of more than 5 σ occur. As for the low-redshift clusters,the measured temperatures show no systematic trends for singlesubsamples, i.e. the spectroscopic fitting procedure also seemsreliable for distant clusters and there appears to be no major sys-tematic e ff ects that have to be corrected. Furthermore, accordingto the mostly small deviations in units of the assumed error, theestimated error budget is likely to be realistic.Luminosities agree on average more strongly for the high-z clusters than for the local sample, for instance the Ettori04and OHara07 results are consistent for 12 of the 18 clusters incommon (see middle panel of Fig. C.3). In contrast to the lo-cal overlap sample, no significant systematic trends between thedi ff erent studies are visible. The deviations from the mean valueplotted in the middle panel of Fig. C.4 are smaller than 1 σ for53% of the results and below 2 σ for 62% of the measurements.We have found that 12% of the results deviate by more than 5 σ .The distribution of deviations implies that the error budget mighthave been previously underestimated by the di ff erent studies, al-though by no means as significantly as for the local systems.The masses derived by Ettori04, Maughan06, and Kotov05plotted in the top panel of Fig. C.3 are consistent within the er-rors for all shared clusters, all results deviate by less then 1 σ from the mean value. The estimated errors are therefore likelyrealistic. However, the small size of the overlap sample of onlyfour clusters does not allow us to peform a robust analysis of thesystematic di ff erences between the di ff erent studies. Appendix D: Local scaling relations for thecombined cluster sample
Fig. D.1, D.2, and D.3 show the z < . ff erent BCES fitting schemes in comparisonto the relations derived by Pratt09 which were adopted for theevolution study in our work. M ( M O • ) kT(keV) Arnaud05Vikhlinin06Pratt09Zhang08Mantz09Zhang07Pacaud07M=0.24 ± ± M O• (BCES(X|Y))BCES(Y|X)BCES orthogonalPratt09 Fig. D.1.
Local cluster sample: M–T relation. The red line showsthe BCES(T | M) best-fit relation for the combined cluster sample,and the grey lines the BCES(M | T) and BCES orthogonal rela-tions. The blue line shows the Pratt09 relation (see Sect. 3.1). L X ( e r g s - ) kT(keV) Pratt09Zhang08Mantz09Zhang07Pacaud07L=0.11 ± ± erg s -1 (BCES(Y|X))BCES (X|Y)BCES orthogonalPratt09 Fig. D.2.
Local cluster sample: L X –T relation. The red lineshows the BCES(L | T) best-fit relation for the combined clustersample, and the grey lines the BCES(T | L) and BCES orthogonalrelations. The blue line shows the Pratt09 relation (see Sect. 3.1). M ( M O • ) L X (10 erg s -1 ) Pratt09Zhang08Mantz09Zhang07Pacaud07M=1.19 ± erg s -1 ) ± M O• (BCES(X|Y))BCES (Y|X)BCES orthogonalPratt09 Fig. D.3.
Local cluster sample: M–L X relation. The red lineshows the BCES(L | M) best fit relation from the combined clus-ter sample, the grey lines the BCES(M ||