An analytic investigation of the scatter in the integrated X-ray properties of galaxy groups and clusters

0.47 MB·English

by  Michael L. Balogh

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed3February2008 (MNLATEXstylefilev2.2)
An analytic investigation of the scatter in the integrated
X–ray properties of galaxy groups and clusters.
Michael L. Balogh1, Arif Babul2, G. Mark Voit3, Ian G. McCarthy2,4,
Laurence R. Jones5, Geraint F. Lewis6, Harald Ebeling7
6 1Department of Physics, University of Waterloo, Waterloo, ON, Canada N2L 3G1, email: [email protected]
0 2Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada V8P 1A1
0 3Department of Physics and Astronomy, Michigan State University,East Lansing MI 48824 USA
2 4Department of Physics and Astronomy, University of Durham, Durham, UK, DH13LE
n 5Department of Physics and Astronomy, University of Birmingham, Birmingham, UK B15 2TT
a 6Institute of Astronomy, School of Physics, A29, Universityof Sydney, NSW 2006, Australia
J 7Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, Hawaii 96822 USA

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v 3February2008
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7 ABSTRACT
1
We revisit the scaling relationships between the dark matter mass and observed X–
1
ray luminosity and temperature of galaxy clusters and groups in the local Universe.
5
Specifically, we compare recent observations with analytic models of the intracluster
0
medium in which the gas entropy distribution has been shifted by a variable amount,
/
h K◦, to investigate the origin of the scatter in these scaling relations, and its influ-
p ence on the luminosity and temperature functions. We find that variations in halo
-
concentration or formation epoch (which might determine the time available for low
o
r entropygas to cool out) areinsufficient to explainthe amountofscatter in the mass–
t luminosity relation. Instead, a range of entropy floors at a fixed halo mass, spanning
s
a approximately ∼ 50 keV cm2 to ∼ 700 keV cm2, is required to match the data. This
: rangeislikelyrelatedtothe varianceinheatingand/orcoolingefficiencyfromhaloto
v
halo.Wedemonstratethatthesemodelsareconsistentwiththeobservedtemperature
i
X and luminosity functions of clusters, with a normalization of σ8 ∼ 0.8 in agreement
r with WMAP measurements (for h = 0.7 and Ωm = 0.3); in particular the scatter in
a
the mass–luminosity relation has an important influence on the shape of the lumi-
nosity function, and must be accounted for to provide a consistent result. Finally, we
present predictions for the redshift evolution of these scaling relations and luminos-
ity/temperature functions. Comparison with recent data at z <0.7 shows reasonable
agreement with a model that assumes a median entropy floor of K◦ = 200 keV cm2.
Whenobservationsareextendedto groupscales(kT <∼1keV),this evolutionwillhave
the potential to discriminate between an entropy floor that is independent of redshift
(for example, in a preheating scenario) and one that depends on the cooling time of
the halo.
Key words: galaxies: clusters — X-rays:galaxies:clusters— intergalactic medium
1 INTRODUCTION normalization of these relations relative to model predic-
tions have been studied extensively (Edge & Stewart 1991;
The X-ray properties of clusters are tracers of both the
Markevitch1998;Horner et al.1999;Nevalainen et al.2000;
gravitational potential and the thermodynamic history of
Arnaudet al. 2005), little attention has been paid to their
the gas. Since the mass is likely dominated by collision-
intrinsicscatter(butseeRowley et al.2004;Kay et al.2004;
lessdarkmatter,forwhichwehaveawell–developedtheory
Smith et al. 2005).
(e.g. Evrard et al. 2002), we might hope to learn about the
relevant baryonic physics through detailed X-ray observa- The presence of scatter in the M-L and M-T relations
tions. In particular, clusters obey fairly well defined scaling suggests intrinsic variations in the structure of clusters,
relations between mass and X-ray temperature (M-T) and which may be due either to variations in the dark matter
mass and X-ray luminosity (M-L). Although the slope and distributionofthehalosthemselves,and/orinthegasprop-
2 Balogh et al.
erties.Inparticular,thescatterislikelytobedrivenbyvari- tion in the underlying physical processes. The data we use
ations in the core properties of clusters, since the scatter is are uncorrected for any cooling-core component, since it is
known to be significantly reduced if the central regions are preciselythiscoreregion thatinterestsus.Wewillcompare
excluded from the analysis (e.g. Markevitch 1998). these observations with analytic, hydrostatic models which
In the case of the dark matter component of relaxed allow modifications to the entropy distribution of the gas
clusters,thereareindicationsthateither(orboth)thehalo (BBLP,VBBB), to determine the range of model parame-
concentrationparameterandtheinnerslopeofthehalopro- tersthatarerequiredtoreproducetheobservedscatter.Al-
file varies from cluster to cluster, perhaps due to cosmic though the hydrostatic nature of these models means they
environmental effects such as the extent of tidal torquing arenotideallysuitedtoexploreindetailtheeffectsofactive
the dark matter experiences during collapse or the merger cooling or heating in clusters, the range of model parame-
history of the halo (e.g. Jing 2000; Bullock et al. 2001; tersrequired tomatch thedatacan berelated indirectly to
Wechsler et al.2002;Zhao et al.2003;Williams et al.2004). theseprocesses.Wewillalsomakeaself-consistent compar-
Theamountofsubstructureanddynamicalstateofthedark ison with the temperature and luminosity functions, which
matter will also vary and depend upon the merging history provideanindependenttestofthemodelsassumingthedark
of the halo. mattermassfunctionisknown(Evrard et al.2002).Finally,
There are also reasons to expect substantial variation wewillpresenttheredshiftevolutionofalltheseobservable
inthegaspropertiesofclusters,independentlyoftheirdark quantities to put further constraints on the model parame-
matter distribution. Models that seek to account for the ters.
mean M-L and M-T relations require some form of en- Throughout this paper we use a cosmology with Ωm =
tropy modification in the central regions of the systems 0.3, ΩΛ =0.7 and H◦ =70 km s−1 Mpc−1. The theoretical
(Thomas et al.2002;Viana et al.2003).Pureheatingmod- models are described in § 2, and the comparison with ob-
elshavebeenverysuccessfulinexplainingtheaveragetrends served X-ray properties is presented in § 3. Predictions for
of these scaling relations, especially when the heating tar- the evolution of these models, and some comparison with
getsthelowestentropygas(Kaiser1991;Balogh et al.1999; early data, are given in § 5. We summarize our conclusions
Babul et al.2002,hereafterBBLP).Itseemslikelythatthe anddiscusstheimplicationsandlimitationsofourfindings,
efficiencyofwhateverphysicalmechanism isresponsible for in § 6.
theheating(e.g.heattransport,AGNenergyinjection,etc)
will vary from cluster to cluster. In this case, one expects
bothcoolingandthefeedbackittriggerstoeliminategasbe-
2 THE MODELS
lowanentropythresholdthatdependsonhalomassandred-
shift(Voit & Bryan2001;Voit et al.2002,hereafterVBBB), 2.1 Dark matter profile shapes
although some gas may exist below this threshold if it is in
theprocess of cooling out. In a realistic cooling model (e.g. Theaveragetheoreticalshapesofdarkmatterhalosarewell
McCarthy et al.2004),scattercanbeintroducedbyappeal- motivated by N-bodysimulations to havea form given by
ing to a range in time available for cooling in each halo. ρ∝r−n1(1+c200r)−n2 (1)
Despite this, McCarthy et al. show that such a range alone
cannotaccountforthescatterintheM-Lrelation;itisalso (Navarroet al. 1996, NFW), where n1, n2, and c200 are fit-
necessary to introducevariations in an initial heating level. tingparameters.Theprimarydeterminantofahalo’sstruc-
However,theydonotconsidertheeffectofvariationsassoci- ture is its virial mass, Mvir. This is commonly defined, us-
ated with the underlying dark matter potential, which will ing spherical collapse models, as the mass within a fixed
contribute some scatter independently of the gas entropy overdensity ∆ that depends on cosmology and redshift; for
distribution. ΛCDM, ∆ ∼ 100 at z = 0 (Eke et al. 1996). However, ob-
Shorter–lived changes to the equilibrium temperature servationsaremoretypicallymadeat∆=200,500orlarger.
and luminosity of a cluster may also be associated with Theradiusandmasscorrespondingtotheoverdensity∆will
merger events (e.g. Ritchie & Thomas 2002; Randall et al. bedenoted R∆ and M∆, respectively.
2002; Rowley et al. 2004). Departures from equilibrium in High resolution simulations show that relaxed clusters
the potential can change the luminosity, but have little ef- haveremarkably uniform profiles. Wewill assume an NFW
fect on the gas temperature (Rowley et al. 2004). However, profile with n1 = 1 and n2 = 2 as our fiducial model. The
the shocks associated with mergers can have a temporary range of values of these parameters reported in the liter-
but significant influence on both the temperature and lu- ature (e.g. Moore et al. 1998; Lewis et al. 2000) appear to
minosity of the gas (Ritchie & Thomas 2002; Rowley et al. be mostly dueto differences in resolution and the fact that
2004). thefittingformula isnot aperfect description of theprofile
Recently, semi– and fully–numerical simulations which (Hayashiet al.2004).Withinagivensimulation, anindica-
includebothcoolingandfeedbackfromstarformation have tion of the amount of variation in halo shapes is best given
been shown to produce clusters with X–ray properties that by the distribution of concentration parameters, c200. This
scale with mass in a way that is in reasonable agree- parameter has a systematic dependence on mass and red-
ment with the observations (e.g. Muanwong et al. 2001; shift (e.g. Ekeet al. 1998; Bullock et al. 2001; Power 2003;
Thomas et al. 2002; Borgani et al. 2002; Viana et al. 2003; Wechsleret al. 2002); we will take the parameterization of
Kay et al. 2004; Rowley et al. 2004; Borgani et al. 2005; Ekeet al. (2001), assuming σ8 =0.8. Most importantly for
Ostrikeret al. 2005). In this paper we will re-examine the our purposes, c200 shows considerable scatter at fixed mass
observed M-L and M-T relations, focusing on the scatter and redshift; Dolag et al. (2004) have recently shown that
in these relations and how it relates to the expected varia- thedistributionofc200 valuesinsimulatedclusters(selected
Scatter in the X-ray scaling laws 3
common (e.g. Ponman et al. 1999) redefinition of entropy
as K ≡ kT n−2/3, where T and n are the electron tem-
e e e e
perature and density, respectively. This is related to the
thermodynamic entropy by a logarithm and an additive
constant, and is given in units of keV cm2. We will usu-
ally quote this quantity in dimensionless units, relative to
K100 =100keV cm2.
We use the formalism of VBBB to compute the hy-
drostatic equilibrium gas distributions under different as-
sumptions about the dark matter potential and the ther-
modynamic history of the gas. Once the halo potential is
specified, the entropy distribution and appropriate bound-
aryconditionsareallthat arerequiredtofully describethe
gasdensityandtemperatureprofiles.Westartwithaninitial
profilewherethegasdensitytracesthedarkmatterdensity,
and solve for the temperature profile needed to satisfy the
assumption of hydrostatic equilibrium. This is a good ap-
proximationatlargeradii,wheretheentropyscalesapprox-
imately as K ∝ r1.1, as found from analytic modelling and
numerical simulations (Lewis et al. 2000; Tozzi & Norman
2001; Voit et al. 2003; McCarthy et al. 2005). The gas dis-
Figure 1. Top panel: The relation between c200 (concentra- tributionatsmallerradiiisdominatedbytheentropymod-
tion) and M200 as a function of M200 is shown as the shaded ifications thatwediscuss below, sothechoiceof initial pro-
region, where the distribution is due to the 1−σ (heavy shad- fileisrelatively unimportant.The normalization ofthisini-
ing)and3−σ(lightshading)distributionofconcentrationsfrom tial, unmodified model, is chosen by assuming the gas frac-
Dolagetal.(2004).The horizontal lineshowsafixed concentra- tion within R200 is equal to the global baryon fraction of
tion of 4, for reference. Bottom panel: The relation between 12.9 per cent (for h = 0.7 and Ω = 0.3, Burles et al. 2001;
M500 and M200 as a function of M200 is shown as the shaded Spergel et al. 2003).
region, assuming the same distribution of concentrations above.
We will explore modifications to this default profile, in
Thehorizontal,solidlineshowstheresultforamodelwithafixed
the form of the shifted–entropy models of VBBB. In this
concentration, c200=4.
case, theentropy distribution is shifted by an additivecon-
stant, which provides a good approximation to pre-heated
models. This entropy shifting causes the gas to expand be-
onlybasedonanoverdensitycriterion)isapproximatelylog-
normal with a width that is nearly independent of mass. yondR200;followingVBBBwethereforechooseasourpres-
sureboundarycondition theaccretion pressureat themax-
This scatter is at least partly due to the presence of sub-
imum radial extent of the gas. The amount by which the
structure,triaxiality,anddeparturefrom equilibriuminthe
entropy distribution is shifted will be left as a free parame-
sample of simulated clusters chosen from the simulations.
We will therefore consider a range of concentrations corre- ter;forourbasemodelweuse K◦ =2K100,which isknown
to provide a reasonable match to the median global scaling
spondingtothe±3σ rangepredictedfromthisdistribution.
relations of clusters (McCarthy et al. 2004).
Althoughdifferentvaluesofσ8 willchangethevalueofc200
at fixedmassbyasmallamount,it will nothaveanimpor- We will also consider the case where K◦ is set by the
entropy of gas that can efficiently cool in time t. This en-
tant effect our discussion in this paper, which is based on
thevariation in c200 and not its absolute value. tropy,whichwewillcallKcool,dependsonmassandredshift
(Voit & Bryan2001),butalsoonthetimeavailableforcool-
InFigure1weshowhowtheconcentrationanditsscat-
ing (McCarthy et al. 2004). Figure 2 shows the maximum
terdependondarkhalomassM200 inourmodel.Theaver-
age concentration and its scatter both decline with increas- value of K◦ predicted in this cooling–based model, calcu-
lated assuming the gas can cool for a Hubble time. The
ingmass. Inthebottom panelwe show howthisaffects the
ratio M200/M500. The scatter in concentration at a fixed mean value of Kcool decreases with increasing redshift due
primarily to the decrease in time available for cooling. Of
mass corresponds to a ∼10 per cent scatter in M200/M500.
course, simply shifting the entropy distribution by a con-
Thedependenceofconcentrationonmassmeanstheslopeof
stant value is not an accurate representation of the effects
measured correlations between observables(likeX-raytem-
of radiative cooling. In reality, an inner entropy gradient
perature and luminosity) and mass will depend on which
(ratherthanafloor)maybeexpected,withsomegasbelow
massisused.Throughout thispaperwewillpresentourre-
the entropy threshold (VBBB; McCarthy et al. 2004). This
sultsasafunctionofM200,andthisrelation canbeusedto
canbeparticularlyimportantforclusters,wherethecentral
deducethecorresponding valueof M500.
entropy in a more realistic model can drop well below the
value of Kcool. Furthermore, the time available for cooling
will likely vary from cluster to cluster, but must always be
2.2 Entropy distributions
lessthanaHubbletime.Bothoftheseeffectstendtoreduce
The shape of the gas profile in a cluster of given mass is the value of K◦; thus our prediction in Figure 2 represents
determined by the entropy distribution of that gas, which a strict maximum value for this quantity.
is sensitive to its thermodynamic history. We adopt the It is important to note that the entropy modifications
4 Balogh et al.
(e.g., Ponman et al. 2003; Pratt & Arnaud 2005). As such,
our estimates of K◦ for low temperature systems should be
treated with some caution. In this paper, our conclusions
rest primarily on thedata for the high mass clusters.
2.3 Prediction of observable quantities
TocomputeX-rayobservablesfromtheanalyticgasprofiles
we use the cooling functions of Raymondet al. (1976) for
gas with one third solar metallicity. To avoid the need to
make bolometric corrections to the data, which depend on
an accurate measurement of the gas temperature, we com-
pute the model luminosities within the observed ROSAT
energy bands 0.1–2.4 keV and 0.5–2.0 keV. Total luminosi-
ties are obtained by integrating out to a minimum surface
brightness of 1 × 10−15 ergs s−1 cm−2 arcmin−2, similar
to that of the WARPS survey (Scharf et al. 1997), unless
statedotherwise1.Thesensitivityofourresultstothislimit
are explicitly noted.
The model temperatures we compute are emission-
weighted by the0.1-2.4 keV luminosity, again excludingre-
Figure 2. The entropy Kcool corresponding to the maximum gions below the WARPS surface brightness limit; however
entropy for gas that can cool in a Hubble time, as a function of thechoice of energy band and surface brightness limit have
halovirialtemperatureandredshift.Thehorizontal, dashed line a negligible effect on the calculated temperatures for our
showstheKcool=2K100 lineforreference. purposes. Simulations suggest that spectral temperatures,
as measured observationally, can be 10–20 per cent higher
thanemission–weightedtemperatures(Mathiesen & Evrard
that we explore in the present study are all confined to the 2001; Rasia et al. 2005; Vikhlinin 2005). For relaxed clus-
clustercore.Therefore,ourapproachistoeffectivelybracket ters, the difference is probably closer to the lower end
therange of central entropy levels required for theheating- (∼ 10 per cent) of this range (Rasia et al. 2005). How-
based and cooling-based models to explain the scatter in ever, this systematic error is not of major concern for the
theobservational data. Both models implicitly assume that present study, as the statistical errors associated with the
the entropy distribution at large radii is essentially identi- observedtemperature,mass,andluminosityofagivenclus-
cal to that found in clusters formed in cosmological numer- ter are typically twice as large as this. For example, in
ical simulations. Gas at large distance from the centre is McCarthy et al.(2004)wederivedsimilarconstraintsonthe
not easily affected by cooling or non-gravitational heating parameters of heating and cooling models of the intraclus-
processes once the cluster is assembled (e.g. Borgani et al. ter medium (ICM) from independent analyses of the M-L
2005; Ostriker et al. 2005). However, heating of the gas be- andluminosity–temperature(L-T)relations,whichsuggests
fore it isaccreted intoa cluster can smooth out thedensity that small systematic errors in temperature measurements
distribution of infalling gas, and this increases the entropy donot havea noticeable influenceon our results.
jumpattheaccretionshock(Voit et al.2003;Borgani et al. Our primary source of data is the HIFLUGCS clus-
2005). In this case, the entropy distribution at large radii ter sample (Reiprich & B¨ohringer 2002). This survey is an
willbelargerthanthatwehaveassumed,andalowervalue X-ray flux-limited sample of nearby clusters based on the
of central entropy will be required to explain the observa- ROSAT All Sky Survey, in which cluster masses are deter-
tions. This underscores the need for detailed comparisons mined from thedensityprofiles, assuming hydrostaticequi-
of theoretical models to the spatially-resolved entropy dis- librium and isothermal temperature profiles. It is common
tributions of large, representative samples of clusters. At practisewhenconsideringX-rayscalingrelationstousetem-
present, however, only a relatively small number of clus- peratures and luminosities that are corrected for a cooling
tershaveaccurately-determinedentropyprofilesfromChan- flowcomponent,andit is knownthat thisreducesthescat-
dra and XMM-Newton data. Analysis of this small dataset terintheserelations(e.g.Markevitch1998).However,since
seems to confirm that for high mass clusters (kT >∼ 4 it is precisely this scatter that is the focus of our study, we
keV) the entropy distribution at large radii does indeed will present all the data as observed, without this correc-
trace the spatial distribution and normalization predicted tion. We take these raw cluster luminosities, in the 0.1–2.4
by ‘adiabatic’ hydrodynamic simulations (McCarthy et al.
2004). This conclusion is strengthened by the fact that for
1 We note that Scharfetal. (1997) attempt to correct for flux
high mass clusters there also appears to be excellent agree-
below the surface brightness limit. However, the flux correction
ment between the observed projected temperature profiles
to the data (typically afactor ∼1.4) isan underestimate at the
at large radii and those predicted by hydrodynamic simu-
lowest luminosities, since it assumes a surface brightness profile
lations(e.g., DeGrandi & Molendi2002;Loken et al. 2002;
slope of β=0.67, appropriate for high and moderate luminosity
Vikhlinin et al. 2005). Thus, our estimates of K◦ for such clusters but not for low luminosity groups. Thus, the lowest lu-
systemsshouldberobust.However,forcoolersystems,there minosity data points may still systematically underestimate the
are preliminary indications of excess entropy at large radii luminositiesbyafactor<∼2.
Scatter in the X-ray scaling laws 5
models (see § 2) results in a significant reduction of lumi-
nosity only below M200 ∼ 1013.5M⊙. The shifted–entropy
modelreproducestheslopeandnormalizationoftherelation
well. However, the scatter in the data is much larger than
expected from the observational uncertainties, as has been
noted before (e.g. Fabian et al. 1994; Markevitch 1998).
Ourcalculationsshowthattheobservedscatterismuch
largerthancan beexpectedfrom variationsinhaloconcen-
tration. This is shown by the shaded region in Figure 3,
which represents the 1− and 3−σ range of luminosities
predicted at a given mass, from the dispersion in simulated
cluster concentrations alone. Approximately 25 per cent of
theobserved clusters lie well outside the3σ range resulting
fromvariationsinhaloconcentration.Eventhoughourmod-
els assume spherical, smooth, virialized halos, these effects
are partly accounted for by the variation in concentration
parameter,whichisdeterminedfromsimulatedclustersthat
areclumpy,non–sphericalandinavarietydynamicalstates.
Furthermore, the predicted scatter in the scaling relations
isnotsignificantlylarger inmodelswhichconsidermorere-
alistic potential shapes (Rowley et al. 2004; Ostriker et al.
2005). It is therefore unlikely that the observed scatter in
Figure 3. The relation between X-ray luminosityin the 0.1-2.4
keV band and halo mass M200. The filled circles are local (z < this relation can be entirely attributed to variations in the
0.2) data from the HIFLUGCS sample (Reiprich&Bo¨hringer shape of thedark matter distributions.
2002). 1−σ error bars on the masses are shown; only clusters Some of the scatter in the M–L relation could be due
with relative errors of < 50 per cent are plotted. The shaded toshort–timescale eventslikemergers,which causechanges
region shows the shifted entropy model with K◦ = 2K100 and in both the luminosity and temperature of the gas (e.g.
a 1σ (heavy shaded region) and 3σ (lighter shaded) range of Ritchie& Thomas 2002; Randall et al. 2002; Rowley et al.
halo concentrations. The dashed line represents the model with
2004). However, it has been shown that these changes tend
K◦=Kcool,usingthemostprobablevalueofc200 ateachmass. to move galaxies along the L–T relation, and do not con-
tribute significantly to its scatter; since the observational
scatter in the L–T relation is comparable in magnitude to
keVband,fromtheReiprich & B¨ohringer(2002)catalogue.
that in the M–L relation of Figure 3 (Fabian et al. 1994;
Since many of the temperatures and masses in this cata-
Markevitch 1998), there must be an important source of
logue have been subjected to a significant cooling-flow cor-
scatter other than mergers. Furthermore, as major merg-
rection, we will only keep the ∼ 80 clusters (out of 106)
ers in massive clusters are expected to have been relatively
for which uncorrected temperatures are available, from the
rare in the past ∼ 2 Gyr (e.g. Kauffmann & White 1993;
catalogue of Horner(2001).Thedynamical mass estimates,
Lacey & Cole 1994), and the luminosity and temperature
M200 and M500, are derived from the gas temperature, as-
boosts typically last for ∼ 0.5 Gyr or less following a ma-
suming hydrostatic equilibrium. To be fully consistent, we
jor merger, these events are relatively rare and unlikely to
make a small adjustment to these masses (M ∝T) so they
be responsible for all the observed scatter in an unbiased
agree with theoriginal (uncorrected) temperatures.
cluster sample.
On the other hand, the X–ray luminosity of a cluster
of given mass is very sensitive to the entropy floor level,
3 SCATTER IN THE M-T AND M-L SCALING as shown in Figure 4. A range of entropy levels 0.5–5 K100
RELATIONS approximatelycoversthescatterintheobservations,athigh
3.1 The M-L relation masses. For afew clusters evenhigherlevels of K◦ ≈7K100
arerequiredtomatchtheirlowluminosities,somethingthat
In Figure 3 we present the M–L relation from the data of was also observed in our earlier comparison with Sunyaev–
Reiprich & B¨ohringer (2002), excluding those clusters with Zeldovich measurements (McCarthy et al. 2003). This is a
mass uncertainties greater than 50 per cent. Note that the remarkably large range in central entropy levels, and thus
mass we plot here is M200. Observationally, M500 can be substantial variations in heating or cooling efficiency must
more precisely determined, and this can reduce the purely exist from cluster to cluster, even if some of the observed
observational scatter in this Figure (Rowley et al. 2004). scatter can be attributed to substructure, departures from
However, when we construct the temperature and luminos- equilibrium, or merger–induced shocks.
ity function in § 4 we will have to use M200, since this was Inthecooling–based model,theentropyfloorisrelated
the mass used to derive the mass function from numerical to the cooling time (as in Figure 2). This model predicts
simulations (Evrard et al. 2002).For consistency,therefore, thatK◦ dependsonmass,increasingbyafactorof∼5from
we have used M200 throughout the paper. We show both ∼0.8K100 to ∼4K100 over the temperature range of inter-
observed and model luminosities in the 0.1–2.4 keV band, est. The resulting M–L relation is shown as thedashed line
to minimize errors in bolometric corrections to the obser- inFigure3;theslopehereissteeperthanforanyofthefixed
vations. The limiting surface brightness cut applied to the entropy–floormodelsshowninFigure4.Note,however,that
6 Balogh et al.
Figure 4. The mass–luminosity relation, with data as in Fig- Figure 5. The relation between X-ray temperature and M200.
ure3,andfivemodelsoffixedconcentrationbutdifferententropy The points are local (z < 0.2) data, with masses from the HI-
thresholds,aslabelled. FLUGCSsample(Reiprich&Bo¨hringer2002)andtemperatures
(uncorrected for cooling flows) from Horner (2001). 1−σ error
bars on the masses are shown; only galaxies with relative errors
whileK◦ mayvarybyafactorof∼5overthismassrange,a of < 50 per cent are plotted. The shaded region represents the
similarorlargerrangeofK◦isrequiredtoexplainthedistri- prediction of the shifted-entropy model with K◦ =2K100 and a
bution of luminosities at fixed mass. Although the cooling realistic1σ(heavyshading)or3σ(lightershading)scatterinthe
model can easily accommodate scatter toward more lumi- haloconcentration parameter.
nous clusters (since Kcool is only the maximum entropy of
gasthatcancool),thereisnosimplemechanismtoaccount
for thescatter of clusters toward lower luminosities. There-
The same would also be truefor themasses in theM–L re-
fore the simple interpretation that the entropy threshold is
lation, where we do see significant scatter (Figure 3), and
duesolelytothecoolingoflowentropygasisnotlikelytobe
thiscouldindicatethattheintrinsicscatterintheluminosi-
correct,aconclusionalsoreachedbyMcCarthy et al.(2004)
ties is larger than the intrinsic scatter in the temperatures.
and Borgani et al. (2005) using more sophisticated models
On the other hand, the fact that X–ray derived masses are
that account for the presence of gas that cools below the
directly proportional to the temperature introduces a cor-
K threshold.
cool relation that could reduce the scatter in the M–T relation
alone. It would be useful to have a larger sample of clus-
terswithaccuratelensingmassesandX–rayobservationsto
3.2 The M-T relation
improveourunderstandingof thescatter in theserelations.
In Figure5 we compare themass-temperature relation pre- The model predictions in Figures 5 and 6 show that
dictedbyourmodelswiththeReiprich & B¨ohringer(2002) the predicted temperature is relatively insensitive to both
catalogue, again excluding those clusters with mass uncer- thehalostructure(i.e.concentration)andtheentropyfloor,
tainties greater than 50 per cent, and using temperatures for K◦ >0.5K100. Recall that the distribution of halo con-
from Horner (2001). The shifted–entropy model provides a centrations partly arises from substructure, triaxiality and
good description of the data over two orders of magnitude departuresfromequilibriuminsimulatedclustersandthere-
in mass, although it is not statistically thebest fit. fore our predicted scatter approximately includes these ef-
UnliketheM–Lrelation,theobservationshereshowre- fects. However, in our model these concentrations are ap-
markablylittlescatter,andthisscatterisconsistentwiththe plied to spherical, smooth halos and the scatter in pre-
publishedmeasurementuncertainties.Recently,Smith et al. dictedtemperaturesisthereforenotaslargeasinnumerical
(2005) used strong gravitational lensing to measure the and analytic models that do not make these assumptions
massesoftenX–rayluminousclustersandfoundthatthere (Rowley et al. 2004; Ostriker et al. 2005). The insensitivity
isintrinsicscatterintheM–Trelation,duemostlytomerg- of our predicted temperatures to the value of the entropy
ing, non-equilibrium systems. Although this scatter is not floor is due to the fact that the higher central tempera-
apparentinFigure5,itispossiblethatitisunderrepresented ture associated with larger K◦ is offset by the flattening
here, as M200 is obtained from theX-ray data by assuming of the central density profile, which means the luminosity–
isothermalityandimposingafunctionalformforthesurface weighted temperature is dominated by the temperature at
brightness profile (Rasia et al. 2005). This parametrization larger radius. Thus, even a model with K◦ = 0.5K100 pre-
couldhavetheeffectofhomogenizingclusterswitharangeof dicts temperatures that are just within the scatter of the
cooling core sizes, geometries, and amount of substructure. observations. For the same reason, the Kcool model predic-
Scatter in the X-ray scaling laws 7
Figure 6.AsFigure4,butforthemass–temperaturerelation.
Figure 7. The shaded regionis the observed temperature func-
tion does not differ significantly from those shown here, so tion at z = 0 from Ikebeetal. (2002), but with temperatures,
we haveomitted it from the figurefor thesake of clarity. uncorrected for cooling flows, taken from Horner (2001). The
In summary, we have found that a variation of nearly datapointsarefromHenry(2000).Inthebottom panelweshow
the shifted-entropy model with three different levels of heating,
anorderofmagnitudeinK◦ isrequiredtoexplainthelarge
scatter in the M −L relation. Variations in dark matter assuming σ8 = 0.85 as determined from WMAP (Spergeletal.
2003; Tegmarketal. 2004). Models are convolved with a 10 per
halo shape (concentration) alone are insufficient. Encour-
cent uncertainty in temperature which flattens the temperature
agingly, the wide range of entropy levels required does not
function.Inthetoppanel,weshowthesamemodels,normalized
conflict with the small observed scatter in the M −T rela- at4–5keVwithdifferentvaluesofσ8,asindicated.
tion, because of the temperature’s insensitivity to the core
properties. Although both theory (e.g. Rowley et al. 2004;
Ostrikeret al. 2005) and observations (Smith et al. 2005) luminosity relations2. The form of the temperature and lu-
suggest that there may be additional scatter in the M-T minosity functions are then completely determined by the
relation thatisnot apparentunderthesimplifying assump- correlation between virial mass and the X-ray observable
tions of both the models and data presented here, it is still (BBLP3).
lessthanthescatterintheM–Lrelation,whichisdominated
by variations in theentropy distribution of thegas.
4.1 The temperature function
The observed z = 0.15 temperature functions from
Ikebeet al. (2002) and Henry (2000) are shown in Fig. 7.
For consistency, the temperatures are taken from Horner
4 SCATTER AND THE LUMINOSITY AND
(2001),and therefore uncorrected for anycooling flowcom-
TEMPERATURE FUNCTIONS
ponent, though this makes little difference in practise. To
Ifthemassspectrumofdarkmatterhalosisknownprecisely, compare with these data, we show predicted temperature
thentheobserved shape and normalization of thetempera- functions from the shifted-entropy models with a range of
ture and luminosity functions provides an independent test normalizations that approximately accounts for the scatter
ofthetheoreticalmodels,thatdoesnotdependonanobser- in the M–L relation. We have smoothed the models with
vationaldetermination ofcluster mass. Scatterin themean 10 per cent Gaussian random noise on the temperatures,
relationsplaysanimportantrolehere,andcaninfluencethe to mimic the scatter in the mean relation (which is consis-
shape of these functions. tentwithbeingduetoobservationaluncertainties).First,in
We construct the theoretical luminosity and temper- the bottom panel, we show three models with different K◦
ature functions using the dark matter mass function of
Evrard et al. (2002), based on the fitting formalism of
2 WenotethatM500isbetterdeterminedobservationally,andit
Jenkins et al.(2001),whichprovidesauniversaldescription
wouldbe useful to have a mass function from numerical simula-
of the mass function to within about 10 per cent. The ad-
tionsfilteredonthisscale.
vantage of the Evrard et al. (2002) mass function is that it 3 However, both the observed and model luminosity functions
is expressed in terms of M200, the same mass that we use shown in Figure 9 of BBLP are incorrect due to errors in the
to compare with theobserved mass-temperature and mass- bolometriccorrectionandcosmologyconversion.
8 Balogh et al.
butthesamenormalizationσ8 =0.85,asmeasuredfromthe
WMAPdata(andadjustedfortheslightlynon-concordance
values of cosmological parameters that we have adopted
Spergel et al. 2003; Tegmark et al. 2004). At the hot end
of the temperature function, the data is best matched by
themodelswith thehighest entropyfloors, whileat theop-
positeextremethelow-entropymodelsfarebetter.However,
we caution that at low temperatures (<∼ 1 keV) there may
becompletenessissuesthatcouldartificiallyflattenthetem-
peraturefunctionand,therefore,yieldvaluesofK◦ thatare
systematically lower than that of the average system (e.g.
Osmond & Ponman 2004).
In thetop panelof Fig. 7,we again showthepredicted
temperature functions for a range of K◦ values, but with
σ8 adjusted to give the same number density of clusters at
T =4–5 keV,whereobservational data from differentstud-
ies are in best agreement (Ikebeet al. 2002). The best-fit
value of σ8 is ∼ 3.5 per cent higher if the normalization to
the data is made over the range kT = 6–7 keV. The range
of K◦ (which are all reasonably consistent with the mass–
temperaturerelation)correspondstoarangeofbest-fitval-
uesofσ8 rangingfrom0.76to0.9.Thisiscomparabletothe Figure8. Thelinesineachpanelshowdifferenttheoreticalmod-
observational uncertainty on this parameter (Spergel et al. elsfortheluminosityfunction.Thesolid linesshowmodelswith
2003),andthereforewecannotusethetemperaturefunction entropy floors of K◦ = 0.5K100 (upper line) and K◦ = 7K100
alone to provide a sensitive test of the size of the entropy (lower line). This approximately brackets the range of entropies
floor.Thisisagain simplybecausetemperatureisrelatively requiredtoexplainthescatterintheM-Lrelation(Figure3).The
insensitive to theentropy of the central gas. dotted lines show the effect of varying σ8 as indicated, keeping
K◦=2K100 fixed;lowervaluesofσ8 reducethenumberoflumi-
nous clusters by a small amount but have no effect on the faint
4.2 The Luminosity Function endoftheluminosityfunction.Finally,thedashedlineshowsthe
K◦ = 2K100 model but omitting the surface brightness thresh-
Tocomputetheluminosity function,theintrinsicscatterin
old,whichincreasesthepredictionatlowluminosities.Leftpanel:
theM-Lrelationmustbetakenintoaccount.Becauseofthe Theopensquareswitherrorbarsaretheobservedlocalluminos-
steepness of the mass function, even a small distribution of ity function from the REFLEX survey (Bo¨hringeretal. 2001).
halo masses corresponding to a given luminosity can have Right panel: Similar, but where the data are from the WARPS
an important effect on the number density of clusters at (Jones etal. 2001, Jones et al., in prep.), in a different energy
thatluminosity.Unfortunately,thevalueofK◦inourmodel band.
doesnothaveauniquephysicalmotivation,andthuswedo
not have a prediction for the scatter as a function of mass.
thresholdof1×10−15 ergss−1cm−2arcmin−2 (Scharf et al.
However, we can see from the data in Figure 3 that the
1997)usedintheothermodels;thesesurfacebrightnesscor-
observations are approximately covered by models with a
rections are relevant only for the least luminous clusters in
rangeofentropyfloors0.5<K◦/K100 <7,soinFigure8we
thesample.
showthepredictionoftheluminosityfunctionforthesetwo
Thus we have shown that consistency between the X–
extremes. The true luminosity function should lie between
ray scaling relations (M–L and M–T) and the luminosity
these limits, with a shape that depends on the distribution
functioncan beachievedinthesemodels;however,abetter
of entropy levels at each luminosity.
understanding of the entropy–floor distribution as a func-
The observed luminosity functions at z = 0.15 from
tion of mass is required to make a firm prediction of the
the WARPS (Jones et al. 2001, Jones et al., in prep.) and
luminosity function shape.
REFLEX (B¨ohringer et al. 2001) surveysare shown in Fig-
ure 8. The solid lines show our model, for K◦ = 0.5K100
(upperline)andK◦ =7K100 (lowerline).Thisrangebrack-
ets the observational data, although the data do lie nearer
5 EVOLUTION
the model with high entropy.This is especially true for the
low-luminosity clusters, with L <∼ 1044ergs s−1. This may WenowturntotheredshiftevolutionoftheX-rayscalingre-
indicate that lower–mass clusters have higher central en- lationsandthetemperatureandluminosityfunctions.These
tropies,on average; thismayalso beevidentfrom Figure3, predictionscanprovideanotherinterestingtestofthediffer-
althoughtherearefewclusterswithaccuratemassmeasure- ence between the fixed–K◦, preheating models and models
mentsat these low luminosities. where K◦ =Kcool.
We also show, as the dotted lines, the effect of vary- In Fig. 9 we show the predicted evolution in the mass-
ing σ8 between 0.76 and 0.9 (assuming K◦ = 2K100). This temperatureandmass-luminosityrelationsforthetwomod-
has only a small effect on the bright end of the luminosity els. The data are the same z ∼ 0.15 data shown in Fig-
function, and no effect on the faint end. The dashed line ures 3 and 5, and the model predictions are shown at
shows the effect of removing thelimiting surface brightness z = 0,0.15,0.4 and 0.7. For massive clusters, T >∼ 4 keV,
Scatter in the X-ray scaling laws 9
Figure9.TheobservedM–LandM–TrelationsshowninFigs.3 Figure10.Theobservedtemperature–luminosityrelationatz=
and 5 are reproduced as the solid circles. In the bottom panels 0isshownasthesolid circles.Thethinandthicklinesrepresent
weshowthedefaultmodelpredictions,wheretheentropyflooris the Kcool and K◦ =2K100 models, respectively. Predictions are
independentofredshift,atz=0(solidline)andz=0.15,0.4,0.7 shownforz=0(solid line)andz=0.15,0.4,0.7(dotted lines).
(dottedlines).Thecurvesinthetoppanelsarethemodelswhere
the entropy floor Kcool is related to the cooling time of the gas
andthusevolves withredshift. than the factor ∼ 4 scatter in luminosity at fixed temper-
ature for local clusters, it is probably premature to claim
these observations rule out either model until the scatter
andtheselection biasesthatresult from it(i.e.Malmquist–
the predicted evolution in both the M–T and M–L correla-
likebias)arerobustlyintegratedintothemodelpredictions.
tions is mostly in the normalization, with little change in
Themost leverage will comefrom low temperaturesystems
theslope.
The amount of evolution in the M–L relation is small athighredshift;theK◦ =2K100 modelpredictsverystrong
(negative) evolution in the luminosities of these groups, as
relativetotheobservedscatteratz=0.Ontheotherhand
the fixed entropy floor becomes very large relative to the
predicted evolution in theM–T relation is more noticeable;
characteristic entropy.
clusters at a given temperature are predicted to be 40–60
The evolution of thetemperature and luminosity func-
per cent less massive at z ∼ 0.7. The sense and magnitude
of the evolution are comparable to recent XMM–Newton tions requires a knowledge of the mass function at z > 0.
andChandradata(Kotov & Vikhlinin2005;Maughan et al. Since this has not yet been precisely measured from simu-
lations, we take the local mass function from Evrard et al.
2006). Both the K◦ = 2K100 and the K◦ = Kcool models
(2002)andevolveittohigherredshiftusingPress-Schechter
modelspredictasimilar amountofevolutionfortheM−T
theory. In Figure 11 we show the temperature function for
relation, so thisis notauseful way todiscriminatebetween
them. both models (normalized at kT ∼4–5 keV, with σ8 ∼ 0.8),
at z =0,0.4 and z =0.7, compared with z =0.4 data from
Interestingly, although the evolution in the M–T and
Henry (2000)4. Over the range of observed temperatures,
M–Lrelationsappearsimilarforbothmodels,thepredicted
kT >3keV,theevolutionin thetwomodelsissimilar, and
evolution of the L–T scaling law is in opposite directions,
ingoodagreementwiththeobservations.Themodelsbegin
as shown in Figure 10. At high temperatures, the amount
to diverge at lower temperatures, where the fixed–entropy
of evolution in the predicted relation is very small. For the
model predicts a little less evolution.
fixed–floormodel,theevolutionisnegligible,whiletheKcool
The predicted evolution of the luminosity function is
model predicts that high redshift clusters will be about
showninFigure12.Unfortunately,ourmodeldoesnotpre-
30 per cent more luminous at fixed temperature, due to
dict the scatter in the M–L relation nor its evolution, so
the fact that the entropy floor is lower at higher redshift.
Observations of distant clusters seem to indicate a much for illustration we have just shown the K◦ = 2K100 model,
which provides a reasonable match to the bright end of the
stronger evolution, with clusters at z ∼0.7 being up to 2.5
local luminosity function. The models are again compared
times brighter than local clusters of the same temperature
with data at similar redshifts, from the WARPS survey
(Vikhlinin et al.2002;Lumbet al.2004;Kotov & Vikhlinin
(Jones et al.2001,Jonesetal.,inprep.).TheWARPSdata
2005; Maughan et al. 2006). In principle, this observation
hasthepotentialtoruleoutbothmodelsshownhere.How-
ever,wenotethattheamountofevolutionobservedissensi- 4 Anupdated versionofthe observed high-redshifttemperature
tivetothedetailsoftheanalysis(e.g.Ettori et al.2004a,b). function,basedonmoredata, ispresentedinHenry(2004).The
Sinceanyevolutioninthemeanscalingrelationismuchless resultsareconsistentwiththoseshownhere.
10 Balogh et al.
Figure 11. The predicted temperature functions at z = 0,0.4 Figure 12. The luminosity functions at z =0,0.4 and z =0.7,
and z = 0.7, for the fixed–entropy models (thick lines) and the forthefixed–entropy(2K100)models(thick lines)andtheKcool
Kcool models (thin lines). The data are at z ∼ 0.4, from Henry models (thin lines). Data at z = 0.4 and 0.7 are shown, from
(2000). the WARPS survey, updated with XMM-Newton and Chandra
luminositiesforclustersatz>0.6.
atz>0.6havebeenupdatedwithaccurateluminositiesmea-
we focus on the scatter in the observed scaling relations,
sured from XMM-Newton and Chandra observations, and
and how this compares with the scatter expected due to
are corrected to the rest-frame energy range 0.5–2.0 keV.
a) a range of halo structures (concentrations); b) the time
Incontrast with thetemperaturefunction, theobserved lu-
availableforcooling–onlyprocessesorc)heating/coolingef-
minosity function evolution is modest, with a factor ∼ 3
ficiency.Sinceclustertemperaturesarerelativelyinsensitive
decrease in the number of massive clusters between z = 0
tovariationsintheentropydistribution(andhencethescat-
and z =0.7.
terintheM–Trelationissmall),wegainthemostbyfocus-
The Kcool model predicts very little evolution in the
ing on the M–L relation and the luminosity function. Our
luminosity function, and thus overpredicts the number of
main findingsare as follows:
high redshift clusters. The K◦ = 2K100 model appears to
be in much better agreement with the data, especially for (i) The variations in dark matter halo concentration ex-
thebrightestclusters.Aswiththelocalluminosityfunction, pected from simulations are not large enough to account
the disagreement at the faint end may indicate that lower for thescatter in theobserved M–L relation of clusters and
luminosity clusters have larger central entropies. However, groups.
as we have already seen (Figure 8), the shape of the lumi- (ii) Simple models of the intracluster medium in which
nosity function is very sensitive to the amount of scatter thecoreentropyismodifiedtohaveaminimumvaluerequire
in the M–L relation, and we have no theoretical or empiri- the value of this floor to be between about 0.5K100 and
cal knowledge about how this scatter evolves. We also note 7K100 to match the slope, normalization and scatter in the
that the high–luminosity end of the luminosity function is observed M–T and M–L scaling relations. The constraint
still poorly determined, with a variation in observed num- comes mostly from theM–L relation, as the temperature is
ber abundance at fixed luminosity measured from different insensitive to thevalueof K◦.
surveys at z > 0.3 being about a factor ∼ 2 (Mullis et al. (iii) The shape of the luminosity function is sensitive to
2004). thescatterintheM–Lrelation.Theobservationsliebetween
the models with K◦ =0.5K100 and K◦ =7K100, but closer
to the higher–entropy model. The scatter in entropy levels
as a function of halo mass must be accounted for if the
6 DISCUSSION AND CONCLUSIONS
parametersσ8 orK◦ aretobeaccurately deducedfrom the
In this paper, we have revisited the observed X-ray scal- luminosity function alone.
ingrelationshipsbetweenmassandtemperature(M–T)and (iv) Themodeltemperaturesareingoodagreementwith
mass and luminosity (M–L), simultaneously with the tem- theobservedtemperaturefunction,assumingthemassfunc-
perature and luminosity functions. We have attempted to tion of Evrard et al. (2002). However, the insensitivity of
comparetheobservablequantitiesasdirectlyaspossible(i.e. temperature to K◦, and the uncertainty on the normaliza-
without bolometric orcooling–flow corrections) with asim- tion parameter σ8, means this does not put strong con-
plesuite of models in which a fiducialgas entropydistribu- straints on thevalue(or range of values) of K◦.
tion is shifted by a value 0.5<K◦/K100 <7. In particular, (v) TheamountofgasthatcancoolinaHubbletimesets