Linking X-ray AGN with dark matter halos: a model compatible with AGN luminosity function and

14 Pages·2013·1 MB·English

A&A561,A58(2014) Astronomy
DOI:10.1051/0004-6361/201321689 &
(cid:13)c ESO2013 Astrophysics
Linking X-ray AGN with dark matter halos: a model compatible
with AGN luminosity function and large-scale clustering properties
GertHütsi1,2,MaratGilfanov1,3,andRashidSunyaev1,3
1 Max-Planck-InstitutfürAstrophysik,Karl-Schwarzschild-Str.1,85741Garching,Germany
e-mail:[email protected]
2 TartuObservatory,61602Tõravere,Estonia
3 SpaceResearchInstituteofRussianAcademyofSciences,Profsoyuznaya84/32,117997Moscow,Russia
Received12April2013/Accepted25October2013
ABSTRACT
Aims.Ourgoalistofindaminimalisticmodelthatdescribestheluminosityfunctionandlarge-scaleclusteringbiasofX-ray-selected
activegalacticnucleiinthegeneralframeworkoftheconcordanceΛCDMmodel.
Methods.Weassumethatasimplepopulation-averagedscalingrelationbetweentheAGNX-rayluminosityL andthehostdarkmat-
X
terhalomassM exists.Withsucharelation,theAGNX-rayluminosityfunctioncanbecomputedfromthehalomassfunction.Using
h
theconcordanceΛCDMhalomassfunctionforthelatter,weobtaintheM −L relationrequiredtomatchtheredshift-dependentAGN
h X
X-rayluminosityfunctionknownfromX-rayobservations.
Results.Wefindthatwithasimplepower-law-scalingM ∝LΓ(z),ourmodelcansuccessfullyreproducetheobservedX-rayluminos-
h X
ityfunction.Furthermore,weautomaticallyobtainpredictionsforthelarge-scaleAGNclusteringamplitudesandtheirdependenceon
theluminosityandredshift,whichseemtobecompatiblewithAGNclusteringmeasurements.Ourmodelalsoincludestheredshift-
dependentAGNdutycycle,whichpeaksattheredshiftz(cid:39)1,anditspeakvalueisconsistentwithunity,suggestingthatonaverage
thereisnomorethanoneAGNperdarkmatterhalo.ForatypicalX-ray-selectedAGNatz∼1,ourbest-fitM −L scalingimplies
h X
lowEddingtonratioL /L ∼10−4−10−3(2–10keVband,nobolometriccorrectionapplied)andcorrespondinglyhighmass-growth
X Edd
e-foldingtimes,suggestingthattypicalX-rayAGNaredominantlyfueledviarelativelyinefficient“hot-halo”accretionmode.
Keywords.galaxies:active–X-rays:galaxies–large-scalestructureofUniverse
1. Introduction over(cid:39)448arcmin2 skyarea,∼4MsChandraDeepFieldSouth
(Xueetal.2011)over(cid:39)465arcmin2,and∼3MsXMM-Newton
That every galaxy of significant size harbors a supermassive
deepsurveycoveringChandraDeepFieldSouth(Comastrietal.
blackhole(SMBH)(withmasses∼106−109M )isprobablyone
(cid:12) 2011).
ofthemostremarkablediscoveriesofmodernastrophysics.The
Despitethesmallskyareascoveredbythosedeepsurveys,a
growth of SMBHs, as manifested by the active galactic nuclei
relativelylargenumberofdetectedAGNhasallowedreasonably
(AGN), has been observed over a broad range of electromag-
gooddeterminationoftheAGNluminosityfunction(LF)andits
neticenergies,fromradiotohardX-raysandgamma-rays.The
evolution over cosmic time (Ueda et al. 2003; Hasinger et al.
observed AGN X-ray emission, believed to stem from the up-
2005; Aird et al. 2010). Even though these narrow survey ge-
scatteringofsofteraccretiondiskphotonsviainverseCompton
ometriesarenotverysuitableformeasuringthespatialcluster-
mechanismbythehotelectroncorona,hasproventobethemost
effectivewayofselectinglargesamplesofAGNoverlargecos- ing properties of AGN, somewhat wider and shallower surveys
mologicalvolumes1.Thisefficiencycomesfromthefactthatto likeX-Boötes,XMM-LSS,AEGIS,andXMM-COSMOShave
allowed measuring the two-point clustering statistics (Murray
absorbX-raysoneneedssignificantcolumndensitiesofabsorb-
et al. 2005; Gandhi et al. 2006; Coil et al. 2009; Gilli et al.
ing material, which is particularly true in the hard X-ray band,
whereN ∼1024cm−2isrequired. 2009; Allevato et al. 2011), albeit with relatively large un-
H
With XMM-Newton2 and Chandra3 X-ray observatories, certainty on large scales. The large-scale clustering measure-
mentsareexpectedtoimprovesignificantlywiththeupcoming
two of the most advanced X-ray instruments in existence,
Spectrum-X-Gamma/eROSITA45 space mission (Predehl et al.
more than 20 deep extragalactic X-ray surveys have been
2010),whichisplannedtocoverentireskydowntothelimiting
performed over varying sky areas and limiting sensitivities
sensitivityof∼10−14 erg/s/cm2 inthe0.5−2keVband(Merloni
(Brandt & Hasinger 2005). The most noticeable among these
etal.2012;Kolodzigetal.2013b).Agoodsummaryofthecur-
are ∼2 Ms Chandra Deep Field North (Alexander et al. 2003)
rentstatusandfutureprospectsfortheX-ray-selectedAGNclus-
tering can be found in Cappelluti et al. (2012); Krumpe et al.
1 In comparison, the deepest optical spectroscopic surveys typically
(2013).
giveafactorof∼10timeslessAGNperdeg2,andonlyutradeepoptical
variabilitystudiesareabletogeneratecomparableAGNskydensities
(e.g.,Brandt&Hasinger2005).
2 http://xmm.esac.esa.int 4 http://www.mpe.mpg.de/erosita/,
3 http://chandra.harvard.edu 5 http://hea.iki.rssi.ru/SRG/
ArticlepublishedbyEDPSciences A58,page1of14
A&A561,A58(2014)
Even though the evolution of AGN over cosmic time is an There is evidence that AGN clustering strength might scale
interesting subject on its own, the remarkable discoveries of with X-ray luminosity (Krumpe et al. 2012; Koutoulidis et al.
correlations between SMBH masses and host galaxy proper- 2012). The same trend is also seen in semianalytical galaxy–
ties (see, e.g., Kormendy & Richstone 1995; Ferrarese & Ford AGN co-evolution models of Fanidakis et al. (2012), where in
2005, for reviews), hinting at the underlying AGN-galaxy co- the“hot-halo”AGNfuelingmodetheyfindquiteatightcorrela-
evolutionmechanisms,havesignificantlybroadenedtheimpor- tion between dark matter halo mass (and thus clustering bias)
tance and actuality of those topics in the astronomical com- and AGN luminosity. It should be mentioned, however, that
munity. Although controversial in terms of observational data otherauthorsanalyzingsimilarsamplesofX-raydetectedAGN,
(e.g., Ferrarese 2002; Kormendy & Bender 2011), in currently such as Coil et al. (2009) and Mountrichas et al. (2013), find
favoredcolddarkmatter(CDM)cosmologies,wherethestruc- noevidenceofsuchluminositydependenceofAGNclustering.
ture grows in a hierarchical fashion starting from the smallest Thisisnotsurprisinggiventheratherpooraccuracyofcluster-
scales(i.e.,thebottom-upscenario),onemayexpectthatSMBH ingmeasurementsintheX-rayband.Indeed,currentlyavailable
masses M should also correlate with the masses of the host samplesofX-ray-selectedAGNtypicallycontainbetweenafew
BH
darkmatterhalosM .Indeed,thisiswhatonecommonlyfindsin hundredandafewthousandobjects.
h
semi-analyticAGN-galaxyco-evolutionmodels,e.g.,Fanidakis Motivated by these ideas, we assume that a relation exists
et al. (2012). There is a significant scatter in the M –M re- between the dark matter halo mass and the AGN X-ray lumi-
BH h
lation, suggesting that there are other important parameters be- nosityitharbors,andtherefore,theX-rayluminosityfunctionof
yond M that determine M , for example, the merger history AGN can be computed from the mass function of dark matter
h BH
of the dark matter halo. However, it is still reasonable to use halos. In comparison to the QSO samples with typical sizes of
M as a single proxy for M , linking the latter to the quan- ∼100000objects,thecurrentlyavailableX-ray-selectedsamples
h BH
tities for which we have good theoretical predictions available, are about two orders of magnitude smaller, which which sug-
suchashalomassfunction(MF)andclusteringbias.Theblack geststhatthesemodelsshouldbekeptfrombeingtoocomplex.
holemass alonedoesnotdetermine theAGNluminosity, since Thus,weinitiallyassumethatthe M −L scalingisgivenbya
h X
theparticularprocessthatfuelstheblackholeisalsoimportant. simpledeterministicrelation.Thesearchforthefunctionalform
However,insomecircumstances,forexampleinthe“hot-halo” of this relation that is consistent with the concordance ΛCDM
fuelingmodel,thelattermayleadtoatightcorrelationbetween halomassfunctionandtheobservedX-rayAGNLFispresented
AGNluminosityandthehosthalomass. inSect.2.Itturnsoutthatthiswayweindeedobtainanaccept-
Future wide-field X-ray surveys like eROSITA promise to ablefittotheX-rayAGNLFand,asabonus,automaticallyob-
give us very precise AGN clustering bias measurements as a tainapredictionaboutX-rayAGNclusteringproperties,which
function of redshift and luminosity (Kolodzig et al. 2013a). seemstoagreewithavailableobservationaldata.Thisandother
Since clustering bias provides direct information on how AGN consequences of our simple model are discussed in Sect. 3. In
populate dark matter halos, this also helps in determining the Sect.4weinvestigatehowanincreasedlevelofthemodelcom-
dominant X-ray AGN fueling mode: “hot-halo” vs. “cold-gas” plexity influences our previous results. In particular, we extend
accretion. As an example, semi-analytic models of Fanidakis our model by allowing a scatter in the Mh−LX scaling relation
et al. (2012) suggest that the hot-halo fueling mode has quite and luminosity dependence of the AGN duty cycle. We bring
strong a dependence between the AGN X-ray luminosity and ourconclusionsinSect.5.
hosthalomass,withbrighterAGNpopulatingpreferablyhigher Throughout this paper we assume a flat ΛCDM cosmology
mass (and thus more strongly biased) halos. For the cold gas withΩm =0.27,Ωb =0.045,h=0.70,andσ8 =0.80.
accretion, on the other hand, which is presumably responsible
formostofthequasaractivity,thehosthalomassandAGNlu-
2. MappinghaloMFtoX-rayAGNLF
minosity are only weakly dependent, with most of the activity
occurringinsomewhatlowermasshalos. In the recent semianalytic galaxy–AGN co-evolution model of
StudiescombiningtheAGNLFandtheirclusteringstrength Fanidakisetal.(2012),theauthorsconsidertwomainAGNfuel-
havebeencarriedoutbyseveralauthorsusingmostlyoptically ingmodes:“hot-halo”and“starburst”mode.Someoftheresults
selected quasars (QSOs), such as Martini & Weinberg (2001); producedbythesemodels,whichwerepresentedinKoutoulidis
Haiman & Hui (2001); Wyithe & Loeb (2005); Lidz et al. et al. (2012), are plotted in Fig. 5. Here the dark-shaded stripe
(2006); Wyithe & Loeb (2009); Bonoli et al. (2010); Shankar represents the “hot-halo” mode, while the light-shaded region
etal.(2010b);Shankaretal.(2010a);Conroy&White(2013). correspondstoforthesumof“starburst”and“hot-halo”modes.
The major differences between X-ray selected AGN and opti- The starburst mode, where the SMBH fueling is induced by
cally selected QSOs is that QSOs represent the brightest and mergersordiskinstabilitiesandwherethereisalotofcoldgas
therarestendoftheAGNpopulation,whiletheX-ray-selected available, is arguably the one responsible for the QSO activity.
AGN,asdetectedinthedeepX-raysurveys,aremuchmorenu- Ontheotherhand,onceSMBHgetsincorporatedintogroup-or
merousandthusmoretypicalexamplesofAGN.Theotherim- cluster-sizeddarkmatterhalo,whichisfilledwithhotlowden-
portant difference is that despite QSOs being much rarer, their sity gas, the AGN fueling takes on a different character. In this
clustering strength is somewhat lower than the clustering am- case, in addition to metallicity, the gas cooling rate is largely
plitudes typically obtained for the X-ray selected AGN: QSO dictatedbythegasdensityandtemperature,whichundertheas-
clustering is compatible with being practically independent of sumption of hydrostatic equilibrium are directly related to the
luminosityandhavingasimilarstrengthasthedarkmatterhalos hostdarkmatterhalomass.Thegascoolingmustinpracticebe
withmasses∼1012h−1M ,whiletheclusteringbiasofX-rayse- moderatedbythefeedbackeffectsfromtheSMBH.Thus,inthe
(cid:12)
lected AGNsuggests thatthey populategroup-sized halos with “hot-halo”fuelingmodethemassofthehosthaloplaysanim-
typicalmassesof∼1013 h−1M .Furthermore,theQSOactivity portantrole indeterminingthe amountof availablefueland its
(cid:12)
iscompatiblewithbeingdrivenbymergersoflowermassdark accretionrate, leadingto theexistence ofa rathertight relation
matterhaloswithplentyofavailablecoldgasorbytheaccretion betweenthehostdarkmatterhalomassandAGNluminosity,as
diskinstabilities. illustratedbyFig.5.
A58,page2of14
G.Hütsietal.:LinkingX-rayAGNwithdarkmatterhalos
Motivatedbytheseconsiderations,westartbyassumingthat
1.4
there is a following simple scaling relation between the halo
massandAGNX-rayluminosity
1.2
(cid:32)L (cid:33)Γ(z)
Mh = M0 LX0 , (1) z) 1
where we take L0 = 1041 erg/s. Having the above scaling rela- (ty 0.8
u
tion,wecanimmediatelywritefortheAGNX-rayLF: d 0.6
f
dn dM dn
(L ,z)= f (z) h(L ,z) [M (L ,z),z], (2) 0.4
dL X duty dL X dM h X
X X h
where dn/dM is the concordance ΛCDM model halo MF and 0.2
h
fduty the “duty cycle”, in our case defined as a fraction of ha- 0 fduty(z)=c12z/[(z - c2)2 + c12] + c3
los that contain AGN in its active state. For the halo MF we
Γ(z) = c + c z
usetheanalyticformgivenbySheth&Tormen(1999)andthe 4 5
hard band (2−10 keV) measurements from Aird et al. (2010) 0.8 Γ(z) = c4 + c5z + c6z2
for the X-ray AGN LF. The Aird et al. (2010) LF data points
innineredshiftbinsareshowninFig.3.Weassume,somewhat 0.7
simplistically, that LF measurements in all luminosity bins are
statisticallyindependentwiththeerrorsfollowingGaussiandis- ) 0.6
z
tribution; i.e., the likelihood function gets factorized and has a (
simpleanalyticform.Tosampletheparametriclikelihoodfunc- Γ 0.5
tionweusedMarkovchainMonteCarlo(MCMC)methods,in
particular the Metropolis-Hastings algorithm (Metropolis et al. 0.4
1953;Hastings1970).Forfindingthemaximumlikelihoodpoint
inparameterspaceweusedthedownhillsimplexmethodasde- 0.3
scribedinPressetal.(1992).
TogainanideaoftheacceptableanalyticalformforΓ(z)and 0.2
f (z),weinitiallyallowthesequantitiestohavefreevaluesin 0.5 1 1.5 2 2.5 3
duty
alloftheredshiftbins:Γ(z), f (z),i=1...9.Thus,including z
i duty i
M wehave19freeparametersintotal.Figure1showsbestfit
0
voaultuMesCaMlonCgcwailtchul1aσtiorengsi.oWnsefosereΓ(thzia)taΓn(dzf)duatrye(zqi)uoitbetawineelldfifrtotemd Flaiwg.i1n.dBiceesstΓfi(tztii)nfgrodmutEyqc.y(c1l)e(vloawlueerspfadnuteyl()zii)n(nuipnpeerredpsahnieftl)bainnsd,sphoowwenr
i
withredcrosses.Intheupperpanelthepowerlawindexisassumedto
byasimplelinearform.
As a next step, we fix Γ(z) to have a linear redshift depen- be in parametric form Γ(z) = c4 +c5z while the amplitude factors in
eachredshiftbinsareassumedtobefreeparameters,i.e.,includingM
denceΓ(z) = c +c zandallow f (z)toremainfreeparame- 0
4 5 duty i the fitting problem has 12 free parameters in total. With solid black
ters,soourfreeparametersare M0,c4,c5, fduty(zi)(i = 1...9), lineweshowthebestfittingthreeparameteranalyticfunctionwiththe
i.e.12intotal.Thebestfitvaluesfor fduty(zi)areshownwithred detailed form as specified in the legend. In the lower panel fduty(z) is
crossesintheupperpanelofFig.1.Itturnsoutthatthesepoints fixed to the above analytic form while the power law indices in each
canbeapproximatedreasonablywellwiththeanalyticform, redshiftbinΓ(z)areallowedtovaryfreely,i.e.,thistimewehave13
i
freeparameters.Solidblackandgraylinesshowlinearandquadratic
c2z polynomial approximations to best fitting Γ(z). The dotted lines and
f (z)= 1 +c , (3) i
duty (z−c )2+c2 3 grayshadedregionsrepresentbestfitvaluesalongwith1σintervalsin
2 1 caseallΓ(z)and f (z)areallallowedtobefree(19freeparameters).
i duty i
i.e.,aLorentzianprofilemultipliedbyzandaconstantoffsetc
3
added6. andquadraticapproximationforΓ(z)ModelIandModelII,re-
To check for the self-consistency of the above analytic de-
spectively. We considered models with f constrained by the
scriptionwenowfix f (z)tohaveformasgiveninEq.(3),but duty
duty condition f ≤ 1,aswellasunconstrainedmodels,with f
allow the power law indices in all nine redshift bins to be free duty duty
allowedtotakeanypositivevalues.Obviously,constrainedmod-
parameters,i.e.,inthiscasewehave13freeparameters:M ,c ,
0 1 els correspond to the case when there is at most one AGN per
c , c , Γ(z)(i = 1...9). The best fit values for Γ(z) are plot-
2 3 i i darkmatterhalowhichisturnedonwithaprobabilitygivenby
tedasredcrossesinthelowerpanelofFig.1,wherewithsolid
f ,whereasintheunconstrainedmodelsanarbitrarynumber
lineswealsoshowthebestlinearandquadraticapproximations. duty
ofAGNperdarkmatterhaloisallowed.Itisinterestingthatin
EventhoughthelinearapproximationforΓ(z)seemstocapture
unconstrained models, f does not reach values much above
the redshift dependence of the power law index quite well, the duty
unity, as illustrated by the shaded region in the upper panel of
quadratic form with its one extra parameter, as it turns out, is
Fig.1.
statisticallywelljustified.
InTable1weshowthebestfittingvaluesforalltheparame-
Thus, our analytic MF-to-LF mapping is given by Eqs. (2),
tersofModelsIandII,withandwithoutanadditionalconstraint
(1), and (3). In the following we call the models with linear
on f . The calculated χ2 values along with expectations are
duty
6 In our earlier paper Hütsi et al. (2012), where we also used the giveninTable2,wherewealsoshowresultsforempiricalmod-
MF-to-LFmappingintheformofEq.(2),wehadadifferentanalytic elsusedtodescribeobservedX-rayLFs,namelypureluminosity
form for f (z), which turns out to give somewhat poorer fit to the evolution(PLE),luminosityanddensityevolution(LADE),and
duty
observedLF. luminosity-dependent density evolution (LDDE) models, with
A58,page3of14
A&A561,A58(2014)
1.0
2
c
0.8
0.4
3
c
0.2
0.90
4
c
0.84
0.24
5
c
0.32
6 0.03
c
0.00
7 27.6
c
27.2
0.4 0.6 0.8 1.0 0.2 0.4 0.84 0.90 0.32 0.24 0.00 0.03 27.2 27.6
c c c c c c c
1 2 3 4 5 6 7
Fig.2.Marginalized2Derrorellipsesrepresenting1σand2σconfidencelevelsalongwith1Dmarginalizedprobabilitydistributions(toppanels
ineachofthecolumns)forallthefreeparametersofunconstrainedModelII.Theblackdotsandredcrossesshowthemaximumlikelihoodpoints
forunconstrainedandconstrained(i.e., f ≤1)ModelII,respectively.
duty
Table1.BestfittingparametervaluesforModelsIandII.Bothconstrainedandunconstrainedcases.
fduty(z)= (z−cz2c)212+c21 +c3 ModelI(c6≡0) ModelI(c6≡0) ModelII ModelII
Mh=M0(cid:16)LLX0(cid:17)c4+c5z+c6z2 + fduty≤1 + fduty≤1
L0=1041erg/s bestfit bestfit bestfit bestfit
c
1 0.47 0.48 0.45 0.43
c
2 0.83 0.94 0.80 0.90
c
3 0.11 0.15 0.14 0.21
c
4 0.84 0.83 0.87 0.87
c
5 −0.21 −0.20 −0.28 −0.28
c6 – – 0.023 0.023
c7≡lnM0 27.3 27.4 27.4 27.5
theiranalyticformstakenfromAirdetal.(2010).Toperforma fixed,ourmodelsmakeseveralotherpredictions,whicharedis-
faircomparisonbetweendifferentanalyticfittingforms,wehave cussedinthenextsection.
recalculatedthebestfitmodelsforthelatterusingthesamefit- To illustrate resulting parameter uncertainties, we show in
tingmachineryasforourmodels.Itcanbeseenthat,exceptfor Fig.21σand2σmarginalizederrorregionsalongwith1Dprob-
PLE,allofthosemodelsprovidestatisticallyvaliddescriptions ability distribution functions for all the seven parameters of
of the observational data. Even though our MF-to-LF mapping unconstrained Model II. The black dots and red crosses mark
does not provide as good a fit as LADE or LDDE models, it the maximum likelihood points for unconstrained and con-
is quite remarkable that this more physically motivated model strained (i.e., f ≤ 1) cases, respectively. We see that all
duty
indeed seems to work. As a bonus, once MF-to-LF mapping is the model parameters seem to be sufficiently well determined.
A58,page4of14
G.Hütsietal.:LinkingX-rayAGNwithdarkmatterhalos
1e-44
z = 0.0-0.2 z = 0.2-0.5 z = 0.5-0.8
1e-46
1e-48
1e-50
1e-52 PLE
LADE
LDDE
1e-54
new model
1e-44
z = 0.8-1.0 z = 1.0-1.2 z = 1.2-1.5
]
1 1e-46
-
)
s
/
g
er 1e-48
(
3
-
c
p 1e-50
M
[

x
L 1e-52
d
/
n
d 1e-54
1e-44
z = 1.5-2.0 z = 2.0-2.5 z = 2.5-3.5
1e-46
1e-48
1e-50
1e-52
1e-54
1e+41 1e+43 1e+45 1e+41 1e+43 1e+45 1e+41 1e+43 1e+45
L [erg/s]
x
Fig.3. Best fitting X-ray AGN LFs in nine redshift bins along with observational data from Aird et al. (2010). In addition to our constrained
ModelIIfitswealsoshowthebestfittingPLE,LADE,andLDDEmodelswiththecorrespondinganalyticformstakenfromAirdetal.(2010).
Table2.Overallperformanceofvariousfittingforms. quadraticterminΓ(z).However,asseenfromtheresultsshown
inTable2,theneedforparameterc isstatisticallyjustified,ata
6
Model #Param #D.o.f. χ2 Expectedχ2 confidencelevelcorrespondingto∼2.5σ.
(1σCL) In Fig. 3 we plot the LFs in nine redshift bins for the best
PLE 7 72 96.2 72±12.0 fitting PLE, LADE, LDDE models, and for our constrained
LADE 8 71 67.4 71±11.9 Model II. Original data points with error bars, as determined
LDDE 9 70 57.8 70±11.8 by Aird et al. (2010), are also shown. Our model slightly but
systematically overshoots the highest luminosity data point at
ModelI 6 73 77.2 73±12.1
highredshifts.Thisisdiscussedinmoredetailinthefollowing
ModelI+,f ≤1 6 73 83.4 73±12.1
duty section.
ModelII 7 72 71.6 72±12.0
ModelII+,f ≤1 7 72 77.0 72±12.0
duty
3. Implicationsandshortcomingsofthemodel
The strongest degeneracy is seen between parameters c and In this section we investigate some of the consequences of the
5
c , which might leave some doubts about the usefulness of the MF-to-LFmappingmodelspresentedabove.
6
A58,page5of14
A&A561,A58(2014)
0.01 wherethepeakinnumberdensitymovestowardlowerredshifts
as one considers intrinsically less luminous AGN. However, at
least in the case of Aird et al. (2010) LF data, the conclusion
0.001
aboutthebehaviorofthemaximuminn(z)curvemaydependon
theparticularchoiceofthefunctionalformusedtofittheX-ray
0.0001 LFdata.Indeed,inourLFmodel,then(z)maximaforthelow-
]
-3c estluminositybinsstayalmostatthesameposition,whileforthe
p highestluminositybinweseegradualmovementtowardhigher
M 1e-05
redshifts. This does not seem to be entirely incompatible with
[
) thepatternoftheoriginalLFdatapointsinFig.10inAirdetal.
z
n( 1e-06 (2010).Furthermore,inourmodeltherateofchangeofn(z)has
quiteastrongdependenceonluminosity,withthelowestlumi-
nosityAGNhavingthefastestriseinnumberdensitygoingfrom
1e-07 highredshiftsdowntoz∼1,theAGNintheintermediatelumi-
Ueda et al. 2003, logLx = 41.5-43., 43.-44.5, 44.5-46. nositybinhavingonlyverymildincreaseandthehighestlumi-
1e-08 Aird ente wal .m 20o1d0el, llooggLLxx == 4411..55--4433..,, 4433..--4444..55,, 4444..55--4466.. nosity bin showing a mild drop in the volume density with the
Γ=2.0 redshift.Thelatterisrelatedtothemodelbehaviorathighlumi-
ΓΓ==11..86 nosityandredshift,discussedbelow.
6 Γ=1.4
3.2. Apossibleshortcomingofthemodel
5
AcarefulexaminationofFig.3revealsthatourmodeloverpre-
4 dictsthenumbersofmostluminousobjectsathighredshift(see
z) thethreelowerpanelsinFig.3).Thismaypointattheshortcom-
(
b ing of the model. Such a shortcoming could be caused by sev-
3
eralreasons.Onepossiblecauseisthelackofpropertreatment
of the Eddington luminosity limit. The Eddington ratio analy-
2 sislateroninthissectionsuggeststhatthisfactormaybecome
importantinthehalomassrangelogMh >∼13.5−14.5atthered-
1 shiftz>∼2−3.Ontheotherhand,itmayalsobeanindicationof
data from Allevato et al. 2011 theshortcomingoftheunderlyinghalomassfunction,somewhat
0 overpredictingtheabundanceofmassivehalosathighredshifts.
0 0.5 1 1.5 2 2.5 3 3.5 4 Another possible reason may be a failure of the sim-
z ple power-law scaling between the luminosity and halo mass,
Eq. (1),at highhalo masses andredshifts. Indeed,due to obvi-
Fig.4.Upperpanel:comovingX-rayAGNnumberdensitiesasafunc-
tion of redshift for three 2−10 keV band luminosity ranges as shown ous considerations of time available for the black hole growth,
inthelegend.Theluminosityisincreasingfromtoptobottom.Along the most massive halos at high redshifts may harbor somewhat
withourconstrainedModelIIwehavealsoplottedtheresultsarising lessmassiveblackholesthanimpliedbyEq.(1).Theparameters
fromtheLFmodelsofUedaetal.(2003)andAirdetal.(2010).Lower of the halo mass–luminosity scaling were determined from the
panel: comparison of bias parameters from the constrained Model II globalLFfitthatisdeterminedbythebulkoftheLFdatapoints
withtheobservationaldatafromAllevatoetal.(2011).Thefourlines andisnotverysensitivetothedataattheextremitiesbecausethe
assumedifferenteffectivephotonindices(Γ=1.4,1.6,1.8,2.0)forthe contribution of high luminosity points to χ2 is relatively small,
approximatetransformationbetweensoftandhardX-raybands.
owing to their small number and also to their relatively larger
uncertainties.
Despitethisshortcoming,however,themodelprovidesasta-
tisticallyvalidoveralldescriptionoftheX-rayLFdata(Table2).
3.1. ComovingAGNdensity
Its(possible)failureatthehighluminosityendonlyaffectsvery
Westartbyinvestigatingthecomovingnumberdensityofhard- fewofthemostmassivehalosathighredshiftsanddoesnotdi-
bandselectedX-rayAGNanditsdependenceonredshift.Inthe minishtheoverallpredictivepowerofthemodelforthebulkof
upperpanelofFig.4wehaveplottedcomovingnumberdensity typicalAGN,asdiscussedinthefollowingsections.
for three X-ray luminosity intervals as specified in the legend.
Herethehighest(lowest)groupofcurvescorrespondtothelow-
3.3. AGNclustering
est (highest) luminosities. Along with constrained Model II re-
sults,forcomparisonwehavealsoplottednumberdensitiesre- Since our model has a fixed scaling relation between the halo
sulting from Ueda et al. (2003) and Aird et al. (2010) LF fits. massandAGNX-rayluminosity,itcanstraightforwardlypredict
In the case of Ueda et al. (2003) we directly use their best fit theAGNclusteringstrength.Indeed,theAGNclusteringbiascan
LFparameters,whereasfortheAirdetal.(2010)LDDEmodel, becomputedas
weusedbestfitparametersobtainedintheprevioussection.The (cid:82)
b [M (L ,z),z] dn (L ,z)dL
comparisonofourresultswithAirdetal.(2010)LDDEparam- b(z)= h h(cid:82) X dLX X X, (4)
eterizationshouldgivesomeideaaboutthepossiblelevelofun- dn (L ,z)dL
certainty in the so-obtained n(z), since both of the models give dLX X X
statisticallyvalidfitstoLFdata. wherethedarkmatterclusteringbiasistakenfromtheanalyti-
This type of figure is often used to illustrate the antihierar- calmodelofShethetal.(2001),andthe L integrationbounds
X
chical growth of SMBHs (as manifested by the AGN activity) are appropriately adjusted to take the details of how the X-ray
A58,page6of14
G.Hütsietal.:LinkingX-rayAGNwithdarkmatterhalos
AGNsamplewasselectedintoaccount.Inparticular,foraflux- 1e+15
z = 1.0
limited sample L (z) = 4πD2(z)F , where D (z) is the z = 0.5
X,min L X,lim L z = 1.5
luminositydistancetoredshiftz. Bandara et al. Mh-MBH relation + fEdd=3.e-4
InthelowerpanelofFig.4wecomparethethus-computed
1e+14
clustering bias as a function of redshift with the AGN clus-
tering data from Allevato et al. (2011). Allevato et al. (2011)
used soft band X-ray data from COSMOS field, which cov- ]
o
ers 2.13 deg2 and contains a total of 780 AGN with available M
[ 1e+13
redshifts, its spectroscopic completeness being ∼53%. In their h
M
analysisAllevatoetal.(2011)appliedamagnitudecutI <23
AB
andredshiftcutz<4,resultingin593objectsintheirfinalsam-
ple,witheffectivespectroscopiccompletenessof∼65%. 1e+12
SincetheclusteringbiasinourmodeldependsonAGNlu-
minosity, with more luminous AGN having higher bias param-
eters, we have to adjust the effective flux limit appropriately in datapoints from Koutoulidis et al. 2012
ordertomakeafaircomparison.SincetheselectionofAllevato 1e+11
1e+41 1e+42 1e+43 1e+44 1e+45
etal.(2011)samplewasdoneinthesoftband,wetransformed
L [erg/s]
their flux limit to the corresponding limit in the hard band. We x
do this in a simple way by assuming an effective population-
Fig.5. Comparison of the M −L scaling relation with the observa-
averagedspectralindexΓ7.Theresultsforfourdifferentvalues tionaldatafromKoutoulidisehtal.X(2012).Tocapturetheredshiftrange
ofΓareshowninthelowerpanelofFig.4.Becauseinthehard of majority of AGN used in this analysis, we have plotted the model
bandonestartstoseemoreandmoresourcesthatareobscuredin scalingrelationsforthreeredshifts:z = 0.5,1.0,1.5.Thedottedlines
thesoftband,theeffectivespectralindexaround∼1.6appearsto withsurroundinggraybandsdisplaytheresultsfromthesemianalytic
bethemostappropriateone.Consideringthis,weseethatexcept galaxy–AGN co-evolution model of Fanidakis et al. (2012), as pre-
for the highest redshift bin, our predictions for the bias param- sentedinKoutoulidisetal.(2012),showingtheir“hot-halo”(darkgray
region)and“allAGN”case,i.e.,thesumof“hot-halo”and“starburst”
eter agree reasonably well with the measurements. The highest
modes (light gray region). The long-dashed line shows the M −L
redshift bin is represented by one point only, and it remains to h X
scaling one obtains once converting the amplitude-adjusted M −M
beseenhowsignificantthedeviationis.Onehastokeepinmind h BH
relation of Bandara et al. (2009) by using the fixed Eddington ratio
thatthemeasurementofthelinearbiasparameter,andespecially f ≡L /L (M )=3×10−4.
Edd X Edd BH
theproperestimationoftheerrorsduetocosmicvariance(e.g.,
accidentalsuperclustersinsmallfieldscouldsignificantlyinflu-
ence the results) from the fields as small as ∼2 deg2, may be
quiteacumbersometask. isclosertotheonesinwhichthemedianoftheEddingtonratio
distributiondecreaseswithredshift.
3.4. AGNdutycycle
3.5. M –L scalingrelation
h X
In both constrained and unconstrained models, the AGN duty
cycle (Eq. (2)) peaks at the redshift z (cid:39) 1 (Fig. 1). Although Asemphasizedabove,thecrucialcomponentinourmodelisthe
unconstrainedmodels(whenthedutycyclewasallowedtotake assumptionabouttheexistenceofascalingrelationbetweenthe
anypositivevalue)resultinasomewhatbetterdescriptionofthe halomassandAGNX-rayluminosity.Koutoulidisetal.(2012)
observed X-ray LF data, it is interesting that the peak value of have recently attempted to infer this type of scaling by using
thedutycycleinthesemodelsisneverthelessclosetounity,with AGN clustering data from several deep X-ray fields: Chandra
theupperlimitof fduty <∼ 1.5(1σconfidence).Thisimpliesthat DeepFieldSouthandNorth,theAEGIS,theextendedChandra
onaveragethereisatmost∼oneAGNperdarkmatterhalo. Deep Field South, and the COSMOS field. In Fig. 5 we show
Thevalueforthedutycycleclosetounityisaconsequence their measurements with error bars along with our constrained
ofthefactthatattheredshiftz∼1,thevolumedensityofX-ray ModelIIpredictionsforredshifts0.5,1.0,and1.5,i.e.approx-
AGN is comparable to the volume density of the dark matter imately covering the redshift range of AGN used in their anal-
halos.Indeed,apower-law M −L scalingrelationdoesnotin- ysis. The dotted lines with surrounding gray bands display the
h X
troduce any new scale, therefore the only way to obtain agree- results from the semianalytic galaxy–AGN co-evolution model
ment with the observed LF is by matching the MF character- ofFanidakisetal.(2012)(aspresentedinFig.9ofKoutoulidis
istic mass M with a characteristic AGN X-ray luminosity L . etal.2012),showingtheir“hot-halo”(darkgreyregion)and“all
∗ ∗
In this MF to LF mapping, the role of the f is to tally their AGN” case, i.e., the sum of “hot-halo” and “starburst” modes
duty
volume densities. As it turns out, the comoving number den- (light gray region). Although the uncertainties are rather large,
sities of M (see Appendix A) and L objects are comparable we see that our model is able to capture the major trend where
∗(cid:104) ∗ (cid:105)
at z ∼ 1: L dn (L = L ,z(cid:39)1)(cid:39) M dn (M = M ,z(cid:39)1) ; the more luminous AGN tend to populate more massive halos.
XdLX X ∗ hdMh h ∗ ThequantitativeagreementbetweenourM −L relationandre-
i.e.,thereisapproximatelyasmanydarkmatterhalosasAGN. h X
sults of Koutoulidis et al. (2012) is truly remarkable since they
Correspondingly, f ∼ 1.Thedutycycledropsto f ∼ 0.2
duty duty
attheredshiftz = 0(Fig.1),amanifestationofthewellknown arebasedontotallyindependentarguments.
The long-dashed line in Fig. 5 shows the M −L scaling
paucity of AGN in the local Universe. It is interesting to note h X
thatthedutycycleatz=0isnotaslowaspredictedbythesim- one obtains when converting the Mh–MBH relation of Bandara
et al. (2009) by assuming a constant Eddington ratio f ≡
plecontinuityequationmodelswithafixedEddingtonratio,but Edd
L /L (M )of3×10−4.Inreality,becausetheBandaraetal.
X Edd BH
7 Not to be confused with the power law index Γ(z) appearing in (2009)MBH–Mhscalingrelationusedstronglensingmassesde-
Eq.(1). rivedfromtheSLACSlenssamplewithmeanredshiftz (cid:39) 0.2,
A58,page7of14
A&A561,A58(2014)
1e+46 our results for L –M with 1σ uncertainties as determined in
X h
Sect. 2 for the constrained Model II. The middle panel shows
1e+45 the amplitude-adjusted Bandara et al. (2009) M –M relation
BH h
with1σuncertainties.InthelowerpanelweshowtheEddington
]
/s 1e+44 ratiosasdeterminedbycombiningtheresultsfromthetwoup-
g
r permostpanels.TheX-rayluminositiesandEddingtonratiosre-
e
[x 1e+43 fer to the 2–10 keV band and no bolometric corrections were
L z = 0.0 applied.ForatypicalTypeIAGNspectrum,thebolometriccor-
z = 0.5 rectionfactorshouldbeinthe∼5−20range(Lussoetal.2012).
1e+42 z = 1.0
z = 2.0 Aswecansee,theEddingtonratiosforhighredshift(z>∼2)
z = 3.0 AGN are predicted to be steeply rising functions of halo mass,
1e+41
while at redshifts z ∼ 1, i.e. close to the peak in X-ray AGN
numberdensity,thedependenceof f on M flattensout,and
Edd h
1e+09 at lower redshifts the trend is mildly reversed; i.e., the AGN in
] lessmassivehaloshavesomewhathigherEddingtonratios.For
o
M z ∼ 3, the 2–10 keV band f reaches values of ∼0.1 for halo
Edd
[H 1e+08 masses ∼2×1013M(cid:12); i.e. the bolometric Eddington ratio will
B approach∼1atthismass.Athighermasses,ourM −L scaling
M h X
relation would imply fEdd >∼ 1. Obviously, this is because our
1e+07
simple M −L scaling relation does not impose the Eddington
h X
Bandara et al. (2009) MBH-Mh relation luminosity limit. This may be considered as one of the short-
+ scaling amplitude evolution from Croton (2009)
comingsofthemodel,asdiscussedinSect.3.2.Onehastonote,
1e+06
however,thatthisdoesnotaffecttheoverallperformanceofthe
model, in the major part of the parameter space. Indeed, such
0.01 massivehalosatredshiftsz ∼ 3areextremelyrare,correspond-
ing to ∼4−5σ peaks in the initial fluctuation field. The bulk of
d
Ed theobjectsareshiningatmuchlowerEddingtonratios,(cid:28)1.
L 0.001
/x Havingspecified MBH–Mh scalingrelation,wecanalsocal-
L culate how SMBHs with different masses grow throughout the
cosmictime.Takingthedutycycleintoaccount,thepopulation-
0.0001
averagede-foldingtimeoftheSMBHgrowthis
1e-0 15e+12 1e+13 1e+14 τ≡ MBH = ηMBHc2 (5)
M˙ f f f L
M [M ] duty duty bol X
h o
Fig.6.Upperpanel:averageL −M relations(Eq.(1))alongwith1σ whereηistheaccretionefficiency,definedas
X h
uncertaintiesforfivedifferentredshiftsasgiveninthelegend.Middle
panel: amplitude-adjusted MBH−Mh scaling relation with 1σ uncer- LX = fb−o1lηM˙c2 (6)
tainty from Bandara et al. (2009). Lower panel: Eddington ratios in
2−10 keV band for five redshifts arising from LX−Mh and MBH−Mh LX refers to the X-ray luminosity in the 2–10 keV band used
relationsgivenintheuppertwopanels. throughout this paper, and f ∼ 5−20 is the bolometric cor-
bol
rectionfactorforthe2–10keVband.Thee-foldingtimecanbe
expressedfurtherthoughtheEddingtonratio:
we have adjusted the amplitude of the scaling relation to the ηM c2 η
redshift z ∼ 1, using results from Croton (2009) semi-analytic τ= BH =0.45 Gyr. (7)
f f f L f f f
models.InBandaraetal.(2009)theauthorsobtain M ∝ Mγ duty bol Edd Edd duty bol Edd
withγ = 1.55±0.31,whichagreesquitewellwiththeBHanalytihc InthelowerpanelofFig.7weplotthee-foldingtimeinunitsof
andsemi-analyticmodelsofWyithe&Loeb(2003)andCroton theHubbletime,i.e.τH(z),vsredshiftforseveralvaluesofthe
(2009),whichinturnprovidevaluesγ = 5/3 (cid:39) 1.67and1.39, black hole mass. Shown in the upper panel is the evolution of
respectively.InCroton(2009)andWyithe&Loeb(2003)mod- the population average 2−10 keV band X-ray luminosity with
els the amplitude of the scaling relation increases by factors of redshift. AGN harboring black holes with masses less than a
∼2 and ∼4, respectively, while going from z = 0 to z = 1. ∼few×108M aretoofaintathighredshiftsandfalloutsidethe
(cid:12)
However, the power law index γ is independent of redshift in luminosityrangeoftheX-rayLFdataofAirdetal.(2010)used
bothmodels. inthisstudy.Thereforetheyarenotdirectlyconstrainedbythe
X-ray data. At these redshifts we only show with dotted lines
the extrapolations of the central model curves and omit the 1σ
3.6. AGNBlackholemasses,Eddingtonratios,
uncertaintyregions.
andthegrowthofSMBHs
We see that the X-ray luminosity of more massive SMBHs
Byassumingthe M –M scalingofBandaraetal.(2009)with declines faster with the redshift than for less massive ones.
BH h
amplitude adjusted according to Croton (2009) semi-analytic For ∼108M black holes, L stays almost constant, while for
(cid:12) X
model, we can convert our L –M relation to f –M scaling even lower mass SMBHs, the evolutionary trend is slightly re-
X h Edd h
relation. This is detailed in Fig. 6 where we have plotted vari- versed; i.e., the objects are getting somewhat brighter as one
ous scalings for five different redshifts. The upper panel gives movestowardlowerredshifts.Asonemighthaveexpectedfrom
A58,page8of14
G.Hütsietal.:LinkingX-rayAGNwithdarkmatterhalos
1e+46 Westartwithintroducingscatterinthe M −L relation.We
h X
assumethatscattercanbedescribedbythelognormalprobabil-
1e+45 itydistribution8:
/s]g 1e+44 f(L |M ,z)= 1√ exp−(cid:2)lnLX−µ(Mh,z)(cid:3)2, (8)
er X h L σ 2π  2σ2 
[ X
x 1e+43
L whosewidthσisassumedtobeindependentofM andz.Eq.(2)
h
fortheluminosityfunctionisnowreplacedby
1e+42
(cid:90)
dn dn
(L ,z)= f (L ,z) f(L |M ,z) (M ,z)dM . (9)
1e+41 dL X duty X X h dM h h
X h
There are several obvious possibilities for µ(M ,z), depending
h
on whether one chose the (i) mode; (ii) median; or (iii) mean
10 of the lognormal probability distribution function to obey our
H previousdeterministic Mh−LX relationdefinedbyEq.(1).This
τ gives
1  (cid:32) (cid:33) 1 
MMBBHH = = 1 100109 MMoo µ(Mh,z) = σ2+lnL0 MMh Γ(z), (10)
MBH = 108 Mo 0
0.1 0 M B0H.5 = 107 Mo 1 1z.5 2 2.5 3 µ(Mh,z) = lnL0(cid:32)MMh0(cid:33)Γ1(z), (11)
eF-ifgo.l7d.inTghteimheasrdτba≡ndMM˙XBBHH-rainyulunmitsinoofsitthieesH(uupbpbeler tpiamneel()loawnedrSpManBeHl) µ(Mh,z) = −σ22 +lnL0(cid:32)MMh0(cid:33)Γ1(z) (12)
forvariousblackholemassesasshowninthelegend.Forthelowmass
SMBHsathigherredshifts,wheretheyaretoofainttobeconstrainedby
forthemode,median,andmean,respectively.Sincethisanalysis
theobservationaldata,wehaveshowntheextrapolationsofthemodel
islargelyinspiredbycomparisonwithobservations,whichpick
curveswithdottedlinesandhaveomittedthecorresponding1σuncer-
upthemostprobableobjects,itisnaturaltochoosethemodeof
taintybands.
thedistribution,Eq.(10).
It is easy to see that in case of small scatter, all the above
descriptionsareequivalent,andinthelimitσ→0
the results plotted in the bottom panel of Fig. 6, for the typ-
itcimalesXa-rraeyl-osneglecintecdoAmGpaNri,sothnetocotrhreesHpounbdbilnegtiSmMe.BWHiteh-ftohledienxg- f(LX|Mh,z)−→δL−L0(cid:32)MMh(cid:33)Γ1(z),
0
ception of most massive SMBHs at high redshifts, the typical
X-ray-selectedAGNarenotintherapidmassgrowthregimein andwerecoverthedeterministicrelationgiveninEq.(1).
thez∼0−3redshiftrange. To calculate the AGN clustering bias, Eq. (4) now general-
ThecrossingofthemodelcurvesinthelowerpanelofFig.7 izesto
attheredshiftofz∼0.85isartificialanddoesnotseemtohave (cid:82) (cid:82)
alanwyipnhdyicsiecsaolfmtheeanMinBgH.−ItMishadnridveMnhb−yLtXhesceaqliunaglirteylaotfiotnhse,pwohwicehr b(z)= b(cid:82)h((cid:82)Mfh(,Lz)f|M(LX,|zM)hd,nz)(MddMnh,(zM)dh,Mz)ddLMhdLX, (13)
leadstotheτHbeingindependentofMBHatthatparticularz. X h dMh h h X
Thus, with the particular form of the MBH-Mh scaling rela- Similar to Eq. (4), the integration limits for LX have to be ad-
tion from Bandara et al. (2009) with amplitude evolution from justedappropriately,takingthewaysAGNsamplewasselected
Croton (2009), our model implies relatively low Eddington ra- intoaccount.
tios, fEdd ∼ 10−4 − 10−3. In the redshift range z ∼ 0−3, for InFig.8,whichistheanalogofthepreviousFig.2,weshow
a typical X-ray-selected AGN, accretion proceeds in the “hot- the results of introducing scatter in the M −L mapping. The
h X
halomode”characterizedbyarelativelylowmassaccretionrate
andlonge-foldingtimeforSMBHmassgrowth.Obviously,this 8 Thisassumption,alongwiththepowerlawMBH–Mhscalingrelation
conclusioncriticallydependsontheassumed M −M scaling usedthroughoutthispaper,leadstoanapproximatelylognormalprob-
BH h
abilitydistributionfunctionfortheEddingtonratio f ,whichagrees
relation.Indeed,asX-rayluminosityforthegivenhalomassis Edd
quitewellwiththemeasurementsofLussoetal.(2012).However,once
directlydeterminedbyourmodelfromtheL −M mapping,the
X h thecompletenesscorrectionistakenintoaccount,atredshiftsz<∼1,the
Eddingtonratioscalesinverselywiththeblackholemass,while
f distributionscanbecomeclosertoapower-lawshape,instead(see
e-foldingtimeislinearlyproportionaltotheblackholemass. Edd
alsoAirdetal.2012).WithinourmodelframeworktheEddingtonratio
distributioncanbedeterminedoncefixingtheM –M relation.Inthis
BH h
paper we have assumed one particular form for the M –M scaling
BH h
4. Extensionsofthemodel relation,i.e.,asimplepowerlaw.Inprinciple,therearequitelargeob-
servationaluncertaintiesregardingtheparticularformforthisrelation,
Theminimalisticmodelconsideredintheprevioussectionshas
andthusbyallowingmorefreedomhere,oneshouldcertainlybeableto
twomajorsimplifications:adeterministicMh−LXrelationanda obtainasignificantlybettermatchwiththeclaimedpowerlawlike fEdd
luminosity-independent duty cycle. In this section we lift these distributionsatlowerredshifts.However,thismoredetailedanalysisis
assumptionsandexploreaclassofmoresophisticatedmodels. beyondthescopeofthecurrentwork.
A58,page9of14
A&A561,A58(2014)
1.0 0.4 0.6
2
c
0.8
0.4 0.4 0.6 0.8 1.0
3
c
0.2
0.4 0.6 0.2 0.4
0.9
4
c
0.8
0.4 0.6 0.8 0.9
0.24
5
c
0.32
0.4 0.6 0.32 0.24
6 0.03
c
0.00
0.4 0.6 0.00 0.03
27.6
7
c
27.2
0.4 0.6 27.2 27.6
0.6
8
c
0.3
0.0
0.4 0.6 0.8 1.0 0.2 0.40.8 0.9 0.32 0.24 0.00 0.03 27.2 27.6 0.0 0.3 0.6
c c c c c c c c
1 2 3 4 5 6 7 8
Fig.8.AnalogofFig.2withoneadditionalparameter,c ≡σ,describingthescatterinthelognormaldistribution.
8
Table3.Bestfittingparametervaluesandχ2valuesforvariousmodels,discussedindetailinthemaintext.
c1 c2 c3 c4 c5 c6 c7≡lnM0 c8≡σ c9≡α #D.o.f. χ2
f (z)andfreescatter 0.425 0.892 0.219 0.870 −0.283 0.0252 27.5 0.0458 0.(fix) 71 71.6
duty
f (z)and0.25dexscatter 0.445 0.907 0.195 0.893 −0.291 0.0227 27.6 0.25ln10(fix) 0.(fix) 72 75.7
duty
f (z)and0.50dexscatter 0.485 0.936 0.157 0.936 −0.295 0.0148 27.8 0.50ln10(fix) 0.(fix) 72 81.6
duty
f (L ,z),noscatter 0.372 0.820 0.224 0.994 −0.299 0.0319 27.0 0.(fix) 0.306 71 46.9
duty X
f (L ,z)andfreescatter 0.389 0.831 0.201 1.20 −0.389 0.0422 27.3 1.39 0.414 70 43.6
duty X
scatterisparametrizedviaasingleparameter–thewidthofthe to 0.25 and 0.50 dex (i.e., σ = 0.25ln10 and σ = 0.50ln10,
log-normal distribution σ ≡ c . All other conventions are ex- respectively). The best parameter values for the case of uncon-
8
actly the same as in Fig. 2. The best fitting values for all the strained f forthesetworunsareshowninthesecondandthird
duty
parameters are given in the first row of Table 3 for the case of rowsinTable3.Itturnsoutthatthedataandmodelareableto
unconstrained f . As is obvious from Fig. 8 and Table 3, no comfortablyaccommodateM −L scatterupto∼0.50dex.
duty h X
scatterisrequiredbytheX-rayLFdata,sincethebest-fitvalue Wenowinvestigatetheeffectoftheluminositydependence
ofσisclosetozero.Correspondingly,theχ2andthebestfitval- ofthedutycycle.Onepossibilityofsuchdependencewassug-
uesofotherparametersarenearlyidenticaltotheonesobtained gestedbyobservations(Bundyetal.2008),aswellasbysemi-
inthedeterministiccase(cf.Table1).Bycomparingwiththere- empiricalmodels(Merloni&Heinz2008;Shankaretal.2013).
sultsshowninFig.2,itisinterestingtonotethatafterallowing WeconsiderL -dependenceinthepowerlawform:
X
for the scatter in the Mh−LX relation, no new strong parameter (cid:32)L (cid:33)α
degeneracies appear and all parameters are as well determined f (L ,z)= X f (z), (14)
duty X duty
asbefore. (cid:101)L
0
ItisalsoclearfromFig.8(bottomrow)thatmoderatescat-
ter, up to σ ∼ 0.5 is still consistent with the LF data, within where fduty(z) is given by Eq. (3) and (cid:101)L0 is fixed at
∼1σconfidence.Toinvestigatemodelswithevengreaterscatter, (cid:101)L = 1043 erg/s. As before, we consider the case of
0
wealsoperformedtwoMCMCrunswherethescatterwasfixed unconstrained f .
duty
A58,page10of14