High redshift quasars and the supermassive black hole mass budget: constraints on quasar formation models

0.37 MB·English

by  J. M. Bromley

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
High redshift quasars and the supermassive black hole
mass budget: constraints on quasar formation models
J. M. Bromley,1 R. S. Somerville2 and A. C. Fabian1
1 Institute of Astronomy, Universityof Cambridge, Madingley Rd., Cambridge CB3 0HA
2 Space Telescope Science Institute, 3700 San MartinDr., Baltimore, MD 21218
2February2008
4
0 ABSTRACT
0 We investigate the constraints on models of supermassive black hole (SMBH) and
2 quasar formation obtainable from two recent observational developments: the discov-
n eryofluminousquasarsatz ∼6,andestimatesofthelocalmassdensityofSMBHs.If
a ∼90percentofthismasswasaccretedatredshiftsz<∼3,assuggestedbytheobserved
J quasarluminosityfunctions,thesejointconstraintsposeachallengeformodels,which
2 must account for the observed luminous quasar population at z ∼ 6 within a very
2 limited ‘mass budget’. We investigate a class of models based within the hierarchical
structureformationscenario,inwhichmajormergersleadtoblackholeformationand

3 fuelling,andtheresultingquasarsshineattheirEddington-limitedrateuntiltheirfuel
v
isexhausted.Weshowthatthesimplestsuchmodel,inwhichaconstantfractionofthe
8
gas within the halo is accreted in each major merger, cannot satisfy both constraints
0
simultaneously. When this model is normalized to reproduce the number density of
0
luminous quasars at z ∼ 6, the mass budget is grossly exceeded due to an overabun-
1
1 danceoflowermassSMBHs.We explorearangeofmodificationsto thesimple model
3 designed to overcome this problem. We show that both constraints can be satisfied if
0 the gas accretion fraction scales as a function of the halo virial velocity. Similar scal-
/ ings have been proposed in order to reproduce the local M• −σ relation. Successful
h
modelscanalsobeconstructedbyrestrictingtheformationofseedblackholestored-
p
- shifts above zcrit ∼ 11.5 or to haloes above a velocity threshold vcrit ∼ 55 kms−1, or
o assuming that only a fraction of major mergers result in formation of a seed SMBH.
tr We also briefly discuss the issue of trying to assume a ‘universal M• −σ relation’
s withintheframeworkofsimplePress–Schechtermodels,andfurthershowthatafixed
a
universal relation between SMBH mass and host halo mass is unlikely.
:
v
i Key words: accretion – galaxies: interactions – galaxies: active – quasars:general
X
r
a
1 INTRODUCTION the numbers of observed quasars as well as in the range of
luminosities and redshifts probed.The2dFQuasar redshift
In recent times, a growing wealth of observations has re- survey(2QZ;Boyleetal.2000)andtheearlierLargeBright
vealed a strong correlation between the mass of supermas- QuasarSurvey(LBQS;Hewett,Foltz& Chaffee1995) have
sive black holes (SMBHs) and the luminosity of the stellar identified over 23000 quasars at redshifts 0.3<∼z<∼3.0. The
bulges in their host galaxies, and an even stronger correla- Sloan Digital Sky Survey (SDSS; Gunn et al. 1998) has
tion between SMBHmass andthebulgevelocitydispersion observed over 650 confirmed quasars in the redshift range
(Magorrian et al. 1998; van der Marel 1999; Gebhardt et 3.0<∼z < 5.7 (Schneider et al. 2003) and a small num-
al. 2000; Ferrarese & Merritt 2000; Tremaine et al. 2002). ber at z>∼5.7 which have been sufficient to constrain for
Thishasinturnproducedincreasinglyaccurateestimatesof the first time the density of bright quasars at z ∼ 6 (Fan
theoveralllocalSMBHmassdensity.IftheseSMBHsare,as et al. 2001a, 2003). These objects must host SMBHs with
nowwidelybelieved,therelicsofearlierquasaractivity(e.g. masses on the order of several 109 M⊙ (Fan et al. 2001a,
Rees 1984 and references therein) then these estimates can 2003; Willott, McLure & Jarvis 2003). Moreover, evidence
place strong constraints on the nature and evolution of the has been found even in these highest redshift objects for
quasarpopulationastheyeffectivelylimitthe‘massbudget’ molecular gas (Bertoldi et al. 2003a; Walter et al. 2003),
available to account for quasar activity. dust (Bertoldi et al. 2003b), and metals (Freudling, Corbin
& Korista 2003), underlining the fact that even at these
The last few years have also seen dramatic progress in
(cid:13)c 0000RAS
2 J. M. Bromley, R. S. Somerville and A. C. Fabian
earlyepochsquasarsareassociated withgalaxies developed tailedmodellingofthestarformation,gascoolingandfeed-
enough to haveexperienced significant star formation. backprocesseswhichalterandcompetefortheintergalactic
In addition to the optical surveys already mentioned, gas thought to fuel quasars. In this Paper, we avoid these
observations of quasars and other active galactic nuclei complications by focusing on constraints at very high red-
(AGN) in other wavebands have also been steadily im- shift z ∼ 6, when cooling times are short and only a small
proving. This includes substantial work done determining fraction of the baryons is locked up in stars, so these com-
the soft X-ray quasar/AGN luminosity function (Miyaji, peting processes should be relatively unimportant. As well,
Hasinger & Schmidt 2000, 2001) and also several imag- wedonotattempttomodelthecomplexinterconnectionof
ing surveys of AGN selected in the hard X-ray band (e.g. galaxyandBH/quasarformationthroughfeedback.Instead,
Mushotzky et al. 2000; Giacconi et al. 2001; Hasinger et we consider a suite of models with simple parametrized
al. 2001; Alexander et al. 2003; Barger et al. 2003) where recipes for SMBH formation and fuelling, which point the
samples are unbiased by any line-of-sight absorption which way for more sophisticated and physical modelling. As we
may be present. Thus through multiwaveband observations discuss in the next section, attempts to combine data for
progress is being made in piecing together the history of observed AGN in X-rayand optical wavebandsnow appear
AGN and their SMBH. to be able to account for the observed present day SMBH
Ithaslongbeenestablishedthatquasars,andindeedall mass density, and moreover seem to suggest that most of
formsofAGN,aretheresultofgaseousmatteraccretingon it was assembled at fairly low redshifts z<∼3. This enables
toaSMBHatthecentreofagalaxy(Zel’dovich &Novikov us to place rough limits on the ‘mass budget’ available at
1964; Salpeter 1964; Lynden-Bell 1969; Bardeen 1970) and higherredshifts.CombiningthiswiththeSDSSobservations
hence must somehow be linked to galaxy formation. Since of the quasar numberdensity at z ∼6 allows us to identify
that time a great number of models have been published successfulmodelswithinthebroadclasswehaveconsidered,
exploring quasar formation and evolution within the con- and to rule out other model variants.
text of the modern Cold Dark Matter (CDM) paradigm of
structureformation (includinge.g.Efstathiou &Rees1988;
Our results will in some sense reflect a ‘best case sce-
Haehnelt & Rees 1993; Haehnelt, Natarajan & Rees 1998;
nario’asinclusionofstarformationandfeedbackwouldpre-
Cattaneo, Haehnelt & Rees 1999; Kauffmann & Haehnelt
sumablyonlyacttomakefuellessplentifulforquasars,and
2000;Cavaliere&Vittorini2000;Monaco,Salucci&Danese
thus strengthen our constraints. As a result however it is
2000; Nulsen&Fabian2000; Haiman &Loeb 2001; Wyithe
difficulttoplacestrongconstraintsonthelowerredshiftbe-
& Loeb 2002a, 2003; Hatziminaoglou et al. 2003; Granato
haviour of the models as the characteristic upper peak and
et al. 2003; for a brief overview of several of these mod-
subsequentdeclineofthequasarpopulationnumberdensity
els see Haehnelt 2003). However, none of the above works
atlowredshiftsislikelytobestronglyrelatedtofueldeple-
have concentrated on successfully reproducing the highest
tion due to star formation (see e.g. Kauffmann & Haehnelt
redshiftpopulationspecificallywithintheconstraintsofthe
2000; Di Matteo et al. 2003). Nevertheless we can demand
limited‘massbudget’impliedbytheestimatedlocalSMBH
that the bright quasar population we track in our models
mass density.
never decreases in number below that of the observed pop-
Any potential model of quasar formation finds itself
ulation at lower redshifts. In practice this constitutes a rel-
faced with three major unknowns. Firstly: where, how and
atively weak constraint.
with what mass do the initial ‘seed’ BHs form? Secondly:
what events trigger their subsequent fuelling and growth,
how much fuel do they supply and how efficiently is it con- InSection2wedetailourargumentslimitingtheSMBH
verted into radiative energy? Thirdly: how does feedback mass density at high redshift and present the criteria we
(from star formation or the AGN activity itself) regulate shallusetojudgewhichofourquasarmodelsaresuccessful.
and possibly even check their growth? These uncertainties In Section 3 we present the basis for our family of quasar
translate inevitably to a relatively large number of free pa- models and the methods of our implementation. We then
rameters in models. In this Paper we aim to reduce this present in Section 4.1 a realization of the simplest member
numberbyconcentrating on thehigh redshift quasarpopu- of this family of models. We show why such a model is not
lation,allowingustoconcentrateonjustthefirsttwoissues: workableanduseittodemonstratethechallengesasuccess-
formationandfuelling.Inordertoinvestigateveryluminous fulmodelmustfacebeforegoingontoexploreothermodels
quasars such as those discovered at z ∼6, a very large vol- withinthefamilyinSection4.2.InSection5webrieflycom-
ume must be simulated so as to capture these extremely pare our work to other recent results in the literature, and
rare objects. For this reason, we rely, as in the majority of discusstheimplications ofimposingafixedscalingbetween
theworkcitedabove,ontheanalyticPress–Schechter(Press black hole mass and halo virial mass or velocity at all red-
& Schechter 1974) and extended Press–Schechter (Lacey & shifts. We summarize and conclude in Section 6. Through-
Cole 1993) formalism to predict the merger history of dark out this paper we assume a ‘standard’ ΛCDM cosmology
matter haloes. We consider a class of models based on the with matter density Ω0 = 0.3, vacuum density ΩΛ = 0.7,
basic assumption that seed black holes are created in ma- Hubble parameter H0 = 70 kms−1Mpc−1, and normaliza-
jor mergers, and that theseevents trigger an episode of gas tion σ8 = 1, although our conclusions are not significantly
accretion in galaxies with apre-existingSMBH.Wefurther alteredforalowervalueofσ8=0.9(thepredictedhighred-
assumethatSMBHsaccreteandradiateattheirEddington- shift SMBH mass densities would typically change by 5–10
limited rate untiltheir fuel supply is exhausted, and that a per cent for most of the models considered here, and in no
fixedfractionoftheaccretedmassisconvertedtoradiation. cases would change so much as to alter our conclusions as
To rigorously explore this problem would require de- to the‘success’ of a model).
(cid:13)c 0000RAS,MNRAS000,000–000
High redshift quasars 3
2 OBSERVATIONAL CONSTRAINTS to a B-band magnitude limit of M ≤ −27.2. We then in-
B
tegrate the best-fitting luminosity function2 to obtain the
Weshall make useof two primary constraints in evaluating
numberdensityofquasarsabovethismagnitudelimit,which
the models considered in this Paper: the number density at a redshift of z = 2 is equal to 2.86 × 10−8 Mpc−3
of luminous quasars at z ∼ 6 and an upper limit on the
(where we have corrected for the lower Hubble parameter
total mass density of SMBHs at z ∼6 (the ‘mass budget’). of H0 = 50 kms−1Mpc−1 used by Boyle et al. (2000) in
We will also consider a lower limit on the quasar number
theirΛCDM fit).
density at lower redshift 2<z<6 as a weaker constraint.
∼ ∼ Boththelowandhighredshiftquasarnumberdensities
In this section, we walk through the arguments we used to
are based on results from optical surveys, and it is worth
obtainthesequantitiesfromresultsintheliterature.Wealso
noting that as a result there is always a danger that ob-
describe several necessary conversions in detail.
scured sources have been omitted from our tally. However
The observed quasars at redshifts of z ∼ 6 lie at the
thereismountingevidencethattheverybrightestmembers
very limits of current detection and as such represent only
ofthequasarpopulation,whichweareconcernedwithhere,
theverybrightesttailofthetotalhighredshiftquasarpop-
are preferentially unobscured (see e.g. Fabian 2003 and ref-
ulation.Thusourmodelstoowillbeconcernedmostlywith
erences therein). A further potential source of error is that
these brightest, rarest objects. Our best information about
thecorrectionsandtemplateswehaveusedfromElvisetal.
the highest redshift quasar population comes from a sam- (1994)arebasedonasampleoflowredshift(z<2)quasars,
ple of z>5.8 quasars identified by Fan et al. (2001a, 2003) ∼
andlittleisknownabouthowwellthesetemplatesapplyto
basedoni-z colourselection fromSloanDigital SkySurvey
high redshift objects, although an investigation byKuhnet
(SDSS)imaging.Thesampleconsistsof6quasarsinthered-
al. (2001) suggests that at least the rest frame optical and
shift range 5.7<z <6.6 and is complete down to an abso-
UV energy distributions change very little for quasars out
lutemagnitudeofM1450 <−27.1(ABsystem;Oke&Gunn to redshifts as far as z∼3−4.
1983;Fukugitaetal.1996).Ithasameanredshiftofz¯=6.08
The final quantity we wish to determine is an esti-
qanudasparrsedaitcttshtisherendusmhibftertodebnesi5ty.2o±f b2r.1ig×ht1(0M−11045M0 <pc−−327(.a1l)l mate of, or at least an upper limit on, the total mass
density in SMBHs at z ∼ 6. The local SMBH mass den-
thesefigurestakeintoaccountadjustmentsforourassumed
sity may be estimated by two independent methods. One
cosmology).Tocomparethiswiththeresultsfromourmod-
method makes use of the M• − σ relation plus the ob-
elsweneedtorelatethemagnitudelimit ofthesampletoa
served galaxy luminosity/velocity function to directly sum
minimum bolometric luminosity for thequasars. To dothis
up the mass present in remnant SMBHs at the present
westartwithagivenbolometricluminosityandconvertthis
day (Aller & Richstone 2002; Yu & Tremaine 2002). These
toarest frame 2500˚Amonochromatic luminosity usingthe
estimates now seem to be converging on a value around
bolometriccorrectionsofElvisetal.(1994).Usingthemean ρ•(z = 0) = 3−5×105 M⊙Mpc−3. Specifically, Yu &
stpoeactluramlitneomsiptylaatettohfetrheestsafrmame aeuwtahvoerlsenwgeththoefn14co5n0v˚Aerwthtihcihs Tremaine(2002)findρ•(z=0)=2.9±0.5×105 M⊙Mpc−3
while an earlier calculation of Merritt & Ferrarese (2001a)
aMll1o4w50s(uAsBtosycsrteeamte)afosrimouurlamteoddemlso1n.oWcherfiomndatic magnitude findsInρ•t(hze=ot0h)er∼ap4.p6ro×ac1h0,5fiMrs⊙taMttpecm−p3t.edbySol tan(1982),
LBol =(1.0±0.075)×1036.66−0.4M1450,AB ergs−1 (1) the cumulative mass that must have been accreted by
SMBHs in order to produce the observed quasar luminos-
andhencethattheSDSSsamplemagnitudelimitofM1450 < ity function is summed up. In the past, the best available
−27.1 corresponds to a minimum quasar luminosity of
LBol =3.03×1047 ergs−1 in our models. desitffiemreantetsmoeftthhoedSsMhaBvHenmoatssaldweanyssitbyeρe•nfirnomgotohdesaegrteweomveenrty.
To determine the number density of quasars as bright
Improved estimates of the optical quasar luminosity func-
as the SDSS z ∼ 6 sample at lower redshift, we use the
tion (e.g. Boyle et al 2000; Fan et al. 2001b) are now avail-
luminosity functions determined from the 2QZ and LBQS
able,andextrapolating tohigherredshiftsandfaintermag-
by Boyle et al. (2000). These results are quoted in absolute
nitudeshasproducedamuchmorecompleteestimateofthe
B-band Vega magnitudes so we first make use of the Elvis
total contribution from accretion by unobscured AGN (e.g.
et al. (1994) bolometric corrections to find,
Chokshi & Turner 1992). The most recent calculations (Yu
LBol =(1.0±0.06)×1036.59−0.4MB,VEGA ergs−1. (2) &preTsernemt daianyeis20a0r2o)unfidndρatchc(azt=th0e)t=ota1l.8c9on×tr1i0b5utMio⊙nMbypct−h3e
where we have assumed an offset of 0.12 mag between the (assumingaccretedmatterisconvertedtoradiationwithan
Vega and AB magnitude systems for quasar-like spectra efficiency of around 10 per cent) and indicate that around
(Schmidt,Schneider&Gunn1995;thisoffsetismuchlarger 90percentofthistotalisduetoaccretionatredshiftsz<3.
∼
than that typically used for magnitude conversions of stel- However account must also be taken of the obscured
lar sources because of the large difference between stellar population of quasars whose demographics we do not yet
and quasar spectra). This implies that our minimum lumi- understandso well. Wedo knowtherearelarge numbersof
nosity of interest, LBol = 3.03×1047 ergs−1, corresponds obscuredweakAGNsuchasSeyfertIIs,whichwillalsocon-
1 In fact the SDSS sample absolute magnitudes are originally
calculated for the rest frame 1280˚A and then the authors make 2 Notethat the αand β parameters for the luminosityfunction
theirowncorrectionstoarriveatthe1450˚Avalues,howeverthe Φ(MB,z) inBoyle et al. (2000) are both missingminus signs in
effectsofthiscorrectiononourowncalculationswillbenegligible theforminwhichthearticlewasoriginallypublished(B.Boyle,
comparedtoourinherentuncertainties. privatecommunication).
(cid:13)c 0000RAS,MNRAS000,000–000
4 J. M. Bromley, R. S. Somerville and A. C. Fabian
tributeto thetruevalueof ρacc (e.g. Alexanderet al. 2001; if theysatisfy theseconstraints subject to themore relaxed
Brandt et al. 2001; Rosati et al. 2002), and indeed are re- condition105 M⊙Mpc−3 ≥ρ•(z∼6)>5×104 M⊙Mpc−3.
quiredtoexplaintheX-raybackground(Fabian&Iwasawa
1999).Inadditiontheremayalsobeasignificantpopulation
of faint highly reddened quasars (Barkhouse & Hall 2001;
3 MODEL FRAMEWORK
Wilkes et al. 2002; Richards et al. 2003a), and there is also
some evidence for a small population of totally obscured, In the hierarchical picture of structure formation, based
Type II quasars (e.g. Crawford et al. 2002; Norman et al. on the CDM model, structures build up progressively over
2002;Wilmanetal.2003;Zakamskaetal.2003)althoughthe time, the smallest objects collapsing first and then gradu-
contributionofthelattertotheSMBHmassdensityisprob- allymergingtogethertocreatelargerandlargerstructures.
ably negligible. Attempts to include the effects of obscured Themuchlessabundantbaryonicmatterinitiallytracesthe
sourcesincalculationsofρacc usingX-rayobservations(e.g. darkmatter.Butwhereasthedarkmatterisdissipationless
Fabian & Iwasawa 1999; Elvis, Risaliti & Zamorani 2002) thebaryonsareable,throughshock-heatingandsubsequent
or by combining multiwaveband observations (e.g. Salucci cooling, tocollapse down toform much moretightly bound
et al. 1999; Barger et al. 2001) have in the past resulted in structures – the first (proto)galaxies – within the potential
valuesofρacc(z=0) that can actually exceed theestimates wellofthecollapseddarkmatter‘haloes’.Thesegalaxiesare
of the total local mass density ρ•(z=0) unless implausibly thenbroughttogetherbythesubsequenthierarchical merg-
high radiation efficiencies are assumed. ers of their dark matter haloes where they can form early
However,thisoverpredictionmayinparthavebeendue groups and clusters or perhaps merge together themselves.
toincorrectlyassumingthatobscuredandunobscuredAGN Webaseourmodelonthepremisethatquasarfuellingis
populations peak at the same redshift (e.g. Barger et al. accomplishedthroughtidalstrippingofangularmomentum
2001; Cowie et al. 2003), and a picture is now emerging from galactic gas during such major galaxy mergers. This
inwhichtheaccretionfromthecombinedAGNpopulations popularpremisehassubstantial observational andtheoreti-
wouldseemtoexactlyaccountforthedirectestimatesofthe calsupport(Negroponte&White1983;Barnes&Hernquist
mass in SMBH remnants at the present day (Fabian 2003). 1991,1996;Bekki&Noguchi1994;Mihos&Hernquist1994,
The high redshift evolution of the obscured AGN popula- 1996;Bahcall etal.1997; McLureetal.1999), althoughthe
tion is much less well known than that of the unobscured mechanismresponsibleforthefinalinflowofthegasonceit
sources(seee.g.Hasinger2002).Howeverbecausetheperiod has arrived within the last 100 pc from the galactic centre
z = 0−3 represents around 84 per cent of the time since is still unknown (for an alternative to the merger hypoth-
thebigbang,wemightneverthelessexpectthatthebulkof esis see e.g. Kawakatu & Umemura 2002; Granato et al.
themass densityfrom obscured sources would bedeposited 2003). In order to track the mergers of galaxies within our
intheepochz<∼3,asistruefortheunobscuredsources.In- modelweconstructMonteCarlorealizationsofdarkmatter
deed the (already fairly high) estimate of ρacc(z =0) made ‘merger trees’ following the algorithm of Somerville & Ko-
byBargeretal.(2001) frommultiwavebandobservationsof latt(1999).Thisalgorithmallowsustoreconstructatypical
Chandra sources only considers the contribution made by evolutionary history for a given final dark matter halo (the
accretion for z ≤3, so if this it to be consistent with other ‘root’ halo) and follow the hierarchical build up of its mass
measurementstherecannotbeverymuchcontributionfrom overtimethroughaccretionandmergerofsmallerhaloesas
higher redshifts. described by the extended Press–Schechter probability dis-
It seems then that almost all of the observed local tribution(Lacey&Cole1993).Foraroothaloofgivenmass
SMBH mass density is accounted for by radiative quasar andredshift thisprocess isaveraged overalarge numberof
accretion, leaving very little room for any quiescent ac- different realizations, allowing us to calculate the average
cretion modes. Moreover the vast majority of this mass contribution (and its time dependence) made by the pro-
must be deposited at redshifts z<∼3, meaning that only a genitor haloes to all quantities of interest. The simulations
very small fraction of the present day SMBH mass den- are repeated over a grid of different root halo masses and
sity ρ•(z = 0) can be in place at high redshifts, i.e. the results are combined by weighting them according to
ρ•(z>∼6) ≪ ρ•(z = 0). If we take the upper bound on theprobability of findinga halo of that mass at the chosen
the current estimates on the local SMBH mass density, i.e. outputredshift.
ρ•(z = 0) ∼ 5×105 M⊙Mpc−3, and then as a first ap- We track the history of each halo back in time until
proximation assume that around 90 per cent of this comes we find a progenitor halo with a circular velocity below
fromaccretioneventsbelowaredshiftofz∼6(certainlywe 30 kms−1 or a mass below 1.7×108 M⊙; the latter mass
knowfortheunobscuredsourcesthiswouldbetrueevenfor corresponds to acircular velocity of 30kms−1 at a redshift
z<3) thenwearriveatan upperlimit onthehighredshift ofz ∼20,soweeffectivelytrackallstructureswithcircular
SMBH mass density of ρ•(z∼6)∼5×104 M⊙Mpc−3. velocities as small as 30kms−1 back to redshifts of z ∼20
Thus in this paper we shall consider a model ‘success- and only progressively larger structures beyond. Structures
ful’ if it both reproduces the SDSS z ∼ 6 population of smallerthan30kms−1 areunlikelytobeabletoaccretegas
bright quasars with a SMBH mass density ρ•(z ∼ 6) ≤ efficientlyinthepresenceofaphotoionizingbackground(see
5×104 M⊙Mpc−3 and continues to grow this population e.g.Gnedin2000),andsincetherecentresultsfromWMAP
of brightquasars sothat theydonotdrop below thevalues indicatethat reionization islikely tohaveoccurred as early
predicted bytheBoyle et al.(2000) luminosity functionsat as z = 14−20 (Kogut et al. 2003) our chosen resolution
z =2. Because of the uncertainties involved and the some- should be valid out to similar redshifts.
whatadhoc approachwehavetakentoreachourmassden- Oncewehavecreatedamergertree,weassociate bary-
sity limit we shall consider models ‘marginally successful’ onic material with each newly formed dark matter halo.
(cid:13)c 0000RAS,MNRAS000,000–000
High redshift quasars 5
This is assumed to fall towards the centre of the resulting diation pressure exactly balances the gravitational pull on
potential well and reside there as a self-supported gaseous infalling matter,
(proto)galaxy. The initial amount of baryonic matter avail- 4πGcm
able to a pristine dark matter halo is set by the universal L=1.15 pM• (4)
σ
e
baryon fraction (here we use f = 0.13, corresponding to
Ω =0.019h−2,which isconsistebnt with theobservations of wheremp isthemassoftheproton,σetheThompsoncross-
b sectionoftheelectron,M•thecurrentmassoftheblackhole
Levshakov et al. 2002). This ratio remains fairly constant
and the factor of 1.15 accounts for themean atomic weight
throughout the history of a halo since the only mechanism
perelectron foratypicalhydrogenandheliumgasmixture.
for baryon loss in our models is the conversion of accreted
Setting these two expressions equal yields an ordinary dif-
matter to quasar radiation, which is fairly negligible. We
ferential equation with thesolution
defertheconsideration ofprocesses thatmay competewith
quasars for the gas in these galaxies (e.g. star formation, M•(t)=M•(0)et/κ (5)
feedback and consequent heating) to a subsequent paper
whereM•(0)istheinitialblackholemassandthetime-scale
(Bromley, Somerville & Fabian 2004, in preparation) so as
κ is given by
to minimize the number of parameters in our model – al-
though we note that this means our results will only apply κ=(1.15)−1 cσe ε . (6)
4πGm 1−ε
to high redshifts where gas is plentiful. p
Whether such galaxies subsequently merge when their We can derive a similar expression for the amount
respectivedarkmatterhaloesmergewilldependontherate of unaccreted gas remaining mfuel(t), and by setting
at which they can lose their orbital energy via dynamical mfuel(tExh) = 0 find the amount of time tExh for which a
friction against the background dark matter. When track- quasarshinesbeforeallofthegasavailabletoitisexhausted:
ing halo-halo mergers within the merging tree we label the 1
central galaxy of the most massive dark matter halo as the mfuel(t)=mfuel(0)− 1−ε[M•(t)−M•(0)] (7)
new central galaxy of the system and all the other galax-
ies become ‘satellites’. We compute the dynamical friction tExh=κln 1+(1−ε)mfuel(0) . (8)
time-scaleforthese‘satellite’ galaxies usingtheapproxima- (cid:20) M•(0) (cid:21)
tion of Binney & Tremaine (1987), modified to account for For computational simplicity, we only record activity from
non-circular orbits (Lacey & Cole 1993); see Somerville & accretion on to central galaxies. Thus if a quasar is ac-
Primack (1999) fordetails. Ifthistime-scale isshorterthan creted into a new halo, becoming a satellite galaxy, it will
thetime tothenext‘branch’of thetree(i.e.thenexthalo- be‘turnedoff’afterasmaller time-scalethantheonegiven
halo merger), then the satellite is merged with the central above. We find that this approximation has a negligible ef-
galaxy. Otherwise the satellite remains until the next halo- fect on ourresults.
halo merger, when its ‘merger clock’ is reset and the calcu- The exact value of the parameter ε in all these equa-
lation is repeated. tions will depend on the nature of the accretion disc and
Whenever two galaxies merge, any central black holes thespin of theblack hole. Taking intoaccount photon cap-
they contain are also assumed to merge instantly. In addi- turebytheblackholeitself,theorysuggestsεcannotexceed
tion, if a ‘major’ galaxy merger occurs (defined here as a 0.057 for a static black hole and 0.4 for a maximally rotat-
mergerinwhichthesatellite galaxyaccountsforatleast 30 ing black hole, although in fact accretion will act to bring
per cent of the total system mass), then some fraction fm the black hole spin to its ‘canonical’ value which reduces
of the gas is stripped of its angular momentum and falls to this maximum to ε < 0.3 (see Thorne 1974; in principle
thecentre,whereit can fuela centralSMBH. Thequantity theeffective efficiency could exceed this limit since the disc
fm is left as a free parameter of themodel. luminosity could be augmented by magnetohydrodynamic
Once accretion on to a particular black hole has been processes such as magnetic coupling with the SMBH, mag-
‘turnedon’bysuchagalaxy-galaxymerger,weassumethat netic stress in the disc or torque on its inner boundary e.g.
the accretion proceeds at the Eddington limit until the fu- Gammie1999;Krolik2001;Li2002;Wang,Lei&Ma2003).
ellinggassuppliedbythemergerisexhausted.Theluminos- MostSMBHmodelsassumetheefficiencyofatypicalaccre-
ity of an accreting black hole L is governed by the rate at tiondisctobewellbelowthismaximum,ataroundε≈0.1,
which accreting matter is supplied m˙fuel, and the efficiency and weshall follow this convention,although in Section 3.2
with which this accreted matter is converted to radiation – wedodiscusshowthebehaviourofourmodelisaffectedby
typically quoted in terms of the fraction ε of its total rest takingdifferent values.
mass energy that is liberated. Thus L = εm˙fuelc2, where c Clearly if one quasar differs from another only in that
is the speed of light. That fraction of the accreted matter itshinesbrightlyenoughtobevisiblefortwiceaslongthen
which is not converted to radiation adds tothe mass of the it is twice as likely to be observed. Thus since we expect
black hole, and so the black hole growth rate M˙• is just the bright lifetime of a typical quasar to be much smaller
M˙• =(1−ε)m˙fuel.Combiningthesetwoexpressionswefind thanourmodel’sintrinsictimesteps(thetimesbetweenhalo
the quasar luminosity is related to the black hole growth and galaxy mergers) we group our results intoredshift bins
rate by of width dz = 0.5 and calculate the fraction of the time
spanned by the bin that each quasar spends shining above
ε
L= M˙•c2. (3) our magnitude limit. When calculating number densities
1−ε
from our results we then weight each quasar by this fac-
However, for Eddington limited accretion the black hole tor, in addition to the probabilistic weight of its root halo
must be shining at the Eddington luminosity where the ra- discussed earlier.
(cid:13)c 0000RAS,MNRAS000,000–000
6 J. M. Bromley, R. S. Somerville and A. C. Fabian
The question remains as to how the initial ‘seed’ black 3.2 The radiative efficiency of quasars
holes are formed. Several previous models have consid-
The efficiency of converting accreted material to radiation,
ered the possibility that SMBHs form from much smaller
(∼ 10−100 M⊙) black holes, remnants from either stan- given by ε, is the controlling parameter in the equation for
Eddington limited black hole accretion. A high value of ε
dard galactic star formation (e.g. Haiman & Loeb 2001)
means that most of the accreting matter is turned to radi-
or from the very first stars – the Population III stars (e.g.
ation, and so only a small mass accretion rate is possible
Volonteri,Haardt&Madau2003).Howevertherearevarious
before the limiting Eddington luminosity is achieved. Con-
problems associated with such scenarios (see e.g. Haehnelt
versely,alowervalueofεmeansmoremassmustbeaccreted
2003). In particular some mechanism is required to facili-
to produce the same amount of energy in radiation and so
tate their migration to the centre of their host galaxy. We
themassgrowthrateforablackholeshiningattheEdding-
instead focus on the possibility that SMBHs form directly
ton luminosity is higher.
from the collapse of a large gaseous cloud in the centre of
a (proto)galaxy. This was first discussed many years ago The probability of observing an accreting black hole
(Rees 1984; Haehnelt & Rees 1993; Silk & Rees 1998); and depends upon the brightness of the accretion event and its
although it has obvious problems with the need to avoid duration.IfblackholesaccreteattheEddingtonluminosity,
fragmentation during the collapse, it nevertheless remains then their brightness is a function only of their mass. Thus
a possibility and a candidate mechanism has been sketched the number of accreting black holes above some luminosity
out byHaehnelt & Rees(1993). Weshall furtherassume in depends on the number of black holes above a particular
our model that this collapse is also connected to the major mass limit – which will depend on the rate at which black
merger of galaxies – that is, the same tidally stripped gas holes have been able to grow prior to the epoch of obser-
which falls into the centre of a post-merger galaxy and can vation – while the duration of a given accretion event will
fuel quasar accretion will also be viewed as the source of depend inversely on the black hole accretion rate if fuel is
the initial SMBH’s formation. Precisely how effective (how limited. This means the overall effect of the value of ε on
likely it is to happen in a given major merger) this is and observations can becomplicated.
how much of the gas this initial formation consumes (how In our models we find that an increased ε tends to de-
efficient it is) will be left as free parameters of the model. creasethenumberdensityofquasars(aboveagivenbright-
We shall however assume that the process is quiescent – nesslimit)athighredshifts,butincreasethenumberdensity
thatis, that noradiation is producedin thisinitial collapse atlowerredshifts.Thusεaffectsthesteepnessoftherisein
– although at high redshifts this is unlikely to make much quasar numberswith time.
differencetoourpredictionsasallbutthebrightestofthese If fuel is plentiful then a lower efficiency model should
‘formation flashes’ would betoofaint tobeobservableany- grow faster. However one would expect that once the rate
way. atwhichmajormergerscanresupplyaSMBHwithfuelbe-
comesthelimitingfactorthenthenumberofbrightquasars
in a low and a high efficiency model would become equal.
In the slower accreting (high efficiency) model they would
shineforlonger howeverand sohaveahigherchanceof be-
3.1 Model summary ing observed. This indeed would seem to be the reason for
the increased numbers of bright quasars at low redshift in
A brief summary of the steps involved in our modelling is
ourhigher efficiency models.
as follows:
Howeverat high redshifts thesituation becomes rather
• Create hierarchical merger trees describing the forma- complex,andtherearethreemain factors whichneedtobe
tion history of dark matter haloes. take into account. Firstly the equalizing between high and
• Associate baryons with each newly formed halo, form- low efficiency models discussed above may be less effective,
ing (proto)galaxies. since even though at redshifts z ∼ 6 the vast majority of
• Ateachlevelinthehierarchy,representingthemergerof quasar growth is still limited by therate of major mergers,
progenitor dark matter haloes, compute whether and when thetimerequiredtoaccretethefuelsuppliedjustbyasingle
each of the constituent galaxies will merge (based on dy- major merger can in fact represent a very large fraction of
namical friction). thetotalobservedredshiftbinatsuchredshifts.Secondlyit
• IneverygalaxymergeranyexistingSMBHsaremerged. isthecasethatinalowefficiencymodelthetotalamountof
• Ineverymajorgalaxymergersomefractionoftheavail- matteraccretedbyaSMBH(andhenceitsmaximumlumi-
able baryons is assumed to be stripped of its angular mo- nosity)willbeslightlylargerthaninahighefficiencymodel
mentum and fall to the centre, where it can either form a sincelessofthemass-energyofthefuelisconvertedtoradi-
new ‘seed’ SMBH or fuel an already existing SMBH. ation. While this is a small effect, it nevertheless can make
• OnceprovidedwithfuelaSMBHshinesattheEdding- a difference when looking (as we do) at the very brightest
tonluminosityandaccretesatthecorrespondingEddington endofthequasarpopulation.Lastlyanadditionalcomplica-
rate untilthis fuel is exhausted. tion arises if an accreting SMBH becomes a satellite before
exhausting its fuel and then merges with the SMBH of a
The mass ratio that constitutes a major merger is fixed at new central galaxy. In this case the mass contribution will
0.3 and the radiation efficiency of accretion is fixed at 0.1. belarger in thefaster growing (low efficiency) models since
The efficiency and effectiveness of SMBH formation, and the satellite SMBH will have reached a larger mass. These
the form of the accreted fraction of baryons are left as free factors would seem to combine to create the drop in high
parameters. redshift quasars we see in our higherefficiency models.
(cid:13)c 0000RAS,MNRAS000,000–000
High redshift quasars 7
Indeed the last of these complications would seem to boundary can also be made successful (by a suitable choice
representapossiblebarriertoaccuratelyfollowinghigheffi- of the free parameters) for values in the range 20 – 40 per
ciency models within our current framework. Although our cent(althoughthisisoftenattheexpenseofsteepeningthe
standardchoiceofε=0.1ishardlyaffectedbythesimplified riseinquasarnumberswithtimewhichcouldhaveimplica-
treatment in our models whereby only accretion in central tionsforthelowredshiftquasarpopulationnotinvestigated
galaxies is followed, it would appear that once efficiencies here)3.
as high as ε = 0.2 are considered SMBH growth becomes Concerning the assumptions made in our model, there
so slow that significant numbers of bright quasars become arethreemainpointswhichneedtobeaddressed.Firstlywe
satellitesbeforeexhaustingtheirfuelandsothissimplifying note that our assumption that quasars always shine at the
assumption begins to break down. Eddingtonluminosity rightuptothepointwheretheirfuel
Thussincea higherefficiency modelboth increases the isexhaustedissomewhatunphysical.Realisticallyaquasar’s
low redshift quasar numbers, where our model is already lightcurveislikelytorisesteeplyasitisfirstactivated,ap-
poorlyconstrained,andalsomayneedamoredetailedtreat- proach the Eddington luminosity and then decay away as
mentofsatellitegalaxies,wewilldelayfurtherinvestigation fuel becomes scarce; however the exact form such a curve
of such models to asubsequent paper(Bromley et al. 2004, would takeisnot knownandwill affect theoverall shapeof
in preparation). thequasarluminosityfunction(seee.g.discussioninCatta-
neo 2001). Nevertheless the effects of a decaying tail in the
quasarlight curvearelikely to bemost important in repro-
3.3 Caveats and assumptions
ducingthefaintendoftheluminosityfunction,andsincewe
Inthis section we brieflydiscuss some caveats in ourmodel areconcentratinginthispaperonthebrightquasarsvisible
and how we have tried to deal with them, and also outline at high redshift, the simple approximation we use is prob-
thepossibleconsequencesoftheassumptionswehavemade. ably sufficient. Were we to include such an effect then we
Webeginbyexaminingsomedetailsinthealgorithmsofour suspectitwouldrequireifanythingslightly morenumerous
simulation. or more massive quasars to still account for the observed
It is usual in Monte Carlo based merger tree models high redshift quasar numbers, making the constraints from
to weight each root halo by the standard Press–Schechter theSMBHmassdensitypresentedinthispapereventighter.
distribution (Press & Schechter 1974) which provides the Secondly we need to consider the effects of our forma-
expected number density of haloes of a given mass at a tion and accretion scenarios. We have assumed that forma-
given epoch. However this distribution has several known tion of an initial seed SMBH requires a major merger, as
discrepancies when compared to N-body simulations (see doessubsequentaccretion.Aconsequenceofthisisthatun-
e.g. Somerville et al. 2000). In an attempt to correct some- lessahaloacquiresitsSMBHfrom aminormerger,itmust
whatforthese,weusethecorrectedmassfunctionofSheth experience at least two major mergers in order to activate
& Tormen (1999) to perform this weighting in our models a quasar. Thus at high redshifts, when few SMBHs have
(other‘improved’mass functions, e.g. Jenkinset al. (2001), beenformed,quasaractivitywill alreadybebiasedtowards
would yield similar results). Unfortunately,although asim- haloes that are in ‘merger-rich’ environments. This is in
ilar correction can be applied to the conditional mass func- contrast both to previous models which have seeded haloes
tion used to generate the merger histories, the resulting with black holes completely independently of mergers, and
probability functions violate the Markov condition and are tothosewhichhavenotdifferentiatedbetweenaccretionand
nolongersuitableforgeneratingself-consistentmergertrees. formation but treated them as the same process (examples
Therefore,tominimizetheeffectsofsuchinaccuraciesinthe ofbothtypesofmodelarediscussedinSection5).Whenwe
rest of the merger tree, we resimulate each tree over a grid examinemodelvariantswithareducedSMBHformationef-
of output redshifts, placing a root halo at each redshift bin ficiency(i.e.wherenotallthegasfreedinthemajormerger
wewish torecord dataover(asusualaveraging each overa is usedin creating theinitial SMBH),we havecontinuedin
grid of halo masses). the same spirit by assuming that the gas not consumed by
Similarly, on the topic of the merger tree algorithms, the initial formation is expelled from the galaxy by some
it should be noted that due to the way our algorithm is outflowprocess(ofcourseitmaybethatthefraction ofgas
implemented (see Somerville & Kolatt 1999 for details) it able to reach the galactic centre is slightly less anyway in
is sometimes thecase that thevery earliest progenitor on a thecasewherenopriorSMBHexists).Forreasonably large
given‘branch’ofthemergertreemayhaveacircularvelocity efficienciesthisdoesnotseemtoounlikely;mostofthemod-
belowourresolutionlimit.Sinceourchoiceofresolutionwas els we shall show to be successful have initial SMBHs with
based on thenotion that smaller haloes would beunableto meanmassesintherange∼5×103−5×105 M⊙ (atz=6)
acquirebaryonsinaphotoionizingbackground,westripthe whichwouldindeedreleaseasubstantialamountofgravita-
progenitor of all its baryons in such cases so as to remain tionalbindingenergyuponformationandcouldconceivably
consistent with this premise. expeltheremaininggasbackoutofthecentralinnerparsec
Finally wenotethatsincesmallmassratiomergersare oftheirgalaxy.Howeverastheformationefficiencybecomes
much more common, the results of the model can be fairly
sensitive to the exact definition of a ‘major’ merger. In our
3 Itshouldbenotedthatwhileitappearsusualinquasarmodels
model we have chosen to set the boundary of the minor-
toconsider thefractionofthetotal systemmassasthebasisfor
major divide as the point where a merging satellite galaxy
the minor-majordivide (as we do here), many galaxy formation
accounts for 30 per cent of the total mass in the system. models (which use the divide in determining starbursts) instead
However we have found that all of the models we inves- usetheratioofbaryonicmasses toclassifymergersas major vs.
tigate in this paper which prove successful for our chosen minor;e.g.Somerville&Primack(1999); Coleetal.(2000).
(cid:13)c 0000RAS,MNRAS000,000–000
8 J. M. Bromley, R. S. Somerville and A. C. Fabian
smaller this rapidly becomes less and less likely, so clearly thenthiswouldweakenourconstraintsonthehighredshift
we should treat as suspect a model which relies on partic- ‘mass budget’.
ularly small efficiencies (which it would be hard to justify
physicallyanyway).AttheendofSection4.3webrieflydis-
cuss the effects on the model if we remove this assumption
andinsteadallowgasunusedinSMBHformationtoaccrete 4 MODEL RESULTS
on to the newly formed SMBH in the next timestep. How-
We now examine several different variants of the models
ever, since our timesteps are the intervals between galaxy
withintheframeworkjustdescribed.Webeginwiththesim-
and halo mergers they are not evenly spaced, this is only a
plest case, which turnsout tobe unableto satisfy our joint
very crude treatment and we do not spend much time on
constraints, and then add additional parameters in an at-
it. A better approach would have been to allow accretion
tempt to findone or more successful variants.
of leftover gas directly after the formation; but this would
require a sufficient understanding of the formation process
to be able to estimate the time-scales involved and with-
4.1 The simplest case
outtheinclusion ofstarformation andotherprocesses that
wouldcompeteforthisgasthetreatmentislikelytoremain Torealizeaparticularmodelweneedtospecifythefraction
unrealistic anyway. of gas accreted on to central SMBHs in major mergers, f
m
Lastly we examine the assumptions we have made to (which may beafunction of other variables), and theeffec-
arrive at theconstraints used to judge our models. We first tiveness and efficiency of the formation of the initial seed
consider the number density of quasars at z ∼6 calculated SMBH. Perhaps the simplest scenario is to take f to be
m
fromtheSDSSdata.Whilewehaveassumedthatthevalue equaltoafixedvalueandassumeacompletelyeffectiveand
found by Fan et al. (2003) represents the true density of efficient SMBH formation process – that is, in every major
the population, our models are not in fact very sensitive to mergerin which theremnant doesnot contain aSMBH,all
changes in this value by a factor of a few. Although none the tidally stripped gas falling to the centre is assumed to
of the six quasars in the SDSS z ∼6 sample show multiple formanewSMBH.Thesinglefreeparameter,f ,maythen
m
images(Fanetal.2003;seealsoRichardsetal.2003b),nev- be fixed by requiring the model to reproduce the observed
erthelessthediscoveryofgalaxiesneartothelineofsightof numberdensityof bright quasars at redshift z ∼6. Wefind
twoofthesequasarssuggeststhattwoofthesamplemayin that a value f =0.0095 achieves this.
m
factbemagnifiedbylensing(Shioyaetal.2002;Whiteetal. We show several results from this model in Figs. 1a–d.
2003; Wyithe2003). If the intrinsic luminosity of these two In the top panels we plot the number density of luminous
quasarsbeforemagnificationisinfactbelowtheSDSSmag- (M1450 < −27.1) quasars and of SMBHs as a function of
nitude cut, this would reduce the number density of bright redshift. In Fig. 1a, the vertical arrows indicate the typical
(M1450 <−27.1) quasars at z ∼6 by a factor of ∼ 32. This uncertainty on the mean, calculated from both the scatter
figureisconsistent withanearlier studybyWyithe&Loeb over different realizations of halo merger histories and the
(2002b) which concluded lensing of the SDSSsample is un- errors in our luminosity binning due to the uncertainties in
likely to have increased the true number density at z = 6 thequasarbolometriccorrection.Becauseitisnotclearthat
by much more than a factor of ∼1.5. Changes at this level thesamplingerrorsofdifferentluminositybinsareindepen-
wouldnothoweveraffectourmodelssignificantly,andsowe dent the errors are not combined in quadrature but added
do not feel that the possible presence of lensing will affect directly (reflecting the worst case scenario). Although the
our conclusions. errors arebased on theassumption that thevariance of the
In constraining our high redshift SMBH ‘mass budget’ underlying distribution can be approximated by the stan-
we have further assumed that all the accreted mass from dard estimator for a Gaussian, comparing several different
quasars combined with the mass of their initial SMBHs is runs of the model suggest they provide a fairly good indi-
accountedforbytheobservedcentralSMBHsinpresentday cation of thetrueerrors. In thebottom panels we show the
galaxies – that is, that the evolution of SMBHs is lossless. contribution to the mass density of SMBHs at z = 6 as a
Howevertherearetwomechanismsthatmightlead tomass functionofSMBHmass,andtheaveragemassofthecentral
loss intheevolution of centralgalactic SMBHs.Firstly it is SMBH as a function of host halo mass, also at z = 6 and
possiblethatthemergerofSMBHsisalossyprocess,aspo- withan indication ofhowmuchofthismassisduetoblack
tentiallyasignificantamountofthemass-energyofabinary hole mergers and formation (seeds). The latter plot is cre-
SMBHsystemcouldbereleasedasgravitationalradiationin ated by averaging the properties of the SMBH in final root
thelast stages of itsmerger (seee.g. Yu& Tremaine 2002). haloesofdifferentmassesatz∼6(actuallyz =5.75)overa
Additionally, it is possible that the time-scale for SMBH large number of realizations (note that by construction the
merging is sufficiently long that there is a high probability SMBH of a ‘root halo’ always corresponds to the SMBH in
of accreting a third SMBH, which is likely to lead to the thecentral galaxy).
ejection of the smallest SMBH or in some cases even the Wecanseethat,whennormalizedatz=6asdescribed
ejection of all three SMBHs via the ’slingshot’ mechanism above, the model overpredicts the number density of lumi-
(seeHaehnelt&Kauffmann2002;Volonterietal.2003,and nous quasars NQSO at lower redshift, by a factor of ∼ 25
references therein). Any such ejected SMBHs that escape by z = 2. As we have discussed, this is likely to be due to
agglomeration inthenucleusoftheirhostgalaxywould not our neglect of star formation and feedback. However, the
beincluded in thekind of censusused toestimate thelocal more serious problem is that when this model is normal-
SMBHmassdensity,andthuseffectivelyconstituteanother izedtoreproduceNQSO atz =6,itinevitablyandseriously
mode of mass loss. If either of these processes is significant overproducestheSMBHmassbudget,alreadyproducingby
(cid:13)c 0000RAS,MNRAS000,000–000
High redshift quasars 9
z = 6 a total mass density of 3.6×105M⊙Mpc−3, in ex- Spread of Bright (M1450 < -27.1) Quasar Number Densities Across Realizations
cess of the observationally derived value of Yu & Tremaine 4.5
d
(2002) for z =0. an 89.75% of realizations produced no bright quasars
B 4
Part of the problem is that only a small fraction of en
SMBHsare‘activated’bymajormergersatagiventime.To o Giv 3.5
abnelEumddininogutsonenloiumgihtetdoqmuaaksearitminutsottphoesSseDssSSazS∼M6BHsamofplaet, buting t 3
least M• = 2×109 M⊙. Only about 1/100 of the galaxies Contri 2.5
withsuchmassiveSMBHsareactiveatagiventimeatsuch ons 2
redshifts.Howeverthemainreasonthatthemodelsobadly alizati 1.5
exceedstheSMBHmassbudgetcanbeseeninFig.1c:most Re
ofthetotalmassdensityiscontributedbylowmassSMBHs, e of 1
g
objects which even when accreting at the Eddington rate enta 0.5
would not be luminous enough to be detected by the SDSS erc
P
at this redshift. 0
10-11 10-10 10-9 10-8 10-7 10-6
Theshort-dashedlineinFig.1dshowsthemeaninitial Bright (M1450 < -27.1) Quasar Comoving Number Density at z = 6 (in Mpc-3)
massforSMBHs(i.e.themasstheycontainaftertheirinitial
gaseouscollapse) asafunctionofthemassofthedarkmat-
Figure 2. The distribution of the number density of bright
terhalotheSMBHfinallyendsupinatz∼6.Interestingly
(M1450 < −27.1) quasars at a redshift of z = 6 obtained in
this appears to become almost flat as the host halo mass
individualrealizationsofthemodel,groupedintermsoftheper-
increases, although the mechanism for this is not obvious.
centage of realizations producing values within the range of a
ToafirstapproximationtheinitialSMBHmasswilljustbe given logarithmic bin. The model uses the same parameters as
a function of the mass of the halo it first formed in. Thus that shown in Fig. 1, and once all the realizations are averaged
onepossibility isthatthisflatteningofthecurverepresents togethermatchestheobservednumberdensityatz=6.Notethat
anupperlimittothelikelymassofthosehaloeswhichform only∼10percentoftherealizationsproducednon-zeronumber
their own SMBHs, larger haloes being very likely to sim- densities. Had our simulation included rarer, larger mass haloes
plyacquiretheirSMBHfrom suchsmallerhaloes whichare thenthelowerdensitybinswouldlikelybecomemorepopulated.
However the mean isdominated by the2–4 per cent of the real-
incorporated into their collapse (due to the nature of hier-
izationsproducingvaluesintherange10−7−10−9 (whichwould
archicalstructureformation).Howeverthismeanbehaviour
notbeaffectedbyconsideringlargerhaloes).Evensignificantin-
does not preclude the possibility of significant outliers, and
creasesinthelowerdensitybinsfromrunningthesimulationover
it is another less apparent failure of this simple model that
largerhaloeswouldnotchangethis(andsowouldnotaffectour
someoftheinitialseedSMBHshaveextremelylargemasses results).
– in some cases as large as ∼109 M⊙.
While we are assuming that fragmentation is somehow
sufficientlysuppressedtoallowtheinitialSMBHtocollapse, the formation efficiency to fseed ∼ 1.8×10−4 in order to
neverthelessitseemsunlikelythatsuchaprocesscouldcon- reducetheSMBHmassdensityat z∼6toourtargetvalue
tinuetoscaleuptosuchlargemasses.Possiblytheformation (the model is ‘marginally successful’ with fseed ∼ 0.036).
process would in fact be rather inefficient in its use of gas. Such a low efficiency is probably unlikely in any case (as
Alternativelyitmaybethat,becausewehaveneglectedthe discussed in Section 3.3), but also the required increase in
star formation and feedback which probably will consume f is sufficiently large that the models even more severely
m
muchofthegasinthehalo,theamountofgasthatisavail- overpredict the number density of luminous quasars at low
able at thefirst major merger is unrealistically large. luminosity (by a factor of ∼ 220). It remains to be seen
Asimplewaytocorrectthisproblemwouldbetomake whether the competing effects of star formation and feed-
themorephysicalassumption that thecollapse process will back can account for these discrepancies. For the moment
not be 100 per cent efficient, but rather only some fraction weproceedundertheassumptionthatthiscanonlybepart
of thetotal gas involvedin thecollapse (fseed) will actually of thesolution.
end up forming a SMBH. A hard upper limit to the initial It is worth mentioning that because quasars such as
mass(Mseed,max)couldalso beimposedtocoverthecaseof those found in the SDSS z > 5.8 sample (which are scat-
formation in extremelylarge early galaxies. Wewill assume tered over 2870 deg2) are so intrinsically rare, we find we
the gas not used in the collapse is lost in an outflow (or must average over around 700 Monte Carlo realizations of
in any case does not become available for later accretion), our model before we get good convergence. For example,
in this case one might hope that this correction would also a single realization of the model is the result of a Press–
help to correct the other problem – the excess total SMBH Schechter weighted average of a grid of root haloes of 50
mass density. This is because haloes which form SMBHs different masses, producing ∼ 6.17 ×105 separate galaxy
but then experience little or no subsequent accretion will merger events between redshifts z = 5.75−6.25, of which
make a smaller contribution to the total SMBH mass den- on average ∼26000 are major and of these only ∼380 will
sity.Howeverinpracticethisiscounterbalancedtoastrong generate quasars with M1450 <−27.1. In fact about 90 per
degree by the need to increase f in order to maintain the centofthe700realizationsatz=6didnotproduceasingle
m
agreement with the observed bright quasar numberdensity quasar above this magnitude limit. The distribution of the
at z =6, because the SMBHs responsible for these quasars quasarnumberdensitiesfoundintheother∼10percentof
must now be grown from smaller seeds. As a result we find therealizations is shown in Fig. 2.
that with a cap of Mseed,max = 106M⊙, we must reduce It is natural to ask whether the rather strong conclu-
(cid:13)c 0000RAS,MNRAS000,000–000
10 J. M. Bromley, R. S. Somerville and A. C. Fabian
Comoving Number Density for QSOs M1450 < -27.1 Comoving Number Density of SMBHs
10-5 10-1
10-6 10-2
10-3 > 107
-3Mpc 10-7 Boyle et al. fit -3Mpc 10-4
nsity / 10-8 Fan et al. fit nsity / 10-5 > 108
e e
mber D 10-9 SDSS z > 5.8 sample mber D 10-6
u u
N 10-10 N
10-7
10-11 10-8 > 109
10-12 10-9
0 1 2 3 4 5 6 7 2 3 4 5 6 7
Redshift Redshift
Distribution of SMBH masses at z = 6 Average SMBH Properties vs Final Host Halo Mass, z = 6
106 109
Total SMBH Mass
3er Mpc 105 Total: 3.6 x 105 110078 SMBHI nMitaiasls C Aoqlluaipresde bMya BssH o Mf SerMgeBrHs
p
nsity / Solar Units 110034 Mass (solar units) 111000456
g Mass De 102 Average 110023
movin 101 101
Co 100
100 10-1
105.5 106 106.5 107 107.5 108 108.5 109 109.5 1010 1010.5 109 1010 1011 1012 1013 1014
SMBH Mass / Solar Units Host DM Halo Final Mass (solar units)
Figure1.Resultsfromthesimplemodel,withaconstantaccretionfractionfm=0.0095andcompletelyefficientandeffectiveformation
of seed SMBHs. (a) Upper Left Panel: The comoving number density of bright (M1450 < −27.1) quasars (dashed line) as a function
of redshift, shown withthe observational estimates (solid lines; see text for details). (b) Upper Right Panel: The cumulative comoving
number density of SMBHs larger than 107, 108 and 109 M⊙ as a function of redshift. (c) Lower Left Panel: The mass distribution of
SMBHsintermsoftheircontributiontothetotal SMBHmassdensityatz=6.(d)Lower Right Panel: AverageSMBHmassatz=6
as afunction of host halomass (solid)and the contribution to this frommergers (longdashed) and initialcollapse(short dashed), the
remainingcontributionbeingfromaccretion.
sions reached above are specific to the particular approach lowered and the seed mass capped, the number density of
wehavetakentoform seed SMBH.Clearly whencompared luminous quasars increases too rapidly, overproducing the
to ‘seeding’ mechanisms which use small seeds, our method observed values at lower redshift much more severely than
would place more BH mass in galaxies which acquire an our original model (increasing the seed mass reduces this
initial black hole but then see little accretion. However, we problem but exacerbates the overproduction of the SMBH
have seen that the end result was not very different when mass density). Thus, although we restrict our attention to
we reduced the mass of the seeds, as more accretion was the ‘gaseous collapse’ model for seed BH formation in the
required to continue to match the observed quasar num- rest of this paper, it is likely that our conclusions will also
ber density at z = 6. One could argue that for this reason bepertinent to models considering other mechanisms.
our result should not be very sensitive to the details of the
recipeforcreatingseedBHs.Toconfirmthiswehaverunour
simple model with an alternative scenario in which haloes
4.2 Model variants
are seeded with 10 M⊙ BHs as soon as the virial velocity
reaches some critical value. The results are shown in Fig. 3 We have shown that the model with the minimal number
forvcrit =40kms−1,illustratingthataslongaswenormal- of free parameters cannot satisfy our requirements. In this
ize our model to NQSO at z = 6, the problem of exceeding section we explore other models with a minimal number of
the SMBH mass budget remains. This is true regardless of additionalparameters.Oursimplemodelhadtwomainfail-
the choice of the critical velocity. Although the total mass ings: firstly, it overproduced the total mass in SMBHs, de-
densityisindeedreduced,itstilljustexceedsthevaluethat positing too much mass in small SMBHs; and secondly the
we deem ‘marginally successful’. Also, as with the previous masses of the seed SMBHs could be excessively large. As
experiment in which the efficiency of seed formation was already mentioned, the latter can be easily resolved by de-
(cid:13)c 0000RAS,MNRAS000,000–000