A simple model for the evolution of super-massive black holes and the quasar population

0.46 MB·English

by  Asim Mahmood

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
A simple model for the evolution of super-massive black
holes and the quasar population
Asim Mahmood1⋆, Julien E. G. Devriendt1,2 and Joseph Silk1
1Department of Astrophysics, Universityof Oxford, Keble Road, Oxford, OX13RH, U.K
2Observatoire Astronomique de Lyon, 9 Avenue Charles Andr´e, 69561 Saint-Genis Laval cedex, France
4 Received...;accepted ...
0
0
2
ABSTRACT

n An empirically motivated model is presented for accretion-dominated growth of the
a super massive black holes (SMBH) in galaxies, and the implications are studied for
J the evolution of the quasar population in the universe. We investigate the core as-

3 pects of the quasar population, including space density evolution, evolution of the
characteristic luminosity, plausible minimum masses of quasars, the mass function of

1 SMBH and their formation epoch distribution. Our model suggests that the charac-
v
teristicluminosityinthe quasarluminosityfunctionarisesprimarilyasaconsequence
3
ofacharacteristicmassscaleabovewhichthereisasystematicseparationbetweenthe
0
blackholeandthehalomergingrates.Atlowermassscales,blackholemergingclosely
0
tracksthemergingofdarkhaloes.Whencombinedwithadeclining efficiencyofblack
1
0 hole formationwith redshift, the model canreproduce the quasarluminosity function
4 overa wide rangeof redshifts.The observedspacedensity evolutionof quasarsis well
0 described by formation rates of SMBH above ∼ 108M⊙. The inferred mass density
/ of SMBH agrees with that found independently from estimates of the SMBH mass
h
function derived empirically from the quasar luminosity function.
p
- Keywords: cosmology:darkmatter–galaxies:formation–galaxies:active–quasars:
o
r general
t
s
a
:
v
i 1 INTRODUCTION Nuclear black holes probably trace the merger of their
X
hostgalaxiesandcoalesceduringmergers.Ifoneenvisagesa
r Supermassive black holes consisting of at least tens of mil-
simple scenario where a dark matter halo harbours a single
a lions of solar masses are now believed to be lurking in the
galaxyandhalomergingleadstogalaxymergingalongwith
nucleiofmostgalaxies. Thesedarkobjectsarealsothecen-
their SMBHs, a coalescence dominated growth naturally
tral engines of the most powerful sources of light in the
leadstoalinearrelationbetweentheblackholeandthehost
universe: the quasars. Discerning the evolution history of
halomasses(seee.g.Haehnelt,Natarajan&Rees1998,here-
SMBHsandquasarsisafundamentalchallengeforthethe-
after HNR98). The observed correlation between the black
ories of structure formation in the universe. Quasars and
holemassesandthebulgevelocitydispersion (M σ re-
other forms of Active Galactic Nuclei (AGN) are thought BH−
lation)ofgalaxies(Ferrarese & Merritt2000),however,sug-
to be powered by accretion on to the SMBHs. By virtue of
geststhattheaccretedcomponentofmassfarsupersedesthe
their enormous brightness these objects are now observable
coalesced one. The observed scaling is concomitant with a
up to sufficiently high redshifts, corresponding to thedawn
scenariowheretheenergyfeedbackfromtheblackholeplays
of the era of galaxy and star formation in the universe. In
a crucial role in regulating the amount of matter accreted
thecontextofhierarchicalstructureformation,itisbelieved
on to it (Silk & Rees 1998). During the accretion process
that mergers of galaxies play a crucial role in igniting the
the mass of the SMBH builds up over some characteristic
quasars. The merger inducedgravitational instabilities fun-
time-scale. The ratio of this characteristic time-scale to the
nelahugeamountofgastothegalacticcores.Asafraction
Hubble time determines the duty cycle of a quasar. Accre-
of the kinetic energy of this gas is released in the form of
tion shuts down when the black hole is so massive that its
electromagnetic radiation, luminosities close totheEdding-
energyfeedbackexceedsthebindingenergyofitsfuelreser-
tonluminosity(∼1046 ergs−1 foratypical∼108M⊙SMBH voir.Indeed,aspointedoutbyHNR98andWyithe& Loeb
masses) are produced.
(2003),theobservedscalingcanberetrievediftheblackhole
were to supply the gas with an energy equal to its binding
energyoverthedynamicaltimeofgalaxies.Theseintricately
⋆ E-mail:[email protected]
(cid:13)c 0000RAS
2 A. Mahmood, J. Devriendt and J. Silk
woven,yetneatsetofideassketchanoutlineforsimplemod- minosity functions. We attribute this break luminosity to
els of quasar and SMBH evolution. a systematic decoupling between halo and galaxy merger
Several analytic and semi-analytic models based on rates in haloes above a characteristic mass along the line
mergers of dark haloes have been proposed to explain the suggested by CV00 that galaxy merging becomes rarer in
high redshift evolution of quasar luminosity functions (see sufficiently massive haloes, where individual galaxies re-
for e.g. Haehnelt & Rees 1993, hereafter HR93; HNR98; tain their properties. Furthermore, in a similar context,
Haiman & Loeb 1998; Kauffmann & Haehnelt 2000, here- Haiman, Ciotti & Ostriker (2003) have argued that a de-
after KH00; Cavaliere & Vittorini 2000, hereafter CV00; creaseinthequasarluminositydensitytowardslowredshifts
Hosokawa 2002; Wyithe & Loeb 2003, hereafter WL03; arises due to a decline in the formation rates of spheroids.
Hatziminaoglou et al. 2003). Simplified analytic models as- Here we propose a characteristic mass scale M∗ for haloes
sumethatallthecoldbaryonsinahaloarelockedinasingle which marks a drop in the formation rates of black holes
galaxyandgalaxymergerstracethehalomergers.Morere- andthuspossiblyspheroidsaswell.Asthelowredshiftuni-
alistically, the progenitor galaxies in a newly formed halo verse is an era of group assemblage, our value of M∗ which
sinktowardsthecentreoveradynamicalfriction time-scale corresponds to a group mass scale seems natural to explain
(Binney&Tremaine1989;Lacey&Cole1993)andmergeto the steep decline observed in the quasar luminosity density
form a new baryonic core. However, as the dynamical fric- at low z.
tion time-scale is generally much shorter than the Hubble Ourmodelentailsaphenomenologicaltreatmentofthe
time, the approximation that these mergers are instanta- ideasdescribedabove.Notwithstandingthecrudenessofthe
neous, works rather well. Galactic mergers are assumed to approach, it yields interesting results for the the quasar
igniteanAGNlastingforacharacteristictimet .There- luminosity functions and their evolving number density,
qso
sultingluminositiesareestimatedasEddingtonluminosities broadly consistent with the observations. It also provides
corresponding to central black hole masses, which in turn a simple explanation for the evolution of the characteristic
are determined through their correlation with the circular luminosity of quasars. Furthermore, the minimum SMBH
velocities of thenewly formed haloes. masses for quasars suggested by the model (by comparison
The uncertainty in the mass of cold gas available for with the observed space density evolution of quasars) are
accretionposesakeydifficultyforthemodels.Therearetwo &108M⊙,whichisconsistent withtheoriginal estimatesof
central aspects of this uncertainty: the first is to ascertain Soltan(1982).Finallythemodelcanbelinkedquitesimply
howmuchcoldgasisavailableingalaxiesofagivenmassat withtheevolutionofcoldgasmassesingalaxiesandstarfor-
agivenepoch(Kauffmann & Haehnelt2000)andthesecond mationratesintheuniverseandthusyieldsestimatesofthe
is to understand how effectively this gas is transported via contributions to the ionising background at high redshifts,
thegalaxymergerstobecomeavailable foraccretion bythe dueto star formation and quasar activity.
forming SMBH (Cavaliere & Vittorini 2000). Thestructureofthepaperisasfollows: insection 2we
The predictions of these models can be tested against reviewsomeimportantaspectsofanalyticalmodellingofhi-
thebackdropof afairly well constrained observational data erarchical structure formation which provides the essential
set.Themost strikingaspectofthisdataisthatthequasar toolsforourmodel;section3discussestherelationbetween
population peaks at z 2.5 and exhibits a steep decline the black hole masses and their host haloes in a scenario
∼
towards lower redshifts (Schmidt, Schneider & Gunn 1995; withadecliningefficiencyofSMBHformationwithredshift.
Warren, Hewett & Osmer 1994; Shaver et al. 1996; Boyle Section 4 extends the argument of major mergers of haloes
et al. 2000). At redshifts higher than 2.5 the number to compute the SMBH formation rates and the luminosity
∼
count appears to decline again, however, the observations functions of the ensuing quasars. In section 5 we compare
at these redshift could be marred by obscuration and ex- theSMBHformationratesfromourmodelagainstthosein-
tinction effects. Simple analytical models mentioned above ferredfromtheempiricaldoublepower-lawfittothequasar
canwellexplainthehighredshiftobservedluminosityfunc- luminosityfunctions.Insection6weestimatethemassfunc-
tions (z & 2). They, however, systematically fail at lower tionsofSMBHandthecomovingcosmological massdensity
redshifts (z . 2) and this epoch in the history of the uni- in the SMBH at various redshifts. Section 7 describes how
verseissomewhatopaqueinourunderstandingoftheevolu- to obtain simple estimates of the cold gas density and star
tionofquasars.EventhoughtheworksofCV00,HR93and formationratedensityintheuniverse;section8extrapolates
KH00 shed important light in this regard, no comprehen- these star formation rates to high redshift to compute the
sive scheme is yet available to systematically quantify the ionising background. These are thencompared with a simi-
various aspects of quasar evolution at z .2. larextrapolation tohigh redshifts, ofthequasarluminosity
Thepresentworkisprimarilymotivatedbythelowred- density.Wesummarise and discuss our results in section 9.
shift anomaly between observations and theoretical predic-
tions. To bridge this gap we introduce a toy model cen-
tred around two main ideas: the first pertains to the effect
2 ELEMENTS FROM EXTENDED
of declining black formation efficiency at low redshifts and
PRESS-SCHECHTER THEORY
thesecond involvesa declining black hole formation ratein
haloes above some characteristic mass M∗. The first idea OurapproachlargelytakesadvantageoftheextendedPress-
is derived from a model investigated in Haehnelt & Rees Schechter(EPS)formalismwhichprovidesessentialtoolsfor
(1993) where it is argued that the conditions for SMBH computing the merger probabilities and rates of bound ob-
formation are more favourable at higher redshifts as com- jects (Bond et al. 1991; Bower 1991; Lacey & Cole 1993,
pared to low redshifts. The second idea is based on the hereafter LC93). Inthis scenario theinitial densityfluctua-
existence of a break luminosity in the empirical quasar lu- tionsintheuniverseareassumed tobeGaussian innature.
(cid:13)c 0000RAS,MNRAS000,000–000
A simple model for the evolution of super-massive black holes and the quasar population 3
The growth of these fluctuations proceeds linearly initially 2.1 Merger rates, survival probabilities and
and is analysed through the linearised ideal fluid equations formation epoch distributions
(e.g. Silk & Bouwens 2001). The fluctuations subsequently
The EPS formalism lacks a self sufficient way to compute
turn non-linear and eventually collapse to form bound ob-
theformation rates of haloes. This is because effectively all
jects as the density reaches a critical threshold. A compar-
haloes at anyepoch are continuously accreting mass and in
ison of EPS results with the results of N-body simulations
thissensearenewlycreated.Thereisnocleardistinctionbe-
asgiveninLacey & Cole(1994),demonstratesthatthepre-
tweentheobjectsaccretinglessmassandtheobjectswhich
scription works rather well.
accrete substantial mass. For computing the major merger
The linear growth of fluctuations is described through
rates, therefore, a criterion has to be introduced by hand.
the linear growth factor D(t). For a critical cosmology this
Onehastoassumethatifnoneoftheprogenitorsofanewly
factorissimplyequaltothescalefactoraforthatepoch.For
formed object had mass greater than half (or some other
the ΛCDM cosmology used in this paper (Ω0 =0.3, Ω0 =
m Λ fraction) of the object mass, the event is a major merger.
0.7, Ω0 = 0.02/h2, σ = 0.9,h = 0.7), we compute it as
b 8 Alternatively Sasaki (1994) proposed to derive the forma-
giveninNavarro, Frenk& White(1997).ThemassMofan
tion rates by assuming that the destruction probability of
inhomogeneity is represented through a sharp k-space fil-
objects (i.e. the probability that they get incorporated into
ter which is used to smooth the matter density field at this
higher mass objects) has no characteristic mass scale. This
mass scale. As suggested in LC93, M can be related to k
through a qualitative mapping M = 6π2ρ k−3. Here ρ iss rate is then simply the positive term in the time derivative
0 s 0 ofthePSmassfunction.InSasaki’sprescriptionthesurvival
themeanbackgrounddensityofmatterintheuniverse.Us-
probability of haloes from one redshift to another is simply
ing the dark matter power-spectrum P(k) parametrized as
the ratio of the growth factors at these redshift. The prod-
in Efstathiou, Bond & White (1992), the variance for mass
uct of formation rate and the survival probability defines
scale M can then besimply calculated as
the formation epoch distribution. Interestingly for a criti-
ks calcosmology, Sasaki’sformation epochdistributioncanbe
σ2 (M) = P(k) 4π k2 dk. (1)
integrated analytically over all redshifts to yield the exact
Z0 mass function of haloes.
The mass function of haloes is now written in terms of Inthepresentworkwecomputethemajormergerrates
σ2(M) as basedontheprescriptionsdescribedinLC93andKS96.The
probability that a given halo of mass M at time t had,
2 ρ¯ ω 1 d σ 1 ω2 2 2
NPS(M,t)= πM0 D(t) σ2dM exp −2D2(t)σ2 , at some earlier time t1, a progenitor with mass below some
r (cid:12) (cid:12) (cid:20) (cid:21) thresholdmassMc(<M2)iscomputedbyintegratingequa-
(cid:12) (cid:12) (2)
(cid:12) (cid:12) tion 3 over S1 from S(Mc) to . If the difference between
where ω is the critical densit(cid:12)y for col(cid:12)lapse. ∞
the times t and t is substantial, then M can reach the
1 2 1
The formation rates, major merger rates and survival
value M through either a single merger occurring anytime
2
probabilities have been discussed by several authors (e.g.
between t and t or it can do so through more than one
1 2
Sasaki1994;Kitayama&Suto1996,hereafterKS96).Inthis
merger/accretion episodesbetweenthetwotimes.However,
context, the excursion set approach of Bond et al. (1991)
ifthedifferencebetweenthetwotimesisverysmallthenthe
(also LC93) yields two important conditional probabilities.
entire change in mass (∆M =M M ) can be treated as
2 1
The first is the conditional probability that a halo of given −
arising from a single merging event. Letting ∆ω =ω ω
1 2
mass M at timet ,had aprogenitor of mass M at earlier −
2 2 1 we thuswrite
time t . This is given as
1
Mc
P1(S1,ω1|S2,ω2)dS1 = √12π(S(1ω1−S2ω)23)/2 P(M1<Mc,t1|M2,t2) = Z0 P1 dM1 (5)
− ∆ω
×exp(cid:20)−(2ω(S11−−ωS22)2)(cid:21)dS1, (3) = erf" 2(Sc−S2)#,
where S is the variance correspopnding to mass M . For
whereω isthecriticaldensityattimet andω attimet . c c
1 1 2 2
small ∆ω taking the series expansion in equation 5 up to
Also,S isthevariancecorrespondingtomassM andS is
1 1 2
firstorderonly,wecomputethemajor mergerratesofdark
thevariancecorrespondingtoM .Therelationbetweenthe
2
haloes as
criticaldensityω(t)atanytimetisgivenasω(t)=ω /D(t),
0
where ω is the present day critical density ( 1.686 for a
0 ∼ 2 1 dω
SCDTMhecossemcoonlodgiys)t.heprobabilitythatahaloofmassM1 at RMM(M,t)=rπ S(M/2)−S(M) ×(cid:12)(cid:12)dt(cid:12)(cid:12)NPS(M,t),
(cid:12) (cid:12) (6)
timet1,willresultinalargerhaloofmassM2 atlatertime p (cid:12) (cid:12)
wherewehavechosenM =M/2.Noteth(cid:12)atw(cid:12)henS tends
t can bewritten as c c
2
to S, the right hand side of equation 5 tends to one, if ∆ω
1 ω (ω ω ) S 3/2 issmall butfinite.This implies that at anygiven epoch,all
P2(S2,ω2|S1,ω1)dS2 = √2π 2 ω11− 2 (cid:20)S2(S11−S2)(cid:21) theexisting objects continuously accrete mass.
(S ω S ω )2 As pointed out in LC93, when the halo mass for a tra-
×exp(cid:20)−2S21S12(−S1−1 S22)(cid:21)dS2. (4) jmecetaonrsythfaaltlsotnoeaofstmhaellpearrevnatluhealaotesanhaedartlhieisrstmimaell,eritmoanslsy.
Through appropriate transformations theconditional prob- Asaresult,itisalwayspossiblethatthelargest coevalpro-
abilities can also be written in terms of halo masses. genitoralreadyhasmorethanhalfthemassofthefinalhalo
(cid:13)c 0000RAS,MNRAS000,000–000
4 A. Mahmood, J. Devriendt and J. Silk
Figure 1. The formation epoch distribution of DM haloes of Figure 2. The evolution of cumulative formationrates of dark
differentmassesasgivenbyequation8. matterhaloeswithmassesaboveMmin.Theserates(particularly
forMmin =1012M⊙, the typical quasar host halomass) donot
produce as steep a decline at low redshifts as that measured for
quasarnumbercounts.
at this epoch. Such an event cannot be characterised as a
major merger as wedefineit. Theapproach of KS does not
account for this possibility. Although this marginally alters
theratesfortherangeofhalomassesweareconcernedwith,
theirnumberdensity.Hence,wesubsequentlydiscussthead-
it is nevertheless interesting to see how one can systemati-
ditionalinputsthat needtobeincorporated intothemodel
cally account for this possibility and therefore, we propose
in order to reproduce the observed evolution of the quasar
a solution to this problem in appendixA.
population.
Equation 4 can be used to calculate the survival prob-
abilities as discussed in LC93 and KS96. These are based
on the assumption that an object retains its identity up to
2.2 Halo circular velocities
the time it doubles its mass. The probability that a halo
of mass M at present time t1, will not undergo such a Foranearlyisothermalhalo,themappingbetweenthehalo
merger/accretion, before time t2, which would result in its circular velocity and its mass can be obtained using there-
final mass being greater than 2M, is given as lations given in Somerville & Primack (1999)
2M
Psurv(M,t1,t2)= P2(M2,t2 M,t1)dM2. (7) v2= GM ΩΛH2 r2 (z), (9)
ZM | c rvir(z) − 3 0 vir
As discussed previously, the survival probability of
where z is the collapse redshift and r is the virial
haloes multiplied with their merger rates, yields their for- vir
radius. r can be computed by noting that M =
mation epochdistribution.Thisdistributiontellsusthatin vir
4π/3r3 (z)ρ (z)wherethevirialdensityρ (z)isgiven
a given population of mass M objects at time t, what frac- vir vir vir
as
tionwereformedatanearliertimet andhaveretainedtheir
f
identity since then. Thus the formation epoch distribution ρ (z)=∆ (z)ρ0(1+z)3 Ω0 . (10)
in termsof formation time t can bewritten as vir c c Ω(z)
f
F(M,tf t)dMdtf = RMM(M,tf) Psurv(M,tf t) Here, Ω0 and Ω(z) are the matter densities of the present
→ × → day and redshift z universe respectively; ρ0 = 2.8
× dM dtf. (8) 1011 h2 M⊙Mpc−3 isthepresent critical densictyof theun×i-
WeshowinFig1theformationepochdistributionofhaloes verseand a fittingexpression for ∆ (z) is given as ∆ (z)=
c c
ofdifferentmassesasafunctionofredshift,andinFig2the 18π2+82x 38x2,wherex=Ω (z) 1(Bryan & Norman
m
formationratesofobjectswithmassesabove1011,1012 and 1998). Thus−the expression for virial−radius in terms of for-
1013 M⊙ respectively.ItcanbeseenfromFig2thatmerger mation redshift z and halo mass M is given as
rates of haloes alone do not produce a significant evolution
inthenumberdensityatlowz,andthereforeanaivemodel Ω0 ∆ (z) −1/3
r =2.14 10−5M1/3 m c (1+z)−1 Mpc.
directly linking quasar triggering to merger rates of haloes vir × (cid:20)Ωm(z) 18π2 (cid:21)
would fail to match the steep decline which is observed in (11)
(cid:13)c 0000RAS,MNRAS000,000–000
A simple model for the evolution of super-massive black holes and the quasar population 5
We model the declining formation efficiency of SMBH
masses through a function ǫ(z) such that M = ǫ(z) v5
BH c
and consider a phenomenological form of ǫ(z) as
z
ǫ(z)=ǫ c exp . (14)
0× − −z∗
(cid:20) (cid:18) (cid:19)(cid:21)
Hereǫ0andcareconstantsandz∗isacharacteristicredshift.
We choose a value z∗ 1 based on the observed peak in
≈
the star formation rate (SFR) density in the universe at
redshiftsbetween1and2(seee.g.Ascasibaretal.2002;Silk
&Rees1998).ThisisbecauseapeakintheSFRdensityisa
generalindicatorofagasrichenvironmentingalaxies.Itis,
however,stillamatterofdebatewhetherSFRdensitypeaks
at z 1 2 or reaches a broad maximum there. To obtain
∼ −
a steep decline we choose c = 1.1 such that the minimum
value of ǫ(z) is roughly 10 percent of the maximum value
1.1 ǫ .ThislatterischosentomatchtheobservedM σ
0 BH
× −
correlation (Ferrarese 2002)
σ α
MBH =1.66×108 200 km s−1 M⊙, (15)
(cid:16) (cid:17)
where we adopt α = 5 consistent with a feedback regu-
latedgrowthofSMBHmasses.Thisisslightlydifferentfrom
Figure 3.Phenomenological redshiftevolution oftheformation the value α = 4.58 0.52 inferred from SMBH mass mea-
±
efficiency of black holes (see text for the exact function). The surements, the difference possibly having its origin in the
characteristic redshift of decline is z∗ = 1. A steep decline is observed non-linearity of vc σ relation (Ferrarese 2002;
obtainedaroundz=0wherevaluesbecomeaslowastenpercent WL03). For simplicity, we co−nsider the linearly fitted rela-
ofthemaximumvalue. tionforellipticalsσ 0.65v toobtainǫ =6 10−5 (such
c 0
≈ ×
that M ǫ v5 at high redshifts).
BH ∼ 0 c
Theserelationstogetherenableustowritethehalocircular Figure3showsaplotofefficiencyversustheformation
velocity as redshift,i.e.howequation14behaves.Wecannowwritethe
black hole masses in termsof formation redshift z and halo
Ω0 ∆ (z) 1/6
v =1.38 10−2M1/3 m c (1+z)1/2 kms−1 . masses M as
c × (cid:20)Ωm(z) 18π2 (cid:21) (12) M = ǫ(z) 5 10−10 M 5/3
BH × × M⊙
(cid:18) (cid:19)
Ω0 ∆ (z) 5/6
3 SUPER MASSIVE BLACK HOLE MASSES × (cid:20)Ωmm(z) 18cπ2 (cid:21) (1+z)5/2 M⊙. (16)
In a self-regulated growth of SMBH masses, the correla-
tion between the galaxy circular velocities (v ) and SMBH
c
masses (M ) is written as (Silk & Rees 1998) 4 FORMATION RATES OF SMBH AND
BH
QUASAR LUMINOSITY FUNCTIONS
MBH ∝vc5, (13) We now use the MBH −Mhalo mapping (equation 16) to
transform the halo major merger rates into the black hole
where the constant of proportionality depends on the frac-
formation rates in terms of the formation redshift z as
tionofcoldgasf availableforaccretionbythecoalesced
cold dM
seed black hole and weakly on the presence of an accretion R1 (M ,z)=R [M (M ,z),z] (M ,z).
BH BH MM halo BH dM BH
disc(HNR98;Burkert&Silk2001).Suchascalingiscorrob- BH
(17)
orated by the investigations of Ferrarese & Merritt (2000)
The superscript 1 in R1 is to differentiate it from a mod-
and Gebhardt et al. (2000). BH
ified rate R , which takes into account a decline in black
Incorporating f in an analytic model is not simple. BH
cold hole formation rates in haloes above a characteristic mass
Firstly, we do not know how much cold gas exists in haloes
M∗. This is achieved through a phenomenological function
of different masses at different epochs and secondly, even
F (M ) such that
if we knew the mass of cold gas, it is impossible to ana- µ halo
lytically compute the contributions coming from different R (M ,z)=R1 (M ,z) F [M (M ,z)].
BH BH BH BH × µ halo BH
mass progenitors during a merger. For the sake of simplic- (18)
ity, therefore, we get around this problem by adopting an For the form of F (M ), we found that a simple expo-
µ halo
empirically motivated decline in black hole formation effi- nential function exp( M/M∗), with a cutoff at M∗ is too
−
ciency with redshift. This can be attributed to a general steep to be consistent with the observed luminosity func-
declineintheamountofcold gasinthegalaxiesatlowred- tions. We therefore adopt a function with a following be-
shifts.Suchanapproachhasbeenconsideredpreviously,for haviour: F(M) 1 for M <<M∗ and F(M) (M∗/M)µ
→ →
example, by Haehnelt & Rees(1993). for M >> M∗, where µ is a positive number. Instead of a
(cid:13)c 0000RAS,MNRAS000,000–000
6 A. Mahmood, J. Devriendt and J. Silk
Figure4.Thisfigureshowsthesecondofourtwophenomenolog- Figure 5.Themodellifetimeofquasars givenastqso =k tdyn,
icalfunctions.Weusethisfunctiontointroduceadropinthefor- where tdyn is the halo dynamical time and the constant of pro-
mationratesofblackholeinhaloesabovethecharacteristicmass portionalityisaboutk≈0.035(e.g.WL03)
scaleM∗,thevalueofwhichhasbeensetbycomparisonwiththe
quasarluminosityfunction(figure21)toaboutM∗=1013.25M⊙. .
The redshift evolution of the luminosity corresponding to this
mapping and the fact that t t we can simply
characteristicmassisplottedagainsttheempiricalcharacteristic qso ≪ Hubble
luminosityofthedoublepowerlawfit,infig7. write thequasar luminosity function as
dM
Φ(L ,z) t R [M (L ),z] BH(L ,z). (21)
sharpexponentialcut-off,suchafunctionhassofter‘power- B ≃ qso× BH BH B × dLB B
law’ drop at thehigh mass end:
The quasar lifetime, t is not a very well constrained
qso
1
F (M)= . (19) quantity. Two relevant time-scales for quasars are the
µ 1+(M/M∗)µ Salpeter (or e-folding) time-scale and the dynamical time-
Fµ(M) for values of µ = 1 and µ = 2 is plotted in figure 4 scale of galaxies. The Salpeter time-scale is given by tSal=
alongItwiisthimthpeoretxapnotnteontmiaelnfutinocntiaotntFh(isMp)o=intextph(a−tMin/mMa∗s)-. 4bl×ack10h7o(lǫersadd/e0fi.1n)edwthherroeugǫrhadthiesrtehlaetiroandiLatbiovle=effiǫrcaideMn˙ccy2,ocf
sive haloes (M >M∗), rarer galaxy mergers could possibly is the speed of light and M˙ is the accretion rate of mat-
alter the M M mapping. Furthermore, the AGN ter on to the black hole. On the other hand it has been
BH halo
could enter the−sub-Eddington regime due to lack of gas noted,forexample,inHNR98andWL03,thattheobserved
masstransportedthroughraregalaxy mergers.However,to MBH vc relation naturally arises if the black hole injects
−
keep our model simple we assume that even for ineffective the gas with an energy equal to its binding energy, over a
galaxymergers,theM M scalingisasgiveninequa- localdynamicaltime-scaleofthegalaxy.Inthisrespect,the
BH halo
tion16.Ifamerger-ignite−dblackholeshinesattheEdding- localdynamicaltimeofagalaxyisalsoanaturaltime-scale.
ton luminosity L /L = 1, we can write the B band In the present model, following WL03, we adopt a quasar
bol Edd
luminosities of AGNsas − lifetimeproportionaltothelocaldynamicaltimeofgalaxies
innewlyformedhaloes.Thelocaldynamicaltimeofgalaxies
M
L =f 1.3 1039 BH J s−1, (20) being themselves proportional to the halo dynamical time,
B B× × (cid:20)108M⊙(cid:21) we get tqso tdyn, where tdyn is the halo dynamical time
∝
where f is the fraction of bolometric luminosity emitted and so we write
B
in the B-band. Realistically, as the accretion proceeds, the Ω0 ∆ (z) −1/2
black hole mass builds up over the characteristic accretion t =k 1.5 109 m c (1+z)−3/2yr, (22)
time. This process is accompanied by a parallel increase in qso × × (cid:20)Ωm(z) 18π2 (cid:21)
thequasarluminositywhichdeterminesitslightcurve.How- where the constant of proportionality is k 0.035 (e.g.
∼
ever, a simplification is possible if one considers that the WL03).
black hole mass instantaneously increases to post merger In figure 6 we plot the luminosity functions as given
values given by equation 16. The black hole then shines at by equation 21 (solid curves). The parameters are µ = 2,
the Eddington luminosity, corresponding to this new mass, M∗ =1013.25 M⊙ andfB =0.06. Theshort-dashedcurveis
over characteristic time-scale t which lies in the range thedoublepowerlawfitasgiveninPei(1995).Wetransform
qso
t /t 10−2 to 10−3. Thus using the L M the data and the double power-law fit to a Λ cosmology by
qso Hubble B BH
∼ −
(cid:13)c 0000RAS,MNRAS000,000–000
A simple model for the evolution of super-massive black holes and the quasar population 7
Figure6.Modelluminosityfunctions(solidcurve)plottedagainsttheobservedluminosityfunctions(filledcircle)ofquasars.Thedata
forz.4isfromPei(1995)andforz≈4.4isfromKennefick, Djorgovski&Carvalho(1995).Themodelparametersareǫ0=6×10−6,
µ=2,M∗ =1013.25M⊙ andtqso ∼0.035tdyn.Theshort-dashedcurves show the doublepower lawfitof Pei(1995).Thelong-dashed
verticallineisthemodelbreakluminositycorrespondingtoM∗ andtheshort-dashedverticallineisthecharacteristicluminosityinthe
doublepower-lawfit.
noting that Φ(L,z) dV−1d−2 and L d2, where dV is Gaussian form
∝ L ∝ L
the volume element and d is the luminosity distance (e.g.
Hosokawa 2002). The doubLlepower-law is given as Lz =L∗(1+z)−(1+α)exp(cid:20)−(z2−σz∗2∗)2(cid:21), (24)
Φobs(LB,z)= (LB/Lz)βΦl∗+/L(LzB/Lz)βh, (23) wherelog(L∗/L⊙)=13.03±0.10,z∗ =2.75±0.05,α=−0.5
and σ∗ =0.93 0.03. In figure 6, the vertical short-dashed
where log(Φ∗/Gpc−3) = 2.95 0.15, βl = 1.64 0.18 and lineisthechara±cteristicluminositygivenbyequation24and
± ±
β = 3.52 0.11. The characteristic luminosity L has a theverticallong-dashedlineshowsthebreakluminosityfor
h z
±
(cid:13)c 0000RAS,MNRAS000,000–000
8 A. Mahmood, J. Devriendt and J. Silk
Figure7.Theevolutionofbreakluminosityinourmodel(solid
line) compared with the characteristic luminosity in the double
Figure9.Cumulativeformationratesofblackholeswithmasses
power law luminosity function fit (dashed line). Also plotted is above107,108 and109 M⊙ respectively.
the break luminosity in a scenario where ǫ= ǫ0 is fixed (dotted
line).ThecharacteristicmassisM∗=1013.25M⊙.
The characteristic luminosity and the break luminosity fol-
loweach otherclosely uptoz 2.5. Thisisespecially clear
≈
in figure 7, where we additionally show (dotted curve) the
break luminosity corresponding to M∗ in a fixed black hole
formation efficiency scenario.
It has been suggested in WL03 that rather than be-
inganintrinsicfeatureofthequasarsthemselves,thebreak
in the luminosity function is present only at low redshifts.
Ourmodelsuggestsotherwise.Wefindthatthebreakinthe
luminosity function exists as a consequence of the charac-
teristic mass and is present over a wide range of redshifts.
Furthermore,thedoublepowerlawintheform givenabove
under-predicts the quasar luminosity function at z = 4.4.
On theother hand,inserting thebreak luminosity given by
ourmodelintothedoublepowerlawdescriptionofthedata
over-predictsthequasarluminosityfunction.Apossibleim-
plication of this is that perhaps the black hole formation
efficiency declines beyondz 4.
∼
The tuning of parameters for our model luminosity
functionisadmittedlycrude.Also,itcanbeseenfromequa-
tions 16 and 20 that there exists a degeneracy between the
parametersǫ andf .Highervaluesofǫ requiresystemat-
0 B 0
ically lower values of f . Nevertheless our fiducial numbers
B
are in broad agreement with the values obtained by other
Figure 8. This figure shows the redshift evolution of formation investigations,andweemphasizeonceagainthatthemodel
ratesofblackholeswithmasses106,107,108 and109 M⊙ respec- presentedhereonly attemptstodiscern thebroad trendsof
tively. The dashed lines are from equation 25 and the solid line SMBH and quasar population evolution.
arefromequation18
5 FORMATION RATES OF SMBH FROM THE
ourmodel,correspondingtoM∗ =1013.25M⊙ haloes.Itcan OBSERVED LUMINOSITY FUNCTIONS
be seen from the figure that the model can systematically
accountfortheobserveddataoverawiderangeofredshifts. Due to their emission of gravitational waves, the merger
Atz <1,wherethebreakluminosityofthemodelisslightly ratesofSMBHswillbedetecteddirectlybyinstrumentssuch
greater than the empirically fitted characteristic luminos- as the Laser Interferometer Space Antenna (LISA). In our
ity, we find that the bright end is slightly over predicted. model,SMBHmerger/formationratesaregivenbyequation
(cid:13)c 0000RAS,MNRAS000,000–000
A simple model for the evolution of super-massive black holes and the quasar population 9
Figure10. ThespacedensityevolutionofquasarswithminimumblackholemassMmininthemodel,plottedagainsttheobservational
data. Filled circles describe the observed space density evolution of quasars with absolute B-band magnitudes MB < −24 and open
triangles describe the observed space density evolution of quasars with absolute B-band magnitudes MB < −26. The data points are
plotted asinKauffmann&Haehnelt2000 (seereferencestherein). AlsoshownarethebestfittothehighredshiftdatafromFanetal.
(2001)(dashed lines)andSchmidt,Schneider &Gunn(1995)(dotted lines).Themodelparametersaresameasinfigure6.
18. One can also use the double power-law fit to the lumi- computedbyintegratingequation18fromaminimumblack
nosityfunctiondatatoestimatetheseratesasafunctionof hole mass Mmin to a tentative upper limit of 1011M⊙.
∼
SMBH mass. These can bewritten as Note that the decline in the cumulative formation rates of
1 SMBHtowardslowredshiftsismuchsteeperascomparedto
Robs(M ,z) = Φ [L (M ,z),z]
BH BH tqso(z) × obs B BH thedropin thecumulativeformation ratesof haloes(figure
2).
dL
× dMBBH(MBH,z). (25) SMBTHhemansusmesbaebrodveenssoitmyeemvoinluimtiounmomfatshseMquasacrasnwniotwh
min
In figure 8 we plot SMBH formation rates for different bewritten as
masses using these two previous methods. The solid curves
Mmax
arecomputedfrom equation18,i.e.ourfiducialmodel, and N (M >M ,z) = dM t (z)
QSO BH min BH qso
the short-dashed curves are results from equation 25. The ZMmin ×{
differentmasses considered arelabelled in thefigure.Itcan R (M ,z) . (26)
BH BH
× }
be seen that there is a significant discrepancy at high red-
Figure10showsthethenumberdensityevolutionofquasars
shifts between the double-power law predictions and our
with minimum black hole masses as labelled in the fig-
model’s.If,asdiscussedpreviously(section4),theefficiency
ure. Also plotted is the observed data for the space den-
ǫdeclinesbeyondz 4,asteeperdeclineintheSMBHfor-
∼ sityevolutionofquasarswithminimumB-bandmagnitudes
mationratesathighredshiftswouldbeobtained.Thiswould
M < 24 (filled circles) and M < 26 (open triangles)
of course bring the two estimates in better agreement. In B − B −
up to a redshift of 10. As can be seen, the figure sketches
otherwords,thiswouldmeanthatourmodelonlygivesthe
a consistent picture with the original estimate of minimum
maximum limiting value of SMBH formation rates at these
redshifts. quasar black hole masses of ∼108M⊙ by Soltan (1982).
In any case, it should be noted from the figure that
theformation ratescorrespondingto106 and107 M⊙ black
holes are predicted to peak at very high redshifts (z & 10)
6 THE MASS FUNCTION AND
and havegreater values (at least a couple of orders of mag-
COSMOLOGICAL MASS DENSITY OF
nitudelarger)thantheirmoremassivecounterparts.There-
SMBHS
fore, their detection (or lack of detection) by LISA would
yield strongconstraints onourunderstandingoftheforma- TheformationratesofSMBHscanbeusedtocomputetheir
tion history ofSMBH andtheirplausibleminimum masses. formation epoch distribution (see section 2.1). These can
Figure9showsthecumulativeformationratesofSMBH then be integrated over redshift to yield the SMBH mass
(cid:13)c 0000RAS,MNRAS000,000–000
10 A. Mahmood, J. Devriendt and J. Silk
function. The formation epoch distribution of SMBHs can
bewritten as
dt
FA (M ,z z) = R (M ,z ) (z ) (27)
BH BH f → BH BH f × dz f
f
P [M (M ,z ),z z] .
surv halo BH f f
× →
However, there are two problems with this equation. The
first is that the survival probabilities of SMBHs are only
crudelycorrelated tothehalosurvivalprobabilities: thede-
clining merger rates of SMBH in haloes with mass above
M∗ suggest thatthesurvivalprobabilitymaybegreaterfor
SMBHs in these haloes. The second problem is more fun-
damental and lies in the fact that equation 16 is not only
a function of halo mass but also of formation redshift. This
canbeunderstoodinthefollowingway:massesofSMBHfor
agivenmasshalowillbehigherathighredshifts(thisholds
even if the formation efficiency is taken as constant such
that ǫ(z) = ǫ ), so that when we compute the destruction
0
probabilityP ofobjects(notethatP =1 P ),we
dest surv dest
−
authorise un-realistic situations where a halo with a higher
mass black hole goes on to form a lower mass black hole.
Hence we underestimate the survival probability. Equation
Figure 11. The mass function of SMBHs. Open squares show
27,therefore,only givesalowerlimit onthenumbercounts
the local estimates given in Saluccietal. (1999). The limiting
of objects of different masses. An upperlimit can be set by
z=2.5massfunctionsfromourmodelhavebeenplottedassolid
line (using NA ) and long-dashed line (using NB ). We also taking Psurv = 1. This amounts to computing mass func-
BH BH
showtheupperlimitmassfunctionatz=1(usingNB )andat tions by simply integrating the formation rates up to the
BH
z=6, whereboth limitingmassfunctions (NA and NB )are redshiftofobservation.Suchanapproachhasalsobeencon-
BH BH
verysimilar. sidered in Haehnelt & Rees (1993). Thuswe write
dt
FB (M ,z z)=R (M ,z ) (z ), z >z.
BH BH f → BH BH f × dz f f
f
(28)
We denote the corresponding SMBH mass functions as
NA (M ,z) and NB (M ,z) respectively. These are
BH BH BH BH
computed as
z
NA,B(M ,z)= dz FA,B(M ,z z) . (29)
BH BH f BH BH f →
Z∞
In figure 11 we plot the SMBH mass functions for sev-
eralredshifts andcomparethem with theestimates of local
SMBH mass function as given in Salucci et al. (1999). At
redshifts greater than z 3 there appears to be no signifi-
∼
cantdifferenceinthetwolimitingmassfunctions(NA and
BH
NB ).Wecannowwritethecomovingcosmological SMBH
BH
mass density as
∞
ρA,B(M >M ,z)= dM M NA,B(M ,z)
BH min BH { BH× BH BH }
ZMmin
(30)
Figure 12 shows the redshift evolution of cosmological
massdensityoftheSMBHwithminimummasses108,3 108
×
and 109 M⊙ (top to bottom) respectively. The solid lines
are computed using NB and dotted lines are computed
Figure 12. Comoving cosmological mass density of SMBHs by BH
using NA . It can be seen that mass function NA gives
using the SMBH mass function NA (dotted lines) and NB BH BH
BH BH an unrealistic decline in the mass density of SMBH which
(solidlines).MinimumSMBHmassesMmin are(toptobottom)
108,3×108 and109 M⊙.Thetwocurves(solidanddotted)can shouldincreasemonotonically.Onceagain,thereasonisthat
be considered as (upper and lower) limits on densities for each the survival probability of the SMBHs are underestimated
Mmin. and thus the number counts are underestimated as well1.
1 We point out that in a recent paper
Bromley,Somerville&Fabian (2003) have found a similar
non-monotonic evolution of SMBH mass density, computed by
using the M −vc relation combined with the Press-Schechter
massfunction.
(cid:13)c 0000RAS,MNRAS000,000–000