Yue Shen- Supermassive Black Holes in the Hierarchical Universe: A General Framework and...

8/3/2019 Yue Shen- Supermassive Black Holes in the Hierarchical Universe: A General Framework and Observational Tests

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The Astrophysical Journal, 704:89108, 2009 October 10 doi:10.1088/0004-637X/704/1/89

C 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

SUPERMASSIVE BLACK HOLES IN THE HIERARCHICAL UNIVERSE: A GENERAL FRAMEWORK ANDOBSERVATIONAL TESTS

Yue Shen

Princeton University Observatory, Princeton, NJ 08544, USAReceived 2009 March 17; accepted 2009 August 4; published 2009 September 21

ABSTRACT

We present a simple framework for the growth and evolution of supermassive black holes (SMBHs) in thehierarchical structure formation paradigm, adopting the general idea that quasar activity is triggered in majormergers. In our model, black hole accretion is triggered during major mergers (mass ratio 0.3) of host darkmatter halos. The successive evolution of quasar luminosities follows a universal light-curve (LC) form duringwhich the growth of the SMBH is modeled self-consistently: an initial exponential growth at a constant Eddingtonratio of order unity until it reaches the peak luminosity, followed by a power-law decay. Assuming that the peakluminosity correlates with the post-merger halo mass, we convolve the LC with the triggering rate of quasar activityto predict the quasar luminosity function (LF). Our model reproduces the observed LF at 0.5 < z < 4.5 forthe full luminosity ranges probed by current optical and X-ray surveys. At z < 0.5, our model underestimatesthe LF at Lbol < 10

45 erg s1, allowing room for the active galactic nuclei (AGNs) activity triggered by secularprocesses instead of major mergers. At z > 4.5, in order to reproduce the observed quasar abundance, the typicalquasar hosts must shift to lower mass halos, and/or minor mergers can also trigger quasar activity. Our modelreproduces both the observed redshift evolution and luminosity dependence of the linear bias of quasar/AGNclustering. Due to the scatter between instantaneous luminosity and halo mass, quasar/AGN clustering weaklydepends on luminosity at low-to-intermediate luminosities; but the linear bias rises rapidly with luminosity at thehigh luminosity end and at high redshift. In our model, the Eddington ratio distribution is roughly lognormal, whichbroadens and shifts to lower mean values from high luminosity quasars (Lbol 10

46 erg s1) to low-luminosityAGNs (Lbol 10

45 erg s1), in good agreement with observations. The model predicts that the vast majority of108.5 M SMBHs were already in place by z = 1, and 50% of them werein place by z = 2; but the less massive(107 M) SMBHswere assembledmore recently, likely more through secular processes than by major mergersinaccordance with the downsizing picture of SMBH assembly since the peak of bright quasar activity around z 23.

Key words: black hole physics cosmology: observations galaxies: active large-scale structure of universe quasars: general surveys

Online-only material: color figures

1. INTRODUCTION

The cosmic assembly and evolution of supermassive blackholes (SMBHs) is a central topic in modern cosmology: in thelocal universe, SMBHs reside in almost every bulge dominantgalaxy (e.g., Kormendy & Richstone 1995; Richstone et al.1998), and they likely played important roles during their co-evolution with galaxy bulges (e.g., Silk & Rees 1998; Wyithe& Loeb 2002, 2003; King 2003; Di Matteo et al. 2005; Crotonet al. 2006), as inferred from observed scaling relations betweenbulge properties and the mass of the central black hole (BH, e.g.,Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al.2000; Graham et al. 2001; Tremaine et al. 2002; Marconi & Hunt

2003). It is also generallyacceptedthat thelocaldormant SMBHpopulation was largely assembled via gas accretion during aluminous quasi-stellar object (QSO) phase1 (e.g., Salpeter1964;Zeldovich & Novikov 1964; Lynden-Bell 1969; Soltan 1982;Small & Blandford 1992; Salucci et al. 1999; Yu & Tremaine2002; Shankar et al. 2004; Marconi et al. 2004). The statisticsof the local dormant SMBHs and distant active QSOs therefore

1 In this paper, we use the term QSO to refer to all actively accreting

SMBHs. We adopt the convention that Lbol = 1045 erg s1 is the dividing line

between quasars and AGNs. This division is slightly lower than the traditionalSeyfert/quasar division (Schmidt & Green 1983) of MB = 23 or

Lbol 1012 L.

provide clues to the buildup of the local BH density acrosscosmic time, and have implications for the hierarchical structureformation paradigm, as well as for the interactions betweenaccreting SMBHs and galaxy bulges.

One of the leading hypotheses for the QSO-triggering mech-anism is galaxy mergers (e.g., Hernquist 1989; Carlberg 1990;Kauffmann & Haehnelt 2000; Hopkins et al. 2008, and ref-erences therein). In the hierarchical cold dark matter (CDM)paradigm, small structures merge to form large structures, andthemerger rate of dark matterhalospeaks at early times,in broadconsistency with the observed peak of bright quasar activities.Gas-rich major mergers (i.e., those with comparable mass ra-tios) between galaxies provide an efficient way to channel a

large amount of gas into the central region to trigger starburstsand possibly feed rapid black hole growth. Observationally, thismerger hypothesis is supported by the ULIRGquasar connec-tion (e.g., Sanders & Mirabel 1996; Canalizo & Stockton 2001),the signature of recent mergers in quasar hosts (e.g., Bennertet al. 2008), and the small-scale overdensities of galaxies aroundluminous quasars or quasar binaries (e.g., Fisher et al. 1996;Bahcall et al. 1997; Serber et al. 2006; Hennawi et al. 2006;Myers et al. 2008; Strand et al. 2008). These observations donot necessarily prove that mergers are directly responsible fortriggering QSO activity, but they at least suggest that QSO ac-tivity is coincident with mergers in many cases.

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The last decade has seen a number of models for QSOevolution based on the merger hypothesis (e.g., Haiman &Loeb 1998; Kauffmann & Haehnelt 2000; Wyithe & Loeb 2002,2003; Volonteri et al. 2003; Hopkins et al. 2008), which agreewith the bulk of QSO observations reasonably well. Motivatedby the success of these merger-based QSO models, we revisitthis problem in this paper with improved observational dataon SMBHs and QSOs from dedicated large surveys, and withupdated knowledge of the merger rate as inferred from recentnumerical simulations. The present work is different fromprevious studies in many aspects in both methodology and theobservational tests used. Our goal is to check if a simple merger-based cosmological QSO framework can reproduce all theobserved statistics of SMBHs (both active and dormant), and toput constraints on the physical properties of SMBH growth. Ourmodel is observationally motivated; therefore, we do not restrictourselves to theoretical predictions of SMBH/QSO propertiesfrom either cosmological or merger event simulations.

The paper is organized as follows. In Section 1.1, we reviewsome aspects of local SMBH demographicsand QSO luminosityfunction (LF); in Section 1.2, we briefly review the current statusof quasar clustering observations; Sections 1.3 and 1.4 discuss

the halo and subhalo merger rate from numerical simulations.Our model formulism is presented in Section 2, and we presentour fiducial model and compare with observations in Section 3.We discuss additional aspects of our model in Section 4 andconclude in Section 5.

We use friends-of-friends (fof) mass with a link lengthb = 0.2 to define the mass of a halo (roughly correspondingto spherical overdensity mass with 200 times the meanmatter density), since the fitting formulae for the halo massfunction are nearly universal with this mass definition (Shethet al. 2001; Jenkins et al. 2001; Warren et al. 2006; Tinker et al.2008). For simplicity, we neglect the slight difference betweenfof mass and virial mass (Appendix A; and see White 2002,for discussions on different mass definitions). Throughout the

paper, L always refers to the bolometric luminosity, and we usesubscripts B or X to denote B band or X-ray luminosity whenneeded. We adopt a flat CDM cosmology with 0 = 0.26, = 0.74, b = 0.044, h = 0.7, 8 = 0.78, and ns = 0.95.We use the Eisenstein & Hu (1999) transfer function to computethe linear power spectrum, and use the fitting formulae for haloabundance from Sheth & Tormen (1999) and for halo linear biasfrom Sheth et al. (2001) based on the ellipsoidal collapse modeland tested against numerical simulations.

1.1. Supermassive Black Hole Demographics

In the local universe, the dormant BH mass function (BHMF)can be estimated by combining scaling relations between BHmass and galaxy bulge properties, such as the M relation

(e.g., Gebhardt et al. 2000; Ferrarese & Merritt 2000; Tremaineet al. 2002) or the MLsph(Msph) relation (e.g., Magorrian et al.1998; Marconi & Hunt 2003), with bulge luminosity or stellarmass/velocity dispersion functions (e.g., Salucci et al. 1999;Yu & Tremaine 2002; Marconi et al. 2004; Shankar et al.2004; McLure & Dunlop 2004; Yu & Lu 2004, 2008; Tundoet al. 2007; Shankar et al. 2009b). Although some uncertaintiesexist on the usage of these scaling relations at both the highand low BH mass end (e.g., Lauer et al. 2007; Tundo et al.2007; Hu 2008; Greene et al. 2008; Graham 2008; Graham &Li 2009), the total BH mass density is estimated to be 4 105 M Mpc

3 with an uncertainty of a factor 1.5 (e.g.,Yu & Lu 2008; Shankar et al. 2009b), but the shape of the local

BHMF is more uncertain (see the discussion in Shankar et al.2009b).

It is now generally accepted that local SMBHs were onceluminous QSOs (Salpeter 1964; Zeldovich & Novikov 1964;Lynden-Bell 1969). An elegant argument tying the active QSOpopulation to the local dormant SMBH population was proposedby Soltan (1982): if SMBHs grow mainly through a luminousQSO phase, then the accreted luminosity density of QSOs toz = 0 should equal the local BH mass density:

,acc =

0

dt

dzdz

0

(1 )L

c2(L, z)dL , (1)

where (L, z) is the bolometric LF per L interval. Given theobserved QSO LF, a reasonably good match between ,acc and can be achieved if the average radiative efficiency 0.1(e.g., Yu & Tremaine 2002; Shankar et al. 2004; Marconi et al.2004; Hopkins et al. 2007; Shankar et al. 2009b). The exactvalue of , however, is subject to some uncertainties from theLF and local BH mass density determination.

Assuming that SMBH growth is through gas accretion, anextended version of the Soltan argument can be derived using

the continuity equation2

(cf. Small & Blandford 1992):

n(M, t)

t+

[n(M, t)M]

M= 0, (2)

where n(M, t) is the BHMF per dM, and M is the meanaccretion rate of BHs with mass (M, M + dM) averagedover both active and inactive BHs. Given the LF, and a modelconnecting luminosity to BH mass, one can derive the BHMFat all times by integrating the continuity equation (e.g., Marconiet al. 2004; Merloni 2004; Shankar et al. 2009b).

Since the local BHMF and the QSO LF together determinethe assembly history of the cosmic SMBH population, anycosmological model of AGN/quasars must first reproduce

the observed LF (e.g., Haiman & Loeb 1998; Kauffmann &Haehnelt 2000; Wyithe & Loeb 2002, 2003; Volonteri et al.2003; Lapi et al. 2006; Hopkins et al. 2008). This is also one ofthe central themes of this paper.

There has been great progress in the measurements of theQSO LF over wide redshift and luminosity ranges for the pastdecade, mostly in the optical band (e.g., Fan et al. 2001, 2004;Wolf et al. 2003; Croom et al. 2004; Richards et al. 2005,2006; Jiang et al. 2006; Fontanot et al. 2007), and the soft/hard X-ray band (e.g., Ueda et al. 2003; Hasinger et al. 2005;Silverman et al. 2005; Barger et al. 2005). While wide-fieldoptical surveys provide the best constraints on the bright end ofthe QSO LFs, deep X-ray surveys can probe the obscured faintend AGN population. Although it has been known for a while

that the spatial density of optically selected bright quasars peaksat redshift z 23, the spatial density of fainter AGNs selectedin the X-ray seems to peak at lower redshift (e.g., Steffen et al.2003; Ueda et al. 2003; Hasinger et al. 2005), a trend nowconfirmed in other observational bands as well (e.g., Bongiornoet al. 2007; Hopkins et al. 2007, and references therein)the

2 Note that here the source term, i.e., the creation of a BH with mass M bymechanisms other than gas accretion, is generally neglected (e.g., Small &Blandford 1992; Merloni 2004; Shankar et al. 2009b). One possible suchmechanism is BH coalescence, which modifies the shape of the BHMF butdoes not change the total BH mass density in the classical case (i.e., neglectingmass loss from gravitational radiation). Dry mergers at low redshift may act asa surrogate of shaping the BHMF.


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manifestation of the so-called downsizing of the cosmic SMBHassembly.

Hopkins et al. (2007) compiled QSO LF data from surveysin various bands (optical, X-ray, IR, etc.). Assuming a generalAGN spectral energy distribution (SED) and a column densitydistribution for obscuration, Hopkins et al. (2007) were ableto fit a universal bolometric LF based on these data. Weadopt their compiled bolometric LF data in our modeling.The advantage of using the bolometric LF data is that bothunobscured and obscured SMBH growth are counted in themodel, but we still suffer from the systematics of bolometriccorrections. The nominal statistical/systematic uncertainty levelof the bolometric LF is 20%30% (cf. Shankar et al. 2009b).

1.2. Quasar Clustering

A new ingredient in SMBH studies, due to dedicated large-scale optical surveys such as the 2QZ (Croom et al. 2004) andSloan Digital Sky Survey (SDSS; York et al. 2000), is quasarclustering. While quasar clustering studies can be traced backto more than two decades ago (e.g., Shaver 1984), statisticallysignificant results only came very recently (e.g., Porciani et al.2004; Croom et al. 2005; Porciani & Norberg 2006; Myers et al.

2006, 2007a, 2007b; Shen et al. 2007; da Angela et al. 2008;Padmanabhan et al. 2009; Shen et al. 2008b, 2009; Ross et al.2009).

Within the biased halo clustering picture (e.g., Kaiser 1984;Bardeen et al. 1986; Mo & White 1996), the observed QSO two-point correlation function implies that quasars live in massivedark matter halos and are biased tracers of the underlying darkmatter. For optically luminous quasars (Lbol 10

45 erg s1),the host halo mass inferred from clustering analysis is a fewtimes 1012 h1 M. Thus quasar clustering provides indepen-dent constraints on how SMBHs occupy dark matter halos,where the abundance and evolution of the latter population canbe well studied in analytical theories and numerical simulations(e.g., Press & Schechter1974; Bond et al. 1991; Lacey & Cole

1993; Mo & White 1996; Sheth & Tormen 1999; Sheth et al.2001; Springel et al. 2005b). By comparing the relative abun-dance of quasars and their host halos, one can constrain theaverage quasar duty cycles or lifetimes (e.g., Cole & Kaiser1989; Martini & Weinberg 2001; Haiman & Hui 2001) to be 108 yr, which means that bright quasars are short-lived.

More useful constraints on quasar models come from studiesof quasar clustering as a function of redshift and luminosity.The redshift evolution of quasar clustering constrains the evo-lution of duty cycles of active accretion, while the luminositydependence of quasar clustering puts constraints on quasar lightcurves (LC). Successful cosmological quasar models thereforenot only need to account for quasar abundances (LF), but alsoneed to explain quasar clustering properties, both as the function

of redshift and luminosity. We will use quasar clustering obser-vations as additional tests on our quasar models, as has beendone in recent studies (e.g., Hopkins et al. 2008; Shankar et al.2009a; Thacker et al. 2009; Bonoli et al. 2009; Croton 2009).

The luminosity dependence of quasar bias can be modeled asfollows. Assume at redshift z the probability distribution of hosthalo mass given bolometric luminosity L is p(M|L, z), then thehalo averaged bias factor is

bL(L, z) =

bM(M, z)p(M|L, z)dM , (3)

where bM(M, z) is the linear bias factor for halos with massM at redshift z. The derivation of the probability distribution

p(M|L, z) dP(M|L, z)/dM is described in later sections(Equation (20)).

1.3. Dark Matter Halo Mergers

In the hierarchical universe, small density perturbations growunder gravitational forces and collapse to form virialized halos(mostly consisting of dark matter). Smaller halos coalesce andmerge into larger ones. Early work on the merger history

of dark matter halos followed the extended PressSchechter(EPS) theory (e.g., Press & Schechter 1974; Bond et al. 1991;Lacey & Cole 1993), which is based on the spherical collapsemodel (Gunn & Gott 1972) with a constant barrier for thecollapse threshold (the critical linear overdensity c 1.68is independent of mass, albeit it depends slightly on cosmologyin the CDM universe). Although the (unconditional) halomass function n(M, z) predicted by the EPS theory agrees withnumerical N-body simulations reasonably well, it is well knownthat it overpredicts (underpredicts) the halo abundance at thelow (high) mass end (Sheth & Tormen 1999, and referencestherein). Motivated by the fact that halo collapses are generallytriaxial, Sheth & Tormen (1999), based on earlier work by Bond& Myers (1996), proposed the ellipsoidal collapse model with

a moving barrier where the collapse threshold also depends onmass. By imposing this mass dependence on a collapse barrier,the abundance of small halos is suppressed and fitting formulaefor the (unconditional) halo mass function are obtained, whichagree with N-body simulations much better than the sphericalEPS predictions (e.g., Sheth & Tormen 1999; Sheth et al. 2001;Jenkins et al. 2001; Warren et al. 2006; Tinker et al. 2008).

In addition to the unconditional halo mass function, theconditional mass function n(M1, z1|M0, z0) gives the massspectrum of progenitor halos at an earlier redshift z1 of adescendant haloM0 at redshiftz0. Thisconditional mass functionthus contains information about the merger history of individualhalos and can be used to generate halo merger trees. A simpleanalytical form for n(M1, z1|M0, z0) can be obtained in the

spherical EPS framework (e.g., Lacey & Cole 1993); forthe ellipsoidal collapse model, an exact but computationallyconsuming solution of the conditional mass function can beobtained by solving the integral equation proposed by Zhang& Hui (2006). Recently, Zhang et al. (2008) derived analyticalformulae for n(M1, z1|M0, z0) in the limit of small look-backtimes for the ellipsoidal collapse model.

Alternatively, the halo merger rate can be retrieved directlyfrom large volume, high resolution cosmological N-body simu-lations, which bypasses the inconsistencies between the spher-ical EPS theory and numerical simulations. Using the prod-uct of the Millennium Simulation (Springel et al. 2005b),Fakhouri & Ma (2008) quantified the mean halo merger rateper halo for a wide range of descendant (at z = 0) halo mass

1012 M0 1015 M, progenitor mass ratio 103 1,and redshift 0 z 6, and they provided an almost universalfitting function for the mean halo merger rate to an accuracy 20% within the numerical simulation results:

B(M0, , z)

n(M0, z)= 0.0289

M0

1.2 1012 M

0.083 2.01

exp

0.098

0.409dc, z

dz

0.371. (4)

Here B(M0, , z) is the instantaneous merger rate at redshift z,for halos with mass (M0, M0 + dM0) and with the progenitor


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mass ratio in the range (, + d ) [B(M0, , z) is in units of

the number of mergers Mpc3 M1 dz1d 1], n(M0, z) isthe

halo mass function (in units of Mpc3M1), M2/M1 1,

where M1 M2 are the masses of the two progenitors, andc,z = c/D(z) is the linear density threshold for sphericalcollapse with D(z) being the linear growth factor. We adoptEquation (4) to estimate the halo merger rate in our modeling.The mass definition used here is fof mass Mfof with a link length

b = 0.2. For the range of mass ratios that we are interestedin (i.e., major mergers), the halo merger rate derived from thespherical EPS model can overpredict the merger rate by up to afactor of 2 (Fakhouri & Ma 2008).

1.4. Galaxy (Subhalo) Mergers

Galaxies reside in the central region of dark matter haloswhere the potential well is deep and gas can cool to form stars.When a secondary halo merges with a host halo, it (along withits central galaxy) becomes a satellite within the virial radiusof the host halo. It will take a dynamical friction time for thesubhalo to sink to the center of the primary halo, where thetwo galaxies merge. The subsequent galaxy (subhalo) mergerrate is therefore different from the halo merger rate discussed

in Section 1.3, and a full treatment with all the dynamics (tidalstripping and gravitational shocking of subhalos) and baryonicphysics is rather complicated.

The simplest argument for the galaxy (subhalo) mergertimescale is given by dynamical friction (Chandrasekhar 1943;Binney & Tremaine 1987)

df fdforb

ln

Mhost

Msatdyn, (5)

where Mhost and Msat are the masses for the host and satel-lite halos, respectively, ln ln(Mhost/Msat) is the Coulomblogarithm, orb is a function of the orbital energy and an-gular momentum of the satellite, fdf is an adjustable param-

eter and dyn r/ Vc(r) is the halo dynamical timescale,usually estimated at the halo virial radius rvir. This formula(Equation (5)) is valid at the small satellite mass limitMhost/Msat 1, although itis alsousedin cases ofMhost Msatin many semi-analytical models (SAMs) with the replacement ofln = (1/2)ln[1+(Mhost/Msat)

2] (i.e., the original definition ofthe Coulomb logarithm) or ln = ln(1+ Mhost/Msat). However,in recent years deviations from the predictions by Equation (5)in numerical simulations have been reported for both theMsat Mhost and the Msat Mhost regimes (e.g., Taffoni et al.2003; Monaco et al. 2007; Boylan-Kolchin et al. 2008; Jianget al. 2008). In particular, even in the regime Msat/Mhost 1,where the original Chandrasekhar formula is supposed to work,Equation (5) substantially underestimates the merger timescale

from simulations by a factor of a few (getting worse at lowermass ratios). This is because Equation (5) is derived by treat-ing the satellite as a rigid body, while in reality the satellitehalo loses mass via tidal stripping3 and hence the duration ofdynamical friction is greatly extended.

Given the limitations of analytical treatments, we thereforeretreat to numerical simulation results. Fitting formulae forthe galaxy merger timescales within merged halos have beenprovided by several groups (Taffoni et al. 2003; Monaco et al.2007; Boylan-Kolchin et al. 2008; Jiang et al. 2008), and the

3 The galaxy associated with the subhalo, on the other hand, does not sufferfrom mass stripping, since it sits in the core region of the subhalo.

merger rate of subhalos has also been directly measured fromsimulations (Wetzel et al. 2009; Stewart et al. 2008).

The fitting formula formerger in Jiang et al. (2008), who usedhydro/N-body simulations, has the following form:

merger(, z) =0.940.6 + 0.6

2 0.43

1

ln[1 + (1/ )]

rvir

Vc, (6)

where is the circularity parameter ( = (1 e2

)1/2

, and eis the orbit eccentricity). In deriving this equation, Jiang et al.have removed the energy dependence of the orbit, i.e., they fixrc = rvir, where rc is thecircularorbit that hasthe same energy asthe satellites orbit. In their simulation, the ratio ofrc/rvir rangesfrom 0.6 to 1.5 and has a median value 1. The distributionof circularity in their simulation (see their Equation (7)) hasa median value 0.5, so in what follows we take = 0.5. Theredshift dependence ofmerger is thus the same as that of the halo

dynamical time dyn = rvir/Vc [vir(z)]1/2(1 + z)3/2 (see

Appendix A).Alternatively, the fitting formula ofmerger in Boylan-Kolchin

et al. (2008) is given by

merger(, z) = 0.216 exp(1.9) 1.3 ln[1 + (1/ )]

rc(E)rvir

dyn. (7)

Combining the halo merger rate (Equation (4)) and galaxy(subhalo) merger timescale (Equations (6) or (7)), we derivethe instantaneous merger rate of galaxies within a merged haloof mass M0 for the progenitor mass ratio and at redshift z:

Bgal(M0, , z) = B[M0, , ze(z, )]dze

dz, (8)

where ze(z, ) is a function of (z, ) and is determined by

tage(z) tage(ze) = merger(, ze), (9)

where tage(z) is the cosmic time at redshift z. Again, hereBgal(M0, , z) is the number of mergers per volume per halomass per mass ratio per redshift. We find that dze/dz is almostconstant at all redshifts for = 0.11 and monotonicallydecreases as increases. This constant can be approximatedby its asymptotic value at large z when (z) 1:

dze

dz {1 + 0.22[ ln(1 + 1/ )]1}2/3, (10)

for the fitting formula ofmerger in Jiang et al. (2008), or

dze

dz {1 + 0.09[ 1.3 ln(1 + 1/ )]1}2/3, (11)

for the fitting formula ofmerger in Boylan-Kolchin et al. (2008).Unfortunately, current studies on the galaxy merger timescale

have not converged yet, and different groups do report similarbut quantitatively different results, at least partly caused by thedifferent definitions and setups in their numerical simulations(e.g., hydro/N-body versus pure N-body simulations, halo find-ing algorithms and definition of mergers, etc.). These fitting for-mulae aregenerally good withina factorof 2 uncertainty. Thisuncertainty in the galaxy merger timescale within DM halos willlead to quite substantial differences in the galaxy merger rate athigh redshift. To see this, we show in Figure 1 the comparisonof different fitting formulae for merger and their consequences.


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Figure 1. Merger timescales and (sub)halomerger rates. Top: comparison of thesubhalo merger timescale from three dynamical friction prescriptions. Bottom:the redshift evolution of the halo merger rate (black solid lines), and the subhalomerger rate using the Boylan-Kolchin et al. (2008) (black dashed lines), andJiang et al. (2008; red dashed lines) prescriptions for the merger timescale, forthree (post)merger halo masses.

(A color version of this figure is available in the online journal.)

In the top panel of Figure 1, we show the ratio of merger/dynfor the most likely orbit with circularity = 0.5 and rc = rvirusing the fitting formula in Boylan-Kolchin et al. (2008; solidline), Jiang et al. (2008; dashed line), and a commonly usedSAM prescription: merger = 1.16

0.78[ ln(1 + 1/ )]1rc/rvir(dotted line). For the major merger regime 0.3, the fittingformula in Boylan-Kolchin et al. (2008) agrees with the SAMprescription well, and they both approach the dynamical time atthe high mass ratio end. However, the fitting formula in Jianget al. (2008) yields a factor of 2 longer than dynamical timeat the high mass ratio end.

In the bottom panel of Figure 1, we show the halo and galaxymajor merger rates per unit time (integrated over > 0.3), as afunction of redshift. The black solid lines show the halo majormerger rate, the black dashed lines show the galaxy merger rateusing the fitting formula of merger from Boylan-Kolchin et al.(2008), and the red dashed lines show the galaxy merger rateadopting the fitting formula from Jiang et al. (2008). In general,the galaxy merger rates fall below the halo merger rate at highredshift and take over at lower redshift. At redshift z > 4, thegalaxy merger rate using the Jiang et al. (2008) formula is lowerby almost an order of magnitude than that using the Boylan-Kolchin et al. (2008) formula (as well as the SAM prescription,since it agrees with Equation (7) for > 0.3), which makes itdifficult to produce the quasar abundance at high redshift (seelater sections).

It is worth noting that although the merger rate peaks at someredshift, this trend is hierarchical such that it peaks at later timefor more massive halos. This is somewhat in contradiction to thedownsizing of the QSO LF. This apparent discrepancy is likelycaused by the fact that at low redshift, it becomes progressivelymore difficult for the black hole to accrete efficiently in massivehalos because of the global deficit of cold gas due to previous

mergers experienced by massive halos and/or possible feedbackquenches (e.g., Kauffmann & Haehnelt 2000; Croton et al.2006). We will treat this fraction of the QSO-triggering mergerevent as a function of redshift and halo mass explicitly in ourmodeling.

If we choose to normalize the total merger rate B (Bgal) byn(M0, z), the abundance of descendant halos with mass M0, atthe same redshift z for halo mergers and galaxy mergers, thenBgal/n falls below B/ n at high redshift, then catches up andexceeds B/ n a little, and evolves more or less parallel to B/ nafterward. Therefore, the evolution in Bgal/n is shallower thanthat in B/ n, consistent with the findings by Wetzel et al. (2009)once the different definitions of B/ n and halo mass are takeninto account.

To summarize Sections 1.3 and 1.4, we have compared thehalo merger rate and galaxy merger rate as the functions ofredshift, (post)merger halo mass, and mass ratio with differentprescriptions for the dynamical friction timescale. It is, however,unclear on which rate we should link to the QSO-triggering rate.It is true that it will take a dynamical friction time for the twogalaxiesto merge after their host halo merged. Butthe black holeaccretion might be triggered very early during their first orbitalcrossing, well before the galaxies merge. In what follows, weadopt thehalo merger rate to model the QSO-triggeringrate, andwe discuss the consequences of adopting the alternative subhalomerger rates in Section 4.2.

1.5. BH Fueling During Mergers

Modeling black hole fueling during mergers from first prin-ciples is not trivial: aside from the lack of physical inputs,most hydrodynamical simulations do not yethave the dynamicalrange to even resolve the outer Bondi radius. Although semi-analytical treatments of BH accretion are possible (e.g., Granatoet al. 2004; Monaco et al. 2007), we do not consider suchtreatments in this paper because of the poorly understood accre-tion physics.

Throughout this paper, we adopt the following ansatz:

1. QSO activity is triggered by a gas-rich major merger event(min < 1).

2. The buildup of the cosmic SMBH population is mainlythrough gas accretion during the QSO phase.


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Table 1Notation and Model Parameters

Symbol Description Value/Units

Halo mass ratio 0 < 1M0 Postmerger halo mass

M Black hole mass

B, Bgal Halo(subhalo) merger rate Mpc3 M1 dz

1d 1

(L, z) QSO LF Mpc3dL1

BL QSO-triggering rate Mpc3dL1dz1

Lpeak QSO peak luminosity

z QSO-triggering redshift

fQSO(z, M0) QSO-triggering fraction

The LpeakM0 relation

Slope 5/3

C(z = 0) Normalization at z = 0 log(6 1045) 12

L Scatter 0.28 dex

C(z) = C(z = 0) + 1 log(1 + z) Evolution in normalization 1 = 0.2 = 1/3

The light-curve model

Radiative efficiency 0.1

l Eddington luminosity perM 1.26 1038 erg s1 M1

tSalpeter e-folding timec2

l(1 )0f Fraction of seed BH mass to peak BH mass 103

tpeak Time to reach the peak luminosity ( ln f)tSalpeter0 Eddington ratio before tpeak 3

Power-law slope of the decaying phase 2.5

The QSO-triggering Rate

min Minimum mass ratio 0.25

Mmin Exponential lower mass cut 3 1011 h1 M

Mmax(z) = Mquench(1 + z) Exponential upper mass cut 1012(1 + z)3/2 h1 M

Note. Parameter values are for the fiducial model.

3. The QSO light curve follows a universal form L(Lpeak, t),where the peak luminosity Lpeak is correlated with the massof the merged halo M0.

Once we have specified the QSO light-curve model and esti-mated the QSO-triggering rate from the major merger rate, wecan convolve them to derive the QSO LF. Within this evolution-ary QSO framework, we can derive the distributions of host halomasses and BH masses for any given instantaneous luminosityand redshift, allowing us to compare with observations of quasarclustering and Eddington ratio distributions. The details of thisframework are elaborated in Section 2.

We note that any correlations established with our model areonly for hosts where a gas-rich major merger is triggered andself-regulated BH growth occurs. Host halos which experiencemany minor mergers or dry mergers may deviate from theserelations. We will come back to this point in the discussionsection.

2. MODEL FORMALISM

In this section, we describe our model setup in detail. Somenotations and model parameters are summarized in Table 1 forclarity.

2.1. Determining the Luminosity Function

Major mergers are generally defined as events with the massratio min 0.3 1, but our framework can be easilygeneralized to other values of min. Because it takes more than just a major merger event to trigger QSO activity, we shallmodel the QSO-triggering rate as a redshift and mass-dependent

fraction of the halo merger rate. The QSO LF at a given redshiftz can be obtained by combining the triggering rate and theevolution of QSO luminosity4:

(L, z)dL =z

BL[Lpeak(L, tz tz ), z]dLpeakdz

, (12)

where(L, z)dL is the QSO number density in the luminosityrange (L, L + dL) at redshift z, BL(Lpeak, z

) is the QSO-triggering rate (merger number density per redshift per peakluminosity) at redshift z with peak bolometric luminosityLpeak(L, tz tz ), which is determined by the specific formof the light curve, and tz and tz are the cosmic time at z andz. In integrating this equation, we impose an upper limit ofredshift zmax = 20, but we found that the integral convergeswell below this redshift, since the merger rate decays rapidlyat high redshift. Equation (12) implies that the distribution oftriggering redshift z given L at redshift z is

dP(Lpeak, z|L, z)

dz = p(Lpeak, z

|L, z) BL(Lpeak, z)

dLpeak

dL,

(13)where again Lpeak is tied to L via the light-curve model.Obviously only quasars with Lpeak L can contribute to(L, z).

4 Note that we have implicitly assumed that a second QSO-triggering mergerevent does not occur during the major accretion phase of the QSOareasonable assumption since statistically speaking bright QSOs are short-lived(tQSO tH). Observationally, binary/multiple quasars within a single halowith comparable luminosities are rare occurrences (fbinary 0.1%, e.g.,Hennawi et al. 2006, 2009; Myers et al. 2008), further supporting thisassumption.


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The QSO-triggering rate BL(Lpeak, z) is related to the halomerger rate B(M0, , z) by

BL(Lpeak, z)dLpeak = fQSO(z, M0)

1min

B(M0, , z)ddM0,

(14)where we parameterize the fraction fQSO as

fQSO(z, M0) = F(z) exp

Mmin(z)

M0

M0

Mmax(z)

, (15)

where F(z) describes how fQSO(z, M0) decreases toward lowerredshift, i.e., theoverall reductiondue to theconsumption of coldgas. We also introduce exponential cutoffs of fQSO at both highand low mass such that at each redshift halos with too small M0cannot trigger QSO activity; on the other hand, overly massivehalos cannot cool gas efficiently and BH growth halts, and thegas-rich fraction may become progressively smaller at higherhalo masses (especially at low redshift). We discuss choices forthese parameters later in Section 3.

To proceed further,we must specify the relation betweenLpeakand M0, and choose a light-curve model. We assume that theLpeakM0 correlation is a power law with lognormal scatter, as

motivated by some analytical arguments and hydrodynamicalsimulations (e.g., Wyithe & Loeb 2002, 2003; Springel et al.2005b; Hopkins et al. 2005; Lidz et al. 2006):

dP(Lpeak|M0)

dlog Lpeak= p(Lpeak|M0) = (2

2L)

1/2

exp

[log Lpeak (C + log M0)]

2

22L

,

(16)

where the mean relation islog

Lpeak

erg s1

= C + log

M0

h1 M

, (17)

with normalization Cand the power-law slope . The log scatteraround this mean relation is denoted as L. When the scatterbetween Lpeak and M0 is incorporated, Equation (14) becomes

BL(Lpeak, z) =fQSO

1min

B(M0, , z)d

M0

Lpeakp(Lpeak|M0)dlog M0,

(18)

from which we obtain the probability distribution of postmergerhalo mass M0 at fixed peak luminosity Lpeak and redshift z:

dP(M0|Lpeak, z)

dlog M0= p(M0|Lpeak, z)

fQSO

1min

B(M0, , z)d

M0

Lpeakp(Lpeak|M0).

(19)

We derive the probability distribution of host halo mass at giveninstantaneous luminosity and redshift, p(M0|L, z), as

dP(M0|L, z)

dlog M0= p(M0|L, z)

=

dP(M0|Lpeak, z

)

dlog M0

dP(Lpeak, z|L, z)

dz dz ,

(20)

which can be convolved with the halo linear bias bM(M0, z) toderive the QSO linear bias bL(L, z) at instantaneous luminosityL and redshift z.

Since we have assumed that a second QSO-triggering mergerevent has not occurred, at redshift z the postmerger halo M0should maintain most of its identity as it formed sometimeearlier, i.e., we neglect mass added to the halo via minor mergersor accretion of diffuse dark matter between the halo formationtime and the time when the QSO is observed at redshift z.

Given the halo mass distribution (20), we can further derivethe active halo mass function hosting a QSO with L > Lmin:

dM0

dlog M0=

Lmin

(L, z)dLdP(M0|L, z)

dlog M0, (21)

which can be used to compute the halo duty cycles.

2.2. The Light-Curve Model

For the light-curve model, there are several common choicesin merger-based cosmological QSO models (e.g., Kauffmann& Haehnelt 2000; Wyithe & Loeb 2002, 2003): (1) a lightbulb model in which the QSO shines at a constant value

Lpeak for a fixed time tQSO; (2) an exponential decay modelL = Lpeak exp(t/tQSO); (3) a power-law decay model L =Lpeak(1 + t/tQSO)

with > 0. Once a light-curve model is

chosen, the total accreted mass during the QSO phase is simply5

M,relic =1

c2

0

L(Lpeak, t)dt. (22)

The evolution of the luminosity Eddington ratio L/LEddis (neglecting the seed BH mass)

(t) =L(Lpeak, t)

lM(t)=

L(Lpeak, t)l(1)

c2 t

0L(Lpeak, t

)dt, (23)

where l 1.26 1038 erg s1 M1 is the Eddington luminosityperM.

Unfortunately, none of the three LC models is self-consistentwithout having 1 at the very beginning of BH growth,simply because they neglect the rising part of the light curve.To remedy this, we must model the evolution of luminosity andBH growth self-consistently (e.g., Small & Blandford 1992; Yu& Lu 2004, 2008). We consider a general form of light curvewhere the BH first grows exponentially at a constant luminosityEddington ratio 0 (e.g., Salpeter1964) to Lpeak at t = tpeak, andthen the luminosity decays monotonically as a power law (e.g.,Yu & Lu 2008):

L(Lpeak, t) =

Lpeak exp

l(1 )0

c2(t tpeak)

, 0 t tpeak

Lpeak

t

tpeak

, t tpeak,

(24)

5 Throughout the paper, we assume constant radiative efficiency . At the verylate stage of evolution or under certain circumstance (i.e., hot gas accretionwithin a massive halo), a SMBH may accrete via radiatively inefficientaccretion flows (RIAFs) with very low and the mass accretion rate (e.g.,Narayan & Yi 1995). We neglect this possible accretion state since the massaccreted during this state is most likely negligible (e.g., Hopkins et al. 2006).


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96 SHEN Vol. 704

where tpeak is determined by

Lpeak = l0M,0 exp

l(1 )0

c2tpeak

, (25)

where M,0 is the seed BH mass at the triggering time t = 0.In Equation (24), determines how rapidly the LC decays.Larger values lead to more rapid decay. Thus, this modelaccommodates a broad range of possible light curves. With thislight-curve model (24) we have

M(Lpeak, t) =

Lpeak

l0exp

l(1 )0

c 2(t tpeak)

, 0 t tpeak

Lpeak

l0+

(1 )Lpeaktpeak

c2

1

1

t

tpeak

1 1

, t tpeak,

(26)

where we require > 1 so that a BH cannot grow infinite mass.The evolution of the luminosity Eddington ratio is therefore

(t) =

0, 0 t tpeak

(t/tpeak)

10

+l(1)tpeakc2(1)

t

tpeak

1 1

, t tpeak, (27)

where during the decaying part of the LC the Eddington ratiomonotonically decreases to zero. We assume that the seed BHmass is a fraction f of the peak mass M(tpeak) Lpeak/(l0),and we have

tpeak

= c2

l(1 )0ln f = ( ln f)t

Salpeter. (28)

In what follows, we set f = 103. Thus, the seed BH isnegligible compared to the total mass accreted, and tpeak 6.9tSalpeter, where tSalpeter is the (e-folding) Salpeter timescale(Salpeter 1964). Although 0 = 1 is the formal definition ofEddington-limited accretion, it assumes the electron scatteringcross section, and in practice super-Eddington accretion cannotbe ruled out (e.g., Begelman 2002). So we consider 0 withinthe range log 0 [1, 1].

The relic BH mass, given Equation (26), is simply

M,relic =Lpeak

l0+

(1 )Lpeaktpeak

( 1)c2=

Lpeak

l0

1

ln f

1

.

(29)If instead we choose an exponential model for the decaying partof the LC (e.g., Kauffmann & Haehnelt 2000), L(t > tpeak) =Lpeak exp[(t tpeak)/tQSO], then the relic mass becomes

M,relic =Lpeak

l0+

(1 )LpeaktQSO

c2. (30)

Therefore, LC models with large values of mimic theexponentially decaying model, and we do not consider theexponential LC model further.

If the decaying parameter is independent of mass, thenEquations (26) and (29) imply M,relic Lpeak. In other words,

Figure 2. Example light curves for = 0.1 and 0 = 3.0, for = 1.5, 2.5, 3.5.Larger values of lead to more rapid decay. The time that the QSO spendsabove half of its peak luminosity is short, < 100 Myr for large values of.

the scalings between Lpeak and M0, and between M,relic and M0,should be the same for QSO-triggering merger remnants (e.g.,

early-type galaxies). Some theoretical models and simulationspredict Lpeak M

4/30 (e.g., King 2003; Di Matteo et al. 2005;

Springel et al. 2005a), where momentum is conserved in self-regulated feedback, while others invoking energy conservation

predicts Lpeak M5/3

0 (e.g., Silk & Rees 1998; Wyithe &Loeb 2003). Two recent determinations of the local black holemasshalo mass relation gave M,relic M

1.650 (Ferrarese 2002)

and M,relic M1.270 (Baes et al. 2003). Within uncertainties

M,relic Lpeak seems to be a reasonable scaling; therefore,we do not consider further mass dependence of . Given thedefinition of tpeak (Equation (28)), our light-curve model (24)is then universal in the sense that it scales with Lpeak in a self-similar fashion.

As an example, Figure 2 shows three light curves with = 1.5, 2.5, 3.5 for = 0.1 and 0 = 3.0, in which casetpeak 115 Myr. The time that a QSO spends above 50% ofits peak luminosity is typically 100 Myr for sharp decayingcurves, but it can be substantially longer for extended decayingcurves.

2.3. BH Mass and Eddington Ratio Distributions

Now we have specified the light-curve model and the peakluminosityhalo mass relation, and tied the LF (L, z) tothe QSO-triggering merger rate. We continue to derive theinstantaneous BH mass and Eddington ratio distributions atfixed instantaneous luminosity L and redshift z, which can be


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compared with observationally determined distributions (e.g.,Babic et al. 2007; Kollmeier et al. 2006; Shen et al. 2008a;Gavignaud et al. 2008).

Suppose that at redshift z we observe quasars with instanta-neous luminosity L. These quasars consist of objects triggeredat different earlier redshift z and are at different stages of theirevolution when witnessed at z (e.g., Equation (12)). There is acharacteristic earlier redshift zc determined by

tage(z) tage(zc) = tpeak, (31)

where tage is the cosmic time. Quasars triggered between [zc, z]are all in the rising part of the LC, while quasars triggeredearlier than zc are all in the decaying part of the LC. In orderto contribute to (L, z), a quasar should have peak luminos-ity Lpeak = L if it is triggered right at zc, and it should haveLpeak > L otherwise. Therefore, the triggering redshift dis-

tribution dP(Lpeak, z|L, z)/dz (Equation (13)) peaks around6

z zc, since BL(Lpeak, z) decreases rapidly when Lpeak in-

creases. Moreover, at the observing redshift z, all the contribut-ing quasars triggered between [zc, z] will have the Eddingtonratio L,zz = 0 and BH mass M,L,zz = L/(l0), while allthe contributing quasars triggered before zc will have the Ed-

dington ratio L,z z < 0 and BH mass M,L,z z > L/(l0).The probability distribution of instantaneous BH mass M at theobserving redshift z and luminosity L is therefore

dP(M|L, z)

dlog M=

dPdz

dz

dlog M

1 z

zc

dPdz

dz , M

L

l0, (32)

where dP(z|L, z)/dz is given by Equation (13) and dz /dM isdetermined by the decaying halfof the LC (Equation (27)), i.e.,the triggering redshift z( zc) is a monotonically increasingfunction of M (note that here M refers to the instantaneousBH mass observed at z). The denominator in the above equationis to normalize the distributionwe literally augment thedistribution with the pileup of objects with M = L/(l0)

triggered within [zc, z]. The fraction of such objects is about20% for L 1047 erg s1 and becomes negligible at lowerluminosities. This procedure removes the spike ( function) atthe constant BH mass M = L/(l0) in the distribution, whichis an artifact of our model.7 Equation (32) gives the equivalentdistribution of Eddington ratios at z and at instantaneousluminosity L.

Given the LF (Equation (12)) and the distribution of M(Equation (32)), we can determine the active BHMF (i.e., themass function of active QSOs above some luminosity cut Lmin)as

dM

dlog M=

Lmin

(L, z)dLdP(M|L, z)

dlog M. (33)

3. THE REFERENCE MODEL

We are now ready to convolve the LC model with the QSO-triggering rate (Equation (18)) to predict the LF (L, z) usingEquation (12). Before we continue, let us review the modelparameters and available observational/theoretical constraints.

6 Except for low luminosities when the lower halo mass cut Mmin in theQSO-triggering rate starts to kick in, the distribution dP(Lpeak, z

|L, z)/dz

instead has a dip around zc.7 We have tested with the alternative BH mass distribution at instantaneousluminosity L and redshift z where the contribution of objects with

M = L/(l0) is described by a function with normalizationz

zcdPdz

dz , and

we find that other derived distributions based on this BH mass distribution arealmost identical to those using Equation (32).

1. The QSO-triggering rate (min, Mmin(z), Mmax(z), F(z)).We choose our fiducial value of min 0.3 followingthe traditional definition of major mergers (1:3 or 1:4).For the minimum halo mass Mmin below which efficientaccretion onto the BH is hampered, we simply chooseMmin(z) = 3 10

11 h1 M. We model the maximum halomass cut above which the gas-rich merger fraction drops asa function of redshift:

Mmax(z) = Mquench(1 + z) , (34)

with > 0. Therefore, at lower redshift halos have asmaller upper threshold. Since at z < 0.5 rich groupto cluster size halos no longer host luminous QSOs, wetentatively set Mquench = 1 10

12 h1 M. The parametersMmin and Mmax control the shape of the LF at both the faintand bright luminosity ends. The global gas-rich mergerfraction, F(z), is more challenging to determine. Directobservationsof thecold gas fraction as a function of redshiftand stellar mass (and therefore halo mass) are still limitedby large uncertainties. Moreover, how gas-rich a mergerneeds to be in order to trigger efficient BH accretion is not

well constrained. For these reasons, we simply modelF

(z)as a two-piece function such that F(z) = 1 when z > 2,the peak of the bright quasar population, and F(z) linearlydecreases to F0 at z = 0. We note again that fQSO(z, M0)should be regarded as the fraction of major mergers thattrigger QSO activity.

2. The LpeakM0 relation (C, , L). Some merger eventsimulations (e.g., Springel et al. 2005a; Hopkins et al.2005; Lidz et al. 2006) reveal a correlation between thepeak luminosity and the mass of the (postmerger) halo:Lpeak 3 10

45(M0/1012 h1 M)

4/3 erg s1 at z = 2,with a lognormal scatter L = 0.35 dex. Other analyticalarguments predict = 5/3, where energy is conservedduring BH feedback (e.g., Silk & Rees 1998; Wyithe &

Loeb 2003). We choose values between [4/3, 5/3], sincethe above simulations are also consistent with the = 5/3slope. Furthermore, we allow the normalization parameterCto evolve with redshift, C(z) = C(z = 0)+log[(1+ z)1 ],with 1 > 0. Thus, for the same peak luminosity, the hosthalos become less massive at higher redshift. This decreaseof the characteristic halo mass with redshift is also modeledby several other authors, although we do not restrict to1 = 1 (e.g., Lapi et al. 2006), or more complicatedprescriptions (e.g., Wyithe & Loeb 2003; Croton 2009),since the form of the LM0 (or MM0) relation is alsodifferent in various prescriptions. A smaller characteristichalo mass is easier to account for the QSO abundance athigh redshift, since there are more mergers of smaller halos,

but it also reduces the clustering strength. The intrinsicscatter of the LpeakM0 relation, L, will have effects onboth the LF and clustering. Larger values of L lead tohigher QSO counts, but will also dilute the clusteringstrength due to the up-scattering of lower mass halos (e.g.,White et al. 2008). The LpeakM0 relation establishes abaseline for the mapping from halos to SMBHs. Althoughin our fiducial model we adopt the above rather simplescaling [ (1 + z)1 ] for the redshift evolution in C(z),we will discuss alternative parameterizations for C(z) inSections 4.2 and 4.4.

3. The light-curve model (0, , tpeak, ). Our chosen fidu-cial value for tpeak is tpeak = 6.9 tSalpeter (f = 10

3), but


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our results are insensitive to the exact value of f. The ra-diative efficiency is 0.1 from the Soltan argument(Equation (1)). We choose 0 between [0.1, 10] and > 1.1.

Normally, to find the best-model parameters one needs to per-forma 2 minimization between modelpredictions and observa-tions. We do notperform such exercisehere because it is difficultto assign relative weights to different sets of observations (i.e.,

LF, quasar clustering, Eddington ratio distributions, etc.), andwe do not know well enough the systematics involved. Instead,we experiment with varying the model parameters within rea-sonable ranges to achieve a global good (as judged by eye)fit to the overall observations. Our fiducial model has the fol-lowing parameter values: min = 0.25, F0 = 1.0, = 1.5, = 5/3, C(z = 0) = log(6 1045) 12, 1 = 0.2,L = 0.28, 0 = 3.0, and = 2.5, which are all within reason-able ranges. In particular, we found that the parameterization ofF(z) is unnecessary because the effect of cold gas consumptionis somewhat described by Mmax(z) already. Below we describein detail the predictions of this reference model and comparisonwith observations, and we defer the model variants and caveatsto Section 4.

Figure 3 shows the model LF d/dlog L L ln(10)(L, z)in solid lines, given by Equation (12), where we overplot thecompiled bolometric LF data in Hopkins et al. (2007). In com-puting Equation (12), we integrate up to z = 20 but theintegral converges well before that. The model underpredictsthe counts at the faint luminosity end (L < 1045 erg s1) forz < 0.5, leaving room for low-luminosity AGNs triggered bymechanisms other than a major merger event (i.e., by secu-lar processes). At redshift z 4.5, the model also underes-timates the QSO abundance. This could be alleviated if thecharacteristic halo mass shifts to even lower values, i.e., evenlarger values of the parameter C at z 4.5 (see discussions inSections 4.2 and 4.4). Alternatively, the high-z SMBH pop-ulation may be different in the sense that it is not tied to

0.3 major merger events directly. Also, the halo merger rate(Equation (4)) has been extrapolated to such high redshiftsfor the massive halos considered here. Despite these facts,the model correctly reproduces the LF from z 0.5 toz 4.5. The turnover below L 1044 erg s1 at z 2.5is caused by the lower mass cutoff Mmin = 3 10

11 h1 M.At lower redshift, this turnover flattens due to the grad-ual pileup of evolved high-peak luminosity quasars wellaftertpeak.

We show some of the predicted distributions in Figure 4 forthis reference model. The upper panels show the distributionsof the triggering redshift z for instantaneous luminositiesL = 1045, 1046, 1047 erg s1 at z = 1 (upper left) and z = 2(upper right). As expected, the distribution of the triggering

redshift peaks around the characteristic redshift zc given byEquation (31). The bottom panels show the distributions ofhost halo mass M0 for the three instantaneous luminositiesat the two redshifts. For bright quasars (L > 1046 erg s1),the distributions of M0 are roughly lognormal, with the widthand mean determined mainly through the LpeakM0 relationbut also slightly modified through the convolutions with otherdistributions (see Equations (13), (19), and (20)). For faintQSOs (L 1045 erg s1), however, the halo mass distributionhas a broad high-mass tail contributed by evolved high-peakluminosity quasars triggered earlier. This contamination ofmassive halos hosting low luminosity QSOs will increase theclustering bias of the faint quasar population.

Figure 5 shows our model predictions for the redshift andluminosity dependence of quasar bias, compared with obser-vations. The model-predicted redshift evolution of quasar biasis in good agreement with observations (e.g., Porciani et al.2004; Croom et al. 2005; Porciani & Norberg 2006; Myers et al.

2006, 2007a, 2007b; Shen et al. 2007; da Angela et al. 2008;Padmanabhan et al. 2009; Shen et al. 2008b, 2009; Ross et al.2009). But it has some difficulties in accounting for the large

bias at z 4, resulting from the need to reproduce the quasarabundance at such high redshift (i.e., the characteristic halomass shifts to lower values). We discuss possible solutions tothis problem in Section 4.2. Our model also predicts the lu-minosity dependence of quasar clustering. At low luminosities,quasar bias depends weakly on luminosity, which is consis-tent with observations (e.g., Porciani & Norberg 2006; Myers

et al. 2007a; da Angela et al. 2008; Shen et al. 2009). This sim-ply reflects the fact that quasars are not light bulbs, i.e., thereis non-negligible scatter around the mean LpeakM0 relation,and some faint quasars live in massive halos because of theevolving light curve (e.g., Adelberger & Steidel 2005; Hopkinset al. 2005; Lidz et al. 2006). On the other hand, the modelpredicts that the bias increases rapidly toward high luminosi-

ties and/or at high redshift. This is consistent with the findingsby Shen et al. (2009) that the most luminous quasars clustermore strongly than intermediate luminosity quasars at z < 3.Our model-predicted luminosity-dependent quasar clusteringbroadly agrees with the measurements in Shen et al. (2009)for 0.4 < z < 2.5, but we caution that their measurements aredone for samples with a broad redshift and luminosity range inorder to build up statistics (e.g., see their Figure 2); hence, adirect comparison is somewhat difficult. At redshift 3, Adel-berger & Steidel (2005) and Francke et al. (2008) measured theclustering of low-luminosity AGNs using cross-correlation withgalaxies. Their data are shown in Figure 5 with the open squareand triangles for the measurements in Francke et al. (2008) andAdelberger & Steidel (2005), respectively, which are broadly

consistent with our predictions. However, the clustering of opti-cally bright quasars is apparently at odds with the low bias valuederived from the AGNgalaxy cross-correlation in Adelberger& Steidel (2005), likely caused by the fact that the latter AGNsample spans a wide redshift range 1.6 z 3.7 and luminos-ity range, and that the uncertainty in their bias determination islarge.

Figure 6 shows our model predictions for the Eddington ratiodistributions at various redshifts and instantaneous luminosities.Aside from the cutoff at 0 = 3 (our model setup), theseEddington ratio distributions are approximately lognormal, andbroaden toward fainter luminosities. This again reflects thenature of the light curveat lower luminosities, there are moreobjects at their late evolutionary stages and shining at lower

Eddington ratios. These predictions are in good agreement withthe observed Eddington ratio distributions for bright quasars(L 1046 erg s1), where the distribution is narrow and peaksat high mean values (log [1, 0]) (e.g., Woo & Urry 2002;Kollmeier et al. 2006; Shen et al. 2008a), as well as for faintAGNs (L 1045 erg s1) where the distribution is broader andpeaks at lower mean values (log [3, 1]) (e.g., Babicet al. 2007; Gavignaud et al. 2008). However, the predictedmean values and widths are not necessarily exactly the same asthose determined from observations, since uncertainties in theBH mass estimators used in these observations may introduceadditional scatter and biases in the observed Eddington ratiodistributions (e.g., Shen et al. 2008a). The Eddington ratio


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Figure 3. Predicted bolometric LFs (L, z) d/dlog L in our fiducial model (solid lines). Overplotted are the complied bolometric LF data from Hopkins et al.(2007) for optical (black circles), soft X-ray (green squares), and hard X-ray (red triangles) samples. The dashed lines are the predicted LF for a model where thenormalization in the mean LpeakM0 relation is increased by an additional amount at z > 3.5, tuned to fit the LF at z 4.5 (see Section 4.2 for details). The bluedotted lines are the predictions based on an alternative prescription for the redshift evolution in the normalization of the LpeakM0 relation as discussed in Section 4.4.


distributions of faint AGNs (L 1044 erg s1) are particularlybroad at low redshift, since more high-Lpeak objects have hadenough time to evolve. On the other hand, at high redshift z 2,the minimum allowable Eddington ratio is set by the cosmic ageat that redshift, e.g., min 10

4 at z = 2 using Equation (27).This explains the narrowing of the lower end of the Eddingtonratio distribution for L = 1044 erg s1, seen in the right panelof Figure 6.

Since we have reproduced the observed LF and predictedthe Eddington ratio distributions as a function of luminosity,we can use Equation (33) to derive the active BHMF in QSOs

above some minimum luminosity. In Figure 7, we show thetotal BHMF (setting Lmin = 0 in Equation (33)) assembledat several redshifts, where the gray shaded region indicatesthe estimates of the local dormant BHMF based on variousgalaxy bulgeBH scaling relations (Shankar et al. 2009b). Ourmodel prediction for the local BHMF is incomplete at the low-mass end, mainly because our model does not include low-luminosity AGN activity (presumably linked to M 10

7 MBH growth) possibly triggered by secular processes (see furtherdiscussion in Section 4.2), and also because we set the minimalhalo mass that can trigger QSO activity during major mergers


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Figure 4. Distributions of triggering redshift (upper panels) and halo mass (lower panels) in our fiducial model, for three instantaneous luminosities and two values ofobserving redshifts. The distributions of triggering redshift generally peak around zc (marked by arrows) given by Equation (31). For lower luminosity, the distributionof the triggering redshift is more extended. As a consequence, there is a significant population of faded low-luminosity QSOs which have massive hosts.

Mmin

= 3 1011 h1 M

in order to reproduce the faint-end LFat high redshift. At the high-mass end, the predicted slope in thetotal BHMF broadly agrees with (although is shallower than)that for the local dormant BHMF. The majority of the presentday 108.5 M BHs were already in place by z = 1, but lessthan 50% of them were assembled by z = 2. This is somewhatin disagreement with McLure & Dunlop (2004), who claimedthat the majority of > 108.5 M SMBHs are already in place atz 2 based on virial BH mass estimates of optically selectedbright quasars. We suspect this discrepancy is caused by: (1)the fact that virial BH mass estimates tend to systematicallyoverestimate the true BH masses due to a Malmquist-type bias(see discussions in Shen et al. 2008a); (2) the high-mass end

slope in the local dormant BHMF is steeper than our modelpredictions.On theotherhand, ourmodelpredictions arein goodagreement with the predictions by Shankar et al. (2009b) forthe high-mass end. It is possible to make our model predictionagree with the local BHMF at the high-mass end by imposingsome cutoff in the light curve (Equation (24)) such that BHscannot grow too massive, but a more accurate observationaldetermination of the high-massend of thelocal BHMF is neededto resolve these issues.

Figure 8 shows the halo duty cycles, defined as the ratio of thenumber density of active halos hosting QSOs brighter than Lminto that of all halos, as a function of halo mass. We have used theSheth et al. (2001) halo mass function and Equation (21) for the


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Figure 5. Predictions for the linear bias of quasar clustering in our fiducialmodel. Top: bias evolution with redshift. The data points are from themeasurements in Shen et al. (2009) and three lines show the predicted biasfor quasars with instantaneous luminosity L = 1045, 1046, 1047 erg s1.The median luminosity of quasars used in the clustering analysis for thesix redshift bins are: log(L/erg s1) = 45.6, 46.3, 46.6, 46.9, 46.9, 47.1.Bottom: predicted luminosity dependence of quasar bias. The filled circlesare from the measurements in Shen et al. (2009) for the 10% most luminousand the remainder of a sample of quasars spanning 0.4 < z < 2.5. The open

circles are the measurement of quasar clustering for 2.9 < z < 3.5 in Shen et al.(2009). The open square is from Francke et al. (2008) using cross-correlationof X-ray-selected AGNs with high redshift galaxies, and the open triangles arefrom Adelberger & Steidel (2005) using cross-correlation of optical AGNs withgalaxies. Note that for the quasar clustering data, we plot biases derived fromboth with and without including negative correlation function data in the fitting(see Table 1 of Shen et al. 2009).


active halo mass function. For quasars (L > 1045 erg s1) andfor typical halo mass M0 2 10

12 h1 M, the duty cycle is 0.15, 0.1, 0.03, 0.01 at z = 3, 2, 1, 0.5.

We can also compare the model predicted BH mass distribu-tions for quasars within certain luminosity ranges with the virial

BH mass estimates. For this purpose, we use the virial BH massestimates from Shen et al. (2008a) for the SDSS DR5 quasarcatalog (Schneider et al. 2007). These optical quasars are flux-limited to i = 19.1 at z 3, and we have used Equation (1)in Shen et al. (2009) to convert i-band magnitude to bolometricluminosity. For quasars/AGNs where the SMBHs are still ac-tively accreting, what we observe is the instantaneous BH massrather than the relic BH mass. In Figure 9, the solid lines showthe model predictions for the BH mass distributions of SDSSquasars at z = 1 and z = 2, weighted by the LF; the dashedlines show the distributions based on virial BH masses. It isremarkable that not only the distributions of our model predic-tions are broader, but also the peaks are shifted to lower masses

( 0.6 dex) compared with those from virial mass estimates,as already discussed extensively in Shen et al. (2008a, i.e., theMalmquist-type bias in virial mass estimates).

4. DISCUSSION

4.1. Comparison with Previous Work

The merger basis of our framework, as advocated by many

authors (e.g., Kauffmann & Haehnelt 2000; Wyithe & Loeb2002, 2003; Volonteri et al. 2003), provides a physical originfor the QSO population, and distinguishes the current studyfrom other works which focus on BH growth using the QSO LFas an input (e.g., Yu & Tremaine 2002; Yu & Lu 2004, 2008;Marconi et al. 2004; Merloni 2004; Shankar et al. 2004, 2009b).Our simple framework, although semi-analytical in nature,accommodates a wide range of updated and new observations ofQSO statistics; these new observations include quasar clusteringand Eddington ratio distributions. Most of the early quasarmodels (e.g., Haiman & Loeb 1998; Kauffmann & Haehnelt2000; Wyithe & Loeb 2002, 2003; Volonteri et al. 2003) focusedmainly on the LF (partly because other observations were notavailable at that time), and most of them assumed simplified

light-curve models whichwere unable to reproduce the observedEddington ratio distributions.Hopkins et al. (2008) presented a merger-based quasar model

that utilizes a variety of quasar observations for comparison,including the latest clustering measurements. Our model frame-work is different from theirs in two major aspects: (1) theyestimated the major merger rate from the combination of empir-ical halo occupation models of galaxies and merging timescaleanalysis, while we used directly the halo merger rate fromsimulationsboth approaches have their own advantages anddisadvantages; (2)the light curvein their model is extracted fromtheir merger-event simulations (Hopkins et al. 2005), while wehave adopted a parameterization of the light curve which is fitby observations. Their light-curve model is clearly more physi-

cally motivated than ours, yet it needs to be confirmed in futuresimulations with higher resolution and better understandingsof BH accretion physics. On the other hand, the virtue of ourframework is that it allows fast and easy estimations of modelpredictions and parameter adjustments to fit updated observa-tions.

4.2. Caveats in the Reference Model

As already mentioned in Section 3, our fiducial model isnot an actual 2 fit to the overall observations (LF, clustering,and Eddington ratio distributions) because of the ambiguityof assigning relative weights to individual observational datasets. Instead, we have experimented with varying the modelparameters within reasonable ranges, to achieve a global good

(as judged by eye) fit to observations. If consider only the LF asobservational constraints, there are model degeneracies betweenthe luminosity decaying rate andthe normalization of themeanLpeakM0 relation C, i.e., if QSOluminosity decaysmore slowly,the typical host halos need to shift to more massive (and lessabundant) halos in order not to overpredict the LF; likewise, ifthe scatter around the mean LpeakM0 relation L increases, wecan reduce Cto match the LF. However, these degeneracies arebroken once the clustering observations are taken into account.A large scatterL or slow varying light curve (small values of)cannot fit the large bias at high redshift and the luminositydependence of clustering. Our model is also more complicatedthan previous models (e.g., Haiman & Loeb 1998; Kauffmann


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Figure 6. Predictions for the Eddington ratio distributions in our fiducial model for three different redshifts. The distribution is narrower and has higher peak valuestoward higher luminosities.These distributions are in goodagreementwith the observed distributions for bothbright quasars and low-luminosity AGNs (e.g., Kollmeieret al. 2006; Babic et al. 2007; Shen et al. 2008a; Gavignaud et al. 2008).

Figure 7. BHMFs assembled at various redshifts in our fiducial model. Thegray shaded region shows the estimates for the local BHMF from Shankaret al. (2009b). The red line shows the prediction for the local BHMF, whichis incomplete at M 107.5 M by a factor of a few because we did notinclude contributions from AGNs triggered by secular processes or minormergers (as reflected in the failure to reproduce the LF at the low-luminosity

end L < 1045 erg s1 at z < 0.5, see Figure 3). The yellow and green linesshow the predicted local BHMF at M > 10

8 h1 M after correcting forBH coalescence; these corrections are likely upper limits (see Section 4.3 fordetails).


& Haehnelt 2000; Wyithe & Loeb 2002, 2003; Volonteri et al.2003; Shankar et al. 2009b) in the sense that we have moreparameters, which is required for a flexible enough frameworkto accommodate a variety of observations.

For our fiducial model, we have used the halo merger rate asthe proxy for the QSO-triggering rate. Alternatively, if we use

thesubhalo merger rate (the delayed version of halo merger rate;Section 1.4), it makes little difference below z = 3, but it furtherunderestimates the LF at z > 3, as expected from Figure 1. Thebolometric LF data we used here have uncertainties from bothmeasurements and bolometric corrections at the 20%30%level (compare Hopkins et al. 2007; Shankar et al. 2009b), notenough to reconcile the discrepancy at z > 4.5. There areseveral ways to modify our model to match observations ofLF at z > 4.5: (1) decrease the threshold min above whichmergers can trigger QSO activity; (2) modify the LpeakM0relation such that at fixed peak luminosity, QSOs shift to evensmaller halos at z > 4.5, which can be achieved by includingsome higher order terms in the redshift evolution of C; and (3)increase the scatter L so that more abundant low-mass haloscan contribute to the LF by upscattering. As an example of sucha model, we modify the redshift evolution of the mean LpeakM0relation at high redshift such that C C + log[( 1+z

4.5)7/2]

for z > 3.5. With this additional term of evolution in themean LpeakM0 relation, the typical host halo shifts to lower

masses (M0 4 1011 h1 M for log L/erg s

1 = 46at z = 6, compared to M0 10

12 h1 M in our fiducialmodel), and the resulting LF (dashed lines in Figure 3) fit theobservations well. This modification has little effects on theLF at z < 3.5, as well as other predicted QSO properties. Analternative parameterization of the redshift evolution in C isfurther discussed in Section 4.4.

However, there are no independent constraints on the mass ofhalos hosting quasars at z > 4.5 (such as those inferred fromquasar clustering), and the merger scenario of quasar activitymay be different at such high redshift. Hence we do not attemptto fully resolve this issue in this paper.

There is also slight tension between quasar clustering and LFat z 4 in our model, as already noted by several studies (Whiteet al. 2008; Wyithe & Loeb 2009; Shankar et al. 2009a). On theone hand, we need smaller halo masses to account for quasarabundance; we also need larger halos to account for the strongclustering on the other. To fully resolve this issue, we need betterunderstanding of the halo bias, and reliable fitting formulae forit, derived from simulations for the relevant mass and redshift


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Figure 8. Halo duty cycles, i.e., fraction of halos hosting QSOs brighter than Lmin , as a function of halo mass, computed using the halo mass function from Sheth et al.(2001) and the active halo mass function from Equation (21).

ranges,8 as well as better measurements of quasar clusteringat high redshift with future larger samples. Nevertheless, ourfiducial model is still consistent with both LF and clustering

observations within the errors.Our model also underpredicts the LF at the low-luminosity

end (L < 1045 erg s1) at z < 0.5. This is somewhat expected,since our model does not include contributions from AGNstriggered by mechanisms other than a major merger. Thefuel budget needed to feed a low-luminosity AGN (L 10

8 M) at high redshift (z 2) to low-mass BHs (M < 10

8 M) locally (e.g., McLure & Dunlop2004; Heckman et al. 2004), along with the cosmic downsiz-

ing in LF evolution (e.g., Steffen et al. 2003; Ueda et al. 2003;Hasinger et al.2005; Hopkins et al.2007; Bongiornoet al.2007).The typical transition from merger-driven BH growth to secu-larly drivenBH growthlikely occursaround BH mass 107 M(e.g., Hopkins & Hernquist 2009), corresponding to L 1045 erg s1 at the Eddington limit. Since the specific mergerrate increases toward higher redshift, while the rate of secu-lar processes is almost constant with time and these processesare relatively slow and inefficient for BH growth, it is conceiv-able that secular processes will only become important at latetimes (z < 0.5 for instance) in building up the low-mass end ofthe SMBH population. The implementation of secularly drivenAGN activity in our model will be presented in future work.

Finally, we mention that our model predicts a turnover in the

LF at L 1044

erg s1

at z 3. This simply reflects the lowermass cutoff at Mmin = 3 10

11 h1 M in our model. Futuredeeper surveys for low-luminosity AGNs at z 3 are necessaryto probe this luminosity regime, and to impose constraints onhow efficiently SMBHs can form in low-mass halos.

4.3. The Effects of BH Coalescence on the BHMF

The BHMFs discussed in Section 3 and Figure 7 are the massfunctions from accretion only, i.e., we have neglected the effectsof BH coalescence. As discussedin Section 1.1, BH coalescencewill redistribute the BHMF but does not change the total BHmass density (neglecting mass loss via gravitational radiation)

since essentially all the mass ended up in BHs were accreted.There are two routes in which the coalescence with pre-existingBHs might become important within our model framework.10

First, both galaxies in themerging pair of halos areelliptical,i.e.,they already experienced a QSO phase in the past and formedmassive nuclear BHs. In this case, the current merger event willbe a dry merger and form a SMBH binary without triggeringa new QSO. Second, during the major merger, one galaxy iselliptical and the other one is spiral, in which case a QSO willbe triggered, and themass of theold BH of theprevious ellipticalwill add to the new BH system.

A complete exploration of these routes and their conse-quences (including the effects of BH ejection, mass loss throughgravitational radiation, etc.) can be better achieved with MonteCarlo realizations of halo merger trees and our model prescrip-tions for QSO triggering and BH growth, which we plan to in-vestigate in a future paper. Here we can approximately estimatethe maximal impact of BH coalescence on the BHMFs in thetwo cases, assuming that BH coalescence always occurs withina Hubble time and all the mass in the pre-existing BHs is addedto the final BH, i.e., no BH ejection or mass loss via gravitationalradiation. We focus on the high-mass end (M 10

8 M) of

the local BHMF since our model prediction is incomplete at thelower-mass end.Although the dry merger case is not subject to the major

merger condition forQSO triggering, we still restrict to minhere because in minor mergers: (1) it will take too long for thetwo pre-existing BHs to become a close binary, and (2) the massincrement due to BH coalescence is insignificant. Observationaldetermination of the major dry merger rate is difficult, andthe current best estimate is: on average, present-day spheroidalgalaxies with MV < 20.5 (corresponding to M 10

8 M)have undergone 0.52 major dry mergers since z 0.7 (Bellet al. 2006). If we assume all the M 10

8 M BHs undergoone 1:1 dry merger after the QSO phase, the local BHMF willredistribute as the yellow line in Figure 7. In this case, the

abundance of the most massive (M > a few 10

9

M) BHs isenhanced by up to a factor of 2 at M = 1010 h1 M.

In the half-dry major merger case, the fraction of the pre-existing BH massto the finalBH mass is(1+ )5/3 0.070.7,where 1/4 4 is the major merger mass ratio (in ourreference model) between the two halos. This is because thefinal BH mass after a QSO phase scales as the 5/3 power tothe halo mass in our model. Averaging over possible valuesof , the fractional increment due to the pre-existing BH is 30%. If all QSO-triggering mergers are this kind of half-dryevent, the predicted z = 0 BHMF (red line in Figure 7) willredistribute from lower mass to higher mass (the green line).The enhancement at the high-mass end of the local BHMF iscomparable to the dry major merger case. In practice, QSO-

triggering mergers can occur between two spirals, hence theactual correction due to pre-existing BHs in half-dry mergersshould be smaller.

Combining these two cases we conclude that the impactof BH coalescence on the BHMF is probably insignificantcompared with other uncertainties and systematics in currentobservations and our model framework. Similar conclusionswere also achieved in several independent work (Volonteri et al.

10 In contrast to the two cases discussed below, we assume that a major mergerbetween two spirals will lead to negligible BH mass contribution from thepre-existing BHs, since in our model setting, spiral galaxy has not yetexperienced a major merger and hence significant BH growth.


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2003; Yu & Lu 2008; Shankar et al. 2009b) albeit with differencein details.

4.4. Implications for BH Scaling Relations

In our formalism, there is a generic scaling relation betweenthe relic BH mass and halo mass, which has the same slope andscatter as the LpeakM0 relation (from Equations (17) and (29)):

M,relic

108 h1 M 0.6(1 + z)1

M0

1012 h1 M

5/3

. (35)

The local MM0 relation for dormant BHs reported in Ferrarese(2002) and Baeset al. (2003) has a slope inthe range 1.31.8,and a normalization lower by a factor of 35 than ourpredictions. This is probably due to the fact that we neglectedcontinued growth of halos by minor mergers and diffuse matteraccretion since the major merger event, if most of the localmassive BHs were assembled at z 1 (as our model predicts).The fact that we did not impose a cutoff in the LC (24) mayalso lead to overly massive BHs. We note that although thenormalization (and perhaps slope as well) of the local MM0

relation may depend on galaxy morphological type (e.g., Zasovet al. 2004; Courteau et al. 2007; Ho 2007), early-type galaxies(S0 and ellipticals) appear to occupy the upper envelope in theMM0 relation. Our model scaling relations are only valid forearly-type galaxies which are presumably merger remnants.

However, it should be pointed out that the BHhalo scalingrelation relies on the assumed LpeakM0 relation. The simpleprescription for its evolution in our reference model alreadyshows some difficulties in reproducing the QSO LF at z 4.5(Figure 3; see Section 4.2). Hence our fiducial model is notvery appropriate for predicting the redshift evolution of theBHhalo scaling relation. To make this point more clear, letus consider a more physically motivated prescription for theLpeakM0 relation in which L V

5vir, where Vvir is the halo

virial velocity (Equation (A5)), and the normalization of theLpeakM0 relation evolves as (Wyithe & Loeb 2002, 2003)

C(z) = C(z = 0) +5

2log(1 + z) +

5

6log

vir(z)

vir(0)

, (36)

e.g., the evolution in C(z) is more rapid than our fiducialsetting. With this new implementation forthe LpeakM0 relation,we found that a good global fit can be achieved with thefollowing parameter adjustments (other parameters are thesame as in our reference model): Mquench = 3 10

12 h1 M,

C(z = 0) = log(0.81045)12 and L = 0.4(1+z)1/2. Since

the redshift evolution in the normalization C(z) is now faster,the starting value C(0) is reduced; consequently the exponential

upper cut of halo mass, Mquench, increases in order to accountfor QSO counts at low redshift. The scatter in the LpeakM0relation needs a redshift evolution to achieve adequate fits forboth clustering and LF over a wide redshift range. The predictedLF is shown as the dotted lines in Figure 3, which does a muchbetter job at z 4.5 than our fiducial model. Other predictedproperties are slightly degraded (but still are reasonably goodfits to observations) than those predicated by our fiducial model.

This new LpeakM0 relation, together with the approxima-tion that the halo virial velocity Vvir vc, the galaxy circu-lar velocity, and the assumption that the local vc relation(Ferrarese 2002) does not evolve, result in a constant M re-lation (Wyithe & Loeb 2003). Neglecting the scatter, the local

mean vc relation in Ferrarese (2002) is

log

vc

km s1

= 0.84 log

km s1

+ 0.55. (37)

Thus, the new LpeakM0 relation predicts a constant Mrelation (e.g., Equations (17), (26), (29), (36), and (A6)):

M

108 M= 0.9 5.0

200 km s1

4.2

, (38)

with a slope and normalization (bounded by M,peak and M,relic)consistent with local estimates (Gebhardt et al. 2000; Ferrarese& Merritt 2000; Tremaine et al. 2002; Lauer et al. 2007).This consistency, however, is built on a couple of assumptionsand approximations, and neglecting successive evolution afterthe self-regulation of BH and bulge growth. Given all thesecomplications, it is beyond the scope of thecurrent study to fullysettlethis issue. We simplyremind thereader that a non-evolvingM relation is generally allowed within our framework.

5. CONCLUSIONS

We have developed a general cosmological framework forthe growth and cosmic evolution of SMBHs in the hierarchicalmerging scenario. Assuming that QSO activity is triggered bymajor mergers of host halos, and that the resulting light curvefollows a universal form with its peak luminosity correlatedwith the (post)merger halo mass, we model the QSO LF andSMBH growth self-consistently across cosmic time. We testedour model against a variety of observations of SMBH statistics:the QSO LF, quasar clustering, quasar/AGN BH mass andEddington ratio distributions. A global good fit is achieved withreasonable parameters. We summarize our model specifics asfollows.

1. The QSO-triggering rate is determined by the 0.25halo merger rate with exponential cutoffs at both thelow and high halo mass ends Mmin = 3 10

11 h1 M,Mmax(z) = 10

12(1 + z)3/2 h1 M.

2. The universal light curve follows an initial exponentialSalpeter growth with the constant Eddington ratio 0 = 3for a few e-folding times to reach the peak luminosityLpeak,which is correlated with the (post)merger halo mass asLpeak = 6 10

45(1 + z)1/3(M0/1012 h1 M)

5/3 erg s1,with a lognormal scatterL = 0.28 dex. It then decays as apower law with slope = 2.5.

Our simple model successfully reproduces the LF, quasarclustering, and Eddington ratio distributions of quasars and

AGNs at 0.5 < z < 4.5, supporting the hypothesis that QSOactivity is linked to major merger events within this redshiftrange. However, there are still many unsettled issues. Belowwe outline several possible improvements of our simple model,which will be addressed in future work.

Our model underpredicts the LF at the faint luminosity endL < 1045 erg s1 at z < 0.5, which is linked to the growthof the less massive 107 M SMBHs. This is indicative of apopulation of low-luminosity AGNs triggered by mechanismsother than major mergers at the low redshift universeeitherby minor mergers or secular processes. We need to incorporatethis ingredient in our SMBH model, in order to match the localBHMF at the low-mass end (M 10

7 M).


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In our modeling, we have neglected the possibilities of aclosely following second major merger event and the triggeringof two simultaneous QSOs during a single major merger event.Therefore, our model does not include more than one QSOswithin a single halo. We will use Monte Carlo realizationsof halo merger trees to assess the probability of such rareoccurrences and see if they can account for the small ( 0.1%)binary/multiple quasar fraction observed (Hennawi et al. 2006,2009; Myers et al. 2008).

Our model can be improved to include the radio loudness ofQSOs as well. If radio loudness requires both a massive hosthalo (to provide the hot IGM) and a massive SMBH (to launchthe kinetic jet), we can statistically populate radio-loud QSOsin halos within our model framework. It can be tested againstthe clustering of radio-loud quasars (e.g., Shen et al. 2009) andthe radio-loud fractio

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