Draft version July 11, 2013
Preprint typeset using LATEX style emulateapj v. 04/17/13

UNDERSTANDING BLACK HOLE MASS ASSEMBLY VIA ACCRETION AND MERGERS AT LATE TIMES INCOSMOLOGICAL SIMULATIONSAndrea Kulier1 ,Jeremiah P. Ostriker1,2 , Priyamvada Natarajan3 , Claire N. Lackner1,4 , and Renyue Cen1
Draft version July 11, 2013

ABSTRACT
Accretion and merger triggered accretion episodes are thought to primarily contribute to the massaccumulation history of supermassive black holes throughout cosmic time. While this might be thedominant growth mode at high redshifts, at lower redshifts and for the most massive black holes,
mergers themselves might add signiﬁcantly to the mass budget. In this paper, we explore this in two
speciﬁc yet disparate environments — black holes harbored in member galaxies in a massive cluster
and those in a void region. We use merger trees derived from hydrodynamical cosmological simulationsof a cluster and void region to examine the growth of supermassive black holes at late times from4 > z > 0. Mass gains from gas accretion and BH-BH mergers are tracked as are black holes thatremain unmerged and “orbiting” due to insuﬃcient dynamical friction in a merger remnant, as well as
those that are ejected due to gravitational recoil. We ﬁnd that gas accretion remains the dominantsource of mass accumulation in almost all of the SMBHs produced; mergers contribute an average of
3.3 ± 0.2% for all SMBHs in the cluster, and 1.3 ± 0.2% in the void from z = 4 to 0. However, mergers
are signiﬁcant for massive SMBHs, with the contribution from mergers reaching a maximum of 20%
in the cluster for black holes with mass around 109 M⊙ . The fraction of central black hole mass frommergers generally increases for larger values of the host galaxy mass: in the void, the fraction is 2% at
M∗ = 1010 M⊙ , 5% at 1011 M⊙ , and 9% at M∗ ≈ 1012 M⊙ , and in the cluster it is 4% at M∗ = 1010 M⊙ ,13% at 1011 M⊙ , and 21% at 1012 M⊙ . We also ﬁnd that the total mass in orbiting SMBHs is negligible
in the void except for the few most massive BHs, but signiﬁcant in the cluster, with a median valueof Morbiting & 107 M⊙ per galaxy for galaxies with stellar mass M∗ > 1011.5 M⊙ . We ﬁnd that 40% of
SMBHs and ≈ 14% of the total SMBH mass1 is found orbiting in the cluster region at z = 0. The
existence of such a large unmerged mass fraction requires modiﬁcation of the Soltan argument wherethese orbiting BHs are unaccounted for. We estimate the correction to the Soltan argument due tosuch orbiting SMBHs as well as SMBHs ejected via gravitational slingshot eﬀects to be in the range
1.6 − 15%, with a mean value of 7.4 ± 3.7% in the estimate of the inventory of the integrated accreted
mass density of SMBHs. Quantifying the growth due to mergers at these late times, we calculate thetotal energy output and strain due to gravitational waves emitted by merging SMBHs.
Subject headings: black hole physics — galaxies: nuclei — quasars: general
1. INTRODUCTION

Observations strongly suggest that supermassive black
holes are harbored in the centers of almost all massive galaxies (Kormendy & Richstone 1995). The massesof these central black holes are observed to correlate with multiple properties of their host galaxies.
The most well-known and studied of these relations
is the M − σ relation between the mass of the blackhole and the velocity dispersion of its host galaxy
spheroid (Gebhardt et al. 2000; Ferrarese & Merritt
2000; Tremaine et al. 2002; Gültekin et al. 2009). Relations between the black hole mass and the stellar
mass and luminosity of the bulge have also been derived
from observations (Marconi & Hunt 2003; Häring & Rix
Electronic address: akulier@princeton.edu
1 Department of Astrophysical Sciences, Princeton University,
Princeton, NJ 08544, USA
2 Department of Astronomy, Columbia University, NYC, NY
10027, USA
3 Department of Astronomy, 260 Whitney Avenue, Yale University, New Haven, CT 06511, USA
4 Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, the Universityof Tokyo, Kashiwa, Japan
1 where the total includes central, orbiting, and ejected SMBHs

2004; Beiﬁori et al. 2012). It has also been claimed
that the mass of the central black hole is correlated with galactic properties on scales larger than thebulge, such as the total stellar mass of the galaxy(Cisternas et al. 2011; Bennert et al. 2011; Beiﬁori et al.
2012), and the mass of the entire host halo (Ferrarese
2002; Volonteri et al. 2011; see however Kormendy et al.
2011; Kormendy & Bender 2011). There is also evidence
of increased scatter in some of these relations for low
mass galaxies (Greene et al. 2010b; Volonteri et al. 2011;
McConnell & Ma 2012). The existence of these correlations suggests interplay between black hole growth andstar formation activity in galactic nuclei.
There appears to be a connection between AGN activity and star formation on galactic scales larger than
the nucleus as well. The total AGN activity follows
roughly the same trend as a function of cosmic redshift
as the global star formation rate (Heckman et al. 2004;
Merloni et al. 2004; Brusa et al. 2009; Shankar et al.
2009). A correlation has also been found between theluminosity of individual AGN and the star formation luminosity of their host galaxies (Netzer 2009; Chen et al.
2013). It is becoming clear at high redshifts that the
global star formation rate tracking AGN activity does not

2

Kulier et al.

imply that these processes occur in tandem in all individual galaxies (Treister et al. 2013, in preparation). While
such relations imply that the evolution of a galaxy andits central black hole are intertwined, their exact origin
remains uncertain. Several mechanisms have been proposed. One possibility is that the evolution of a galaxy
regulates the growth of its central black hole by determining the amount of gas that ﬁnally reaches the blackhole (Booth & Schaye 2011). However, a SMBH can also
regulate the evolution of its host galaxy through energy
input via AGN feedback, which may be able to suppress star formation (Maiolino et al. 2012). Dubois et al.
(2012) have run recent cosmological simulations including SMBH growth via accretion and mergers, as well as
outﬂows and heating from AGN feedback, and conclude
that AGN feedback is necessary to expel gas and suppress
star formation. They also argue that AGN feedback isable to transform late-type galaxies into early-types andexplain galaxy scaling relations such as the fundamental
plane (Dubois et al. 2013a).
It is not known which of the former two processes isdominant. A third explanation of the scaling relations
is that some other factor — for instance, the total gas
reservoir — regulates both the growth of the SMBH andthe evolution of the galaxy. Bournaud et al. (2012) observed a correlation between giant clumps of gas andstars, which are indicative of violent disk instabilities,
and AGN activity at z ∼ 0.7. They propose that theevolution of disk instabilities, which cause gas inﬂow to
the central bulge and SMBH, could produce the observed
correlation between star formation and AGN activity(Bournaud et al. 2011). Finally, a purely statistical explanation has been proposed for the existence of theseobserved correlations based on the idea that repeatedmergers of galaxies, which also lead to the eventual mergers of their central black holes, can cause the SMBHs
and their hosts to have correlated masses as a consequence of the central limit theorem (Hirschmann et al.
2010). One or a combination of all these mechanisms
could be responsible for the observed scaling relations
between SMBHs and their hosts. The argument basedon the central limit theorem has recently been gaining
strength as the evidence accumulates from both observations (van Dokkum et al. 2008) and theory (Oser et al.
2010) that the most massive galaxies grow their stellar components substantially via minor mergers during
recent cosmic epochs. This is especially important for
massive BCGs (brightest cluster galaxies) in rich clusters(Hausman & Ostriker 1978; Lin et al. 2013). Evidence
that BCGs have grown via mergers has been reinforced
by a new statistical test (Lin et al. 2010). The impact of
the increased frequency of minor mergers for the assembly of the stellar component suggests that they mightplay an important role in black hole growth as well at
late times.
It is widely believed that black hole growth occurs
primarily via accretion or merger triggered accretion
episodes over cosmic time, although secular evolution
driven by stellar evolutionary processes also appears
to be important (Ciotti & Ostriker 1997; Ostriker et al.
2010), especially for z < 2. In massive galaxies, whichhave undergone multiple mergers, it is believed that mostof the SMBH growth occurs in short accretion episodes
fueled by gas ﬂowing into the central region due to themerger or galactic cooling ﬂows. SMBHs hosted in small
galaxies that have not undergone any major mergers
must, however, be supplied with gas through some different mechanism, such as dynamical relaxation and perhaps secular evolution processes (Sesana 2012a). Whilethese may indeed be the primary channel of growth during early times, it is becoming evident that the actual
merger of black holes might contribute appreciably tothe ﬁnal mass inventory of the most massive black holes
at low redshifts. This is suggested by the current evidence that minor mergers are a signiﬁcant componentof the late-time stellar mass growth of the most massive
galaxies.
In this paper, we examine how gas inﬂows, mergers,
and the large-scale environment determine black hole
growth. To this end we track the growth histories of
black holes in a typical over-dense cluster environment
and an under-dense void environment. In particular, weknow that the merger history in these environments at
late times is divergent. Therefore, in this work we focus
on quantifying the role that mergers play in the massassembly history at late times. This has important consequences for the expected gravitational radiation fromsuch events at low redshifts as well the observational
implications for the number of wandering black holes.
Since BH-BH mergers will be accompanied by the gravitational slingshot eﬀect (Peres 1962; Bekenstein 1973;
Fitchett & Detweiler 1984), late time mergers may ejecta signiﬁcant amount of mass in BHs from merged galaxies. An inventory of BHs at the present epoch will, ofcourse, not count this mass and thus will underestimate
the z = 0 cosmic mass density locked up in SMBHs.
Furthermore, we expect that some SMBHs at any given
time will be orbiting in the outskirts of the galactic potential — as the result of a lower velocity gravitational
wave recoil or a recent merger of its host with a more
massive galaxy. These orbiting BHs will also not befound in observations of galactic centers. The correction
to the SMBH mass density from these two populations,which we estimate here, implies that the Soltan argument (Soltan 1982) will necessarily overestimate the eﬃciency of energy ouput from accreting BHs. This is dueto the fact that the energy output from accretion that is
directly observed is incommensurate with the SMBH cosmic mass density estimated only from the census of BHs
in galactic centers. Neglecting the “unmerged” SMBH
population introduces errors in estimates of the inferred
accretion eﬃciency for the population of BHs.
The outline of the paper is as follows: we ﬁrst brieﬂysummarize relevant previous work on the evolution of supermassive black holes in §2. We then describe the key
aspects of the problem that are tackled here — the roleof late time mergers — in §3. In §4, the methodology
and cosmological simulations used are detailed, followedby the analysis and results in §5. We conclude with a
discussion of the implications and observational consequences of accounting for the role of minor mergers at
late times on our current understanding of black hole
growth.
2. PREVIOUS WORKThe evolution of supermassive black holes through cosmic time has been modeled using diﬀerent approaches bytracking the dark matter halos they reside in, including

SMBH Mass Assembly Via Accretion and Mergers
Monte Carlo merger trees (e.g., Haehnelt & Kauﬀmann
2000; Lippai et al. 2009; Natarajan 2011), PressSchechter theory (e.g., Yoo et al. 2007), and cosmological simulations (e.g., Di Matteo et al. 2003; Sijacki et al.
2009). A more complete review of the early models of
the growth of SMBHs is given in Natarajan (2004).
Simulations of the evolution of black hole scaling relations have been done both semi-analytically(e.g., Kisaka & Kojima 2010) and numerically (e.g.,
Johansson et al. 2009). Most studies have focused on understanding black hole growth in individual galaxies residing in average over-density environments, typically theﬁeld. There have been few studies of black hole mass assembly in extremely over-dense cluster and under-dense
void environments. Using dark matter merger trees derived from Press-Schechter theory, Yoo et al. (2007) examined the expected growth of black holes due to mergers
in a large cluster with halo mass Mh = 1015 h−1 M⊙ atlate times. They found that most SMBHs in the cluster with masses & 107.5 M⊙ underwent mergers, with themost massive SMBHs increasing their mass by a factor of∼ 2 since z = 2. Although the central galaxy generally
contained the most massive SMBH at z = 0, Yoo et al.
(2007) found that for some cluster assembly histories, themost massive SMBH may be hosted by a satellite galaxy.
Current simulations do not agree on the evolutionof BH-galaxy scaling relations with redshift. Someﬁnd that SMBHs at higher z should be more massive than they would be if they followed the current
M − σ relation (Hopkins et al. 2009; Dubois et al. 2012),
while others ﬁnd that they should be less massive(Malbon et al. 2007; Di Matteo et al. 2008). Observations at redshifts up to z ∼ 4 seem to indicate that
the M − σ relation changes such that SMBHs are more
massive at ﬁxed velocity dispersion than those in thepresent day (McLeod & Bechtold 2009; Woo et al. 2008;
Greene et al. 2010a ; see however Shields et al. 2003;
Zhang et al. 2012). Others observations have lookedat the M − M∗,spheroid and M − M∗,host relationships,
and similarly have argued for evidence supporting an increase in black hole mass at ﬁxed host mass with increasing redshift (Bennert et al. 2011; Merloni et al. 2010;
Zhang et al. 2012).
Complementary constraints on BH growth models
can be derived from X-ray luminosity functions ofAGN. Since X-rays can escape even the most obscured
Compton-thick galactic nuclear sources, they oﬀer aunique probe of actively growing BHs (Salvaterra et al.
2007). Several studies that have attempted to explain
the origin of the cosmic X-ray background have also provided insights into both the obscured and unobscured
accreting populations of BHs over cosmic time (c.f. models by Gilli et al. 2007; Treister et al. 2009). Relevant toour study are the constraints obtained by Volonteri et al.
(2006) on the accretion history of SMBHs at z < 3 using observations of the faint X-ray background combinedwith optical and hard X-ray luminosity functions. They
found that a model in which the Eddington ratio is afunction of the AGN luminosity — as suggested by previous simulations — ﬁts the observational constraints
somewhat better than models with a constant Eddington
ratio or an Eddington ratio decreasing with redshift.
As for the population of “wandering” black holes,
the buildup of populations of orbiting and ejected3

black holes as a result of galaxy mergers and gravitational wave recoils has been examined most recently
by Rashkov & Madau (2013). Although their approachbears similarities to ours, they studied the evolutionof intermediate-mass black holes (IMBHs), which arethought to be the ancestral seeds of the supermassive
black holes currently found in the centers of galaxies.
They populated the N-body Via Lactea II cosmological simulation of a Milky Way-size halo (Diemand et al.
2008) with seed IMBHs, which they then allowed to
evolve via mergers and gravitational wave recoils. They
found that even when assuming “maximal” numbers of
BHs would escape the galaxy due to gravitational wave
kicks, a sizable population of leftover IMBHs should beorbiting in the halo of a galaxy with the mass of the
Milky Way. We focus on slightly diﬀerent mass scales
in this work and examine the status of the growth ofthe central SMBH and the SMBH wanderers in a typicalcluster rather than a galaxy scale halo.
3. EXPLORING THE ROLE OF LOW-REDSHIFT MERGERSIn this paper, we examine the consequences of mergers
for the mass assembly history of SMBHs at late times
and the observational consequences thereof. The energy
released during black hole assembly over cosmic time
has been estimated by integrating the energy output
of observed AGN (Soltan 1982). This argument madeby Soltan employs the observed luminosity function ofquasars and requires an estimate of the total mass inblack holes per unit cosmic volume, which was assumed
thus far to not be signiﬁcantly altered by mergers among
black holes (Menou & Haiman 2004). However, mergers
in cluster environments at late times are ubiquitous andthese events will produce gravitational waves, and theamount of energy released in them is important to know
accurately for detection experiments (McWilliams et al.
2012). Using the Soltan argument, the SMBH mass density has been found to be consistent with the luminosity
density of QSOs if these QSOs have a mass-to-energy
conversion eﬃciency of ǫ ≃ 0.1 (Yu & Tremaine 2002).
Taking mergers explicitly into account, including ejections will alter the observational consequences at low
redshifts and that is precisely what we explore in detailin this paper. Thus far the total energy that is emitted in gravitational waves has been estimated using primarily Monte-Carlo dark matter merger trees; however,these approaches do not take into careful account the enhanced merger rates in cluster environments where themost massive galaxies live (Sesana 2012b).
We note that not all galactic mergers will result inblack hole mergers, due to a ﬁnite dynamical friction time
for the black hole to sink to the center of the galactic potential. Also, there exists the “ﬁnal parsec problem” (seee.g., Milosavljević & Merritt 2003), which is shorthandfor the physical diﬃculties in bridging the gap between
the binary separation reached by dynamical friction between the SMBH and the ambient stellar population andthe much smaller separation at which gravitational radiation takes over as the primary angular momentum loss
mechanism for the BH binary. We will not address thisproblem in this paper, but will assume that black holes
eﬀectively merge at galactic centers on the dynamical
friction timescale.
In this work, we use recent cosmological simulations

4

Kulier et al.

(Cen 2011a,b, 2012a,b, 2013) to explore the relevance of
diﬀerent processes on the growth of SMBHs and their implications for the setting up of the various observed scaling relations in two signiﬁcantly diﬀerent environments.
In our book-keeping we keep track of the total mass inblack holes in the following four categories at redshifts
between z = 4 and 0:• The mass in central black holes acquired fromaccretion of gaseous matter.
• The mass in central black holes acquired frommergers with smaller black holes.
• The mass in orbiting black holes that have not yet
fallen to the center of the galactic potential.
• The mass in ejected black holes that have been
kicked out due to gravitational radiation recoil.
The ﬁnal two categories will be assessed as corrections
to the normal Soltan-type arguments for estimating therevised eﬃciency of BH-associated energy generation.
4. METHODOLOGY AND DESCRIPTION OFCOSMOLOGICAL SIMULATIONSAs the basis for our analysis of SMBH evolution, we
use the large-scale hydrodynamical galaxy simulations of
Cen (2011a,b, 2012a,b, 2013). Detailed descriptions of
the simulations can be found in the papers referenced
above; we provide a brief overview in this section. We
use galactic (not dark matter based) merger trees derived
from these simulations as described in Lackner et al.
(2012). The simulations are performed with the AMR(Adaptive Mesh Reﬁnement) Eulerian hydrodynamics
code, Enzo (Bryan 1999; O’Shea et al. 2004; Joung et al.
2009). They consist of a low-resolution box of 120 h−1
Mpc on a side, and two high-resolution regions within
this box, one containing a cluster with mass ∼ 3 × 1014
M⊙ , and the other a void. These represent +1.8σ and−1.0σ ﬂuctuations in the cosmic density ﬁeld, respectively. These two extreme regions bracket the cosmic
average environment. The cluster region box has dimensions 21×24×20 h−3 Mpc3 , and the void box has dimensions 31 × 31 × 35 h−3 Mpc3 . The dark matter particle
mass is 1.07 × 108 h−1 M⊙ , while the stellar particle massis generally around 106 M⊙ . The resolution in the cluster and void regions is always better than 460 h−1 pc
physical.
The simulation includes prescriptions for UV background (Haardt & Madau 1996), shielding from UV radiation by neutral hydrogen (Cen et al. 2005), metallicitydependent radiative cooling (Cen et al. 1995), formationof star particles from gas (Cen & Ostriker 1992), andsupernovae feedback (Cen et al. 2005). It does not include feedback from AGN, which may be partly the reason that the largest galaxies in the simulation box havetoo many stars in comparison with observed relations.
This is a well known problem in hydrodynamical simulations (Oser et al. 2010; Guo et al. 2010) and one that
does exist in our version.

The simulations use the following cosmological parameters, consistent with Komatsu et al. (2009): ΩM =0.28, Ωb = 0.046, ΩΛ = 0.72, σ8 = 0.82, H0 = 100h−1
Mpc−1 = 70 km s−1 Mpc−1 , and n = 0.96. These arealso the values we adopt throughout this paper in our
calculations.
Galaxies in the simulation box are identiﬁed by using
the HOP algorithm on stellar particles (Eisenstein & Hu
1999). The following are some of the physical parameters calculated and tracked for each galaxy: position,
velocity, total mass, stellar mass, gas mass, mean formation time, mean stellar metallicity, mean gas metallicity,
star formation rate, and luminosities and colors for ﬁve
SDSS bands, among others. In general, broad agreement
is found between the simulation results and observations(Cen 2011a).
A merger tree is created from this simulation. In
the cluster box, there are 38 redshift slices fromz = 4 to z = 0 with ∆z = 0.05 for z < 1.35,
and slices at z = 1.5, 1.6, 1.75, 1.9, 2.0, 2.2, 2.5, 2.8, 3.1,
and 4.
In the void box, there are 14 redshift
slices between z = 0 and z = 4, at z =0, 0.05, 0.15, 0.2, 0.4, 0.5, 0.6, 0.8, 1.0, 1.6, 1.9, 2.5, 3.1 and
4 (Lackner et al. 2012). We seed the galaxies in thismerger tree with central black holes and trace their evolution based on the evolution of their host galaxies. Further details are provided below.
4.1. Black Hole Evolution Prescriptions

Since our focus is on massive galaxies, we use galaxies from the simulation only if they have stellar masses
M∗ > 109 M⊙ , due to the resolution limit of the simulation. In the cluster box, all galaxies above this mass
limit have descendants at z = 0. In the void box, thegrouping algorithm identiﬁes some groupings of particles
above this mass limit as galaxies that do not exist in
the next redshift slice. For our purposes, we ignore anygroupings of particles that do not have a descendant at
z = 0.
We place a seed black hole in any galaxy that reaches
the lower mass limit of 109 M⊙ at any redshift. The seedblack hole is taken to have mass nM∗ , where n is selectedfrom a log-normal distribution with median value 10−3 .
This matches the approximate observed mass ratio of the
SMBH to its host galaxy bulge (Volonteri 2012), and isbased on the assumption that of the gas added to (andretained by) galaxies, approximately one part in a thousand is accreted onto the central black hole, with mostof the remainder transformed into stars (Li et al. 2007).
It is widely believed that AGN feedback helps to limit
the growth of the central black hole, and some simulations (Hopkins & Hernquist 2009; Ostriker et al. 2010;
Park & Ricotti 2012) indicate that this is likely. We notethat this assignment of BH seed masses as early as z = 4
pre-supposes the existence of a scaling relation at thisepoch akin to what is empirically measured at z = 0.
SMBHs are then allowed to grow through BH-BH
mergers and accretion of gaseous material from thegalaxy. We calculate the accreted and merged massby considering all parent black holes of a single blackhole to be “merged mass” except for the most massive
one. We adopt a simple prescription for accretion inwhich the mass accreted by the central black hole ineach redshift slice is proportional to the amount of stars

SMBH Mass Assembly Via Accretion and Mergers
formed in the galaxy in that redshift slice, with somescatter in the assumed proportionality factor. Becausestars are sometimes ejected from galaxies in the simulation, it is possible for the total stellar mass of agalaxy to occasionally decrease. If this is the case, we
assume zero accretion onto the black hole in that redshift
slice. Otherwise, the mass accreted by the black hole istaken to be n′ ∆Minsitu , where ∆Minsitu is the change ingalaxy stellar mass due to star formation in that redshift slice, and n′ is a proportionality factor chosen fromthe same log-normal distribution as the proportionality
factor for the seeding prescription. We choose this accretion rate in order to match the observed correlation
between AGN activity and galactic star formation rate
(Heckman et al. 2004; Merloni et al. 2004; Brusa et al.
2009; Shankar et al. 2009). Because the luminosity of
an AGN is determined by its growth rate due to accretion, these data imply that AGN grow at a rate roughly
proportional to the rate of star formation in their hostgalaxy, with a proportionality factor of about 10−3 . We
adjust the scatter of the log-normal distribution fromwhich the proportionality factor for the seed masses and
BH accretion rate is chosen so as to produce a scatter of∼ 0.4 in the cluster M• − M∗ relation at z = 0, similar toobserved values reported in the literature (Bennert et al.
2011; Hu 2009; McConnell & Ma 2012).
Because we ﬁx the relations of seed mass to host galaxy
mass and accreted mass to star formation to be the sameat all redshifts, we essentially force the M• − M∗ to beconstant with redshift. Accounting for ejected and orbiting BHs can cause some evolution in the relation bylowering the central BH mass compared to the M• − M∗relation, but, as will be described in the results (§5), thechange in mass from these eﬀects is on average rather
small. Thus our model does not account for the evolution
in the BH-galaxy scaling relations seen in some previous
work (e.g., Merloni et al. 2010; Zhang et al. 2012).
When a smaller galaxy merges into a larger one, we
calculate the dynamical time for the black hole from thesmaller galaxy to move to the center of the newly formed
merged galaxy. We use the dynamical friction formula ofBinney & Tremaine (1987):
2
108 M⊙Reσ
19 Gyr. (1)tDF =ln(1 + M∗ /M• ) 5 kpc
200 km/s M•
We obtain values for the velocity dispersion σ and eﬀective radius Re of our simulated galaxies by using observedﬁts to the mass from SDSS data (Nipoti et al. 2009), andredshift dependences from Oser et al. (2010):

0.73
M∗
Re = 2.5 kpc(1 + z)−1.44 ,(2)1011 M⊙0.2
M∗(1 + z)0.44 .
(3)σ(Re ) = 190 km/s
1011 M⊙If a galaxy is involved in another merger before a satellite black hole has merged with the central one, we recalculate the dynamical friction time. If the satellite blackhole is in the more massive galaxy, we take the new dynamical friction time to be the smaller of the remaining
time to black hole merger and the dynamical friction timethat would be calculated for the post-merger galaxy. For

5a less massive galaxy merging into a more massive one,its satellite and central black holes are taken to have a dynamical friction time calculated for the new post-merger
galaxy. In this way, we keep track of both central andsatellite SMBHs in galaxies.
If a black hole is ejected or displaced from the center ofa galaxy due to a gravitational wave recoil (see §4.2.1 below), the galaxy may have no central SMBH. If a galaxy
lacking a central BH produces and additional 109 M⊙ instar formation, we seed it again with a black hole withmass proportional to the mass in stars formed, where
the proportionality constant is chosen from the same lognormal distribution as for the seeds and accreted massdescribed earlier.
4.2. Low redshift merging and gravitational radiation

We calculate the gravitational radiation luminosity expected based on the BH-BH mergers that occur in our
simulation. This energy is equivalent to the mass lostfrom the black hole pair as they are merging. For
each BH-BH merger, we apply an approximation for
the energy emitted in gravitational waves taken from
Barausse et al. (2012):
Erad=[1 − ẼISCO (ã)]ν+M
4ν 2 [4p0 + 16p1 ã(ã + 1) + ẼISCO (ã − 1)].

(4)

Here M ≡ m1 + m2 is the total mass of the two black
holes and ν ≡ m1 m2 /M 2 is the symmetric mass ratio.
The constants p0 and p1 come from a polynomial ﬁt to
the energy emitted by inspiraling binary black holes, andhave the values:
p0 = 0.04827 ± 0.00039,

p1 = 0.01707 ± 0.00032. (5)Here ẼISCO is the energy per unit mass at the innermost
stable circular orbit and is given by:s
2
ẼISCO (ã) = 1 − eq (ã),(6)
3r̃ISCO
whereeqr̃ISCO(ã) =3 + Z2p
− sign(ã) (3 − Z1 )(3 + Z1 + 2Z2 ),hi
Z1 =1 + (1 − ã2 )1/3 (1 + ã)1/3 + (1 − ã)1/3 ,q(7)
Z2 = 3ã2 + Z12 .

In all the above equations, ã is deﬁned asã ≡L̂ · (S 1 + S 2 )|a1 | cos β + q 2 |a2 | cos γ=,
M2(1 + q)2(8)

where q ≡ m2 /m1 < 1 is the mass ratio of the two black
holes, |a1 | and |a2 | are the spin magnitudes, and β andγ are the angles between the orbital angular momentum
unit vector L̂ and the spins of the ﬁrst and second blackhole, respectively.
By construction, this formula is accurate in both thetest-particle limit and for equal-mass binaries. The expected spin vector distribution of SMBHs and its evolution are not well agreed upon in the literature (see

6

Kulier et al.

e.g., Volonteri et al. 2005; Barausse 2012; Dubois et al.
2013b). Thus, we make the most “simple” possible assumption by selecting spin parameter values from a random uniform distribution between 0 and 1, and the cosineof the angles between the spins and the orbital angularmomentum from a uniform random distribution between-1 and 1. This is similar to the assumptions made by
Schnittman & Buonanno (2007) when calculating the expected recoil velocity distribution of SMBHs.
4.2.1. Gravitational Wave Recoils

When two orbiting supermassive black holes merge,
the gravitational radiation produced can impart a
linear (“slingshot”) momentum to the SMBH resulting from the merger (Peres 1962; Bekenstein 1973;
Fitchett & Detweiler 1984). Recent numerical simulations of general relativity show that such “kicks”
can in some cases be large enough to exceed the escape velocity of the host galaxy and eject the resulting SMBH (Herrmann et al. 2007; Koppitz et al. 2007;
Campanelli et al. 2007a,b; Lousto & Zlochower 2011).
Gravitational wave recoils with velocities insuﬃcient toeject the SMBH can still displace it from the center ofthe galaxy, to which it may return via dynamical friction.
As we noted earlier, ejections of SMBHs are particularly important as they could alter the quasar eﬃciency
determined from the classic “Soltan argument” (Soltan
1982). This argument assumes that the infall of mass
onto quasars over the history of the universe can be
integrated over the local galaxy population to give theobserved mass density of SMBHs in the local universe.
Since quasar luminosities are proportional to their mass
growth rates — L = ǫṀ c2 , where ǫ is the eﬃciency factor — observations of the total luminosity from quasars
can be used to ﬁnd the eﬃciency factor ǫ (Yu & Tremaine
2002; Marconi et al. 2004). However, if a signiﬁcant fraction of SMBHs are ejected from their hosts, then there isless mass in galactic center SMBHs today than was produced in quasars in the past, and the eﬃciency obtained
from the Soltan argument is an overestimate.
We account for the eﬀects of gravitational wave “kicks”
on our population of SMBHs, including both ejections
and displacements from the center of the galaxy. We
use the ﬁtting formula based on numerical simulations
from Lousto et al. (2012) for the velocity imparted to an
SMBH resulting from a merger:
V recoil (q, α) = vm ê1 + v⊥ (cos ξê1 + sin ξê2 ) + vk n̂k ,η 2 (1 − q)
[1 + Bm η],(1 + q)i
η2 h kk(α2 − qα1 ) ,v⊥ = H(1 + q)ihvk = 16η 2 /(1 + q) V1,1 + VA S̃z + VB S̃z2 + VC S̃z3
vm = Am⊥× |α⊥
2 − qα1 | cos(φ∆ − φ1 ).

equal mass and spin contribution to the recoil velocity inthe orbital plane and S̃ = 2(α2 +q 2 α1 )/(1+q)2 . φ∆ −φ1⊥is the angle between ∆⊥ = M (S ⊥2 /m2 − S 1 /m1 ) andsome ﬁducial direction at merger. The coeﬃcients are obtained numerically and are H = 6.9×103, Am = 1.2×104 ,Bm = −0.93, V1,1 = 3677.76 km/s, VA = 2481.21
km/s, VB = 1792.45 km/s, and VC = 1506.52 km/s
(Lousto et al. 2012). This formula is similar to thoseobtained by previous authors, e.g. Campanelli et al.
(2007a); Baker et al. (2008); van Meter et al. (2010). We
assume randomly distributed spin magnitudes and spin
directions with respect to the orbital angular momentum
as described in §4.2.
To calculate the trajectory of the kicked SMBH, we follow a similar prescription as Madau & Quataert (2004):
we assume the density proﬁle of the galaxy is a truncatedisothermal sphere, with a core radius equal to the radius of gravitational inﬂuence of the post-merger SMBH,2
RBH ≈ GMBH /σ 2 , so that ρ(r) = σ 2 /[2πG(r2 + RBH)].
The velocity dispersion σ is obtained from Equation 3.
If the |V recoil | from gravitational wave radiation is foundto be larger than the isothermal sphere escape speed
2σ(ln(Re /RBH ))1/2 , where the eﬀective radius Re is obtained from Equation 2, we assume the black hole isejected from the galaxy. If the kick is insuﬃcient to eject
the black hole, we calculate the time for the displaced
SMBH to return to the center of the galaxy via dynamical friction. We approximate the orbits of the kicked BHs
as purely radial, and calculate their radii as a function oftime numerically. This generally results in a dynamical
friction timescale that is signiﬁcantly shorter than thatfor circular orbits, which is the assumption we use whencalculating the dynamical friction timescale of a SMBH
after the merger of its host galaxy with a more massive
galaxy.
We solve numerically for the radial position of the
SMBH as a function of time on a radial orbit using theequation of dynamical friction
GM (r)
d2 r
=−
r̂
dt2
r2


2x −x24πG2 ρMBH ln Λerf(x) − √ e
v̂,
−
v2
π

(10)

√
where M (r) is the mass within radius r, x = v/ 2σ,and the Coulomb logarithm ln Λ is taken to be equal to1, for reasons described in Madau & Quataert (2004) andMaoz (1993). It should be noted that the dynamical friction times for radial orbits are highly dependent on theassumed central density (or equivalently, core radius),
and so are only a rough approximation.
4.3. Scaling of Cluster and Void Boxes(9)

Here η = q/(1 + q)2 , where q = m1 /m2 is the mass ratioof the smaller to larger black hole, αi = S i /m2i is thedimensionless spin of black hole i, k and ⊥ refer to components parallel and perpendicular to the orbital angularmomentum, respectively, ê1 and ê2 are orthogonal unit
vectors in the orbital plane, ξ is the angle between the un-

Our simulation considerably overproduces stellar masscompared to the amount of dark matter present. This is acommon problem in cosmological simulations (Oser et al.
2010; Guo et al. 2010). As described above, our prescriptions for the growth of the central BH are such that themass of the BH must be nearly proportional to the stellar
mass of its host. Thus the excess stellar mass per unit
cosmic volume in our simulations also implies an excessof BH mass. We attempt to allow for this in a simple way
by scaling down our results for the cluster and void boxes

SMBH Mass Assembly Via Accretion and Mergers
3
2

0.3
0.25

Mmergers /Mtot

10-1
Central Mass from Gas Accretion
Central Mass from Mergers
Total Central Mass
Mass in Orbiting BHs
M• − M∗ Fit

log(M• /108 M⊙ )

-2
-3
3
2

Void Box
Cluster Box
Global AverageM∗•log( 10M
8 M ) = 1.12851 log( 1011 )+ (0.348448)⊙

Void Box

-3
1

-1

0

1

2

3

Fig. 2.— The number-weighted mean fraction of total central
SMBH mass gained via BH mergers as a function of SMBH mass
at z = 0. Error bars are 1σ. Values are shown for BHs in the voidand cluster boxes as well as for an approximate “global average”
combination of the two (see §4.3).

-2

0

-2

log(M• /108 M⊙ )

-1

-1

0.1

-3

0

-2

0.15

0

9

1

0.2

0.05

M∗
•
log( 10M
8 M ) = 1.10725 log( 1011 )+ (0.445239)⊙

Cluster Box

72

log(M∗ /1011 M⊙ )Fig. 1.— M• − M∗ relation at z = 0. The points are medianmasses and the error bars represent one quartile. We plot themedian BH mass per galaxy in four categories: the total massof the central SMBH (M• ), the amount of mass in the central
SMBH that was accreted from galactic gas, the amount of central
SMBH mass from mergers with less massive SMBHs, and the total
amount of mass found in SMBHs that are orbiting in the galaxy.
The black dashed line is a ﬁt to M• − M∗ for all the galaxies ineach simulation box (not only the displayed median points), andthe resulting relation is displayed on the plot.
Gray solid lines are two diﬀerent M• − M∗ relations from observational data; the upper one is from Bennert et al. (2011), the lower
one from Cisternas et al. (2011). The orange dot-dashed line is a ﬁt
to M• − σ data from the Virgo cluster from Ferrarese et al. (2006),
converted to M• − M∗ using Equation 3. The star symbol represents the position of M87 and its SMBH (Gebhardt & Thomas
2009; Forte et al. 2012) marked as a reference.

in proportion to the excess of star formation eﬃciency.
In the cluster, the stellar mass is 3 × 1013 M⊙ withinr200 , the radius within which the mean density is equal
to 200 times the critical density. The dark matter mass
within this same radius is 3 × 1014 M⊙ (Lackner et al.
2012). This implies a stellar to dark matter ratioof 0.1 within the virial radius. Comparing this tothe stellar-halo mass relation found by Leauthaud et al.
(2012) using weak lensing and halo occupation distribution methods, combined with the fraction of halo
mass that is in gas given by Pratt et al. (2009), one
ﬁnds that our simulation overproduces stars for a cluster of this mass by a factor of roughly 4 to 6. Determinations of the stellar-halo mass relation found by
matching simulated dark matter halos to observed galaxy
mass functions (Guo et al. 2010; Behroozi et al. 2010;
Moster et al. 2013) ﬁnd somewhat lower values than

Leauthaud et al. (2012), implying an even larger excess
for our simulation. Recent observational determinationsof low redshift cluster stellar masses using WISE and2MASS and halo masses using Chandra ﬁnd values of thestellar to dark matter ratio in the range ≈ 0.01 − 0.03
(Lin et al. 2012). This corresponds to an overproductionof stellar mass by a factor of 3 to 10. Given these results,
we choose a scaling factor for the cluster of 1/5. Thisfactor is applied to both galaxy and black hole masses.
Further, all predicted observables are scaled down in theway in which they are proportional to the BH or galaxy
mass; e.g, the luminosity due to accretion of gas by theBH, which is directly proportional to the growth of theBH mass, is scaled by 1/5, whereas the square of thegravitational wave strain h2c , which is proportional to
5/3M• (see Equations 11 and 12), is scaled by (1/5)5/3 .
Given the lack of observational data for the star formation eﬃciency speciﬁcally in the void, we assume arbitrarily that the stellar to dark matter mass ratio in thevoid is half of the mean ratio of the universe. We calculate the mean universal value using the z = 0 stellar
mass density from Muzzin et al. (2013), who combined
the Kroupa IMF with measurements from Cole et al.
(2001), Bell et al. (2003), and Baldry et al. (2012) to obtain ρ∗ = 3.07 × 108 M⊙ Mpc−3 . This results in a meanstellar to dark matter mass ratio of 0.00967, and thus aratio of 0.00484 for the void. The stellar to dark matterratio for the entire void box in our simulation is 0.012(Lackner et al. 2012), so we scale the results we obtainfor the void by a factor of 2/5 in the same manner that
we do for the cluster.
It should be noted that these scalings for the void andcluster box are simple and do not reﬂect the more complex trends in stellar mass production in our simulation.
In particular, our simulation does not overproduce stellar mass equally for all galaxies, but instead produces agalaxy stellar mass function whose shape does not match
observations. The simulation overproduces very massive galaxies and underproduces low-mass galaxies, especially in the cluster box. Also, the most massive galax-

8

Kulier et al.

ies are over-merged. Detailed comparison of our galaxy
stellar mass function with observations can be found in
Lackner et al. (2012).
The cluster and void boxes in the cosmological simulation we use are +1.8σ and -1σ ﬂuctuations in the cosmic density ﬁeld, respectively, and were chosen so as tobracket the “global average” of various physical quantities (Cen 2011a). To approximate this global average, we
combine the rescaled quantities from the void and cluster boxes in a weighted average. We choose weights suchthat the average stellar mass density at z = 0 is equal
to 3.07 × 108 M⊙ Mpc−3 , the observed local stellar massdensity. Although we only count galaxies with stellar
masses greater than 109 M⊙ , galaxies with masses below
this mass limit should contribute negligibly to the total
stellar mass density (Brinchmann et al. 2004). This results in a weight of 19% (per unit volume) for the cluster
box and 81% for the void box. We apply these weights
at all redshifts.
All the results given in this paper for the void and cluster boxes are scaled by the aforementioned factors of 2/5
and 1/5, respectively, unless explicitly stated otherwise.
The global average is the weighted average of the scaled
void and cluster values. Because the scaling factors areconstant for all masses and redshifts, it is simple to reobtain the original results from our model, or to calculate
results for a diﬀerent preferred normalization.
We also run multiple realizations of our randomized
model and use them to compute one-sigma errors on the
results we obtain. However, it should be noted that although our black hole growth is modeled with random
scatter, we are always using the same galaxy merger trees
obtained from the simulations of Cen (2011a).
5. RESULTS

In this section, we present the results of following the
SMBH evolution in both the cluster and void environments. Figure 1 shows the black hole mass in diﬀerent categories versus the galaxy stellar mass in the voidand cluster boxes. Our simulations reproduce the empirically observed trend of central black hole mass withstellar mass. Shown for comparison are M• −M∗ relations
from Ferrarese et al. (2006), Cisternas et al. (2011), andBennert et al. (2011). We ﬁnd that in the cluster environment for the most massive black holes a larger fraction of their mass growth occurs due to direct black holemergers compared to lower mass black holes. In the voidbox, where there are on average fewer mergers, the median contribution of direct mergers to the mass inventoryof BHs is zero except for the most massive bin, in which
it is still lower than for BHs of the same mass in thecluster box.
For the same reason, the amount of mass in BHs orbiting in galaxies in the void box is negligible with the
exception of the largest mass bin, as opposed to in thecluster box, where the orbiting mass is substantial for
galaxies with M∗ & 1011 M⊙ . It should be noted that
due to the way we scale down our galaxy and BH masses(§4.3), the number of orbiting black holes will be overestimated for the scaled-down mass range, since the dynamical friction time (Equation 1) for the BHs to spiral in is
calculated based on the unscaled mass of galaxies in ourmodel. However, even for the original masses a substantial number of orbiting BHs are predicted for galaxies

of reasonable masses; in the cluster box, ∼ 3 orbiting
SMBHs are expected for each galaxy with (unscaled)mass 1012 M⊙ , with the number increasing for larger
masses. For the void box, the number for a 1012 M⊙
galaxy is ∼ 0.6 orbiting BHs per galaxy; galaxies withthis original mass are among the most massive in thevoid.
The diﬀerence between the fraction of BH mass contributed by mergers in the void and cluster boxes isshown more clearly in Figure 2. The number-weighted
mean mass fraction from mergers is essentially zero for
SMBHs with mass less than 107 M⊙ in both boxes andincreases for larger masses. For the cluster, the fraction reaches a maximum of 20% for BHs with masses
around 109 M⊙ . For BHs with larger masses, the fraction
decreases to about 12%, although the values have highscatter. In the void, the fraction reaches about 8% for themost massive BHs (M• & 109.5 M⊙ ). The fraction of central black hole mass from mergers can also be examined
as a function of the host galaxy mass. The contribution
from mergers is negligible for host masses M∗ . 109.5 M⊙
in both the void and cluster boxes. It increases with hostmass such that in the void the fraction of BH mass frommergers is 2% at M∗ = 1010 M⊙ , 5% at 1011 M⊙ , and
9% for the most massive galaxies in the void, which havemasses 1011.5 M⊙ . M∗ . 1012 M⊙ . In the cluster box,
the fraction from mergers is 4% at M∗ = 1010 M⊙ , 13% at
1011 M⊙ , and 21% at 1012 M⊙ . The fraction drops slightly
in the cluster for M∗ > 1012.5 M⊙ , likely due to the long
dynamical friction time in such massive galaxies thatcauses orbiting BHs to accumulate instead of merging
with the central BH. The number-weighted mean masscontribution from mergers for all SMBHs is 1.3 ± 0.2%
in the void box and 3.3 ± 0.2% in the cluster box.
Between redshifts z = 4 and z = 0, the mean massweighted merger ratio for SMBH mergers in both thevoid and cluster is between 1 : 4 and 1 : 5. Our results
can be compared to the mean mass-weighted merger ratio
found by Oser et al. (2012) for individual galaxies with
masses 4.5 × 1010h−1 M⊙ . M∗ . 3.6 × 1011h−1 M⊙ fromz = 2 to z = 0 using cosmological re-simulations. Our
average values for the merger ratios are the same for2 > z > 0 as for 4 > z > 0, and are similar to the valueof ∼ 1 : 5 obtained by Oser et al. (2012) for galaxies.
Our values are also roughly consistent with the range ofvalues given for the median mass-weighted merger ratioof galaxy bulges in Hopkins et al. (2010), who studied
galaxy bulges with masses 109 M⊙ . M∗ . 1012 M⊙ at
z = 0 using semi-empirical models. Since SMBH mergers
are subsequent to galaxy mergers and SMBHs and theirhosts are connected by scaling relations, the similarity of
the results is not unexpected.
The mass distribution of central black holes that we
obtain is shown in Figures 3 and 4. Figure 3 shows thefraction of black holes with mass greater than some M• ,whereas Figure 4 shows the fraction of total black holemass contained in black holes with mass > M• . As can beseen in Figure 3, the mass function of black holes is quite
similar in the void and cluster boxes at all redshifts. The
time evolution in the fraction of BHs with mass abovesome value is relatively small in both the void and cluster box. A slight trend can be seen in both boxes in which
the fraction of high mass black holes ﬁrst increases from

SMBH Mass Assembly Via Accretion and Mergers

1

91
z = 1.9z = 0.5

Void Box

0.8

0.8

0.6

0.6

0.4

0.4

z = 1.9z = 0.5f• (> M )

0.2

0.2

Void Boxz = 0.2

Cluster Box

z=0

0

0
1

1
z=0

z = 0.2

Cluster Box

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
0

0
-5 -4 -3 -2 -1 0

1

2

3 -5 -4 -3 -2 -1 0

1

2

3

-5

-4

-3

-2

-1

0

1

2

3

log(M• /108 M⊙ )Fig. 3.— The fraction of central black holes with mass greater than some value M• . The left four panels show the fraction for the voidand cluster boxes at z = 1.9, 0.5, 0.2 and 0. The right two panels show the same curves, but grouped together for the void and cluster
boxes, showing all four redshifts.

1

z = 1.9z = 1.0
z = 0.5z = 0.2
z=0

0.8
0.6
0.4
0.2

Void Box01

fM (> M )

0.8
0.6
0.4
0.2
Cluster Box
0

0.8
0.6
0.4
0.2
Global Average
0
-5

-4

-3

-2

-1

0

1

2

3

log(M• /108 M⊙ )Fig. 4.— The fraction of total black hole mass contained in black
holes with mass above some M• , shown at diﬀerent redshifts.

z = 1.9 to z ≈ 0.2, but then decreases between z ≈ 0.2
and z = 0 as more low mass black holes are added tothe population. As can be seen in Figure 4, the corresponding fraction of total mass contained in high mass
black holes increases over time up to z = 0.2, and thenstops. Figure 4 also shows that in both the cluster andvoid box, the total mass is overwhelmingly concentrated
in the most massive black holes. At low redshifts, half ofthe mass is found in SMBHs with M• & 108.5 M⊙ in thevoid, and in SMBHs with M• & 109 M⊙ in the cluster,
despite the fact that such black holes are less than 5% of
the total population in both boxes.
As described in §4.3, our simulation is known to signiﬁcantly overproduce galaxies with large stellar mass inboth the void and cluster boxes. It produces galaxies
that are more massive than those observed, and thesegalaxies are also over-merged. It also has a tendency tounderproduce galaxies with low stellar mass. Since thegrowth of our BHs closely follows the growth of their
host galaxies by design, we would expect that the mass
distribution of our SMBH population would be similarly
biased toward high masses and away from low masses.
Although we scale down the masses in the cluster and thevoid by a constant factor to match the observed stellar
to dark matter mass ratio in these regions, and scale our“global average” so that the stellar mass density matches
observations at z = 0, the shape of the BH population
obtained with our model is likely to be skewed toward
more massive black holes than in reality. Comparing the
global average mass function we obtain at z = 0 to thatfrom observations (Shankar et al. 2004; Marconi et al.
2004; Hopkins et al. 2007; Tundo et al. 2007), this indeed seems to be the case. Our mass function is a factorof ∼ 5 lower than the mean of these observations for
107 M⊙ . M• . 109 M⊙ . For 109 M⊙ . M• . 1010 M⊙ ,

Kulier et al.

our mass function is within the large 1σ range for someof the observations at these high masses, but the slope isnoticeably much shallower than that observed in any of
the four works; i.e., there is an excess of massive BHs.
The exact eﬀects of these known shortcomings in our
simulation on our model results are diﬃcult to assess.
The number of orbiting BHs and amount of orbiting mass
predicted may be too high, since the dynamical friction
timescale is larger for more massive galaxies, and especially since these timescales are calculated using the unscaled values of the galaxy masses. However, the dynamical friction timescale is also longer for satellite BHs
with lower masses, which would be more plentiful were
our SMBH mass function not skewed toward high masses.
These have opposing eﬀects on the predicted size of theorbiting BH population, and we cannot determine which
is dominant. The fraction of BH mass from mergers (Figure 2) is also aﬀected in opposing ways. The fact thatthe galaxies in the simulation suﬀer from over-merging
implies that the central black holes do as well; however,
the ratio of mass from mergers to total mass is also dependent on the shape of the BH mass function, which
we know to be too low at low masses and too high at
high masses. Other results will be aﬀected by the overmerging as well; it increases the gravitational wave luminosity and strain we predict, as well as the number of
BHs ejected by gravitational wave recoil, results that aredescribed below.
The growth of SMBHs over time follows diﬀerent
trends in the void and cluster boxes. Figure 5 shows
the rate of mass growth via accretion and mergers for
SMBHs as a function of redshift. In both the cluster andvoid, accretion is the dominant source of growth for the
total mass contained in SMBHs at all redshifts. Nevertheless, mergers become more important at recent times
compared to accretion in both boxes. In the cluster, theaccretion rate follows an overall downward trend withtime, decreasing by a factor of ∼ 3 since z = 2, while
the merger rate stays roughly consant. In the void, theaccretion rate increases slightly (by a factor of . 2) sincez = 2, while the merger rate increases by about an orderof magnitude (although the scatter is large).
Because mergers become more signiﬁcant over time,in both the void and cluster the total black hole massfrom mergers is added slightly later on average than thatfrom accretion. In the void, half of the total mass fromaccretion at z = 0 has been added by z ≈ 0.6, whereas
for mergers half the mass has been added by z ≈ 0.4. In
the cluster, both half the merged and accreted mass areadded before that in the void, and the diﬀerence between
the two is larger; half the accreted mass is added by z ≈1.2, while half the merged mass is added after z ≈ 0.8.
These SMBH growth trends parallel those of galaxies.
Galactic mass assembly in the cluster tends to happen
earlier than in the void, and in both the cluster and void
mass buildup by accretion peaks at earlier times than bymergers (Lackner et al. 2012).
The connection between galaxy growth via galaxy
mergers and BH growth via BH-BH mergers can also
be seen in the top panel of Figure 6. This panel shows
the number of mergers, weighted by the merger mass
ratio, per central BH in a Hubble time. The merger history of the BH population approximately follows thatof the galaxy population — which is to be expected, as

dM /dt/Comoving Volume [M⊙ yr−1 h3 Mpc−3 ]

10

0.001

0.0001

1e-05

1e-06

1e-07

Accretion in Cluster Box
Accretion in Void Box
Mergers in Cluster Box
Mergers in Void Box

1e-08
0

0.5

1

1.5

2

2.5

3

3.5

z

Fig. 5.— Growth rate of mass contained in central SMBHs perunit cosmic volume from accretion and mergers. The dashed lines
show the rate of mass increase due to gas accretion onto central
black holes; the solid lines show the rate of mass increase due tomergers with smaller black holes. Red lines represent the cluster
and blue lines represent the void.

BH mergers are subsequent to galaxy mergers, althoughsometimes delayed by dynamical friction. One can also
see in this panel that in the cluster for z > 0.5, on averagea central BH is predicted to experience the equivalent inmass increase of a ∼ 1 : 5 to ∼ 1 : 4 merger in a Hubble
time. In the void, we predict an average of the equivalentof a ∼ 1 : 10 merger in a Hubble time at all redshifts.
While this implies a diﬀerence of ∼ 2 in the growth due tomergers per central galaxy between the void and cluster,
this is much less than the ∼ 1.5 − 3 order-of-magnitude
diﬀerence in the mass growth rate due to mergers perunit comoving volume seen in Figure 5.
The bottom panel of Figure 6 shows a related quantity,
the fraction of central BHs ejected from their host galaxies in a Hubble time as a function of redshift. This follows
the trend of the BH merger rate closely, since mergers
cause ejections via gravitational wave recoil. The fraction ejected per Hubble time in the cluster is fairly constant over time at ∼ 0.07. In the void, the scatter is verylarge, but for z < 1 is ∼ 0.04.
Figure 7 shows the expected bolometric luminosity
density emitted by the SMBHs due to gas accretion as
a function of redshift for the cluster and void boxes aswell as the global average. We assume an eﬃciency ofǫ = 0.1, so that Lbol = 0.1Ṁacc c2 . The quoted results
can be scaled for other assumed values of the eﬃciency.
Because the luminosity density is taken to be directly
proportional to the rate of gas accretion, the shapes of
the curves for the void and cluster boxes are the same as
those for the accretion rate in Figure 5.
The expected global average luminosity output isroughly constant with redshift for z > 0.5, decreasing forthe most recent times. In the lower panel, we compare
our estimated global average luminosity to observations.
Shown also is the observed QSO luminosity density from

SMBH Mass Assembly Via Accretion and Mergers
1

11

1e+10

Cluster Box
Void Box
Global Average

0.1

1e+08
Lbol /Comoving Volume [L⊙ h3 Mpc−3 ]Events per Galaxy/Central BH in a Hubble Time

1e+09

Galaxy Merger Rate in Cluster Box
Galaxy Merger Rate in Void Box
BH Merger Rate in Cluster Box
BH Merger Rate in Void Box

0.01

Ejection Rate in Cluster Box
Ejection Rate in Void Box

0.1

1e+07

1e+06
1e+09

1e+08

1e+07
Observed Quasar Luminosity Density
10−4 × SFRD × c2
Our Calculated Global Average

0.01

1e+06
0
0

0.5

1

1.5

2

2.5

3

3.5

0.5

1

1.5

2

2.5

3

3.5

4

z

z
Fig. 6.— Top Panel: Merger rates of galaxies and SMBHs. The
merger rates are expressed as the number of mergers per galaxy
or central BH, weighted by merger ratio (e.g., a merger with ratio
1:2 is counted as 1/2), divided by the length of the simulation
timestep and multiplied by the Hubble time at that redshift. The
resulting value is a dimensionless number, giving the number ofmergers weighted by ratio per Hubble time. Dotted lines represent
the merger rate for galaxies and solid lines represent the merger
rate for black holes.
Bottom Panel: The ejection rate of black holes from their hostgalaxes, expressed as the rate of ejections in each simulation
timestep multiplied by the Hubble time, giving the number of ejections per Hubble time.

Fig. 7.— Top Panel: Estimated SMBH bolometric luminosity
E = 0.1Ṁacc c2 per unit comoving volume due to gas accretion as
a function of z for the void and cluster boxes and their weighted“global average”.
Bottom Panel: SMBH bolometric luminosity from diﬀerent
sources. In magenta is the observed QSO luminosity densityfrom Hopkins et al. (2007). In black, for reference, is an estimateof the QSO luminosity density using the observed SFRD from
Hopkins & Beacom (2006), based on an accretion rate onto theblack hole of 10−3 Ṁ∗ and an eﬃciency factor of ǫ = 0.1; the grayarea represents an error of 1σ. The blue line is our calculation
for the global average bolometric luminosity, also shown in the top
panel.

Hopkins et al. (2007), as well as ǫ × 0.001 × SFRD × c2 ,using the observed star formation density (Madau diagram) from Hopkins & Beacom (2006). The observed
QSO luminosity density has a roughly similar shape as
the “expected trend” calculated from the observed SFRD,
although it is somewhat lower. The luminosity density
calculated from our model matches the overall magnitude of the observed luminosity density well, but does
not replicate the shape. The insuﬃcient downturn at
low redshift in our result may be due to the fact that thecosmological simulation we use is known to especiallyoverproduce stars at low z (Lackner et al. 2012), which
would cause the proportional BH accretion rate to remain too high as well. Our scaling of the void to cluster
is also quite simplistic, taking the void to cluster ratio
to be constant with redshift. If this is not the case, theshape of our predicted trend would also change.
We also calculate the expected total energy density
emitted in gravitational wave radiation as a result of BH-

BH mergers. The results are shown in Figure 8 for the
global average. The luminosity follows a general increase
with time, with the increase being signiﬁcantly steeper
for z & 2.5. The gravitational wave luminosity, whilevery similar as a function of redshift to the rate of mass
increase due to mergers (shown for the void and cluster
separately in Figure 5), is not directly proportional, since
the energy in emitted gravitational wave radiation is also
dependent on the mass ratio of the merging black holes(see Equations 4-8).
Further, we calculate the gravitational wave strain produced by our population of black holes using the method
described in Sesana (2012b). For a population of merging
black hole binaries, where the black holes in each binary
have masses M1 and M2 with M1 > M2 , the characteristic amplitude of the gravitational wave signal hc is given
byZZZ
4
1 dEgw (M)d3 n
h2c (f )=(11)
dzdM1 dqπfdzdM1 dq 1 + z d ln fr

12Kulier et al.
0.8
0.7
0.6
σlog(M• )EGW /∆t [ergs s−1 h3 Mpc−3 ]

1e+42

1e+41

0.5
0.4
0.3
0.2

1e+40
0

0.5

1

1.5

2

2.5

3

3.5

z

Fig. 8.— The global average gravitational wave luminosity density resulting from BH-BH mergers.

where the energy emitted per logarithmic frequency interval is
π 2/3 5/3 2/3
dEgw
=
M fr
(12)
d ln fr
3
where M = (M1 M2 )3/5 /(M1 + M2 )1/5 is the chirp massof the binary and fr = (1 + z)f is the rest frame frequency of the gravitational radiation. The amplitude A
is deﬁned by
−2/3

f
.
(13)
hc (f ) = A
yr−1
We ﬁnd a global average strain amplitude of log A =−14.82 ± 0.13 (1σ error). This is below the current
observational upper limit of A = 6 × 10−15 found by
van Haasteren et al. (2011). Our result falls between
the recent theoretical results of Sesana (2012b) and
McWilliams et al. (2012); the former predicts a 3σ rangeof 1.1 × 10−16 < A < 4.2 × 10−15 , while the latter
predicts an expected strain of A = 4.1 × 10−15 with a
2σ lower limit of 1.1 × 10−15 . Our mean amplitude ishigher than that of Sesana (2012b) and lower than thatof McWilliams et al. (2012), but within 2σ of each.
5.1. Scatter in the M• − M∗ relation with mass

We divide the galaxies in the cluster box at z = 0 into
bins in stellar mass, and ﬁt separate M• − M∗ relations
to each bin. We then calculate the scatter around therelation in each bin, σ, given byP2[log10 (M•,i ) − α − βxi ]
i
σ2 =
,
(14)
Ndof
where α and β are the coeﬃcients of the ﬁt. We plot the
scatter for each bin in Figure 9. The relation is found tobe tighter for black holes with larger masses. The voidbox, not shown here, exhibits a similar trend. Such a
decrease in scatter with increasing M∗ is the expected
result of galaxy and subsequent BH-BH mergers, whichtighten the M• − M∗ relation due to the central limit
theorem (Hirschmann et al. 2010).
McConnell & Ma (2012) have done a similar calculation using a large data set of SMBHs with measured

0.1
-3

-2

-1

0

1

2

log(M∗ /1011 M⊙ )Fig. 9.— Scatter around the M• − M∗ relation as a function ofM∗ for the cluster box at z = 0. Points represent the scatter σaround a separate M• − M∗ relation ﬁt in each bin. Error bars are
one standard deviation of σ.

masses, and have obtained the scatter in various black
hole-galaxy relations. While they were unable to ﬁnd asigniﬁcant decrease in the scatter as a function of mass
when taking into consideration their large error bars,
our results are nevertheless not inconsistent with theirM• − Mbulge relation.
5.2. Corrections to the Soltan argument
The Soltan argument for obtaining the accretion eﬃciency onto SMBHs depends on our ability to accurately
measure the mass density in SMBHs at the present day.
Since we observe such black holes only at the centers ofgalaxies, if some black holes have been ejected via gravitational wave recoil into the outer regions of the galaxy
or out of the galaxy entirely, they will not be accounted
for in the observed present day SMBH mass density. Thisis also true for those still in orbit after a merger. By taking into account such recoils in our model, we keep trackof the fraction of mass in orbiting and ejected BHs thatmust be a correction to the Soltan argument. We reportthe values and 1σ errors for the logarithm of the mass andnumber fractions of orbiting and ejected BHs. We also
give ranges for some of the results we obtain in multiple
realizations of the model to emphasize that some of theseresults can vary widely between our diﬀerent realizations
and so should be taken as uncertain.
We ﬁnd that at z = 0, in the void, the fraction of
all BHs that are orbiting in the outskirts of their hostgalaxies is log fn = −1.51 ± 0.08, with a range between
our diﬀerent realizations of −1.67 to −1.35. The fraction of the total SMBH mass in orbiting black holes islog fm = −2.47±0.21, with a large range −2.92 to −2.12.
For the cluster, the amount of orbiting mass is muchgreater than in the void, as can already be seen in Figure 1. In fact, there is a very large population of orbiting
SMBHs in the cluster, with log fn = −0.39 ± 0.01 (about40%) of BHs at z = 0 orbiting; however, most of theseBHs are quite small (since less massive BHs have larger
dynamical friction times), and so the fraction of mass inorbit is log fm = −0.87 ± 0.07. It should be noted thatthese orbiting black holes are highly concentrated in theSMBH Mass Assembly Via Accretion and Mergers
most massive galaxies in our cluster box — as could be
expected from the fact that a larger galaxy will have alonger dynamical friction time. In fact, approximately
one third of all orbiting BHs in the cluster are found in
the most massive galaxy. However, as mentioned above,
our largest galaxies are considerably overmassive before
scaling (the most massive being M∗ = 1.75 × 1013 M⊙ ),and so the number and fraction by mass of orbiting BHs
is probably exaggerated. Combining the cluster and void,these values correspond to a global average fraction oforbiting BHs of log fn = −0.53 ± 0.01. The fraction ofmass that is unaccounted for by observing galaxy centers
is log fm = −0.98 ± 0.07, with range −1.13 to −0.81.
We also predict the fraction of black holes that arecompletely ejected from their host galaxies due to gravitational wave recoil. At z = 0, we ﬁnd that in the voidlog fn = −1.61 ± 0.16, where fn is the fraction of all
black holes that are not associated with any galaxy as aresult of being ejected, equivalent to a loss of fm of thetotal SMBH mass, where log fm = −1.59 ± 0.26 (range−2.09 to −0.97). In the cluster log fn = −1.38 ± 0.06
and log fm = −1.39 ± 0.20 (range −1.85 to −0.94). It
should be noted that while the fraction of black holes
ejected is roughly constant for all our runs of the model,
the fraction of mass ejected can vary signiﬁcantly from
one to the other, so the values we obtain are not verycertain. The global average fraction of ejected BHs islog fn = −1.43 ± 0.05, with a range −1.59 to −1.34, andthe fraction of mass ejected is log fm = −1.41 ± 0.17 witha range −1.80 to −1.02.
In combination, we predict that the total correction
to the Soltan argument from both unaccounted-for orbiting and ejected BHs is within the range 1.6 − 15%(log fm from −1.80 to −0.81), with an average in the log
of log fm = −1.19 ± 0.25, or linearly 7.4 ± 3.7%.
6. CONCLUSIONS

We have used the results of a set of hydrodynamicgalaxy simulations of a void and cluster region in the universe to predict the evolution of the supermassive black
holes that reside in these galaxies. We ﬁnd signiﬁcantlate time growth of black holes in massive galaxies, although this growth is likely exaggerated in our simulation. Our predicted M• − M∗ relation agrees well withobserved trends, as could be expected since our accretion
rate onto the black holes was set to the observed value of
10−3 Ṁ∗ (with some scatter). We calculate the contribution to the mass of the central black hole from accretionof gas and mergers with smaller black holes subsequent to
the mergers of their two host galaxies. We ﬁnd that in thecluster, the total BH mass from mergers is added later on
average than the mass from accretion, with half the massfrom accretion added before z ≈ 1.2 and half the massfrom mergers added after z ≈ 0.8. In the void, half thetotal mass accreted onto the BH population is accreted
before z ≈ 0.6, but half the merged mass is added afterz ≈ 0.4. Mergers contribute a negligible amount to themass of black holes with M . 107 M⊙ in both the cluster
and the void region. In the void, the number-weighted
mean fraction of mass from mergers rises with black holemass up to ≈ 8% for BHs with M• & 109.5 M⊙ . In thecluster, the fraction from mergers reaches a maximum
value of 20% for M• ≈ 109 M⊙ , and decreases for larger
mass BHs. For all BHs with some mass contribution

13

from mergers, the mean fraction of mass from mergers
is larger in the cluster box than for galaxies of the same
mass in the void box, although values for high-mass BHs
have large scatter (Figure 2).
Additionally, we predict the mass in black holes orbiting in galaxies due to a galaxy-galaxy merger or gravitational wave recoil. While essentially negligible inthe void box except for the most massive few galaxies(M∗ ∼ 1011.5 M⊙ ), a signiﬁcant amount of such mass is
expected in cluster galaxies with M∗ & 1011 M⊙ at z = 0.
In the cluster, approximately 40% of the BHs and 14% of
the BH mass is orbiting. We predict around 3 orbiting
black holes on average for a galaxy in the cluster withunscaled mass around 1012 M⊙ , or scaled mass around2 × 1011 M⊙ . Such orbiting black holes are expected to
produce observational signatures such as stellar tidal disruption ﬂares that are oﬀ-center in the galaxy (Li et al.
2012; Liu & Chen 2013). They are also a candidate to explain observed ultra-luminous X-ray sources (Islam et al.
2004; Volonteri & Perna 2005; McWilliams et al. 2012).
We compute the expected energy emitted in gravitational wave radiation due to black hole mergers, shownin Figure 8. More energy is expected at smaller redshifts
due to the larger amount of mass added via SMBH mergers during this time. We calculate the total strain amplitude from gravitational waves to be log A = −14.82 ±0.13. We also compute the bolometric luminosity fromaccretion, shown in Figure 7, which is directly proportional to the mass increase from accretion. Our expected
trend has roughly the same magnitude as the observed
luminosity density (∼ 2 × 108 L⊙ h3 Mpc−3 ), but does
not decrease suﬃciently for z < 1 to match the observed
trend.
We keep track of SMBHs that end up orbiting in agalaxy due to insuﬃcient dynamical friction and SMBHs
that are ejected from their hosts by gravitational wave
recoils. These two populations would not be accounted
for in attempts to calculate the local mass density of
SMBHs by measuring the masses of black holes at thecenters of galaxies. As such, they are a correction to the
Soltan argument. We ﬁnd that such SMBHs comprise
between 1.6% and 15% of the total mass in SMBHs, witha mean of 7.4 ± 3.7%.
We also ﬁnd a modestly decreasing variance around theM• − M∗ relation with increasing mass in both the cluster and the void, shown for the cluster in Figure 9. Thisis a result of the fact that more massive BHs have undergone more mergers, which tighten the scaling relation
due to the central limit theorem. Although current observational data are not suﬃcient to conﬁrm or disprovethe existence of this decrease in scatter, our results are
consistent with the latest observational ﬁndings within
the errors.
Therefore, late time mergers and their environment
have interesting and observationally detectable consequences for the mass assembly history of supermassive
black holes.
Computing resources were in part provided by theNASA High-End Computing (HEC) Program throughthe NASA Advanced Supercomputing (NAS) Division atAmes Research Center. This work is supported in partby grant NASA NNX11AI23G. AK is supported by a

14

Kulier et al.

NSF Graduate Research Fellowship.
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