Draft version November 1, 2018
Preprint typeset using LATEX style emulateapj v. 7/8/03
HOW MERGERS MAY AFFECT THE MASS SCALING RELATION BETWEEN GRAVITATIONALLY BOUND
SYSTEMS
Chien Y. Peng1,2
(Received 2003 March 13)
Draft version November 1, 2018
ABSTRACT
Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy stellar bulge masses
(Mbulge) and there are several ideas to explain the origin of this relationship. This study isolates
the role of galaxy mergers from considerations of other detailed physics to more clearly show how a
linear BH–galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how primordial BHs
were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this
circumstance, the MBH-Mgal relation is a passive attractor that eventually converges to a tight linear
relation because of two basic statistical effects: a central limit-like tendency for galaxy mergers which
is much stronger for major mergers than for minor mergers, and a convergence towards a linear relation
that is due mainly to minor mergers. A curious consequence of this thought experiment is that, if
galaxy bulges are formed by major mergers, then merging statistics naturally show that MBH would
correlate more strongly with bulge dominated galaxies, because of stronger central-seeking tendencies,
than with disk dominated galaxies. Even if some other physics is ultimately responsible for causing
a linear MBH-Mbulge relationship, this thought experiment shows that, counter to intuition, random
merging of galaxies tends to strengthen rather than weaken a pre-existing, linear, correlation. This
idea may be generalized to other gravitationally bound systems (dark matter halo, compact nuclear
objects) that retain their physical identities after experiencing mergers.
Subject headings: galaxies: bulges — galaxies: formation — galaxies: evolution – galaxies: statistics
1. INTRODUCTION
In recent years, there have been several surpris-
ing discoveries of fundamental scaling relations be-
tween supermassive black hole masses (MBH) and
large scale galaxy bulge properties: stellar veloc-
ity dispersion σ∗, bulge mass Mbulge, the pro-
file slope of galaxies, and the inner core ra-
dius (Gebhardt et al. 2000; Ferrarese & Merritt 2000;
Kormendy & Richstone 1995; Magorrian et al. 1998; Ho
1999; Kormendy & Gebhardt 2001; Graham et al. 2001;
Marconi & Hunt 2003; Häring & Rix 2004; Barth et al.
2005; Lauer et al. 2006; Greene & Ho 2006; Woo et al.
2006). At masses lower than Mbulge. 1010M⊙, the cen-
tral object with which galaxies correlate may be either in-
termediate mass BHs in dwarf galaxies (Filippenko & Ho
2003; Barth et al. 2004, 2005; Greene & Ho 2006) or
central massive star clusters (Ferrarese et al. 2006;
Graham & Driver 2007). The small amount of intrin-
sic scatter between black holes and bulges is often inter-
preted to suggest a causal connection between the two
— that the growth of one might somehow regulate the
other (e.g. Silk & Rees 1998; Di Matteo et al. 2005).
When did the fundamental BH scaling relationships
come about? At higher redshifts, observations are still
in the early stages, complicated in part by the chal-
lenges of measuring the BH mass and the bulge prop-
erties in the same galaxy. However, recent evidence
from the study of quasar host galaxies indicates that a
fundamental MBH-Mbulge correlation might have been
present as early as z ∼ 4 (Peng et al. 2006a,b). Fur-
1 Space Telescope Science Institute, 3700 San Martin Drive, Bal-
timore, MD 21218; cyp@stsci.edu.
2 STScI (Giacconi) Fellow
thermore, there also appears to be an evolution in the
MBH-Mbulge ratio by a factor of ∼ 4 in the same studies,
which points to the possibility that the BH masses may
have matured more quickly than their surrounding stellar
bulge mass in the past. In addition, other observations
that use [Oiii] and CO emission line widths to infer the
gravitational potential of quasar bulges (Shields et al.
2003, 2006; Salviander et al. 2006, 2007; Ho 2007) sug-
gest similar trends3, and residual traces of evolution re-
main detectable even below z = 1 (Treu et al. 2004, 2007;
Woo et al. 2006). Despite the general agreements ob-
servationally (but, see Li et al. 2006; Borys et al. 2005),
there remain several weaknesses that still complicate the
interpretation: a significant factor of 2 systematic un-
certainty in the normalization of the BH mass calibra-
tion, the possibility that the normalization of the virial
BH mass estimate (Kaspi et al. 2000; Vestergaard 2002;
Onken et al. 2004; Kaspi et al. 2005; Greene & Ho 2005;
Peterson & Bentz 2006; Vestergaard & Peterson 2006)
may evolve with time, because of the fact that the bulge
masses are not directly measured, and the possibility
that the host galaxy mass may be biased low in high
redshift quasars (Lauer et al. 2007) because of the steep
decline in the luminosity function of galaxies. With re-
gard to the latter selection bias, it is worthwhile to note,
however, that low redshift (z . 0.3) quasars host galax-
ies (McLure & Dunlop 2001), which have similarly high
MBH andMgal to the high redshift objects in Peng et al.
(2006a,b), do not exhibit a mass dependent bias even
when the scatter is large. Despite these uncertainties,
the finding of an existing strong MBH-Mbulge correla-
3 A lack of evolution in the stellar velocity dispersion im-
plies that bulge mass decreases with look-back time (see, e.g,
Robertson et al. 2006b).
http://arxiv.org/abs/0704.1860v5
2 PENG
tion at z & 1 is likely to be more secure.
Controversies about the evolution aside, it is not only
puzzling that the MBH-bulge correlations should ex-
ist, but that they would have a small intrinsic scatter
of 0.3 dex in MBH and that the correlation with the
bulge mass is practically linear (Marconi & Hunt 2003;
Häring & Rix 2004; Lauer et al. 2006). These curious
facts have received wide theoretical attention over the
years and could be explained in a number of ways. For
instance, both the MBH-Mbulge and MBH-σ∗ relation
can be produced by gas accretion onto a nuclear disk
(Burkert & Silk 2001; Cen 2007) or turbulent dissipa-
tion of gas (Escala 2006) followed by star formation, by
a combination of BH accretion, galaxy mergers, and star
formation (Li et al. 2006; Kauffmann & Haehnelt 2000;
Haehnelt & Kauffmann 2000), by gravitational collapse
of inner parts of a galaxy that forms the bulge from
a rotating isothermal sphere (Adams et al. 2001, 2003),
by stellar capture (Miralda-Escudé & Kollmeier 2005) in
the accretion disk. The effect of dissipation in galaxy
merging on the BH and fundamental plane scaling rela-
tions has also been visited: Ciotti & van Albada (2001)
speculated, and Robertson et al. (2006a) show, that dis-
sipational mergers can produce the MBH-σ∗ slope and
maintain the fundamental plane of elliptical galaxies. On
the other hand, whereas dissipationlessmergers of ellipti-
cal galaxies also tend to preserve the observed fundamen-
tal plane (Boylan-Kolchin et al. 2006; Robertson et al.
2006a), dissipationless mergers of disk galaxies tend to
produce a relation that more closely parallels the virial
plane.
Despite the aforementioned models, the explanation
that has spawned immense activity is the theory of feed-
back from an active galactic nucleus (AGN). The AGN
feedback idea rests observations that quasars typically
radiate at a fixed fraction (10%−100%) of the Edding-
ton luminosity. If such radiation is produced by a massive
enough BH, this energy budget is in principle sufficient
to quench star formation and terminate the BH growth
itself (e.g. Silk & Rees 1998; Di Matteo et al. 2005;
Cattaneo et al. 2005; Springel et al. 2005; Hopkins et al.
2007, 2006a, and references therein). This idea has been
incorporated into cosmological merger simulations (e.g.
Granato et al. 2004; Fontanot et al. 2006; Croton et al.
2006), and is seen to have profound possibilities for ex-
plaining a wide array of other galaxy evolution puzzles,
including the evolution in the galaxy and quasar luminos-
ity functions, mass functions, star formation rates, the
X-ray background, and the bimodality of galaxy colors
(e.g. see Hopkins et al. 2006a,b, and references therein).
Even though AGN feedback is promising for explaining
many aspects of galaxy evolution, it is possible that some
other mechanism (or mechanisms) may either have signif-
icant influence on the scaling relation between MBH and
Mbulge, or in fact be more fundamentally the cause. It is
therefore worthwhile to look for such factors and to fully
study their effects. Dating back years before AGN feed-
back physics became a popular idea, one such fundamen-
tal factor considered by many was galaxy merging (e.g.
Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann
2000; Ciotti & van Albada 2001; Nipoti et al. 2003;
Islam et al. 2003, 2004; Volonteri et al. 2003). Even
without feedback, those simulations already find it pos-
sible to produce the BH scaling relations, albeit to dif-
fering degrees of agreement with the observations. How-
ever, the reason why a tight linear correlation should
emerge is not immediately apparent even when consider-
ing no other physics besides merging. In fact, as accorded
correctly by intuition, a correlation indeed cannot arise
spontaneously from chaotic combinations of galaxies and
black hole masses in general. As will be shown below,
what does make a tight correlation emerge is the fact
that the galaxy mass function decreases with increasing
mass. This circumstance brings about a number of inter-
esting implications that will be explored in future stud-
ies. In this study, the modest goal is to show that when
the focus is switched from the BH-bulge relation, to un-
derstanding the more general BH-galaxy relation, some
insights may be gained into understanding the growth
and evolution of the MBH-Mbulge relation itself.
This study therefore revisits the issue of galaxy merg-
ing from the standpoint of a thought experiment. This
toy model identifies three basic reasons for why the
MBH-Mbulge relation may appear the way it does, even
if the BH masses and their host galaxy masses were
completely uncorrelated initially, or if they started out
with a steep powerlaw correlation. The over-arching
premise is that the galaxy mass function has a Schechter
(1976) powerlaw form, so that there is a decline in galaxy
number density with increasing mass, especially above
masses M∗. When this circumstance is met, it can
be shown that a “linear attractor” and a central limit-
like tendency can work efficiently to produce a linear
MBH-Mgal relation, and to reduce the scatter over time.
However, while it is tempting to generalize this result
to the MBH-σ∗ relation, it is not as simple to do be-
cause it is not clear how the stellar velocity dispersion
scales during galaxy mergers, something that depends on
physics not considered in this study (e.g. dissipational
vs. dissipationless mergers, star formation, AGN feed-
back – see Robertson et al. 2006b; Hopkins et al. 2007;
Di Matteo et al. 2007; Sijacki et al. 2007).
The main emphasis of the current study is thus
only to present a pure statistical exercise, and as such
will not invoke merger trees or external physics – be-
sides which many and much more sophisticated mod-
els have already been run (e.g. Volonteri et al. 2003;
Islam et al. 2004; Granato et al. 2004; Fontanot et al.
2006; Robertson et al. 2006b; Ciotti et al. 2007). In a
sense, this work examines a common thread shared by
all previous cosmological simulations. While it is tempt-
ing to invoke realism by introducing detailed physics
from the get-go, e.g. star formation, BH accretion, and
AGN feedback, isolating the effects of simple statistics
enables a cleaner exposition of why the convergence be-
havior should be expected. As such, the current study
is not a critique of other, more physical, models which
can also explain the same correlation, or to pass criti-
cal judgment about which is more or less relevant. In-
stead, the main message of this study is that, regard-
less of what other physics may ultimately produce the
MBH-Mbulge relation or weaken it, an existing corre-
lation should strengthen if galaxies continue to merge
thereafter, whether by major or by minor mergers. In
other words, merging alone, in the absence of all other
physics, is a sufficient condition to bring about a tight,
linear, MBH-Mbulge relation over time, and is always
“pulling behind the scenes” to bring about such a corre-
MASS SCALING RELATIONS AND GALAXY MERGERS 3
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Fig. 1.— (a) Two examples of initial correlations between MBH and Mgal. (b) An arbitrary number distribution of MBH at each mass
slice M1 or M2, where µ is the mean value of the distribution.
lation. The issue of interest to follow up is to what extent
this, or other as yet identified mechanisms, may matter
in the end, and to predict more realistic scatter in the
MBH-Mgal relation (as opposed to only MBH-Mbulge)
under this hypothesis using more realistic merger trees
and physics. We will address this subject in a followup
study.
The following discussion will begin by presenting a
heuristic view to explain why a linear MBH-Mgal re-
lation is a natural consequence of random merging (Sec-
tion 2), followed by Monte-Carlo simulations to illustrate
the effect (Section 3), and lastly by a discussion and con-
clusion. In much of the discussion below, the relationship
under consideration is more generally the MBH-Mgal re-
lation, whereas BHs are thought to correlate more tightly
with the spheroid component of Mgal, that is Mbulge. It
will be seen that the tighter correlation between MBH
and Mbulge is a special case of the MBH-Mgal relation
and can be understood in the same framework if this hy-
pothesis represents the dominant mechanism by which
BHs correlate with galaxy masses.
2. HEURISTIC PICTURES
2.1. The Central Limit Theorem of Galaxy Mergers
As galaxies undergo merging, it can be shown that
the scatter in the MBH-Mgal relation diminishes with
increasing number of mergers as a consequence of the
central limit theorem (see Appendix). To see this most
easily, consider first major mergers, which by definition
occur between galaxies of roughly equal mass, often de-
fined to be within the range M1/M2 ≤ 4 (Figure 1a).
Their similarity in mass also means that the number dis-
tribution (i.e. mass function) ofMBH at a specific galaxy
mass, M1, is similar to that at M2, i.e. they are drawn
from parent distributions that have roughly the same
shape (Figure 1b). However, the mean (µ) of the distri-
butions might be offset by an amount that depends on
the steepness of the MBH-Mgal correlation: the steeper
the correlation (Figure 1a, upper ellipse), the larger is
the offset.
Furthermore, consider that in the limit of no initial
MBH-Mgal correlation (Figure 1a, lower ellipse), µ is
constant with galaxy mass. The average of any two
BH masses randomly drawn from the parent distribu-
tion (e.g. Fig. 1b) therefore has a tendency toward
the mean value µ of the parent, by the central limit
theorem. This tendency also means that the resulting
BH mass distribution from mergers, obtained by sum-
ming BHs drawn from the same distribution, will have a
smaller log-normal scatter, σ(log(µ)) = σ(µ)/µ, than the
original log-normal distribution, because the fractional
mass error, σ 〈MBH,1 + MBH,2〉 / 〈MBH,1 + MBH,2〉,
is smaller, which means the scatter in the MBH vs.
Mgal relation, logarithmic on both axes, decreases.
In the instance where the initial MBH mass distribu-
tion (Figure 1b) is normal, with a scatter σBH,init, the
resulting scatter of all galaxies that have undergone
one full cycle of major mergers is σ (log µBH,merge) =
σ (log µBH,init) /
2 (see Appendix). Therefore an en-
semble of galaxies that has undergone Nmaj mergers
should have a scatter in the MBH-Mgal relation that
is reduced by ∼ 20.5Nmaj, compared with the initial rela-
tion. However, a log-normal MBH distribution will have
a different convergence rate.
Clearly, this central limit theorem behavior applies to
a finite parent distribution of any shape, but the size of
the scatter and the rate at which the scatter decreases
both depend on the shape of the distribution and the
steepness of the initial MBH-Mgal correlation. In the
situation where the correlation between MBH and Mgal
is steep (e.g. Fig. 1a, upper ellipse) the effective cu-
mulative mass function of BHs (Figure 1b) residing in
galaxies with M2 ≤Mgal ≤ M1 has a wider σBH,init
than if the MBH-Mgal relation is shallow (Fig. 1a, lower
ellipse). For the same reason, minor mergers also have
wider σBH,init, as the galaxy mass differences are larger,
than do major mergers. Therefore, galaxies that have
only undergone major mergers would produce an MBH-
Mgal relation that is significantly tighter than for galax-
ies that have only experienced minor mergers, for the
same merger rate, and if the initial mass correlation is
not flat. This effect is seen in the Monte-Carlo simula-
tions below.
2.2. Galaxy Merging From a Schechter Mass
Distribution
If the mass density of galaxies follows a Schechter pow-
erlaw form (Schechter 1976),
Φ(M) = Φ0
, (1)
4 PENG
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log (M     )gal
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Fig. 2.— (a) No initial correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy
at a much lower Mgal than with one that is comparable to itself (due to the bottom heavy mass function), but roughly of comparable
MBH (due to non-correlation between MBH-Mgal). Thus the net evolutionary vector is steep. In contrast, a galaxy at location 2 is likely
to merge with another that is comparable in both MBH and Mgal to itself, so the evolution vector is shallower. (b) An initial, strong
powerlaw, correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy comparable in Mgal
but at a much lower MBH than itself. Thus, the net evolutionary vector is shallow. In contrast, a galaxy at location 2 is likely to merge
with another of comparable in MBH and Mgal to itself, so the evolution vector is steeper.
then it is possible also to show that a linear MBH-Mgal
relation naturally emerges over time, so that the relation,
MBH(z) = Γ(z)Mgal(z)β, (2)
eventually takes on β = 1. The value of Γ, which lo-
cally is measured to be Γ(0) ∼ 1/800 for bulges (e.g.
Häring & Rix 2004), is otherwise arbitrary in the discus-
sion below. Γ is degenerate with respect to assumptions
about the initial scatter of the MBH-Mgal relation, the
initial slope β, and the initial normalization Γ(∞), for
which there are currently insufficient observational con-
straints; it will not be addressed further in this study.
The other assumption used here is that the probability
for two galaxies to merge comes from Monte-Carlo sam-
pling of a Schechter mass function (Eq. 1). In actuality,
galaxies do not merge randomly, especially at late times.
However, complete randomness is only used to facilitate
the discussion, and is not a pre-requisite, since the rea-
soning depends only on the fact that minor mergers oc-
cur more frequently than major mergers. This assump-
tion does mean that, depending on the relative balance
of major vs. minor mergers, the effects described here,
namely convergence toward linearity versus central-limit
behavior, may be more relevant at some epoch in time
than at others
The reason that a linear correlation emerges through
galaxy mergers is illustrated in Figure 2. Figure 2a shows
the situation in which the initial BH and galaxy mass
distributions are completely uncorrelated, so that β = 0.
The lower part of the diagram shows hypothetical mass
functions with two different “faint end” slopes, A and B,
which will be individually considered in the Monte-Carlo
simulation below. If there is no correlation between the
MBH and Mgal, then the ratio MBH/Mgal will be, on
average, larger for low mass galaxies than for high mass
galaxies as can be seen by comparing the MBH/Mgal ra-
tio at any two locations, for example, those labeled “1”
and “2.” Therefore, as galaxies merge, a massive object
at the extreme end of the mass function, located at posi-
tion 1, on average, is more likely to merge with another
having a much larger MBH/Mgal, thereby evolving the
merger product in a steep upward direction, as exagger-
ated by the vertical arrow. In contrast, an object at
position 2 is likely to merge with objects comparable in
both MBH and Mgal to itself, so the net evolutionary
vector has a shallower slope. Therefore, the cumulative
effect of mergers along the mass spectrum is to steepen
the massive end of the MBH-Mgal relation relative to
the lower extreme, even as the lower end grows in MBH
and Mgal on average.
In the other extreme, Figure 2b, if the primordial rela-
tion between the BH and galaxy masses is steep, corre-
sponding to β ≥ 1, the opposite behavior occurs. Galax-
ies at location 1 generally have a larger MBH/Mgal ra-
tio than galaxies that have lower mass. Therefore, the
MBH/Mgal ratio for massive galaxies would tend to de-
crease through mergers. The net effect on massive galax-
ies is to evolve the merger remnant more quickly to the
right on average than a lower mass galaxy at location 2.
Because of a mirror symmetry between Figures 2a and
2b, the natural equilibrium state of the MBH-Mgal re-
lation is at β = 1, so that further merging of galaxies
would evolve remnants along the linear relation, with a
constant ratio Γ=MBH/Mgal. Also because of the con-
MASS SCALING RELATIONS AND GALAXY MERGERS 5
vergence toward this “attractor” state the scatter in the
relationship would necessarily decrease over time through
galaxy merging.
Lastly, it is worth mentioning that the presence of a
break in the galaxy mass function at M∗ is a sufficient,
but not necessary, condition for convergence toward lin-
earity. A pure powerlaw with M∗= ∞ would produce a
similar behavior, however, the convergence is slower for
flatter slopes (α → −1). Furthermore, while the con-
vergence behaviors just described is quite strong for a
Schechter mass function with α = −1, the convergence
would fail for a pure powerlaw with the same slope, be-
cause of a lack of a break in the mass function.
3. MONTE CARLO SIMULATIONS
To illustrate the idea discussed above, and to quantify
how quickly a linear MBH-Mgal relation might emerge,
it is useful to consider several numerical simulations for
the situations shown in Figure 2. For each of the two
scenarios, no-correlation (Fig. 2a) and steep correlation
(Fig. 2b), it is also instructive to consider two differ-
ent initial mass functions, A and B, shown in the lower
half of Figure 2. The two powerlaw slopes explored be-
low are α = −1.5 and α = −0.5, respectively. These
choices are motivated by observations of the luminosity
functions for high redshift galaxies (e.g. Gabasch et al.
2006; Giallongo et al. 2005) under the assumption that
light traces mass.
3.1. Simulation Set-up and Definitions
Definition of the number of major and minor mergers.
— As implied in Section 2, how closely a galaxy lies
to the linear part of the MBH-Mgal relation, and how
tight the final scatter is for an ensemble of galaxies, will
depend on the cumulative merger history. Therefore, the
most useful way to understand the simulation results is
to define the number of major mergers, Nmaj, as the
cumulative sum of all such events over the entire tree for
a given galaxy, not just in the most immediate, that is,
the main, branch. For example, even if a galaxy has never
experienced a single major merger on the main branch
in its lifetime, it could still lie close to the final, linear
part of the MBH-Mgal relation because the progenitors
of the main branch, and their progenitors, and so forth,
could have experienced a number of major mergers.
Evolution of the mass function. — One issue to con-
sider is how the galaxy mass function might evolve, and
whether the path of evolution might affect the final con-
vergence. The effect of galaxy masses growing with time
is to both increase M∗ and steepen the “faint end slope”
(α) of the Schechter function. As galaxy populations
grow in mass and number density, the rate of change in
M∗ and α would affect the rate of convergence to the
final MBH-Mgal normalization, slope, and scatter. Pre-
dicting the rate of change in the MBH-Mgal relation re-
quires realistic merger trees and accounting for other de-
tailed physics such as feedback, which will be addressed
in a followup study. For the current purposes of show-
ing that convergence toward a linear MBH-Mgal relation
does naturally occur, it suffices to consider two scenarios:
a replenishment scenario, in which the Schechter mass
function is continuously, and randomly, replenished as
galaxies merge, and a depletion scenario, in which no new
galaxies are formed to take the place of those that have
merged. Combined with considerations about the initial
mass function slopes, the simulations will have covered
the gamut of sensible possibilities and conditions that
may be present at early and late cosmic epochs.
Initial scatter of the MBH-Mgal relation and the initial
mass function M∗. — In the simulation, the distribution
of galaxies is first drawn randomly from the Schechter
mass function, after which a BH mass is assigned, fol-
lowing Equation 2, by drawing from a log-normal Gaus-
sian distribution with a generous Gaussian dispersion of
σBH = 2 dex, i.e. a scatter of a factor 100 in mass.
The exact choice of the initial scatter is directly propor-
tional to, but otherwise only partially determines, the
final scatter in the MBH-Mgal relation. Other factors
that determine the final scatter depend on how long the
simulation runs, and on the initial value of M∗. Cur-
rently, there are some observational constraint on the
rate of mergers, which will be considered in a followup
study. In this study, the results are merely normalized ar-
bitrarily to match the final MBH-Mgal relation observed
today.
The simulation “clock.” — The progress of time is not
well defined in Monte-Carlo simulations, so it is useful
to define merging cycles for the sake of keeping track of
the simulation progress. Each full cycle is defined as be-
ing complete after the number of merger events equals
the number of galaxies present at the beginning of that
particular cycle. Galaxies that are produced or merged
retain their states for the following cycle. Because some
galaxies may merge multiple times by being drawn re-
peatedly, not all galaxies will be involved in mergers af-
ter each full cycle. The exact definition of the simulation
clock is unimportant, as it is only the relative number of
major vs. minor mergers on average that determines the
degree of convergence toward a linear MBH-Mgal rela-
tion, where a major merger is defined as having a mass
ratio of at most 4:1.
3.2. Replenishment Scenarios
The replenishment scenario is one of the two simple
ways considered to emulate the progress of galaxy evo-
lution. Here, by definition, the rate of galaxy mergers
equals the rate of galaxy number production. The way a
galaxy is newly produced is by being selected randomly
from an initial mass function, parameterized by α and
M∗, which does not evolve with time. In contrast, galax-
ies “grow” only by merging with another member in the
galaxy pool existing at the time, and hierarchical merg-
ing is the only avenue for mass growth. Therefore, as
galaxies merge, the cumulative mass function does un-
dergo evolution. However, the total number of galax-
ies remains constant because of the 1:1 ratio of merg-
ing:replenishment.
Figures 3 and 4 illustrate the results for initial con-
ditions β = 0 (i.e. no MBH-Mgal correlation) and
β = 2 (steep MBH-Mgal correlation), respectively, show-
ing only a small subset of the data points. In each figure,
two different initial mass functions, α = −0.5 (Figs. 3a,
4a) and α = −1.5 (Figs. 3b, 4b) are considered. In each
of the Figures, the initial distribution of the MBH-Mgal
relation (or lack thereof) is shown with crosses. The open
colored data points show the MBH-Mgal development of
galaxies that have undergone Nmaj ≥ 5 major merger
episodes, after 10 (blue triangles), 100 (green squares),
6 PENG
Fig. 3.— No initial correlation (β = 0) in the MBH-Mgal relation, in the replenishment scenario. The black crosses represent the initial
distribution of points, and the solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data
points. The colored data points represent objects that have undergone at least five major merger episodes after 10 (blue triangles), 100
(green squares), and 1000 (red circles) complete “merging cycles.” The crosses are the primordial distribution, corresponding to the initial
mass function. The shaded region illustrates the locus of all points after 1000 cycles; the density of points doubles with each contour level.
The cumulative histograms after the corresponding merger sequences are shown below the data points. (a) An initial Schechter powerlaw
slope of α = −0.5. (b) An initial Schechter powerlaw slope of α = −1.5.
Fig. 4.— Similar to Figure 3, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 3 for details.
MASS SCALING RELATIONS AND GALAXY MERGERS 7
Fig. 5.— Similar to Figure 3b, showing the effect of central-limit
tendencies with increasing number of major mergers for galaxies
after 1000 merger cycles. The colored data points illustrate objects
that have undergone 1 ≤ Nmaj ≤ 4, (blue triangles), 5 ≤ Nmaj ≤
14 (green squares), Nmaj > 14 (red circles) major mergers. The
greyscale contours shows the locus of all the points.
Fig. 6.— Similar to Figure 3, except the BH mass is drawn from
a Schechter law of α = −1.5 instead of a Gaussian distribution.
See Figure 3 for details.
and 1000 (red circles) merger cycles have transpired.
These data points effectively illustrate the progress of
the MBH-Mgal evolution for objects that might be mor-
phologically identified as early-type galaxies of each cy-
cle. For clarity, the contour levels represent the locus of
points after 1000 merger cycles, and the levels are spaced
at multiples of 2 in density. The luminosity functions of
the galaxy pool at the end of the merger cycles are shown
in the lower half of each diagram in corresponding col-
ors and locale in mass. Lastly, a linear reference line is
overplotted in the Figures with normalization given by
R0=800 (Häring & Rix 2004), and the simulations are
scaled/shifted arbitrarily to match; it is not a fit to the
data points.
As shown in Figures 3 and 4, the convergence towards
a tight linear relation is fairly quick. After five major
merger episodes a linear relation starts to emerge regard-
less of the initial conditions of the mass function or the
form of the MBH-Mgal correlation. One reason for this
quick convergence is the central-limit behavior of major
mergers which is shown in Figure 5, in which the increas-
ing number of mergers is represented by different symbols
and shades. The one notable case where the convergence
toward linearity is slower than the other scenarios is Fig-
ure 4b, where the effect is only evident at 1010.5 M⊙ or
greater, even as the scatter has decreased markedly. In
general, if the MBH-Mgal correlation is steep initially,
the tail at low mass remains steep after a large number
of major mergers has occurred, even as the massive end
converges toward linearity.
Lastly, the qualitative convergence effects do not de-
pend on the assumption about the distribution of BH
mass at each galaxy mass. Figure 6 shows an example
that is in direct analog to Figure 3b, except that the BH
mass is instead drawn from a Schechter mass function
with α = −1.5.
3.3. Depletion Scenarios
The other extreme of the merger simulations is to con-
sider what effect galaxy depletion from a finite reservoir
has on the MBH-Mgal relation. Because the number
density of galaxies builds up over time, the depletion
scenario is expected to not be realistic. Nevertheless, it
is useful for illustrating how the MBH-Mgal convergence
is affected by a different evolution in the mass function
as compared with the replenishment scenario.
The depletion scenarios are constructed by creating a
large sample of 5× 105 objects, initially having no corre-
lation between BH and galaxy masses (Figure 7) or with
a β = 2 correlation between the two (Figure 8). The BH
masses are assigned to the galaxies with a log-normal
distribution of dispersion σ = 2 centered around Equa-
tion 2. In each scenario, galaxies are created to have
initial mass functions of α = −0.5 (Figures 7a and 8a) or
α = −1.5 (Figures 7b, 8b). Then, as galaxies merge, no
new ones are created to replace them. As a consequence,
the mass function evolves by growing in M∗, the num-
ber density decreases, and a sharp truncation develops
at low masses (see lower half of Figures 7 and 8). As
the number of merging cycles increases, the scatter de-
creases quickly and converges toward a linear relation, as
illustrated by the solid line. Once again, as shown in Fig-
ure 8 (especially 8b), the convergence is much slower for
steep α and steep β compared with other scenarios. And
while the convergence trends are noticeable, because of
a dearth of minor galaxies with which to merge at late
times (red circles), the slope is virtually “frozen in,” and
the subsequent convergence is due mostly to the central-
limit theorem.
4. DISCUSSION AND CONCLUSION
This study has revisited the issue of how galaxy merg-
ing may affect the MBH-Mgal scaling relation from
the standpoint of basic mass addition and statistics,
thereby clearly isolating the merger cause from other de-
8 PENG
Fig. 7.— No initial correlation (β = 0) in the MBH-Mgal relation, depletion scenario. The contours represent the initial distribution
of points, and a solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The
colored data points represent objects that have undergone at least 1 major merger episodes after 1 complete merging cycles (blue triangles),
10 (green squares), and 14 (red circles). The cumulative histograms after the corresponding merger sequences are shown below the data
points. a) An initial Schechter powerlaw slope of α = −0.5. b) An initial Schechter powerlaw slope of α = −1.5.
Fig. 8.— Similar to Figure 7, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 7 for details.
MASS SCALING RELATIONS AND GALAXY MERGERS 9
tailed physics that must otherwise affect galaxy evolu-
tion. Through Monte-Carlo simulations, a tight, linear,
MBH-Mgal correlation appears to emerge when galaxies
have undergone five or more major mergers (along the
entire tree, not just the main branch), and many minor
ones, for practically all reasonable initial correlations be-
tweenMBH andMgal, or a lack of one. The main reasons
for these behaviors are seen to be the following:
1. The galaxy mass function decreases with increasing
mass.
2. Major mergers have a strong central-limit ten-
dency, so that regardless of the initial MBH-Mgal
correlation, the scatter should decrease with an
increasing number of events. While this ten-
dency also acts on minor mergers, the drive toward
smaller scatter is weaker because minor mergers oc-
cur between galaxies that are vastly discrepant in
both galaxy and BH mass as compared with major
mergers, by definition. The corollary is that the
steeper a correlation between MBH (y-axis) and
Mgal, the stronger the central-limit tendency for
major mergers compared with minor. However,
major mergers alone are not enough to cause the
MBH-Mgal relation to converge to linearity over
time because the ratio MBH/Mgal is not changed
much.
3. Minor mergers are primarily responsible for causing
the MBH-Mgal relation to converge toward a linear
— that is, MBH = ΓMgal— relation because the
mass function of galaxies follows a Schechter pow-
erlaw. Without minor mergers, the MBH-Mbulge
relation can be “frozen” to a slope that is not
necessarily linear. This linear attractor causes a
convergence toward a tighter MBH-Mgal relation;
however, it is less efficient at reducing the scatter
compared with the central-limit seeking tendency
of major mergers, as shown in Figure 5.
It is curious that galaxy merging itself might produce
a linear MBH-Mgal relation. However, a natural ques-
tion that does arise is, “When is the merging statistics
presented in this study relevant?” On the surface, it
is easy to conclude that because the reasoning refers
to a two component model it ought to apply to “dry”
mergers, but perhaps not to a three component model
involving stars, gas, and BH. Thus, the implication is
also that it ought not apply to galaxies undergoing gas-
rich mergers, that is, early cosmic history. However, it
is not clear that such a skepticism is warranted. For
example, in the entire discussion thus far, the abscissa,
Mgal, might just as well refer to Mgal=Mstellar+Mgas,
instead of just Mstellar. If BHs do not grow much by
accretion and that the gas does not get removed from
the definition of Mgal during mergers, then the MBH-
Mgal correlation can emerge from statistical merging.
The argument holds true even if Mgas transforms arbi-
trarily into Mstellar, as long as the sum is conserved. In
the limit where BHs do grow most of their mass dur-
ing AGN accretion, as might be implied by Soltan (e.g.
1982); Yu & Tremaine (e.g. 2002), so that ∆MBH∝Mgas
and ∆MBH≫MBH, then the correlation between MBH
and Mgal comes out by construction rather than by sta-
tistical merging. However, statistical reasoning would
still be a “supporting actor” to reduce the scatter and
to forcibly steer the MBH-Mgal relation in the preferred
linear direction. Likewise, even if BH growth, or other
physics (e.g. gravitational radiation, three body BH
ejection – Merritt et al. 2004; Volonteri & Perna 2005;
Ciotti et al. 2007, and references therein), were a “heat-
ing” source, that is, one that randomizes a tight linear
MBH-Mbulge relation, the linear and central limit attrac-
tors would cause a re-convergence if galaxies continue to
merge thereafter by both major and minor mergers. In
summary, while it is entirely possible that the MBH-
Mbulge relation has origins outside of basic statistics,
galaxy merger statistics can still affect the final outcome
of a MBH-Mgal correlation in both the scatter and the
slope. In any event, statistical reasoning is a fundamen-
tally robust explanation for why random galaxy merging
does not corrupt a pre-existing MBH-Mbulge relation,
which is important to bear in mind in the context of the
MBH-Mbulge or MBH-Mgal relation in a hierarchically
forming universe.
While the MBH-Mbulge relation might have other
origins, it is nonetheless interesting and revealing to
follow through the consequences of statistical merg-
ing. For instance, simple statistics naturally explains
why black holes appear to correlate most strongly
with galaxy bulges, rather than more generally with a
galaxy as a whole, which might include a stellar disk
(Kormendy & Gebhardt 2001): bulge masses, assum-
ing they were assembled through major mergers, have
a stronger central-seeking tendency than disk galaxies,
whose growth history might involve more minor merger
events. As such, the MBH-Mbulge relation is a special
case of a more fundamental MBH-Mgal relation. Revers-
ing the argument, the observational fact thatMBH corre-
lates most strongly with bulge masses, coupled with the
central-limit theorem reasoning, implies that the merger
trees of elliptical galaxies were more dominated by ma-
jor merger events than were disk galaxies. Conversely,
the fact that the scatter in the MBH-Mgal relation is
observed to be much larger for disk dominated galaxies
implies, statistically, that their progenitors, and progen-
itors thereof, have undergone more minor mergers.
The possibility that a more fundamental correlation is
between MBH and Mgal (rather than Mbulge) also has
practical implications for what slope and scatter would
be measured by observations. First, because the slope
changes with mass even for objects that experienced the
same number of major mergers (e.g. Figures 3 and 7),
the deviation from linearity and the intrinsic scatter will
depend exactly on how the data are cut. Simply defining
a sample of objects based on a mass selection cut will
bias one’s measurement of the slope and scatter. Fur-
thermore, defining a sample based on morphology criteria
may also implicitly preselect samples that have certain
major vs. minor merger histories. Observationally, it is
therefore crucial, when comparing intrinsic scatter and
slope of the MBH-Mbulge relation to be specific about
sample selection parameter space, morphology, bulge-
to-disk ratios, or other criteria, lest the conclusions be
caused by subtle but trivial selection biases.
Another consequence of this thought experiment is
that the ratio Γ=MBH/Mbulge approaches an asymp-
10 PENG
totic value with time from having a smaller ratio in
the past. On the surface, this appears contrary to
the findings of Peng et al. (2006a,b); Woo et al. (2006);
Shields et al. (2003, 2006) based on quasar host galaxy
studies that the ratio Γ decreases over time. If the
quasar host galaxy studies are correct and are not sig-
nificantly affected by biases pointed out by Lauer et al.
(2007), then some other physics not considered here is
responsible for causing a decline in the normalization of
Γ with time (e.g. see Croton 2006; Hopkins et al. 2007;
Fontanot et al. 2006). For instance, the abscissa is am-
biguous about what mass Mgal corresponds. If gas mass
is a significant fraction of a galaxy’s mass, then form-
ing stars out of the gas reservoir would decrease Γ over
time, if the abscissa Mgal represents the galaxy’s stellar
bulge mass. Secular growth of galaxy bulges by accret-
ing stars in galaxy disks would also decrease Γ, at the
expense of increasing the scatter. Major mergers of pure
stellar bulges, however, would not cause Γ to decrease
over time.
In hindsight, the results of this study could have been
anticipated from Islam et al. (2003, 2004); Ciotti et al.
(2007), given that the initial conditions used in those
studies are a special case of this one where the initial
BH scatter σBH → 0 (Figure 3a or 3b) (M. Volonteri
and L. Ciotti 2007, private communication). Just as rel-
evant, Croton (2006); Ciotti et al. (2007) show that once
the MBH-Mbulge relation is in place, it is fairly imper-
vious to being randomized by galaxy merging. And be-
cause Islam et al. (2003) also uses realistic cosmological
merger trees, they confirm that the arguments presented
here ought to remain relevant. However, the reasons be-
hind the MBH-Mbulge convergence behavior are difficult
to extract from previous studies because of the use of pri-
ors, the use of identical BH seeds, the inclusion of other
physics, and the focus on only the BH-bulge coevolution
(i.e. major mergers). The latter, especially, is worth
examining further, because the prior that one chooses
about whether the BH correlates with just its bulge or
with the entire galaxy can lead to differing interpreta-
tions.
In particular, one conclusion from Islam et al. (2003,
2004) is that the MBH-Mbulge relation converges to a
slightly non-linear slope of β = 0.9; hence they reason
that other physics, perhaps BH growth through accre-
tion of gas, is required in order to increase the slope
closer to linearity. The reasoning presented in the cur-
rent study, however, would stipulate that linearity is an
asymptotic outcome of mergers, but deviations from lin-
earity come from the possibility that the low mass galax-
ies have not yet achieved the asymptotic limit, because
of a weaker convergence. At low masses, the slope de-
viates from unity in either direction depending on the
initial mass function of the galaxies (e.g. compare Fig-
ure 3a with 3b), on the mass cut of the study, and on the
relative incidence of minor versus major mergers.
An interesting consequence to consider is how the
MBH-Mgal relation might differ between high and low
density galaxy clusters. However, one of the unrealis-
tic side-effects of using Monte-Carlo simulations to de-
termine merger rates is that the normalization of the
MBH-Mbulge relation is the same in all density regimes.
This is because the normalization factor, Γ, depends only
on the ratio of major to minor merger events which, in
the Monte-Carlo universe, is not affected by a simple
rescaling of the mass function. However, in the real uni-
verse, the relative rates of major and minor mergers can
change with density and, as such, may result in different
normalization and scatter in the MBH-Mgal relation.
Lastly, because of the ambiguity in what Mgal corre-
sponds, depending on whether it refers to the total stellar
mass, gas mass, dark matter halo mass, or a combination
thereof, the degree of scatter and linearity would clearly
differ, as a result of different initial mass functions and
merger histories. Because the scenarios considered above
depend on a linear addition of masses, the arguments
therefore may not apply to gas masses that are not grav-
itationally bound to a galaxy. These and other issues will
be addressed in a future study, which will incorporate the
use of realistic merger trees.
ACKNOWLEDGMENT
Over the course of this study, I have greatly en-
joyed lively discussions with many friends and colleagues,
including Jenny Greene, Hans-Walter Rix, Luis Ho,
Eric Bell, Michael Santos, Robert Kennicutt, Rachel
Somerville, Aaron Barth, Christy Tremonti, Darren Cro-
ton, Tommaso Treu, and David Koo. I also thank Jim
Rose, Wayne Christiansen, Marta Volonteri, Luca Ciotti,
and Alister Graham for comments and discussions on
past and future studies. This study greatly benefited
from discussions with Phil Hopkins on issues related to
AGN feedback over the past several months. I also grate-
fully acknowledge insightful comments and suggestions
from the referee, and Science Editor Chung-Pei Ma, and
the support of STScI through the Institute (Giacconi)
Fellowship Program.
APPENDIX
This appendix shows that theMBH-Mbulge relation follows a central-limit-like behavior when galaxies undergo major
mergers. Specifically, this means that if the initial parent distribution of progenitorMBH that undergoes merging is, for
simplicity, normally distributed about a mean BH mass µ, thus having a logarithmic dispersion σ(log(µinit)), then a new
distribution ofMBHs after merging will have a log-normal dispersion that scales as: σ (log (µmerge)) ∼ σ(log(µinit))/
First, the mean, µmerge, of the resulting BH distribution after two BHs merge from the initial parent distribution is:
µmerge = 〈MBH,1 +MBH,2〉 , (1)
where MBH,1 and MBH,2 are drawn from the same parent distribution for major galaxy mergers. Then,
log (µmerge) = log 〈MBH,1 +MBH,2〉 , (2)
MASS SCALING RELATIONS AND GALAXY MERGERS 11
From propagation of errors the log-normal error is: σ(log(x)) = σ(x)/x, then,
σ (log (µmerge)) =
σ 〈MBH,1 +MBH,2〉
〈MBH,1 +MBH,2〉
, (3)
By definition, a distribution obtained by averaging the mass of merging BH pairs is a normal distribution with a mean
of the initial parent distribution, µinit:
µinit =
〈MBH,1 +MBH,2〉
µmerge
. (4)
Because M,1 and M,2 are drawn from a normalized distribution around a parent mean µ, the new distribution of
σ(µmerge) ≡ σ 〈M,1 + M,2〉 is:
σ (µmerge) ∼
σ(µinit)√
× 2. (5)
Substituting Eqs. A4 and A5 into A3 yields:
σ (log(µmerge)) ∼
σ(µinit)
µinit
. (6)
Using the fact that, σ(log(µ)) = σ(µ)/µ, Equation A6 becomes:
σ (log(µmerge)) ∼
σ (log(µinit))√
. (7)
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  Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy
stellar bulge masses (Mbulge) and there are several ideas to explain the origin
of this relationship. This study isolates the role of galaxy mergers from
considerations of other detailed physics to more clearly show how a linear
BH-galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how
primordial BHs were seeded inside galaxies, if the galaxy mass function
declines with increasing mass. Under this circumstance, the MBH-Mgal relation
is a passive attractor that eventually converges to a tight linear relation
because of two basic statistical effects: a central limit-like tendency for
galaxy mergers which is much stronger for major mergers than minor mergers, and
a convergence toward a linear relation that is due mainly to minor mergers. A
curious consequence of this thought experiment is that, if galaxy bulges are
formed by major mergers, then merging statistics naturally show that MBH would
correlate more strongly with bulge dominated galaxies, because of stronger
central-seeking tendencies, than with disk dominated galaxies. Even if some
other physics is ultimately responsible for causing a linear MBH-Mbulge
relationship, this thought experiment shows that, counter to intuition, random
merging of galaxies that harbor random BH masses tends to strengthen rather
than weaken a pre-existing, linear, correlation. This idea may be generalized
to other gravitationally bound systems (dark matter halo, compact nuclear
objects) that retain their physical identities after experiencing mergers.

Draft version November 1, 2018
Preprint typeset using LATEX style emulateapj v. 7/8/03
HOW MERGERS MAY AFFECT THE MASS SCALING RELATION BETWEEN GRAVITATIONALLY BOUND
SYSTEMS
Chien Y. Peng1,2
(Received 2003 March 13)
Draft version November 1, 2018
ABSTRACT
Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy stellar bulge masses
(Mbulge) and there are several ideas to explain the origin of this relationship. This study isolates
the role of galaxy mergers from considerations of other detailed physics to more clearly show how a
linear BH–galaxy mass relation (MBH-Mgal) can naturally emerge regardless of how primordial BHs
were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this
circumstance, the MBH-Mgal relation is a passive attractor that eventually converges to a tight linear
relation because of two basic statistical effects: a central limit-like tendency for galaxy mergers which
is much stronger for major mergers than for minor mergers, and a convergence towards a linear relation
that is due mainly to minor mergers. A curious consequence of this thought experiment is that, if
galaxy bulges are formed by major mergers, then merging statistics naturally show that MBH would
correlate more strongly with bulge dominated galaxies, because of stronger central-seeking tendencies,
than with disk dominated galaxies. Even if some other physics is ultimately responsible for causing
a linear MBH-Mbulge relationship, this thought experiment shows that, counter to intuition, random
merging of galaxies tends to strengthen rather than weaken a pre-existing, linear, correlation. This
idea may be generalized to other gravitationally bound systems (dark matter halo, compact nuclear
objects) that retain their physical identities after experiencing mergers.
Subject headings: galaxies: bulges — galaxies: formation — galaxies: evolution – galaxies: statistics
1. INTRODUCTION
In recent years, there have been several surpris-
ing discoveries of fundamental scaling relations be-
tween supermassive black hole masses (MBH) and
large scale galaxy bulge properties: stellar veloc-
ity dispersion σ∗, bulge mass Mbulge, the pro-
file slope of galaxies, and the inner core ra-
dius (Gebhardt et al. 2000; Ferrarese & Merritt 2000;
Kormendy & Richstone 1995; Magorrian et al. 1998; Ho
1999; Kormendy & Gebhardt 2001; Graham et al. 2001;
Marconi & Hunt 2003; Häring & Rix 2004; Barth et al.
2005; Lauer et al. 2006; Greene & Ho 2006; Woo et al.
2006). At masses lower than Mbulge. 1010M⊙, the cen-
tral object with which galaxies correlate may be either in-
termediate mass BHs in dwarf galaxies (Filippenko & Ho
2003; Barth et al. 2004, 2005; Greene & Ho 2006) or
central massive star clusters (Ferrarese et al. 2006;
Graham & Driver 2007). The small amount of intrin-
sic scatter between black holes and bulges is often inter-
preted to suggest a causal connection between the two
— that the growth of one might somehow regulate the
other (e.g. Silk & Rees 1998; Di Matteo et al. 2005).
When did the fundamental BH scaling relationships
come about? At higher redshifts, observations are still
in the early stages, complicated in part by the chal-
lenges of measuring the BH mass and the bulge prop-
erties in the same galaxy. However, recent evidence
from the study of quasar host galaxies indicates that a
fundamental MBH-Mbulge correlation might have been
present as early as z ∼ 4 (Peng et al. 2006a,b). Fur-
1 Space Telescope Science Institute, 3700 San Martin Drive, Bal-
timore, MD 21218; cyp@stsci.edu.
2 STScI (Giacconi) Fellow
thermore, there also appears to be an evolution in the
MBH-Mbulge ratio by a factor of ∼ 4 in the same studies,
which points to the possibility that the BH masses may
have matured more quickly than their surrounding stellar
bulge mass in the past. In addition, other observations
that use [Oiii] and CO emission line widths to infer the
gravitational potential of quasar bulges (Shields et al.
2003, 2006; Salviander et al. 2006, 2007; Ho 2007) sug-
gest similar trends3, and residual traces of evolution re-
main detectable even below z = 1 (Treu et al. 2004, 2007;
Woo et al. 2006). Despite the general agreements ob-
servationally (but, see Li et al. 2006; Borys et al. 2005),
there remain several weaknesses that still complicate the
interpretation: a significant factor of 2 systematic un-
certainty in the normalization of the BH mass calibra-
tion, the possibility that the normalization of the virial
BH mass estimate (Kaspi et al. 2000; Vestergaard 2002;
Onken et al. 2004; Kaspi et al. 2005; Greene & Ho 2005;
Peterson & Bentz 2006; Vestergaard & Peterson 2006)
may evolve with time, because of the fact that the bulge
masses are not directly measured, and the possibility
that the host galaxy mass may be biased low in high
redshift quasars (Lauer et al. 2007) because of the steep
decline in the luminosity function of galaxies. With re-
gard to the latter selection bias, it is worthwhile to note,
however, that low redshift (z . 0.3) quasars host galax-
ies (McLure & Dunlop 2001), which have similarly high
MBH andMgal to the high redshift objects in Peng et al.
(2006a,b), do not exhibit a mass dependent bias even
when the scatter is large. Despite these uncertainties,
the finding of an existing strong MBH-Mbulge correla-
3 A lack of evolution in the stellar velocity dispersion im-
plies that bulge mass decreases with look-back time (see, e.g,
Robertson et al. 2006b).
http://arxiv.org/abs/0704.1860v5
2 PENG
tion at z & 1 is likely to be more secure.
Controversies about the evolution aside, it is not only
puzzling that the MBH-bulge correlations should ex-
ist, but that they would have a small intrinsic scatter
of 0.3 dex in MBH and that the correlation with the
bulge mass is practically linear (Marconi & Hunt 2003;
Häring & Rix 2004; Lauer et al. 2006). These curious
facts have received wide theoretical attention over the
years and could be explained in a number of ways. For
instance, both the MBH-Mbulge and MBH-σ∗ relation
can be produced by gas accretion onto a nuclear disk
(Burkert & Silk 2001; Cen 2007) or turbulent dissipa-
tion of gas (Escala 2006) followed by star formation, by
a combination of BH accretion, galaxy mergers, and star
formation (Li et al. 2006; Kauffmann & Haehnelt 2000;
Haehnelt & Kauffmann 2000), by gravitational collapse
of inner parts of a galaxy that forms the bulge from
a rotating isothermal sphere (Adams et al. 2001, 2003),
by stellar capture (Miralda-Escudé & Kollmeier 2005) in
the accretion disk. The effect of dissipation in galaxy
merging on the BH and fundamental plane scaling rela-
tions has also been visited: Ciotti & van Albada (2001)
speculated, and Robertson et al. (2006a) show, that dis-
sipational mergers can produce the MBH-σ∗ slope and
maintain the fundamental plane of elliptical galaxies. On
the other hand, whereas dissipationlessmergers of ellipti-
cal galaxies also tend to preserve the observed fundamen-
tal plane (Boylan-Kolchin et al. 2006; Robertson et al.
2006a), dissipationless mergers of disk galaxies tend to
produce a relation that more closely parallels the virial
plane.
Despite the aforementioned models, the explanation
that has spawned immense activity is the theory of feed-
back from an active galactic nucleus (AGN). The AGN
feedback idea rests observations that quasars typically
radiate at a fixed fraction (10%−100%) of the Edding-
ton luminosity. If such radiation is produced by a massive
enough BH, this energy budget is in principle sufficient
to quench star formation and terminate the BH growth
itself (e.g. Silk & Rees 1998; Di Matteo et al. 2005;
Cattaneo et al. 2005; Springel et al. 2005; Hopkins et al.
2007, 2006a, and references therein). This idea has been
incorporated into cosmological merger simulations (e.g.
Granato et al. 2004; Fontanot et al. 2006; Croton et al.
2006), and is seen to have profound possibilities for ex-
plaining a wide array of other galaxy evolution puzzles,
including the evolution in the galaxy and quasar luminos-
ity functions, mass functions, star formation rates, the
X-ray background, and the bimodality of galaxy colors
(e.g. see Hopkins et al. 2006a,b, and references therein).
Even though AGN feedback is promising for explaining
many aspects of galaxy evolution, it is possible that some
other mechanism (or mechanisms) may either have signif-
icant influence on the scaling relation between MBH and
Mbulge, or in fact be more fundamentally the cause. It is
therefore worthwhile to look for such factors and to fully
study their effects. Dating back years before AGN feed-
back physics became a popular idea, one such fundamen-
tal factor considered by many was galaxy merging (e.g.
Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann
2000; Ciotti & van Albada 2001; Nipoti et al. 2003;
Islam et al. 2003, 2004; Volonteri et al. 2003). Even
without feedback, those simulations already find it pos-
sible to produce the BH scaling relations, albeit to dif-
fering degrees of agreement with the observations. How-
ever, the reason why a tight linear correlation should
emerge is not immediately apparent even when consider-
ing no other physics besides merging. In fact, as accorded
correctly by intuition, a correlation indeed cannot arise
spontaneously from chaotic combinations of galaxies and
black hole masses in general. As will be shown below,
what does make a tight correlation emerge is the fact
that the galaxy mass function decreases with increasing
mass. This circumstance brings about a number of inter-
esting implications that will be explored in future stud-
ies. In this study, the modest goal is to show that when
the focus is switched from the BH-bulge relation, to un-
derstanding the more general BH-galaxy relation, some
insights may be gained into understanding the growth
and evolution of the MBH-Mbulge relation itself.
This study therefore revisits the issue of galaxy merg-
ing from the standpoint of a thought experiment. This
toy model identifies three basic reasons for why the
MBH-Mbulge relation may appear the way it does, even
if the BH masses and their host galaxy masses were
completely uncorrelated initially, or if they started out
with a steep powerlaw correlation. The over-arching
premise is that the galaxy mass function has a Schechter
(1976) powerlaw form, so that there is a decline in galaxy
number density with increasing mass, especially above
masses M∗. When this circumstance is met, it can
be shown that a “linear attractor” and a central limit-
like tendency can work efficiently to produce a linear
MBH-Mgal relation, and to reduce the scatter over time.
However, while it is tempting to generalize this result
to the MBH-σ∗ relation, it is not as simple to do be-
cause it is not clear how the stellar velocity dispersion
scales during galaxy mergers, something that depends on
physics not considered in this study (e.g. dissipational
vs. dissipationless mergers, star formation, AGN feed-
back – see Robertson et al. 2006b; Hopkins et al. 2007;
Di Matteo et al. 2007; Sijacki et al. 2007).
The main emphasis of the current study is thus
only to present a pure statistical exercise, and as such
will not invoke merger trees or external physics – be-
sides which many and much more sophisticated mod-
els have already been run (e.g. Volonteri et al. 2003;
Islam et al. 2004; Granato et al. 2004; Fontanot et al.
2006; Robertson et al. 2006b; Ciotti et al. 2007). In a
sense, this work examines a common thread shared by
all previous cosmological simulations. While it is tempt-
ing to invoke realism by introducing detailed physics
from the get-go, e.g. star formation, BH accretion, and
AGN feedback, isolating the effects of simple statistics
enables a cleaner exposition of why the convergence be-
havior should be expected. As such, the current study
is not a critique of other, more physical, models which
can also explain the same correlation, or to pass criti-
cal judgment about which is more or less relevant. In-
stead, the main message of this study is that, regard-
less of what other physics may ultimately produce the
MBH-Mbulge relation or weaken it, an existing corre-
lation should strengthen if galaxies continue to merge
thereafter, whether by major or by minor mergers. In
other words, merging alone, in the absence of all other
physics, is a sufficient condition to bring about a tight,
linear, MBH-Mbulge relation over time, and is always
“pulling behind the scenes” to bring about such a corre-
MASS SCALING RELATIONS AND GALAXY MERGERS 3
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Fig. 1.— (a) Two examples of initial correlations between MBH and Mgal. (b) An arbitrary number distribution of MBH at each mass
slice M1 or M2, where µ is the mean value of the distribution.
lation. The issue of interest to follow up is to what extent
this, or other as yet identified mechanisms, may matter
in the end, and to predict more realistic scatter in the
MBH-Mgal relation (as opposed to only MBH-Mbulge)
under this hypothesis using more realistic merger trees
and physics. We will address this subject in a followup
study.
The following discussion will begin by presenting a
heuristic view to explain why a linear MBH-Mgal re-
lation is a natural consequence of random merging (Sec-
tion 2), followed by Monte-Carlo simulations to illustrate
the effect (Section 3), and lastly by a discussion and con-
clusion. In much of the discussion below, the relationship
under consideration is more generally the MBH-Mgal re-
lation, whereas BHs are thought to correlate more tightly
with the spheroid component of Mgal, that is Mbulge. It
will be seen that the tighter correlation between MBH
and Mbulge is a special case of the MBH-Mgal relation
and can be understood in the same framework if this hy-
pothesis represents the dominant mechanism by which
BHs correlate with galaxy masses.
2. HEURISTIC PICTURES
2.1. The Central Limit Theorem of Galaxy Mergers
As galaxies undergo merging, it can be shown that
the scatter in the MBH-Mgal relation diminishes with
increasing number of mergers as a consequence of the
central limit theorem (see Appendix). To see this most
easily, consider first major mergers, which by definition
occur between galaxies of roughly equal mass, often de-
fined to be within the range M1/M2 ≤ 4 (Figure 1a).
Their similarity in mass also means that the number dis-
tribution (i.e. mass function) ofMBH at a specific galaxy
mass, M1, is similar to that at M2, i.e. they are drawn
from parent distributions that have roughly the same
shape (Figure 1b). However, the mean (µ) of the distri-
butions might be offset by an amount that depends on
the steepness of the MBH-Mgal correlation: the steeper
the correlation (Figure 1a, upper ellipse), the larger is
the offset.
Furthermore, consider that in the limit of no initial
MBH-Mgal correlation (Figure 1a, lower ellipse), µ is
constant with galaxy mass. The average of any two
BH masses randomly drawn from the parent distribu-
tion (e.g. Fig. 1b) therefore has a tendency toward
the mean value µ of the parent, by the central limit
theorem. This tendency also means that the resulting
BH mass distribution from mergers, obtained by sum-
ming BHs drawn from the same distribution, will have a
smaller log-normal scatter, σ(log(µ)) = σ(µ)/µ, than the
original log-normal distribution, because the fractional
mass error, σ 〈MBH,1 + MBH,2〉 / 〈MBH,1 + MBH,2〉,
is smaller, which means the scatter in the MBH vs.
Mgal relation, logarithmic on both axes, decreases.
In the instance where the initial MBH mass distribu-
tion (Figure 1b) is normal, with a scatter σBH,init, the
resulting scatter of all galaxies that have undergone
one full cycle of major mergers is σ (log µBH,merge) =
σ (log µBH,init) /
2 (see Appendix). Therefore an en-
semble of galaxies that has undergone Nmaj mergers
should have a scatter in the MBH-Mgal relation that
is reduced by ∼ 20.5Nmaj, compared with the initial rela-
tion. However, a log-normal MBH distribution will have
a different convergence rate.
Clearly, this central limit theorem behavior applies to
a finite parent distribution of any shape, but the size of
the scatter and the rate at which the scatter decreases
both depend on the shape of the distribution and the
steepness of the initial MBH-Mgal correlation. In the
situation where the correlation between MBH and Mgal
is steep (e.g. Fig. 1a, upper ellipse) the effective cu-
mulative mass function of BHs (Figure 1b) residing in
galaxies with M2 ≤Mgal ≤ M1 has a wider σBH,init
than if the MBH-Mgal relation is shallow (Fig. 1a, lower
ellipse). For the same reason, minor mergers also have
wider σBH,init, as the galaxy mass differences are larger,
than do major mergers. Therefore, galaxies that have
only undergone major mergers would produce an MBH-
Mgal relation that is significantly tighter than for galax-
ies that have only experienced minor mergers, for the
same merger rate, and if the initial mass correlation is
not flat. This effect is seen in the Monte-Carlo simula-
tions below.
2.2. Galaxy Merging From a Schechter Mass
Distribution
If the mass density of galaxies follows a Schechter pow-
erlaw form (Schechter 1976),
Φ(M) = Φ0
, (1)
4 PENG
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log (M     )gal
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Fig. 2.— (a) No initial correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy
at a much lower Mgal than with one that is comparable to itself (due to the bottom heavy mass function), but roughly of comparable
MBH (due to non-correlation between MBH-Mgal). Thus the net evolutionary vector is steep. In contrast, a galaxy at location 2 is likely
to merge with another that is comparable in both MBH and Mgal to itself, so the evolution vector is shallower. (b) An initial, strong
powerlaw, correlation in the MBH-Mgal relation. A galaxy at location 1 is more likely to merge with another galaxy comparable in Mgal
but at a much lower MBH than itself. Thus, the net evolutionary vector is shallow. In contrast, a galaxy at location 2 is likely to merge
with another of comparable in MBH and Mgal to itself, so the evolution vector is steeper.
then it is possible also to show that a linear MBH-Mgal
relation naturally emerges over time, so that the relation,
MBH(z) = Γ(z)Mgal(z)β, (2)
eventually takes on β = 1. The value of Γ, which lo-
cally is measured to be Γ(0) ∼ 1/800 for bulges (e.g.
Häring & Rix 2004), is otherwise arbitrary in the discus-
sion below. Γ is degenerate with respect to assumptions
about the initial scatter of the MBH-Mgal relation, the
initial slope β, and the initial normalization Γ(∞), for
which there are currently insufficient observational con-
straints; it will not be addressed further in this study.
The other assumption used here is that the probability
for two galaxies to merge comes from Monte-Carlo sam-
pling of a Schechter mass function (Eq. 1). In actuality,
galaxies do not merge randomly, especially at late times.
However, complete randomness is only used to facilitate
the discussion, and is not a pre-requisite, since the rea-
soning depends only on the fact that minor mergers oc-
cur more frequently than major mergers. This assump-
tion does mean that, depending on the relative balance
of major vs. minor mergers, the effects described here,
namely convergence toward linearity versus central-limit
behavior, may be more relevant at some epoch in time
than at others
The reason that a linear correlation emerges through
galaxy mergers is illustrated in Figure 2. Figure 2a shows
the situation in which the initial BH and galaxy mass
distributions are completely uncorrelated, so that β = 0.
The lower part of the diagram shows hypothetical mass
functions with two different “faint end” slopes, A and B,
which will be individually considered in the Monte-Carlo
simulation below. If there is no correlation between the
MBH and Mgal, then the ratio MBH/Mgal will be, on
average, larger for low mass galaxies than for high mass
galaxies as can be seen by comparing the MBH/Mgal ra-
tio at any two locations, for example, those labeled “1”
and “2.” Therefore, as galaxies merge, a massive object
at the extreme end of the mass function, located at posi-
tion 1, on average, is more likely to merge with another
having a much larger MBH/Mgal, thereby evolving the
merger product in a steep upward direction, as exagger-
ated by the vertical arrow. In contrast, an object at
position 2 is likely to merge with objects comparable in
both MBH and Mgal to itself, so the net evolutionary
vector has a shallower slope. Therefore, the cumulative
effect of mergers along the mass spectrum is to steepen
the massive end of the MBH-Mgal relation relative to
the lower extreme, even as the lower end grows in MBH
and Mgal on average.
In the other extreme, Figure 2b, if the primordial rela-
tion between the BH and galaxy masses is steep, corre-
sponding to β ≥ 1, the opposite behavior occurs. Galax-
ies at location 1 generally have a larger MBH/Mgal ra-
tio than galaxies that have lower mass. Therefore, the
MBH/Mgal ratio for massive galaxies would tend to de-
crease through mergers. The net effect on massive galax-
ies is to evolve the merger remnant more quickly to the
right on average than a lower mass galaxy at location 2.
Because of a mirror symmetry between Figures 2a and
2b, the natural equilibrium state of the MBH-Mgal re-
lation is at β = 1, so that further merging of galaxies
would evolve remnants along the linear relation, with a
constant ratio Γ=MBH/Mgal. Also because of the con-
MASS SCALING RELATIONS AND GALAXY MERGERS 5
vergence toward this “attractor” state the scatter in the
relationship would necessarily decrease over time through
galaxy merging.
Lastly, it is worth mentioning that the presence of a
break in the galaxy mass function at M∗ is a sufficient,
but not necessary, condition for convergence toward lin-
earity. A pure powerlaw with M∗= ∞ would produce a
similar behavior, however, the convergence is slower for
flatter slopes (α → −1). Furthermore, while the con-
vergence behaviors just described is quite strong for a
Schechter mass function with α = −1, the convergence
would fail for a pure powerlaw with the same slope, be-
cause of a lack of a break in the mass function.
3. MONTE CARLO SIMULATIONS
To illustrate the idea discussed above, and to quantify
how quickly a linear MBH-Mgal relation might emerge,
it is useful to consider several numerical simulations for
the situations shown in Figure 2. For each of the two
scenarios, no-correlation (Fig. 2a) and steep correlation
(Fig. 2b), it is also instructive to consider two differ-
ent initial mass functions, A and B, shown in the lower
half of Figure 2. The two powerlaw slopes explored be-
low are α = −1.5 and α = −0.5, respectively. These
choices are motivated by observations of the luminosity
functions for high redshift galaxies (e.g. Gabasch et al.
2006; Giallongo et al. 2005) under the assumption that
light traces mass.
3.1. Simulation Set-up and Definitions
Definition of the number of major and minor mergers.
— As implied in Section 2, how closely a galaxy lies
to the linear part of the MBH-Mgal relation, and how
tight the final scatter is for an ensemble of galaxies, will
depend on the cumulative merger history. Therefore, the
most useful way to understand the simulation results is
to define the number of major mergers, Nmaj, as the
cumulative sum of all such events over the entire tree for
a given galaxy, not just in the most immediate, that is,
the main, branch. For example, even if a galaxy has never
experienced a single major merger on the main branch
in its lifetime, it could still lie close to the final, linear
part of the MBH-Mgal relation because the progenitors
of the main branch, and their progenitors, and so forth,
could have experienced a number of major mergers.
Evolution of the mass function. — One issue to con-
sider is how the galaxy mass function might evolve, and
whether the path of evolution might affect the final con-
vergence. The effect of galaxy masses growing with time
is to both increase M∗ and steepen the “faint end slope”
(α) of the Schechter function. As galaxy populations
grow in mass and number density, the rate of change in
M∗ and α would affect the rate of convergence to the
final MBH-Mgal normalization, slope, and scatter. Pre-
dicting the rate of change in the MBH-Mgal relation re-
quires realistic merger trees and accounting for other de-
tailed physics such as feedback, which will be addressed
in a followup study. For the current purposes of show-
ing that convergence toward a linear MBH-Mgal relation
does naturally occur, it suffices to consider two scenarios:
a replenishment scenario, in which the Schechter mass
function is continuously, and randomly, replenished as
galaxies merge, and a depletion scenario, in which no new
galaxies are formed to take the place of those that have
merged. Combined with considerations about the initial
mass function slopes, the simulations will have covered
the gamut of sensible possibilities and conditions that
may be present at early and late cosmic epochs.
Initial scatter of the MBH-Mgal relation and the initial
mass function M∗. — In the simulation, the distribution
of galaxies is first drawn randomly from the Schechter
mass function, after which a BH mass is assigned, fol-
lowing Equation 2, by drawing from a log-normal Gaus-
sian distribution with a generous Gaussian dispersion of
σBH = 2 dex, i.e. a scatter of a factor 100 in mass.
The exact choice of the initial scatter is directly propor-
tional to, but otherwise only partially determines, the
final scatter in the MBH-Mgal relation. Other factors
that determine the final scatter depend on how long the
simulation runs, and on the initial value of M∗. Cur-
rently, there are some observational constraint on the
rate of mergers, which will be considered in a followup
study. In this study, the results are merely normalized ar-
bitrarily to match the final MBH-Mgal relation observed
today.
The simulation “clock.” — The progress of time is not
well defined in Monte-Carlo simulations, so it is useful
to define merging cycles for the sake of keeping track of
the simulation progress. Each full cycle is defined as be-
ing complete after the number of merger events equals
the number of galaxies present at the beginning of that
particular cycle. Galaxies that are produced or merged
retain their states for the following cycle. Because some
galaxies may merge multiple times by being drawn re-
peatedly, not all galaxies will be involved in mergers af-
ter each full cycle. The exact definition of the simulation
clock is unimportant, as it is only the relative number of
major vs. minor mergers on average that determines the
degree of convergence toward a linear MBH-Mgal rela-
tion, where a major merger is defined as having a mass
ratio of at most 4:1.
3.2. Replenishment Scenarios
The replenishment scenario is one of the two simple
ways considered to emulate the progress of galaxy evo-
lution. Here, by definition, the rate of galaxy mergers
equals the rate of galaxy number production. The way a
galaxy is newly produced is by being selected randomly
from an initial mass function, parameterized by α and
M∗, which does not evolve with time. In contrast, galax-
ies “grow” only by merging with another member in the
galaxy pool existing at the time, and hierarchical merg-
ing is the only avenue for mass growth. Therefore, as
galaxies merge, the cumulative mass function does un-
dergo evolution. However, the total number of galax-
ies remains constant because of the 1:1 ratio of merg-
ing:replenishment.
Figures 3 and 4 illustrate the results for initial con-
ditions β = 0 (i.e. no MBH-Mgal correlation) and
β = 2 (steep MBH-Mgal correlation), respectively, show-
ing only a small subset of the data points. In each figure,
two different initial mass functions, α = −0.5 (Figs. 3a,
4a) and α = −1.5 (Figs. 3b, 4b) are considered. In each
of the Figures, the initial distribution of the MBH-Mgal
relation (or lack thereof) is shown with crosses. The open
colored data points show the MBH-Mgal development of
galaxies that have undergone Nmaj ≥ 5 major merger
episodes, after 10 (blue triangles), 100 (green squares),
6 PENG
Fig. 3.— No initial correlation (β = 0) in the MBH-Mgal relation, in the replenishment scenario. The black crosses represent the initial
distribution of points, and the solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data
points. The colored data points represent objects that have undergone at least five major merger episodes after 10 (blue triangles), 100
(green squares), and 1000 (red circles) complete “merging cycles.” The crosses are the primordial distribution, corresponding to the initial
mass function. The shaded region illustrates the locus of all points after 1000 cycles; the density of points doubles with each contour level.
The cumulative histograms after the corresponding merger sequences are shown below the data points. (a) An initial Schechter powerlaw
slope of α = −0.5. (b) An initial Schechter powerlaw slope of α = −1.5.
Fig. 4.— Similar to Figure 3, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 3 for details.
MASS SCALING RELATIONS AND GALAXY MERGERS 7
Fig. 5.— Similar to Figure 3b, showing the effect of central-limit
tendencies with increasing number of major mergers for galaxies
after 1000 merger cycles. The colored data points illustrate objects
that have undergone 1 ≤ Nmaj ≤ 4, (blue triangles), 5 ≤ Nmaj ≤
14 (green squares), Nmaj > 14 (red circles) major mergers. The
greyscale contours shows the locus of all the points.
Fig. 6.— Similar to Figure 3, except the BH mass is drawn from
a Schechter law of α = −1.5 instead of a Gaussian distribution.
See Figure 3 for details.
and 1000 (red circles) merger cycles have transpired.
These data points effectively illustrate the progress of
the MBH-Mgal evolution for objects that might be mor-
phologically identified as early-type galaxies of each cy-
cle. For clarity, the contour levels represent the locus of
points after 1000 merger cycles, and the levels are spaced
at multiples of 2 in density. The luminosity functions of
the galaxy pool at the end of the merger cycles are shown
in the lower half of each diagram in corresponding col-
ors and locale in mass. Lastly, a linear reference line is
overplotted in the Figures with normalization given by
R0=800 (Häring & Rix 2004), and the simulations are
scaled/shifted arbitrarily to match; it is not a fit to the
data points.
As shown in Figures 3 and 4, the convergence towards
a tight linear relation is fairly quick. After five major
merger episodes a linear relation starts to emerge regard-
less of the initial conditions of the mass function or the
form of the MBH-Mgal correlation. One reason for this
quick convergence is the central-limit behavior of major
mergers which is shown in Figure 5, in which the increas-
ing number of mergers is represented by different symbols
and shades. The one notable case where the convergence
toward linearity is slower than the other scenarios is Fig-
ure 4b, where the effect is only evident at 1010.5 M⊙ or
greater, even as the scatter has decreased markedly. In
general, if the MBH-Mgal correlation is steep initially,
the tail at low mass remains steep after a large number
of major mergers has occurred, even as the massive end
converges toward linearity.
Lastly, the qualitative convergence effects do not de-
pend on the assumption about the distribution of BH
mass at each galaxy mass. Figure 6 shows an example
that is in direct analog to Figure 3b, except that the BH
mass is instead drawn from a Schechter mass function
with α = −1.5.
3.3. Depletion Scenarios
The other extreme of the merger simulations is to con-
sider what effect galaxy depletion from a finite reservoir
has on the MBH-Mgal relation. Because the number
density of galaxies builds up over time, the depletion
scenario is expected to not be realistic. Nevertheless, it
is useful for illustrating how the MBH-Mgal convergence
is affected by a different evolution in the mass function
as compared with the replenishment scenario.
The depletion scenarios are constructed by creating a
large sample of 5× 105 objects, initially having no corre-
lation between BH and galaxy masses (Figure 7) or with
a β = 2 correlation between the two (Figure 8). The BH
masses are assigned to the galaxies with a log-normal
distribution of dispersion σ = 2 centered around Equa-
tion 2. In each scenario, galaxies are created to have
initial mass functions of α = −0.5 (Figures 7a and 8a) or
α = −1.5 (Figures 7b, 8b). Then, as galaxies merge, no
new ones are created to replace them. As a consequence,
the mass function evolves by growing in M∗, the num-
ber density decreases, and a sharp truncation develops
at low masses (see lower half of Figures 7 and 8). As
the number of merging cycles increases, the scatter de-
creases quickly and converges toward a linear relation, as
illustrated by the solid line. Once again, as shown in Fig-
ure 8 (especially 8b), the convergence is much slower for
steep α and steep β compared with other scenarios. And
while the convergence trends are noticeable, because of
a dearth of minor galaxies with which to merge at late
times (red circles), the slope is virtually “frozen in,” and
the subsequent convergence is due mostly to the central-
limit theorem.
4. DISCUSSION AND CONCLUSION
This study has revisited the issue of how galaxy merg-
ing may affect the MBH-Mgal scaling relation from
the standpoint of basic mass addition and statistics,
thereby clearly isolating the merger cause from other de-
8 PENG
Fig. 7.— No initial correlation (β = 0) in the MBH-Mgal relation, depletion scenario. The contours represent the initial distribution
of points, and a solid line shows the local MBH-Mgal relationship from Häring & Rix (2004) – it is not a fit to the data points. The
colored data points represent objects that have undergone at least 1 major merger episodes after 1 complete merging cycles (blue triangles),
10 (green squares), and 14 (red circles). The cumulative histograms after the corresponding merger sequences are shown below the data
points. a) An initial Schechter powerlaw slope of α = −0.5. b) An initial Schechter powerlaw slope of α = −1.5.
Fig. 8.— Similar to Figure 7, except for a steep initial correlation (β = 2) in the MBH-Mgal relation. See Figure 7 for details.
MASS SCALING RELATIONS AND GALAXY MERGERS 9
tailed physics that must otherwise affect galaxy evolu-
tion. Through Monte-Carlo simulations, a tight, linear,
MBH-Mgal correlation appears to emerge when galaxies
have undergone five or more major mergers (along the
entire tree, not just the main branch), and many minor
ones, for practically all reasonable initial correlations be-
tweenMBH andMgal, or a lack of one. The main reasons
for these behaviors are seen to be the following:
1. The galaxy mass function decreases with increasing
mass.
2. Major mergers have a strong central-limit ten-
dency, so that regardless of the initial MBH-Mgal
correlation, the scatter should decrease with an
increasing number of events. While this ten-
dency also acts on minor mergers, the drive toward
smaller scatter is weaker because minor mergers oc-
cur between galaxies that are vastly discrepant in
both galaxy and BH mass as compared with major
mergers, by definition. The corollary is that the
steeper a correlation between MBH (y-axis) and
Mgal, the stronger the central-limit tendency for
major mergers compared with minor. However,
major mergers alone are not enough to cause the
MBH-Mgal relation to converge to linearity over
time because the ratio MBH/Mgal is not changed
much.
3. Minor mergers are primarily responsible for causing
the MBH-Mgal relation to converge toward a linear
— that is, MBH = ΓMgal— relation because the
mass function of galaxies follows a Schechter pow-
erlaw. Without minor mergers, the MBH-Mbulge
relation can be “frozen” to a slope that is not
necessarily linear. This linear attractor causes a
convergence toward a tighter MBH-Mgal relation;
however, it is less efficient at reducing the scatter
compared with the central-limit seeking tendency
of major mergers, as shown in Figure 5.
It is curious that galaxy merging itself might produce
a linear MBH-Mgal relation. However, a natural ques-
tion that does arise is, “When is the merging statistics
presented in this study relevant?” On the surface, it
is easy to conclude that because the reasoning refers
to a two component model it ought to apply to “dry”
mergers, but perhaps not to a three component model
involving stars, gas, and BH. Thus, the implication is
also that it ought not apply to galaxies undergoing gas-
rich mergers, that is, early cosmic history. However, it
is not clear that such a skepticism is warranted. For
example, in the entire discussion thus far, the abscissa,
Mgal, might just as well refer to Mgal=Mstellar+Mgas,
instead of just Mstellar. If BHs do not grow much by
accretion and that the gas does not get removed from
the definition of Mgal during mergers, then the MBH-
Mgal correlation can emerge from statistical merging.
The argument holds true even if Mgas transforms arbi-
trarily into Mstellar, as long as the sum is conserved. In
the limit where BHs do grow most of their mass dur-
ing AGN accretion, as might be implied by Soltan (e.g.
1982); Yu & Tremaine (e.g. 2002), so that ∆MBH∝Mgas
and ∆MBH≫MBH, then the correlation between MBH
and Mgal comes out by construction rather than by sta-
tistical merging. However, statistical reasoning would
still be a “supporting actor” to reduce the scatter and
to forcibly steer the MBH-Mgal relation in the preferred
linear direction. Likewise, even if BH growth, or other
physics (e.g. gravitational radiation, three body BH
ejection – Merritt et al. 2004; Volonteri & Perna 2005;
Ciotti et al. 2007, and references therein), were a “heat-
ing” source, that is, one that randomizes a tight linear
MBH-Mbulge relation, the linear and central limit attrac-
tors would cause a re-convergence if galaxies continue to
merge thereafter by both major and minor mergers. In
summary, while it is entirely possible that the MBH-
Mbulge relation has origins outside of basic statistics,
galaxy merger statistics can still affect the final outcome
of a MBH-Mgal correlation in both the scatter and the
slope. In any event, statistical reasoning is a fundamen-
tally robust explanation for why random galaxy merging
does not corrupt a pre-existing MBH-Mbulge relation,
which is important to bear in mind in the context of the
MBH-Mbulge or MBH-Mgal relation in a hierarchically
forming universe.
While the MBH-Mbulge relation might have other
origins, it is nonetheless interesting and revealing to
follow through the consequences of statistical merg-
ing. For instance, simple statistics naturally explains
why black holes appear to correlate most strongly
with galaxy bulges, rather than more generally with a
galaxy as a whole, which might include a stellar disk
(Kormendy & Gebhardt 2001): bulge masses, assum-
ing they were assembled through major mergers, have
a stronger central-seeking tendency than disk galaxies,
whose growth history might involve more minor merger
events. As such, the MBH-Mbulge relation is a special
case of a more fundamental MBH-Mgal relation. Revers-
ing the argument, the observational fact thatMBH corre-
lates most strongly with bulge masses, coupled with the
central-limit theorem reasoning, implies that the merger
trees of elliptical galaxies were more dominated by ma-
jor merger events than were disk galaxies. Conversely,
the fact that the scatter in the MBH-Mgal relation is
observed to be much larger for disk dominated galaxies
implies, statistically, that their progenitors, and progen-
itors thereof, have undergone more minor mergers.
The possibility that a more fundamental correlation is
between MBH and Mgal (rather than Mbulge) also has
practical implications for what slope and scatter would
be measured by observations. First, because the slope
changes with mass even for objects that experienced the
same number of major mergers (e.g. Figures 3 and 7),
the deviation from linearity and the intrinsic scatter will
depend exactly on how the data are cut. Simply defining
a sample of objects based on a mass selection cut will
bias one’s measurement of the slope and scatter. Fur-
thermore, defining a sample based on morphology criteria
may also implicitly preselect samples that have certain
major vs. minor merger histories. Observationally, it is
therefore crucial, when comparing intrinsic scatter and
slope of the MBH-Mbulge relation to be specific about
sample selection parameter space, morphology, bulge-
to-disk ratios, or other criteria, lest the conclusions be
caused by subtle but trivial selection biases.
Another consequence of this thought experiment is
that the ratio Γ=MBH/Mbulge approaches an asymp-
10 PENG
totic value with time from having a smaller ratio in
the past. On the surface, this appears contrary to
the findings of Peng et al. (2006a,b); Woo et al. (2006);
Shields et al. (2003, 2006) based on quasar host galaxy
studies that the ratio Γ decreases over time. If the
quasar host galaxy studies are correct and are not sig-
nificantly affected by biases pointed out by Lauer et al.
(2007), then some other physics not considered here is
responsible for causing a decline in the normalization of
Γ with time (e.g. see Croton 2006; Hopkins et al. 2007;
Fontanot et al. 2006). For instance, the abscissa is am-
biguous about what mass Mgal corresponds. If gas mass
is a significant fraction of a galaxy’s mass, then form-
ing stars out of the gas reservoir would decrease Γ over
time, if the abscissa Mgal represents the galaxy’s stellar
bulge mass. Secular growth of galaxy bulges by accret-
ing stars in galaxy disks would also decrease Γ, at the
expense of increasing the scatter. Major mergers of pure
stellar bulges, however, would not cause Γ to decrease
over time.
In hindsight, the results of this study could have been
anticipated from Islam et al. (2003, 2004); Ciotti et al.
(2007), given that the initial conditions used in those
studies are a special case of this one where the initial
BH scatter σBH → 0 (Figure 3a or 3b) (M. Volonteri
and L. Ciotti 2007, private communication). Just as rel-
evant, Croton (2006); Ciotti et al. (2007) show that once
the MBH-Mbulge relation is in place, it is fairly imper-
vious to being randomized by galaxy merging. And be-
cause Islam et al. (2003) also uses realistic cosmological
merger trees, they confirm that the arguments presented
here ought to remain relevant. However, the reasons be-
hind the MBH-Mbulge convergence behavior are difficult
to extract from previous studies because of the use of pri-
ors, the use of identical BH seeds, the inclusion of other
physics, and the focus on only the BH-bulge coevolution
(i.e. major mergers). The latter, especially, is worth
examining further, because the prior that one chooses
about whether the BH correlates with just its bulge or
with the entire galaxy can lead to differing interpreta-
tions.
In particular, one conclusion from Islam et al. (2003,
2004) is that the MBH-Mbulge relation converges to a
slightly non-linear slope of β = 0.9; hence they reason
that other physics, perhaps BH growth through accre-
tion of gas, is required in order to increase the slope
closer to linearity. The reasoning presented in the cur-
rent study, however, would stipulate that linearity is an
asymptotic outcome of mergers, but deviations from lin-
earity come from the possibility that the low mass galax-
ies have not yet achieved the asymptotic limit, because
of a weaker convergence. At low masses, the slope de-
viates from unity in either direction depending on the
initial mass function of the galaxies (e.g. compare Fig-
ure 3a with 3b), on the mass cut of the study, and on the
relative incidence of minor versus major mergers.
An interesting consequence to consider is how the
MBH-Mgal relation might differ between high and low
density galaxy clusters. However, one of the unrealis-
tic side-effects of using Monte-Carlo simulations to de-
termine merger rates is that the normalization of the
MBH-Mbulge relation is the same in all density regimes.
This is because the normalization factor, Γ, depends only
on the ratio of major to minor merger events which, in
the Monte-Carlo universe, is not affected by a simple
rescaling of the mass function. However, in the real uni-
verse, the relative rates of major and minor mergers can
change with density and, as such, may result in different
normalization and scatter in the MBH-Mgal relation.
Lastly, because of the ambiguity in what Mgal corre-
sponds, depending on whether it refers to the total stellar
mass, gas mass, dark matter halo mass, or a combination
thereof, the degree of scatter and linearity would clearly
differ, as a result of different initial mass functions and
merger histories. Because the scenarios considered above
depend on a linear addition of masses, the arguments
therefore may not apply to gas masses that are not grav-
itationally bound to a galaxy. These and other issues will
be addressed in a future study, which will incorporate the
use of realistic merger trees.
ACKNOWLEDGMENT
Over the course of this study, I have greatly en-
joyed lively discussions with many friends and colleagues,
including Jenny Greene, Hans-Walter Rix, Luis Ho,
Eric Bell, Michael Santos, Robert Kennicutt, Rachel
Somerville, Aaron Barth, Christy Tremonti, Darren Cro-
ton, Tommaso Treu, and David Koo. I also thank Jim
Rose, Wayne Christiansen, Marta Volonteri, Luca Ciotti,
and Alister Graham for comments and discussions on
past and future studies. This study greatly benefited
from discussions with Phil Hopkins on issues related to
AGN feedback over the past several months. I also grate-
fully acknowledge insightful comments and suggestions
from the referee, and Science Editor Chung-Pei Ma, and
the support of STScI through the Institute (Giacconi)
Fellowship Program.
APPENDIX
This appendix shows that theMBH-Mbulge relation follows a central-limit-like behavior when galaxies undergo major
mergers. Specifically, this means that if the initial parent distribution of progenitorMBH that undergoes merging is, for
simplicity, normally distributed about a mean BH mass µ, thus having a logarithmic dispersion σ(log(µinit)), then a new
distribution ofMBHs after merging will have a log-normal dispersion that scales as: σ (log (µmerge)) ∼ σ(log(µinit))/
First, the mean, µmerge, of the resulting BH distribution after two BHs merge from the initial parent distribution is:
µmerge = 〈MBH,1 +MBH,2〉 , (1)
where MBH,1 and MBH,2 are drawn from the same parent distribution for major galaxy mergers. Then,
log (µmerge) = log 〈MBH,1 +MBH,2〉 , (2)
MASS SCALING RELATIONS AND GALAXY MERGERS 11
From propagation of errors the log-normal error is: σ(log(x)) = σ(x)/x, then,
σ (log (µmerge)) =
σ 〈MBH,1 +MBH,2〉
〈MBH,1 +MBH,2〉
, (3)
By definition, a distribution obtained by averaging the mass of merging BH pairs is a normal distribution with a mean
of the initial parent distribution, µinit:
µinit =
〈MBH,1 +MBH,2〉
µmerge
. (4)
Because M,1 and M,2 are drawn from a normalized distribution around a parent mean µ, the new distribution of
σ(µmerge) ≡ σ 〈M,1 + M,2〉 is:
σ (µmerge) ∼
σ(µinit)√
× 2. (5)
Substituting Eqs. A4 and A5 into A3 yields:
σ (log(µmerge)) ∼
σ(µinit)
µinit
. (6)
Using the fact that, σ(log(µ)) = σ(µ)/µ, Equation A6 becomes:
σ (log(µmerge)) ∼
σ (log(µinit))√
. (7)
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