The Millennium Galaxy Catalogue: The local supermassive black hole mass function in early- and late-type galaxies

Abstract

We provide a new estimate of the local supermassive black hole mass function
using (i) the empirical relation between supermassive black hole mass and the
Sersic index of the host spheroidal stellar system and (ii) the measured
(spheroid) Sersic indices drawn from 10k galaxies in the Millennium Galaxy
Catalogue. The observational simplicity of our approach, and the direct
measurements of the black hole predictor quantity, i.e. the Sersic index, for
both elliptical galaxies and the bulges of disc galaxies makes it
straightforward to estimate accurate black hole masses in early- and late-type
galaxies alike. We have parameterised the supermassive black hole mass function
with a Schechter function and find, at the low-mass end, a logarithmic slope
(1+alpha) of ~0.7 for the full galaxy sample and ~1.0 for the early-type galaxy
sample. Considering spheroidal stellar systems brighter than M_B = -18 mag, and
integrating down to black hole masses of 10^6 M_sun, we find that the local
mass density of supermassive black holes in early-type galaxies rho_{bh,
early-type} = (3.5+/-1.2) x 10^5 h^3_{70} M_sun Mpc^{-3}, and in late-type
galaxies rho_{bh, late-type} = (1.0+/-0.5) x 10^5 h^3_{70} M_sun Mpc^{-3}. The
uncertainties are derived from Monte Carlo simulations which include
uncertainties in the M_bh-n relation, the catalogue of Sersic indices, the
galaxy weights and Malmquist bias. The combined, cosmological, supermassive
black hole mass density is thus Omega_{bh, total} = (3.2+/-1.2) x 10^{-6} h_70.
That is, using a new and independent method, we conclude that (0.007+/-0.003)
h^3_{70} per cent of the universe's baryons are presently locked up in
supermassive black holes at the centres of galaxies.

Full text

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arXiv:0704.0316v1 [astro-ph] 3 Apr 2007
Mon. Not. R. Astron. Soc. 000, 1–15 (2006)Printed 1 February 2008 (MN LATEX style file v2.2)
The Millennium Galaxy Catalogue: The local supermassive
black hole mass function in early- and late-type galaxies
Alister W. Graham1⋆, Simon P. Driver2, Paul D. Allen1,2and Jochen Liske3
1Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
2SUPA†, School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK
3European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany.
Received 2006 Jan 01; Accepted 2006 December 31
ABSTRACT
We provide a new estimate of the local supermassive black hole mass function
using (i) the empirical relation between supermassive black hole mass and the S´ ersic
index of the host spheroidal stellar system and (ii) the measured (spheroid) S´ ersic
indices drawn from 10k galaxies in the Millennium Galaxy Catalogue. The obser-
vational simplicity of our approach, and the direct measurements of the black hole
predictor quantity, i.e. the S´ ersic index, for both elliptical galaxies and the bulges of
disc galaxies makes it straightforward to estimate accurate black hole masses in early-
and late-type galaxies alike. We have parameterised the supermassive black hole mass
function with a Schechter function and find, at the low-mass end, a logarithmic slope
(1+α) of ∼ 0.7 for the full galaxy sample and ∼1.0 for the early-type galaxy sample.
Considering spheroidal stellar systems brighter than MB= −18 mag, and integrating
down to black hole masses of 106M⊙, we find that the local mass density of super-
massive black holes in early-type galaxies ρbh,early−type = (3.5 ± 1.2) × 105h3
Mpc−3, and in late-type galaxies ρbh,late−type = (1.0 ± 0.5) × 105h3
The uncertainties are derived from Monte Carlo simulations which include uncer-
tainties in the Mbh–n relation, the catalogue of S´ ersic indices, the galaxy weights and
Malmquist bias. The combined, cosmological, supermassive black hole mass density is
thus Ωbh,total= (3.2±1.2)×10−6h70. That is, using a new and independent method,
we conclude that (0.007 ± 0.003)h3
70per cent of the universe’s baryons are presently
locked up in supermassive black holes at the centres of galaxies.
70M⊙
70M⊙ Mpc−3.
Key words: black hole physics — galaxies: bulges — galaxies: fundamental param-
eters — galaxies: luminosity function, mass function — galaxies: structure — surveys
1 INTRODUCTION
Two purely photometric properties of galaxies, or rather
their spheroidal1
components, are known to correlate
strongly with a galaxy’s supermassive black hole (SMBH)
mass Mbh. The first property is optical luminosity (Kor-
mendy 1993; Franceschini et al. 1998; Magorrian et al. 1998).
Due to the observation that SMBHs are associated with the
‘bulge’ of a galaxy, and not the disc, it is necessary to per-
form a bulge/disc decomposition if one is to properly treat
lenticular and late-type galaxies. At present, apart from (Er-
win, Graham & Caon 2002; their figure 3, with only eight
⋆AGraham@astro.swin.edu.au
† Scottish Universities Physics Alliance (SUPA)
1By the term ‘spheroidal’, we mean an entire elliptical galaxy or
the dynamically hot component of a disc galaxy.
elliptical and five disc galaxies), no optically2calibrated rela-
tion that pertains to both elliptical galaxies and the bulges
of disc galaxies is available. On the other hand, one can
simply exclude the disc+bulge galaxies and only work with
elliptical galaxies (e.g., McLure & Dunlop 2002, 2004).
In an effort to include disc galaxies, some authors have
assigned some fixed fraction, such as three-tenths, of a disc
galaxy’s total light to that of the bulge. However, given
there is a known trend of decreasing bulge-to-total lumi-
nosity ratio with increasing morphological type (e.g., Hub-
ble 1926, 1936; Kent 1985; Simien & de Vaucouleurs 1986;
Andredakis, Peletier & Balcells 1995), the above approach
introduces a systematic bias such that the SMBH masses
2A useful near-infrared Mbh–Lspheroidrelation is presented by
Marconi & Hunt (2003), updated in Graham (2007).

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2
Graham et al.
are over-estimated in the late-type disc galaxies and under-
estimated in the early-type disc galaxies, skewing the mid-
to low-mass end of the SMBH mass function. Another ap-
proach has been to use the average bulge-to-total flux ratios
derived from past R1/4-bulge + exponential-disc decomposi-
tions (Simien & de Vaucouleurs 1986). However, Andredakis
& Sanders (1994) showed that Sb and Sc galaxies are, on av-
erage, better described with an exponential-bulge than an
R1/4-bulge. Andredakis et al. (1995) subsequently showed
that an R1/n-bulge was more appropriate, with the S´ ersic
index n shown to decrease with increasing disc galaxy type.
While an R1/4-bulge plus exponential-disc results in an over-
estimate of the B/D ratio when the bulge has a S´ ersic index
n < 4 (c.f. Figures 15 and 1 in Graham 2001), the use of
an exponential-bulge + exponential-disc decomposition re-
sults in an under-estimate of the bulge-to-disc (B/D) flux
ratio when the bulge actually possesses a S´ ersic profile with
n > 1 (Graham 2001, his figure 13). Therefore, R1/n-bulge
plus exponential-disc fits are required. The various Mbh-L
relations in the literature predict SMBH masses that differ
by factors of two to ten depending on the luminosity. This
obviously inhibits the use of the Mbh-L relation at present.
An investigation of this problem is not undertaken here but
presented in Graham (2007).
The second3photometric quantity known to correlate
with Mbh is the concentration of the stars in the host
spheroidal stellar system (Graham et al. 2001, 2003). This
concentration is monotonically related to the shape, i.e. the
S´ ersic index n, of the spheroid’s light-profile (Trujillo, Gra-
ham & Caon 2001, their equation 6). Moreover, the Mbh–n
relation is known to be as tightly correlated as the Mbh–σ
relation and have the same small degree of scatter (see No-
vak, Faber & Dekel 2005 for a recent comparison of these
relations).
Using an expanded galaxy set (27 galaxies) with up-
dated distances and black hole masses, Graham & Driver
(2007a) have recently shown that the Mbh–n relation (see
Fig.1) is curved rather than linear. Fitting a quadratic equa-
tion, they obtained
log(Mbh)=7.98(±0.09) + 3.70(±0.46) log(n/3)
−3.10(±0.84)[log(n/3)]2, (1)
with an intrinsic scatter ǫintrinsic = 0.18+0.07
tal absolute scatter in logMbh is 0.31 dex, which compares
favourably with the value of 0.34 dex from the logMbh-σ
data and relation in Tremaine et al. (2002). The parameter
in front of the second order term in Eq.1 is inconsistent with
a value of zero at the 99.99 per cent confidence level.
We have explored here whether the departure from a
linear relation may have been driven by an increased and
uneven scatter, i.e. outliers4, at the high-mass end of the
Mbh–n relation, or whether the curvature is inherent in the
rest of the data set. In Fig.2 we show the results of fitting
−0.06dex. The to-
3A third photometric quantity that has been predicted to cor-
relate well with SMBH mass is the central stellar density of the
host spheroid (Graham & Driver 2007a, their section 6).
4As noted in Graham & Driver (2007a, their Section 3.3), there
is reason to suspect that the highest SMBH mass, pertaining to
NGC 4486, may have been overestimated, perhaps by a factor of
4.
a log-quadratic relation after the removal of the five highest
mass data points from Fig.1. The coefficient in front of the
quadratic term is again found to be inconsistent with a value
of zero, this time at the 99 per cent level, and all three
coefficients remain consistent, at the 1σ level, with the values
given in Eq.1.
Support for the Mbh–n relation stems from its appli-
cation to galaxies not included in its construction. For ex-
ample, De Francesco et al. (2007) have recently measured a
S´ ersic index n = 4.1 for NGC 3998, from which one would
predict log(Mbh/M⊙) = 8.43, in perfect agreement with the
mass they derived from a kinematical study of the nuclear
gas. In another example, Guhathakurta et al. (2006, their
Figure S2) report a S´ ersic index n = 2 for M31, which has a
SMBH mass equal to 3.5×107M⊙(Ferrarese & Ford 2005)
and is therefore also in good agreement with the data in
Fig 1. Furthermore, for a sample of 11 narrow-line Seyfert
galaxies (Ryan et al. 2007), the Mbh–n relation predicts
SMBH masses in agreement with those derived using the size
of the broad line region and the continuum flux, and suggests
a problem with the (Mbh–luminosity)-derived masses.
The Mbh-n relation also implies a maximum mass to
which SMBHs have formed. The broad turn-over seen in
Fig.1 peaks at n = 11.9, where the predicted 1σ range of
SMBH masses spans 0.8 to 3.8×109M⊙(Graham & Driver
2007a, their Eq.8). In Graham & Driver (2005, their fig-
ure 1), one can see that for n >= 5 there is not that much
difference in profile shape, and hence there should not be
much difference in SMBH mass for galaxies with n >= 5.
Above n ∼ 12, increasing n to infinity has almost no effect
on the profile shape, and hence the SMBH masses should
all be the same. Some kind of asymptotic-like Mbh–n func-
tion is therefore in some sense demanded by the form of the
S´ ersic R1/nmodel. At the high-n end of the distribution,
we do not believe that galaxies with n > 11.9 have smaller
SMBH masses than those with n < 11.9, and we note that
the log-quadratic Mbh–n relation (held fixed for n > 11.9)
appears more logical than say a rising linear Mbh–n rela-
tionship. In passing, we note that the highest SMBH mass
which has been directly measured in a quasar using rever-
beration mapping is only 2.6×109M⊙(S50836+71, Kaspi
et al. 2007) and is thus consistent with our predicted up-
per 1-σ range. The second highest quasar SMBH mass is
0.9×109M⊙(3C273).
In this paper we employ the Mbh–n relation in Eq.1 to
derive the SMBH mass function using data from the Mil-
lennium Galaxy Catalogue (MGC), which is described in
Section 2. Preliminary results have been presented in Driver
et al. (2006b, 2006c). Using the bulge-disc decompositions of
the brightest 104MGC galaxies (Allen et al. 2006), in Sec-
tion 3 we construct the SMBH mass function. In Section 4
we compare our results with previous efforts to measure the
SMBH mass density, ρbh,0, using other means. A summary
of our analysis is provided in Section 5.
Throughout this paper, unless specified otherwise, we
use ΩΛ = 0.7,ΩM = 0.3 and h70 = H0/(70 km s−1Mpc−1).

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MGC: SMBH mass function
3
2THE MILLENNIUM GALAXY CATALOGUE,
AND OUR SPHEROID SAMPLE
The MGC is a medium-deep (BMGC = 24 mag)5imag-
ing survey of the nearby universe with a median seeing
of 1.27′′and it has 96.1 per cent complete (99.8 per cent
for BMGC < 19.2 mag) redshift information for the MGC-
BRIGHT sample of 10,095 objects with BMGC < 20 mag
(Driver et al. 2005). The imaging data was acquired with the
2.5 m Isaac Newton Telescope which surveyed 37.5 square
degrees6in a 35 arcmin wide strip along the equatorial sky
from 10h to 14h 50′(Liske et al. 2003). Each field was ob-
served for 750 seconds through a Kitt Peak National Obser-
vatory B-band filter (4407˚ A). The survey reaches a depth
of µlimit = 26 mag arcsec−2, with objects catalogued down
to BMGC = 24 mag. For comparison, the Sloan Digital Sky
Survey Data Release 5 (Adelman-McCarthy et al. 2007) has
a median (r-band) seeing of 1.4′′and an effective exposure
time of 54.1 seconds per band, leading to a g-band (4686˚ A)
magnitude limit of 22.2 mag (roughly on an AB system, and
with B − g roughly one-third of a mag, Blanton & Roweis
2007).
The MGC redshift information has come from a number
of sources, as previously detailed in Driver et al. (2005, their
Table 1). The median redshift is 0.12 and sample selection
effects are well understood (Driver et al. 2005; Liske et al.
2006).
Allen et al. (2006) have performed an R1/n-bulge plus
exponential-disc decomposition, using GIM2D (Marleau &
Simard 1998; Simard et al. 2002), for all 10,095 objects. In
addition to the best-fitting bulge S´ ersic index n, GIM2D
derives the (not necessarily symmetric) upper and lower un-
certainty, δn, on the S´ ersic index. In Fig.3 we show these
uncertainties in n as a function of n. The distribution is
such that 68 per cent of the galaxies have an error on n
of less than ∼20 per cent. Repeat observations, under dif-
ferent seeing conditions and on different chips of the wide-
field camera, exist for 682 galaxies. Fig.15 in Allen et al.
(2006) shows the ability of GIM2D to consistently recover
the S´ ersic index of the spheroid component. The mean offset
and standard deviation in the quantity ∆logn from repeat
observations was reported there to be −0.002 and 0.132 dex,
respectively. However, that distribution has longer tails than
expected from a Gaussian. This is due to 11 per cent of the
objects whose galaxy ‘type’ (see Allen et al. 2006) was in dis-
agreement. Excluding these objects7, the half-width of the
68 percentile is 0.0729 dex, which corresponds to an 18 per
cent mismatch in the value of n. This figure agrees well with
the formal GIM2D error observed in Fig.3, and also with the
20 per cent uncertainty used in Graham & Driver (2007a)
and commonly reported in the literature (e.g. MacArthur,
5BMGCis the Galactic extinction corrected, SExtractor (BEST,
Vega) apparent magnitude (Liske et al. 2003).
6The actual usable area of sky reduced to 30.88 square degrees
after excluding the “bad” regions (around bright stars, diffraction
spikes, CCD defects, CCD gaps, CCD edges, vignetted corner,
etc.).
7The half-width of the 68 percentile of the full distribution (i.e.,
all 682 galaxies) is 0.108 dex, which translates to a 28 per cent
mismatch in the value of n, slightly higher than the formal GIM2D
(1σ) error. This means that ∼11 per cent of the data may have a
larger uncertainty than is assigned by GIM2D.
Courteau & Holtzman 2003). In this paper we adopted the
GIM2D-derived values for both n and δn, as given in the cat-
alogue ’mgc gim2d’ which is publicly available at the MGC
website8.
From the 10,095 galaxies in the MGC with BMGC < 20
mag, we restrict the sample to those 7,745 objects with
0.013 < z < 0.18. Many of these are disc-only systems
and therefore rejected as potential black hole hosts. In pass-
ing we note that, once the attenuating effects of dust have
been dealt with, the MGC displays a uniformly flat distri-
bution in cos(i), where i is the inclination of the disk, as one
would expect for a uniformly distributed sample of galax-
ies (Driver et al. 2007b, their figure 5). We further refine
our galaxy sample by imposing the requirement that both
the bulges and the elliptical galaxies, collectively referred to
as ‘spheroids’, have half-light radii greater than 0.333 arc-
sec (1 pixel) and that the bulge-to-total luminosity ratio
(B/T) is greater than 0.01 (based on the lower values ob-
served by Graham 2001, his figure 15). This helped avoid
bright nuclear components such as star clusters that may
have been fitted with the R1/nmodel in GIM2D. We also
required that the absolute spheroid luminosity be brighter
than −18 B-mag (discussed further in Section 3.2.4). Fi-
nally, we required that the galaxies core colour be red, such
that (u − r)core > 2.00 mag, denoting the transition in the
colour bimodality for the MGC (Driver et al. 2006a). We re-
fer to the red sample as ‘Sample 3’. However in Section 5.2
we show that the effect of including galaxies with blue cores
(Sample 1 and 2) does not significantly alter our results on
the SMBH mass density.
We additionally construct two mutually exclusive sub-
samples, which we label ’early-types’ and ’late-types’. The
distinction is based on whether a galaxy’s B/T ratio is
greater than or less than 0.4. Our choice of 0.4 is lower
than values of 0.5 or 0.6 which have often been used in the
past. Figure 4 shows the B/T ratio for ∼3k MGC galaxies
brighter than BMGC = 19 mag and which satisfy the above
criteria and which have also been classified by eye into three
morphological bins (Driver et al. 2006a): early-type galaxies
(E/S0), early-type spiral galaxies (Sabc) and late-type spi-
ral galaxies (Sd/Irr). If one was to use a cut at B/T = 0.5 or
0.6 to identify the early-type galaxies, then one would miss
roughly half of them.
2.1Colour correction
Before proceeding, we note that Eq. 1 was constructed in
the R-band, while the MGC galaxies have been imaged in
the B-band. The presence of radial, B − R colour gradients
(e.g., La Barbera et al. 2005, and references therein) may
therefore result in different values for the S´ ersic index n in
the two bands.
In general, colour gradients are known to be fairly small
in observations of local, early-type galaxies more luminous
than ∼ −17 B-mag (e.g., Peletier et al. 1990; Taylor et al.
2005, and references therein). The SDSS-VAGC (Blanton et
al. 2005) has S´ ersic indices in the ugriz passbands in their
low-redshift catalogue. Although these indices are somewhat
8http://www.eso.org/∼jliske/mgc/

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4
Graham et al.
different to ours, in that they are derived from a single R1/n-
galaxy model rather than an R1/n-bulge plus exponential-
disc model, we should be able to get some insight from these
data for the early-type galaxies, or at least the (disc-less)
elliptical galaxies. An analysis of Fig.5, which shows the dif-
ference between the SDSS r- and g-band S´ ersic index, re-
veals that the median value of log(nr/ng) for galaxies with
nr > 2.0 — i.e. predominantly the early-type galaxies — is
only 0.003 dex (after removal of the few obvious outliers with
absolute values greater than 0.3 dex). We therefore apply
no correction to our B-band S´ ersic indices of the early-type
(B/T > 0.4) galaxies.
The single R1/n-galaxy models that have been applied
to the SDSS data are not suitable for quantifying possible
changes, with wavelength, to the S´ ersic indices of bulges in
late-type galaxies. Instead, we use the B- and R-band S´ ersic
indices from the bulges of 86 disc galaxies given by Graham
(2003), 79 of which have indices in both bands. The average
(± std.dev.) of the 79 values of log(nR/nB) is 0.09 (±0.15).
Splitting the sample into Sa–Sb and Sc–Sd–Sm galaxies gave
the same small offset of 0.09 dex for each grouping. This
suggests that the difference in n is not a function of spi-
ral galaxy type nor bulge size. In addition to radial stellar
population gradients across the bulges, a plausible contrib-
utor to this difference is dust. If the dust in spiral galaxies
is more abundant at their centres, it will redden their cen-
tres more than their outskirts, reducing the central parts of
the B-band light profile relative to the R-band light profile
and thereby yielding smaller S´ ersic indices for the bulge in
the B-band. The similar B- and R-band S´ ersic indices for
the early-type galaxies suggests that dust is not a signifi-
cant issue in these systems. MacArthur et al. (2003) also
provide B- and R-band S´ ersic indices for an independent
sample of 47 and 43 late-type spiral galaxies, respectively
— with 42 in common. In those instances where MacArthur
et al. (2003) provided multiple S´ ersic indices for the same
galaxy in the same passband, we averaged the logarithm of
the S´ ersic indices. The average (± std.dev.) of the 42 val-
ues of log(nR/nB) is 0.08 (±0.17), in good agreement with
the data from Graham (2003). In deriving the SMBH mass
function using Eq. 1 (established in the R-band), we will
therefore apply a positive correction of 0.09 dex to the log-
arithm of the B-band S´ ersic indices from the bulges of our
MGC late-type galaxies.
The presence of two distinct populations in Fig.5, rather
than one continuous distribution, suggests that the step-like
correction we will apply to the S´ ersic indices of our early-
and late-type galaxies (0.0 dex and 0.09 dex respectively)
may be more appropriate than a continuous correction based
on some parameter such as the B/T ratio.
3 SMBH MASS FUNCTION AND SPACE
DENSITY
To derive the SMBH mass function, we first modify the
S´ ersic indices of the late-type systems by 0.09 dex to convert
from the B-band to the R-band, in accord with the previ-
ous Section. We then derive individual black hole masses for
each spheroid using equation 1. For each black hole we de-
termine an associated space-density weighting based on the
MGC blue and red spheroid luminosity functions as derived
Table 1. Supermassive black hole mass function data (corrected
for Malmquist bias) for the full, early- and late-type galaxy sam-
ple (Sample 3, see Section 5.2) shown in Fig. 6. The uncertainties
given are the upper and lower quartiles (i.e. ±25 per cent) from
extensive Monte Carlo realisation of the combined errors.
log10Mbh
M⊙
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
Note: number densities are scaled to per unit logMbhinterval and
not per 0.25logMbhinterval.
φ(10−4h3
70Mpc−3dex−1)
Early-type
0.11+0.08
−0.05
0.21+0.08
−0.06
0.00+0.10
−0.00
0.31+0.11
−0.09
0.19+0.16
−0.12
0.00+0.28
−0.00
0.37+0.47
−0.42
2.48+0.51
−0.52
4.61+0.54
−0.51
6.06+0.58
−0.58
6.02+0.67
−0.63
8.31+1.03
−0.90
15.55+1.54
−1.27
17.56+1.79
−1.50
13.02+1.57
−1.55
5.81+1.75
−1.71
1.96+1.31
−2.24
0.00+1.13
−0.00
0.00+0.03
−0.00
All galaxies
0.42+0.13
0.00+0.13
0.00+0.16
0.55+0.17
1.29+0.20
0.00+0.27
0.76+0.51
3.23+0.54
4.97+0.54
6.58+0.59
6.40+0.72
9.45+1.15
16.29+1.74
20.06+2.26
16.79+1.85
7.53+2.13
3.26+1.41
0.00+1.88
0.00+0.16
Late-type
0.34+0.11
−0.10
0.00+0.11
−0.00
0.02+0.12
−0.02
0.25+0.12
−0.11
1.13+0.13
−0.12
0.00+0.13
−0.00
0.37+0.14
−0.13
0.73+0.14
−0.13
0.36+0.15
−0.14
0.53+0.16
−0.15
0.36+0.17
−0.16
1.15+0.21
−0.19
0.71+0.26
−0.22
2.56+0.56
−0.42
3.73+0.49
−0.48
1.78+0.45
−0.49
1.47+0.33
−0.91
0.00+0.67
−0.00
0.00+0.12
−0.00
−0.11
−0.00
−0.00
−0.16
−0.17
−0.00
−0.42
−0.54
−0.52
−0.61
−0.68
−1.01
−1.36
−1.89
−1.70
−2.22
−3.34
−0.00
−0.00
in Driver et al. (2007a). The weight is the space density,
φ(L), of the appropriate spheroid type (red or blue, divided
at (u−r)core = 2.0 mag) in the specified luminosity interval,
L, divided by the number of galaxies which contributed to
that interval, N(L). The red and blue spheroid luminosity
functions of Driver et al. (2007a) were derived via ‘vanilla’
step-wise maximum likelihood (see Efstathiou, Ellis & Pe-
terson 1988) with K-corrections and e-corrections as defined
in Driver et al. (2007a). The red and blue spheroid functions
show markedly different forms and the justification for the
segregation into two colour types is also given in Driver et
al. (2007a).
The SMBH mass functions are then derived by sum-
ming the distribution of black hole mass times weights, i.e.,
φ(Mbh) =?W(L)Mbh, where W(L) = φ(L)/N(L). This
was constructed for black holes derived from all galaxies,
early-types only (B/T > 0.4) and late-types only (B/T ≤
0.4). See Fig.6 for a graphic representations and Table 1 for
a tabulated version of these distributions.
There are a number of sources of potential error. We
model these via Monte Carlo simulations, following both in-
dividually and collectively the uncertainties on: the param-
eters defining equation 1; the S´ ersic indices; the luminosity
function weights; and the Malmquist bias9. Errors not mod-
elled at this stage because they are considered secondary
are uncertainties in the: K-corrections; e-corrections; dust
attenuation; spectroscopic incompleteness; photometric er-
rors effecting the magnitudes and hence weights; luminosity
dependent cosmic variance; and the choice of colour cut in
9Here we define Malmquist bias to be the systematic bias in our
final measurements due to the presence of errors in our data.

Page 5

MGC: SMBH mass function
5
the derived red and blue spheroid luminosity functions. Note
that the global cosmic variance for the MGC survey area was
derived in Driver et al. (2005) by comparison to mock cat-
alogues produced by the Durham group10; this amounts to
an overall uncertainty of 6 per cent in all 4 × 10,001 Monte
Carlo simulations.
The Monte Carlo simulations consist of repeating the
above analysis 10,001 times with the input values per-
turbed by the listed errors individually and collectively, i.e.,
4 × 10,001 Monte Carlo simulations in all. All error dis-
tributions are assumed to be Gaussian. The impact of the
error(s) is(are) then assessed by comparing the median and
standard deviations of the derived distributions to the orig-
inal estimate.
Table 2 summarises the cumulative sum of the SMBH
mass functions, ρbh,0, with error estimates from each of the
above sources. In all cases we see that the dominant error
is from the uncertainty in equation 1. This is perhaps not
surprising since equation 1 is based on a quadratic fit to
only 27 systems for which both credible black hole masses
and S´ ersic indices exist. Improvements in this method will
therefore come from a larger calibration sample (similarly
for the M − σ and M − L relations) and this should be
attainable with next generation facilities. Perhaps surprising
is that the errors in the S´ ersic distribution have relatively
little impact. In part this is indicative of the size of the
sample but also helped by the quadratic nature of the M−n
relation preventing high n values leading to unreasonably
large Mbh values. The errors from the luminosity function
(i.e., statistics and cosmic variance) appear negligible, with
cosmic variance the most significant.
Finally, we correct for the Malmquist bias in a rather
straightforward manner. This is achieved by measuring ρbh,0
assuming no errors and then remeasuring it but allowing
all of the above errors to be perturbed simultaneously via
Monte Carlo simulations. The difference between the two
estimated values (raw and the median of the Monte Carlo
simulations) provides a crude measurement of the system-
atic offset caused by the errors inherent in our data. To
provide Malmquist bias corrected values we then subtract
this systematic from our original values (∼ 1 to 3 per cent
downward correction). These final Malmquist bias corrected
values are those shown in Tables 1 and 2.
Fig.6 shows the resulting black hole mass functions,
where the error bars indicate the combined errors derived
from the Monte Carlo analysis but excluding the cosmic
variance error (as this is a hidden systematic and not a
random error). Our resulting SMBH mass functions (and
number density and mass density) depend on the Hubble
constant. As was shown in Graham & Driver (2007a), the
Mbh–n relation is independent of the Hubble constant. This
is because the S´ ersic index n does not depend on galaxy dis-
tance, and while the SMBH mass does depend on distance,
the overwhelming majority of galaxies that were used to
construct the Mbh–n relation had their distances obtained
by Tonry et al. (2001) using surface brightness fluctuations
— a technique that provides distances without assuming
some Hubble constant. The mass function dependence on
the Hubble constant arises from the 1/Vmax weighting given
10http://star-www.dur.ac.uk/∼cole/mocks/
Table 2. The space density of matter in supermassive black holes.
The errors are 1-σ values, after excluding 3-σ outliers. The densi-
ties given in Table 1, rather than the fitted empirical models, have
been integrated down to SMBH masses of 106M⊙. The final col-
umn shows the density normalised against the critical density. The
reason why the density varies with h3and not h2is explained in
Section 3. Factoring in the intrinsic scatter, ∆, from the Mbh–n
log-quadratic relation, the numbers in this Table should be in-
creased by exp[(∆ln10)2/2], which equals 1.09 if ∆ = 0.18 dex
(Graham & Driver 2007a).
No.
Bulges
Early- and Late-type (B/T > 0.01)
17694.87 ± 1.84 ± 0.07 ± 0.01 ± 0.29
16764.53 ± 1.63 ± 0.07 ± 0.01 ± 0.27
15434.41 ± 1.65 ± 0.07 ± 0.01 ± 0.26
ρbh,0a±δ1b±δ2c±δ3d±δ4e
h3
Ωbh
70105M⊙Mpc−3
10−6h70
Sample 1
Sample 2
Sample 3
3.6±1.4
3.3±1.2
3.2±1.2
Early-type (B/T > 0.4)
3.78 ± 1.28 ± 0.06 ± 0.01 ± 0.23
3.57 ± 1.21 ± 0.06 ± 0.01 ± 0.21
3.46 ± 1.14 ± 0.06 ± 0.01 ± 0.21
Sample 1
Sample 2
Sample 3
1539
1485
1352
2.8±1.0
2.6±0.9
2.5±0.9
Late-type (0.01 < B/T ≤ 0.4)
1.14 ± 0.56 ± 0.04 ± 0.01 ± 0.07
0.95 ± 0.49 ± 0.03 ± 0.01 ± 0.06
Sample 1
Sample 2
Sample 3
230
191
191
0.8±0.4
0.7±0.4
as above
aMalmquist bias corrected.
bMonte Carlo simulation of the errors in the Mbh–n relation
assuming a Gaussian error distribution.
cMonte Carlo simulation of the error in n as specified by GIM2D
and assuming a Gaussian error distribution.
dMonte Carlo simulations of the error in the individual galaxy
weights assuming a Gaussian error distribution.
eMGC global cosmic variance of 6 per cent for the effective ∼ 30
sq degree region with 0.013 < z < 0.18.
to each galaxy; it is this volume term which introduces an
h3dependence.
3.1Parameterisation of the SMBH mass function
We have fitted two empirical models to our SMBH mass
functions over the mass range 106< Mbh/M⊙ < 109. The
first is a mild variation of the commonly-used 3-parameter
Schechter (1976) function, and is given by
φ(Mbh) = φ∗
?Mbh
M∗
?α+1
exp
?
1 −
?Mbh
M∗
??
,(2)
where φ(M∗) = φ∗ (per unit dlog(Mbh) per Mpc3). The
turnover of the mass function and the maximum density
occur at the SMBH mass
Mmax = (α + 1)M∗,α + 1 > 0(3)
where the associated maximum density is
φmax = φ∗(α + 1)α+1exp[−α].(4)
The logarithmic slope at the low-mass end of the mass
function is given by the exponent 1 + α; a value of α =
−1 therefore corresponds to a flat distribution, and larger
values correspond to a decreasing function as the SMBH
mass decreases. For the early-type galaxy samples the slope
(1+α) is ∼1 (see Table 3). For the (early+late)-type samples
the slope is approximately two-thirds; the shallower decline

Page 6

6
Graham et al.
is due to the contribution of SMBHs in late-type galaxies.
These, and the other, best-fitting parameters are reported
in Table 3 and the fits themselves are shown in Fig.6.
We explored the suitability of a second model having
an additional fourth parameter such that the mass term in
the exponential of equation 2 is raised to the power of β to
give
φ(Mbh) = φ∗
?Mbh
M∗
?α+1
exp
?
1 −
?Mbh
M∗
?β?
(5)
(Aller & Richstone, their equation 10). Our data, how-
ever, did not justify the need for this additional parameter.
For the (early+late)-type samples, the value of β equalled
1.0±0.1. For the early-type sample, it ranged from 0.3 to
0.5, but did not give rise to significantly better fits. For this
reason we do not show the fits or the parameters from this
model. (Neither equations 2 nor 5 provided an acceptable de-
scription to the SMBH mass function in our late-type galaxy
sample.)
3.2 Integrating the SMBH mass function
3.2.1 SMBH number density
The total SMBH number density can be obtained by inte-
grating equation 2 with respect to log(Mbh) (because φ is
expressed in units of h3
Integrating over 106< Mbh/M⊙ < 109, the number density
is given by the expression
70Mpc−3per decade in SMBH mass).
?log(Mbh/M⊙)=9
log(Mbh/M⊙)=6
?Mbh=109M⊙
Mbh=106M⊙
φ∗e1
ln(10)
φ(Mbh) dlogMbh
=
φ(Mbh)
ln(10)Mbh
dMbh
=
?
γ
?
α + 1,109M⊙
M∗
?
− γ
?
α + 1,106M⊙
M∗
??
, (6)
where γ(a,x) is the incomplete gamma function (e.g., Press
et al. 1992) defined by
γ(a,x) =
?x
0
e−tta−1dt. (7)
For the early- and late-type galaxy values of φ∗,M∗ and
α (given in Table 3), the number density of SMBHs with
masses between one million and one billion solar masses is
2.3×10−3h3
ber density is 2.1 × 10−3h3
These values are an order of magnitude smaller than
those reported in Shankar et al. (2004). The difference can
be attributed to their rising (Mbh–L)-derived SMBH mass
function (for all galaxy types) as one moves towards lower-
masses. In contrast, our (Mbh–n)-derived mass function
shows the opposite behaviour at the low-mass end.
70Mpc−3. For the early-type galaxies, the num-
70Mpc−3.
3.2.2SMBH mass density
In so far as equation 2 represents the mass function over the
SMBH mass range 106< Mbh/M⊙ < 109, the SMBH mass
density for SMBHs having such masses can be obtained from
ρbh=
?log(Mbh/M⊙)=9
log(Mbh/M⊙)=6
α + 2,109M⊙
M∗
φ(Mbh)Mbh dlogMbh
=φ∗M∗e1
ln(10)
?
γ
?
?
− γ
?
α + 2,106M⊙
M∗
??
. (8)
Using the best-fitting parameters in Table 3, we obtain
ρbh = 4.3 × 105h3
4.0 × 105h3
function over all masses increases the above values to 4.8 ×
105h3
We have, however, opted not to use the above formula,
but to instead acquire the SMBH mass densities directly
from our data, rather than from the fitted model.
Computing the local mass density, and integrating down
to SMBH masses of 106M⊙, we obtain ρbh,early−type= (3.5±
1.2) × 105h3
105h3
The above densities correspond to Ωbh,early−type =
ρbh,early−type/ρcrit
= (2.5 ± 0.9) × 10−6h70,
Ωbh,late−type = (0.7 ± 0.4) × 10−6h70 (see Table 2), where
ρcrit = 3H2
For reference, using an independent technique, Fukugita &
Peebles (2004, their equation 75) give 2.5+2.5
SMBHs in early-type galaxies, and 1.3+1.2
SMBHs in late-type galaxies.
In deriving the (linear) SMBH mass density, ρbh,0, (i.e.,
not logρbh,0) from the convolution of the distribution func-
tion of n with the logMbh–logn relation, one needs to allow
for the intrinsic scatter in this log−log correlation. If there
is a Gaussian distribution of intrinsic scatter in the SMBH
mass at any fixed logn, with a standard deviation which is
independent of logn and equal to ∆, then the SMBH mass
density should be increased by the factor exp[(∆ln10)2/2]
(Yu & Tremaine 2002, their equation 12). Such a Gaus-
sian distribution, however, is not true for the scatter of
points about the log-quadratic Mbh–n relation shown in
Fig.1; the scatter is clearly greater at the high-mass end.
In fact, removal of the two highest mass SMBHs results in a
log-quadratic relation consistent with zero intrinsic scatter
(Graham & Driver 2007a). It is thus questionable whether
one should apply this multiplicative term and because of
this uncertainty, in what follows, we have not. However, one
should note that if there were in fact no measurement er-
rors and ∆ = 0.31 dex (the total absolute scatter about the
logMbh–log n relation in Fig.1), then the corrective factor to
apply to ρbh,0 would be 1.29. If ∆ = 0.18 dex (the intrinsic
scatter after factoring in the suspected measurement errors),
then the multiplicative factor drops to 1.09. If there is no
intrinsic scatter, then the multiplicative factor is simply 1.
70M⊙ Mpc−3(all-types), and ρbh =
70M⊙ Mpc−3(early-types). Integrating the mass
70M⊙ Mpc−3and 4.2×105h3
70M⊙ Mpc−3, respectively.
70M⊙ Mpc−3while ρbh,late−type= (1.0 ± 0.5) ×
70M⊙ Mpc−3. These results are presented in Table 2.
and
0/8πG is the critical density for flat space11.
−1.2×10−6for the
−0.7× 10−6for the
3.2.3SMBH baryon fraction
The above numbers can be expressed in terms of the baryon
fraction of the universe. Already a picture is emerging
in which dark accretion onto SMBHs may be negligible.
Shankar et al. (2004) claim that the local SMBH mass func-
tion can be accounted for from mass accreted by X-ray se-
11For H0= 70 km s−1Mpc−1, ρcrit∼ 1.36 × 1011M⊙Mpc−3.

Page 7

MGC: SMBH mass function
7
Table 3. Best-fitting parameters from the empirical SMBH mass function given in
equation 2. The number density, φ∗, is per decade in SMBH mass. The late-type
galaxy sample is not shown here due to the poor match between the empirical model
and the data (see Fig.6). Sample 2 excludes bulges in disc galaxies if the core colour is
bluer than (u−r)c= 2.00 mag. Sample 3 (our primary sample) excludes blue bulges
and blue elliptical galaxies if (u − r)c≤ 2.00 mag (see Section 5.2).
Data sample logφ∗
log(M∗/M⊙)α
h3
70Mpc−3dex−1
Early- and Late-type, (B/T > 0.01):
Sample 1: no colour cut
Sample 2: (u − r)c[bulges] > 2.00 mag
Sample 3: (u − r)c[all] > 2.00 mag
Early-type, (B/T > 0.4):
Sample 1: no colour cut
Sample 2: (u − r)c[bulges] > 2.00 mag
Sample 3: (u − r)c[all] > 2.00 mag
-2.76
-2.76
-2.81
8.45
8.43
8.46
-0.32
-0.29
-0.30
-2.68
-2.67
-2.74
8.26
8.25
8.29
0.07
0.10
0.00
lected AGN, i.e. no significant ‘dark’ accretion is required
for SMBH growth (see also Cao 2007). If correct, this would
imply that the growth of SMBHs from the accretion of mas-
sive black hole remnants of Population III stars may be small
(Madau & Rees 2001; Islam, Taylor & Joseph 2003; Wyithe
& Loeb 2004). Moreover, baryonic-fuelling is consistent with
a picture that links the BH mass to the host spheroid’s bary-
onic, or at least stellar, properties, such as luminosity (e.g.,
Erwin, Graham & Caon 2002; McLure & Dunlop 2002; Mar-
coni & Hunt 2003 and references therein), concentration
(Graham et al. 2001; Graham & Driver 2007a) and mass
(Magorrian et al. 1998; H¨ aring & Rix 2004). Assuming the
above standard picture in which SMBHs form via the ac-
cretion of baryons (Blandford 2004), then in terms of the
invariant baryon fraction of the total mass-energy density,
such that Ωbaryon = 0.0453h−2
Table 2, row 3; see also Blake et al. 2007), SMBHs at the
centres of galaxies today contain (0.007±0.003)h3
of the universe’s baryon inventory.
Letting Mspheroid denote the stellar mass of a spheroid,
H¨ aring & Rix (2004) have shown that Mbh/Mspheroid =
0.0009 when Mbh = 106M⊙ and = 0.0019 when Mbh =
109M⊙, confirming the results given in Merritt & Ferrarese
(2001, their Section 3; see also Laor 2001). Taking the av-
erage (or using the peak in our SMBH mass function at
log(Mbh) = 8.3, see Fig.6) one obtains Mbh equals 0.14 per
cent of Mspheroid (or 0.16 per cent). Dividing Ωbh/Ωbaryon
by this value one obtains Ωspheroid ∼ 6 per cent (or ∼5 per
cent) of Ωbaryon. This is of course just a rough estimate, and
a more precise value obtained using the actual spheroid lu-
minosity function will be presented in Driver et al. (2007a,
2007b).
70(Tegmark et al. 2006, their
70per cent
3.2.4Comments on the low mass end
Figure 7 shows how the SMBH space density is built up
as one includes (intrinsically) fainter spheroids, for our
three galaxy bins. As was noted in Section 2, we excluded
spheroids fainter than MB = −18 mag in our analysis. For
the early-type galaxy sample, Figure 7 reveals that integrat-
ing down to an absolute magnitude of −16 B-mag gives a
SMBH mass density that is consistent (at the 2-σ level) with
the result obtained using only those spheroids brighter than
−18 B-mag. We do however caution that the SMBH masses
are less reliable in this regime, and as such we do not wish
to place too much emphasis on this result.
It is also worth noting that the central massive object
in spheroids fainter than MB ∼ −18 mag is often a nuclear
star cluster (Ferrarese et al. 2006, left panel of their fig-
ure 2; Balcells et al. 2007). It is therefore questionable as to
whether or not one should include such faint spheroids in an
analysis of this kind. Moreover, if they are not individually
modelled, the presence of additional nuclear star clusters
can bias the S´ ersic R1/nfit to give spuriously high values of
the index n. The effect is to over-estimate the contribution
from SMBHs in faint spheroids. Given the high frequency
of nucleated bulges in spiral galaxies (e.g., Carollo, Stiavelli,
& Mack 1998; Balcells et al. 2003), there is thus reason to
doubt the rising SMBH space density curve shown in Fig.7
for the late-type galaxies.
4COMPARISON WITH PAST ESTIMATES
McLure & Dunlop (2004) provided two estimates of the
SMBH mass function (in early-type galaxies) using two tech-
niques. At the low-mass end (< 108M⊙) their estimates do
not agree; the mass function they obtained using the Mbh–
L relation gave noticeably higher SMBH number densities
than they obtained using the Mbh–σ relation (Fig.8, lower
panel). Both the method and data that we have used is dif-
ferent to that used in McLure & Dunlop (2004) and thus
provides an independent check on the SMBH mass func-
tion. Our analysis, using the Mbh–n relation, shows a mass
function which declines with decreasing SMBH masses that
are less than 108M⊙, and therefore better matches McLure
& Dunlop’s mass function constructed using the Mbh–σ re-
lation rather than the Mbh–L relation.
In plotting the mass functions from McLure & Dunlop
(2004) in Fig.8, we have multiplied their (Mbh–L)-derived
masses by 1.70. This increase stems from the conversion of
their R-band Mbh–L relation to the K-band. Starting from

Page 8

8
Graham et al.
equation 6 in McLure & Dunlop (2002)12, one has
log(Mbh/M⊙) = −0.5MR− 2.91
= −0.5[MK + 2.7 − MK,⊙+ 3.28 − 5log(70/50)] − 2.91
= 1.25log(LK/LK,⊙) − 5.53 (9)
Following McLure & Dunlop (2002), we have assumed an
average R−K colour of 2.7. We have used MK,⊙ = 3.28 mag
(Binney & Merrifield 1998). McLure & Dunlop (2002) used
the SMBH masses derived using Tonry et al.’s (2001) surface
brightness fluctuation distances — which are independent
of the Hubble constant. However they did not use these h-
independent distances in deriving the absolute magnitude of
the bulges, but used redshift derived distances and H0 = 50
km s−1Mpc−1. Equation 9 gives SMBH masses that are 0.23
dex larger than equation 1 in McLure & Dunlop (2004), and
hence the factor of 1.70 (see Graham & Driver 2007b for
more details).
In the lower panel of Fig.8, one can see that our analysis
suggests there may be a greater (up to a factor of ∼2) num-
ber density of SMBHs with masses log(Mbh/M⊙) ∼ 8.5 than
reported in McLure & Dunlop (2004). At smaller masses,
8 > log(Mbh/M⊙) > 6, our mass functions roughly display
the same decline with decreasing SMBH mass as McLure &
Dunlop’s analysis based on the Mbh–σ relation. Such a de-
clining SMBH mass function (for early-type galaxies) is also
in qualitative agreement with Shankar et al.’s (2004) mass
function derived using a bivariate velocity dispersion distri-
bution. In passing we note that we have increased Shankar
et al.’s SMBH masses by 14 per cent, after adjusting for
the correct dependence on the Hubble constant (see Gra-
ham & Driver 2007b). On the other hand, our data dis-
agree with mass functions that rise monotonically (which
includes asymptotically) towards lower-mass SMBHs, such
as the (Mbh–L)-derived mass functions in McLure & Dunlop
(2002), Aller & Richstone (2002) and Shankar et al. (2004).
The upper panel of Fig.8 reveals a clear disagreement,
at masses below ∼ 108M⊙, between our (total) mass func-
tion and that derived from Mbh–L relations. We are not
aware of any (Mbh–σ)-derived mass function for late-type
galaxies, and therefore we do not show any such curve in
this panel. Some of the mismatch from the (Mbh–L)-derived
SMBH masses may stem from the methods used to assign
bulge flux in lenticular and spiral galaxies (see Graham &
Driver 2007b and Graham 2007).
Over the past few years a number of papers have con-
structed the local SMBH mass function and estimated the
local SMBH mass density, ρbh,0. We have listed several of
these in Table 4. It can be seen that our estimates of ρbh,0
agree particularly well with the estimate in Fukugita & Pee-
bles (2004), and is consistent (within the 1-σ error bounds)
with several other recent studies. It is worth noting, however,
that our estimates are a factor of ∼2 greater than reported
in Wyithe (2006) and Aller & Richstone (2002). Due to the
declining number density with decreasing SMBH mass, our
12Transformation of equation 5 in McLure & Dunlop (2002)
would be considerably more difficult because it was constructed
from an inactive plus active galaxy sample, a fraction of which
have distances that depend on the Hubble constant and also on
the adopted cosmology given that some of the active galaxies have
redshifts which extend out to ∼ 0.5.
adopted integration across the complete mass spectrum of
SMBHs, rather than truncating at 106M⊙, does not signifi-
cantly increase our estimate of ρbh,0.
In an effort to try and understand why our SMBH mass
densities may be slightly higher than reported by some,
Graham & Driver (2007b) has examined two representative
studies; one which used the Mbh–σ relation (Aller & Rich-
stone 2002) and the other the Mbh–L relation (Shankar et al.
2004). Graham & Driver identified a number of corrections
which can be made and showed that the mass density from
Aller & Richstone (2002) is more than a factor of ∼ 2 too
low. As with the analysis by McLure & Dunlop, which was
a factor of 1.7 too low, the reason is because of over-looked
dependencies on the Hubble constant. These have been cor-
rected in Figure 8, but not incorporated into Table 4.
5 POTENTIAL BIASES IN OUR DATA
5.1A Linear Mbh–n relation
While we believe the quadratic relation between Mbhand n
(equation 1) is the optimal expression to use when predicting
Mbhfrom n, we have explored how ρbh,0would change if we
adopted the linear relation presented in Graham & Driver
(2007a, their equation 2). Although a linear fit to the data
in Figure 1 results in predictions of larger SMBH masses at
the high-n end, over the range ∼2 < n < ∼8 it actually
predicts lower SMBH masses (see Figure 3 in Graham &
Driver 2007a). Because the bulk of the MGC data falls in
this interval (see Figure 3)13, use of the linear Mbh–n rela-
tion results in a 34 per cent lower SMBH mass density. This
value is consistent, i.e. within the 1-σ uncertainty, with our
estimate ρbh,0 = (4.4 ± 1.7) × 105h3
as noted in Graham & Driver (2007a), this linear relation
has an intrinsic scatter of 0.31 dex, and so the estimate of
ρbh,0 should be increased by 29 per cent, to give a value of
3.8 × 105h3
We have additionally removed the three high-n data
points from Figure 1 and obtained a new linear relation with
the remaining 24 data points. Doing so results in a relation
with zero intrinsic scatter and a mass density that is 43
per cent higher than obtained with the quadratic relation.
Specifically, we obtain a value of (6.3 ± 2.5) × 105h3
Mpc−3, which is only marginally higher than our upper 1-σ
value of 6.1×105h3
(4.0 ± 1.2) × 105h3
1.3) × 105h3
The associated SMBH mass functions are shown in Fig-
ure 9 and given in Table 5. We caution that we do not believe
70M⊙ Mpc−3. However,
70M⊙ Mpc−3.
70M⊙
70M⊙Mpc−3. We also find ρbh,early−type=
70M⊙ Mpc−3and ρbh,late−type = (2.3 ±
70M⊙ Mpc−3.
13We speculate that the apparent shortage of S´ ersic indices
greater than ∼8 in the MGC data (compared to Figure 1) may
arise from GIM2D’s difficulty in deriving such large values while
producing a galaxy magnitude that is close to the value obtained
using SExtractor (due to the large wings of high-n profiles). Con-
firmation of this speculation would require simulations beyond
the scope of this paper. We therefore offer it as nothing more
than speculation, but do note that if correct, a variety of plausi-
ble stretches to the MGC values of n that we tested (to recover the
possible true distribution) resulted in a ∼20 per cent increase to
the value of ρbh,0(i.e., smaller than our current 1-σ uncertainties
on this value).

Page 9

MGC: SMBH mass function
9
Table 4. Local SMBH mass density estimates. The difference between Samples 1 and 3 (see
Section 5.2) in our study is that the latter excludes galaxies with (u − r)core < 2.00 mag, such
as blue pseudo-bulges. The factor h3
because the Mbh–n relation is independent of the Hubble constant. The majority of the densities
from other papers have been transformed to H0 = 70 km s−1Mpc−1using h2rather than h3,
as indicated in each paper. However, as discussed in Graham & Driver (2007b), this may not
always be appropriate.
70= [H0/(70 km s−1Mpc−1)]3is appropriate for our study
Studyρbh,0(E/S0)
70105M⊙Mpc−3
(3.8 ± 1.3)h70
(3.5 ± 1.2)h70
...
(3.4+3.4
3.3
3.1+0.9
−0.8
2.8 ± 0.4
...
1.8 ± 0.6
2.0 ± 0.2
...
6.2
ρbh,0(Sp)
70105M⊙Mpc−3
(1.1 ± 0.6)h70
(1.0 ± 0.5)h70
...
(1.7+1.7
1.3
1.1+0.5
−0.5
...
...
0.6 ± 0.5
0.9 ± 0.2
...
2.0
ρbh,0(total)
70105M⊙ Mpc−3
(4.9 ± 1.9)h70
(4.4 ± 1.7)h70
2.28 ± 0.44
(5.1+3.8
4.6+1.9
−1.4
4.2+1.1
−1.1
...
2.2+3.9
−1.4
2.4 ± 0.8
2.9 ± 0.4
4.6h−1
8.2
h2
h2
h2
This study (Sample 1)
This study (Sample 3)
Wyithe (2006)
Fukugita & Peebles (2004)a
Marconi et al. (2004)
Shankar et al. (2004)b
McLure & Dunlop (2004)
Wyithe & Loeb (2003)
Aller & Richstone (2002)c
Yu & Tremaine (2002)d
Merritt & Ferrarese (2001)e
Salucci et al. (1999)
−1.7)h−1
70
−0.8)h−1
70
−1.9)h−1
70
70
aSee their equation 75.
bBased on their (Mbh–L)-derived mass function, and in agreement with their (Mbh–σ)-derived
values.
cTaken from their Table 2.
dBased on their (Mbh–σ)-derived mass function.
eSee also Ferrarese (2002).
these are more accurate than those provided in Table 1, but
that they provide an (extreme) upper limit to the SMBH
mass function and space density. As was noted in Section 2,
the evidence for a curved Mbh–n relation is inherent in the
low-n data. Excluding the five highest n galaxies results in
a quadratic relation that is fully consistent with equation 1.
Removing the two galaxies with the highest SMBH masses
(one of which is suspected to be biased high: NGC 4486)
produces the same result. The linear relation obtained after
removing the three high n galaxies is thus entirely driven by
two galaxies (NGC 4486 and NGC 4649).
5.2 Blue Spheroids and Pseudobulges
We divided each of the early-type (E/S0), late-type (Sp)
and (early + late)-type galaxy bins into three (colour) sam-
ples. Sample 1 has had no modifications to it, while Sam-
ple 2 excludes bulges (in disc galaxies) if the (u − r) core
colour is bluer than 2.00 mag. The core-colour is derived
from Sloan Digital Sky Survey point-spread-function mag-
nitudes which have been shown to maximise the colour bi-
modality, see Driver et al. (2006a). This colour-cut may help
avoid possible pseudo-bulges (e.g., Erwin et al. 2003; Kor-
mendy & Kennicutt 2004; Kormendy & Fisher 2005) which
may or may not have SMBHs. Sample 3 is further reduced
to exclude all systems (i.e., bulges and elliptical galaxies)
with (u − r)c < 2.00 mag (see Fig.10).
It is not clear whether supermassive black holes ex-
ist in (every) pseudo-bulge. The tight correlation between
the properties of dynamically hot spheroids and their black
hole mass has been tied to their joint formation process. As
pseudo-bulges are believed to have formed via a distinctly
Table 5. Supermassive black hole mass function data (corrected
for Malmquist bias) for the full, early- and late-type galaxy sam-
ple (Sample 3, see Section 5.2) shown in Figure 9. The linear
Mbh–n relation (Section 5.1), obtained after excluding the three
galaxies with the highest S´ ersic index, has been used here to pro-
vide an upper estimate to the SMBH mass function. The uncer-
tainties given are the upper and lower quartiles (i.e. ±25 per cent)
from extensive Monte Carlo realisation of the combined errors.
log10Mbh
M⊙
5.00
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
Note: number densities are scaled to per unit logMbhinterval and
not per 0.25logMbhinterval.
φ(10−4h3
70Mpc−3dex−1)
Early-type
0.06+0.04
−0.04
0.00+0.07
−0.00
0.00+0.08
−0.00
0.42+0.10
−0.09
0.00+0.13
−0.00
0.45+0.18
−0.15
0.00+0.42
−0.00
2.96+0.64
−0.67
4.48+0.57
−0.59
8.22+0.55
−0.55
7.21+0.69
−0.65
12.03+0.87
−0.83
15.88+1.01
−0.89
11.85+0.87
−0.80
9.52+0.90
−0.94
4.48+0.89
−0.76
2.51+0.48
−0.43
1.50+0.33
−0.33
0.42+0.27
−0.25
All galaxies
0.12+0.12
0.00+0.16
1.13+0.17
0.84+0.17
0.00+0.19
1.28+0.23
0.00+0.39
4.33+0.67
4.68+0.58
8.76+0.61
8.37+0.75
13.20+0.96
16.23+1.14
13.93+0.96
12.89+0.93
5.45+1.11
3.69+0.57
1.50+0.41
0.94+0.29
Late-type
0.08+0.12
−0.08
0.00+0.15
−0.00
1.26+0.14
−0.13
0.44+0.14
−0.13
0.11+0.14
−0.13
0.84+0.15
−0.14
0.00+0.15
−0.00
1.34+0.15
−0.15
0.19+0.17
−0.16
0.54+0.18
−0.16
1.13+0.18
−0.18
1.17+0.19
−0.19
0.35+0.23
−0.22
2.11+0.32
−0.29
3.39+0.27
−0.27
1.05+0.32
−0.31
1.18+0.17
−0.15
0.00+0.14
−0.00
0.52+0.10
−0.09
−0.07
−0.00
−0.16
−0.15
−0.00
−0.21
−0.00
−0.70
−0.57
−0.60
−0.68
−0.88
−1.01
−0.90
−0.98
−1.09
−0.50
−0.39
−0.25

Page 10

10
Graham et al.
different formation process (i.e., from a redistribution of
stars due to perpendicular-to-the-disc resonances with a bar)
it is plausible that pseudo-bulges do not possess SMBHs. De-
spite this conjecture, we note that bars can fuel AGN growth
(e.g., Ann & Thakur 2005; Combes 2006; Ohta et al. 2006).
Moreover, the Milky Way has a SMBH mass that appears
consistent with both the Mbh–n and Mbh–σ relations and
probably has a pseudo-bulge — suggested from its rotation
(e.g., Minniti et al. 1992), flattening (e.g., Sharples, Walker
& Cropper 1990; Lopez-Corredoira et al. 2005), presence
of a bar (e.g., Lopez-Corredoira et al. 2006) and peanut-
shaped structure (e.g., Alard 2001, see also Debattista et al.
2006), but see Zoccali et al. (2006) and Matteucci (2006).
The situation is therefore ambiguous, which has resulted in
our decision to create three subsamples.
To unequivocally distinguish dynamically hot bulges
from rotating pseudo-bulges requires kinematical informa-
tion. In the absence of this, an effective method is to use
colour, with the pseudo-bulges typically exhibiting blue
colours comparable to a young (metal rich) disc population
and the genuine bulges exhibiting red colours consistent with
an old population. We do however note that in some cases
the bulges are poorly resolved and the central colour is thus
a blend of the bulge and disc, which might result in some
real bulges erroneously falling blueward of our colour cut.
However the fact that the colour bimodality is clearly evi-
dent in our data suggests this is probably not a major issue,
and we note that a positive correlation between bulge and
disc colour is known to exist (e.g., Peletier & Balcells 1996).
Including blue spheroids with (u−r)c < 2.00 mag, Sam-
ple 1 gives densities of ρbh,early−type= (3.8±1.3)×105h3
Mpc−3and ρbh,late−type = (1.1 ± 0.6) × 105h3
(Table 2).
70M⊙
70M⊙ Mpc−3
6 SUMMARY
We have used the log-quadratic relation between supermas-
sive black hole mass and the host spheroid’s S´ ersic index
n (i.e. the Mbh–n relation, Graham & Driver 2007a) to
predict the SMBH masses in galaxies from the Millennium
Galaxy Catalogue (MGC) for which a two-dimensional,
seeing-corrected, R1/n-bulge plus exponential-disc decom-
position has been performed (Allen et al. 2006).
Applying the appropriate volume corrections, the num-
ber density of SMBHs with masses greater than 106M⊙,
and in spheroids more luminous than MB = −18 mag, is
(2.3 ± 0.1) × 10−3h3
We have constructed the local (0.013 < z < 0.18, ¯ z =
0.12) SMBH mass function using direct estimates of the
Mbh predictor-quantity n (Fig.6). Using every spheroid,
the SMBH mass function appears well represented by a 3-
parameter Schechter function having a logarithmic slope at
the low-mass end of around two-thirds. Excluding the late-
type galaxies, identified as systems with bulge-to-total ratios
smaller than 0.4, the SMBH mass function was again well
fit with a Schechter function, this time having a low-mass
slope of 1, and a turnover mass at ∼ 2 × 108M⊙.
Integrating down to a SMBH mass of 106M⊙, the local
mass density of SMBHs from early- and late-type galaxies
combined is ρbh,0 = (4.4 ± 1.7) × 105h3
value is slightly greater than, but still consistent with, most
70Mpc−3.
70M⊙ Mpc−3. This
values obtained via independent means over recent years
(Table 4, but see Graham & Driver 2007b). Our mass den-
sity estimate can be split into ρbh,early−type= (3.5 ± 1.2) ×
105h3
Mpc−3, based upon a B/T flux ratio cut at 0.4.
In terms of the critical density of the universe, we obtain
Ωbh,total = (3.2 ± 1.2) × 10−6h70, or Ωbh,early−type = (2.5 ±
0.9) × 10−6h70 and Ωbh,late−type = (0.7 ± 0.4) × 10−6h70.
Given that 4.5h−2
form of baryons (Tegmark et al. 2006), the above figures
imply (0.007±0.003)h3
70per cent of the total baryon content
of the universe is locked up in SMBHs at the centres of
galaxies.
70M⊙Mpc−3and ρbh,late−type= (1.0±0.5)×105h3
70M⊙
70per cent of the critical density is in the
7 ACKNOWLEDGMENTS
A.G. would like to thank St Andrews University’s School of
Physics & Astronomy for its hospitality during a 3 week visit
to work on this paper. We are happy to thank Francesco
Shankar for kindly supplying us with the data from his
(Mbh–σ)-derived SMBH mass function shown in Shankar
et al. (2004, their Fig.7). We are additionally grateful to
Francoise Combes for her helpful comments about the Milky
Way. The Millennium Galaxy Catalogue consists of imaging
data from the Isaac Newton Telescope at the Spanish Ob-
servatorio del Roque de Los Muchachos of the Instituto de
Astrof´ ısica de Canarias, and spectroscopic data from the
Anglo Australian Telescope, the ANU 2.3 m, the ESO New
Technology Telescope, the Telescopio Nazionale Galileo, and
the Gemini Telescope. This research has made use of the
NASA/IPAC Extragalactic Database (NED) which is oper-
ated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with the National Aeronau-
tics and Space Administration. The survey has been sup-
ported through grants from the United Kingdom Particle
Physics and Astronomy Research Council, and the Aus-
tralian Research Council through Discovery Project Grant
DP0451426. The data and data products are publicly avail-
able from http://www.eso.org/∼jliske/mgc/ or on request
from J. Liske or S.P. Driver.
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Graham et al.
Figure 1. Relationship between SMBH mass and host spheroid
S´ ersic index. The data points were taken from Graham & Driver
(2007a, their table 1) and the curve shows the best quadratic fit
(equation 1).
Figure 3. GIM2D-derived uncertainty in n, namely δn (which
need not be symmetrical about n), is converted to a fractional un-
certainty and shown as a function of n for the full MGC spheroid
sample. Sixty eight per cent of the data has an error on n of less
than 20 per cent. The bottom-left envelope of points in the left
panel trace the lower limit on the value of n (equal to 0.2) that
we set in GIM2D (see Allen et al. 2006). A second envelope arises
from the 0.09 dex colour-correction which is applied to the bulges
of the late-type galaxies (see Section 2.1).
00.20.4 0.60.81
0
10
20
30
40
50
B/T
N
00.20.40.6 0.81
0
10
20
30
40
50
B/T
N
00.20.40.60.81
0
10
20
30
40
50
B/T
N
Figure 4. Distribution of GIM2D-derived, bulge-to-total flux ra-
tios (B/T) for those MGC galaxies with an (eyeball) morpho-
logical classification (Driver et al. 2006a). Galaxies with a core
colour bluer than (u − r)c = 2.00 mag have been excluded from
the top panel. Spheroids fainter than MB= −17 mag and effec-
tive half-light radii smaller than 0.333 arcseconds (1 pixel) have
been excluded from all panels. The dashed line at B/T = 0.4 de-
notes our adopted divide between early- and late-type galaxies.

Page 13

MGC: SMBH mass function
13
Figure 2. Left panel: Variant of Fig.1 after removing the five data points with the highest SMBH masses. The curved line corresponds
to logMbh= a +blog(n/3) +c[log(n/3)]2. Projections of the ∆χ2= 1.00,4.00 and 6.63 contours, shown in the central and right panels,
onto each axis gives the 68.3, 95.4, and 99 per cent confidence interval on each of the parameters a,b and c. The optimal log-quadratic
relation (shown in the left panel) is consistent with zero intrinsic scatter and reveals that the coefficient, c, in front of the quadratic term
is inconsistent with a value of zero at the ∼ 2.5 σ level. Subsequently, one can conclude that the log-quadratic relation in Eq.1 is not the
result of increased scatter, i.e. outliers, at the high-mass end of the Mbh–n relation (see Graham & Driver 2007a for more details).
Figure 5. Logarithmic difference between the r- and g-band
S´ ersic indices from individual galaxies in the SDSS-VAGC (Blan-
ton et al. 2005), in which a single R1/nfunction was fitted to
each galaxy and the S´ ersic index restricted to lie between 0.2 and
6. The peak of the distribution for galaxies with g-band S´ ersic
indices greater than 2.0 (i.e., predominantly bulge-dominated,
early-type galaxies) is 0.003 dex. The peak of the distribution for
galaxies with ng < 2.0 (i.e., predominantly late-type galaxies) is
0.06 dex. The choice of ng = 2.0 originates from the S´ ersic index
bimodality plots in Driver et al. (2006a). While the right-hand
panel is based on all the SDSS-VAGC data, the left hand panel
shows a random sample of 75,000 with ng > 2.0 and 75,000 with
ng < 2.0. Such a restriction prevents washing away the structure
in the left panel with too many data points.
Figure 6.
samples, e.g. S3 = Sample 3, see Section 5.2. Early-type galax-
ies are those with B/T > 0.4 (see Fig.4). The best-fitting 3-
parameter models (equation 2) are shown. Model parameters
are given in Table 3. The number density shown along the ver-
tical axis is expressed in units of h3
SMBH mass. The fitting is performed over the SMBH mass range
109> Mbh/M⊙ > 106. Residuals about the models are shown
beneath each fit.
Black hole mass functions for our three (colour)
70Mpc−3per decade in

Page 14

14
Graham et al.
Figure 7. Cumulative SMBH space density, ρbh. Our early-type
galaxies are those with B/T > 0.4, while our late-type galaxies
have 0.01 < B/T ≤ 0.4. The cutoff at an absolute magnitude
of −18 B-mag marks the boundary to which we can trust our
data. The horizontal shading is centred on the (H0corrected, i.e.
41 per cent increased) solution for (early+late)-type galaxies by
Shankar et al. (2004, their section 3.2), and shows their quoted
1-σ uncertainty.
109876
-6
-5
-4
-3
-2
(a) Early + Late types
109876
-6
-5
-4
-3
-2
(b) Early types
Figure 8. Our observed SMBH mass functions, along with our
fits to Sample 3 (solid curves), overlaid with others estimates
of the mass function. Shown in the top panel are curves from
Shankar et al. (2004; S04) and Aller & Richstone (2002): (AR02)a
is based on the Madgwick et al. (2002) luminosity function, and
(AR02)b is based on the Marzke et al. (1994) luminosity function.
Shown in the lower panel are curves from Shankar et al. (2004),
and also McLure & Dunlop (2004; MD04) who used two inde-
pendent techniques (see Section 4). The (Mbh–σ)-derived mass
functions can be seen to turn down at low-masses, in agreement
with our (Mbh–n)-derived mass function. We have corrected all
curves to H0 = 70 km s−1Mpc−1, following the prescription
given by Graham & Driver (2007b).

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MGC: SMBH mass function
15
Figure 9. Our (extreme) high-mass SMBH mass function con-
structed using the linear Mbh–n relation which excluded the three
highest-n galaxies (see Section 5.1). The solid curves are the opti-
mal fits shown in Figure 6. The dashed curve in the middle panel
is the (Mbh–L)-derived mass function given in McLure & Dunlop
(2004).
Figure 10. The (u − r)c core colour distribution of the three
spheroid samples. Only galaxies with B/T > 0.01 and MB< −18
mag are shown. A colour cut at (u − r)c = 2.00 mag reflects the
central minimum observed in the bimodal distribution of core
colours shown in Liske et al. (2007, in prep.).