local supermassive black holes, relics of active galactic nuclei and the x-ray background

Mon. Not. R. Astron. Soc. 351, 169–185 (2004) doi:10.1111/j.1365-2966.2004.07765.x

Local supermassive black holes, relics of active galactic nucleiand the X-ray background

A. Marconi,1� G. Risaliti,1,2 R. Gilli,1 L. K. Hunt,3 R. Maiolino1 and M. Salvati11INAF-Osservatorio Astrofisico di Arcetri, Largo Fermi 5, I-50125 Firenze, Italy2Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA3INAF-Istituto di Radioastronomia, Sez. Firenze, Largo Fermi 5, I-50125 Firenze, Italy

Accepted 2004 February 23. Received 2004 February 18; in original form 2003 December 2

ABSTRACTWe quantify the importance of mass accretion during active galactic nuclei (AGN) phasesin the growth of supermassive black holes (BHs) by comparing the mass function of blackholes in the Local Universe with that expected from AGN relics, which are black holes grownentirely with mass accretion during AGN phases. The local BH mass function (BHMF) isestimated by applying the well-known correlations between BH mass, bulge luminosity andstellar velocity dispersion to galaxy luminosity and velocity functions. We find that differentcorrelations provide the same BHMF only if they have the same intrinsic dispersion. The densityof supermassive black holes in the Local Universe that we estimate is ρBH = 4.6+1.9

−1.4h20.7 ×

105 M� Mpc−3. The relic BHMF is derived from the continuity equation with the onlyassumption that AGN activity is due to accretion on to massive BHs and that merging is notimportant. We find that the relic BHMF at z = 0 is generated mainly at z < 3 where the majorpart of the growth of a BH takes place. Moreover, BH growth is antihierarchical in the sense thatsmaller BHs (M BH < 107 M�) grow at lower redshifts (z < 1) with respect to more massiveones (z ∼ 1–3). Unlike previous work, we find that the BHMF of AGN relics is perfectlyconsistent with the local BHMF, indicating that local BHs were mainly grown during AGNactivity. This agreement is obtained while satisfying, at the same time, the constraints imposedfrom the X-ray background (XRB). The comparison between the local and relic BHMFs alsosuggests that the merging process is not important in shaping the relic BHMF, at least at lowredshifts (z < 3), and allows us to estimate the average radiative efficiency (ε), the ratio betweenemitted and Eddington luminosity (λ) and the average lifetime of active BHs. Our analysisthus suggests the following scenario: local BHs grew during AGN phases in which accretingmatter was converted into radiation with efficiencies ε = 0.04–0.16 and emitted at a fractionλ = 0.1–1.7 of the Eddington luminosity. The average total lifetime of these active phasesranges from � 4.5 × 108 yr for M BH < 108 M� to � 1.5 × 108 yr for M BH > 109 M�, butcan become as large as ∼109 yr for the lowest acceptable ε and λ values.

Key words: black hole physics – galaxies: active – galaxies: evolution – galaxies: nuclei –quasars: general – cosmology: miscellaneous.

1 I N T RO D U C T I O N

Since the discovery of quasars (Schmidt 1963), it has been sug-gested that active galactic nuclei (AGN) are powered by mass ac-cretion on to a supermassive black hole (BH) with mass in the range106–1010 M� (Salpeter 1964; Zel’dovich & Novikov 1964; Lynden-Bell 1969). This paradigm combined with the observed evolution ofAGNs implies that a significant number of galaxies in the Local Uni-

�E-mail: [email protected]

verse should host a supermassive BH (e.g. Soltan 1982; Cavaliere& Padovani 1989; Chokshi & Turner 1992).

Supermassive BHs have now been detected in ∼40 galaxiesthrough gas and stellar dynamical methods (Kormendy & Gebhardt2001; Merritt & Ferrarese 2001b). Some galaxies are quiescent (e.g.M32, van der Marel et al. 1997; NGC 3250, Barth et al. 2001) andsome are mildly or strongly active (e.g. M87, Marconi et al. 1997;Macchetto et al. 1997; Centaurus A, Marconi et al. 2001; CygnusA, Tadhunter et al. 2003). The BH mass MBH correlates with someproperties of the host galaxy, such as spheroid (bulge) luminos-ity Lbul and mass Mbul (Kormendy & Richstone 1995; Magorrian

C© 2004 RAS

170 A. Marconi et al.

et al. 1998), light concentration (Graham et al. 2001) and the centralstellar velocity dispersion σ � (Ferrarese & Merritt 2000; Gebhardtet al. 2000). The latter correlation was thought to be the tightest, butMarconi & Hunt (2003) have recently shown that all the correla-tions have similar intrinsic dispersion when considering only galax-ies with secure BH detections (see also McLure & Dunlop 2002;Erwin, Graham & Caon 2003). Overall, the dispersion is of the orderof 0.3 in log MBH at a given value of Lbul, Mbul or σ �. The existenceof any correlations of BH and host galaxy bulge properties has im-portant implications for theories of galaxy formation in general andbulge formation in particular. Indeed, several attempts at explain-ing the origin of these correlations and the difficulties/constraintsthat they pose to galaxy formation models can be found in the lit-erature (e.g. Silk & Rees 1998; Cattaneo, Haehnelt & Rees 1999;Haehnelt & Kauffmann 2000; Ciotti & van Albada 2001; Cavaliere& Vittorini 2002; Adams et al. 2003, and references therein).

To date, all newly found BHs have masses (or upper limits) inagreement with those expected from the above correlations, sug-gesting that all galaxies host a massive BH in their nucleus. Byapplying the correlations between MBH and host galaxy properties,it is then possible to estimate the mass function of local BHs or,more simply, their total mass density (ρBH) in the Local Universe(e.g. Salucci et al. 1999; Marconi & Salvati 2002; Yu & Tremaine2002; Ferrarese 2002; Aller & Richstone 2002).

It is important to verify if local BHs are exclusively relics ofAGN activity or if other mechanisms, such as merging, play a role.Hereafter we will call ‘AGN relics’, or simply ‘relics’, those blackholes which grew up from small seeds (1–103 M�) following massaccretion during AGN phases. For instance, an AGN relic of 109 M�is different from a BH of the same mass but which was formed bythe merging of many smaller BHs.

A simple comparison between local and relic BHs was performedby Salucci et al. (1999) and Fabian & Iwasawa (1999), who deter-mined ρBH from the observed X-ray background (XRB) emission. Arevised estimate was obtained by Elvis, Risaliti & Zamorani (2002),who found ρBH = (7.5 –16.8) × 105(ε/0.1)−1 × 105 M� Mpc−3, indisagreement with the estimate from local BHs [ρBH = (2.5 ± 0.4)h2

65 × 105 M� Mpc−3; Yu & Tremaine 2002; Aller & Richstone2002]. Elvis et al. (2002) thus suggested that, in order to reconcilethis discrepancy, massive BHs should have large accretion efficien-cies (i.e. larger than the canonically adopted value of ε = 0.1),hence they should be rapidly rotating. A more detailed comparisonwas performed by Marconi & Salvati (2002), who found an agree-ment between the black hole mass functions (BHMFs) of local andrelic black holes. Recently, however, Yu & Tremaine (2002) andFerrarese (2002) found a disagreement at large masses (M BH �108 M�) where more AGN relics are expected relative to localBHs. As previously stated, a possibility to reconcile this discrep-ancy is to assume high accretion efficiencies, but, clearly, this issueis still much debated.

The relation between AGN relics and local BHs is also beingstudied in the framework of coeval evolution of BH and host galaxy.Several physical models have been proposed in which the fuelling ofthe BH, hence the AGN activity, is triggered by merging events [inthe context of the hierarchical structure formation paradigm, see forinstance Kauffmann & Haehnelt (2000), Volonteri, Haardt & Madau(2003), Menci et al. (2003), Wyithe & Loeb (2003), Hatziminaoglouet al. (2003) and Haehnelt (2003)] or is simply directly related to thestar formation history of the host galaxy (e.g. Di Matteo et al. 2003;Haiman, Ciotti & Ostriker 2004; Granato et al. 2004). The BH hasthen a feedback on the host galaxy through the energy released inthe AGN phase (e.g. Silk & Rees 1998; Blandford 1999; Begelman

2003). As a result of this double interaction (galaxy feeding theBH, and AGN feedback on the galaxy), these models can in gen-eral reproduce both the observed BH–host galaxy correlations andthe AGN luminosity functions [see e.g. Haehnelt, Natarajan & Rees(1998), Monaco, Salucci & Danese (2000) and Nulsen & Fabian(2000), and previous references]. However, this big effort in mod-elling cannot uniquely answer the question if local BHs are relics ofAGNs, since a wide range of models with many different underly-ing assumptions cannot be ruled out with the available observationalconstraints.

The aim of this paper is to investigate the assumption that massiveblack holes in nearby galaxies are relics of AGN activity by com-paring the local BHMF with that of AGN relics. We remark thatin this paper we do not build a physical model of the coevolutionbetween central BH and host galaxy but we compare differentialand integrated mass densities (local BHs) with differential and inte-grated energy densities (AGNs), with the only assumption that AGNactivity is caused by mass accretion on to the central BH.

We refine the analysis by Marconi & Salvati (2002) and evaluatethe discrepancies found by other authors between local and relicBHMFs. In Section 2 we estimate the local BHMF by applyingthe known correlations between MBH and host galaxy properties tothe galaxy luminosity and stellar velocity dispersion function. Wealso check the self-consistency of the results, and show that differ-ent MBH–host galaxy properties relations provide the same BHMFwithin the uncertainties. In Section 3 we use the continuity equationto estimate the BHMF of AGN relics; and in Section 4, we comparelocal and relic BHMFs, and find that local BHs are consistent withAGN relics. We then show (Section 5) that the energetic constraintsinferred from the X-ray background (XRB) are also satisfied andthat there is no discrepancy between ρBH of local BHs and that ex-pected from the XRB. In Sections 6, 7 and 8 we discuss constraintson the accretion efficiency ε and on the Eddington ratio λ = L/L Edd

(where L is the AGN luminosity and LEdd is the Eddington luminos-ity of the active BH), and we estimate the accretion history and theaverage lifetime of massive BHs. We summarize our results and wedraw our conclusions in Section 9.

In this paper we adopt the current ‘standard’ cosmologicalmodel, with H 0 = 100 h km s−1 Mpc−1 and h = 0.7, �m = 0.3and � = 0.7.

2 T H E M A S S F U N C T I O N O F L O C A LB L AC K H O L E S

The first step of the analysis presented here consists in the determi-nation of the mass function of local BHs, i.e. black holes residing innearby galaxies. The sample of galaxies with dynamically measuredBH masses is small (∼40) and not selected with well-defined crite-ria. Thus it is useless for a direct determination of the local blackhole mass function. However, the BHMF can be derived by apply-ing the existing relations between MBH and host galaxy properties togalaxy luminosity or velocity functions. After we have described theadopted formalism, we will verify the consistency of the MBH–σ �

and MBH–Lbul relations in providing the same BHMF within the un-certainties. We will then estimate the BHMF for early and all galaxytypes, showing that different galaxy luminosity/velocity functionsprovide BHMFs that are in agreement within the uncertainties.

2.1 Formalism

We describe here the simple formalism that is commonly used toderive the BHMF from galaxy luminosity or velocity functions.

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Local black holes, AGN relics and XRB 171

We define φ(x) dx as the number of galaxies per unit comovingvolume with observable x (e.g. luminosity L, or stellar velocity dis-persion σ ) in the range x to x + dx. The observable y (e.g. MBH, theBH mass) is related to x through the log-linear relation log y = a+ b log x and �(y) is the intrinsic dispersion in log y at constantlog x. Assuming a log-normal distribution then

P(log y | log x) = 1√2π�y

exp

{−1

2

[log y − a − b log x

�(y)

]2}

,

(1)

where P(log y | log x) d log y is the probability that y is in the rangelog y to log y + d log y for a given log x. Thus the number of galaxieswith x and y in the ranges x to x + dx and y to y + dy is

�(y, x) dx dy = P(log y | log x)

y ln 10dyφ(x) dx . (2)

Then �(y) dy, the number of galaxies with y in the range y to y +dy, is thus the convolution of φ and P:

�(y) =∫ +∞

0

P(log y | log x)

y ln 10φ(x) dx . (3)

In the limit of zero intrinsic dispersion,

�(y) = x1−bφ(x)

10ab, (4)

with log y = a + b log x . After substituting y with MBH and, forinstance, x with L, the spheroid luminosity, the total mass densityin BHs is simply

ρBH =∫ +∞

0

M�(M) dM

=∫ +∞

0

dM

∫ +∞

0

dL1

ln 10P(log M | log L)φ(L),

(5)

and in the zero intrinsic dispersion case

ρBH,0 = 10a

∫ +∞

0

Lbφ(L) dL, (6)

where we have written M = M BH for simplicity.Inverting the order of integration in equation (5) and integrating

on MBH one finally gets

ρBH = exp{

12 [�(MBH) ln 10]2

}ρBH,0. (7)

A non-null �(MBH) increases ρBH by a factor ofexp{ 1

2 [�(MBH) ln 10]2} with respect to the zero dispersioncase, a fact already noted by Yu & Tremaine (2002).

2.2 Consistency of the MBH–σ� and MBH–Lbul relations

Several authors (e.g. Salucci et al. 1999; Marconi & Salvati 2002;Yu & Tremaine 2002; Ferrarese 2002; Aller & Richstone 2002) haveadopted the method just described to determine the mass functionof local BHs. In the most recent works, the MBH–σ � correlation hasbeen preferred to MBH–Lbul on the grounds that it is tighter and,moreover, Yu & Tremaine (2002) found a factor of 2 discrepancybetween the values of ρBH determined by applying the MBH–σ �

and MBH–Lbul relations. Thus, we first investigate this inconsistencyof the MBH–σ � and MBH–Lbul relations by examining the BHMFsderived by applying the two relations to the velocity and luminosityfunctions obtained from the same sample of galaxies.

We consider the Sloan Digital Sky Survey (SDSS) sample of9000 early-type galaxies from Bernardi et al. (2003a), for which the

luminosity and velocity functions were determined independently(Bernardi et al. 2003b; Sheth et al. 2003). By ‘independently’ wemean that the velocity function was derived by directly measuringσ �

of all galaxies and not by applying the Faber–Jackson (FJ) relationto the galaxy luminosity function (e.g. Gonzalez et al. 2000; Aller& Richstone 2002; Ferrarese 2002). Sheth et al. (2003) comparedtheir velocity function with that derived applying the FJ relation tothe luminosity function and found that it is fundamental to take intoaccount the intrinsic dispersion of the relation in order to obtain thecorrect velocity function. It is consequently expected that the samemight apply to the determination of the BHMF.

To derive the BHMF using the MBH–σ � relation we apply equa-tion (3). For the galaxy velocity function φ we use the parametrizedform for early-type galaxies by Sheth et al. (2003). For the MBH–σ �

relation we use the coefficients obtained by fitting the ‘Group 1’galaxies of Marconi & Hunt (2003) (i.e. galaxies with ‘secure’ BHmass determinations):

log MBH = (8.30 ± 0.07) + (4.11 ± 0.33)(log σ� − 2.3). (8)

The slope is in agreement with that of Tremaine et al. (2002) (4.02± 0.32) but there is a larger normalization (8.13 ± 0.06, i.e. ∼0.2 inlog MBH). This choice of the coefficients for the MBH–σ � relation ismade for consistency, since, in the following, we use the coefficientsof the MBH–Lbul relation determined for the same ‘Group 1’ galax-ies by Marconi & Hunt (2003). The local BHMFs derived with theMBH–σ � relation assuming intrinsic dispersions �(M BH) = 0 and�(M BH) = 0.30 are shown in Fig. 1(a). The shaded area and errorbars indicate the 16th and 84th percentiles of 1000 Monte Carlorealizations of the local BHMF. The realizations were obtained byrandomly varying the input parameters assuming that they are nor-mally distributed with 1σ uncertainties given by their measurementerrors. For the velocity function we have considered only the erroron φ�, the number density at σ �, since the other errors are stronglycorrelated among themselves (Sheth et al. 2003). The 16th and 84thpercentiles indicate the ±1σ uncertainties on the logarithm of thelocal BHMF, whose values from the Monte Carlo realizations arenormally distributed at a given MBH.

To derive the BHMF using the MBH–Lbul relation we again applyequation (3). The galaxy luminosity function φ by Bernardi et al.(2003b) is given as a function of the total galaxy light. Since the cor-relation of MBH is with bulge light, we need to apply a correction�m in the case of S0 galaxies to transform from total to bulge lumi-nosity. Following Yu & Tremaine (2002), the luminosity functionfor the bulges of S0 galaxies is directly given by

φb(m) = fS0

fE + fS0φ(m − �m), (9)

where φ is the luminosity function of early-type galaxies, m is theabsolute bulge magnitude, �m = m − m total, and f E, f S0 are the frac-tions of E and S0 galaxies with respect to the total galaxy population.The galaxy type fractions and �m used in this paper are shown inTable 1. The morphological type fractions are from Fukugita, Hogan& Peebles (1998) (their Sbc fraction has been evenly split betweenSab and Scd). The 1σ uncertainties are a conservative estimate wemade after comparing various determinations of the morphologi-cal type fractions available in the literature [see the discussion inFukugita et al. (1998)]. The �m for the B band are those estimated byAller & Richstone (2002) by rebinning the Simien & de Vaucouleurs(1986) data in the appropriate bins of galaxy types. The �m for theH band are based on data from Hunt, Pierini & Giovanardi (2004),and the values for S0 galaxies were taken from the analysis byMarconi & Hunt (2003). The �m values are little dependent on the

C© 2004 RAS, MNRAS 351, 169–185


Figure 1. (a) Local BHMF for early-type galaxies based on the SDSS sample of Bernardi et al. (2003a). The shaded area and error bars (labelled ‘barred’)indicate the 16th and 84th percentiles of 1000 Monte Carlo realizations of the local BHMF and correspond to ±1σ uncertainties (see text). The � values indicatethe assumed intrinsic dispersions of the MBH–σ � or MBH–Lbul relations. (b) Local BHMF for early-type galaxies derived using the luminosity functions fromdifferent surveys (Marzke et al. 1994; Kochanek et al. 2001; Nakamura et al. 2003; Bernardi et al. 2003b) and the MBH–Lbul relation with �(M BH) = 0.3. Theshaded area and error bars have the same meaning as in (a).

Table 1. Adopted galaxy properties per morphological types.

E S0 Sab Scd Notes

Morph. fraction 0.11 ± 0.03 0.21 ± 0.05 0.43 ± 0.07 0.19 ± 0.07 1m(bulge) − m(total) 0 0.64 ± 0.30 1.46 ± 0.56 2.86 ± 0.59 2m(bulge) − m(total) 0 0.60 ± 0.50 1.78 ± 1.01 2.78 ± 1.21 3

Notes. 1; from Fukugita et al. (1998) (see text for details). 2; B band; estimate by Aller & Richstone (2002) basedon data from Simien & de Vaucouleurs (1986). 3; H band; based on data from Hunt et al. (2004).

photometric band at least within the considerable scatter. Thereforein the following analysis we will always use the B-band data regard-less of the photometric band in which φ was measured. Applyingthe above corrections to the early-type r� band luminosity functionby Bernardi et al. (2003b) and the colour transformation r � − K =2.8 ± 0.2, we obtain the luminosity function of the bulges of earlytypes in the K band. Cole et al. (2001) estimate K = z� − 2.12,which, combined with the average value for the early-type galaxiesr � − z� = 0.68 (Bernardi et al. 2003b), provides the required colour.Fukugita, Shimasaku & Ichikawa (1995) estimate r � − z� = 0.79,0.63, 0.70, 0.65, 0.57 for E, S0, Sab, Sbc, Scd, respectively, and thisjustifies our conservative choice of the scatter, which also allows usto apply the same colour correction to all morphological types. Forthe MBH–Lbul relation we use the coefficients obtained by fitting the‘Group 1’ galaxies of Marconi & Hunt (2003) in the K band:

log MBH = (8.21 ± 0.07) + (1.13 ± 0.12)(log L K ,bul − 10.9),(10)

where LK,bul is in units of LK �. The local BHMF derived withthe MBH–Lbul relation assuming intrinsic dispersions �(M BH) =0.30 and 0.50 are plotted in Fig. 1(a). As before, the 16th and84th percentiles of 1000 Monte Carlo realizations of the lo-cal BHMF indicate ±1σ uncertainties (shaded area and errorbars).

As is clear from Fig. 1(a), the effect of including the intrinsic dis-persion in the MBH–σ � and MBH–Lbul correlations is that of softening

the high-mass decrease of the BHMF, thus increasing the total den-sity. However, the most important result is that, in order to providethe same BHMF, the relations MBH–σ � and MBH–Lbul must have thesame intrinsic dispersion to within 0.1 in log. Yu & Tremaine (2002)found that ρBH[M BH –L bul] ∼ 2ρBH[M BH –σ �] by using �(M BH) =0 for MBH–σ � and �(M BH) = 0.5 for MBH–Lbul. This discrepancycan be entirely ascribed to the effect quantified by equation (7). In-deed, the densities ρBH of the BHMFs plotted in Fig. 1(a) are ρBH =2.7+0.5

−0.4 × 105 M� Mpc−3 [for MBH–σ � with �(M BH) = 0, inagreement with Yu & Tremaine (2002)] and ρBH = 5.5+2.0

−1.3 × 105

M� Mpc−3 [for MBH –Lbul with �(M BH) = 0.5]. Conversely,with �(M BH) = 0.30, one obtains 3.4+0.6

−0.5 × 105 M� Mpc−3

[for M BH–σ �] and 3.8+1.2−1.0 × 105 M� Mpc−3 [for M BH–L bul],

and these two values are in excellent agreement. The densities inmassive BHs were evaluated in the log(M BH/M�) = 6–10 rangeand the same range will be considered throughout the rest of thispaper.

One advantage of using the MBH–σ � relation is that the uncertain-ties on the derived BHMF are smaller than in the case of MBH–Lbul.This is because with the MBH–σ � relation one does not have to applyany correction for the bulge fraction. On the other hand, measuringstellar velocity dispersions is much more difficult than measuringgalaxy luminosities; thus it is clear that the two relations MBH–σ �

and MBH–Lbul should complement each other.In summary, the use of the same intrinsic dispersion for the MBH–

σ � and MBH–Lbul relations provides perfectly consistent BHMFs

C© 2004 RAS, MNRAS 351, 169–185


with the same mass densities ρBH. This is a confirmation ofthe results by Marconi & Hunt (2003), who showed that, whenconsidering only secure BH measurements, the MBH–σ �, MBH –Lbul

and MBH−Mbul relations have similar intrinsic dispersions (∼0.3 inlog). When not taking into account the intrinsic dispersion of theMBH–σ � relation, the local BHMF is systematically underestimatedat the high-mass end, where the disagreement with the BHMF ofAGN relics has been claimed.

2.3 The black hole mass function for early-type galaxies

Having established that both MBH–σ � and MBH–Lbul relations pro-vide consistent BHMFs, we can now evaluate the effects of usingluminosity functions from different galaxy surveys and photometricbands in determining the BHMF in early-type galaxies.

In Fig. 1(b) we compare the local BHMFs for early-type galax-ies obtained from different galaxy luminosity functions, in differ-ent photometric bands. We use the luminosity functions by Marzkeet al. (1994), Kochanek et al. (2001), Nakamura et al. (2003) andBernardi et al. (2003b), with details of the derivation specified inthe following.

(i) Bernardi et al. (2003b): this is the same BHMF as plotted inpanel (a) derived using the MBH–Lbul relation.

(ii) Marzke et al. (1994): we use the luminosity functions permorphological type from the CfA survey. The luminosities are inZwicky magnitudes, MZ, and we apply the colour transformationsdirectly measured by Kochanek et al. (2001) computing MZ − K forall the objects used for the luminosity functions (K is obtained fromthe 2MASS catalogue): M Z − K = 4.1 ± 0.65, 3.95 ± 0.65, 3.79± 0.56, 3.34 ± 0.64 for E, S0, Sa–Sb, Sc–Sd, respectively. The Kmagnitudes used are isophotal magnitudes K 20 and the correctionto total magnitudes is K tot = K 20 − (0.2 ± 0.04) (Kochanek et al.2001). Then we use the bulge–total correction and the M BH–L K,bul

relation from Table 1.(iii) Kochanek et al. (2001): we use the luminosity function in

the K band, apply the correction from K 20 to K tot described previ-ously, and use the morphological fractions and bulge–total correc-tions from Table 1.

Figure 2. (a) Local BHMFs for all galaxy types derived using different galaxy surveys and shown with the same notation as in Fig. 1(a). (b) Best estimate ofthe local BHMF obtained by combining all the Monte Carlo realizations of the BHMF that were used for panel (a). The solid line represents the 50th percentileand the shaded area is delimited by the 16th and 84th percentile levels.

(iv) Nakamura et al. (2003): we use the luminosity function inthe SDSS r � band, apply the colour correction K = r � − (2.8 ±0.2) described previously, and use the bulge–total corrections ofTable 1.

As in the previous section, uncertainties are estimated with 1000Monte Carlo realizations of the local BHMF where, in the case of thegalaxy luminosity functions, we use only errors on ��, the galaxynumber density. All the local BHMFs for early-type galaxies are inremarkable agreement within the uncertainties. The discrepancy atthe low-mass end (M BH < 108 M�) between the BHMF derivedwith the Bernardi et al. (2003b) luminosity functions and the others isnot significant. It occurs in a region where the luminosity functionsof early-type galaxies are extrapolated and is due to the differentfunctional forms adopted to fit the data (Gaussian for Bernardi et al.2003b, Schechter functions for the others).

2.4 The black hole mass function for all galaxy types

Here we estimate the local BHMF by considering also late mor-phological types. We use all the luminosity and velocity functionsdescribed in the previous section [with the exception of the one byBernardi et al. (2003b), which is only for early-type galaxies]. Forthe velocity function of late-type galaxies we take the estimate madeby Sheth et al. (2003) and shown in their fig. 6. When necessary, weapply the bulge–total correction with the numbers in Table 1 and wethen use either the MBH–Lbul or the MBH–σ � relation as describedin Section 2.2.

Fig. 2(a) shows the local BHMF for all galaxies, while Table 2reports the estimated BH mass densities ρBH for both early-typeand all galaxies. These densities are computed for h = 0.7. All theBHMFs and ρBH values are in agreement within the errors. Finally,in Fig. 2(b) we present our best estimate of the local BHMF obtainedby merging all the random realizations of the BHMFs shown inFig. 2(a) and considering the 16th, 50th and 84th percentile levels.Our best estimate of the local density in massive BHs is consequentlyρBH = 4.6+1.9

−1.4 × 105 M� Mpc−3. Roughly 70 per cent of the total BHdensity resides in early-type galaxies. Our estimate of the densityin local BHs is in agreement with Merritt & Ferrarese (2001a) and

C© 2004 RAS, MNRAS 351, 169–185


Table 2. Mass density of local BHs for early and all galaxy types (columns3 and 4, respectively). Column 1 indicates the galaxy luminosity or velocityfunction that was combined with the MBH–host galaxy property in colomn2 to determine the local BHMF. All densities are computed with �(M BH)= 0.3. Uncertainties are the 16th and 84th percentiles of the Monte Carlorealizations described in the text. These correspond to ± 1σ uncertainties.

Adopted luminosity MBH–host ρBH(105 M� Mpc−3)or velocity function galaxy property E + S0 All

Sheth et al. (2003) MBH–σ � 3.4+0.6−0.5 5.0+1.7

−1.1

Bernardi et al. (2003b) MBH–LK,bul 3.8+1.2−1.0 –

Nakamura et al. (2003) MBH–LK,bul 2.5+1.2−0.8 4.4+2.0

−1.3

Kochanek et al. (2001) MBH–LK,bul 3.3+1.0−0.7 4.5+1.4

−1.1

Marzke et al. (1994) MBH–LK,bul 3.4+2.7−1.5 4.5+3.1

−1.7

All L or V functions – – 4.6+1.9−1.4

with Ferrarese (2002), though, in the latter case, the shape of ourBHMF is very different at the high-mass end. Our estimate is afactor �1.8 larger than those by Yu & Tremaine (2002) and Aller &Richstone (2002), and the reasons for this discrepancy are outlinedin Section 4.

3 T H E M A S S F U N C T I O N O F AG N R E L I C S

Once the mass function of local BHs has been determined, the sub-sequent step is the estimate of the BHMF of AGN relics, i.e. BHsin galactic nuclei that were grown exclusively during active phasesfrom small (1–103 M�) seeds. We will first describe the continuityequation that will be used to relate the relic BHMF, N(M, t), to theAGN luminosity function φ(L, z), under the assumption that AGNare powered by mass accretion on to massive BHs. The physicalquantity directly related to the mass accretion on to the BH is the to-tal intrinsic AGN luminosity L. Since AGN luminosity functions aredetermined in limited energy bands, we will provide suitable bolo-metric corrections. We will then briefly describe the adopted AGNluminosity functions obtained with our bolometric corrections. Fi-nally we will present our estimates of the relic BHMFs derived fromdifferent AGN luminosity functions.

3.1 The continuity equation

The black hole mass function (BHMF) of AGN relics, N(M, t), canbe estimated from the continuity equation (Cavaliere, Morrison &Wood 1971; Small & Blandford 1992) with which, under simpleassumptions, it is possible to relate N(M, t) to the AGN luminosityfunction. If N(M, t) dM is the comoving number density of BHs withmass in the range M to M + dM at cosmic time t, the continuityequation can be written as

∂N (M, t)

∂t+ ∂

∂M[N (M, t)〈M(M, t)〉] = 0, (11)

where 〈M(M, t)〉 is the ‘average’ accretion rate on the BH of massM. We adopt the working assumption that AGNs are powered byaccretion on to BHs, and that the BH growth takes place duringphases in which the AGN is accreting at a fraction λ of the Eddingtonlimit (L = λ L Edd) converting mass into energy with an efficiencyε. We can thus simply relate the AGN luminosity function φ(L , t)[φ(L , t) d log L is the comoving number density of AGNs in therange log L to log L + d log L at cosmic time t] to the BH mass

function:

φ(L, t) d log L = δ(M, t)N (M, t) dM, (12)

where δ(M, t) is the fraction of BHs with mass M that are active attime t, i.e. the BH duty cycle. If a BH is accreting at a fraction λ ofthe Eddington rate, its emitted luminosity is

L = λMc2

tEdd= εMacc c2, (13)

where tEdd is the Eddington time, ε is the accretion efficiency andMacc is the matter falling on to the black hole. The growth rate ofthe BH, M , is thus given by M = (1 − ε)Macc, since a fraction ε ofthe accreted mass is converted into energy and thus escapes the BH.Since 〈M(M, t)〉 = δ(M, t)M(M, t), combining equations (12) and(13) we can write

N (M, t)〈M(M, t)〉 = (1 − ε)

εc2 ln 10[φ(L, t)]L=λMc2/tEdd

dL

dM, (14)

which can be placed in equation (11) where the only unknown func-tion is N(M, t). If ε and λ are constant, we can then write

∂N (M, t)

∂t= − (1 − ε)λ2c2

εt2Edd ln 10

[∂φ(L, t)

∂L

]L=λMc2/tEdd

, (15)

which can be easily integrated given the AGN luminosity functionand the initial conditions.

For the initial conditions, we assume that, at the starting redshiftz s, δ[M , t(z s)] = 1. This can be interpreted by saying either that atzs all black holes are active or that we are following the evolutiononly of those BHs which were active at zs. Thus

M N (M, ts) = [φ(L, ts)]L=λMc2/tEdd. (16)

We will show that the final results are little sensitive to the choiceof the initial conditions, provided that z s > 3, since most of the BHgrowth takes place at lower redshifts.

Equation (15), which represents the continuity equation in thecase of constant ε and λ, can be trivially integrated on M to derivethe relation used by various authors (Padovani, Burg & Edelson1990; Chokshi & Turner 1992; Yu & Tremaine 2002; Ferrarese2002):

ρBH = 1 − ε

εc2UT. (17)

Note the factor (1 − ε), which is needed to account for the part of theaccreting matter that is radiated away during the accretion process.Here UT is the total comoving energy density from AGNs (not to beconfused with the total observed energy density) and is given by

UT =∫ zs

0

dzdt

dz

∫ L2

L1

Lφ(L, z) d log L; (18)

here φ(L, z) d log L is used if the AGN luminosity function is definedper logarithmic luminosity bin, otherwise it should be φ(L, z) dL.

The right-hand element of the continuity equation, which con-tains the source function, is null, meaning that we neglect any pro-cess which, at time t, might ‘create’ or ‘destroy’ a BH with massM. Indeed, in the merging process of two BHs, M 1 + M 2 → M 12,BHs with M1 and M2 are destroyed while a BH with M12 is cre-ated. We decided to neglect merging of BHs because, at present,the merging rate, γ (M1, M2, z), is very uncertain and dependent onthe assumptions of the model with which it is computed. More-over, the aim of this paper is to assess if local BHs are indeed relicsof AGN activity, and by neglecting merging we can assess the im-portance of mass accretion during AGN phases. Yu & Tremaine(2002) have used an integral version of the continuity equation in

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order to take into account merging, but their approach does not allowa direct comparison of the local and relic BHMFs. Furthermore, thediscrepancy they find between local BHs and AGN relics becomesworse if merging is important. Granato et al. (2004) in their physicalmodel of coevolution of BH and host galaxy find that the growthof the BH is due to mass accretion. Similarly, Haiman et al. (2004)find that merging of BHs is important only at high redshifts while,at lower z, BH growth is dominated by mass accretion. These rea-sons support our choice of neglecting the merging process. Indeed,merging of BHs might well be very important in shaping the BHMFat high redshifts (z > 3 − 5) but, as we will see, the relic BHMFat z = 0 is very little dependent on its shape and normalization at z∼ 3. Moreover, if the relic BHMF at z = 0 is changed significantlyby merging, then its remarkable agreement with the local BHMFin the merging-free case would be a mysterious coincidence (seeSection 4).

3.2 The bolometric corrections

The BH mass accretion rate is given by the AGN luminosity L which,in turn, can be estimated from the luminosity in a given band b, Lb,by applying a suitable bolometric correction f bol,b = L/Lb. At thebeginning of Section 3 we have referred to L as the total intrin-sic AGN luminosity, and before presenting the adopted bolometriccorrections it is necessary to define the meaning of intrinsic.

The integral of the observed spectral energy distribution (SED)of an AGN provides the total observed luminosity, Lobs. Thus thequasar bolometric corrections by Elvis et al. (1994) provide Lobs

because they are based on the average observed SED. However,Lobs does not give an accurate estimate of the BH mass accretionrate because it often includes reprocessed radiation (i.e. radiationabsorbed along other lines of sight and re-emitted isotropically).The accretion rate is better related to the total luminosity directlyproduced by the accretion process, which we call the total intrinsicluminosity L. In AGNs, L is given by sum of the optical–ultraviolet(UV) and X-ray luminosities radiated by the accretion disc and hotcorona, respectively. Conversely, it is well known that the infrared(IR) radiation is reprocessed from the UV (e.g. Antonucci 1993).Thus, in order to estimate L, one has to remove the IR bump fromthe observed SEDs of unobscured AGNs. In radio-loud AGNs theremight be an important contribution from synchrotron radiation to theIR. However, this will not affect our average bolometric correctionssince the number of radio-loud AGNs is small, ∼10 per cent of thewhole AGN population.

In previous works, several authors have used the bolometric cor-rections by Elvis et al. (1994) but, as explained above, these providethe observed and not the intrinsic AGN luminosity. Removing theIR bump from the Elvis et al. (1994) spectral templates, the 1 µmto X-ray range includes about two-thirds of the total energy; thusone obtains, for instance, L/ν B L νB = 7.9 ± 2.9 instead of thecommonly used L/ν B L νB = 11.8 ± 4.3. However, the Elvis et al.(1994) quasar templates, obtained from an X-ray-selected sampleof quasars, are X-ray ‘bright’ (see the discussion in Elvis et al.2002) and thus underestimate the bolometric corrections in the X-ray energy bands. Moreover, the dependence on luminosity of thebolometric corrections cannot be obtained. For these reasons wewill derive new estimates of the bolometric corrections as describedin the following.

First, we construct a template spectrum. In the optical–UV regionour template consists of a broken power law with α = −0.44 (L ν ∝να) in the range 1 µm < λ < 1300 Å, and α = −1.76 in the range1200–500 Å (e.g. Telfer et al. 2002; Vanden Berk et al. 2001). At

λ > 1 µm our big blue bump is truncated assuming α = 2 as inthe Rayleigh–Jeans tail of a blackbody. The X-ray spectrum beyond1 keV consists of a single power law plus a reflection component.The power law has the typical photon index � = 1.9 (e.g. Georgeet al. 1998; Perola et al. 2002) and an exponential cut-off at E c =500 keV. Following Ueda et al. (2003), the reflection component isgenerated with the PEXRAV model (Magdziarz & Zdziarski 1995) inthe XSPEC package with a reflection solid angle of 2π, inclinationangle of cos i = 0.5 and solar abundances. We then rescale the X-rayspectrum to a given αOX (Zamorani et al. 1981), which is defined as

αOX = − log[Lν(2500 Å)/Lν(2keV)]

log[ν(2500 Å)/ν(2keV)], (19)

and we finally connect with a simple power law the point at 500 Åwith that at 1 keV. To account for the dependence of αOX on lumi-nosity, we use the relation by Vignali, Brandt & Schneider:

αOX = −0.11 log Lν(2500Å) + 1.85. (20)

Finally, we assume that the template spectra, hence the bolometriccorrections, are independent of redshift.

The template spectrum for a L = 1012 L� AGN is plotted inFig. 3(a) and is compared with the radio-quiet and radio-loud quasarmedian spectra by Elvis et al. (1994). Our template is in very goodagreement in the optical–UV part but is obviously missing the IRbump, which is due to reprocessed UV radiation and thus not con-sidered here. Our template is fainter in X-rays but this is expectedsince, as already discussed, the quasar sample by Elvis et al. (1994)is X-ray bright.

In Fig. 3(b) we plot the bolometric corrections L/ν B L νB ,L/L(0.5–2 keV) and L/L(2 –10 keV) derived from the above tem-plates. The error bars represent the 16th and 84th percentiles from1000 Monte Carlo realizations of the spectral templates where wehave assumed the following 1σ uncertainties for the input parame-ters: ±0.1 for the spectral slopes (α; Telfer et al. 2002; Vanden Berket al. 2001) in the 1 µm–500 Å range and ±0.05 for the αOX, con-stant at all luminosities. The latter is a conservative assumption (see,e.g. Yuan et al. 1998) but is made to account for possible, but un-accounted for, systematic errors. As in the case of the local BHMF,the 16th and 84th percentiles correspond to ±1σ uncertainties onthe logarithm of the bolometric corrections. The 50th percentilescan be fitted with a third-degree polynomial to obtain the followingconvenient relations:

log[L/L(2–10 keV)] = 1.54 + 0.24L + 0.012L2 − 0.0015L3,

log[L/L(0.5–2 keV)] = 1.65 + 0.22L + 0.012L2 − 0.0015L3,

log(L/νB LνB ) = 0.80 − 0.067L + 0.017L2 − 0.0023L3, (21)

where L = (log L − 12) and L is the bolometric luminosity in unitsof L�. Hence, the bolometric corrections have log-normal distri-butions with average values given by the above equations and ±1σ

scatters (at fixed L) which can be derived from Fig. 3(b). Scattersare given by ∼0.05 for the B band and ∼0.1 for the X-rays, takento be independent of L for simplicity. It is worth noting that theB-band bolometric correction is in agreement with that by Elvis et al.(1994) whose average value and scatter are shown by the hatchedarea in Fig. 3(b). The scatter of the Elvis et al. (1994) bolometriccorrection is the standard deviation of their quasar sample whileours are uncertainties on the average values. The average αOX in thelog(L/L�) = 11.5–12.5 range is 1.43, the same value as estimatedby Elvis et al. (2002) after correcting for the biases due to X-ray oroptical selection of the parent quasar samples.

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Figure 3. (a) Template spectrum for a L = 1012 L� AGN obtained as described in the text (solid line). The dashed and dotted lines represent the radio-quietand radio-loud templates by Elvis et al. (1994). (b) Bolometric corrections derived from the spectral templates built as described in the text. The error barsrepresent the 16th and 84th percentiles of the 1000 Monte Carlo realizations described in the text. The hatched area represents the B-band bolometric correction(with ± 1σ scatter) by Elvis et al. (1994).

3.3 The luminosity function of active galactic nuclei

In order to ensure a consistent treatment, the AGN luminosity func-tion φ(L, t) used in the continuity equation must describe the evo-lution of the entire AGN population. The luminosity L is the totalluminosity radiated from the accreting mass and is estimated withthe bolometric corrections described in the previous section.

The AGN luminosity functions available in the literature do notdescribe the entire AGN population because they are the result ofsurveys performed in ‘narrow’ spectral bands, with given flux limitsand selection criteria. For example, the luminosity function of Boyleet al. (2000) includes only quasars selected for their blue colour butit is now known that there is a population of ‘red’ quasars which ismissed. The luminosity function of soft X-ray-selected (0.5–2 keV)AGNs of Miyaji, Hasinger & Schmidt (2000) misses most of thesources with significant X-ray absorption (N H > 1022 cm−2); itincludes mostly broad-line AGNs (∼80 per cent of the total) butit is known that, at least locally, there is a dominant population ofobscured AGNs (e.g. Maiolino & Rieke 1995). Current hard X-raysurveys (2–10 keV) are less sensitive to obscuration than optical andsoft X-ray ones, thus the very recent luminosity function by Uedaet al. (2003) probes by far the largest fraction of the whole AGNpopulation. However, it is restricted only to Compton-thin AGNs(N H < 1024 cm−2), while it is known that, at least locally, there is asignificant fraction of Compton-thick objects (e.g. Risaliti, Maiolino& Salvati 1999). In summary, when using the luminosity functionsavailable in the literature, one should know that they describe partof the AGN population, and thus account only for a fraction of thelocal BHMF.

In this paper, we consider the AGN luminosity functions by Boyleet al. (2000) (band B), Miyaji et al. (2000) (0.5–2 keV) and Uedaet al. (2003) (2–10 keV). Thus, the AGN luminosity function thatwill be used in the continuity equation is given by

φ(L) dL = φ(Lx ) dLx (22)

where x is either the B, the 0.5–2 keV or the 2–10 keV band andL = f bol,x Lx.

The AGN luminosity functions obtained with the bolometric cor-rections described in the previous section are compared in Fig. 4(a)at selected redshifts. The luminosity functions by Boyle et al. (2000)and Miyaji et al. (2000) are in rough agreement at the high-L end,meaning that they may be sampling the same quasar population.Indeed, most of the objects in the Miyaji et al. (2000) sample arebroad-line AGN, which, at high L, become the quasars observedby Boyle et al. (2000). The disagreement at low luminosities is be-cause the Boyle et al. (2000) luminosity function is extrapolated forMB > − 23 i.e. for L < 1012 L�. In contrast the luminosity func-tion by Ueda et al. (2003) samples a larger fraction of the AGNpopulation at all luminosities.

In Fig. 4(b) we compare the differential comoving energy densi-ties computed in the luminosity ranges 41.5 < log (L X/erg s−1) <

48 (0.5–2 and 2–10 keV bands) and −28 < MB < − 21 (B band).For the Ueda et al. (2003) luminosity function we also plot the dif-ferential comoving energy density for objects with L > 1012 L�.High-luminosity objects provide ∼50 per cent of the total energydensity in the Ueda et al. (2003) luminosity function and their emis-sion has a redshift distribution similar to that of the AGNs by Miyajiet al. (2000) and Boyle et al. (2000). Clearly, lower-luminosity ob-jects contribute significantly at z < 1.5 and this is an important resultof the recent Chandra and XMM surveys (e.g. Hasinger 2003; Fioreet al. 2003; Ueda et al. 2003). The total comoving energy densitiesin the redshift range z = 0–3 (i.e. the integrals of the quantities plot-ted in the figure) are 4.2 × 10−16, 6.4 × 10−16 and 1.5 × 10−15 ergcm−3, for Boyle et al. (2000), Miyaji et al. (2000) and Ueda et al.(2003), respectively. The high-L objects in the Ueda et al. (2003)luminosity function provide 8 × 10−16 erg cm−3, i.e. ∼50 per centof the total energy density.

By applying equation (17), the mass densities in BHs can bewritten as

ρBH = ρ

(1 − ε

9ε

)× 105 M� Mpc−3, (23)

where ρ is 0.6 (from the Boyle et al. 2000, AGN luminosity func-tion), 0.9 (Miyaji et al. 2000), 2.2 (Ueda et al. 2003) and 1.2 (Ueda

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Figure 4. (a) AGN luminosity functions at selected redshifts obtained with the bolometric corrections described in the text. Different symbols indicate differentAGN luminosity functions, while different line styles distinguish between redshifts. (b) Differential comoving energy densities of AGNs from different surveysobtained with the bolometric corrections described in the text. In the case of the Ueda et al. (2003) luminosity function, we also plot the comoving energydensity for high-luminosity objects (L > 1012 L�). U is the total comoving energy density in the redshift range z = 0–3, in units of 10−16 erg cm−3 and 〈z〉 isthe average redshift computed following equation (29).

Figure 5. (a) Relic BHMFs at z = 0 (solid lines) calculated assuming ε = 0.1 and λ = 1 compared with the relic BHMFs at z s = 3 (dotted lines). Differentsymbols indicate the different AGN luminosity functions used. (b) Relic BHMFs obtained from the Ueda et al. (2003) AGN luminosity function but withdifferent λ, zs and δ with respect to panel (a). For comparison, the thick line with no symbols indicates the relic BHMF from Ueda et al. (2003) plotted in (a).As in the previous panels, solid lines are the relic BHMFs at z = 0, while dotted lines are those at zs.

et al. 2003, high-L objects). The value derived from the Ueda et al.(2003) luminosity function (2.2 × 105 M� Mpc−3) is already inmarginal agreement with ρBH = 4.5+1.8

−1.4 × 105 M� Mpc−3, i.e. theestimate of the local BH density given in Section 2.4.

3.4 The relics of active galactic nuclei

Given the AGN luminosity functions (Boyle et al. 2000; Miyaji et al.2000; Ueda et al. 2003), we can integrate the continuity equationassuming that λ = 1 (AGNs emitting at the Eddington luminosity),ε = 0.1 and z s = 3. Though we know that the above luminosity

functions do not describe the whole AGN population, initially wedo not apply any correction for the objects that have been missed.

The relic BH mass functions are shown in Fig. 5(a). As expected,the AGNs traced by the Ueda et al. (2003) luminosity function leavemore relics than those traced by the luminosity functions of Boyleet al. (2000) and Miyaji et al. (2000). It is also notable that the relicBHMFs at z = 0 (solid lines) are about two orders of magnitudelarger than those at z s = 3 (dotted lines), meaning that most of theBH growth took place for z < z s = 3. The dashed lines represent therelic BHMF obtained by considering only AGNs with L > 1012 L�.[In practice this was obtained by multiplying the AGN luminosityfunctions by exp(−1012/L).] The comparison between the dashed

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Figure 6. (a) Local and relic BHMFs. The shaded area represents the ± 1σ uncertainty on the logarithm of local BHMF (see Fig. 2b). (b) Relic BHMFobtained using the Ueda et al. (2003) hard X-ray luminosity function and accounting for the missing Compton-thick AGNs (see text). The error bars are the16th and 84th percentile levels of 1000 realizations of the relic BHMF (see text). The local BHMF is plotted with the same notation as in the panel (a).

and solid lines indicates that today’s high-mass BHs (M BH > 108

M�) grew during quasar phases (L > 1012 L�).The BHMFs in Fig. 5(a) were estimated assuming λ = 1 and ε =

0.1. It can be seen from equation (15) that the efficiency ε is a simplescaling factor, and an increase in ε will decrease the level of the relicBHMF. However, a variation of the Eddington fraction λ will nothave a simple scaling effect but it will produce a combination ofscaling and translation along the M-axis since λ also enters the L =λ Mc2/t Edd relation. In Fig. 5(b) the same relic BHMF derived usingUeda et al. (2003) (thick line with no symbols) is compared with theBHMF obtained assuming λ = 0.1 (line with empty squares). Aspreviously, solid and dotted lines indicate the BHMF at z = 0 and atz = z s respectively. Decreasing λ has the net effect of increasing thenumber of BHs at high masses (M > 108.5 M�) while decreasingthat at lower masses. In the same figure, we also show the BHMFsobtained by assuming that the starting redshift of integration is z s = 5(line with empty triangles) and z s = 2.5 (line with empty diamonds)and that the fraction of active BHs at z0 is δ = 0.1 instead of 1 (linewith stars). The relic BHMF at z = 0 is almost independent of theinitial conditions. Changing the starting assumption, i.e. zs or δ(zs),does not have any appreciable effect provided that z s � 2.5. This isa consequence of the fact that the main BH growth takes place atz < 3 when there is more time available (∼85 per cent of the age ofthe Universe). At larger redshifts there is too little time for the BHgrowth.

In summary, relic BHs grew mainly at low redshifts (z < 3) andhigh-mass BHs were produced during quasar activity. The initialconditions for the continuity equation influence little the relic BHMFat z = 0.

4 L O C A L B L AC K H O L E S A N D AG N R E L I C S

In this section we compare the local and relic BHMFs. At first we willconsider the relic BHMFs derived in the previous section, withouttrying to account for the AGN population missing in the adoptedluminosity functions. After showing that, as expected, the Ueda et al.(2003) luminosity function encompasses the largest fraction of theAGN population, thus providing a better match to the local BHMF,

we will focus on it and try to account for the missing AGNs in a waythat also satisfies the constraints imposed by the X-ray background.

In Fig. 6(a) we compare the local BHMF derived in Section 2.4(Fig. 2b) with the relic BHMFs obtained adopting the Boyle et al.(2000), Miyaji et al. (2000) and Ueda et al. (2003) AGN luminosityfunctions (ε = 0.1, λ = 1) without any correction for the miss-ing AGN population. As shown in the previous section, these relicBHMFs are insensitive to the adopted initial conditions, and thusthe only parameters on which they depend are ε and λ.

The first result that can be evinced from the figure is that thereis no discrepancy at high masses. All the relic BHMFs are slightlysmaller than the local BHMF. As explained previously, this is notsignificant because all the adopted luminosity functions are missingpart of the AGN population. Conversely Yu & Tremaine (2002) andFerrarese (2002) using the Boyle et al. (2000) luminosity functionfound a significant discrepancy because their relic BHMF at highmasses is larger than the local one. The reasons for the solution ofthis discrepancy are the following.

(i) We have taken into account the intrinsic dispersion of theM BH–σ � and MBH–Lbul correlations, thus softening the high-massdecrease of the local BHMF.

(ii) We have adopted zero-points in the MBH–σ � and MBH–Lbul

correlations that are a factor ∼1.6 (∼0.2 in log) larger than thoseused by Yu & Tremaine (2002). These zero-points (and slopes) werederived from the analysis of Marconi & Hunt (2003), and their largervalue is a consequence of the rejection of galaxies with unreliableBH masses.

(iii) We have adopted bolometric corrections that, on average,are about two-thirds of those adopted by previous authors. Thisis because we have not taken into account the IR emission in theestimate of the bolometric luminosity. The IR bump is produced byreprocessed UV radiation, and thus its use in the determination ofL would result in overestimated accretion rates.

We now focus on the Ueda et al. (2003) AGN luminosity functionsince we have established that, as expected, it is the one that providesthe most complete description of the AGN population. The aim isto assess whether the relic BHMF can account for the whole local

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BHMF, i.e. if local BHs can be entirely explained as being AGNrelics.

The Ueda et al. (2003) luminosity function only describes thepopulation of Compton-thin AGNs, i.e. objects with absorbing col-umn densities log N H < 24 [ cm−2]. To estimate the correction forthe missing Compton-thick sources, we follow Ueda et al. (2003)and assume that there are as many AGNs in the 24–25 bin of log N H

as in the 23–24 one, as indicated by the N H distribution of Risalitiet al. (1999). Thus, with the N H distribution estimated by Ueda et al.(2003), the luminosity function has to be multiplied by 1.3 in orderto account for the Compton-thick AGNs. In reality, the correctingfactor is luminosity-dependent but, since it is limited to the 1.2–1.4range, we chose an average value for simplicity. With this contribu-tion from Compton-thick sources, Ueda et al. (2003) are able to pro-vide a reasonable fit of the X-ray background spectrum. (We will re-turn to this issue in the following section.) While AGNs at log N H >

25 are not important contributors of the X-ray background, they docontribute to the relic BHMF. Thus, we assume that for log N H > 25there are as many AGNs as in the 23–24 or 24–25 bins of log N H. Thisassumption is also justified by the Risaliti et al. (1999) N H distribu-tion. The correcting factor then becomes 1.6. Therefore, to compen-sate for the missing obscured AGNs, we apply a correcting factor of1.6 independently of luminosity. In Fig. 6(b) the corrected Ueda et al.(2003) luminosity function is used to determine the relic BHMF.We also show the uncertainties (16th and 84th percentiles, i.e. ±1σ errors on the log of the relic BHMF as previously) estimatedwith the usual 1000 Monte Carlo realizations of the relic BHMF.These were obtained by varying the number density of the luminos-ity function (±30 per cent 1σ error on the number density to avoidcorrelated errors on the other parameters), the hard X-ray bolomet-ric correction (±0.1 1σ error on log f bol,X), and the factor used tocorrect for the missing Compton-thick AGNs (±0.3 1σ error).

The relic and local BHMF agree well within the uncertainties.Thus, adopting the best possible description of the whole AGNpopulation, the mass function of relic BHs is in excellent agreementwith the mass function of local BHs. Local BHs are thus AGN relicsand were mainly grown during active phases in the life of the hostgalaxy. This agreement has been obtained with ε = 0.1 and λ =1, indicating that (i) efficiencies higher than commonly adopted forAGNs are not required, and (ii) the main growth of BHs occurs inphases during which the AGN is emitting close to the Eddingtonlimit. In Section 6 we will explore the locus in the λ–ε plane thatis permitted by the comparison from the local and relic BHMFs.As discussed in Section 3.1 we have neglected merging in our esti-mate of the relic BHMF. However, the good agreement between thelocal and relic BHMFs suggests that the merging process does notsignificantly affect the build-up of the BHMF, at least in the z < 3redshift range.

5 C O N S T R A I N T S F RO M T H EX - R AY BAC K G RO U N D

From the X-ray background (XRB) light it is possible to estimate theexpected mass density of relic BHs (Salucci et al. 1999; Fabian &Iwasawa 1999; Elvis et al. 2002), which can then be compared withthe mass density of local BHs. From this comparison, Elvis et al.(2002) inferred that massive BHs must be rapidly rotating for thehigh efficiency needed (ε > 0.15) to match the XRB and local BHmass densities. We will show that, as a result of the redshift distri-bution inferred from the new X-ray surveys, the match between theXRB and local BH mass densities can be obtained without requiringlarge efficiencies (ε > 0.1; see also Fabian 2003; Comastri 2003).

The XRB provides another type of constraint. It has been shown(Setti & Woltjer 1989; Madau, Ghisellini & Fabian 1994; Comastriet al. 1995; Gilli, Risaliti & Salvati 1999) that the XRB spectrumcan be reproduced by summing the spectra of the whole AGN pop-ulation after suitable corrections to take into account the absorptionalong the line of sight. The AGN population and its redshift distri-bution are derived from AGN luminosity functions but, in order tofit the XRB spectrum, one has to include a correction for the missing(obscured) AGNs. This correction is the same as that which mustbe adopted here to determine the relic BHMF from the whole AGNpopulation. Thus, the ratio R12 between AGNs included and missedin the adopted luminosity functions is a parameter that determinesboth the relic BHMF and the XRB spectrum (and the X-ray sourcecounts). When rescaling the AGN luminosity function with the(1 + R12) factor to match the local BHMF, one should also ver-ify that it is also possible to match the XRB spectrum and sourcecounts at the same time. In practice, one can use the R12 value re-quired by the XRB synthesis models in order to fit the XRB spectrumand source counts. It is beyond the scope of this paper to produce amodel synthesis of the XRB, but we will show that with R12 valuesfound in the literature we can match the local and relic BHMF, andalso satisfy the constraints imposed by the X-ray background.

The local density in massive BHs expected from the observedX-ray background (XRB) light can be estimated with the relation

ρBH = 1 − ε

εc2(1 + 〈z〉)U �

T, (24)

where U�T is the total observed (as opposed to comoving) AGN

energy density and 〈z〉 is the average source redshift. The factor(1 − ε) is needed to take into account the fact that not all the accretingmass falls into the BH. U�

T can be estimated from the observed X-raybackground light as

U �T = fbol,X fobsc

4π

cIX, (25)

where IX is the total observed surface brightness of the X-ray back-ground (i.e. the integral of the XRB spectrum), f obsc is the correctionto take into account source obscuration in the X-rays (i.e. f obscIX

would be the total XRB surface brightness if AGNs were not ob-scured), and f bol,X is the X-ray bolometric correction (for more de-tails, see Fabian & Iwasawa 1999; Salucci et al. 1999; Elvis et al.2002). Elvis et al. 2002 estimate U �

T ∼ (1.5 –3.4) × 10−15 erg cm−3.We first verify that the above formula is consistent with the

scheme followed in this paper, and then establish how 〈z〉 must becomputed. The observed background surface brightness at energyE is

I (E) = 1

4π

∫ z0

0

dz(1 + z)

4πD2L

dV

dz

×∫ L2

L1

f (E(1 + z))Lφ(L, z) dL, (26)

where DL is the luminosity distance, V is the comoving volume, Lis the source luminosity in the energy band x, φ is the luminosityfunction in the same band, and f (E) is the source spectrum at energyE normalized to have unit luminosity in the band x. Integrating onE to find the total surface brightness in band x one finds

Ix = c

4π

∫ zmax

0

dz1

(1 + z)

dt

dz

∫ L2

L1

Lφ(L, z) dL. (27)

Applying the bolometric correction [L = fbol,X L with φ(L, z) dL =φ(L, z) dL], the obscuration correction (f obsc) and comparing with

C© 2004 RAS, MNRAS 351, 169–185


equation (18), one finds that

(1 + 〈z〉) 4πIT

c= (1 + 〈z〉)U �

T = UT, (28)

where I T = f bol,X f obsc Ix and UT is the total comoving (as opposedto observed) energy density. The average redshift 〈z 〉 is then

(1 + 〈z〉) =∫ zmax

0dz(dt/dz)

∫ L2

L1Lφ(L, z) dL∫ zmax

0dz[1/(1 + z)](dt/dz)

∫ L2

L1Lφ(L, z) dL

. (29)

Using the luminosity function of Ueda et al. (2003) one finds 〈z〉= 1.1, lower than the value 〈z〉∼ 2 assumed by previous authors. Forcomparison, the high-luminosity objects (L > 1012 L�) in the Uedaet al. (2003) luminosity function have 〈z〉 = 1.3, while Miyaji et al.(2000) and Boyle et al. (2000) have 〈z 〉 = 1.4 and 1.5, respectively.With the U�

T estimate by Elvis et al. (2002) we get

ρBH = (4.7–10.6) ×(

1 − ε

9ε

)× 105 M� Mpc−3, (30)

which is perfectly consistent with the estimate from the local BHMFρBH = (3.2 –6.5) × 105 M� Mpc−3 without requiring efficiencieslarger than the ‘canonical’ value ε = 0.1. This agreement has alsobeen remarked upon by Fabian (2003) and Comastri (2003). Thecritical point is clearly the value of 〈z〉, the average redshift of X-raysources emitting the XRB, which has been significantly reduced byChandra and XMM surveys (e.g. Hasinger 2003; Fiore et al. 2003;Steffen et al. 2003; Ueda et al. 2003). The minimum value of ε, al-lowed for consistency between the two estimates of ρBH, is ε = 0.07,just slightly larger than in the non-rotating BH case. Conversely, themaximum allowed efficiency is ε = 0.27, substantially below thatin the maximally rotating Kerr BH case (see Section 6). It is in-triguing to find that efficiencies smaller than those expected fromnon-rotating BHs or larger than those expected from maximallyrotating Kerr BHs are excluded.

We have thus verified that the expected density of BH rem-nants inferred from the XRB is consistent with the local one with-out requiring efficiencies larger than the canonically adopted valueε = 0.1. We now verify if, with the obscured/unobscured ratios R12

adopted in XRB synthesis models, it is possible to reproduce alsothe local BHMF. For the scope of this paper it suffices to noticethat the Ueda et al. (2003) luminosity function, with the correctionfor the missing Compton-thick AGNs that we also adopt, is usedby the same authors to reproduce successfully the XRB spectrumand source counts. Thus the agreement of the local and the relicBHMF in Fig. 6(b) is obtained by also meeting the constraints fromthe XRB, which, in practice, provide an estimate of the numberof AGNs missed by the luminosity function. The same comparisoncould also be done using the Miyaji et al. (2000) luminosity functioncombined with the background model of Gilli, Salvati & Hasinger(2001). However, though that model is successful in reproducing theXRB spectrum and source counts, recent Chandra and XMM sur-veys have shown that the model redshift distribution is not correct(e.g. Hasinger 2003).

In summary, our analysis shows that it is possible to meet theXRB constraints in terms of both ρBH and the local BHMF.

6 AC C R E T I O N E F F I C I E N C YA N D E D D I N G TO N R AT I O

The relic BHMF derived from the Ueda et al. (2003) AGN lumi-nosity function, corrected for the Compton-thick AGNs, provides

a good match to the local BHMF and also satisfies the XRB con-straints. The match is obtained for ε = 0.1 and λ = 1. Here weinvestigate the locus in the λ–ε plane where an acceptable matchof the local and relic BHMFs can be found, i.e. we determine theacceptable λ and ε values.

To quantify the comparison between the local and relic BHMFswe recall that all the realizations of the BHMFs have a log-normaldistribution at given M and we consider the following expression:

k2 =∫

[log NL(M) − log NR(M)]2

σL(M)2 + σR(M)2d log M

/∫d log M, (31)

where N L(M) and N R(M) are the local and relic BHMFs, and σ L(M)and σ R(M) are their 1σ uncertainties. The integration is performedin the log(M/M�) = 6–10 range. Thus k2 is the average squaredeviation between the logarithms of the two BHMFs measured inunits of the total standard deviation. If, for instance,

log NL(M) = log NR(M) + n√

σL(M)2 + σR(M)2,

then k2 = n2, i.e. the two functions differ, on average, by n timesthe total standard deviation.

The ε and λ values that correspond to the minimum k2 (k2min =

0.42) are marked by the filled diamond in Fig. 7 and are ε = 0.08and λ = 0.5. To have an acceptable match between the local andrelic BHMF, we require that k2 � 1 and this constraint identifiesthe region limited by the solid line. For comparison, the dashed linelimits the region where k2 < 0.72, which corresponds to the aver-age square deviation in the ‘canonical’ case, ε = 0.1 and λ = 1,which is marked by the cross. The allowed efficiencies include thenon-rotating Schwarzschild BH (ε = 0.054, Schwarzschild 1916;Shapiro & Teukolsky 1983) and are well below the maximally rotat-ing Kerr BH (ε = 0.42, Kerr 1963; Shapiro & Teukolsky 1983). Thebest agreement between the local and relic BHMFs is obtained forefficiencies larger than that of the non-rotating BH case, suggest-ing that, on average, BHs should be rotating. Hughes & Blandford(2003) found that BHs are typically spun-down by mergers, andthis would limit the importance of mergers in the growth of BHs, inagreement with our assumption in the continuity equation.

Figure 7. Locus where accretion efficiency ε and Eddington ratio λ providethe best match between the local and relic BHMFs. The solid and dashedlines limit the regions where the logarithms of the local and relic BHMFdiffer, on average, by less than 1σ and 0.7σ (see text). The filled diamondmarks the k2 minimum and corresponds to ε = 0.08 and λ = 0.5. The crossindicates ε = 0.1 and λ = 1. The dashed lines are theoretical efficiencies fornon-rotating (ε = 0.054) and maximally rotating BHs (ε = 0.42).

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The allowed Eddington ratios, λ = L/L Edd, are in the range0.1 < λ < 1.7, indicating that BH growth takes place during lu-minous accretion phases close to the Eddington limit. McLure &Dunlop (2004), using a large sample of SDSS quasars, have re-cently estimated that the average λ varies from 0.1 at z ∼ 0.2 to 0.4at z ∼ 2. Accounting for the different bolometric corrections (theyused f bol,B = 9.8, ∼1.5 times larger than the value adopted by us),λ varies from 0.15 at z ∼ 0.2 to 0.6 at z ∼ 2. These values are inexcellent agreement with the constraints posed on λ in Fig. 7.

The results in Fig. 7 do not imply that accreting BHs cannothave ε and λ values outside the region of the best match betweenthe local and relic BHMFs. Indeed, those limits are only for averageefficiencies and Eddington ratios during phases in which black holesare significantly grown.

We conclude by noting that ε represents only the radiative effi-ciency. If there is significant release of mechanical energy, the trueefficiency might be higher. For instance, in M87 the kinetic en-ergy carried away by the jet is much larger than the radiated energy(Owen, Eilek & Kassim 2000) and, in general, the jets of radio-loudAGNs can carry away up to half of the total power in kinetic energy(Rawlings & Saunders 1991; Celotti, Padovani & Ghisellini 1997;Tavecchio et al. 2000). The release of mechanical and radiative en-ergy is important for the feedback on the galaxy, which is thought tobe one of the causes behind the M BH–σ � and MBH–Lbul correlations(e.g. Silk & Rees 1998; Blandford 1999; Begelman 2003; Granatoet al. 2004). Taking into account mechanical energy, the expressionε/(1 − ε) should be transformed to εR/(1 − εR − εM), where εR andεM are the radiative and mechanical efficiencies. Then, similarly towhat has been done in this section, one could place constraints onboth εR and εM, but this is beyond the scope of this paper.

7 G ROW T H A N D AC C R E T I O N H I S TO RYO F M A S S I V E B L AC K H O L E S

Having established that the obscuration-corrected AGN luminosityfunction by Ueda et al. (2003) with ε = 0.1 and λ = 1 provides arelic BHMF that is fully consistent with the local BHMF and theX-ray background, we can analyse the growth history of massiveBHs and, in the next section, the average lifetime of their activephases.

The redshift dependence of the total density in massive BHs isgiven by (equations 17 and 18)

ρBH(z′) = 1 − ε

εc2

∫ zs

z′dz

dt

dz

∫ L2

L1

Lφ(L, z) d log L (32)

and is plotted in Fig. 8(a). The relics of the AGNs traced with theluminosity functions by Boyle et al. (2000) and Miyaji et al. (2000)reach 50 per cent of the z = 0 mass density around z ∼ 1.7, whilethe AGNs traced by the Ueda et al. (2003) luminosity function doso at z ∼ 1.4. This is a consequence of the larger number of low-luminosity AGNs that are present at z ∼ 1 (see also Fig. 4b). Indeed,when separating the contributions from low-mass (M BH < 108 M�)and high-mass BHs, which contribute 50 per cent each of the z = 0BH density, it is clear that low-mass BHs grow later than high-massBHs. These, as in the case of the AGNs traced by Boyle et al. (2000)and Miyaji et al. (2000), reach 50 per cent of their final mass atz ∼ 1.7.

This issue can be further investigated by computing the averagegrowth history of a BH with given starting, or final, mass. Thegrowth history of a BH with given starting mass MBH,0 at zs can beestimated as follows. The average accretion rate at z (or t) is given

from equation (15) as

〈M(M, t)〉 = 1

tEdd ln 10

(1 − ε)λ

εN (M, t)[φ(L, t)]L=λMc2/tEdd

. (33)

Thus one can solve the following differential equation to obtain the‘average’ growth history of BHs:

dM = 〈M(M, t)〉 dt

dzdz. (34)

In Fig. 8(b) we plot the average growth history of BHs with differentmasses at z s = 3. A supermassive BH like that of M87 or Cygnus A(M BH ∼ 3 × 109 M�; Marconi et al. 1997; Tadhunter et al. 2003)was already quite massive (∼108 M�) at zs, while a smaller BHlike that of Centaurus A (Marconi et al. 2001) was less massive,around M BH = 106 M�. A supermassive BH with M BH ∼ 109 M�at z s = 3 should now be over 1010 M�. Indeed, the existence ofvery massive BHs at high redshifts is suggested by the detection ofvery luminous quasars and, in particular, those detected at z ∼ 6 bythe SDSS survey (e.g. Fan 2003). For instance, MBH for the farthestquasar known (z ∼ 6) is estimated as M BH = 3 × 109 M� (Willott,McLure & Jarvis 2003), and one would expect its local counterpartto be more massive than 1010 M�. However, these quasars havenot been detected yet, and it is not clear if this is because thesehypermassive BHs are very rare or they simply do not exist. In thelatter case there should be a physical reason that prevents a BH fromgrowing beyond 1010 M�, possibly the feedback on the host galaxymentioned in the previous sections (see also Netzer 2003).

From Fig. 8(b) we can also infer a confirmation of what wasalready found in Fig. 8(a), namely that more massive BHs growearlier. The symbols in the figure, filled squares, empty squares andstars, mark the points when a BH reaches 90, 50 and 5 per cent ofits z = 0 mass, respectively. It is clear that for z < z s = 3 all BHsgain more than 95 per cent of their final mass, but BHs that arenow more massive than 108 M� had already gained 50 per cent oftheir z = 0 mass at z ∼ 2. Conversely, BHs that now have massesaround M BH ∼ 107 M� grew very recently, at z < 1. Again, thisis a consequence of the luminosity function by Ueda et al. (2003)in which the distribution of lower-luminosity AGNs peaks at z ∼ 1.Thus, the luminosity function of Ueda et al. (2003) points towardsan antihierarchical growth of BHs in the sense that the largest BHswere formed earlier.

If the correlations between M BH and host galaxy properties werevalid at higher redshifts [as is suggested by the results of Shieldset al. (2003)], this would immediately imply that also the most mas-sive galaxies should form earlier, in contrast with the predictions ofcurrent semi-analytic models of galaxy formation [see the introduc-tion of Granato et al. (2004) for more details and references]. Thedetection of high-mass galaxies in submillimetre surveys is indeedmore consistent with the ‘monolithic’ scenario in which massiveellipticals form at relatively high redshifts (e.g. Genzel et al. 2003;Granato et al. 2004, and references therein).

The BH accretion history (i.e. the total accretion rate at given zper unit comoving volume) can be estimated using equation (14) as

MAGN(z) =∫ M2

M1

N [M, t(z)]〈M[M, t(z)]〉 dM . (35)

In Fig. 9 we plot the cosmic accretion history of BHs [computedusing the Ueda et al. (2003) luminosity function, ε = 0.1 and λ = 1]and we compare it with the estimate by Barger et al. (2001) and withthe cosmic star formation rate by Chary & Elbaz (2001).

Our estimate of the cosmic accretion rate on to BHs agrees wellwith that by Barger et al. (2001), probably because most of the high-redshift AGNs used by Ueda et al. (2003) come from the Barger et al.

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Figure 8. (a) Growth history of total BH mass density (with ε = 0.1 and λ = 1). Different symbols indicate the use of different AGN luminosity functions.The dashed and dot-dashed lines indicate the growth history of ρBH in low-mass (M < 108 M�) and high-mass BHs, computed using the Ueda et al. (2003)luminosity function. The densities are normalized to the final value at z = 0, which is reported in the upper part of the plot. The units for ρBH are 105 M�Mpc−3. (b) Average growth history of BHs with given starting mass at z s = 3 computed using the Ueda et al. (2003) luminosity function and ε = 0.1, λ = 1.The symbols, filled squares, empty squares and stars, indicate when the BH reaches 90, 50 and 5 per cent of its final mass, respectively.

Figure 9. Cosmic BH accretion history compared with the estimate byBarger et al. (2001) and with the cosmic star formation rate by Chary &Elbaz (2001).

(2001) sample. Differently from us, Barger et al. (2001) estimate thebolometric AGN luminosities by integrating the observed spectralenergy distributions. Thus the agreement with our analysis shouldbe viewed as a consistency check on the bolometric corrections andon the corrections for obscured AGNs that we applied.

The cosmic accretion history has a similar redshift dependenceas the cosmic star formation rate, which we report here in the formestimated by Chary & Elbaz (2001). To aid the eye, the dashedline represents the estimate of the cosmic accretion history rescaledby 4000. The comparison suggests that indeed the two rates havea similar redshift dependence and justifies the assumption that theaccretion on to a BH is proportional to the star formation rate, atleast at a cosmic level. This fundamental assumption is made inseveral models of coeval evolution of BH and galaxy and, together

with the feedback from the AGN, explains the observed M BH–σ �

and MBH–Lbul correlations (e.g. Granato et al. 2004; Haiman et al.2004).

8 T H E L I F E T I M E O F AC T I V E B L AC K H O L E S

We now estimate the average total lifetime of active BHs with theformalism used in this paper and the corrected AGN luminosityfunction by Ueda et al. (2003).

In Fig. 10(a) we plot the BH duty cycle δ(M, t) at selected redshiftsderived from equation (12) and computed using the Ueda et al.(2003) luminosity function, ε = 0.1 and λ = 1. From Section 3.1,the duty cycle δ(M, t) is the fraction of BHs with mass M active attime t or redshift z. According to the definitions used in this paper, weconsider a BH to be active if it is emitting at the adopted fraction λ ofthe Eddington luminosity. Hence, objects that are usually classifiedas ‘active’ but which are emitting well below their Eddington limit(e.g. M87 or Centaurus A) should not be counted among the activeBHs whose fraction is given by the duty cycle. BHs more massivethan ∼109 M� are very rarely active in the Local Universe (onlyone out of 10 000) while they become more numerous at higherredshifts (by a factor ∼100 at z = 2). Conversely, lower-mass BHsare usually a factor of 10 more numerous. A unit duty cycle at z =3 is the initial condition assumed for the solution of the continuityequation (see Sections 3.1 and 3.4). The values of the duty cycle weobtain at z ∼ 0–1 are consistent with the average values of (3–6) ×10−3 estimated by Haiman et al. (2004).

The average total duration of the accretion process, i.e. the totallifetime of an AGN that has left a relic of mass M, is estimated byfirst solving equation (34) to obtain M(z, M0), the growth historyof a BH with mass M0 at zs. Then the total ‘active’ time is simplygiven by

τ (M) =∫ zs

0

δ[M(z, M0), z]dt

dzdz. (36)

In Fig. 10(b) we plot the average total lifetime of AGNs (solidline) as a function of the relic BH mass at z = 0 computed using

C© 2004 RAS, MNRAS 351, 169–185


Figure 10. (a) BH duty cycle δ(M, z) at given redshifts computed with the Ueda et al. (2003) luminosity function and ε = 0.1, λ = 1. (b) Average totallifetime of active BHs (AGNs) as a function of the relic BH mass at z = 0 computed with ε = 0.1, λ = 1, the Ueda et al. (2003) (solid line) and Boyle et al.(2000) (dashed line) luminosity function. The dotted line is the corresponding Salpeter time. The scale of the y-axis on the right is the average lifetime in unitsof the Salpeter time. The lines with the empty squares and triangles are the average lifetimes computed from the Ueda et al. (2003) luminosity function withε = 0.04, λ = 0.1, and ε = 0.15, λ = 1.6, respectively.

the Ueda et al. (2003) AGN luminosity function and ε = 0.1,λ = 1. The average lifetime of AGNs that leave a relic BH mass of>109 M� is of the order of 1.5 × 108 yr, while for smaller relicmasses (<108 M�) longer active phases are needed (τ BH ∼ 4.5 ×108 yr). Considering two limiting cases from Fig. 7, (ε, λ) = (0.04,0.1) and (0.15, 1.6), the average lifetimes can increase up to 109 yr.We remark that these numbers are the average lifetimes for z < z s

with z s = 3.The derived total lifetimes are consistent with the main result

from this paper that most of the BH masses are assembled via massaccretion. Indeed, if a BH grows by accreting matter with efficiencyε and emitting at a fraction λ of its Eddington luminosity, its e-folding time is given by tSal, the Salpeter time (Salpeter 1964),

tSal = εtEdd

(1 − ε)λ= 4.2 × 107yr

(1 − ε

9ε

)−1

λ−1. (37)

Thus, the initial BH mass has been e-folded ∼3 times from zs = 3for M BH > 109 M� and more than 10 times for M BH < 108 M�.The apparently long lifetimes are thus the natural consequence ofthe fact that to grow a BH from small-mass [∼(1 –10) M�] orintermediate-mass [(100 –1000) M�, e.g. Schneider et al. 2002]seeds, several e-folding times must pass easily implying τ BH > 108

yr. For instance, to grow a BH from 103 to 109 M� with λ = 1 and ε

= 0.1, one would need ∼14tSal, i.e. ∼ 6 × 108 yr (see also Haiman &Loeb 2001). Indeed, in the model by Granato et al. (2004) of coevalevolution of BH and galaxy, the time needed to grow most of theBH mass is ≈3 × 108 yr (note however that they use ε = 0.15 andλ = 3, with which the Salpeter time is roughly halved with respectto our paper).

The reason why the growth of smaller BHs (<108 M�) requireslonger time is not obvious but can be understood from Fig. 8(b).Consider, for instance, BHs with masses ∼107 and ∼1010 M� atz = 0. Their average growth history is traced by the lower and uppertracks in the figure. The larger BH has M BH ∼ 109 M� at z = 3and has to e-fold its mass by three times. Conversely, the smallerBH has M BH ∼ 102 M� at z = 3 and has to e-fold its mass by atleast ten times. The masses of large BHs (>108 M�) increase by

smaller factors from their z = 3 values, compared to the masses ofsmaller BHs. Thus they require shorter active phases.

In general, literature estimates of AGN lifetimes range from 106 to108 yr and are still highly uncertain (see the review by Martini 2003).Models where BHs grow by a combination of gas accretion traced byshort-lived (∼107 yr) quasar activity and merging in hierarchicallymerging galaxies are consistent with a wide range of observationsin the redshift range 0 < z < 5 (Haehnelt 2003). However, it isnot surprising to find a discrepancy with our analysis since we donot consider merging and the BH growth history that we find isantihierarchical. Our estimates agree better with models in whichBHs are mainly grown by gas accretion (e.g. Haiman et al. 2004;Granato et a. 2004).

With an estimate of AGN lifetime based on BH demograph-ics similar to the one presented here, Yu & Tremaine (2002) findτ = (0.3–1.3) × 108 yr for the luminous quasars using the AGNluminosity function by Boyle et al. (2000). The main reason our es-timate is larger is that the Ueda et al. (2003) luminosity function hasa larger number density of objects, which results in longer lifetimes.Indeed, the quasar lifetimes computed using the luminosity functionby Boyle et al. (2000) are ∼8 × 107 yr (dashed line in Fig. 10b), inagreement with Yu & Tremaine (2002).

Comparing to observation-based estimates, the lifetimes from thispaper are in agreement with the results from the length of radio jets(see Martini 2003 for more details). More recently, Miller et al.(2003) found from SDSS data that a very high fraction of galaxieshost an AGN (∼20–40 per cent), suggesting lifetimes longer thanpreviously thought (i.e. >108 yr). Finally, our estimate is in agree-ment with the upper limit of 109 yr set by the time-scale over whichthe quasar luminosity density rises and falls (see e.g. Osmer 2003).

In summary, we estimate that local high-mass BHs (M >

109 M�) have been active, in total, ∼ 1.5 × 108 yr. On the contrary,the assembly of lower-mass BHs has required active phases lastingat least three times that long (∼4.5 × 108 yr). These average totallifetimes can be as large as ∼109 yr if one considers the smaller effi-ciency and fraction of Eddington luminosity that are still compatiblewith local BHs (ε = 0.04, λ = 0.1).

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9 C O N C L U S I O N S

We have quantified the importance of mass accretion during AGNphases in the growth of supermassive black holes (BHs) by compar-ing the mass function of black holes in the Local Universe with thatexpected from AGN relics, which are black holes grown entirelyduring AGN phases.

The local black hole mass function (BHMF) has been estimatedby applying the well-known correlations between BH mass, bulgeluminosity and stellar velocity dispersion to galaxy luminosity andvelocity functions. We have found that different BH–galaxy corre-lations provide the same BHMF only if they have the same intrinsicdispersion, confirming the findings of Marconi & Hunt (2003). Thedensity of supermassive black holes in the Local Universe is ρBH =4.6+1.9

−1.4h20.7 × 105 M� Mpc−3.

The relic BHMF is derived from the continuity equation with theonly assumption that AGN activity is due to accretion on to mas-sive BHs and that merging is not important. We find that the relicBHMF at z = 0 is generated mainly at z < 3 where the major partof BH growth takes place. The relic BHMF at z = 0 is very littledependent on its value at z s = 3 since the main growth of BHs tookplace at z < 3. Moreover, the BH growth is antihierarchical in thesense that smaller BHs (M BH < 108 M�) grow at lower redshifts(z < 1) with respect to more massive ones (z ∼ 1–3). If the corre-lations between BH mass and host galaxy properties hold at higherredshifts, this would represent a potential problem for hierarchicalmodels of galaxy formation.

Unlike previous work, we find that the BHMF of AGN relicsis perfectly consistent with the local BHMF, indicating that localblack holes were mainly grown during AGN activity. This agree-ment is obtained while satisfying, at the same time, the constraintsimposed from the X-ray background in terms of both BH mass den-sity and fraction of obscured AGNs. The reasons for the solutionof the discrepancy at high masses found by other authors are thefollowing.

(i) We have taken into account the intrinsic dispersion of theM BH–σ � and MBH–Lbul correlations in the determination of the localBHMF.

(ii) We have adopted the coefficients of the M BH–σ � and MBH–Lbul (K-band) correlations derived by Marconi & Hunt (2003) afterconsidering only ‘secure’ BH masses.

(iii) We have derived improved bolometric corrections that donot take into account reprocessed IR emission in the estimate of thebolometric luminosity.

The comparison between the local and relic BHMFs also suggeststhat the merging process at low redshifts (z < 3) is not importantin shaping the relic BHMF, and allows us to estimate the averageradiative efficiency (ε), the ratio between emitted and Eddingtonluminosity (λ) and the average lifetime of active BHs.

Our analysis thus suggests the following scenario: Local blackholes grew during AGN phases in which accreting matter was con-verted into radiation with efficiencies ε = 0.04–0.16 and emitted at afraction λ = 0.1–1.7 of the Eddington luminosity. The average totallifetime of these active phases ranges from �4.5 × 108 yr for M BH

< 108 M� to � 1.5 × 108 yr for M BH > 109 M� but can becomeas large as ∼109 yr for the lowest acceptable ε and λ values.

AC K N OW L E D G M E N T S

We thank Reinhard Genzel, Linda Tacconi and Andrew Bakerfor useful suggestions. AM and LKH acknowledge support by

MIUR (Cofin 01-02-02). AM, GR and RM acknowledge supportby INAOE, Mexico, during the 2003 Guillermo–Haro Workshopwhere part of this work was performed. This research has made useof NASA’s Astrophysics Data System Bibliographic Services.

R E F E R E N C E S

Adams F. C., Graff D. S., Mbonye M., Richstone D. O., 2003, ApJ, 591, 125Aller M. C., Richstone D., 2002, AJ, 124, 3035Antonucci R., 1993, ARA&A, 31, 473Barger A. J., Cowie L. L., Bautz M. W., Brandt W. N., Garmire G. P.,

Hornschemeier A. E., Ivison R. J., Owen F. N., 2001, AJ, 122, 2177Barth A. J., Sarzi M., Rix H., Ho L. C., Filippenko A. V., Sargent W. L. W.,

2001, ApJ, 555, 685Begelman M. C., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser.

Vol. 1, Coevolution of Black Holes and Galaxies. Carnegie Obser-vatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Bernardi M. et al., 2003a, AJ, 125, 1817Bernardi M. et al., 2003b, AJ, 125, 1849Blandford R. D., 1999, in Merritt D. R., Valluri M., Sellwood J. A., eds,

Galaxy Dynamics. Astron. Soc. Pac., San Francisco, p. 87Boyle B. J., Shanks T., Croom S. M., Smith R. J., Miller L., Loaring N.,

Heymans C., 2000, MNRAS, 317, 1014Cattaneo A., Haehnelt M. G., Rees M. J., 1999, MNRAS, 308, 77Cavaliere A., Padovani P., 1989, ApJ, 340, L5Cavaliere A., Vittorini V., 2002, ApJ, 570, 114Cavaliere A., Morrison P., Wood K., 1971, ApJ, 170, 223Celotti A., Padovani P., Ghisellini G., 1997, MNRAS, 286, 415Chary R., Elbaz D., 2001, ApJ, 556, 562Chokshi A., Turner E. L., 1992, MNRAS, 259, 421Ciotti L., van Albada T. S., 2001, ApJ, 552, L13Cole S. et al., 2001, MNRAS, 326, 255Comastri A., 2003, in Centrella J., ed., AIP Conf. Proc., The Astrophysics

of Gravitational Wave Sources. Am. Inst. Phys., New York, p. 151Comastri A., Setti G., Zamorani G., Hasinger G., 1995, A&A, 296, 1Di Matteo T., Croft R. A. C., Springel V., Hernquist L., 2003, ApJ, 593, 56Elvis M. et al., 1994, ApJS, 95, 1Elvis M., Risaliti G., Zamorani G., 2002, ApJ, 565, L75Erwin P., Graham A. W., Caon N., 2003, in Ho L. C., ed., Carnegie Obs. As-

trophys. Ser. Vol. 1, Coevolution of Black Holes and Galaxies. CarnegieObservatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Fabian A. C., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser.Vol. 1, Coevolution of Black Holes and Galaxies. Carnegie Obser-vatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Fabian A. C., Iwasawa K., 1999, MNRAS, 303, L34Fan X., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser. Vol. 1, Coevo-

lution of Black Holes and Galaxies. Carnegie Observatories, Pasadena(www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Ferrarese L., 2002, in Lee C.-H., Chang H.-Y., eds, Current High-EnergyEmission Around Black Holes. World Scientific, Singapore, p. 3

Ferrarese L., Merritt D., 2000, ApJ, 539, L9Fiore F. et al., 2003, A&A, 409, 79Fukugita M., Shimasaku K., Ichikawa T., 1995, PASP, 107, 945Fukugita M., Hogan C. J., Peebles P. J. E., 1998, ApJ, 503, 518Gebhardt K. et al., 2000, ApJ, 539, L13Genzel R., Baker A. J., Tacconi L. J., Lutz D., Cox P., Guilloteau S., Omont

A., 2003, ApJ, 584, 633George I. M., Turner T. J., Netzer H., Nandra K., Mushotzky R. F., Yaqoob

T., 1998, ApJS, 114, 73Gilli R., Risaliti G., Salvati M., 1999, A&A, 347, 424Gilli R., Salvati M., Hasinger G., 2001, A&A, 366, 407Gonzalez A. H., Williams K. A., Bullock J. S., Kolatt T. S., Primack J. R.,

2000, ApJ, 528, 145Graham A. W., Erwin P., Caon N., Trujillo I., 2001, ApJ, 593, L11

C© 2004 RAS, MNRAS 351, 169–185


Granato G. L., De Zotti G., Silva L., Bressan A., Danese L., 2004, ApJ, 600,580

Haehnelt M. G., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser.Vol. 1, Coevolution of Black Holes and Galaxies. Carnegie Obser-vatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Haehnelt M. G., Kauffmann G., 2000, MNRAS, 318, L35Haehnelt M. G., Natarajan P., Rees M. J., 1998, MNRAS, 300, 817Haiman Z., Loeb A., 2001, ApJ, 552, 459Haiman Z., Ciotti L., Ostriker J. P., 2004, ApJ, in press (astro-ph/0304129)Hasinger G., 2003, in Holt S. S., Reynolds C., eds, AIP Conf. Proc. Vol. 666,

The Emergence of Cosmic Structure: Thirteenth Astrophysics Confer-ence. Am. Inst. Phys., New York, p. 227

Hatziminaoglou E., Mathez G., Solanes J., Manrique A., Salvador-Sole E.,2003, MNRAS, 343, 692

Hughes S. A., Blandford R. D., 2003, ApJ, 585, L101Hunt L. K., Pierini D., Giovanardi C., 2004, A&A, 414, 905Kauffmann G., Haehnelt M., 2000, MNRAS, 311, 576Kerr R. P., 1963, Phys. Rev. Lett., 11, 237Kochanek C. S. et al., 2001, ApJ, 560, 566Kormendy J., Gebhardt K., 2001, in Wheeler J. C., Martel H., eds, 20th Texas

Symp. on Relativistic Astrophysics. Am. Inst. Phys., Melville, NY, p.363

Kormendy J., Richstone D., 1995, ARA&A, 33, 581Lynden-Bell D., 1969, Nat, 223, 690Macchetto F., Marconi A., Axon D. J., Capetti A., Sparks W., Crane P., 1997,

ApJ, 489, 579McLure R. J., Dunlop J. S., 2002, MNRAS, 331, 795McLure R. J., Dunlop J. S., 2004, MNRAS, submitted (astro-ph/0310267)Madau P., Ghisellini G., Fabian A. C., 1994, MNRAS, 270, L17Magdziarz P., Zdziarski A. A., 1995, MNRAS, 273, 837Magorrian J. et al., 1998, AJ, 115, 2285Maiolino R., Rieke G. H., 1995, ApJ, 454, 95Marconi A., Hunt L. K., 2003, ApJ, 589, L21Marconi A., Salvati M., 2002, in Maiolino R., Marconi A., Nagar N., eds,

Issues in Unification of Active Galactic Nuclei. Astron. Soc. Pac., SanFrancisco, p. 217

Marconi A., Axon D. J., Macchetto F. D., Capetti A., Soarks W. B., CraneP., 1997, MNRAS, 289, L21

Marconi A., Capetti A., Axon D. J., Koekemoer A., Macchetto D., SchreierE. J., 2001, ApJ, 549, 915

Martini P., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser.Vol. 1, Coevolution of Black Holes and Galaxies. Carnegie Obser-vatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Marzke R. O., Geller M. J., Huchra J. P., Corwin H. G., 1994, AJ, 108, 437Menci N., Cavaliere A., Fontana A., Giallongo E., Poli F., Vittorini V., 2003,

ApJ, 587, L63Merritt D., Ferrarese L., 2001a, MNRAS, 320, L30Merritt D., Ferrarese L., 2001b, in Knapen J. H., et al., eds, The Central

Kiloparsec of Starbursts and AGN. Astron. Soc. Pac., San Francisco,p. 335

Miller C. J., Nichol R. C., Gomez P., Hopkins A., Bernardi M., 2003, ApJ,597, 142

Miyaji T., Hasinger G., Schmidt M., 2000, A&A, 353, 25

Monaco P., Salucci P., Danese L., 2000, MNRAS, 311, 279Nakamura O., Fukugita M., Yasuda N., Loveday J., Brinkmann J., Schneider

D. P., Shimasaku K., SubbaRao M., 2003, AJ, 125, 1682Netzer H., 2003, ApJ, 583, L5Nulsen P. E. J., Fabian A. C., 2000, MNRAS, 311, 346Osmer P. S., 2003, in Ho L. C., ed., Carnegie Obs. Astrophys. Ser.

Vol. 1, Coevolution of Black Holes and Galaxies. Carnegie Obser-vatories, Pasadena (www.ociw.edu/ociw/symposia/series/symposium1/proceedings.html)

Owen F. N., Eilek J. A., Kassim N. E., 2000, ApJ, 543, 611Padovani P., Burg R., Edelson R. A., 1990, ApJ, 353, 438Perola G. C., Matt G., Cappi M., Fiore F., Guainazzi M., Maraschi L.,

Petrucci P. O., Piro L., 2002, A&A, 389, 802Rawlings S., Saunders R., 1991, Nat, 349, 138Risaliti G., Maiolino R., Salvati M., 1999, ApJ, 522, 157Salpeter E. E., 1964, ApJ, 140, 796Salucci P., Szuszkiewicz E., Monaco P., Danese L., 1999, MNRAS, 307, 637Schmidt M., 1963, Nat, 197, 1040Schneider R., Ferrara A., Natarajan P., Omukai K., 2002, ApJ, 571, 30Schwarzschild K., 1916, Sitzungsber. Dtsch. Akad. Wiss. Berlin, Kl. Math.

Phys. Tech., 189Setti G., Woltjer L., 1989, A&A, 224, L21Shapiro S. L., Teukolsky S. A., 1983, Black Holes, White Dwarfs, and

Neutron Stars. Wiley, New YorkSheth R. K. et al., 2003, ApJ, 594, 225Shields G. A., Gebhardt K., Salviander S., Wills B. J., Xie B., Brotherton

M. S., Yuan J., Dietrich M., 2003, ApJ, 583, 124Silk J., Rees M. J., 1998, A&A, 331, L1Simien F., de Vaucouleurs G., 1986, ApJ, 302, 564Small T. A., Blandford R. D., 1992, MNRAS, 259, 725Soltan A., 1982, MNRAS, 200, 115Steffen A. T., Barger A. J., Cowie L. L., Mushotzky R. F., Yang Y., 2003,

ApJ, 596, L23Steidel C. C., Adelberger K. L., Giavalisco M., Dickinson M., Pettini M.,

1999, ApJ, 519, 1Tadhunter C., Marconi A., Axon D., Wills K., Robinson T. G., Jackson N.,

2003, MNRAS, 342, 861Tavecchio F. et al., 2000, ApJ, 543, 535Telfer R. C., Zheng W., Kriss G. A., Davidsen A. F., 2002, ApJ, 565, 773Tremaine S. et al., 2002, ApJ, 574, 740Ueda Y., Akiyama M., Ohta K., Miyaji T., 2003, ApJ, 598, 886Vanden Berk D. E. et al., 2001, AJ, 122, 549van der Marel R. P., de Zeeuw P., Rix H.-W., Quinlan G. D., 1997, Nat, 385,

610Vignali C., Brandt W. N., Schneider D. P., 2003, AJ, 125, 433Volonteri M., Haardt F., Madau P., 2003, ApJ, 582, 559Willott C. J., McLure R. J., Jarvis M. J., 2003, ApJ, 587, L15Wyithe J. S. B., Loeb A., 2003, ApJ, 595, 614Yu Q., Tremaine S., 2002, MNRAS, 335, 965Yuan W., Brinkmann W., Siebert J., Voges W., 1998, A&A, 330, 108Zamorani G. et al., 1981, ApJ, 245, 357Zel’dovich Y. B., Novikov I. D., 1964, Sov. Phys. Dokl., 158, 811

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local supermassive black holes, relics of active galactic nuclei and the x-ray background

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