The Evolution of Black Hole Mass and Spin in AGN

# The Evolution of Black Hole Mass and Spin in Active Galactic Nuclei

A.R. King, J.E. Pringle & J.A. Hofmann
Theoretical Astrophysics Group, University of Leicester, Leicester LE1 7RH
Institut für Theoretische Physik und Astrophysik, Leibnizstrasse 15, 24118 Kiel, Germany
July 12, 2019
###### Abstract

Observations show that the central black hole in galaxies has a mass only of the stellar bulge mass. Thus whatever process grows the black hole also promotes star formation with far higher efficiency. We interpret this in terms of the generic tendency of AGN accretion discs to become self–gravitating outside some small radius  pc from the black hole. We argue that mergers consist of sequences of such episodes, each limited by self–gravity to a mass , with angular momentum characteristic of the small part of the accretion flow which formed it. In this picture a major merger with gives rise to a long series of low–mass accretion disc episodes, all chaotically oriented with respect to one another. Thus the angular momentum vector oscillates randomly during the accretion process, on mass scales times smaller than the total mass accreted in a major merger event.

We show that for essentially all AGN parameters, the disc produced by any accretion episode of this type has lower angular momentum than the hole, allowing stable co– and counter–alignment of the discs through the Lense–Thirring effect. A sequence of randomly–oriented accretion episodes as envisaged above then produces accretion discs stably co– or counter–aligned with the black hole spin with almost equal frequency. Accretion from these discs very rapidly adjusts the hole’s spin parameter to average values (the precise range depending slightly on the disc vertical viscosity coefficient ) from any initial conditions, but with significant fluctuations () about these. We conclude (a) AGN black holes should on average spin moderately, with the mean value decreasing slowly as the mass increases; (b) SMBH coalescences leave little long–term effect on ; (c) SNBH coalescence products in general have modest recoil velocities, so that there is little likelihood of their being ejected from the host galaxy; (d) black holes can grow even from stellar masses to at high redshift ; (e) jets produced in successive accretion episodes can have similar directions, but after several episodes the jet direction deviates significantly. Rare examples of massive holes with larger spin parameters could result from prograde coalescences with SMBH of similar mass, and are most likely to be found in giant ellipticals. We compare these results with observation.

###### keywords:
accretion, accretion discs – black holes, galaxies – active
volume: 000pagerange: The Evolution of Black Hole Mass and Spin in Active Galactic NucleiReferencespubyear: 2006

## 1 Introduction

Astronomers generally agree that the nuclei of most galaxies contain supermassive black holes (SMBH), but as yet have little clear idea of how they grow. Cosmological large–scale structure simulations strongly suggest that galaxy mergers are the basic motor of growth, and predict that a black hole typically acquires a mass of order its own mass in such mergers (e.g. di Matteo et al., 2005, Li et al., 2007, and references therein). This is reasonable, as growth rates of this order are needed in order to reach the hole masses observed in the nearby universe. If one assumes further that all the mass gain carries the same sense of specific angular momentum, the hole always spins rapidly up to Kerr parameters close to unity (e.g. Volonteri et al., 2005; Volonteri & Rees, 2005). The resulting high radiative efficiency of accretion means that the Eddington limit severely restricts the rate of black hole mass growth, creating difficulties in understanding observations of high masses at early cosmic times.

However it is sigmificant that is only a very small part of the total mass involved in a major galaxy merger, in line with the observation that the black hole mass in nearby galaxies is typically only of the galaxy bulge mass. Moreover, current cosmological simulations are entirely unable to resolve the hydrodynamics of matter accretion on to the central black hole. The mass inflow time through an accretion disc of 1 pc radius is already  yr (Shlosman, Begelman & Frank, 1990), a factor greater than the timescale on which rapid hole growth occurs. The best cosmological simulations resolve only lengthscales at least 100 times larger. Inevitably the simulations have to resort to sub–resolution recipes in trying to model the accretion process. For example it is usual to assume accretion at the Bondi rate from some large radius, whereas in reality the infalling gas must possess angular momentum which will complicate things.

In this picture the very small mass which accretes on to the central black hole must have almost zero angular momentum about it. Moreover, since the hole is growing its mass as rapidly as possible, it must be close to its Eddington limit and thus feeding back energy and momentum into its surroundings. Indeed simple models of the resulting momentum–driven feedback on to the host galaxy correctly reproduce the relation between black hole mass and velocity dispersion without any free parameter (King, 2003; 2005, Murray et al., 2005). Along with this, there must be copious star formation from material close to the hole, i.e. at distances  pc (see below) where the angular momentum neglected in estimating the Bondi accretion rate becomes significant in supporting the captured gas against gravity. This star formation also injects energy and momentum into the gas near the hole. Hence although it is very difficult to predict the gas flow near the black hole it seems more likely to be chaotic than well–ordered, and rather inefficient in growing the hole compared with forming stars, thus keeping the latter a fairly constant fraction of the stellar bulge mass.

An obvious reason for the relative inefficiency of hole growth versus star formation is that an accretion disc becomes self–gravitating at radii where its mass exceeds , where is the disc scaleheight and the central (here black hole) mass (e.g. Pringle, 1981, Frank et al. 2002). For AGN this means that discs must be self–gravitating outside some radius  pc (Shlosman et al., 1990; Collin–Souffrin & Dumont, 1990; Huré et al., 1994). Cooling times in these regions are so short that self–gravity is likely to result in star formation (Shlosman & Begelman, 1989; Collin & Zahn, 1999). As discussed by King & Pringle, 2007 (hereafter KP07) we expect that most of the gas initially at radii is either turned into stars, or expelled by those stars which do form, on a rapid (almost dynamical) timescale, while the gas which is initially at radii forms a standard accretion disc, which slowly drains on to the black hole and powers the AGN. KP07 show that for local, low–luminosity AGN, fuelling by well separated episodes of this type explains observational features such as the luminosity function for moderate–mass black holes, and the presence and location of a ring of young stars observed about the Galactic Centre.

This paper deals not with such low–luminosity AGN but instead with high–luminosity events causing major black–hole mass growth. In these cases our discussion suggests a picture in which, within an event, the flow on to the hole is episodic, via a rapid succession of accretion discs limited by self–gravity (as in the well–separated episodes in low–luminosity AGN), and thus with masses . The number of such episodes is evidently . Given that the flow is constantly stirred up by energy and momentum input, and that we are concerned only with the very small fraction of the gas with almost zero angular momentum, it is reasonable to expect the disc orientations to be essentially random.

We thus arrive at a picture of AGN accretion close to that suggested by Sanders (1984) (see also Heckman et al., 2004, Greene & Ho, 2007, and KP07). This picture reproduces several characteristic features of nearby AGN. In particular the random orientations of the accretion episodes mean that the radio jets show no correlation with the grand–design structure of the host, as observed for low–redshift AGN (Kinney et al., 2000), and inferred at higher redshift by Sajina et al (2007). Moreover the picture implies a luminosity function in broad agreement with that observed for moderate mass central black holes (KP07). Finally this picture provides a plausible explanation for the ring(s) of young stars seen around the black hole in the centre of our own Galaxy (Genzel et al., 2003; Lu et al., 2006).

Since observational properties such as the luminosity function of AGN and quasars are sensitive only to the total mass increase , and not to its internal angular momentum budget, our picture predicts the same values for these observables as found in previous studies (e.g. Wyithe & Loeb, 2003) and is therefore in broad agreement with expected galaxy merger rates.

Clearly the proper hydrodynamic treatment of the kind of complex event we envisage must await advances in computing power. However we can gain some insight by simple arguments.

## 2 SMBH spinup

To understand how a supermassive black hole grows we have to know its spin rate. This governs the accretion efficiency and thus, through the Eddington limit, the maximum growth rate for the mass. The range here is very large. Given an adequate mass supply, a low Kerr spin parameter and consequent low accretion efficiency allows very much faster mass growth than a rapidly spinning hole with (cf. King & Pringle, 2006).

Accretion on to a black hole affects its spin in two ways. First, it torques the hole by adding matter carrying the specific angular momentum of the innermost stable circular orbit (ISCO), and second, the Lense–Thirring (LT) effect creates a viscously mediated torque between the hole and any accretion disc not completely co– or counter–aligned with the hole spin. Gas within the disc follows precessing orbits because of the LT effect, and the resulting viscous dissipation tries to produce an axisymmetric situation by adjusting the direction of the angular momentum vectors of both the hole spin and the disc.

The LT effect is very important in AGN, as the accretion episodes all initially have random directions with respect to the hole spin. As it has a large lever arm, it operates more quickly than the accretion torque, which cannot greatly change the hole spin until the hole has significantly increased its mass. Accordingly, the LT effect determines whether the disc and hole angular momenta are co– or counter–aligned long before the accretion torque has had a real effect. Early studies of the LT effect (e.g. Scheuer & Feiler, 1996) concluded that it always produced co–alignment. Accordingly studies of SMBH growth (e.g. Volonteri et al., 2005; Volonteri & Rees, 2005) argued that the main effect of the accretion torque was always to produce spinup. However King et al. (2005; hereafter KLOP) showed on very general grounds that an initially retrograde accretion disc with total angular momentum would end up stably counter–aligned with a hole of angular momentum provided that and the initial angle between the angular momenta satisfied

 cosθ<−Jd2Jh. (1)

(Throughout this paper and the Kerr parameter denote the absolute values of these quantities.)

Thus Scheuer & Feiler’s (1996) conclusion that co–alignment always occurs depended on their implicit assumption that the outer disc was fixed, i.e. . By contrast, KLOP’s result implies that counter–alignment occurs in a fraction

 f=12(1−Jd2Jh) (2)

of accretion episodes.

In their analysis, KLOP considered a disc of fixed total angular momentum misaligned with the hole. In reality any misaligned disc must be warped at some radius , in general rather smaller than its outer edge. Only the part of the disc around exerts an LT torque on the hole, so the correct interpretation of in such cases is the total angular momentum passing through during the alignment process. This interpretation agrees with numerical simulations of the alignment process in such cases (Lodato & Pringle, 2006). Of course if the whole disc is warped is simply the total angular momentum of the disc as before.

Equation (2) shows that a random sequence of episodes with must have , so that co– and counter–alignment are almost equally common. Since the ISCO for retrograde accretion carries higher specific angular momentum than for prograde accretion if the Kerr parameter is significant (in the ratio 11:3 for maximal spins) we see that spindown is more effective than spinup, ultimately producing a slowly–spinning hole (Hughes & Blandford, 2003 reach a similar conclusion for the effect of repeated coalescences of spinning holes, for the same reason; see also Colbert & Wilson, 1995). King & Pringle (2006) used this argument to show how suitably random accretion episodes could allow the growth of very large SMBH () at high redshift from even stellar–mass seed black holes.

In this paper we show that the fact that self–gravity cuts off the accretion discs in AGN feeding episodes implies in almost all cases. A sequence of repeated episodes of this type with random orientations therefore spins the hole down to low spin parameters, and we find a mean value .

## 3 SMBH accretion episodes

We first require a simple description of the properties of the accretion disc during a feeding episode of the type described above. One can regard an evolving disc as passing through a sequence of steady states, so we follow KP07 in adopting the disc properties derived by Collin–Souffrin & Dumont (1990) in the context of AGN. These are essentially the same as those derived by Shakura & Sunyaev (1973) for steady discs in close binaries. The disc surface density is

 Σ=7.5×106(α10.03)−4/5(ϵ0.1)−3/5 ×(L0.1LE)3/5M1/58(RRs)−3/5gcm−2. (3)

Here is the Shakura & Sunyaev (1973) viscosity parameter, is the accretion efficiency, with the luminosity and accetion rate related by

 L=ϵ˙Mc2. (4)

Further

 LE=1.4×1046M8ergs−1, (5)

is the Eddington luminosity, is the black hole mass, , in units of M, is the radius and

 Rs=2.96×1013M8cm (6)

is the Schwarzschild radius of the central black hole.

The mass of the disc inside radius is then

 M(

The disc semi–thickness obeys

 ×(L0.1LE)1/5M−1/108(RRs)1/20. (8)

The disc becomes self–gravitating when its mass reaches

 Md=Msg≃HRM, (9)

(e.g. Pringle, 1981) which occurs at radii , where

 RsgRs=1.13×103(α10.03)14/27(ϵ0.1)8/27 ×(L0.1LE)−8/27M−26/278 . (10)

In line with our discussion in the Introduction we identify as the mass in an individual accretion episode. The evolution timescale of the disc, and thus the characteristic duration of an episode, is given by , i.e.

 τsg=1.12×106(α0.03)−2/27(ϵ0.1)22/27 ×(L0.1LE)−22/27M−4/278yr . (11)

In line with our discussion above we expect a major merger event to consist of a chaotic sequence of such episodes. These may overlap, or be slightly separated in time. Our conclusions are independent of the the time profile of the accretion, and depend only on the total mass and angular momentum accreted. We differ from other treatments such as those of Volonteri & Rees (2005) and Volonteri et al. (2007) in not following their assumption that all the mass in a given major merger accretes at the same angle, even at the scale of the inner accretion disc.

## 4 The Evolution of Black Hole Mass and Spin

We can now see how a sequence of accretion episodes limited by disc self–gravity will affect the black hole mass and spin. We compare the angular momentum of the warped disc with the angular momentum of the hole. The interpretation of as the total angular momentum passing through shows that

 Jd\laMsg(GMRwarp)1/2 (12)

(i.e. the total accreted disc mass multiplied by the disc specific angular momentum at ), provided that is less than the outer disc radius . In this latter case we replace with . The warp radius is given by (cf KLOP)

 RwarpRs=990(α10.03)1/8(α20.03)−5/8a5/8 ×(ϵ0.1)1/4(L0.1LE)−1/4M1/88. (13)

Here is the Schwarzschild radius and the dimensionless Kerr spin parameter, with the dimensionless viscosity coefficient relevant to vertical disc motions, and we note that Lodato & Pringle (2007) have recently argued that never exceeds a value in a warped disc.

If the condition fails we instead take

 Jd=Msg(GMRsg)1/2, (14)

which implies that can never be larger than given by (12).

Given the hole angular momentum

 Jh=2−1/2Ma(GMRs)1/2, (15)

we find from (12)

 Jd2Jh≤0.061a−11/16(α10.03)−5/432(α20.03)−5/16 ×(ϵ0.1)−13/216(L0.1LE)13/216M8−37/432. (16)

We see that the ratio determining the tendency of the disc to co– or counter–align under LT torques is always unless the spin parameter is small itself, effectively independently of all other parameters except for a weak dependence on . With retrograde accretion episodes stably counter–align with the hole angular momentum, spinning it down. Since spindown is more efficient than spinup, the general effect of a sequence of randomly–oriented accretion episodes as envisaged in the Introduction is to decrease . Thus one might imagine that the end result is a value of oscillating around zero, and both King & Pringle (2006) and Volonteri et al (2007) assume this. However as decreases for a given , the probability of accreting in a stably retrograde fashion (given by in (2)) approaches zero. Evidently must on average decrease only to the point where the expected spinup in prograde accretion is equal to the expected spindown in retrograde accretion, i.e.

or equivalently

One has to use the equations of Bardeen (1970) to evaluate the derivatives in this equation, which then defines the mean value .

### 4.1 Numerical Simulations

We simulate the effect of repeated accretion episodes of the type dicussed here as follows. We assume that each episode has mass and accretes at the Eddington rate appropriate to the current black hole mass . The total angular momentum of each episode is given by the recipe (12), replacing by if . The direction of is chosen at random from an isotropic distribution. The disc and hole are assumed to co– or counter–align according to the criterion (1). In contrast to Volonteri et al (2007) we do not assume that all the mass accreting in a major merger has the same orientation of angular momentum, but only the mass contained within each self–gravitating disc episode, allowing each episode to be randomly oriented.

Figures 1, 2 show the results of two such simulations. The figures omit the epochs when the hole is not accreting, so the horizontal axes measure the accretion time rather than the total elapsed time. Since the accretion is always at the current Eddington rate, the accretion time is the shortest possible time for the hole to acquire its mass through accretion.

The two simulations differ only in the value of the ‘vertical’ viscosity coefficient , which is 0.03 in Fig. 1 and 1 in Fig 2. The two simulations use the same random sequence of accretion episode orientations in order to highlight the effect of changing . The main difference is that the mean value is somewhat lower in the second case (see below).

As one can see, the main features inferred above do appear. Although the value of fluctuates widely, its mean does indeed quickly settle to the value predicted by (18) in each case. We can fit this as

 ¯a≃AM−0.0488,  RwarpRsg (19)

where (for ), (for ) is fixed by continuity at the changeover between the two power–law regimes in each case. In Fig. 1, this occurs at , which in the case with higher vertical viscosity the changeover is only at a mass of almost . We see that for holes of masses we expect in the first case and in the second, with excursions about these means. Figures 3, 4 show the normalized distributions of as a function of mass in the case .

## 5 SMBH Coalescences

We have so far discussed the evolution of SMBH mass and spin via accretion. Figures 1, 2 show that the Kerr parameter always has a strong tendency to move towards the mean value from any initial value as the hole mass doubles through accretion. Thus although a coalescence of two SMBH with comparable masses would presumably cause a discontinuous departure from the mean accretion–driven trend of Figs. 1, 2, accretion would drive it back towards the mean trend as it doubles the mass of the coalesced hole. In other words coalescences have little long–term effect on , although they can strongly affect the spin for a relatively short time (see the end of the Discussion).

## 6 SMBH Recoil Velocities

The relatively modest spin rates we find mean that SMBH coalescences even with similar masses (i.e. in major mergers) are likely to produce fairly small gravitational radiation recoil velocities  km s (González et al., 2006). These are well below the escape velocity from the merged host. This agrees with the observational evidence that most massive galaxies do have nuclear SMBH. We note that Bogdanović et al. (2007) reach similar conclusions, but for entirely orthogonal reasons. Specifically, they argue that the effect of gas accretion is to cause any SMBH to align with the angular momentum of the accretion flow. They further argue that this flow is of such large scale and well–defined angular momentum that in any incipient SMBH coalescence both holes have their spins aligned with the flow and thus with each other. It is known that co–aligned spins reduce the recoil velocity to modest values  km s, similar to those for non–spinning holes as we found above.

Our arguments differ from these in three ways. First, we have argued above that self–gravity severely limits the angular momentum of any accretion flow so that , so that co– and counter–alignment occur almost equally often, rather than the hole and disc always co–aligning as asserted by Bogdanović et al. Second, we have also argued that the accretion flow on to an SMBH cannot have as large a scale as claimed by Bogdanović et al., who suggest a linear size of  pc, because the viscous timescale from such radii exceeds a Hubble time. Finally Bogdanović et al.’s picture predicts that the SMBH spin and the accretion flow are both aligned with the large–scale structure of the galaxy. One would thus expect any jet structure to show a similar alignment othogonal to the galaxy disc apparently feeding it. However this disagrees with the observational evidence discussed by Kinney et al (2000; see also Nagar & Wilson 1999): in a sample of nearby () Seyfert galaxies the direction of the jet from the central black hole is unrelated to the orientation of the disc of the host galaxy. As mentioned above Sajina et al. (2007) have recently inferred a similar result at higher redshift.

## 7 Maximum Mass Growth Rate for SMBH

In an earlier paper (King & Pringle 2006) we considered the problem of growing the largest SMBH masses at cosmological redshift , i.e. only  yr after the Big Bang. We concluded that growth to these masses from even stellar initial values was possible provided that the spin rate, and thus the radiation efficiency of mass accretion, was kept sufficiently low (). From the discussion of the last section we see that this condition is always satisfied by accretion from randomly–oriented accretion episodes. The main limit on black hole mass growth is therefore the mass supply rather than the Eddington limit. Figure 1 shows this explicitly, giving as a function of the accretion time.

## 8 SMBH spin directions

We have so far considered the magnitude of the black–hole angular momentum vector rather than its direction. The latter may be important, as the central part of the accretion disc always has its axis parallel or antiparallel to it, and this is probably the direction of any jet produced by the system. Observations of various types support this conclusion. Schmitt et al (2003) show that the extended [O III] emission in a sample of nearby Seyfert galaxies is well aligned with the radio, and it is usual to assume that the orientation of the [O III] emission is governed by the geometry of the inner torus, of typical radius 0.1 – 1.0 pc (e.g. Antonucci, 1993). Similarly, Verdoes–Kleijn and de Zeeuw (2005) and Sparks et al (2000) find evidence that dust discs tend to be perpendicular to radio and optical jets respectively.

As the torques between the disc and the hole are all internal, the outcome of an accretion event is that the hole spin vector aligns along the total a.m. vector (cf KLOP). We can therefore track the direction of over time. From eqn (16) we see that for , and even for near the mean value (19) we have for the two cases . By simple vector addition we have that in such an accretion episode the vector deviates from its original direction by an angle given by

 sinψ=JdJhsinθ\la0.26,16 (20)

for any original misalignment angle , so that for . Thus successive accretion episodes have a tendency to produce jets in similar directions, even more so if happens to be larger than average. This may explain why some ‘double–double’ radio sources are seen with apparently near–common projected axes for successive jet events. Kharb et al. (2006, Figs. 1, 5) give a beautiful example of this. Ultimately the effect of even a few episodes is that the spin axis loses all memory of its original direction. There is clearly no correlation at all with structures in the host galaxy. Coalescences have similar disorienting effects.

## 9 Discussion

We have argued that accretion on to supermassive black holes in AGN occurs through events with total mass , consisting of long sequences of randomly oriented disc accretion episodes with masses . The mass of these disc episodes is strongly constrained by the generic tendency of such discs to become self–gravitating and form stars outside quite small radii  pc. The self–gravity constraint also limits the disc angular momentum. This is always less than the maximum that the hole’s mass would allow it to have. A hole with significant spin therefore stably co– and counter–aligns initially prograde and retrograde discs respectively, producing a net spindown until its angular momentum is reduced to a relatively modest value, with fluctuations about the average given by (19). We note from (19) that itself decreases slowly as the black hole mass decreases.

We can compare these results with those for coalescences of SMBH as a consequence of the mergers of their host galaxies. Hughes and Blandford (2003) show that the average secular evolution of black hole spin under coalescences is

 a=a0(M0M)2.4 (21)

where is the original spin parameter of a hole within initial mass . Thus coalescences doubling the mass of a hole on average reduce by a factor . This is an even faster decline of spin than given by assuming fixed angular momentum but growing mass, which would produce , reflecting the fact that retrograde coalescences are more effective in reducing spin than prograde ones are in growing it. Evidently coalescences can reduce quite quickly to zero. However we have seen that accretion very rapidly pushes back towards the mean value (cf eqns 18, 19) from any starting point. Hence on average coalescences have little net effect on the mean value of . We conclude that repeated accretion episodes and coalescences on average drive the black holes in AGN towards fairly low values of , with statistical fluctuations around them of order .

Given this result, we should consider the observational evidence constraining values of for SMBH. Much of this is derived from the Soltan (1982) argument that the totality of background light in the low–redshift Universe is compatible with black–hole growth with radiative efficiency (e.g. Wang et al., 2006; Treister & Urry, 2006; Hopkins, Richard & Hernquist, 2007), which formally requires . However is such a slow function of except near (see Figure 5) that this conclusion is vulnerable to the systematic effects inherent in trying to estimate the efficiency from observation. Evidently we have to find from observation to an accuracy of a factor in order to exclude any value of other than those very near unity. Since retrograde and prograde accretion appear equally probable, the effective efficiency for the Soltan argument is actually the symmetrized curve

 ϵ±=12[ϵ(a)+ϵ(−a)] (22)

in Figure (5), whose vertical range is now compressed to . In this case only a very accurate estimate of can give real information about at all.

Streblyanska et al. (2005) suggest a different method of estimating the typical spin for SMBH. They find that the composite X–ray spectrum of a sample of AGN shows evidence for a broad iron line. They fit lines produced with assumed emissivity and line centre energy from both non–rotating and rotating black holes, and argue that the outer radius of the emission region for the non–rotating hole is unreasonably small in the case of the Type I AGN in their sample, although the statistical fit is marginally preferred over a rotating hole. The best–fitting rotating hole for this group has an inner emission radius of , which requires . Since holes with such values of would automatically have higher efficiencies and thus presumably stronger iron lines, it is difficult to know what fraction of the SMBH in the sample actually have such fairly large spin parameters (note that, as discussed below, our arguments only refer to the statistical mean , and individual SMBH can differ significantly from this).

We remark as a general caveat that the search for any effect depending on , or the interpretation of a particular observational effect as being due to , is likely to produce a bias towards . This effect is particularly marked because of the very high efficiency of prograde accretion for (see Fig 5). We note that Brenneman & Reynolds (2006) estimate in the specific case of the broad iron line in MCG-06-30-15, remarkably close to the maximum allowed value.

Of course since our treatment is statistical it does not rule out larger spin values in individual cases. The most important way of producing large values of in individual cases is in coalescences of two SMBH. Hughes & Blandford (2003) show that this can produce significant for prograde coalescences of two black holes with similar mass. This suggests that giant ellipticals offer a promising site for significant values of . Another conceivable way of producing exceptions to our general conclusions is that we have assumed accretion from thin discs, in which cooling is efficient, so that the total disc mass is limited to times that of the hole. In some situations it may happen that cooling is inefficient and our conclusions do not hold. More work is needed here, although such cases are likely to be rare. Finally one might argue that in some cases an accretion event with mass may contrive to retain some memory of an overall angular momentum, despite the fact that it must have a near–radial orbit within the host galaxy in order to accrete at all (cf. Kendall et al., 2003: recall that accretion within a Hubble time requires disc radii  pc). In this case individual episodes no longer have uncorrelated angular momenta as we assumed. This is effectively what is assumed by Volonteri & Rees (2005), and Volonteri et al (2007) for all episodes.

## 10 Acknowledgments

ARK acknowledges a Royal Society Wolfson Research Merit Award. JH acknowledges a scholarship from the Deutscher Akademischer Austausch Dienst (DAAD) and partial support from the Deutsche Forschungsgesellschaft (DFG) through grant SFB 439.

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