# Linking the fate of massive black hole binaries to the active galactic nuclei luminosity function

###### Abstract

Massive black hole binaries are naturally predicted in the context of the hierarchical model of structure formation. The binaries that manage to lose most of their angular momentum can coalesce to form a single remnant. In the last stages of this process, the holes undergo an extremely loud phase of gravitational wave emission, possibly detectable by current and future probes. The theoretical effort towards obtaining a coherent physical picture of the binary path down to coalescence is still underway. In this paper, for the first time, we take advantage of observational studies of active galactic nuclei evolution to constrain the efficiency of gas-driven binary decay. Under conservative assumptions we find that gas accretion toward the nuclear black holes can efficiently lead binaries of any mass forming at high redshift () to coalescence within the current time. The observed “downsizing” trend of the accreting black hole luminosity function further implies that the gas inflow is sufficient to drive light black holes down to coalescence, even if they bind in binaries at lower redshifts, down to for binaries of , and for binaries of . This has strong implications for the detection rates of coalescing black hole binaries of future space-based gravitational wave experiments.

###### keywords:

quasars: supermassive black holes - galaxies: interactions - galaxies: nuclei - galaxies: active - black hole physics - gravitational waves^{†}

^{†}pagerange: Linking the fate of massive black hole binaries to the active galactic nuclei luminosity function–References

## 1 Introduction

Massive black hole (MBH) pairs are expected to form during galaxy mergers (Begelman, Blandford & Rees, 1980, BBR, hereafter). If the nuclei of the two merging galaxies manage to survive against the tidal forces in play for long enough (e.g. Callegari et al., 2009; Van Wassenhove et al., 2014), dynamical friction can efficiently bring the two MBHs to the centre of the galactic remnant, forcing them to bind in a MBH binary (BHB).

¿From the binary formation on, dynamical friction becomes less and less efficient (BBR), and other dynamical processes are needed to further evolve the binary. In particular, the interaction with single stars and with nuclear gas have been thoroughly investigated (see Dotti, Sesana & Decarli, 2012, for a recent review). The sole effect of gravitational wave emission forces the two MBHs to coalesce within the Hubble time if any physical process manages to shrink the semi-major axes of the BHB down to:

(1) |

where is the total mass of the binary, is its mass ratio, and is a function of the binary eccentricity (Peters, 1964). The assessment of how effective the various processes are to evolve the binary down to is usually referred to as the ’last parsec’ problem.

Attempts to determine the fate of BHBs (whether they manage to reach and to coalesce or they remain bound in double systems forever) have been made first considering gas-poor environments, where BHB dynamics is assumed to be driven by three-body interactions with single stars. In principle, only stars whose orbits intersect the BHB can efficiently interact with it. In an extended stellar system, however, only a small fraction of the phase space (the so called binary “loss cone”) is populated by such orbits. Stars interacting with the BHB remove energy and angular momentum from the binary, getting ejected from the loss cone. The binary evolution timescale is hence related to the rate at which new stars are fed into the loss cone (e.g. Makino & Funato, 2004). Physical mechanisms able to efficiently refuel the loss cone are required in order for the binary to coalesce in less that an Hubble time. Possible mechanisms that have been proposed so far are: the presence of massive perturbers (such as giant molecular clouds Perets & Alexander, 2008), deviations from central symmetry (e.g. Khan, Just & Merritt 2011; Preto et al. 2011; Gualandris & Merritt 2011, but see also Vasiliev, Antonini & Merritt 2014) and gravitational potential evolving with time (e.g. Vasiliev, Antonini & Merritt, 2014).

Similarly, the effect of the interaction between BHBs and nuclear gas has been explored analytically as well as numerically (see Dotti, Sesana & Decarli, 2012, for an up to date review). While the full details of the gas/binary interaction are still under debate, mainly due to the complexity of the system, a clear issue remains to be addressed. Similarly to the stellar-driven case, the migration timescale of the BHB primarily depends on how much gas is able to spiral toward the MBHs and interact with them, instead of, e.g. turn into stars (e.g. Lodato et al., 2009). This problem is remarkably similar to the fueling problem of Active Galactic Nuclei (AGN), i.e. how gas manages to lose most of its angular momentum in order to sustain the observed nuclear activity.

Differently from the stellar driven case, however, observational studies of the AGN population allow to constrain the properties of the gas flowing onto a MBH (in particular its mass accretion rate). Decades of multi-wavelength surveys of accreting MBHs have provided a relatively robust picture of the AGN luminosity function evolution (see e.g. Hasinger, Miyaji & Schmidt, 2005; Hopkins, Richards & Herquist, 2007; Buchner et al., 2015). Coupling such evolution with the observationally determined MBH mass function via a continuity equation (Cavaliere, Morrison & Wood, 1971; Small & Blandford, 1992) allows to further infer the evolution of the nuclear inflow rates as a function of MBH mass and redshift (e.g. Merloni & Heinz 2008, Shankar et al. 2013).

In this work we will assume that the fueling of BHBs is consistent with that of single MBHs in the same mass and redshift interval. In this way we can estimate the incremental reservoir of angular momentum that a BHB can interact with during its cosmic evolution, constraining the binary fate, at least in a statistical fashion, directly from observations.

## 2 Model

### 2.1 Gas driven BHB dynamics

To get an order of magnitude estimate of the binary coalescence timescale we propose a very simple zeroth-order model for the interaction between a BHB and a circum-binary accretion disc. We assume that the BHB is surrounded by an axisymmetric, geometrically thin accretion disc co-rotating with the BHB. Under this assumption, the gas inflow is expected to be halted by the binary, whose gravitational torque acts as a dam, at a separation (e.g. Artymowicz & Lubow, 1994), where is the binary semi-major axis. At this radius the gravitational torque between the binary and the disc is perfectly balanced by the torques that determine the large scale () radial gas inflow. Then, we can write the variation of the binary orbital angular momentum magnitude () as:

(2) |

where is the total binary mass, and is the accretion rate within the disc. Considering the definition of , where is the binary reduced mass, from eq. 2 we derive:

(3) |

where does not evolve in time consistently with the assumption that the binary interacts with the disc only through the gravitational torques, that stop the gas inflow preventing the binary components to accrete.

As a note of caution, we stress that the circum-binary discs could be, in principle, counter-rotating with respect to the BHBs they orbit (Nixon, King & Pringle, 2011). In this configuration the gas interacts with the BHB at , instead of (Nixon et al., 2011), and the specific angular momentum transfer per unit time is uncertain, depending on how strongly the secondary MBH is able to perturb the gas inflow. As an example, if all the gas passing through bounds to the secondary MBH, the binary angular momentum diminishes by two times the angular momentum carried by the gas. In this scenario, the BHB evolves on the same timescale regardless of the BHB-disc relative orientation. Moreover, Rödig & Sesana (2014) demonstrated that BHBs embedded in self-gravitating retrograde discs may secularly tilt their orbital plane toward a coplanar prograde equilibrium configuration. For these reasons we will focus our investigation on the prograde case in the following.

As a first order of magnitude estimate of the coalescence timescale we can make the simplifying (although quite common) assumption of an Eddington limited accretion event and integrate eq. 3. Further assuming a fixed radiative efficiency (, see section 2.2) we obtain

(4) |

where and define the binary separation range where the MBH-gas interaction drives the binary orbital decay.

Eq. 4 shows that, in order to coalesce, the binary has to interact with an amount of matter of the order of its reduced mass, with only a weak dependence on the exact ratio between the initial and final separation. A conservative estimate for this ratio can be obtained setting equal to the radius at which the two MBHs bind in a binary

(5) |

where we have assumed the relation (Gultekin et al., 2009). The final separation can be conservatively estimated as pc, in order for the BHB to coalesce due to the emission of gravitational waves in yr. Under these assumptions and for a binary mass .

### 2.2 Gas inflow onto BHBs: observational constraints

We can now, for the first time, try to relax any a priori assumption on the accretion rate in the circum-binary disc (such as the Eddington limit used in eq. 4), assuming an observational driven prescription for the evolution of as a function of MBH and redshift. In particular, we adopt the average accretion rates obtained assuming that the MBH evolution is governed by a continuity equation, where the MBH mass function at any given time can be used to predict that at any other time, provided the distribution of accretion rates as a function of black hole mass is known. The continuity equation can be written as:

(6) |

where ( is the black hole mass in solar units), is the MBH mass function at time , and is the average accretion rate of a MBH of mass at time . The average accretion rate can be defined through a “fueling” function, , describing the distribution of accretion rates for objects of mass at time : . Such a fueling function is not known a priori, and observational determinations thereof have been able so far to probe robustly only the extremes of the overall population. However, the fueling function can be derived by inverting the integral equation that relates the luminosity function () of the AGN population with its mass function. Indeed we can write:

(7) |

where we have called and , with the radiative efficiency. This is assumed to be constant and its average value can be estimated by means of the Soltan argument (Soltan, 1982), which relates the mass density of remnants MBH in the local Universe with the integrated amount of accreted gas during the AGN phases, as identified by the luminosity function.

Gilfanov & Merloni (2014) reviewed the most recent assessments of the Soltan argument. Adopting as a starting point the bolometric AGN luminosity function of Hopkins et al. (2007), the estimate of the (mass-weighted) average radiative efficiency, , can be expressed as:

(8) |

where is the local () MBH mass density in units of 4.2 (Marconi et al., 2004); is the mass density of black holes at the highest redshift probed by the bolometric luminosity function, , in units of the local one, and encapsulates our uncertainty on the process of BH formation and seeding in proto-galactic nuclei; is the fraction of the MBH mass density (relative to the local one) grown in heavily obscured, Compton Thick AGN; finally, is the fraction of BH mass contained in “wandering” objects, that have been ejected from a galaxy nucleus, for example, in the aftermath of a merging event because of the anisotropy in the emission of gravitational waves (e.g. Lousto & Zlochower, 2013, and references therein). More recent estimates of the fraction of MBH mass density accumulated in heavily obscured, Compton-Thick AGN (Buchner et al., 2015) suggest that . Neglecting in eq. 8 (i.e. assuming a negligibly small seed BH mass density), the average radiative efficiency will vary approximately between 0.075 and 0.1 for . Therefore in the following we will use the results obtained by performing a numerical inversion of eq. 7, based on a minimization scheme that used the Hopkins et al. (2007) AGN bolometric luminosity function as a constraint, and assuming a fixed radiative efficiency in the range . The average Eddington ratios (bolometric luminosity normalized to the Eddington limit) and accretion rates obtained in this way are shown as a function of redshift in the left and right panels of figure 1, respectively. We note that increasing the radiative efficiency value implies a decrease in the average accretion rates, especially for higher MBH masses at higher redshifts where the MBH evolution is relatively more important. This is consistent with the adopted calculation scheme where the AGN luminosity function is assumed as a constraint to derive the accretion rates estimates.

## 3 Results

The observational constraints on the average value of (function of M and z) discussed in section 2.2 allow us to numerically integrate eq. 3 to determine the migration timescale for any BHB. We can further translate into an estimate of the minimum redshift at which a BHB of mass must form in order to coalesce within a given redshift . The results of the numerical integration are shown in figure 2 for a binary mass ratio (lower panel) and (upper panel).

No ’last parsec’ problem seems to exist for binaries of total mass Mand formed at . More massive binaries () M do not coalesce within the present time if formed at , and the extreme cases of M coalesce within only if formed at . Binaries forming at higher redshift coalesce in shorter times, since the average accretion rates increase with within the redshift interval considered in this analysis. For example, all the binaries forming at coalesce within .

As a note of caution, we stress that the assumption that all the gas inflow is stopped by the binary is oversimplifying. As a matter of fact, numerical 2-D and 3-D simulations (independently of the exact treatment of gravity or hydrodynamics) demonstrated that the deviations from axisymmetry close to the binary, driven by its gravitational potential, allow for periodic inflows of gas within (Hayasaki, Mineshige & Ho, 2008; Cuadra et al., 2009; Rödig et al., 2011; Sesana et al., 2012; Shi et al., 2012; Noble et al., 2012; D’Orazio, Haiman & MacFadyen, 2013; Farris et al., 2014). To put firm upper limits to the MBH migration timescale, we assume that only a fraction of the gas inflow interacts dynamically with the binary, while the remaining of the gas fails to strongly interact gravitationally with the binary, and falls onto one of the MBHs unimpeded. A simple timescale estimate for the binary shrinkage, under the assumption of MBHs accreting at a fixed fraction of the Eddington limit, can be obtained replacing with in eq. 4. Reducing the fraction of interacting matter increases the migration timescale, hence increasing the minimum required for the BHB to coalesce within a given as shown in figure 3.

As expected, decreasing the evolution of every binary slows down, but the general trends discussed while commenting the cases remain valid. BHBs with total mass Mat are particularly affected, because of the redshift at which their typical Eddington ratio peaks (see figure 1). Still, these BHBs manage to coalesce between , as well as their more massive counterparts.

## 4 Conclusions

We estimated the gas driven orbital decay of BHBs from the instant at which they bind in a binary down to their final coalescence. For the first time we propose an observationally driven approach, that has the advantage of not being affected neither by any assumption on the (largely unknown) feeding process driving the accretion, nor by the fraction of the gas inflow that turns into stars at large scales before interacting with the BHB.

Our investigation proves that 1) high redshift BHBs of any mass coalesce on a very short time; 2) Low mass BHBs () formed at low redshift manage to merge anyway within , since their accretion history peaks at lower redshifts. These findings are particularly relevant since the coalescence of low mass BHBs is one of the sources of gravitational waves detectable by future space based gravitational wave interferometers, such as the mission concept eLISA (Amaro Seoane, 2013).

We have worked under very conservative assumptions:

- We assumed that binaries in the late stages of galaxy mergers
are fueled as much as MBHs in comparable isolated galaxies, without assuming
any merger driven boost in the accretion. The merger process itself is
considered, however, an efficient reshuffler of the gas angular
momentum at galactic scales, driving efficiently gas inflows all the
way down to the two MBHs, as confirmed by observations
(e.g. Kennicutt & Keel, 1984; Keel et al., 1985; Alonso et al., 2007; Koss et al., 2011; Ellison et al., 2011; Silverman et al., 2011; Satyapal et al., 2014) as well as by a wealth of
numerical works performed on different kind of mergers
(e.g. Di Matteo, Springel & Hernquist, 2005; Johansson, Burkert & Naab 2009; Hopkins & Quataert, 2010; Callegari et al., 2011; Van Wassenhove et al., 2012; Capelo et al., 2014).

- We have assumed that all the gas accretion onto MBHs is
radiatively efficient, and that the mass accretion rate at few gravitational
radii (, where basically all the luminosity is emitted) is equal to the
that at thousands of , where the gas interacts with the secondary MBH. We
stress that a significant fraction of the accretion flow, however, could be
ejected in the form of fast outflows, as often found in numerical simulations
(e.g. Proga, 2003; Narayan et al., 2012).

Under our conservative assumptions, gas driven migration of high mass () BHBs formed at low redshift could be inefficient. Such
binaries are of particular interest, being the only ones observable through
pulsar timing (Hobbs et al., 2010, and references therein). The morphological and
dynamical characteristic of their hosts suggest, however, that interactions
with stars could play a significant role in the binaries shrinking. The hosts
of very massive MBHs often show triaxial profiles (e.g. Faber et al., 1997; Kormendy et al., 2009, and references
therein). The lack of spherical and axial symmetry in
the potential of the hosts allows the single stars to modify substantially
their angular momentum components. Stars can hence re-fill the loss cone of
the binaries at rates significantly higher than those expected in spherical
systems^{1}^{1}1In spherical potentials the collisional refilling of the
binary loss cone can lead binaries to coalescence within
yr only
if the total mass of the binary is , see
e.g. Section 8.3 in Merritt (2013)., possibly leading BHBs to a fast
coalescence (see Vasiliev, 2014, for a recent discussion).

## Acknowledgments

We acknowledge the anonimous Referee, Alberto Sesana and Eugene Vasiliev for useful comments and fruitful discussions.

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