## Abstract

We compute the effect of an orbiting gas disc in promoting the coalescence of a central supermassive black hole binary. Unlike earlier studies, we consider a finite mass of gas with explicit time dependence: we do not assume that the gas necessarily adopts a steady state or a spatially constant accretion rate, i.e. that the merging black hole was somehow inserted into a pre-existing accretion disc. We consider the tidal torque of the binary on the disc, and the binary's gravitational radiation. We study the effects of star formation in the gas disc in a simple energy feedback framework.

The disc spectrum differs in detail from that found before. In particular, tidal torques from the secondary black hole heat the edges of the gap, creating bright rims around the secondary. These rims do not in practice have uniform brightness either in azimuth or time, but can on average account for as much as 50 per cent of the integrated light from the disc. This may lead to detectable high-photon-energy variability on the relatively long orbital time-scale of the secondary black hole, and thus offer a prospective signature of a coalescing black hole binary.

We also find that the disc can drive the binary to merger on a reasonable time-scale only if its mass is at least comparable with that of the secondary black hole, and if the initial binary separation is relatively small, i.e. a0≲ 0.05 pc. Star formation complicates the merger further by removing mass from the disc. In the feedback model we consider, this sets an effective limit to the disc mass. As a result, binary merging is unlikely unless the black hole mass ratio is ≲0.001. Gas discs thus appear not to be an effective solution to the ‘last parsec’ problem for a significant class of mergers.

## 1 INTRODUCTION

In recent years, the process of shrinking a supermassive black hole (SMBH) binary by interaction with a circumbinary gaseous disc has been the topic of intense theoretical research. Initially, this reflected attempts to overcome the ‘last parsec’ problem, i.e. the fact that dynamical friction with the stellar background is ineffective in shrinking the binary below separations smaller than 1pc (Begelman, Blandford & Rees 1980; Milosavljević & Merritt 2001), to the point where gravitational radiation can complete the coalescence of the two holes. More recently, there has been interest in finding an electromagnetic counterpart to the gravitational wave emission expected during the final stages of black hole coalescence (Ivanov, Papaloizou & Polnarev 1999; Armitage & Natarajan 2002; Milosavljević & Phinney 2005; Dotti et al. 2006; Loeb 2007; Cuadra et al. 2009).

In general, most work on this subject has assumed driving by an accretion disc with constant mass inflow rate supplied from distances far from the binary, and thus effectively assumed an infinite mass supply. However it is clear that in reality, where the gas is part of the galaxy merger producing the SMBH binary, the mass of gas ending up in the disc must be finite. Thus, in general the disc does not settle to a steady state with a constant accretion rate, and its structure differs from a standard disc because it is affected by the tidal torque exerted by the binary. Moreover, the rate or even the success of the shrinkage must depend on the total disc mass (as hinted by Cuadra et al. 2009). Here, we investigate the problem by taking a disc of finite mass which is explicitly time varying. We find significant differences from the results of assuming a steady-state accretion disc.

We model the disc evolution in terms of a simple diffusion equation, including the effects of the tidal torques from the secondary and calculating the binary orbit evolution self-consistently. Our initial condition is a finite disc mass concentrated at radii of the order of the initial binary separation. While still highly idealized, we regard this choice as more realistic than the assumption of a steady state, constant- disc, into which a second black hole has somehow been inserted.

This paper is organized as follows. In Section 2, we describe the general features of binary shrinkage by a gas disc. In Section 3, we describe our method for following both the binary and disc evolution. In Section 4, we describe the results. We constrain the disc mass required to bring the binary to coalescence and describe the expected appearance of such discs. In Section 5, we draw our conclusions.

## 2 DISC-ASSISTED BINARY SHRINKAGE

The process of disc assisted binary shrinkage (hardening) is for low mass ratios dynamically similar to the process of planetary migration within a protostellar disc, and many studies have alluded to this analogy. For a low (but finite) mass companion, the tidal interaction between the binary and the gaseous disc produces a gap in the disc at the location of the secondary. The exchange of angular momentum across this gap is mediated through tides by the ‘satellite’ (i.e. the secondary black hole) which behaves as a fluid element in the disc. In this case, the binary shrinks on the disc's viscous time-scale tν (Armitage & Natarajan 2002). For a 108 M primary black hole, the viscous time at R∼ 1 pc is a few times 108 years, so the binary can shrink in a reasonable time before gravitational radiation emission takes over as the main cause of shrinkage.

However, things are different if the angular momentum associated with the binary orbit is comparable to or larger than the local disc angular momentum (for a circular binary orbit, this is equivalent to requiring the secondary mass to be comparable to or larger than the local disc mass). Then, the viscous torques in the disc are not able to redistribute efficiently the excess angular momentum transferred by the tides to the disc, and the binary shrinking slows down significantly.

We define Mp and Ms as the masses of the primary and secondary black holes. In general, the binary orbit evolution occurs on a time-scale (Syer & Clarke 1995; Ivanov et al. 1999):

1
which for MsMd(a) can be significantly longer than the disc viscous time. In the equation above, the local ‘disc mass’ is defined as Md(a) = 4πΣ(a)a2, where Σ(a) is the surface density of the unperturbed disc at the binary separation a. In such a configuration, disc-assisted shrinking from parsec scales would take much longer than ∼109 years and would effectively preclude binary coalescence. Here, the companion acts as a dam in the accretion flow. The disc inside the binary orbit is rapidly accreted, since it cannot be refilled from the outer circumbinary disc. Outside the binary orbit, accreting gas builds up at the circumbinary disc's inner edge. This moves inwards on a time-scale ∼tshrink, much longer than the viscous time.

If we were now additionally to make the usual assumption that the disc is continuously fuelled from the outside, then the surface density at the inner edge of the circumbinary disc would increase until it was large enough to speed up the binary hardening, as described by Ivanov et al. (1999). The disc structure would be such that at large radii, while at smaller radii νΣ∝R−1/2, typical of a ‘decretion’ disc (Pringle 1991), where ν is the disc viscosity. However, in reality, with a finite gas mass, the surface density at the inner edge of the circumbinary disc is unable to grow steadily at the rate described by Ivanov et al. (1999). Evolution proceeds more slowly and a numerical solution is needed. We discuss these solutions in Section 3. The overall normalization of the surface density differs from that predicted by Ivanov et al. (1999), while the scaling with radius does follow the νΣ∝R−1/2 decretion-like solution.

The qualitative appearance of the system as the binary evolves towards coalescence is clear. In a major merger, two black holes of comparable mass are brought together at a distance ≲0.1 pc (Escala et al. 2005), together with a finite amount of gas which we assume to settle rapidly to a fairly wide disc configuration. Part of the gas can accrete directly on to the primary black hole, producing a burst of active galactic nucleus (AGN) like activity. Given the large amount of gas mass available, this burst is likely to be Eddington limited. As a result, the disc around the primary black hole may be dispersed by the strong outflows arising from such a super-Eddington flow. Since the secondary hole acts as a dam for the gas outside the binary, the disc around the primary is not readily refilled. For protostellar accretion, the extent to which the dam is porous is still under discussion (e.g. Lubow & D'Angelo 2006). Since AGN discs are typically much thinner than protostellar discs, the porosity of the dam for the AGN discs considered here is likely to be much reduced, and we neglect this effect for the purposes of this paper. Thus, after this initial episode the system is characterized by a massive circumbinary disc truncated at the binary separation, and possibly by a low-density circumprimary disc left over from the initial Eddington-limited episode. The system appears bright at long wavelengths [optical/infrared (IR)], qualitatively similar to the description of Milosavljević & Phinney (2005).

At this point, there are two possibilities: either (i) the circumbinary disc mass is too low, the coalescence fails (or takes inordinately long), and the two black holes are left orbiting at a fraction of a parsec distance, or (b) the disc mass is large enough to drive the binary to coalescence on a reasonable time-scale. In this event, the peak of the disc spectral energy distribution (SED) progressively moves to higher energies and the disc becomes more luminous, until gravitational radiation losses bring the black holes into coalescence. If the remnant circumprimary disc is still present, it will suddenly brighten as it is squeezed by the merging binary, as qualitatively described by Armitage & Natarajan (2002). After coalescence, the dam on the accretion flow is removed, so the inner disc hole is refilled and the disc spectrum finally moves to high energies.

## 3 TIME-DEPENDENT MODEL

### 3.1 Coupled disc/SMBH binary evolution

The evolution of an accretion disc in the presence of an embedded satellite can be described by a diffusive evolution model for the disc, including the tidal term arising from the secondary, i.e.

2
where λ=Λ/(ΩR)2, with Λ the specific tidal torque, and
3
(We choose this formulation rather than the standard one with Λ, for later convenience.) In equation (3), is the angular velocity at radius R, a is the radial position of the satellite, q=Ms/Mp is the mass ratio between the secondary and the primary black hole and p=Ra. This simplified form of the specific torque is commonly used in literature (see e.g. Armitage et al. 2002; Armitage & Bonnell 2002; Armitage & Natarajan 2002; Lodato & Clarke 2004). We smooth the torque term for Ra, where it would have a singularity (see equation 3). We use the same smoothing prescription as in Syer & Clarke (1995) and Lin & Papaloizou (1986), i.e. for |Ra| < max [H, RH], where H is the disc thickness and RH is the size of the Hill sphere (Roche lobe) of the secondary. Finally, we note that if the mass ratio q is large the formalism above implies a non-vanishing torque at relatively large distances from the secondary. However, the tidal torque is actually provided by the cumulative effect of a series of Lindblad resonances (Goldreich & Tremaine 1980), and should therefore vanish beyond the outer and the inner Lindblad resonance, located at p= 2±2/3a. We therefore smoothly truncate the torque beyond these radii.

The back reaction of the disc on the secondary orbital motion follows from angular momentum conservation, in the form

4
where the integral is taken over the whole disc surface and Ωs is the angular velocity of the secondary. It follows that the behaviour of the system described here is determined by two dimensionless parameters. First,
5
measures the relative strength of the second and first terms on the RHS of equation (2), while
6
gives a measure of the magnitude of the RHS of equation (4).

The parameter A has the simple physical interpretation (Lin & Papaloizou 1979) A= (Δ/R)3, where Δ is the gap width. To open a gap, the gravitational effect of the satellite has to overcome the pressure of the disc and the viscosity, which both oppose gap opening. We neglect the effects of pressure in this paper.

The second important parameter, B, represents the ratio of local disc mass to satellite mass. If the satellite is more massive than the disc (i.e. B≪ 1) its inertia makes migration slow. Conversely, if B≫ 1, the secondary behaves like a fluid element of the disc and migrates on the local viscous time-scale.

### 3.2 Gravitational wave torques

We also include in our calculation the torques arising from gravitational wave radiation. For this purpose, we add in equation (4) the gravitational wave term causing the binary separation to evolve at a rate

7
The corresponding time-scale for a coalescence induced purely by gravitational wave emission is thus
8

### 3.3 Energy dissipation from tides

Most previous studies of satellite–disc interaction using a formalism like that above do not discuss in detail the extra energy dissipation introduced by the tidal term. As a consequence, the disc spectrum in the presence of a massive satellite is taken to be that of a standard accretion disc truncated at a radius of the order of the binary separation (e.g. see Milosavljević & Phinney 2005).

In reality, this point is subtle because the tidal interaction is not formally a dissipative term. However, the formalism of equations (2–4) enforces angular momentum conservation, and simultaneously assumes that the disc and secondary orbits remain circular. This implicitly assumes the presence of some dissipative term to damp out any eccentric motion. To compute this extra dissipation, we ignore the viscous term in equation (2) and follow the evolution of the system as driven purely by the tidal term. The total energy of the disc (kinetic plus potential) is

9
and its time derivative is
10
where in the last step we have integrated by parts. We now compute the change in the energy Es=−GMpMs/2a of the secondary. Using equation (4), we have
11
We note that the energy changes of the disc and the satellite differ by a factor Ω−Ωs. Therefore, the total energy loss due to the effect of tidal torques in equations (2–4) is given by
12
This energy loss is positive definite, as it should be, because both Ωs−Ω and λ change sign across the binary orbit. The local dissipation term is thus
13
This term is only significant in a narrow range of radii at the disc edges, which are hotter than a simple truncated disc. We include this additional dissipation term in calculating the disc spectrum.

This analysis assumes that the binary and disc orbits remain circular, and so would predict a bright circular disc rim. In reality, the binary is likely to become somewhat eccentric. We discuss this point further in the concluding section of this paper.

### 3.4 Viscosity

The final piece of physics needed in order to solve equations (2–4) is a prescription for the disc viscosity. We use the standard α-prescription (Shakura & Sunyaev 1973) for a gas-pressure-dominated disc

14
where cs=kTcmH is the sound speed at the disc mid-plane, Tc is the mid-plane temperature, μ= 0.67 is the mean molecular weight and mH is the mass of the proton. The mid-plane temperature follows from solving the vertical radiative flux equation for an optically thick disc, assuming the opacity κ to be either electron scattering or Kramers opacity, whichever the larger, i.e.
15
where σB is Stefan–Boltzmann constant and Teff=Tc1/4 is the effective temperature. The optical depth τ is given by τ=κΣ and the disc thickness H is calculated from the vertical hydrostatic balance including both gas and radiation pressure. This setup, and in particular the requirement that the disc viscosity is proportional to the gas pressure (equation 14), ensures that the disc is thermally stable. We take α= 0.1 throughout (King, Pringle & Livio 2007).

### 3.5 Gravitational instability and star formation

The outer parts of AGN discs can be gravitationally unstable beyond ∼0.01–0.1 pc (Kolykhalov & Sunyaev 1979; Lodato & Bertin 2003; Goodman 2003; Lodato 2007; King & Pringle 2007). The gravitational stability parameter Q (Toomre 1964) is defined as

16
where κΩ is the epicyclic frequency. If Q≲ 1, the disc is unstable. Further evolution of the disc is controlled by the ratio of the cooling time-scale, tcool, to the dynamical time-scale β=tcoolΩ. For β≲ 1, the disc fragments into gravitationally bound objects (Gammie 2001; Rice, Lodato & Armitage 2005). For longer cooling times, β≳ 3, the instability saturates at a finite amplitude and the disc displays a long-lived spiral structure (Lodato & Rice 2004, 2005). In protostellar accretion discs, the parameter β is generally much larger than unity (Rafikov 2005). However, in AGN discs the opposite occurs (i.e. β≪ 1), and these discs are expected to fragment rather easily (Goodman 2003; Nayakshin 2006; Levin 2007; Nayakshin, Cuadra & Springel 2007).

The evolution of a fragmenting β≪ 1 disc is uncertain. The prescription of a constant cooling time throughout the collapse of a fragmenting region is an oversimplification. If the collapse of self-gravitating fragments is not prevented, most of the mass quickly (almost dynamically) ends up in stars (e.g. Nayakshin et al. 2007) and this might inhibit accretion altogether. Conversely, the energy input arising from the star formation process (either via protostellar accretion or from nuclear burning in the newly formed stars) could keep the disc close to marginal stability (Goodman 2003; Thompson, Quataert & Murray 2005; Nayakshin 2006).

In this paper, we explore this latter possibility. In particular, at each time-step we compute the gravitational stability parameter Q (equation 16) for every radial bin in the disc. If Q < 1, star formation is allowed. We define the local star formation rate per unit area, . The appropriate value of this rate is found by requiring that the corresponding energy liberation rate is high enough to keep the disc in the state of marginal stability, assumed to be Q= 1. The internal heating of the disc is then dominated by the feedback from star formation, and hence the energy equation becomes

17
The free parameter of the model, ε, is the efficiency of mass-to-energy conversion in star formation. The star formation rate per unit surface is added as a sink term on the RHS of equation (2).

A reasonable estimate of ε is obtained by calculating the total amount of energy liberated by nuclear reactions and supernovae assuming a Salpeter initial mass function (IMF) from 0.1 to 100 M. This yields ε∼ 10−3 (Thompson et al. 2005). However, ε could in principle be lower or higher than this value. A higher value ε∼ 0.01 could be reached if the IMF of stars is top-heavy, as found for the star formation event in the central 0.5 pc of the Milky Way (Nayakshin & Sunyaev 2005; Paumard et al. 2006). A lower value of ε might be appropriate if the stellar disc thickens quickly and adopts a geometrically thicker configuration than the gaseous disc (Nayakshin et al. 2007). Most of feedback energy would then be escaping from the disc rather than being deposited in the mid-plane.

Another important effect is the increased pressure in the disc (Thompson et al. 2005) from winds and supernovae. However, we limit our attention here to the energy effects of star formation only, and leave pressure effects to a future paper.

## 4 TIME-DEPENDENT CALCULATIONS

Here, we describe the results of our simulations. We have run two sets, first without star formation (Section 4.2), and then taking feedback from star formation into account (Section 4.3).

### 4.1 Initial conditions and disc setup

We solve the set of equations (2–4) using standard techniques (see e.g. Lodato & Clarke 2004). We use a logarithmically spaced radial grid extending from the innermost stable orbit at 6Rg, where Rg=GMp/c3 is the primary gravitational radius, to an outer radius of 20a0, where a0 is the initial binary separation. We typically use 100 mesh points. We use a zero-torque boundary condition at both our innermost grid point (i.e. close to the primary black hole) and at our outermost one, at 20a0. The outer boundary condition does not really play a role since the disc never extends that far out.

At each time during the simulation, we compute the effective temperature Teff(R) at the disc surface, including both the standard viscous dissipation term and the additional tidally induced term described above. From Teff, we calculate the disc SED and the total luminosity by integrating a series of blackbody spectra with the appropriate radial-dependent temperature over the disc surface.

All our calculations start with similar initial conditions. We take the primary black hole to have a mass of Mp= 108 M. We have run a number of simulations with different secondary mass, total disc mass Md and different initial binary separation a0. The disc is initialized with a constant surface density over a narrow range of radii between 0.8a0 and 2a0, where the density normalization is chosen to give the required total disc mass. This is intended to give a simple realization of the concept that both black hole and gas arrive at small radii as part of the same merger process. Although clearly an idealization, we regard such initial configuration more plausible than the standard assumption that the black hole is somehow parachuted into an accretion disc that has already had time to relax into a steadily accreting state. We have considered the two initial separations a0= 0.01 and 0.05 pc. During a gas-rich merger of two galaxies each hosting a black hole, the two holes can typically reach a distance of ≲0.1 pc by dynamical friction against the stellar and gaseous background (Escala et al. 2005; Dotti et al. 2006). Thus, our initial conditions are rather optimistic as to how far in a black hole can progress. Even so we find problems with merging on reasonable time-scales even from such small initial separations. This must imply stringent limits on the possibility of gas-driven coalescences.

### 4.2 Discs without star formation

In the first set of simulations, we neglect the effects of star formation and allow the disc to become unstable (Q < 1) without taking any further action. Although obviously unrealistic, these simulations are useful for comparing with those where star formation is included, and also to illustrate the state of the disc at decoupling, i.e. at the point where the binary coalescence starts to be driven by gravitational wave emission rather than by disc torques, as a function of the main disc parameters. By insisting that no gas is removed by star formation, we overestimate the degree to which the gas can facilitate a black hole merger.

Our results should be compared with those of Milosavljević & Phinney (2005), who consider the same general setup but do not follow the time-dependent evolution of the system. Instead they simply truncate the disc at the outer radius where it becomes gravitationally unstable. We therefore briefly summarize their main findings. They argue that, contrary to what happens when the mass ratio q≪ 1 (Armitage & Natarajan 2002), inner disc evolution for non-extreme mass ratios is much faster than the binary orbital evolution, so that after a short time the only gas present in the system is a circumbinary disc. We are interested in the state of the disc at decoupling, i.e. at the moment where binary evolution driven by gravitational wave losses becomes faster than the disc viscous evolution and the disc inner edge cannot follow the secondary black hole as it spirals in. Milosavljević & Phinney (2005) assume the circumbinary disc at decoupling to be well described by a constant- steady-state solution, truncated at the inner edge. Using the requirement that the disc is thin at the inner edge, they argue that , where is the Eddington accretion rate evaluated at the innermost stable orbit, LEdd is the Eddington luminosity and η is the accretion efficiency. The resulting SED is therefore approximately given by a standard multicolour blackbody spectrum, but lacking the high-energy component and therefore peaking at optical–IR wavelengths. After decoupling, the binary torques are removed and the disc is free to flow to the bottom of the potential well, producing in a bright flare, and thus recovering the high-energy part of the standard disc spectrum. As we will see, this is not the complete picture.

#### 4.2.1 Initial binary separation equal to 0.01 pc

We consider first the case where the initial binary separation is a0= 0.01 pc, although the general behaviour of the system is similar for all the simulations. At this distance from a 108 M primary, the time-scale for a merger induced purely by gravitational radiation is tgw≈ 2 × 107q−1 yr.

The luminosity as a function of time for all our simulations is shown in Fig. 1, where the left-hand panel refers to q= 0.3, the middle panel refers to q= 0.1 and the right-hand panel refers to q= 0.01. The various lines indicate different values for the total disc mass Md (see the caption for details). We consider a range of cases varying from a disc mass comparable to the secondary mass Md/Ms= 1 to one where it is much smaller Md/Ms= 0.01. When the disc mass is much larger than the secondary mass, the secondary black hole is simply swept into the central black hole by the accretion flow.

Figure 1

Luminosity evolution of circumbinary discs for an initial separation of a0= 0.01 pc. The secondary/primary mass ratio is Ms/Mp=q= 0.3 (left-hand panel), 0.1 (middle panel) and 0.01 (right-hand panel). For each panel, the various lines refer to different disc masses: Md/Ms= 1 (solid lines), 0.5 (dotted line), 0.1 (short-dashed line) and 0.01 (long-dashed line).

Figure 1

Luminosity evolution of circumbinary discs for an initial separation of a0= 0.01 pc. The secondary/primary mass ratio is Ms/Mp=q= 0.3 (left-hand panel), 0.1 (middle panel) and 0.01 (right-hand panel). For each panel, the various lines refer to different disc masses: Md/Ms= 1 (solid lines), 0.5 (dotted line), 0.1 (short-dashed line) and 0.01 (long-dashed line).

We start with roughly half of the gas inside the orbit of the secondary black hole and half outside. Initially, the matter inside the binary orbit accretes on to the primary black hole, resulting in a relatively bright AGN, where the luminosity obviously scales with the total disc mass. For the choice of parameters made here, the peak luminosity is at most of the order of the Eddington luminosity for our most massive disc. The inner disc evolves faster than the binary orbit and the surface density, and the disc luminosity drops rapidly. However, a low-density disc is still present at decoupling in all of our simulations. Clearly, the amount of mass left over in the inner disc at decoupling is strongly model dependent. However, the presence of a small, but non-negligible, mass in the inner disc might have important observational consequences. As explained above, even if such a low-density disc does not provide a significant contribution to the SED, it can be squeezed when angular momentum loss by gravitational radiation becomes dominant, producing a sudden burst of luminosity. After this, the outer disc can finally flow to the innermost region and provide a longer lasting AGN-like appearance, even though the post-merger luminosity is generally sub-Eddington.

Fig. 2 shows the evolution of the binary separation with time. The initial evolution is rather slow, being driven mostly by the disc torques. It suddenly accelerates when the gravitational radiation induced evolution takes over and the outer disc decouples from the binary evolution. We see from Fig. 2 that the radius at which the disc decouples increases with decreasing disc mass. A similar result was also found by Milosavljević & Phinney (2005), who parametrized the disc mass terms of the mass flux at the disc inner edge (see below). Note that for disc masses smaller than roughly 0.1 of the secondary mass, the merger time becomes comparable with the gravitational radiation time-scale, indicating that the disc gives only a limited contribution towards accelerating the merger.

Figure 2

Evolution of the binary separation a for the initial value of a0= 0.01 pc. The three plots have the same parameters as the corresponding plots in Fig. 1. Similarly, the line styles are the same as in Fig. 1.

Figure 2

Evolution of the binary separation a for the initial value of a0= 0.01 pc. The three plots have the same parameters as the corresponding plots in Fig. 1. Similarly, the line styles are the same as in Fig. 1.

Fig. 3 shows a series of snapshots of the surface density in the outer disc for the case where q= 0.1 and Md/Ms= 1 (which corresponds to the solid line in the middle panels of Figs 1 and 2). The lines refer to t= 0.5 Myr (solid line), 0.75 Myr (short-dashed line), 1 Myr (long-dashed line), 1.5 Myr (dot–dashed line) and, 1.6 Myr (dotted line). This clearly shows that in the situation considered in this paper the surface density at the inner edge does not grow with time (unlike the Ivanov et al. 1999 solution). The eventual growth of Σ at the inner edge only occurs at late times, when the binary orbit accelerates due to gravitational wave emission (cf. Fig. 2) and the tidal torque is effectively removed.

Figure 3

Surface density of the outer disc for the case q= 1, Md/Ms= 1. The lines refer to t= 0.5 Myr (solid line), 0.75 Myr (short-dashed line), 1Myr (long-dashed line), 1.5 Myr (dot–dashed line) and 1.6 Myr (dotted line). The last two cases refer to epochs when the binary orbit accelerates because of gravitational wave emission (cf. Fig. 2), and the torque on the inner disc is removed.

Figure 3

Surface density of the outer disc for the case q= 1, Md/Ms= 1. The lines refer to t= 0.5 Myr (solid line), 0.75 Myr (short-dashed line), 1Myr (long-dashed line), 1.5 Myr (dot–dashed line) and 1.6 Myr (dotted line). The last two cases refer to epochs when the binary orbit accelerates because of gravitational wave emission (cf. Fig. 2), and the torque on the inner disc is removed.

Fig. 4 shows a number of properties of the disc at decoupling, which is the moment at which the binary separation evolution becomes faster than the disc viscous time close to the binary orbit. The top panels show the disc surface density profile (left-hand panel) and the profile of the local mass flux in the disc in units of the Eddington rate at the innermost stable orbit , where we assume mass-to-energy conversion efficiency of η= 0.1 (we use this scaling for ease of comparison with the results of Milosavljević & Phinney 2005.). The bottom plots show the aspect ratio H/R (left-hand panel) and the effective temperature profile (right-hand panel). These plots refer to q= 0.1 and the three lines correspond to Md/Ms= 1 (solid line), 0.5 (dotted line) and 0.1 (dashed line).

Figure 4

Properties of the disc at decoupling, i.e. when the binary evolution via gravitational wave radiation becomes faster than viscous evolution at the secondary's orbit. The initial separation was a0= 0.01 pc. The secondary/primary mass ratio is q= 0.1. The different lines refer to different values of the initial disc mass, with the same notation as Fig. 1. Shown here are the disc surface density profile in cgs units (upper-left panel), the mass flux through the disc in units of the Eddington rate at the innermost stable orbit (upper-right panel), the aspect ratio H/R (lower-left panel) and the effective temperature in Kelvin (lower-right panel).

Figure 4

Properties of the disc at decoupling, i.e. when the binary evolution via gravitational wave radiation becomes faster than viscous evolution at the secondary's orbit. The initial separation was a0= 0.01 pc. The secondary/primary mass ratio is q= 0.1. The different lines refer to different values of the initial disc mass, with the same notation as Fig. 1. Shown here are the disc surface density profile in cgs units (upper-left panel), the mass flux through the disc in units of the Eddington rate at the innermost stable orbit (upper-right panel), the aspect ratio H/R (lower-left panel) and the effective temperature in Kelvin (lower-right panel).

There are several interesting features in these plots. We first discuss the radial profiles of . The main thing to note is that the outer circumbinary disc does not have a constant , as a standard steady accretion disc model would require. The tidal torque at the inner edge of the circumbinary disc changes its overall structure, such that if a steady state is reached the mass flux scales as . This kind of profile is typical of viscous disc solutions in the presence of a torque at the inner edge (Pringle 1991; Syer & Clarke 1995; Ivanov et al. 1999), and as we will discuss below it affects the shape of the SED of the disc. We stress that the choice of scaling to the inner Eddington rate is motivated by convenience, and the mass flux can significantly exceed the reference value (the Eddington value for the innermost stable orbit) without the luminosity becoming super-Eddington, since the disc is truncated at a radius much larger than the innermost stable orbit. Milosavljević & Phinney (2005) still argue that the limit is required in order to keep the disc thin at its inner edge, at least for relatively large q. Although our most massive disc does have at its inner edge, we find that the disc aspect ratio H/R is ≪1 (see bottom-left panel), therefore in principle allowing a much larger inner mass flux without violating the thin disc requirement. The straight (red) line in the top-right panel of Fig. 4 shows the scaling between the inner edge of the circumbinary disc at decoupling and the mass flux predicted by Milosavljević & Phinney (2005), where the normalization has been chosen to match our results. Although our simulations do reproduce this scaling quite well, our results imply a binary separation at decoupling larger by a factor 2 than Milosavljević & Phinney (2005). At decoupling, we have a= 335, 520, 1400Rg, for the three cases shown in Fig. 4, i.e. Md/Ms= 1, 0.5 and 0.1, respectively.

The top-left panel in Fig. 4 shows the surface density profile at decoupling. Here, we note that although the inner circumprimary disc has been significantly depleted in mass and its surface density is thus low, it has still not disappeared. As emphasized above, even a small amount of mass in the circumprimary disc produces a bright burst as it is squeezed by the final evolution of the binary orbit, producing the sudden peak in the light curves plotted in Fig. 1. The bottom-right panel in Fig. 4 shows the effective temperature profile at decoupling. Here, the most significant feature is the presence of two bright spots at the edges of the gap around the secondary. This results from the extra dissipation term discussed in Section 3.3, and provides a significant, and previously neglected, contribution to the overall SED.

The resulting SED for the three cases of Fig. 4 is shown in the top panel of Fig. 5. These are calculated as multicolour blackbody spectra for the appropriate effective temperature profiles. The general shape is similar to that predicted by Milosavljević & Phinney (2005), although there are significant differences. The disc is very bright at optical/IR wavelengths and lacks the high-energy emission from the inner disc. The inner disc emission is generally negligible because of the low density of this disc, although some high-energy emission from the inner disc can be seen for the lowest disc mass (dashed line). The bottom plot in Fig. 5 shows the contribution to the total SED of the various disc components, for Md/Ms= 1. The short-dashed line shows the contribution of the outer circumbinary disc only, without including the extra dissipation term at the gap edges. The long-dashed line shows the spectrum arising from the whole disc, but neglecting again the dissipation at the gap edges. This emphasizes the contribution of the inner disc. The solid line shows the total SED, including the emission from the gap edges, which in turn results in a significant contribution at relatively higher energies. The red line, for comparison, shows the expected SED of a truncated disc with constant . This is the shape of the SED as predicted by Milosavljević & Phinney (2005). As one can see, the long-wavelength part of the SED of a constant- disc is shallower than the one we calculate. The steepening of the SED in the presence of a massive satellite had been discussed by Syer & Clarke (1995). Then, we have νFν∝ν12/7, rather than the value of νFν∝ν4/3 found for a steady accretion disc.

Figure 5

Top: SED of the disc (in cgs units) at decoupling for various disc masses: Md/Ms= 1 (solid line); 0.5 (dotted line) and 0.1 (dashed line). The initial separation was a0= 0.01 pc. Bottom: contributions to the overall SED for disc mass Md/Ms= 1 shown in the top plot. The solid line indicates the full spectrum; the short-dashed line shows the contribution of only the outer disc, without including the extra tidal dissipation term and the long-dashed line includes the emission of the inner disc but still does not include the hot gap edges. Comparing the long-dashed and solid lines thus shows the importance of the hot gap emission at high radiation frequencies. For reference, the red line shows the SED of a standard constant- accretion disc: its spectrum is shallower than the one found here, which more closely approximates the structure of a decretion disc.

Figure 5

Top: SED of the disc (in cgs units) at decoupling for various disc masses: Md/Ms= 1 (solid line); 0.5 (dotted line) and 0.1 (dashed line). The initial separation was a0= 0.01 pc. Bottom: contributions to the overall SED for disc mass Md/Ms= 1 shown in the top plot. The solid line indicates the full spectrum; the short-dashed line shows the contribution of only the outer disc, without including the extra tidal dissipation term and the long-dashed line includes the emission of the inner disc but still does not include the hot gap edges. Comparing the long-dashed and solid lines thus shows the importance of the hot gap emission at high radiation frequencies. For reference, the red line shows the SED of a standard constant- accretion disc: its spectrum is shallower than the one found here, which more closely approximates the structure of a decretion disc.

Note that the frequency range of the hot-edge emission is comparable to that of the inner disc. Indeed Fig. 4 shows that the effective temperature of the edge is similar to that of the inner disc. However, the area covered by the hot edges is much larger than the inner disc and provides a much larger luminosity. The total luminosity of the hot edges is roughly equal to half of the bolometric luminosity.

After the black holes coalesce and the tidal torque at the inner edge is removed, the gas will eventually freely flow to the inner disc, providing a luminous high-energy component to the SED, which would gradually flatten to reach a standard constant- solution in the inner disc.

#### 4.2.2 Initial binary separation equal to 0.05 pc

For a larger initial binary separation of a0= 0.05 pc, the gravitational radiation time-scale is far longer and would not produce coalescence in a Hubble time without any further binary hardening mechanism. Here again, we take the primary black hole mass to be 108 M. The mass ratio between the secondary and the primary is q= 0.1 and we consider, as in the previous section, different values for the disc/secondary mass ratio. Fig. 6 shows the evolution of the binary separation for Md/Ms= 1 (solid line), 0.5 (dotted line) and 0.1 (dashed line). Here again, we see the same trend as observed in for the smaller initial separation a0= 0.01 pc. We note that disc torques are able to induce coalescence within a Hubble time only when the disc mass is comparable to the secondary black hole. At the end of our simulations (∼1 Gyr), the binary had merged only for Md=Ms. The final state of the disc at decoupling follows the same trend as before.

Figure 6

Evolution of the binary separation with time when the initial separation is taken to have the larger value of a0= 0.05 pc. The primary mass here is 108 M, the secondary/primary mass ratio is q= 0.1 and the disc/secondary mass ratio is Md/Ms= 1 (solid line), 0.5 (dotted line) and 0.1 (dashed line). Only the more massive discs produce a merger within 1 Gyr.

Figure 6

Evolution of the binary separation with time when the initial separation is taken to have the larger value of a0= 0.05 pc. The primary mass here is 108 M, the secondary/primary mass ratio is q= 0.1 and the disc/secondary mass ratio is Md/Ms= 1 (solid line), 0.5 (dotted line) and 0.1 (dashed line). Only the more massive discs produce a merger within 1 Gyr.

### 4.3 The effects of star formation

From the analysis of the results of the previous section, we may draw the following general conclusion. Even starting from an initial separation of a0= 0.05 pc, a circumbinary disc is able to bring the SMBH binary to coalescence in a reasonable time only if the disc mass is of the order of, or slightly smaller than, the secondary black hole mass. However, in the previous section we did not include the effects of gravitational instability and simply allowed the disc to have a value of Q smaller than one in its outer parts, beyond ≈0.05 pc. Here, we include the effects of gravitational instability by requiring that, whenever the disc is unstable, it is kept in a state of marginal stability by the feedback induced by star formation. We thus remove from the calculation the amount of mass needed to be converted into stars to provide such feedback, as detailed in Section 3.5. The effect of removing gas from the disc is to reduce the efficacy of the disc in merging the black holes.

We first consider the case where the secondary/primary black hole mass ratio is q= 0.1, the initial mass ratio between the disc and the secondary is Md/Ms= 0.5 and the initial binary separation is 0.01 pc. We also set the star formation feedback efficiency to ε= 10−3. In Fig. 7, we show the resulting evolution of the binary separation (solid line) and that in the corresponding calculation without allowing for star formation (dashed line). Star formation significantly reduces the ability of the disc to induce the merger. This is because star formation depletes the gas reservoir in the outer parts of the disc, reducing its surface density, and so significantly reduces the disc density just outside the secondary orbit.

Figure 7

Evolution of the binary separation with time in calculations with (solid curve; see Section 4.3) and without (dashed; see Section 4.2) star formation in the disc. The latter case is the one shown in Figs 2 and 4. The primary mass is 108 M, the secondary/primary mass ratio is q= 0.1, the disc/secondary mass ratio is Md/Ms= 0.5 and the initial separation is a0= 0.01 pc. Note that removal of gas by star formation significantly slows down binary merging.

Figure 7

Evolution of the binary separation with time in calculations with (solid curve; see Section 4.3) and without (dashed; see Section 4.2) star formation in the disc. The latter case is the one shown in Figs 2 and 4. The primary mass is 108 M, the secondary/primary mass ratio is q= 0.1, the disc/secondary mass ratio is Md/Ms= 0.5 and the initial separation is a0= 0.01 pc. Note that removal of gas by star formation significantly slows down binary merging.

Fig. 8 shows the evolution of the total star formation rate within the disc. We can see that over the first 105 years the disc undergoes significant star formation. The total mass converted into stars is of the order of ≈2 × 106 M , that is roughly half of the initial disc mass. Finally, we show in Fig. 9 the main properties of the disc after ≈6 Myr from the beginning of the simulation, after most of the star formation has died out but before the secondary has migrated significantly. The top panel shows the disc surface density, the middle panel shows the aspect ratio H/R and the bottom panel shows the value of the stability parameter Q. We can thus see that our star formation prescription indeed causes the disc to hover near marginal stability in its outer parts.

Figure 8

Total star formation rate for the simulation shown also in Fig. 7. Over the entire course of the simulation the total amount of gas turned into stars is ≈2 × 106 M.

Figure 8

Total star formation rate for the simulation shown also in Fig. 7. Over the entire course of the simulation the total amount of gas turned into stars is ≈2 × 106 M.

Figure 9

Disc properties after 6 Myr. At this time, star formation has considerably depleted the disc mass and little disc evolution has taken place. Because of this, the results are relatively independent of initial disc mass. The initial separation was a0= 0.01 pc, the secondary/primary black hole mass ratio is q= 0.1, the disc mass is Md/Ms= 0.5 and the star formation feedback efficiency is ε= 10−3. Top panel: disc surface density in cgs units. Middle panel: aspect ratio H/R. We see that typically H/R≈ 10−3. Bottom panel: gravitational stability parameter Q.

Figure 9

Disc properties after 6 Myr. At this time, star formation has considerably depleted the disc mass and little disc evolution has taken place. Because of this, the results are relatively independent of initial disc mass. The initial separation was a0= 0.01 pc, the secondary/primary black hole mass ratio is q= 0.1, the disc mass is Md/Ms= 0.5 and the star formation feedback efficiency is ε= 10−3. Top panel: disc surface density in cgs units. Middle panel: aspect ratio H/R. We see that typically H/R≈ 10−3. Bottom panel: gravitational stability parameter Q.

Summarizing our findings for the simple energy feedback star formation model, we note the following. In this model, the evolution of the system does not depend significantly on the initial disc mass (except when it is so low as to be gravitationally stable at all radii, and would not be able to drive the merger). Thus, different initial disc masses only imply differing amounts of star formation rate at early times, and the surface density after a few million years is very similar in all cases.

The effect of changing the efficiency of star formation feedback ε is not trivial. Naively, one would expect to simply have more star formation as one decreases its value (cf. equation 17), thus leading to a smaller disc density Σ. However, the reduced Σ requires a smaller temperature in order to be gravitationally stable, which in turn implies a much smaller cooling rate and a lower star formation rate. We have run different simulations with ε= 10−3 and 10−4 and found only found small differences.

Finally, in Fig. 10 the solid line shows the evolution of the binary orbit in the presence of star formation when the initial separation is 0.05 pc, the secondary/primary mass ratio is q= 0.1 and the initial disc mass is equal to the secondary mass. This should be compared with the equivalent run where star formation is not included (dotted line). Clearly, while in principle there is initially enough gas mass in the disc to bring the holes to coalescence, the effect of star formation depletes the disc severely and the amount of gas left over is not enough to lead to coalescence within a Hubble time.

Figure 10

Same as Fig. 7, but for the initial separation a0= 0.05 pc and the disc/secondary mass ratio is Md/Ms= 1. As before, solid curve shows the case with star formation included, whereas the dotted shows that where star formation ignored. Star formation severely depletes the disc, and the remaining disc mass is not large enough to induce a black hole merger within a Hubble time.

Figure 10

Same as Fig. 7, but for the initial separation a0= 0.05 pc and the disc/secondary mass ratio is Md/Ms= 1. As before, solid curve shows the case with star formation included, whereas the dotted shows that where star formation ignored. Star formation severely depletes the disc, and the remaining disc mass is not large enough to induce a black hole merger within a Hubble time.

## 5 DISCUSSION AND CONCLUSIONS

We consider the merging of two black holes in the centre of a galaxy driven by a gaseous disc. We make the assumption that the secondary black hole (mass Ms) arrives close to the already present primary (mass Mp) as a result of a galactic merger of some kind (cf. King & Pringle 2007), bringing with it a certain amount of gas (mass Md). This contrasts with the usual assumption that the merger process manages to parachute the secondary black hole into an already existing steady-state accretion disc of infinite extent and mass.

If the amount of gas is large (MdMs) and does not mainly turn into stars then the secondary black hole is simply swept into the primary by the resulting accretion flow. Thus, we consider what happens when the amount of gas is comparable to, or smaller than, the secondary mass. We first ignore the possibility of star formation within the accretion disc flow (Section 4.2) in order to demonstrate the time-dependent properties of the interactions between disc and secondary black hole. We then, more realistically (Section 4.3), take account of formation of stars from the disc gas when it becomes self-gravitating, and take account of the feedback from those stars on the gas in the disc.

We have computed the properties and the SED of the disc at and around the time of decoupling (i.e. when gravitational radiation takes over as the dominant merging effect) in cases where the merger occurs. Our findings differ from earlier results (Milosavljević & Phinney 2005) for three main reasons.

• The disc is not in a steady state with uniform , but closer to a decretion disc with . This makes the SED steeper (νFν∝ν1.7, rather than the steady accretion disc value of νFν∝ν1.3).

• The region inside the secondary orbit might not be empty, and the small gas density can still lead to a strong burst of luminosity during the final stages of the merger, as predicted for low mass ratios by Armitage & Natarajan (2002).

• The enhanced dissipation resulting from the interaction between the disc and the black hole heats the disc edges either side of the secondary significantly. This is a new and general result, implying an extra high-photon-energy contribution to the SED.

The existence of this harder emission weakens the prediction of Milosavljević & Phinney (2005) that the presence of a binary can be easily inferred by a shift of the peak emission to low photon energies. More positively, it offers a new way of inferring the presence of a binary through a combination of this result with possible detectable periodic variability (Haiman, Kocsis & Menou 2009). Although in this paper we have assumed that the secondary and disc orbits are exactly circular, in reality the system may be driven to somewhat eccentric orbits. For disc masses large enough to shrink the binary significantly (as assumed here), dissipation in the disc is likely to keep the eccentricity low, and so the general picture remains similar to that we have described. However, the dissipation at the inner edge of the disc will not be uniform in azimuth or in time. This implies variability at photon energies which one would normally associate with the innermost disc region around a single black hole. However, the typical time-scale of this disc-gap emission is not that of the innermost orbits around the primary black hole, but instead the much longer orbital time-scale of the secondary. Thus, the detection at high photon energies of variability on the long time-scales typical of the outer disc would be a clear signature of the presence of a black hole binary.

Evidently detailed modelling of such effects requires the use of two-dimensional (or even three-dimensional) disc models, as it depends on the effectiveness of disc waves generated by tides to carry away the excess tidal energy.

Circumbinary discs are gravitationally unstable at R≈ 0.1 pc, just as their cousins around single SMBH are. It is not yet clear if star formation ‘catastrophe’ can be avoided due to the action of thermal or momentum feedback from massive stars in the disc (e.g. Goodman 2003; Thompson et al. 2005). In the case of the thermal feedback, as shown here, gas is converted into stars too quickly compared to the binary evolution time-scales. This has a severe impact on the ability of the disc to induce a black hole merger in a reasonable time. We find that the rapid effect of star formation sets an upper limit to the effective mass of a gaseous disc. Because of this, mergers within a Gigayear require initial separations closer than a0= 0.01 pc rather than 0.05 pc, implying that gas-driven mergers of this sort are unable to solve the ‘final parsec’ problem. To avoid the disc becoming self-gravitating, the disc must have Q≳ 1 and so must have a mass at most Md/MpH/R. We also require that the disc mass be at least comparable to the secondary mass to produce a merger. Our modelling of star formation leads to disc aspect ratios H/R of the order of a few times 10−3, and so we conclude that disc-assisted mergers only work for mass ratios q≲ 0.001 if only thermal feedback is included. Whether such low-mass secondaries would be able to produce a tight pair in the first place is a matter of debate (Callegari et al. 2009).

Our treatment of star formation is very simplistic by necessity. The inclusion of other effects neglected here, such as momentum feedback and dynamical support from the newly born stars embedded in the disc, might stabilize the disc at higher surface densities. Additionally, the process of fragmentation might result in the formation of clouds and clumps in the disc, rather than stars, whose dynamics are still unclear. Both effects would results in a relatively more massive gaseous disc at ∼0.1 pc. Finally, dynamical friction from the newly born stars (not included here) might help in bringing the holes together. While it is hard to make a precise estimate, we believe our calculations with and without star formation (see Figs 7 and 10) bracket the range of possible time-scales for disc-induced binary mergers.

Nevertheless, it seems quite likely that the ‘final parsec’ problem is difficult to overcome in all cases, particularly for non-extreme mass ratios. This would imply the relative ubiquity of secondary black holes in galactic nuclei.

We thank the referee for a careful and constructive review of the paper.

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