THE FINAL-PARSEC PROBLEM IN THE COLLISIONLESS LIMIT

We consider the evolution of a binary SBH composed of two point masses, m₁ and m₂, which are on a Keplerian orbit with semimajor axis a and eccentricity e. The total mass of the binary, ${M}_{\mathrm{bin}}\equiv {m}_{1}+{m}_{2}$, is a small fraction (10⁻²–10⁻³) of the total mass of surrounding stellar distribution. The mass ratio of the binary is q ≡ m₂/m₁ ≤ 1.

A star passing at a distance ≲a from the binary undergoes a complex scattering interaction and is ejected with a typical velocity $\sim \sqrt{{{GM}}_{\mathrm{bin}}/a}$ (the characteristic orbital velocity of the binary); for a hard binary this is higher than the average velocity of the stellar population, thus the star carries away energy and angular momentum from the binary. The precise definition of a hard binary varies among different studies; here we adopt that a binary is hard if its semimajor axis is smaller than

$$\begin{eqnarray}&&{a}_{{\rm{h}}}\equiv \displaystyle \frac{q}{4{(1+q)}^{2}}\;{r}_{\mathrm{infl}},\end{eqnarray} \tag{ 1 }$$

where, in turn, the radius of influence r_infl is defined as the radius enclosing the mass in stars equal to twice the total mass of the binary. This definition depends on the evolutionary phase, as the slingshot process reduces the density of stars in the galactic nucleus. The most rapid depletion occurs just after the binary formation, and after the binary becomes hard the depletion slows down considerably. For consistency with merger simulations, in which it is not possible to assign any particular value to r_infl before the two galactic nuclei have merged, we adopted the convention to evaluate the influence radius after the hard binary has formed (see Merritt & Szell 2006 for a discussion on different definitions of r_infl).

Another important quantity is the hardening rate S ≡ d(1/a)/dt. In what follows, we will frequently compare S with the reference value S_full that would occur if the distribution of stars in phase space were not affected by the presence of the binary—in other words, if the loss cone was "full." Unfortunately this value also does not have a commonly accepted definition. For instance, if the stars were uniformly distributed in space, with density ρ and with isotropically directed velocities all of the same magnitude v, then the hardening rate can be expressed as

$$\begin{eqnarray}&&{S}_{\mathrm{uniform}}={H}_{1}\displaystyle \frac{G\rho }{v},\end{eqnarray} \tag{ 2 }$$

where the dimensionless coefficient H₁ can be measured from scattering experiments (e.g., Quinlan 1996; Sesana et al. 2006), and has a value H₁ ≈ 18 in the hard-binary limit. A more widely used definition applies to systems with a uniform density and a Maxwellian velocity distribution with dispersion σ:

$$\begin{eqnarray}f(x,v) & \equiv & \rho \;\tilde{f}(v)=\displaystyle \frac{\rho }{{(2\pi \;{\sigma }^{2})}^{3/2}}\;\mathrm{exp}(-\displaystyle \frac{{v}^{2}}{2{\sigma }^{2}}),\\ {S}_{\mathrm{Maxwell}} & \equiv & {\displaystyle \int }_{0}^{\infty }{dv}\;4\pi {v}^{2}\;\tilde{f}(v)\;{S}_{\mathrm{uniform}}(v)=H\displaystyle \frac{G\rho }{\sigma },\end{eqnarray} \tag{ 3 }$$

where $H\equiv {H}_{1}\sqrt{2/\pi }\;\approx \;14.5$.

The assumption of a uniform-density background is clearly an oversimplification, and a more robustly defined quantity is obtained by averaging the single-velocity hardening rate (2) over the actual distribution function of stars f(x, v) in the entire galaxy. If we assume that the latter is isotropic (i.e., depends only on the energy E, but not on the angular momentum), then

$$\begin{eqnarray}&&{S}_{\mathrm{iso}}\equiv {\displaystyle \int }_{0}^{\infty }{dv}\;4\pi {v}^{2}\;f(E)\;\displaystyle \frac{{H}_{1}\;G}{v}=4\pi \;{H}_{1}\;G{\displaystyle \int }_{{{\rm{\Phi }}}_{0}}^{0}{dE}\;f(E).\end{eqnarray} \tag{ 4 }$$

Here Φ₀ is the depth of potential well of the stellar cusp (excluding the potential of the SBH), so that the integration includes energies corresponding to stars that are unbound to the binary, but still bound to the entire galaxy (this is a rather ad hoc convention, but it is justified by the fact that few stars remain bound to the binary for any significant time after it becomes hard). Another derivation of this quantity from a somewhat different perspective can be found e.g., in Paper I, Equations (7)–(9) or in Section 8.3 of Merritt (2013). Note that the above value has no relation to the total mass of stars in the galaxy, which is given by $\int {dE}\;f(E)\;g(E)$, where g(E) is the density of states and rapidly rises with E.

Yet another way to define a reference hardening rate is to substitute the density and velocity dispersion evaluated at r_infl into Equation (3) (Sesana & Khan 2015). This definition has the advantage of being easily computed from quantities that are accessible to an N-body snapshot, without the need to calculate the distribution function; on the other hand, it reflects only the local parameters near the galaxy center and not the entire population of stars that can interact with the binary. Since the velocity dispersion can itself be expressed as ${\sigma }^{2}\sim {{GM}}_{\mathrm{bin}}/{r}_{\mathrm{infl}}$, we may write

$$\begin{eqnarray}&&{S}_{\mathrm{infl}}\equiv {\mathcal{A}}\;{({{GM}}_{\mathrm{bin}})}^{1/2}\;{r}_{\mathrm{infl}}^{-5/2}.\end{eqnarray} \tag{ 5 }$$

Here the dimensionless factor A can be chosen to match the previous definition (Equation (4)); its value depends weakly on galaxy structure and lies in the range 3–5.

In what follows, we will refer to either S_iso or S_infl as the "full-loss-cone hardening rate." We stress that there is no fundamental reason to expect that the actual hardening rate will be close to S_full, since the distribution of stars at low angular momenta is not isotropic, and even the value of S_full need not remain constant in time—it merely serves as a useful reference value.

The loss of energy and the evolution of eccentricity due to the emission of GWs are given by the following expressions (Peters 1964):

$$\begin{eqnarray}&&\displaystyle \frac{d(1/a)}{{dt}}=\displaystyle \frac{1/a}{{T}_{\mathrm{GW}}},\quad {T}_{\mathrm{GW}}\equiv \displaystyle \frac{5}{64}\displaystyle \frac{{c}^{5}{a}^{4}}{{G}^{3}{M}^{3}}\displaystyle \frac{{(1+q)}^{2}}{q}f(e),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&f(e)\equiv \displaystyle \frac{{(1-{e}^{2})}^{7/2}}{1+\displaystyle \frac{73}{24}{e}^{2}+\displaystyle \frac{37}{96}{e}^{4}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{de}}^{2}}{{dt}}=-\displaystyle \frac{{G}^{3}{M}^{3}}{{c}^{5}{a}^{4}}\displaystyle \frac{q}{{(1+q)}^{2}}\displaystyle \frac{2{e}^{2}(304+121{e}^{2})}{15{(1-{e}^{2})}^{5/2}}.\end{eqnarray} \tag{ 8 }$$

Throughout the paper, we will mostly present the results in dimensionless N-body units, with the mass and the scale radius of the galaxy model both set to unity. The models may be scaled to a given galaxy using any two out of three fundamental scales (length, time and mass). To simplify the discussion, we reduce this freedom to a one-parameter family of galaxies in which the length and mass scales are related through the ${M}_{\bullet }-\sigma$ relation in the following form (e.g., Merritt et al. 2009, Figure 12):

$$\begin{eqnarray}&&{r}_{\mathrm{infl}}={r}_{0}\times {({M}_{\bullet }/{10}^{8}\;{M}_{\odot })}^{\kappa },\quad {r}_{0}=30{\rm{}}\;{\rm{pc}},\ \kappa =0.56.\end{eqnarray} \tag{ 9 }$$

The Monte Carlo method used in this work is an extension of the Raga code (Vasiliev 2015). We follow the motion of N particles in the combined potential of the stellar distribution, ${{\rm{\Phi }}}_{\star }({\boldsymbol{r}})$, and the two point masses orbiting each other, centered at origin. The orbit of the binary is assumed to be Keplerian, aligned in the $x-y$ plane, and elongated in x direction; we make no attempt to follow either the change of the orbital plane, which we know to be small from the N-body simulations (although it could be quite significant over long timescale in triaxial galaxies, see Cui & Yu 2014), or the precession of its periapsis, which we assume is not particularly important for dynamics. The evolution is broken down into many small intervals of time (episodes) of duration T_epi ≫ T_bin; during each episode we keep the stellar potential and the parameters of the binary orbit unchanged.

Each particle is moving independently from the others under the influence of three forces: the gradient of the smooth static stellar potential, represented as a spherical-harmonic expansion with spline-interpolated coefficients as functions of radius; the time-dependent force from the binary; and random velocity kicks that model the effect of two-body encounters for a system composed of a certain number of stars, N_⋆, which does not need to be related to the number of particles in the simulation, and can even be set to infinity (thus switching off the two-body relaxation). The velocity perturbations are computed from the local (position-dependent) drift and diffusion coefficients, using the standard formalism from the relaxation theory (e.g., Merritt 2013, Chapter 5) and an isotropic approximation for the distribution function of stars (i.e., with dependence on energy only). Thus our method descends from the Spitzer's formulation of Monte Carlo approach (Spitzer & Hart 1971) and does not rely on orbit-averaging, as other contemporary Monte Carlo codes that use the formalism of Hénon (1971).

During the encounter of a particle with the binary, defined as the time when the distance of the particle from origin is less than 10a, we record the changes in energy and angular momentum of the particle that arise due to the motion in time-dependent potential of the binary (not including the perturbations from relaxation or the torque from the non-spherical stellar potential). After each particle's trajectory has been calculated over the entire episode, we sum up these changes for all particles that experienced one or more encounters with the binary, and change the binary's energy and angular momentum by the same amount with the opposite sign, while keeping its orbital phase unchanged from the end of the episode to the start of the next one. Thus the stars do not exert force on the black holes directly, but we use conservation laws to impose the changes to the binary orbit parameters. The binary orbit parameters are optionally modified after each episode according to the expressions for gravitational-wave emission (6), (8). The stellar potential and the diffusion coefficients that account for two-body relaxation are also updated at the end of each episode, reflecting the changes in the stellar distribution in the course of evolution (here the most important effect is the gradual erosion of the central stellar cusp).

It is instructive to compare our method with the other schemes used for an approximate treatment of the joint evolution of the binary and the stellar distribution. Quinlan & Hernquist (1997) developed a program scfbdy which combines elements from the self-consistent field (SCF) method (Hernquist & Ostriker 1992) with a direct integration of black hole–star interactions. The two black holes were integrated using nbody2 (Aarseth 1999)—a direct-summation code with neighbor scheme, adaptive timestep, and optional two-body regularization, taking into account the forces from all stars individually. The motion of star particles, on the other hand, was computed in the gravitational potential of the two black holes and the smooth potential of the other stars, represented by a basis-set expansion (Hernquist & Ostriker 1992). Their approach is quite similar to ours, with the following differences: (a) the binary's center of mass is not fixed at origin, although the center of stellar potential expansion is; (b) the star particles exert "real" forces on the black holes, which lead directly to the changes in the binary orbit, instead of relying on the Newton's third law as in our method; (c) two-body relaxation is not modeled. They used a rather small update interval for the stellar potential, which could lead to artificial relaxation due to fluctuations in the expansion coefficients (Hernquist & Barnes 1990; Weinberg 1996; Sellwood 2015); in our code we recompute the potential less frequently and use N_samp ≫1 sampling points from each particle's trajectory stored during each episode, to further reduce discreteness noise. Nevertheless, any finite-N system is not entirely free of relaxation (e.g., Weinberg 1998), so our simulations without explicitly added two-body relaxation should be regarded as an upper limit for the evolution rate expected in truly collisionless systems. Finally, Quinlan & Hernquist (1997) only dealt with nearly spherical systems, although in principle their scheme can equally well work for non-spherical systems after switching on the corresponding terms in the spherical-harmonic expansion (as in our method).

Hemsendorf et al. (2002) used a similar approach in their code EuroStar: the two black holes and a subset of star particles with angular momenta lower than a certain threshold are integrated with a direct-summation method, adapted from Aarseth's nbody6 code, while the rest of the star particles are followed in the smooth field by a modified version of the SCF method.

Sesana et al. (2006, 2008) used a large suite of three-body scattering experiments to derive the expressions for the rate of hardening and eccentricity growth, extending the results of Quinlan (1996) to a wider range of mass ratios and eccentricities. Sesana et al. (2007) and Sesana (2010) developed a hybrid approach for the evolution of the binary, combining the analytical fits to these scattering experiments with a time-dependent model for the loss cone draining and repopulation. They did not simulate the effect of two-body relaxation and Brownian motion of the binary center of mass explicitly, but included some prescriptions to take it into account. The changes in the stellar potential due to ejection of stars were ignored.

Meiron & Laor (2012) introduced another scheme based on the conservation laws to find the changes in the binary's orbit. The motion of stars in the static spherically symmetric potential of the stellar cusp plus the time-dependent potential of the binary is followed over a short interval of time, after which the changes in the total energy and angular momentum of all stars are translated into the forces that should be applied to the black holes. Unlike our method, in which we assume a Keplerian orbit for the binary and adjust its parameters on a timescale much longer than the orbital period, their approach requires rather short update intervals to follow the motion of the black holes directly.

Compared to the previous studies, our approach most closely resembles that of Quinlan & Hernquist (1997) and Hemsendorf et al. (2002), in that we also use the spherical harmonic expansion technique to represent the smooth potential of the stellar cluster, which is itself a nearly collisionless system, while considering the interaction between stars and the binary as a succession of three-body scattering events, as in the series of papers by Sesana et al. The treatment of the binary is somewhat more approximate in our method—we assume its center-of-mass to be at rest, and ignore the changes in its orbital plane and orientation of the line of nodes. The evolution of its most important parameters—binding energy and eccentricity—is deduced from the orbits of stars using conservation laws, in a similar manner to Meiron & Laor (2012), but using much longer update intervals, spanning many orbital periods. The most important new feature of our method with respect to the previous studies is the ability to efficiently model the two-body relaxation with adjustable magnitude, allowing us both to compare our Monte Carlo simulations with direct N-body simulations (using the same relaxation rate), and to extend them into nearly collisionless regime, by switching off the relaxation.

Fokker–Planck methods have also been applied to this problem, always in the spherical geometry (Milosavljević & Merritt 2003b; Merritt & Wang 2005; Merritt et al. 2007). The Fokker–Planck approach has the advantage of a potentially much finer resolution of the loss-cone region in (E, L) space. A disadvantage is the need to orbit-average the diffusion coefficients. Our Monte Carlo algorithm avoids that limitation; on the other hand, the orbit-averaged approximation only breaks down near the loss cone, and at least in the spherical geometry, a boundary-layer treatment of the loss cone is available that is based on the local (not orbit-averaged) Fokker–Planck equation and which automatically accounts for empty versus full loss cones (Cohn & Kulsrud 1978). The secondary slingshot is also more naturally accounted for in a Monte Carlo algorithm, although it has also been implemented in Fokker–Planck treatments (Merritt et al. 2007). The primary advantage of our method over Fokker–Planck algorithms is the ability to model stellar systems regardless of their shape, including the chaotic orbits that arise in non-spherical geometries. Limitations of our approach include neglect of Brownian motion (justified to some extent below in Section 4.3) and the assumption of near-equilibrium of the stellar distribution, which means that it is not expected to perform well in highly dynamic situations such as mergers. It is also worth noting that the method scales linearly with N and already at N = 10⁵ outperforms the GPU-accelerated direct-summation code by a factor of 20.

We now consider the long-term evolution of binary SBH in galaxies with a realistically large number of stars N_⋆. Figure 3 shows the hardening rates S computed on the interval 50 ≤ t ≤ 100 for a series of γ = 1 models with different N_⋆, shape and binary mass ratio. It appears that in both spherical and axisymmetric cases the hardening rates continue to drop with increasing N_⋆, but triaxial models tend to a nonzero limiting value of S as ${N}_{\star }\to \infty$. This was already suggested in Paper I based on N-body simulations, but the number of particles N ≤ 10⁶ was not enough to establish it clearly; additional simulations with N = 2 × 10⁶ and Monte Carlo models support this conclusion.

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Figure 4 shows simulations extended to a much longer interval of time, approximately 0.4 Gyr for a model scaled to a galaxy with ${M}_{\mathrm{bin}}={10}^{8}\;{M}_{\odot }$. For each geometry we show the simulation without relaxation (i.e., in the collisionless limit), as well as for a moderately small relaxation rate (smaller than is achievable in N-body simulations). It is immediately clear that in the collisionless limit there is a striking difference between three geometries: in the spherical case, the binary stalls at a semimajor axis barely smaller than ${a}_{{\rm{h}}}$, and in the axisymmetric case it shrinks roughly a factor of ten below ${a}_{{\rm{h}}}$, but ultimately also stalls. By contrast, in the triaxial case the binary continues to shrink, although the hardening rate decreases with time. In the simulations that include relaxation, however, the situation is quite different—the binary never stalls in any geometry. For the triaxial case, the hardening rate in the N_⋆ = 5 × 10⁶ simulation is already not much higher than in the collisionless limit, but the axisymmetric systems differ dramatically from their collisionless counterpart: in both N_⋆ = 5 × 10⁶ and N_⋆ = 10⁸ systems the hardening rate decreases at early times, as in the collisionless limit, but unlike the latter, it then attains a non-zero lower limit.

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We also considered a model with a stronger initial degree of triaxiality (axis ratio 1:0.8:0.6); initially it had a higher hardening rate, which then dropped to a level comparable to that of the less flattened model. While a higher hardening rate is naturally explained by a larger reservoir of chaotic orbits in a more triaxial model (see next section), its subsequent decrease is a result of the loss of triaxiality in the central parts of the model (Figure 5). Interestingly, the model with a milder initial flattening retained its shape better. This evolution toward axisymmetry clearly occurs due to the binary: if we replace it with a single SBH, the shape stays nearly constant on much longer intervals of time than considered here. It is commonly assumed that even single SBHs inevitably destroy triaxiality (e.g., Gerhard & Binney 1985). Merritt & Quinlan (1998) simulated growth of black holes in triaxial models formed via cold collapse and found that central SBHs containing more than ∼3% the mass of their host galaxies induced evolution toward axisymmetry. However, later studies have shown that this does not necessary happen in isolated stationary systems: using a more flexible, orbit-superposition algorithm, Poon & Merritt (2002) found that self-consistent and apparently stable models could be constructed in which the SBH mass was a substantial fraction of the nuclear mass, a result confirmed in other studies (Holley-Bockelmann et al. 2002; Poon & Merritt 2004). Vasiliev (2015) demonstrated, using the same Monte Carlo code, that relaxation is the main driving force behind shape evolution: in the collisionless regime the shape remains nearly constant. The likely reason is that the diffusion of chaotic orbits, which tends to erase triaxiality, is greatly facilitated by the noise from two-body relaxation (e.g., Kandrup et al. 2000). The shape evolution seen in the collisionless simulations with a binary SBH—as opposed to a single SBH—may be caused by resonant perturbations of chaotic orbits by the time-dependent gravitational field (Kandrup et al. 2003), although more work is needed to explore this effect.

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In addition, we performed simulations of models with a steep cusp (γ = 2) and otherwise the same parameters. They demonstrated a similar behavior, with only the triaxial model continuing to shrink indefinitely in the collisionless case, although the hardening rate was dropping with time more rapidly than in the γ = 1 model. The loss of triaxiality was mild, much like the γ = 1 model with the same axis ratios (1:0.9:0.8).

Before we proceed to a theoretical explanation of the hardening rates, we discuss the properties of orbits in our models.

We have examined the types of orbits that bring stars into interaction with the binary. For each particle that arrives at a distance less than 10a from the center of mass of the binary, we record its phase-space coordinates at its first approach. If a particle is scattered more than once by the binary, we only consider the first interaction, because it is difficult to define unambiguously when one interaction ends and the next one begins. Then we compute a trajectory starting from these initial conditions in a smooth stellar potential plus a single point mass ${M}_{\mathrm{bin}}$ at the origin, for a time corresponding to 100 orbital periods. This allows us to use powerful analysis tools applicable to orbits in smooth stationary potentials. To distinguish regular and chaotic orbits, we use the Lyapunov exponent, and to determine if an orbit is centrophilic, we follow the algorithm described in the appendix of Paper I.

After the binary became sufficiently hard ($a\lesssim 0.3\;{a}_{{\rm{h}}}$), most of the orbits that interact with it are unbound to the binary (have energies higher than the depth of stellar potential well Φ₀). In the non-spherical cases, almost all such orbits (≳90%) are found to be chaotic, and for the triaxial system, a similarly large fraction of them are truly centrophilic, i.e., may attain arbitrarily low values of angular momentum. There are no genuinely centrophilic orbits in axisymmetric systems, because the variation of angular momentum is bounded from below by its conserved z-component, but for chaotic orbits the average value of L is typically much larger than its minimum achievable value L_z, thus they constitute a reservoir of "usable" orbits (loss region) that is much larger than the loss cone itself.

Regarding the overall orbital structure of the model, the chaotic orbits are a minority (∼10%, depending on the degree of flattening and triaxiality). As discussed above, the shape of triaxial models gradually evolves toward axisymmetry from the inside out, but globally it remains sufficiently triaxial to support a substantial population of centrophilic orbits—their fraction is roughly proportional to the deviation from axisymmetry, and their total mass is thus considerably larger than the mass of the binary.

The evolution of binary in the collisionless limit can be qualitatively described with a rather simple model for the draining of the population of orbits that can interact with the binary—similar to the one presented in Paper I, but without a detailed analysis of properties of orbits in a particular simulation.

We begin with the triaxial case, and assume that at each energy there is initially a fixed fraction η of chaotic centrophilic orbits, which occupy the low-angular-momentum region of phase space ${L}^{2}\leqslant \eta {L}_{\mathrm{circ}}^{2}(E)$. To account for their gradual depletion, we introduce the fraction of surviving orbits ξ(E, t), so that the mass of chaotic orbits is

$$\begin{eqnarray}&&{M}_{\mathrm{ch}}={\displaystyle \int }_{{{\rm{\Phi }}}_{0}}^{0}{dE}\;4{\pi }^{2}\;{T}_{\mathrm{rad}}(E)\;\eta {L}_{\mathrm{circ}}^{2}(E)\;\xi (E,t)\;f(E),\end{eqnarray} \tag{ 11 }$$

where we have neglected the L-dependence of radial orbital period T_rad. The hardening rate due to interaction with these low-angular-momentum orbits is given by a generalization of Equation (4):

$$\begin{eqnarray}&&S(t)\equiv \displaystyle \frac{d(1/a)}{{dt}}=4\pi \;{H}_{1}\;G{\displaystyle \int }_{{{\rm{\Phi }}}_{0}}^{0}{dE}\;f(E)\;\xi (E,t).\end{eqnarray} \tag{ 12 }$$

On the other hand, the same interactions eject stars and decrease the fraction of surviving orbits. Equating the energy carried away by stars with the change in the binary's binding energy gives

$$\begin{eqnarray}&&\displaystyle \frac{3}{2}\displaystyle \frac{G\mu }{a}{{dM}}_{\mathrm{ch}}\approx -\displaystyle \frac{{{Gm}}_{1}{m}_{2}}{2}d(\displaystyle \frac{1}{a})\end{eqnarray} \tag{ 13 }$$

with $\mu \equiv {m}_{1}{m}_{2}/{M}_{\mathrm{bin}}$ is the binary reduced mass (e.g., Merritt 2004, Equation (25)). Thus

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{ch}}}{{dt}}\approx -\displaystyle \frac{1}{3}\displaystyle \frac{{m}_{1}{m}_{2}}{\mu }\;a\;\displaystyle \frac{d}{{dt}}(\displaystyle \frac{1}{a})\approx -\displaystyle \frac{1}{3}{M}_{\mathrm{bin}\;}a\;S.\end{eqnarray} \tag{ 14 }$$

If we assume that the stars interacting with the binary become unbound to the galaxy (i.e., neglect the secondary slingshot), then the draining of the loss region occurs independently at each energy, and we can write the following equation for the evolution of surviving fraction of stars:

$$\begin{eqnarray}&&\displaystyle \frac{d\xi (E,t)}{{dt}}=-\xi (E,t)\displaystyle \frac{{H}_{1}}{6\pi }\displaystyle \frac{2{{GM}}_{\mathrm{bin}}a(t)}{\eta {L}_{\mathrm{circ}}^{2}(E){T}_{\mathrm{rad}}(E)}.\end{eqnarray} \tag{ 15 }$$

We take $y\equiv {a}_{{\rm{h}}}/a$ as the new independent variable instead of t. Substituting (12) into the above equation, we integrate it over y with the initial condition y = 1, ξ(E, y = 1) = 1 and obtain

$$\begin{eqnarray}\xi (E,y) & = & \mathrm{exp}(-\displaystyle \frac{{H}_{1}{{GM}}_{\mathrm{bin}}}{3\pi \eta {L}_{\mathrm{circ}}^{2}(E){T}_{\mathrm{rad}}(E)}h(y)),\\ h(y) & \equiv & {\displaystyle \int }_{1}^{y}\displaystyle \frac{{dy}^{\prime} }{y^{\prime} \;S(y^{\prime} )}.\end{eqnarray} \tag{ 16 }$$

Here h(y) is another unknown function related to S(y). Substituting the above expression back into (12), we can perform the integration over E, keeping h as a fixed parameter, and obtain an equation of the kind S(y) = g(h(y)) with some function g. Then we express h from the resulting equation as a function of S and y, and finally differentiate h by y to obtain an ordinary differential equation for S(y).

To illustrate this approach and acquire qualitative insight, we consider the asymptotic behavior of the system at large t for the case of a power-law density profile, ρ(r) ∝ r^−γ. Pure power-law profiles are unphysical in the sense that the total mass is infinite and the gravitational potential can take arbitrarily large positive values, but the contribution of stars at large radii to the hardening rate is negligible, so that all quantities of interest are finite. We choose to define the gravitational potential of the stellar cusp so that its value at the center is Φ₀ = 0; since we only consider stars that are not bound to the binary, their energies lie in the range $0\leqslant E\lt \infty$. Moreover, we may neglect the potential of the SBHs, because at late times most of the surviving loss-region stars are located at far larger distances than r_infl. Then all dynamical quantities have power-law dependence on energy: $f(E)\propto {E}^{-(6-\gamma )/(4-2\gamma )}$, ${L}_{\mathrm{circ}}(E)\propto {E}^{(4-\gamma )/(4-2\gamma )}$, ${T}_{\mathrm{rad}}(E)\propto {E}^{\gamma /(4-2\gamma )}$. Carrying out the integration in (12), we obtain $S(y)\propto {(h(y)/\eta )}^{-(2+\gamma )/(8-\gamma )}$, and ultimately get the asymptotic dependence of the hardening rate on a:

$$\begin{eqnarray}&&S(a)\propto {[\eta /\mathrm{ln}({a}_{{\rm{h}}}/a)]}^{\displaystyle \frac{2+\gamma }{6-2\gamma }}.\end{eqnarray} \tag{ 17 }$$

In other words, the hardening rate due to depletion of centrophilic orbits drops with time (or a⁻¹) very slowly, and has a moderate dependence on the fraction of chaotic orbits in the model η. This expression is also valid in the limiting case of a singular isothermal sphere (γ = 2).

We now follow a similar argument for the axisymmetric case. Here we again assume that orbits with ${L}^{2}\lt \eta {L}_{\mathrm{circ}}^{2}(E)$ are chaotic, but only those with ${L}_{z}\lt {L}_{\mathrm{LC}}\equiv \sqrt{2{{GM}}_{\mathrm{bin}}a}$ can interact with the binary. The mass of such "useful" chaotic orbits in the case of a hard enough binary, when ${L}_{\mathrm{LC}}\lt \sqrt{\eta }{L}_{\mathrm{circ}}$, is given by

$$\begin{eqnarray}&&{M}_{\mathrm{ch}}={\displaystyle \int }_{{{\rm{\Phi }}}_{0}}^{0}{dE}\;4{\pi }^{2}\;{T}_{\mathrm{rad}}(E)\;2\sqrt{\eta }{L}_{\mathrm{circ}}(E){L}_{\mathrm{LC}}(a)\;\xi (E,t)f(E),\end{eqnarray} \tag{ 18 }$$

and the expression for their depletion rate, analogous to (15), is

$$\begin{eqnarray}&&\displaystyle \frac{d\xi (E,t)}{{dt}}=-\xi (E,t)\displaystyle \frac{{H}_{1}}{6\pi }\displaystyle \frac{\sqrt{2{{GM}}_{\mathrm{bin}}a(t)}}{2\sqrt{\eta }{L}_{\mathrm{circ}}(E){T}_{\mathrm{rad}}(E)}.\end{eqnarray} \tag{ 19 }$$

The solution is obtained along the same lines as for the triaxial case, with the different definition of $h(y)\equiv \int {dy}/(\sqrt{y}\;S(y))$. We again consider the asymptotical evolution described by (12) and (19) for an idealized scale-free model and obtain

$$\begin{eqnarray}S(a)\propto \{\begin{array}{ll}{(\eta a/{a}_{{\rm{h}}})}^{\displaystyle \frac{2+\gamma }{4-2\gamma }}, & \ \gamma \lt 2,\\ \mathrm{exp}[-C{a}_{{\rm{h}}}/(\eta a)], & \ \gamma =2.\end{array}\end{eqnarray} \tag{ 20 }$$

Unlike the triaxial case, the hardening rate drops quickly with decreasing a. Moreover, in realistic non-scale-free systems the mass of chaotic orbits (18) that are both not yet depleted (ξ(E, t) ∼1) and able to interact with the binary (L_z < L_LC(a)) is finite, and after some time drops below the mass of the binary itself, which means that the further evolution virtually ceases.

These findings are beautifully illustrated by the results of collisionless Monte Carlo models in Figure 4, bottom panel: the hardening rate in the triaxial case drops very gently with ${a}_{{\rm{h}}}/a$ and closely follows its asymptotic expression (17), while in the axisymmetric case it decays much faster, following (20), and then drops nearly to zero when all chaotic orbits are depleted. Since our galaxy models are not scale-free and gradually lose the triaxiality, the hardening rate in the collisionless triaxial model is better described by a somewhat steeper dependence on a, namely S ∝ a^ν with ν ranging from ∼1/3 for mildly triaxial γ = 1 models to ≳1/2 for more strongly triaxial models or for steeper cusps (γ = 2), see Table 1. The fact that steeper cusps result in a stronger slowdown of hardening rate can be explained by a more rapidly declining dependence of the distribution function on energy: once the centrophilic orbits close to the binary are depleted, there are less available stars at larger distances in models with steeper density profiles. Figure 6 illustrates the above arguments about the size of the loss region and its depletion.

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Table 1.  Long-term Evolution of Hardening Rate in Triaxial Models

Model	Sinfl	${a}_{{\rm{h}}}$	μ	ν
Default (γ = 1, q = 1,
x:y:z = 1:0.9:0.8)	20	0.0125	0.38	0.32
q = 1/9	30	0.004	0.33	0.32
x:y:z = 1:0.8:0.6	20	0.0125	0.96	0.51
γ = 2	1700	0.0022	0.25	0.62
Merger (γ = 1), N = 128 k	15	0.017	0.21	0
Merger, N = 256 k	15	0.017	0.43	0.18
Merger, N = 512 k	15	0.017	0.45	0.25
Merger, N = 1024 k	15	0.017	0.87	0.47

Note. First column is the model type, second and third are the full-loss-cone hardening rate and the radius of hard binary, in model units; last two columns are the dimensionless parameters of Equation (21), computed as best-fit values to the hardening rate found in collisionless Monte Carlo simulations.

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Finally, we consider the effects of relaxation on the long-term behavior of the hardening rate. From the above discussion, it is clear that it may only matter for spherical and axisymmetric systems, because in the triaxial case the draining is the main mechanism that keeps the loss cone filled. In general, the relaxation rate scales as ${N}_{\star }^{-1}$, but it does not trivially translate into the hardening rate because of several complications. First of all, the steady-state flux of stars into the loss cone has a different dependence on the size of the loss cone L_LC and the relaxation rate in the limits of empty and full loss cone regimes: in the former case, at a fixed energy it scales as ${[{N}_{\star }\mathrm{ln}({L}_{\mathrm{circ}}/{L}_{\mathrm{LC}})]}^{-1}$, and in the latter—as ${({L}_{\mathrm{LC}}/{L}_{\mathrm{circ}})}^{2}$. Integrated over all energies with an appropriate boundary condition at each energy, the hardening rate has a weaker than ${N}_{\star }^{-1}$ scaling, which furthermore depends on the evolutionary stage: as a gets smaller, a larger fraction of total flux comes from the full-loss-cone region. Second, the flux of stars into the loss cone is actually higher than the steady-state models would predict, because initially there are large gradients in the phase space: stars with L ≲ L_LC are absent while at larger L their distribution is nearly unchanged. Time-dependent models for the loss cone repopulation (Milosavljević & Merritt 2003b) predict a flux that is several times higher than the steady-state value at early times.

Overall, we find that in the spherical case the hardening rate drops with N_⋆ rather mildly at N_⋆ ≲ 2 × 10⁵, but for larger N_⋆ approaches the asymptotic ${N}_{\star }^{-1}$ scaling. In N-body simulations, however, it does not drop quite as fast at large N_⋆. One possible reason might be the wandering (Brownian motion) of the binary (Quinlan & Hernquist 1997; Chatterjee et al. 2003), which effectively increases the size of the loss cone from ${L}_{\mathrm{LC}}^{2}=2{{GM}}_{\mathrm{bin}}a$ to ${L}_{\mathrm{LC}}^{2}=2{{GM}}_{\mathrm{bin}}{r}_{\mathrm{wand}}$. The wandering radius scales as ${r}_{\mathrm{wand}}\propto {({m}_{\star }/{M}_{\mathrm{bin}})}^{1/2}\propto {N}_{\star }^{-1/2}$ (e.g., Merritt 2001), thus for realistically large N_⋆ it should remain below a for the most part of evolution, and will not substantially increase the hardening rate.

Relaxation in the axisymmetric case occurs at the same rate as in the spherical system, but the effective size of the loss cone corresponds to the angular momentum of the chaotic region of the phase space $\sqrt{\eta }{L}_{\mathrm{circ}}$, which is then drained into the loss cone proper by the non-spherical torques and not by relaxation. This suggests that in the empty-loss-cone regime the steady-state flux is moderately (a factor of few) larger than in the spherical case (Magorrian & Tremaine 1999; Vasiliev & Merritt 2013), because it depends only logarithmically on the effective size of the loss region. Time-dependent flux is again higher than the steady-state value, by larger factors than in spherical systems (Vasiliev & Merritt 2013, Figure 13, left panel). Given these uncertain complications, it is hard to derive more quantitative theoretical estimates for axisymmetric systems. The results of Monte Carlo simulations suggest that the hardening rate drops at least as ${N}_{\star }^{-1/2}$ in the range ${10}^{6}\leqslant {N}_{\star }\leqslant {10}^{9}$, and probably even steeper at larger N_⋆, which means that it is too slow for realistic galaxies to bring the binary to GW-dominated regime in a reasonable time.

Motivated by the above arguments, we write the stellar-dynamical hardening rate S_⋆ in an evolving model as

$$\begin{eqnarray}&&{S}_{\star }(a)=\mu \;{S}_{\mathrm{infl}}\;{(a/{a}_{{\rm{h}}})}^{\nu },\end{eqnarray} \tag{ 21 }$$

where the dimensionless coefficient μ ≲1 defines the initial value of S_⋆ at the moment of hard binary formation, expressed in the units of "full-loss-cone rate" (5), and the exponent ν describes its decay with a. Theoretical models of the previous section and the results of Monte Carlo simulations suggest that in the collisionless case, $\mu ={\mathcal{O}}(1)$ and $\nu \simeq 0.3\div0.6$ in triaxial models (see Table 1), while in the presence of relaxation ν ≃ 0 and μ ≪ 1, scaling roughly as $\mu \simeq {({N}_{\star }/{10}^{5})}^{-1}$ in the spherical and $\mu \simeq {({N}_{\star }/{10}^{5})}^{-1/2}$ in the axisymmetric cases.

As the binary hardens, GW emission becomes more and more effective. The instantaneous hardening rate due to GW is S_GW = 1/(aT_GW), where T_GW is given by Equation (6). We denote a_GW to be the value of a at which S_GW = S_⋆. Using the definitions of ${a}_{{\rm{h}}}$ (1) and S_infl (5) with ${\mathcal{A}}=4$, we obtain a simple estimate of the coalescence time for the case $\nu =0$ (i.e., S_⋆ = const):

$$\begin{eqnarray}{T}_{\mathrm{coal}} & \approx & 1.6\times {10}^{8}{\rm{}}\;{\rm{year}}\times {(\displaystyle \frac{{r}_{\mathrm{infl}}}{30{\rm{}}\;{\rm{pc}}})}^{2}{(\displaystyle \frac{{M}_{\mathrm{bin}}}{{10}^{8}\;{M}_{\odot }})}^{-1}\\ & & \times {\mu }^{-4/5}\;{(\displaystyle \frac{4q}{{(1+q)}^{2}})}^{-1/5}f{(e)}^{1/5}.\end{eqnarray} \tag{ 22 }$$

Let us now consider a qualitative model for the evolution of the binary driven by both stellar-dynamical and GW hardening.

$$\begin{eqnarray}\displaystyle \frac{d(1/a)}{{dt}} & = & {S}_{\star }(a)+{S}_{\mathrm{GW}}(a,e)\\ & = & {S}_{\star ,{\rm{h}}}{(\displaystyle \frac{a}{{a}_{{\rm{h}}}})}^{\nu }+{S}_{\mathrm{GW},{\rm{h}}}{(\displaystyle \frac{{a}_{{\rm{h}}}}{a})}^{5}\displaystyle \frac{f(e)}{f({e}_{{\rm{h}}})},\end{eqnarray} \tag{ 23 }$$

where the values with subscript "h" refer to the moment of hard binary formation, and the eccentricity dependence f(e) is given by Equation (7). We further define a dimensionless parameter ${\epsilon }_{\mathrm{GW}}\equiv {S}_{\mathrm{GW},{\rm{h}}}/{S}_{\star ,{\rm{h}}}$. In all realistic situations, ${\epsilon }_{\mathrm{GW}}\ll 1$, meaning that at the early stage of evolution the hardening is driven by stellar encounters and not by GW emission. If we assume that the eccentricity remains constant throughout the evolution, then the above equation yields the following time to coalescence

$$\begin{eqnarray}{T}_{\mathrm{coal}} & = & \displaystyle \frac{1}{{S}_{\star ,{\rm{h}}}{a}_{{\rm{h}}}}{\displaystyle \int }_{1}^{\infty }\displaystyle \frac{{dy}}{{y}^{-\nu }+{\epsilon }_{\mathrm{GW}}{y}^{5}}\\ & = & \displaystyle \frac{1}{4{\epsilon }_{\mathrm{GW}}\;{S}_{\star ,{\rm{h}}}{a}_{{\rm{h}}}}\;{}_{2}{F}_{1}(1,\displaystyle \frac{4}{5+\nu };\ \displaystyle \frac{9+\nu }{5+\nu };\ -\displaystyle \frac{1}{{\epsilon }_{\mathrm{GW}}}).\end{eqnarray} \tag{ 24 }$$

Here ${}_{2}{F}_{1}$ is the Gauss' hypergeometric function and $y\equiv {a}_{{\rm{h}}}/a$. In the limit ${\epsilon }_{\mathrm{GW}}\to 0$, the asymptotic value is

$$\begin{eqnarray}{T}_{\mathrm{coal}} & \simeq & 1.7\times {10}^{8}{\rm{}}\;{\rm{year}}\times {(\displaystyle \frac{{r}_{\mathrm{infl}}}{30{\rm{}}\;{\rm{pc}}})}^{\displaystyle \frac{10+4\nu }{5+\nu }}{(\displaystyle \frac{{M}_{\mathrm{bin}}}{{10}^{8}\;{M}_{\odot }})}^{-\displaystyle \frac{5+3\nu }{5+\nu }}\\ & & \times {\mu }^{-\displaystyle \frac{4}{5+\nu }}\;{(\displaystyle \frac{4q}{{(1+q)}^{2}})}^{\displaystyle \frac{3\nu -1}{5+\nu }}f{(e)}^{\displaystyle \frac{1+\nu }{5+\nu }}\ {20}^{\nu }.\end{eqnarray} \tag{ 25 }$$

The ratio between ${a}_{{\rm{h}}}$ and a_GW, which describes how much the binary must shrink by stellar-dynamical processes before the GW emission takes over, is

$$\begin{eqnarray}\displaystyle \frac{{a}_{{\rm{h}}}}{{a}_{\mathrm{GW}}} & \simeq & 160\times {(\displaystyle \frac{{r}_{\mathrm{infl}}}{30\;{\rm{pc}}})}^{\displaystyle \frac{5}{10+2\nu }}{(\displaystyle \frac{{M}_{\mathrm{bin}}}{{10}^{8}\;{M}_{\odot }})}^{-\displaystyle \frac{5}{10+2\nu }}\\ & & \times {[\mu \;f(e)]}^{\displaystyle \frac{1}{5+\nu }}\;{(\displaystyle \frac{4q}{{(1+q)}^{2}})}^{\displaystyle \frac{4}{5+\nu }}{0.4}^{\nu }.\end{eqnarray} \tag{ 26 }$$

The assumption of a constant eccentricity is not quite correct, though, as it both increases by stellar encounters, as described by Equation (10), and decreases due to GW emission, as given by Equation (8). In the final stage of evolution, when one may neglect S_⋆ compared to S_GW, the following quantity is conserved:

$$\begin{eqnarray}&&{a}^{-1}\;{e}^{12/19}\;{(304+121{e}^{2})}^{870/2299}\;{(1-{e}^{2})}^{-1}=\mathrm{const}.\end{eqnarray} \tag{ 27 }$$

Evolutionary tracks may be computed by numerical integration of the coupled system of Equations (8), (10), (23) for $a(t),e(t)$. Figure 7 shows several examples of evolutionary tracks computed using these equations, together with the results of Monte Carlo simulations of triaxial models with different initial eccentricity and mass ratio. We used the initial eccentricity as a free parameter in the evolutionary tracks, and adjusted to match the coalescence times from the Monte Carlo simulations: since the eccentricity varies rather erratically at early stages of evolution, it is hard to match these two curves without any tuning, but with this one free parameter one gets a quite good agreement at late stages of evolution.

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Figure 8 shows the coalescence times computed for triaxial models with ${M}_{\mathrm{bin}}={10}^{8}\;{M}_{\odot }$, using the simple estimate (22) and numerically computed evolutionary tracks, as a function of initial eccentricity. For e = 0, the coalescence time ${T}_{\mathrm{coal}}^{e=0}$ can be obtained analytically (Equation (25)). For arbitrary eccentricity, it may be approximated as

$$\begin{eqnarray}{T}_{\mathrm{coal}} & \approx & {T}_{\mathrm{coal}}^{e=0}\times (1-{e}^{2})[k+(1-k){(1-{e}^{2})}^{4}],\\ k & = & 0.4+0.1\;{\mathrm{log}}_{10}({M}_{\mathrm{bin}}/{10}^{8}\;{M}_{\odot }).\end{eqnarray} \tag{ 28 }$$

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It is instructive to compare the above expression to the simple estimate (22), which predicts ${T}_{\mathrm{coal}}\propto {(1-{e}^{2})}^{7/10}$ for the case of constant hardening rate and no evolution of eccentricity. For e = 0 the latter underestimates the coalescence time by a factor ∼3, but at higher eccentricities the detailed evolutionary tracks come closer to the simple estimate, because the slowdown of stellar-dynamical hardening is compensated by the increase in eccentricity at the same evolutionary stage. The rather weak trend of the eccentricity-dependent factor with ${M}_{\mathrm{bin}}$ reflects the larger ratio between ${a}_{{\rm{h}}}$ and a_GW for smaller ${M}_{\mathrm{bin}}$ (Equation (26)), thus for them the stellar-dynamical increase in e is larger and consequently brings GW on stage earlier.

The coalescence time very weakly depends on ${M}_{\mathrm{bin}}$—if we adopt the ${M}_{\bullet }-\sigma$ relation in the form (9), then ${T}_{\mathrm{coal}}\propto {M}_{\mathrm{bin}}^{0\div0.1}$. It also only moderately depends on the initial eccentricity: despite the much steeper dependence of the GW hardening rate on e, most of the time is spent on the stellar-dynamical hardening stage. As a consequence, the efficiency of stellar-dynamical hardening μ is as important as the eccentricity. In the triaxial case, the efficiency is high enough and it decays slowly enough that for all reasonable parameters the coalescence time is shorter than the Hubble time. It also rather mildly depends on the fraction of chaotic orbits η, which itself is determined by triaxiality; the caveat is that the latter changes in the course of evolution, but generally even a slight triaxiality is enough to support the required population of centrophilic orbits.

By contrast, in the collisionless axisymmetric case, the hardening rate slows down so rapidly that the binary stalls at a separation too large for efficient GW emission, unless the eccentricity is very high. If we take into account relaxation-driven repopulation of the loss cone, then Equation (22) suggests that the coalescence time may be shorter than the Hubble time for ${M}_{\mathrm{bin}}\lesssim {10}^{8}\;{M}_{\odot }$ and moderate eccentricity (under the optimistic assumption that the hardening rate, determined from Monte Carlo simulations to be proportional to ${N}_{\star }^{-1/2}$, stays at this relatively high value for a much longer time than these simulations were run). But in a realistic galaxy, even relatively minor perturbations from axisymmetry would create a sufficient reservoir of centrophilic orbits, whose draining maintains a much higher hardening rate than the relaxation can provide.

In the present work, we have considered a stellar-dynamical solution to the final-parsec problem—the evolution of binary SBHs driven by encounters with stars in a galactic nucleus, as its orbit shrinks from the radius of a hard binary (${a}_{{\rm{h}}}\sim 1$ pc) to the separation ${a}_{\mathrm{GW}}\sim {10}^{-2}\;{a}_{{\rm{h}}}$ at which GW emission becomes effective. The central difficulty is to ensure a continuous supply of stars onto low-angular-momentum orbits, where they can be scattered by the binary and carry away its energy and angular momentum. This region of phase space, dubbed the loss cone, is quickly depleted by the binary once it becomes hard, so an efficient refilling mechanism is required to enable continued hardening. In a perfectly spherical galaxy, this can only be achieved by two-body relaxation, which is not sufficient to bring the binary to coalescence in a Hubble time, except in the smallest galaxies (Merritt et al. 2007) —hence the problem. We have focused on the additional mechanisms of loss-cone repopulation that exist in non-spherical (axisymmetric and triaxial) galaxies.

We addressed this problem using a variety of methods, but primarily with Raga,⁶ a novel stellar-dynamical Monte Carlo code that is able to follow the evolution of non-spherical stellar systems under the influence of two-body relaxation, the magnitude of which can be adjusted—unlike conventional N-body simulations in which the relaxation is essentially determined by the number of particles. We extended the code to include interactions between stars and the massive binary, by following the trajectories of particles in the superposition of the smooth galactic potential plus the time-dependent potential of the two SBHs as they orbited one another, and used conservation laws to determine the reciprocal changes in the binary orbital parameters.

We used a large suite of direct-summation N-body simulations to verify that the Monte Carlo code accurately describes the evolution of the binary for a wide range of parameters—mass ratio, eccentricity, number of particles and the shape of the galaxy model. Then we extended the Monte Carlo simulations into the range of ${N}_{\star }$ much larger than is presently accessible for conventional N-body simulations, including the intriguing collisionless limit (${N}_{\star }=\infty$), which is nearly achieved in real galaxies. We determined the scaling laws and asymptotic behavior of the stellar-dynamical hardening rate from rather simple analytic arguments, and confirmed these findings with the Monte Carlo simulations. Taking into account GW emission, the evolution of binary semimajor axis a and eccentricity e can be described by a simple system of differential equations, which we used to determine the coalescence time as a function of initial parameters of the binary and the galaxy; these evolutionary tracks were again verified by Monte Carlo simulations for a few test cases.

Our main results can be summarized as follows.

• 1.  
  The binary continues to shrink as long as there are stars in the so-called loss region—the region of phase space from which stars can precess into the loss cone proper (that is, ${L}^{2}\lt {L}_{\mathrm{LC}}^{2}\equiv 2{{GM}}_{\mathrm{bin}}a$) due to non-spherical torques. In the spherical case, the loss cone and loss region are the same. In the triaxial case, the loss region consists of mostly chaotic orbits, whose total mass is roughly proportional to the mass of galaxy with a coefficient $\eta \ll 1$ determined by the degree of flattening and triaxiality, and unless the galaxy is nearly axisymmetric, the mass of stars in the loss region is much larger than ${M}_{\mathrm{bin}}$. In the axisymmetric case, the volume of the loss region shrinks with a as $\sqrt{\eta a}$, halfway between the spherical and triaxial cases.
• 2.  
  The stellar-dynamical hardening rate S_⋆ is always smaller than the value S_full corresponding to a full loss cone. This is explained by a gradual depletion (draining) of the loss region, which occurs faster at high binding energies. As a rough estimate, decreasing a by a factor of two requires ejection of stars with total mass $\sim {M}_{\mathrm{bin}}$. In the collisionless limit, the hardening rate declines with a very slowly for a triaxial system, because the total mass of loss region stars is typically large compared to ${M}_{\mathrm{bin}}$. By contrast, in an axisymmetric system, the volume of the loss region also shrinks with a, and it depletes much faster; thus the hardening rate declines rapidly and drops nearly to zero at $a\lesssim 0.1{a}_{{\rm{h}}}$. This is not enough to bridge the gap to the GW-dominated regime.
• 3.  
  Taking into account relaxation-driven repopulation of the loss region does not change the results in the triaxial case very much, because the hardening is dominated by draining of stars that are initially in the loss region, but accounting for relaxation does change the dynamics in the axisymmetric and spherical cases dramatically. After the initial population of the loss region is nearly depleted, it is refilled by two-body relaxation, maintaing the hardening rate at a nearly constant level that depends on ${N}_{\star }$. The axisymmetric case offers a didactic example of how much a system with ${N}_{\star }\sim {10}^{6}$, typical of present-day, high-fidelity N-body simulations, or even ${N}_{\star }\sim {10}^{8}$, can differ from the collisionless limit in its long-term evolution.
• 4.  
  Coalescence times estimated for triaxial galaxies weakly depend on the mass of the binary, its mass ratio, or the degree of triaxiality (provided that the departure from axisymmetry is larger than a few percent). Coalescence times fall in the range from a few Gyr for almost circular binaries, to $\lesssim {10}^{8}$ year for very eccentric ones. For a given combination of the binary mass ${M}_{\mathrm{bin}}$ and its radius of influence r_infl, the coalescence time can be computed using Equation (25) for a circular orbit, and using Equation (28) for an eccentric orbit, where the typical values of the dimensionless parameters $\mu \sim 0.2\div1$ and $\nu \sim 0\div0.5$ are listed in Table 1. This time is up to a factor of few times greater than the simple estimate (22) that does not account for the decrease of the hardening rate with time.

Consider first the spherical case, which has been extensively studied by N-body simulations (e.g., Quinlan & Hernquist 1997; Milosavljević & Merritt 2001; Hemsendorf et al. 2002; Berczik et al. 2005; Merritt et al. 2007), methods based on scattering experiments (Quinlan 1996; Sesana 2010; Meiron & Laor 2012 e.g.,), or Fokker–Planck models (Milosavljević & Merritt 2003b; Merritt et al. 2007). It is generally accepted that in the spherical case the binary quickly depletes the loss cone and its evolution nearly stalls at a value of a just a few times smaller than ${a}_{{\rm{h}}}$. A number of effects may moderately decrease the stalling radius or increase the relaxation rate. The secondary slingshot—the re-ejection of stars that have once interacted with the binary but did not gain enough energy to escape the galaxy—leads to a gradual ($\propto \mathrm{log}t$) increase of $1/a$ at late times (Milosavljević & Merritt 2003b), and this trend was indeed found in our collisionless Monte Carlo simulations. Once these stars are completely eliminated, evolution of the binary finally stalls in the purely collisionless case; Merritt (2006) and Sesana et al. (2007) found that the stalling radius is only a few times smaller than ${a}_{{\rm{h}}}$ over a wide range of binary mass ratios and cusp density profiles. Similarly, time-dependent solution of the Fokker–Planck equation describing the relaxation of stars in angular momentum yields a higher rate of loss-cone repopulation at early times than the steady-state flux, due to sharper gradients in the phase space (Milosavljević & Merritt 2003b); this is taken into account automatically in the Monte Carlo scheme, and does not substantially affect the overall evolution.

Brownian motion is not accounted for in our method, and this may be the reason for the discrepancy between Monte Carlo and N-body hardening rates at high N in spherical systems. Quinlan & Hernquist (1997) and Chatterjee et al. (2003) argued that wandering of the binary may explain the very weak dependence of hardening rate on N found in their simulations. However, as summarized by Makino & Funato (2004), several factors complicate the interpretation of their result: the N-body algorithm used in that paper is less collisional than traditional N-body schemes, but not entirely free of relaxation, and the unequal masses of particles mean that the granularity of the potential depends on the evolutionary stage. The amplitude of Brownian motion of the binary, while larger than for a single SBH of the same mass (Merritt 2001), scales roughly as ${N}^{-1/2}$, and in real galaxies would be smaller than the size of the loss cone for most part of the evolution. Using somewhat different arguments, Milosavljević & Merritt (2003b) estimated the timescale of loss-cone refilling by Brownian motion and concluded that this effect is unlikely to substantially affect the evolution.

Consider next the case of non-spherical galaxies, which seem to offer a more promising way to solve the final-parsec problem via collisionless dynamics. Yu (2002) estimated draining rates for the loss regions in axisymmetric and triaxial galaxies, using arguments similar to those in our Section 4.3. Yu (2002) concluded that the loss region in triaxial galaxies is not likely to be depleted if the flattening parameter, responsible for the fraction of centrophilic orbits, is $\gtrsim 0.05$, although the plots in which she showed evolution timescales demonstrate the effect of gradual depletion of the loss region only for the case of small flattening. For axisymmetric systems, Yu (2002) estimated that the loss region (or "loss wedge," in the terminology of Magorrian & Tremaine 1999) can be depleted rather quickly in many cases, and the relaxation-limited evolution timescales are still longer than the Hubble time. Thus our conclusions qualitatively agree with that study.

Merritt & Poon (2004) considered self-consistent triaxial models of galactic nuclei with SBHs which had a significant fraction of centrophilic orbits. They solved the evolutionary equations similar to (12) and (15), and found that the hardening rate is nearly independent of time, but scales with the square of the fraction of chaotic orbits. Analysis of the data plotted in their Figure 6 suggests that $1/a\propto {t}^{0.7\div0.9}$, more in line with our findings in the present study and in Paper I. For the case of a singular isothermal cusp ($\gamma =2$), Equation (17) suggests that the asymptotic hardening rate scales as the squared fraction of chaotic orbits η, in agreement with their results. Their Equation (55) implies a hardening rate $\simeq {\eta }^{2}{S}_{\mathrm{full}}$, and our Monte Carlo simulation of a γ = 2 Dehnen model has on average a similar hardening rate with η = 0.2, even though it declines with time. Thus the basic conclusion of that paper, that even a moderate amount of triaxiality is sufficient to drive the binary to coalescence in less than a Hubble time, is corroborated by our simulations and asymptotic analysis, even though some details differ (most importantly, their neglect of the slowdown of the hardening rate).

Most other studies to date have assumed or inferred that the loss cone must be kept nearly full in non-spherical systems, using various arguments. Holley-Bockelmann & Sigurdsson (2006) explored the properties of orbits in a triaxial galaxy with a central SBH. They estimated that the time required to change the angular momenta of particles due to collisionless torques was much shorter than the Hubble time, and concluded that the loss cone must remain full. However, this argument does not take into account the draining of the loss region, and they did not analyze the evolution of their model under this process. More recently, Li et al. (2014) performed a similar analysis for a nearly axisymmetric model, which in fact was slightly triaxial in the central part. They computed the mass of stars belonging to orbits that are able to come into the loss cone with a size corresponding to the initial stage of binary evolution, which was several times larger than ${M}_{\mathrm{bin}}$. From this they concluded that the loss cone should remain well populated during the subsequent evolution, but their estimate did not take into account that the volume of the loss region also shrinks along with the binary semimajor axis in the axisymmetric case. It is unclear whether the slight triaxiality of their model would be sufficient to support enough truly centrophilic orbits during the entire evolution.

Sesana (2010) considered a hybrid model for binary evolution, based on the hardening and eccentricity growth rates computed from scattering experiments. He assumed that after the initial phase of formation of a hard binary, accompanied by the erosion of the stellar cusp at $r\lesssim {r}_{\mathrm{infl}}$, the subsequent evolution occurs in the full-loss-cone regime, i.e., the hardening rate is given by S_uniform (Equation (2)) with density and velocity dispersion computed at r_infl. Thus his evolutionary tracks are similar to our calculations in Section 4.4 with parameters $\mu =1$ (the loss cone is full) and $\nu =0$ (the hardening rate does not decrease with time). As Figure 8 and Equation (22) show, in this case the coalescence time is shortened by a factor of few with respect to the more conservative assumptions proposed in our study.

More recently, Sesana & Khan (2015) compared the evolutionary tracks from the hybrid model of Sesana (2010) with those obtained by N-body simulations of mergers (Khan et al. 2012). They found reasonable agreement for the eccentricity evolution and hardening rate, even though the latter was somewhat lower and declined with time in N-body simulations. They ascribed this to the gradual decrease of the density and corresponding increase of r_infl. As we have argued in Section 4.3, the decline of hardening rate is mostly caused by the depletion of centrophilic orbits at all radii, rather than simply the decrease of density at the influence radius. For instance, in our collisionless simulations of isolated models with γ = 1(2), the actual hardening rate dropped by a factor of 5 (10) by the end of simulations, while the density at r_infl, and correspondingly the full-loss-cone hardening rate S_full, decreased by less than 30%. The coalescence times quoted in Sesana & Khan (2015) are longer than ours due to a different ${M}_{\mathrm{bin}}-{r}_{\mathrm{infl}}$ relation adopted in that paper. We stress, however, that their estimates are based on the hardening rates from collisional N-body simulations, while coalescence times for collisionless systems, advocated in the present study, are up to a few times longer for the same galaxy parameters.

Studies based on N-body simulations have generally observed little or no dependence of the hardening rate on N in non-spherical galaxy models that were formed via mergers (Khan et al. 2011, 2012; Preto et al. 2011) or created as isolated models (Berczik et al. 2006; Khan et al. 2013). This result has been interpreted as an indication that the loss cone remains nearly full, although Berczik et al. (2006) reported that the hardening rate was gradually decreasing with time, suggesting that the reservoir of centrophilic orbits was being depopulated. A similar trend can be seen in other merger simulations (e.g., Preto et al. 2011, Figure 1, or Khan et al. 2012, Figure 2); in the latter paper, the models with steeper cusps displayed a systematically more rapid decline of hardening rate with time. All these trends are in agreement with our findings, even though the previous studies did not highlight them. We note, however, that in our simulations the hardening rate always turns out to be substantially lower than S_full for large enough N. This might seem to be in contrast with other studies that report a hardening rate comparable to S_full for non-spherical systems. However, a more detailed examination suggests that the apparent discrepancy can be attributed to different definitions of the full-loss-cone rate, and to different normalizations of the stellar density profile. For instance, in Khan et al. (2013) and Holley-Bockelmann & Khan (2015) the flattened models were created by adiabatic squeezing of the original density profile, and hence their scale radii are roughly a factor of two smaller than ours, as can be seen in Figure 1 of the latter paper. From Equation (5), it is apparent that this translates to a hardening rate ∼6 times higher than ours. We have re-simulated their models with our N-body code and found generally good agreement with their results.

Most importantly, as we have argued in Paper I and this study, the hardening rates in N-body simulations are dominated by, or at least have a significant contribution from, collisional effects even for $N\sim {10}^{6}$, thus it is not easy to extrapolate these results to real galaxies. Using the Monte Carlo method, we were able to reach the collisionless regime, which turned out to be very different for axisymmetric and triaxial galaxies, while in N-body simulations of Paper I they looked nearly the same.

It is also instructive to compare loss-cone theory in the single and binary SBH cases. The first obvious difference is the much larger size of the loss cone in the case of a binary. As a consequence, the relaxation-driven repopulation of the loss cone almost always occurs in the empty-loss-cone regime; on the other hand, collisionless changes in angular momentum due to non-spherical torques occur on the same dynamical timescale as the depletion of the loss cone, so that it always remains partially populated in non-spherical systems. A second important factor is that the size of the loss cone decreases as the binary shrinks, while in the case of a single SBH it can only grow. Third, the mass of stars needed to be delivered into the loss cone of the binary is a few times larger than the mass of (the lighter component of) the binary, while for single SBHs the accreted mass in stars is typically small compared to the SBH mass.

These three factors explain the fundamental difference between collisionless spherical and axisymmetric systems, on the one hand, and triaxial ones, on the other hand. For the latter ones, the evolution is almost entirely driven by draining of centrophilic orbits, whose total mass is much larger than ${M}_{\bullet }$ and furthermore does not depend on the size of the loss cone. Interestingly, in the case of a single SBH most of captured stars arrive from regular pyramid orbits inside r_infl, while in the case of the binary the loss region consists mainly of chaotic orbits outside r_infl; this explains the slightly different time dependence of the draining rates. In the axisymmetric case, the volume of the loss region composed of chaotic orbits that can be delivered into the loss cone by collisionless torques, shrinks along with the binary semimajor axis, and its orbit population is nearly depleted before the binary reaches the GW-dominated regime. The subsequent evolution is determined by the rate at which this loss region is repopulated by relaxation. Since the volume of this region is still much larger than the volume of the loss cone proper, it is more easily repopulated in axisymmetric than in spherical systems. The same is true for single SBHs; the fact that for them the difference between axisymmetric and triaxial systems is much less than between spherical and axisymmetric ones (Vasiliev 2014a, Figure 4) stems largely from the adopted isotropic (non-depleted) initial conditions for the relaxation. On the other hand, in the case of a massive binary the phase space is already depleted out to much larger values of angular momentum than the current loss cone boundary, thus it takes longer for the relaxation to resupply the loss region.

In short, loss cone theory in non-spherical systems is a delicate interplay between collisional and collisionless effects, and the outcome depends on the evolutionary history of the loss cone, as well as the changes in the global structure of the system (its shape and phase-space gradients). Only using a combination of various approaches—N-body simulations, orbit analysis, Monte Carlo methods and scaling arguments—can one hope to understand the behavior of realistic stellar systems.

The evolution of binary SBHs in gas-poor galaxies is determined by the rate of slingshot interactions with stars in the loss cone—the low-angular-momentum region of the phase space. The fact that the loss cone is quickly depleted in idealized spherical systems gave rise to the final-parsec problem. Repopulation of the loss cone occurs both due to collisional and collisionless effects; the latter are only relevant in non-spherical systems. We have developed a Monte Carlo method that can efficiently deal with both collisionless and collisional evolution, and used it to show that in the collisionless limit, the repopulation is efficient if the galaxy is even slightly triaxial. To the extent that mergers result in galactic shapes that are not exactly axisymmetric, our results imply that the final-parsec problem does not exist in most galaxies.