We investigate a purely stellar dynamical solution to the Final Parsec Problem. Galactic nuclei resulting from major mergers are not spherical, but show some degree of triaxiality. With N-body simulations, we show that equal-mass massive black hole binaries (MBHBs) hosted by them will continuously interact with stars on centrophilic orbits and will thus inspiral-in much less than a Hubble time-down to separations at which gravitational-wave (GW) emission is strong enough to drive them to coalescence. Such coalescences will be important sources of GWs for future space-borne detectors such as the Laser Interferometer Space Antenna (LISA). Based on our results for equal-mass mergers, and given that the hardening rate of unequal-mass binaries is similar, we expect that LISA will see between ∼10 and ∼ few × 10 2 such events every year, depending on the particular massive black hole (MBH) seed model as obtained in recent studies of merger trees of galaxy and MBH co-evolution. Orbital eccentricities in the LISA band will be clearly distinguishable from zero with e 0.001-0.01.

Galaxy centers are residing places for Super Massive Black Holes (SMBHs). Galaxy mergers bring SMBHs close together to form gravitationally bound binary systems which, if able to coalesce in less than a Hubble time, would be one of the most promising sources of gravitational waves for the Laser Interferometer Space Antenna (LISA). In spherical galaxy models, SMBH binaries stall at a separation of approximately one parsec, leading to the "final parsec problem" (FPP). On the other hand, it has been shown that merger-induced triaxiality of the remnant in equal-mass mergers is capable of supporting a constant supply of stars on so-called centrophilic orbits that interact with the binary and thus avoid the FPP. In this paper, using a set of direct N -body simulations of mergers of initially spherically symmetric galaxies with different mass ratios, we show that the merger-induced triaxiality is also able to drive unequal-mass SMBH binaries to coalescence. The binary hardening rates are high and depend only weakly on the mass ratios of SMBHs for a wide range of mass ratios q. There is, however, an abrupt transition in the hardening rates for mergers with mass ratios somewhere between q ∼ 0.05 and 0.1, resulting from the monotonic decrease of merger-induced triaxiality with mass ratio q, as the secondary galaxy becomes too small and light to significantly perturb the primary, i.e., the more massive one. The hardening rates are significantly higher for galaxies having steep cusps in comparison with those having shallow cups at centers. The evolution of the binary SMBH leads to relatively shallower inner slopes at the centers of the merger remnants. The stellar mass displaced by the SMBH binary on its way to coalescence is ∼ 1 − 5 times the combined mass of binary SMBHs. The coalescence time scales for SMBH binary with mass ∼ 10 6 M ⊙ are less than 1 Gyr and for those at the upper end of SMBH masses 10 9 M ⊙ are 1-2 Gyr for less eccentric binaries whereas less than 1 Gyr for highly eccentric binaries. SMBH binaries are thus expected to be promising sources of gravitational waves at low and high redshifts. Subject headings: Stellar dynamics -black hole physics -Galaxies: kinematics and dynamics -Galaxy: center.

We present a clear N-body realization of the growth of a Bahcall-Wolf f ∝ E 1/4 (ρ ∝ r −7/4 ) density cusp around a massive object ("black hole") at the center of a stellar system. Our N-body algorithm incorporates a novel implementation of Mikkola-Aarseth chain regularization to handle close interactions between star and black hole particles. Forces outside the chain were integrated on a GRAPE-6A/8 special-purpose computer with particle numbers up to N = 0.25 × 10 6 . We compare our N-body results with predictions of the isotropic Fokker-Planck equation and verify that the time dependence of the density (both configuration and phase-space) predicted by the Fokker-Planck equation is well reproduced by the N-body algorithm, for various choices of N and of the black hole mass. Our results demonstrate the feasibility of direct-force integration techniques for simulating the evolution of galactic nuclei on relaxation time scales.

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Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian Ñow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., testparticle integrations in Ðxed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential ; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force Ðeld with only "" along-track ÏÏ errors ; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N~2), where N is the number of steps per period.

We present, for the first time, a clear N-body realization of the strong mass segregation solution for the stellar distribution around a massive black hole. We compare our N-body results with those obtained by solving the orbit-averaged Fokker-Planck (FP) equation in energy space. The N-body segregation is slightly stronger than in the FP solution, but both confirm the robustness of the regime of strong segregation when the number fraction of heavy stars is a (realistically) small fraction of the total population. In view of recent observations revealing a dearth of giant stars in the sub-parsec region of the Milky Way, we show that the time scales associated with cusp re-growth are not longer than (0.1 − 0.25) × T rlx (r h ). These time scales are shorter than a Hubble time for black holes masses M • 4 × 10 6 M ⊙ and we conclude that quasi-steady, mass segregated, stellar cusps may be common around MBHs in this mass range. Since EMRI rates scale as M −α • , with α ∈ [ 1 4 , 1], a good fraction of these events should originate from strongly segregated stellar cusps.

Stars near the Galactic center reach a few percent of light speed during pericenter passage, which makes post-Newtonian effects potentially detectable. We formulate the orbit equations in Hamiltonian form such that the O(v 2 /c 2 ) and O(v 3 /c 3 ) post-Newtonian effects of the Kerr metric appear as a simple generalization of the Kepler problem. A related perturbative Hamiltonian applies to photon paths. We then derive a symplectic integrator with adaptive time-steps, for fast and accurate numerical calculation of post-Newtonian effects. Using this integrator, we explore relativistic effects. Taking the star S2 as an example, we find that general relativity would contribute tenths of mas in astrometry and tens of km s −1 in kinematics. (For eventual comparison with observations, redshift and time-delay contributions from the gravitational field on light paths will need to be calculated, but we do attempt these in the present paper.) The contribution from stars, gas, and dark matter in the Galactic center region is still poorly constrained observationally, but current models suggest that the resulting Newtonian perturbation on the orbits could plausibly be of the same order as the relativistic effects for stars with semi-major axes 0.01 pc (or 250 mas). Nevertheless, the known and distinctive time dependence of the relativistic perturbations may make it possible to disentangle and extract both effects from observations.

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