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We assess the contribution of dynamical hardening by direct three-bodyscattering interactions to the rate of stellar-mass black hole binary (BHB)mergers in galactic nuclei. We derive an analytic model for the single-binaryencounter rate in a nucleus with spherical and disk components hosting asuper-massive black hole (SMBH). We determine the total number of encounters$N_{\rm GW}$ needed to harden a BHB to the point that inspiral due togravitational wave emission occurs before the next three-body scattering event.This is done independently for both the spherical and disk components. Using aMonte Carlo approach, we refine our calculations for $N_{\rm GW}$ to includegravitational wave emission between scattering events. For astrophysicallyplausible models we find that typically $N_{\rm GW} \lesssim$ 10.  We find two separate regimes for the efficient dynamical hardening of BHBs:(1) spherical star clusters with high central densities, low velocitydispersions and no significant Keplerian component; and (2) migration traps indisks around SMBHs lacking any significant spherical stellar component in thevicinity of the migration trap, which is expected due to effective orbitalinclination reduction of any spherical population by the disk. We also find aweak correlation between the ratio of the second-order velocity moment tovelocity dispersion in galactic nuclei and the rate of BHB mergers, where thisratio is a proxy for the ratio between the rotation- and dispersion-supportedcomponents. Because disks enforce planar interactions that are efficient inhardening BHBs, particularly in migration traps, they have high merger ratesthat can contribute significantly to the rate of BHB mergers detected by theadvanced Laser Interferometer Gravitational-Wave Observatory.

On the rate of black hole binary mergers in galactic nuclei due to dynamical hardening