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The Role of the Kozai-Lidov Mechanism in Black Hole Binary Mergers in Galactic Centers

In order to understand the rate of merger of stellar-mass black hole binaries (BHBs) by gravitational wave (GW) emission it is important to determine the major pathways to merger. We use numerical simulations to explore the evolution of BHBs inside the radius of influence of supermassive black holes (SMBHs) in galactic centers. In this region the evolution of binaries is dominated by perturbations from the central SMBH. In particular, as first pointed out by Antonini and Perets, the Kozai-Lidov (KL) mechanism trades relative inclination of the BHB to the SMBH for eccentricity of the BHB, and for some orientations can bring the BHB to an eccentricity near unity. At very high eccentricities, GW emission from the BHB can become efficient, causing the members of the BHB to coalesce. We use a novel combination of two N-body codes to follow this evolution. We are forced to simulate small systems to follow the behavior accurately. We have completed 400 simulations that range from $\sim$ 300 stars around a $10^{3}$ M$_{\odot}$ black hole to $\sim$ 4500 stars around a $10^{4}$ M$_{\odot}$ black hole. These simulations are the first to follow the internal orbit of a binary near a SMBH while also following the changes to its external orbit self-consistently. We find that this mechanism could produce mergers at a maximum rate per volume of $\sim 100$ Gpc$^{-3}$ yr$^{-1}$ or considerably less if the inclination oscillations of the binary remain constant as the BHB inclination to the SMBH changes, or if the binary black hole fraction is small.