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Gravitational-wave merging events from the dynamics of stellar mass binary black holes around the massive black hole in a galactic nucleus

We study the dynamical evolution of the stellar mass binary black holes (BBHs) in a galactic nucleus that contains a massive black hole (MBH). For a comprehensive study of their merging events, we consider simultaneously the non-resonant and resonant relaxations of the BBHs, the binary-single encounters of the BBHs with the field stars, the Kozai-Lidov (KL) oscillation and the close encounters between the BBHs and the central MBH, which usually lead to binaries' tidal disruptions. As the BBHs are usually heavier than the background stars, they sink to the center by mass segregation, making the KL oscillation an important effect in merging BBHs. The binary-single encounters can not only lead to softening and ionization of the BBHs, it can also make them hardening, that increases the merging rates significantly. The mergers of BBHs are mainly contributed by galaxies containing MBHs less massive than $10^8 M_\odot$ and the total event rates are likely in orders of $1$--$10$ Gpc$^{-3}$ yr$^{-1}$, depending on the detailed assumptions of the nucleus clusters. About $3-10\%$ of these BBH mergers are with eccentricity $\ge 0.01$ when their gravitational wave oscillating frequencies enter the LIGO band ($10$\,Hz). Our results show that merging the BBHs within galactic nuclei can be an important source of the merging events detected by the Advanced LIGO/Virgo detectors, and they can be distinguished from BBH mergers from the galactic fields and globular clusters when enough events are accumulated.