Recent theoretical studies suggest that stellar-mass binary black holes (BBHs) would merge more efficiently due to the Kozai-Lidov mechanism if these binaries form in the vicinity of supermassive black holes (SMBHs). Since SMBHs are likely rotating rapidly, we continue our earlier study on the generalization of the Kozai-Lidov formalism to include the spin of the SMBH and study the evolution of a nearby BBH. We find that the eccentricity and orbital inclination of the BBH is significantly affected, because the spin (i) forces the orbital plane of the center-of-mass of the BBH around the SMBH to precess (the Lense-Thirring effect) and (ii) imposes an additional gravitomagnetic force on the BBH. As a result, the merger time of the BBH could be significantly different. We calculate the waveform from the BBH in one representative example and study its detectability by a milli-Hertz GW detector, such as the Laser Interferometer Space Antenna (LISA). We find that the signal is distinguishable from that in the case without spin. Our results imply that the BBHs in the LISA band could potentially be used to probe the spin of the SMBHs in galaxy centers.

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