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Supermassive black holes (SMBHs) play an important role in galaxy evolution.Binary and triple SMBHs can form after galaxy mergers. A third SMBH mayaccelerate the SMBH merging process, possibly through the Kozai mechanism. Weuse N -body simulations to analyze oscillations in the orbital elements ofhierarchical triple SMBHs with surrounding star clusters in galaxy centers. Wefind that SMBH triples spend only a small fraction of time in the hierarchicalmerger phase (i.e., a binary SMBH with a distant third SMBH perturber). Most ofthe time, the enclosed stellar mass within the orbits of the innermost or theoutermost SMBH is comparable to the SMBH masses, indicating that the influenceof the surrounding stellar population cannot be ignored. We search forEccentric Kozai-Lidov (EKL) oscillations for which (i) the eccentricity of theinner binary and inclination are both oscillate and are anti-phase or in-phaseand (ii) the oscillation period is consistent with EKL timescale. We find thatEKL oscillations are short-lived and rare: the triple SMBH spends around 3% ofits time in this phase over the ensemble of simulations, reaching around 8% inthe best-case scenario. This suggests that the role of the EKL mechanism inaccelerating the SMBH merger process may have been overestimated in previousstudies. We follow-up with three-body simulations, using initial conditionsextracted from the simulation, and the result can to some extent repeat theobserved EKL-like oscillations. This comparison provides clues about why thoseEKL oscillations with perturbing stars are short-lived.

Analysis of Kozai Cycles in Equal-Mass Hierarchical Triple Supermassive Black Hole Mergers in the Presence of a Stellar Cluster