Analysis of Kozai cycles in equal-mass hierarchical triple supermassive black hole mergers in the presence of a stellar cluster

Supermassive black holes (SMBHs) play an important role in galaxy evolution. Binary and triple SMBHs can form after galaxy mergers. A third SMBH may accelerate the SMBH merging process, possibly through the Kozai mechanism. We use N-body simulations to analyse oscillations in the orbital elements of hierarchical triple SMBHs with surrounding star clusters in galaxy centres. We find that SMBH triples spend only a small fraction of time in the hierarchical merger phase (i.e. a binary SMBH with a distant third SMBH perturber). Most of the time, the enclosed stellar mass within the orbits of the innermost or the outermost SMBH is comparable to the SMBH masses, indicating that the influence of the surrounding stellar population cannot be ignored. We search for Eccentric Kozai-Lidov (EKL) oscillations for which (i) the eccentricity of the inner binary and inclination are both oscillate and are antiphase or in-phase and (ii) the oscillation period is consistent with EKL time-scale. We find that EKL oscillations are short-lived and rare: the triple SMBH spends around 3 per cent of its time in this phase over the ensemble of simulations, reaching around 8 per cent in the best-case scenario. This suggests that the role of the EKL mechanism in accelerating the SMBH merger process may have been overestimated in previous studies. We follow-up with three-body simulations, using initial conditions extracted from the simulation, and the result can to some extent repeat the observed EKL-like oscillations. This comparison provides clues about why those EKL oscillations with perturbing stars are short-lived.