The steady state of intermediate-mass black holes near a supermassive black hole

Aims. Our aim is to investigate the properties of a cluster of intermediate-mass black holes (IMBHs) surrounding a supermassive black hole (SMBH). Methods. We simulated clusters of equal-mass IMBHs ( m IMBH = 10 3 M ⊙ ) initialised in a shell between 0.15 ≤ r pc ≤ 0.25 centred about a SMBH. We explored the influence of the cluster population and SMBH on the merger rate, the ejection rate, and the escape velocity. For M SMBH = 4 × 10 6 M ⊙ , we used both a Newtonian and post-Newtonian formalism, going up to the 2.5th order and including cross terms. We ran 40 and 60 simulations per cluster population for either formalism, respectively. For the other two SMBH masses ( M SMBH = 4 × 10 5 M ⊙ and M SMBH = 4 × 10 7 M ⊙ ), we modelled the system only taking into account relativistic effects. In the case of M SMBH = 4 × 10 5 M ⊙ , 30 simulations were run per population. For M SMBH = 4 × 10 7 M ⊙ we ran ten simulations per population. The simulations ended once a black hole escaped the cluster, a merger occured, or the system evolved until 100 Myr. Results. The post-Newtonian formalism accelerates the loss rate of IMBHs compared to the Newtonian formalism. Ejections occur more often for lighter SMBHs while more massive ones increase the rate of mergers. Although relativistic effects allow for circularisation, all merging binaries have e ≳ 0.97 when measured 1 − 2 kyr before the merging event. The strongest gravitational wave signals are often sourced by IMBH-SMBH binaries that eventually merge. Strong signals were suppressed during our Newtonian calculations since, here, the IMBH typically stalls in the vicinity of the SMBH, before being generally ejected via the slingshot mechanism or experiencing a head-on collision. Weaker and more frequent signals are expected from gravitational wave radiation emitted in a flyby. In our post-Newtonian calculations, 30/406 (7.4%) of the gravitational wave events capable of being observed with LISA and μ Ares were detected as gravitational wave capture binaries with the remaining being in-cluster mergers. Throughout our investigation, no IMBH-IMBH binaries were detected.