Extreme-mass ratio inspirals in strong segregation regime -- to inspiral or to get ejected?

Extreme-mass ratio inspirals (EMRIs) of stellar-mass black holes (BHs) are among the main targets for upcoming low-frequency gravitational wave (GW) detectors such as the Laser Interferometer Space Antenna (LISA). In the classical scenario, EMRIs are formed when BHs scatter off each other and are driven onto highly eccentric orbits that gradually inspiral due to GW emission. If the cluster is in a state of strong mass segregation, the BHs are expected to reside in a steep cusp around the central massive black hole (MBH), which would facilitate more efficient EMRI formation. However, strong mass segregation may also lead to an increased rate of ejections due to close encounters between the BHs. Here, we test the relevance of such ejections for EMRI formation by numerically solving a two-dimensional Fokker-Planck equation. Our formalism includes the effects of two-body relaxation, GW dissipation, and ejections. We find that the EMRI formation rate can be suppressed due to ejections by more than an order of magnitude for strongly segregated BH cusps with density index $\gamma\gtrsim 2.25$ around central MBHs of mass $M_{\bullet} \lesssim 10^6 M_\odot $. The EMRI formation rate levels off up to a maximum value of $\simeq 200~{\rm Gyr}^{-1}$ due to ejections, which is roughly an order of magnitude lower than the usual scenarios ignoring ejections for steep BH cusps around low mass MBHs. Our analysis brings forth the significance of strong scatterings for EMRI formation in galactic nuclei.