Revised event rates for extreme and extremely large mass-ratio inspirals

One of the main targets of the Laser Interferometer Space Antenna (LISA) is the detection of extreme mass-ratio inspirals (EMRIs) and extremely large mass-ratio inspirals (X-MRIs). Their orbits are expected to be highly eccentric and relativistic when entering the LISA band. Under these circumstances, the inspiral time-scale given by Peters' formula loses precision and the shift of the last-stable orbit (LSO) caused by the massive black hole spin could influence the event rates estimate. We re-derive EMRIs and X-MRIs event rates by implementing two different versions of a Kerr loss-cone angle that includes the shift in the LSO, and a corrected version of Peters' time-scale that accounts for eccentricity evolution, 1.5 post-Newtonian hereditary fluxes, and spin-orbit coupling. The main findings of our study are summarized as follows: (1) implementing a Kerr loss-cone changes the event rates by a factor ranging between 0.9 and 1.1; (2) the high-eccentricity limit of Peters' formula offers a reliable inspiral time-scale for EMRIs and X-MRIs, resulting in an event rate estimate that deviates by a factor of about 0.9 to 3 when compared to event rates computed with the corrected version of Peters' time-scale and the usual loss-cone definition. (3) Event rates estimates for systems with a wide range of eccentricities should be revised. Peters' formula overestimates the inspiral rates of highly eccentric systems by a factor of about 8 to 30 compared to the corrected values. Besides, for e$_0 \lesssim$0.8, implementing the corrected version of Peters' formula would be necessary to obtain accurate estimates.