Hanging on the cliff: Extreme mass ratio inspiral formation with local two-body relaxation and post-Newtonian dynamics

¹ Dipartimento di Fisica, Università degli Studi di Trento, Via Sommarive 14, 38123 Povo, Italy
² Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
³ INFN – Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
⁴ INAF – Osservatorio Astronomico di Brera, Via Brera 20, 20121 Milano, Italy

^⋆ Corresponding author; d.mancieri@campus.unimib.it

Received: 19 September 2024
Accepted: 20 January 2025

Abstract

Extreme mass ratio inspirals (EMRIs) are anticipated to be primary gravitational wave sources for the Laser Interferometer Space Antenna (LISA). They form in dense nuclear clusters when a compact object is captured by the central massive black holes (MBHs) as a consequence of the frequent two-body interactions occurring between orbiting objects. The physics of this process is complex and requires detailed statistical modelling of a multi-body relativistic system. We present a novel Monte Carlo approach to evolving the post-Newtonian (PN) equations of motion of a compact object orbiting an MBH. The approach accounts for the effects of two-body relaxation locally on the fly, without leveraging on the common approximation of orbit-averaging. We applied our method to study the function S(a₀), describing the fraction of EMRI to total captures (including EMRIs and direct plunges, DPs) as a function of the initial semi-major axis a₀ for compact objects orbiting central MBHs with M_• ∈ [10⁴ M_⊙, 4 × 10⁶ M_⊙]. The past two decades have consolidated a picture in which S(a₀)→0 at large initial semi-major axes, with a sharp transition from EMRIs to DPs occurring around a critical scale a_c. A recent study challenges this notion for low-mass MBHs, finding EMRIs forming at a ≫ a_c, which were called ‘cliffhangers’. Our simulations confirm the existence of cliffhanger EMRIs, which we find to be more common then previously inferred. Cliffhangers start to appear for M_• ≲ 3 × 10⁵ M_⊙ and can account for up to 55% of the overall EMRIs forming at those masses. We find S(a₀)≫0 for a ≫ a_c, reaching values as high as 0.6 for M_• = 10⁴ M_⊙, much higher than previously found. We test how these results are influenced by different assumptions on the dynamics used to evolve the system and treatment of two-body relaxation. We find that the PN description of the system greatly enhances the number of EMRIs by shifting a_c to larger values at all MBH masses. Conversely, the local treatment of relaxation has a mass-dependent impact, significantly boosting the number of cliffhangers at low MBH masses compared to an orbit-averaged treatment. These findings highlight the shortcomings of standard approximations used in the EMRI literature and the importance of carefully modelling the (relativistic) dynamics of these systems. The emerging picture is more complex than previously thought, and should be considered in future estimates of rates and properties of EMRIs detectable by LISA.