Binary black hole merger in the extreme mass ratio limit

TL;DR

This paper addresses the challenge of modeling the inspiral-to-plunge transition in extreme-mass-ratio binary black holes by embedding a test-marticle description (mass $μ$) in a Schwarzschild background within an Effective One Body framework. It couples a Padé-resummed radiation-reaction force to a conservative EOB Hamiltonian in the ν→0 limit and computes gravitational waves by solving time-domain Regge-Wheeler-Zerilli perturbation equations with a delta-function source representing the particle’s trajectory. The authors validate their numerical setup against known circular-orbit results, deliver waveforms for the inspiral–plunge transition, and identify a quasi-universal, quasi-geodesic plunge around $r\simeq 5.15M$ followed by a quasi-normal-mode ringdown, with energy and angular-momentum fluxes in good agreement with the emitted radiation. These results provide high-precision EMRI-scale waveforms that can calibrate and test EOB-based analytical models and bridge PN approximations with perturbative NR results in the extreme-mass-ratio regime, informing future DN06 comparisons.

Abstract

We discuss the transition from quasi-circular inspiral to plunge of a system of two nonrotating black holes of masses $m_1$ and $m_2$ in the extreme mass ratio limit $m_1m_2\ll (m_1+m_2)^2$. In the spirit of the Effective One Body (EOB) approach to the general relativistic dynamics of binary systems, the dynamics of the two black hole system is represented in terms of an effective particle of mass $μ\equiv m_1m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M\equiv m_1+m_2$ and submitted to an ${\cal O}(μ)$ radiation reaction force defined by Padé resumming high-order Post-Newtonian results. We then complete this approach by numerically computing, à la Regge-Wheeler-Zerilli, the gravitational radiation emitted by such a particle. Several tests of the numerical procedure are presented. We focus on gravitational waveforms and the related energy and angular momentum losses. We view this work as a contribution to the matching between analytical and numerical methods within an EOB-type framework.

Figures (6)

• Figure 1: Left panel: Plunge relative orbit from $r=7M$. The circle represented with a thick dashed line is the LSO at $r=6M$. The thick solid line circle represents the light ring at $r=3M$. Right panel: Radial velocity $v_r=\dot{\hat{r}}$ and orbital angular frequency $\omega=\dot{\varphi}$ versus coordinate time.
• Figure 2: Test of the code: waveform (on a logarithmic scale in the inset) for a particle plunging radially on the black hole along the $z$-axis from $r=10M$
• Figure 3: Dominant $\ell=m=2$ even-parity Left panel and and $\ell=2$, $m=1$ odd-parity Right panel gravitational wave multipoles generated by the quasi-circular plunge of a particle with $\nu=0.01$ initially at $r=7M$.
• Figure 4: Left panel: angular momentum flux computed from the gravitational waveform compared to the angular momentum loss assumed in the dynamics with $\nu=0.01$. Right panel: Instantaneous gravitational wave frequency for two values of $\nu$. The signal is " approximately universal" after $u/(2M)\simeq 280$, which roughly corresponds to $r\lesssim 5.15$ ("quasi-geodesic plunge"). After $u/(2M)\simeq 300$ starts the QNM phase.
• Figure 5: Consistency checks of the sources. Left panel: relative difference between waveforms obtained with different expressions for $S^{(\rm e)}_{22}$. Right panel: check of the smallness of the effects linked to the terms $\propto\dot{\hat{P}}_{\varphi}$ and $\propto\dot{\hat{H}}$ in the odd-parity source $S^{(\rm o)}_{21}$ from (\ref{['src:odd']}).