Extreme Mass-Ratio Binary Black Hole Merger: Characteristics of the Test-Particle Limit

Abstract

We study binary black hole mergers in the extreme mass-ratio limit. We determine the energy, angular momentum, and linear momentum of the post-merger, remnant black hole. Unlike previous works, we perform our analysis directly in the test-particle limit by solving the Regge-Wheeler-Zerilli wave equation with a source that moves along a geodesic. We rely on the fact that toward the merger, small mass-ratio binary systems follow a quasiuniversal geodesic trajectory. This formalism captures the final premerger stages of small mass-ratio binaries and thus provides a straightforward universal description in a region inaccessible to numerical relativity simulations. We present a general waveform template that may be used in the search for gravitational wave bursts from small and intermediate mass-ratio binary systems. Finally, this formalism gives a formal proof that the recoil velocity is quadratic in the symmetric mass ratio $\nu$. Specifically, the velocity is given by $V/c\approx 0.0467 \nu^2$. This result is about $4\%$ larger than previously estimated. Most of this difference stems from the inclusion of higher multipoles in our calculation.