Asymptotically matched quasi-circular inspiral and transition-to-plunge in the small mass ratio expansion

TL;DR

This paper develops a comprehensive analytical framework for modeling the inspiral and transition-to-plunge phases of small mass ratio binary systems in general relativity, crucial for accurate gravitational waveform predictions.

Contribution

It derives equations for quasi-circular inspiral and transition-to-plunge motions at all perturbative orders, establishing a matching scheme for these phases in the self-force formalism.

Findings

Derived equations for inspiral at multiple orders in Kerr background.

Formulated transition-to-plunge equations as Painlevé transcendental equations.

Validated the matching scheme for various coefficients and orders.

Abstract

In the small mass ratio expansion and on the equatorial plane, the two-body problem for point particles in general relativity admits a quasi-circular inspiral motion followed by a transition-to-plunge motion. We first derive the equations governing the quasi-circular inspiral in the Kerr background at adiabatic, post-adiabatic and post-post-adiabatic orders in the slow-timescale expansion in terms of the self-force and we highlight the structure of the equations of motion at higher subleading orders. We derive in parallel the equations governing the transition-to-plunge motion to any subleading order, and demonstrate that they are governed by sourced linearized Painlev\'e transcendental equations of the first kind. The first ten perturbative orders do not require any further developments in self-force theory, as they are determined by the second-order self-force. We propose a scheme that matches the slow-timescale expansion of the inspiral with the transition-to-plunge motion to all perturbative orders in the overlapping region exterior to the last stable orbit where both expansions are valid. We explicitly verify the validity of the matching conditions for a large set of coefficients involved, on the one hand, in the adiabatic or post-adiabatic inspiral and, on the other hand, in the leading, subleading or higher subleading transition-to-plunge motion. This result is instrumental for deriving gravitational waveforms within the self-force formalism beyond the innermost stable circular orbit.