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.
Cited by 3
Albertini et al., Comparing second-order gravitational self-force, numerical relativity, and effective one body waveforms from inspiralling, quasicircular, and nonspinning black hole binaries
Phys. Rev. D 106, 084061 (2022) [Crossref]
Albalat et al., Success of the small mass-ratio approximation during the final orbits of binary black hole simulations
Phys. Rev. D 107, 084021 (2023) [Crossref]
Wardell et al., Gravitational Waveforms for Compact Binaries from Second-Order Self-Force Theory
Phys. Rev. Lett. 130, 241402 (2023) [Crossref]
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- European Space Agency
- Fonds De La Recherche Scientifique - FNRS (FNRS) (through Organization: Fonds National de la Recherche Scientifique [FNRS])
- Grantová Agentura České Republiky (through Organization: Grantová agentura České republiky / Czech Science Foundation [GAČR])