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

by Geoffrey Compère, Lorenzo Küchler

This is not the latest submitted version.

Submission summary

As Contributors: Geoffrey Compère · Lorenzo Küchler
Arxiv Link: (pdf)
Code repository:
Date submitted: 2022-05-11 10:23
Submitted by: Küchler, Lorenzo
Submitted to: SciPost Physics
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


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.

Current status:
Has been resubmitted

Author comments upon resubmission

We improved the manuscript addressing all of the referee's comments and made the requested changes.

Reports on this Submission

Anonymous Report 1 on 2022-5-11 (Invited Report)


The authors have addressed all my comments, so I am happy to recommend this for publication, with two small requested changes that can be made during the production stage.

Requested changes

1 - In Eqs. (201-3), the “PA” and “PL” parts of the superscripts should be set in Roman.

2- In the conclusions, “EOB waveforms [39–42] (derived using methods from [23, 43])” makes it sound like the EOB waveforms are derived using those methods. Something like “EOB waveforms [39–42] or other waveforms [44–46]. (These waveforms are currently are calibrated in part using extreme mass-ratio waveforms calculated using methods from [23, 43].)” would be clearer.

  • validity: high
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: good

Author:  Lorenzo Küchler  on 2022-05-18  [id 2491]

(in reply to Report 1 on 2022-05-11)

We would like to thank the referee for her/his comments which improved the quality of the paper and for the positive assessment of our manuscript.

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