Universality in the Transition from Inspiral to Plunge for Extreme-Mass-Ratio-Inspirals: High-Accuracy Analytic Solutions and Catastrophe Theory
Abstract
We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasi-circular, inclined orbits in Kerr spacetime from the perspective of catastrophe theory. Our goal is to uncover the mathematical structures underlying the universality of the transition dynamics, which remains governed by the same Painlevé I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlevé I equation selected by the physical boundary conditions of slowly evolving quasi-circular inspiral at early times. We argue that these conditions uniquely select the tritronquée solution of Painlevé I. We then compare existing high-accuracy analytic approximations of the tritronquée solution with direct numerical integrations of the Painlevé I equation, finding comparable accuracy and improved stability under differentiation and integration for the analytic solution. In the second part of this work, we show that the equilibrium structure of the Kerr radial effective potential admits a natural interpretation in terms of catastrophe theory. Equatorial orbits are associated with the fold catastrophe, while inclined orbits are described by the cusp catastrophe. In both cases, the transition to plunge corresponds to slow evolution across fold lines of the catastrophe manifold, providing a geometric explanation for the universal appearance of the Painlevé I equation in the transition dynamics.
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BibTeX
@misc{metidieri2026universality,
title = {{Universality in the Transition from Inspiral to Plunge for Extreme-Mass-Ratio-Inspirals: High-Accuracy Analytic Solutions and Catastrophe Theory}},
author = {Ariadna Ribes Metidieri and Béatrice Bonga and Badri Krishnan and José Luis Jaramillo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.13786},
eprint = {2606.13786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13786},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universality in the Transition from Inspiral to Plunge for Extreme-Mass-Ratio-Inspirals: High-Accuracy Analytic Solutions and Catastrophe Theory
AU - Ariadna Ribes Metidieri
AU - Béatrice Bonga
AU - Badri Krishnan
AU - José Luis Jaramillo
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2606.13786
UR - https://arxiv.org/abs/2606.13786
AB - We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasi-circular, inclined orbits in Kerr spacetime from the perspective of catastrophe theory. Our goal is to uncover the mathematical structures underlying the universality of the transition dynamics, which remains governed by the same Painlevé I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlevé I equation selected by the physical boundary conditions of slowly evolving quasi-circular inspiral at early times. We argue that these conditions uniquely select the tritronquée solution of Painlevé I. We then compare existing high-accuracy analytic approximations of the tritronquée solution with direct numerical integrations of the Painlevé I equation, finding comparable accuracy and improved stability under differentiation and integration for the analytic solution. In the second part of this work, we show that the equilibrium structure of the Kerr radial effective potential admits a natural interpretation in terms of catastrophe theory. Equatorial orbits are associated with the fold catastrophe, while inclined orbits are described by the cusp catastrophe. In both cases, the transition to plunge corresponds to slow evolution across fold lines of the catastrophe manifold, providing a geometric explanation for the universal appearance of the Painlevé I equation in the transition dynamics.
ER -