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arXiv · 2606.13786

Universality in the Transition from Inspiral to Plunge: High-Accuracy Analytic Solutions and Catastrophe Theory

Ariadna Ribes Metidieri·Béatrice Bonga·Badri Krishnan·José Luis Jaramillo
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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 Painlev\'e I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlev\'e 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 tritronqu\'ee solution of Painlev\'e I. We then compare existing high-accuracy analytic approximations of the tritronqu\'ee solution with direct numerical integrations of the Painlev\'e 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 Painlev\'e I equation in the transition dynamics.

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Ariadna Ribes Metidieri, Béatrice Bonga, Badri Krishnan, José Luis Jaramillo. 2026-06-11. Universality in the Transition from Inspiral to Plunge: High-Accuracy Analytic Solutions and Catastrophe Theory. https://arxiv.org/abs/2606.13786

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