- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Universality in the transition from inspiral to plunge for extreme-mass-ratio-inspirals: High-accuracy analytic solutions and catastrophe theory

Ariadna Ribes Metidieri1,2,3,*, Béatrice Bonga3,†, Badri Krishnan3,2,4,‡, and José Luis Jaramillo5,§

  • *Contact author: ariadna.metidieri@nbi.ku.dk
  • †Contact author: bbonga@science.ru.nl
  • ‡Contact author: badri.krishnan@ru.nl
  • §Contact author: Jose-Luis.Jaramillo-Martin@ube.fr
PDF

Phys. Rev. D 114, 064084 – Published 28 September, 2026

DOI: https://doi.org/10.1103/5y8s-7kqp

Abstract

We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasicircular, 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 quasicircular inspiral at early times. We argue that these conditions uniquely select the so-called Boutroux’s 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.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. LIGO Scientific, Virgo, and KAGRA Collaborations, arXiv:2605.27225.
  2. M. Evans et al., arXiv:2109.09882.
  3. D. Reitze et al., Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833.
  4. M. Maggiore et al. (ET Collaboration), J. Cosmol. Astropart. Phys. 03 (2020) 050.
  5. B. Sathyaprakash et al., Classical Quantum Gravity 29, 124013 (2012); 30, 079501(E) (2013).
  6. M. Colpi et al. (LISA Collaboration), arXiv:2402.07571.
  7. J. Luo et al. (TianQin Collaboration), Classical Quantum Gravity 33, 035010 (2016).
  8. W.-R. Hu and Y.-L. Wu, Natl. Sci. Rev. 4, 685 (2017).
  9. S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. L. Berry, E. Berti, P. Amaro-Seoane, A. Petiteau, and A. Klein, Phys. Rev. D 95, 103012 (2017).
  10. A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000).
  11. E. A. Huerta, P. Kumar, J. R. Gair, and S. T. McWilliams, Phys. Rev. D 90, 024024 (2014).
  12. L. Küchler, G. Compère, L. Durkan, and A. Pound, SciPost Phys. 17, 056 (2024).
  13. A. Apte and S. A. Hughes, Phys. Rev. D 100, 084031 (2019).
  14. H. Lim, G. Khanna, A. Apte, and S. A. Hughes, Phys. Rev. D 100, 084032 (2019).
  15. G. Compère, K. Fransen, and C. Jonas, Classical Quantum Gravity 37, 095013 (2020).
  16. G. Compère and L. Küchler, SciPost Phys. 13, 043 (2022).
  17. G. Compère and L. Küchler, Phys. Rev. Lett. 126, 241106 (2021).
  18. A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000).
  19. J. L. Jaramillo and B. Krishnan, arXiv:2211.03405.
  20. A. Adali and S. Tanveer, J. Differ. Equations 261, 3843 (2016).
  21. M. Berry, Phys. Bull. 32, 84 (1981).
  22. T. Poston and I. Stewart, Catastrophe Theory and Its Applications, Dover Books on Mathematics (Dover Publications, New York, 1996).
  23. J. L. Jaramillo and B. Krishnan, arXiv:2206.02117.
  24. B. Bonga, J. Feldbrugge, and A. Ribes Metidieri, Phys. Rev. D 111, 063061 (2025).
  25. A. M. Serra and O. Bulashenko, Phys. Rev. D 112, 024041 (2025).
  26. J. M. Ezquiaga, R. K. L. Lo, and L. Vujeva, Phys. Rev. D 112, 043544 (2025).
  27. N. Loutrel, Classical Quantum Gravity 40, 215004 (2023).
  28. M. Boschini, N. Loutrel, D. Gerosa, and G. Fumagalli, Phys. Rev. D 111, 024008 (2025).
  29. M. Gadioux and H. S. Reall, Phys. Rev. D 108, 084021 (2023).
  30. M. Gadioux, R. A. Hennigar, and H. S. Reall, Phys. Rev. D 110, 084029 (2024).
  31. O. Meneses Rojas, Singularity theory and strong gravity: Caustics and wave fronts as probes into black holes, Theses, Université Bourgogne Europe, 2025.
  32. J. L. Jaramillo, B. Krishnan, and C. F. Sopuerta, Int. J. Mod. Phys. D 32, 2342005 (2023).
  33. J. Jaramillo, L. Al Sheikh, J. Besson, B. Krishnan, M. Lenzi, R. Macedo, O. Meneses-Rojas, B. Raffaelli, C. Sopuerta, and C. Vitel, J. Phys. Conf. Ser. 3177, 012009 (2026).
  34. B. Carter, Phys. Rev. 174, 1559 (1968).
  35. Y. Mino, Phys. Rev. D 67, 084027 (2003).
  36. C. Dyson and M. van de Meent, Classical Quantum Gravity 40, 195026 (2023).
  37. S. Drasco and S. A. Hughes, Phys. Rev. D 73, 024027 (2006); 88, 109905(E) (2013); 90, 109905(E) (2014).
  38. W. Schmidt, Classical Quantum Gravity 19, 2743 (2002).
  39. L. Küchler, G. Compère, and A. Pound, Classical Quantum Gravity 43, 015018 (2026).
  40. L. Honet, L. Küchler, A. Pound, and G. Compère, Phys. Rev. D 113, 044051 (2026).
  41. M. Kesden, Phys. Rev. D 83, 104011 (2011).
  42. DLMF, NIST Digital Library of Mathematical Functions, edited by f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, https://dlmf.nist.gov/ (Release 1.2.6 of 2026-03-15).
  43. P. Boutroux, Ann. Sci. l’École Norm. Supér. 3e, 99 (1914).
  44. M. Huang and S. Tanveer, Duke Math. J. 163, 665 (2012), https://projecteuclid.org/journals/duke-mathematical-journal/volume-163/issue-4/Proof-of-the-Dubrovin-conjecture-and-analysis-of-the-tritronqu%C3%A9e/10.1215/00127094-2429589.full.
  45. L. Küchler (private communication).
  46. Black Hole Perturbation Toolkit (https://bhptoolkit.org).
  47. R. Thom, Stabilité structurelle et morphogenèse (W. A. Benjamin, New York, 1972).
  48. S. Kowalevski, Acta Math. 12, 177 (1889).
  49. R. Conte and M. Musette, The Painlevé Handbook (Springer, Cham, 2020).
  50. E. Zeeman, Catastrophe Theory: Selected Papers, 1972-1977, Advanced Book Program (Addison-Wesley Publishing Company, Reading, Massachusetts, 1977).
  51. V. I. Arnol’d, Funct. Anal. Appl. 6, 254 (1972).
  52. S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences (Clarendon Press, Oxford, 1983).
  53. N. Warburton, L. Barack, and N. Sago, Phys. Rev. D 87, 084012 (2013).
  54. G. Compère, Y. Liu, and J. Long, Phys. Rev. D 105, 024075 (2022).
  55. L. C. Stein and N. Warburton, Phys. Rev. D 101, 064007 (2020).
  56. R. Haberman, SIAM J. Appl. Math. 37, 69 (1979).
  57. R. W. Batterman, The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence (Oxford University Press, New York, 2001).
  58. A. J. K. Chua, M. L. Katz, N. Warburton, and S. A. Hughes, Phys. Rev. Lett. 126, 051102 (2021).
  59. M. L. Katz, A. J. K. Chua, L. Speri, N. Warburton, and S. A. Hughes, Phys. Rev. D 104, 064047 (2021).

Outline

Information

Phys. Rev. D 114, 064084– Published 28 September, 2026
  • Received 26 June 2026
  • Accepted 1 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation