- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

New approach to the calculation of extreme-mass-ratio inspirals with a spinning secondary

Viktor Skoupý*

  • Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, CZ-180 00 Prague, Czech Republic

  • *Contact author: viktor.skoupy@matfyz.cuni.cz
PDF

Phys. Rev. D 114, 084030 – Published 9 October, 2026

DOI: https://doi.org/10.1103/kpj2-bftm

Abstract

Extreme-mass-ratio inspirals (EMRIs) are among the most promising sources for future space-based gravitational-wave (GW) detectors, such as the Laser Interferometer Space Antenna. To fully leverage the scientific potential, the GW templates required for parameter estimation must be modeled with high accuracy for eccentric precessing binary systems with nonzero spins. This work introduces a practical and efficient framework for incorporating the effects of secondary spin in fully generic, eccentric, and off-equatorial EMRIs to the first postadiabatic order. We utilize recently found analytic solutions for the trajectories of spinning bodies in Kerr spacetime to significantly simplify the calculation of the corresponding asymptotic GW fluxes. Furthermore, thanks to the recently proven flux-balance laws, we show how to express the rates of change of the constants of motion, including the Carter-Rüdiger constant, using asymptotic Teukolsky amplitudes and purely geodesic functions that are already established in the literature. Finally, we show how this framework performs in the case of nearly spherical inspirals and numerically demonstrate that the linear-in-spin part of the phases is gauge independent. A Wolfram Mathematica implementation of the code developed in this work is publicly available in the kerrspinningfluxes package.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. P. Amaro-Seoane et al. (LISA Collaboration), Laser interferometer space antenna, arXiv:1702.00786.
  2. M. Colpi et al., LISA definition study report, arXiv:2402.07571.
  3. J. Luo et al. (TianQin Collaboration), TianQin: A space-borne gravitational wave detector, Classical Quantum Gravity 33, 035010 (2016).
  4. W.-H. Ruan, C. Liu, Z.-K. Guo, Y.-L. Wu, and R.-G. Cai, The LISA-Taiji network, Nat. Astron. 4, 108 (2020).
  5. L. Barack and C. Cutler, Using LISA EMRI sources to test off-Kerr deviations in the geometry of massive black holes, Phys. Rev. D 75, 042003 (2007).
  6. J. R. Gair, M. Vallisneri, S. L. Larson, and J. G. Baker, Testing general relativity with low-frequency, space-based gravitational-wave detectors, Living Rev. Relativity 16, 7 (2013).
  7. 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, Science with the space-based interferometer LISA. V: Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012 (2017).
  8. E. Poisson, A. Pound, and I. Vega, The motion of point particles in curved spacetime, Living Rev. Relativity 14, 7 (2011).
  9. L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rep. Prog. Phys. 82, 016904 (2019).
  10. A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, in Handbook of Gravitational Wave Astronomy (Springer, Singapore, 2021).
  11. T. Hinderer and E. E. Flanagan, Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital motion, Phys. Rev. D 78, 064028 (2008).
  12. M. van de Meent and N. Warburton, Fast self-forced inspirals, Classical Quantum Gravity 35, 144003 (2018).
  13. P. Lynch, M. van de Meent, and N. Warburton, Eccentric self-forced inspirals into a rotating black hole, Classical Quantum Gravity 39, 145004 (2022).
  14. P. Lynch, M. van de Meent, and N. Warburton, Self-forced inspirals with spin-orbit precession, Phys. Rev. D 109, 084072 (2024).
  15. J. Mathews and A. Pound, Post-adiabatic waveform-generation framework for asymmetric precessing binaries, Phys. Rev. D 112, 104078 (2025).
  16. O. Burke, G. A. Piovano, N. Warburton, P. Lynch, L. Speri, C. Kavanagh, B. Wardell, A. Pound, L. Durkan, and J. Miller, Assessing the importance of first postadiabatic terms for small-mass-ratio binaries, Phys. Rev. D 109, 124048 (2024).
  17. C. Chapman-Bird, A. J. K. Chua, S. Hughes, M. Katz, Z. Nasipak, M. Pigou, L. Speri, and N. Warburton, fastemriwaveforms (2025), 10.5281/zenodo.15630565.
  18. A. J. K. Chua, M. L. Katz, N. Warburton, and S. A. Hughes, Rapid generation of fully relativistic extreme-mass-ratio-inspiral waveform templates for LISA data analysis, Phys. Rev. Lett. 126, 051102 (2021).
  19. M. L. Katz, A. J. K. Chua, L. Speri, N. Warburton, and S. A. Hughes, Fast extreme-mass-ratio-inspiral waveforms: New tools for millihertz gravitational-wave data analysis, Phys. Rev. D 104, 064047 (2021).
  20. L. Speri, M. L. Katz, A. J. K. Chua, S. A. Hughes, N. Warburton, J. E. Thompson, C. E. A. Chapman-Bird, and J. R. Gair, Fast and Fourier: Extreme mass ratio inspiral waveforms in the frequency domain, Front. Appl. Math. Stat. 9, 1266739 (2023).
  21. C. E. A. Chapman-Bird, L. Speri, Z. Nasipak, O. Burke, M. L. Katz, A. Santini, S. Kejriwal, P. Lynch, J. Mathews, H. Khalvati, J. E. Thompson, S. Isoyama, S. A. Hughes, N. Warburton, A. J. K. Chua, and M. Pigou, Efficient waveforms for asymmetric-mass eccentric equatorial inspirals into rapidly spinning black holes, Phys. Rev. D 112, 104023 (2025).
  22. N. Sago, T. Tanaka, W. Hikida, K. Ganz, and H. Nakano, Adiabatic evolution of orbital parameters in Kerr spacetime, Prog. Theor. Phys. 115, 873 (2006).
  23. S. Akcay, S. R. Dolan, C. Kavanagh, J. Moxon, N. Warburton, and B. Wardell, Dissipation in extreme-mass ratio binaries with a spinning secondary, Phys. Rev. D 102, 064013 (2020).
  24. J. Mathews, A. Pound, and B. Wardell, Self-force calculations with a spinning secondary, Phys. Rev. D 105, 084031 (2022).
  25. A. M. Grant, Flux-balance laws for spinning bodies under the gravitational self-force, Phys. Rev. D 111, 084015 (2025).
  26. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  27. S. Drasco and S. A. Hughes, Gravitational wave snapshots of generic extreme mass ratio inspirals, Phys. Rev. D 73, 024027 (2006).
  28. T. Tanaka, Y. Mino, M. Sasaki, and M. Shibata, Gravitational waves from a spinning particle in circular orbits around a rotating black hole, Phys. Rev. D 54, 3762 (1996).
  29. W.-B. Han, Gravitational radiations from a spinning compact object around a supermassive Kerr black hole in circular orbit, Phys. Rev. D 82, 084013 (2010).
  30. E. Harms, G. Lukes-Gerakopoulos, S. Bernuzzi, and A. Nagar, Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole, Phys. Rev. D 93, 044015 (2016); 100, 129901(A) (2019).
  31. E. Harms, G. Lukes-Gerakopoulos, S. Bernuzzi, and A. Nagar, Spinning test body orbiting around a Schwarzschild black hole: Circular dynamics and gravitational-wave fluxes, Phys. Rev. D 94, 104010 (2016).
  32. G. Lukes-Gerakopoulos, E. Harms, S. Bernuzzi, and A. Nagar, Spinning test-body orbiting around a Kerr black hole: Circular dynamics and gravitational-wave fluxes, Phys. Rev. D 96, 064051 (2017).
  33. A. Nagar, F. Messina, C. Kavanagh, G. Lukes-Gerakopoulos, N. Warburton, S. Bernuzzi, and E. Harms, Factorization and resummation: A new paradigm to improve gravitational wave amplitudes. III. The spinning test-body terms, Phys. Rev. D 100, 104056 (2019).
  34. G. A. Piovano, A. Maselli, and P. Pani, Extreme mass ratio inspirals with spinning secondary: A detailed study of equatorial circular motion, Phys. Rev. D 102, 024041 (2020).
  35. V. Skoupý and G. Lukes-Gerakopoulos, Spinning test body orbiting around a Kerr black hole: Eccentric equatorial orbits and their asymptotic gravitational-wave fluxes, Phys. Rev. D 103, 104045 (2021).
  36. N. Warburton, T. Osburn, and C. R. Evans, Evolution of small-mass-ratio binaries with a spinning secondary, Phys. Rev. D 96, 084057 (2017).
  37. G. A. Piovano, R. Brito, A. Maselli, and P. Pani, Assessing the detectability of the secondary spin in extreme mass-ratio inspirals with fully relativistic numerical waveforms, Phys. Rev. D 104, 124019 (2021).
  38. V. Skoupý and G. Lukes-Gerakopoulos, Adiabatic equatorial inspirals of a spinning body into a Kerr black hole, Phys. Rev. D 105, 084033 (2022).
  39. M. Mapelli, Y. Bouffanais, F. Santoliquido, M. A. Sedda, and M. C. Artale, The cosmic evolution of binary black holes in young, globular, and nuclear star clusters: Rates, masses, spins, and mixing fractions, Mon. Not. R. Astron. Soc. 511, 5797 (2022).
  40. L. V. Drummond and S. A. Hughes, Precisely computing bound orbits of spinning bodies around black holes. I. General framework and results for nearly equatorial orbits, Phys. Rev. D 105, 124040 (2022).
  41. L. V. Drummond and S. A. Hughes, Precisely computing bound orbits of spinning bodies around black holes. II. Generic orbits, Phys. Rev. D 105, 124041 (2022).
  42. V. Skoupý, G. Lukes-Gerakopoulos, L. V. Drummond, and S. A. Hughes, Asymptotic gravitational-wave fluxes from a spinning test body on generic orbits around a Kerr black hole, Phys. Rev. D 108, 044041 (2023).
  43. G. A. Piovano, C. Pantelidou, J. Mac Uilliam, and V. Witzany, Spinning particles near Kerr black holes: Orbits and gravitational-wave fluxes through the Hamilton-Jacobi formalism, Phys. Rev. D 111, 044009 (2025).
  44. V. Witzany, J. Steinhoff, and G. Lukes-Gerakopoulos, Hamiltonians and canonical coordinates for spinning particles in curved space-time, Classical Quantum Gravity 36, 075003 (2019).
  45. V. Witzany, Hamilton-Jacobi equation for spinning particles near black holes, Phys. Rev. D 100, 104030 (2019).
  46. L. V. Drummond, P. Lynch, A. G. Hanselman, D. R. Becker, and S. A. Hughes, Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations, Phys. Rev. D 109, 064030 (2024).
  47. V. Skoupý, G. A. Piovano, and V. Witzany, Spherical inspirals of spinning bodies into Kerr black holes, Phys. Rev. D 112, 124054 (2025).
  48. L. V. Drummond, S. A. Hughes, V. Skoupý, P. Lynch, and G. A. Piovano, Shifted-geodesic approximation for spinning-body gravitational wave fluxes, arXiv:2603.12189.
  49. V. Skoupý and V. Witzany, Analytic solution for the motion of spinning particles in Kerr spacetime, Phys. Rev. Lett. 134, 171401 (2025).
  50. V. Witzany, V. Skoupý, L. C. Stein, and S. Tanay, Actions of spinning compact binaries: Spinning particle in Kerr matched to dynamics at 1.5 post-Newtonian order, Phys. Rev. D 111, 044032 (2025).
  51. V. Skoupý, kerrspinningfluxes version v1.1.0 (2026), 10.5281/zenodo.18377281.
  52. M. Mathisson, Republication of: New mechanics of material systems, Gen. Relativ. Gravit. 42, 1011 (2010).
  53. A. Papapetrou, Spinning test particles in general relativity. 1, Proc. R. Soc. A 209, 248 (1951).
  54. W. G. Dixon, Dynamics of extended bodies in general relativity. I. Momentum and angular momentum, Proc. R. Soc. A 314, 499 (1970).
  55. W. Tulczyjew, Motion of multipole particles in general relativity theory, Acta Phys. Pol. 18, 393 (1959), https://www.actaphys.uj.edu.pl/fulltext?series=T&vol=18&no=5&page=393.
  56. R. Rüdiger, Conserved quantities of spinning test particles in general relativity. I, Proc. R. Soc. A 375, 185 (1981).
  57. R. Rüdiger, Conserved quantities of spinning test particles in general relativity. II, Proc. R. Soc. A 385, 229 (1983).
  58. B. Carter, Global structure of the kerr family of gravitational fields, Phys. Rev. 174, 1559 (1968).
  59. R. Fujita and W. Hikida, Analytical solutions of bound timelike geodesic orbits in Kerr spacetime, Classical Quantum Gravity 26, 135002 (2009).
  60. M. van de Meent, Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime, Classical Quantum Gravity 37, 145007 (2020).
  61. J.-A. Marck, Solution to the equations of parallel transport in kerr geometry; tidal tensor, Proc. R. Soc. A 385, 431 (1983).
  62. G. A. Piovano, Particles with precessing spin in Kerr spacetime: Analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane, Phys. Rev. D 113, 064024 (2026).
  63. S. A. Hughes, N. Warburton, G. Khanna, A. J. K. Chua, and M. L. Katz, Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency, Phys. Rev. D 103, 104014 (2021).
  64. J. Mathews, B. Wardell, A. Pound, and N. Warburton, Post-adiabatic self-force waveforms: Slowly spinning primary and precessing secondary, Phys. Rev. D 113, 064034 (2026).
  65. L. Honet, J. Mathews, G. Compère, A. Pound, B. Wardell, G. A. Piovano, M. van de Meent, and N. Warburton, Spin-aligned inspiral waveforms from self-force and post-Newtonian theory, arXiv:2510.16112.
  66. V. Skoupý and V. Witzany, Post-Newtonian expansions of extreme mass ratio inspirals of spinning bodies into Schwarzschild black holes, Phys. Rev. D 110, 084061 (2024).
  67. Black hole perturbation toolkit, http://bhptoolkit.org/ (2026).
  68. N. Warburton, B. Wardell, O. Long, S. Upton, P. Lynch, Z. Nasipak, and L. C. Stein, Kerrgeodesics (2023), 10.5281/zenodo.8108254.
  69. B. Wardell, N. Warburton, K. Cunningham, L. Durkan, B. Leather, Z. Nasipak, C. Kavanagh, A. Ottewill, M. Casals, T. Torres, J. Neef, and S. Barsanti, Teukolsky (2025), 10.5281/zenodo.15745889.
  70. W. Schmidt, Celestial mechanics in Kerr spacetime, Classical Quantum Gravity 19, 2743 (2002).
  71. N. Warburton, L. Barack, and N. Sago, Isofrequency pairing of geodesic orbits in Kerr geometry, Phys. Rev. D 87, 084012 (2013).

Outline

Information

Phys. Rev. D 114, 084030– Published 9 October, 2026
  • Received 17 March 2026
  • Accepted 14 September 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation