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Extreme-mass-ratio inspirals in relativistic accretion discs

Francisco Duque1, Laura Sberna2, Andrew Spiers2, and Rodrigo Vicente3

  • 1Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg 1, D-14476 Potsdam, Germany
  • 2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
  • 3Gravitation Astroparticle Physics Amsterdam (GRAPPA), University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands

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Phys. Rev. D 113, 084028 – Published 13 April, 2026

DOI: https://doi.org/10.1103/jcv5-ssfd

Abstract

We compute relativistic Lindblad torques for circular, equatorial extreme-mass-ratio inspirals (EMRIs) embedded in relativistic thin accretion discs, including spinning black hole configurations. We find that relativistic effects can amplify the magnitude of these torques by orders of magnitude in the strong-field regime, and that the torque can even reverse direction as the EMRI approaches the innermost stable circular orbit (ISCO). However, we show that the location of this reversal is highly spin-dependent, shifting progressively closer to the ISCO, where gravitational-wave emission completely dominates the inspiral, as the spin of the central black hole increases. Spin also modifies the radial dependence of the Lindblad torques. We investigate whether Lindblad torques can be approximated by parametrized power laws of the form TLR=A(rs/10M)nr (or combinations thereof), and find significant spin- and disc-dependent variations in the slope parameter nr. For instance, for spin a/M=0.9, we find nr=3.6 in the strong-field regime, compared to the Newtonian value of nr=4.5. Given current forecasts of parameter recovery for “golden,” loud EMRIs in accretion discs (Δnr∼0.5), we predict LISA could distinguish between different disc configurations through their relativistic Lindblad torque signatures, providing the first direct probe of the midplane structure of the inner region of accretion discs, which is inaccessible to electromagnetic observations.

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References (92)

  1. N. Neumayer, A. Seth, and T. Boeker, Nuclear star clusters, Astron. Astrophys. Rev. 28, 4 (2020).
  2. M. C. Miller and V. M. Lauburg, Mergers of stellar-mass black holes in nuclear star clusters, Astrophys. J. 692, 917 (2009).
  3. M. A. Abramowicz and P. C. Fragile, Foundations of black hole accretion disk theory, Living Rev. Relativity 16, 1 (2013).
  4. P. Gondolo and J. Silk, Dark matter annihilation at the galactic center, Phys. Rev. Lett. 83, 1719 (1999).
  5. R. Brito, V. Cardoso, and P. Pani, Superradiance: New frontiers in black hole physics, Lect. Notes Phys. 906, 1 (2015).
  6. L. Hui, Wave dark matter, Annu. Rev. Astron. Astrophys. 59, 247 (2021).
  7. B. Kocsis, N. Yunes, and A. Loeb, Observable signatures of EMRI black hole binaries embedded in thin accretion disks, Phys. Rev. D 84, 024032 (2011).
  8. E. Barausse, V. Cardoso, and P. Pani, Environmental effects for gravitational-wave astrophysics, J. Phys. Conf. Ser. 610, 012044 (2015).
  9. L. Speri, A. Antonelli, L. Sberna, S. Babak, E. Barausse, J. R. Gair, and M. L. Katz, Measuring accretion-disk effects with gravitational waves from extreme mass ratio inspirals, Phys. Rev. X 13, 021035 (2023).
  10. P. S. Cole, G. Bertone, A. Coogan, D. Gaggero, T. Karydas, B. J. Kavanagh, T. F. M. Spieksma, and G. M. Tomaselli, Distinguishing environmental effects on binary black hole gravitational waveforms, Nat. Astron. 7, 943 (2023).
  11. Y. Chen et al. (NANOGrav Collaboration), Galaxy tomography with the gravitational wave background from supermassive black hole binaries, arXiv:2411.05906.
  12. M. Garg, A. Derdzinski, L. Zwick, P. R. Capelo, and L. Mayer, The imprint of gas on gravitational waves from LISA intermediate-mass black hole binaries Mon. Not. R. Astron. Soc. 517, 1339 (2022).
  13. G. Caneva Santoro, S. Roy, R. Vicente, M. Haney, O. J. Piccinni, W. Del Pozzo, and M. Martinez, First constraints on compact binary environments from LIGO-Virgo data, Phys. Rev. Lett. 132, 251401 (2024).
  14. S. Roy and R. Vicente, Compact binary coalescences in dense gaseous environments can pose as ones in vacuum, Phys. Rev. D 111, 084037 (2025).
  15. L. Zwick, J. Takátsy, P. Saini, K. Hendriks, J. Samsing, C. Tiede, C. Rowan, and A. A. Trani, Environmental effects in stellar mass gravitational wave sources I: Expected fraction of signals with significant dephasing in the dynamical and AGN channels, Astrophys. J. 991, 131 (2025).
  16. G. M. Tomaselli, Scattering of wave dark matter by supermassive black holes, Phys. Rev. D 111, 063075 (2025).
  17. R. Chen, R. S. Chandramouli, F. Pozzoli, R. Buscicchio, and E. Barausse, Muffled murmurs: Environmental effects in the LISA stochastic signal from stellar-mass black hole binaries, Phys. Rev. D 112, 084053 (2025).
  18. L. Zwick, A. Derdzinski, M. Garg, P. R. Capelo, and L. Mayer, Dirty waveforms: Multiband harmonic content of gas-embedded gravitational wave sources, Mon. Not. R. Astron. Soc. 511, 6143 (2022).
  19. A. Derdzinski, D. D’Orazio, P. Duffell, Z. Haiman, and A. MacFadyen, Evolution of gas disc–embedded intermediate mass ratio inspirals in the LISA band, Mon. Not. R. Astron. Soc. 501, 3540 (2021).
  20. J. S. Santos, V. Cardoso, J. Natário, and M. van de Meent, Gravitational waves from b-EMRIs: Doppler shift and beaming, resonant excitation, helicity oscillations and self-lensing, Phys. Rev. Lett. 135, 211402 (2025).
  21. L. Hu, R.-G. Cai, and S.-J. Wang, Distinctive gwbs from eccentric inspiraling SMBH binaries with a DM spike, J. Cosmol. Astropart. Phys. 02 (2025) 067.
  22. E. Barausse, Relativistic dynamical friction in a collisional fluid, Mon. Not. R. Astron. Soc. 382, 826 (2007).
  23. V. Cardoso, K. Destounis, F. Duque, R. P. Macedo, and A. Maselli, Black holes in galaxies: Environmental impact on gravitational-wave generation and propagation, Phys. Rev. D 105, L061501 (2022).
  24. R. Vicente, T. K. Karydas, and G. Bertone, A fully relativistic treatment of emris in collisionless environments, Phys. Rev. Lett. 135, 211401 (2025).
  25. H. Khalvati, A. Santini, F. Duque, L. Speri, J. Gair, H. Yang, and R. Brito, Impact of relativistic waveforms in LISA’s science objectives with extreme-mass-ratio inspirals, Phys. Rev. D 111, 082010 (2025).
  26. R. Brito and S. Shah, Extreme mass-ratio inspirals into black holes surrounded by scalar clouds, Phys. Rev. D 108, 084019 (2023).
  27. V. Cardoso, K. Destounis, F. Duque, R. Panosso Macedo, and A. Maselli, Gravitational waves from extreme-mass-ratio systems in astrophysical environments, Phys. Rev. Lett. 129, 241103 (2022).
  28. S. Datta, Probing horizon scale quantum effects with Love, Classical Quantum Gravity 39, 225016 (2022).
  29. F. Duque, C. F. B. Macedo, R. Vicente, and V. Cardoso, Axion weak leaks: Extreme mass-ratio inspirals in ultra-light dark matter, Phys. Rev. Lett. 133, 121404 (2024).
  30. C. Dyson, T. F. Spieksma, R. Brito, M. van de Meent, and S. Dolan, Environmental effects in extreme mass ratio inspirals: Perturbations to the environment in Kerr, Phys. Rev. Lett. 134, 211403 (2025).
  31. L. Polcar and V. Witzany, Towards relativistic inspirals into black holes surrounded by matter, Phys. Rev. D 112, 104003 (2025).
  32. S. Xin and E. R. Most, Relativistic scalar dark matter drag forces on a black hole binary, arXiv:2507.18934.
  33. Y. Guo, Z. Zhong, Y. Chen, V. Cardoso, T. Ikeda, and L. Zhou, Ultralight boson ionization from comparable-mass binary black holes, arXiv:2509.09643.
  34. J. C. Aurrekoetxea, J. Marsden, K. Clough, and P. G. Ferreira, Self-interacting scalar dark matter around binary black holes, Phys. Rev. D 110, 083011 (2024).
  35. 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).
  36. D. Mancieri, L. Broggi, M. Vinciguerra, A. Sesana, and M. Bonetti, Eccentricity distribution of extreme mass ratio inspirals, Phys. Rev. D 113, 043062 (2026).
  37. M. Colpi et al., LISA definition study report, arXiv:2402.07571.
  38. Y. Gong, J. Luo, and B. Wang, Concepts and status of Chinese space gravitational wave detection projects, Nat. Astron. 5, 881 (2021).
  39. A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, Handbook of Gravitational Wave Astronomy (Springer, Singapore, 2022).
  40. A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, in Handbook of Gravitational Wave Astronomy (Springer, Singapore, 2022).
  41. F. Duque, S. Kejriwal, L. Sberna, L. Speri, and J. Gair, Constraining accretion physics with gravitational waves from eccentric extreme-mass-ratio inspirals, Phys. Rev. D 111, 084006 (2025).
  42. H. Sun, Y.-P. Li, Z. Pan, and H. Yang, Probing formation channels of extreme mass-ratio inspirals, arXiv:2509.00469.
  43. A. Franchini, M. Bonetti, A. Lupi, G. Miniutti, E. Bortolas, M. Giustini, M. Dotti, A. Sesana, R. Arcodia, and T. Ryu, Quasi-periodic eruptions from impacts between the secondary and a rigidly precessing accretion disc in an extreme mass-ratio inspiral system, Astron. Astrophys. 675, A100 (2023).
  44. I. Linial and B. D. Metzger, EMRI + TDE=QPE: Periodic X-ray flares from star–disk collisions in galactic nuclei, Astrophys. J. 957, 34 (2023).
  45. J. Chakraborty et al., Prospects for EMRI/MBH parameter estimation using quasi-periodic eruption timings: Short-timescale analysis, Astrophys. J. 992, 120 (2025).
  46. P. Goldreich and S. Tremaine, Disk-satellite interactions, Astrophys. J. 241, 425 (1980).
  47. P. Artymowicz, On the wave excitation and a generalized torque formula for Lindblad resonances excited by external potential, Astrophys. J. 419, 155 (1993).
  48. H. Tanaka, T. Takeuchi, and W. R. Ward, Three-dimensional interaction between a planet and an isothermal gaseous disk. I. Corotation and Lindblad torques and planet migration, Astrophys. J. 565, 1257 (2002).
  49. H. Tanaka and W. R. Ward, Three-dimensional interaction between a planet and an isothermal gaseous disk. II. Eccentricity waves and bending waves, Astrophys. J. 602, 388 (2004).
  50. H. Tanaka and K. Okada, Three-dimensional interaction between a planet and an isothermal gaseous disk. III. Locally isothermal cases, Astrophys. J. 968, 28 (2024).
  51. C. W. Fairbairn and R. R. Rafikov, Eccentric planet-disc interactions: Orbital migration and eccentricity evolution, Mon. Not. R. Astron. Soc. 537, 1779 (2025).
  52. N. Yunes, B. Kocsis, A. Loeb, and Z. Haiman, Imprint of accretion disk-induced migration on gravitational waves from extreme mass ratio inspirals, Phys. Rev. Lett. 107, 171103 (2011).
  53. S. Kejriwal, L. Speri, and A. J. K. Chua, Impact of correlations on the modeling and inference of beyond vacuum-GR effects in extreme-mass-ratio inspirals, Phys. Rev. D 110, 084060 (2024).
  54. C. M. Hirata, Lindblad resonance torques in relativistic discs: I. Basic equations, Mon. Not. R. Astron. Soc. 414, 3198 (2011).
  55. C. M. Hirata, Lindblad resonance torques in relativistic discs: II. Computation of resonance strengths, Mon. Not. R. Astron. Soc. 414, 3212 (2011).
  56. I. D. Novikov and J. B. Zeldovich, Physics of relativistic collapse, in International Conference on Relativistic Theories of Gravitation (London, 1965), Vol. 1, Report No. AD-629097, AD-628885.
  57. A. Hegade K. R., C. F. Gammie, and N. Yunes, A relativistic treatment of accretion disk torques on extreme mass-ratio inspirals around non-spinning black holes, Phys. Rev. D 112, 124012 (2025).
  58. A. Hegade K. R., C. F. Gammie, and N. Yunes, A relativistic treatment of accretion disk torques on extreme mass ratio inspirals around spinning black holes, Phys. Rev. D 112, 124068 (2025).
  59. J. M. Bardeen and J. A. Petterson, The lense-thirring effect and accretion disks around Kerr black holes, Astrophys. J. Lett. 195, L65 (1975).
  60. P. Natarajan and J. E. Pringle, The alignment of disk and black hole spins in active galactic nuclei, Astrophys. J. Lett. 506, L97 (1998).
  61. K. Chatterjee, N. Kaaz, M. Liska, A. Tchekhovskoy, and S. Markoff, Misaligned magnetized accretion flows onto spinning black holes: Magneto-spin alignment, outflow power, and intermittent jets, Phys. Rev. D 112, 063013 (2025).
  62. A. Lobban and A. King, AGN light echoes and the accretion disc self-gravity limit, Mon. Not. R. Astron. Soc. 511, 1992 (2022).
  63. T. F. M. Spieksma and E. Cannizzaro, In the grip of the disk: Dragging the companion through an AGN, Mon. Not. R. Astron. Soc. 546, stag021 (2026).
  64. Y. Wang, Z. Zhu, and D. N. C. Lin, Stellar/BH population in AGN discs: Direct binary formation from capture objects in nuclei clusters, Mon. Not. R. Astron. Soc. 528, 4958 (2024).
  65. S. A. Teukolsky, Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. Lett. 29, 1114 (1972).
  66. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  67. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  68. S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon Press Oxford, 1992), reprint: 2009.
  69. S. A. Teukolsky and W. Press, Perturbations of a rotating black hole. III-interaction of the hole with gravitational and electromagnetic radiation, Astrophys. J. 193, 443 (1974).
  70. 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).
  71. M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Rev. Relativity 6, 6 (2003).
  72. Z. Nasipak, Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit, arXiv:2507.07746.
  73. https://github.com/FranciscoDuque/Relativistic-Lindblad-Torques
  74. W. R. Ward, Protoplanet migration by nebula tides, Icarus 126, 261 (1997).
  75. D. Gangardt, A. A. Trani, C. Bonnerot, and D. Gerosa, pAGN: The one-stop solution for AGN disc modeling, Mon. Not. R. Astron. Soc. 530, 3689 (2024).
  76. P. Goldreich and S. Tremaine, The excitation of density waves at the Lindblad and corotation resonances by an external potential., Astrophys. J. 233, 857 (1979).
  77. I. D. Novikov and K. S. Thorne, Astrophysics and black holes, in Astrophysics of Black Holes (1973), pp. 343–550, https://ui.adsabs.harvard.edu/abs/1973blho.conf..343N/abstract.
  78. N. I. Shakura and R. A. Sunyaev, Black holes in binary systems. Observational appearance, Astron. Astrophys. 24, 337 (1973), https://ui.adsabs.harvard.edu/abs/1973A%26A....24..337S/abstract.
  79. W. J. Potter, A full relativistic thin disc—the physics of the plunging region and the value of the stress at the ISCO, Mon. Not. R. Astron. Soc. 503, 5025 (2021).
  80. J. A. García, A. C. Fabian, T. R. Kallman, T. Dauser, M. L. Parker, J. E. McClintock, J. F. Steiner, and J. Wilms, High-density effects in X-ray reflection models from accretion disks, Mon. Not. R. Astron. Soc. 462, 751 (2016).
  81. Y. Wu, Y.-X. Chen, and D. N. C. Lin, Chaotic type I migration in turbulent discs, Mon. Not. R. Astron. Soc. 528, L127 (2023).
  82. C. E. A. Chapman-Bird et al., The fast and the frame-dragging: Efficient waveforms for asymmetric-mass eccentric equatorial inspirals into rapidly spinning black holes, Phys. Rev. D 112, 104023 (2025).
  83. E. Berti and M. Volonteri, Cosmological black hole spin evolution by mergers and accretion, Astrophys. J. 684, 822 (2008).
  84. A. R. King, J. E. Pringle, and J. A. Hofmann, The evolution of black hole mass and spin in active galactic nuclei, Mon. Not. R. Astron. Soc. 385, 1621 (2008).
  85. C. Dyson and D. J. D’Orazio, Spiral density waves and torque balance in the Kerr geometry, arXiv:2601.19123.
  86. L. Copparoni, E. Barausse, L. Speri, L. Sberna, and A. Derdzinski, Implications of stochastic gas torques for asymmetric binaries in the LISA band, Phys. Rev. D 111, 104079 (2025).
  87. P. F. Hopkins and E. Quataert, An analytic model of angular momentum transport by gravitational torques: From galaxies to massive black holes, Mon. Not. R. Astron. Soc. 415, 1027 (2011).
  88. F. J. Sánchez-Salcedo, R. O. Chametla, and A. Santillán, Torques on low-mass bodies in retrograde orbit in gaseous disks, Astrophys. J. 860, 129 (2018).
  89. M. Silva, H. G. Blake-Goszyk, and C. M. Hirata, Resonant interactions from dynamical perturbers on generic orbits around an extreme mass ratio inspiral, Phys. Rev. D 112, 084012 (2025).
  90. S.-J. Paardekooper, C. Baruteau, and W. Kley, A torque formula for non-isothermal type I planetary migration—II. Effects of diffusion, Mon. Not. R. Astron. Soc. 410, 293 (2011).
  91. R. P. Nelson and J. C. B. Papaloizou, The interaction of giant planets with a disc with MHD turbulence—IV. Migration rates of embedded protoplanets, Mon. Not. R. Astron. Soc. 350, 849 (2004).
  92. C. Baruteau, S. Fromang, R. P. Nelson, and F. Masset, Corotation torques experienced by planets embedded in weakly magnetized turbulent discs, Astron. Astrophys. 533, A84 (2011).

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Phys. Rev. D 113, 084028– Published 13 April, 2026
  • Received 22 October 2025
  • Accepted 2 March 2026
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