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Constraining deviations from the Teukolsky equation with GW250114

Sebastian H. Völkel1,2,* and Nicola Franchini3,†

  • *Contact author: sebastian.voelkel@uni-tuebingen.de
  • †Contact author: nicola.franchini@tecnico.ulisboa.pt

Phys. Rev. D 114, 104038 – Published 10 November, 2026

DOI: https://doi.org/10.1103/zzzw-vpgx

Abstract

The recent gravitational-wave detection GW250114 by the LIGO-Virgo-KAGRA (LVK) Collaboration provides unprecedented opportunities for precision tests of general relativity (GR) through black hole ringdown. In this study, we provide the first bounds on theory-agnostic deviations from the Teukolsky equation as described by the beyond-Teukolsky formalism. It directly connects deviations in the perturbation equations on the level of the effective potential in the Teukolsky equation with changes in the quasinormal mode (QNM) spectrum. We incorporate information on the final mass and spin from a full LVK inspiral-merger-ringdown analysis as parametrized priors in our analysis, reflecting theoretical uncertainties. Using publicly available LVK posterior information on agnostic damped sinusoid parameters, we then demonstrate how much beyond-Teukolsky potentials can be constrained. The high signal-to-noise ratio (SNR) allows us to avoid the expensive full Bayesian analysis of all parameters and to work directly with a simplified likelihood for the fundamental QNM only. This strategy is promising for future events with even higher SNR and allows, in principle, for a quick and simple test of GR without performing the full data analysis procedure, and provides a basis for complementary theory-specific tests. We report that current bounds on deviation parameters are in agreement with the Teukolsky equation.

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References (110)

  1. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9, 031040 (2019).
  3. R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GWTC-2: Compact binary coalescences observed by LIGO and Virgo during the first half of the third Observing Run, Phys. Rev. X 11, 021053 (2021).
  4. R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run, Phys. Rev. D 109, 022001 (2024).
  5. R. Abbott et al. (KAGRA Collaboration, VIRGO Collaboration, and LIGO Scientific Collaboration), GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the Third Observing Run, Phys. Rev. X 13, 041039 (2023).
  6. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and 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, Astrophys. J. Lett. 1004, L22 (2026).
  7. S. L. Detweiler, Black holes and gravitational waves. III—The resonant frequencies of rotating holes, Astrophys. J. 239, 292 (1980).
  8. F. Echeverria, Gravitational wave measurements of the mass and angular momentum of a black hole, Phys. Rev. D 40, 3194 (1989).
  9. L. S. Finn, Detection, measurement and gravitational radiation, Phys. Rev. D 46, 5236 (1992).
  10. O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Garrison, and R. Lopez-Aleman, Black hole spectroscopy: Testing general relativity through gravitational wave observations, Classical Quantum Gravity 21, 787 (2004).
  11. E. Berti, V. Cardoso, and C. M. Will, On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73, 064030 (2006).
  12. G. Carullo, W. Del Pozzo, and J. Veitch, Observational black hole spectroscopy: A time-domain multimode analysis of GW150914, Phys. Rev. D 99, 123029 (2019); 100, 089903(E) (2019).
  13. R. Brito, A. Buonanno, and V. Raymond, Black-hole spectroscopy by making full use of gravitational-wave modeling, Phys. Rev. D 98, 084038 (2018).
  14. S. Ma, K. Mitman, L. Sun, N. Deppe, F. Hébert, L. E. Kidder, J. Moxon, W. Throwe, N. L. Vu, and Y. Chen, Quasinormal-mode filters: A new approach to analyze the gravitational-wave ringdown of binary black-hole mergers, Phys. Rev. D 106, 084036 (2022).
  15. M. Isi and W. M. Farr, Analyzing black-hole ringdowns, arXiv:2107.05609.
  16. K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Relativity 2, 2 (1999).
  17. H.-P. Nollert, TOPICAL REVIEW: Quasinormal modes: The characteristic ‘sound’ of black holes and neutron stars, Classical Quantum Gravity 16, R159 (1999).
  18. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  19. R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83, 793 (2011).
  20. E. Berti, K. Yagi, H. Yang, and N. Yunes, Extreme gravity tests with gravitational waves from compact binary coalescences: (II) Ringdown, Gen. Relativ. Gravit. 50, 49 (2018).
  21. N. Franchini and S. H. Völkel, Testing general relativity with black hole quasi-normal modes, in Recent Progress on Gravity Tests, edited by C. Bambi and A. Cárdenas-Avendaño, Springer Series in Astrophysics and Cosmology (Springer, Singapore, 2024), 10.1007/978-981-97-2871-8_9.
  22. E. Berti et al., Black hole spectroscopy: From theory to experiment, Classical Quantum Gravity 43, 123001 (2026).
  23. R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  24. S. A. Teukolsky, Rotating black holes—separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. Lett. 29, 1114 (1972).
  25. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  26. W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164, 1776 (1967).
  27. S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).
  28. B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26, 331 (1971).
  29. D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34, 905 (1975).
  30. A. Buonanno, G. B. Cook, and F. Pretorius, Inspiral, merger and ring-down of equal-mass black-hole binaries, Phys. Rev. D 75, 124018 (2007).
  31. E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa, and B. Bruegmann, Inspiral, merger and ringdown of unequal mass black hole binaries: A Multipolar analysis, Phys. Rev. D 76, 064034 (2007).
  32. V. Baibhav, M. H.-Y. Cheung, E. Berti, V. Cardoso, G. Carullo, R. Cotesta, W. Del Pozzo, and F. Duque, Agnostic black hole spectroscopy: Quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory, Phys. Rev. D 108, 104020 (2023).
  33. A. Buonanno, L. E. Kidder, and L. Lehner, Estimating the final spin of a binary black hole coalescence, Phys. Rev. D 77, 026004 (2008).
  34. L. Rezzolla, E. Barausse, E. Nils Dorband, D. Pollney, C. Reisswig, J. Seiler, and S. Husa, On the final spin from the coalescence of two black holes, Phys. Rev. D 78, 044002 (2008).
  35. W. Tichy and P. Marronetti, The Final mass and spin of black hole mergers, Phys. Rev. D 78, 081501(R) (2008).
  36. E. Barausse and L. Rezzolla, Predicting the direction of the final spin from the coalescence of two black holes, Astrophys. J. Lett. 704, L40 (2009).
  37. J. Healy, C. O. Lousto, and Y. Zlochower, Remnant mass, spin, and recoil from spin aligned black-hole binaries, Phys. Rev. D 90, 104004 (2014).
  38. I. Kamaretsos, M. Hannam, and B. S. Sathyaprakash, Is black-hole ringdown a memory of its progenitor?, Phys. Rev. Lett. 109, 141102 (2012).
  39. L. London, D. Shoemaker, and J. Healy, Modeling ringdown: Beyond the fundamental quasinormal modes, Phys. Rev. D 90, 124032 (2014); 94, 069902(E) (2016).
  40. M. H.-Y. Cheung, E. Berti, V. Baibhav, and R. Cotesta, Extracting linear and nonlinear quasinormal modes from black hole merger simulations, Phys. Rev. D 109, 044069 (2024); 110, 049902(E) (2024); 112, 049901(E) (2025).
  41. C. Pacilio, S. Bhagwat, F. Nobili, and D. Gerosa, Flexible mapping of ringdown amplitudes for nonprecessing binary black holes, Phys. Rev. D 110, 103037 (2024).
  42. L. Magaña Zertuche et al., High-precision ringdown surrogate model for nonprecessing binary black holes, Phys. Rev. D 112, 024077 (2025).
  43. G. Carullo, Ringdown amplitudes of nonspinning eccentric binaries, J. Cosmol. Astropart. Phys. 10 (2024) 061.
  44. K. Mitman et al., Probing the ringdown perturbation in binary black hole coalescences with an improved quasinormal mode extraction algorithm, Phys. Rev. D 112, 064016 (2025).
  45. F. Nobili, S. Bhagwat, C. Pacilio, and D. Gerosa, Ringdown mode amplitudes of precessing binary black holes, Phys. Rev. D 112, 044058 (2025).
  46. L. Gao et al., Robustness of extracting quasinormal mode information from black hole merger simulations, Phys. Rev. D 112, 024025 (2025).
  47. L. Sberna, P. Bosch, W. E. East, S. R. Green, and L. Lehner, Nonlinear effects in the black hole ringdown: Absorption-induced mode excitation, Phys. Rev. D 105, 064046 (2022).
  48. M. H.-Y. Cheung et al., Nonlinear Effects in Black Hole Ringdown, Phys. Rev. Lett. 130, 081401 (2023).
  49. K. Mitman et al., Nonlinearities in Black Hole Ringdowns, Phys. Rev. Lett. 130, 081402 (2023).
  50. T. May, S. Ma, J. L. Ripley, and W. E. East, Nonlinear effect of absorption on the ringdown of a spinning black hole, Phys. Rev. D 110, 084034 (2024).
  51. M. De Amicis, S. Albanesi, and G. Carullo, Inspiral-inherited ringdown tails, Phys. Rev. D 110, 104005 (2024).
  52. M. De Amicis et al., Late-time tails in nonlinear evolutions of merging black holes, Phys. Rev. Lett. 135, 171401 (2025).
  53. S. Ma, M. A. Scheel, J. Moxon, K. C. Nelli, N. Deppe, L. E. Kidder, W. Throwe, and N. L. Vu, Merging black holes with Cauchy-characteristic matching: Computation of late-time tails, Phys. Rev. D 112, 024003 (2025).
  54. R. F. Rosato and P. Pani, Universality of late-time ringdown tails, Phys. Rev. D 112, 024080 (2025).
  55. M. De Amicis, E. Cannizzaro, G. Carullo, and L. Sberna, Dynamical quasinormal mode excitation, Phys. Rev. D 113, 024048 (2026).
  56. A. Kuntz, Green function of the Pöschl-Teller potential, SciPost Phys. 20, 120 (2026).
  57. P. Arnaudo, J. Carballo, and B. Withers, Beyond quasinormal modes: A complete mode decomposition of black hole perturbations, Phys. Rev. D 114, 024025 (2026).
  58. M. De Amicis, E. Cannizzaro, G. Carullo, A. Kuntz, and L. Sberna, Dynamical quasinormal mode excitation II: Propagation and convergence in Schwarzschild, arXiv:2605.16492.
  59. P. Arnaudo and B. Withers, Bouncing singularities in Schwarzschild: A geometric origin of the QNM convergence region, arXiv:2605.16489.
  60. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Tests of general relativity with GW150914, Phys. Rev. Lett. 116, 221101 (2016); 121, 129902(E) (2018).
  61. R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog, Phys. Rev. D 103, 122002 (2021).
  62. R. Abbott et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), Tests of general relativity with GWTC-3, Phys. Rev. D 112, 084080 (2025).
  63. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), GWTC-4.0: Tests of General Relativity. III. Tests of the Remnants, arXiv:2603.19021.
  64. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), GWTC-5.0: Tests of general relativity, arXiv:2607.19293.
  65. G. Carullo, D. Laghi, N. K. Johnson-McDaniel, W. Del Pozzo, Oscar J. C. Dias, M. Godazgar, and J. E. Santos, Constraints on Kerr-Newman black holes from merger-ringdown gravitational-wave observations, Phys. Rev. D 105, 062009 (2022).
  66. S. Maenaut, G. Carullo, P. A. Cano, A. Liu, V. Cardoso, T. Hertog, and Tjonnie G. F. Li, Ringdown analysis of rotating black holes in effective field theory extensions of general relativity, Phys. Rev. D 113, 044039 (2026).
  67. A. K.-W. Chung and N. Yunes, Black hole ringdown spectroscopy of curvature-coupled scalar and pseudoscalar fields, Phys. Rev. D 114, L021505 (2026).
  68. H. Zhong, M. Isi, K. Chatziioannou, and W. M. Farr, Multidimensional hierarchical tests of general relativity with gravitational waves, Phys. Rev. D 110, 044053 (2024).
  69. A. G. Abac et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), GW250114: Testing Hawking’s area law and the Kerr nature of black holes, Phys. Rev. Lett. 135, 111403 (2025).
  70. A. G. Abac et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), Black hole spectroscopy and tests of general relativity with GW250114, Phys. Rev. Lett. 136, 041403 (2026).
  71. K. Chandra, R. Gamba, and D. Chiaramello, From source properties to strong-field tests: A multipronged analysis of GW250114 with an effective one-body model for generic orbits, arXiv:2512.04593.
  72. L. Grimaldi, E. Maggio, L. Pompili, and A. Buonanno, Plunge-merger-ringdown tests of general relativity with GW250114, Phys. Rev. D 113, L061506 (2026).
  73. L. Pierini and L. Gualtieri, Quasi-normal modes of rotating black holes in Einstein-dilaton Gauss-Bonnet gravity: The first order in rotation, Phys. Rev. D 103, 124017 (2021).
  74. L. Pierini and L. Gualtieri, Quasinormal modes of rotating black holes in Einstein-dilaton Gauss-Bonnet gravity: The second order in rotation, Phys. Rev. D 106, 104009 (2022).
  75. P. Wagle, N. Yunes, and H. O. Silva, Quasinormal modes of slowly-rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D 105, 124003 (2022).
  76. M. Srivastava, Y. Chen, and S. Shankaranarayanan, Analytical computation of quasinormal modes of slowly rotating black holes in dynamical Chern-Simons gravity, Phys. Rev. D 104, 064034 (2021).
  77. P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Quasinormal modes of rotating black holes in higher-derivative gravity, Phys. Rev. D 108, 124032 (2023).
  78. P. A. Cano, K. Fransen, T. Hertog, and S. Maenaut, Universal Teukolsky equations and black hole perturbations in higher-derivative gravity, Phys. Rev. D 108, 024040 (2023).
  79. P. A. Cano, L. Capuano, N. Franchini, S. Maenaut, and S. H. Völkel, Higher-derivative corrections to the Kerr quasinormal mode spectrum, Phys. Rev. D 110, 124057 (2024).
  80. F. S. Khoo, J. L. Blázquez-Salcedo, B. Kleihaus, and J. Kunz, Quasinormal modes of rotating black holes in shift-symmetric Einstein-scalar-Gauss–Bonnet theory, Eur. Phys. J. C 85, 1366 (2025).
  81. J. L. Blázquez-Salcedo, F. S. Khoo, B. Kleihaus, and J. Kunz, Quasinormal modes of rapidly rotating Einstein-Gauss-Bonnet-dilaton black holes, Phys. Rev. D 111, L021505 (2025).
  82. A. K.-W. Chung and N. Yunes, Quasinormal mode frequencies and gravitational perturbations of black holes with any subextremal spin in modified gravity through METRICS: The scalar-Gauss-Bonnet gravity case, Phys. Rev. D 110, 064019 (2024).
  83. A. K.-W. Chung, K. K.-H. Lam, and N. Yunes, Quasinormal mode frequencies and gravitational perturbations of spinning black holes in modified gravity through METRICS: The dynamical Chern-Simons gravity case, Phys. Rev. D 111, 124052 (2025).
  84. T. Alapati and S. Shankaranarayanan, Axial quasinormal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second order in spin and coupling, Phys. Rev. D 112, 024053 (2025).
  85. V. Cardoso, M. Kimura, A. Maselli, E. Berti, Caio F. B. Macedo, and R. McManus, Parametrized black hole quasinormal ringdown: Decoupled equations for nonrotating black holes, Phys. Rev. D 99, 104077 (2019).
  86. R. McManus, E. Berti, Caio F. B. Macedo, M. Kimura, A. Maselli, and V. Cardoso, Parametrized black hole quasinormal ringdown. II. Coupled equations and quadratic corrections for nonrotating black holes, Phys. Rev. D 100, 044061 (2019).
  87. M. Kimura, Note on the parametrized black hole quasinormal ringdown formalism, Phys. Rev. D 101, 064031 (2020).
  88. S. H. Völkel, N. Franchini, and E. Barausse, Theory-agnostic reconstruction of potential and couplings from quasinormal modes, Phys. Rev. D 105, 084046 (2022).
  89. S. H. Völkel, N. Franchini, E. Barausse, and E. Berti, Constraining modifications of black hole perturbation potentials near the light ring with quasinormal modes, Phys. Rev. D 106, 124036 (2022).
  90. N. Franchini and S. H. Völkel, Parametrized quasinormal mode framework for non-Schwarzschild metrics, Phys. Rev. D 107, 124063 (2023).
  91. S. Thomopoulos, S. H. Völkel, and H. P. Pfeiffer, Ringdown spectroscopy of phenomenologically modified black holes, Phys. Rev. D 112, 064054 (2025).
  92. P. A. Cano, L. Capuano, N. Franchini, S. Maenaut, and S. H. Völkel, Parametrized quasinormal mode framework for modified Teukolsky equations, Phys. Rev. D 110, 104007 (2024); 113, 069902(E) (2026).
  93. R. Tang, N. Franchini, S. H. Völkel, and E. Berti, Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation, Phys. Rev. D 113, 104052 (2026).
  94. D. Li, P. Wagle, Y. Chen, and N. Yunes, Perturbations of spinning black holes beyond general relativity: Modified Teukolsky equation, Phys. Rev. X 13, 021029 (2023).
  95. A. Hussain and A. Zimmerman, Approach to computing spectral shifts for black holes beyond Kerr, Phys. Rev. D 106, 104018 (2022).
  96. S. Hirano, M. Kimura, M. Yamaguchi, and J. Zhang, Parametrized black hole quasinormal ringdown formalism for higher overtones, Phys. Rev. D 110, 024015 (2024).
  97. C. De Simone, Discrete symmetries of modified Teukolsky equations, Phys. Rev. D 114, 084063 (2026).
  98. C. De Simone, S. H. Völkel, K. D. Kokkotas, and S. Capozziello, Parametrized beyond-Teukolsky framework in the time domain, Phys. Rev. D 114, 064060 (2026).
  99. S. Völkel and N. Franchini, sebastianvoelkel/parametrized_qnm_framework: Parametrized qnm framework, 10.5281/zenodo.14001739 (2024).
  100. V. Varma, S. E. Field, M. A. Scheel, J. Blackman, D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer, Surrogate models for precessing binary black hole simulations with unequal masses, Phys. Rev. Res. 1, 033015 (2019).
  101. D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC Hammer, Publ. Astron. Soc. Pac. 125, 306 (2013).
  102. J. Goodman and J. Weare, Ensemble samplers with affine invariance, Commun. Appl. Math. Comput. Sci. 5, 65 (2010).
  103. A. Maselli, P. Pani, L. Gualtieri, and E. Berti, Parametrized ringdown spin expansion coefficients: A data-analysis framework for black-hole spectroscopy with multiple events, Phys. Rev. D 101, 024043 (2020).
  104. G. Carullo, Enhancing modified gravity detection from gravitational-wave observations using the parametrized ringdown spin expansion coeffcients formalism, Phys. Rev. D 103, 124043 (2021).
  105. J.-N. Chen, L.-B. Wu, and Z.-K. Guo, Extended parametrized spin expansion formalism for ringdown analysis with GW250114, arXiv:2606.22580.
  106. S. H. Völkel and E. Barausse, Bayesian metric reconstruction with gravitational wave observations, Phys. Rev. D 102, 084025 (2020).
  107. L. Rezzolla and A. Zhidenko, New parametrization for spherically symmetric black holes in metric theories of gravity, Phys. Rev. D 90, 084009 (2014).
  108. E. Seidel and S. Iyer, Black hole normal modes: A WKB approach. 4. Kerr black holes, Phys. Rev. D 41, 374 (1990).
  109. K. D. Kokkotas, Normal modes of the Kerr black hole, Classical Quantum Gravity 8, 2217 (1991).
  110. K. Dey, E. Barausse, M. Crisostomi, and R. Trotta, Impact of sky localization uncertainty on ringdown inference, arXiv:2605.18595.

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