- 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

Quantifying systematic biases in black hole spectroscopy

Sebastian H. Völkel* and Arnab Dhani†

  • *Contact author: sebastian.voelkel@aei.mpg.de
  • †Contact author: arnab.dhani@aei.mpg.de
PDF

Phys. Rev. D 112, 084076 – Published 29 October, 2025

DOI: https://doi.org/10.1103/g6sz-dw28

Abstract

How long after the merger of two black holes can one rely on linear perturbation theory, and how many quasinormal modes are in the ringdown? Such questions suggest that black hole spectroscopy suffers from systematic uncertainties that potentially spoil ringdown analyses, both from high-accuracy simulations and in data from gravitational wave detectors. In this work, we demonstrate that linear-signal analysis is a powerful tool for quantifying biases, allowing for detailed explorations that are computationally too expensive for traditional Bayesian injection and recovery approaches. We quantify the validity of the Fisher information matrix and bias formula by comparing it to robust but slow Bayesian sampling. Working with flat noise in the time domain, statistical errors and systematic biases can mostly be detected analytically. Due to its efficiency, we provide detailed parameter space analyses for potentially unmodeled small contributions from overtones, quadratic modes, and tails. We find linear signal analysis well suited for predicting biases in simple ringdown models at intermediate signal-to-noise ratios (SNRs) of order 50 when unmodeled effects are small. It is also valuable in explaining ongoing issues in extracting quasinormal modes from high-precision simulations, as one can understand them as high-SNR signals. Therefore, this approach offers promising prospects for improving ringdown models by efficiently identifying and incorporating systematic uncertainties, ultimately enhancing the accuracy and robustness of black hole spectroscopy.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (105)

  1. S. L. Detweiler, Black holes and gravitational waves. III—The resonant frequencies of rotating holes, Astrophys. J. 239, 292 (1980).
  2. 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).
  3. 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).
  4. T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
  5. F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24, 737 (1970).
  6. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  7. K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Relativity 2, 2 (1999).
  8. H.-P. Nollert, Topical review: Quasinormal modes: The characteristic ‘sound’ of black holes and neutron stars, Classical Quantum Gravity 16, R159 (1999).
  9. E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26, 163001 (2009).
  10. R. A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. 83, 793 (2011).
  11. E. Berti et al., Black hole spectroscopy: From theory to experiment, arXiv:2505.23895.
  12. W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164, 1776 (1967).
  13. S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).
  14. B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26, 331 (1971).
  15. D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34, 905 (1975).
  16. A. Bachelot and A. Motet-Bachelot, Les résonances d’un trou noir de Schwarzschild, Ann. L’Inst. Henri Poincare Sect. (A) Phys. Theor. 59, 3 (1993).
  17. H. R. Beyer, On the completeness of the quasinormal modes of the Poschl-Teller potential, Commun. Math. Phys. 204, 397 (1999).
  18. C. V. Vishveshwara, Scattering of gravitational radiation by a Schwarzschild black-hole, Nature (London) 227, 936 (1970).
  19. R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations, Phys. Rev. D 5, 2419 (1972).
  20. E. W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34, 384 (1986).
  21. C. Gundlach, R. H. Price, and J. Pullin, Late time behavior of stellar collapse and explosions: 1. Linearized perturbations, Phys. Rev. D 49, 883 (1994).
  22. C. Gundlach, R. H. Price, and J. Pullin, Late time behavior of stellar collapse and explosions: 2. Nonlinear evolution, Phys. Rev. D 49, 890 (1994).
  23. L. Barack, Late time dynamics of scalar perturbations outside black holes. 2. Schwarzschild geometry, Phys. Rev. D 59, 044017 (1999).
  24. R. F. Rosato and P. Pani, On the universality of late-time ringdown tail, Phys. Rev. D 112, 024080 (2025).
  25. H.-P. Nollert and R. H. Price, Quantifying excitations of quasinormal mode systems, J. Math. Phys. (N.Y.) 40, 980 (1999).
  26. S. Hod, The radiative tail of realistic gravitational collapse, Phys. Rev. Lett. 84, 10 (2000).
  27. S. Hod, Mode coupling in rotating gravitational collapse: Gravitational and electromagnetic perturbations, Phys. Rev. D 61, 064018 (2000).
  28. 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).
  29. 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).
  30. L. London, D. Shoemaker, and J. Healy, Modeling ringdown: Beyond the fundamental quasinormal modes, Phys. Rev. D 90, 124032 (2014); 94, 069902(E) (2016).
  31. M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky, Black hole ringdown: The importance of overtones, Phys. Rev. X 9, 041060 (2019).
  32. E. Finch and C. J. Moore, Searching for a ringdown overtone in GW150914, Phys. Rev. D 106, 043005 (2022).
  33. M. Isi and W. M. Farr, Revisiting the ringdown of GW150914, arXiv:2202.02941.
  34. 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).
  35. S. Ma, L. Sun, and Y. Chen, Black hole spectroscopy by mode cleaning, Phys. Rev. Lett. 130, 141401 (2023).
  36. 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).
  37. P. J. Nee, S. H. Völkel, and H. P. Pfeiffer, Role of black hole quasinormal mode overtones for ringdown analysis, Phys. Rev. D 108, 044032 (2023).
  38. M. Isi and W. M. Farr, Comment on Analysis of ringdown overtones in GW150914, Phys. Rev. Lett. 131, 169001 (2023).
  39. G. Carullo, R. Cotesta, E. Berti, and V. Cardoso, Reply to Comment on “Analysis of ringdown overtones in GW150914”, Phys. Rev. Lett. 131, 169002 (2023).
  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).
  41. A. Maselli, S. Yi, L. Pierini, V. Vellucci, L. Reali, L. Gualtieri, and E. Berti, Black hole spectroscopy beyond Kerr: Agnostic and theory-based tests with next-generation interferometers, Phys. Rev. D 109, 064060 (2024).
  42. M. Giesler et al., Overtones and nonlinearities in binary black hole ringdowns, Phys. Rev. D 111, 084041 (2025).
  43. K. Mitman et al., Probing the ringdown perturbation in binary black hole coalescences with an improved quasi-normal mode extraction algorithm, Phys. Rev. D 112, 064016 (2025).
  44. F. Nobili, S. Bhagwat, C. Pacilio, and D. Gerosa, Ringdown mode amplitudes of precessing binary black holes, Phys. Rev. D 112, 044058 (2025).
  45. S. Thomopoulos, S. H. Völkel, and H. P. Pfeiffer, Ringdown spectroscopy of phenomenologically modified black holes, Phys. Rev. D 112, 064054 (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. Magaña Zertuche et al., High-precision ringdown surrogate model for non-precessing binary black holes, Phys. Rev. D 112, 024077 (2025).
  48. N. Oshita, Ease of excitation of black hole ringing: Quantifying the importance of overtones by the excitation factors, Phys. Rev. D 104, 124032 (2021).
  49. H. Motohashi, Resonant excitation of quasinormal modes of black holes, Phys. Rev. Lett. 134, 141401 (2025).
  50. N. Oshita and V. Cardoso, Reconstruction of ringdown with excitation factors, Phys. Rev. D 111, 104043 (2025).
  51. N. Oshita, E. Berti, and V. Cardoso, Unstable chords and destructive resonant excitation of black hole quasinormal modes, Phys. Rev. Lett. 135, 031401 (2025).
  52. R. K. L. Lo, L. Sabani, and V. Cardoso, Quasinormal modes and excitation factors of Kerr black holes, Phys. Rev. D 111, 124002 (2025).
  53. E. Berti and V. Cardoso, Quasinormal ringing of Kerr black holes. I. The excitation factors, Phys. Rev. D 74, 104020 (2006).
  54. Z. Zhang, E. Berti, and V. Cardoso, Quasinormal ringing of Kerr black holes. II. Excitation by particles falling radially with arbitrary energy, Phys. Rev. D 88, 044018 (2013).
  55. A. Dhani, Importance of mirror modes in binary black hole ringdown waveform, Phys. Rev. D 103, 104048 (2021).
  56. A. Dhani and B. S. Sathyaprakash, Overtones, mirror modes, and mode-mixing in binary black hole mergers, arXiv:2107.14195.
  57. X. J. Forteza and P. Mourier, High-overtone fits to numerical relativity ringdowns: Beyond the dismissed n=8 special tone, Phys. Rev. D 104, 124072 (2021).
  58. K. Mitman et al., Nonlinearities in black hole ringdowns, Phys. Rev. Lett. 130, 081402 (2023).
  59. G. B. Cook, Aspects of multimode Kerr ringdown fitting, Phys. Rev. D 102, 024027 (2020).
  60. G. Carullo and M. De Amicis, Late-time tails in nonlinear evolutions of merging black hole binaries, arXiv:2310.12968.
  61. V. Cardoso, G. Carullo, M. De Amicis, F. Duque, T. Katagiri, D. Pereniguez, J. Redondo-Yuste, T. F. M. Spieksma, and Z. Zhong, Hushing black holes: Tails in dynamical spacetimes, Phys. Rev. D 109, L121502 (2024).
  62. M. De Amicis, S. Albanesi, and G. Carullo, Inspiral-inherited ringdown tails, Phys. Rev. D 110, 104005 (2024).
  63. T. Islam, G. Faggioli, G. Khanna, S. E. Field, M. van de Meent, and A. Buonanno, Phenomenology and origin of late-time tails in eccentric binary black hole mergers, Phys. Rev. D 112, 024061 (2025).
  64. S. Ma, K. C. Nelli, J. Moxon, M. A. Scheel, N. Deppe, L. E. Kidder, W. Throwe, and N. L. Vu, Einstein–Klein–Gordon system via Cauchy-characteristic evolution: Computation of memory and ringdown tail, Classical Quantum Gravity 42, 055006 (2025).
  65. M. De Amicis et al., Late-time tails in nonlinear evolutions of merging black holes, arXiv:2412.06887.
  66. 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).
  67. R. O’Shaughnessy, L. London, J. Healy, and D. Shoemaker, Precession during merger: Strong polarization changes are observationally accessible features of strong-field gravity during binary black hole merger, Phys. Rev. D 87, 044038 (2013).
  68. H. Zhu et al., Black hole spectroscopy for precessing binary black hole coalescences, Phys. Rev. D 111, 064052 (2025).
  69. M. R. Sinha, L. Sun, and S. Ma, Impact of detector calibration accuracy on black hole spectroscopy, arXiv:2506.15979.
  70. M. H.-Y. Cheung et al., Nonlinear effects in black hole ringdown, Phys. Rev. Lett. 130, 081401 (2023).
  71. J. Redondo-Yuste, G. Carullo, J. L. Ripley, E. Berti, and V. Cardoso, Spin dependence of black hole ringdown nonlinearities, Phys. Rev. D 109, L101503 (2024).
  72. 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).
  73. S. Yi, A. Kuntz, E. Barausse, E. Berti, M. H.-Y. Cheung, K. Kritos, and A. Maselli, Nonlinear quasinormal mode detectability with next-generation gravitational wave detectors, Phys. Rev. D 109, 124029 (2024).
  74. N. Khera, S. Ma, and H. Yang, Quadratic mode couplings in rotating black holes and their detectability, Phys. Rev. Lett. 134, 211404 (2025).
  75. O. Steppohn, S. H. Völkel, and T. Dietrich, Black hole spectroscopy of collapsing and merging neutron stars, arXiv:2508.15534.
  76. M. Crisostomi, K. Dey, E. Barausse, and R. Trotta, Neural posterior estimation with guaranteed exact coverage: The ringdown of GW150914, Phys. Rev. D 108, 044029 (2023).
  77. C. Pacilio, S. Bhagwat, and R. Cotesta, Simulation-based inference of black hole ringdowns in the time domain, Phys. Rev. D 110, 083010 (2024).
  78. L. S. Finn and D. F. Chernoff, Observing binary inspiral in gravitational radiation: One interferometer, Phys. Rev. D 47, 2198 (1993).
  79. A. Krolak, J. A. Lobo, and B. J. Meers, Estimation of the parameters of the gravitational wave signal of a coalescing binary system, Phys. Rev. D 48, 3451 (1993).
  80. K. D. Kokkotas, G. Tsegas, and A. Krolak, Statistical analysis of the estimators of the parameters of the gravitational wave signal from a coalescing binary, Classical Quantum Gravity 11, 1901 (1994).
  81. E. E. Flanagan and S. A. Hughes, Measuring gravitational waves from binary black hole coalescences: 2. The waves’ information and its extraction, with and without templates, Phys. Rev. D 57, 4566 (1998).
  82. C. Cutler and M. Vallisneri, LISA detections of massive black hole inspirals: Parameter extraction errors due to inaccurate template waveforms, Phys. Rev. D 76, 104018 (2007).
  83. A. Gupta et al., Possible causes of false general relativity violations in gravitational wave observations, SciPost Phys. Comm. Rep. 5 (2024).
  84. R. S. Chandramouli, K. Prokup, E. Berti, and N. Yunes, Systematic biases due to waveform mismodeling in parametrized post-Einsteinian tests of general relativity: The impact of neglecting spin precession and higher modes, Phys. Rev. D 111, 044026 (2025).
  85. 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).
  86. S. Völkel and N. Franchini, sebastianvoelkel/parametrized_qnm_framework: Parametrized qnm framework (2024).
  87. 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).
  88. M. Isi and W. M. Farr, Analyzing black-hole ringdowns, arXiv:2107.05609.
  89. D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC Hammer, Publ. Astron. Soc. Pac. 125, 306 (2013).
  90. A. Dhani, S. H. Völkel, A. Buonanno, H. Estelles, J. Gair, H. P. Pfeiffer, L. Pompili, and A. Toubiana, Systematic biases in estimating the properties of black holes due to inaccurate gravitational-wave models, Phys. Rev. X 15, 031036 (2025).
  91. V. Kapil, L. Reali, R. Cotesta, and E. Berti, Systematic bias from waveform modeling for binary black hole populations in next-generation gravitational wave detectors, Phys. Rev. D 109, 104043 (2024).
  92. S. Yi, F. Iacovelli, S. Marsat, D. Wadekar, and E. Berti, Systematic biases from the exclusion of higher harmonics in parameter estimation on LISA binaries, arXiv:2502.12237.
  93. L. Capuano, M. Vaglio, R. S. Chandramouli, C. L. Pitte, A. Kuntz, and E. Barausse, Systematic bias in LISA ringdown analysis due to waveform inaccuracy, arXiv:2506.21181.
  94. X. J. Forteza, S. Bhagwat, S. Kumar, and P. Pani, Novel ringdown amplitude-phase consistency test, Phys. Rev. Lett. 130, 021001 (2023).
  95. E. Berti, J. Cardoso, V. Cardoso, and M. Cavaglia, Matched-filtering and parameter estimation of ringdown waveforms, Phys. Rev. D 76, 104044 (2007).
  96. M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Testing the no-hair theorem with GW150914, Phys. Rev. Lett. 123, 111102 (2019).
  97. R. Cotesta, G. Carullo, E. Berti, and V. Cardoso, Analysis of ringdown overtones in GW150914, Phys. Rev. Lett. 129, 111102 (2022).
  98. Y.-F. Wang, C. D. Capano, J. Abedi, S. Kastha, B. Krishnan, A. B. Nielsen, A. H. Nitz, and J. Westerweck, A gating-and-inpainting perspective on GW150914 ringdown overtone: Understanding the data analysis systematics, arXiv:2310.19645.
  99. A. Toubiana, L. Pompili, A. Buonanno, J. R. Gair, and M. L. Katz, Measuring source properties and quasinormal mode frequencies of heavy massive black-hole binaries with LISA, Phys. Rev. D 109, 104019 (2024).
  100. The LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration, Black hole spectroscopy and tests of general relativity with GW250114, arXiv:2509.08099.
  101. A. Correia, Y.-F. Wang, J. Westerweck, and C. D. Capano, Low evidence for ringdown overtone in GW150914 when marginalizing over time and sky location uncertainty, Phys. Rev. D 110, L041501 (2024).
  102. Y.-F. Wang, C. D. Capano, J. Abedi, S. Kastha, B. Krishnan, A. B. Nielsen, A. H. Nitz, and J. Westerweck, A gating-and-inpainting perspective on GW150914 ringdown overtone: Understanding the data analysis systematics, arXiv:2310.19645.
  103. LIGO Scientific, VIRGO, and Kagra Collaborations et al., GW230814: Investigation of a loud gravitational-wave signal observed with a single detector, arXiv:2509.07348.
  104. M. Maggiore et al. (ET Collaboration), Science case for the Einstein Telescope, J. Cosmol. Astropart. Phys. 03 (2020) 050.
  105. S. Bhagwat, X. J. Forteza, P. Pani, and V. Ferrari, Ringdown overtones, black hole spectroscopy, and no-hair theorem tests, Phys. Rev. D 101, 044033 (2020).

Outline

Information

Phys. Rev. D 112, 084076– Published 29 October, 2025
  • Received 6 August 2025
  • Accepted 1 October 2025
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