- 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

Uncovering subdominant multipole asymmetries in binary black-hole mergers

Jannik Mielke*, Angela Borchers, and Frank Ohme

  • *Contact author: jannik.mielke@aei.mpg.de
PDF

Phys. Rev. D 114, 044013 – Published 5 August, 2026

DOI: https://doi.org/10.1103/f32p-y3mk

Abstract

In dynamically formed binaries, the spins of the black holes tend to be misaligned with the system’s orbital angular momentum. This causes the spins to precess and leads to an asymmetric emission of gravitational waves. The resulting gravitational-wave multipole asymmetries directly source the recoil of the remnant black hole and are the critical element in fully describing precession. Recoil and precession are of significant astrophysical importance, but multipole asymmetries contribute only minimally to the overall signal strength. Consequently, most current gravitational-wave models either do not incorporate asymmetries at all, or only consider the dominant ones. Here we highlight the importance of subdominant multipole asymmetries for an accurate recoil velocity calculation and discuss their detectability with third generation detectors. Neglecting subdominant asymmetries leads to velocity differences of up to 210  km/s and can, in particular, introduce systematic biases in the inference of masses and the spin geometry. We further discuss universal characteristics of subdominant multipole asymmetries in order to prepare the ground for potential future asymmetry models. In the inspiral regime, the average antisymmetric frequencies can be described by a multiple of the orbital frequency. During ringdown, however, they become equal to their corresponding symmetric frequencies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (86)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), 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).
  2. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), 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).
  3. R. Abbott et al. (LIGO ScientificCollaborations ), 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).
  4. R. Abbott et al. (LIGO ScientificCollaborations ), 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).
  5. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW190412: Observation of a binary-black-hole coalescence with asymmetric masses, Phys. Rev. D 102, 043015 (2020).
  6. J. C. Bustillo, S. H. W. Leong, and K. Chandra, GW190412: Measuring a black-hole recoil direction through higher-order gravitational-wave modes, Nat. Astron. 9, 1530 (2025).
  7. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW190814: Gravitational waves from the coalescence of a 23 solar mass black hole with a 2.6 solar mass compact object, Astrophys. J. Lett. 896, L44.
  8. A. G. Abac et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration ), GW241011 and GW241110: Exploring binary formation and fundamental physics with asymmetric, high-spin black hole coalescences, Astrophys. J. Lett. 993, L21 (2025).
  9. M. Hannam et al., General-relativistic precession in a black-hole binary, Nature (London) 610, 652 (2022).
  10. V. Varma, S. Biscoveanu, T. Islam, F. H. Shaik, C.-J. Haster, M. Isi, W. M. Farr, S. E. Field, and S. Vitale, Evidence of large recoil velocity from a black hole merger signal, Phys. Rev. Lett. 128, 191102 (2022).
  11. P. Kolitsidou, J. E. Thompson, and M. Hannam, Impact of anti-symmetric contributions to signal multipoles in the measurement of black-hole spins, Phys. Rev. D 111, 024050 (2025).
  12. A. Borchers, F. Ohme, J. Mielke, and S. Ghosh, Observability of spin precession in the presence of a black-hole remnant kick, Phys. Rev. D 110, 024037 (2024).
  13. T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Spin-induced orbital precession and its modulation of the gravitational waveforms from merging binaries, Phys. Rev. D 49, 6274 (1994).
  14. L. E. Kidder, Coalescing binary systems of compact objects to post-5/2-Newtonian order. V. Spin effects, Phys. Rev. D 52, 821 (1995).
  15. F. Pretorius, Binary black hole coalescence, arXiv:0710.1338.
  16. B. Brügmann, J. A. González, M. Hannam, S. Husa, and U. Sperhake, Exploring black hole superkicks, Phys. Rev. D 77, 124047 (2008).
  17. M. Campanelli, C. O. Lousto, Y. Zlochower, and D. Merritt, Maximum gravitational recoil, Phys. Rev. Lett. 98, 231102 (2007).
  18. J. A. González, M. Hannam, U. Sperhake, B. Brügmann, and S. Husa, Supermassive recoil velocities for binary black-hole mergers with antialigned spins, Phys. Rev. Lett. 98, 231101 (2007).
  19. M. Ruiz, M. Alcubierre, D. Núñez, and R. Takahashi, Multipole expansions for energy and momenta carried by gravitational waves, Gen. Relativ. Gravit. 40, 2467 (2008).
  20. J. Mielke, S. Ghosh, A. Borchers, and F. Ohme, Revisiting the relationship of black-hole kicks and multipole asymmetries, Phys. Rev. D 111, 064009 (2025).
  21. D. Gerosa, M. Kesden, E. Berti, R. O’Shaughnessy, and U. Sperhake, Resonant-plane locking and spin alignment in stellar-mass black-hole binaries: A diagnostic of compact-binary formation, Phys. Rev. D 87, 104028 (2013).
  22. S. Vitale, R. Lynch, R. Sturani, and P. Graff, Use of gravitational waves to probe the formation channels of compact binaries, Classical Quantum Gravity 34, 03LT01 (2017).
  23. B. Farr, D. E. Holz, and W. M. Farr, Using spin to understand the formation of LIGO’s black holes, Astrophys. J. 854, L9 (2018).
  24. M. Mapelli, Formation channels of single and binary stellar-mass black holes, in Handbook of Gravitational Wave Astronomy (Springer, Singapore, 2021).
  25. D. Gerosa and M. Fishbach, Hierarchical mergers of stellar-mass black holes and their gravitational-wave signatures, Nat. Astron. 5, 749 (2021).
  26. P. Mahapatra, A. Gupta, M. Favata, K. G. Arun, and B. S. Sathyaprakash, Remnant black hole kicks and implications for hierarchical mergers, Astrophys. J. Lett. 918, L31 (2021).
  27. C. Araújo-Álvarez, H. W. Y. Wong, A. Liu, and J. Calderón Bustillo, Kicking time back in black hole mergers: Ancestral masses, spins, birth recoils, and hierarchical-formation viability of GW190521, Astrophys. J. 977, 220 (2024).
  28. P. Mahapatra, D. Chattopadhyay, A. Gupta, F. Antonini, M. Favata, B. S. Sathyaprakash, and K. G. Arun, Reconstructing the genealogy of LIGO-Virgo black holes, Astrophys. J. 975, 117 (2024).
  29. A. Borchers, C. S. Ye, and M. Fishbach, Gravitational-wave kicks impact spins of black holes from hierarchical mergers, Astrophys. J. 987, 146 (2025).
  30. S. H. W. Leong and J. C. Bustillo, Kick and spin: New probes for multi-messenger black-hole mergers in agns, arXiv:2512.08382.
  31. M. J. Graham et al., A light in the dark: Searching for electromagnetic counterparts to black hole-black hole mergers in LIGO/Virgo o3 with the zwicky transient facility, Astrophys. J. 942, 99 (2023).
  32. M. A. Shaikh, V. Varma, A. Ramos-Buades, H. P. Pfeiffer, M. Boyle, L. E. Kidder, and M. A. Scheel, Defining eccentricity for spin-precessing binaries, Classical Quantum Gravity 42, 195012 (2025).
  33. J. E. Thompson, E. Hamilton, L. London, S. Ghosh, P. Kolitsidou, C. Hoy, and M. Hannam, Phenomenological gravitational-wave model for precessing black-hole binaries with higher multipoles and asymmetries, Phys. Rev. D 109, 063012 (2024).
  34. S. Ghosh, P. Kolitsidou, and M. Hannam, First frequency-domain phenomenological model of the multipole asymmetry in gravitational-wave signals from binary-black-hole coalescence, Phys. Rev. D 109, 024061 (2024).
  35. E. Hamilton et al., Phenomxpnr: An improved gravitational wave model linking precessing inspirals and nr-calibrated merger-ringdown, Phys. Rev. D 113, 084055 (2026).
  36. H. Estellés, A. Buonanno, R. Enficiaud, C. Foo, and L. Pompili, Adding equatorial-asymmetric effects for spin-precessing binaries into the seobnrv5phm waveform model, Phys. Rev. D 113, 044049 (2026).
  37. 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).
  38. P. Schmidt, M. Hannam, S. Husa, and P. Ajith, Tracking the precession of compact binaries from their gravitational-wave signal, Phys. Rev. D 84, 024046 (2011).
  39. M. Boyle, R. Owen, and H. P. Pfeiffer, Geometric approach to the precession of compact binaries, Phys. Rev. D 84, 124011 (2011).
  40. M. Boyle, L. E. Kidder, S. Ossokine, and H. P. Pfeiffer, Gravitational-wave modes from precessing black-hole binaries, arXiv:1409.4431.
  41. E. W. Leaver, An analytic representation for the quasi-normal modes of Kerr black holes, Proc. R. Soc. A 402, 285 (1985).
  42. O. M. Moreschi, Supercentre of mass system at future null infinity, Classical Quantum Gravity 5, 423 (1988).
  43. O. M. Moreschi and S. Dain, Rest frame system for asymptotically flat space-times, J. Math. Phys. (N.Y.) 39, 6631 (1998).
  44. L. Magaña Zertuche et al., High precision ringdown modeling: Multimode fits and bms frames, Phys. Rev. D 105, 104015 (2022).
  45. J. N. Goldberg, A. J. Macfarlane, E. Newman, F. Rohrlich, and E. Sudarshan, Spin-s spherical harmonics and edth, J. Math. Phys. (N.Y.) 8, 2155 (1967).
  46. K. S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52, 299 (1980).
  47. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, 1st ed. (W. H. Freeman, San Francisco, 1973).
  48. E. Racine, A. Buonanno, and L. Kidder, Recoil velocity at second post-Newtonian order for spinning black hole binaries, Phys. Rev. D 80, 044010 (2009).
  49. A. Borchers and F. Ohme, Inconsistent black hole kick estimates from gravitational-wave models, Classical Quantum Gravity 40, 095008 (2023).
  50. M. Boyle, D. Iozzo, and L. C. Stein, moble/scri: v1.2 (2020).
  51. D. Merritt, M. Milosavljevic, M. Favata, S. A. Hughes, and D. E. Holz, Consequences of gravitational radiation recoil, Astrophys. J. Lett. 607, L9 (2004).
  52. D. Gerosa and M. Fishbach, Hierarchical mergers of stellar-mass black holes and their gravitational-wave signatures, Nat. Astron. 5, 749 (2021).
  53. S. Ma, M. Giesler, V. Varma, M. A. Scheel, and Y. Chen, Universal features of gravitational waves emitted by superkick binary black hole systems, Phys. Rev. D 104, 084003 (2021).
  54. S. H. W. Leong, A. F. Tomé, J. Calderón Bustillo, A. del Río, and N. Sanchis-Gual, Gravitational-wave signatures of the mirror asymmetry in binary black hole mergers: Measurability and correlation to gravitational-wave recoil, Phys. Rev. D 112, 084078 (2025).
  55. J. Calderón Bustillo, A. del Rio, N. Sanchis-Gual, K. Chandra, and S. H. W. Leong, Testing mirror symmetry in the universe with LIGO-Virgo black-hole mergers, Phys. Rev. Lett. 134, 031402 (2025).
  56. SXS Collaboration, The sxs catalog of simulations v3.0.0, 10.5281/zenodo.15415231 (2025).
  57. M. Boyle, K. Mitman, M. Scheel, and L. Stein, The SXS package (2025), 10.5281/ZENODO.15547069.
  58. SXS Collaboration, Binary black-hole simulation SXS:BBH:1941 (2025), 10.26138/SXS:BBH:1941.
  59. SXS Collaboration, Binary black-hole simulation SXS:BBH:3392 (2025), 10.26138/SXS:BBH:3392.
  60. M. Punturo et al., The Einstein telescope: A third-generation gravitational wave observatory, Classical Quantum Gravity 27, 194002 (2010).
  61. S. Hild et al., Sensitivity studies for third-generation gravitational wave observatories, Classical Quantum Gravity 28, 094013 (2011).
  62. A. Abac et al. (ET Collaboration), The science of the Einstein telescope, J. Cosmol. Astropart. Phys. 03 (2026) 081.
  63. G. Ashton et al., Bilby: A user-friendly Bayesian inference library for gravitational-wave astronomy, Astrophys. J. 241, 27 (2019).
  64. J. S. Speagle, Dynesty: A dynamic nested sampling package for estimating Bayesian posteriors and evidences, Mon. Not. R. Astron. Soc. 493, 3132 (2020).
  65. C. Hoy and V. Raymond, Pesummary: The code agnostic parameter estimation summary page builder, SoftwareX 15, 100765 (2021).
  66. J. E. Thompson, C. Hoy, E. Fauchon-Jones, and M. Hannam, On the use and interpretation of signal-model indistinguishability measures for gravitational-wave astronomy, Phys. Rev. D 112, 064011 (2025).
  67. M. A. Scheel et al., The SXS Collaboration’s third catalog of binary black hole simulations, Classical Quantum Gravity 42, 195017 (2025).
  68. SXS Collaboration, Binary black-hole simulation SXS:BBH:1207 (2025), 10.26138/SXS:BBH:1207.
  69. M. Boyle, PostNewtonian.jl (2024), https://github.com/moble/PostNewtonian.jl.
  70. E. Hamilton, L. London, and M. Hannam, Ringdown frequencies in black holes formed from precessing black-hole binaries, Phys. Rev. D 107, 104035 (2023).
  71. L. Magaña Zertuche and E. Finch, qnmfits, https://github.com/sxs-collaboration/qnmfits (2025), gitHub repository.
  72. L. C. Stein, qnm: A Python package for calculating Kerr quasinormal modes, separation constants, and spherical-spheroidal mixing coefficients, J. Open Source Software 4, 1683 (2019).
  73. G. B. Cook and M. Zalutskiy, Gravitational perturbations of the Kerr geometry: High-accuracy study, Phys. Rev. D 90, 124021 (2014).
  74. G. B. Cook, Aspects of multimode Kerr ringdown fitting, Phys. Rev. D 102, 024027 (2020).
  75. K. Mitman, I. Pretto, H. Siegel, M. A. Scheel, S. A. Teukolsky, M. Boyle, N. Deppe, L. E. Kidder, J. Moxon, K. C. Nelli, W. Throwe, and N. L. Vu, Probing the ringdown perturbation in binary black hole coalescences with an improved quasi-normal mode extraction algorithm, Phys. Rev. D 112, 064016 (2025).
  76. 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).
  77. L. Magaña Zertuche et al., High-precision ringdown surrogate model for nonprecessing binary black holes, Phys. Rev. D 112, 024077 (2025).
  78. H. Zhu et al., Black hole spectroscopy for precessing binary black hole coalescences, Phys. Rev. D 111, 064052 (2025).
  79. H. Zhu, H. Siegel, and K. Mitman, precession-ringdown, https://github.com/HengruiZhu99/precession_ringdown (2025), gitHub repository.
  80. F. Nobili, S. Bhagwat, C. Pacilio, and D. Gerosa, Ringdown mode amplitudes of precessing binary black holes, Phys. Rev. D 112, 044058 (2025).
  81. C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
  82. J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  83. P. Virtanen et al., Scipy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  84. LIGO Scientific, Virgo, and KAGRA Collaborations, LVK Algorithm Library—LALSuite, Free software (GPL) (2018), 10.7935/GT1W-FZ16.
  85. S. E. Field, C. R. Galley, J. S. Hesthaven, J. Kaye, and M. Tiglio, Fast prediction and evaluation of gravitational waveforms using surrogate models, Phys. Rev. X 4, 031006 (2014).
  86. J. Mielke, subdominant-asymmetries, gitHub repository, https://github.com/janmie1701/subdominant_asymmetries (2026).

Outline

Information

Phys. Rev. D 114, 044013– Published 5 August, 2026
  • Received 6 March 2026
  • Accepted 14 July 2026
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