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A complete measurement of a black-hole recoil through higher-order gravitational-wave modes

Nature Astronomy volume 9, pages 1530–1540 (2025)Cite this article

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Abstract

General relativity predicts that gravitational waves (GWs) carry linear momentum. Consequently, the remnant black hole of a black-hole merger can inherit a recoil velocity or ‘kick’ of crucial implications in, for example, black-hole formation scenarios. While the kick magnitude is determined by the mass ratio and spins of the source, estimating its direction requires a measurement of the two orientation angles of the source. While the orbital inclination angle is commonly reported in GW observations, the scientific potential of the azimuthal one has not been exploited so far. Here we show how the presence of more than one GW emission mode allows one to constrain this angle and, consequently, the kick direction of a real GW event. We analyse the GW190412 signal, which contains higher-order modes, with a numerical relativity surrogate waveform model for black-hole mergers. We rule out kick magnitudes below the typical escape velocity of dense globular clusters vesc ≈ 50 km s−1 with a Bayes factor of ~21 (or ~95% probability). The kick forms angles \({\theta }_{{\mathrm{KL}}}^{-100M}=3{2}_{-14}^{+35}\,\text{deg}\) with the orbital angular momentum defined at a reference time tref = −100 M before merger (with M denoting the system mass in geometric units), \({\theta }_{{{KN}}}=4{4}_{-17}^{+19}\,\text{deg}\) with the line of sight and \({\phi }_{{{KN}}}^{-100M}=6{9}_{-38}^{+33}\,\text{deg}\) with the projection of the latter onto the former, all quoted at a 90% credible level. We anticipate that complete characterization of black-hole recoils will aid in evaluating candidate multi-messenger observations of black-hole mergers in active galactic nuclei, by testing the consistency of observed signals with proposed electromagnetic emission mechanisms.

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Fig. 1: Sketch of our BH merger reference frame.
Fig. 2: GW190412 as observed on Earth and 180 deg away: impact of HMs.
Fig. 3: Azimuthal angle around GW190412: impact of HMs and precession.
Fig. 4: Magnitude and direction estimates of the GW190412 recoil.

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Data availability

The GW strain data used in this work and the corresponding noise power spectral densities are publicly available via the Gravitational Wave Open Science Center (GWOSC) Event Page for GW190412 (https://gwosc.org/eventapi/html/O3_Discovery_Papers/GW190412/v2/) and via the data release webpage for GW190412 in LIGO-DCC (https://dcc.ligo.org/LIGO-P190412/public). The parameter inference code Bilby, the waveform model NRSur7dq4 and its remnant model NRSur7dq4Remnant are publicly available. The NR waveforms used in our injection studies are available via the public SXS Waveform Catalog. Codes used to evolve our samples to different reference frequencies can be made available upon request to the corresponding authors.

Code availability

Codes used to evolve our samples to different reference frequencies can be made available upon request to the corresponding authors.

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Acknowledgements

We thank N. Sanchis-Gual, B. McKernan and S. Ford for their comments on the manuscript. We thank S. Ossokine and N. Johnson-McDaniel for their advice to perform parameter estimation at a given reference time tref and V. Varma for discussions on the importance of this choice. We also thank A. Borchers for useful discussions regarding the accuracy of kick estimations of various waveform models. Finally, we thank T. Dent, T. Li and T. Mömbacher for useful discussions. The analysed data and the corresponding power spectral densities are publicly available at the online Gravitational Wave Open Science Center109. J.C.B. is supported by a fellowship from ‘la Caixa’ Foundation (ID100010434) and from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 847648. The fellowship code is LCF/BQ/PI20/11760016. J.C.B. is also supported by the research grant PID2020-118635GB-I00 and by a Ramón y Cajal Fellowship RYC2022-036203-I from the Spain-Ministerio de Ciencia e Innovación. K.C. acknowledges the generous support provided through NSF grant numbers PHY-2207638, AST-2307147, PHY-2308886 and PHY-2309064. We acknowledge using the IUCAA LDG cluster Sarathi for the computational and numerical work. We acknowledge computational resources provided by the CIT cluster of the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY0823459, and the support of the NSF CIT cluster for the provision of computational resources for our parameter inference runs. This material is based upon work supported by NSF’s LIGO Laboratory, which is a major facility fully funded by the National Science Foundation. We acknowledge the use of computing facilities supported by grants from the Croucher Innovation Award from the Croucher Foundation Hong Kong. This research has made use of data or software obtained from the Gravitational Wave Open Science Center (gwosc.org), a service of the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. This material is based upon work supported by NSF’s LIGO Laboratory, which is a major facility fully funded by the National Science Foundation, as well as the Science and Technology Facilities Council (STFC) of the UK, the Max-Planck-Society (MPS) and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal and Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan. This Article has LIGO DCC number P2200332.

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Author notes

  1. These authors contributed equally: Juan Calderón Bustillo, Samson H. W. Leong, Koustav Chandra.

Authors and Affiliations

  1. Instituto Galego de Física de Altas Enerxías, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

    Juan Calderón Bustillo

  2. Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Juan Calderón Bustillo & Samson H. W. Leong

  3. Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai, India

    Koustav Chandra

  4. Institute for Gravitation and the Cosmos, Department of Physics, Pennsylvania State University, University Park, PA, USA

    Koustav Chandra

Authors
  1. Juan Calderón Bustillo
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  2. Samson H. W. Leong
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Contributions

J.C.B. designed and led the study, designed the analyses and the statistical framework and wrote the manuscript. S.H.W.L. developed tools to extract kick estimates, evolve parameter inference samples and run the injection campaign presented in the Supplementary Information. K.C. performed parameter inference runs on GW190412.

Corresponding authors

Correspondence to Juan Calderón Bustillo, Samson H. W. Leong or Koustav Chandra.

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Nature Astronomy thanks Gregorio Carullo, Isobel Romero-Shaw and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Figs. 1–4 and Tables 1 and 2.

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Calderón Bustillo, J., Leong, S.H.W. & Chandra, K. A complete measurement of a black-hole recoil through higher-order gravitational-wave modes. Nat Astron 9, 1530–1540 (2025). https://doi.org/10.1038/s41550-025-02632-5

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