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A practical Bayesian method for gravitational-wave ringdown analysis with multiple modes

Nature Astronomy (2026)Cite this article

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Abstract

Gravitational-wave (GW) ringdown signals from black holes encode crucial information about the gravitational dynamics in the strong-field regime, which offers unique insights into the properties of black holes. Improving the sensitivity of GW detectors will enable the extraction of several quasi-normal modes from ringdown signals. However, incorporating several modes drastically enlarges the parameter space, posing computational challenges to data analysis. Inspired by the \({\mathcal{F}}\)-statistic method in the continuous GW searches, here we develop an algorithm that enhances the parameter-marginalization scheme, dubbed FIREFLY, which is tailored for accelerating the ringdown signal analysis. FIREFLY analytically marginalizes the amplitude and phase parameters of quasi-normal modes to reduce the computational cost and to speed up the standard Bayesian inference with full parameters from hours to minutes while achieving consistent posterior and evidence. The acceleration becomes more pronounced when more quasi-normal modes are considered. Rigorously based on Bayesian inference and importance sampling, our method is statistically interpretable, flexible in prior choice and compatible with various advanced sampling techniques and, thus, provides a new perspective for accelerating future GW data analysis.

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Fig. 1: Workflow of FIREFLY.
Fig. 2: Posteriors of the N = 3 scenario in the overtone analysis.
Fig. 3: P-P plots for the overtone analysis.
Fig. 4: Evidences given by FIREFLY and the full-parameter sampling.
Fig. 5: Joint posteriors of the final mass and the final spin with different starting times in the higher-mode analysis.

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Data availability

The data used in this study can be found in the public SXS Gravitational Waveform Database at https://data.black-holes.org/ (object IDs SXS:BBH:0305 and SXS:BBH:0065). Source data are provided with this paper.

Code availability

The code FIREFLY is publicly available via GitHub at https://github.com/Yiming-astro/Firefly-ringdown. This repository provides all materials needed to reproduce the results of this study.

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Acknowledgements

We thank H. W. Lee, H. Nakano and N. Uchikata for comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 123B2043 and 12573042), the Beijing Natural Science Foundation (1242018), the National SKA Program of China (Grant No. 2020SKA0120300), the Max Planck Partner Group Program funded by the Max Planck Society and the High-performance Computing Platform of Peking University. H.-T.W. and L.S. are supported by the Fundamental Research Funds for the Central Universities at Dalian University of Technology and Peking University, respectively. J.Z. is supported by the National Natural Science Foundation of China (Grant No. 12405052) and the Startup Research Fund of Henan Academy of Sciences (Grant No. 241841220).

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Author notes

  1. These authors contributed equally: Yiming Dong, Ziming Wang.

Authors and Affiliations

  1. Department of Astronomy, School of Physics, Peking University, Beijing, China

    Yiming Dong & Ziming Wang

  2. Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing, China

    Yiming Dong, Ziming Wang & Lijing Shao

  3. School of Physics, Dalian University of Technology, Dalian, China

    Hai-Tian Wang

  4. Institute for Gravitational Wave Astronomy, Henan Academy of Sciences, Zhengzhou, China

    Junjie Zhao

  5. National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China

    Lijing Shao

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Contributions

Y.D. and Z.W. contributed equally to the work, namely to the design and programming of the FIREFLY code and writing the initial draft of the paper. H.-T.W. contributed to the \({\mathcal{F}}\)-statistic-based method and developed preliminary sampling codes for the ringdown analysis. J.Z. contributed to the Bayesian analysis and sampling implementation. L.S. contributed to the design and methodology of FIREFLY and writing the paper and supervised and coordinated the whole project. All co-authors discussed the results and provided input to the data analysis and the content of the paper.

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Correspondence to Lijing Shao.

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Dong, Y., Wang, Z., Wang, HT. et al. A practical Bayesian method for gravitational-wave ringdown analysis with multiple modes. Nat Astron (2026). https://doi.org/10.1038/s41550-025-02766-6

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