A research team led by Prof. Lijing Shao in the Kavli Institute for Astronomy and Astrophysics at Peking University has proposed a new Bayesian framework for gravitational-wave ringdown analysis and developed an open-source algorithm, FIREFLY, to apply to real gravitational-wave data. Since their first discovery about ten years ago, gravitational waves have become instrumental in understanding astrophysical processes, fundamental laws of physics, as well as black-hole spacetimes. Above all, useful information is eventually extracted from (usually rather noisy) data with dedicated statistical techniques.
Employing an effective strategy yields multiplied outcomes. Based on Bayes’ theorem and the principle of importance sampling, FIREFLY optimizes the sampling strategies and achieves orders-of-magnitude speedup over traditional Bayesian methods for black-hole ringdown signals with multiple characteristic modes, the so-called “quasi-normal modes”, while still preserving statistical interpretability and users’ flexibility in prior choices. This new study provides an efficient computational tool for studying black hole physics and gravitational theories and offers a practical methodology for analyzing numerous events in the future gravitational-wave observations.
Binary black hole mergers are among the most important classes of gravitational-wave sources. The ringdown phase is the final stage of a binary black hole merger, describing the process by which the merger remnant transitions from a perturbed state to a stable equilibrium. According to black-hole perturbation theory, the ringdown signal can be decomposed into a superposition of a series of quasi-normal modes, whose amplitudes, frequencies, and other parameters encode the physical properties of the post-merger black hole, such as its mass and spin. The analysis of these quasi-normal modes is known as “black hole spectroscopy”, and their measurement provides a unique window into fundamental aspects of black holes, such as testing the no-hair theorem (black holes are hairless, only endowed with mass, spin and charge), or the Hawking’s area theorem (black holes’ total surface area never shrinks).
Figure 1: Schematic of the ringdown signal from a binary black hole merger, where the signal is often modeled as a superposition of multiple quasi-normal modes (Y. Dong & Z. Wang)
With the advent of more sensitive next-generation ground-based gravitational-wave observatories, such as the Einstein Telescope and Cosmic Explorer, as well as space-borne missions including LISA, TianQin, and Taiji, it is expected that more quasi-normal modes will be extracted from ringdown signals, opening new opportunities for tests of black hole physics. However, from the standpoint of data analysis, introducing more quasi-normal modes into the model causes the “curse of dimensionality” problem, where the parameter space grows rapidly. The dramatic increase in computation cost eventually limits comprehensive analyses of ringdown signals.
To address these challenges, the research team led by Prof. Shao proposed a new Bayesian algorithm for gravitational-wave ringdown analysis, dubbed FIREFLY. By introducing auxiliary inference and a two-step importance-sampling strategy, the method reduces the dimensionality of the computationally expensive, traditional sampling problem from its full parameter space to a set of sub-parameter spaces. Extensive simulation studies showed that, when analyzing ringdown signals with multiple quasi-normal modes, FIREFLY achieves orders-of-magnitude speedups while producing high-fidelity posterior samples. In the context of next-generation ground-based gravitational-wave detectors, FIREFLY delivers tens- to hundreds-fold reductions in the computational cost compared with traditional Bayesian methods, enhancing the detection and analysis for a large number of events with multiple modes.
Figure 2: Comparison of posterior samples obtained with FIREFLY (green) and the traditional Bayesian method (blue). It can be seen that FIREFLY produces extremely high-fidelity results for all inferred parameters, including mass Mf, spin cf and other amplitude and phase parameters (Y. Dong, Z. Wang, et al. 2026)
In addition, FIREFLY is designed from first principles of statistical inference, making itself compatible with other acceleration strategies, such as optimized samplers. The method requires only a specific mathematical structure in the likelihood function, and can be extended to the data analysis for many other types of gravitational-wave sources. This work provides a new analytical framework for dealing with the data challenges in the future gravitational-wave observations, expected to strongly advance cutting-edge research in black-hole physics and gravitational theory.
PhD students, Yiming Dong and Ziming Wang, both from the Department of Astronomy, School of Physics, Peking University, are co–first authors of the paper, “A practical Bayesian method for gravitational-wave ringdown analysis with multiple modes”, which was published in Nature Astronomy on January 15, 2026. Prof. Lijing Shao is the corresponding author, and Associate Professor Hai-Tian Wang of Dalian University of Technology and Associate Researcher Junjie Zhao of Henan Academy of Sciences are co-authors.
This research was supported by the National Natural Science Foundation of China's Young Scientists Fund (Doctoral Program) and other funding sources.
Article Link:https://www.nature.com/articles/s41550-025-02766-6
