gr-qcSep 13, 2026

Parameter Estimation of Ringdown Quasinormal Modes with Autoencoder

Authors: Momoka Iida, Hayato Motohashi, Hirotaka Takahashi

Organizations: Graduate School of Integrative Science and Engineering, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan · Research Center for Space Science, Advanced Research Laboratories, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan · Department of Physics, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan · Graduate School of Information and Data Science and Department of Design and Data Science, Tokyo City University, 3-3-1 Ushikubo-Nishi, Tsuzuki-ku, Yokohama, Kanagawa 224-8551, Japan · 

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Abstract

Ringdown gravitational waves from binary black hole mergers can be modeled as superpositions of quasinormal modes (QNMs), whose frequencies and excitation factors encode properties of the remnant Kerr black hole. Reliable extraction of multiple QNM components is challenging because of mode overlap and noise. We develop an autoencoder-based framework for multi-component QNM analysis, in which the latent space is trained to represent the physical parameters of individual modes, enabling waveform denoising and parameter estimation within a common framework. Using controlled model waveforms constructed as finite sums of Kerr QNMs with recently established high-precision frequencies and excitation factors, including their nontrivial spin dependence near resonant excitation, we assess the method across partitioned spin intervals. The model achieves good in-domain waveform reconstruction and parameter recovery for the two longest-lived components of eight-component input waveforms, while its performance degrades when the validation spins lie far outside the training range. In a selected spin interval, the framework also recovers the 32 parameters of an eight-component waveform with good overall agreement. These results demonstrate the feasibility of physics-informed autoencoder-based inference for a prescribed multi-component ringdown waveform family and motivate further tests with progressively more realistic signals.

Figures & tables

Figure 1: Analysis flow and autoencoder architecture.
Spin range ( a/Ma/M )Training samplesValidation samples
[0.0, 0.15][0.0,\,0.15]6,0001,500
[0.10, 0.60][0.10,\,0.60]20,0005,000
[0.55, 0.80][0.55,\,0.80]10,0002,500
[0.75, 0.95][0.75,\,0.95]8,0002,000
[0.92, 0.98][0.92,\,0.98]4,8001,200
[0.975, 0.998][0.975,\,0.998]8,0002,000
Table 1: Same-spin-range configuration: ranges of spin parameter a/Ma/M and numbers of training and validation samples.
Training spin range ( a/Ma/M )Validation spin range ( a/Ma/M )Training samplesValidation samples
[0.1, 0.6][0.1,\,0.6][0.55, 0.8][0.55,\,0.8]25,0002,000
[0.75, 0.95][0.75,\,0.95][0.0, 0.15][0.0,\,0.15]15,0002,000
Table 2: Different-spin-range configuration: ranges of spin parameter a/Ma/M and numbers of training and validation samples.
Figure 2: Characteristic behavior of BB in the spin interval 0.0–0.15
Spin range ( a/Ma/M )Match scorennRe (Bn)(B_{n}) (abs.)Im (Bn)(B_{n}) (abs.)Re (ωn)(\omega_{n}) [rad/s] (rel.)Im (ωn)(\omega_{n}) [rad/s] (rel.)
[0.0, 0.15][0.0,\,0.15]0.999±1.6×10−30.999\pm 1.6\times 10^{-3}03.6×10−23±2.8×10−233.6\times 10^{-23}\pm 2.8\times 10^{-23}5.8×10−24±5.1×10−245.8\times 10^{-24}\pm 5.1\times 10^{-24}−5.6×10−4±1.4×10−2-5.6\times 10^{-4}\pm 1.4\times 10^{-2}9.7×10−5±1.3×10−39.7\times 10^{-5}\pm 1.3\times 10^{-3}
11.6×10−22±1.3×10−221.6\times 10^{-22}\pm 1.3\times 10^{-22}1.4×10−23±1.4×10−231.4\times 10^{-23}\pm 1.4\times 10^{-23}−6.3×10−4±1.7×10−2-6.3\times 10^{-4}\pm 1.7\times 10^{-2}1.4×10−4±2.3×10−31.4\times 10^{-4}\pm 2.3\times 10^{-3}
[0.10, 0.60][0.10,\,0.60]0.999±1.1×10−30.999\pm 1.1\times 10^{-3}01.5×10−23±1.5×10−231.5\times 10^{-23}\pm 1.5\times 10^{-23}3.0×10−23±3.3×10−233.0\times 10^{-23}\pm 3.3\times 10^{-23}1.1×10−3±5.9×10−31.1\times 10^{-3}\pm 5.9\times 10^{-3}−2.4×10−4±1.5×10−3-2.4\times 10^{-4}\pm 1.5\times 10^{-3}
17.2×10−23±7.2×10−237.2\times 10^{-23}\pm 7.2\times 10^{-23}2.0×10−22±2.2×10−222.0\times 10^{-22}\pm 2.2\times 10^{-22}1.2×10−3±6.8×10−31.2\times 10^{-3}\pm 6.8\times 10^{-3}−3.0×10−4±1.8×10−3-3.0\times 10^{-4}\pm 1.8\times 10^{-3}
[0.55, 0.80][0.55,\,0.80]0.999±1.9×10−30.999\pm 1.9\times 10^{-3}06.1×10−23±5.9×10−236.1\times 10^{-23}\pm 5.9\times 10^{-23}3.5×10−23±3.2×10−233.5\times 10^{-23}\pm 3.2\times 10^{-23}3.4×10−4±4.0×10−33.4\times 10^{-4}\pm 4.0\times 10^{-3}−2.1×10−4±2.4×10−3-2.1\times 10^{-4}\pm 2.4\times 10^{-3}
15.2×10−22±5.0×10−225.2\times 10^{-22}\pm 5.0\times 10^{-22}1.3×10−22±1.2×10−221.3\times 10^{-22}\pm 1.2\times 10^{-22}3.7×10−4±4.3×10−33.7\times 10^{-4}\pm 4.3\times 10^{-3}−2.2×10−4±2.5×10−3-2.2\times 10^{-4}\pm 2.5\times 10^{-3}
Table 3: Same-spin-range configuration: mean values and standard deviations of the match score, relative errors ( Re(ωn),Im(ωn)\mathrm{Re}(\omega_{n}),\mathrm{Im}(\omega_{n}) ), and absolute errors ( Re(Bn),Im(Bn)\mathrm{Re}(B_{n}),\mathrm{Im}(B_{n}) ) in the validation data.
Figure 3: Resonance around a spin of 0.9.
Figure 4: Validation results for spins in the range 0.0–0.15. (a) Top: input waveform; Bottom: output waveform (Denoised) and original waveform (Original) (b) Parameters and match score (c) Relationship between match score and spin parameter (d) Distributions of true and estimated parameters (e) Relationship between parameter estimation errors and true values.
Figure 5: Validation results for spins in the range 0.1–0.6.
Figure 6: Validation results for spins in the range 0.55–0.8.
Figure 7: Validation results for spins in the range 0.75–0.95.
Figure 8: Validation results for spins in the range 0.92–0.98.
Figure 9: Validation results for spins in the range 0.975–0.998.
Figure 10: Validation results for spins in the range 0.995–max.
Figure 11: Different-spin-range configuration with training on spins 0.1–0.6 and validation on spins 0.55–0.8.
Figure 12: Different-spin-range configuration with training on spins 0.75–0.95 and validation on spins 0.0–0.15.
Figure 13: Relationship between spin and the peak amplitude of the standardized input data used for training.
Figure 17
Figure 17: Comparison between the distributions of the true parameters (gray) and the estimated parameters for all 32 parameters.
Figure 18: Parameter-estimation errors for the 32 inferred parameters. The absolute errors are shown for the excitation factors, and the relative errors are shown for the QNM frequencies.
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