Resonance in black hole ringdown: Benchmarking quasinormal mode excitation and extraction (2509.06411v1)

Abstract: We investigate how resonant excitation near exceptional points manifests in Kerr black hole ringdown waveforms and examine its extraction. Using waveforms generated by localized initial data, where quasinormal mode amplitudes are given solely by excitation factors, we establish a controlled benchmark for overtone extraction. Applying an iterative fitting method with mirror modes, we analyze a mild resonance in the $(l,m)=(2,2)$ multipole and a sharp resonance in the $(3,1)$ multipole occurring as part of a sequence of successive resonances. For $(2,2)$, we extract the fundamental mode, the first three overtones, and the fundamental mirror mode with relative errors below $10\%$, and show that residual waveforms exhibit the expected damped sinusoids together with distinctive resonance signatures. For $(3,1)$, we demonstrate that resonances can not only amplify but also reduce QNM excitations, reshaping the overtone hierarchy and rendering the sharp resonance more pronounced in ringdown. Our results clarify the imprint of resonance in Kerr ringdown and highlight both the robustness and limitations of current extraction techniques, providing a foundation for more reliable extraction of higher overtones and for applications to observational data analysis.

• The paper benchmarks QNM amplitude extraction using an iterative fitting method to reveal resonance-induced alterations in the overtone hierarchy.
• It applies the Teukolsky formalism and mirror mode analysis to accurately extract QNM signals from localized source data in Kerr black holes.
• The study highlights practical implications for black hole spectroscopy by demonstrating improved extraction techniques in the presence of resonances.

Resonant Excitation and Extraction of Quasinormal Modes in Kerr Black Hole Ringdown

Introduction and Motivation

The ringdown phase of gravitational wave signals from perturbed black holes encodes information about the strong-field regime of spacetime, with the signal well described by a superposition of quasinormal modes (QNMs). Each QNM is characterized by a complex frequency determined by the background geometry, and an amplitude that encodes both the excitation process and the spacetime structure. Recent work has highlighted the dominant role of excitation factors—quantities depending only on the background spacetime—in determining the overtone hierarchy, especially for prograde modes. This has significant implications for black hole spectroscopy, as reliable extraction of QNM amplitudes across multiple overtones could enable more precise probing of black hole parameters beyond frequency measurements.

A particularly nontrivial feature arises when QNM frequencies approach each other in the complex plane, leading to avoided crossings and resonant amplification of excitation factors near exceptional points—a haLLMark of non-Hermitian systems. These resonances can manifest as lemniscate (figure-eight) patterns in the complex plane and have been proposed as a resolution to anomalies in the Kerr QNM spectrum. However, the practical extraction of overtone amplitudes, especially in the presence of such resonances, remains challenging due to ambiguities in ringdown start time, overfitting, and the limitations of standard fitting metrics.

This work benchmarks the extraction of QNM amplitudes in the presence of resonant excitation, using waveforms generated by localized initial data where amplitudes are determined solely by excitation factors. The paper applies an iterative fitting method with mirror modes to analyze both mild and sharp resonances in the Kerr QNM spectrum, providing a controlled environment for overtone extraction and clarifying the imprint of resonance on ringdown signals.

Theoretical Framework: Teukolsky Formalism and QNM Decomposition

The analysis is grounded in the Teukolsky formalism for linear perturbations of Kerr black holes, where the master variable $\rho^{-4}\Psi_4$ satisfies a separable equation in Boyer-Lindquist coordinates. The radial Green's function is constructed from "in" and "up" solutions, with QNM frequencies identified as the poles of the Green's function (i.e., where the Wronskian vanishes). The ringdown waveform at infinity can be written as an integral over real frequencies, which, by contour deformation, yields a sum over QNMs plus a power-law tail.

The QNM amplitude for each mode factorizes into a source-dependent term and an excitation factor $B_{lmn}$, the latter given by

$$B_{lmn} = \left. \frac{B^\text{ref}_{lm}}{2\omega} \left(\frac{dB^\text{inc}_{lm}}{d\omega}\right)^{-1} \right|_{\omega=\omega_{lmn}}$$

where $B^\text{ref}_{lm}$ and $B^\text{inc}_{lm}$ are asymptotic amplitudes of the radial Teukolsky solutions. The excitation factors are sensitive to the choice of normalization, master variable, and especially the integration constant in the tortoise coordinate, which can introduce nontrivial phase and amplitude shifts.

A key technical point is the treatment of mirror modes (negative-frequency QNMs), which are related to the original (positive-frequency) modes by complex conjugation and index reversal. The symmetry properties of the Teukolsky equation ensure that the excitation factors for mirror modes are complex conjugates of those for the original modes.

Figure 1: QNM frequencies (left) and excitation factors (right) for the $(l,m)=(2,2)$ mode as a function of spin parameter $a/M$. The thick curves highlight the region $0.85 \leq a/M \leq 0.95$, showing resonant amplification of the fifth and sixth overtones.

Waveform Construction and Benchmarking Setup

To provide a controlled benchmark for overtone extraction, the paper employs waveforms generated by a localized delta-function source. In this setup, the QNM amplitudes are determined solely by the excitation factors, eliminating uncertainties from source dependence. The waveform at infinity is given by

$$\Psi_{4,lm}(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega\, e^{-i\omega t} \frac{1}{2i\omega} \frac{B^\text{ref}_{lm}}{B^\text{inc}_{lm}}$$

which, via the residue theorem, yields a sum over QNMs:

$$\Psi_{4,lm}(t) = -\sum_{n=-\infty}^{\infty} B_{lmn} e^{-i\omega_{lmn} t} + \text{(power-law tail)}$$

This construction allows direct comparison between extracted amplitudes and theoretically computed excitation factors.

Figure 2: Time-domain ringdown waveforms of the $(l,m)=(2,2)$ mode induced by a localized source, for various spin parameters $a/M$.

Iterative Fitting Methodology

The extraction of QNM amplitudes is performed using an iterative fitting algorithm, which sequentially fits and subtracts the dominant mode from the waveform, then fits the residual for the next-longest-lived mode, and so on. This approach mitigates overfitting and is robust to the presence of mirror modes, which become significant in asymmetric mass ratio cases.

The fitting function at each iteration is a sum of damped sinusoids:

$$\psi^\text{fit}(t) = \sum_{n \in \{\mathrm{QNM}\}} C_n e^{-i\omega_n t}$$

with complex amplitudes $C_n$ determined by maximizing the overlap with the data. The stability of the extraction is assessed by varying the fitting window and monitoring the flatness of the amplitude plateau.

The method also includes explicit fitting and subtraction of the leading power-law tail, which is essential for isolating the QNM content in the late-time signal.

Results: Resonant Excitation and Overtone Extraction

Mild Resonance: $(l,m)=(2,2)$ at $a/M \simeq 0.9$

The iterative fitting procedure successfully extracts the fundamental mode, the first three overtones, and the fundamental mirror mode with relative errors below 10%. The extracted amplitudes exhibit stable plateaus as a function of fitting start time, confirming the robustness of the method. For higher overtones (fifth and sixth), the extraction becomes less accurate, with errors of order unity, due to contamination from the power-law tail and error propagation from lower overtones.

Residual waveforms after subtraction of extracted QNMs display the expected damped sinusoids, and the combined contribution of the fifth and sixth overtones reproduces the observed oscillation in the residual, consistent with the resonance structure in the excitation factors.

Sharp Resonance: $(l,m)=(3,1)$ at $a/M \simeq 0.9722$

In the case of a sharp resonance, the excitation factors for the $n=+5$ and $+6$ overtones are strongly amplified, while the $n=+4$ overtone is suppressed by several orders of magnitude due to a preceding mild resonance. This leads to a reversal of the usual overtone hierarchy: the $n=+5$ and $+6$ modes dominate the late-time signal, while $n=+4$ is subdominant. This behavior is directly reflected in the time-domain waveform and has important implications for overtone extraction in the presence of resonances.

The sum of the excitation factors for the resonant pair does not exhibit dramatic variation across the resonance, consistent with the lemniscate trajectory in the complex plane. However, the time-domain impact is substantial, underscoring the necessity of direct waveform analysis rather than relying solely on excitation factor magnitudes.

Implementation Considerations and Practical Implications

• Excitation Factor Ambiguities: Careful attention must be paid to the conventions used for excitation factors, especially the integration constant in the tortoise coordinate. Inconsistent choices can lead to apparent discrepancies in extracted amplitudes.
• Iterative Fitting Robustness: The iterative method is effective for extracting up to the third or fourth overtone in controlled settings. Extraction of higher overtones is limited by tail contamination and error propagation.
• Resonance-Induced Hierarchy Changes: Resonances can both amplify and suppress QNM excitations, leading to nontrivial overtone hierarchies. Extraction algorithms should adapt the order of mode extraction based on amplitude hierarchy rather than damping time alone.
• Benchmarking with Localized Sources: The use of localized initial data provides a valuable benchmark for testing extraction algorithms, as the true amplitudes are known from excitation factors.

Future Directions

• Algorithmic Improvements: Enhanced fitting algorithms, possibly incorporating regularization or Bayesian approaches, may enable more reliable extraction of higher overtones and better handling of tail contamination.
• Application to Observational Data: Insights from this controlled paper can inform the analysis of real gravitational wave signals, where excitation factors still largely determine overtone amplitudes.
• Exploration of Alternative Waveforms: Investigating other initial data configurations or including more realistic source terms could further test the generality of the findings.

Conclusion

This paper provides a detailed benchmark of QNM overtone extraction in the presence of resonant excitation, clarifying the imprint of resonance on Kerr ringdown waveforms. The results demonstrate both the robustness and limitations of current extraction techniques, highlight the importance of excitation factor conventions, and reveal that resonances can fundamentally alter overtone hierarchies. These findings lay the groundwork for more reliable extraction of higher overtones and have direct implications for black hole spectroscopy and the analysis of gravitational wave data.