Detectability of avoided crossings in black hole ringdowns

Source: arXiv:2605.16199 · Published 2026-05-15 · By Hayato Imafuku, Naritaka Oshita, Hiroki Takeda

TL;DR

This paper addresses the observational challenge of detecting and resolving black hole quasinormal modes (QNMs) that undergo avoided crossings (ACs), where two QNM frequencies come close in the complex plane and their amplitudes interfere destructively with enhanced magnitude and near opposite phases. While black hole ringdown spectroscopy aims to distinguish multiple QNMs to test strong-field gravity, resolving closely spaced QNM pairs near ACs is difficult due to their near degeneracy and interference effects. The authors develop a Bayesian parameter estimation framework using three waveform models—(1) a simple superposition of two damped sinusoids (2DS), (2) an AC-motivated two-mode interference model (2AC), and (3) an effective exceptional-point (EP) double-pole model with a linear growth term—and systematically evaluate how well two nearly degenerate QNM frequencies and amplitudes can be separately inferred from noisy simulated data.

They find that individual QNM frequencies become spectroscopically unresolved below fractional frequency separations |δω/ω1|≈0.1 even at high SNR (~100), consistent with analytical expectations that resolvability deteriorates quadratically with decreasing mode separation. The 2DS and 2AC models yield comparable frequency resolvability, but the 2AC and EP models better capture the characteristic enhanced and destructively interfering amplitude structure near ACs. The EP model, valid close to exceptional points where modes coalesce, recovers a nonzero linear time-growth signature unique to AC interference, serving as an effective detection indicator despite unresolved modes. Application to a realistic Kerr black hole overtone pair exhibiting an AC confirmed these trends.

Overall, the results indicate that although spectroscopic resolution of individual QNMs near ACs is limited with current methods, complementary waveform descriptions that incorporate AC-specific interference effects may enable detection of collective signatures of mode coalescence in black hole ringdowns.

Key findings

• Individual QNM frequencies are unresolved (σf/|δf| > 1) for fractional separations |δω/ω1| <~ 0.1 at SNR ~ 100 in controlled injection studies (Fig. 2).
• The resolvability ratio σf/|δf| and στ/|δτ| scales approximately as 1/(ρ |δ|^2), where ρ is SNR and δ the frequency or damping time separation (Eqs. 34, 35), explaining quadratic deterioration with decreasing separation.
• The 2AC model yields posterior median frequency estimates closer to true values than the basic 2DS model for small separations but similar uncertainties (Fig. 1).
• The EP model recovers a nonzero linear time-growth amplitude coefficient D near exceptional points, distinguishing nearly degenerate QNM interference signatures (Fig. 3).
• Amplitude parameters in the 2DS model are systematically underestimated for small separations due to degeneracy and destructive interference, while 2AC and EP models infer amplitudes more accurately (Fig. 3).
• Applied to a Kerr black hole overtone pair (2,2,5) and (2,2,6) with |δω/ω1| ~0.075 and SNR ~ 100, individual mode frequencies remain unresolved, but EP model recovers nonzero linear growth consistent with AC signatures (Fig. 4).
• Lower ringdown modes must be removed for effective AC signature detection in overtone examples; contamination from slowly damped unrelated modes could obscure AC features.
• Posterior distributions for phases and geometric parameters are fixed or marginalized; inferred frequency and damping posteriors widen as mode separation decreases.

Threat model

The adversary is effectively the measurement noise (stationary Gaussian) imposed by gravitational-wave detectors at design sensitivity, which limits the capability to resolve closely spaced QNM frequencies. The analysis assumes perfect knowledge or removal of contaminating signals from other modes unrelated to the avoided crossing and assumes fixed geometrical parameters. The adversary cannot introduce non-Gaussian noise or systematic calibration errors. The key limitation is the inherent indistinguishability of nearly degenerate modes due to their fundamental interference and noise floor rather than active tampering.

Methodology — deep read

1. Threat Model & Assumptions: The study assumes an optimistic injection scenario where two QNMs approach each other exhibiting avoided crossing (AC) interference, and all other QNMs and late-time tails are negligible or removed. The black hole parameters (mass, spin) are known approximately, and geometric parameters (sky location, polarization) are fixed to simplify inference. The adversary in the sense of observational noise is modeled as stationary Gaussian detector noise at design sensitivity. The goal is to assess detectability and resolvability of nearly degenerate QNM pairs, not robustness to adversarial noise or model mismatch.

2. Data: Signals are simulated ringdown waveforms injected into colored Gaussian noise modeled after a three-detector network (LIGO Hanford, LIGO Livingston, Virgo) at O4 design sensitivity. The injected signals are generated primarily from the 2AC waveform model, which incorporates AC interference features, with controlled variations of frequency and damping time differences |δω/ω1| = 0.3, 0.1, 0.01, and 0.001. Each injection has SNR ~ 100 (and additional tests at SNR 10 and 1000). The injections correspond to Schwarzschild or Kerr QNM parameters appropriate for 60 M⊙ black holes. Geometric parameters are held fixed via delta-function priors. For a realistic Kerr AC example, overtone modes (2, 2, 5) and (2, 2, 6) of spin j=0.9 are used.

3. Architecture/Algorithm: Three waveform template families are used for Bayesian inference in the time domain:

• 2DS: simple superposition of two damped sinusoids, each parameterized by amplitude, frequency, damping time, and phase.
• 2AC: AC-inspired two-mode model explicitly including enhancement and destructive interference parameters, expressed as a function of frequency separation δω.
• EP: effective exceptional point model approximating the two modes as a single double-pole mode with a linear growth (time) component, parameterized by two amplitudes and phases plus a single frequency and damping time.

These models are mathematically connected; the 2AC model generalizes 2DS with AC-specific amplitude structure, and the EP model approximates the 2AC model in the δω→0 limit.

4. Training Regime: Bayesian inference is performed with pyRing and CPNest nested sampling, using 2048 live points and max MCMC chain length 2048. Priors on waveform model parameters are uniform with physically motivated ranges (e.g., damping times 0.5–50 ms, frequencies 21–500 Hz). Geometric parameters are fixed to injected values. Likelihoods assume Gaussian stationary noise with known PSDs.

5. Evaluation Protocol: Posterior distributions over waveform parameters are obtained. Resolvability of frequency (f1,f2) and damping (τ1,τ2) pairs is quantified using a Rayleigh-type criterion σf/|δf|<1 (analogously for τ), where σf is maximum posterior uncertainty. Amplitude parameter recovery and bias are quantified by fractional deviations from injected values. Results are analyzed for controlled injections with systematically varied δω and for a realistic Kerr overtone AC example. Analytical Fisher matrix estimates are also developed to interpret resolvability scaling.

6. Reproducibility: The pyRing package and CPNest nested sampler used are open source, though no dedicated code release for this project is mentioned. Injection parameters and model priors are described in detail for replication. Some inferred posteriors and figures are provided. The dataset is simulated and synthetic, but based on realistic detector noise and QNM theory.

Example End-to-End: For a frequency-shifted injection with |δω/ω1|=0.01 and SNR=100, the 2AC model waveform is generated with f1=201.2 Hz, f2=199.2 Hz, τ1=τ2=3.322 ms, amplitude A to give SNR∼100, and α derived from mass scaling. This injection is added to simulated O4 network Gaussian noise. PyRing performs nested sampling over the 2AC model parameters (A, α, phases, frequencies, damping times) holding geometry fixed. Posterior distributions over frequencies show broadened peaks overlapping between modes, yielding σf/|δf|>1, meaning unresolved modes. Amplitude parameter posteriors center near injected values with smaller bias than 2DS. The EP model fit infers a nonzero linear growth coefficient D. This illustrates that even with optimistic high SNR, individual frequencies are unresolved whereas collective AC signatures remain detectable.

Technical innovations

• Formulation and numerical comparison of three waveform models—2DS, 2AC, and EP—for describing black hole ringdown QNMs near frequency avoided crossings.
• Bayesian parameter estimation framework in the time domain for assessing resolvability of nearly degenerate QNM pairs under realistic O4 detector noise assumptions.
• Derivation and use of AC-inspired 2AC waveform model explicitly incorporating destructive interference and amplitude enhancement dependent on complex frequency separation δω.
• Introduction of EP effective double-pole waveform model capturing characteristic linear time-growth mode interference as a complementary signature of almost coalescing QNMs.
• Analytical Fisher matrix scaling relations explaining quadratic loss of mode resolvability as mode separation decreases, clarifying observational limits.

Datasets

• Simulated ringdown waveforms with two QNM modes exhibiting AC phenomena—SNR ~ 10 to 1000 injections—based on Schwarzschild and Kerr black hole parameters (M=60 M⊙, spin j=0 or 0.9)
• O4 design sensitivity noise PSDs for LIGO Hanford, LIGO Livingston, and Virgo detectors used to generate Gaussian noise realizations

Baselines vs proposed

• 2DS model frequency resolvability at SNR ~ 100 becomes unresolvable (σf/|δf| > 1) below |δω/ω1| ≈ 0.1; 2AC model shows similar resolvability behavior (Fig. 2).
• For amplitude inference at small separations, 2DS model systematically underestimates amplitudes by up to ~50%, whereas 2AC and EP models maintain <10% bias (Fig. 3).
• EP model linearly growing amplitude parameter D inferred to be consistent with zero when no AC present, but significantly nonzero for AC cases, serving as sensitive AC signature.
• In Kerr overtone (2,2,5)/(2,2,6) injection (|δω/ω1| ~0.075), both 2DS and 2AC models fail to resolve individual modes (σf/|δf| > 5), while EP model effectively detects linear growth (Fig. 4).

Figures from the paper

Figures are reproduced from the source paper for academic discussion. Original copyright: the paper authors. See arXiv:2605.16199.

Fig 4: The top panels show the posterior distributions of the frequencies (left) and damping times (right) inferred using the

Fig 5: and 6 present the posterior distributions of the

Fig 6: Similar to Fig. 5, but for the damping-time-shifted injections.

Fig 4 (page 11).

Fig 5 (page 11).

Fig 6 (page 11).

Fig 7 (page 11).

Fig 8 (page 11).

Limitations

• Analysis assumes that contamination from other QNMs or late-time tails is negligible or removed, which is optimistic and may not hold in realistic ringdown signals.
• Geometric parameters (sky location, polarization, coalescence time) are fixed to injected values, ignoring uncertainties that could further degrade resolvability.
• Noise modeled as stationary, Gaussian, and perfectly known PSDs; real detector noise non-stationarities and calibration uncertainties are not incorporated.
• No tests performed for robustness to model misspecification, such as deviations from Kerr, or uncertainties in black hole mass and spin parameters.
• Exceptional point (EP) model is only a valid approximation in the very small complex frequency separation limit; fails for larger separations.
• The study does not consider observational constraints from full ringdown including earlier inspiral/merger phases nor attempts joint inference.

Open questions / follow-ons

• How robust are AC and EP-related waveform signatures to contamination from multiple simultaneously excited QNMs and overlapping late-time tails in realistic astrophysical ringdowns?
• What is the impact of including prior uncertainties in black hole mass, spin, and sky location on the resolvability of closely spaced QNMs exhibiting avoided crossings?
• Can advanced or new detector networks with improved sensitivity and bandwidth significantly improve the resolvability of nearly degenerate QNMs?
• Is it possible to design alternative waveform models or inference strategies that better separate or characterize exceptional-point-like behavior beyond the EP approximation?

Why it matters for bot defense

From a bot-defense or CAPTCHA perspective, this paper offers lessons about detecting subtle interference effects in signals composed of closely spaced, nearly degenerate components, analogous to challenging-to-distinguish user actions or challenge-responses. The work highlights how straightforward decomposition models (like the 2DS model) can underestimate amplitudes due to near-degeneracy and destructive interference, stressing the need for waveform or feature parameterizations adapted to the underlying interference physics (like the 2AC and EP models). Conceptually, this reinforces that individual component resolution in a composite signal may be limited, but collective interference patterns can serve as distinct signatures useful for detection or classification.

Practitioners in bot-defense could draw analogies in developing challenge templates or detection heuristics that recognize collective interference/artifact signatures instead of relying solely on isolating single, clean features. The Bayesian methodology shows how carefully designed prior distributions and waveform parameterizations adapted to characteristic interference phenomena improve inference quality. However, the difficulty resolving nearly degenerate modes despite high SNR illustrates fundamental limits, implying detection systems must combine multiple complementary signatures robust to noise and contamination. While physically specialized to black hole ringdowns, the insights about interference-induced degeneracies and effective double-pole model approximations inform general detection and spectral resolution challenges in security biometrics or bot-behavior analysis.

Cite

@article{arxiv2605_16199,
  title={ Detectability of avoided crossings in black hole ringdowns },
  author={ Hayato Imafuku and Naritaka Oshita and Hiroki Takeda },
  journal={arXiv preprint arXiv:2605.16199},
  year={ 2026 },
  url={https://arxiv.org/abs/2605.16199}
}

Read the full paper

• arXiv:2605.16199 (abstract)
• arXiv:2605.16199 (PDF)