Model-agnostic search of gravitational wave echoes in LVK data

Di Wu; Jing Ren; Qing-Guo Huang; Xi-Li Zhang

REVIEW 3 major objections 4 minor 100 references

Model-agnostic search of gravitational wave echoes in LVK data

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper presents a model-agnostic search for long-lived quasinormal modes behind gravitational wave echoes, applies it to three high-SNR black hole mergers, and finds no statistically significant evidence of postmerger echoes, instead set

desk verdict Solid incremental echo-search paper with a clean null result, but the 'model-independent' claim is stronger than the UniEw template validation supports. read the letter →

arxiv 2512.24730 v2 pith:BLRLGIAB submitted 2025-12-31 gr-qc astro-ph.HE

Di Wu , Xi-Li Zhang , Qing-Guo Huang , Jing Ren This is my paper · ORCID

classification gr-qcastro-ph.HE

keywords gravitationalwaveechoesquasinormalmodesultracompactobjectsmodel-independentsearchphase-marginalizedlikelihoodLIGO-Virgo-KAGRABayesianupperlimitsGW150914

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish a robust, model-agnostic way to search for gravitational wave echoes—repeating wavelets that would follow a black hole ringdown if the remnant has a reflective surface just outside the would-be horizon. The key move is a generalized phase-marginalized likelihood that coherently combines frequency bins belonging to the same quasinormal mode across a network of detectors, preserving partial phase information that earlier amplitude-only searches discarded. The authors validate the pipeline by injecting a benchmark echo waveform into real O1 noise and demonstrating reliable recovery. Applied to GW150914, GW231226, and GW250114, the search finds no statistically significant evidence for postmerger echoes and derives 90% upper limits on the network SNR and the average strain amplitude of long-lived quasinormal modes. A sympathetic reader would care because this turns an uncertain theoretical signal into a concrete, falsifiable search, yielding the first model-independent constraints on late-time echoes.

What carries the argument

The carrying mechanism is the generalized phase-marginalized likelihood (Eq. 10): for each quasinormal mode n, the data from all detectors are combined into a single complex frequency-domain series, and the likelihood marginalizes over one constant phase per mode, yielding an I0 Bessel function of the absolute value of the coherent overlap between data and the search template, minus a network-optimal-SNR penalty. The search template is the 'UniEw' model, a uniform comb of equally spaced Lorentzian lines with common spacing Δf and damping time τ, which approximates the long-lived QNM spectrum without committing to a specific ultracompact-object microstructure. This reduces the problem to seve

What would settle it

Inject a numerically simulated echo waveform from a spinning ultracompact object with frequency-dependent reflectivity—so that the slow phase term δn + 2πftd is not actually slow—into real detector noise, run the pipeline, and check whether the injection is recovered with the claimed SNR; failure would show the search is blind to non-UniEw signals. Alternatively, a future detection of a comb-like line with cross-detector coherence and a matching phase pattern would falsify the null result.

Watch

Extended reading notes

Core claim

The paper's central claim is that the late-time echo signal, when the interior reflection is strong, can be captured by a simple uniform comb of equally spaced, long-lived quasinormal modes with a Lorentzian amplitude profile, and that a phase-marginalized likelihood which coherently combines all frequency bins and all detectors for each mode is substantially more sensitive than previous per-bin phase-marginalized approaches. The new likelihood (Eq. 10) marginalizes over a single constant phase per mode, producing a zeroth-order Bessel function of the complex matched-filter statistic, and it is shown to suppress instrumental line contamination, improve detection significance at long duration

Load-bearing premise

The entire search rests on the assumption that a real echo signal's phase around each mode is dominated by the Lorentzian line-shape term, so the slowly varying phase can be safely marginalized; if a physical waveform's phase evolves differently, the template will not match it and the null result would not exclude it.

Editorial extensions

If this is right

If a merger remnant has a reflective surface close to the horizon, its late-time signal should appear as a comb of long-lived QNMs; this search provides the first model-agnostic probe of that regime.
The new likelihood coherently combines frequency bins and detectors, so its detection significance and the stability of its upper limits improve with observation duration, whereas the old per-bin likelihood degrades.
The 90% upper limits, e.g., SNR90% ≈ 4.8 and A90% ≈ 1.3 × 10^-24 for GW231226, imply that any echo signal in these events must have an average strain amplitude below roughly 10^-24.
Most instrumental lines are rejected by the phase-coherent combination, but a few transient, single-detector line features survive and are identified as non-astrophysical, illustrating the search's sensitivity to signal-like structures.
The pipeline is applicable to future high-ringdown-SNR events and longer observation durations, where the phase-coherent gain is largest and the constraints will tighten.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the same phase-coherent likelihood formalism does not depend on echo-specific physics and could be applied to other long-lived narrow-band gravitational-wave sources, such as boson clouds around spinning black holes.
Extension: the template's neglect of the slowly varying phase term δn + 2πftd is the main risk; if a real UCO's phase evolution is not dominated by the Lorentzian term—due to overlapping modes or frequency-dependent reflectivity—the search could systematically miss the signal, so the reported upper limits should be read as limits on UniEw-like signals.
Extension: a testable prediction is that the strongest constraints should come from events with the lowest noise floor and longest usable postmerger stretch; applying the pipeline to GW250114 with longer T or to future O5 events should push SNR90% below 4.
Extension: the per-event upper limits could be combined hierarchically across the full LVK catalog to produce the first model-independent population bound on near-horizon reflectivity of compact remnants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a competent, incremental data-analysis paper from a group that has been building this echo-comb search program for several years. The new piece is real: a multi-detector extension of the phase-marginalized likelihood from Ref. [70], which coherently combines frequency bins within each QNM across the Hanford-Livingston network. The derivation (Eqs. 6–10) is clean, the reduction to the old per-bin likelihood in the low-resolution limit is correct, and the validation on O1 background with 150 time-slide realizations is genuine evidence that the pipeline handles real noise. Applying the method to GW250114 is also new, and the null result for all three events is consistent with the presented background distributions. Credit where earned: the code and data are on GitHub, the notching procedure is clearly described, and the paper is honest about many of its own limitations—UniEw is repeatedly called a 'simplified model' and a 'leading-order description.'

The soft spots are real but not fatal. The main one is the self-referential template validation: the injection waveform is generated from the same UniEw model that the search implements, so the sensitivity validation only tests one phase-clean, constant-reflectivity case. The quoted upper limits therefore inherit the phase ansatz of Eq. (5). Calling these 'model-independent constraints' in the abstract and conclusions is stronger than the evidence supports—'template-independent' would be more accurate. The stress-test note is right about this, and it is not a minor quibble: the whole gain of the new likelihood comes from trusting that phase structure.

There are also smaller statistical issues that should be fixed before publication. The reported p-values are single-trial; with five events and multiple durations, the smallest p-value (~0.18) is not corrected for the look-elsewhere effect. The GW250114 background uses only 150 realizations, mixing pre- and post-event segments. The 90% upper limits are not calibrated by injection coverage studies, so it is hard to know whether they are over- or under-covering. And the mysterious broad quasi-periodic artifact in the old-likelihood GW150914 114 s case is left unexplained; the paper notes it, but a reader cannot tell how much it affects the null result. None of this obviously breaks the central null result—the Bayes factors are all unremarkable—but it does mean the upper-limit numbers should be treated as preliminary.

Who is this for? People working on echo searches and UCO phenomenology, especially those who want a direct comparison with prior comb searches. It deserves a serious referee; the methodology is sound enough to warrant peer review, and the flaws are addressable in revision. I would accept it with major comments on calibration and trial factors, not desk-reject it.

Referee Report

3 major / 4 minor

Summary. The paper extends the phase-marginalized likelihood of Ref. [70] to a two-detector network, coherently combining frequency bins within each QNM while marginalizing over a per-mode constant phase. It uses the simplified UniEw template (Eq. 16) and a Bayesian search pipeline with an iterative notching procedure, validates it on O1 background with 150 time-slide realizations and a single constant-reflectivity injection, and then applies it to GW150914, GW231226, and GW250114. No significant evidence for echoes is found; 90% upper limits are set on network SNR and average initial amplitude (e.g., SNR90%≈4.8 and A90%≈1.3e-24 for GW231226 at T=145 s with the new likelihood). The likelihood derivation (Eqs. 6-10) is internally consistent and correctly reduces to the old likelihood in the low-resolution limit.

Significance. If the claims are fully supported, this is a useful contribution: it provides an efficient coherent search statistic, demonstrates robustness of the notched pipeline on O1 data, and gives concrete null constraints on long-lived QNM amplitudes for a recently detected O4 event. The paper ships analysis code, summary posterior data, and uses public LVK data, which is a strength. However, the significance is reduced by the gap between the title/abstract's 'model-independent' framing and the actual use of a simplified UniEw template. The reported upper limits are conditional on the phase ansatz of Eq. (5), and the validation exercises only one phase-clean benchmark signal. The central method is sound, but the scope of the claims needs tightening and the robustness to model mismatch needs to be demonstrated.

major comments (3)

[Secs. II-III, V-VI; Eqs. (5), (16); Fig. 7] The 'model-independent' claim is stronger than the evidence supports. The likelihood's coherent gain and the reported upper limits (Fig. 7, Table IV) depend on the phase model of Eq. (5), which is encoded in the UniEw template (Eq. 16). The only injection test (Sec. IV) uses a constant reflectivity Rwall=0.99 and a ringdown excitation, i.e., precisely the phase-clean case. The paper itself calls UniEw a 'simplified model' and 'leading-order description'. A model-mismatch robustness study is needed: inject waveforms with frequency-dependent reflectivity, overlapping modes, or different excitation phases, and quantify detection efficiency and bias in the inferred SNR/amplitude. Alternatively, the abstract and Secs. V-VI should explicitly state that the constraints apply only within the UniEw template family.
[Sec. II, Eqs. (3), (5), (7)] The marginalization treats δ'_n = δ_n + 2π f t_d as constant across all bins of a mode. But using t_d≈1/Δf and the mode cutoff f_cut ≤ Δf/2, the phase variation 2π f_cut t_d can be O(π). Thus the constant-phase assumption is not automatically valid. For signals whose phase retains the second term of Eq. (5), the coherent sum could partially cancel even when the amplitude model is correct. The single injection does not probe this regime. The authors should either derive and verify a condition such as 2π f_cut t_d << 1 for the searched parameter space, or extend the phase treatment.
[Secs. IV-V; Fig. 7; Table II] The upper-limit calibration is based on a loud injection (network SNR≈16, logB≈40 at T=49 s). The quoted 90% limits are around SNR≈5, but there is no injection-recovery study at near-threshold amplitudes. Without demonstrating that the posterior-based upper limits have correct coverage for weak signals, the strength of the constraints is not fully established. A small injection campaign at SNR values bracketing the claimed limits would address this.

minor comments (4)

[Sec. V.A] The text says the strain-data preparation follows the procedure 'outlined in Sec. VI'; this should be Sec. III (pipeline description).
[Title and Abstract] The title uses 'Model-independent' while the abstract uses 'model-agnostic' and the body repeatedly describes UniEw as simplified/leading-order. Please harmonize the terminology and qualify the claims consistently.
[Footnote 3 and Eq. (16)] The admitted notation typo conflating A and A' should be corrected in the main text rather than only explained in a footnote.
[Appendix A / Fig. 9] The broad quasi-periodic artifact seen for GW150914 at 114 s is said to require 'further investigation'. This is fine, but it would be useful to state explicitly in Sec. V.A that unresolved artifacts are included in the background distribution and therefore do not bias the p-values.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: likelihood and upper limits are explicitly built on a labeled simplified template; the remaining concern is model-mismatch coverage, not circularity.

full rationale

The derivation chain is self-contained at the level of the search statistic: Eq. (2) is the standard UCO transfer function, Eqs. (4)-(5) are stated as a pole approximation around each QNM, and Eqs. (6)-(10) derive the network phase-marginalized likelihood from the Gaussian likelihood by phase marginalization. The search template Eq. (16) is explicitly labeled a simplified and leading-order description of long-lived QNMs (Sec. III), so the paper does not disguise the template as an exact first-principles waveform. The injection test uses a benchmark waveform from Eq. (2) with constant reflectivity Rwall=0.99 and ringdown excitation (Sec. IV), which is an internal self-consistency check rather than independent validation of the phase model against frequency-dependent reflectivity or overlapping modes. The null results for GW150914, GW231226, and GW250114 are computed from real data against time-slide backgrounds with p-values (Sec. V), and the upper limits are stated as limits on the UniEw search template parameters, not on an independently defined physical amplitude: 'we constrain its strength by setting upper limits on the network SNR and amplitude of the UniEw search template, which approximates the QNMs at leading order.' Thus no fitted parameter is renamed as a prediction, and no result equals its input by construction. The central limitation—that a real echo whose phase evolution differs from Eq. (5) could partially cancel in the coherent sum and bias the limits—is a model-mismatch/correctness risk rather than circularity. The reliance on Refs. [68,70] for the phase approximation and UniEw model is a self-citation, but the likelihood derivation is explicit and the real-data analysis is independent of those fitted values; I therefore rate this as minor self-citation, not load-bearing circularity.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central claim is a null result and a constraint, so there are no invented physical entities. The parameter space of the search (Δf, q0, A′, τ, fmin, fmax, φHL,0, T) is scanned/inferred; this is a template family parameterization, not an independently justified physical model. The strongest methodological assumption is that the UniEw template family captures the target signal: the injection study uses the same model, so it does not provide independent evidence for template fidelity. The time-slide background for GW250114 (using post-event segments) is an additional data-selection assumption that weakens the p-value interpretation.

free parameters (9)

η (interior-reflection efficiency parameter) = η ∈ [1, 4] scanned
Controls the range of frequency spacing Δf via MΔf = R̄/η (Eq. 18). The scan range is chosen to cover Planck-scale reflective surfaces; the 'model-agnostic' nature of the search is achieved by scanning this parameter rather than fitting it, but it is still a manually chosen range that defines the sensitivity space.
Δf (frequency spacing between QNMs) = posterior medians 3.8119 Hz (inj T=49s new)
Primary search parameter; related to echo time delay by Δf ≈ 1/td. Injected, not fitted, in the injection test; in the real search it is inferred from the null data.
q0 (relative offset of QNM comb) = posterior ~0.982-0.98 (injection)
Uniform prior [0,1]; fitted in the analysis.
A′ (average frequency-domain amplitude) = posterior ~1.2-1.5 ⟨|n_j|⟩ (injection); upper limits A90 ~1e-24 real
Prior uniform in [⟨|n|⟩/100, 10⟨|n|⟩]; the quantity converted to time-domain average amplitude A = A′/τ for the upper limit statements. Fitted to data.
τ (damping time of QNMs) = posterior log10(1/τ) ~ -1.3 (injection)
Log-uniform prior in [1/T, Δf_max]; fitted. The combination τΔf is interpreted as inverse log-reflectivity.
fmin, fmax (frequency band edges) = posterior fmin ~140 Hz, fmax ~242 Hz (injection); priors [50, 1.1 f_RD]
Uniform priors with the constraint fmax-fmin > 10Δf; fitted to data.
φHL,0 (relative detector response phase) = posterior ~0 (injection)
Uniform prior in [−π/2, 3π/2]; absorbs uncertainty in the arrival-time lag between H and L. Fitted to data.
T (analysis duration) = two benchmark values per event (e.g., 57.2 s and 114.4 s)
Not fitted but scanned over two values; changing T changes the frequency resolution and hence the sensitivity of the search.
line notch threshold (normalized strain amplitude 6) = 6
Manual threshold for deciding which instrumental lines to notch; variation of this threshold affects background and sensitivity, not explored.

assumptions (6)

domain assumption Standard GW data-analysis assumptions: stationary Gaussian noise in each frequency bin, independent across bins and detectors, PSD known.
The likelihood (Eqs. 6-8) is built on Gaussian likelihood per frequency bin and detector with one-sided PSD P_I,j. The paper states non-Gaussian effects are handled by notching, but the likelihood is Gaussian.
domain assumption The late-time echo waveform is dominated by a single long-lived QNM per resonance, with amplitude and phase profile given by Eqs. (4)-(5).
Central modeling assumption: the signal is a superposition of equally spaced Lorentzian-like narrow lines with slowly varying phase; this is the foundation of the UniEw template (Eq. 16).
domain assumption The source sky position and waveform phase evolution permit approximating the detector response ratio R_I/R_J as a constant amplitude A_JI and phase φ_JI,j linear in frequency (Eqs. 13-14).
Used to combine the two LIGO detectors coherently with a single relative phase parameter φHL,0 and fixed ΔtHL from the main event analysis; response errors are claimed to be small for long-lived QNMs.
domain assumption The remnant mass M and spin χ from the main-event analysis correctly set the prior bounds f_RD (Eq. 19) and Δf range (Eq. 18).
Prior range of the search band and comb spacing depends on the best-fit M, χ; if these are wrong, the sensitivity space changes, though the search remains 'agnostic' within the prior.
domain assumption Time-slide background realizations are statistically independent samples from the same noise distribution as the postmerger search segments.
Used for p-values: for GW250114, background segments include both pre- and post-event data (because of data-quality limits); this introduces a potential mismatch with the on-source segment (selection bias for the noise distribution).
ad hoc to paper The simplified UniEw template family adequately represents the target long-lived QNM signal.
This is the key template assumption. It is validated only with injections from the same family (Eq. 2 with constant Rwall = 0.99), so the method's sensitivity to other physically motivated echo waveforms (e.g., frequency-dependent reflectivity, spinning UCO with 2-component response) is untested.

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Cite this review

Pith. "Pith review of Model-agnostic search of gravitational wave echoes in LVK data." pith.science (2026). https://pith.science/paper/BLRLGIAB

@misc{pith2026251224730,
  author       = {Pith},
  title        = {Pith review of: Model-agnostic search of gravitational wave echoes in LVK data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLRLGIAB}},
  note         = {Machine review of arXiv:2512.24730}
}

read the original abstract

Gravitational wave echoes offer a unique probe of the near-horizon structure of astrophysical black holes, beyond the standard "black hole spectroscopy." Theoretical waveform predictions, however, remain uncertain, motivating robust searches that avoid specific echo modeling. We present a model-agnostic search framework targeting long-lived quasinormal modes (QNMs) expected from strong interior reflection. By employing a generalized phase-marginalized likelihood that coherently combines data for each QNM across a detector network, our method enhances sensitivity to the signals. To handle real detector noise, we implement an optimized notching procedure to suppress instrumental spectral lines and refine the Bayesian parameter settings. We validate the performance of this framework using injection studies on O1 background data, demonstrating reliable signal recovery in realistic noise conditions. We then apply this method to three binary black hole merger events with high ringdown signal-to-noise ratios (SNRs): GW150914 from O1, GW231226 from O4a, and the recently reported O4 event GW250114. No statistically significant evidence for postmerger echoes is found. Consequently, we derive 90% upper limits on the network SNR and the average initial strain amplitude of the long-lived QNMs. These results provide model-agnostic constraints on late-time echoes from LVK data, complementing existing searches for other echo signatures.

Figures

Figures reproduced from arXiv: 2512.24730 by the authors.

**Figure 2.** The injected echo waveform in the frequency domain (top panel) and time domain (bottom panel). The top [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** The log Bayes factor distributions (top), the overall posterior distributions of the inferred network SNR [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Corner plots for the overall posterior distributions for the spacing ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Comparison of background log Bayes factor distributions with and without notch filtering for detector noise [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Background search results for the three events. Left: the background log Bayes factor distributions. Right: [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Postmerger search results for the three events. Left: the posterior distributions of the inferred network SNR. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Normalized amplitude and relative phase for three representative examples of prominent instrumental lines [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Illustration of non-Gaussian artifacts identified in detector noise. (a) Example of a narrow spectral line in [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: Background log Bayes factor distributions (left panel) and the overall posterior distributions of the inferred [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: The inferred network SNR posteriors for Gaussian noise using two likelihoods in the case of [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: Corner plots of the overall posterior for the amplitude, width, spacing, and mode number derived from [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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