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The paper derives the first observational bounds on theory-agnostic deviations from the Teukolsky equation from the GW250114 ringdown, finding all deviation parameters consistent with general relativity.

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load-bearing objection First real-data beyond-Teukolsky bounds, but the quoted 60–100 km scales are prior-dominated and the λ=1 bounds double-count ringdown information. the 4 major comments →

arxiv 2607.26561 v1 pith:PGJPF24C submitted 2026-07-29 gr-qc

Constraining deviations from the Teukolsky equation with GW250114

Sebastian H. V\"olkel , Nicola Franchini This is my paper
classification gr-qc PACS 04.30.-w04.70.-s
keywords black hole ringdownTeukolsky equationquasinormal modestests of general relativityGW250114beyond-Teukolsky formalismgravitational wave astronomyblack hole spectroscopy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper sets out to answer a sharp question: if the Teukolsky equation that governs black-hole ringdown in general relativity were slightly wrong, how wrong could its effective potential be, given the exceptionally loud gravitational-wave event GW250114? Using the beyond-Teukolsky formalism, which parametrizes small complex corrections ζ_k to the Teukolsky potential and computes their linear effect on the quasinormal-mode spectrum, the authors combine an inspiral-merger-ringdown estimate of the remnant mass and spin with the ringdown-only measurement of the fundamental (2,2,0) mode. They introduce two theory-agnostic priors that rescale the IMR posterior covariance or box the maximum-likelihood values, controlled by a hand-set factor λ. Their central result is that every ζ_k is consistent with zero, with one-standard-deviation uncertainties corresponding to length scales of roughly 60–100 km — the first such constraints on the perturbation equation itself. These bounds are an order-of-magnitude statement about how much the Kerr perturbation equations can deviate at the scale of tens of kilometers and still match the loudest ringdown observed to date.

Core claim

The central claim is that the ringdown of GW250114, analyzed through the beyond-Teukolsky framework, provides the first observational bounds on deviations from the Teukolsky equation. The authors demonstrate that all complex deviation parameters ζ_k, which encode small modifications δV(r) to the Teukolsky potential in powers of (r/r_+)^k, are consistent with zero within both optimistic and pessimistic priors on the remnant mass and spin. The one-standard-deviation uncertainties on Re(ζ_k) and Im(ζ_k), converted to length scales via sqrt(σ_ζ)·M, fall in the range of roughly 60–100 km, in agreement with the independent ParSpec bound of about 80 km. The method itself — a simplified Gaussian lik

What carries the argument

The machinery is the beyond-Teukolsky framework combined with a simplified Gaussian likelihood. The framework modifies the Teukolsky equation by adding a small potential δV(r) = (1/Δ) Σ_{k=-K}^{4} α_k (r/r_+)^k, with dimensionless complex parameters ζ_k = α_k/M^2. At linear order, the quasinormal-mode frequency shifts as ω = ω_GR + (1/M) Σ ζ_k d^k_ω, where the coefficients d^k_ω are precomputed for each (ℓ,m,n). The analysis then approximates the LVK posteriors for (M, χ) and for the fundamental mode frequency and damping time as multivariate Gaussians, and samples the likelihood of the measured ω_{220} against the model prediction, varying one ζ_k at a time. The remnant mass and spin are an

Load-bearing premise

The entire constraint rests on the assumption that the remnant black hole's mass and spin, as predicted by the GR-based inspiral-merger-ringdown analysis, are close enough to the true values that the IMR-informed priors (with the hand-set width λ) cover the actual (M, χ); if a beyond-GR theory shifted the remnant parameters by more than that width, the reported ζ_k bounds would be biased toward zero and would not reflect the true deviations.

What would settle it

Take the public GW250114 posteriors, inject a simulated ringdown signal with a known nonzero ζ_k and a remnant mass and spin shifted by ~1% away from the GR IMR maximum-likelihood values, then run the paper's pipeline; if the recovered ζ_k posterior fails to exclude zero or recover the injected value, the IMR-anchored priors are demonstrably insufficient to constrain beyond-Teukolsky deviations.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, deviations in the effective potential of Kerr black-hole perturbation theory are bounded at the tens-of-kilometers scale for this event — the sharpest such bound to date.
  • The simplified likelihood pipeline can be rerun quickly on future high-SNR events without a full Bayesian analysis, making it a practical screening test for beyond-GR theories.
  • The consistency with the independent ParSpec bound (~80 km) suggests that different agnostic parametrizations are converging on the same length-scale ceiling for deviations.
  • The reported ζ_k constraints can be translated into bounds on specific theories, such as higher-derivative gravity, once their predicted potential deviations are mapped to ζ_k.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The hand-set λ parameter is the real dial of the analysis: λ=1 double-counts ringdown information already present in the IMR posterior, while large λ weakens the priors to the point where the ζ_k bounds may be dominated by prior ignorance. A fully Bayesian joint fit of M, χ, and ζ_k would determine where between these extremes the true constraint lies.
  • If a beyond-GR theory predicts a final mass or spin that differs from the GR IMR value by more than the λ-scaled width, the ζ_k posteriors will be systematically shifted. A direct test would be to inject a simulated signal with a known nonzero ζ_k and a shifted remnant and check whether the pipeline recovers the injection.
  • The one-at-a-time variation of ζ_k means the bounds are conditional; simultaneous marginalization would likely be far less informative, but mapping the joint posterior to local properties of the effective potential near its maximum, as done in the non-rotating case with WKB, could restore interpretability.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a lightweight method to constrain theory-agnostic deviations from the Teukolsky equation using the fundamental (l=m=2, n=0) ringdown mode of the high-SNR event GW250114. The authors combine a Gaussian approximation of the LVK full-IMR posterior for the remnant mass and spin (Prior 1, Eq. 8; Prior 2, Eq. 12) with a Gaussian approximation of the LVK agnostic damped-sinusoid posterior for ω_220, and sample the likelihood (Eq. 10) with MCMC. They vary one complex beyond-Teukolsky parameter ζ_k at a time and report that all ζ_k are consistent with GR (ζ_k=0). They translate the marginalized uncertainties into characteristic length scales √σ_ζ M ≈ 60–100 km and claim the first bounds on the beyond-Teukolsky framework.

Significance. If the bounds are robust, this would be the first observational constraint on the beyond-Teukolsky formalism of Ref. [82], complementing the independent ParSpec analysis of the same event. The methodological idea of using a simplified Gaussian likelihood for high-SNR ringdowns is pragmatic and could be applied to future loud events. The paper is unusually transparent about the limited constraining power of a single mode and about the prior-dependence of the results, presenting both 'optimistic' (λ=1) and 'pessimistic' (λ=25 or λ=5) choices. It also correctly notes that the linearized framework has limited accuracy for large ζ_k. These strengths make the work potentially useful to the ringdown community.

major comments (4)
  1. [Sec. II.C, Eq. (8) and Fig. 2] The Gaussian prior on (M,χ) is taken from the NRSur7dq4 full-IMR posterior, which already contains the ringdown information used later in the likelihood Eq. (10). For λ=1, the prior and likelihood are not independent, so the ζ_k posteriors are largely a re-projection of the GR-based IMR mapping rather than new information. The paper acknowledges this qualitatively ('underestimate the statistical errors') but does not quantify the impact on the quoted 60–100 km length scale. An injection study or an inspiral-only prior is needed to establish what these bounds actually measure.
  2. [Sec. III, length-scale paragraph] The '60–100 km' characteristic scales are quoted only from the λ=1 'optimistic' bounds, which are the most affected by the double-counting problem above. The λ=25 'pessimistic' bounds are not translated to length scales, so the headline quantitative claim does not reflect the full range of prior choices. The paper should either report the length-scale range for both prior choices or explicitly state that the 60–100 km figure is conditional on the optimistic prior.
  3. [Sec. II.C, Eq. (12) and λ choices] The scaling parameter λ is set ad hoc (λ=1/25 for the Gaussian prior, λ=1/5 for the box prior) with no calibration to known beyond-GR theories or to plausible shifts in final mass/spin. The relationship between a covariance multiplier and a percentage box is not justified. Consequently, even the 'pessimistic' bounds remain uncalibrated; they are not a systematic treatment of theoretical uncertainty. A prescription for λ based on theory-specific estimates or on injection tests is required to support the claim of providing 'bounds'.
  4. [Sec. II.A, Eqs. (6)–(7) and Figs. 2–3] The beyond-Teukolsky framework assumes |ζ_k| ≪ 1 for the linear expansion Eq. (7) to be valid, but the marginalized posteriors in Figs. 2 and 3 extend to |ζ| ≳ 1. The text states that linear corrections are about 1% accurate for the considered ranges, but this is not demonstrated for the actual posterior support. If the posterior overlaps the non-perturbative regime, the reported bounds cannot be interpreted as bounds on the linear deviation parameters. The paper should identify the region of validity and restrict its reporting to that region.
minor comments (5)
  1. [Sec. II.B] The phrase 'The high signal-to-noise (SNR) ratio' is redundant; 'SNR' already includes 'ratio'.
  2. [Sec. II.A, Eq. (6)] The truncation value K is not stated in the text; the figures show k = -2,...,4, so K=2. Please state this explicitly and justify the truncation.
  3. [Sec. II.C] The symbol λ is used both as a covariance multiplier (Eq. 8) and as a percentage width (Eq. 12). Clarify this in the notation, or use a different symbol for the box prior.
  4. [General] The phrase 'the here presented analysis' is awkward; suggest 'the present analysis'.
  5. [Sec. IV] The conclusion mentions future applications but does not quantify the expected improvement for next-generation detectors; a short estimate would strengthen the outlook.

Circularity Check

0 steps flagged

No significant circularity: the beyond-Teukolsky input is prior work with independent content, and the λ=1 double-counting of ringdown information is explicitly disclosed as a limitation rather than a hidden equivalence.

full rationale

The claimed derivation chain is: (i) adopt the beyond-Teukolsky relation Eq. (7) from Ref. [82] with coefficients from Ref. [88]; (ii) approximate the LVK IMR (M,χ) posterior and the LVK damped-sinusoid ω220 posterior as Gaussians (Eqs. 8 and 10); (iii) sample the likelihood with informed priors and report ζ_k bounds. Step (i) is a parameter-free perturbative mapping between potential deviations δV and QNM frequency shifts; its stated assumptions (small |ζ_k|, linear order) do not include the target data bounds, and the coefficients are provided in a public repository, so the authors' self-citation here is not load-bearing circularity. Step (iii) is an empirical bound rather than a first-principles prediction. The only near-circular element is that for λ=1 the (M,χ) prior comes from the NRSur7dq4 full-IMR analysis, which already includes ringdown information that is reused in the ringdown likelihood Eq. (10); the paper explicitly flags this: 'the bounds for λ=1 can be understood as “optimistic” bounds, since they completely ignore the theoretical uncertainties in final mass and spin, and underestimate the statistical errors since they use the full IMR information.' This is a disclosed statistical double-counting/optimism caveat, not an equivalence by construction, and the λ=25 'pessimistic' prior plus the external ParSpec comparison (~80 km) provide a cross-check less dependent on the optimistic prior. I therefore find no circular step that reduces the central claim to its inputs; the score of 2 reflects the density of self-citations and the optimistic-prior caveat, not a circular derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The analysis contributes no new physics entities and its free-parameter budget is modest: one hand-chosen hyperparameter λ, an ad hoc series truncation K=2 and prior range. The constraining power, however, is not self-generated: it is leased from the GR-based IMR remnant estimates (axiom 3) and from the assumed linear beyond-Teukolsky mapping (axiom 1), both external to this paper's data analysis.

free parameters (3)
  • λ (theoretical-error scaling) = λ = 1 and 25 (Gaussian prior); λ = 1 and 5 (uniform-box prior)
    Hand-chosen hyperparameter scaling the assumed uncertainty of the GR-based IMR remnant mass/spin; sets how informative the priors are (Sec. II.C). It is not fitted to data and there is no principled scale, which is why the quoted bounds range between 'optimistic' and 'pessimistic'.
  • ζ_k (k = -2,...,4) = inferred posteriors ≈ 0 (within quoted widths)
    The target deviation parameters, varied one at a time with uniform priors on [-8,8]; the prior range and the K=2 truncation of the series in Eq. (6) are ad hoc but checked against the narrow posteriors.
  • Prior truncation K=2 in Eq. (6) = k from -2 to 4
    The framework allows any K>0; restricting to k=-2..4 limits the probed deformation space and is a modeling choice (Sec. II.A).
axioms (5)
  • domain assumption Beyond-Teukolsky linear response: ω = ω_GR + (1/M) Σ ζ_k d^ω_k (Eq. 7), with d^ω_k from Ref. [82]/repo [88]
    The central model transfer function. Assumes first-order perturbation theory in the potential deformation is accurate over the sampled ζ range; accuracy level (~1%) is imported from Ref. [82], not re-derived here.
  • ad hoc to paper |ζ_k| << 1 holds over the posterior support
    Sec. II.A states |ζ_k| << 1 as the framework's validity condition, but posterior tails in Figs. 2-3 reach |ζ| ~ O(1), where the condition is violated yet the linear mapping is still applied unconditionally.
  • domain assumption GR-based IMR (NRSur7dq4) final mass/spin are accurate up to λ-scaled 'theoretical errors'
    Load-bearing for the entire analysis: the ζ_k constraints exist only because (M, χ) are anchored to the GR IMR values (Prior 1, Eq. 8; Prior 2, Eq. 12, Sec. II.C). A beyond-GR shift larger than the λ-scaled width would bias the bounds.
  • domain assumption LVK agnostic ringdown posteriors (two damped sinusoids, t=10M) are unbiased and well-approximated by multivariate Gaussians
    Sec. II.B uses the LVK [62] f220/τ220 chains as the measurement and approximates their (skewed) distribution by a Gaussian; Fig. 1 shows visible deviations for the fundamental mode that are accepted as 'approximate'.
  • standard math Teukolsky equation is the correct GR description of Kerr ringdowns
    Standard background of black-hole perturbation theory (Refs. [25,26]); not in dispute.

pith-pipeline@v1.3.0-daily-deepseek · 12735 in / 22903 out tokens · 209547 ms · 2026-08-01T13:19:55.198010+00:00 · methodology

0 comments
read the original abstract

The recent gravitational-wave detection GW250114 by the LIGO-Virgo-KAGRA (LVK) Collaboration provides unprecedented precision for testing general relativity (GR) through black hole ringdowns. In this study, we provide the first bounds on theory-agnostic deviations from the Teukolsky equation as described by the beyond-Teukolsky formalism. It directly connects deviations in the perturbation equations on the level of the effective potential in the Teukolsky equation with changes in the quasinormal mode (QNM) spectrum. We incorporate information on the final mass and spin from a full LVK inspiral-merger-ringdown analysis as parametrized priors in our analysis, reflecting theoretical uncertainties. Using publicly available LVK posterior information on agnostic damped sinusoid parameters, we then demonstrate how much beyond-Teukolsky potentials can be constrained. The high signal-to-noise ratio (SNR) allows us to avoid the expensive full Bayesian analysis of all parameters and to work directly with a simplified likelihood for the fundamental QNM only. This strategy is promising for future events with even higher SNR and allows, in principle, for a quick and simple test of theories beyond GR without performing the full data analysis procedure. We report that current bounds on deviation parameters are in agreement with the Teukolsky equation.

Figures

Figures reproduced from arXiv: 2607.26561 by Nicola Franchini, Sebastian H. V\"olkel.

Figure 1
Figure 1. Figure 1: FIG. 1. Marginalized posterior distributions from the LVK [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Optimistic ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Bounds on [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

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