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REVIEW 3 major objections 4 minor 1 cited by

Multimode ringdown modelling with $\texttt{qnmfits}$ and $\texttt{KerrRingdown}$

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Two independently written codes for extracting black-hole ringdown mode amplitudes agree to machine precision.

desk verdict A useful and honest pair of ringdown-fitting codes, but the advertised machine-precision cross-validation is asserted without evidence; easy fix, worth refereeing. read the letter →

arxiv 2502.03155 v1 pith:Q6BEAJRT submitted 2025-02-05 gr-qc astro-ph.IM

Lorena Magaña Zertuche , Leda Gao , Eliot Finch , Gregory B. Cook This is my paper · ORCID
classification gr-qcastro-ph.IM PACS 04.30.-w04.70.-s
keywords gravitationalwavesringdownquasinormalmodesnumericalrelativityleast-squaresfittingmodemixingcodecross-validationKerrblackholes
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

The paper aims to establish that quasinormal-mode amplitudes of a black-hole ringdown can be extracted reliably by two independent, open-source fitting tools: qnmfits in Python and KerrRingdown in Mathematica. The authors model the ringdown as a sum of damped sinusoids with Kerr quasinormal-mode frequencies, account for spherical-spheroidal mode mixing, and fit the complex amplitudes by least squares. They report that for multiple test cases, including aligned-spin and precessing numerical-relativity waveforms, both codes give identical mismatches and identical complex mode amplitudes. A sympathetic reader would care because a cross-checked, easy-to-use toolset lets the ringdown community compare extractions of mode content from numerical simulations without worrying about implementation artifacts.

What carries the argument

The load-bearing object is the mode-mixed QNM expansion $h_{\ell m}(t) = \sum_{\ell' n \pm} A^\pm_{\ell' mn} e^{-i\omega^\pm_{\ell' mn}(t-t_0)} C_{\ell\ell' m}(a\omega^\pm_{\ell' mn})$, together with the mismatch $\mathcal{M}$ of Eq. (5) that the least-squares fit minimizes. The spherical-spheroidal mixing coefficients $C_{\ell\ell' m}$ are what let a single spheroidal QNM contribute to several spherical-harmonic data channels, and the fact that the two codes reach the same minimum through different linear-algebra routes is what makes their agreement nontrivial. Both codes label modes by the sign of the real frequency, distinguishing regular from mirror modes, and obtain their frequency and mixing data from tabulated sources.

What would settle it

Run both packages on a synthetic ringdown generated from the model itself with known amplitudes, including at least one mode where the two codes depend on different data sources; if the recovered complex amplitudes or mismatches differ by more than round-off (roughly a relative difference above $10^{-13}$), the machine-precision claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that multimode least-squares ringdown fitting can be performed reliably in two independent software implementations: given a numerical-relativity waveform decomposed into spherical harmonics and a user-chosen set of Kerr quasinormal modes, qnmfits and KerrRingdown return the same complex amplitudes $A^\pm_{\ell mn}$ and the same mismatch $\mathcal{M}$ up to machine precision, across a variety of fit configurations. The two codes were written independently and use different numerical routes, one calling a standard linear-algebra solver and the other solving the normal equations, so the agreement is presented as cross-validation of the implementations and of the underlying QNM-frequency and mixing-coefficient data. The demonstration covers mode-mixing fits, overtone fits, and a greedy-selected twenty-mode all-sky fit, on both an aligned-spin and a precessing waveform.

Load-bearing premise

The load-bearing premise is that the QNM frequency and mixing-coefficient tables on which the two codes rely agree in sign and scaling conventions, even though they come from different sources and the paper does not display that table-level comparison.

Editorial extensions

If this is right

  • Users can fit the same numerical-relativity ringdown with either package and use the agreement between them as a built-in sanity check.
  • The packages make QNM amplitude extraction fast enough to scan start times and mode choices, since the least-squares fits are essentially instantaneous.
  • The greedy-fit algorithm gives a practical way to rank which quasinormal modes matter for a given binary system, informing which overtones, mirror modes, or higher multipoles should be included in a model.
  • Fitting the remnant mass and spin is built into both codes, so the user can jointly obtain remnant parameters and mode amplitudes rather than using fixed values.
  • The demonstration on a precessing waveform mapped to the superrest frame indicates the codes work for CCE-type waveforms, not only simple aligned-spin data.

Reading between the lines

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

  • Because the two codes draw QNM frequencies and mixing coefficients from different datasets, their machine-precision agreement also implies that those datasets are mutually consistent in phase and normalization; the paper does not show that table-level comparison, so publishing it would strengthen reproducibility.
  • A natural next step is to run the same cross-check on synthetic waveforms with exactly known injected amplitudes, which would quantify the accuracy of the method itself rather than only the consistency of the two implementations.
  • The paper itself notes that least-squares fits carry no uncertainty estimate and can be biased for rapidly decaying or subdominant QNMs, so the machine-precision agreement establishes consistency between the codes, not automatic physical accuracy of every recovered amplitude.
  • The fast least-squares fits could serve as a cheap frequentist companion to Bayesian ringdown analyses, giving an immediate first estimate of mode amplitudes and of whether a mode merits a full posterior investigation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This note introduces two open-source packages, qnmfits (Python) and KerrRingdown (Mathematica), for performing multimode least-squares fits of quasinormal-mode amplitudes to numerical-relativity ringdown waveforms. The theoretical model (Eqs. 1–6) is the standard decomposition of spherical-harmonic waveform modes into spheroidal QNMs, including regular and mirror modes, with spherical–spheroidal mixing coefficients. The paper demonstrates the codes on two SXS waveforms, one aligned-spin and one precessing, showing the mismatch versus start time and normalized QNM amplitudes for three mode sets. The central claim is that the two independently written packages have been verified against each other and produce identical results up to machine precision for both the mismatch and complex QNM amplitudes (Section 3, final paragraph).

Significance. If the cross-validation claim can be substantiated, this is a genuinely useful software contribution: it provides two independent, DOI-issued, open-source implementations of a standard and widely used method, along with a helpful clarification of the regular/mirror versus prograde/retrograde convention. The demonstration on an aligned-spin and a precessing CCE waveform is a sensible first illustration, and the authors are explicit about the least-squares method's limitation of not providing fit uncertainties. However, the paper's main advertised result—the machine-precision agreement of the two codes—is currently only asserted, not demonstrated, and the convention consistency between the two QNM data sources is not established. The contribution therefore reads more as a software announcement than as a fully checkable cross-validation study.

major comments (3)
  1. [Section 3, final paragraph] The statement that qnmfits and KerrRingdown 'produce identical results, up to machine precision, for both the mismatch and complex QNM amplitudes for a variety of test cases' is the central claim advertised in the abstract and introduction, yet no supporting evidence appears in the manuscript. There is no comparison table, verification plot, list of the test cases, or test-suite output. Because the two codes use different linear-algebra strategies (numpy.linalg.lstsq versus a normal-equations implementation), numerical agreement is not automatic. Please add a reproducibility section or appendix reporting the measured differences (for example, maximum absolute and relative deviations in M and in A±_{ℓmn} across the test cases) together with the definition of the machine-precision threshold used.
  2. [Sections 2.1 and 2.2] qnmfits obtains QNM frequencies and spherical–spheroidal mixing coefficients from the qnm package (with Ref. [38] for the (2,2,8) mode), while KerrRingdown imports them from the dataset of Ref. [40]. Complex QNM amplitudes are convention-dependent: a different phase convention for the spheroidal harmonics, a different sign convention for mirror modes, or a different normalization of C_{ℓℓ′m}(aω) would shift fitted amplitudes by mode-dependent factors. The manuscript does not state the conventions of Ref. [40] nor demonstrate that they are mutually consistent with those of the qnm package. Please include a direct comparison of the input data for representative modes (frequencies and mixing coefficients), or state and verify the convention mapping between the two sources.
  3. [Section 3] The cross-validation claim refers to 'a variety of test cases', but the demonstration section presents only two NR waveforms and does not specify what the full test suite was. The details of waveform preprocessing (resampling, alignment, mapping to the superrest frame) are also not given, which makes the demonstration hard to reproduce independently. Please specify the test cases and preprocessing steps used for the verification, or point to a persistent test-suite artifact in the code repositories.
minor comments (4)
  1. [Eq. (5)] The numerator of the mismatch expression appears to have a typographical error: what is printed as '|p' should presumably be a squared absolute value (|Σℓm⟨aℓm|bℓm⟩|²). Please correct the formula.
  2. [Fig. 2 caption] The caption's phrase 'right panels' is ambiguous because the right side contains two stacked panels; please refer to 'top right' and 'bottom right' panels for clarity.
  3. [Section 3, mode lists] For m = 0 modes, the notation '(ℓ, 0, n, ±)' may be confusing because the regular and mirror modes coincide; a brief remark explaining how the ± label is interpreted for m = 0 would help the reader.
  4. [References [35,37,38,40]] For reproducibility, please include the specific version or DOI for the qnm package, the access date for Ref. [38], and the version of Ref. [40] used in the fits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cross-validation claim is an implementation check, not a derived prediction.

full rationale

The paper's central content is two independently written fitting codes, qnmfits and KerrRingdown, and its main claim is that the two packages produce identical results up to machine precision for mismatch and complex QNM amplitudes on test cases. This is an empirical cross-check of implementations, not a derivation of a physical result from premises. The QNM frequencies and spherical-spheroidal mixing coefficients are treated as inputs obtained from external or publicly available sources, and the complex amplitudes are least-squares fit outputs rather than predictions made from fitted parameters. Some cited inputs are authored by the present authors, such as the superrest-frame and greedy-algorithm work in Ref. [34] and the Kerr-mode dataset in Ref. [40], but these are used as data and tools, not as theorems that force the paper's conclusions. The paper does not present a fitted parameter as a prediction, nor does it define any quantity in terms of the result it claims to derive. The absence of explicit verification tables weakens the reproducibility of the cross-validation claim, but that is a completeness concern, not circularity. No circular step can be exhibited from the text, so the appropriate score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim of the paper is about the correctness and agreement of two fitting codes. The main assumptions are the standard QNM superposition model and the consistency of externally provided QNM data. The only fitted quantities are the amplitudes and the user-selected start time, which are the intended outputs of the codes.

free parameters (2)
  • QNM complex amplitudes A±_ℓmn = varies per fit
    The amplitudes are the unknown quantities the codes fit for, one per QNM in the model (Eq. 4). They are the intended outputs, not hidden assumptions.
  • Ringdown start time t0 = varies (e.g., peak strain or user choice)
    The start time is a user-selected parameter in the analysis; results in Fig. 2 are shown as a function of t0, and the choice affects the fitted amplitudes.
assumptions (4)
  • domain assumption The ringdown waveform is a linear superposition of damped sinusoid quasinormal modes with fixed frequencies (Eq. 1).
    This is the standard QNM expansion assumption in black hole perturbation theory, invoked in Section 2.2.
  • standard math Spherical harmonics form a complete basis, so spheroidal harmonics can be expanded as in Eq. 3.
    Used in Section 2.2 to relate NR spherical-harmonic modes to spheroidal QNMs.
  • standard math The least-squares mismatch minimization (Eq. 5) is equivalent to the linear solve implemented for the amplitudes.
    The codes minimize the sum of squared residuals; this is a standard linear least-squares problem.
  • domain assumption QNM frequencies and mixing coefficients from the qnm package and Ref. [40] are correct and mutually consistent.
    The two codes use different sources for these inputs (Section 2.1), and the claimed cross-code agreement depends on this consistency.

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Cite this review

Pith. "Pith review of Multimode ringdown modelling with $\texttt{qnmfits}$ and $\texttt{KerrRingdown}$." pith.science (2026). https://pith.science/paper/Q6BEAJRT

@misc{pith2026250203155,
  author       = {Pith},
  title        = {Pith review of: Multimode ringdown modelling with $\textttqnmfits$ and $\textttKerrRingdown$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6BEAJRT}},
  note         = {Machine review of arXiv:2502.03155}
}
abstract

In the last decade, the ringdown community has made large strides in understanding the aftermath of binary black hole mergers through the study of numerical simulations. In this note, we introduce two flavors of fitting algorithms, that have been verified against each other, for the extraction of quasinormal mode amplitudes from ringdown waveforms - $\texttt{qnmfits}$ in Python and $\texttt{KerrRingdown}$ in Mathematica.

Figures

Figures reproduced from arXiv: 2502.03155 by the authors.

Figure 1
Figure 1. Selected modes of the Kerr QNM spectrum. BH QNM frequencies are conventionally represented as complex numbers, with the real part giving the angular frequency of the mode and the imaginary part giving (minus) the inverse of the damping time: ωℓmn = 2πfℓmn − i/τℓmn. Here we plot fℓmn and 1/τℓmn, each scaled by the remnant BH mass M to make a dimensionless quantity. The spectrum of a Kerr BH also depends on the dimens… view at source ↗
Figure 2
Figure 2. Left panel: Mismatch as a function of the ringdown start time for both simulations considered in this work, and for each set. Right panels: QNM amplitudes (normalized by the fundamental mode) at a chosen ringdown start time for both simulations considered in this work. The top panel shows the positive-frequency (prograde) modes from set 1. The bottom panel shows the overtone amplitudes considered in set 2. In the le… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasinormal modes from numerical relativity with Bayesian inference

    gr-qc 2025-10 conditional novelty 6.0 of 10

    A catalog-trained Gaussian-process noise model converts numerical-relativity waveforms into Bayesian quasinormal-mode fits with analytic posteriors and mode significances, demonstrated on CCE simulations.

Reference graph

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Reviewed August 9, 2026 · model on record in the stance chip above.