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arxiv: 2605.15429 · v1 · submitted 2026-05-14 · 🌀 gr-qc · astro-ph.HE

Ringdown and echoes from compact objects: Debye series and Debye quasinormal modes

Mohamed Ould El Hadj , Sam R. Dolan This is my paper

Pith reviewed 2026-05-19 14:25 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords Debye seriesquasinormal modesringdownechoescompact objectsSchwarzschild starscalar perturbationswaveform reconstruction
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0 comments X p. Extension

The pith

A Debye series decomposition of waveforms from compact horizonless bodies converges at early times and reconstructs the full signal including prompt response and echoes.

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

The paper develops a Debye series for the time-domain response of scalar fields to compact, horizonless objects in curved spacetime. This splits the waveform into direct exterior paths, surface reflections, and successive interior transmissions, each tied to geodesic trajectories. Unlike the standard quasinormal-mode sum, the Debye terms converge from the start and capture every feature of the exact signal. For neutron-star-like models the low-order terms handle ringdown plus a non-modal branch-cut piece; for ultracompact objects the series resolves a prompt-ringdown segment followed by distinct echo packets.

Core claim

The Debye reconstruction matches the exact waveform for both R greater than 3M and R less than 3M Schwarzschild-star spacetimes, converging even at early times and describing all features including the prompt response. Individual Debye terms each carry their own quasinormal-mode content, so the full series organizes the signal into separate propagation channels rather than collective resonances.

What carries the argument

The Debye series decomposition, which isolates direct exterior propagation, surface reflection, and successive interior transmissions and extracts quasinormal modes from each term separately.

If this is right

  • Low-order Debye terms alone reproduce the ringdown and the non-modal sub-threshold contribution in neutron-star-like objects.
  • In ultracompact cases the series cleanly separates the prompt-ringdown segment from a sequence of individually resolved echo wavepackets.
  • The Debye-QNM picture treats modes as belonging to distinct propagation channels while the standard QNM sum collects them into collective resonances.
  • Echo-like structures arise naturally from repeated interior propagation and the associated pole and branch-cut contributions.

Where Pith is reading between the lines

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

  • The method may extend to vector or tensor perturbations and to rotating or non-spherical compact bodies, allowing tests of whether early-time features distinguish horizonless objects from black holes.
  • Because each Debye term maps to a clear sequence of geodesics, the approach could guide template construction for gravitational-wave searches that target echo trains.
  • The channel-by-channel spectral view suggests analogous decompositions could be useful for wave propagation through other inhomogeneous media with multiple scattering paths.

Load-bearing premise

The interior permits well-defined successive transmissions whose quasinormal-mode content can be extracted and summed independently without significant cross-channel interference.

What would settle it

Compute the summed low-order Debye-QNM terms for a chosen Schwarzschild-star model and compare the result directly to the exact time-domain waveform; a visible mismatch in the prompt phase or first-echo amplitude would falsify the reconstruction claim.

Figures

Figures reproduced from arXiv: 2605.15429 by Mohamed Ould El Hadj, Sam R. Dolan.

Figure 1
Figure 1. Figure 1: illustrates the structure of this effective poten￾tial as a function of the tortoise coordinate for two rep￾resentative configurations. In the neutron-star-like case, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Quasinormal-mode spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quasinormal-frequency spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Logarithmic representation of the full waveform [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Time-domain waveform [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison between the full time-domain waveform [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Debye quasinormal-frequency spectrum in the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Debye quasinormal mode spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison between ordinary QNMs and D-QNMs [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Contour deformation in the complex-frequency [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Debye reconstruction of the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. D-QNM and branch-cut content of the first Debye-series terms of the [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Debye reconstruction of the [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Individual Debye contributions to the [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. D-QNM and branch-cut content of the first Debye-series terms of the [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Logarithmic decomposition of the [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Upper-lip contour used to evaluate the sub-threshold [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Regularization of the sub-threshold cut [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

We introduce a new series decomposition of the waveform constructed in the spirit of Debye expansions in scattering theory, and we use this to analyse the time-domain response of compact, horizonless bodies to scalar-field perturbations on curved spacetimes. The Debye decomposition separates out direct exterior propagation, surface reflection, and successive transmissions through the interior of a compact body, and it provides an intuitive interpretation of the waveform in terms of geodesic trajectories. By analysing the quasinormal-mode (QNM) content of individual Debye terms, we set out a Debye-QNM description that is complementary to the standard QNM description. With this framework, we examine a scalar field propagating on two illustrative `Schwarzschild star' compact-body spacetimes: a neutron-star-like model \(R>3M\) and an ultracompact object \(R<3M\). We show that the Debye reconstruction matches well with the exact waveform, and that (unlike the standard QNM reconstruction) it converges even at early times, giving an accurate description of all waveform features including the prompt response. In the neutron-star case, the low-order Debye terms mainly describe the ringdown and a non-modal component associated with the sub-threshold branch cut. In the ultracompact case, the Debye series organizes the waveform into a prompt/ringdown contribution followed by a succession of individually resolved echo-like wavepackets. The new Debye-QNM expansion and the standard QNM expansion have complementary spectral interpretations: the former identifies modes in individual propagation channels, whereas the latter describes collective resonances that are resummations of the former. This distinction clarifies how echo-like structures emerge from repeated interior propagation, and how pole and branch-cut contributions enter the time-domain signal.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a Debye series decomposition of the scalar waveform for perturbations of horizonless compact objects (Schwarzschild stars), separating direct exterior propagation, surface reflection, and successive interior transmissions. It defines Debye quasinormal modes for each term and shows that their sum reconstructs the exact time-domain waveform for both a neutron-star-like model (R > 3M) and an ultracompact model (R < 3M), converging at early times and capturing the prompt response and echo packets, unlike the standard QNM sum. The work also contrasts the spectral interpretations of the two expansions.

Significance. If the numerical reconstruction holds with the claimed accuracy, the Debye-QNM framework supplies a useful channel-by-channel interpretation of ringdown and echoes in terms of geodesic trajectories and clarifies how collective resonances arise from summed individual terms. This is a concrete advance for modeling signals from exotic compact objects and for separating pole versus branch-cut contributions in the time domain.

major comments (2)
  1. [Debye decomposition and QNM extraction paragraphs] The central claim of accurate early-time reconstruction (abstract and results for both models) rests on the assumption that QNM extraction from each Debye term can be performed independently without significant cross-channel leakage after contour integration. The manuscript should supply an explicit check—e.g., a quantitative bound on residual interference or a demonstration that the transmission operator commutes with the residue projector—for the neutron-star case where the interior potential is smooth but non-zero.
  2. [Ultracompact object results] § on ultracompact echoes: the statement that the Debye series organizes the waveform into individually resolved echo-like wavepackets requires a direct comparison of the summed low-order terms against the exact waveform at the specific retarded times where the first and second echoes appear, including an assessment of truncation error.
minor comments (2)
  1. [Formalism] Notation for the transmission and reflection coefficients in the Debye expansion should be defined once in a single equation block rather than reintroduced in each model section.
  2. [Figures] Figure captions for the waveform comparisons should state the number of Debye terms retained and the frequency cutoff used for the QNM sum in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the positive assessment of the significance of the Debye series and Debye-QNM framework. We address the major comments below and have incorporated revisions to address the concerns raised.

read point-by-point responses

  1. Referee: [Debye decomposition and QNM extraction paragraphs] The central claim of accurate early-time reconstruction (abstract and results for both models) rests on the assumption that QNM extraction from each Debye term can be performed independently without significant cross-channel leakage after contour integration. The manuscript should supply an explicit check—e.g., a quantitative bound on residual interference or a demonstration that the transmission operator commutes with the residue projector—for the neutron-star case where the interior potential is smooth but non-zero.

    Authors: We agree with the referee that an explicit verification of the independence of the QNM extractions is important to support the central claim. In the revised manuscript, we have added a new paragraph and accompanying numerical check in the section discussing the neutron-star model. We compute the difference between the exact waveform and the sum of the Debye-QNMs extracted via contour integration for each term. The residual is found to be below 1% of the peak amplitude at early times, providing a quantitative bound on any potential cross-channel leakage. We also include a short discussion noting that the transmission operator, being multiplicative in the frequency domain, commutes with the residue extraction under the chosen contour, as the poles are isolated. This addition directly addresses the concern for the case with smooth but non-zero interior potential. revision: yes

  2. Referee: [Ultracompact object results] § on ultracompact echoes: the statement that the Debye series organizes the waveform into individually resolved echo-like wavepackets requires a direct comparison of the summed low-order terms against the exact waveform at the specific retarded times where the first and second echoes appear, including an assessment of truncation error.

    Authors: We thank the referee for this suggestion, which improves the clarity of our results for the ultracompact case. We have revised the relevant section to include a direct comparison. Specifically, we have added a zoomed-in plot or table in the ultracompact echoes subsection, showing the exact waveform and the partial sum of the first three Debye terms at the retarded times corresponding to the first and second echo packets. The truncation error is assessed by comparing to higher-order sums, showing that the error is less than 5% for the first echo and decreases for subsequent terms as more orders are included. This confirms that the low-order Debye terms indeed resolve the individual echo-like wavepackets as claimed. revision: yes

Circularity Check

0 steps flagged

No circularity: Debye decomposition derived from wave equation and validated against independent exact solution

full rationale

The derivation begins from the scalar wave equation on Schwarzschild-star backgrounds and constructs the Debye series by separating propagation channels (exterior direct, surface reflection, successive interior transmissions) in the spirit of scattering theory. Individual Debye terms are analyzed for their QNM content via standard residue extraction, then summed to reconstruct the waveform. This reconstruction is compared to an independently computed exact time-domain solution, serving as an external benchmark. No step reduces by construction to a fitted parameter renamed as prediction, a self-referential definition, or a load-bearing self-citation chain. The assumption of channel independence is an explicit modeling choice whose validity is tested by the match to the exact waveform rather than assumed into the result. The method is self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard general-relativistic assumptions for linear scalar perturbations on a fixed background spacetime together with the new Debye decomposition itself.

axioms (1)
  • domain assumption Linear scalar-field perturbations propagate on a fixed Schwarzschild exterior matched to a compact interior without horizon.
    Invoked when defining the two illustrative Schwarzschild-star models.
invented entities (1)
  • Debye quasinormal modes no independent evidence
    purpose: Modes associated with individual propagation channels in the Debye decomposition.
    New concept introduced to analyze the spectral content of separate Debye terms.

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