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arxiv: 2501.16433 · v3 · pith:B3IT73GYnew · submitted 2025-01-27 · 🌀 gr-qc · astro-ph.HE· hep-ph

Greybody factors, reflectionless scattering modes, and echoes of ultracompact horizonless objects

Romeo Felice Rosato , Shauvik Biswas , Sumanta Chakraborty , Paolo Pani This is my paper

Pith reviewed 2026-05-23 04:40 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-ph
keywords greybody factorsultracompact objectsechoesreflectionless scattering modeshorizonless objectsscatteringringdownwormholes
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The pith

High-frequency quasi-reflectionless scattering modes produce the echoes observed from ultracompact horizonless objects.

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

The paper examines the greybody factors of ultracompact horizonless objects to trace their connection to echoes in the time-domain gravitational-wave response. These greybody factors contain both low-frequency resonances and high-frequency quasi-reflectionless scattering modes. The central finding is that the high-frequency modes, which turn purely reflectionless under symmetric cavity potentials such as those possible for wormholes, are the direct source of the echoes. This holds both for horizonless objects and for black holes surrounded by matter localized at large distances. The result reframes how echoes arise by shifting attention from low-frequency resonances to the high-frequency scattering behavior.

Core claim

The greybody factor of ultracompact objects features both low-frequency resonances and high-frequency, quasi-reflectionless scattering modes, which become purely reflectionless in the presence of symmetric cavity potentials, as it might be the case for a wormhole. It is these high-frequency (quasi-)reflectionless scattering modes, rather than low-frequency resonances, to be directly responsible for the echoes in the time-domain response of ultracompact objects or of black holes surrounded by matter fields localized at large distances.

What carries the argument

Greybody factors of ultracompact horizonless objects, which encode both low-frequency resonances and high-frequency quasi-reflectionless scattering modes that drive time-domain echoes.

If this is right

  • Echoes arise from high-frequency scattering modes rather than low-frequency resonances in both horizonless objects and black holes with distant matter shells.
  • Symmetric cavity potentials, such as those of wormholes, turn the high-frequency modes into purely reflectionless scattering modes.
  • The same greybody-to-echo link established for black holes carries over to horizonless objects.
  • Time-domain echoes can be predicted from the high-frequency structure of the greybody factor alone.

Where Pith is reading between the lines

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

  • Observational searches for echoes in gravitational-wave data could target the frequency band associated with these reflectionless modes rather than the quasinormal-mode spectrum.
  • Models of ultracompact objects with asymmetric potentials would still exhibit quasi-reflectionless modes but with reduced echo amplitude compared to symmetric cases.
  • The mechanism suggests echoes could appear in other wave equations with localized barriers, such as those describing scalar or electromagnetic perturbations around compact objects.

Load-bearing premise

The greybody factor analysis and its connection to time-domain echoes, previously established for black holes, extends directly to ultracompact horizonless objects without additional modifications to the scattering potential or wave equation.

What would settle it

A numerical computation of the time-domain waveform for a concrete ultracompact object model that produces clear echoes while its greybody factor lacks high-frequency quasi-reflectionless modes, or conversely shows such modes without producing echoes.

Figures

Figures reproduced from arXiv: 2501.16433 by Paolo Pani, Romeo Felice Rosato, Shauvik Biswas, Sumanta Chakraborty.

Figure 1
Figure 1. Figure 1: Scattering scheme of plane waves e ±iωr∗ for a Schwarzschild-like wormhole (top panel) and for a Schwarzschild-like ECO modeled by a reflective potential bar￾rier (bottom panel). See text for further details. ECO Model Surface reflection amplitude Wormhole RECO(ω) = R ′ BH(ω)e −2iωr0 ∗ Constant RECO RECO(ω) = cost Boltzmann RECO RECO(ω) = e −|ω|/TH Table I. Reflection amplitude at the ECO surface for diffe… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between reflectivity (left panel) and GFs (right panel) of a Schwarzschild BH (black solid lines) and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the real part of the QNM frequencies and the resonances/oscillations displayed by the GF [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic representation of a RSM for a Schwarzschild-like wormhole. For special discrete (real) fre￾quencies, a plane wave e −iωr∗ propagates from one universe to the other without being altered by the double-potential barrier. where R1, R2 (T1, T2) are the reflection (transmission) amplitudes for waves originating from the right of the first and second potentials, respectively, whereas R′ 2 is the reflec… view at source ↗
Figure 5
Figure 5. Figure 5: Reflectivity for a double Schwarzschild potential in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Real and imaginary parts of the reflection amplitude [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reflectivity (left panel) and GF (right panel) for an ECO with Boltzmann reflection amplitude at the surface [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reflectivity in some ECO models with constant reflectivity at the effective radius. In the left panel we vary [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: All of the wormhole quantities showed in the plots refer to a wormhole with gluing radius [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fourier transform of the complex reflectivity [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The spectral amplitudes for l = 2 modes emitted by a point particle with mass µ in radial infall with specific energy Ep (see legends in each panel). For comparison, we also show two models ∝ √ Rlm/ω and ∝ √ Rlm. For all cases with Ep > 1, the model ∝ √ Rlm/ω accurately describe the spectral amplitude at all frequencies. quasi-RSMs. This correspondence not only strength￾ens the theoretical understanding o… view at source ↗
Figure 12
Figure 12. Figure 12: GF and reflectivity computed via the transfer-matrix formalism for a double Schwarzschild potential with different [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison between the GF of a wormhole com [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The reflectivity R22 and the GF Γ22 are being presented against the dimensionless frequency Mω for the dominant l = 2 = m modes associated with the scattering of gravitational perturbation from an ECO. For a comparison with the Schwarzschild wormhole case, presented in [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: GF for the s = 0 perturbation for various compact objects. In the wormhole case, we have taken, r0 = (2 + 10−6 )M, while for the Boltzmann ECO case, we have taken r0 = (2 + 10−4 )M. As evident, in all of these cases, the analytical result matches with the numerical result at low frequencies, while at high frequencies there are differences. arXiv:2310.07368 [gr-qc]. [34] E. Barausse, V. Cardoso, and P. Pan… view at source ↗
read the original abstract

Motivated by a recently discovered connection between the greybody factors of black holes and the ringdown signal, we investigate the greybody factors of ultracompact horizonless objects, also elucidating their connection to echoes. The greybody factor of ultracompact objects features both low-frequency resonances and high-frequency, quasi-reflectionless scattering modes, which become purely reflectionless in the presence of symmetric cavity potentials, as it might be the case for a wormhole. We show that it is these high-frequency (quasi-)reflectionless scattering modes, rather than low-frequency resonances, to be directly responsible for the echoes in the time-domain response of ultracompact objects or of black holes surrounded by matter fields localized at large distances.

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

1 major / 0 minor

Summary. The paper investigates the greybody factors of ultracompact horizonless objects, motivated by a prior connection between greybody factors and black-hole ringdown signals. It identifies both low-frequency resonances and high-frequency quasi-reflectionless scattering modes in these factors; the latter become purely reflectionless for symmetric cavity potentials (e.g., wormholes). The central claim is that the high-frequency (quasi-)reflectionless modes, rather than the low-frequency resonances, are directly responsible for the echoes observed in the time-domain response of ultracompact objects or of black holes with matter localized at large distances.

Significance. If the result holds, the work supplies a frequency-domain mechanistic account of echoes that could help interpret gravitational-wave signals from exotic compact objects. It employs standard wave-scattering methods in GR without free parameters or invented entities, extending an existing black-hole result to horizonless cases and thereby offering a concrete, falsifiable link between greybody factors and time-domain echoes.

major comments (1)
  1. [Abstract] Abstract: the central claim that high-frequency quasi-reflectionless modes are directly responsible for echoes in generic ultracompact objects requires explicit verification that the time-domain mapping survives asymmetry in the scattering potential. The abstract states that purely reflectionless modes occur only for symmetric cavity potentials (as in wormholes), while generic objects yield only quasi-reflectionless modes; without a demonstration that the echo mechanism persists under asymmetry, the extension from the symmetric case remains unverified and load-bearing for the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that high-frequency quasi-reflectionless modes are directly responsible for echoes in generic ultracompact objects requires explicit verification that the time-domain mapping survives asymmetry in the scattering potential. The abstract states that purely reflectionless modes occur only for symmetric cavity potentials (as in wormholes), while generic objects yield only quasi-reflectionless modes; without a demonstration that the echo mechanism persists under asymmetry, the extension from the symmetric case remains unverified and load-bearing for the claim.

    Authors: We thank the referee for highlighting this point. The manuscript explicitly treats the generic (asymmetric) case through the quasi-reflectionless modes. Calculations of the greybody factors for asymmetric cavity potentials are compared directly to the corresponding time-domain signals, showing that the high-frequency quasi-reflectionless features produce the echo train. The small residual reflection coefficient at high frequencies damps the echoes but does not eliminate the repeated reflections that characterize them. This establishes that the time-domain mapping survives asymmetry, supporting the central claim for generic ultracompact objects as stated in the abstract. We are prepared to add a clarifying sentence or additional plot if the referee considers it necessary for emphasis. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard scattering analysis.

full rationale

The paper motivates its investigation from a prior connection between greybody factors and ringdown (likely cited externally) but performs independent greybody factor computations and time-domain mapping for ultracompact objects using the wave equation and scattering potential. No equations reduce by construction to fitted parameters, self-definitions, or unverified self-citations; the extension to horizonless cases is presented as an application of existing methods rather than a tautology. The result is self-contained against external benchmarks of black-hole perturbation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard general-relativity wave equations and scattering theory without introducing new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (2)
  • standard math Wave propagation and scattering in stationary, asymptotically flat spacetimes obey the standard Klein-Gordon or Regge-Wheeler-type equations derived from the Einstein equations.
    Invoked implicitly when defining greybody factors and reflectionless modes for both black holes and horizonless objects.
  • domain assumption The connection between frequency-domain greybody factors and time-domain ringdown/echo signals established for black holes carries over to ultracompact horizonless objects.
    This is the motivating premise stated in the first sentence of the abstract.

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Forward citations

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Reference graph

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