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arxiv: 2606.02146 · v1 · pith:VH47VTOTnew · submitted 2026-06-01 · 🌀 gr-qc

Spectral suppression of black hole ringdown tails

Jose Antonio Le\'on Vega , Alejandro Svyatkovskyy Kholyavka , Sayak Datta , Xisco Jim\'enez Forteza This is my paper

Pith reviewed 2026-06-28 13:23 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole ringdownPrice's lawspectral suppressionRegge-Wheeler equationbranch cutbinary mergersgravitational waveslate-time tails
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The pith

Perturbation sources with a carrier frequency and finite spectral width suppress black hole ringdown tails by an exponential factor set by their product.

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

Black hole perturbation theory predicts late-time power-law tails following Price's law, but these are missing from most numerical relativity simulations of binary mergers. The paper demonstrates that this absence stems from the oscillatory character of realistic sources, which have both a central frequency and a characteristic width in their spectrum. The coefficient that excites the branch cut in the Green's function solution is suppressed by a factor involving the product of that frequency and width. For Gaussian sources the suppression is exponential in the square of this product. This same factor also governs whether the black hole responds broadly or selectively to the incoming wave.

Core claim

For a generic perturbation with carrier frequency ν and characteristic width σ, the branch-cut excitation coefficient governing the tail is suppressed by α=σν. For a Gaussian pulse, the suppression ∼e^{-α²/2}. This suppression is exact and confirmed by the time domain Regge Wheeler evolutions. Analytic derivations of leading and next-to-leading order tail coefficients agree with numerical fits at the ten percent level. The results explain the lack of tails in quasi-circular mergers and their presence in head-on and eccentric collisions.

What carries the argument

The branch-cut excitation coefficient in the Green's function for the Regge-Wheeler equation, determined by the Fourier transform of the source spectrum at the branch point.

Load-bearing premise

The perturbing source can be characterized by a single carrier frequency and a width whose Fourier transform directly determines the branch-cut coefficient in the solution of the wave equation.

What would settle it

Time-domain numerical integration of the Regge-Wheeler equation driven by a Gaussian pulse whose measured late-time tail amplitude matches the predicted exponential suppression e^{-(σν)^2 / 2}.

Figures

Figures reproduced from arXiv: 2606.02146 by Alejandro Svyatkovskyy Kholyavka, Jose Antonio Le\'on Vega, Sayak Datta, Xisco Jim\'enez Forteza.

Figure 1
Figure 1. Figure 1: Normalized spectral shape of the leading tail coeffi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ratio of QNM to tail amplitude as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same analysis as in Fig [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

The late-time power law tail predicted by Price's law is a generic feature of black hole perturbation theory, yet it is largely absent in numerical relativity waveforms of binary black hole mergers. We show that this suppression arises from the spectral structure of oscillatory sources. For a generic perturbation with carrier frequency $\nu$ and characteristic width $\sigma$, the branch-cut excitation coefficient governing the tail is suppressed by $\alpha=\sigma\nu$. For a Gaussian pulse, the suppression $\sim e^{-\alpha^2/2}$. This suppression is exact and confirmed by the time domain Regge Wheeler evolutions. The same parameter that controls the transition from broadband to frequency selective black hole response is also responsible for the tail suppression. Moreover, we analytically derive the leading- and next-to-leading-order tail coefficients, finding agreement with numerical fits below the $\sim10\%$ level. Our results provide a first principle explanation for the absence of tails in quasi-circular mergers and their enhancement in head-on and eccentric ones.

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 / 1 minor

Summary. The manuscript claims that the absence of late-time power-law tails (Price's law) in numerical relativity waveforms of binary black hole mergers arises from spectral suppression in oscillatory perturbation sources. For a generic source with carrier frequency ν and width σ, the branch-cut excitation coefficient in the Green's function solution to the Regge-Wheeler equation is suppressed by α=σν; for a Gaussian pulse this yields an exact factor ∼e^{-α²/2}. The paper derives leading- and next-to-leading-order analytic expressions for the tail coefficients, reports ∼10% agreement with time-domain Regge-Wheeler evolutions, and argues that the same parameter α governs the transition from broadband to frequency-selective black-hole response, thereby explaining stronger tails in head-on/eccentric mergers versus quasi-circular ones.

Significance. If the central derivation holds, the result supplies a first-principles, frequency-domain explanation for a long-standing discrepancy between perturbation theory and numerical relativity, without introducing new free parameters. The explicit analytic tail coefficients and their direct numerical confirmation constitute a strength that could be tested against existing and future simulations with controlled source spectra.

major comments (2)
  1. [Abstract / frequency-domain section] Abstract and the frequency-domain analysis: the claim that the suppression is 'exact' and that the branch-cut coefficient is directly proportional to the source Fourier transform evaluated near ω=0 is load-bearing, yet the explicit integral expression for the coefficient (and its reduction to the stated α=σν factor) is not displayed; without it the quantitative 10% agreement cannot be verified independently.
  2. [Numerical results section] Numerical confirmation paragraph: the reported ∼10% agreement between analytic tail coefficients and time-domain fits is central to validating the suppression mechanism, but the manuscript provides neither the fitting window, the precise definition of the error metric, nor the data-exclusion criteria used in the Regge-Wheeler evolutions; this information is required to assess whether the agreement is robust or sensitive to analysis choices.
minor comments (1)
  1. [Abstract] The abstract states agreement 'below the ∼10% level' without specifying whether this refers to relative error on the amplitude, on the power-law index, or on both; a brief clarification would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [Abstract / frequency-domain section] Abstract and the frequency-domain analysis: the claim that the suppression is 'exact' and that the branch-cut coefficient is directly proportional to the source Fourier transform evaluated near ω=0 is load-bearing, yet the explicit integral expression for the coefficient (and its reduction to the stated α=σν factor) is not displayed; without it the quantitative 10% agreement cannot be verified independently.

    Authors: We agree that the explicit integral expression is necessary for independent verification of the suppression factor. In the revised manuscript we will insert the integral definition of the branch-cut excitation coefficient (the source Fourier transform evaluated at ω=0) and explicitly reduce it to the Gaussian factor ∼e^{-α²/2} with α=σν. This addition will make the 'exact' claim for the Gaussian case directly verifiable while preserving the existing analytic and numerical results. revision: yes

  2. Referee: [Numerical results section] Numerical confirmation paragraph: the reported ∼10% agreement between analytic tail coefficients and time-domain fits is central to validating the suppression mechanism, but the manuscript provides neither the fitting window, the precise definition of the error metric, nor the data-exclusion criteria used in the Regge-Wheeler evolutions; this information is required to assess whether the agreement is robust or sensitive to analysis choices.

    Authors: We acknowledge that the fitting details were omitted. In the revision we will add an appendix (or subsection) that specifies the fitting window (late-time regime after the ringdown has decayed), the error metric (relative difference between analytic coefficient and numerical fit), and the data-exclusion criteria (removal of initial transients and early oscillatory contamination). These additions will allow readers to reproduce and assess the robustness of the reported ∼10% agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from Green's function

full rationale

The central result follows directly from the frequency-domain structure of the Green's function solution to the Regge-Wheeler equation: the branch-cut tail coefficient is proportional to the Fourier transform of the source evaluated near ω=0. For a source characterized by carrier frequency ν and width σ, the factor α=σν (and Gaussian suppression e^{-α²/2}) is an immediate consequence of this proportionality and the source spectrum, without any fitting of tail amplitudes or reduction to the target observable. Analytic leading/next-to-leading coefficients are derived from the same framework and compared to independent time-domain evolutions for validation only. No load-bearing self-citations, ansatzes smuggled via prior work, or fitted inputs renamed as predictions appear in the described chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard framework of linear black-hole perturbation theory on a Schwarzschild background; no new free parameters or entities are introduced.

axioms (1)
  • domain assumption Linear perturbations of a Schwarzschild black hole obey the Regge-Wheeler equation
    Standard assumption in black-hole perturbation theory invoked to obtain the Green's function whose branch cut produces the tail.

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discussion (0)

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Reference graph

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