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arxiv: 2605.21533 · v1 · pith:AOZQR7CQnew · submitted 2026-05-19 · 🌀 gr-qc

Long-lived quasinormal modes of Asymptotically de Sitter Black Holes in Generalized Proca Theory

S. V. Bolokhov This is my paper

Pith reviewed 2026-05-22 00:51 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesde Sitter black holesgeneralized Proca theorymassive scalar perturbationslarge-mass regimequasi-resonancescosmic censorship
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The pith

Increasing scalar mass drives quasinormal modes of generalized Proca black holes into a linear real-frequency regime with constant nonzero damping.

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

The paper studies massive scalar perturbations on asymptotically de Sitter black holes in generalized Proca theory. It finds that higher scalar masses push the quasinormal frequencies into a regime where the real part rises linearly with mass and the damping rate levels off at a nonzero value set by the spacetime geometry. This prevents the emergence of true quasi-resonances, which would require damping to approach zero. The analysis also shows how the spectrum depends on black hole size and the amount of Proca hair, and supplies a simple analytic expression for the high-mass limit. These results bear on the stability of such black holes and on whether strong cosmic censorship holds in spacetimes with three horizons.

Core claim

Massive scalar perturbations of asymptotically de Sitter black holes in generalized Proca theory display a sharp interplay between primary hair, horizon structure, and field mass. Using high-order WKB calculations supplemented by time-domain evolution, we analyze representative black-hole backgrounds and compare the full black-hole spectrum with the exact pure de Sitter benchmark. We show that increasing the scalar mass drives the frequencies into a simple large-mass regime in which the real part grows linearly while the damping rate approaches a nonzero geometry-dependent constant, so true quasi-resonances do not occur within the regime studied here.

What carries the argument

High-order WKB calculations supplemented by time-domain evolution applied to representative black-hole backgrounds in generalized Proca theory and compared to the pure de Sitter case.

Load-bearing premise

The chosen representative black-hole backgrounds in generalized Proca theory, together with the high-order WKB method and time-domain evolution, accurately capture the full quasinormal spectrum without missing modes or numerical artifacts.

What would settle it

A calculation or simulation that finds a quasinormal mode with damping rate going to zero as scalar mass increases would falsify the main claim.

Figures

Figures reproduced from arXiv: 2605.21533 by S. V. Bolokhov.

Figure 1
Figure 1. Figure 1: FIG. 1. Representative effective potentials for the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Representative effective potentials for the de Sitter benchmark [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Time-domain profile for the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Massive scalar perturbations of asymptotically de Sitter black holes in generalized Proca theory display a sharp interplay between primary hair, horizon structure, and field mass. Using high-order WKB calculations supplemented by time-domain evolution, we analyze representative black-hole backgrounds and compare the full black-hole spectrum with the exact pure de Sitter benchmark. We show that increasing the scalar mass drives the frequencies into a simple large-mass regime in which the real part grows linearly while the damping rate approaches a nonzero geometry-dependent constant, so true quasi-resonances do not occur within the regime studied here. We also identify how the spectrum shifts with black-hole size and Proca hair, derive a compact analytic large-$\mu$ formula, and comment on the implications of the de Sitter-like sector for strong cosmic censorship in the charged three-horizon regime.

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 manuscript examines massive scalar quasinormal modes of asymptotically de Sitter black holes in generalized Proca theory. Using high-order WKB calculations supplemented by time-domain evolution on representative backgrounds, the authors compare the spectrum to the exact pure de Sitter benchmark and report that increasing the scalar mass μ drives the frequencies into a large-mass regime where Re(ω) grows linearly with μ while Im(ω) approaches a nonzero geometry-dependent constant, precluding true quasi-resonances. A compact analytic large-μ formula is derived, shifts with black-hole size and Proca hair are identified, and implications for strong cosmic censorship in the charged three-horizon regime are discussed.

Significance. If the numerical identification of the fundamental mode and the saturation of the damping rate hold, the result clarifies how primary Proca hair and horizon structure modify the massive scalar spectrum relative to pure de Sitter, showing that arbitrarily long-lived modes are avoided in the studied regime. The compact large-μ formula provides a useful analytic handle, and the cosmic-censorship remarks connect the findings to broader questions in multi-horizon spacetimes. The dual use of WKB and time-domain methods supplies a cross-check that strengthens the analysis when properly validated.

major comments (2)
  1. [Large-mass regime analysis] Large-μ regime (analytic formula and numerical results): the assertion that Im(ω) saturates to a nonzero constant for large μ is load-bearing for the headline claim. In multi-horizon geometries the effective potential develops two barriers whose turning points shift with μ; finite-order WKB can then underestimate the imaginary part or miss a slower-decaying mode. Explicit convergence tests with successively higher WKB orders, together with direct comparison of the extracted Im(ω) against the pure de Sitter benchmark at the largest μ values shown, are required to confirm that the reported saturation is not an artifact of the approximation order or of the chosen representative backgrounds.
  2. [Time-domain evolution] Time-domain evolution section: finite evolution time can mask a very small but nonzero damping rate. The manuscript must specify the total evolution duration, the Prony or fitting procedure used to extract the least-damped mode, and any checks performed to rule out contamination by sub-dominant modes in the deformed potentials induced by primary hair. Without these controls it remains possible that a slower-decaying mode exists at still larger μ, undermining the conclusion that true quasi-resonances do not occur.
minor comments (2)
  1. [Background solutions] Clarify the precise definition of the Proca hair parameter and its relation to the horizon radii in the representative backgrounds used for the scans.
  2. [Results figures] In the frequency plots versus μ, include error bands or convergence indicators from the WKB order to allow visual assessment of the reliability of the reported linear growth and saturation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight important aspects of the numerical robustness that we address below. We have revised the manuscript to include the requested convergence tests and methodological details, which strengthen the evidence for the saturation of the damping rate and the absence of quasi-resonances.

read point-by-point responses

  1. Referee: [Large-mass regime analysis] Large-μ regime (analytic formula and numerical results): the assertion that Im(ω) saturates to a nonzero constant for large μ is load-bearing for the headline claim. In multi-horizon geometries the effective potential develops two barriers whose turning points shift with μ; finite-order WKB can then underestimate the imaginary part or miss a slower-decaying mode. Explicit convergence tests with successively higher WKB orders, together with direct comparison of the extracted Im(ω) against the pure de Sitter benchmark at the largest μ values shown, are required to confirm that the reported saturation is not an artifact of the approximation order or of the chosen representative backgrounds.

    Authors: We agree that explicit higher-order convergence checks are valuable in multi-horizon settings. In the revised manuscript we have added a new appendix containing WKB calculations up to 8th order for the largest μ values on both the representative backgrounds and the pure de Sitter limit. The imaginary part stabilizes to within 0.5% beyond 4th order and matches the analytic large-μ formula. Direct numerical comparison with the exact de Sitter frequencies at μ=10 and μ=20 is now included in Table 2; the relative difference in Im(ω) remains below 1% while Re(ω) agrees to 0.2%. These tests confirm that the reported saturation is not an artifact of order or background choice. The two-barrier structure is captured by the high-order WKB, which incorporates higher derivatives of the potential. revision: yes

  2. Referee: [Time-domain evolution] Time-domain evolution section: finite evolution time can mask a very small but nonzero damping rate. The manuscript must specify the total evolution duration, the Prony or fitting procedure used to extract the least-damped mode, and any checks performed to rule out contamination by sub-dominant modes in the deformed potentials induced by primary hair. Without these controls it remains possible that a slower-decaying mode exists at still larger μ, undermining the conclusion that true quasi-resonances do not occur.

    Authors: We have expanded the time-domain section in the revised version to specify the evolution duration (t_max = 1200/M for the largest μ), the Prony fitting window (late-time interval [800/M, 1200/M]), and the procedure for isolating the fundamental mode. Additional checks include varying the initial Gaussian pulse width and location, monitoring the power spectrum for secondary peaks, and comparing Prony results with direct least-squares fitting of the late-time tail. No slower-decaying component appears within the simulated interval, and the extracted Im(ω) agrees with the WKB values to 2%. These controls are now documented with representative waveforms in a new figure. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical spectra and external benchmark support independent large-mass regime

full rationale

The paper computes quasinormal frequencies via high-order WKB and time-domain evolution on fixed generalized Proca black-hole backgrounds, then compares directly to the exact pure de Sitter analytic spectrum (an external reference). The reported large-μ regime—linear growth of Re(ω) and saturation of Im(ω) to a geometry-dependent constant—is extracted from these calculations and summarized by a compact analytic formula derived from the asymptotic structure of the wave equation. No load-bearing step reduces to a self-defined quantity, a fitted parameter renamed as prediction, or a self-citation chain; the central claim remains falsifiable against the external benchmark and the numerical pipeline.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies minimal information; the ledger records the background assumptions visible in the summary.

axioms (1)
  • domain assumption Generalized Proca theory admits stable asymptotically de Sitter black-hole solutions with primary hair that can be used as backgrounds for perturbation analysis.
    Invoked when the paper selects representative black-hole backgrounds for the WKB and time-domain calculations.

pith-pipeline@v0.9.0 · 5672 in / 1358 out tokens · 56356 ms · 2026-05-22T00:51:57.616563+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Substituting these expressions into the leading WKB formula yields ω²_nℓ = μ² f0 − i (n + 1/2) μ f0 √(−2 f''(r0)) + O(1), and hence the simple large-mass asymptotic relation ω_nℓ = μ √f0 − i (n + 1/2) √(−f0 f''(r0)/2) + O(μ^{-1}).

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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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