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arxiv: 2604.07858 · v1 · submitted 2026-04-09 · 🌀 gr-qc

Quasinormal modes of the thick braneworld in f(T) gravity

Zi-Jie Li , Hai-Long Jia , Qin Tan , Wen-Di Guo This is my paper

Pith reviewed 2026-05-10 17:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesthick branef(T) gravitybrane splittingtorsion gravityasymptotic iteration methodBernstein spectral methodextra dimensions
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The pith

In f(T) gravity the quadratic coefficient alpha induces a split in the thick brane and makes the damping of quasinormal modes depend on both alpha and overtone number.

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

This paper examines a thick brane embedded in five-dimensional f(T) gravity with the specific choice f(T) equal to T plus alpha times T squared. It first restricts alpha to the interval from negative seven forty-eighths to positive one forty-eighth so that the energy density stays positive and the scalar field remains real. Inside that interval the alpha term produces a split in the brane profile. The authors compute the complex quasinormal frequencies of scalar perturbations with the asymptotic iteration method and the Bernstein spectral method, then confirm the spectra by direct time-domain evolution of the wave equation. For negative alpha the decay rate of the lowest mode falls as the absolute value of alpha grows, while higher overtones show the opposite trend.

Core claim

The central claim is that the parameter alpha in the f(T) equals T plus alpha T squared model can induce a brane-splitting structure within the allowed range of alpha between negative seven forty-eighths and positive one forty-eighth. The quasinormal frequencies are obtained using the asymptotic iteration method and the Bernstein spectral method, which agree for low overtones. For negative alpha the decay rate of the first quasinormal mode decreases with increasing absolute value of alpha, while higher overtones show the opposite behavior, and this is confirmed by time-domain evolution.

What carries the argument

The thick brane metric ansatz modified by the quadratic torsion correction alpha T squared, which generates the splitting in the warp factor and scalar field profile.

Load-bearing premise

The five-dimensional metric ansatz for the thick brane together with the conditions that the scalar field stays real and the energy density remains positive must be valid.

What would settle it

An independent numerical solution of the perturbation wave equation on the same background that produces quasinormal frequencies violating the reported trend of decreasing decay rate for the first mode when alpha is negative would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.07858 by Hai-Long Jia, Qin Tan, Wen-Di Guo, Zi-Jie Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Plots of the warp factor [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plots of the energy density for different parameters. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plots of the scalar field for different parameters. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Plots of the scalar potential for different parameters [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of the effective potential for different parameter [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plots of the dual potential for different parameters. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The first four [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Evolution waveform of the Gauss wave packet for differe [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Zero modes (dot) excited by the Gauss pulses and the an [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Evolution waveform of the odd packet with [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Beat effect when a Gauss wave packet is incident with [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original abstract

We investigate the quasinormal modes (QNMs) of a thick brane model in $f(T)$ gravity with $f(T) = T + \alpha T^2$. Requiring the energy density to remain positive and the scalar field to be real constrains the parameter $\alpha$ to the range $[-\frac{7}{48},\frac{1}{48}]$. Within this allowed region, we find that the parameter $\alpha$ can induce a brane-splitting structure. The quasinormal frequencies of the system are computed using both the asymptotic iteration method and the Bernstein spectral method. The two approaches show good agreement in the low-overtone regime. For $\alpha<0$, the decay rate of the first QNM decreases as $|\alpha|$ increases, whereas higher overtones exhibit the opposite behavior. To further examine the influence of model parameters on the QNM spectrum, we also perform numerical time-domain evolution of perturbations, whose results are consistent with the frequency-domain analysis. Our results provide a concrete example of quasinormal spectra in thick brane models within $f(T)$ gravity and may offer useful insights for future observational tests of extra dimensions.

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

3 major / 2 minor

Summary. The manuscript investigates quasinormal modes of a thick brane in f(T) gravity with the specific form f(T) = T + α T². It constrains the parameter α to the interval [-7/48, 1/48] by requiring positive energy density and a real scalar field, reports that α can induce a brane-splitting structure, computes the QNM spectrum using the asymptotic iteration method and the Bernstein spectral method (with agreement reported only in the low-overtone regime), finds that for α < 0 the decay rate of the fundamental mode decreases with increasing |α| while higher overtones exhibit the opposite trend, and cross-checks the results via time-domain evolution of perturbations.

Significance. If the numerical trends are robust, the work supplies a concrete example of how the quadratic correction in f(T) gravity modifies the QNM spectrum of a thick brane, including the possibility of brane splitting. The combination of two frequency-domain techniques plus a time-domain consistency check is a methodological strength that could inform future searches for extra-dimensional signatures in gravitational-wave data.

major comments (3)
  1. [Numerical results and method comparison] The central claim that higher overtones (n ≥ 2) exhibit the opposite decay-rate trend with |α| rests on the Bernstein or AIM results outside the regime where the two methods are stated to agree. No explicit convergence tests, error estimates, or cross-validation for n ≥ 2 are described, leaving the reported sign reversal vulnerable to truncation or asymptotic-matching artifacts rather than physical behavior.
  2. [Background and field equations] The thick-brane metric ansatz, the reduction of the five-dimensional f(T) field equations, and the explicit derivation of the background scalar-field and warp-factor profiles are not supplied in full; these steps are load-bearing for the subsequent perturbation equation and QNM calculation.
  3. [QNM tables and figures] No quantitative error bars, residual norms, or convergence tables are provided for the tabulated or plotted QNM frequencies, contrary to standard practice when reporting trends that change sign with overtone number.
minor comments (2)
  1. [Abstract] The abstract states that the two frequency-domain methods agree in the low-overtone regime but does not qualify the higher-overtone claim or mention the absence of error estimates.
  2. [Throughout] Notation for the overtone index n and the precise definition of the decay rate (Im(ω)) should be introduced once and used consistently in all figures and tables.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the paper accordingly to improve clarity and robustness.

read point-by-point responses

  1. Referee: The central claim that higher overtones (n ≥ 2) exhibit the opposite decay-rate trend with |α| rests on the Bernstein or AIM results outside the regime where the two methods are stated to agree. No explicit convergence tests, error estimates, or cross-validation for n ≥ 2 are described, leaving the reported sign reversal vulnerable to truncation or asymptotic-matching artifacts rather than physical behavior.

    Authors: We acknowledge that the two methods are shown to agree primarily in the low-overtone regime, as stated in the manuscript. The trends for n ≥ 2 are obtained from the Bernstein spectral method and are independently corroborated by the time-domain evolution of perturbations, which exhibits consistent qualitative behavior across the parameter range. To strengthen the presentation, we will add explicit convergence tests, residual norms, and error estimates for higher overtones in the revised version. revision: yes

  2. Referee: The thick-brane metric ansatz, the reduction of the five-dimensional f(T) field equations, and the explicit derivation of the background scalar-field and warp-factor profiles are not supplied in full; these steps are load-bearing for the subsequent perturbation equation and QNM calculation.

    Authors: We agree that including the full derivation of the background solution would enhance readability and self-containment. In the revised manuscript we will provide the metric ansatz, the reduction of the f(T) field equations, and the explicit steps leading to the warp factor and scalar-field profiles, either in the main text or in a dedicated appendix. revision: yes

  3. Referee: No quantitative error bars, residual norms, or convergence tables are provided for the tabulated or plotted QNM frequencies, contrary to standard practice when reporting trends that change sign with overtone number.

    Authors: We will include quantitative error estimates, residual norms, and convergence tables for the QNM frequencies (both low and higher overtones) in the revised manuscript. These additions will substantiate the reported trends and address the concern regarding possible numerical artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: model parameters constrained by physics, spectra computed numerically from equations

full rationale

The derivation begins with the standard f(T) = T + α T² ansatz and the thick-brane metric, then imposes positivity of energy density and reality of the scalar field to bound α ∈ [-7/48, 1/48]. Within this interval the brane-splitting structure and QNM frequencies are obtained by direct numerical solution of the perturbation equations via AIM and Bernstein methods; the time-domain evolution provides an independent cross-check. No parameter is fitted to the QNM data themselves, no self-citation supplies a uniqueness theorem or load-bearing ansatz, and the reported trends (decay-rate reversal with |α| for higher overtones) are outputs of the numerical integration rather than re-statements of the input constraints. The chain is therefore self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on a standard thick-brane metric ansatz in five dimensions and the ad-hoc choice of quadratic f(T); α is the only free parameter and is bounded by positivity rather than fitted to QNM data.

free parameters (1)
  • α
    Quadratic coefficient in f(T); constrained to [-7/48, 1/48] by requiring positive energy density and real scalar field.
axioms (2)
  • domain assumption Thick brane metric ansatz in 5D warped geometry
    Standard construction for thick braneworlds; invoked to define the background.
  • ad hoc to paper f(T) = T + α T² form of the action
    Chosen for the specific model under study.

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discussion (0)

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