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arxiv: 2607.06081 · v1 · pith:GA2RD2NC · submitted 2026-07-07 · gr-qc · hep-th

Quasinormal modes of scalar perturbations in Rastall thick brane

Shan Huang , Chun-Chun Zhu , Tao-Tao Sui This is my paper

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classification gr-qc hep-th PACS 04.50.Kd04.60.Cf11.25.Mj
keywords modesrastallscalarbraneperturbationsquasinormalthickasymptotic
0
0 comments X

The pith

Rastall parameter tunes scalar brane ringing and late-time tails

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

This paper computes the quasinormal mode (QNM) spectrum of graviscalar perturbations in a five-dimensional thick brane model built in Rastall gravity, a modified theory where energy-momentum conservation is violated in proportion to the curvature gradient. The central object is a single master equation for scalar metric perturbations, derived in the longitudinal gauge, which reduces to a one-dimensional Schrödinger-like eigenvalue problem along the extra dimension. The effective potential depends explicitly on the Rastall parameter lambda. The authors compute complex QNM frequencies using three independent numerical methods (Bernstein spectral, direct integration, and time-domain evolution) and find that lambda controls the spectrum in two distinct ways. First, the imaginary parts of the frequencies decrease monotonically with increasing lambda, meaning modes live longer. Second, the real parts of higher overtones show non-monotonic V-shaped behavior, reflecting competition between a lowering and broadening of the potential barrier. Third, the late-time signal decays as a power law t^{-beta} where beta is analytically fixed by the 1/z^2 asymptotic coefficient of the potential, itself a function of lambda; numerical time-domain evolution confirms this prediction. The paper also identifies a stability window in lambda where the potential is non-negative, and checks that outside this window unstable modes can appear, confirmed by exponential growth in time-domain profiles.

Core claim

The Rastall parameter lambda acts as a single dial that simultaneously controls the damping rate, the overtone structure, and the late-time power-law decay exponent of graviscalar quasinormal modes in a thick brane. The tail exponent beta is analytically determined by lambda through the asymptotic 1/z^2 coefficient of the effective potential, and this relation is confirmed by direct numerical evolution.

What carries the argument

The derivation chains through: (1) a longitudinal gauge fixing with the constraint phi + 2*Psi = 0, (2) elimination of the scalar field fluctuation delta_phi in favor of Psi, yielding a single master equation (Eq. 13), (3) a KK decomposition that casts the problem as a Schrodinger-like equation with an effective potential U(z) depending on lambda (Eq. 17-18), and (4) the Ching et al. late-time tail classification, which connects the asymptotic 1/z^2 coefficient of U(z) to the power-law exponent beta.

If this is right

  • The Rastall parameter provides a tunable handle on brane scalar mode lifetimes: larger lambda yields longer-lived metastable excitations, which could affect how scalar signals persist in braneworld phenomenology.
  • The V-shaped non-monotonicity of higher overtone real parts means that simple monotonic interpolation of QNM spectra across modified-gravity parameters is not generally valid; the interplay between barrier height and width can produce spectral turning points.
  • The analytic link between the asymptotic 1/z^2 potential coefficient and the tail exponent beta means that measuring or constraining late-time decay behavior could in principle invert to constrain lambda.
  • The stability window -9/116 <= lambda <= 54/328 provides a concrete parameter range for model-building in Rastall thick branes, outside of which tachyonic or unstable scalar modes must be explicitly checked.
  • The estimated physical frequencies (~10^11 Hz) and damping times (~10^{-12} to 10^{-11} s) place these modes in an ultra-high-frequency regime far from current gravitational-wave detector bands, and their scalar polarization content would require non-standard detector sensitivity.

Load-bearing premise

The derivation of the master equation assumes that the longitudinal gauge with the constraint phi + 2*Psi = 0 and the substitution of delta_phi in terms of Psi fully capture the scalar sector dynamics. The paper states these follow from linearizing the Rastall field equations but does not provide intermediate algebraic steps or an explicit gauge-invariance check, so if the gauge fixing inadvertently eliminates a physical mode or if Rastall's modified conservation introduces a

What would settle it

If the gauge fixing in deriving the master equation (Eq. 13) eliminates a physical scalar degree of freedom, or if Rastall's non-conservation introduces an additional scalar constraint not captured by the single master variable Psi, the entire QNM spectrum and the lambda-dependent tail exponent would be artifacts of an incomplete perturbation sector.

Figures

Figures reproduced from arXiv: 2607.06081 by Chun-Chun Zhu, Shan Huang, Tao-Tao Sui.

Figure 1
Figure 1. Figure 1: FIG. 1. Plots of the warp factor ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The shape of the graviscalar effective potential [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left panel: The relation between the real parts of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Time evolution of Gaussian wave packet at [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Time-domain evolution outside the non-negative [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

We investigate quasinormal modes of the graviscalar sector in a five-dimensional thick brane model in Rastall gravity. By considering a specific flat brane solution supported by a canonical scalar field, we derive a master equation and reduce it to a Schr\"odinger-like eigenvalue problem for the Kaluza-Klein modes. Using the Bernstein spectral method and direct integration in the frequency domain, complemented by numerical time-domain evolutions, we compute the complex quasinormal frequencies for the scalar perturbations. Our results reveal a strong dependence of the QNM spectrum on $\lambda$: the imaginary parts of the frequencies, governing the decay rate, decrease monotonically with increasing $\lambda$, indicating longer-lived modes. The real parts exhibit a more complex, non-monotonic behavior. Furthermore, we analyze the late-time behavior of the perturbations, showing that the asymptotic tail follows a power law whose exponent is determined by the Rastall parameter, in agreement with theoretical predictions for the asymptotic form of the potential. These findings provide a comprehensive dynamical characterization of the scalar sector of Rastall thick branes, offering potential observational signatures for probing modified gravity in extra-dimensional scenarios.

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 / 6 minor

Summary. This paper studies quasinormal modes (QNMs) of graviscalar perturbations in a five-dimensional thick brane model within Rastall gravity. The authors derive a master equation for the scalar sector in the longitudinal gauge, reduce it to a Schrödinger-like eigenvalue problem with a λ-dependent effective potential, and compute QNM frequencies using the Bernstein spectral method, direct integration, and time-domain evolution. The main findings are: (1) the imaginary parts of the QNM frequencies decrease monotonically with increasing Rastall parameter λ (longer-lived modes), (2) the real parts of higher overtones show non-monotonic V-shaped behavior, and (3) the late-time tail follows a power law t^{-β} whose exponent β is analytically determined by λ through the asymptotic 1/z² coefficient of the effective potential. The three numerical methods agree to 5+ significant figures for the fundamental mode, and the analytic tail exponent matches numerical fitting within ~2%.

Significance. The paper addresses a gap in the literature: while tensor-sector QNMs of the Rastall thick brane have been studied, the graviscalar sector has not. The combination of three independent numerical methods (Bernstein spectral, direct integration, time-domain) with mutual agreement to 5+ significant figures is a genuine strength. The analytic derivation of the tail exponent β from the asymptotic potential structure, confirmed by numerical fitting, is a clean result. The phenomenological discussion of frequency scales and polarization content is appropriate and grounded. The central claim—that the scalar QNM spectrum and tail exponent depend on λ—is well-supported by the presented evidence, provided the master equation derivation is correct.

major comments (2)
  1. §II.B, Eq. (13): The master equation derivation proceeds from the linearized Rastall field equations via the longitudinal gauge constraint (Eq. 11) and the substitution (Eq. 12), but the intermediate linearized equations are not shown. The coefficient (10λ−3)/(4λ−3) in front of □₍₄₎Ψ is the load-bearing λ-dependent factor that propagates into the effective potential (Eq. 17) and the asymptotic 1/z² coefficient (Eq. 26) determining the tail exponent β. The GR limit λ→0 gives coefficient (−3)/(−3)=1, which is plausible, but this single check does not by itself confirm the full λ-dependent structure is correct, since errors in the λ-terms could cancel at λ=0. The authors should either provide the key intermediate linearized equations (at least the (μ,ν), (z,μ), and (z,z) components after linearization) or explicitly verify that the λ-dependent terms in Eqs. (13) and (17) reproduce correctly
  2. §II.B, Eqs. (11)–(12): The paper states that the off-diagonal components give the constraint ϕ+2Ψ=0 and the (z,μ) components give Eq. (12), but does not discuss whether the Rastall non-conservation (∇_M T^{MN} = λ∇^N R) introduces additional scalar constraints relative to standard GR that might alter the degree-of-freedom count. Since the entire QNM spectrum depends on the master equation being a complete description of the scalar sector, a brief statement confirming that no physical degree of freedom is lost or spurious constraint introduced by the gauge fixing—ideally with a reference to where this is established for Rastall gravity or an explicit count—would strengthen the derivation. This is particularly important because the Rastall modification changes the constraint structure.
minor comments (6)
  1. Table I, λ=0.07, n=2 row: The time-domain fit gives Re(m/k)=1.898705, Im(m/k)=−0.906472, while the frequency-domain methods give Re(m/k)=1.89787, Im(m/k)=−0.897285. The real parts differ by ~0.001 and the imaginary parts by ~0.009, which is noticeably larger than the agreement for other rows. This should be briefly commented on.
  2. Table I, λ=−0.01, n=2 row: The direct integration method gives Re(m/k)=1.79713, while the Bernstein method gives 1.78713. This appears to be a discrepancy of ~0.01; please check whether this is a typo.
  3. Eq. (17): The compact form uses the notation φ‴ (third derivative) written as 'φ' in the numerator of the third term. This should be clarified, perhaps by writing φ‴ explicitly for clarity.
  4. Fig. 2: The potential is plotted for three λ values (−0.07, 0, 0.16), but the caption does not state what k value is used. Since U/k² is plotted, k is presumably arbitrary, but this should be stated in the figure or in the caption.
  5. §III.B: The time-domain evolution uses a finite-difference scheme on Eq. (23), but the grid spacing, time step, and convergence check are not specified. A brief statement of these numerical parameters would improve reproducibility.
  6. Refs. [56, 58] appear to be closely related prior work on graviscalar QNMs in thick branes (including by overlapping groups). The relationship to the present work—specifically what is new beyond these references—should be stated more clearly, perhaps in the Introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. Both major comments concern the derivation of the master equation (Eq. 13) and the completeness of the scalar-sector constraint analysis. We agree that the intermediate linearized equations and a degree-of-freedom count should be made explicit, and will add both to the revised manuscript.

read point-by-point responses

  1. Referee: §II.B, Eq. (13): The master equation derivation proceeds from the linearized Rastall field equations via the longitudinal gauge constraint (Eq. 11) and the substitution (Eq. 12), but the intermediate linearized equations are not shown. The coefficient (10λ−3)/(4λ−3) in front of □₍₄₎Ψ is the load-bearing λ-dependent factor that propagates into the effective potential (Eq. 17) and the asymptotic 1/z² coefficient (Eq. 26) determining the tail exponent β. The GR limit λ→0 gives coefficient (−3)/(−3)=1, which is plausible, but this single check does not by itself confirm the full λ-dependent structure is correct, since errors in the λ-terms could cancel at λ=0. The authors should either provide the key intermediate linearized equations (at least the (μ,ν), (z,μ), and (z,z) components after linearization) or explicitly verify that the λ-dependent terms in Eqs. (13) and (17) reproduce correctly

    Authors: The referee is correct that the GR limit alone is insufficient to verify the full λ-dependent structure, and we agree that the intermediate linearized equations should be shown. In the revised manuscript, we will add an explicit display of the key linearized components—specifically the (μ,ν), (z,μ), and (z,z) components of the linearized Rastall field equation (2) after imposing the longitudinal gauge (10)—in an appendix or as an expanded derivation in §II.B. This will allow the reader to trace how the coefficient (10λ−3)/(4λ−3) in Eq. (13) arises from the λ-dependent terms in the linearized (μ,ν) equation, and how the off-diagonal (μ≠ν) components yield the constraint ϕ+2Ψ=0 (Eq. 11) and the (z,μ) components yield the expression for δφ (Eq. 12). We will also add an independent consistency check: substituting the explicit background solution (7)–(8) into the linearized equations and verifying that the λ-dependent coefficients in both Eq. (13) and Eq. (17) are reproduced by direct symbolic computation. This goes beyond the λ→0 limit and tests the full λ-structure at the level of the explicit background. revision: yes

  2. Referee: §II.B, Eqs. (11)–(12): The paper states that the off-diagonal components give the constraint ϕ+2Ψ=0 and the (z,μ) components give Eq. (12), but does not discuss whether the Rastall non-conservation (∇_M T^{MN} = λ∇^N R) introduces additional scalar constraints relative to standard GR that might alter the degree-of-freedom count. Since the entire QNM spectrum depends on the master equation being a complete description of the scalar sector, a brief statement confirming that no physical degree of freedom is lost or spurious constraint introduced by the gauge fixing—ideally with a reference to where this is established for Rastall gravity or an explicit count—would strengthen the derivation. This is particularly important because the Rastall modification changes the constraint structure.

    Authors: This is a well-taken point. The Rastall modification does change the constraint structure relative to standard GR, and we should address this explicitly. In the revised manuscript, we will add a degree-of-freedom count for the scalar sector. The argument proceeds as follows. In the longitudinal gauge, the scalar-sector metric perturbations are described by two functions, ϕ and Ψ, plus the scalar-field fluctuation δφ—three functions in total. The off-diagonal (μ≠ν) components of the linearized field equations provide one constraint (Eq. 11), reducing to two independent functions. The (z,μ) components provide a second constraint (Eq. 12), reducing to one. The remaining (μ,ν) and (z,z) components then yield a single second-order equation for the master variable Ψ, which is Eq. (13). Crucially, the Rastall non-conservation ∇_M T^{MN} = λ∇^N R is not an independent equation beyond the field equation (2); it is a consequence of the Bianchi identity applied to (2). Therefore it does not introduce an additional independent constraint on the perturbations beyond those already encoded in the linearized field equations. The degree-of-freedom count is thus the same as in the GR case: one physical scalar degree of freedom, described by the master equation. We will also note that this counting is consistent with the analysis in Ref. [80] (Zhong, Yang, and Liu, JHEP 09, 128 (2022)), where the background solution and its perturbation structure in Rastall gravity are established. We will add this discussion as a paragraph in §II.B. revision: yes

Circularity Check

0 steps flagged

No significant circularity: QNM spectrum and tail exponent derived from independently verifiable analytical potential, not from fitted or self-referential inputs.

full rationale

The paper's central results—the QNM spectrum and the late-time tail exponent β—are not circular. The effective potential U(z) (Eq. 18) is derived analytically from the background solution (Eqs. 7–9, taken from Ref. [80]) and the master equation (Eq. 13). The background solution is an exact analytical solution of the Rastall field equations, not a fitted quantity. The QNM frequencies are computed from U(z) using standard, independent numerical methods (Bernstein spectral, direct integration, time-domain evolution) that do not encode the output frequencies as inputs. The tail exponent β is derived analytically from the asymptotic 1/z² coefficient of U(z) (Eq. 26) via the Ching et al. classification (Refs. [84, 85], external), and then confirmed by independent numerical time-domain fitting (Table III). While Ref. [80] shares authors with the present paper, the cited result is an exact analytical solution that can be independently verified by substitution into the field equations (Eqs. 5–6), so the self-citation is not load-bearing in a circular sense. The derivation chain is self-contained against external benchmarks. The skeptic's concern about missing intermediate algebraic steps in the master equation derivation is a correctness risk, not a circularity issue.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper introduces no new particles, forces, or entities. The Rastall parameter λ and the thick brane background are inherited from prior literature (Rastall 1972; Zhong, Yang & Liu 2022). The graviscalar perturbation framework follows Kobayashi, Koyama & Soda (2002). All axioms are either standard mathematical results or domain assumptions from the Rastall gravity framework.

free parameters (2)
  • λ (Rastall parameter) = varied over [-0.07, 0.16] in main results; also tested at -0.1, -0.2, -0.3, 0.2, 0.4, 0.6
    The Rastall parameter λ is a free parameter of the theory. It is not fitted to observational data in this paper; rather, the QNM spectrum is computed as a function of λ. Observational constraints |λ| ≪ 1 are cited from Ref. [81].
  • k (brane mass scale) = set to 1 in dimensionless computations; k = 10⁻³ eV used for phenomenological estimates
    k sets the mass/length scale of the brane. It is a free parameter of the background solution, not determined by the theory. The phenomenological section uses k = 10⁻³ eV as a representative value.
axioms (4)
  • domain assumption Rastall non-conservation law ∇_M T^MN = λ∇^N R (Eq. 1)
    This is the foundational postulate of Rastall gravity, assumed throughout. It modifies standard energy-momentum conservation.
  • domain assumption The longitudinal gauge (Eq. 10) with constraint ϕ + 2Ψ = 0 (Eq. 11) fully captures scalar perturbation dynamics
    Invoked in §II.B to reduce the coupled scalar perturbation equations to a single master equation. The paper states this follows from off-diagonal components but does not provide a gauge-invariance proof.
  • standard math The Ching et al. asymptotic tail classification [84,85] applies to the Rastall graviscalar potential
    Used in §III.B to derive the power-law tail exponent β from the 1/z² asymptotic form of U(z). This is a standard result for Schrödinger-like potentials.
  • domain assumption κ₅ = 1 (five-dimensional gravitational scale set to unity)
    Stated in §II.A 'for simplicity in calculation.' This is a standard normalization choice and does not affect dimensionless results.

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