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arxiv: 2606.04424 · v1 · pith:XKZT7LORnew · submitted 2026-06-03 · 🌀 gr-qc

Real Part Emergence in Purely Imaginary Quasinormal Modes in Perturbed de Sitter Braneworlds

Hai-Long Jia , Wen-Di Guo , Yun-Tao Gu , Yu-Xiao Liu This is my paper

Pith reviewed 2026-06-28 05:34 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesde Sitter branebraneworldgravitational perturbationscomplex frequenciesstabilityextra dimensionscosmological constant
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The pith

Perturbations on thick de Sitter branes turn purely imaginary quasinormal modes into complex modes with a real frequency part.

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

The paper examines how perturbations affect the quasinormal mode spectrum of gravitational perturbations in a thick de Sitter brane with infinite extra dimensions. In the unperturbed case the modes are purely imaginary, so time-domain signals decay without oscillation. Adding perturbations on the brane causes these modes to acquire a non-zero real part that depends on the perturbation parameters, producing complex frequencies. In the time domain this creates transient oscillations during the intermediate stage of the signal, while the late-time tail stays dominated by the zero mode. Because early-time signals are more readily observed, the induced oscillations could affect how the brane cosmological constant is extracted from gravitational data.

Core claim

In the unperturbed thick de Sitter brane, gravitational quasinormal modes have purely imaginary frequencies, corresponding to non-oscillatory decay. Perturbations on the brane cause these modes to develop a real frequency component that depends on the perturbation parameters, transforming them into complex modes. In the time domain this produces transient oscillatory signatures in the intermediate stage of the waveform, whose fitted frequencies match those of the first newly induced mode, while the late-time waveform remains dominated by the zero mode.

What carries the argument

the perturbation-induced shift of quasinormal frequencies from purely imaginary to complex values on the de Sitter brane

If this is right

  • Time-domain signals develop transient oscillations in the intermediate stage.
  • Fitted frequencies of those oscillations match the first newly induced quasinormal mode.
  • Late-time waveforms remain dominated by the zero mode.
  • The induced oscillations are more likely to be detectable because early-time signals are more observable.
  • This behavior may affect extraction of the cosmological constant on the brane from gravitational signals.

Where Pith is reading between the lines

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

  • The same real-part emergence could appear under perturbations in other braneworld geometries with different backgrounds.
  • Searches for intermediate oscillatory features in gravitational-wave data might indirectly constrain extra-dimensional models.
  • The dependence of the real part on perturbation parameters suggests that stronger perturbations produce larger real frequencies, offering a route to parameter estimation from signals.

Load-bearing premise

The unperturbed thick de Sitter brane has purely imaginary quasinormal frequencies that can be systematically shifted by perturbations to produce a real part.

What would settle it

A numerical computation of the quasinormal spectrum or time-domain evolution for any specific perturbation on the de Sitter brane that finds the modes remain purely imaginary with no intermediate oscillations would falsify the reported real-part emergence.

Figures

Figures reproduced from arXiv: 2606.04424 by Hai-Long Jia, Wen-Di Guo, Yun-Tao Gu, Yu-Xiao Liu.

Figure 1
Figure 1. Figure 1: FIG. 1: Deformations of the bulk scalar field, the warp factor [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Deformations of the energy density and the effective p [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Peak values of the variations of the energy density an [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Ratio between the peak variations of the effective pot [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Specific shapes of the deformation of the effective pot [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Left panel: Variation of the QNM spectrum induced by t [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Left panel: Dependence of the QNMs on the perturbatio [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Time-domain evolutions for odd initial data ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Time-domain evolutions for even initial data ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Time-domain evolutions for odd initial data ( [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison between the time-domain evolution and t [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original abstract

For braneworlds with infinite extra dimensions, an analysis of the stability of the characteristic spectrum is essential for understanding their dynamical properties. In this study, we investigate the stability of the gravitational perturbation spectrum in a thick de Sitter brane. Unlike the flat brane case, the de Sitter brane features purely imaginary quasinormal frequencies, corresponding to time-domain signals that decay without oscillation. Our results demonstrate that, upon introducing perturbations on the brane, the originally purely imaginary modes develop a nonvanishing real part that depends on the perturbation parameters, thereby becoming complex-frequency modes with both real and imaginary components. In the time domain, this behavior manifests as transient oscillatory signatures in the intermediate stage of the signal, whose fitted frequencies are consistent with those of the first newly induced quasinormal mode, while the late-time waveform remains dominated by the zero mode. As early-time signals are more readily observable, such perturbation-induced oscillations are more likely to be detectable and may have an impact on the extraction of the cosmological constant on the brane from gravitational signals.

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 investigates the gravitational perturbation spectrum in thick de Sitter braneworlds with infinite extra dimensions. It claims that the unperturbed quasinormal modes (QNMs) are purely imaginary, leading to non-oscillatory decaying signals. Introducing perturbations on the brane causes these modes to develop a non-vanishing real part dependent on perturbation parameters, resulting in complex QN frequencies. In the time domain, this manifests as transient oscillations in the intermediate stage, consistent with the first induced QNM, while late-time is dominated by the zero mode. The authors suggest implications for detectability and cosmological constant extraction from gravitational signals.

Significance. If substantiated, the finding that perturbations can induce real parts in originally purely imaginary QNMs in de Sitter braneworlds highlights a mechanism for generating oscillatory signatures in gravitational perturbations. This could have implications for the stability analysis and observational signatures in braneworld models, particularly since early-time signals are more observable. The paper provides a potential link between perturbation parameters and the emergence of complex modes.

major comments (2)
  1. Abstract: The abstract states the outcome but supplies no derivation steps, numerical methods, error estimates, or explicit perturbation implementation, preventing verification that the mathematics supports the claim as stated.
  2. The assumption that the thick de Sitter brane possesses purely imaginary quasinormal frequencies in the unperturbed case (abstract, paragraph 3) requires explicit demonstration via the mode equation or reference to a specific prior result with the relevant dispersion relation to confirm it is not dependent on unstated choices in the model setup.
minor comments (1)
  1. The dependence of the new real part on unspecified perturbation parameters leaves open the possibility that the reported effect is tied to choices made in the model setup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment point by point below. The manuscript already contains the relevant derivations and demonstrations in the main text, but we agree that targeted clarifications will strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract: The abstract states the outcome but supplies no derivation steps, numerical methods, error estimates, or explicit perturbation implementation, preventing verification that the mathematics supports the claim as stated.

    Authors: Abstracts are intentionally concise and omit detailed derivations, methods, and error estimates, which appear in the body of the paper (numerical solution of the perturbed mode equation and perturbation implementation are described in Sections 3 and 4). We will revise the abstract to include a brief clause referencing the numerical approach used. revision: yes

  2. Referee: The assumption that the thick de Sitter brane possesses purely imaginary quasinormal frequencies in the unperturbed case (abstract, paragraph 3) requires explicit demonstration via the mode equation or reference to a specific prior result with the relevant dispersion relation to confirm it is not dependent on unstated choices in the model setup.

    Authors: Section 2 derives the unperturbed mode equation explicitly and shows that its solutions are purely imaginary frequencies via the resulting dispersion relation; this follows directly from the de Sitter thick-brane background and is independent of unstated choices. We will add an explicit cross-reference to this derivation (or the relevant prior result) in the revised abstract and introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The paper's central claim—that perturbations on a thick de Sitter brane induce a nonzero real part in originally purely imaginary quasinormal modes—is presented as a numerical or analytic result from the perturbed mode equations, with the unperturbed spectrum taken as an input assumption rather than derived from the perturbation itself. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain is exhibited in the abstract or stated claims; the dependence on perturbation parameters is an output of the model, not a tautology. The derivation therefore remains independent of its target result and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the central claim rests on the domain assumption that the unperturbed de Sitter brane spectrum is purely imaginary.

axioms (1)
  • domain assumption Thick de Sitter brane possesses purely imaginary quasinormal frequencies corresponding to non-oscillatory decay in the unperturbed case.
    Explicitly stated as the baseline before perturbations are introduced.

pith-pipeline@v0.9.1-grok · 5733 in / 1307 out tokens · 31561 ms · 2026-06-28T05:34:45.317942+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

68 extracted references · 60 canonical work pages · 40 internal anchors

  1. [1]

    001 0.0247498 -0.368233 0.0967193 -0.542292 0.122729 -0.459683

  2. [2]

    01 0.191071 -0.411263 0.252136 -0.336975 0.242234 -0.341501

  3. [3]

    Talent Scientific Fund of Lanzhou University

    1 0.321139 -0.131438 0.326107 -0.153556 0.326274 -0.153405 TABLE I: Comparison between the first QNF extracted through t he time-domain fitting results and the frequency-domain results with and without the perturbation for p = 0. We first investigate the fitting results of the time-domain waveform s in the case p = 0, as summarized in Table I. For the unpertu...

    2025

  4. [4]

    Dirichlet-Branes and Ramond-Ramond Charges

    J. Polchinski, Dirichlet Branes and Ramond-Ramond charges , Phys. Rev. Lett. 75, 4724 (1995), [ arXiv:hep-th/9510017]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  5. [5]

    The Hierarchy Problem and New Dimensions at a Millimeter

    N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, The Hierarchy problem and new dimensions at a millimeter , Phys. Lett. B 429, 263 (1998), [ arXiv:hep-ph/9803315]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  6. [6]

    New Dimensions at a Millimeter to a Fermi and Superstrings at a TeV

    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV , Phys. Lett. B 436, 257 (1998), [ arXiv:hep-ph/9804398]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  7. [7]

    Inflation and Large Internal Dimensions

    N. Kaloper and A. D. Linde, Inflation and large internal dimensions , Phys. Rev. D 59, 101303 (1999), [arXiv:hep-th/9811141]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  8. [8]

    A Large Mass Hierarchy from a Small Extra Dimension

    L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension , Phys. Rev. Lett. 83, 3370 (1999), [arXiv:hep-ph/9905221]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  9. [9]

    An Alternative to Compactification

    L. Randall and R. Sundrum, An Alternative to compactification , Phys. Rev. Lett. 83, 4690 (1999), [ arXiv:hep-th/9906064]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  10. [10]

    I. I. Kogan, S. Mouslopoulos, A. Papazoglou, G. G. Ross, a nd J. Santiago, A Three three-brane universe: New phenomenol- ogy for the new millennium? , Nucl. Phys. B 584, 313 (2000), [ arXiv:hep-ph/9912552]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  11. [11]

    Thick domain walls and singular spaces

    M. Gremm, Thick domain walls and singular spaces , Phys. Rev. D 62, 044017 (2000), [ arXiv:hep-th/0002040]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  12. [12]

    Opening up extra dimensions at ultra-large scales

    R. Gregory, V. A. Rubakov, and S. M. Sibiryakov, Opening up extra dimensions at ultra large scales , Phys. Rev. Lett. 84, 5928 (2000), [ arXiv:hep-th/0002072]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  13. [13]

    G. R. Dvali, G. Gabadadze, and M. Porrati, Metastable gravitons and infinite volume extra dimensions , Phys. Lett. B 484, 112 (2000), [ arXiv:hep-th/0002190]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  14. [14]

    Thick de Sitter 3-Branes, Dynamic Black Holes and Localization of Gravity

    A. Wang, Thick de Sitter 3 branes, dynamic black holes and localizati on of gravity , Phys. Rev. D 66, 024024 (2002), [arXiv:hep-th/0201051]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  15. [15]

    Thick brane solutions

    V. Dzhunushaliev, V. Folomeev, and M. Minamitsuji, Thick brane solutions , Rept. Prog. Phys. 73, 066901 (2010), [arXiv:0904.1775]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  16. [16]

    Brane-World Gravity

    R. Maartens and K. Koyama, Brane-World Gravity, Living Rev. Rel. 13, 5 (2010), [ arXiv:1004.3962]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  17. [17]

    Introduction to Extra Dimensions and Thick Braneworlds

    Y.-X. Liu, Introduction to Extra Dimensions and Thick Braneworlds , [ arXiv:1707.08541]

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    D. V. Ahluwalia, J. M. H. da Silva, C. Y. Lee, Y.-X. Liu, S. H. Pereira, and M. M. Sorkhi, Mass dimension one fermions: Constructing darkness , Phys. Rept. 967, 1 (2022), [ arXiv:2205.04754]

    work page arXiv 2022

  19. [19]

    On the Unification Problem in Physics

    T. Kaluza, Zum unit¨ atsproblem der physik , Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1921, 966 (1921), [arXiv:1803.08616]

    work page internal anchor Pith review Pith/arXiv arXiv 1921

  20. [20]

    Kaluza, Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie, Z

    T. Kaluza, Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie, Z. phys. 37, 895 (1926)

    1926

  21. [21]

    J. M. Overduin and P. S. Wesson, Kaluza-Klein gravity, Phys. Rept. 283, 303 (1997), [ arXiv:gr-qc/9805018]. 20

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  22. [22]

    Yang, B.-F

    S.-Q. Yang, B.-F. Zhan, Q.-L. Wang, C.-G. Shao, L.-C. Tu , W.-H. Tan, and J. Luo, Test of the Gravitational Inverse Square Law at Millimeter Ranges , Phys. Rev. Lett. 108, 081101 (2012)

    2012

  23. [23]

    Tan, S.-Q

    W.-H. Tan, S.-Q. Yang, C.-G. Shao, J. Li, A.-B. Du, B.-F. Zhan, Q.-L. Wang, P.-S. Luo, L.-C. Tu, and J. Luo, New Test of the Gravitational Inverse-Square Law at the Submillimet er Range with Dual Modulation and Compensation , Phys. Rev. Lett. 116, 131101 (2016)

    2016

  24. [24]

    Tan, A.-B

    W.-H. Tan, A.-B. Du, W.-C. Dong, S.-Q. Yang, C.-G. Shao, S.-G. Guan, Q.-L. Wang, B.-F. Zhan, P.-S. Luo, and L.-C. Tu, et al. Improvement for Testing the Gravitational Inverse-S quare Law at the Submillimeter Range , Phys. Rev. Lett. 124, 051301 (2020)

    2020

  25. [25]

    J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer, a nd B. R. Heckel, New Test of the Gravitational 1/r 2 Law at Separations down to 52 µ m, Phys. Rev. Lett. 124, 101101 (2020), [ arXiv:2002.11761 [hep-ex] ]

    work page arXiv 2020

  26. [26]

    J. Ke, J. Luo, C. G. Shao, Y. J. Tan, W. H. Tan, and S. Q. Yang , Combined Test of the Gravitational Inverse-Square Law at the Centimeter Range , Phys. Rev. Lett. 126, 211101 (2021)

    2021

  27. [27]

    Quasi-Localization of Gravity by Resonant Modes

    C. Csaki, J. Erlich, and T. J. Hollowood, Quasilocalization of gravity by resonant modes , Phys. Rev. Lett. 84, 5932 (2000), [arXiv:hep-th/0002161]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  28. [28]

    I. H. Brevik, K. Ghoroku, S. D. Odintsov, and M. Yahiro, Localization of gravity on brane embedded in AdS(5) and dS(5 ), Phys. Rev. D 66, 064016 (2002), [ arXiv:hep-th/0204066]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  29. [29]

    Braneworld resonances

    C. Clarkson and S. S. Seahra, Braneworld resonances, Class. Quant. Grav. 22, 3653 (2005), [ arXiv:gr-qc/0505145]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  30. [30]

    Fermion localization on thick branes

    A. Melfo, N. Pantoja, and J. D. Tempo, Fermion localization on thick branes , Phys. Rev. D 73, 044033 (2006), [arXiv:hep-th/0601161]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  31. [31]

    Y.-X. Liu, J. Yang, Z.-H. Zhao, C.-E. Fu, and Y.-S. Duan, Fermion Localization and Resonances on A de Sitter Thick Brane, Phys. Rev. D 80, 065019 (2009), [ arXiv:0904.1785]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  32. [32]

    Y.-X. Liu, H. Guo, C.-E. Fu, and J.-R. Ren, Localization of Matters on Anti-de Sitter Thick Branes , JHEP 02, 080 (2010), [arXiv:0907.4424]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  33. [33]

    Y.-X. Liu, Y. Zhong, Z.-H. Zhao, and H.-T. Li, Domain wall brane in squared curvature gravity , J. High Energy Phys. 06, 135 (2011), [ arXiv:1104.3188]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  34. [34]

    Linearization of a warped $f(R)$ theory in the higher-order frame

    Y. Zhong and Y.-X. Liu, Linearization of a warped f (R) theory in the higher-order frame , Phys. Rev. D 95, 104060 (2017), [arXiv:1611.08237]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  35. [35]

    C.-C. Zhu, Q. Tan, Y.-P. Zhang, and Y.-X. Liu, Evolution of fermion resonance in thick brane , Commun. Theor. Phys. 76, 125403 (2024), [ arXiv:2304.04479]

    work page arXiv 2024

  36. [36]

    C.-C. Zhu, J. Chen, W.-D. Guo, and Y.-X. Liu, Gravitational echoes from braneworlds , JHEP 01, 010 (2025), [arXiv:2406.16256]

    work page arXiv 2025

  37. [37]

    S. S. Seahra, Ringing the Randall-Sundrum braneworld: Metastable gravi ty wave bound states , Phys. Rev. D 72, 066002 (2005), [ arXiv:hep-th/0501175]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  38. [38]

    S. S. Seahra, Metastable massive gravitons from an infinite extra dimensi on, Int. J. Mod. Phys. D 14, 2279 (2005), [arXiv:hep-th/0505196]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  39. [39]

    Quasinormal ringing on the brane

    H. Chung, L. Randall, M. J. Rodriguez, and O. Varela, Quasinormal ringing on the brane , Class. Quant. Grav. 33, 245013 (2016), [ arXiv:1508.02611]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  40. [40]

    Tan, W.-D

    Q. Tan, W.-D. Guo, and Y.-X. Liu, Sound from extra dimensions: Quasinormal modes of a thick br ane, Phys. Rev. D 106, 044038 (2022), [ arXiv:2205.05255]

    work page arXiv 2022

  41. [41]

    Tan, W.-D

    Q. Tan, W.-D. Guo, Y.-P. Zhang, and Y.-X. Liu, Characteristic modes of a thick brane: Resonances and quasi normal modes, Phys. Rev. D 109, 024017 (2024), [ arXiv:2304.09363]

    work page arXiv 2024

  42. [42]

    Q. Tan, Y. Zhong, and W.-D. Guo, Quasibound and quasinormal modes of a thick brane in Rastall gravity, JHEP 07, 252 (2024), [ arXiv:2404.11217]

    work page arXiv 2024

  43. [43]

    Jia, W.-D

    H.-L. Jia, W.-D. Guo, Q. Tan, and Y.-X. Liu, Quasinormal ringing of thick braneworlds with a finite extra dimension, Phys. Rev. D 110, 064077 (2024), [ arXiv:2406.03929]

    work page arXiv 2024

  44. [44]

    Q. Tan, S. Long, W. Deng, and J. Jing, Graviscalar quasinormal modes and asymptotic tails of a thi ck brane , Phys. Lett. B 868, 139667 (2025), [ arXiv:2409.06947]

    work page arXiv 2025

  45. [45]

    Q. Tan, S. Long, W. Deng, and J. Jing, Quasinormal modes and echoes of a double braneworld , JHEP 02, 055 (2025), [arXiv:2410.06945]

    work page arXiv 2025

  46. [46]

    Jia, W.-D

    H.-L. Jia, W.-D. Guo, Y.-X. Liu, and Q. Tan, Quasinormal ringing of de Sitter braneworlds , JHEP 06, 117 (2025), [arXiv:2501.00477]. 21

    work page arXiv 2025

  47. [47]

    W. Deng, S. Long, Q. Tan, Z.-C. Chen, and J. Jing, Scalar-gravitational quasinormal modes and echoes in a five dimensional thick brane, [ arXiv:2508.20937]

    work page arXiv

  48. [48]

    Y.-P. E, C.-C. Zhu, and Y.-X. Liu, Quasinormal modes of thick branes in f (R) gravity, [ arXiv:2512.17208]

    work page arXiv

  49. [49]

    R. A. Konoplya and A. Zhidenko, Asymptotic tails of massive gravitons in light of pulsar tim ing array observations , Phys. Lett. B 853, 138685 (2024), [ arXiv:2307.01110]

    work page arXiv 2024

  50. [50]

    The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background

    G. Agazie et al. [NANOGrav], The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wa ve Background, Astrophys. J. Lett. 951, L8 (2023), [ arXiv:2306.16213]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  51. [51]

    Late time behavior of cosmological perturbations in a single brane model

    K. Koyama, Late time behavior of cosmological perturbations in a singl e brane model , JCAP 09, 010 (2004), [arXiv:astro-ph/0407263]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  52. [52]

    Cosmological Backgrounds of Gravitational Waves

    C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves , Class. Quant. Grav. 35, 163001 (2018), [arXiv:1801.04268]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  53. [53]

    Locally Localized Gravity

    A. Karch and L. Randall, Locally localized gravity, JHEP 05, 008 (2001), [ arXiv:hep-th/0011156]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  54. [54]

    V. I. Afonso, D. Bazeia, and L. Losano, First-order formalism for bent brane , Phys. Lett. B 634, 526 (2006), [arXiv:hep-th/0601069]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  55. [55]

    Y.-X. Liu, H. Guo, C.-E. Fu, and H.-T. Li, Localization of Bulk Matters on a Thick Anti-de Sitter Brane , Phys. Rev. D 84, 044033 (2011), [ arXiv:1101.4145]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  56. [56]

    Spectral Butterfly Effect and Resilient Ringdown in Thick Braneworlds

    H.-L. Jia, W.-D. Guo, Y.-P. Zhang, and Y.-X. Liu, Spectral Butterfly Effect and Resilient Ringdown in Thick Bran eworlds, [arXiv:2604.23364]

    work page internal anchor Pith review Pith/arXiv arXiv

  57. [57]

    Modeling the fifth dimension with scalars and gravity

    O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch, Modeling the fifth-dimension with scalars and gravity , Phys. Rev. D 62, 046008 (2000), [ arXiv:hep-th/9909134]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  58. [58]

    Four-dimensional gravity on a thick domain wall

    M. Gremm, Four-dimensional gravity on a thick domain wall , Phys. Lett. B 478, 434 (2000), [ arXiv:hep-th/9912060]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  59. [59]

    Universal Aspects of Gravity Localized on Thick Branes

    C. Csaki, J. Erlich, T. J. Hollowood, and Y. Shirman, Universal aspects of gravity localized on thick branes , Nucl. Phys. B 581, 309 (2000), [ arXiv:hep-th/0001033]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  60. [60]

    Advanced Methods in Black-Hole Perturbation Theory

    P. Pani, Advanced Methods in Black-Hole Perturbation Theory , Int. J. Mod. Phys. A 28, 1340018 (2013), [arXiv:1305.6759]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  61. [61]

    Fortuna and I

    S. Fortuna and I. Vega, Bernstein spectral method for quasinormal modes and other e igenvalue problems , Eur. Phys. J. C 83, 1170 (2023), [ arXiv:2003.06232]

    work page arXiv 2023

  62. [62]

    J. L. Jaramillo, R. P. Macedo and L. A. Sheikh, Pseudospectrum and Black Hole Quasinormal Mode Instabilit y, Phys. Rev. X 11, 031003 (2021), [ arXiv:2004.06434]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  63. [63]

    Motohashi, Resonant Excitation of Quasinormal Modes of Black Holes , Phys

    H. Motohashi, Resonant Excitation of Quasinormal Modes of Black Holes , Phys. Rev. Lett. 134, 141401 (2025), [arXiv:2407.15191]

    work page arXiv 2025

  64. [64]

    J. P. Cavalcante, M. Richartz, and B. C. da Cunha, Exceptional Point and Hysteresis in Perturbations of Kerr B lack Holes , Phys. Rev. Lett. 133, 261401 (2024), [ arXiv:2407.20850]

    work page arXiv 2024

  65. [65]

    Y. Yang, E. Berti, and N. Franchini, Black Hole Quasinormal Mode Resonances , Phys. Rev. Lett. 135, 201401 (2025), [arXiv:2504.06072]

    work page arXiv 2025

  66. [66]

    Exceptional Points and Resonance in Black Hole Ringdown

    R. Panosso Macedo, T. Katagiri, K. i. Kubota, and H. Moto hashi, Exceptional Points and Resonance in Black Hole Ringdown, [ arXiv:2512.02110]

    work page internal anchor Pith review Pith/arXiv arXiv

  67. [67]

    L. B. Wu, L. Xie, L. M. Cao, M. F. Ji, and Y. S. Zhou, Quasinormal modes of Schwarzschild-de Sitter black holes i n semi-open systems, Sci. China Phys. Mech. Astron. 69, 240415 (2026), [ arXiv:2512.06903]

    work page arXiv 2026

  68. [68]

    Cheng, X

    L. Cheng, X. Guo, Y. Li, J. Tao, and P. Wang, Exceptional Points in Quasinormal Spectra of Hairy Black Ho les, [arXiv:2603.22261]

    work page arXiv