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arxiv: 2606.31330 · v1 · pith:N6BLZ7AOnew · submitted 2026-06-30 · ✦ hep-th · gr-qc

Two-scalar-field f(R) Thick Branes, Gravitational Resonances and Quasinormal Modes

Xin-Yi Pan , Heng Guo , Jing-Xin Gong , Hong-Tao Jiang This is my paper

Pith reviewed 2026-07-01 04:57 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords thick branesf(R) gravitytwo scalar fieldsgravitational resonancesquasinormal modesKaluza-Klein spectrumghost-free branchtensor perturbations
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The pith

In ghost-free f(R) thick branes with two scalars, internal structure produces no narrow real-axis tensor resonances.

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

The paper tests whether tunable energy-density splitting from two scalar fields can create long-lived tensor resonances inside an f(R) thick brane. It enforces positivity of f_R to isolate the smooth ghost-free branch and finds that both the relative-probability spectrum and the phase-shift transmission spectrum lack narrow peaks on the real axis. The same branch yields massive Kaluza-Klein modes whose quasinormal frequencies, obtained by the asymptotic iteration method and by time-domain evolution with a supersymmetric partner potential, all carry negative imaginary parts and quality factors between 0.9 and 1.9. These results indicate that the brane’s internal structure alone does not support long-lived tensor excitations when the effective gravitational coupling remains positive.

Core claim

In the ghost-free branch defined by f_R > 0, neither the relative-probability spectrum nor the phase-shift transmission spectrum exhibits narrow real-axis resonant peaks; the internal brane structure therefore does not generate long-lived tensor resonances. The massive tensor excitations are instead broad, short-lived dissipative modes whose fundamental quasinormal frequencies have negative imaginary parts and quality factors Q ≃ 0.9–1.9. Sharp quasi-localization peaks appear only in the singular branch where f_R vanishes and the tensor potential develops divergent structures; those peaks are interpreted as singular-boundary signals rather than physical resonances of the smooth background.

What carries the argument

The positivity condition on f_R ≡ df/dR, which serves as the effective gravitational coupling and cleanly separates the smooth ghost-free branch from the singular branch containing divergent structures in the tensor potential.

If this is right

  • The internal brane structure does not produce long-lived tensor resonances in the physically admissible ghost-free region.
  • Massive tensor excitations must be described by complex-frequency quasinormal modes rather than real-frequency resonances.
  • Sharp quasi-localization peaks are singular-boundary signals and do not appear in the smooth ghost-free background.
  • The appropriate diagnostic for the Kaluza-Klein spectrum is the complex-frequency plane, not the real-axis transmission spectrum.

Where Pith is reading between the lines

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

  • Models that allow f_R to approach zero or change sign may need separate checks to confirm whether apparent resonances are physical or artifacts of the singular boundary.
  • The same real-axis versus complex-frequency comparison could be applied to other modified-gravity thick-brane constructions to distinguish genuine long-lived modes from singular-boundary effects.
  • Time-domain evolution with a supersymmetric partner potential as a zero-mode filter offers a practical numerical route for extracting quasinormal frequencies when analytic solutions are unavailable.

Load-bearing premise

That enforcing f_R > 0 produces a smooth geometry whose tensor perturbations remain free of singular artifacts.

What would settle it

Observation of narrow real-axis resonant peaks, or of quasinormal modes with positive imaginary parts, inside any configuration that satisfies f_R > 0 throughout the bulk.

read the original abstract

In this paper, we investigate thick brane worlds in $f(R)$ gravity supported by two-scalar-field. The two-scalar sector provides an analytical warped background with tunable energy-density splitting, allowing us to test whether a Bloch-type internal structure can generate long-lived tensor perturbations resonances in the physically admissible region. We impose the positivity of \(f_R\equiv df/dR\), the derivative of the gravitational Lagrangian with respect to the Ricci scalar, which plays the role of an effective gravitational coupling in \(f(R)\) gravity. This separates the smooth ghost-free branch from a singular branch where this effective coupling vanishes. In the ghost-free branch, neither the relative-probability spectrum nor the phase-shift transmission spectrum shows narrow real-axis resonant peaks. These real-axis diagnostics indicate that the internal brane structure alone does not produce long-lived tensor resonances in the ghost-free region. Sharp quasi-localization peaks appear only in the singular branch, where the vanishing effective coupling induces divergent structures in the tensor potential; these peaks should therefore be interpreted as singular-boundary signals rather than ghost-free resonances of the smooth brane background. We then characterize the ghost-free massive Kaluza-Klein modes in the complex-frequency plane. Using the Asymptotic Iteration Method where applicable and time-domain evolutions with a supersymmetric partner potential as a zero-mode filtering tool, we extract the fundamental quasinormal frequencies. The modes have negative imaginary parts and quality factors \(Q\simeq0.9-1.9\), showing that the ghost-free massive tensor excitations are broad, short-lived dissipative modes. Thus the QNM spectrum provides the appropriate complex-frequency description of the Kaluza-Klein dynamics when no narrow real-axis resonances are resolved.

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 constructs analytical two-scalar-field thick brane solutions in f(R) gravity with tunable energy-density splitting. It imposes f_R > 0 to isolate the smooth ghost-free branch from the singular branch where f_R vanishes. In the ghost-free branch, neither the relative-probability spectrum nor the phase-shift transmission spectrum exhibits narrow real-axis resonant peaks. QNMs of the massive KK tensor modes are extracted via the Asymptotic Iteration Method (where applicable) and time-domain evolution employing a supersymmetric partner potential for zero-mode filtering; the resulting modes have negative imaginary parts and quality factors Q ≃ 0.9–1.9, indicating broad, short-lived excitations. The central conclusion is that internal brane structure alone does not generate long-lived tensor resonances in the ghost-free region.

Significance. If the numerical diagnostics hold, the work supplies a concrete, analytically tractable example showing that Bloch-type internal structure in f(R) branes produces no long-lived resonances without crossing into the singular branch. The combination of an exact two-scalar background, AIM extraction, and SUSY-partner time-domain filtering constitutes a reproducible methodological strength that cleanly separates the branches and supports the interpretation of singular-branch peaks as boundary artifacts rather than physical resonances.

major comments (2)
  1. [Abstract and numerical methods section] Abstract and § on numerical methods: the central claim that no narrow real-axis peaks appear in the ghost-free branch rests on the relative-probability and phase-shift spectra together with the AIM/time-domain QNM extraction, yet the manuscript supplies no derivation details, convergence criteria, error estimates, or explicit checks against post-hoc parameter choices; these omissions are load-bearing for the reported absence of resonances and the Q ≃ 0.9–1.9 range.
  2. [Tensor perturbation potential] Tensor-mode potential derivation: the assertion that vanishing f_R induces divergent structures in the tensor potential (used to reinterpret singular-branch peaks) is central to the ghost-free versus singular separation, but the explicit functional dependence of the potential on f_R is not displayed, preventing direct verification of the claimed divergence.
minor comments (2)
  1. [QNM results] The definition of the quality factor Q (whether Re(ω)/|Im(ω)| or an alternative convention) should be stated explicitly when reporting the range 0.9–1.9.
  2. [Figures on spectra] Figure captions for the probability and phase-shift spectra should indicate the numerical resolution and frequency sampling used to conclude the absence of narrow peaks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. The comments highlight areas where additional transparency strengthens the presentation, and we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and numerical methods section] Abstract and § on numerical methods: the central claim that no narrow real-axis peaks appear in the ghost-free branch rests on the relative-probability and phase-shift spectra together with the AIM/time-domain QNM extraction, yet the manuscript supplies no derivation details, convergence criteria, error estimates, or explicit checks against post-hoc parameter choices; these omissions are load-bearing for the reported absence of resonances and the Q ≃ 0.9–1.9 range.

    Authors: We agree that the numerical procedures require more explicit documentation to support the central claims. In the revised manuscript we have expanded the numerical methods section with: the explicit integral expressions and discretization schemes used for the relative-probability and phase-shift spectra; convergence criteria (integration tolerances, grid spacing, and shooting-method stopping conditions); quantitative error estimates obtained from repeated runs at different resolutions; and systematic checks confirming that the absence of narrow real-axis peaks and the reported Q range remain stable under reasonable variations of post-hoc parameters. These additions make the diagnostics reproducible and the conclusions robust. revision: yes

  2. Referee: [Tensor perturbation potential] Tensor-mode potential derivation: the assertion that vanishing f_R induces divergent structures in the tensor potential (used to reinterpret singular-branch peaks) is central to the ghost-free versus singular separation, but the explicit functional dependence of the potential on f_R is not displayed, preventing direct verification of the claimed divergence.

    Authors: We acknowledge the omission. The revised manuscript now displays the full analytic expression for the tensor perturbation potential, written explicitly in terms of f_R, the warp factor, and the scalar-field profiles. The divergent terms that appear when f_R → 0 are isolated and discussed, thereby allowing direct verification of the singular behavior and supporting the branch separation used in the reinterpretation of the peaks. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no circular reductions identified

full rationale

The paper constructs an explicit analytical two-scalar warped background, imposes the standard positivity condition f_R > 0 to isolate the ghost-free branch, and obtains the reported spectra and QNMs as direct numerical outputs from the resulting Schrödinger-like potential via AIM and SUSY-partner time-domain evolution. No load-bearing step equates a prediction or resonance diagnostic to a fitted parameter or prior self-citation by construction; the absence of narrow real-axis peaks and the broad character of the QNMs (negative imaginary parts, low Q) are computed results rather than definitional inputs. The separation into singular versus ghost-free branches follows from the sign of f_R without circular redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger populated from explicit statements in the abstract.

axioms (1)
  • domain assumption positivity of f_R separates ghost-free from singular branch
    Explicitly imposed to define the physically admissible region where effective gravitational coupling remains non-vanishing.

pith-pipeline@v0.9.1-grok · 5856 in / 1293 out tokens · 28823 ms · 2026-07-01T04:57:49.966016+00:00 · methodology

discussion (0)

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Works this paper leans on

145 extracted references · 9 canonical work pages · 3 internal anchors

  1. [1]

    B. P. Abbott et al. (LIGO Scientific, Virgo),GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers,Phys. Rev. X9, 031040 (2019), [arXiv:1811.12907]

    Pith/arXiv arXiv 2019

  2. [2]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, Virgo),GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo,Phys. Rev. X11, 021053 (2021), [arXiv:2010.14527]

    Pith/arXiv arXiv 2021

  3. [3]

    Abbott et al

    R. Abbott et al. (KAGRA, VIRGO, LIGO Scientific),GWTC-3: Compact Binary Coalescences Observed,Phys. Rev. X13, 041039 (2023), [arXiv:2111.03606]. – 24 –

    Pith/arXiv arXiv 2023

  4. [4]

    A. G. Abac et al. (LIGO Scientific, VIRGO, KAGRA),GWTC-4.0: Updating the Gravitational-Wave Transient Catalog, [arXiv:2508.18082]

    Pith/arXiv arXiv

  5. [5]

    B. P. Abbott et al. (LIGO Scientific, Virgo),Tests of general relativity with GW150914, Phys. Rev. Lett.116, 221101 (2016), [arXiv:1602.03841]

    Pith/arXiv arXiv 2016

  6. [6]

    B. P. Abbott et al. (LIGO Scientific, Virgo),Tests of General Relativity with GW170817, Phys. Rev. Lett.123, 011102 (2019), [arXiv:1811.00364]

    Pith/arXiv arXiv 2019

  7. [7]

    B. P. Abbott et al. (LIGO Scientific, Virgo),Tests of General Relativity with the Binary Black Hole Signals from GWTC-1,Phys. Rev. D100, 104036 (2019), [arXiv:1903.04467]

    Pith/arXiv arXiv 2019

  8. [8]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, Virgo),Tests of general relativity with binary black holes from the second LIGO-Virgo catalog,Phys. Rev. D103, 122002 (2021), [arXiv:2010.14529]

    Pith/arXiv arXiv 2021

  9. [9]

    Abbott et al

    R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA),Tests of General Relativity with GWTC-3, [arXiv:2112.06861]

    Pith/arXiv arXiv

  10. [10]

    A. G. Abac et al. (LIGO Scientific, Virgo, KAGRA),GW250114: Testing Hawking’s Area Law and the Kerr Nature of Black Holes,Phys. Rev. Lett.135, 111403 (2025), [arXiv:2509.08054]

    Pith/arXiv arXiv 2025

  11. [11]

    LIGO Scientific, Virgo, KAGRA Collaborations,Black Hole Spectroscopy and Tests of General Relativity with GW250114, [arXiv:2509.08099]

    Pith/arXiv arXiv

  12. [12]

    Guo,Standard siren cosmology with the LISA-Taiji network,Sci

    Z.-K. Guo,Standard siren cosmology with the LISA-Taiji network,Sci. China Phys. Mech. Astron.65, 210431 (2022)

    2022

  13. [13]

    K. D. Kokkotas and B. G. Schmidt,Quasinormal modes of stars and black holes,Living Rev. Rel.2, 2 (1999), [arXiv:gr-qc/9909058]

    Pith/arXiv arXiv 1999

  14. [14]

    H. P. Nollert,TOPICAL REVIEW: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars,Class. Quant. Grav.16, R159 (1999)

    1999

  15. [15]

    Berti, V

    E. Berti, V. Cardoso, and A. O. Starinets,Quasinormal modes of black holes and black branes,Class. Quant. Grav.26, 163001 (2009), [arXiv:0905.2975]

    Pith/arXiv arXiv 2009

  16. [16]

    R. A. Konoplya and A. Zhidenko,Quasinormal modes of black holes: From astrophysics to string theory,Rev. Mod. Phys.83, 793 (2011), [arXiv:1102.4014]

    Pith/arXiv arXiv 2011

  17. [17]

    Cardoso, E

    V. Cardoso, E. Franzin, and P. Pani,Is the gravitational-wave ringdown a probe of the event horizon? Phys. Rev. Lett.116, 171101 (2016), [arXiv:1602.07309]

    Pith/arXiv arXiv 2016

  18. [18]

    Jusufi et al.,Quasinormal modes, quasiperiodic oscillations, and the shadow of rotating regular black holes,Phys

    K. Jusufi et al.,Quasinormal modes, quasiperiodic oscillations, and the shadow of rotating regular black holes,Phys. Rev. D103, 024013 (2021), [arXiv:2008.08450]

    arXiv 2021

  19. [19]

    M. H. Y. Cheung et al.,Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea,Phys. Rev. Lett.128, 111103 (2022), [arXiv:2111.05415]

    arXiv 2022

  20. [20]

    Giesler, M

    M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky,Black Hole Ringdown: The Importance of Overtones,Phys. Rev. X9, 041060 (2019), [arXiv:1903.08284]

    arXiv 2019

  21. [21]

    C. V. Vishveshwara,Scattering of gravitational radiation by a Schwarzschild black-hole, Nature227, 936 (1970)

    1970

  22. [22]

    Nollert,About the significance of quasinormal modes of black holes,Phys

    H.-P. Nollert,About the significance of quasinormal modes of black holes,Phys. Rev. D53, 4397 (1996), [arXiv:gr-qc/9602032]

    Pith/arXiv arXiv 1996

  23. [23]

    Onozawa, T

    H. Onozawa, T. Mishima, T. Okamura, and H. Ishihara,Quasinormal modes of maximally charged black holes,Phys. Rev. D53, 7033 (1996), [arXiv:gr-qc/9603021]. – 25 –

    Pith/arXiv arXiv 1996

  24. [24]

    Andersson and H

    N. Andersson and H. Onozawa,Quasinormal modes of nearly extreme Reissner-Nordstrom black holes,Phys. Rev. D54, 7470 (1996), [arXiv:gr-qc/9607054]

    Pith/arXiv arXiv 1996

  25. [25]

    Onozawa,A Detailed study of quasinormal frequencies of the Kerr black hole,Phys

    H. Onozawa,A Detailed study of quasinormal frequencies of the Kerr black hole,Phys. Rev. D55, 3593 (1997), [arXiv:gr-qc/9610048]

    Pith/arXiv arXiv 1997

  26. [26]

    P. T. Leung et al.,Quasinormal modes of dirty black holes,Phys. Rev. Lett.78, 2894 (1997), [arXiv:gr-qc/9903031]

    Pith/arXiv arXiv 1997

  27. [27]

    Zhang, Y

    Y. Zhang, Y. X. Gui, and F. Li,Quasinormal modes of a Schwarzschild black hole surrounded by quintessence,Gen. Rel. Grav.39, 1003 (2007), [arXiv:gr-qc/0612010]

    Pith/arXiv arXiv 2007

  28. [28]

    A. S. Miranda, J. Morgan, and V. T. Zanchin,Quasinormal modes of plane-symmetric black holes according to the AdS/CFT correspondence,JHEP11, 030 (2008), [arXiv:0809.0297]

    Pith/arXiv arXiv 2008

  29. [29]

    G. T. Horowitz and V. E. Hubeny,Quasinormal modes of AdS black holes and the approach to thermal equilibrium,Phys. Rev. D62, 024027 (2000), [arXiv:hep-th/9909056]

    Pith/arXiv arXiv 2000

  30. [30]

    R. A. Konoplya,Gravitational quasinormal radiation of higher dimensional black holes,Phys. Rev. D68, 124017 (2003), [arXiv:hep-th/0309030]

    Pith/arXiv arXiv 2003

  31. [31]

    Kanti and K

    P. Kanti and K. Tamvakis,Quest for localized 4-D black holes in brane worlds,Phys. Rev. D 65, 084010 (2002), [arXiv:hep-th/0110298]

    Pith/arXiv arXiv 2002

  32. [32]

    Cardoso and J

    V. Cardoso and J. P. S. Lemos,Quasinormal modes of Schwarzschild anti-de Sitter black holes,Phys. Rev. D64, 084017 (2001), [arXiv:gr-qc/0105103]

    Pith/arXiv arXiv 2001

  33. [33]

    Toshmatov, Z

    B. Toshmatov, Z. Stuchlík, J. Schee, and B. Ahmedov,Quasinormal frequencies of black hole in the braneworld,Phys. Rev. D93, 124017 (2016), [arXiv:1605.02058]

    Pith/arXiv arXiv 2016

  34. [34]

    Demésy et al.,Dispersive perfectly matched layers and high-order absorbing boundary conditions for electromagnetic quasinormal modes,J

    G. Demésy et al.,Dispersive perfectly matched layers and high-order absorbing boundary conditions for electromagnetic quasinormal modes,J. Opt. Soc. Am. A40, 1947 (2023)

    1947

  35. [35]

    P. T. Leung, S. Y. Liu, and K. Young,Completeness and orthogonality of quasinormal modes in leaky optical cavities,Phys. Rev. A49, 3057 (1994)

    1994

  36. [36]

    J. Ren, S. Franke, and S. Hughes,Quasinormal Modes, Local Density of States, and Classical Purcell Factors for Coupled Loss-Gain Resonators,Phys. Rev. X11, 041020 (2021)

    2021

  37. [37]

    Nordstrom,On the possibility of unifying the electromagnetic and the gravitational fields, Phys

    G. Nordstrom,On the possibility of unifying the electromagnetic and the gravitational fields, Phys. Z.15, 504 (1914), [arXiv:physics/0702221]

    Pith/arXiv arXiv 1914

  38. [38]

    Kaluza,Zum Unitätsproblem der Physik,Sitzungsber

    T. Kaluza,Zum Unitätsproblem der Physik,Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1921, 966 (1921), [arXiv:1803.08616]

    Pith/arXiv arXiv 1921

  39. [39]

    Klein,Quantum Theory and Five-Dimensional Theory of Relativity,Z

    O. Klein,Quantum Theory and Five-Dimensional Theory of Relativity,Z. Phys.37, 895 (1926)

    1926

  40. [40]

    Akama,An Early Proposal of ‘Brane World’,Lect

    K. Akama,An Early Proposal of ‘Brane World’,Lect. Notes Phys.176, 267 (1982), [arXiv:hep-th/0001113]

    Pith/arXiv arXiv 1982

  41. [41]

    V. A. Rubakov and M. E. Shaposhnikov,Do We Live Inside a Domain Wall?,Phys. Lett. B 125, 136 (1983)

    1983

  42. [42]

    Arkani-Hamed, S

    N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali,The Hierarchy problem and new dimensions at a millimeter,Phys. Lett. B429, 263 (1998), [arXiv:hep-ph/9803315]

    Pith/arXiv arXiv 1998

  43. [43]

    Antoniadis, N

    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. B436, 257 (1998), [arXiv:hep-ph/9804398]. – 26 –

    Pith/arXiv arXiv 1998

  44. [44]

    Randall and R

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

    Pith/arXiv arXiv 1999

  45. [45]

    Randall and R

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

    Pith/arXiv arXiv 1999

  46. [46]

    DeWolfe, D

    O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch,Modeling the fifth-dimension with scalars and gravity,Phys. Rev. D62, 046008 (2000), [arXiv:hep-th/9909134]

    Pith/arXiv arXiv 2000

  47. [47]

    Gremm,Four-dimensional gravity on a thick domain wall,Phys

    M. Gremm,Four-dimensional gravity on a thick domain wall,Phys. Lett. B478, 434 (2000), [arXiv:hep-th/9912060]

    Pith/arXiv arXiv 2000

  48. [48]

    Csaki, J

    C. Csaki, J. Erlich, T. J. Hollowood, and Y. Shirman,Universal aspects of gravity localized on thick branes,Nucl. Phys. B581, 309 (2000), [arXiv:hep-th/0001033]

    Pith/arXiv arXiv 2000

  49. [49]

    Campos,Critical phenomena of thick branes in warped space-times,Phys

    A. Campos,Critical phenomena of thick branes in warped space-times,Phys. Rev. Lett.88, 141602 (2002), [arXiv:hep-th/0111207]

    Pith/arXiv arXiv 2002

  50. [50]

    Bazeia, C

    D. Bazeia, C. Furtado, and A. R. Gomes,Brane structure from scalar field in warped space-time,JCAP02, 002 (2004), [arXiv:hep-th/0308034]

    Pith/arXiv arXiv 2004

  51. [51]

    Dzhunushaliev and V

    V. Dzhunushaliev and V. Folomeev,Spinor brane,Gen. Rel. Grav.43, 1253 (2011), [arXiv:0909.2741]

    Pith/arXiv arXiv 2011

  52. [52]

    Dzhunushaliev and V

    V. Dzhunushaliev and V. Folomeev,Thick brane solutions supported by two spinor fields, Gen. Rel. Grav.44, 253 (2012), [arXiv:1104.2733]

    Pith/arXiv arXiv 2012

  53. [53]

    Kobayashi, K

    S. Kobayashi, K. Koyama, and J. Soda,Thick brane worlds and their stability,Phys. Rev. D 65, 064014 (2002), [arXiv:hep-th/0107025]

    Pith/arXiv arXiv 2002

  54. [54]

    Kehagias and K

    A. Kehagias and K. Tamvakis,Localized gravitons, gauge bosons and chiral fermions in smooth spaces generated by a bounce,Phys. Lett. B504, 38 (2001), [arXiv:hep-th/0010112]

    Pith/arXiv arXiv 2001

  55. [55]

    Melfo, N

    A. Melfo, N. Pantoja, and A. Skirzewski,Thick domain wall space-times with and without reflection symmetry,Phys. Rev. D67, 105003 (2003), [arXiv:gr-qc/0211081]

    Pith/arXiv arXiv 2003

  56. [56]

    Liu, X.-H

    Y.-X. Liu, X.-H. Zhang, L.-D. Zhang, and Y.-S. Duan,Localization of Matters on Pure Geometrical Thick Branes,JHEP02, 067 (2008), [arXiv:0708.0065]

    Pith/arXiv arXiv 2008

  57. [57]

    Y.-X. Liu, L. Zhao, and Y.-S. Duan,Localization of Fermions on a String-like Defect,JHEP 04, 097 (2007), [arXiv:hep-th/0701010]

    Pith/arXiv arXiv 2007

  58. [58]

    Shiromizu, K

    T. Shiromizu, K. Maeda, and M. Sasaki,The Einstein equation on the 3-brane world,Phys. Rev. D62, 024012 (2000), [arXiv:gr-qc/9910076]

    Pith/arXiv arXiv 2000

  59. [59]

    Tanaka,Classical black hole evaporation in Randall-Sundrum infinite brane world,Prog

    T. Tanaka,Classical black hole evaporation in Randall-Sundrum infinite brane world,Prog. Theor. Phys. Suppl.148, 307 (2003), [arXiv:gr-qc/0203082]

    Pith/arXiv arXiv 2003

  60. [60]

    Gregory,Braneworld black holes,Lect

    R. Gregory,Braneworld black holes,Lect. Notes Phys.769, 259 (2009), [arXiv:0804.2595]

    Pith/arXiv arXiv 2009

  61. [61]

    Jaman and K

    N. Jaman and K. Myrzakulov,Braneworld inflation with an effectiveα-attractor potential, Phys. Rev. D99, 103523 (2019), [arXiv:1807.07443]

    Pith/arXiv arXiv 2019

  62. [62]

    Adhikari, M

    R. Adhikari, M. R. Gangopadhyay, and Yogesh,Power Law Plateau Inflation Potential In The RS II Braneworld Evading Swampland Conjecture,Eur. Phys. J. C80, 899 (2020), [arXiv:2002.07061]

    arXiv 2020

  63. [63]

    Geng et al.,Information Transfer with a Gravitating Bath,SciPost Phys.10103 (2021), [arXiv:2012.04671]

    H. Geng et al.,Information Transfer with a Gravitating Bath,SciPost Phys.10103 (2021), [arXiv:2012.04671]. – 27 –

    arXiv 2021

  64. [64]

    H. Geng, S. Lüst, R. K. Mishra, and D. Wakeham,Holographic BCFTs and Communicating Black Holes,JHEP08003 (2021), [arXiv:2104.07039]

    arXiv 2021

  65. [65]

    H. Geng, L. Randall, and E. Swanson,BCFT in a black hole background: an analytical holographic model,JHEP12, 056 (2022), [arXiv:2209.02074]

    arXiv 2022

  66. [66]

    Bhattacharya et al.,Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model,JHEP05, 135 (2021), [arXiv:2103.15852]

    A. Bhattacharya et al.,Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model,JHEP05, 135 (2021), [arXiv:2103.15852]

    arXiv 2021

  67. [67]

    Dzhunushaliev, V

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

    Pith/arXiv arXiv 2010

  68. [68]

    Maartens and K

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

    Pith/arXiv arXiv 2010

  69. [69]

    Herrera-Aguilar et al.,Aspects of thick brane worlds: 4D gravity localization, smoothness, and mass gap,Mod

    A. Herrera-Aguilar et al.,Aspects of thick brane worlds: 4D gravity localization, smoothness, and mass gap,Mod. Phys. Lett. A25, 2089 (2010)

    2089

  70. [70]

    Liu,Introduction to Extra Dimensions and Thick Braneworlds, [arXiv:1707.08541]

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

    Pith/arXiv arXiv

  71. [71]

    D. V. Ahluwalia et al.,Mass dimension one fermions: Constructing darkness,Phys. Rept. 967, 1 (2022), [arXiv:2205.04754]

    arXiv 2022

  72. [72]

    Guo, Y.-T

    H. Guo, Y.-T. Lu, C.-L. Wang and Y. Sun,Localization of scalar field on the brane-world by coupling with gravity, JHEP06, 114 (2024), doi:10.1007/JHEP06(2024)114 [arXiv:2310.01451 [hep-th]]

    work page doi:10.1007/jhep06(2024)114 2024

  73. [73]

    P. M. Sá,Triple unification of inflation, dark energy, and dark matter in two-scalar-field cosmology, Phys. Rev. D102, 103519 (2020), doi:10.1103/PhysRevD.102.103519 [arXiv:2007.07109 [gr-qc]]

    work page doi:10.1103/physrevd.102.103519 2020

  74. [74]

    Luongo, T

    O. Luongo, T. Mengoni and P. M. Sá,Gravitational waves from two scalar fields unifying the dark sector with inflation, Phys. Rev. D113, 023541 (2026), doi:10.1103/pj2r-gnd6

    work page doi:10.1103/pj2r-gnd6 2026

  75. [75]

    C. A. S. Almeida, M. M. Ferreira, A. R. Gomes, and R. Casana,Fermion localization and resonances on two-field thick branes,Phys. Rev. D79, 125022 (2009), [arXiv:0901.3543]

    Pith/arXiv arXiv 2009

  76. [76]

    W. T. Cruz, L. J. S. Sousa, R. V. Maluf, and C. A. S. Almeida,Graviton resonances on two-field thick branes,Phys. Lett. B730, 314 (2014), [arXiv:1310.4085]

    Pith/arXiv arXiv 2014

  77. [77]

    Bulk matter fields on two-field thick branes

    C.-E. Fu, Y.-X. Liu and H. Guo,Bulk matter fields on two-field thick branes, Phys. Rev. D 84, 044036 (2011), doi:10.1103/PhysRevD.84.044036 [arXiv:1101.0336 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.044036 2011

  78. [78]

    Bazeia and A.R

    D. Bazeia and A.R. Gomes,Bloch brane,JHEP05(2004) 012 [hep-th/0403141]

    Pith/arXiv arXiv 2004

  79. [79]

    Hybrid Bloch Brane

    D. Bazeia, E. E. M. Lima and L. Losano,Hybrid Bloch Brane, Eur. Phys. J. C77, 127 (2017), doi:10.1140/epjc/s10052-017-4701-4 [arXiv:1611.09314 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-017-4701-4 2017

  80. [80]

    T. P. Sotiriou and V. Faraoni,f(R)theories of gravity,Rev. Mod. Phys.82, 451 (2010), [arXiv:0805.1726]

    Pith/arXiv arXiv 2010

Showing first 80 references.