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Radial spectra and dynamical signatures of excited boson stars

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The first zero of the fundamental radial mode coincides with the first critical point of the equilibrium branch for every boson star configuration examined.

desk verdict Regular variable method is a genuine advance and the n=0..3 zero-crossing data look credible, but the "all branches" claim outruns the displayed data and "within numerical resolution" needs quantification. read the letter →

arxiv 2608.06067 v1 pith:VBVE5SDN submitted 2026-08-06 gr-qc

Chen-Hao Hao , Wen-Di Guo , Qin Tan , Jieci Wang This is my paper · ORCID
classification gr-qc PACS 04.40.-b
keywords bosonstarsexcitedstatesradialperturbationsstabilityturningpointsEinstein–Klein–Gordonsystemself-interactingscalarfieldeigenvalueproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that for spherically symmetric boson stars, including excited configurations whose scalar profile has radial nodes, the lowest radial perturbation mode crosses zero at exactly the same point along the equilibrium branch where the ADM mass, Noether charge, and binding energy reach their first simultaneous extremum. To get there, the authors reformulate the linearized Einstein–Klein–Gordon pulsation equations in additive variables that remain finite at the nodes of the background field, so the eigenvalue problem can be integrated directly through nodeful backgrounds. They verify the coincidence for ground and excited branches up to seven nodes across several values of the quartic self-interaction strength, and they further show that the fundamental radial eigenvalue evaluated at the thresholds identified in nonlinear spherical evolutions obeys a simple power law in node number and self-interaction coupling. A sympathetic reader would care because the result connects a spectral stability diagnostic with a purely equilibrium turning-point criterion, making stability boundaries of boson stars easier to locate and opening a regular perturbative window onto excited configurations.

What carries the argument

The load-bearing object is the pair of regular additive perturbation variables $f \equiv \varphi_0\,\delta\varphi_1$ and $g \equiv -\varphi_0\,\xi/\omega$, which replace the conventional relative variables $\delta\varphi_1$, $\delta\varphi_2$ that contain inverse powers of $\varphi_0$. In these variables the linearized Einstein–Klein–Gordon system closes as a first-order system in $(f, f', g, g', \mathcal{N})$ with no singular terms at the background nodes, so ordinary shooting can integrate through the nodes of any excited configuration. The eigenvalue $\chi^2$ is then extracted with a two-parameter shooting ($\chi^2$ and $\zeta_1$) and outer boundary conditions on $f$, $\delta q$, and $\Lambda$, with a zero-mode diagnostic based on the rank of the $3\times 2$ boundary map $\mathcal{C}(\chi^2; r_{\mathrm{out}})$ and its singular-value ratio $\epsilon$.

What would settle it

Take the $n=1$, $\eta=28$ branch near the predicted crossing $\omega\approx 0.858$; perturb a configuration with $\omega$ just above and just below that value and evolve it spherically. If the onset of instability does not occur at the crossing, or if an independent integration of the original singular perturbation equations puts the zero at a different $\omega$, the central claim is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the constrained fundamental radial eigenvalue $\chi^2_{0,n}$ of an excited boson star branch crosses zero at the same background frequency as the first simultaneous extremum of the ADM mass, Noether charge, and binding energy. Previously the correspondence between radial mode zero and turning point was established only for the nodeless branch; the obstacle for excited states was that standard perturbation variables become singular at the nodes of the background scalar field. By working with regular additive variables $f$ and $g$, the authors obtain a closed first-order system with no inverse powers of the background field, allowing a shooting method to follow the lowest radial mode continuously along fixed-node branches. For every branch examined ($n=0$ through $n=7$, with quartic couplings $\eta=0$, $7.9577$, $28$, $58$), the zero crossing coincides with the critical point within numerical resolution. Additionally, evaluating the radial eigenvalue at the self-interaction thresholds reported in nonlinear spherical evolutions yields a power-law scaling $\chi^2_{0,n} \propto \eta\, n^{p}$, with the same scaling holding for higher-node configurations not used in the fit.

Load-bearing premise

The result stands only if the reformulated $f,g$ perturbation equations together with the outer boundary conditions reproduce the true physical radial spectrum without introducing spurious or gauge zero modes, and if the rank condition truly distinguishes a zero mode from a near-zero mode.

Editorial extensions

If this is right

  • If the coincidence holds generally, the radial stability boundary for a fixed-node branch can be located from the equilibrium curves alone, without a separate eigenvalue solve.
  • The regular treatment of nodes extends standard radial-mode technology to nodeful solitonic backgrounds, where the old relative variables fail.
  • The empirical relation $\chi^2_{0,n}/\mu^2 \simeq C\, \eta\, n^{p}$ at the nonlinear thresholds can serve as a quick estimator for the coupling at which excited configurations become dynamically long lived, before running full evolutions.
  • The observed softening of the fundamental mode with increasing node number implies that higher excited branches are more easily destabilized, consistent with their transient role in dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the zero-crossing/turning-point correspondence survives outside the tested parameter range, it supports a general stability-exchange principle for boson stars that might also apply to rotating or charged variants, where critical points are still defined by mass/charge extrema.
  • The empirical power law might be derivable from a WKB estimate of the effective potential between nodes; an analytic derivation would turn it into a predictive formula.
  • A direct next test is to compute the lowest nonradial mode on the same branches; if the radial zero remains the first mode to destabilize, the correspondence would extend to higher multipoles, with possible observational signatures in gravitational-wave echoes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a regular radial perturbation framework for spherically symmetric equilibrium boson stars with radial nodes. Introducing additive variables f and g instead of the conventional relative variables, the authors write a closed first-order system for the metric and scalar perturbations that is regular at the zeros of the background scalar field, integrate it through the nodes of excited configurations, and compute the lowest radial eigenvalue chi^2_{0,n} along fixed-node branches. For n=0,...,3 and several values of the quartic self-interaction coupling, they report a zero crossing of chi^2_{0,n} and claim that this zero coincides, within numerical resolution, with the first critical point of the ADM mass, Noether charge, and binding energy. They further evaluate chi^2_{0,n} at the nonlinear stability thresholds of Brito et al. and fit an empirical scaling relation in the node number and coupling, testing it on n=5,...,7.

Significance. If the central claim is correct, the paper provides a useful technical advance: the regular variable formulation permits a direct integration of the radial perturbation equations through the nodes of excited backgrounds, and it ties the constrained radial zero mode to equilibrium turning points for a class of configurations. The perturbation system is written out in full, and the tables for n=0,...,3 give a substantial amount of reproducible numerical data. The empirical correlation with nonlinear threshold models is interesting and, unusually, is tested on configurations not used in the fit. The result is a plausible and potentially useful diagnostic, but the breadth of the central claim currently exceeds the evidence displayed in the manuscript.

major comments (3)
  1. [Sec. IV.A, paragraph after Fig. 3; Tables II-V] The abstract and conclusions state that 'for all branches examined' the first zero of the fundamental radial eigenvalue coincides with the first critical point, but the only displayed zero-crossing data are for n=0,...,3 in Tables II-V and Fig. 3. The sentence 'We have extended the calculation to higher excited states up to n=7 and found the same behavior in all cases examined; these results are therefore not displayed separately' is not sufficient support for the central claim as worded. At minimum, the paper should provide a table or figure listing the zero-crossing frequencies and the corresponding critical frequencies for n=4,...,7, or the abstract and conclusions should be restricted to the branches for which data are shown.
  2. [Sec. III.D and Sec. IV.A] The phrase 'within numerical resolution' is used in the abstract and conclusions but is never quantified. No grid-convergence study, no statement of the shooting accuracy, no values of r_out/R99 used for the plateau, and no thresholds for the singular-value ratio epsilon(0;r_out) in Eq. (59) are given. The tabulated zero frequencies and critical frequencies agree to four or five significant digits, but without an error estimate or a bracket size for the zero crossing, the coincidence claim cannot be assessed quantitatively. Please provide a quantitative measure of the numerical uncertainty in the zero-crossing location for representative cases.
  3. [Sec. III.C, paragraph on eigenvalue search] The identification of chi^2_{0,n} as the fundamental (lowest) radial eigenvalue along the entire branch is asserted rather than demonstrated. The text states that the 'lowest real root' is located on a reference background and followed continuously, with selected anchor points rescanned to verify that no lower root was missed, but no details are given on the scanning density, the mode spacing, or how mode crossings or avoided crossings were excluded. Since the central claim concerns the 'first zero of the constrained fundamental radial eigenvalue,' the completeness of this search is load-bearing. Please provide explicit evidence, such as the scanning interval and the anchor-point rescan results, that no lower real root exists in the omega range traversed.
minor comments (5)
  1. [Eq. (28) and Sec. IV.A] Because Eq. (28) follows from the first law, the extrema of M, Q, and E_b coincide by construction for any regular parameter along the branch. The wording 'independently identified from the ADM mass, the Noether charge, and the binding energy' therefore overstates the independence of these three diagnostics; consider rephrasing to make clear that the non-trivial comparison is between the chi^2 zero and this common equilibrium critical point.
  2. [Table I] The notation '751.4556e-4' in Table I is ambiguous: it should be clear that the first column entry is eta=75 and the second is chi^2=1.4556e-4. Please separate the columns or use explicit notation.
  3. [Eqs. (60)-(61)] The empirical fits are reported with R^2 values but without uncertainties on the fitted prefactors and exponents; given that only four data points are used for a two-parameter fit, these uncertainties should be quoted if the scaling is presented as predictive.
  4. [Fig. 3 caption] The caption states that dashed segments are guides to the eye, but the zero crossing is read from interpolation between solid data points. Please state explicitly how the zero-crossing frequency was extracted from the plotted data.
  5. [Reference [42]] The DOI '10.1103/2dhs-phl4' appears malformed; please verify the bibliographic data.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral zero and the equilibrium critical point are computed by independent numerical procedures, and the empirical scaling is explicitly phenomenological.

full rationale

The central assertion—that the first zero of the fundamental radial eigenvalue coincides with the first critical point of M, Q, and Eb—is a numerical comparison between two independently computed objects: the eigenvalue chi2_0,n from the regular perturbation system (Sec. III) and critical frequencies obtained from equilibrium branch quantities (Sec. II). Neither is defined in terms of the other; the zero-mode rank diagnostic (58) is a spectral statement, while the critical points are located from equilibrium data. The first-law identity (26)-(28) makes the extrema of M, Q, and Eb mutually equivalent, so calling them 'independent' diagnostics is an overstatement, but that equivalence does not feed into the spectral calculation and is not circular. The empirical scaling (60)-(61) is explicitly a fit to threshold data and is labelled phenomenological; it is not used to derive the coincidence claim. Self-citations (e.g., [17], [20], [33], [34]) appear only in background references and are not load-bearing for the derivation. The absence of displayed n=4..7 zero-crossing data is a support/completeness concern, not circularity. No circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central numerical claim relies on standard equilibrium and perturbation equations plus the new regular-variable ansatz. The only fitted quantities are the parameters of the empirical scaling, which are auxiliary to the main claim. No new entities are introduced.

free parameters (4)
  • Empirical prefactor for omega=0.90 scaling = 1.07991e-2
    Fitted to chi^2 threshold data for n=1..4 in Eq. (60); central claim does not depend on it.
  • Empirical exponent for omega=0.90 scaling = 0.237199
    Fitted in Eq. (60).
  • Empirical prefactor for omega=0.92 scaling = 8.98299e-3
    Fitted in Eq. (61).
  • Empirical exponent for omega=0.92 scaling = 0.303924
    Fitted in Eq. (61).
assumptions (5)
  • domain assumption Einstein-Klein-Gordon system with quartic potential V = mu^2|phi|^2 + eta|phi|^4 is the correct model for boson stars
    Used from Eq. (1)-(2); standard model in the field, not derived in this paper.
  • domain assumption Spherical symmetry and harmonic time dependence of the background scalar field
    Assumed in Eq. (4) and (11); restricts to spherically symmetric sector.
  • standard math The first law dM/dp = omega dQ/dp holds along equilibrium branches
    Used in Eq. (26) to equate critical points of M, Q, E_b; cited to [7,50].
  • domain assumption The sign of the lowest radial eigenvalue chi^2_0,n determines stability in the spherical sector
    Stated in Eq. (54); standard criterion for radial stability.
  • ad hoc to paper The additive variables f,g form a complete regular basis for the perturbation space such that no modes are lost or introduced
    Core of the new method; not proven, only asserted in Sec. III A.

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Cite this review

Pith. "Pith review of Radial spectra and dynamical signatures of excited boson stars." pith.science (2026). https://pith.science/paper/VBVE5SDN

@misc{pith2026260806067,
  author       = {Pith},
  title        = {Pith review of: Radial spectra and dynamical signatures of excited boson stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBVE5SDN}},
  note         = {Machine review of arXiv:2608.06067}
}
read the original abstract

We compute the lowest radial mode of spherically symmetric boson stars along equilibrium branches with a fixed number of radial nodes, considering both mini boson stars and quartically self-interacting models. By reformulating the pulsation equations in additive variables that remain regular at the zeros of the background scalar field, the eigenvalue problem can be integrated directly through the nodes of excited configurations. For all branches examined, the first zero of the constrained fundamental radial eigenvalue coincides, within numerical resolution, with the first simultaneous critical point of the Arnowitt--Deser--Misner (ADM) mass, Noether charge, and binding energy. We further evaluate the radial eigenvalue for the threshold models identified in nonlinear spherical evolutions of excited boson stars and find a simple empirical correlation with the node number and self-interaction strength. Our results provide a regular perturbative framework for excited boson stars and clarify the relation between constrained radial modes, equilibrium critical points, and nonlinear stability diagnostics.

Figures

Figures reproduced from arXiv: 2608.06067 by the authors.

Figure 1
Figure 1. FIG. 1. Background properties of boson stars with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. shows the binding energy 𝐸𝑏 under three dif￾ferent 𝜂 for 𝑛 = 0, . . . , 3. Higher-node branches exhibit qualitatively similar background behavior. n = 0 n = 1 n = 2 n = 3 n = 4 0.75 0.80 0.85 0.90 0.95 1.00 0.00 1.00 2.00 3.00 ω M ω = 0.8530 ω = 0.8810 ω = 0.8880 ω = 0.8916 ω = 0.8940 n = 0 n = 1 n = 2 n = 3 n = 4 0.75 0.80 0.85 0.90 0.95 1.00 0.00 1.00 2.00 3.00 ω Q ω = 0.8530 ω = 0.8810 ω = 0.8880 ω = 0.8916 ω = 0… view at source ↗
Figure 3
Figure 3. shows 𝜒 2 0,𝑛 as a function of the background fre￾quency 𝜔 for 𝜂 = 0, 7.9577, 28, and 58. For all cases con￾sidered, the fundamental eigenvalue changes sign along the branch, separating a region with 𝜒 2 0,𝑛 < 0 from one with 𝜒 2 0,𝑛 > 0. Thus, each fixed-node branch exhibits a radial zero mode that marks the boundary between radially unstable and oscillatory configurations within the spherically symmetric perturbat… view at source ↗

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Works this paper leans on

50 extracted references · 13 canonical work pages

  1. [1]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116(2016) no.6, 061102 doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]]

    arXiv 2016

  2. [2]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. X9(2019) no.3, 031040 doi:10.1103/PhysRevX.9.031040 [arXiv:1811.12907 [astro-ph.HE]]

    arXiv 2019

  3. [3]

    Akiyamaet al.[Event Horizon Telescope], Astrophys

    K. Akiyamaet al.[Event Horizon Telescope], Astrophys. J. Lett.875(2019), L1 doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238 [astro-ph.GA]]

    arXiv 2019

  4. [4]

    Akiyamaet al.[Event Horizon Telescope], Astro- phys

    K. Akiyamaet al.[Event Horizon Telescope], Astro- phys. J. Lett.930(2022) no.2, L12 doi:10.3847/2041- 8213/ac6674 [arXiv:2311.08680 [astro-ph.HE]]

    arXiv 2022

  5. [5]

    D. J. Kaup, Phys. Rev.172(1968), 1331-1342 doi:10.1103/PhysRev.172.1331

    doi:10.1103/physrev.172.1331 1968

  6. [6]

    Ruffini and S

    R. Ruffini and S. Bonazzola, Phys. Rev.187(1969), 1767-1783 doi:10.1103/PhysRev.187.1767

    doi:10.1103/physrev.187.1767 1969

  7. [7]

    F. E. Schunck and E. W. Mielke, Class. Quant. Grav. 20(2003), R301-R356 doi:10.1088/0264-9381/20/20/201 [arXiv:0801.0307 [astro-ph]]

    arXiv 2003

  8. [8]

    S. L. Liebling and C. Palenzuela, Living Rev. Rel.26(2023) no.1, 1 doi:10.1007/s41114-023-00043-4 [arXiv:1202.5809 [gr-qc]]

    arXiv 2023

Show all 50 references
  1. [9]

    F. H. Vincent, Z. Meliani, P. Grandclement, E. Gour- goulhon and O. Straub, Class. Quant. Grav.33(2016) no.10, 105015 doi:10.1088/0264-9381/33/10/105015 [arXiv:1510.04170 [gr-qc]]

    2016 arXiv

  2. [10]

    Olivares, Z

    H. Olivares, Z. Younsi, C. M. Fromm, M. De Laurentis, O. Porth, Y. Mizuno, H. Falcke, M. Kramer and L. Rez- zolla, Mon. Not. Roy. Astron. Soc.497(2020) no.1, 521- 535 doi:10.1093/mnras/staa1878 [arXiv:1809.08682 [gr- qc]]

    2020 arXiv

  3. [11]

    Palenzuela, P

    C. Palenzuela, P. Pani, M. Bezares, V. Cardoso, L. Lehner and S. Liebling, Phys. Rev. D96 (2017) no.10, 104058 doi:10.1103/PhysRevD.96.104058 [arXiv:1710.09432 [gr-qc]]

    2017 arXiv

  4. [12]

    Cardoso and P

    V. Cardoso and P. Pani, Living Rev. Rel.22(2019) no.1, 4 doi:10.1007/s41114-019-0020-4 [arXiv:1904.05363 [gr- qc]]

    2019 arXiv

  5. [13]

    J. W. Lee and I. g. Koh, Phys. Rev. D53(1996), 2236-2239 doi:10.1103/PhysRevD.53.2236 [arXiv:hep- ph/9507385 [hep-ph]]

    1996

  6. [14]

    Arbey, J

    A. Arbey, J. Lesgourgues and P. Salati, Phys. Rev. D68(2003), 023511 doi:10.1103/PhysRevD.68.023511 [arXiv:astro-ph/0301533 [astro-ph]]

    2003 arXiv

  7. [15]

    Liang, J

    C. Liang, J. R. Ren, S. X. Sun and Y. Q. Wang, JHEP02(2023), 249 doi:10.1007/JHEP02(2023)249 [arXiv:2207.11147 [gr-qc]]

    2023 arXiv

  8. [16]

    T. X. Ma, C. Liang, J. Yang and Y. Q. Wang, Phys. Rev. D108(2023) no.10, 104011 doi:10.1103/PhysRevD.108.104011 [arXiv:2304.08019 [gr-qc]]

    2023 arXiv

  9. [17]

    C. H. Hao, Y. Q. Wang and J. Wang, Phys. Lett. B 873(2026), 140153 doi:10.1016/j.physletb.2026.140153 [arXiv:2509.19822 [gr-qc]]

    2026

  10. [18]

    L. X. Huang, K. Yang and Y. Q. Wang, [arXiv:2607.02017 [gr-qc]]

    arXiv

  11. [19]

    S. X. Sun, L. Zhao and Y. Q. Wang, JHEP08(2023), 152 doi:10.1007/JHEP08(2023)152 [arXiv:2210.09265 [gr- qc]]

    2023 arXiv

  12. [20]

    C. H. Hao, X. Su and Y. Q. Wang, Eur. Phys. J. C 85(2025) no.3, 348 doi:10.1140/epjc/s10052-025-13937-0 [arXiv:2404.11002 [gr-qc]]

    2025 arXiv

  13. [21]

    C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha and N. Sanchis-Gual, JCAP04 (2021), 051 doi:10.1088/1475-7516/2021/04/051 [arXiv:2102.01703 [gr-qc]]

    2021 arXiv

  14. [22]

    Herdeiro, E

    C. Herdeiro, E. Radu, E. dos Santos Costa Filho and N. Sanchis-Gual, [arXiv:2605.13965 [gr-qc]]

    arXiv

  15. [23]

    Gleiser, Phys

    M. Gleiser, Phys. Rev. D38(1988), 2376 [er- ratum: Phys. Rev. D39(1989) no.4, 1257] doi:10.1103/PhysRevD.38.2376

    1988 doi

  16. [24]

    Jetzer, Nucl

    P. Jetzer, Nucl. Phys. B316(1989), 411-428 doi:10.1016/0550-3213(89)90038-2

    1989 doi

  17. [25]

    Gleiser and R

    M. Gleiser and R. Watkins, Nucl. Phys. B319(1989), 733-746 doi:10.1016/0550-3213(89)90627-5

    1989 doi

  18. [26]

    T. D. Lee and Y. Pang, Nucl. Phys. B315(1989), 477 doi:10.1016/0550-3213(89)90365-9

    1989 doi

  19. [27]

    Alcubierre, J

    M. Alcubierre, J. Barranco, A. Bernal, J. C. Degol- lado, A. Diez-Tejedor, M. Megevand, D. Núñez and O. Sarbach, Class. Quant. Grav.38(2021) no.17, 174001 doi:10.1088/1361-6382/ac0160 [arXiv:2103.15012 [gr-qc]]

    2021 arXiv

  20. [28]

    Kojima, S

    Y. Kojima, S. Yoshida and T. Futamase, Prog. Theor. Phys.86(1991), 401-410 doi:10.1143/PTP.86.401

    1991 doi

  21. [29]

    Yoshida, Y

    S. Yoshida, Y. Eriguchi and T. Futamase, Phys. Rev. D 50(1994), 6235-6246 doi:10.1103/PhysRevD.50.6235

    1994 doi

  22. [30]

    C. F. B. Macedo, P. Pani, V. Cardoso and L. C. B. Crispino, Phys. Rev. D88(2013) no.6, 064046 doi:10.1103/PhysRevD.88.064046 [arXiv:1307.4812 [gr- qc]]

    2013 arXiv

  23. [31]

    Visinelli, Int

    L. Visinelli, Int. J. Mod. Phys. D30(2021) no.15, 2130006 doi:10.1142/S0218271821300068 [arXiv:2109.05481 [gr-qc]]

    2021 arXiv

  24. [32]

    Cardoso, W

    V. Cardoso, W. D. Guo, C. F. B. Macedo and P. Pani, Mon. Not. Roy. Astron. Soc.503(2021) no.1, 563-573 doi:10.1093/mnras/stab404 [arXiv:2009.07287 [gr-qc]]

    2021 arXiv

  25. [33]

    Q. Tan, Y. P. Zhang, W. D. Guo, J. Chen, C. C. Zhu and Y. X. Liu, Eur. Phys. J. C83(2023) no.1, 84 doi:10.1140/epjc/s10052-022-11164-5 [arXiv:2203.00277 [hep-th]]

    2023 arXiv

  26. [34]

    Q. Tan, W. Deng, S. Long and J. Jing, JCAP 05(2025), 044 doi:10.1088/1475-7516/2025/05/044 [arXiv:2409.17760 [gr-qc]]

    2025

  27. [35]

    Seidel and W

    E. Seidel and W. M. Suen, Phys. Rev. D42(1990), 384- 403 doi:10.1103/PhysRevD.42.384

    1990 doi

  28. [36]

    Balakrishna, E

    J. Balakrishna, E. Seidel and W. M. Suen, Phys. Rev. D58(1998), 104004 doi:10.1103/PhysRevD.58.104004 [arXiv:gr-qc/9712064 [gr-qc]]

    1998 arXiv

  29. [37]

    S. H. Hawley and M. W. Choptuik, Phys. Rev. D62(2000), 104024 doi:10.1103/PhysRevD.62.104024 [arXiv:gr-qc/0007039 [gr-qc]]

    2000 arXiv

  30. [38]

    Di Giovanni, D

    F. Di Giovanni, D. Guerra, S. Albanesi, M. Miravet- Tenés and D. Tseneklidou, Phys. Rev. D106 (2022) no.8, 084013 doi:10.1103/PhysRevD.106.084013 [arXiv:2207.06174 [gr-qc]]

    2022 arXiv

  31. [39]

    J. E. Nyhan and B. Kain, Phys. Rev. D105 (2022) no.12, 123016 doi:10.1103/PhysRevD.105.123016 [arXiv:2206.07715 [gr-qc]]

    2022 arXiv

  32. [40]

    Y. P. Zhang, Y. B. Zeng, Y. Q. Wang, S. W. Wei and Y. X. Liu, Eur. Phys. J. C82(2022) no.9, 809 12 doi:10.1140/epjc/s10052-022-10743-w [arXiv:2201.01498 [gr-qc]]

    2022 arXiv

  33. [41]

    Y. P. Zhang, S. X. Sun, Y. Q. Wang, S. W. Wei, P. Laguna and Y. X. Liu, Phys. Rev. Res.6(2024) no.3, 033187 doi:10.1103/PhysRevResearch.6.033187 [arXiv:2310.01178 [gr-qc]]

    2024 arXiv

  34. [42]

    B. X. Ge, E. A. Lim, U. Sperhake, T. Evstafyeva, D. Cors, E. de Jong, R. Croft and T. Helfer, Phys. Rev. D112(2025) no.12, 124080 doi:10.1103/2dhs-phl4 [arXiv:2410.23839 [gr-qc]]

    2025

  35. [43]

    F. V. Kusmartsev, E. W. Mielke and F. E. Schunck, Phys. Rev. D43(1991), 3895-3901 doi:10.1103/PhysRevD.43.3895 [arXiv:0810.0696 [astro- ph]]

    1991 arXiv

  36. [44]

    Kleihaus, J

    B. Kleihaus, J. Kunz and S. Schneider, Phys. Rev. D85(2012), 024045 doi:10.1103/PhysRevD.85.024045 [arXiv:1109.5858 [gr-qc]]

    2012 arXiv

  37. [45]

    Kain, Phys

    B. Kain, Phys. Rev. D103(2021) no.12, 123003 doi:10.1103/PhysRevD.103.123003 [arXiv:2106.01740 [gr-qc]]

    2021 arXiv

  38. [46]

    N. M. Santos, C. L. Benone and C. A. R. Herdeiro, JCAP 06(2024), 068 doi:10.1088/1475-7516/2024/06/068 [arXiv:2404.07257 [gr-qc]]

    2024 arXiv

  39. [47]

    Sanchis-Gual, C

    N. Sanchis-Gual, C. Herdeiro and E. Radu, Class. Quant. Grav.39(2022) no.6, 064001 doi:10.1088/1361- 6382/ac4b9b [arXiv:2110.03000 [gr-qc]]

    2022 arXiv

  40. [48]

    Brito, C

    M. Brito, C. Herdeiro, E. Radu, N. Sanchis-Gual and M. Zilhão, Phys. Rev. D107(2023) no.8, 084022 doi:10.1103/PhysRevD.107.084022 [arXiv:2302.08900 [gr-qc]]

    2023 arXiv

  41. [49]

    Jetzer, Phys

    P. Jetzer, Phys. Lett. B222(1989), 447-452 doi:10.1016/0370-2693(89)90342-0

    1989 doi

  42. [50]

    C. A. R. Herdeiro, J. Kunz, I. Perapechka, E. Radu and Y. Shnir, Phys. Rev. D103(2021) no.6, 065009 doi:10.1103/PhysRevD.103.065009 [arXiv:2101.06442 [gr-qc]]

    2021 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.