Pith. sign in

REVIEW 5 major objections 5 minor 62 references

Effects of gravitational lensing on neutrino oscillation in Hu-Sawicki f(R) gravity

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper argues that gravitational lensing in the Hu-Sawicki f(R) gravity model imprints a calculable dependence on the model parameter $\lambda$, the neutrino mass hierarchy, and the absolute value of the lightest neutrino mass onto…

desk verdict A promising application to Hu-Sawicki f(R) is undermined by an inverted closest-approach relation in the non-radial phase derivation, so the numerical results as plotted do not yet support the claims. read the letter →

arxiv 2506.23905 v4 pith:PNCJJAEY submitted 2025-06-30 hep-ph gr-qc

Ya-Ru Wang , Ze-Wen Li , Shu-Jun Rong This is my paper
classification hep-phgr-qc PACS 14.60.Pq95.30.Sf
keywords neutrinooscillationsgravitationallensingHu-Sawickif(R)gravitymodifiedmasshierarchylightestflavortransitionprobabilitiesweak-fieldapproximation
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 aims to show that when neutrinos are gravitationally lensed by a compact object described by Hu-Sawicki f(R) gravity, a modified-gravity model whose metric adds a $\lambda r^2$ term to Schwarzschild, the flavor-oscillation probabilities carry a calculable imprint of $\lambda$, alongside the neutrino mass hierarchy and the absolute value of the lightest neutrino mass. The authors derive the oscillation phase in covariant form for both radial and lensed non-radial trajectories, then compute two- and three-flavor transition probabilities in the weak-field approximation. They find that inverted mass ordering gives larger amplitudes and shorter periods, that a nonzero lightest neutrino mass changes the oscillation period, and that the $\lambda$ dependence is visible only once lensing is included. They also present a strong-field numerical extension which they interpret as amplifying the sensitivity to $\lambda$. If the calculation is right, lensed neutrinos from compact astrophysical objects could serve as a complementary probe of modified gravity and of neutrino mass parameters.

What carries the argument

The central object is the covariant phase integral $\Phi_k=\int p^{(k)}_\mu dx^\mu$ evaluated with the canonical momentum $p^{(k)}_\mu=m_k g_{\mu\nu}dx^\nu/ds$ for the $k$-th neutrino mass eigenstate. This phase carries the calculation: for non-radial (lensed) propagation it becomes $\Phi_k=(m_k^2/2E_0)$ times an integral over the Hu-Sawicki metric functions $A=1/B=1-2M/r+\lambda r^2$ and the impact parameter $b$, and the weak-field expansion of that integral produces the explicit lensing-modified probabilities. The companion piece is the lensing geometry itself, encoded in the deflection angle $\delta=4M/b+5\pi M^2/(2b^2)+4\pi\lambda/b^3$, which fixes the impact parameters $b_1,b_2$ whose two paths are superposed, with normalization, to give the two- and three-flavor transition probabilities.

What would settle it

Repeat the weak-field two-flavor probability calculation using the alternative phase convention cited in the paper's own Section II.B.1 (Ref [56]) for the same Sun-Earth parameters and λ values; if the resulting curves no longer depend on λ, the claimed dependence is an artifact of the chosen phase convention rather than a property of the spacetime.

Watch

Extended reading notes

Core claim

In the spacetime of the Hu-Sawicki f(R) gravity model, the metric is taken as $ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ with $A(r)=1/B(r)=1-2M/r+\lambda r^2$. For a neutrino on a lensed trajectory in the equatorial plane, the covariant phase $\Phi_k=\int p^{(k)}_\mu dx^\mu$ reduces, under the relativistic approximation, to $\Phi_k=(m_k^2/2E_0)\int \sqrt{AB}(1-b^2A/r^2)^{-1/2}dr$, where $b$ is the impact parameter. Expanding in the weak field $M/r\ll 1$ gives a closed-form phase; the lensing geometry enters through the deflection angle $\delta=4M/b+5\pi M^2/(2b^2)+4\pi\lambda/b^3$, whose two real impact parameters $b_1,b_2$ for a given source-detector configuration are superposed with the normalization used in Eq. (34). The paper's central claim is that the resulting two- and three-flavor probabilities depend distinctly on $\lambda$, on normal versus inverted mass ordering, and on the lightest neutrino mass $m_l$, with the $\lambda$ dependence absent for radial propagation and stronger in the strong-field extension.

Load-bearing premise

The load-bearing premise is that the neutrino phase is correctly computed as the integral of the canonical momentum along the trajectory, the convention of Refs [23,34]; the paper itself notes that Ref [56] obtains a different phase for the same type of radial propagation, and if that alternative convention is the physical one, the λ-dependent lensing modification would change or disappear.

Editorial extensions

If this is right

  • Because the radial phase reduces to the flat-spacetime result, the Hu-Sawicki parameter $\lambda$ leaves no imprint on oscillations unless the neutrino path is lensed.
  • In the weak-field Sun-Earth parameter set, two-flavor probabilities with $\lambda=10^{-26}\,\mathrm{m}^{-2}$ deviate visibly from the Schwarzschild ($\lambda=0$) curves, with inverted mass ordering producing larger amplitudes and shorter periods.
  • A nonzero lightest neutrino mass of 0.01–0.02 eV shortens and distorts the oscillation period, so flavor ratios of lensed neutrinos could in principle distinguish a nonzero absolute mass scale.
  • In the paper's strong-field extension, the oscillation period is smaller and the zero- and nonzero-$\lambda$ curves separate more clearly, which the authors interpret as amplified sensitivity near compact objects.
  • Three-flavor transitions such as $\nu_\mu\to\nu_\tau$ show oscillation-profile differences between the Hu-Sawicki and Schwarzschild spacetime that are especially clear for inverted ordering.

Reading between the lines

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

  • Extension beyond the paper: the same covariant-phase construction could be applied to other modified-gravity metrics, such as spinning black holes or other f(R) forms, turning lensed-neutrino oscillations into a more general gravity discriminator.
  • The strong-field section still uses the weak-field deflection angle truncated at order $M^3/b^3$ while keeping the full phase integral; integrating the exact geodesic deflection would be the natural next step to confirm the claimed amplification.
  • At the TeV–PeV energies where astrophysical neutrinos are actually detected, wave-packet decoherence may wash out the fine oscillation structure; a quantitative coherence-length estimate would show whether the effect is observable in practice.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper computes neutrino oscillation phases in the Hu-Sawicki f(R) gravity metric, for both radial and non-radial (lensed) propagation, and uses them to derive two- and three-flavor oscillation probabilities under weak-field and supposedly strong-field regimes. The central claim is that lensing-affected oscillation probabilities show a clear dependence on the Hu-Sawicki parameter λ, the neutrino mass hierarchy, and the lightest neutrino mass, and that strong-field lensing amplifies these effects.

Significance. If the derivation were correct, the weak-field result would be a new application of the covariant phase formalism to a specific modified-gravity metric, and the fact that the λ-dependence follows from the metric without parameter fitting would be of interest to the neutrino-lensing community. However, the quantitative claims are compromised by algebraic inconsistencies in the core derivation, and the strong-field section does not actually use strong-field geometry. No reproducible code is provided. The potential significance is moderate and conditional on a corrected derivation.

major comments (5)
  1. [II.B.2, Eq. (23)] The photon momentum derived from the mass-shell condition is incorrect. With the metric (9)-(10), g^{rr}=1/B=A, so the mass-shell condition (8) gives p_k^2 = B(E_k^2/A - m_k^2 - J_k^2/D), and for a massless particle p_0 = ±E_0 sqrt(B(1/A - b^2/D)). Equation (23) instead gives p_0 = ±E_0 sqrt(1/A - b^2/D), omitting the factor B. This error propagates into the phase integral (24), the turning-point condition (27), and every subsequent numerical result.
  2. [II.B.2, Eq. (31)] The relation between impact parameter and closest approach is inverted. Setting p_0(r_0)=0 with the correct momentum yields b^2 = D/A = r_0^2/A(r_0), i.e. b = r_0/sqrt(A(r_0)). Equation (31) instead gives b = ±r_0 sqrt(A(r_0)). Since r_0(b) enters the limits of the phase integral (30) and the lens-geometry solution (37), all probabilities in Figs. 2-8 are computed with the wrong closest-approach mapping. The quantitative λ-dependence claimed in Sections III and IV is therefore not established.
  3. [III.A, Eq. (28)] The phase integrand in Eq. (28) is inconsistent with Eq. (24). With A=1/B, the integrand in (24) is sqrt(AB)(1 - b^2 A/r^2)^{-1/2} = (1 - b^2 A/r^2)^{-1/2}. Equation (28) instead writes (1 - b^2/r^2)^{-1/2}(1 - 2M/r + λr^2)^{-1/2}, which is a different function of r for nonzero M and λ. The subsequent weak-field expansion (29) and the antiderivative (30) do not follow from the stated phase formula, so the analytic phase used in the numerics is not derived from the covariant phase (6).
  4. [IV] The claimed strong-field analysis does not use strong-field geometry. The deflection angle used, Eq. (36), is a weak-field, large-impact-parameter expansion, and the lens equation (37) is the weak-field equation solved for b. The only change from Section III is retaining the 5πM^2/b^2 term and leaving the integral (24) unevaluated analytically. The conclusion that 'strong-field gravitational lensing amplifies these effects' (Abstract, Section V) is therefore not supported by the presented calculation.
  5. [II.B.1] The paper acknowledges in Section II.B.1 that Ref. [56] obtains a different radial phase from the one adopted here, and states that it follows the methodology of Refs. [23,34]. Because the central observable is a phase difference, this convention choice is load-bearing: if the alternative phase convention is correct, the λ-dependent lensing modification changes or disappears. The paper should justify the adopted convention on physical grounds or at least quantify the sensitivity of the results to this choice.
minor comments (5)
  1. [II.A, Eq. (3)] The coefficient of the interference term in Eq. (3) appears to contain index errors: P_αβ should involve U*_αi U_βi U_αj U*_βj, but the manuscript writes U_βi U*_βj U_αj U*_βi. Please check and correct this expression.
  2. [Throughout] The manuscript contains numerous typographical errors, including 'balck hole', 'angualr momentum', 'the angel', 'weak-filed', 'Through out', 'probabilites', 'particulary', and garbled symbols in the figure captions (e.g., '¦￿' in Figs. 7 and 8). A thorough proofreading is needed.
  3. [III.A, Eq. (32)] In Eq. (32), the second line reads '-b = r0 ...', which is dimensionally and notationally inconsistent; it should be 'b = -r0 ...' or a similar explicit statement for the negative branch.
  4. [III.B, Eq. (37)] Equation (37) uses x0 and y0, but the text defines xD and yD as the detector coordinates. Please clarify the notation.
  5. [III.A, Eqs. (29)-(30)] The integration leading from Eq. (29) to Eq. (30) is not shown and the integrand in (29) has unclear notation (the fraction inside the square root is typeset ambiguously). The antiderivative should be verified and the steps presented in detail.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the λ, mass-hierarchy, and lightest-mass dependences are computed from the adopted Hu-Sawicki metric by direct substitution into the standard covariant phase integral, with parameters scanned rather than fitted.

full rationale

The paper's derivation is self-contained: it adopts the Hu-Sawicki metric from the literature, substitutes it into the covariant phase formula from Refs. [23,34], and numerically evaluates oscillation probabilities for chosen values of λ, mass ordering, and lightest neutrino mass. No parameter is fitted to the target probabilities; λ and ml are scanned inputs, and the mixing parameters come from external NuFIT results. The observed λ-dependence of the final probabilities is a direct mathematical consequence of having inserted λ into the metric, which is a normal model calculation rather than a circularity. There are no self-citations by the present authors, and the methodology credits are to external works [23,34]. The paper explicitly notes that an alternative phase convention exists in Ref. [56] and states that it follows [23,34]; this is an honest acknowledgment of a load-bearing modeling assumption, not a circular reduction. Any algebraic inconsistency in the b-r0 relation would be a correctness issue, not a circularity issue. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The analysis introduces no new particles or forces. The free parameters are the Hu-Sawicki parameter λ and the lightest neutrino mass ml, both scanned over hand-picked values. The main load-bearing axioms are the assumed metric and the adopted phase convention.

free parameters (2)
  • lambda (Hu-Sawicki parameter) = 0 and 10^-26 m^-2
    Model parameter of f(R) gravity; scanned over two values with no derived bound or physical justification.
  • lightest neutrino mass ml = 0, 0.01, 0.02 eV
    Varied to show sensitivity; not fitted but chosen by hand.
assumptions (3)
  • domain assumption The metric of Eq. (10) describes Hu-Sawicki f(R) spacetime: A(r)=1/B(r)=1-2M/r+λr^2.
    Adopted from Ref [55]; if this is not the correct static solution for Hu-Sawicki f(R), all subsequent phases are invalid.
  • standard math The covariant neutrino phase is Φk=∫p(k)_μ dx^μ with the momentum as in Eq. (7), following Refs [23,34].
    Standard QFT in curved spacetime, though the paper notes Ref [56] derives a different radial phase, so this choice is load-bearing.
  • domain assumption Weak-field expansion M/r≪1 to second order in Eq. (29).
    Used to derive Eq. (30); the strong-field section abandons this expansion for the phase but keeps the weak-field deflection angle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effects of gravitational lensing on neutrino oscillation in Hu-Sawicki f(R) gravity." pith.science (2026). https://pith.science/paper/PNCJJAEY

@misc{pith2026250623905,
  author       = {Pith},
  title        = {Pith review of: Effects of gravitational lensing on neutrino oscillation in Hu-Sawicki f(R) gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNCJJAEY}},
  note         = {Machine review of arXiv:2506.23905}
}
abstract

Gravitational lensing serves as a powerful probe of compact astrophysical objects and dark matter distributions. As relativistic counterparts to photons, neutrinos experiencing lensing offer a complementary means to investigate the properties of curved spacetimes. This paper studies neutrino oscillations within the spacetime geometry described by the Hu-Sawicki f(R) gravity model, focusing on the modifications induced by gravitational lensing. We calculate the oscillation phases for both radial and non-radial neutrino propagation and derive the corresponding flavor transition probabilities for 2-flavor and 3-flavor scenarios under the weak-field approximation. Our analysis demonstrates that the lensing-affected oscillation probabilities exhibit a clear dependence on the Hu-Sawicki model parameter $\lambda$ , the neutrino mass hierarchy, and the absolute value of the lightest neutrino mass. Furthermore, extending the analysis beyond the weak-field regime reveals that strong-field gravitational lensing amplifies these effects. These results, while theoretical, indicate that future high-precision measurements of lensed neutrinos from compact astrophysical objects could, in principle, help test modified gravity models and constrain neutrino parameters, provided that experimental and wave-packet decoherence challenges are overcome.

Figures

Figures reproduced from arXiv: 2506.23905 by the authors.

Figure 1
Figure 1. FIG. 1: Illustration of gravitational lensing caused by an Hu-Sawicki [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Oscillation probability of the two flavor case including the lensing effects of Hu-Sawicki [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Oscillation probability of the two flavor case including the lensing effects of Hu-Sawicki [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Oscillation probability of two flavor case [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Oscillation probability of the three flavor neutrino (normal ordering). From top to [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Oscillation probability of the three flavor neutrino (inverted ordering). From top to [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Oscillation probability of the two flavor case [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Oscillation probability of the three flavor case [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

62 extracted references · 38 canonical work pages

  1. [56]

    H. G. Ellis, J. Math. Phys. 14, 104 (1973) . 20

    work page 1973

  2. [1]

    Using Eq

    Radial propagation If neutrinos propagate radially, the angle φ stays constant dφ = 0 , therefore Jk = 0 . Using Eq. ( 6), the phase in this scenario is [ 23] Φk = Z D S −Ek dt dr 0 + pk dr, (12) where S and D denote the source and detector of the neutrinos, respectively. The light-ray differ- ential is written as dt dr 0 = E0 p0 B A , (13) where E0 and p...

    work page

  3. [2]

    The angle φ in this case is no longer constant along the trajectory, therefore the phase is also influenced by Jk

    Non–radial propagation If the gravitational lensing effect sets in, the neutrinos propagate non-radially. The angle φ in this case is no longer constant along the trajectory, therefore the phase is also influenced by Jk. 6 The phase in Eq. ( 6) now is expressed as Φk = Z D S −Ek dt dr 0 + pk + Jk dφ dr 0 , (18) where the light-ray differentials are dt dr ...

    work page

  4. [3]

    Pontecorvo, Zh

    B. Pontecorvo, Zh. Eksp. Teor. Fiz. 34, 247 (1957), [English translation: Sov. Phys. JETP 6, 429 (1958)]

    work page 1957

  5. [4]

    Fukuda et al

    Y. Fukuda et al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 81, 1562 (1998)

    work page 1998

  6. [5]

    Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 89, 011301 (2002)

    work page 2002

  7. [6]

    Lesgourgues and S

    J. Lesgourgues and S. Pastor, Phys. Rep. 429, 307 (2006)

    work page 2006

  8. [7]

    de Gouvea, Ann

    A. de Gouvea, Ann. Rev. Nucl. Part. Sci. 64, 363 (2014)

    work page 2014

Show all 62 references
  1. [8]

    Esteban, M

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, and T. Schwetz, J. High Energy Phys. 2020, 178 (2020) , arXiv:1910.05215 [hep-ph]

    2020 arXiv

  2. [9]

    Schneider, Extragalactic Astronomy and Cosmology (Springer, Berlin, 2006)

    P. Schneider, Extragalactic Astronomy and Cosmology (Springer, Berlin, 2006)

    2006

  3. [10]

    K. N. Abazajian et al. , Astropart. Phys. 35, 177 (2011)

    2011

  4. [11]

    Bartelmann, Class

    M. Bartelmann, Class. Quantum Grav. 27, 233001 (2010)

    2010

  5. [12]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000)

    2000

  6. [13]

    Massey et al

    R. Massey et al. , Nature 445, 286 (2007)

    2007

  7. [14]

    Perlick, Living Rev

    V. Perlick, Living Rev. Relativ. 7, 9 (2004)

    2004

  8. [15]

    Adrian-Martinez et al

    S. Adrian-Martinez et al. (KM3NeT Collaboration), J. Phys. G 43, 084001 (2016)

    2016

  9. [16]

    M. G. Aartsen et al. (IceCube Collaboration), Science 342, 1242856 (2013)

    2013

  10. [17]

    M. G. Aartsen et al. (IceCube Collaboration), Science 361, 147 (2018)

    2018

  11. [18]

    Ahlers and K

    M. Ahlers and K. Murase, Phys. Rev. D 90, 023010 (2014)

    2014

  12. [19]

    R. M. Crocker, C. Giunti, and D. J. Mortlock, Phys. Rev. D 69, 063008 (2004)

    2004

  13. [20]

    Fornengo, C

    N. Fornengo, C. Giunti, C. W. Kim, and J. Song, Phys. Rev. D 56, 1895 (1997)

    1997

  14. [21]

    D. V. Ahluwalia and C. Burgard, Gen. Rel. Grav. 28, 1161 (1996), arXiv:gr-qc/9603008

    1996 arXiv

  15. [22]

    L. A. Anchordoqui et al. , J. High Energy Astrophys. 1-2, 1 (2014)

    2014

  16. [23]

    Wudka, Mod

    J. Wudka, Mod. Phys. Lett. A 6, 3291 (1991)

    1991

  17. [24]

    Piriz, M

    D. Piriz, M. Roy, and J. Wudka, Phys. Rev. D 54, 1587 (1996), arXiv:hep-ph/9604403

    1996 arXiv

  18. [25]

    C. Y. Cardall and G. M. Fuller, Phys. Rev. D 55, 7960 (1997), arXiv:hep-ph/9610494. 19

    1997 arXiv

  19. [26]

    Fornengo, C

    N. Fornengo, C. Giunti, C. W. Kim, and J. Song, Phys. Rev. D 56, 1895 (1997) , arXiv:hep-ph/9611231

    1997 arXiv

  20. [27]

    Lambiase, G

    G. Lambiase, G. Papini, R. Punzi, and G. Scarpetta, Phys. Rev. D 71, 073011 (2005), arXiv:gr- qc/0503027

    2005

  21. [28]

    R. M. Crocker, C. Giunti, and D. J. Mortlock, Phys. Rev. D 69, 063008 (2004), arXiv:hep-ph/0308168

    2004 arXiv

  22. [29]

    C. M. Zhang and A. Beesham, Int. J. Mod. Phys. D 12, 727 (2003), arXiv:gr-qc/0004047

    2003 arXiv

  23. [30]

    J. Y.-Y. Lin and G. Holder, JCAP 04, 054 (2020), arXiv:1910.03550 [astro-ph.CO]

    2020 arXiv

  24. [31]

    Ren and C.-M

    J. Ren and C.-M. Zhang, Class. Quant. Grav. 27, 065011 (2010), arXiv:1002.0648 [gr-qc]

    2010 arXiv

  25. [32]

    Visinelli, Gen

    L. Visinelli, Gen. Rel. Grav. 47, 62 (2015), arXiv:1410.1523 [gr-qc]

    2015 arXiv

  26. [33]

    Capolupo, G

    A. Capolupo, G. Lambiase, and A. Quaranta, Phys. Rev. D 101, 095022 (2020), arXiv:2003.00516 [hep-th]

    2020 arXiv

  27. [34]

    the detector are located at (xS, yS) and (xD, yD) in this system with radius rS, rD, respectively

    and the adapted versions see [ 39, 40, 58]. the detector are located at (xS, yS) and (xD, yD) in this system with radius rS, rD, respectively. Rotating the system with an angle γ, we arrive at (x′, y′), which are related to the old ones through y′ = −x sin γ + y cos γ and x′ =...

  28. [35]

    Boshkayev, O

    K. Boshkayev, O. Luongo, and M. Muccino, Eur. Phys. J. C 80, 964 (2020), arXiv:2010.08254 [gr-qc]

    2020 arXiv

  29. [36]

    Mandal, Nucl

    S. Mandal, Nucl. Phys. B 965, 115338 (2021)

    2021

  30. [37]

    Swami, K

    H. Swami, K. Lochan, and K. M. Patel, Phys. Rev. D 102, 024043 (2020) , arXiv:2002.00977 [hep-ph]

    2020 arXiv

  31. [38]

    Buoninfante, G

    L. Buoninfante, G. G. Luciano, L. Petruzziello, and L. Smaldone, Phys. Rev. D 101, 024016 (2020), arXiv:1906.03131 [gr-qc]

    2020 arXiv

  32. [39]

    Koutsoumbas and D

    G. Koutsoumbas and D. Metaxas, Gen. Rel. Grav. 52, 102 (2020), arXiv:1909.02735 [hep-ph]

    2020 arXiv

  33. [40]

    R. C. Pantig, L. Mastrototaro, G. Lambiase, and A. Övgün, Eur. Phys. J. C 82, 1155 (2022), arXiv:2208.06664 [gr-qc]

    2022 arXiv

  34. [41]

    Swami, Eur

    H. Swami, Eur. Phys. J. C 82, 974 (2022), arXiv:2202.12310 [hep-ph]

    2022 arXiv

  35. [42]

    Chakrabarty, A

    H. Chakrabarty, A. Chatrabhuti, D. Malafarina, B. Silasan, and T. Tangphati, JCAP 08, 018 (2023), arXiv:2302.01564 [gr-qc]

    2023 arXiv

  36. [43]

    Alloqulov, H

    M. Alloqulov, H. Chakrabarty, D. Malafarina, B. Ahmedov, and A. Abdujabbarov, JCAP 02, 070 (2025), arXiv:2408.12916 [gr-qc]

    2025 arXiv

  37. [44]

    T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010)

    2010

  38. [45]

    De Felice and S

    A. De Felice and S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010)

    2010

  39. [46]

    Capozziello and M

    S. Capozziello and M. De Laurentis, Phys. Rep. 509, 167 (2011)

    2011

  40. [47]

    Hu and I

    W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 (2007)

    2007

  41. [48]

    Chakrabarty et al

    H. Chakrabarty et al. , JCAP 2023, 018 (2023)

    2023

  42. [49]

    Swami, K

    H. Swami, K. Lochan, and K. M. Patel, Phys. Rev. D 102, 024043 (2020)

    2020

  43. [50]

    Pontecorvo, Zh

    B. Pontecorvo, Zh. Eksp. Teor. Fiz. 34, 247 (1957)

    1957

  44. [51]

    Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962)

    1962

  45. [52]

    Pontecorvo, Zh

    B. Pontecorvo, Zh. Eksp. Teor. Fiz. 53, 1717 (1967)

    1967

  46. [53]

    E. K. Akhmedov and A. Y. Smirnov, Phys. Atom. Nucl. 72, 1363 (2009), arXiv:0905.1903 [hep-ph]

    2009 arXiv

  47. [54]

    E. K. Akhmedov and A. Y. Smirnov, Found. Phys. 41, 1279 (2011), arXiv:1008.2077 [hep-ph]

    2011 arXiv

  48. [55]

    Stodolsky, Gen

    L. Stodolsky, Gen. Rel. Grav. 11, 391 (1979)

    1979

  49. [57]

    M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988)

    1988

  50. [58]

    Mohan, N

    G. Mohan, N. Parbin, and U. D. Goswami, Eur. Phys. J. C 85, 413 (2025) , arXiv:2411.19048 [gr-qc]

    2025 arXiv

  51. [59]

    S. I. Godunov and G. S. Pastukhov, Phys. Atom. Nucl. 74, 302 (2011) , arXiv:0906.5556 [hep-ph]

    2011 arXiv

  52. [60]

    Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Rela- tivity (John Wiley and Sons, New York, 1972)

    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Rela- tivity (John Wiley and Sons, New York, 1972)

    1972

  53. [61]

    Chakrabarty, D

    H. Chakrabarty, D. Borah, A. Abdujabbarov, D. Malafarina, and B. Ahmedov, Eur. Phys. J. C 82, 24 (2022), arXiv:2109.02395 [gr-qc]

    2022 arXiv

  54. [62]

    Esteban, M

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. a. P. Pinheiro, and T. Schwetz, JHEP 12, 216 (2024), arXiv:2410.05380 [hep-ph]

    2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.