Pith. sign in

REVIEW 3 major objections 4 minor 84 references

Eddington-inspired Born-Infeld gravity: Constraints from the generalized parton distributions (GPDs)

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Using the proton's internal pressure profile from generalized parton distributions, this paper derives new bounds on the EiBI gravity parameter, $|\kappa|\le 0.10$–$0.3\,\mathrm{m^5\,kg^{-1}\,s^{-2}}$.

desk verdict A clear but seriously incomplete application of Avelino's EiBI bounds: the quark-only pressure does not yield constraints on kappa until the total EMT pressure is used. read the letter →

arxiv 2505.09291 v2 pith:GIM7ADFU submitted 2025-05-14 hep-ph gr-qchep-th

The MMGPDs Collaboration , Muhammad Goharipour , Anoushiravan Moradi , K. Azizi This is my paper · ORCID
classification hep-phgr-qchep-th
keywords Eddington-inspiredBorn-InfeldgravityprotonpressuredistributiongravitationalformfactorgeneralizedpartondistributionsdeeplyvirtualComptonscatteringEiBIparameterconstraintsvonLauecondition
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 uses the proton's internal pressure profile, extracted from the gravitational form factor $D(t)$ through a QCD analysis of generalized parton distributions, to constrain the extra parameter $\kappa$ of Eddington-inspired Born-Infeld (EiBI) gravity. It reports updated bounds $|\kappa| \leq 0.10\text{--}0.3\,\mathrm{m^5\,kg^{-1}\,s^{-2}}$, with the tighter end coming from the pressure profile fitted with $M^2=2\,\mathrm{GeV^2}$ and the looser end from average-pressure inequalities. It also finds that constraints from the first and second pressure moments are not significantly stronger than peak-pressure constraints, contrary to an earlier expectation. The result matters because it puts subatomic structure on the table as a testing ground for modified gravity, with limits competitive with neutron-star bounds.

What carries the argument

The load-bearing machinery is a chain of three pieces: the $D(t)$ gravitational form factor, which gives the pressure $p(r)$ through a Fourier transform and radial derivatives (Eq. 15); the von Laue condition, the stability statement that the volume integral of $p(r)$ vanishes, which lets the average physical pressure be equated with the average effective gravitational pressure $p_G$ from EiBI; and the inequalities $|\kappa| \le |p|^{-1}$ (Eq. 24) and $|\kappa| \le |\langle p\rangle|/\langle p^2\rangle$ (Eq. 33), which turn peak or moment pressures into upper bounds on $\kappa$. The new input is the $D_Q(t)$ parameterization of Eq. (19), whose $M^2$ value changes the bounds by roughly a factor of three to four.

What would settle it

A lattice QCD calculation of the full quark-plus-gluon gravitational form factor $D(t)$ for the proton would settle the matter: if the total pressure profile differs substantially from the quark-only profile used here, the $|\kappa|$ bounds would shift by the same factor, and if the second pressure moment disagrees with the GPD-based value, the moment-based bounds would be shown to be model-dependent.

Watch

Extended reading notes

Core claim

The paper claims that the quark pressure profile of the proton, obtained from the gravitational form factor $D_Q(t)$ in Eq. (19) for two values of the fit parameter $M^2$, can be inserted directly into the EiBI inequalities (24) and (33) to update the bound on $\kappa$. The resulting constraints are $|\kappa| \le 0.44$ and $0.10\,\mathrm{m^5\,kg^{-1}\,s^{-2}}$ from peak pressure, and $|\kappa| \le 1.32$ and $0.30\,\mathrm{m^5\,kg^{-1}\,s^{-2}}$ from pressure moments, for $M^2=1$ and $2\,\mathrm{GeV^2}$, respectively. The stronger profile yields a bound competitive with neutron-star limits, though still weaker than collider-based limits. The paper concludes that precise experimental and theoretical determinations of the proton's mechanical properties are a viable route to testing EiBI gravity and related modifications.

Load-bearing premise

The bounds assume that the quark contribution to the proton pressure, from $D_Q(t)$, is the full pressure that enters the EiBI inequalities; if the omitted gluon pressure is comparable, the derived $|\kappa|$ limits do not constrain the physical pressure that sources gravity.

Editorial extensions

If this is right

  • If the analysis is correct, the EiBI parameter is bounded by $|\kappa| \le 0.10$ to $0.3\,\mathrm{m^5\,kg^{-1}\,s^{-2}}$, on the same scale as limits from neutron-star observations.
  • The bound depends strongly on the pressure model: the $M^2=2\,\mathrm{GeV^2}$ profile gives limits three to four times tighter than $M^2=1\,\mathrm{GeV^2}$, so data that pin down the $t$-dependence of $D(t)$ directly sharpen the gravity constraint.
  • Moment-based constraints do not improve on peak-pressure constraints, so future experimental work should target direct measurements of the first and second pressure moments rather than relying on peak values.
  • Improvements in DVCS data, GPD reconstructions, and lattice QCD will, through the same inequalities, translate into stronger bounds on EiBI gravity and analogous modified theories.

Reading between the lines

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

  • The cleanest check of the bounds is a lattice QCD evaluation of the total (quark plus gluon) $D(t)$: the paper states that its $D_Q(t)$ carries only quark contributions, while the inequalities and the von Laue condition refer to the full energy-momentum tensor.
  • The same inequality chain could be applied to other hadrons with measured mechanical properties, giving independent subatomic bounds on EiBI gravity outside the proton.
  • Because the inequalities depend only on pressure moments, any future $D(t)$ extraction can be passed through them without new gravity input, making this a reusable route for testing modified gravity.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript derives upper bounds on the Eddington-inspired Born-Infeld (EiBI) parameter κ by combining Avelino's inequalities for EiBI gravity, Eqs. (24) and (33), with the proton pressure profile extracted by the MMGPDs Collaboration from a QCD analysis of skewness-dependent GPDs. The authors evaluate the bounds using peak pressure, average peak pressure, and first/second moments of the quark pressure distribution, obtaining limits in the range |κ| ≤ 0.10–2.69 m^5 kg^{-1} s^{-2} depending on the model parameter M^2 and the chosen estimator. They conclude that proton mechanical properties provide competitive constraints on EiBI gravity and motivate improved GPD reconstructions.

Significance. If the derived bounds were established, the paper would demonstrate that subatomic pressure distributions can test modified gravity, complementing astrophysical constraints from neutron stars. The manuscript is clearly written and the arithmetic from the quoted pressure profiles to the stated κ values is straightforward and reproducible. However, the central claim depends on two load-bearing identifications: that the quark-only pressure from Eq. (19) can be used as the total physical pressure in the EiBI inequalities, and that Eq. (24) is the correct finiteness condition for τ. Both identifications are problematic, and the first is explicitly contradicted by the paper's own statement that the gluon contribution is absent. Consequently, the numerical constraints in Eqs. (25)–(36) are not established as bounds on the physical pressure that sources EiBI gravity.

major comments (3)
  1. [Section 'Constraints on EiBI from proton interior pressure profile', Eqs. (24)–(36)] The pressure p(r) used in the bounds is computed from D_Q(t) in Eq. (19), and the text immediately after Eq. (19) states that this is only the quark contribution and does not include the gluon. The EiBI inequalities in Eqs. (24) and (33) involve the physical energy density and pressure of the matter that sources gravity; the von Laue condition in Eq. (18) and the averaged condition in Eq. (22) apply to the total EMT pressure. Without an explicit argument that the gluon pressure is negligible for the peak value and for the first and second moments, the numerical results in Eqs. (25)–(30) and (35)–(36) are bounds on a quark-only pressure, not on the total proton pressure appearing in the EiBI field equations.
  2. [Eq. (24) and Eqs. (25)–(30)] Equation (24) states that |κ| ≤ |p|^{-1} follows from the dominant energy condition and finiteness of τ. But τ in Eq. (12) is [(1+κρ)(1−κp)^3]^{-1/2}, so the reality conditions are 1+κρ > 0 and 1−κp > 0. With the dominant energy condition ρ ≥ |p|, the common bound is |κ| ≤ 1/ρ, not |κ| ≤ 1/|p|. The peak-pressure limits in Eqs. (25)–(30) are therefore not consequences of Eq. (24). In addition, p(r) diverges as r→0 as noted near Fig. 2, so the 'peak pressure' evaluations depend on an unstated radial cutoff and are not well defined as stated.
  3. [Eq. (33) and Eqs. (35)–(36)] The derivation of Eq. (33) from Eq. (32) is not shown. Pointwise, Eq. (32) gives |pG| ≥ |κ| p^2, and averaging yields ⟨|pG|⟩ ≥ |κ| ⟨p^2⟩. Since |⟨pG⟩| ≤ ⟨|pG⟩|, the inequality in Eq. (33), |κ| ≤ |⟨p⟩|/⟨p^2⟩, requires an additional assumption that the pressure anisotropy parameter ξ, Eq. (34), controls the ratio |⟨p⟩|/⟨|pG|⟩. The authors should provide the full derivation or cite the specific step in Ref. [54] that justifies replacing the averaged absolute value by the absolute value of the average.
minor comments (4)
  1. [Abstract and Conclusions] The abstract and conclusions state that the bounds are 'competitive with existing bounds from neutron stars', but the quoted comparison in the Conclusions says they are 'several orders of magnitude weaker than those obtained from collider experiments'; please clarify which comparison is intended.
  2. [Around Eq. (34)] The definition ξ ≡ ⟨p⟩p/⟨p^2⟩ appears to have a typo: as written ξ is dimensionful, despite the text calling it dimensionless. The intended definition should be stated cleanly.
  3. [Throughout] There are several typographical issues, e.g., 'bonds' for 'bounds', 'anlyzing' for 'analyzing', and inconsistent use of 'GFF D(t)' versus 'D-form factor'. These do not affect the physics but should be corrected.
  4. [Figure 2 and related text] The statement that the pressure 'diverges toward r = 0' is important because the central bounds depend on the behavior near r=0; the manuscript should state the limiting behavior of p(r) from Eq. (19) and specify the radial cutoff used to evaluate the peak values in Eqs. (25)–(30).

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the kappa bounds are evaluations of imported inequalities on an external pressure fit; the quark-only caveat is a validity gap, not circularity.

full rationale

The derivation chain is not circular. The proton pressure p(r) is taken from the published MMGPDs analysis of Ref. [62], where the gravitational form factor D_Q(t) of Eq. (19) was fitted to Compton form factor data; no EiBI parameter enters that fit. The inequalities used to constrain kappa, Eqs. (24) and (33), are imported from Avelino's independent derivation (Ref. [54]) from the EiBI field equations, not from the pressure profile itself. The paper simply evaluates those inequalities on the previously obtained pressure curve, so the numerical bounds in Eqs. (25)-(36) are outputs of the calculation rather than inputs disguised as predictions. The self-citation of Ref. [62] supplies the empirical input, but that work is a data-constrained, externally checkable analysis, so it is not load-bearing circularity under the stated rules. The clearest scientific caveat, explicitly acknowledged in the paper, is that D_Q(t) contains only the quark contribution and does not include gluons, while Avelino's inequalities refer to the full matter energy-momentum tensor pressure; this is a physical mismatch that could invalidate the quoted bounds, but it is not a circularity, because the quark pressure is not defined in terms of kappa and the bounds do not reduce to their own inputs by construction. Likewise, the divergence of p(r) at r tending to zero noted near Fig. 2 makes the peak-pressure bounds cutoff-dependent, but that is a numerical robustness concern rather than a circular step. Overall, no equation, fitted parameter, or cited result here is equivalent to the target conclusion by definition.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The calculation rests on the standard Fourier relation between D(t) and pressure, the assumption that proton EMT drives EiBI gravity at femtometer scales, the dominant energy condition, the spin-zero spherical treatment, and the key ad hoc identification of quark-only pressure with physical pressure. The main free parameters are the GPD model parameters and an unspecified peak cutoff.

free parameters (4)
  • M^2 = 1 and 2 GeV^2
    Governing t-dependence of D_Q(t) in Eq. (19); chosen from the GPD model in Ref. [62], not fitted here. Changes the kappa bound by roughly a factor of four.
  • d1^Q(0) = not stated in this paper
    Normalization of D_Q(t) from a fit to CFF data in Ref. [62]; it scales the pressure profile and therefore the kappa limits.
  • alpha = 3
    Fixed exponent in the D_Q(t) parameterization, Eq. (19).
  • peak evaluation radius = not stated
    p(r) diverges at r=0; the bounds in Eqs. (25) to (30) require a finite radius near the center that is not specified.
assumptions (6)
  • standard math Fourier transform relation between D(t) and p(r) and s(r), Eqs. (15) to (17).
    Background result used without proof to convert the form factor into a pressure profile.
  • domain assumption The proton's EMT is the matter source for EiBI gravity at sub-femtometer scales.
    Needed for the entire comparison; taken from EiBI theory and Ref. [54].
  • domain assumption Dominant energy condition rho >= |p| holds inside the proton.
    Used in the paragraph around Eq. (24) to justify the finite-tau condition.
  • domain assumption Proton modeled as a spherically symmetric, spin-zero compact object; only diagonal EMT components matter.
    Inherited from Avelino's treatment in Ref. [54] and used to write Eq. (31).
  • ad hoc to paper Quark-only p(r) can stand in for the total physical pressure p in Eqs. (24) and (33).
    The paper notes after Eq. (19) that gluon contributions are absent, but still applies the von Laue condition and EiBI constraints to the quark pressure.
  • domain assumption Apparent metric inside the proton is nearly Minkowski, so the von Laue condition applies to pT = p + pG.
    Used in the Constraints section to connect the physical and effective gravitational pressures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Eddington-inspired Born-Infeld gravity: Constraints from the generalized parton distributions (GPDs)." pith.science (2026). https://pith.science/paper/GIM7ADFU

@misc{pith2026250509291,
  author       = {Pith},
  title        = {Pith review of: Eddington-inspired Born-Infeld gravity: Constraints from the generalized parton distributions (GPDs)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIM7ADFU}},
  note         = {Machine review of arXiv:2505.09291}
}
abstract

The Eddington-inspired Born-Infeld (EiBI) theory of gravity modifies general relativity in high-density regimes. It offers an alternative framework that avoids cosmological singularities and remodels gravitational dynamics within compact objects. An important feature of EiBI gravity is its additional parameter, $\kappa$, which governs deviations from standard gravitational behavior. In this study, we investigate constraints on $\kappa$ using the internal pressure distribution of the proton, derived from gravitational form factor (GFF) $ D(t) $ obtained through a QCD analysis of generalized parton distributions (GPDs). By comparing pressure profiles extracted from skewness-dependent GPDs with previous determinations based on deeply virtual Compton scattering (DVCS) data, we establish updated bounds on $\kappa$. Our results show that the choice of proton pressure model significantly impacts the constraints, with the strongest limits ($|\kappa| \leq 0.10\text{--}0.3\, \text{m}^5\, \text{kg}^{-1}\, \text{s}^{-2}$). We further demonstrate that constraints obtained based on the first and second moments of the pressure distribution yield competitive bounds compared to those derived from peak pressures or those derived from just the first moment. These findings highlight the importance of precise experimental and theoretical determinations of the proton's mechanical properties in testing alternative theories of gravity. The present study motivates future improvements in GPD reconstructions for stronger constraints on EiBI gravity and related modifications.

Figures

Figures reproduced from arXiv: 2505.09291 by the authors.

Figure 1
Figure 1. FIG. 1. Pressure distribution of the proton, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

84 extracted references · 16 canonical work pages

  1. [54]

    P. P. Avelino, Phys. Lett. B 795, 627 (2019), arXiv:1902.01318 [gr-qc]

    arXiv 2019

  2. [1]

    Brans and R

    C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961)

    1961

  3. [2]

    R. H. Dicke, Phys. Rev. 125, 2163 (1962)

    1962

  4. [3]

    H. A. Buchdahl, Mon. Not. Roy. Astron. Soc. 150, 1 (1970)

    1970

  5. [4]

    A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). 6

    1980

  6. [5]

    T. P. Sotiriou and V . Faraoni, Rev. Mod. Phys. 82, 451 (2010), arXiv:0805.1726 [gr-qc]

    arXiv 2010

  7. [6]

    De Felice and S

    A. De Felice and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc]

    arXiv 2010

  8. [7]

    Milgrom, Astrophys

    M. Milgrom, Astrophys. J. 270, 365 (1983)

    1983

Show all 84 references
  1. [8]

    J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004), [Erratum: Phys.Rev.D 71, 069901 (2005)], arXiv:astro-ph/0403694

    2004 arXiv

  2. [9]

    Famaey and S

    B. Famaey and S. McGaugh, Living Rev. Rel. 15, 10 (2012), arXiv:1112.3960 [astro-ph.CO]

    2012 arXiv

  3. [10]

    Fierz and W

    M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173, 211 (1939)

    1939

  4. [11]

    de Rham, G

    C. de Rham, G. Gabadadze, and A. J. Tolley, Phys. Rev. Lett. 106, 231101 (2011), arXiv:1011.1232 [hep-th]

    2011 arXiv

  5. [12]

    de Rham, Living Rev

    C. de Rham, Living Rev. Rel. 17, 7 (2014), arXiv:1401.4173 [hep-th]

    2014 arXiv

  6. [13]

    G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

    1974

  7. [14]

    Gleyzes, D

    J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, Phys. Rev. Lett. 114, 211101 (2015), arXiv:1404.6495 [hep-th]

    2015 arXiv

  8. [15]

    Langlois, Int

    D. Langlois, Int. J. Mod. Phys. D 28, 1942006 (2019), arXiv:1811.06271 [gr-qc]

    2019 arXiv

  9. [16]

    Ashtekar, Phys

    A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986)

    1986

  10. [17]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21, R53 (2004), arXiv:gr-qc/0404018

    2004 arXiv

  11. [18]

    Perez, Living Rev

    A. Perez, Living Rev. Rel. 16, 3 (2013), arXiv:1205.2019 [gr- qc]

    2013 arXiv

  12. [19]

    Dapor and K

    A. Dapor and K. Liegener, Phys. Lett. B 785, 506 (2018), arXiv:1706.09833 [gr-qc]

    2018 arXiv

  13. [20]

    Ambjorn and R

    J. Ambjorn and R. Loll, Nucl. Phys. B 536, 407 (1998), arXiv:hep-th/9805108

    1998 arXiv

  14. [21]

    Ambjorn, A

    J. Ambjorn, A. Goerlich, J. Jurkiewicz, and R. Loll, Phys. Rept. 519, 127 (2012), arXiv:1203.3591 [hep-th]

    2012 arXiv

  15. [22]

    Loll, Class

    R. Loll, Class. Quant. Grav. 37, 013002 (2020), arXiv:1905.08669 [hep-th]

    2020 arXiv

  16. [23]

    Ambjørn, D

    J. Ambjørn, D. Coumbe, J. Gizbert-Studnicki, A. G ¨orlich, and J. Jurkiewicz, Class. Quant. Grav. 36, 224001 (2019), arXiv:1904.05755 [hep-th]

    2019 arXiv

  17. [24]

    E. P. Verlinde, JHEP 04, 029 (2011), arXiv:1001.0785 [hep-th]

    2011 arXiv

  18. [25]

    E. P. Verlinde, SciPost Phys. 2, 016 (2017), arXiv:1611.02269 [hep-th]

    2017 arXiv

  19. [26]

    A. N. Makarenko, S. Odintsov, and G. J. Olmo, Phys. Rev. D 90, 024066 (2014), arXiv:1403.7409 [hep-th]

    2014 arXiv

  20. [27]

    Kibaro ˘glu, S

    S. Kibaro ˘glu, S. D. Odintsov, and T. Paul, Phys. Dark Univ.44, 101445 (2024), arXiv:2402.08951 [gr-qc]

    2024 arXiv

  21. [28]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Rept. 505, 59 (2011), arXiv:1011.0544 [gr-qc]

    2011 arXiv

  22. [29]

    Clifton, P

    T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513, 1 (2012), arXiv:1106.2476 [astro-ph.CO]

    2012 arXiv

  23. [30]

    Joyce, B

    A. Joyce, B. Jain, J. Khoury, and M. Trodden, Phys. Rept. 568, 1 (2015), arXiv:1407.0059 [astro-ph.CO]

    2015 arXiv

  24. [31]

    Koyama, Rept

    K. Koyama, Rept. Prog. Phys. 79, 046902 (2016), arXiv:1504.04623 [astro-ph.CO]

    2016 arXiv

  25. [32]

    Berti et al

    E. Berti et al. , Class. Quant. Grav. 32, 243001 (2015), arXiv:1501.07274 [gr-qc]

    2015 arXiv

  26. [33]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and V . K. Oikonomou, Phys. Rept. 692, 1 (2017), arXiv:1705.11098 [gr-qc]

    2017 arXiv

  27. [34]

    Akrami et al

    Y . Akrami et al. (CANTATA),Modified Gravity and Cosmol- ogy. An Update by the CANTATA Network, edited by E. N. Sari- dakis, R. Lazkoz, V . Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´an Jim´enez, M. De Laurentis, and G. J. Olmo (Springer,

  28. [35]

    Banados and P

    M. Banados and P. G. Ferreira, Phys. Rev. Lett. 105, 011101 (2010), [Erratum: Phys.Rev.Lett. 113, 119901 (2014)], arXiv:1006.1769 [astro-ph.CO]

    2010 arXiv

  29. [36]

    P. Pani, V . Cardoso, and T. Delsate, Phys. Rev. Lett. 107, 031101 (2011), arXiv:1106.3569 [gr-qc]

    2011 arXiv

  30. [37]

    P. Pani, T. Delsate, and V . Cardoso, Phys. Rev. D 85, 084020 (2012), arXiv:1201.2814 [gr-qc]

    2012 arXiv

  31. [38]

    Harko, F

    T. Harko, F. S. N. Lobo, M. K. Mak, and S. V . Sushkov, Phys. Rev. D 88, 044032 (2013), arXiv:1305.6770 [gr-qc]

    2013 arXiv

  32. [39]

    P. P. Avelino and R. Z. Ferreira, Phys. Rev. D 86, 041501 (2012), arXiv:1205.6676 [astro-ph.CO]

    2012 arXiv

  33. [40]

    P. P. Avelino, Phys. Rev. D85, 104053 (2012), arXiv:1201.2544 [astro-ph.CO]

    2012 arXiv

  34. [41]

    Delsate and J

    T. Delsate and J. Steinhoff, Phys. Rev. Lett.109, 021101 (2012), arXiv:1201.4989 [gr-qc]

    2012 arXiv

  35. [42]

    J. H. C. Scargill, M. Banados, and P. G. Ferreira, Phys. Rev. D 86, 103533 (2012), arXiv:1210.1521 [astro-ph.CO]

    2012 arXiv

  36. [43]

    Y . H. Sham, P. T. Leung, and L. M. Lin, Phys. Rev. D 87, 061503 (2013), arXiv:1304.0550 [gr-qc]

    2013 arXiv

  37. [44]

    Bouhmadi-L ´opez, C.-Y

    M. Bouhmadi-L ´opez, C.-Y . Chen, and P. Chen, Eur. Phys. J. C 75, 90 (2015), arXiv:1406.6157 [gr-qc]

    2015 arXiv

  38. [45]

    Sotani and U

    H. Sotani and U. Miyamoto, Phys. Rev. D 90, 124087 (2014), arXiv:1412.4173 [gr-qc]

    2014 arXiv

  39. [46]

    S.-W. Wei, K. Yang, and Y .-X. Liu, Eur. Phys. J. C 75, 253 (2015), [Erratum: Eur.Phys.J.C 75, 331 (2015)], arXiv:1405.2178 [gr-qc]

    2015 arXiv

  40. [47]

    X.-L. Du, K. Yang, X.-H. Meng, and Y .-X. Liu, Phys. Rev. D 90, 044054 (2014), arXiv:1403.0083 [gr-qc]

    2014 arXiv

  41. [48]

    P. P. Avelino, Phys. Rev. D 93, 044067 (2016), arXiv:1511.03223 [gr-qc]

    2016 arXiv

  42. [49]

    P. P. Avelino, Phys. Rev. D 93, 104054 (2016), arXiv:1602.08261 [gr-qc]

    2016 arXiv

  43. [50]

    Beltran Jimenez, L

    J. Beltran Jimenez, L. Heisenberg, G. J. Olmo, and D. Rubiera- Garcia, Phys. Rept. 727, 1 (2018), arXiv:1704.03351 [gr-qc]

    2018 arXiv

  44. [51]

    Albarran, M

    I. Albarran, M. Bouhmadi-L ´opez, C.-Y . Chen, and P. Chen, Phys. Dark Univ.23, 100255 (2019), arXiv:1811.05041 [gr-qc]

    2019 arXiv

  45. [52]

    N. M. Jim ´enez Cruz and C. Escamilla-Rivera, Int. J. Mod. Phys. D 28, 1950167 (2019), arXiv:1907.06788 [gr-qc]

    2019 arXiv

  46. [53]

    P. P. Avelino, JCAP 11, 022 (2012), arXiv:1207.4730 [astro- ph.CO]

    2012 arXiv

  47. [55]

    V . D. Burkert, L. Elouadrhiri, and F. X. Girod, Nature557, 396 (2018)

    2018

  48. [56]

    Casanellas, P

    J. Casanellas, P. Pani, I. Lopes, and V . Cardoso, Astrophys. J. 745, 15 (2012), arXiv:1109.0249 [astro-ph.SR]

    2012 arXiv

  49. [57]

    A. D. I. Latorre, G. J. Olmo, and M. Ronco, Phys. Lett. B 780, 294 (2018), arXiv:1709.04249 [hep-th]

    2018 arXiv

  50. [58]

    Delhom, V

    A. Delhom, V . Miralles, and A. Pe ˜nuelas, Eur. Phys. J. C 80, 340 (2020), arXiv:1907.05615 [hep-th]

    2020 arXiv

  51. [59]

    Beltr ´an Jim´enez, A

    J. Beltr ´an Jim´enez, A. Delhom, G. J. Olmo, and E. Orazi, Phys. Lett. B 820, 136479 (2021), arXiv:2104.01647 [gr-qc]

    2021 arXiv

  52. [60]

    M. V . Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018), arXiv:1805.06596 [hep-ph]

    2018 arXiv

  53. [61]

    V . D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorc ´e, P. Schweitzer, and P. E. Shanahan, Rev. Mod. Phys.95, 041002 (2023), arXiv:2303.08347 [hep-ph]

    2023 arXiv

  54. [62]

    Goharipour, H

    M. Goharipour, H. Hashamipour, H. Fatehi, F. Irani, K. Azizi, and S. V . Goloskokov (MMGPDs), Phys. Rev. D 112, 014016 (2025), arXiv:2501.16257 [hep-ph]

    2025 arXiv

  55. [63]

    Dehghan, F

    Z. Dehghan, F. Almaksusi, and K. Azizi, JHEP 06, 025 (2025), arXiv:2502.16689 [hep-ph]

    2025 arXiv

  56. [64]

    Kumeri ˇcki, Nature 570, E1 (2019)

    K. Kumeri ˇcki, Nature 570, E1 (2019)

    2019

  57. [65]

    Azizi and U

    K. Azizi and U. ¨Ozdem, Eur. Phys. J. C 80, 104 (2020), arXiv:1908.06143 [hep-ph]

    2020 arXiv

  58. [66]

    A. V . Belitsky and A. V . Radyushkin, Phys. Rept.418, 1 (2005), arXiv:hep-ph/0504030

    2005 arXiv

  59. [67]

    Diehl, Phys

    M. Diehl, Phys. Rept. 388, 41 (2003), arXiv:hep-ph/0307382. 7

    2003 arXiv

  60. [68]

    A. V . Radyushkin, Phys. Rev. D59, 014030 (1999), arXiv:hep- ph/9805342

    1999

  61. [69]

    M. V . Polyakov and C. Weiss, Phys. Rev. D60, 114017 (1999), arXiv:hep-ph/9902451

    1999 arXiv

  62. [70]

    Hashamipour, M

    H. Hashamipour, M. Goharipour, K. Azizi, and S. V . Goloskokov, Phys. Rev. D 107, 096005 (2023), arXiv:2211.09522 [hep-ph]

    2023 arXiv

  63. [71]

    Deser and G

    S. Deser and G. W. Gibbons, Class. Quant. Grav. 15, L35 (1998), arXiv:hep-th/9803049

    1998 arXiv

  64. [72]

    D. N. V ollick, Phys. Rev. D 69, 064030 (2004), arXiv:gr- qc/0309101

    2004

  65. [73]

    D. N. V ollick, Phys. Rev. D 72, 084026 (2005), arXiv:gr- qc/0506091

    2005

  66. [74]

    D. N. V ollick, (2006), arXiv:gr-qc/0601136

    2006 arXiv

  67. [75]

    M. G. Kurbah and F. Ahmed, Eur. Phys. J. C 84, 903 (2024)

    2024

  68. [76]

    G. J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011), arXiv:1101.3864 [gr-qc]

    2011 arXiv

  69. [77]

    P. E. Shanahan and W. Detmold, Phys. Rev. Lett. 122, 072003 (2019), arXiv:1810.07589 [nucl-th]

    2019 arXiv

  70. [78]

    Lorc ´e and P

    C. Lorc ´e and P. Schweitzer, Acta Phys. Polon. B 56, 3 (2025), arXiv:2501.04622 [hep-ph]

    2025 arXiv

  71. [79]

    Kroll, H

    P. Kroll, H. Moutarde, and F. Sabatie, Eur. Phys. J. C 73, 2278 (2013), arXiv:1210.6975 [hep-ph]

    2013 arXiv

  72. [80]

    P. P. Avelino and L. Sousa, Phys. Rev. D 97, 064019 (2018), arXiv:1802.03961 [gr-qc]

    2018 arXiv

  73. [81]

    P. P. Avelino and R. P. L. Azevedo, Phys. Rev. D 97, 064018 (2018), arXiv:1802.04760 [gr-qc]

    2018 arXiv

  74. [82]

    S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2023)

    2023

  75. [83]

    Goharipour, F

    M. Goharipour, F. Irani, M. H. Amiri, H. Fatehi, B. Falahi, A. Moradi, and K. Azizi (MMGPDs), Nucl. Phys. B 1017, 116962 (2025), arXiv:2503.08847 [hep-ph]

    2025 arXiv

  76. [2021]

    arXiv:2105.12582 [gr-qc]

    arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.