Analysing Ultra High Energy Cosmic Rays' Anisotropy in $\boldsymbol{f(R, T)}$ Gravity Theory

Swaraj Pratim Sarmah; Umananda Dev Goswami

REVIEW 3 major objections 5 minor 104 references

Analysing Ultra High Energy Cosmic Rays' Anisotropy in $\boldsymbol{f(R, T)}$ Gravity Theory

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that the predicted arrival-direction anisotropy of ultra-high-energy cosmic rays is sensitive to the assumed theory of gravity: two f(R,T) modified-gravity models give a smaller dipole amplitude than $\Lambda$CDM at…

desk verdict First f(R,T)-UHECR anisotropy paper has a load-bearing problem: the formula it plots (Δ=3η/ξ) is not the dipole of its own diffusion solution, so the ΛCDM-vs-f(R,T) ranking and Auger fits are unsupported. read the letter →

arxiv 2412.17494 v2 pith:U4EY4VDG submitted 2024-12-23 astro-ph.HE

Swaraj Pratim Sarmah , Umananda Dev Goswami This is my paper · ORCID

classification astro-ph.HE PACS 95.30.Sf98.70.Sa04.50.Kd

keywords ultra-high-energycosmicrayscosmic-rayanisotropyf(RT)gravitymodifiedtheoriesofdiffusivepropagationturbulentmagneticfieldsdipoleamplitudeLambda-CDMcosmology

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 tries to establish that the arrival-direction anisotropy of ultra-high-energy cosmic rays is not fixed by astrophysics alone: it depends on the theory of gravity used for the expanding universe. Working with protons diffusing in a turbulent extragalactic magnetic field, the authors compute the dipole anisotropy for an ensemble of sources in two f(R,T) gravity models (minimal exponential and non-minimal R-T coupling) and in the standard $\Lambda$CDM model. They report that both f(R,T) models predict a lower anisotropy amplitude, meaning a more isotropic sky, than $\Lambda$CDM at most energies, and that after adjusting the magnetic-field strength and source separation the f(R,T) curves fall inside the published surface-detector uncertainty bands with reduced chi-squared values near one. The reason a sympathetic reader cares is that if true, UHECR anisotropy becomes a new observable probe of modified gravity rather than a purely propagation-driven signal.

What carries the argument

The load-bearing object is the anisotropy ratio $\$\Delta$ = 3\eta/\xi$, with $\eta=J/J_0$ the modification factor (full diffusive flux divided by the no-energy-loss flux) and $\xi$ the density-enhancement factor that measures how diffusion piles up particles relative to rectilinear propagation. This ratio is evaluated from the transport-equation solution in the expanding universe, whose diffusion scale $\$lambda^{2}$$ depends on the Hubble parameter $H(z)$ of each model. The two f(R,T) forms enter through their Friedmann equations: $f(R,T)=\$\alpha$ R + \$\beta$ $e^{{T}}$$ and $f(R,T)=R + f_0 R $T^{{\delta}}$$, with best-fit parameters taken from background cosmology, and the source ensemble enters through a discrete-source factor $F$ built from the source spacing $d_s$. The mechanism of the paper is to feed each model's $H(z)$ into the flux integrals, form the ratio $\$\Delta$$, and compare the resulting energy-dependent curve with published surface-detector data for two array spacings.

What would settle it

Re-derive the ensemble dipole directly from the angular expansion of the summed single-source fluxes without assuming $\$\Delta$ = 3\eta/\xi$, and compare it with Eq. (34) at, say, 10 EeV for the same $B$ and $d_s$; a difference larger than the plotted uncertainty band would invalidate the paper's amplitudes. Observationally, a full-sky measurement of the dipole at 0.1–10 EeV with per-bin uncertainties near 10% would separate the models because their predicted amplitudes differ by up to a factor of two in that range.

Watch

Extended reading notes

Core claim

The paper's central claim is that the energy-dependent dipole amplitude $\$\Delta$(E)$ of diffusive ultra-high-energy cosmic rays is highly sensitive to the cosmological model. Using the anisotropy expression $\$\Delta$ = 3\eta/\xi$, where $\eta = J/J_0$ is the modification factor and $\xi$ is the density enhancement from diffusion, the authors find that two f(R,T) models, $f(R,T)=\$\alpha$ R + \$\beta$ $e^{{T}}$$ and $f(R,T)=R + f_0 R $T^{{\delta}}$$, produce systematically lower amplitudes than $\Lambda$CDM at most energies between 0.01 and 100 EeV. The $\Lambda$CDM model gives the lowest reduced chi-squared value (0.788) of the three, but the f(R,T) models still give acceptable fits (0.982 and 0.871) once the magnetic field $B$ and source spacing $d_s$ are tuned to 65–70 nG and 25–30 Mpc respectively. The authors therefore claim that modified gravity can account for the observed anisotropy pattern without endorsing standard cosmology, and that anisotropy measurements are, in principle, discriminating between gravitational frameworks.

Load-bearing premise

The paper rests on the formula $\$\Delta$ = 3\eta/\xi$ being the correct dipole amplitude for the discrete multi-source ensemble; the formula is stated without derivation, and if the true ensemble average differs, every predicted curve, fitted magnetic field, and chi-squared comparison changes.

Editorial extensions

If this is right

If the central claim is correct, the dipole amplitude of ultra-high-energy cosmic rays should be quoted together with the assumed cosmological model; the same source and magnetic-field inputs give different amplitudes in different gravity theories.
The fits require stronger magnetic fields and smaller or comparable source spacings for the f(R,T) models than for $\Lambda$CDM (65–70 nG and 25–30 Mpc versus 20 nG and 30 Mpc), so independent measurements of extragalactic magnetic fields and source density become a direct cross-check.
Below roughly 0.1 EeV all three models nearly coincide, so future discriminating power lies mainly in the 0.1–100 EeV range where the model curves separate.
Because the reduced chi-squared values are all near one, the current data do not prefer modified gravity; the claim is compatibility, not superiority.

Reading between the lines

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

The fitted magnetic-field strength and source spacing are free parameters, so this analysis establishes that f(R,T) gravity can accommodate the observed anisotropy, not that it predicts it; the comparison becomes predictive only once $B$ and the source density are fixed by independent observations.
The same pipeline could rank other modified-gravity theories by their predicted dipole amplitude, since $H(z)$ enters only through the expansion time and diffusion scale; the exponential and non-minimal couplings used here are examples, not the full set.
A natural extension is to repeat the calculation for heavier nuclei, since composition changes the effective rigidity and therefore the energy at which the model curves separate; the proton-only baseline may not survive a mixed-composition fit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear X,

The thing to know about 2412.17494: it is a cleanly written, honest first application of f(R,T) gravity to UHECR anisotropy, and the conclusion is almost certainly not supported by the calculation as presented.

What is new and good: the authors extend their earlier f(R) and f(Q) diffusion program to two f(R,T) models, compute H(z) for each, plug into the standard Syrovatskii solution, and compare Δ(E) with Auger SD 750/1500 data. They are transparent that B and ds are tuned per model, they report χ2, and they admit the cosmological parameters are ΛCDM-based. The literature coverage is adequate, and the H(z) figures look consistent with the cited models.

The soft spot is central. The bridge between the transport calculation and the plots is Eq. (34), Δ = 3η/ξ. I checked the stress-test concern and it holds. The formula is asserted and not derived. More importantly, it is not the dipole amplitude of the solution they are using. In the no-loss single-source limit, the standard result from the same Green's function is Δ_std = 3r/(2ct); Eq. (34) gives something proportional to λ^3/(c r^2), a different object. And in the multi-source ensemble, Eq. (30) sums over shell radii only; the resulting density is spherically symmetric about the observer, so the true dipole of that ensemble is zero. The plotted quantity is a monopole flux-enhancement ratio, not an anisotropy amplitude. That means the model ranking, the fitted B and ds, and the χ2 values in Table I are properties of an unjustified scalar formula, not a prediction for Auger's dipole.

Minor points: γ is overloaded (turbulence index in Eq. 3, source spectral index in Section VI), and the 'parameterized to align' procedure is a fit, not an independent test. The authors do acknowledge the ΛCDM parameter issue, which is fair.

I don't think the paper is a throwaway. The diffusion machinery is standard, the f(R,T) H(z) inputs come from published fits, and the question - can modified gravity leave an observable imprint on UHECR anisotropy? - is legitimate. But as it stands the headline claim fails. The fix is to compute the actual ensemble dipole from the angular source distribution, or drop the word 'anisotropy' and present Δ as a flux-enhancement ratio. Either way, the Auger comparison changes.

I would send it to a referee because the topic deserves scrutiny and the paper is a genuine first pass, but I would expect major revision. Probably more useful as a reading-group case study than a citation.

Best.

Referee Report

3 major / 5 minor

Summary. The paper studies the energy-dependent dipole anisotropy of ultra-high energy cosmic rays in two f(R,T) gravity models and compares their predictions with the standard ΛCDM model. It solves the diffusion equation for a discrete ensemble of extragalactic proton sources, computes the "density enhancement" ξ and a flux modification factor η, and then uses the ratio Δ = 3η/ξ (Eq. 34) as the anisotropy amplitude. The authors tune the rms magnetic field B and source spacing ds separately for each model (Figs. 2–3, Table I) to match Pierre Auger SD 750 and SD 1500 data and report χ² values. The main claim is that the f(R,T) models predict lower anisotropy than ΛCDM and, after tuning, can reproduce the observed trend.

Significance. If the calculation were correct, a clean sensitivity of UHECR dipole anisotropy to the underlying gravity model would be interesting and could provide a novel cosmological test. The paper makes a good-faith effort to use the standard diffusion framework and to compare with actual Auger data, and the inclusion of uncertainty bands and χ² statistics is commendable. However, the central observable, Eq. (34), is not the dipole anisotropy of the diffusion solution, and the comparison procedure tunes B and ds separately for each model. As a result, the reported model ranking and the claimed compatibility with Auger data are not supported by the analysis.

major comments (3)

[§V, Eq. (34)] The formula Δ = 3η/ξ is asserted rather than derived, and it is not the dipole amplitude of the transport solution. For the standard diffusion dipole one needs (3D/c)|∇n|/n, which involves the gradient of the density. In the manuscript's own no-loss, static, single-source limit (H=0, λ²=Dt, η=1), Eq. (34) gives Δ = 3√π (Dt)^{3/2} e^{r²/(4Dt)}/(c r²), whereas the exact Green's-function dipole is Δ_std = 3r/(2ct). These expressions have different scalings, so Eq. (34) cannot be the anisotropy of the solution in Eq. (6). The citation to Ref. [104] does not repair this because the formula is not reproduced or derived for the multi-source case.
[§V, Eq. (30)] The multi-source factor F is constructed from shell radii r_i only and therefore contains no angular information; the summed particle density is spherically symmetric about the observer. Its gradient vanishes, so the true dipole amplitude of the ensemble, as constructed, is zero, while Eq. (34) returns nonzero values because Δ is a ratio of scalar flux and density-enhancement quantities. The quantity plotted in Figs. 2–5 is thus a monopole flux-enhancement ratio, not an arrival-direction dipole, and it cannot be compared with the Auger dipole amplitudes.
[§VI, Figs. 2–3 and Table I] The comparison is not a test of model predictions. The magnetic field B and source separation ds are adjusted per model (B=20 nG, ds=30 Mpc for ΛCDM; B=65 nG, ds=30 Mpc for Model I; B=70 nG, ds=25 Mpc for Model II), and the abstract states that these parameters were chosen "to align well with the observational data." Consequently the reported χ² values measure the quality of a fit with model-specific free parameters, not the predictive success of the gravity models or the sensitivity of anisotropy to cosmology. The claim that the f(R,T) models "can effectively reproduce" the Auger data is therefore circular.

minor comments (5)

[§VI] The text says all plots assume "the redshift z=2," but Eq. (28) integrates over z up to zmax; please clarify whether z=2 is the maximum source redshift and whether this is applied consistently.
[§V, Eq. (34)] The symbol Δ is used for the anisotropy but no definition is given in terms of the spherical-harmonic dipole; please state explicitly how Δ relates to the Auger dipole amplitude.
[Table I] The number of degrees of freedom used to compute χ²_red is not stated; without it, reduced χ² values near unity cannot be interpreted.
[§VII] There are typographical errors such as "captures the of anisotropy at higher energies" and "the f(R,T) = R + f0RTδ model... with observational trends"; these should be corrected.
[Fig. 5] The uncertainty bands in Fig. 5 are not defined; please specify which parameters were varied and over what range.

Circularity Check

1 steps flagged · score 4.0 of 10

The fixed-parameter comparison at B=20 nG and ds=30 Mpc is a genuine model calculation, but the paper's post-fit 'alignment' with Auger data and the quoted chi-square values reduce to the tuned magnetic-field/source-spacing parameters; the central ranking claim remains partially independent.

fitted input called prediction [Abstract; Section VI (Fig. 3 and Table I); Section VII (Conclusion)]
"We parameterized the magnetic field and the source distance in anisotropy calculations to align well with the observational data from the Pierre Auger Observatory for all the cosmological models. An uncertainty band is presented along with the chi^2 test for all cosmological models to demonstrate the goodness of fitting. ... Fig. 3 shows the updated theoretical predictions for the two f(R,T) gravity models, where adjustments in the magnetic field (B) and source separation distance (ds) are performed to improve alignment with the Auger SD 750 and SD 1500 datasets."

The two free parameters B and ds are separately adjusted for each f(R,T) model (65 nG/30 Mpc and 70 nG/25 Mpc) until the predicted curve matches the same Auger dataset that is then used to compute chi^2 in Table I and to claim 'goodness of fitting'. The abstract itself states these parameters were chosen 'to align well with the observational data'. Therefore the reported agreement, the uncertainty-band overlap in Fig. 5, and the chi^2 values are in-sample properties of the fitted curve rather than independent predictions: the alignment is forced by the parameter choice. The fixed-parameter comparison at B=20 nG and ds=30 Mpc (Fig. 2/Fig. 4) is not affected by this fit, so the circularity is partial rather than total.

full rationale

The paper's main quantitative claim that f(R,T) models give a lower anisotropy amplitude than LambdaCDM at most energies is computed at common, un-fitted parameters (B=20 nG, ds=30 Mpc), with the cosmological Hubble parameter H(z) taken from external fits (Ref. [99]) and checked against OHD in Fig. 1. That part of the derivation chain is self-contained: the plotted Delta is an output of the diffusion calculation, not an input fitted to the Auger anisotropy data. The circular element is only in the validation narrative: after showing the fixed-parameter comparison, the authors tune B and ds separately for each f(R,T) model to improve agreement with the same Auger points, then present the resulting curves, uncertainty bands, and chi^2 values as demonstrating that the models 'effectively reproduce the observed energy-dependent anisotropy'. Because the tuned parameters are free and are chosen on the same dataset used for the chi^2 test, the claimed reproduction is statistically forced in-sample. This weakens the support for the 'compatibility with Auger' statement but does not invalidate the independent fixed-parameter model ordering. The paper also acknowledges in the conclusion that the cosmological parameters are LambdaCDM-based first approximations, which is an explicit limitation rather than a circular step. A separate correctness concern, not counted here as circularity, is that Eq. (34) defines Delta = 3 eta/xi while the ensemble factor F in Eq. (30) is a spherically symmetric sum over shell radii with no directional information, so the plotted Delta may not be the actual dipole amplitude of the transport solution; this is a validity issue, not a self-referential reduction.

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

The central claim rests on the fitted parameters B and ds, on the f(R,T) model parameters inherited from Ref. [99], and on an assumed H(z) framework. The most fragile unproven input is the anisotropy formula Delta = 3 eta / xi, which is cited but not derived here. No new physical entities are introduced.

free parameters (6)

B (magnetic field RMS strength) = 20 nG common comparison; 65 nG (Model I), 70 nG (Model II) after tuning
Manually adjusted per model in Section VI to align predicted anisotropy with Auger SD 750/1500 data; Fig. 3.
ds (source separation distance) = 30 Mpc common comparison; 25 Mpc (Model II) after tuning
Adjusted per model in Section VI; Fig. 3.
Source spectral index gamma = 2
Assumed for protons in Section VI; no fit or justification is given.
f(R,T) Model I parameters alpha, beta, sigma = 0.8469, -0.4285, -0.0041
Adopted from CC+BAO best fit in Ref. [99], used in Eqs. (20)-(23).
f(R,T) Model II parameters f0, sigma, delta = 2.4285, 0.1836, 0.1002
Adopted from Ref. [99], used in Eqs. (24)-(26).
Background cosmology H0, Omega_m0, Omega_r0 = 72 km/s/Mpc, 0.3, 1e-4
Taken from WMAP7 [100] and used for both f(R,T) models and ΛCDM; the authors call this a first approximation in the conclusion.

assumptions (6)

standard math The diffusion equation (Eq. 4) and its Green's function solution (Eq. 6) describe UHECR transport in an expanding universe.
Standard result from Berezinsky and Gazizov [97]; basis for all flux and anisotropy calculations in the paper.
domain assumption The dipole anisotropy is given by Delta = 3 eta / xi (Eq. 34).
Stated without derivation in Section V; cited to Supanitsky [104], but its applicability to the multi-source ensemble summed through factor F is not shown.
domain assumption Sources are discrete, identical, with average spacing ds, and the sum over sources yields the factor F (Eqs. 29-30).
Source distribution model taken from Refs. [102,103]; assumes isotropically distributed equivalent sources.
domain assumption The diffusion coefficient D(E) follows Eq. (3) with Kolmogorov turbulence coefficients.
Numerical fit from Ref. [4]; used in Eq. (33) to compute the Syrovatskii variable.
domain assumption f(R,T) gravity affects UHECR anisotropy only through H(z) in dt/dz and the Hubble radius, with no direct coupling to magnetic fields or cosmic rays.
Central modeling choice of Sections III-V; it limits the physical interpretation of the model differences.
domain assumption The cosmic rays are protons with a fixed source injection spectrum.
Stated in Section VI; Auger data indicate a composition that evolves with energy, so the proton-only assumption may bias anisotropy predictions.

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Cite this review

Pith. "Pith review of Analysing Ultra High Energy Cosmic Rays' Anisotropy in $\boldsymbol{f(R, T)}$ Gravity Theory." pith.science (2026). https://pith.science/paper/U4EY4VDG

@misc{pith2026241217494,
  author       = {Pith},
  title        = {Pith review of: Analysing Ultra High Energy Cosmic Rays' Anisotropy in $\boldsymbolf(R, T)$ Gravity Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4EY4VDG}},
  note         = {Machine review of arXiv:2412.17494}
}

abstract

In this study, we investigated the anisotropy of diffusive Ultra-High Energy Cosmic Rays (UHECRs) by employing three cosmological models: two models from the $f(R, T)$ gravity theory and the other is the standard $\Lambda$CDM model. The primary objective of this work was to ascertain the role of the $f(R, T)$ gravity theory in comprehending the anisotropy of UHECRs without implicitly endorsing the conventional cosmology. We parameterized the magnetic field and the source distance in anisotropy calculations to align well with the observational data from the Pierre Auger Observatory for all the cosmological models. An uncertainty band is presented along with the $\chi^2$ test for all cosmological models to demonstrate the goodness of fitting. Our findings revealed that the amplitude of the anisotropy is highly sensitive to these cosmological models. Notably, the $f(R, T)$ models exhibited a lower amplitude of anisotropy (i.e., more isotropy), while the $\Lambda$CDM model predicted a comparatively higher amplitude at most of the energies considered.

Figures

Figures reproduced from arXiv: 2412.17494 by the authors.

**Figure 2.** FIG. 2. The anisotropy of UHECRs as a function of energy [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Left: The modified anisotropy of UHECRs as a function of energy [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. The anisotropy of the UHECRs as a function of the magnetic field [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. Left: The unmodified anisotropy of the UHECRs as a function of energy [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

104 extracted references · 36 canonical work pages

[104]

A. D. Supanitsky, JCAP 04, 046 (2021) [arXiv:2007.09063v2]

work page Pith review arXiv 2021
[1]

Blasi, Astron

P. Blasi, Astron. Astrophys. Rev. 21, 70 (2013), [arXiv:1311.7346]

arXiv 2013
[2]

E. G. Berezhko, H. V olk, Astrophys. J. Lett.661, L175 (2007) [arXiv:0704.1715]

work page Pith review arXiv 2007
[3]

J. W. Hewitt, M. Lemoine-Goumard, Comptes rendus Physique 16, 674 (2015)

2015
[4]

Harari, S

D. Harari, S. Mollerach, E. Roulet, Phys. Rev. D 89, 123001 (2014) [arXiv:1312.1366]

arXiv 2014
[5]

Mollerach, E

S. Mollerach, E. Roulet, Phys. Rev. D 105, 063001 (2022) [arXiv:2111.00560v2]

arXiv 2022
[6]

Constraints on the origin of cosmic rays above $10^{18}$ eV from large scale anisotropy searches in data of the Pierre Auger Observatory

P. Abreu et al. (Pierre Auger Collaboration), Astrophys. J. Lett. 762 (2013) L13 [arXiv:1212.3083v1]

work page Pith review arXiv 2013
[7]

Search for EeV Protons of Galactic Origin

R.U. Abbasi (Telescope Array Collaboration), Astropart. Phys. 86, 21 (2017) [arXiv:1608.06306v2]

work page Pith review arXiv 2017

Show all 104 references

[8]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), Science 357, 1266 (2017) [arXiv:1709.07321v1]

2017 arXiv
[9]

Aab et al

A. Aab et al. (The Pierre Auger Collaboration), Phys. Rev. Lett. 125, 121106 (2020) [arXiv:2008.06488]

2020 arXiv
[10]

Aab et al

A. Aab et al. (The Pierre Auger Collaboration), Phys. Rev. D 102, 062005 (2020) [arXiv:2008.06486]

2020 arXiv
[11]

Abreu et al

P. Abreu et al. (Pierre Auger Collaboration), Eur. Phys. J. C 81 (2021) 966 [arXiv:2109.13400v3]

2021 arXiv
[12]

Novotn´y, the Pierre Auger Collaboration, PoS ICRC 324 (2021)

V . Novotn´y, the Pierre Auger Collaboration, PoS ICRC 324 (2021)

2021
[13]

A. M. Hillas, Phys. Lett. A 24, 677 (1967)

1967
[14]

G. R. Blumenthal, Phys. Rev. D 1, 1596 (1970)

1970
[15]

Berezinsky, A

V . Berezinsky, A. Z. Gazizov, S. I. Grigorieva, Phys. Rev. D74, 043005 (2006) [arXiv:hep-ph/0204357]

2006 arXiv
[16]

Lemoine, Phys

M. Lemoine, Phys. Rev. D 71, 083007 (2005) [arXiv:astro-ph/0411173]

2005 arXiv
[17]

R. U. Abbasi et al. (HiRes Collaboration), Phys. Rev. Lett. 100 (2008) 101101 [arXiv:astro-ph/0703099v2]

2008 arXiv
[18]

Abraham et al

J. Abraham et al. (The Pierre Auger Collaboration), Phys. Rev. Lett. 101, 061101 (2008) [arXiv:0806.4302v1]

2008 arXiv
[19]

Abu-Zayyad et al

T. Abu-Zayyad et al. (Telescope Array Collaboration), Astrophys. J. 768, L1 (2013)

2013
[20]

Greisen, Phys

K. Greisen, Phys. Rev. Lett. 16, (1966) 748

1966
[21]

Zatsepin, V .A

G.T. Zatsepin, V .A. Kuz’min, JETP Lett.4, 78 (1966)

1966
[22]

Abraham et al., (Pierre Auger Collaboration), Astropart

J. Abraham et al., (Pierre Auger Collaboration), Astropart. Phys. 31, 399 (2009) [arXiv:0903.1127v2]

2009 arXiv
[23]

Abu-Zayyad et al

T. Abu-Zayyad et al. (Telescope Array Collaboration), Phys. Rev. D 88, 112005 (2013) [arXiv:1304.5614]

2013 arXiv
[24]

Abreu et al

P. Abreu et al. (Pierre Auger Collaboration), JCAP 05, 021 (2023) [arXiv:2209.05926]

2023 arXiv
[25]

Abreu et al., (Pierre Auger Collaboration), Astrophys

P. Abreu et al., (Pierre Auger Collaboration), Astrophys. J. 755, L4 (2012)

2012
[26]

M. G. Aartsen et al., (IceCube Collaboration), Phys. Rev. D 89, 062007 (2014) [arXiv:1311.7048]

2014 arXiv
[27]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), JCAP 10, 022 (2019) [arXiv:1906.07422]

2019 arXiv
[28]

Abreu et al., (Pierre Auger Collaboration), Astrophys

P. Abreu et al., (Pierre Auger Collaboration), Astrophys. J. 760, 148 (2012)

2012
[29]

Abraham et al., (Pierre Auger Collaboration), Phys

J. Abraham et al., (Pierre Auger Collaboration), Phys. Rev. Lett. 104, 091101 (2010) [arXiv:1002.0699v1]

2010 arXiv
[30]

Pierre Auger Collaboration, JCAP 02, 026 (2013)

2013
[31]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 90, 122005 (2014) [arXiv:1409.4809]

2014 arXiv
[32]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 90, 122006 (2014) [arXiv:1409.5083]

2014 arXiv
[33]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 96, 122003 (2017)

2017
[34]

Mollerach, E

S. Mollerach, E. Roulet, Phys. Rev. D 99, 103010 (2019) [arXiv:1903.05722]

2019 arXiv
[35]

Mollerach, E

S. Mollerach, E. Roulet, Prog. Part. Nucl. Phys. 98, 85 (2018) [arXiv:1710.11155]

2018 arXiv
[36]

Deligny, K

O. Deligny, K. Kawata, P. Tinyakov, PTEP2017, 12A104 (2017) [arXiv:1702.07209]

2017 arXiv
[37]

Deligny, Astropart

O. Deligny, Astropart. Phys. 104, 13 (2019) [arXiv:1808.03940]

2019 arXiv
[38]

Aab et al

A. Aab et al. (Pierre Auger Collaboration), Astrophys. J. 868, 4 (2018) [arXiv:1808.03579]

2018 arXiv
[39]

Abeeu et al., Astrophys

P. Abeeu et al., Astrophys. J. 935, 170 (2022) [arXiv:2206.13492]

2022 arXiv
[40]

A. Aab. et al., (Pierre Auger Collaboration), Astrophys. J. Lett., 853:L29 (2018) [arXiv:1801.06160]

2018 arXiv
[41]

R. U. Abbasi et al.,(Telescope Array Collaboration), Astrophys. J. Lett. 898, L28 (2020) [arXiv:2007.00023]

2020
[42]

R. U. Abbasi et al.,(Telescope Array Collaboration), Astrophys. J. 862, 91 (2018)

2018
[43]

R. U. Abbasi et al.,(Telescope Array Collaboration), [arXiv:2110.14827]. 12
[44]

Tinyakov et al., (Telescope Array Collaboration), PoS ICRC, 375 (2021)]

P. Tinyakov et al., (Telescope Array Collaboration), PoS ICRC, 375 (2021)]

2021
[45]

A. G. Reiss et al., Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201]

1998 arXiv
[46]

Perlmutter et al., Astrophys

S. Perlmutter et al., Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133]

1999 arXiv
[47]

D. N. Spergel et al., Astrophys. J. Suppl. S 170, 377 (2007) [arXiv:astro-ph/0603449]

2007 arXiv
[48]

Brax, Rep

P. Brax, Rep. Prog. Phys. 81, 1 (2018)

2018
[49]

E. J. Copeland, M. Sami, S. Tsujikawa, IJMP D 15, 1753-1936 (2006) [arXiv:hep-th/0603057]

2006 arXiv
[50]

U. D. Goswami, H. Nandan, M. Sami, Phys. Rev. D 82, 103530 (2010)

2010
[51]

Nojiri, S.D

S. Nojiri, S.D. Odintsov, V .K. Oikonomou, Phys. Rep.692, 1 (2017)

2017
[52]

Sotiriou, V

P. Sotiriou, V . Faraoni, Rev. Mod. Phys.82, 451 (2010) [arXiv:0805.1726]

2010 arXiv
[53]

De Felice, S

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

2010
[54]

D. J. Gogoi, U. D. Goswami, IJMP D 31, 2250048 (2022) [arXiv:2108.01409]

2022 arXiv
[55]

U. D. Goswami, K Deka, IJMP D 22, 13 (2013) 1350083 [arXiv.1303.5868]

2013 arXiv
[56]

J. B. Jim ´enez et al., Coincident General Relativity, Phys. Rev. D 98, 044048 (2018) [arXiv:1710.03116]

2018 arXiv
[57]

Harko et al., Coupling matter in modified Q gravity, Phys

T. Harko et al., Coupling matter in modified Q gravity, Phys. Rev. D 98, 084043 (2018) [arXiv:1806.10437]

2018 arXiv
[58]

Mandal, P

S. Mandal, P. K. Sahoo, J. R. L. Santos, Phys. Rev. D 102, 024057 (2020) [arXiv:2008.01563]

2020 arXiv
[59]

Mandal, D

S. Mandal, D. Wang, P. K. Sahoo, Phys. Rev. D 102, 124029 (2020) [arXiv:2011.00420]

2020 arXiv
[60]

Frusciante, Phys

N. Frusciante, Phys. Rev. D 103, 044021 (2021) [arXiv:2101.09242]

2021 arXiv
[61]

Sarmah, A

P. Sarmah, A. De, U. D. Goswami, Phys. Dark Universe 40, 101209 (2023) [arXiv:2303.05905]

2023 arXiv
[62]

Bahamonde, C

S. Bahamonde, C. G. B ¨ohmer, M. Wright, Phys. Rev. D 92 104042 (2015)

2015
[63]

Bahamonde, C

S. Bahamonde, C. G. B ¨ohmer, M. Krˇsˇs´ak, Phys. Lett. B 775, 37 (2017)

2017
[64]

Azizi, E

T. Azizi, E. Yaraie, Int. J. Mod. Phys. D 23 (2014) 1450021

2014
[65]

Bertolami, P

O. Bertolami, P. Frazo, J. Pramos, Phys. Rev. D 83 (2011) 044010

2011
[66]

Nesseris, Phys

S. Nesseris, Phys. Rev. D 79 (2009) 044015

2009
[67]

Futamase, K.I

T. Futamase, K.I. Maeda, Phys. Rev. D 39 (1989) 399

1989
[68]

Uzan, Phys

J.P. Uzan, Phys. Rev. D 59 (1999) 123510

1999
[69]

Harko, F.S.N

T. Harko, F.S.N. Lobo, Eur. Phys. J. C 70 (2010) 373

2010
[70]

Harko, F.S.N

T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, Phys. Rev. D 84 (2011) 024020

2011
[71]

Sharif, M

M. Sharif, M. Zubair, J. Cosmol. Astropart. Phys. 03 (2012) 028

2012
[72]

Alvarenga et al., Phys

F.G. Alvarenga et al., Phys. Rev. D 87 (2013) 103526

2013
[73]

Zaregonbadi, M

R. Zaregonbadi, M. Farhoudi, N. Riazi, Phys. Rev. D 94 (2016) 084052

2016
[74]

Rudra, Eur

P. Rudra, Eur. Phys. J. Plus 130 (4) (2015) 66

2015
[75]

Sharif, A

M. Sharif, A. Siddiqa, Gen. Relativ. Gravit. 51 (6) (2019) 74

2019
[76]

J. Bora, U. D. Goswami, Phys. Dark Universe 38, 101132 (2022)

2022
[77]

Mertsch, M

P. Mertsch, M. Ahlers, JCAP 11, 048 (2019) [arXiv:1909.09052v2]

2019 arXiv
[78]

Harari, S

D. Harari, S. Mollerach, E. Roulet, Phys. Rev. D 103, 023012 (2021) [arXiv:2010.10629v2]

2021 arXiv
[79]

Ahlers, P

M. Ahlers, P. Mertsch, Prog. Part. Nucl. Phys. 94, 184 (2017) [arXiv:1612.01873v1]

2017 arXiv
[80]

Mollerach , E

S. Mollerach , E. Roulet, O. Taborda, JCAP 12, 021 (2022) [arXiv:2207.11540v2]

2022 arXiv
[81]

A. U. Abeysekara et al., Astrophys. J 871, 96 (2019)

2019
[82]

Chakraborty et al., (Grapes-3 Collaboration), PoS ICRC 395 (2021)

M. Chakraborty et al., (Grapes-3 Collaboration), PoS ICRC 395 (2021)

2021
[83]

Globus et al., MNRAS 484, 4167 (2019)

N. Globus et al., MNRAS 484, 4167 (2019)

2019
[84]

G. Sigl, M. Lemoine, P. Biermann, Astropart.Phys. 10 (1999) 141

1999
[85]

S. P. Sarmah, U. D. Goswami, Eur. Phys. J. C 84, 419 (2024) [arXiv:2303.16678]

2024 arXiv
[86]

S. P. Sarmah, U. D. Goswami, Astropart. Phys. 163, 103005 (2024)[arXiv:2309.14361]

2024 arXiv
[87]

S. P. Sarmah, U. D. Goswami, [arXiv:2411.00366]

arXiv
[88]

S. P. Sarmah, U. D. Goswami, Nucl. Phys. B 1013, 116851 (2025) [arXiv:2406.11902]

2025 arXiv
[89]

Aab et al., (Pierre Auger Collaboration), Astrophys

A. Aab et al., (Pierre Auger Collaboration), Astrophys. J. 891, 142 (2020) [arXiv:2002.06172]

2020 arXiv
[90]

J. L. Han, Annu. Rev. Astron 255, 111 (2017)

2017
[91]

Hu et al., Astrophys

Y . Hu et al., Astrophys. J.941, 133 (2022)

2022
[92]

Chadayammuri, MNRAS 512, 2 (2022)

U. Chadayammuri, MNRAS 512, 2 (2022)

2022
[93]

Feretti et al., Astron

L. Feretti et al., Astron. Astrophys. Rev. 20, 54 (2012)

2012
[94]

J. P. Vall´ee, New Astro. Rev. 55, 91 (2011)

2011
[95]

Vazza et al., Class

F. Vazza et al., Class. Quantum Grav. 34, 234001 (2017)

2017
[96]

G. Sigl, F. Miniati and T. A. Ensslin, Phys. Rev. D 70, 043007 (2004) [arXiv:astro-ph/0401084]

2004 arXiv
[97]

Berezinsky, A

V . Berezinsky, A. Z. Gazizov, Astrophys. J.643, 8 (2006) [arXiv:astro-ph/0512090]

2006 arXiv
[98]

S. I. Syrovatskii, Soviet Astro. 3, 22 (1959)

1959
[99]

Rudra, K

P. Rudra, K. giri, Nucl. Phys. B 967, 115428 (2021)

2021
[100]

Komatsu et al., Astrophys

E. Komatsu et al., Astrophys. J. Suppl. 192, 18 (2011) [arXiv:1001.4538]

2011 arXiv
[101]

Solanki et al., Phys

R. Solanki et al., Phys. Dark Universe, 32, 100820 (2021), [arXiv:2105.00876]

2021 arXiv
[102]

J. M. Gonz ´alez, S. Mollerach, E. Roulet, Phys. Rev. D 104, 063005 (2021) [arXiv:2105.08138]

2021 arXiv
[103]

Mollerach, E

S. Mollerach, E. Roulet, JCAP 10 013 (2013) [arXiv:1305.6519v1]

2013 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [104]

A. D. Supanitsky, JCAP 04, 046 (2021) [arXiv:2007.09063v2]

work page Pith review arXiv 2021

[2] [1]

Blasi, Astron

P. Blasi, Astron. Astrophys. Rev. 21, 70 (2013), [arXiv:1311.7346]

arXiv 2013

[3] [2]

E. G. Berezhko, H. V olk, Astrophys. J. Lett.661, L175 (2007) [arXiv:0704.1715]

work page Pith review arXiv 2007

[4] [3]

J. W. Hewitt, M. Lemoine-Goumard, Comptes rendus Physique 16, 674 (2015)

2015

[5] [4]

Harari, S

D. Harari, S. Mollerach, E. Roulet, Phys. Rev. D 89, 123001 (2014) [arXiv:1312.1366]

arXiv 2014

[6] [5]

Mollerach, E

S. Mollerach, E. Roulet, Phys. Rev. D 105, 063001 (2022) [arXiv:2111.00560v2]

arXiv 2022

[7] [6]

Constraints on the origin of cosmic rays above $10^{18}$ eV from large scale anisotropy searches in data of the Pierre Auger Observatory

P. Abreu et al. (Pierre Auger Collaboration), Astrophys. J. Lett. 762 (2013) L13 [arXiv:1212.3083v1]

work page Pith review arXiv 2013

[8] [7]

Search for EeV Protons of Galactic Origin

R.U. Abbasi (Telescope Array Collaboration), Astropart. Phys. 86, 21 (2017) [arXiv:1608.06306v2]

work page Pith review arXiv 2017