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arxiv: 2606.28141 · v1 · pith:E6CWHAELnew · submitted 2026-06-26 · 🌌 astro-ph.HE

Transition from Diffusion to Drift-Dominated Cosmic Ray Transport and the Origin of the Knee

Luis Enrique Espinosa Castro , Carmelo Evoli , Pasquale Blasi This is my paper

Pith reviewed 2026-06-29 02:34 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords cosmic ray kneediffusiondrift transportgalactic magnetic fieldcosmic ray escapegrammagetest particle simulationPeV spectrum
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The pith

If parallel diffusion becomes energy-independent above 1 TeV, drift motions can produce the cosmic ray knee at PeV energies.

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

Cosmic ray transport in the Galaxy involves both diffusion along magnetic field lines and drift motions caused by the curvature and gradients of the large-scale field. The paper uses test-particle simulations in a model Galactic magnetic field to compute how particles escape and accumulate grammage between TeV and 10 PeV. In a purely azimuthal field, drifts create a knee in the spectrum at PeV energies but the model underpredicts the grammage. When the field includes a component perpendicular to the disc, parallel diffusion takes over and prevents a knee unless the parallel diffusion coefficient becomes independent of energy above 1 TeV, in which case drifts can again dominate and generate the knee.

Core claim

The paper shows that the knee can arise from the transition to drift-dominated transport when the parallel diffusion coefficient becomes energy independent above approximately 1 TeV. This occurs because drifts due to magnetic field curvature then compete effectively with diffusion for particle escape, leading to an energy-dependent escape time that steepens the spectrum at PeV energies. The result holds only under specific conditions on the large-scale field topology and the turbulence properties.

What carries the argument

Drift motions induced by the curvature and gradients of the large-scale Galactic magnetic field competing with parallel and perpendicular diffusion, with the key transition when parallel diffusion turns energy-independent.

Load-bearing premise

The parallel diffusion coefficient becomes energy independent at energies above roughly 1 TeV.

What would settle it

Direct or indirect evidence that the parallel diffusion coefficient continues to scale with energy above 1 TeV would prevent drift dominance from producing the knee.

Figures

Figures reproduced from arXiv: 2606.28141 by Carmelo Evoli, Luis Enrique Espinosa Castro, Pasquale Blasi.

Figure 1
Figure 1. Figure 1: FIG. 1. Three-dimensional diagrams of coherent magnetic field geometry in our simulations for a fully-azimuthal antisymmetric [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Parallel (red), perpendicular (green) and antisymmetric (blue) coefficients of the diffusion tensor (Eq. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Cosmic ray escape time (red) and grammage (blue) as function of particle energy for a simulation box with [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Left panel: Corresponding escape times as function of particle energy for simulations with [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Left panel: Cosmic ray escape time (red) and grammage (blue) as function of particle energy for a simulation box [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Cosmic ray escape time (red) and grammage (blue) as function of particle energy for a simulation box with [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Cosmic ray escape time (red) and grammage (blue) as function of particle energy for a simulation box with [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Cosmic ray escape time (red) and grammage (blue) as function of particle energy for a simulation box with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

In a magnetic field with a complex topology, as can be the Galactic magnetic field, cosmic ray transport cannot simply be described by diffusion parallel and perpendicular to magnetic field lines, because the gradients and curvature of the large-scale magnetic field induce drift motions. These effects become especially important at high energies. Here we revisit the possibility that the competition between diffusion and drifts may lead to a knee in the cosmic ray spectrum. We carry out test-particle simulations of cosmic ray transport in a mock Galactic magnetic field made of a regular large scale component, with a non-trivial topology and a homogeneous and isotropic turbulent magnetic field, with a spectrum that is assumed to be Kolmogorov-like in the basic setup. These simulations are used to infer the escape time and the grammage accumulated by cosmic rays with energy in the TeV--10 PeV energy range. In the case of a large scale magnetic field with a purely azimuthal structure, the drift due to the curvature of magnetic field lines produces a knee in the PeV range, but the model fails to reproduce the grammage, due to the exceedingly low value of the perpendicular diffusion coefficient. If the large scale magnetic field acquires a component perpendicular to the Galactic disc, the parallel diffusion coefficient becomes quickly dominant in terms of particle escape, and drifts are unable to compete. A knee structure does not appear in such a scenario. However, if the parallel diffusion coefficient becomes energy independent at $E\gtrsim 1$ TeV, a knee may arise around PeV energies due to drift dominance. We discuss two cases in which this situation may occur.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper performs test-particle simulations of cosmic ray propagation in a mock Galactic magnetic field consisting of a large-scale regular component (azimuthal or with a perpendicular tilt) plus a homogeneous isotropic Kolmogorov turbulent field. Escape times and grammage are computed in the TeV–10 PeV range to investigate whether the transition to drift-dominated transport, driven by curvature and gradient drifts, can produce the observed knee. A knee appears in the PeV range only for a purely azimuthal large-scale field when the parallel diffusion coefficient is assumed to become energy-independent above ~1 TeV; the tilted-field case suppresses drifts, and the azimuthal case fails to match observed grammage.

Significance. If the energy-independent parallel diffusion regime can be shown to arise self-consistently, the work would provide a transport-based mechanism for the knee that depends on realistic magnetic-field topology rather than source properties alone. The test-particle approach in a non-trivial field geometry is a concrete step toward quantifying drift versus diffusion competition, but the current results rest on an external modeling assumption whose compatibility with the simulated turbulence remains untested.

major comments (2)
  1. [Abstract] Abstract: The production of a knee at PeV energies is stated to require that the parallel diffusion coefficient becomes energy-independent for E ≳ 1 TeV so that drifts can dominate escape. This condition is introduced as an external requirement rather than shown to emerge from the Kolmogorov turbulence spectrum used in the simulations; standard quasi-linear theory for isotropic Kolmogorov turbulence predicts D∥ ∝ E^{1/3} (or steeper) continuing through the TeV–PeV range, so the knee appears only under an added assumption whose self-consistency is not demonstrated.
  2. [Abstract] Abstract: In the purely azimuthal large-scale field geometry the curvature drift produces a knee, yet the model yields an exceedingly low perpendicular diffusion coefficient that prevents reproduction of the observed grammage; this internal inconsistency undermines the viability of the drift-knee scenario even in the configuration where the spectral feature appears.
minor comments (1)
  1. [Abstract] The abstract supplies only qualitative outcomes and does not report quantitative values for escape times, grammage, knee position, or error estimates, which limits assessment of the numerical robustness of the claimed transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting the key assumptions and limitations in our work. Below we respond point-by-point to the major comments. We agree that the energy-independent parallel diffusion regime is an external assumption and that the azimuthal-field case does not reproduce observed grammage; the manuscript already flags the latter as a limitation. We will revise the text to make these points more explicit while preserving the exploratory nature of the study.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The production of a knee at PeV energies is stated to require that the parallel diffusion coefficient becomes energy-independent for E ≳ 1 TeV so that drifts can dominate escape. This condition is introduced as an external requirement rather than shown to emerge from the Kolmogorov turbulence spectrum used in the simulations; standard quasi-linear theory for isotropic Kolmogorov turbulence predicts D∥ ∝ E^{1/3} (or steeper) continuing through the TeV–PeV range, so the knee appears only under an added assumption whose self-consistency is not demonstrated.

    Authors: We agree that the transition to energy-independent parallel diffusion is introduced as an external modeling assumption rather than derived from the simulated Kolmogorov turbulence. The manuscript does not claim that this regime arises self-consistently within quasi-linear theory; instead, the abstract and discussion section explicitly condition the appearance of the knee on this assumption and outline two possible physical contexts (e.g., non-linear wave-particle interactions or deviations from standard QLT) where such a flattening might occur. We will revise the abstract and conclusions to state more clearly that the knee is obtained only under this added hypothesis and that demonstrating its emergence from the turbulence remains outside the scope of the present test-particle study. revision: partial

  2. Referee: [Abstract] Abstract: In the purely azimuthal large-scale field geometry the curvature drift produces a knee, yet the model yields an exceedingly low perpendicular diffusion coefficient that prevents reproduction of the observed grammage; this internal inconsistency undermines the viability of the drift-knee scenario even in the configuration where the spectral feature appears.

    Authors: The manuscript already states in the abstract and in the results section that the purely azimuthal geometry produces a knee via curvature drift but fails to match the observed grammage because of the very low perpendicular diffusion coefficient. We present this as an intrinsic limitation of that particular large-scale field topology. The referee correctly identifies this as an internal tension that weakens the scenario in the only geometry where the knee appears. We will expand the discussion in the conclusions to emphasize that this mismatch indicates the drift-knee mechanism, as implemented here, is not viable without additional ingredients (such as a stronger perpendicular diffusion or a different field geometry) and will add a quantitative comparison of the computed grammage to observational constraints. revision: yes

Circularity Check

0 steps flagged

No circularity: knee produced conditionally on external assumption about D_parallel(E), not by construction from inputs

full rationale

The paper performs forward test-particle simulations in a mock GMF (regular azimuthal or tilted component plus isotropic Kolmogorov turbulence) to compute escape times and grammage for TeV–PeV particles. The knee appears only under the explicit additional assumption that D_∥ becomes energy-independent above ~1 TeV, allowing drifts to dominate; this assumption is introduced as a condition whose possible origins are discussed separately rather than derived from the simulated turbulence spectrum or field geometry. No equation or result reduces to a fitted parameter, self-citation chain, or redefinition of the target feature. The derivation chain is therefore self-contained against the stated simulation inputs and assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full simulation parameters, turbulence spectrum details, and magnetic field components are not specified, limiting the ledger to explicitly mentioned assumptions.

free parameters (1)
  • energy threshold for energy-independent parallel diffusion
    Introduced to allow drift dominance above ~1 TeV and produce the knee; value chosen to match the desired spectral feature.
axioms (1)
  • domain assumption Turbulent magnetic field follows a Kolmogorov-like spectrum
    Standard assumption invoked for the basic setup of the turbulent component.

pith-pipeline@v0.9.1-grok · 5824 in / 1169 out tokens · 50361 ms · 2026-06-29T02:34:46.205375+00:00 · methodology

discussion (0)

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Works this paper leans on

74 extracted references · 26 linked inside Pith

  1. [1]

    Adriani, G

    O. Adriani, G. C. Barbarino, G. A. Bazilevskaya et al., Science332, 69 (2011), arXiv:1103.4055 [astro-ph.HE]

    Pith/arXiv arXiv 2011

  2. [2]

    Y. S. Yoon, H. S. Ahn, P. S. Allison et al., ApJ728, 122 (2011), arXiv:1102.2575 [astro-ph.HE]

    Pith/arXiv arXiv 2011

  3. [3]

    Aguilar, D

    M. Aguilar, D. Aisa, B. Alpat et al., Phys. Rev. Lett.114, 171103 (2015)

    2015

  4. [4]

    Y. S. Yoon, T. Anderson, A. Barrau et al., ApJ839, 5 (2017), arXiv:1704.02512 [astro-ph.HE]

    Pith/arXiv arXiv 2017

  5. [5]

    Q. An, R. Asfandiyarov, P. Azzarello et al., Science Advances5, eaax3793 (2019), arXiv:1909.12860 [astro-ph.HE]

    arXiv 2019

  6. [6]

    Aguilar, L

    M. Aguilar, L. Ali Cavasonza, G. Ambrosi et al., Phys. Rep.894, 1 (2021)

    2021

  7. [7]

    Alemanno, Q

    F. Alemanno, Q. An, P. Azzarello et al., Phys. Rev. Lett.126, 201102 (2021), arXiv:2105.09073 [astro-ph.HE]

    arXiv 2021

  8. [8]

    Adriani, Y

    O. Adriani, Y. Akaike, K. Asano et al., Phys. Rev. Lett.129, 101102 (2022), arXiv:2209.01302 [astro-ph.HE]

    arXiv 2022

  9. [9]

    Adriani, Y

    O. Adriani, Y. Akaike, K. Asano et al., Phys. Rev. Lett.130, 171002 (2023), arXiv:2304.14699 [astro-ph.HE]

    arXiv 2023

  10. [10]

    Antoni, W

    T. Antoni, W. D. Apel, A. F. Badea et al., Astroparticle Physics24, 1 (2005), arXiv:astro-ph/0505413 [astro-ph]

    Pith/arXiv arXiv 2005

  11. [11]

    W. D. Apel, J. C. Arteaga-Velázquez, K. Bekk et al., Phys. Rev. Lett.107, 171104 (2011), arXiv:1107.5885 [astro-ph.HE]

    Pith/arXiv arXiv 2011

  12. [12]

    W. D. Apel, J. C. Arteaga-Velázquez, K. Bekk et al., Astroparticle Physics47, 54 (2013), arXiv:1306.6283 [astro-ph.HE]

    Pith/arXiv arXiv 2013

  13. [13]

    Amenomori, X

    M. Amenomori, X. J. Bi, D. Chen et al., ApJ678, 1165 (2008), arXiv:0801.1803 [hep-ex]

    arXiv 2008

  14. [14]

    M. G. Aartsen, M. Ackermann, J. Adams et al., Phys. Rev. D100, 082002 (2019), arXiv:1906.04317 [astro-ph.HE]. 15

    arXiv 2019

  15. [15]

    N. M. Budnev, A. Chiavassa, O. A. Gress et al., Astroparticle Physics117, 102406 (2020), arXiv:2104.03599 [astro-ph.HE]

    arXiv 2020

  16. [16]

    R. U. Abbasi, M. Abe, T. Abu-Zayyad et al., ApJ865, 74 (2018), arXiv:1803.01288 [astro-ph.HE]

    arXiv 2018

  17. [17]

    Z. Cao, F. Aharonian, Axikegu et al., Phys. Rev. Lett.132, 131002 (2024), arXiv:2403.10010 [astro-ph.HE]

    arXiv 2024

  18. [18]

    The LHAASO Collaboration, Z. Cao, F. Aharonian et al., arXiv e-prints , arXiv:2505.14447 (2025), arXiv:2505.14447 [astro-ph.HE]

    arXiv 2025

  19. [19]

    Z. Cao, F. Aharonian, Y. X. Bai et al., Phys. Rev. Lett.136, 121001 (2026), arXiv:2511.05013 [astro-ph.HE]

    arXiv 2026

  20. [20]

    Peters, Il Nuovo Cimento22, 800 (1961)

    B. Peters, Il Nuovo Cimento22, 800 (1961)

    1961

  21. [21]

    J. R. Hörandel, Astroparticle Physics21, 241 (2004), arXiv:astro-ph/0402356 [astro-ph]

    Pith/arXiv arXiv 2004

  22. [22]

    Cristofari, P

    P. Cristofari, P. Blasi and E. Amato, Astroparticle Physics123, 102492 (2020), arXiv:2007.04294 [astro-ph.HE]

    arXiv 2020

  23. [23]

    Aharonian, R

    F. Aharonian, R. Yang and E. de Oña Wilhelmi, Nature Astronomy3, 561 (2019), arXiv:1804.02331 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  24. [24]

    S. Kaci, G. Giacinti, F. Aharonian et al., arXiv e-prints , arXiv:2510.01369 (2025), arXiv:2510.01369 [astro-ph.HE]

    arXiv 2025

  25. [25]

    B. T. Zhang and S. Yu, arXiv e-prints , arXiv:2602.08940 (2026), arXiv:2602.08940 [astro-ph.HE]

    arXiv 2026

  26. [26]

    Cocconi, Il Nuovo Cimento3, 1433 (1956)

    G. Cocconi, Il Nuovo Cimento3, 1433 (1956)

    1956

  27. [27]

    De Marco, P

    D. De Marco, P. Blasi and T. Stanev, J. Cosmology Astropart. Phys.2007, 027 (2007), arXiv:0705.1972 [astro-ph]

    Pith/arXiv arXiv 2007

  28. [28]

    Giacinti, M

    G. Giacinti, M. Kachelrieß and D. V. Semikoz, Phys. Rev. D91, 083009 (2015), arXiv:1502.01608 [astro-ph.HE]

    Pith/arXiv arXiv 2015

  29. [29]

    Subedi, W

    P. Subedi, W. Sonsrettee, P. Blasi et al., ApJ837, 140 (2017), arXiv:1612.09507 [physics.space-ph]

    Pith/arXiv arXiv 2017

  30. [30]

    Dundovic, O

    A. Dundovic, O. Pezzi, P. Blasi et al., Phys. Rev. D102, 103016 (2020), arXiv:2007.09142 [astro-ph.HE]

    arXiv 2020

  31. [31]

    V. S. Ptuskin, S. I. Rogovaya, V. N. Zirakashvili et al., A&A268, 726 (1993)

    1993

  32. [32]

    P. A. Isenberg and J. R. Jokipii, ApJ234, 746 (1979)

    1979

  33. [33]

    Candia, E

    J. Candia, E. Roulet and L. N. Epele, Journal of High Energy Physics2002, 033 (2002), arXiv:astro-ph/0206336 [astro-ph]

    Pith/arXiv arXiv 2002

  34. [34]

    Candia and E

    J. Candia and E. Roulet, J. Cosmology Astropart. Phys.2004, 007 (2004), arXiv:astro-ph/0408054 [astro-ph]

    Pith/arXiv arXiv 2004

  35. [35]

    Evoli, D

    C. Evoli, D. Grasso and L. Maccione, J. Cosmology Astropart. Phys.2007, 003 (2007), arXiv:astro-ph/0701856 [astro-ph]

    Pith/arXiv arXiv 2007

  36. [36]

    Giacalone and J

    J. Giacalone and J. R. Jokipii, ApJ520, 204 (1999)

    1999

  37. [37]

    J. R. Jokipii, ApJ146, 480 (1966)

    1966

  38. [38]

    Blasi, Proc

    P. Blasi, Proc. Int. Sch. Phys. Fermi208, 41 (2024), arXiv:2307.11640 [astro-ph.HE]

    arXiv 2024

  39. [39]

    R. C. Tautz and A. Shalchi, ApJ744, 125 (2012)

    2012

  40. [40]

    Rossi and S

    B. Rossi and S. Olbert,Introduction to the physics of space.(1970)

    1970

  41. [41]

    Ptuskin, V

    V. Ptuskin, V. Zirakashvili and E.-S. Seo, ApJ718, 31 (2010), arXiv:1006.0034 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  42. [42]

    K. M. Schure and A. R. Bell, MNRAS435, 1174 (2013), arXiv:1307.6575 [astro-ph.HE]

    Pith/arXiv arXiv 2013

  43. [43]

    Caprioli, Proc

    D. Caprioli, Proc. Int. Sch. Phys. Fermi208, 143 (2024), arXiv:2307.00284 [astro-ph.HE]

    arXiv 2024

  44. [44]

    Alves Batista, J

    R. Alves Batista, J. Becker Tjus, J. Dörner et al., J. Cosmology Astropart. Phys.2022, 035 (2022), arXiv:2208.00107 [astro-ph.HE]

    arXiv 2022

  45. [45]

    Haverkorn, J

    M. Haverkorn, J. C. Brown, B. M. Gaensler et al., ApJ680, 362 (2008), arXiv:0802.2740 [astro-ph]

    Pith/arXiv arXiv 2008

  46. [46]

    Iacobelli, M

    M. Iacobelli, M. Haverkorn, E. Orrú et al., A&A558, A72 (2013), arXiv:1308.2804 [astro-ph.GA]

    Pith/arXiv arXiv 2013

  47. [47]

    Casse, M

    F. Casse, M. Lemoine and G. Pelletier, Phys. Rev. D65, 023002 (2001), arXiv:astro-ph/0109223 [astro-ph]

    Pith/arXiv arXiv 2001

  48. [48]

    Kuhlen, P

    M. Kuhlen, P. Mertsch and V. H. M. Phan, ApJ992, 11 (2025), arXiv:2211.05882 [astro-ph.HE]

    arXiv 2025

  49. [49]

    Kuhlen, P

    M. Kuhlen, P. Mertsch and V. H. M. Phan, ApJ992, 10 (2025), arXiv:2211.05881 [astro-ph.HE]

    arXiv 2025

  50. [50]

    J. W. Bieber and W. H. Matthaeus, ApJ485, 655 (1997)

    1997

  51. [51]

    Evoli and U

    C. Evoli and U. Dupletsa, Proc. Int. Sch. Phys. Fermi208, 257 (2024), arXiv:2309.00298 [astro-ph.HE]

    arXiv 2024

  52. [52]

    Tibaldo, S

    L. Tibaldo, S. W. Digel, J. M. Casandjian et al., ApJ807, 161 (2015), arXiv:1505.04223 [astro-ph.HE]

    Pith/arXiv arXiv 2015

  53. [53]

    Evoli, G

    C. Evoli, G. Morlino, P. Blasi et al., Phys. Rev. D101, 023013 (2020), arXiv:1910.04113 [astro-ph.HE]

    arXiv 2020

  54. [54]

    Weinrich, M

    N. Weinrich, M. Boudaud, L. Derome et al., A&A639, A74 (2020), arXiv:2004.00441 [astro-ph.HE]

    arXiv 2020

  55. [55]

    Ferrière, Mem

    K. Ferrière, Mem. Soc. Astron. Italiana82, 824 (2011)

    2011

  56. [56]

    Unger and G

    M. Unger and G. R. Farrar, ApJ970, 95 (2024), arXiv:2311.12120 [astro-ph.GA]

    arXiv 2024

  57. [57]

    Haverkorn, arXiv e-prints , arXiv:2606.02149 (2026), arXiv:2606.02149 [astro-ph.GA]

    M. Haverkorn, arXiv e-prints , arXiv:2606.02149 (2026), arXiv:2606.02149 [astro-ph.GA]

    Pith/arXiv arXiv 2026

  58. [58]

    J. L. Han, Ap&SS278, 181 (2001), arXiv:astro-ph/0110319 [astro-ph]

    Pith/arXiv arXiv 2001

  59. [59]

    M. S. Pshirkov, P. G. Tinyakov, P. P. Kronberg et al., ApJ738, 192 (2011), arXiv:1103.0814 [astro-ph.GA]

    Pith/arXiv arXiv 2011

  60. [60]

    Xu and J

    J. Xu and J. L. Han, ApJ966, 240 (2024), arXiv:2404.02038 [astro-ph.GA]

    arXiv 2024

  61. [61]

    Shalchi, ApJ881, L27 (2019), arXiv:1908.00694 [astro-ph.SR]

    A. Shalchi, ApJ881, L27 (2019), arXiv:1908.00694 [astro-ph.SR]

    arXiv 2019

  62. [62]

    Schroer, C

    B. Schroer, C. Evoli and P. Blasi, Phys. Rev. D103, 123010 (2021), arXiv:2102.12576 [astro-ph.HE]

    arXiv 2021

  63. [63]

    Giacinti, M

    G. Giacinti, M. KachelrieSS and D. V. Semikoz, J. Cosmology Astropart. Phys.2018, 051 (2018), arXiv:1710.08205 [astro-ph.HE]

    Pith/arXiv arXiv 2018

  64. [64]

    Lemoine, Journal of Plasma Physics89, 175890501 (2023), arXiv:2304.03023 [physics.plasm-ph]

    M. Lemoine, Journal of Plasma Physics89, 175890501 (2023), arXiv:2304.03023 [physics.plasm-ph]

    arXiv 2023

  65. [65]

    Kempski, D

    P. Kempski, D. B. Fielding, E. Quataert et al., MNRAS525, 4985 (2023), arXiv:2304.12335 [astro-ph.HE]

    arXiv 2023

  66. [66]

    Kempski, D

    P. Kempski, D. Li, D. B. Fielding et al., ApJ990, L18 (2025), arXiv:2412.03649 [astro-ph.HE]

    arXiv 2025

  67. [67]

    Dampe Collaboration, Science Bulletin67, 2162 (2022), arXiv:2210.08833 [astro-ph.HE]

    arXiv 2022

  68. [68]

    Grebenyuk, D

    V. Grebenyuk, D. Karmanov, I. Kovalev et al., Advances in Space Research64, 2559 (2019), arXiv:1809.09665 [astro-ph.HE]

    arXiv 2019

  69. [69]

    Recchia and S

    S. Recchia and S. Gabici, A&A692, A20 (2024), arXiv:2312.11397 [astro-ph.HE]

    arXiv 2024

  70. [70]

    Ambrosone, C

    A. Ambrosone, C. Evoli, B. Schroer et al., A&A698, L18 (2025), arXiv:2503.14651 [astro-ph.HE]

    arXiv 2025

  71. [71]

    Yang and F

    R. Yang and F. Aharonian, Phys. Rev. D111, 083040 (2025), arXiv:2410.22199 [astro-ph.HE]

    arXiv 2025

  72. [72]

    Pezzi and P

    O. Pezzi and P. Blasi, MNRAS529, L13 (2024), arXiv:2305.02890 [astro-ph.HE]

    arXiv 2024

  73. [73]

    A. J. Farmer and P. Goldreich, ApJ604, 671 (2004), arXiv:astro-ph/0311400 [astro-ph]

    Pith/arXiv arXiv 2004

  74. [74]

    Blasi, E

    P. Blasi, E. Amato and P. D. Serpico, Phys. Rev. Lett.109, 061101 (2012), arXiv:1207.3706 [astro-ph.HE]

    Pith/arXiv arXiv 2012