Ultra-relativistic shocks in gamma-ray burst and active-galactic-nucleus jets are long-standing candidate sources of ultra-high-energy cosmic rays (UHECRs), yet linear theory and particle-in-cell simulations indicate an intrinsic ceiling on the energy attainable when scattering is mediated only by plasma skin-depth-scale turbulence. Reville & Bell (2014) showed that the cosmic-ray-current-driven instabilities that could amplify magnetic field on much larger scales upstream and downstream of such shocks grow too slowly, at the gyroradius of the highest-energy particles, to provide the scattering needed to lift this ceiling, and called for self-consistent kinetic simulations to confirm or refute this conclusion. Twelve years of follow-up work has qualified the argument (the limit is geometry-dependent, downstream turbulence is not strictly necessary for acceleration, and self-generated coherence scales grow secularly in the longest simulations to date) but has not delivered the self-consistent calculation at genuinely ultra-relativistic, weakly-magnetised, electron-ion shocks that settles whether ions can be accelerated beyond the intrinsic limit to ultra-high energies. The problem is to determine, self-consistently, the nonlinear evolution of the self-generated turbulence and the resulting maximum particle energy at such shocks, and thereby decide whether weakly-magnetised ultra-relativistic shocks are viable UHECR accelerators.

Background

Cosmic rays are charged particles (mostly protons and nuclei) observed at Earth up to energies of about 1020 eV10^{20}\,\mathrm{eV}; those above roughly 1018 eV10^{18}\,\mathrm{eV} are called ultra-high-energy cosmic rays (UHECRs), and their sources are still unidentified. The best-established acceleration mechanism is diffusive shock acceleration (first-order Fermi acceleration): particles that repeatedly scatter back and forth across a shock front gain energy on each crossing cycle, and confinement requires magnetic turbulence that scatters them on either side of the shock. Ultra-relativistic shocks, with Lorentz factor Γsh≫1\Gamma_{\rm sh}\gg 1, occur in the outflows of gamma-ray bursts (GRBs) and active galactic nuclei (AGN), and because any large-scale upstream magnetic field is generically oriented in the shock plane, they are usually weakly magnetised and quasi-perpendicular, with magnetization parameter σ\sigma (the ratio of incoming magnetic to kinetic energy density) far below unity.

In the absence of a strong ordered field, such shocks are mediated by kinetic plasma instabilities, chiefly the Weibel (current-filamentation) instability, an instability of interpenetrating currents that generates magnetic structure at the proton skin depth c/ωppc/\omega_{pp} (the scale over which protons can shield electric fields, with ωpp\omega_{pp} the proton plasma frequency). Particle-in-cell (PIC) simulations, the standard first-principles kinetic plasma technique, show that particles scattering on these skin-depth-scale fields undergo small-angle scattering rather than isotropic diffusion, so the maximum energy grows only as t1/2t^{1/2} rather than linearly (the Bohm limit), and saturates at a value scaling as σ−1/4\sigma^{-1/4} when the ambient field matters (Sironi, Spitkovsky & Arons 2013). Building on Lemoine & Pelletier (2010), Reville & Bell (2014) sharpened this into an intrinsic limit: accelerated particles must be isotropised in the downstream plasma frame before the compressed mean field carries them away, and if scattering operates only at scale λd\lambda_d near the skin depth, the maximum energy is

Emax⁡≃(Γsh100)2(λd10 c/ωpp)(σd10−2)(σu10−8)−1/2 PeV,E_{\max}\simeq \left(\frac{\Gamma_{\rm sh}}{100}\right)^{2}\left(\frac{\lambda_d}{10\,c/\omega_{pp}}\right)\left(\frac{\sigma_d}{10^{-2}}\right)\left(\frac{\sigma_u}{10^{-8}}\right)^{-1/2}\,\mathrm{PeV},

where σu\sigma_u and σd\sigma_d are the upstream and downstream magnetizations. This falls short of UHECR energies by orders of magnitude for plausible GRB parameters. Reville & Bell then asked whether cosmic-ray-driven instabilities could supply the missing large-scale turbulence: the returning (shock-reflected) accelerated particles carry a current that can drive non-resonant Bell-type modes (instabilities in which the cosmic-ray current amplifies magnetic field on scales shorter than the driving particles' gyroradius), both upstream and downstream. Their linear dispersion analysis and order-of-magnitude growth-time estimates, with an efficiency parameter η\eta capturing the returning-particle density and its spatial/energy profile, showed that on the wavelengths comparable to the gyroradius of the highest-energy particles, every such instability grows too slowly to modify the scattering before the flow advects it away. They concluded that the limit cannot be circumvented, disfavouring weakly-magnetised ultra-relativistic shocks as high-energy particle accelerators in general, while explicitly leaving the nonlinear evolution beyond the reach of their linear treatment and calling for future self-consistent, large-scale kinetic simulations as the decisive test.

Problem Statement

Determine, by self-consistent kinetic calculation (large-scale simulation or controlled equivalent), whether weakly-magnetised (σu≲10−3\sigma_u\lesssim10^{-3}) ultra-relativistic (Γsh≫1\Gamma_{\rm sh}\gg1) collisionless shocks can accelerate protons and nuclei beyond the intrinsic limit

Emax⁡≃(Γsh100)2(λd10 c/ωpp)(σd10−2)(σu10−8)−1/2 PeV,E_{\max}\simeq \left(\frac{\Gamma_{\rm sh}}{100}\right)^{2}\left(\frac{\lambda_d}{10\,c/\omega_{pp}}\right)\left(\frac{\sigma_d}{10^{-2}}\right)\left(\frac{\sigma_u}{10^{-8}}\right)^{-1/2}\,\mathrm{PeV},

where Γsh\Gamma_{\rm sh} is the shock Lorentz factor, σu\sigma_u and σd\sigma_d are the upstream and downstream magnetization parameters (the ratios of magnetic to kinetic energy density), λd\lambda_d is the wavelength of the dominant downstream scattering turbulence, and c/ωppc/\omega_{pp} is the proton skin depth. This limit is imposed by scattering on plasma skin-depth-scale turbulence alone; the task is to decide whether it can be circumvented, and thereby whether such shocks are viable sources of ultra-high-energy cosmic rays (UHECRs, meaning cosmic rays above roughly 1018 eV10^{18}\,\mathrm{eV}). This requires resolving, nonlinearly and self-consistently, the questions Reville & Bell (2014) could address only linearly: (i) whether the nonlinear evolution of the cosmic-ray-current-driven instabilities upstream and downstream transfers magnetic energy to wavelengths comparable to the gyroradius of the highest-energy particles fast enough to provide the strong large-scale scattering the linear growth estimates found too slow, including its dependence on the returning-particle spectrum and spatial profile (the efficiency parameter η\eta); (ii) whether pre-existing large-scale ambient field structure, non-uniform injection across the shock surface, or environmental preconditioning of the upstream medium can seed or supply the required large-scale turbulence in realistic source environments; and (iii) whether anisotropic downstream cosmic-ray distributions interacting with the relativistically hot downstream plasma drive instabilities more effective than the linear estimates indicate. The problem is complete when the maximum ion energy at such shocks is established relative to the intrinsic limit, with the turbulence spectrum, coherence scale, and acceleration history tracked up to the gyroradius scale of the highest-energy particles, for representative weakly-magnetised ultra-relativistic environments (e.g. GRB external shocks and relativistic AGN jets); a demonstration that the limit is intrinsic and cannot be circumvented in realistic environments equally solves the problem.

References

  1. Reville, B. & Bell, A. R. 2014, 'On the maximum energy of shock-accelerated cosmic rays at ultra-relativistic shocks', MNRAS 439, 2050, DOI 10.1093/mnras/stu088, https://arxiv.org/abs/1401.2803
  2. Lemoine, M. & Pelletier, G. 2010, 'On electromagnetic instabilities at ultra-relativistic shock waves', MNRAS 402, 321, DOI 10.1111/j.1365-2966.2009.15869.x, https://arxiv.org/abs/0904.2657
  3. Sironi, L., Spitkovsky, A. & Arons, J. 2013, 'The Maximum Energy of Accelerated Particles in Relativistic Collisionless Shocks', ApJ 771, 54, DOI 10.1088/0004-637X/771/1/54, https://arxiv.org/abs/1301.5333
  4. Bell, A. R., Araido, A. T., Matthews, J. H. & Blundell, K. M. 2017, 'Cosmic ray acceleration by relativistic shocks: limits and estimates', MNRAS 473, 2364, DOI 10.1093/mnras/stx2485, https://arxiv.org/abs/1709.07793
  5. Plotnikov, I., Grassi, A. & Grech, M. 2018, 'Perpendicular relativistic shocks in magnetized pair plasma', MNRAS 477, 5238, DOI 10.1093/mnras/sty979, https://arxiv.org/abs/1712.02883
  6. Huang, Z.-Q., Kirk, J. G., Giacinti, G. & Reville, B. 2022, 'The implications of TeV detected GRB afterglows for acceleration at relativistic shocks', ApJ 925, 182, DOI 10.3847/1538-4357/ac3f38, https://arxiv.org/abs/2112.00111
  7. Kirk, J. G., Reville, B. & Huang, Z.-Q. 2022, 'Particle acceleration at ultrarelativistic, perpendicular shock fronts', MNRAS 519, 1022, DOI 10.1093/mnras/stac3589, https://arxiv.org/abs/2212.02349
  8. Huang, Z.-Q., Reville, B., Kirk, J. G. & Giacinti, G. 2023, 'Prospects for ultra-high-energy particle acceleration at relativistic shocks', MNRAS 522, 4955, DOI 10.1093/mnras/stad1356, https://arxiv.org/abs/2304.08132
  9. Bresci, V., Gremillet, L. & Lemoine, M. 2022, 'Saturation of the asymmetric current filamentation instability under conditions relevant to relativistic shock precursors', Physical Review E 105, 035202, DOI 10.1103/PhysRevE.105.035202, https://arxiv.org/abs/2111.04651
  10. Bresci, V., Lemoine, M. & Gremillet, L. 2023, 'Particle acceleration at magnetized, relativistic, turbulent shock fronts', Physical Review Research 5, 023194, DOI 10.1103/PhysRevResearch.5.023194, https://arxiv.org/abs/2303.11394
  11. Grošelj, D., Sironi, L. & Spitkovsky, A. 2024, 'Long-term Evolution of Relativistic Unmagnetized Collisionless Shocks', ApJL 963, L44, DOI 10.3847/2041-8213/ad2c8c, https://arxiv.org/abs/2401.02392
  12. Golant, R., Vanthieghem, A., Grošelj, D. & Sironi, L. 2025, 'Generation of Large-scale Magnetic Fields Upstream of Gamma-Ray Burst Afterglow Shocks', ApJ 986, 211, DOI 10.3847/1538-4357/add404, https://arxiv.org/abs/2410.05388
  13. Jikei, T., Grošelj, D. & Sironi, L. 2026, 'Magnetic Field Amplification and Particle Acceleration in Weakly Magnetized Transrelativistic Electron–Ion Shocks', ApJ 998, 149, DOI 10.3847/1538-4357/ae3723, https://arxiv.org/abs/2512.03169
  14. Wu, Z.-F., Guevara-Montoya, S., Beniamini, P., Giannios, D., Grošelj, D. & Sironi, L. 2026, 'Maximum Energy of Particles Accelerated in Gamma-Ray Burst Afterglow Shocks', ApJ 1003, 242, DOI 10.3847/1538-4357/ae664b, https://arxiv.org/abs/2601.19135