Particle acceleration at ultra-relativistic shocks - Gamma-ray burst afterglow spectra and UHECRs

We consider particle acceleration of the Fermi type at the ultra-relativistic shock bounding an expanding relativistic fireball. We distinguish between the initial shock crossing cycle, in which particles can gain a large factor in energy, and subsequent crossing cycles, where the energy typically only doubles. We show that repeated shock crossings yield a power-law spectrum of accelerated particles with index p similar to 2.2 - 2.3, compatible with afterglow observations. We suggest that ultra-high-energy cosmic rays (UHECRs) might be produced by the initial boost in a relativistic fireball expanding into a pulsar wind bubble created by the progenitor system.