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

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