Signatures of the Origin of High-Energy Cosmic Rays in Cosmological Gamma-Ray Bursts

We derive observational consequences of the hypothesis that cosmic rays (CR's) of energy $>10^{19}eV$ originate in the same cosmological objects producing gamma-ray bursts (GRB's). Inter-galactic magnetic fields $\gtrsim 10^{-12} G$ are required in this model to allow CR's to be observed continuously in time by producing energy dependent delays in the CR arrival times. This results in individual CR sources having very narrow observed spectra, since at any given time only those CR's having a fixed time delay are observed. Thus, the brightest CR sources should be different at different energies. The average number of sources contributing to the total CR flux decreases with energy much more rapidly than in a model of steady CR sources, dropping to one at $E_{crit} \simeq2\times10^{20}$~eV with very weak sensitivity to the inter-galactic magnetic field strength. Below $E_{crit}$, a very large number of sources is expected, consistent with observations. Above $E_{crit}$, a source may be observed with a flux considerably higher than the time-averaged CR flux from all sources, if a nearby GRB occurred recently. If such a source is present, its narrow spectrum may produce a ``gap'' in the total spectrum. These signatures should be detectable by the planned ``Auger'' CR experiment.