Cosmological Origin for Cosmic Rays Above $10^{19}$ eV

The cosmic ray spectrum at $10^{19}{\rm eV}-10^{20}{\rm eV}$, reported by the Fly's Eye and the AGASA experiments, is shown to be consistent with a cosmological distribution of sources of protons, with a power law generation spectrum ${\rm d}\ln N/{\rm d}\ln E=-2.3\pm0.5$ and energy production rate of $4.5\pm1.5\times10^{44}{\rm erg}\ {\rm Mpc}^{-3}\ {\rm yr}^{-1}$. The two events measured above $10^{20}{\rm eV}$ are not inconsistent with this model. Verifying the existence of a ``black-body cutoff'', currently observed with low significance, would require $\sim30$ observation-years with existing experiments, but only $\sim1$ year with the proposed $\sim5000\ {\rm km}^2$ detectors. For a cosmological source distribution, no anisotropy is expected in the angular distribution of events with energies up to $\sim5\times10^{19}{\rm eV}$.