We have developed a numerical model for the temporal evolution of particle and photon spectra resulting from nonthermal processes at the shock fronts formed in merging clusters of galaxies. Fermi acceleration is approximated by injecting power-law distributions of particles during a merger event, subject to constraints on maximum particle energies. We consider synchrotron, bremsstrahlung, Compton, and Coulomb processes for the electrons, nuclear, photomeson, and Coulomb processes for the protons, and knock-on electron production during the merging process. The broadband radio through γ-ray emission radiated by nonthermal protons and primary and secondary electrons is calculated both during and after the merger event. Using ROSAT observations to establish typical parameters for the matter density profile of clusters of galaxies, we find that typical merger shocks are weak and accelerate particles with relatively soft spectra. We consider the prospects for detecting nonthermal radio and γ-ray emission from clusters of galaxies and implications for the origin of ultra-high-energy cosmic rays and the diffuse γ-ray background. Our results suggest that only a few of the isotropically distributed unidentified EGRET sources are due to shocks formed in cluster mergers and that only a minor contribution to the diffuse extragalactic γ-ray background can originate from cluster merger shocks. Cluster merger shocks can accelerate protons to approx<10~(19) eV for the standard parameters considered here. We predict that GLAST will detect several cluster mergers, and depending on the mean magnetic fields in the intracluster medium, the Low Frequency Array could detect anywhere from several to several hundred.
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