Galactic Diffuse $\gamma$-ray Emission at TeV Energies and the Ultra-High Energy Cosmic Rays

Using the cosmic ray (CR) data available in the energy interval $(10 - 2 \times 10^{7})$GeV\mathrm{/}particle,wehavecalculatedtheprofileoftheprimary$\gamma$-ray spectrum produced by the interaction of these CR with thermal nuclei of the ISM. Normalized to the EGRET measurements, this allows an estimate of the galactic diffuse $\gamma$-ray background due to intermediate and high energy CR at TeV energies. On the other hand, over the last few years, several particles with energies above ${10}^{20}$ eV (beyond the Greisen-Zatsepin-Kuzmin cut-off) have been detected. These particles are very likely extragalactic protons originated at distances not greater than $30-50$ Mpc [e.g., 1]. The propagation of these ultra-high energy protons (UHEP) through the intergalactic medium leads to the development of $\gamma$-ray cascades and an ultimate signature at TeV energies. To assess the statistical significance of this $\gamma$-ray signature by the UHEP, we have also simulated the development of electromagnetic cascades triggered by the decay of a ${10}^{19}$ eV ${\pi }^o$ in the intergalactic medium after an UHEP collision with a cosmic microwave background photon.