Excess GeV radiation and cosmic ray origin

Recent EGRET observations of the diffuse Galactic gamma-ray emission reveal a spectrum which is incompatible with the assumption that the cosmic ray spectra measured locally hold throughout the Galaxy: the spectrum above 1 GeV, where the emission is supposedly dominated by pi^0-decay, is harder than that derived from the local cosmic ray proton spectrum. We demonstrate that in case of a SNR origin of cosmic ray nucleons part of this gamma-ray excess may be attributed to the dispersion of the spectral indices in these objects. In global averages, as are gamma-ray line-of-sight integrals, this dispersion leads to a positive curvature in the composite spectrum, and hence to modified pi^0-decay gamma-ray spectra.