We present a search for high energy gamma-ray emission from 9 nearby starburst galaxies and M31 with the EGRET instrument aboard CGRO. Though the diffuse gamma-ray emission from starburst galaxies was suspected to be detectable, we find no emission from NGC 253, M82 nor from the average of all 9 galaxies. The 2 sigma upper limit for the EGRET flux above 100 MeV for the averaged survey observations is 1.8 x 10-8 ph cm-2 s-1. From a model of the expected radio and gamma-ray emission, we find that the magnetic field in the nuclei of these galaxies is > 25 micro Gauss, and the ratio of proton and electron densities is < 400. The EGRET limits indicate that the rate of massive star formation in the survey galaxies is only about an order of magnitude higher than in the Milky Way. The upper limit to the gamma-ray flux above 100 MeV for M31 is 1.6 x 10-8 ph cm-2 s-1. At the distance of M31, the Milky Way flux would be over twice this value, indicating higher gamma-ray emissivities in our Galaxy. Therefore, since the supernova rate of the Milky Way is higher than in M31, our null detection of M31 supports the theory of the supernova origin of cosmic rays in galaxies.
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