Cosmic-ray protons interacting with gas at the mean density of the interstellar medium (ISM) in starburst galaxies lose energy rapidly via inelastic collisions with ambient nuclei. The resulting pions produce secondary electrons and positrons, high-energy neutrinos, and γ-ray photons. We estimate the cumulative γ-ray emission from starburst galaxies. We find a total integrated background above 100 MeVof F_γ ≈ 10~(-6) GeV cm~(-2) s~(-1) sr~(-1) and a corresponding specific intensity at GeV energies of vI_v ≈ 10~(-7) GeV cm~(-2) s~(-1) sr~(-1). Starbursts may thus account for a significant fraction of the extragalactic γ-ray background. We show that the FIR-radio correlation provides a strong constraint on the γ-ray emission from starburst galaxies because pions decay into both γ-rays and radio-emitting electron/ positron pairs. We identify several nearby systems where the potential for observing γ-ray emission is the most favorable (M82, NGC 253, and IC 342), predict their fluxes, and predict a linear FIR- γ-ray correlation for the densest starbursts. If established, the FIR-γ-ray correlation would provide strong evidence for the "calorimeter" theory of the FIR-radio correlation and would imply that cosmic rays in starburst galaxies interact with gas at approximately the mean density of the ISM, thereby providing an important constraint on the physics of the ISM in starbursts.
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