In this work we use a multi-messenger approach to determine if the high
energy diffuse neutrino flux observed by the IceCube Observatory can originate
from γ-ray sources powered by Cosmic Rays interactions with gas. Typical
representatives of such sources are Starburst and Ultra-Luminous Infrared
Galaxies. Using the three most recent calculations of the non-blazar
contribution to the extragalactic γ-ray background measured by the
Fermi-LAT collaboration, we find that a hard power-law spectrum with spectral
index α≤2.12 is compatible with all the estimations for the allowed
contribution from non-blazar sources, within 1σ. Using such a spectrum
we are able to interpret the IceCube results, showing that various classes of
hadronically powered γ-ray galaxies can provide the dominant
contribution to the astrophysical signal above 100 TeV and about half of the
contribution to the energy flux between 10-100 TeV. With the addition of
neutrinos from the Galactic plane, it is possible to saturate the IceCube
signal at high energy. Our result shows that these sources are still well
motivated candidates.Comment: Accepted for publication on JCA