High energy neutrinos from astrophysical sources: An upper bound

We show that cosmic-ray observations set a model-independent upper bound of ${E}_{\ensuremath{\nu}}^{2}{\ensuremath{\Phi}}_{\ensuremath{\nu}}<2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}{\mathrm{G}\mathrm{e}\mathrm{V}/\mathrm{c}\mathrm{m}}^{2}\mathrm{}\mathrm{s}\mathrm{}\mathrm{sr}$ to the intensity of high-energy neutrinos produced by photo-meson (or $p\ensuremath{-}p)$ interactions in sources of size not much larger than the proton photo-meson (or $p\ensuremath{-}p)$ mean-free-path. This bound applies, in particular, to neutrino production by either AGN jets or GRBs. The upper limit is two orders of magnitude below the intensity predicted in some popular AGN jet models and therefore contradicts the theory that the cosmic gamma-ray background is due to photo-pion interactions in AGN jets. The upper bound is consistent with our predictions from GRB models. The predicted intensity from GRBs is ${E}^{2}dN/dE\ensuremath{\sim}0.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}{\mathrm{G}\mathrm{e}\mathrm{V}/\mathrm{c}\mathrm{m}}^{2}\mathrm{}\mathrm{s}\mathrm{}\mathrm{sr}$ for ${10}^{14}\mathrm{eV}<E<{10}^{16}\mathrm{eV};$ we also derive the expected intensity at higher energy.