  We obtain the maximum diffuse neutrino intensity predicted by hadronic
photoproduction models of the type which have been applied to the jets of
active galactic nuclei or gamma ray bursts. For this, we compare the proton and
gamma ray fluxes associated with hadronic photoproduction in extragalactic
neutrino sources with the present experimental upper limit on cosmic ray
protons and the extragalactic gamma ray background, employing a transport
calculation of energetic protons traversing cosmic photon backgrounds. We take
into account the effects of the photon spectral shape in the sources on the
photoproduction process, cosmological source evolution, the optical depth for
cosmic ray ejection, and discuss the possible effects of magnetic fields in the
vicinity of the sources. For photohadronic neutrino sources which are optically
thin to the emission of neutrons we find that the cosmic ray flux imposes a
stronger bound than the extragalactic gamma ray background in the energy range
between 10^5 GeV and 10^11 GeV, as previously noted by Waxman & Bahcall (1999).
We also determine the maximum contribution from the jets of active galactic
nuclei, using constraints set to their neutron opacity by gamma-ray
observations. This present upper limit is consistent with the jets of active
galactic nuclei producing the extragalactic gamma ray background hadronically,
but we point out future observations in the GeV-to-TeV regime could lower this
limit. We also briefly discuss the contribution of gamma ray bursts to
ultra-high energy cosmic rays as it can be inferred from possible observations
or limits on their correlated neutrino fluxes.
