High energy astrophysical neutrinos: The upper bound is robust

We elucidate the physical basis for the upper bound on high energy neutrino fluxes implied by the observed cosmic ray flux. We stress that the bound is valid for neutrinos produced either by $p,\gamma$ reactions or by $p-p\left(n\right)\mathrm{}$ reactions in sources which are optically thin for high energy protons to photo-meson and nucleon-meson interactions. We show that the upper bound is robust and conservative. The Waxman-Bahcall bound overestimates the most likely neutrino flux by a factor $\sim 5/\tau ,$ for small optical depths $\tau .$ The upper limit cannot be plausibly evaded by invoking magnetic fields, optically thick active galactic nuclei (AGNs), or large hidden fluxes of extragalactic protons. We describe the implications of the bound for future experiments including the AMANDA, ANTARES, Auger, ICECUBE, NESTOR, and OWL/AIRWATCH detectors.