  The observed fluxes of cosmic rays and gamma rays are used to infer the
maximum allowed high-energy neutrino flux allowed for Gamma Ray Bursts (GRBs),
following Mannheim, Protheroe, and Rachen (2000). It is shown that if GRBs
produce the ultrahigh-energy cosmic rays, they should contribute (a) at least
10% of the extragalactic gamma ray background between 3 MeV and 30 GeV,
contrary to their observed energy flux which is only a minute fraction of this
flux, and (b) a cumulative neutrino flux a factor of 20 below the AMANDA
(Neutrino 2000) limit on isotropic neutrinos. This could have two implications,
either GRBs do not produce the ultrahigh energy cosmic rays or that the GRBs
are strongly beamed and emit most of their power at energies well above 100 GeV
implausibly increasing the energy requirements, but consistent with the
marginal detections of a few low-redshift GRBs by MILAGRITO, HEGRA-AIROBICC,
and the Tibet-Array. All crucial measurements to test the models will be
available in the next few years. These are measurements of (i) high-energy
neutrinos with AMANDA-ICECUBE or an enlarged ANTARES/NESTOR ocean detector,
(ii) GRB redshifts from HETE-2 follow-up studies, and (iii) GRB spectra above
10 GeV with low-threshold imaging air Cherenkov telescopes such as MAGIC and
the space telescopes AGILE and GLAST.
