Upper Limit on the Cosmic Gamma-Ray Burst Rate from High Energy Diffuse Neutrino Background

We derive upper limits on the ratio $f_{GRB/CCSN}(z) \equiv R_{GRB}(z)/R_{CCSN}(z) \equiv f_{GRB/CCSN}(0)(1+z)^\alpha$, the ratio of the rate, $R_{GRB}$, of long-duration Gamma Ray Bursts (GRBs) to the rate, $R_{CCSN}$, of core-collapse supernovae (CCSNe) in the Universe ($z$ being the cosmological redshift and $\alpha\geq 0$), by using the upper limit on the diffuse TeV--PeV neutrino background given by the AMANDA-II experiment in the South Pole, under the assumption that GRBs are sources of TeV--PeV neutrinos produced from decay of charged pions produced in $p\gamma$ interaction of protons accelerated to ultrahigh energies at internal shocks within GRB jets. For the assumed ``concordance model'' of cosmic star formation rate, $R_{SF}$, with $R_{CCSN}(z) \propto R_{SF}(z)$, our conservative upper limits are $f_{GRB/CCSN}(0)\leq 5.0\times10^{-3}$ for $\alpha=0$, and $f_{GRB/CCSN}(0)\leq 1.1\times10^{-3}$ for $\alpha=2$, for example. These limits are already comparable to (and, for $\alpha\geq 1$ already more restrictive than) the current upper limit on this ratio inferred from other astronomical considerations, thus providing a useful independent probe of and constraint on the CCSN-GRB connection. Non-detection of a diffuse TeV--PeV neutrino background by the up-coming IceCube detector in the South pole after three years of operation, for example, will bring down the upper limit on $f_{GRB/CCSN}(0)$ to below few $\times10^{-5}$ level, while a detection will confirm the hypothesis of proton acceleration to ultrahigh energies in GRBs and will potentially also yield the true rate of occurrence of these events in the Universe.