We derive upper limits on the ratio $f_{\mathrm{GRB}/\mathrm{CCSN}}(z)\equiv R_{\mathrm{GRB}}(z)/R_{\mathrm{CCSN}}(z)\equiv f_{\mathrm{GRB}/\mathrm{CCSN}}(0)(1+z{)}^{\alpha }$, the ratio of the rate, $R_{\mathrm{GRB}}$, of long-duration gamma-ray bursts (GRBs) to the rate, $R_{\mathrm{CCSN}}$, of core-collapse supernovae (CCSNe) in the Universe ($z$ being the cosmological redshift and $\alpha \ge 0$), 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, $R_{\mathrm{SF}}$, with $R_{\mathrm{CCSN}}(z)\propto R_{\mathrm{SF}}(z)$, our conservative upper limits are $f_{\mathrm{GRB}/\mathrm{CCSN}}(0)\le 5.0\times {10}^{-3}$ for $\alpha =0$, and $f_{\mathrm{GRB}/\mathrm{CCSN}}(0)\le 1.1\times {10}^{-3}$ for $\alpha =2$, for example. These limits are already comparable to (and, for $\alpha \ge 1$, 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. Nondetection of a diffuse TeV–PeV neutrino background by the upcoming IceCube detector in the South Pole after three years of operation, for example, will bring down the upper limit on $f_{\mathrm{GRB}/\mathrm{CCSN}}(0)$ to below a few $\times {10}^{-5}$ 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.

• Received 1 November 2007

DOI:https://doi.org/10.1103/PhysRevD.77.043008