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

Abstract

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 $\alpha \ge 0$), 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, $R_{SF}$, with $R_{CCSN}(z)\propto R_{SF}(z)$, our conservative upper limits are $f_{GRB/CCSN}(0)\le 5.0\times 10^{-3}$ for $\alpha =0$, and $f_{GRB/CCSN}(0)\le 1.1\times 10^{-3}$ 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. 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 $\times 10^{-5}$ 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.

Two references added, an overall constant numerical factor corrected, Figures and relevant portions of abstract and main text slightly changed, main conclusions unchanged, 18 pages Latex with 4 Figures, version accepted for publication in PRD

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