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 \times 10 ^ { -3 } for \alpha = 0 , and f _ { { GRB } / { CCSN } } ( 0 ) \leq 1.1 \times 10 ^ { -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 \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 .