The IceCube collaboration recently reported the potential detection of two
cascade neutrino events in the energy range 1-10 PeV. We study the possibility
that these PeV neutrinos are produced by gamma-ray bursts (GRBs), paying
special attention to the contribution by untriggered GRBs that elude detection
due to their low photon flux. Based on the luminosity function, rate
distribution with redshift and spectral properties of GRBs, we generate, using
Monte-Carlo simulation, a GRB sample that reproduce the observed fluence
distribution of Fermi/GBM GRBs and an accompanying sample of untriggered GRBs
simultaneously. The neutrino flux of every individual GRBs is calculated in the
standard internal shock scenario, so that the accumulative flux of the whole
samples can be obtained. We find that the neutrino flux in PeV energies
produced by untriggered GRBs is about 2 times higher than that produced by the
triggered ones. Considering the existing IceCube limit on the neutrino flux of
triggered GRBs, we find that the total flux of triggered and untriggered GRBs
can reach at most a level of ~10^-9 GeV cm^-2 s^-1 sr^-1, which is insufficient
to account for the reported two PeV neutrinos. Possible contributions to
diffuse neutrinos by low-luminosity GRBs and the earliest population of GRBs
are also discussed.Comment: Accepted by ApJ, one more figure added to show the contribution to
the diffuse neutrino flux by untriggered GRBs with different luminosity,
results and conclusions unchange
