The increasingly deep limit on the neutrino emission from gamma-ray bursts
(GRBs) with IceCube observations has reached the level that could put useful
constraints on the fireball properties. We first present a revised analytic
calculation of the neutrino flux, which predicts a flux an order of magnitude
lower than that obtained by the IceCube collaboration. For benchmark model
parameters (e.g. the bulk Lorentz factor is \Gamma=10^{2.5}, the observed
variability time for long GRBs is t_v=0.01 s and the ratio between the energy
in accelerated protons and in radiation is \eta_p=10 for every burst) in the
standard internal shock scenario, the predicted neutrino flux from 215 bursts
during the period of the 40-string and 59-string configurations is found to be
a factor of ~3 below the IceCube sensitivity. However, if we accept the
recently found inherent relation between the bulk Lorentz factor and burst
energy, the expected neutrino flux increases significantly and the spectral
peak shifts to lower energy. In this case, the non-detection then implies that
the baryon loading ratio should be \eta_p<10 if the variability time of long
GRBs is fixed to t_v=0.01 s. Instead, if we relax the standard internal shock
scenario but keep to assume \eta_p=10, the non-detection constrains the
dissipation radius to be R>4x10^{12} cm assuming the same dissipation radius
for every burst and benchmark parameters for fireballs. We also calculate the
diffuse neutrino flux from GRBs for different luminosity functions existing in
the literature. The expected flux exceeds the current IceCube limit for some
luminosity functions, and thus the non-detection constrains \eta_p<10 in such
cases when the variability time of long GRBs is fixed to t_v=0.01 s.Comment: Accepted by ApJ, 14 pages, 5 figures, typos corrected, scheduled for
the June 10, 2012, v752 - 1 issu