Icecube nondetection of gamma-ray bursts: Constraints on the fireball properties

The increasingly deep limit on the neutrino emission from gamma-ray bursts (GRBs) with IceCube observations has reached a level that could place useful constraints on the fireball properties. We first present a revised analytic calculation of the neutrino flux that predicts a flux of one order of magnitude lower than that obtained by the IceCube Collaboration. For the benchmark model parameters (e.g., the bulk Lorentz factor is Γ = 10^2.5, the observed variability time for the long GRBs is t ^ob _v = 0.01 s, and the ratio between the energy in the accelerated protons and in the radiation is η_p = 10 for every burst) in the standard internal shock scenario, the predicted neutrino flux from 215 bursts during the period of the 40- and 59-string configurations is a factor of 3 below the IceCube sensitivity. However, if we accept the recently found inherent relation between the bulk Lorentz factor and the burst energy, then the expected neutrino flux significantly increases and the spectral peak shifts to a lower energy. In this case, the nondetection implies that the baryon-loading ratio should be η_p ≲ 10 if the variability time of the long GRBs is fixed to t ^ob _v = 0.01 s. Instead, if we relax the standard internal-shock scenario but still assume η_p = 10, then the nondetection constrains the dissipation radius, R ≳ 4 × 10 ¹² cm, assuming the same dissipation radius for every burst and benchmark parameters for the fireballs. We also calculate the diffuse neutrino flux from the GRBs for different luminosity functions from the literature. The expected flux exceeds the current IceCube limit for some of the luminosity functions, and, thus, the nondetection constrains η_p ≲ 10 when the variability time of the long GRBs is fixed at t ^ob _v = 0.01 s.