Neutrino and Cosmic-Ray Release from Gamma-Ray Bursts: Time-Dependent Simulations

We revisit the neutrino and ultra high-energy cosmic ray (UHECR) production from gamma-ray bursts (GRBs) with time-dependent simulations for the proton-induced cascades. This method can generate self-consistent photon, neutrino and escaped neutron spectra. To obtain the integrated background spectra, we take into account the distributions of the burst luminosity and pulse duration timescale. A benchmark case with standard GRB luminosity function, a bulk Lorentz factor $\Gamma=300$ and a proton to gamma-ray luminosity fraction $f_{\rm p}=10$, is consistent with both the neutrino upper-limits and the observed UHECR intensity at $\sim 10^{20}$ eV, while requiring a different type of UHECR source at the ankle. For the benchmark case the GRBs in the bright end of the luminosity function, which contribute most of the neutrinos, have their photon spectrum substantially distorted by secondary photons. Such bright GRBs are few in number, and reducing their $f_p$ eliminates the distortion, while reducing the neutrino production. Even if we neglect the contribution of the brightest GRBs, the UHECR production rate at GZK energies is almost unchanged. These nominal GRB models, especially with $L_{\rm iso} \lesssim 10^{53} ~\mbox{erg} ~\mbox{s}^{-1}$, appear to meet the current constraints as far as being candidate UHECR sources above the ankle energy.