The role of hadronic cascades in GRB models of efficient neutrino production

We investigate the effects of hadronic cascades on the gamma-ray burst (GRB) prompt emission spectra in scenarios of efficient neutrino production. By assuming a fiducial GRB spectrum and a power-law proton distribution extending to ultra-high energies, we calculate the proton cooling rate and the neutrino emission produced through photopion processes. For this, we employ a numerical code that follows the formation of the hadronic cascade by taking into account non-linear feedback effects, such as the evolution of the target photon field itself due to the contribution of secondary particles. We show that in cases of efficient proton cooling and subsequently efficient high-energy neutrino production, the emission from the hadronic cascade distorts and may even dominate the GRB spectrum. Taking this into account, we constrain the allowable values of the ratio $\eta_p=L_p/L_{\gamma}$, where $L_{p}$ and $L_{\gamma}$ are the isotropic equivalent proton and prompt gamma-ray luminosities. For the highest value of $\eta_{p}$ that does not lead to the dominance of the cascading emission, we then calculate the maximum neutrino luminosity from a single burst and show that it ranges between $(0.01- 0.6)L_{p}$ and $(0.5-1.4)L_{\gamma}$ for various parameter sets. We discuss possible implications of other parameters, such as the magnetic field strength and the shape of the initial gamma-ray spectrum, on our results. Finally, we compare the upper limit on $\eta_p$ derived here with various studies in the field, and we point out the necessity of a self-consistent treatment of the hadronic emission in order to avoid erroneously high neutrino fluxes from GRB models.