The Limited Contribution of Low- and High-luminosity Gamma-Ray Bursts to Ultra-high-energy Cosmic Rays

The acceleration site for ultra-high-energy cosmic rays (UHECRs) is still an open question despite extended research. In this paper, we reconsider the prompt phase of gamma-ray bursts (GRBs) as a possible candidate for this acceleration and constrain the maximum proton energy in optically thin synchrotron and photospheric models, using properties of the prompt photon spectra. We find that neither of the models favors acceleration of protons to 10²⁰ eV in high-luminosity bursts. We repeat the calculations for low-luminosity GRBs (llGRBs) considering both protons and completely stripped iron and find that the highest obtainable energies are <10¹⁹ eV and <10²⁰ eV for protons and iron respectively, regardless of the model. We conclude therefore that for our fiducial parameters, GRBs, including low-luminosity bursts, contribute little to nothing to the UHECRs observed. We further constrain the conditions necessary for an association between UHECRs and llGRBs and find that iron can be accelerated to 10²⁰ eV in photospheric models, given very efficient acceleration and/or a small fractional energy given to a small fraction of accelerated electrons. This will necessarily result in high prompt optical fluxes, and the detection of such a signal could therefore be an indication of successful UHECR acceleration at the source.