Delayed GeV--TeV Photons from Gamma-Ray Bursts Producing High-Energy Cosmic Rays

A scenario in which cosmic rays (CRs) above $10^{20}{\rm eV}$ are produced by cosmological gamma-ray bursts (GRBs) is consistent with observations provided that deflections by the inter-galactic magnetic field (IGMF) delay and spread the arrival time of the CRs over $\geq50{\rm yr}$. The energy lost by the CRs as they propagate and interact with the microwave background is transformed by cascading into secondary GeV-TeV photons. We show that a significant fraction of these photons can arrive with delays much smaller than the CR delay if much of inter-galactic space is occupied by large-scale magnetic ``voids'', regions of size $\gtrsim5{\rm Mpc}$ and field weaker than $10^{-15}{\rm G}$. Such voids might be expected, for example, in models where a weak primordial field is amplified in shocked, turbulent regions of the intergalactic medium during the formation of large-scale structure. For a field strength $\sim4\times10^{-11}{\rm G}$ in the high field regions, the value required to account for observed galactic fields if the IGMF were frozen in the protogalactic plasma, the delay of CRs produced by a burst at a distance of $100{\rm Mpc}$ is $\sim100{\rm yr}$, and the fluence of secondary photons above $10{\rm GeV}$ on hour--day time scales is $I(>E)\sim10^{-6}E_{\rm TeV}^{-1}{\rm cm}^{-2}$. This fluence is close to the detection threshold of current high-energy $\gamma$-ray experiments. Detection of the delayed flux would support the GRB-CR association and would also provide information on the IGMF structure.