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 bycosmological gamma-ray bursts (GRBs) is consistent with observations providedthat deflections by the inter-galactic magnetic field (IGMF) delay and spreadthe arrival time of the CRs over $\geq50{\rm yr}$. The energy lost by the CRsas they propagate and interact with the microwave background is transformed bycascading into secondary GeV-TeV photons. We show that a significant fractionof these photons can arrive with delays much smaller than the CR delay if muchof inter-galactic space is occupied by large-scale magnetic ``voids'', regionsof size $\gtrsim5{\rm Mpc}$ and field weaker than $10^{-15}{\rm G}$. Such voidsmight be expected, for example, in models where a weak primordial field isamplified in shocked, turbulent regions of the intergalactic medium during theformation of large-scale structure. For a field strength$\sim4\times10^{-11}{\rm G}$ in the high field regions, the value required toaccount for observed galactic fields if the IGMF were frozen in theprotogalactic 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 $I(>E)\sim10^{-6}E_{\rm TeV}^{-1}{\rmcm}^{-2}$. This fluence is close to the detection threshold of currenthigh-energy $\gamma$-ray experiments. Detection of the delayed flux wouldsupport the GRB-CR association and would also provide information on the IGMFstructure.Comment: ApJ Letters in press; 10 pages (LaTeX file, 2 PostScript figures, aaspp4.sty style file, tar'ed, gzip'ed and uuencoded