Time structure of gamma-ray signals generated in line-of-sight interactions of cosmic rays from distant blazars

Blazars are expected to produce both gamma rays and cosmic rays. Therefore, observed high-energy gamma rays from distant blazars may contain a significant contribution from secondary gamma rays produced along the line of sight by the interactions of cosmic-ray protons with background photons. Unlike the standard models of blazars that consider only the primary photons emitted at the source, models which include the cosmic-ray contribution predict that even ~10 TeV photons should be detectable from distant objects with redshifts as high as z> 0.1. Secondary photons contribute to signals of point sources only if the intergalactic magnetic fields are very small, below ~10 femtogauss, and their detection can be used to set upper bounds on magnetic fields along the line of sight. Secondary gamma rays have distinct spectral and temporal features. We explore the temporal properties of such signals using a semi-analytical formalism and detailed numerical simulations, which account for all the relevant processes, including magnetic deflections. In particular, we elucidate the interplay of time delays coming from the proton deflections and from the electromagnetic cascade, and we find that, at multi-TeV energies, secondary gamma-rays can show variability on timescales of years for femtogauss magnetic fields.