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

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