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.
