A generalized study of linear electromagnetic cascades in astrophysical sources

High-energy gamma rays can trigger electromagnetic cascades via pair production on ambient photons, reprocessing their energy to lower frequencies. A classic example is the cascade from the gamma rays produced by ultra-high-energy cosmic rays in extragalactic photon fields, whose universal spectral shape was first described by Berezinsky in the 1970s. Recently, internal cascades, developing within the gamma-ray sources themselves, have gained a prominent role, as the IceCube data suggest that most detected neutrinos originate in gamma-ray-opaque environments. We analyze under what conditions these internal cascades can approach a universal spectrum. Since the Berezinsky treatment breaks down if synchrotron losses dominate, we present a generalized theory incorporating synchrotron-dominated cascades. We show the emergence of universal cascade spectrum among various examples of high-energy sources containing non-thermal cosmic rays, and discuss the conditions for its appearance.