High energy electromagnetic cascades in extragalactic space: physics and   features
  Using the analytic modeling of the electromagnetic cascades compared with more precise numerical simulations we describe the physical properties of electromagnetic cascades developing in the universe on CMB and EBL background radiations. A cascade is initiated by very high energy photon or electron and the remnant photons at large distance have two-component energy spectrum, $\propto E^{-2}$ ($\propto E^{-1.9}$ in numerical simulations) produced at cascade multiplication stage, and $\propto E^{-3/2}$ from Inverse Compton electron cooling at low energies. The most noticeable property of the cascade spectrum in analytic modeling is 'strong universality', which includes the standard energy spectrum and the energy density of the cascade $\omega_{\rm cas}$ as its only numerical parameter. Using numerical simulations of the cascade spectrum and comparing it with recent Fermi LAT spectrum we obtained the upper limit on $\omega_{\rm cas}$ stronger than in previous works. The new feature of the analysis is "$E_{\max}$ rule". We investigate the dependence of $\omega_{\rm cas}$ on the distribution of sources, distinguishing two cases of universality: the strong and weak ones. 
V. Berezinsky and O. Kalashev
astro-ph.HE