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 cosmic microwave background and extragalactic background light 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 the 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 ${\omega }_{\mathrm{cas}}$ 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 ${\omega }_{\mathrm{cas}}$ stronger than in previous works. The new feature of the analysis is the “$E_{\max }$ rule.” We investigate the dependence of ${\omega }_{\mathrm{cas}}$ on the distribution of sources, distinguishing two cases of universality: the strong and weak ones.

• Received 26 May 2016

DOI:https://doi.org/10.1103/PhysRevD.94.023007