A generalized study of linear electromagnetic cascades in astrophysical sources

TL;DR

This work develops a generalized theory of electromagnetic cascades inside astrophysical sources, extending Berezinsky's IC-dominated picture to scenarios where synchrotron losses dominate. By identifying three cascade regimes—equal-reproduction, soft-radiation, and cooling-only—the authors show that a universal cascade spectrum with a low-energy slope near 3/2 and a high-energy slope near 2 can emerge in many environments, particularly when synchrotron losses prevail. They validate the analytic expectations with AM^3 simulations across AGN coronae, GRBs, blazars, and TDEs, and delineate clear conditions under which universality holds or breaks, such as target-field broadbandness or broad energy-range injections. The results have practical implications for interpreting high-energy emission and neutrino production in gamma-ray–opaque sources, and offer diagnostic handles on source magnetization, compactness, and acceleration mechanisms.

Abstract

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.

Figures (8)

• Figure 1: IC-dominated cascade for a monochromatic photon target. Lepton (left) and photon (right) distributions within the interaction region are shown; the setup is described in the main text. Top: luminosity; bottom: interaction and escape timescales (IC scattering, escape, and $\gamma\gamma$ pair production). Thin black lines denote the primary injected species; thick lines show the cascade components. The different regimes of the pair cascade in the left plot are marked by distinct background colors. In the right plot, the same colors indicate the energy ranges where photon injection is dominated by the corresponding regime. In the equal-reproduction regime, pairs and photons split in energy equally. In the cooling-only regime, no pairs are injected; existing pairs cool with an index $s_e=2$, producing photons with $s_\gamma=3/2$. In the escape-dominated regime, pairs escape before cooling significantly, and their spectrum is suppressed.
• Figure 2: Synchrotron-dominated cascade for a monochromatic photon target. Same as Fig. \ref{['fig:ic_cascade']} for a synchrotron-dominated case; the timescales now include also synchrotron radiation and SSA. The setup is described in the main text. At high energies, we have the soft-radiation regime, with the photons attenuated by pair production and the pairs settling into the universal $s_e=3$ state, producing photons with $s_\gamma=2$. Below the threshold for pair production, the pairs enter the cooling-only regime, with $s_e=2$, and producing photons with $s_\gamma=3/2$. In the escape-dominated regime, here achieved for a very narrow low-energy range, pairs escape nearly freely.
• Figure 3: Electromagnetic cascade in an AGN-corona-inspired hadronic model. Lepton (left) and photon (right) spectra in the coronal benchmark from Table \ref{['tab:corona_parameters']}. Top: luminosity; bottom: interaction and escape timescales (synchrotron radiation, IC scattering, SSA, Compton scattering, and $\gamma\gamma$ pair production). Pair-production–driven components are colored; non-cascade particles are black. Background shading marks cascade regimes for pairs, and for photons the synchrotron emission from leptons in the corresponding regime. Most pairs are injected at $\varepsilon_e \simeq \gamma_{p,\rm br} m_p$ and promptly enter the soft-radiation cascade, with synchrotron-dominated losses, yielding a flat $s_\gamma=2$ spectrum. The non-cascade pairs come mostly from BH, and is subdominant (the sharp BH drop reflects numerical-method matching in AM$^3$ ).
• Figure 4: Electromagnetic cascade in a GRB-inspired lepto-hadronic model. Same as Fig. \ref{['fig:corona']} for a GRB benchmark. Leptonic and hadronic contributions are distinguished by line style. Background shading indicates the cascade regimes for the pairs: a high-energy soft-radiation regime (set by pair production and synchrotron cooling) and a low-energy cooling-only regime without pair injection. For photons, the background shading highlights the energy range where photons are produced by leptons in the corresponding cascade regime. Particles which do not come from $\gamma\gamma$ (non-cascade) are shown in black.
• Figure 5: Electromagnetic cascade in a blazar-inspired lepto-hadronic model. Same as Fig. \ref{['fig:grb']} for a blazar benchmark. The numerical values of the parameters are collected in Table \ref{['tab:blazar_parameters']}. Particles which do not come from $\gamma\gamma$ (non-cascade) are shown in black. The pairs transition from the soft-radiation cascade at high energies to the escape-dominated regime at low energies, producing the characteristic cascade with $s_\gamma=2$.