Diffusion of ultra-high energy protons in galaxy clusters and secondary X- and gamma-ray emissions

In this work we simulate the propagation of ultra-high energy (UHE) protons in the magnetised intergalactic medium of galaxy clusters (GCs). Differently from previous works on the subject, we trace proton trajectories in configurations of the intra-cluster magnetic field (ICMF), which have been extracted from a constrained magnetic-SPH simulation of the local universe. Such an approach allows us to take into account the effects of several features of the ICMFs, e.g. irregular geometrical structure and field fluctuations due to merger shocks, which cannot be investigated analytically or with usual numerical simulations. Furthermore, we are able to simulate a set of clusters, which have properties quite similar to those of GCs observed in the nearby universe. We estimate the time that UHE protons take to get out of the clusters and found that in the energy range 5 × 1018 ≲ E ≲ 3 × 1019 eV proton propagation takes place in the Bohm scattering diffusion regime passing smoothly to a small pitch angle diffusion regime at larger energies. We apply our results to estimate the secondary gamma- and hard X-ray (HXR) emissions produced by UHE protons in a rich GC. We show that the main emission channel is due to the synchrotron HXR radiation of secondary electrons originated by proton photo-pair production scattering onto the CMB. This process may give rise to a detectable signal if a relatively powerful AGN, or a dead quasar, accelerating protons at UHEs is harboured by a rich GC in the local universe.