tokens: spectrum would be expected from diffusive shock accelerations, but harder spectra are possible when the particle energy is converted from the electromagnetic energy of the sources, for example, in pulsar magnetospheres (Arons 2003 ), or in reconnection processes (Guo et al. 2014 ). Besides, an interplay between the acceleration and dissipation processes at the acceleration site could also lead to a hard injection spectrum (Dermer et al. 2012 ; Murase et al. 2012 ). Murase et al. ( 2016 ) and Bechtol et al. ( 2015 ) demonstrated that a pp scenario with a spectrum normalized to the IceCube datapoint at ∼10 TeV would overproduce γ rays above the flux of the non-blazar component of the isotropic γ -ray background. In this work, we will instead consider the case of an injection spectrum index equal or smaller than two, but normalized to the IceCube datapoint at 30 TeV (which is about three times lower than that at 10 TeV). Specifically, in Section 3 , we focus on a benchmark case of an injection spectrum and maximum energy . We discuss the effects of different choices of dN / dE and in Section 4 . 2.2. Cosmic-Ray Diffusion Galaxy clusters are known as efficient containers of cosmic rays (Völk et al. 1996 ; Berezinsky et al. 1997 ). Depending on the particle’s Larmor radius r L and the field’s coherence length l c , the propagation of cosmic rays in the cluster turbulent magnetic field can be divided into semi-diffusive () and diffusive () regimes (Kotera Lemoine 2008 ). When a particle’s energy reaches , the particle enters the semi-diffusive regime with a diffusion coefficient ofHowever, most particles have energy much lower than , and thus stay in the diffusive regime. The diffusion coefficient in this regime can be written as (Brunetti Jones 2014 )where is the spectrum index for Kolmogorov diffusion, and are the typical amplitude and scale of magnetic field fluctuations in massive clusters (Brunetti Jones 2015 ). The diffusion time of cosmic rays in the cluster magnetic field can be estimated asCosmic rays with diffusion time longer than the Hubble time would be completely confined by the cluster magnetic field, defining the lower limit of the energy of cosmic rays we expect to observe from the cluster as Particles with would have a better chance to escape from the cluster magnetic field. To estimate this escape probability, we first write down the probability that a particle with diffusion coefficient D reaches radius r after time t :assuming that the particle starts from r = 0 and the field is homogenous. The probability that a particle can successfully escape from a cluster of size within the Hubble time is then (Kotera Lemoine 2008 ) Cosmic rays accelerated in the strong shocks in the accretion region are expected to have a spectrum index . The flux of high-energy cosmic rays from a cluster with mass at redshift z = 0.1 can be calculated aswhere d L is the luminosity distance, and f π is the pion production rate that will be introduced
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