UHECR (E_p ∼ 10¹⁸−10²⁰ eV) injection model. (a) Distribution of the angle between the initial emission axis and the momentum vector of secondary UHECR and cascade photons at ε_γ > 10 GeV, after propagation in the magnetized region. (b) PDF of the time-delay distribution of secondary UHECRs and γ-rays emerging from the magnetized region with respect to the rectilinear travel time. (c) The solid line shows the secondary UHE proton spectrum after propagation through the galaxy-cluster region, and the dotted line shows the associated cascade γ-ray flux. Further propagation through the IGM produces the cosmogenic γ-ray flux at Earth, shown by the dashed line. The fluxes are normalized to 10 injected protons.