Cascade photons as test of protons in UHECR

An isotropic component of high energy $\gamma$-ray spectrum measured by Fermi LAT constrains the proton component of UHECR. The strongest restriction comes from the highest, $(580-820)$ GeV, energy bin. One more constraint on the proton component is provided by the IceCube upper bound on ultrahigh energy cosmogenic neutrino flux. We study the influence of these restrictions on the source properties, such as evolution and distribution of sources, their energy spectrum and admixture of nuclei. We also study the sensitivity of restrictions to various Fermi LAT galactic foreground models (model B being less restrictive), to the choice of extragalactic background light model and to overall normalization of the energy spectrum. We claim that the $\gamma$-ray-cascade constraints are stronger than the neutrino ones, and that however many proton models are viable. The basic parameters of such models are relatively large $\gamma_g$ and not very large $z_{\max}$. The allowance for H$e^4$ admixture also relaxes the restrictions. However we foresee that future CTA measurements of $\gamma$-ray spectrum at $E_\gamma \simeq (600 - 800)$ GeV, as well as resolving of more individual $\gamma$-ray sources, may rule out the proton-dominated cosmic ray models.