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Ultra-high cosmic rays (UHECRs) with energies >10^19 eV emitted atcosmological distances will be attenuated by cosmic microwave and infraredbackground radiation through photohadronic processes. Lower energyextra-galactic cosmic rays (~10^18-10^19 eV) can only travel a linear distancesmaller than ~Gpc in a Hubble time due to the diffusion if the extra-galacticmagnetic fields are as strong as nano Gauss. These prevent us from directlyobserving most of the UHECRs in the universe, and thus the observed UHECRintensity reflects only the emissivity in the nearby universe within hundredsof Mpc. However, UHECRs in the distant universe, through interactions with thecosmic background photons, produce UHE electrons and gamma-rays that in turninitiate electromagnetic cascades on cosmic background photons. This secondarycascade radiation forms part of the extragalactic diffuse GeV-TeV gamma-rayradiation and, unlike the original UHECRs, is observable. Motivated by newmeasurements of extragalactic diffuse gamma-ray background radiation byFermi/LAT, we obtained upper limits placed on the UHECR emissivity in thedistant universe by requiring that the cascade radiation they produce notexceed the observed levels. By comparison with the gamma-ray emissivity ofcandidate UHECR sources (such as GRBs and AGNs) at high-redshifts, we find thatthe obtained upper limit for a flat proton spectrum is ~10^1.5 times largerthan the gamma-ray emissivity in GRBs and ~10 times smaller than the gamma-rayemissivity in BL Lac objects. In the case of iron nuclei composition, thederived upper limit of the UHECR emissivity is a factor of 3-5 times higher.Robust upper limit on the cosmogenic neutrino flux is further obtained, whichis marginally reachable by the Icecube detector and the next-generationdetector JEM-EUSO.

Constraining the Emissivity of Ultrahigh Energy Cosmic Rays in the Distant Universe with the Diffuse Gamma-ray Emission