  Ultra-high cosmic rays (UHECRs) with energies >10^19 eV emitted at
cosmological distances will be attenuated by cosmic microwave and infrared
background radiation through photohadronic processes. Lower energy
extra-galactic cosmic rays (~10^18-10^19 eV) can only travel a linear distance
smaller than ~Gpc in a Hubble time due to the diffusion if the extra-galactic
magnetic fields are as strong as nano Gauss. These prevent us from directly
observing most of the UHECRs in the universe, and thus the observed UHECR
intensity reflects only the emissivity in the nearby universe within hundreds
of Mpc. However, UHECRs in the distant universe, through interactions with the
cosmic background photons, produce UHE electrons and gamma-rays that in turn
initiate electromagnetic cascades on cosmic background photons. This secondary
cascade radiation forms part of the extragalactic diffuse GeV-TeV gamma-ray
radiation and, unlike the original UHECRs, is observable. Motivated by new
measurements of extragalactic diffuse gamma-ray background radiation by
Fermi/LAT, we obtained upper limits placed on the UHECR emissivity in the
distant universe by requiring that the cascade radiation they produce not
exceed the observed levels. By comparison with the gamma-ray emissivity of
candidate UHECR sources (such as GRBs and AGNs) at high-redshifts, we find that
the obtained upper limit for a flat proton spectrum is ~10^1.5 times larger
than the gamma-ray emissivity in GRBs and ~10 times smaller than the gamma-ray
emissivity in BL Lac objects. In the case of iron nuclei composition, the
derived 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, which
is marginally reachable by the Icecube detector and the next-generation
detector JEM-EUSO.
