  Recent observations of isotropic diffuse backgrounds by Fermi and IceCube
allow us to get more insight into distant very-high-energy (VHE) and
ultra-high-energy (UHE) gamma-ray/neutrino emitters, including cosmic-ray
accelerators/sources. First, we investigate the contribution of intergalactic
cascades induced by gamma-rays and/or cosmic rays (CRs) to the diffuse
gamma-ray background (DGB) in view of the latest Fermi data. We identify a
possible VHE Excess from the fact that the Fermi data are well above
expectations for an attenuated power law, and show that cascades induced by VHE
gamma rays (above ~10 TeV) and/or VHECRs (below ~10^19 eV) may significantly
contribute to the DGB above ~100 GeV. The relevance of the cascades is also
motivated by the intergalactic cascade interpretations of extreme TeV blazars
such as 1ES 0229+200, which suggest very hard intrinsic spectra. This
strengthens the importance of future detailed VHE DGB measurements. Then, more
conservatively, we derive general constraints on the cosmic energy budget of
high-energy gamma rays and neutrinos based on recent Fermi and IceCube
observations of extragalactic background radiation. We demonstrate that these
multi-messenger constraints are useful and the neutrino limit is very powerful
for VHE/UHE hadronic sources. Furthermore, we show the importance of
constraints from individual source surveys by future imaging atmospheric
Cherenkov telescopes such as Cherenkov Telescope Array, and demonstrate that
the cascade hypothesis for the VHE DGB can be tested by searching for distant
emitters of cascaded gamma rays.
