Upper Limit on the Cosmological Gamma-ray Background
  We show that the current extragalactic gamma-ray background (EGB) measurement
below 100 GeV sets an upper limit on EGB itself at very high energy (VHE) above
100 GeV. The limit is conservative for the electromagnetic cascade emission
from VHE EGB interacting with the cosmic microwave-to-optical background
radiation not to exceed the current EGB measurement. The cascade component fits
the measured VHE EGB spectrum rather well. However, once we add the
contribution from known source classes, the Fermi VHE EGB observation exceeds
or even violates the limit, which is approximated as E^2dN/dE < 4.5x10^-5
(E/100 GeV)^-0.7 MeV/cm^2/s/sr. The upper limit above 100 GeV is useful in the
future to probe the EGB origin and the new physics like axion-like particles
and Lorentz-invariance violation.
