  Structure formation in the intergalactic medium (IGM) produces large-scale,
collisionless shock waves, where electrons can accelerate to highly
relativistic energies. Such electrons can Compton scatter cosmic microwave
background photons up to gamma-ray energies. We study the radiation emitted in
this process using a hydrodynamic cosmological simulation of a LCDM universe.
This radiation, extending beyond TeV energies, has roughly constant energy flux
per decade in photon energy, in agreement with the predictions of Loeb & Waxman
(2000). Assuming that a fraction xi_e=0.05 of the shock thermal energy is
transferred to relativistic electrons, as inferred from collisionless
non-relativistic shocks in the interstellar medium, we find that the radiation
energy flux, e^2(dJ/de)~ 50-160 eV cm^-2 s^-1 sr^-1, constitutes ~10% of the
extragalactic gamma-ray background. The associated point-sources are too faint
to account for the ~60 unidentified EGRET gamma-ray sources, but GLAST should
resolve several sources associated with large-scale IGM structures for
xi_e~0.03, and many more sources for larger xi_e. The intergalactic origin of
the radiation can be verified through a cross-correlation with, e.g., the
galaxy distribution that traces the same structure. Its shock-origin may be
tested by a cross-correlation with radio synchrotron radiation, emitted as the
same electrons gyrate in post-shock magnetic fields. We predict that GLAST and
Cherenkov telescopes such as MAGIC, VERITAS and HESS should resolve gamma-rays
from nearby (redshifts z < 0.01) rich galaxy clusters, perhaps in the form of a
\~5-10 Mpc diameter ring-like emission tracing the cluster accretion shock,
with luminous peaks at its intersections with galaxy filaments detectable even
at z~0.025.
