  Inverse-Compton cascades initiated by energetic gamma rays (E>100 GeV)
enhance the GeV emission from bright, extragalactic TeV sources. The absence of
this emission from bright TeV blazars has been used to constrain the
intergalactic magnetic field (IGMF), and the stringent limits placed upon the
unresolved extragalactic gamma-ray background (EGRB) by Fermi has been used to
argue against a large number of such objects at high redshifts. However, these
are predicated upon the assumption that inverse-Compton scattering is the
primary energy-loss mechanism for the ultra-relativistic pairs produced by the
annihilation of the energetic gamma rays on extragalactic background light
photons. Here we show that for sufficiently bright TeV sources
(isotropic-equivalent luminosities >10^{42} erg/s) plasma beam instabilities,
specifically the "oblique" instability, present a plausible mechanism by which
the energy of these pairs can be dissipated locally, heating the intergalactic
medium. Since these instabilities typically grow on timescales short in
comparison to the inverse-Compton cooling rate, they necessarily suppress the
inverse-Compton cascades. As a consequence, this places a severe constraint
upon efforts to limit the IGMF from the lack of a discernible GeV bump in TeV
sources. Similarly, it considerably weakens the Fermi limits upon the evolution
of blazar populations. Specifically, we construct a TeV-blazar luminosity
function from those objects presently observed and find that it is very well
described by the quasar luminosity function at z~0.1, shifted to lower
luminosities and number densities, suggesting that both classes of sources are
regulated by similar processes. Extending this relationship to higher
redshifts, we show that the magnitude and shape of the EGRB above ~10 GeV is
naturally reproduced with this particular example of a rapidly evolving
TeV-blazar luminosity function.
