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.
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