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The interaction of TeV photons from blazars with the extragalactic backgroundlight produces a relativistic beam of electron-positron pairs streaming throughthe intergalactic medium (IGM). The fate of the beam energy is uncertain. Bymeans of two- and three-dimensional particle-in-cell simulations, we study thenon-linear evolution of dilute ultra-relativistic pair beams propagatingthrough the IGM. We explore a wide range of beam Lorentz factors gamma_b>>1 andbeam-to-plasma density ratios alpha<<1, so that our results can be extrapolatedto the extreme parameters of blazar-induced beams (gamma_b~10^6 andalpha~10^(-15), for the most powerful blazars). For cold beams, we show thatthe oblique instability governs the early stages of evolution, but itsexponential growth terminates - due to self-heating of the beam in thetransverse direction - when only a negligible fraction~(alpha/gamma_b)^(1/3)~10^(-7) of the beam energy has been transferred to theIGM plasma. Further relaxation of the beam proceeds through quasi-longitudinalmodes, until the momentum dispersion in the direction of propagation saturatesat DeltaP_{b,parallel}/gamma_b m_e c ~ 0.2. This corresponds to a fraction ~10%of the beam energy being ultimately transferred to the IGM plasma, irrespectiveof gamma_b or alpha. If the initial dispersion in beam momentum satisfiesDeltaP_{b,parallel}/gamma_b m_e c > 0.2 (as typically expected forblazar-induced beams), the fraction of beam energy deposited into the IGM ismuch smaller than ~10%. It follows that at least ~90% of the beam energy isstill available to power the GeV emission produced by inverse Comptonup-scattering of the Cosmic Microwave Background by the beam pairs.

Relativistic Pair Beams from TeV Blazars: A Source of Reprocessed GeV Emission rather than IGM Heating