The interaction of TeV photons from blazars with the extragalactic background light produces a relativistic beam of electron-positron pairs streaming through the intergalactic medium ( IGM ) .
The fate of the beam energy is uncertain .
By means of two- and three-dimensional particle-in-cell simulations , we study the non-linear evolution of dilute ultra-relativistic pair beams propagating through the IGM .
We explore a wide range of beam Lorentz factors \gamma _ { b } \gg 1 and beam-to-plasma density ratios \alpha \ll 1 , so that our results can be extrapolated to the extreme parameters of blazar-induced beams ( \gamma _ { b } \sim 10 ^ { 6 } and \alpha \sim 10 ^ { { -15 } } , for the most powerful blazars ) .
For cold beams , we show that the oblique instability governs the early stages of evolution , but its exponential growth terminates – due to self-heating of the beam in the transverse direction – when only a negligible fraction \sim ( \alpha / \gamma _ { b } ) ^ { 1 / 3 } \sim 10 ^ { -7 } of the beam energy has been transferred to the IGM plasma .
Further relaxation of the beam proceeds through quasi-longitudinal modes , until the momentum dispersion in the direction of propagation saturates at \Delta p _ { b, \parallel } / \gamma _ { b } m _ { e } c \sim 0.2 .
This corresponds to a fraction \sim 10 \% of the beam energy being ultimately transferred to the IGM plasma , irrespective of \gamma _ { b } or \alpha .
If the initial dispersion in beam momentum satisfies \Delta p _ { b 0 , \parallel } / \gamma _ { b } m _ { e } c \gtrsim 0.2 ( as typically expected for blazar-induced beams ) , the fraction of beam energy deposited into the IGM is much smaller than \sim 10 \% .
It follows that at least \sim 90 \% of the beam energy is still available to power the GeV emission produced by inverse Compton up-scattering of the Cosmic Microwave Background by the beam pairs .