2012ApJ...758..101S

Highly beamed relativistic e ^±-pair energy distributions result in double photon collisions of the beamed gamma rays from TeV blazars at cosmological distances with the isotropically distributed extragalactic background light (EBL) in the intergalactic medium. The typical energies k₀≃ 10^–7 in units of m_e c ² of the EBL are more than 10 orders of magnitude smaller than the observed gamma-ray energies k₁ ≥ 10⁷. Using the limit k ₀{Lt} k₁, we demonstrate that the angular distribution of the generated pairs in the lab frame is highly beamed in the direction of the initial gamma-ray photons. For the astrophysically important case of power-law distributions of the emitted gamma-ray beam up to the maximum energy M interacting with Wien-type N(k₀)∝k^q₀ exp (- k₀/Θ) soft photon distributions with total number density N₀, we calculate analytical approximations for the electron production spectrum. For distant objects with luminosity distances d_L{Gt} r₀= (σ_TN₀)^–1 = 0.49N.^–1₀ Mpc (with Thomson cross section σ_T), the implied large values of the optical depth τ₀= d_L/r₀ indicate that the electron production spectra differ at energies inside and outside the interval [(Θln τ₀)^–1, τ₀/Θ], given the maximum gamma-ray energy M {Gt} Θ^–1. In the case M {Gt} Θ^–1, the production spectrum is strongly peaked near E ≃ Θ^–1, being exponentially reduced at small energies and decreasing with the steep power law ∝E.^-1 - p ^ up to the maximum energy E = M - (1/2).