Left: photon target energy at threshold ϵ_min for the pair-production reaction as a function of the γ-ray energy E_γ. The black solid line shows the standard case. The other lines indicate the modified threshold, which result from the LIV-modified kinematics, for different values of the parameter E_LIV (in units of 10¹⁹ GeV). Right: EBL (Dominguez et al. 2011, solid red; Kneiske & Dole 2010, solid blue) and CMB (dashed red) local density ϵn(ϵ) (horizontal axis) as a function of the energy ϵ (vertical axis), to be compared with the energy threshold in the left panel.