Threshold anomalies of ultra-high energy cosmic photons due to Lorentz invariance violation

From special relativity, photon annihilation process \HepProcess{{\Pgg}{\Pgg}{\to}{\Pep}{\Pem}} prevents cosmic photons with energies above a threshold to propagate long distance in space due their low energy background photons. However, modifications of the dispersion relation from Lorentz invariance violation~(LIV) can cause novel phenomena beyond relativity happen. In this paper, we point out that these include optical transparency, reduction and reappearance ultra-high space. The recent observation near PeV events by LHAASO Collaboration reveals necessity consider anomalies. Future observations extragalactic sources testify LIV properties