Limits on the Lorentz Invariance Violation from UHECR astrophysics

In this paper, Lorentz Invariance Violation (LIV) is introduced in the calculations of photon propagation in the Universe. LIV is considered in the photon sector and the mean free path of the $\gamma \gamma \rightarrow e^{+} e^{-}$ interaction is calculated. The corresponding photon horizon including LIV effects is used to predict major changes in the propagation of photons with energy above $10^{18}$ eV. The flux of GZK photons on Earth considering LIV is calculated for several source models of ultra-high energy cosmic ray (UHECR). The predicted flux of GZK gamma-rays is compared to the new upper limits on the photon flux obtained by the Pierre Auger Observatory in order to impose upper limits on the LIV coefficients of order $n =$ 0, 1 and 2. The limits on the LIV coefficients derived here are more realistic than previous works and in some cases more restrictive. The analysis resulted in LIV upper limits in the photon sector of $\delta_{\gamma,0}^{limit} \sim -10^{-20}$, $\delta_{\gamma,1}^{limit} \sim -10^{-38} \; \mathrm{eV^{-1}}$ and $\delta_{\gamma,2}^{limit} \sim -10^{-56} \; \mathrm{eV^{-2}}$ in the astrophysical scenario which best describes UHECR data.