Expected Signature For the Lorentz Invariance Violation Effects on $γ-γ$ Absorption

Abstract: There are still some {significant and} unanswered questions about the {incredible} {very high energy (VHE)} $\gamma$-ray signatures. To help understand the mechanism, focusing on the linear and quadratic perturbation mode for the subluminal regime, the present paper revisited the expected signature for the Lorentz invariance violation effects on $\gamma-\gamma$ absorption in TeV spectra of Gamma-ray bursts (GRBs). We note that there is {a critical energy} for the pair production process, which is sensitive to the assumed quantum gravity energy scale. We suggest that a {reemergence of the energy spectrum of $\gamma$-rays} at the several tens of TeV is a rough observational diagnostic for the Lorentz invariance violation (LIV) effects. The expected spectra characteristics are applied to a GRB 221009A. The results show that the cosmic opacity with LIV effects considered here {can} roughly reproduce the observed $\gamma$-ray spectra for the source, which enabled us to constrain the upper limit of the values of energy scale at $E_{\rm QG,~1}\leq3.35\times10^{20}$ GeV for the linear perturbation and $E_{\rm QG,~2}\leq9.19\times10^{12}$ GeV for the quadratic perturbation. {These scenarios would update the bound of the LIV coefficient with $\xi_{\rm 1}^{\prime}\geq 3.62\times10^{-2}$ for the linear perturbation, and $\xi_{\rm 2}^{\prime}\geq 1.33\times10^{6}$ for the quadratic perturbation in the standard model extension (SME) framework, respectively. }