Lorentz violation from cosmological objects with very high energy photon emissions

Lorentz violation (LV) is predicted by some quantum gravity theories, where photon dispersion relation is modified, and the speed of light becomes energy-dependent. Consequently, it results in a tiny time delay between high energy photons and low energy ones. Very high energy (VHE) photon emissions from cosmological distance can amplify these tiny LV effects into observable quantities. Here we analyze four VHE $\gamma$-ray bursts (GRBs) from Fermi observations, and briefly review the constraints from three TeV flares of active galactic nuclei (AGNs) as well. One step further, we present a first robust analysis of VHE GRBs taking the intrinsic time lag caused by sources into account, and give an estimate to quantum gravity energy $\sim 2 \times 10^{17}$ GeV for linear energy dependence, and $\sim 5 \times 10^9$ GeV for quadratic dependence. However, the statistics is not sufficient due to the lack of data, and further observational results are desired to constrain LV effects better.