Testing Quantum Gravity in the Multi-Messenger Astronomy Era

Quantum gravity (QG) remains elusive despite almost century-long efforts to combine general relativity and quantum mechanics. All the approaches triggered and powered by purely theoretical considerations eventually failed with a prevailing feeling of a complete lack of guidance from the experimental side. Currently, however, this circumstance is beginning to change considerably. We have entered the era of multi-messenger astronomy. The electromagnetic window to the universe-so far the only one-has been tremendously enlarged in the energy range beyond gamma rays up to ultra-high-energy photons and has been complemented by other messengers: high-energy cosmic rays, cosmic neutrinos, and gravitational waves (GWs). This has created a unique environment in which to observationally constrain various phenomenological QG effects. In this paper, we focus on the LIV phenomenology manifested as energy-dependent time-of-flight delays and strong lensing time delays. We review results regarding time-of-flight delays obtained with GRBs. We also recall the idea of energy-dependent lensing time delays, which allow one to constrain LIV models independently of the intrinsic time delay. Lastly, we show how strongly a gravitationally lensed GW signal would place interesting constraints on the LIV.