Gravitational time advancement under gravity's rainbow

Under gravity's rainbow, we investigate its effects on the gravitational time advancement, which is a natural consequence of measuring proper time span for a photon's round trip. This time advancement can be complementary to the time delay for testing the gravity's rainbow, because they are sensitive to different modified dispersion relations (MDRs). Its observability on ranging a spacecraft far from the Earth by two radio and a laser links is estimated at superior conjunction (SC) and inferior conjunction (IC). We find that (1) the IC is more favorable than the SC for measurement on the advancement caused by the rainbow; (2) a specific type of MDR has a significantly larger effect on the advancement than others in both SC and IC cases; and (3) a combination of available optical clocks and the realization of planetary laser ranging in the future will benefit distinguishing the gravity's rainbow from GR by measuring the gravitational time advancement.