Time delay of light signals in an energy-dependent spacetime metric

In this paper we review the problem of time delay of photons propagating in a spacetime with a metric that explicitly depends on the energy of the particles (gravity-rainbow approach). We show that corrections due to this approach—which is closely related to the double special relativity proposal—produce for small redshifts ($z\ll 1$) smaller time delays than in the generic Lorentz invariance violating case.