In-vacuo dispersion
In-vacuo dispersion is an effect related to the hypothesis of a quantum gravity scale modified dispersion relation (MDR), which is described by the Hamiltonian of a relativistic particle characterizing its propagation in a quantum spacetime. Deviations from the general relativistic dispersion relations are often (but not always) associated with either the violation or deformation of relativistic symmetries. The core idea is to describe the effects of quantum gravity as a deformation of the classical trajectory of particles—a deformation that grows with the particle’s energy and is suppressed by the quantum gravity energy scale [AEMN98, Ad22].
Since, in the context of quantum gravity, the relevant energies are ultra-relativistic, one can often focus on the massless case. For certain types of MDRs, the speed of a massless particle becomes energy-dependent in vacuum, which is the origin of the effect’s name.
For this reason, if particles are emitted from a given source with an energy difference, they arrive at a detector with a time difference proportional to the product of the energy difference and the propagation distance, but suppressed by the quantum gravity energy scale. This is a quantum spacetime-induced time delay effect, analogously to what the propagation of light through a medium (\(\eta\) – phenomenological parameter, \(M_P\) –Planck mass, \(D(z)\) – redshift-dependent distance factor)
\[ \Delta t = \eta D(z) \frac{\Delta E}{M_P} \]
If these particles propagate over cosmological distances, even a modification suppressed by the Planck scale could produce a measurable effect with modern telescopes.
Different formulations of the time delay effect have been proposed based on various hypotheses, such as the coupling between the energy-momentum of the particle and the spacetime metric, the local symmetries of spacetime, and the redshift dependence of the quantum gravity scale [Ad22].