  In this letter, we describe a general mechanism for emergence of a rainbow
metric from a quantum cosmological model. This idea is based on QFT on a
quantum space-time. Under general assumptions, we discover that the quantum
space-time on which the field propagates can be replaced by a classical
space-time, whose metric depends explicitly on the energy of the field: as
shown by an analysis of dispersion relations, quanta of different energy
propagate on different metrics, similar to photons in a refractive material
(hence the name "rainbow" used in the literature). In deriving this result, we
do not consider any specific theory of quantum gravity: the qualitative
behavior of high-energy particles on quantum space-time relies only on the
assumption that the quantum space-time is described by a wave-function $\Psi_o$
in a Hilbert space $\mathcal{H}_G$.
