  In recent years Planck-scale modifications of the dispersion relation have
been attracting increasing interest also from the viewpoint of possible
applications in astrophysics and cosmology, where spacetime curvature cannot be
neglected. Nonetheless the interplay between Planck-scale effects and spacetime
curvature is still poorly understood, particularly in cases where curvature is
not constant. These challenges have been so far postponed by relying on an
ansatz, first introduced by Jacob and Piran. We here propose a general strategy
of analysis of the effects of modifications of dispersion relation in FRW
spacetimes, applicable both to cases where the relativistic equivalence of
frames is spoiled ("preferred-frame scenarios") and to the alternative
possibility of "DSR-relativistic theories", theories that are fully
relativistic but with relativistic laws deformed so that the modified
dispersion relation is observer independent. We show that the Jacob-Piran
ansatz implicitly assumes that spacetime translations are not affected by the
Planck-scale, while under rather general conditions the same Planck-scale
quantum-spacetime structures producing modifications of the dispersion relation
also affect translations. Through the explicit analysis of one of the effects
produced by modifications of the dispersion relation, an effect amounting to
Planck-scale corrections to travel times, we show that our concerns are not
merely conceptual but rather can have significant quantitative implications.
