Planck-scale-modified dispersion relations in homogeneous and isotropic spacetimes

TL;DR

This paper derives a general Planck-scale-modified dispersion relation for particles in homogeneous, isotropic spacetimes, leading to observable effects like redshift and lateshift in a deformed FLRW universe.

Contribution

It introduces a covariant framework for Planck-scale deformations of dispersion relations compatible with homogeneous and isotropic spacetimes, specifically deriving a FLRW Hamiltonian akin to $$-Poincare9.

Findings

Derived a general dispersion relation for homogeneous, isotropic spacetimes.

Presented a Planck-scale deformed Hamiltonian for FLRW geometry.

Identified observable effects such as redshift and lateshift.

Abstract

The covariant understanding of dispersion relations as level sets of Hamilton functions on phase space enables us to derive the most general dispersion relation compatible with homogeneous and isotropic spacetimes. We use this concept to present a Planck-scale deformation of the Hamiltonian of a particle in Friedman-Lema\^itre-Robertson-Walker (FLRW) geometry that is locally identical to the $\kappa$-Poincar\'e dispersion relation, in the same way as the dispersion relation of point particles in general relativity is locally identical to the one valid in special relativity. Studying the motion of particles subject to such Hamiltonian we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe.