Dispersion Relations in $\kappa$-Noncommutative Cosmology

TL;DR

This paper investigates how noncommutative geometry modifies wave dispersion relations in curved spacetime, specifically in cosmological models, revealing a new energy-dependent variation of light speed influenced by cosmic expansion.

Contribution

It introduces a general framework for noncommutative deformations in curved backgrounds using Drinfeld twist, applied to cosmology with a novel expression for light speed variation.

Findings

Derived a new formula for the variation of the speed of light in noncommutative cosmology.

Showed the variation depends linearly on photon energy, cosmological time, and Hubble parameter.

Applicable to any twist and curved background within the noncommutative differential geometry approach.

Abstract

We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist $-$giving $\kappa$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lema\^{i}tre-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.