Scale-dependent homogeneity measures for causal dynamical triangulations

TL;DR

This paper introduces two scale-dependent measures to analyze the homogeneity of quantum geometries in causal dynamical triangulations, revealing scale-dependent inhomogeneity and homogeneity properties in quantum spacetime.

Contribution

It proposes novel volumetric and spectral measures for quantum spacetime homogeneity and applies them to causal triangulations, revealing scale-dependent inhomogeneity and homogeneity.

Findings

Quantum spacetime shows inhomogeneity at small scales.

High homogeneity observed at large scales.

Power-law scaling of inhomogeneity with spatial volume.

Abstract

I propose two scale-dependent measures of the homogeneity of the quantum geometry determined by an ensemble of causal triangulations. The first measure is volumetric, probing the growth of volume with graph geodesic distance. The second measure is spectral, probing the return probability of a random walk with diffusion time. Both of these measures, particularly the first, are closely related to those used to assess the homogeneity of our own universe on the basis of galaxy redshift surveys. I employ these measures to quantify the quantum spacetime homogeneity as well as the temporal evolution of quantum spatial homogeneity of ensembles of causal triangulations in the well-known physical phase. According to these measures, the quantum spacetime geometry exhibits some degree of inhomogeneity on sufficiently small scales and a high degree of homogeneity on sufficiently large scales. This inhomogeneity appears unrelated to the phenomenon of dynamical dimensional reduction. I also uncover evidence for power-law scaling of both the typical scale on which inhomogeneity occurs and the magnitude of inhomogeneity on this scale with the ensemble average spatial volume of the quantum spatial geometries.