Scale-dependent homogeneity measures for causal dynamical triangulations

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.

25 pages, 19 figures

References in corpus (25)

• BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales
• Spectral Dimension of the Universe
• Emergence of a 4D World from Causal Quantum Gravity
• Reconstructing the Universe
• Nonperturbative Quantum Gravity
• Dynamically Triangulating Lorentzian Quantum Gravity
• Cosmic homogeneity demonstrated with luminous red galaxies
• Toward an Understanding of Foreground Emission in the BICEP2 Region
• The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity
• A non-perturbative Lorentzian path integral for gravity
• Planckian Birth of the Quantum de Sitter Universe
• The Nonperturbative Quantum de Sitter Universe
• Spectral geometry as a probe of quantum spacetime
• Semiclassical Universe from First Principles
• A second-order phase transition in CDT
• Non-perturbative 3d Lorentzian Quantum Gravity
• Second- and First-Order Phase Transitions in CDT
• The Semiclassical Limit of Causal Dynamical Triangulations
• Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations
• Renormalization Group Flow in CDT
• The Scale of Cosmic Isotropy
• Measuring the transition to homogeneity with photometric redshift surveys
• A Validation of Causal Dynamical Triangulations
• A first look at transition amplitudes in (2+1)-dimensional causal dynamical triangulations
• Geodesic distances in Liouville quantum gravity

Cited by in corpus (8)

• Quantum Gravity from Causal Dynamical Triangulations: A Review
• Fractal Spacetimes in Stochastic Gravity? -- Views from Anomalous Diffusion and the Correlation Hierarchy
• Measuring the Homogeneity (or Otherwise) of the Quantum Universe
• Setting the physical scale of dimensional reduction in causal dynamical triangulations
• Scaling analyses of the spectral dimension in 3-dimensional causal dynamical triangulations
• Approximate Killing symmetries in non-perturbative quantum gravity
• On the Nature of Spatial Universes in 3D Lorentzian Quantum Gravity
• Making the case for causal dynamical triangulations