Energetics of the Quantum Graphity Universe

TL;DR

This paper investigates the energy evolution of the quantum graphity model, analyzing how space emerges from high-energy states to low-energy geometric configurations, with analytic and numerical results for different graph states.

Contribution

It introduces a detailed analysis of the energetics in quantum graphity, including analytic results and an approximation method for larger graphs, considering both complete and empty initial states.

Findings

Derived the slope of the energy curve in the high-energy domain.

Plotted the energy curve exactly for small number of vertices.

Used epitaxial approximation to study larger graphs.

Abstract

Quantum graphity is a background independent model for emergent geometry, in which space is represented as a complete graph. The high-energy pre-geometric starting point of the model is usually considered to be the complete graph, however we also consider the empty graph as a candidate pre-geometric state. The energetics as the graph evolves from either of these high-energy states to a low-energy geometric state is investigated as a function of the number of edges in the graph. Analytic results for the slope of this energy curve in the high-energy domain are derived, and the energy curve is plotted exactly for small number of vertices $N$. To study the whole energy curve for larger (but still finite) $N$, an epitaxial approximation is used. It is hoped that this work may open the way for future work to compare predictions from quantum graphity with observations of the early universe, making the model falsifiable.