Energetics of the quantum graphity universe

Quantum graphity is a background independent model for emergent geometry, in which space is represented as a dynamical graph. The high-energy pregeometric starting point of the model is usually considered to be the complete graph; however, we also consider the empty graph as a candidate pregeometric 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 determined exactly for small number of vertices N. To study the whole energy curve for larger (but still finite) N, an epitaxial approximation is introduced. This work may open the way to compare predictions from quantum graphity with observations of the early Universe, making the model falsifiable.