Birth and growth of a two-dimensional universe

A master equation for the evolution of a two-dimensional universe is derived based on the simplicial quantum gravity regarding the evolution as the Markov process of a space-time lattice. Three typical phases, expanding, elongating, and collapsing phase, which have been found in the numerical simulation, are studied together with their boundaries, analytically. Asymptotic solutions of the evolution equation for statistical quantities, such as the averaged area, the boundary length, correlations of their fluctuations, are obtained for each phase and boundary. After introducing a physical time, the cosmological significance of each phase is discussed.

Figure 1

Collapsing phase (P1). Expanding phase (P2). Elongating phase (P3). The boundary state is on the boundary B2, and the Polymer state is in the elongating phase P3.