Thermodynamic solution of the homogeneity, isotropy and flatness puzzles (and a clue to the cosmological constant)

Abstract

We obtain the analytic solution of the Friedmann equation for fully realistic cosmologies including radiation, non-relativistic matter, a cosmological constant λ and arbitrary spatial curvature κ. The general solution for the scale factor a(τ), with τ the conformal time, is an elliptic function, meromorphic and doubly periodic in the complex τ-plane, with one period along the real τ-axis, and the other along the imaginary τ-axis. The periodicity in imaginary time allows us to compute the Hawking temperature and the global gravitational entropy of cosmological spacetimes, just as Gibbons, Hawking and Perry did for black holes and de Sitter spacetime. The global gravitational entropy favours universes like our own which are spatially flat, homogeneous, and isotropic, with a small positive cosmological constant.