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We obtain the analytic solution of the Friedmann equation for fully realisticcosmologies including radiation, non-relativistic matter, a cosmologicalconstant $\lambda$ and arbitrary spatial curvature $\kappa$. The generalsolution for the scale factor $a(\tau)$, with $\tau$ the conformal time, is anelliptic function, meromorphic and doubly periodic in the complex $\tau$-plane,with one period along the real $\tau$-axis, and the other along the imaginary$\tau$-axis. The periodicity in imaginary time allows us to compute thethermodynamic temperature and entropy of such spacetimes, just as Gibbons andHawking did for black holes and the de Sitter universe. The gravitationalentropy favors universes like our own which are spatially flat, homogeneous,and isotropic, with a small positive cosmological constant.

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