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

TL;DR

This paper derives an exact analytic solution for the Friedmann equation in realistic cosmologies, revealing the thermodynamic properties of such universes and providing insights into the conditions favoring our universe's flatness, homogeneity, and positive cosmological constant.

Contribution

It presents a novel elliptic function solution for the scale factor in cosmology including all key components and explores the thermodynamics and implications for the universe's geometry and cosmological constant.

Findings

The solution is an elliptic function with specific periodicities.

Thermodynamic temperature and entropy can be computed from the solution.

Gravitational entropy favors flat, homogeneous, isotropic universes with a positive cosmological constant.

Abstract

We obtain the analytic solution of the Friedmann equation for fully realistic cosmologies including radiation, non-relativistic matter, a cosmological constant $\lambda$ and arbitrary spatial curvature $\kappa$. The general solution for the scale factor $a(\tau)$, with $\tau$ the conformal time, is an elliptic 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 the thermodynamic temperature and entropy of such spacetimes, just as Gibbons and Hawking did for black holes and the de Sitter universe. The gravitational entropy favors universes like our own which are spatially flat, homogeneous, and isotropic, with a small positive cosmological constant.