Gravitational entropy and the flatness, homogeneity and isotropy puzzles

We suggest a new explanation for the observed large scale flatness, homogeneity and isotropy of the universe. The basic ingredients are elementary and well-known, namely Einstein's theory of gravity and Hawking's method of computing gravitational entropy. The new twist is provided by the boundary conditions we recently proposed for "big bang" type singularities dominated by conformal matter, enforcing $CPT$ symmetry and analyticity. Here, we show that, besides allowing us to describe the big bang, these boundary conditions allow new gravitational instantons, enabling us to calculate the gravitational entropy of cosmologies which include radiation, dark energy and space curvature of either sign. We find the gravitational entropy of these universes, $S_g \sim S_\Lambda^{1/ 4} S_r$, where $S_\Lambda$ is the famous de Sitter entropy and $S_r$ is the total entropy in radiation. To the extent that $S_g$ exceeds $S_\Lambda$, the most probable universe is flat. By analysing the perturbations about our new instantons, we argue it is also homogeneous and isotropic on large scales.