Friedmann Cosmology and Almost Isotropy

Abstract

In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally isotropic in a sense which allows both weak gravitational lensing in all directions and strong gravitational lensing in localized angular regions at most points. We then prove that such a manifold is Gromov Hausdorff close to a length space $Y$ which is a collection of space forms joined at discrete points. Within the paper we define a concept we call an "exponential length space" and prove that if such a space is locally isotropic then it is a space form.