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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2003
 DOI:
 10.48550/arXiv.math/0302244
 arXiv:
 arXiv:math/0302244
 Bibcode:
 2003math......2244S
 Keywords:

 Mathematics  Differential Geometry;
 Astrophysics;
 Astrophysics  Cosmology and Extragalactic Astrophysics;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 Mathematics  Metric Geometry;
 53C20 (Primary);
 85F05 53C23 (Secondary)
 EPrint:
 45 pages, 7 figures (eps), AIM Workshop on General Relativity 2002, added some details to the explanation, fixed typos/grammar