Conformal Geometry and the Universe

The conformal approach to Euclidean geometry introduced by David Hestenes, uses null vectors in an enlarged space to represent points. Here we show how these same techniques can be extended to the curved spacetimes relevant in cosmology. An extended example is given of the transfer of a result from 2d non-Euclidean geometry to 4d de Sitter space (the Origin Lemma in the Poincaré disc), and implications of this conformal approach for asymptotically de Sitter universes, such as the one we appear to live in, are discussed. In a simplified approach, this suggests that our current universe should be approximately spatially flat but with closed spatial sections. This prediction of approximate flatness is achieved without invoking inflation, but needs refining for realistic universe histories.