Geometry of standard cosmology. The initial singularity

Abstract

This note is derived from lectures on cosmology given at the Complutense University of Madrid, basically aimed at a mathematical audience. The author derives the geometrical properties of a cosmological model from a set of seven postulates H1 to H7. At large scales a cosmic fluid is represented by a vector field Z whose integral curves define a congruence of geodesics. The cosmic time and the Hubble parameter are defined in terms of such congruence. The first three hypotheses concern the homogeneity and isotropy of the model. The age of the Universe and the singularity problem are dealt with in hypothesis H4, and H5 is on the behaviour of Z and on the positivity of the Ricci curvature. Einstein’s equations appear only at the end with H6.