Implications of the singularity theorem for the size of a nonsingular universe

Abstract

A general property of universes without initial singularity is investigated based on the singularity theorem, assuming the null convergence condition and the global hyperbolicity. As a direct consequence of the singularity theorem, the universal covering of a Cauchy surface of a non-singular universe with a past trapped surface must have the topology of S 3 . In addition, we find that the affine size of a non-singular universe, defined through the affine length of null geodesics, is bounded above. In the case where a part of the non-singular spacetime is described by Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, we find that this upper bound can be understood as the affine size of the corresponding closed de Sitter universe. We also evaluate the upper bound of the affine size of our universe based on the trapped surface confirmed by recent observations of baryon acoustic oscillations, assuming that our universe has no initial singularity.