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

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 nonsingular universe with a past trapped surface must have the topology of $S^3$. In addition, we find that the affine size of a nonsingular universe, defined through the affine length of null geodesics, is bounded above. In the case where a part of the nonsingular spacetime is described by Friedmann-Lemaître-Robertson-Walker 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.