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

TL;DR

This paper explores the properties and size limitations of nonsingular universes using the singularity theorem, revealing topological constraints and an upper bound on the universe's affine size, with implications for cosmological models.

Contribution

It demonstrates that nonsingular universes with certain conditions must have a spherical topology and establishes an upper bound on their affine size, linking theory with observational data.

Findings

Nonsingular universes with past trapped surfaces have spherical topology.

The affine size of such universes is bounded above.

Observational data suggests our universe's size is within this bound.

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 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-Lema\^itre-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.