Singularity theorems demonstrate the inevitable breakdown of the concept of continuous, classical spacetime under highly general conditions. Quantum gravity is expected to intervene to avoid singularities, and models so far hint toward several regularized geometries, in which limited spacetime regions requiring full quantum gravitational description can be safely covered by an extension of some suitable spacetime geometry. Motivated by these premises, in recent years, a systematic, quantum gravity agnostic study has been carried out to catalog all the conceivable nonsingular, continuous, and globally hyperbolic geometries arising from evading Penrose's focusing theorem in gravitational collapse. In this study, we extend this inquiry by systematically examining all potential nonsingular, continuous, and globally hyperbolic extensions into the past of Friedmann-Lemaì‚tre-Robertson-Walker metrics. As in the black-hole case, our investigation reveals a remarkably limited set of alternative scenarios. The stringent requisites of homogeneity and isotropy drastically restrict the viable singularity-free geometries to merely three discernible nonsingular cosmological spacetimes: a bouncing universe (where the scale factor reaches a minimum in the past before reexpanding), an emergent universe (where the scale factor reaches and maintains a constant value in the past), and an asymptotically emergent universe (where the scale factor diminishes continually, asymptotically approaching a constant value in the past). We also discuss the implications of these findings for the initial conditions of our Universe, and the arrow of time.
