Topological Big Bangs: Reflection, Itty-Bitty Blenders, and Eternal Trumpets

We discuss and formalize topological means by which the initial singularity might be mollified, at the level of the spacetime manifold's structure, in classical cosmological models of a homogeneous expanding universe. One construction, dubbed a "reflective" topological big bang, generalizes Schrodinger's elliptic de Sitter space and is built to be compatible with the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) picture of the large-scale universe, only minimally modifying it via some nontrivial topology at an earliest "moment" in the universe's history. We establish a mathematical characterization of the admissible topological structures of reflective topological big bangs, and we discuss implications for a standard concern in cosmology, the horizon problem. We present a nonreflective example that we've christened the Itty-Bitty Blender spacetime: this spacetime and its universal cover, the Eternal Trumpet spacetime, exhibit interesting potential structures of spacetimes avoiding the Hawking and Penrose singularity theorems. While these toy models provide a proof-of-concept picture, several questions remain regarding the capacity to realize these structures under physical energy conditions.