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

TL;DR

The paper investigates topological strategies to mollify the cosmological initial singularity by embedding nontrivial manifold structures into the early universe while preserving FLRW-like evolution at later times. It classifies reflective topological big bangs via orientable manifolds arising from nonorientable (n−1)-surfaces and shows a direct correspondence with orientation double covers; it then introduces two toy models— the Itty-Bitty Blender (nonreflective with localized CTCs) and its universal cover, the Eternal Trumpet (globally hyperbolic)—to illustrate how causal pathologies can be confined or eliminated and to examine horizon problems under unconventional early-universe geometries. The results demonstrate that minimal reflective topology alone does not resolve the horizon problem without altering early-time dynamics, while nonreflective and unwound constructions can avoid certain singularity theorems yet raise energy-condition considerations and questions about physical realizability. Overall, the work opens a landscape of possible topological mollifications of the Big Bang, highlighting both mathematical classifications and the need for further work to connect these ideas with energy conditions, inflation, and observable consequences.

Abstract

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.

Figures (6)

• Figure 1: De Sitter space as a submanifold of Minkowski space for $n = 2$. On the right, a pair of points to be identified is shown in a vertical cross-section.
• Figure 2: The quotient of de Sitter space obtained via antipodal identification. The dotted line is an example of a continuous curve that passes through the reflection surface, emerging on the opposite side.
• Figure 3: A few simple chains of covering relations giving admissible topologies of orientable reflective topological big bangs admitting flat metrics in $n=4$. Here, $K^2$ refers to the Klein bottle and $M_s$ the (open) Möbius band.
• Figure 4: Coordinates on the solid toroidal region $r<1$. $\alpha$ specifies the shaded cross section (left), and $(r,\theta)$ are polar coordinates within the cross section.
• Figure 5: A top-down view of a section of the torus, showing rotation of timecones on the region $r < 1$ as $r$ varies, according to equation (\ref{['eqn:nulls']}). If one continued to $r > 1$, the timecones would asymptote to rotating a full 90 degrees outward.

Theorems & Definitions (5)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2