Creation of a Compact Topologically Nontrivial Inflationary Universe

TL;DR

The paper argues that low-scale inflation (V(\phi) \ll 1) faces exponential suppression in creation probabilities for traditional closed or infinite flat/open universes, but compact flat/open universes with nontrivial topology can evade this barrier. It shows, via chaotic mixing in small, multiply connected spaces and a Wheeler–DeWitt quantum cosmology treatment of a toroidal universe, that inflation can originate from Planck-scale patches without exponential suppression; kinetic-energy contributions can modulate small-a behavior but do not reintroduce a barrier under typical conditions. These results have potential implications for string cosmology and initial-condition problems, suggesting topology selection during baby-universe formation could favor compact, nontrivial geometries. The work also discusses observational prospects and the need for further exploration of more general anisotropic/topological configurations and their measure in eternal inflation contexts.

Abstract

If inflation can occur only at the energy density V much smaller than the Planck density, which is the case for many inflationary models based on string theory, then the probability of quantum creation of a closed or an infinitely large open inflationary universe is exponentially suppressed for all known choices of the wave function of the universe. Meanwhile under certain conditions there is no exponential suppression for creation of topologically nontrivial compact flat or open inflationary universes. This suggests, contrary to the standard textbook lore, that compact flat or open universes with nontrivial topology should be considered a rule rather than an exception.

Figures (2)

• Figure 1: Two eigenmodes of the Wheeler-DeWitt equation for the wave function of the flat compact toroidal universe. The scale factor $a$ is given in units of $V^{-1/6}$. The description of the evolution of the universe in terms of classical space-time begins at $a\gtrsim V^{-1/6}$ ($a \gtrsim 1$ in our figure), when the semiclassical approximation becomes valid and the "cosmic clock" starts ticking.
• Figure 2: A typical solution of the Wheeler-DeWitt equation in the case when the universe initially was dominated by kinetic energy of the scalar field, with $V(\phi) = const \ll 1$ and $K \gg 1$. The scale factor is shown in units of $\left({K\over V}\right)^{{1/6}}$. Inflationary regime occurs at $a > \left({K\over V}\right)^{{1/6}}$. The wave function $\Psi(a)$ vanishes at $a \to 0$.