The eternal fractal in the universe

TL;DR

This work addresses the gauge-invariance problem in characterizing eternal inflation and its large-scale fractal structure. It introduces a coordinate-independent eternal fractal set $E$ on the initial surface and shows its fractal dimension $\dim E$ is gauge-invariant and computable from the FP spectrum, with $\dim E=3-\gamma$ where $\gamma$ is the dominant eigenvalue in the scale-factor gauge. The paper derives a gauge-invariant nonlinear diffusion equation for $\bar X(\phi)=1-X(\phi)$ that governs the probability of attaining eternal points, and demonstrates consistency between this equation and the FP criterion $\gamma_V>0$, including an exact solution in a toy model. It also establishes topological bounds on merging of thermalized regions through $\dim E$, linking fractal geometry to percolation-like behavior and outlining conditions under which all thermalized regions must merge or remain causally separated. The results provide a robust, coordinate-free framework for understanding the global topology of eternally inflating spacetimes and set the stage for applying these ideas to broader inflationary models and their observational implications.

Abstract

Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow existence of points that never thermalize. I explore the fractal geometry of the resulting spacetime, using coordinate-independent quantities. The formalism of stochastic inflation can be used to obtain the fractal dimension of the set of eternally inflating points (the ``eternal fractal''). I also derive a nonlinear branching diffusion equation describing global properties of the eternal set and the probability to realize eternal inflation. I show gauge invariance of the condition for presence of eternal inflation. Finally, I consider the question of whether all thermalized regions merge into one connected domain. Fractal dimension of the eternal set provides a (weak) sufficient condition for merging.

Figures (3)

• Figure 1: Probability $\bar{X}\left( \phi \right)$ to have no eternal points.
• Figure 2: Illustrative potential for eternal inflation.
• Figure 3: Effective potential of Eq. (\ref{['eq:U']}).