TL;DR
This paper investigates the fractal geometry of the universe predicted by eternal inflation models, deriving mathematical descriptions of the eternal set and conditions for the merging of thermalized regions.
Contribution
It introduces a gauge-invariant formalism to compute the fractal dimension of the eternal inflation set and models the global properties with a nonlinear branching diffusion equation.
Findings
Derived the fractal dimension of the eternal set.
Established gauge invariance of eternal inflation conditions.
Provided a criterion for the merging of thermalized regions.
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.
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