Eternal fractal in the universe

Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow the 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 of realizing eternal inflation. I show gauge invariance of the condition for the presence of eternal inflation. Finally, I consider the question of whether all thermalized regions merge into one connected domain. The fractal dimension of the eternal set provides a (weak) sufficient condition for merging.