The Self-Organized Critical Multiverse

Recently a dynamical selection mechanism for vacua based on search optimization was proposed in the context of false-vacuum eternal inflation on the landscape. The search algorithm, defined by local vacuum transitions, is optimal in regions of the landscape where the dynamics are tuned at criticality, with de Sitter vacua having an average lifetime of order their Page time. The purpose of this paper is to shed light on the nature of the dynamical phase transition at the Page lifetime. We focus on a finite region of the landscape, which exchanges volume with the rest of the landscape and as such acts as an open system. Through a change of variables the master equation governing the comoving volume of de Sitter vacua is mapped to a stochastic equation for coupled overdamped stochastic oscillators -- the well-known Ornstein-Uhlenbeck process. The rest of the landscape, which acts as an environment, is assumed to result in a non-vanishing driving term for all sites in the region with uncorrelated, white noise fluctuations (though not necessarily Gaussian). We first show that the static susceptibility of the oscillators diverges as the average lifetime of de Sitter vacua approaches the Page time. Thus, optimal regions of the landscape are most susceptible to volume influx from their environing landscape. We then show that the displacement fluctuations for the oscillators exhibit a $1/f$ power spectrum over a broad range of frequencies, precisely at the critical Page lifetime distribution. A $1/f$ power spectrum is a hallmark of non-equilibrium systems at criticality. In analogy with sand avalanches in the abelian sandpile or neuronal avalanches in the brain, de Sitter vacua at criticality can be thought of as undergoing scale invariant volume fluctuation avalanches.