Large deviations in the early Universe

Abstract

Fluctuations play a critical role in cosmology. They are relevant across a range of phenomena from the dynamics of inflation to the formation of structure. In many cases, these fluctuations are coarse grained and follow a Gaussian distribution as a consequence of the central limit theorem. Yet, some classes of observables are dominated by rare fluctuations and are sensitive to the details of the underlying microphysics. In this paper, we argue that the large deviation principle can be used to diagnose when one must appeal to the fundamental description. Concretely, we investigate the regime of validity for the Fokker-Planck equation that governs stochastic inflation. For typical fluctuations, this framework leads to the central limit-type behavior expected of a random walk. However, fluctuations in the regime of the large deviation principle are determined by instantonlike saddle points accompanied by a new energy scale. When this energy scale is above the UV cutoff of the effective field theory, the tail is only calculable in the microscopic description. We explicitly demonstrate this phenomenon in the context of determining the phase transition to eternal inflation, the distribution of scalar field fluctuations in de Sitter, and the production of primordial black holes.