Stochastic inflation and primordial density perturbations

Inflation in the very early universe generates a very large, homogeneous and isotropic universe, but also a distribution of primordial density perturbations. Quantum fluctuations during inflation are inevitably imprinted onto the primordial radiation after inflation through their effect on the local expansion history. This process can successfully describe the almost Gaussian, almost scale-invariant distribution of small density perturbations seen on large cosmological scales in the cosmic microwave background radiation. Recently interest has turned to the possibility of inflation generating much larger density fluctuations on smaller astrophysical scales, possibly leading to the formation of primordial black holes. I will discuss recent progress using a stochastic description of inflation to study nonlinear and nonperturbative evolution during inflation, and the resulting non-Gaussian distribution of large density perturbations after inflation, and some of the outstanding challenges.