Stochastic inflation beyond slow roll
  We study the stochastic formalism of inflation beyond the usual slow-roll
approximation. We verify that the assumptions on which the stochastic formalism
relies still hold even far from the slow-roll attractor. This includes
demonstrating the validity of the separate universe approach to evolving
long-wavelength scalar field perturbations beyond slow roll. We also explain
that, in general, there is a gauge correction to the amplitude of the
stochastic noise. This is because the amplitude is usually calculated in the
spatially-flat gauge, while the number of e-folds is used as the time variable
(hence one works in the uniform-$N$ gauge) in the Langevin equations. We show
that these corrections vanish in the slow-roll limit, but we also explain how
to calculate them in general. We compute them in difference cases, including
ultra-slow roll and the Starobinsky model that interpolates between slow roll
and ultra-slow roll, and find the corrections to be negligible in practice.
This confirms the validity of the stochastic formalism for studying quantum
backreaction effects in the very early universe beyond slow roll.
