Stochastic Inflation in Numerical Relativity

Abstract: A set of 3+1 equations for stochastic inflation incorporating all metric and scalar matter degrees of freedom, first presented in previous work, are re-derived in a gauge invariant manner. We then present numerical implementations of these stochastic equations, cast in the BSSN formulation of Numerical Relativity, demonstrating their efficacy in both a slow-roll and an ultra slow-roll scenario. We find the evolution is correctly reproduced for all the dynamical variables, and the energy and momentum constraints are well-satisfied. This demonstrates that the stochastic equations are theoretically and numerically robust and ready to be applied to a wider inflationary landscape. Our simulations result in real space realizations of the fully non-linear stochastic dynamics with gradient information retained. As a generalisation of standard stochastic inflation, inflationary numerical relativity and lattice cosmology, this work opens up the possibility for reliable predictions of non-perturbative phenomena and provides precise initial conditions for subsequent cosmological eras.