Nonlinear evolution of long-wavelength metric fluctuations in inflationary models

Abstract

Stochastic inflation can be viewed as a sequence of two-step processes. In the first step a stochastic impulse from short-distance quantum fluctuations acts on long waves---the interaction. In the second step the long waves evolve semiclassically---the propagation. Both steps must be developed to address whether fluctuations for cosmic structure formation may be non-Gaussian. We describe a formalism for following the nonlinear propagation of long-wavelength metric and scalar-field fluctuations. We perform an expansion in spatial gradients of the Arnowitt-Deser-Misner equations and we retain only terms up to first order. At each point the fields obey evolution equations like those in a homogeneous universe, but now described by a local scale factor ${e}^{\ensuremath{\alpha}}$ and Hubble expansion rate $H$. However, the different points are joined together through the momentum constraint equation. The gradient expansion is appropriate for inflation if the long-wave fields are smoothed over scales below ${e}^{\ensuremath{-}\ensuremath{\alpha}}{H}^{\ensuremath{-}1}$. Our equations are naturally described in the Einstein-Hamilton-Jacobi framework, which governs an ensemble of inhomogeneous universes, and which may be interpreted as a semiclassical approximation to the quantum theory. We find that the Hubble parameter, which is a function of the local values of the scalar field, obeys a separated Hamilton-Jacobi equation that also governs the semiclassical phase of the wave functional. In our approximation, time hypersurface changes leave the equations invariant. However, the stochastic impulses that change the field initial conditions are most simply given on uniform expansion factor hypersurfaces whereas propagation is most easily solved on uniform Hubble hypersurfaces, in terms of $\ensuremath{\alpha}({x}^{j},H)$, the nonlinear analog of $\ensuremath{\zeta}$ of linear perturbation theory; we therefore pay special attention to hypersurface shifting. In particular, we describe the transformation process for the fluctuation probability functional. Exact general solutions are found for the case of a single scalar field interacting through an exponential potential. For example, we show that quantum corrections to long-wavelength evolution of the metric are characteristically small using exact Green's-function solutions of the Wheeler-DeWitt equation for this potential. Approximate analytic solutions to our classical system for slowly evolving multiple scalar fields are also easy to obtain in this formalism, contrasting with previous numerical approaches.