Nonlinear evolution of long-wavelength metric fluctuations in inflationary models

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