We develop a non-linear framework for describing long-wavelength perturbations in multiple-field inflation. The basic variables describing inhomogeneities are defined in a non-perturbative manner, are invariant under changes of time slicing on large scales and include both matter and metric perturbations. They are combinations of spatial gradients generalising the gauge-invariant variables of linear theory. Dynamical equations are derived and supplemented with stochastic source terms which provide the long-wavelength initial conditions determined from short-wavelength modes. Solutions can be readily obtained via numerical simulations or analytic perturbative expansions. The latter are much simpler than the usual second-order perturbation theory. Applications are given in a companion paper.

In this paper we review the separate universe approach for cosmological perturbations and point out that it is essentially the lowest order approximation to a gradient expansion. Using this approach, one can study the nonlinear evolution of inhomogeneous spacetimes and find the conditions under which the long wavelength curvature perturbation can vary with time. When there is one degree of freedom or a well-defined equation of state the nonlinear long wavelength curvature perturbation remains constant. With more degrees of freedom it can vary and this variation is determined by the nonadiabatic pressure perturbation, exactly as in linear theory. We identify combinations of spatial vectors characterizing the curvature perturbation which are invariant under a change of time hypersurfaces.

After simplifying and improving the non-Gaussian formalism we developed in previous work, we derive a quantitative expression for the three-point correlator (bispectrum) of the curvature perturbation in general multiple-field inflation models. Our result describes the evolution of non-Gaussianity on superhorizon scales caused by the nonlinear influence of isocurvature perturbations on the adiabatic perturbation during inflation. We then study a simple quadratic two-field potential and find that when slow roll breaks down and the field trajectory changes direction in field space, the nonGaussianity can become large. However, for the simple models studied to date, the magnitude of this non-Gaussianity decays away after the isocurvature mode is converted into the adiabatic mode.

We consider massive λφ 4 theory in de Sitter background. The mass of the scalar field φ is chosen small enough, such that the amplification of superhorizon momentum modes leads to a significant enhancement of infrared correlations, but large enough such that perturbation theory remains valid. Using the Closed-TimePath approach, we calculate the infrared corrections to the two-point function of φ to 2-loop order. To this approximation, we find agreement with the correlation found using stochastic methods. When breaking the results down to individual Feynman diagrams obtained by the two different methods, we observe that these agree as well.