We calculate the scalar gravitational and matter perturbations in the context of slow-roll inflation with multiple scalar fields, that take values on a (curved) manifold, to first order in slow roll. For that purpose a basis for these perturbations determined by the background dynamics is introduced and multiple field slow-roll functions are defined. To obtain analytic solutions to first order, the scalar perturbation modes have to be treated in three different regimes. Consistency of the various approximations fixes their matching times. Multiple field effects in the gravitational potential are due to the rotation of the basis and to the particular solution caused by the coupling to the field perturbation perpendicular to the field velocity. They can contribute even to leading order if the corresponding multiple field slow-roll function is sizable during the last 60 e-folds. The analytical results are illustrated and checked numerically with the example of a quadratic potential.

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.

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.