Scalar perturbations during multiple-field slow-roll inflation

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 this purpose a basis for these perturbations determined by the background dynamics is introduced and multiple-field slow-roll functions are defined. To obtain analytical solutions to first order, the scalar perturbation modes have to be treated in three different regimes. Matching is performed by identifying leading order asymptotic expansions analytically in different regions. The possible sources for multiple-field effects in the gravitational potential are the particular solution caused by the coupling to the field perturbation perpendicular to the field velocity, and the rotation of the basis. The former can contribute even to leading order if the corresponding multiple-field slow-roll function is sizable during the last 60 e-folds. Making some simplifying assumptions, the evolution of adiabatic and isocurvature perturbations after inflation is discussed. The analytical results are illustrated and checked numerically with the example of a quadratic potential.