We study cosmological perturbations in general inflation models with multiple scalar fields and arbitrary kinetic terms, with special emphasis on the multifield extension of Dirac-Born-Infeld (DBI) inflation. We compute the second-order action governing the dynamics of linear perturbations in the most general case. Specializing to DBI inflation, we show that the adiabatic and entropy modes propagate with a common effective sound speed and are thus amplified at sound horizon crossing. In the small sound speed limit, we find that the amplitude of the entropy modes is much higher than that of the adiabatic modes. We also derive, in the general case, the third-order action which is useful for studying primordial non-Gaussianities generated during inflation. In the DBI case, we compute the dominant contributions to non-Gaussianities, which depend on both the adiabatic and entropy modes.

We study the linear perturbations of multi-field inflationary models governed by a Lagrangian which is a general function of the scalar fields and of a global kinetic term combining their spacetime gradients with an arbitrary field space metric. Our analysis includes k -inflation, Dirac-Born-Infeld inflation and its multi-field extensions which have been recently studied. For this general class of models, we calculate the action to second order in the linear perturbations. We decompose the perturbations into an adiabatic mode, parallel to the background trajectory, and entropy modes. We show that all the entropy modes propagate with the speed of light whereas the adiabatic mode propagates with an effective speed of sound. We also identify the specific combination of entropy modes which sources the curvature perturbation on large scales. We then study in some detail the case of two scalar fields: we write explicitly the equations of motion for the adiabatic and entropy modes in a compact form and discuss their quantum fluctuations and primordial power spectra. *