The spectrum of cosmological perturbations produced by a multi-component   inflaton to second order in the slow-roll approximation
  We derive general analytic formulae for the power spectrum and spectral index of the curvature perturbation produced during inflation driven by a multi-component inflaton field, up to the second order in the slow-roll approximation. We do not assume any specific properties of the potential or the metric on the scalar field space, except for the slow-roll condition, Einstein gravity, and the absence of any permanent isocurvature modes. 
Takahiro T. Nakamura and Ewan D. Stewart