A covariant approach to general field space metric in multi-field   inflation
  We present a covariant formalism for general multi-field system which enables us to obtain higher order action of cosmological perturbations easily and systematically. The effects of the field space geometry, described by the Riemann curvature tensor of the field space, are naturally incorporated. We explicitly calculate up to the cubic order action which is necessary to estimate non-Gaussianity and present those geometric terms which have not yet known before. 
Jinn-Ouk Gong, Takahiro Tanaka
astro-ph.CO gr-qc hep-ph hep-th