  Employing the covariant formalism, we derive the evolution equations for two
scalar fields with non-canonical field space metric up to third order in
perturbation theory. These equations can be used to derive predictions for
local bi- and trispectra of multi-field cosmological models. Our main
application is to ekpyrotic models in which the primordial curvature
perturbations are generated via the non-minimal entropic mechanism. In these
models, nearly scale-invariant entropy perturbations are generated first due to
a non-minimal kinetic coupling between two scalar fields, and subsequently
these perturbations are converted into curvature perturbations. Remarkably, the
entropy perturbations have vanishing bi- and trispectra during the ekpyrotic
phase. However, as we show, the conversion process to curvature perturbations
induces local non-Gaussianity parameters $f_{NL}$ and $g_{NL}$ at levels that
should be detectable by near-future observations. In fact, in order to obtain a
large enough amplitude and small enough bispectrum of the curvature
perturbations, as seen in current measurements, the conversion process must be
very efficient. Interestingly, for such efficient conversions the trispectrum
parameter $g_{NL}$ remains negative and typically of a magnitude ${\cal
O}(10^2) - {\cal O}(10^3),$ resulting in a distinguishing feature of
non-minimally coupled ekpyrotic models.
