  The computation of the primordial power spectrum in multi-field inflation
models requires us to correctly account for all relevant interactions between
adiabatic and non-adiabatic modes around and after horizon crossing. One
specific complication arises from derivative interactions induced by the
curvilinear trajectory of the inflaton in a multi-dimensional field space. In
this work we compute the power spectrum in general multi-field models and show
that certain inflaton trajectories may lead to observationally significant
imprints of `heavy' physics in the primordial power spectrum if the inflaton
trajectory turns, that is, traverses a bend, sufficiently fast (without
interrupting slow roll), even in cases where the normal modes have masses
approaching the cutoff of our theory. We emphasise that turning is defined with
respect to the geodesics of the sigma model metric, irrespective of whether
this is canonical or non-trivial. The imprints generically take the form of
damped superimposed oscillations on the power spectrum. In the particular case
of two-field models, if one of the fields is sufficiently massive compared to
the scale of inflation, we are able to compute an effective low energy theory
for the adiabatic mode encapsulating certain relevant operators of the full
multi-field dynamics. As expected, a particular characteristic of this
effective theory is a modified speed of sound for the adiabatic mode which is a
functional of the background inflaton trajectory and the turns traversed during
inflation. Hence in addition, we expect non-Gaussian signatures directly
related to the features imprinted in the power spectrum.
