  In this paper we discuss a multi-field model of inflation in which generally
all fields are non-minimally coupled to the Ricci scalar and have non-canonical
kinetic terms. The background evolution and first-order perturbations for the
model are evaluated in both the Jordan and Einstein frames, and the respective
curvature perturbations compared. We confirm that they are indeed not the same
- unlike in the single-field case - and also that the difference is a direct
consequence of the isocurvature perturbations inherent to multi-field models.
This result leads us to conclude that the notion of adiabaticity is not
invariant under conformal transformations. Using a two-field example we show
that even if in one frame the evolution is adiabatic, meaning that the
curvature perturbation is conserved on super-horizon scales, in general in the
other frame isocurvature perturbations continue to source the curvature
perturbation. We also find that it is possible to realise a particular model in
which curvature perturbations in both frames are conserved but with each being
of different magnitude. These examples highlight that the curvature
perturbation itself, despite being gauge-invariant, does not correspond
directly to an observable. The non-equivalence of the two curvature
perturbations would also be important when considering the addition of Standard
Model matter into the system.
