  We give a short and self-contained proof for rates of convergence of the
Allen-Cahn equation towards mean curvature flow, assuming that a classical
(smooth) solution to the latter exists and starting from well-prepared initial
data. Our approach is based on a relative entropy technique. In particular, it
does not require a stability analysis for the linearized Allen-Cahn operator.
As our analysis also does not rely on the comparison principle, we expect it to
be applicable to more complex equations and systems.
