When can preheating affect the CMB?

We discuss the principles governing the selection of inflationary models for which preheating can affect the CMB. This is a (fairly small) subset of those models which have nonnegligible entropy/isocurvature perturbations on large scales during inflation. We study new models which belong to this class-two-field inflation with negative nonminimal coupling and hybrid/double/supernatural inflation models where the tachyonic growth of entropy perturbations can lead to the variation of the curvature perturbation, R, on super-Hubble scales. Finally, we present evidence against recent claims for the variation of R in the absence of substantial super-Hubble entropy perturbations.