On the number of $e$-folds in the Jordan and Einstein frames

We investigate how inflationary predictions are affected by the difference in the number of $e$-folds between the Jordan and Einstein frames. We study several test models in relation to a Jordan frame defined by the common Higgs-inflation-like non-minimal coupling to gravity and consider two different formulations of gravity: metric and Palatini. We find that the difference is quite contained in case of a metric Jordan frame while can be quite remarkable in case of a Palatini Jordan frame. We also discuss a way to overcome the discrepancy by introducing a frame invariant physical distance and a consequently frame invariant number of $e$-folds.