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We show that in single field inflationary models the super-horizon evolutionof curvature perturbations on comoving slices $\mathcal{R}$, which can causethe production of primordial black holes (PBH), is not due to entropyperturbations, but to the background evolution effect on the conversion betweenentropy and curvature perturbations. We derive a general relation between thetime derivative of comoving curvature perturbations and entropy perturbations,in terms of a conversion factor depending on the background evolution. Contraryto previous results derived in the uniform density gauge assuming the gradientterm can be neglected on super-horizon scales, the relation is valid on anyscale for any minimally coupled single scalar field model, also on sub-horizonscales where gradient terms are large.  We apply it to the case of quasi-inflection inflation, showing that whileentropy perturbations are decreasing, $\mathcal{R}$ can grow on super-horizonscales, due to a large increase of the conversion factor. This happens in thetime interval during which a sufficiently fast decrease of the equation ofstate $w$ transforms into a growing mode that in slow-roll models would be adecaying mode. The same mechanism also explains the super-horizon evolution of$\mathcal{R}$ in globally adiabatic systems, for which entropy perturbationsvanish on any scale, such as ultra slow-roll inflation and its generalizations.

Are primordial black holes produced by entropy perturbations in single field inflationary models?