How (Not) to Palatini

Abstract: We revisit the problem of defining non-minimal gravity in the first order formalism. Specializing to scalar-tensor theories, which may be disguised as higher-derivative' models with the gravitational Lagrangians that depend only on the Ricci scalar, we show how to recast these theories as Palatini-like gravities. The correct formulation utilizes the Lagrange multiplier method, which preserves the canonical structure of the theory, and yields the conventional metric scalar-tensor gravity. We explain the discrepancies between the na\"ive Palatini and the Lagrange multiplier approach, showing that the na\"ive Palatini approach really swaps the theory for another. The differences disappear only in the limit of ordinary General Relativity, where an accidental redundancy ensures that the na\"ive Palatini works there. We outline the correct decoupling limits and the strong coupling regimes. As a corollary we find that the so-calledModified Source Gravity' models suffer from strong coupling problems at very low scales, and hence cannot be a realistic approximation of our universe. We also comment on a method to decouple the extra scalar using the chameleon mechanism.