How (not) to use the Palatini formulation of scalar-tensor gravity

We revisit the problem of defining nonminimal 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\"{}ve Palatini and the Lagrange multiplier approach, showing that the na\"{}ve 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\"{}ve Palatini approach works there. We outline the correct decoupling limits and the strong coupling regimes. As a corollary we find that the so-called ``modified 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.