Legendre scalarization in gravity and cosmology

We propose a new formulation of f ( R ) gravity, dubbed scalarized f ( R ) gravity, in which the Legendre transform is included as a dynamical term. This leads to a theory with second-order field equations that describes general relativity with a self-interacting scalar field, without requiring the introduction of conformal frames. We demonstrate that the quadratic version of scalarized f ( R ) gravity reduces to general relativity with a massive scalar field, and we explore its implications for Friedmann cosmology. Our findings suggest that scalarized f ( R ) gravity may lead to simplified descriptions of cosmological applications, while the proposed formulation could offer a new perspective on the relationship between f ( R ) gravity and scalar–tensor theories.