Legendre scalarization in gravity and cosmology

<jats:title>Abstract</jats:title><jats:p>We propose a new formulation of <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) gravity, dubbed scalarized <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) 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 <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) gravity reduces to general relativity with a massive scalar field, and we explore its implications for Friedmann cosmology. Our findings suggest that scalarized <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) gravity may lead to simplified descriptions of cosmological applications, while the proposed formulation could offer a new perspective on the relationship between <jats:italic>f</jats:italic>(<jats:italic>R</jats:italic>) gravity and scalar–tensor theories.</jats:p>