Multi-scalar-tensor equivalents for modified gravitational actions

A general scheme of constructing scalar-tensor equivalents to modified gravitational actions are studied using the algebra of exterior differential forms and the first order formalism that allows an independent connection and coframe. By introducing appropriate constraints on the connection, pseudo-Riemannian cases as well as non-Riemannian cases are discussed for various gravitational models. The issue of the dynamical degree of freedom for the resulting scalar fields is discussed at the level of the field equations. Explicit scalar-tensor equivalents for gravitational models based on $f(R)$ models, the quadratic curvature Lagrangians and the models involving the gradients of the scalar curvature are presented. In particular, explicit scalar-tensor equivalence for gravitational Lagrangians popular in some cosmological models are constructed.