Gravitational scalar-tensor theory

We consider a new form of gravity theories in which the action is written in terms of the Ricci scalar and its first and second derivatives. Despite the higher derivative nature of the action, the theory is ghost-free under an appropriate choice of the functional form of the Lagrangian. This model possesses 2 + 2 physical degrees of freedom, namely 2 scalar degrees and 2 tensor degrees. We exhaust all such theories with the Lagrangian of the form f(R,{({{\nabla }}R)}²,\square R), where R is the Ricci scalar, and then we show some examples beyond this ansatz. In the course of the analysis, we prove the equivalence between these examples and a subclass of generalized bi-Galileon theories.