Gravitational scalar-tensor theory
  We consider a new form of theories of gravity 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 free from ghost 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
$f(R, (\nabla R)^2, \Box R)$, where $R$ is the Ricci scalar, and then show some
examples beyond this ansatz. In course of analysis, we prove the equivalence
between these examples and generalized bi-Galileon theories.
