Scalar-tensor theories of gravitation have recently regained a great interest after the discovery of the Chameleon mechanism and of the Galileon models. The former allows, in principle, to reconcile the presence of cosmological scalar fields with the constraints from experiments at the Solar System scale. The latter open up the possibility of building inflationary models that, among other things, do not need ad hoc potentials. Further generalizations have finally led to the most general tensor-scalar theory, recently dubbed the "Fab Four", with only first and second order derivatives of the fields in the equations of motion and that self-tune to a vanishing cosmological constant. This model has a very rich phenomenology that needs to be explored and confronted with experimental data in order to constrain a very large parameter space. In this paper, we present some results regarding a subset of the theory named "John", which corresponds to a non-minimal derivative coupling between the scalar field and the Einstein tensor in the action. We show that this coupling gives rise to an inflationary model with very unnatural initial conditions. Thus, we include a non-minimal, but non-derivative, coupling between scalar field and Ricci scalar, a term named "George" in the Fab Four terminology. In this way, we find a more sensible inflationary model, and, by performing a post-newtonian expansion of spherically symmetric solutions, we derive the set of equations that constrain the parameter space with data from experiments in the solar system.
Bruneton, J-P., Rinaldi, M., Kanfon, A., Hees, A., Schlögel, S., & Fuzfa, A. (2012). Fab Four: When John and George play gravitation and cosmology. Advances in Astronomy, 2012(430694). https://doi.org/10.1155/2012/430694