Fab Four: When John and George play gravitation and cosmology
  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.
