Extended quintessence with an exponential coupling
  We study a class of extended quintessence cosmologies where the scalar field playing the role of the dark energy is exponentially coupled to the Ricci scalar. We find that the dynamics induced by the effective gravitational potential in the Klein-Gordon equation dominates the motion of the field in the early universe. The resulting "R-boost" trajectory is characterized by a kinetic dark energy density, given by [3\rhomnr0(1+z)]^2 [ 32 rhor0 omegaJBD0]^(-1), where omegaJBD0, rhor0 and rhomnr0 are calculated at present, and represent the Jordan Brans Dicke parameter, the density of relativistic matter and of those species which are non-relativistic at redshift z, respectively. We show that such a trajectory represents an attractor, equivalent to a tracking solution with equation of state w=-1/3, providing a large basin of attraction for the initial dark energy density regardless of the properties of the potential energy yielding acceleration today. We derive the up to date constraints from Big Bang Nucleosynthesis (BBN) on the present scenario, and we show that they are largely satisfied for interesting trajectories of the dark energy scalar field in the early universe. We compute the cosmological perturbation spectra in these cosmologies. For a fixed value of omegaJBD0, the projection and Integrated Sachs-Wolfe effects on the cosmic microwave background anisotropy are considerably larger in the exponential case with respect to a quadratic non-minimal coupling, reflecting the fact that the effective gravitational constant depends exponentially on the dynamics of the field. 
Valeria Pettorino, Carlo Baccigalupi and Gianpiero Mangano