Dark energy as a scalar field non-minimally coupled to gravity

The cosmological constant is not the only possibility in order to describe the accelerated expansion of the Universe. A different approach is to modify the gravitational sector of the Einstein equations. In scalar-tensor theories the gravitational interaction is affected by both a scalar and a tensor field. The dependence of gravity from the scalar field is obtained through a non-minimal coupling function which multiplies the Ricci scalar in the Lagrangian. In this thesis we consider a specific shape of the coupling function that reduces to the minimal coupling case and to the induced gravity case for specific choices of the parameters. We consider two shapes for the potential: one leads to an effectively massless Klein-Gordon equation while the other is motivated by the fact that it is a viable potential for the chaotic inflation in superconformal theory. For the former we consider also the conformal coupling case, which is the required coupling in order to obtain a conformally invariant theory. We derive the fundamental equation at the background and linear perturbations level and then we recover the initial condition for the perturbations. In order to study the evolution for the background and linear fluctuations within non-minimally coupling we modified the publicly available Einstein-Boltzmann code CLASS. The evolution of the dark energy density parameter and the equation of state are shown. Furthermore we pay attention to the actual value of the post-Newtonian parameters in order to see which choices of the parameters satisfies the Solar System constraints. We present the results obtained for CMB anisotropies, linear matter power spectrum, metric and scalar field perturbations. As for the background we confront them with the ΛCDM model for both the potential considered