Cosmological tests of modified gravity models with non-zero gravitational slip

Aim of this thesis is to understand how cosmological observations can be used to test departures from general relativity. In particular, we focus on modifications of gravity which predict a non vanishing gravitational slip, i.e. predict that the two scalar potentials appearing in the perturbed metric (in the Newtonian, or Poisson gauge) can differ by a small amount. In GR the only contribution to such a difference comes from the anisotropic stress sourced by matter which, at late time and at linear order, is negligible and the relation between the two potentials is trivial Φ = Ψ. The presence of the gravitational slip affects large scale structures and, in many theories of gravity, it is related to the non-standard propagation of tensor modes. This quantity can be reconstructed in a model independent way by combining different cosmological probes. In this work we parametrize deviations from GR by introducing the phenomenological functions μ(a, k) = G matter /G, Σ(a, k) = Glight /G and η(a, k) = Φ/Ψ. Then, we use the effective field theory formalism of Dark Energy to obtain the analytic expression for these quantities, in the quasi-static approximation, for two classes of theories: Horndeski and beyond Horndeski models. We pay particular attention to the subclasses of these theories which exhibit non-zero gravitational slip, i.e. η ≠1. Since the three phenomenological functions can be reconstructed from cosmological observations, it is possible to constrain these theories and, eventually ruling out some of them, by analysing the analytic expressions for these functions for each class of models. We finally account for the recent strong constraint on the speed of tensor modes from the GW170817 event