My PhD Thesis focused on the derivation of constraints on modified gravity models by combining mass profiles of galaxy clusters, reconstructed through gravitational lensing and internal dynamics as traced by the motion of member galaxies. Modifications of gravity change the relation between the matter density fluctuations and the scalar potentials Phi, Psi in the metric describing the space-time of a galaxy cluster. An accurate measurement of these potentials can be used to investigate deviations from General Relativity (GR) predictions according to which Phi=Psi. Mass profiles from dynamics track only the potential Phi, while the lensing mass profiles reflect the contribution of the sum of the potentials. In the first part of the Thesis I analyse the galaxy clusters MACS J1206 and RXJ 2248 for which high-quality imaging and spectroscopic data have been obtained from the CLASH and CLASH-VLT projects. I combine previous determinations of lensing and dynamics mass profiles for the galaxy cluster MACS J1206 to constrain generic deviation from GR, parametrized by the ratio eta=Psi/Phi, under the assumptions of spherical symmetry and dynamical relaxation. I obtain eta=1.01 with 30% statistical uncertainties in the cluster outskirt at the 68% C.L., in agreement with GR and competitive with constraints from other analyses. I then implement a modified version of the MAMPOSSt procedure (Mamon+13), a method to reconstruct galaxy clusters mass profiles from dynamics by solving the Jeans' equation. I introduce a parametrization of the potential which accounts for a large variety of modified gravity models. As a case study, I determined the total mass profiles from dynamics of the aforementioned clusters in f(R) gravity, combining then the information from the lensing mass reconstruction to constrain the additional degree of freedom of the theory lambda. The larger is lambda the stronger is the departure from GR. In the case of MACSJ 1206 the kinematic+lensing analysis is in agreement with GR, lambda<1.61 Mpc at 95% C.L., while for RXJ 2248 instead a possible tension with standard gravity appears when adding lensing information lambda>0.14 Mpc at 95% C.L. In order to better understand the nature of this apparent tension, in the last part of the Thesis I preform an analysis of ΛCDM cosmological simulations of galaxy clusters aimed at investigating the impact of systematic effects. I found that 70% of clusters produce spurious detection of modified gravity when no selection criteria are applied. I then defined two observational parameters, connected to the main systematics affecting the analysis, which help in selecting the suitable clusters for the application of our method. Moreover, I predict that a dozen “regular” clusters with reliable mass profile reconstructions are already sufficient to obtain the tightest constraints on GR deviations reachable with our analysis. This work is particularly interesting in view of the next generation imaging and spectroscopic surveys that will provide data for hundreds of clusters.
