MODIFIED THEORIES OF GRAVITY

The recent observational data in cosmology seem to indicate that the universe is currently expanding in an accelerated way. This unexpected conclusion can be explained assuming the presence of a non-vanishing yet extremely fine tuned cosmological constant, or invoking the existence of an exotic source of energy, dark energy, which is not observed in laboratory experiments yet seems to dominate the energy budget of the Universe. On the other hand, it may be that these observations are just signalling the fact that Einstein's General Relativity is not the correct description of gravity when we consider distances of the order of the present horizon of the universe. In order to study if the latter explanation is correct, we have to formulate new theories of the gravitational interaction, and see if they admit cosmological solutions which fit the observational data in a satisfactory way. A necessary condition for the viability of a theory of ``modified gravity'' is that it has to reproduce to high precision the results of General Relativity in experimental setups where the latter is well tested. Quite in general, modifying General Relativity introduces new degrees of freedom, which are responsible for the different large distance behavior. For a modified gravity theory to be phenomenologically viable, it is necessary that the extra degrees of freedom are efficiently screened on terrestrial and astrophysical scales. One of the known mechanisms which can screen the extra degrees of freedom is known as the Vainshtein mechanism, which involves derivative self-interaction terms for these degrees of freedom. In this thesis, we consider a class of nonlinear massive gravity theories known as dGRT Massive Gravity. These theories are candidates as viable models to modify gravity at very large distances, and, apart from the mass, they contain two free parameters. We investigate the effectiveness of the Vainshtein screening mechanism in this class of theories. There are two branches of static and spherically symmetric solutions, and we consider only the branch in which the Vainshtein mechanism can occur. We truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak f\mbox{}ield limit for the gravitational potentials, while keeping all non-linearities of the mode which is involved in the screening. We determine analytically the number and properties of local solutions which exist asymptotically on large scales, and of local (inner) solutions which exist on small scales. We analyze in detail in which cases the solutions match in an intermediate region. Asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while non asymptotically flat solutions can connect both with inner solutions which display the Vainshtein mechanism, or with solutions which display a self-shielding behaviour of the gravitational field. We show furthermore that there are some regions in the parameter space where global solutions do not exist, and characterise precisely in which regions of the phase space the Vainshtein mechanism takes place.