Modified Theories of Gravity
  The recent observational data in cosmology seem to indicate that the universe
is currently expanding in an accelerated way. An intriguing interpretation of
these data is that they may just be signalling 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 this thesis we consider
two models which modify General Relativity at very large distances, the
Cascading DGP and the dRGT massive gravity, and investigate their
phenomenological viability. We start with a general introduction to standard
cosmology and we introduce the late time acceleration problem and the
cosmological constant problem. We then provide a pedagogical introduction to
the DGP model, of which the Cascading DGP is an extension, and to the dRGT
massive gravity. Concerning the Cascading DGP, we show that the thin limit of
the 4D brane inside the (already thin) 5D brane is well defined, at least for
the class of configurations that we consider, and confirm that gravity is
regularized in these set-ups. We give a geometrical interpretation of the
presence of the critical tension, and comment on the difference between the
results in the literature and our results, which we support with a numerical
calculation. Regarding the dRGT massive gravity, we focus on the branch of
solutions in which the Vainshtein mechanism can occur. We determine
analytically the number and properties of local solutions which exist
asymptotically on large scales (but still below the gravitational Compton
wavelength), and of local (inner) solutions which exist on small scales. We
characterize exactly the properties of global solutions in every point of the
phase space, and characterize precisely in which regions the Vainshtein
mechanism takes place. We also provide numerical solutions which confirm our
analysis.
