Recently, research in cosmology has seen a growing interest in theories of gravity beyond General Relativity (GR). From an observational point of view, there are two main reasons for this. Firstly, the law of gravity has never been directly tested on scales larger than the Solar System. Hence, by understanding better the various signatures that different gravity models can leave on cosmological observables, one can improve the chances of identifying any departures from GR, or alternatively, extend the model's observational success into a whole new regime. Secondly, theories of modified gravity can arise also as an alternative to the cosmological constant (or any other form of dark energy) to explain the current accelerated expansion of the Universe. Using my results from suitably modified Boltzmann, N-body codes and semi-analytical models of structure formation, I will describe the way modified gravity models typically impact a series of cosmological observables. I will use two popular models as examples, which are known as Galileon and Nonlocal Gravity. In the Galileon model, the modifications to gravity on large scales are driven by nonlinear derivative interactions of a scalar field, which can nevertheless be suppressed in the Solar System by means of a mechanism known as Vainshtein screening. This model can provide a good fit to the latest CMB, BAO and SNIa data, although with different cosmological parameters than the standard LCDM model. Specifically, unlike LCDM, the Galileon model predicts nonzero neutrino masses (over 5sigma) and the constraints on the Hubble rate are compatible with its local determinations. The results from my N-body simulations and Halo Occupation Distribution analysis also show that the model can describe the measured clustering amplitude of Luminous Red Galaxies and that the screening mechanism can be very efficient in "hiding" the modifications to gravity on small scales. However, these results also show that the observational viability of the model may be under pressure due to the combined constraints derived from the sign of the ISW effect and from Solar System tests. In the Nonlocal model, the acceleration of the universe is driven by terms that involve the inverse of a derivative operator acting on curvature tensors. This model is also likely to pass large-scale structure constraints with the same flying colors as Galileon gravity, but the lack of a screening mechanism in this model makes it unclear on whether or not it is able to satisfy Solar System constraints. These steps I will describe for the cases of the Galileon and Nonlocal models can be viewed as guidelines for one to place constraints on other (or not yet invented) models of modified gravity. References: The results I will describe are based on the following papers: arXiv:1208.0600, arXiv:1302.6241, arXiv:1306.3219, arXiv:1308.3699, arXiv:1401.1497, arXiv:1404.1365, arXiv:1406.0485, arXiv:1408.1084
