Application of the excursion set theory to modified gravity

Gravity could be modified compared to General Relativity on very large scales, as suggested by the discovery of the accelerated expansion of the Universe. One possible explanation would be the existence, on cosmological scales, of scalar fields, which have hidden their presence until now thanks to particular screening mechanisms and which would infuence the growth of large scale structures by their coupling to Cold Dark Matter. Since a complete analysis of the structure formation problem would require complete N-body simulations, which are really complex from a computational point of view, the goal of this thesis is to find a set of semi-analytical approximations that allows to obtain reliable results. First of all we will investigate the spherical collapse of such structures, in order to understand how this new scale-dependent fifth force, mediated by the scalar field, changes the critical initial overdensity necessary to collapse and also how different models modify this value. In a second step, we will use the excursion set theory to relate the statistics of the density field at early times to the number of virialized, gravitationally bound structures. Since the fifth force is stronger for smaller objects, a smaller initial overdensity is required to collapse and we expect to find a larger number of haloes with respect to General Relativity.