Various astrophysical data sets support the current standard model of cosmology, in which our universe is well-described on large scales by a cosmological constant Lambda and cold dark matter (CDM). The Lambda-CDM paradigm rests on two assumptions: (i) the cosmological principle; and that (ii) Einstein's General Relativity is the correct theory of gravity in the classical limit. The former implies that regardless of our location in the universe, its properties look the same if smoothed on large enough scales. The latter dictates how the universe as a whole and the structures within it evolve, gravitation being the dominant force at large distances. Under these premises, to explain the observed late-time accelerated expansion of the universe we need an exotic form of energy with large negative pressure, named dark energy. Lambda is the simplest candidate for this obscure ingredient, and is currently associated with the energy density of the vacuum. Cold dark matter is the second most abundant constituent of the universe, even though it has not been detected yet. This slowly moving collection of particles forms the scaffolding of the stunning, luminous structures we see with our telescopes. Although both dark components are so far in the realm of speculation, a cosmological constant suffers from important theoretical shortcomings.

An alternative is to question the validity of General Relativity on cosmological scales. In fact, cosmic acceleration could stem from gravity behaving differently on the largest scales, eliminating the need for dark energy. Moreover, modifications to General Relativity lead to changes in the formation of structures compared to standard gravity. In particular, the accretion history of collapsed objects, as well as their abundance as a function of mass and time are key probes for such departures.

The first part of this thesis presents the statistical analysis of number counts data for the most massive galaxy clusters in the universe, which can put tight constraints on possible departures from General Relativity. The increasing quality of upcoming surveys requires fast and accurate predictions of the cosmological observables. In the second part, I develop methods to evaluate in efficient and unbiased way relevant quantities that describe the distribution of matter and its evolution in Lambda-CDM and f(R) cosmologies.