The role of Gravity in the comprehension of the early and late time Universe

In this thesis we study modified theories of gravity as a solution to longstanding open problems in Cosmology. In particular our main contribution relies in the determination of a viable form of f(R) that allows us to explain the current accelerated phase of the Universe. To set the initial conditions on the f(R) functions, we involve the use of the so called cosmography of the Universe, i.e. the technique of fixing constraints on the observable Universe by comparing expanded observables with current data. This powerful approach is essentially model independent and correspondingly we got a model independent reconstruction of f(z) classes within the interval z ∈ [0,1]. To allow the Hubble rate to evolve around z ≤ 1, we considered three relevant frameworks of effective cosmological dynamics, i.e. the ΛCDM model, the CPL parametrization and a polynomial approach to dark energy. Finally, cumbersome algebra permits us to pass from f(z) to f(R) and the general outcome of our work is the determination of a viable f(R) function, that effectively describes the observed Universe dynamics. Furthermore, we study a form of f(R) that can justify the relic Dark Matter abundance and the asymmetry between matter and antimatter. In particular we study two models of f(R) theories of gravitation that, with the opportune choice of free parameters, introduce a little perturbation to the scale factor of the Universe in the radiation dominated (RD) phase predicted by General Relativity (GR), i.e., a(t) ≈ t^(1/2). This little perturbation generates a Ricci scalar different by zero, i.e., R 6= 0 that reproduces the correct magnitude for the asymmetry factor η computed in the frame of the theories of gravitational baryogenesis and gravitational leptogenesis. Besides, the evolution of relics particles (WIMPs) in the f(R) context is analysed in order to explain PAMELA data. Furthermore, theories where the Planck scale is dynamically generated from dimensionless interactions are studied here in the context of the inflationary scenario. We first study the minimal single-field realisation in the low-energy effective field theory limit, finding the predictions ns ≈ 0.96 for the spectral index and r ≈ 0.13 for the tensor-to-scalar ratio, which can be reduced down to ≈ 0.04 in presence of large couplings. Next we consider agravity as a dimensionless quantum gravity theory finding a multi-field inflation that converges towards an attractor trajectory that predicts ns ≈ 0.96 and 0.003 < r < 0.13, interpolating between the quadratic and Starobinsky inflation.