In the ΛCDM model, the late-time accelerated expansion of the Universe is explained via a dark energy fluid in the form of a cosmological constant. Such a cosmological constant dominates the energy budget of the Universe today, and yet, it is still a poorly understood species because it is not observed yet. A competitive theoretical approach to understand this is via the so-called f (R) extended theories of gravity, which explain the late acceleration epoch of the Universe resorting to a geometrical modification of the field equations. We illustrate how f (R) theories are constructed and how both the analysis of the cosmological expansion and the growth of matter density perturbations in these theories may differ from the standard Einsteinian results. We study the evolution of matter density perturbations in a viable f (R) model (Hu-Sawicki model) and explain why the Hu-Sawicki model is indeed a viable alternative to ΛCDM by discussing the Dynamical System approach as a method used to obtain the cosmological background solutions. A complete comparison of density perturbations in both the ΛCDM model and Hu-Sawicki model is presented
