Modified theories of gravity and cosmology
One usually call modified gravity when the Einstein-Hilbert action is modified/extended. One can modify it by changing the matter/energy (right hand side of the Einstein field equations) or the parts related to the geometry (left hand side of the Einstein field equations). Usually, changing the matter/energy is not considered as being part of modified gravity since the geometrical gravitational theory is the same but new different kind of sources are considered. This is an alternative way to understand some issues related to cosmology.For example, quintessence models introduce a scalar field to understand the late-time accelerating behaviour of the Universe. In this perspective, the accelerated expansion at late times is due to some field sourcing the right hand side of the Einstein field equations. This, however, is not the only possible approach to achieve a theoretical description of cosmic acceleration. Another option is to consider cosmic acceleration as a breakdown of general relativity at cosmological scales. In other words, instead of introducing some new matter fluid, one changes the left hand side of the Einstein field equations, i.e. the pure gravitational sector, in order to obtain the needed late time accelerated expansion. Modifications of general relativity have a history almost as long as the one of general relativity itself. With the development of the semiclassical approaches to quantum gravity and, successively, of unification schemes like supergravity and M-theory, it was realised that in many cases low energy versions of these theories correspond to modifications of general relativity, leading to an increased interest in such models, such that nowadays the exploration of modifications of GR occupies a significant part of the research in relativistic gravitation.
I have been interested on studying different modified theories of gravity with the aim of understanding the dark energy and dark matter components of the Universe. The cosmological equations describing the dynamics of a homogeneous and isotropic Universe are systems of ordinary differential equations, and one of the most elegant ways these can be investigated is by casting them into the form of dynamical systems. This allows the use of powerful analytical and numerical methods to gain a quantitative understanding of the cosmological dynamics derived by the models under study. We published a long review in this topic in Physics Reports