A Panorama of Viable $F(R)$ Gravity Dark Energy Models

In this work we shall study the late-time dynamics of several $F(R)$ gravity models. By appropriately expressing the field equations in terms of the redshift and of a statefinder function, we shall solve numerically the field equations using appropriate physical motivated initial conditions. We consider models which, by construction, are described by a nearly $R^2$-model at early epochs and we fine tune the parameters to achieve viability and compatibility with the latest Planck constraints at late times. Such models provide a unified description of inflation and dark energy era and notably a common feature of all the models is the presence of dark energy oscillations. Furthermore, we show that, in contrast to general relativistic fluids and scalar field descriptions, a large spectrum of different dark energy physics is generated by simple $F(R)$ gravity models, varying from phantom, to nearly de Sitter and to quintessential dark energy eras.