In this section we apply f (R) theories to dark energy. Our interest is to construct viable f (R) models that can realize the sequence of radiation, matter, and accelerated epochs. In this section we do not attempt to find unified models of inflation and dark energy based on f (R) theories.

Originally the model () was proposed to explain the late-time cosmic acceleration [113, 120, 114, 143] (see also [456, 559, 17, 223, 212, 16, 137, 62] for related works). However, this model suffers from a number of problems such as matter instability [215, 244], the instability of cosmological perturbations [146, 74, 544, 526, 251], the absence of the matter era [28, 29, 239], and the inability to satisfy local gravity constraints [469, 470, 245, 233, 154, 448, 134]. The main reason why this model does not work is that the quantity is negative. As we will see later, the violation of the condition gives rise to the negative mass squared for the scalaron field. Hence we require that to avoid a tachyonic instability. The condition is also required to avoid the appearance of ghosts (see Section 7.4). Thus viable f (R) dark energy models need to satisfy [568]

where is the Ricci scalar today.In the following we shall derive other conditions for the cosmological viability of f (R) models. This is based on the analysis of [26]. For the matter Lagrangian in Eq. (2.1) we take into account non-relativistic matter and radiation, whose energy densities and satisfy

respectively. From Eqs. (2.15) and (2.16) it follows that

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