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## I. INTRODUCTION 

The physical origin of the dark energy (DE) which is responsible for an accelerated ex- pansion of the current Universe is one of the largest mysteries not only in cosmology but also in fundamental physics [1]. Although the standard spatially flat Λ-Cold-Dark-Matter (ΛCDM) model is consistent with all kinds of current observational data [2], some tentative deviations from it have been reported recently [3, 4] which, if proven to be not due to system- atic and other errors, may eventually rule out an exact cosmological constant. Furthermore, in the ΛCDM model, the cosmological term is regarded as a new fundamental constant whose observed value is much smaller than any other energy scale known in physics. So, its understanding in fundamental physics is lacking today, although some non-perturbative effects may generate such a small quantity [5]. On the other hand, we know that “primor- dial DE,” which is responsible for inflation in the early universe [6–8], is not identical to the cosmological constant, in particular, it is not stable and eternal. Hence it is natural to seek for non-stationary models of the current DE, too. 

Among them, f ( R ) gravity which modifies and generalizes the Einstein gravity by in- corporating a new phenomenological function of the Ricci scalar R , f ( R ), provides a self- consistent and non-trivial alternative to ΛCDM model, see e.g. Ref. [9] for a recent review. This theory is a special class of the scalar-tensor theory of gravity with the vanishing Brans- Dicke parameter ω$_{BD}$ [10, 11]. It contains a new scalar degree of freedom dubbed ”scalaron” in Ref. [6], thus, it is a non-perturbative generalization of the Einstein gravity. 

This additional degree of freedom imposes a number of conditions on viable functional forms of f ( R ). In particular, in order to have the correct Newtonian limit for R ≫ R$_{0}$ ≡ R ( t$_{0}$ ) ∼ H 2 0 where t$_{0}$ is the present moment and H$_{0}$ is the Hubble constant, as well as the standard matter-dominated stage with the scale factor behaviour a ( t ) ∝ t 2 / 3 driven by cold dark matter and baryons, the following conditions should be fulfilled: 

<img src='content_image/358725.jpg'>

where the prime denotes the derivative with respect to the argument R . In addition, the stability condition f ′′ ( R ) > 0 has to be satisfied that guarantees that the standard matter- dominated Friedmann stage remains an attractor with respect to an open set of neighboring isotropic cosmological solutions in f ( R ) gravity. In quantum language, this condition means that scalaron is not a tachyon. Note that the other stability condition, f ′ ( R ) > 0, which 

