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means that gravity is attractive and graviton is not a ghost, is automatically fulfilled in this regime. Specific functional forms that satisfy all these conditions have been proposed in Refs. [12–14] etc., and much work has been done on their cosmological consequences. 

In the previous paper [15] we calculated evolution of matter density fluctuations in viable f ( R ) models [12, 14] in the limiting case R ≫ R$_{0}$ during the matter-dominated stage and found an analytic expression for them. In this paper we extend the previous analysis and perform numerical calculations of the evolution of both background space-time and density fluctuations for the particular f ( R ) model of Ref. [14] without such restriction on R . As a result, we have found the phantom boundary crossing at an intermediate redshift z ≲ 1 for the background space-time metric and an anomalous behaviour of the growth index of fluctuations. 

The rest of the paper is organized as follows. In § 2 we introduce evolution equations for the homogeneous and isotropic background and present results of numerical integra- tion. In § 3 we report numerical solutions for the evolution of density fluctuations and other observables. Section 4 is devoted to conclusions and discussion. 

## II. EVOLUTION OF THE BACKGROUND UNIVERSE 

We adopt the following action with a four-parameter family of f ( R ) models: 

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<img src='content_image/337520.jpg'>

where n, λ, R$_{s}$ , and M are model parameters and S$_{m}$ is the action of the matter content which is assumed to be minimally coupled to gravity (thus, the action (2) is written in the Jordan frame). This is the model of Ref. [14] modified by the last term in (3) borrowed from the inflationary model of Ref. [6]. This term is introduced for several purposes associated with high-curvature behaviour of the theory. One of them, as explained in Ref. [14] , is to avoid excessive growth of the scalaron mass, m 2 s = 1 / 3 f ′′ ( R ) in the regime (1), towards the early Universe, t → 0. The other one is to remove the additional and undesirable “Big Boost” singularity which can arise in the original models [12–14] as was shown in Ref. [16] (see Refs. [15, 17, 18] for more discussion on this point). The value of M should be 

