The dark energy equation of state, , is directly related to the one used in SN Ia observations. From Eqs. (4.92) and (4.93) it is given by

where the last approximate equality is valid in the regime where the radiation density is negligible relative to the matter density . The viable f (R) models approach the CDM model in the past, i.e., as . In order to reproduce the standard matter era () for , we can choose in Eqs. (4.92) and (4.93). Another possible choice is , where is the present value of . This choice may be suitable if the deviation of from 1 is small (as in scalar-tensor theory with a nearly massless scalar field [583, 93]). In both cases the equation of state can be smaller than before reaching the de Sitter attractor [306, 31, 587, 435], while the effective equation of state is larger than . This comes from the fact that the denominator in Eq. (4.97) becomes smaller than in the presence of the matter fluid. Thus f (R) gravity models give rise to the phantom equation of state of dark energy without violating any stability conditions of the system. See [210, 417, 136, 13] for observational constraints on the models (4.83) and (4.84) by using the background expansion history of the universe. Note that as long as the late-time attractor is the de Sitter point the cosmological constant boundary crossing of reported in [52, 50] does not typically occur, apart from small oscillations of around the de Sitter point.There are some works that try to reconstruct the forms of f (R) by using some desired form for the evolution of the scale factor or the observational data of SN Ia [117, 130, 442, 191, 621, 252]. We need to caution that the procedure of reconstruction does not in general guarantee the stability of solutions. In scalar-tensor dark energy models, for example, it is known that a singular behavior sometimes arises at low-redshifts in such a procedure [234, 271]. In addition to the fact that the reconstruction method does not uniquely determine the forms of f (R), the observational data of the background expansion history alone is not yet sufficient to reconstruct f (R) models in high precision.

Finally we mention a number of works [115, 118, 119, 265, 319, 515, 542, 90] about the use of metric f (R) gravity as dark matter instead of dark energy. In most of past works the power-law f (R) model has been used to obtain spherically symmetric solutions for galaxy clustering. In [118] it was shown that the theoretical rotation curves of spiral galaxies show good agreement with observational data for , while for broader samples the best-fit value of the power was found to be [265]. However, these values are not compatible with the bound derived in [62, 160] from a number of other observational constraints. Hence, it is unlikely that f (R) gravity works as the main source for dark matter.

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