In [26] it was shown that needs to be close to 0 during the radiation domination as well as the matter domination. Hence the viable f (R) models are close to the CDM model in the region . The Ricci scalar remains positive from the radiation era up to the present epoch, as long as it does not oscillate around . The model (, ) is not viable because the condition is violated.

As we will see in Section 5, the local gravity constraints provide tight bounds on the deviation parameter in the region of high density (), e.g., for [134, 596]. In order to realize a large deviation from the CDM model such as today () we require that the variable changes rapidly from the past to the present. The f (R) model given in Eq. (4.81), for example, does not allow such a rapid variation, because evolves as in the region . Instead, if the deviation parameter has the dependence

it is possible to lead to the rapid decrease of as we go back to the past. The models that behave as Eq. (4.82) in the regime are The models (A) and (B) have been proposed by Hu and Sawicki [306] and Starobinsky [568], respectively. Note that roughly corresponds to the order of for . This means that for . In the next section we will show that both the models (A) and (B) are consistent with local gravity constraints for .In the model (A) the following relation holds at the de Sitter point:

where and is the Ricci scalar at the de Sitter point. The stability condition (4.80) gives [587] The parameter has a lower bound determined by the condition (4.86). When , for example, one has and . Under Eq. (4.86) one can show that the conditions (4.56) are also satisfied.Similarly the model (B) satisfies [568]

with When we have and , which is the same as in the model (A). For general , however, the bounds on in the model (B) are not identical to those in the model (A).Another model that leads to an even faster evolution of is given by [587]

A similar model was proposed by Appleby and Battye [35]. In the region the model (4.89) behaves as , which may be regarded as a special case of (4.82) in the limit thatThe models (A), (B) and (C) are close to the CDM model for , but the deviation from it appears when decreases to the order of . This leaves a number of observational signatures such as the phantom-like equation of state of dark energy and the modified evolution of matter density perturbations. In the following we discuss the dark energy equation of state in f (R) models. In Section 8 we study the evolution of density perturbations and resulting observational consequences in detail.

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