Einstein gravity, without the cosmological constant, corresponds to and , so that the term in Eq. (2.7) vanishes. In this case we have and hence the Ricci scalar is directly determined by the matter (the trace ). In modified gravity the term does not vanish in Eq. (2.7), which means that there is a propagating scalar degree of freedom, . The trace equation (2.7) determines the dynamics of the scalar field (dubbed “scalaron” [564]).

The field equation (2.4) can be written in the following form [568]

where and Since and , it follows that Hence the continuity equation holds, not only for , but also for the effective energy-momentum tensor defined in Eq. (2.9). This is sometimes convenient when we study the dark energy equation of state [306, 568] as well as the equilibrium description of thermodynamics for the horizon entropy [53].There exists a de Sitter point that corresponds to a vacuum solution () at which the Ricci scalar is constant. Since at this point, we obtain

The model satisfies this condition, so that it gives rise to the exact de Sitter solution [564]. In the model , because of the linear term in , the inflationary expansion ends when the term becomes smaller than the linear term (as we will see in Section 3). This is followed by a reheating stage in which the oscillation of leads to the gravitational particle production. It is also possible to use the de Sitter point given by Eq. (2.11) for dark energy.We consider the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime with a time-dependent scale factor and a metric

where is cosmic time. For this metric the Ricci scalar is given by where is the Hubble parameter and a dot stands for a derivative with respect to . The present value of is given by where describes the uncertainty of [264].The energy-momentum tensor of matter is given by , where is the energy density and is the pressure. The field equations (2.4) in the flat FLRW background give

where the perfect fluid satisfies the continuity equation We also introduce the equation of state of matter, . As long as is constant, the integration of Eq. (2.17) gives . In Section 4 we shall take into account both non-relativistic matter () and radiation () to discuss cosmological dynamics of f (R) dark energy models.Note that there are some works about the Einstein static universes in f (R) gravity [91, 532]. Although Einstein static solutions exist for a wide variety of f (R) models in the presence of a barotropic perfect fluid, these solutions have been shown to be unstable against either homogeneous or inhomogeneous perturbations [532].

http://www.livingreviews.org/lrr-2010-3 |
This work is licensed under a Creative Commons License. Problems/comments to |