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” ).
The field equation (2.4) can be written in the following form [306, 568] as well as the equilibrium description of thermodynamics for the horizon entropy .
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. 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.
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 givef (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 .
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