2.1 Equations of motion
The field equation can be derived by varying the action (2.1) with respect to
:
where
.
is the energy-momentum tensor of the matter fields defined by the
variational derivative of
in terms of
:
This satisfies the continuity equation
as well as
, i.e.,
.
The trace of Eq. (2.4) gives
where
and
.
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].