2.1 Equations of motion

The field equation can be derived by varying the action (2.1View Equation) with respect to gμν:
Σ ≡ F(R )R (g) − 1-f(R )g − ∇ ∇ F (R ) + g □F (R) = κ2T (M), (2.4 ) μν μν 2 μν μ ν μν μν
where F(R ) ≡ ∂f ∕∂R. T (μMν) is the energy-momentum tensor of the matter fields defined by the variational derivative of ℒM in terms of μν g:
2 δℒM T(μMν )= − √------μν-. (2.5 ) − g δg
This satisfies the continuity equation
∇ μT(M )= 0, (2.6 ) μν
as well as Σμν, i.e., μ ∇ Σμν = 0.2 The trace of Eq. (2.4View Equation) gives
3□F (R ) + F(R )R − 2f (R ) = κ2T , (2.7 )
where T = g μνT (M) μν and □F = (1∕√ −-g)∂ (√ −-ggμν∂ F ) μ ν.

Einstein gravity, without the cosmological constant, corresponds to f(R ) = R and F (R ) = 1, so that the term □F (R) in Eq. (2.7View Equation) vanishes. In this case we have R = − κ2T and hence the Ricci scalar R is directly determined by the matter (the trace T). In modified gravity the term □F (R) does not vanish in Eq. (2.7View Equation), which means that there is a propagating scalar degree of freedom, φ ≡ F (R ). The trace equation (2.7View Equation) determines the dynamics of the scalar field φ (dubbed “scalaron” [564Jump To The Next Citation Point]).

The field equation (2.4View Equation) can be written in the following form [568Jump To The Next Citation Point]

2( (M ) (D)) Gμν = κ Tμν + Tμν , (2.8 )
where Gμν ≡ R μν − (1∕2)gμνR and
κ2T(D) ≡ gμν(f − R )∕2 + ∇ μ∇ νF − gμν□F + (1 − F )R μν. (2.9 ) μν
Since ∇ μG μν = 0 and (M ) ∇ μT μν = 0, it follows that
∇ μTμ(Dν )= 0. (2.10 )
Hence the continuity equation holds, not only for Σ μν, but also for the effective energy-momentum tensor T (μDν) defined in Eq. (2.9View Equation). This is sometimes convenient when we study the dark energy equation of state [306Jump To The Next Citation Point568Jump To The Next Citation Point] as well as the equilibrium description of thermodynamics for the horizon entropy [53Jump To The Next Citation Point].

There exists a de Sitter point that corresponds to a vacuum solution (T = 0) at which the Ricci scalar is constant. Since □F (R ) = 0 at this point, we obtain

F (R )R − 2f(R ) = 0. (2.11 )
The model 2 f (R) = αR satisfies this condition, so that it gives rise to the exact de Sitter solution [564Jump To The Next Citation Point]. In the model 2 f(R) = R + αR, because of the linear term in R, the inflationary expansion ends when the term αR2 becomes smaller than the linear term R (as we will see in Section 3). This is followed by a reheating stage in which the oscillation of R leads to the gravitational particle production. It is also possible to use the de Sitter point given by Eq. (2.11View Equation) for dark energy.

We consider the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime with a time-dependent scale factor a(t) and a metric

2 μ ν 2 2 2 ds = gμνdx dx = − dt + a (t)dx , (2.12 )
where t is cosmic time. For this metric the Ricci scalar R is given by
R = 6(2H2 + H˙), (2.13 )
where H ≡ ˙a∕a is the Hubble parameter and a dot stands for a derivative with respect to t. The present value of H is given by
H = 100h kmsec −1Mpc − 1 = 2.1332h × 10−42 GeV, (2.14 ) 0
where h = 0.72 ± 0.08 describes the uncertainty of H0 [264].

The energy-momentum tensor of matter is given by Tμ(νM )= diag (− ρM ,PM ,PM ,PM ), where ρM is the energy density and P M is the pressure. The field equations (2.4View Equation) in the flat FLRW background give

2 2 3F H = (F R − f)∕2 − 3H F˙ + κ ρM , (2.15 ) − 2F H˙ = ¨F − H F˙ + κ2(ρM + PM ), (2.16 )
where the perfect fluid satisfies the continuity equation
˙ρ + 3H (ρ + P ) = 0. (2.17 ) M M M
We also introduce the equation of state of matter, wM ≡ PM ∕ρM. As long as wM is constant, the integration of Eq. (2.17View Equation) gives ρM ∝ a−3(1+wM ). In Section 4 we shall take into account both non-relativistic matter (w = 0 m) and radiation (w = 1∕3 r) 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 [91532Jump To The Next Citation Point]. 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].


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