4.1 Dynamical equations

We introduce the following variables
F˙ f R κ2ρr x1 ≡ − ----, x2 ≡ − ----2-, x3 ≡ ---2-, x4 ≡ -----2, (4.61 ) HF 6FH 6H 3F H
together with the density parameters
2 Ωm ≡ κ--ρm- = 1 − x1 − x2 − x3 − x4, Ωr ≡ x4, ΩDE ≡ x1 + x2 + x3. (4.62 ) 3F H2
It is straightforward to derive the following equations
dx1-= − 1 − x − 3x + x2 − x x + x , (4.63 ) dN 3 2 1 1 3 4 dx2 x1x3 dN--= -m---− x2 (2x3 − 4 − x1 ) , (4.64 ) dx x x ---3= − -1-3-− 2x3(x3 − 2) , (4.65 ) dN m dx4- dN = − 2x3x4 + x1x4, (4.66 )
where N = ln a is the number of e-foldings, and
d-lnF- Rf,RR- m ≡ d lnR = f , (4.67 ) ,R r ≡ − d-lnf- = − Rf,R-= x3. (4.68 ) d ln R f x2
From Eq. (4.68View Equation) the Ricci scalar R can be expressed by x3∕x2. Since m depends on R, this means that m is a function of r, that is, m = m (r). The ΛCDM model, f(R) = R − 2Λ, corresponds to m = 0. Hence the quantity m characterizes the deviation of the background dynamics from the ΛCDM model. A number of authors studied cosmological dynamics for specific f (R) models [160Jump To The Next Citation Point382Jump To The Next Citation Point488252Jump To The Next Citation Point31Jump To The Next Citation Point19828072411592351279483321432].

The effective equation of state of the system is defined by

w ≡ − 1 − 2H˙∕(3H2 ), (4.69 ) eff
which is equivalent to weff = − (2x3 − 1)∕3. In the absence of radiation (x4 = 0) the fixed points for the above dynamical system are
P : (x ,x ,x ) = (0,− 1,2), Ω = 0, w = − 1, (4.70 ) 1 1 2 3 m eff P2 : (x1,x2,x3 ) = (− 1,0,0), Ωm = 2, weff = 1 ∕3, (4.71 ) P : (x ,x ,x ) = (1,0, 0), Ω = 0, w = 1∕3, (4.72 ) 3 1 2 3 m eff P4 : (x1,x2,x3 ) = ((− 4,5,0), Ωm = 0, ) weff = 1 ∕3, (4.73 ) 3m 1 + 4m 1 + 4m P5 : (x1,x2,x3 ) = ------,− ---------2 ,--------- , (4.74 ) 1 + m 2 (1 + m ) 2(1 + m ) m-(7-+-10m-) --m--- Ωm = 1 − 2 (1 + m )2 , weff = − 1 + m , (4.75 ) ( ) P6 : (x1,x2,x3 ) = 2(1-−-m-) ,--1-−-4m---,− (1-−-4m-)(1-+-m-)- , 1 + 2m m (1 + 2m ) m (1 + 2m ) 2 − 5m − 6m2 Ωm = 0, weff = --------------. (4.76 ) 3m (1 + 2m )
The points P5 and P6 are on the line m(r) = − r − 1 in the (r,m ) plane.

The matter-dominated epoch (Ωm ≃ 1 and we ff ≃ 0) can be realized only by the point P5 for m close to 0. In the (r,m) plane this point exists around (r,m) = (− 1,0). Either the point P1 or P6 can be responsible for the late-time cosmic acceleration. The former is a de Sitter point (weff = − 1) with r = − 2, in which case the condition (2.11View Equation) is satisfied. The point P6 can give rise to the accelerated expansion (weff < − 1 ∕3) provided that -- m > (√3 − 1)∕2, or − 1∕2 < m < 0, or √ -- m < − (1 + 3)∕2.

In order to analyze the stability of the above fixed points it is sufficient to consider only time-dependent linear perturbations δxi(t) (i = 1,2,3) around them (see [170171Jump To The Next Citation Point] for the detail of such analysis). For the point P5 the eigenvalues for the 3 × 3 Jacobian matrix of perturbations are

∘ --------------------------------- − 3m5 ± m5 (256m3 + 160m2 − 31m5 − 16) 3(1 + m ′5), --------------------5--------5--------------, (4.77 ) 4m5 (m5 + 1)
where m5 ≡ m (r5) and m ′5 ≡ dm(r5) dr with r5 ≈ − 1. In the limit that |m5 | ≪ 1 the latter two eigenvalues reduce to ∘ ------- − 3∕4 ± − 1∕m5. For the models with m5 < 0, the solutions cannot remain for a long time around the point P5 because of the divergent behavior of the eigenvalues as m5 → − 0. The model f(R ) = R − α∕Rn (α > 0,n > 0) falls into this category. On the other hand, if 0 < m5 < 0.327, the latter two eigenvalues in Eq. (4.77View Equation) are complex with negative real parts. Then, provided that m ′5 > − 1, the point P5 corresponds to a saddle point with a damped oscillation. Hence the solutions can stay around this point for some time and finally leave for the late-time acceleration. Then the condition for the existence of the saddle matter era is
dm m (r) ≃ +0, ----> − 1, at r = − 1. (4.78 ) dr
The first condition implies that viable f (R) models need to be close to the ΛCDM model during the matter domination. This is also required for consistency with local gravity constraints, as we will see in Section 5.

The eigenvalues for the Jacobian matrix of perturbations about the point P1 are

∘ ------------ − 3, − 3± --25-−-16∕m1--, (4.79 ) 2 2
where m1 = m (r = − 2). This shows that the condition for the stability of the de Sitter point P1 is [440243250Jump To The Next Citation Point26Jump To The Next Citation Point]
0 < m (r = − 2) ≤ 1. (4.80 )
The trajectories that start from the saddle matter point P5 satisfying the condition (4.78View Equation) and then approach the stable de Sitter point P 1 satisfying the condition (4.80View Equation) are, in general, cosmologically viable.

One can also show that P6 is stable and accelerated for (a) ′ m 6 < − 1, √ -- ( 3 − 1)∕2 < m6 < 1, (b) m ′6 > − 1, √ -- m6 < − (1 + 3)∕2, (c) m ′6 > − 1, − 1∕2 < m6 < 0, (d) m ′6 > − 1, m6 ≥ 1. Since both P5 and P6 are on the line m = − r − 1, only the trajectories from m ′5 > − 1 to m ′ < − 1 6 are allowed (see Figure 2View Image). This means that only the case (a) is viable as a stable and accelerated fixed point P6. In this case the effective equation of state satisfies the condition we ff > − 1.

From the above discussion the following two classes of models are cosmologically viable.

From Eq. (4.56View Equation) the viable f (R) dark energy models need to satisfy the condition m > 0, which is consistent with the above argument.


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