4.2 Viable f (R) dark energy models

We present a number of viable f (R) models in the (r,m ) plane. First we note that the ΛCDM model corresponds to m = 0, in which case the trajectory is the straight line (i) in Figure 2View Image. The trajectory (ii) in Figure 2View Image represents the model f (R ) = (Rb − Λ )c [31Jump To The Next Citation Point], which corresponds to the straight line m(r) = [(1 − c)∕c]r + b − 1 in the (r,m ) plane. The existence of a saddle matter epoch demands the condition c ≥ 1 and bc ≃ 1. The trajectory (iii) represents the model [26Jump To The Next Citation Point382Jump To The Next Citation Point]
f(R ) = R − αRn (α > 0, 0 < n < 1 ), (4.81 )
which corresponds to the curve m = n(1 + r)∕r. The trajectory (iv) represents the model 2 m (r) = − C (r + 1)(r + ar + b), in which case the late-time accelerated attractor is the point P6 with √ -- ( 3 − 1)∕2 < m < 1.
View Image

Figure 2: Four trajectories in the (r,m ) plane. Each trajectory corresponds to the models: (i) ΛCDM, (ii) f(R) = (Rb − Λ)c, (iii) f (R) = R − αRn with α > 0,0 < n < 1, and (iv) 2 m (r) = − C (r + 1)(r + ar + b). From [31Jump To The Next Citation Point].

In [26Jump To The Next Citation Point] it was shown that m needs to be close to 0 during the radiation domination as well as the matter domination. Hence the viable f (R) models are close to the ΛCDM model in the region R ≫ R0. The Ricci scalar remains positive from the radiation era up to the present epoch, as long as it does not oscillate around R = 0. The model f(R ) = R − α ∕Rn (α > 0, n > 0) is not viable because the condition f,RR > 0 is violated.

As we will see in Section 5, the local gravity constraints provide tight bounds on the deviation parameter m in the region of high density (R ≫ R0), e.g., m (R) ≲ 10 −15 for R = 105R0 [134Jump To The Next Citation Point596Jump To The Next Citation Point]. In order to realize a large deviation from the ΛCDM model such as m (R) > 𝒪 (0.1) today (R = R0) we require that the variable m changes rapidly from the past to the present. The f (R) model given in Eq. (4.81View Equation), for example, does not allow such a rapid variation, because m evolves as m ≃ n (− r − 1) in the region R ≫ R0. Instead, if the deviation parameter has the dependence

m = C (− r − 1)p, p > 1, C > 0, (4.82 )
it is possible to lead to the rapid decrease of m as we go back to the past. The models that behave as Eq. (4.82View Equation) in the regime R ≫ R0 are
2n (A ) f (R) = R − μRc --(R∕Rc-)---- with n, μ,Rc > 0, (4.83 ) ([R∕Rc )2n + 1 ] ( 2 2)−n (B ) f (R) = R − μRc 1 − 1 + R ∕R c with n, μ,Rc > 0. (4.84 )
The models (A) and (B) have been proposed by Hu and Sawicki [306Jump To The Next Citation Point] and Starobinsky [568Jump To The Next Citation Point], respectively. Note that Rc roughly corresponds to the order of R0 for μ = 𝒪 (1). This means that p = 2n + 1 for R ≫ R 0. In the next section we will show that both the models (A) and (B) are consistent with local gravity constraints for n ≳ 1.

In the model (A) the following relation holds at the de Sitter point:

(1 + x2n)2 μ = -2n−1------d2n------, (4.85 ) xd (2 + 2x d − 2n)
where xd ≡ R1 ∕Rc and R1 is the Ricci scalar at the de Sitter point. The stability condition (4.80View Equation) gives [587Jump To The Next Citation Point]
2x4dn − (2n − 1)(2n + 4)x2dn + (2n − 1)(2n − 2) ≥ 0. (4.86 )
The parameter μ has a lower bound determined by the condition (4.86View Equation). When n = 1, for example, one has √ -- xd ≥ 3 and √ -- μ ≥ 8 3∕9. Under Eq. (4.86View Equation) one can show that the conditions (4.56View Equation) are also satisfied.

Similarly the model (B) satisfies [568Jump To The Next Citation Point]

(1 + x2)n+2 ≥ 1 + (n + 2)x2 + (n + 1)(2n + 1)x4, (4.87 ) d d d
with
xd (1 + x2d)n+1 μ = --------2-n+1---------------2-. (4.88 ) 2[(1 + xd) − 1 − (n + 1)xd]
When n = 1 we have √ -- xd ≥ 3 and √ -- μ ≥ 8 3∕9, which is the same as in the model (A). For general n, however, the bounds on μ in the model (B) are not identical to those in the model (A).

Another model that leads to an even faster evolution of m is given by [587Jump To The Next Citation Point]

(C) f(R ) = R − μR tanh (R ∕R ) with μ,R > 0. (4.89 ) c c c
A similar model was proposed by Appleby and Battye [35Jump To The Next Citation Point]. In the region R ≫ Rc the model (4.89View Equation) behaves as f (R ) ≃ R − μRc [1 − exp (− 2R ∕Rc )], which may be regarded as a special case of (4.82View Equation) in the limit that p ≫ 15. The Ricci scalar at the de Sitter point is determined by μ, as
xd cosh2(xd) μ = ----------------------- . (4.90 ) 2 sinh (xd)cosh(xd ) − xd
From the stability condition (4.80View Equation) we obtain
μ > 0.905, xd > 0.920. (4.91 )

The models (A), (B) and (C) are close to the ΛCDM model for R ≫ Rc, but the deviation from it appears when R decreases to the order of Rc. This leaves a number of observational signatures such as the phantom-like equation of state of dark energy and the modified evolution of matter density perturbations. In the following we discuss the dark energy equation of state in f (R) models. In Section 8 we study the evolution of density perturbations and resulting observational consequences in detail.


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