9.2 Background cosmological dynamics

We discuss the background cosmological evolution of dark energy models based on Palatini f (R) gravity. We shall carry out general analysis without specifying the forms of f (R). We take into account non-relativistic matter and radiation whose energy densities are ρ m and ρ r, respectively. In the flat FLRW background (2.12View Equation) we obtain the following equations
2 FR − 2f = − κ ρm, (9.9 ) ( ˙ )2 6F H + -F- − f = κ2 (ρm + 2ρr), (9.10 ) 2F
together with the continuity equations, ρ˙m + 3H ρm = 0 and ˙ρr + 4H ρr = 0. Combing Eqs. (9.9View Equation) and (9.10View Equation) together with continuity equations, it follows that
˙ -3κ2H-ρm-- -FR--−-2f- R = F,RR − F = − 3H F,RR − F , (9.11 ) 2 H2 = 2-κ-(ρm-+-ρr) +-FR--−-f , (9.12 ) 6F ξ
where
[ ] 3 F,R(F R − 2f ) 2 ξ ≡ 1 − ---------------- . (9.13 ) 2 F (F,RR − F )

In order to discuss cosmological dynamics it is convenient to introduce the dimensionless variables:

2 y ≡ F-R-−--f, y ≡ -κ-ρr--, (9.14 ) 1 6F ξH2 2 3F ξH2
by which Eq. (9.12View Equation) can be written as
2 -κ--ρm- = 1 − y − y . (9.15 ) 3F ξH2 1 2
Differentiating y1 and y2 with respect to N = lna, we obtain [253Jump To The Next Citation Point]
dy1-= y [3 − 3y + y + C (R )(1 − y )], (9.16 ) dN 1 1 2 1 dy2 ----= y2[− 1 − 3y1 + y2 − C (R )y1], (9.17 ) dN
where
C (R ) ≡ ----RF˙-----= − 3---(F-R-−-2f-)F,RR----. (9.18 ) H (FR − f) (F R − f)(F,RR − F )
The following constraint equation also holds
1 − y − y F R − 2f -----1----2 = − ---------. (9.19 ) 2y1 F R − f
Hence the Ricci scalar R can be expressed in terms of y1 and y2.

Differentiating Eq. (9.11View Equation) with respect to t, it follows that

˙ ˙ ˙ ˙ -H- = − 3-+ 3y − 1y − --F-- − --ξ--+ --F-R---, (9.20 ) H2 2 2 1 2 2 2HF 2H ξ 12F ξH3
from which we get the effective equation of state:
2-˙H- 1- --F˙- -ξ˙-- --F˙R---- weff = − 1 − 3H2 = − y1 + 3y2 + 3HF + 3H ξ − 18F ξH3 . (9.21 )
The cosmological dynamics is known by solving Eqs. (9.16View Equation) and (9.17View Equation) with Eq. (9.18View Equation). If C (R ) is not constant, then one can use Eq. (9.19View Equation) to express R and C (R ) in terms of y1 and y2.

The fixed points of Eqs. (9.16View Equation) and (9.17View Equation) can be found by setting dy ∕dN = 0 1 and dy ∕dN = 0 2. Even when C(R ) is not constant, except for the cases C(R ) = − 3 and C (R) = − 4, we obtain the following fixed points [253Jump To The Next Citation Point]:

1.
Pr: (y1,y2) = (0, 1) ,
2.
Pm: (y1,y2) = (0,0) ,
3.
Pd: (y1,y2) = (1, 0) .

The stability of the fixed points can be analyzed by considering linear perturbations about them. As long as dC ∕dy1 and dC ∕dy2 are bounded, the eigenvalues λ1 and λ2 of the Jacobian matrix of linear perturbations are given by

1.
P r: (λ ,λ ) = (4 + C(R ),1) 1 2 ,
2.
Pm: (λ1, λ2) = (3 + C (R ),− 1) ,
3.
Pd: (λ1,λ2) = (− 3 − C(R ),− 4 − C (R )) .

In the ΛCDM model (f(R ) = R − 2Λ) one has weff = − y1 + y2∕3 and C(R ) = 0. Then the points Pr, Pm, and Pd correspond to we ff = 1∕3, (λ1,λ2) = (4,1) (radiation domination, unstable), w = 0 eff, (λ ,λ ) = (3,− 1) 1 2 (matter domination, saddle), and w = − 1 eff, (λ1,λ2 ) = (− 3,− 4) (de Sitter epoch, stable), respectively. Hence the sequence of radiation, matter, and de Sitter epochs is in fact realized.

View Image

Figure 8: The evolution of the variables y1 and y2 for the model f(R ) = R − β∕Rn with n = 0.02, together with the effective equation of state w eff. Initial conditions are chosen to be − 40 y1 = 10 and −5 y2 = 1.0 − 10. From [253Jump To The Next Citation Point].

Let us next consider the model n f(R ) = R − β∕R with β > 0 and n > − 1. In this case the quantity C (R ) is

1+n C (R ) = 3n-R----−--(2 +-n-)β-. (9.22 ) R1+n + n(2 + n)β
The constraint equation (9.19View Equation) gives
β 2y1 --1+n = ----------------------------. (9.23 ) R 3y1 + n(y1 − y2 + 1) − y2 + 1
The late-time de Sitter point corresponds to R1+n = (2 + n )β, which exists for n > − 2. Since C (R ) = 0 in this case, the de Sitter point Pd is stable with the eigenvalues (λ1,λ2) = (− 3,− 4 ). During the radiation and matter domination we have β∕R1+n ≪ 1 (i.e., f(R ) ≃ R) and hence C (R ) = 3n. Pr corresponds to the radiation point (we ff = 1∕3) with the eigenvalues (λ1,λ2 ) = (4 + 3n, 1), whereas Pm to the matter point (weff = 0) with the eigenvalues (λ1,λ2 ) = (3 + 3n, − 1 ). Provided that n > − 1, Pr and Pm correspond to unstable and saddle points respectively, in which case the sequence of radiation, matter, and de Sitter eras can be realized. For the models m n f(R ) = R + αR − β ∕R, it was shown in [253Jump To The Next Citation Point] that unified models of inflation and dark energy with radiation and matter eras are difficult to be realized.

In Figure 8View Image we plot the evolution of weff as well as y1 and y2 for the model f(R ) = R − β ∕Rn with n = 0.02. This shows that the sequence of (Pr) radiation domination (weff = 1∕3), (Pm) matter domination (we ff = 0), and de Sitter acceleration (weff = − 1) is realized. Recall that in metric f (R) gravity the model f (R) = R − β∕Rn (β > 0, n > 0) is not viable because f ,RR is negative. In Palatini f (R) gravity the sign of f ,RR does not matter because there is no propagating degree of freedom with a mass M associated with the second derivative f,RR [554].

In [21253Jump To The Next Citation Point] the dark energy model f(R ) = R − β ∕Rn was constrained by the combined analysis of independent observational data. From the joint analysis of Super-Nova Legacy Survey [39], BAO [227] and the CMB shift parameter [561], the constraints on two parameters n and β are n ∈ [− 0.23,0.42] and β ∈ [2.73,10.6] at the 95% confidence level (in the unit of H0 = 1[253]. Since the allowed values of n are close to 0, the above model is not particularly favored over the ΛCDM model. See also [1161485224647] for observational constraints on f (R) dark energy models based on the Palatini formalism.


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