9.3 Matter perturbations

We have shown that f (R) theory in the Palatini formalism can give rise to the late-time cosmic acceleration preceded by radiation and matter eras. In this section we study the evolution of matter density perturbations to confront Palatini f (R) gravity with the observations of large-scale structure [359Jump To The Next Citation Point356Jump To The Next Citation Point357598380597Jump To The Next Citation Point]. Let us consider the perturbation δρm of non-relativistic matter with a homogeneous energy density ρm. Koivisto and Kurki-Suonio [359] derived perturbation equations in Palatini f (R) gravity. Using the perturbed metric (6.1View Equation) with the same variables as those introduced in Section 6, the perturbation equations are given by
( ) ( ) Δ F˙ 1 3F˙2 --ψ + H + --- A + --- ----+ 3H F˙ α a2 2F 2F 2F [ ( 2 ) ( ) ] = -1- 3H2 − 3 ˙F-− R-− Δ-- δF + 3F˙+ 3H δ˙F − κ2δρ , (9.24 ) 2F 4F2 2 a2 2F m [ ( ) ] 1 3F˙ 2 H α − ˙ψ = 2F- δ˙F − H + 2F- δF − ˙Fα + κ ρmv , (9.25 ) 1- ˙ χ˙+ H χ − α − ψ = F (δF − F χ), (9.26 ) ( ) ( 2 ) A˙+ 2H + F˙- A + 3H˙ + 3F¨ + 3H-F˙ − 3-˙F- + -Δ- α + 3-F˙˙α 2F F 2F F 2 a2 2 F [ ( ) ( ) ] 1 2 2 3F˙2 Δ 6F˙ = 2F- κ δρm + 6H + 6H˙ + -F2- − R − a2- δF + 3H − F-- δ˙F + 3δ¨F , (9.27 ) R δF − FδR = − κ2δρm, (9.28 )
where the Ricci scalar R can be understood as R (T ).

From Eq. (9.28View Equation) the perturbation δF can be expressed by the matter perturbation δρ m, as

F κ2 δρ δF = -,R-----m, (9.29 ) R 1 − m
where m = RF,R ∕F. This equation clearly shows that the perturbation δF is sourced by the matter perturbation only, unlike metric f (R) gravity in which the oscillating mode of δF is present. The matter perturbation δρm and the velocity potential v obey the same equations as given in Eqs. (8.86View Equation) and (8.87View Equation), which results in Eq. (8.89View Equation) in Fourier space.

Let us consider the perturbation equations in Fourier space. We choose the Longitudinal gauge (χ = 0) with α = Φ and ψ = − Ψ. In this case Eq. (9.26View Equation) gives

δF- Ψ − Φ = F . (9.30 )
Under the quasi-static approximation on sub-horizon scales used in Section 8.1, Eqs. (9.24View Equation) and (8.89View Equation) reduce to
( ) k2 1 k2 2 -2Ψ ≃ --- -2δF − κ δρm , (9.31 ) a 2F a ¨ ˙ k2- δm + 2H δm + a2Φ ≃ 0. (9.32 )
Combining Eq. (9.30View Equation) with Eq. (9.31View Equation), we obtain
2 2 ( ) 2 2 ( ) k--Ψ = − κ--δρm 1 − --ζ--- , k-Φ = − κ-δρm- 1 + --ζ--- , (9.33 ) a2 2F 1 − m a2 2F 1 − m
where
2 2 ζ ≡ k--F,R-= k---m. (9.34 ) a2 F a2R
Then the matter perturbation satisfies the following Eq. [597Jump To The Next Citation Point]
( ) κ2ρm ζ ¨δm + 2H ˙δm − ----- 1 + ------ δm ≃ 0. (9.35 ) 2F 1 − m
The effective gravitational potential defined in Eq. (8.98View Equation) obeys
κ2ρm-a2- Φe ff ≃ − 2F k2 δm. (9.36 )
In the above approximation we do not need to worry about the dominance of the oscillating mode of perturbations in the past. Note also that the same approximate equation of δm as Eq. (9.35View Equation) can be derived for different gauge choices [597Jump To The Next Citation Point].

The parameter ζ is a crucial quantity to characterize the evolution of perturbations. This quantity can be estimated as ζ ≈ (k∕aH )2m, which is much larger than m for sub-horizon modes (k ≫ aH). In the regime ζ ≪ 1 the matter perturbation evolves as 2∕3 δm ∝ t. Meanwhile the evolution of δm in the regime ζ ≫ 1 is completely different from that in GR. If the transition characterized by ζ = 1 occurs before today, this gives rise to the modification to the matter spectrum compared to the GR case.

In the regime ζ ≫ 1, let us study the evolution of matter perturbations during the matter dominance. We shall consider the case in which the parameter m (with |m | ≪ 1) evolves as

m ∝ tp, (9.37 )
where p is a constant. For the model n f(R ) = R − μRc (R ∕Rc) (n < 1) the power p corresponds to p = 1 + n, whereas for the models (4.83View Equation) and (4.84View Equation) with n > 0 one has p = 1 + 2n. During the matter dominance the parameter ζ evolves as ζ = ±(t∕t )2p+2∕3 k, where the subscript “k” denotes the value at which the perturbation crosses ζ = ±1. Here + and − signs correspond to the cases m > 0 and m < 0, respectively. Then the matter perturbation equation (9.35View Equation) reduces to
d2δm- 1-dδm- 3[ (3p+1)(N− Nk)] dN 2 + 2 dN − 2 1 ± e δm = 0. (9.38 )

When m > 0, the growing mode solution to Eq. (9.38View Equation) is given by

( √ -- ) √ -- --6e(3p+1)(N−Nk)∕2 -˙δm-- --6-(3p+1)(N−Nk )∕2 δm ∝ exp 3p + 1 , fδ ≡ H δ = 2 e . (9.39 ) m
This shows that the perturbations exhibit violent growth for p > − 1∕3, which is not compatible with observations of large-scale structure. In metric f (R) gravity the growth of matter perturbations is much milder.

When m < 0, the perturbations show a damped oscillation:

− (3p+2)(N −N )∕4 1 3p + 1 δm ∝ e k cos(x + 𝜃), fδ = − -(3p + 2) − -------xtan (x + 𝜃 ), (9.40 ) 4 2
where √ -- x = 6e(3p+1)(N−Nk)∕2∕(3p + 1), and 𝜃 is a constant. The averaged value of the growth rate fδ is given by f¯δ = − (3p + 2)∕4, but it shows a divergence every time x changes by π. These negative values of f δ are also difficult to be compatible with observations.

The f (R) models can be consistent with observations of large-scale structure if the universe does not enter the regime |ζ| > 1 by today. This translates into the condition [597Jump To The Next Citation Point]

|m (z = 0)| ≲ (a0H0 ∕k )2. (9.41 )
Let us consider the wavenumbers −1 −1 0.01h Mpc ≲ k ≲ 0.2h Mpc that corresponds to the linear regime of the matter power spectrum. Since the wavenumber − 1 k = 0.2h Mpc corresponds to k ≈ 600a0H0 (where “0” represents present quantities), the condition (9.41View Equation) gives the bound |m (z = 0)| ≲ 3 × 10 −6.

If we use the observational constraint of the growth rate, f ≲ 1.5 δ [418605211], then the deviation parameter m today is constrained to be −5 |m (z = 0)| ≲ 10-−4 10 for the model n f (R ) = R − λRc(R ∕Rc ) (n < 1) as well as for the models (4.83View Equation) and (4.84View Equation[597Jump To The Next Citation Point]. Recall that, in metric f (R) gravity, the deviation parameter m can grow to the order of 0.1 by today. Meanwhile f (R) dark energy models based on the Palatini formalism are hardly distinguishable from the ΛCDM model [356386385597]. Note that the bound on m (z = 0) becomes even severer by considering perturbations in non-linear regime. The above peculiar evolution of matter perturbations is associated with the fact that the coupling between non-relativistic matter and a scalar-field degree of freedom is very strong (as we will see in Section 10.1).

The above results are based on the fact that dark matter is described by a cold and perfect fluid with no pressure. In [358Jump To The Next Citation Point] it was suggested that the tight bound on the parameter m can be relaxed by considering imperfect dark matter with a shear stress. Although the approach taken in [358] did not aim to explain the origin of a dark matter stress Π that cancels the k-dependent term in Eq. (9.35View Equation), it will be of interest to further study whether some theoretically motivated choice of Π really allows the possibility that Palatini f (R) dark energy models can be distinguished from the ΛCDM model.


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