8.1 Matter density perturbations

Let us consider the perturbations of non-relativistic matter with the background energy density ρm and the negligible pressure (Pm = 0). In Fourier space Eqs. (6.17View Equation) and (6.18View Equation) give
( k2 ) δρ˙m + 3H δρm = ρm A − 3H α − --v , (8.86 ) a2 v˙= α, (8.87 )
where in the second line we have used the continuity equation, ρ˙m + 3H ρm = 0. The density contrast defined in Eq. (6.32View Equation), i.e.
δ = δρm-+ 3Hv, (8.88 ) m ρm
obeys the following equation from Eqs. (8.86View Equation) and (8.87View Equation):
¨ ˙ k2- ¨ ˙ δm + 2H δm + a2(α − χ˙) = 3B + 6H B, (8.89 )
where B ≡ Hv − ψ and we used the relation ˙ 2 2 A = 3(H α − ψ) + (k ∕a )χ.

In the following we consider the evolution of perturbations in f (R) gravity in the Longitudinal gauge (6.33View Equation). Since χ = 0, α = Φ, ψ = − Ψ, and A = 3 (H Φ + ˙Ψ ) in this case, Eqs. (6.11View Equation), (6.13View Equation), (6.15View Equation), and (8.89View Equation) give

[( ) k2 1 2 k2 -2Ψ + 3H (H Φ + Ψ˙) = − --- 3H + 3 ˙H − -2- δF − 3H δ˙F a 2F a ] ˙ ˙ ˙ 2 + 3H F Φ + 3F (H Φ + Ψ ) + κ δρm , (8.90 ) Ψ − Φ = δF-, (8.91 ) F ( ) ¨ ˙ k2- 2 κ2- ˙ ˙ ˙ ¨ ˙ δF + 3H δF + a2 + M δF = 3 δρm + F (3H Φ + 3Ψ + Φ ) + (2F + 3H F )Φ, (8.92 ) 2 δ¨ + 2H ˙δ + k--Φ = 3B¨ + 6H B˙, (8.93 ) m m a2
where B = Hv + Ψ. In order to derive Eq. (8.92View Equation), we have used the mass squared M 2 = (F ∕F − R )∕3 ,R introduced in Eq. (5.2View Equation) together with the relation δR = δF∕F,R.

Let us consider the wavenumber k deep inside the Hubble radius (k ≫ aH). In order to derive the equation of matter perturbations approximately, we use the quasi-static approximation under which the dominant terms in Eqs. (8.90View Equation) – (8.93View Equation) correspond to those including k2∕a2, δρm (or δm) and M 2. In General Relativity this approximation was first used by Starobinsky in the presence of a minimally coupled scalar field [567], which was numerically confirmed in [403]. This was further extended to scalar-tensor theories [93Jump To The Next Citation Point171586Jump To The Next Citation Point] and f (R) gravity [586Jump To The Next Citation Point597Jump To The Next Citation Point]. Precisely speaking, in f (R) gravity, this approximation corresponds to

{ } k2- k2- k2- 2 { 2 2 2 2 } a2|Φ|,a2|Ψ |, a2|δF |,M |δF | ≫ H |Φ|,H |Ψ|,H |B|,H |δF| , (8.94 )
and
˙ ˙ ˙ |X | ≲ |HX |, where X = Φ, Ψ, F,F ,δF,δF . (8.95 )
From Eqs. (8.90View Equation) and (8.91View Equation) it then follows that
( ) ( ) -1- a2-2 -1- a2- 2 Ψ ≃ 2F δF − k2κ δρm , Φ ≃ − 2F δF + k2 κ δρm . (8.96 )
Since (k2∕a2 + M 2)δF ≃ κ2δρm ∕3 from Eq. (8.92View Equation), we obtain
k2- κ2δρm--2-+-3M--2a2∕k2- k2- κ2δρm--4-+-3M-2a2∕k2-- a2Ψ ≃ − 2F 3 (1 + M 2a2∕k2), a2Φ ≃ − 2F 3(1 + M 2a2∕k2 ). (8.97 )
We also define the effective gravitational potential
Φ ≡ (Φ + Ψ )∕2. (8.98 ) eff
This quantity characterizes the deviation of light rays, which is linked with the Integrated Sachs–Wolfe (ISW) effect in CMB [544Jump To The Next Citation Point] and weak lensing observations [27Jump To The Next Citation Point]. From Eq. (8.97View Equation) we have
-κ2 a2- Φeff ≃ − 2F k2δρm. (8.99 )

From Eq. (6.12View Equation) the term Hv is of the order of H2 Φ∕(κ2ρm ) provided that the deviation from the ΛCDM model is not significant. Using Eq. (8.97View Equation) we find that the ratio 3Hv ∕(δρm ∕ρm ) is of the order of (aH ∕k)2, which is much smaller than unity for sub-horizon modes. Then the gauge-invariant perturbation δm given in Eq. (8.88View Equation) can be approximated as δm ≃ δρm ∕ρm. Neglecting the r.h.s. of Eq. (8.93View Equation) relative to the l.h.s. and using Eq. (8.97View Equation) with δρm ≃ ρm δm, we get the equation for matter perturbations:

¨δm + 2H δ˙m − 4πGe ffρmδm ≃ 0, (8.100 )
where Geff is the effective (cosmological) gravitational coupling defined by [586597Jump To The Next Citation Point]
G 4 + 3M 2a2∕k2 Ge ff ≡ ------------------. (8.101 ) F 3 (1 + M 2a2∕k2)

We recall that viable f (R) dark energy models are constructed to have a large mass M in the region of high density (R ≫ R0). During the radiation and deep matter eras the deviation parameter m = Rf,RR ∕f,R is much smaller than 1, so that the mass squared satisfies

( ) 2 R- 1- M = 3 m − 1 ≫ R. (8.102 )
If m grows to the order of 0.1 by the present epoch, then the mass M today can be of the order of H0. In the regimes M 2 ≫ k2∕a2 and M 2 ≪ k2∕a2 the effective gravitational coupling has the asymptotic forms G ≃ G∕F eff and G ≃ 4G ∕(3F ) eff, respectively. The former corresponds to the “General Relativistic (GR) regime” in which the evolution of δm mimics that in GR, whereas the latter corresponds to the “scalar-tensor regime” in which the evolution of δm is non-standard. For the f (R) models (4.83View Equation) and (4.84View Equation) the transition from the former regime to the latter regime, which is characterized by the condition M 2 = k2∕a2, can occur during the matter domination for the wavenumbers relevant to the matter power spectrum [306Jump To The Next Citation Point568Jump To The Next Citation Point587Jump To The Next Citation Point270Jump To The Next Citation Point589Jump To The Next Citation Point].

In order to derive Eq. (8.100View Equation) we used the approximation that the time-derivative terms of δF on the l.h.s. of Eq. (8.92View Equation) is neglected. In the regime M 2 ≫ k2∕a2, however, the large mass M can induce rapid oscillations of δF. In the following we shall study the evolution of the oscillating mode [568Jump To The Next Citation Point]. For sub-horizon perturbations Eq. (8.92View Equation) is approximately given by

( 2 ) 2 δ¨F + 3H δ˙F + k--+ M 2 δF ≃ κ-δ ρm. (8.103 ) a2 3
The solution of this equation is the sum of the matter induce mode δF ≃ (κ2 ∕3)δρ ∕ (k2 ∕a2 + M 2) ind m and the oscillating mode δFosc satisfying
( ) ¨ ˙ k2- 2 δFosc + 3H δFosc + a2 + M δFosc = 0. (8.104 )

As long as the frequency ∘ ------------ ω = k2∕a2 + M 2 satisfies the adiabatic condition |˙ω| ≪ ω2, we obtain the solution of Eq. (8.104View Equation) under the WKB approximation:

1 (∫ ) δFosc ≃ ca− 3∕2√----cos ωdt , (8.105 ) 2ω
where c is a constant. Hence the solution of the perturbation δR is expressed by [568Jump To The Next Citation Point587Jump To The Next Citation Point]
2 ( ∫ ) δR ≃ --1------κ-δρm----+ ca −3∕2 ---1√----cos ωdt . (8.106 ) 3f,RR k2∕a2 + M 2 f,RR 2ω

For viable f (R) models, the scale factor a and the background Ricci scalar R(0) evolve as a ∝ t2∕3 and R(0) ≃ 4∕(3t2) during the matter era. Then the amplitude of δRosc relative to R (0) has the time-dependence

2 |δRosc|∝ ------M--t------. (8.107 ) R (0) (k2∕a2 + M 2)1∕4
The f (R) models (4.83View Equation) and (4.84View Equation) behave as p m (r ) = C (− r − 1) with p = 2n + 1 in the regime R ≫ Rc. During the matter-dominated epoch the mass M evolves as −(p+1) M ∝ t. In the regime M 2 ≫ k2∕a2 one has |δRosc|∕R (0) ∝ t−(3p+1)∕2 and hence the amplitude of the oscillating mode decreases faster than R(0). However the contribution of the oscillating mode tends to be more important as we go back to the past. In fact, this behavior was confirmed in the numerical simulations of [587Jump To The Next Citation Point36]. This property persists in the radiation-dominated epoch as well. If the condition (0) |δR | < R is violated, then R can be negative such that the condition f,R > 0 or f,RR > 0 is violated for the models (4.83View Equation) and (4.84View Equation). Thus we require that |δR| is smaller than R (0) at the beginning of the radiation era. This can be achieved by choosing the constant c in Eq. (8.106View Equation) to be sufficiently small, which amounts to a fine tuning for these models.

For the models (4.83View Equation) and (4.84View Equation) one has F = 1 − 2n μ(R ∕Rc)−2n−1 in the regime R ≫ Rc. Then the field ϕ defined in Eq. (2.31View Equation) rapidly approaches 0 as we go back to the past. Recall that in the Einstein frame the effective potential of the field has a potential minimum around ϕ = 0 because of the presence of the matter coupling. Unless the oscillating mode of the field perturbation δϕ is strongly suppressed relative to the background field ϕ (0), the system can access the curvature singularity at ϕ = 0 [266Jump To The Next Citation Point]. This is associated with the condition (0) |δR | < R discussed above. This curvature singularity appears in the past, which is not related to the future singularities studied in  [46154]. The past singularity can be cured by taking into account the R2 term [37Jump To The Next Citation Point], as we will see in Section 13.3. We note that the f (R) models proposed in [427] [e.g., f(R ) = R − αRc ln(1 + R ∕Rc)] to cure the singularity problem satisfy neither the local gravity constraints [580] nor observational constraints of large-scale structure [194].

As long as the oscillating mode δRosc is negligible relative to the matter-induced mode δRind, we can estimate the evolution of matter perturbations δm as well as the effective gravitational potential Φeff. Note that in [192Jump To The Next Citation Point434] the perturbation equations have been derived without neglecting the oscillating mode. As long as the condition |δRosc| < |δRind| is satisfied initially, the approximate equation (8.100View Equation) is accurate to reproduce the numerical solutions [192589Jump To The Next Citation Point]. Equation (8.100View Equation) can be written as

( ) d2 δm 1 3 d δm 3 4 + 3M 2a2∕k2 ----2 + --− --weff ---- − --Ωm --------2--2--2-= 0, (8.108 ) dN 2 2 dN 2 3(1 + M a ∕k )
where N = ln a, weff = − 1 − 2H˙∕(3H2 ), and Ωm = 8πG ρm ∕(3F H2 ). The matter-dominated epoch corresponds to we ff = 0 and Ωm = 1. In the regime M 2 ≫ k2∕a2 the evolution of δm and Φeff during the matter dominance is given by
δm ∝ t2∕3, Φe ff = constant, (8.109 )
where we used Eq. (8.99View Equation). The matter-induced mode δRind relative to the background Ricci scalar (0) R evolves as |δRind|∕R (0) ∝ t2∕3 ∝ δm. At late times the perturbations can enter the regime M 2 ≪ k2∕a2, depending on the wavenumber k and the mass M. When M 2 ≪ k2∕a2, the evolution of δ m and Φe ff during the matter era is [568Jump To The Next Citation Point]
(√33−1)∕6 (√33−5)∕6 δm ∝ t , Φeff ∝ t . (8.110 )
For the model m (r ) = C (− r − 1)p, the evolution of the matter-induced mode in the region M 2 ≪ k2 ∕a2 is given by (0) −2p+(√33−5)∕6 |δRind|∕R ∝ t. This decreases more slowly relative to the ratio (0) |δRosc|∕R [587Jump To The Next Citation Point], so the oscillating mode tends to be unimportant with time.
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