Matter density perturbations

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. (6.17

) and (6.18

) 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. (6.32

), i.e.

obeys the following equation from Eqs. (8.86

) and (8.87

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 (6.33 ). Since $χ = 0$ , $α = Φ$ , $ψ = − Ψ$ , and $A = 3 (H Φ + ˙Ψ )$ in this case, Eqs. (6.11 ), (6.13 ), (6.15 ), and (8.89 ) 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. (8.92

), we have used the mass squared $M 2 = (F ∕F − R )∕3 ,R$ introduced in Eq. (5.2

) 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. (8.90 ) – (8.93 ) 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 [93 , 171 , 586 ] and f (R) gravity [586 , 597 ]. Precisely speaking, in f (R) gravity, this approximation corresponds to

and

From Eqs. (8.90

) and (8.91

) it then follows that

Since $(k2∕a2 + M 2)δF ≃ κ2δρm ∕3$ from Eq. (8.92

), we obtain

We also define the effective gravitational potential

This quantity characterizes the deviation of light rays, which is linked with the Integrated Sachs–Wolfe (ISW) effect in CMB [544

] and weak lensing observations [27

]. From Eq. (8.97

) we have

From Eq. (6.12 ) the term $Hv$ is of the order of $H2 Φ∕(κ2ρm )$ provided that the deviation from the $Λ$ CDM model is not significant. Using Eq. (8.97 ) 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. (8.88 ) can be approximated as $δm ≃ δρm ∕ρm$ . Neglecting the r.h.s. of Eq. (8.93 ) relative to the l.h.s. and using Eq. (8.97 ) with $δρm ≃ ρm δm$ , we get the equation for matter perturbations:

where $Geff$ is the effective (cosmological) gravitational coupling defined by [586 , 597

]

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

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 (4.83

) and (4.84

) 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 [306

, 568

, 587

, 270

, 589

In order to derive Eq. (8.100 ) we used the approximation that the time-derivative terms of $δF$ on the l.h.s. of Eq. (8.92 ) 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 [568 ]. For sub-horizon perturbations Eq. (8.92 ) is approximately given by

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

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

where $c$ is a constant. Hence the solution of the perturbation $δR$ is expressed by [568

, 587

]

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

The f (R) models (4.83

) and (4.84

) 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 [587

, 36 ]. 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 (4.83

) and (4.84

). 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. (8.106

) to be sufficiently small, which amounts to a fine tuning for these models.

For the models (4.83 ) and (4.84 ) one has $F = 1 − 2n μ(R ∕Rc)−2n−1$ in the regime $R ≫ Rc$ . Then the field $ϕ$ defined in Eq. (2.31 ) 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 $ϕ = 0$ [266 ]. 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 [461 , 54 ]. The past singularity can be cured by taking into account the $R2$ term [37 ], 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 [192 , 434 ] the perturbation equations have been derived without neglecting the oscillating mode. As long as the condition $|δRosc| < |δRind|$ is satisfied initially, the approximate equation (8.100 ) is accurate to reproduce the numerical solutions [192 , 589 ]. Equation (8.100 ) can be written as

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

where we used Eq. (8.99

). 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 [568

]

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 $(0) |δRosc|∕R$ [587

], so the oscillating mode tends to be unimportant with time.