10.4 Evolution of matter density perturbations

Let us next study the evolution of perturbations in non-relativistic matter for the action (10.10View Equation) with the potential U (ϕ) and the coupling F (ϕ) = e−2Q ϕ. As in metric f (R) gravity, the matter perturbation δm satisfies Eq. (8.93View Equation) in the Longitudinal gauge. We define the field mass squared as M 2 ≡ U,ϕϕ. For the potential consistent with local gravity constraints [such as (10.23View Equation)], the mass M is much larger than the present Hubble parameter H0 during the radiation and deep matter eras. Note that the condition M 2 ≫ R is satisfied in most of the cosmological epoch as in the case of metric f (R) gravity.

The perturbation equations for the action (10.10View Equation) are given in Eqs. (6.11View Equation) – (6.18View Equation) with f = F (ϕ)R, 2 ω = (1 − 6Q )F, and V = U. We use the unit 2 κ = 1, but we restore 2 κ when it is necessary. In the Longitudinal gauge one has χ = 0, α = Φ, ψ = − Ψ, and ˙ A = 3(H Φ + Ψ) in these equations. Since we are interested in sub-horizon modes, we use the approximation that the terms containing k2 ∕a2, δρm, δR, and M 2 are the dominant contributions in Eqs. (6.11View Equation) – (6.19View Equation). We shall neglect the contribution of the time-derivative terms of δϕ in Eq. (6.16View Equation). As we have discussed for metric f (R) gravity in Section 8.1, this amounts to neglecting the oscillating mode of perturbations. The initial conditions of the field perturbation in the radiation era need to be chosen so that the oscillating mode δϕosc is smaller than the matter-induced mode δϕind. In Fourier space Eq. (6.16View Equation) gives

( ) k2- M--2 -1- a2 + ω δϕind ≃ 2ω F,ϕδR. (10.36 )
Using this relation together with Eqs. (6.13View Equation) and (6.18View Equation), it follows that
2QF k2 δϕind ≃ --2---2--------2--------2-2Ψ. (10.37 ) (k ∕a )(1 − 2Q )F + M a
Combing this equation with Eqs. (6.11View Equation) and (6.13View Equation), we obtain [596Jump To The Next Citation Point547] (see also [84632Jump To The Next Citation Point631])
k2 κ2 δρ (k2∕a2)(1 − 2Q2)F + M 2 -2-Ψ ≃ − -----m -------2--2-------2-----, (10.38 ) a 2F (k ∕a )F + M k2 κ2 δρm (k2∕a2)(1 + 2Q2)F + M 2 a2-Φ ≃ − --2F-- ----(k2∕a2-)F-+-M-2-----, (10.39 )
where we have recovered κ2. Defining the effective gravitational potential Φeff = (Φ + Ψ )∕2, we find that Φeff satisfies the same form of equation as (8.99View Equation).

Substituting Eq. (10.39View Equation) into Eq. (8.93View Equation), we obtain the equation of matter perturbations on sub-horizon scales [with the neglect of the r.h.s. of Eq. (8.93View Equation)]

¨δm + 2H δ˙m − 4πGe ffρmδm ≃ 0, (10.40 )
where the effective gravitational coupling is
G (k2∕a2)(1 + 2Q2 )F + M 2 Geff = ---------2--2--------2-----. (10.41 ) F (k ∕a )F + M
In the regime M 2∕F ≫ k2∕a2 (“GR regime”) this reduces to Ge ff = G ∕F, so that the evolution of δm and Φ eff during the matter domination (Ω = ρ ∕(3F H2 ) ≃ 1 m m) is standard: δ ∝ t2∕3 m and Φe ff ∝ constant.

In the regime 2 2 2 M ∕F ≪ k ∕a (“scalar-tensor regime”) we have

G G 4 + 2ω Ge ff ≃ --(1 + 2Q2 ) = --------BD-, (10.42 ) F F 3 + 2ωBD
where we used the relation (10.11View Equation) between the coupling Q and the BD parameter ωBD. Since ω = 0 BD in f (R) gravity, it follows that G = 4G ∕(3F ) eff. Note that the result (10.42View Equation) agrees with the effective Newtonian gravitational coupling between two test masses [93175Jump To The Next Citation Point]. The evolution of δm and Φeff during the matter dominance in the regime M 2∕F ≪ k2∕a2 is
√ ------- √ ------- δm ∝ t( 25+48Q2−1)∕6, Φeff ∝ t( 25+48Q2− 5)∕6. (10.43 )
Hence the growth rate of δm for Q ⁄= 0 is larger than that for Q = 0.

As an example, let us consider the potential (10.23View Equation). During the matter era the field mass squared around the potential minimum (induced by the matter coupling) is approximately given by

( ) (2− p)∕(1− p) M 2 ≃ ---1-−-p----Q2 ρm- U , (10.44 ) (2ppC)1∕(1− p) U0 0
which decreases with time. The perturbations cross the point 2 2 2 M ∕F = k ∕a at time t = tk, which depends on the wavenumber k. Since the evolution of the mass during the matter domination is given by M ∝ t− 21−−pp, the time t k has a scale-dependence: t ∝ k− 3(41−−pp) k. More precisely the critical redshift z k at time tk can be estimated as [596Jump To The Next Citation Point]
[ ( )2 (1−p) p ] 41−p z ≃ -k----1-- --2-pC---------1------U0- − 1, (10.45 ) k a0H0 |Q| (1 − p)1−p (3F Ω (0m))2−pH20 0
where the subscript “0” represents present quantities. For the scales 30a0H0 ≲ k ≲ 600a0H0, which correspond to the linear regime of the matter power spectrum, the critical redshift can be in the region z > 1 k. Note that, for larger p, z k decreases.

When t < tk and t > tk the matter perturbation evolves as δm ∝ t2∕3 and √ ------2 δm ∝ t( 25+48Q −1)∕6, respectively (apart from the epoch of the late-time cosmic acceleration). The matter power spectrum P δm at time t = tΛ (at which ¨a = 0) shows a difference compared to the ΛCDM model, which is given by

( )2 (√25+48Q2−1− 2) √------ --Pδm(tΛ)-- tΛ- 6 3 (1−p)(-245+−4p8Q2−5) P ΛCDM (tΛ) = tk ∝ k . (10.46 ) δm

The CMB power spectrum is also modified by the non-standard evolution of the effective gravitational potential Φ eff for t > t k. This mainly affects the low multipoles of CMB anisotropies through of the ISW effect. Hence there is a difference between the spectral indices of the matter power spectrum and of the CMB spectrum on the scales (− 1 −1 0.01h Mpc ≲ k ≲ 0.2h Mpc[596]:

∘ ---------- (1 −-p)(--25 +-48Q2-−-5-) Δns (tΛ ) = 4 − p . (10.47 )
Note that this covers the result (8.116View Equation) in f (R) gravity (√-- Q = − 1∕ 6 and p = 2n∕ (2n + 1)) as a special case. Under the criterion Δns (tΛ) < 0.05 we obtain the bounds p > 0.957 for Q = 1 and p > 0.855 for Q = 0.5. As long as p is close to 1, the model can be consistent with both cosmological and local gravity constraints. The allowed region coming from the bounds Δn (t ) < 0.05 s Λ and (10.35View Equation) are illustrated in Figure 9View Image.
View Image

Figure 9: The allowed region of the parameter space in the (Q, p) plane for BD theory with the potential (10.23View Equation). We show the allowed region coming from the bounds Δns (tΛ) < 0.05 and fδ < 2 as well as the the equivalence principle (EP) constraint (10.35View Equation).

The growth rate of δm for t > tk is given by ˙ ∘ ---------2 fδ = δm∕(H δm ) = ( 25 + 48Q − 1)∕4. As we mentioned in Section 8, the observational bound on fδ is still weak in current observations. If we use the criterion fδ < 2 for the analytic estimation ∘ ---------- fδ = ( 25 + 48Q2 − 1)∕4, we obtain the bound Q < 1.08 (see Figure 9View Image). The current observational data on the growth rate fδ as well as its growth index γ is not enough to place tight bounds on Q and p, but this will be improved in future observations.


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