8.2 The impact on large-scale structure

We have shown that the evolution of matter perturbations during the matter dominance is given by δm ∝ t2∕3 for M 2 ≫ k2∕a2 (GR regime) and √-- δm ∝ t( 33−1)∕6 for M 2 ≪ k2∕a2 (scalar-tensor regime), respectively. The existence of the latter phase gives rise to the modification to the matter power spectrum [14674544Jump To The Next Citation Point526Jump To The Next Citation Point251] (see also [597Jump To The Next Citation Point49349494446278435] for related works). The transition from the GR regime to the scalar-tensor regime occurs at 2 2 2 M = k ∕a. If it occurs during the matter dominance (R ≃ 3H2), the condition M 2 = k2∕a2 translates into [589Jump To The Next Citation Point]
m ≃ (aH ∕k)2, (8.111 )
where we have used the relation 2 M ≃ R∕ (3m ) (valid for m ≪ 1).

We are interested in the wavenumbers k relevant to the linear regime of the galaxy power spectrum [577578]:

0.01h Mpc − 1 ≲ k ≲ 0.,h Mpc −1, (8.112 )
where h = 0.72 ± 0.08 corresponds to the uncertainty of the Hubble parameter today. Non-linear effects are important for k ≳ 0.2h Mpc − 1. The current observations on large scales around k ∼ 0.01h Mpc −1 are not so accurate but can be improved in future. The upper bound k = 0.2h Mpc −1 corresponds to k ≃ 600a0H0, where the subscript “0” represents quantities today. If the transition from the GR regime to the scalar-tensor regime occurred by the present epoch (the redshift z = 0) for the mode k = 600a H 0 0, then the parameter m today is constrained to be
−6 m (z = 0) ≳ 3 × 10 . (8.113 )
When m (z = 0) ≲ 3 × 10−6 the linear perturbations have been always in the GR regime by today, in which case the models are not distinguished from the ΛCDM model. The bound (8.113View Equation) is relaxed for non-linear perturbations with −1 k ≳ 0.2h Mpc, but the linear analysis is not valid in such cases.

If the transition characterized by the condition (8.111View Equation) occurs during the deep matter era (z ≫ 1), we can estimate the critical redshift z k at the transition point. In the following let us consider the models (4.83View Equation) and (4.84View Equation). In addition to the approximations 2 2 (0) 3 H ≃ H 0Ω m (1 + z ) and 2 R ≃ 3H during the matter dominance, we use the the asymptotic forms m ≃ C (− r − 1)2n+1 and r ≃ − 1 − μRc ∕R with C = 2n(2n + 1)∕μ2n. Since the dark energy density today can be approximated as ρ (0D)E ≈ μRc ∕2, it follows that 2 (0) μRc ≈ 6H 0Ω DE. Then the condition (8.111View Equation) translates into the critical redshift [589Jump To The Next Citation Point]

[ ( ) ]1∕(6n+4) k 2 2n(2n + 1 )(2Ω(D0E))2n+1 zk = a-H-- ----μ2n----(Ω0-)2(n+1) − 1. (8.114 ) 0 0 m
For n = 1, μ = 3, Ω (m0) = 0.28, and k = 300a0H0 the numerical value of the critical redshift is zk = 4.5, which is in good agreement with the analytic value estimated by (8.114View Equation).

The estimation (8.114View Equation) shows that, for larger k, the transition occurs earlier. The time tk at the transition has a k-dependence: t ∝ k− 3∕(6n+4) k. For t > t k the matter perturbation evolves as (√33−1)∕6 δm ∝ t by the time t = tΛ corresponding to the onset of cosmic acceleration (¨a = 0). The matter power spectrum Pδm = |δm |2 at the time tΛ shows a difference compared to the case of the ΛCDM model [568Jump To The Next Citation Point]:

(√-- ) ( )2 -336−1− 23 √-- ---Pδm-(tΛ)--- = tΛ- ∝ k 633n−+54 . (8.115 ) P δmΛCDM (tΛ) tk
We caution that, when zk is close to zΛ (the redshift at t = tΛ), the estimation (8.115View Equation) begins to lose its accuracy. The ratio of the two power spectra today, i.e., ΛCDM Pδm(t0)∕Pδm (t0) is in general different from Eq. (8.115View Equation). However, numerical simulations in [587Jump To The Next Citation Point] show that the difference is small for n of the order of unity.

The modified evolution (8.110View Equation) of the effective gravitational potential for z < z k leads to the integrated Sachs–Wolfe (ISW) effect in CMB anisotropies [544Jump To The Next Citation Point382Jump To The Next Citation Point545Jump To The Next Citation Point]. However this is limited to very large scales (low multipoles) in the CMB spectrum. Meanwhile the galaxy power spectrum is directly affected by the non-standard evolution of matter perturbations. From Eq. (8.115View Equation) there should be a difference between the spectral indices of the CMB spectrum and the galaxy power spectrum on the scale (8.112View Equation[568Jump To The Next Citation Point]:

√ --- Δns = --33-−-5. (8.116 ) 6n + 4
Observationally we do not find any strong signature for the difference of slopes of the two spectra. If we take the mild bound Δns < 0.05, we obtain the constraint n > 2. Note that in this case the local gravity constraint (5.60View Equation) is also satisfied.

In order to estimate the growth rate of matter perturbations, we introduce the growth index γ defined by [484Jump To The Next Citation Point]

˙ fδ ≡ -δm-- = (&tidle;Ωm )γ, (8.117 ) H δm
where &tidle;Ωm = κ2ρm ∕(3H2 ) = FΩm. This choice of Ω&tidle;m comes from writing Eq. (4.59View Equation) in the form 2 2 3H = ρDE + κ ρm, where ˙ 2 ρDE ≡ (F R − f )∕2 − 3H F + 3H (1 − F ) and we have ignored the contribution of radiation. Since the viable f (R) models are close to the ΛCDM model in the region of high density, the quantity F approaches 1 in the asymptotic past. Defining ρDE and &tidle;Ωm in the above way, the Friedmann equation can be cast in the usual GR form with non-relativistic matter and dark energy [568Jump To The Next Citation Point270Jump To The Next Citation Point589Jump To The Next Citation Point].

The growth index in the ΛCDM model corresponds to γ ≃ 0.55 [612395Jump To The Next Citation Point], which is nearly constant for 0 < z < 1. In f (R) gravity, if the perturbations are in the GR regime (2 2 2 M ≫ k ∕a) today, γ is close to the GR value. Meanwhile, if the transition to the scalar-tensor regime occurred at the redshift zk larger than 1, the growth index becomes smaller than 0.55 [270]. Since 0 < &tidle;Ωm < 1, the smaller γ implies a larger growth rate.

View Image

Figure 4: Evolution of γ versus the redshift z in the model (4.83View Equation) with n = 1 and μ = 1.55 for four different values of k. For these model parameters the dispersion of γ with respect to k is very small. All the perturbation modes shown in the figure have reached the scalar-tensor regime (M 2 ≪ k2∕a2) by today. From [589Jump To The Next Citation Point].
View Image

Figure 5: The regions (i), (ii) and (iii) for the model (4.84View Equation). We also show the bound n > 0.9 coming from the local gravity constraints as well as the condition (4.87View Equation) coming from the stability of the de Sitter point. From [589Jump To The Next Citation Point].

In Figure 4View Image we plot the evolution of the growth index γ in the model (4.83View Equation) with n = 1 and μ = 1.55 for a number of different wavenumbers. In this case the present value of γ is degenerate around γ0 ≃ 0.41 independent of the scales of our interest. For the wavenumbers k = 0.1h Mpc −1 and k = 0.01h Mpc −1 the transition redshifts correspond to zk = 5.2 and zk = 2.7, respectively. Hence these modes have already entered the scalar-tensor regime by today.

From Eq. (8.114View Equation) we find that zk gets smaller for larger n and μ. If the mode k = 0.2h Mpc −1 crossed the transition point at zk > 𝒪(1) and the mode k = 0.01h Mpc −1 has marginally entered (or has not entered) the scalar-tensor regime by today, then the growth indices should be strongly dispersed. For sufficiently large values of n and μ one can expect that the transition to the regime 2 2 2 M ≪ k ∕a has not occurred by today. The following three cases appear depending on the values of n and μ [589Jump To The Next Citation Point]:

All modes have the values of γ0 close to the ΛCDM value: γ0 = 0.55, i.e., 0.53 ≲ γ ≲ 0.55 0.
All modes have the values of γ0 close to the value in the range 0.40 ≲ γ0 ≲ 0.43.
The values of γ0 are dispersed in the range 0.40 ≲ γ0 ≲ 0.55.

The region (i) corresponds to the opposite of the inequality (8.113View Equation), i.e., m (z = 0) ≲ 3 × 10−6, in which case n and μ take large values. The border between (i) and (iii) is characterized by the condition m (z = 0) ≈ 3 × 10− 6. The region (ii) corresponds to small values of n and μ (as in the numerical simulation of Figure 4View Image), in which case the mode −1 k = 0.01h Mpc entered the scalar-tensor regime for zk > 𝒪 (1).

The regions (i), (ii), (iii) can be found numerically by solving the perturbation equations. In Figure 5View Image we plot those regions for the model (4.84View Equation) together with the bounds coming from the local gravity constraints as well as the stability of the late-time de Sitter point. Note that the result in the model (4.83View Equation) is also similar to that in the model (4.84View Equation). The parameter space for n ≲ 3 and μ = 𝒪(1) is dominated by either the region (ii) or the region (iii). While the present observational constraint on γ is quite weak, the unusual converged or dispersed spectra found above can be useful to distinguish metric f (R) gravity from the ΛCDM model in future observations. We also note that for other viable f (R) models such as (4.89View Equation) the growth index today can be as small as γ ≃ 0.4 0 [589]. If future observations detect such unusually small values of γ0, this can be a smoking gun for f (R) models.

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