We are interested in the wavenumbers relevant to the linear regime of the galaxy power spectrum [577, 578]:

where corresponds to the uncertainty of the Hubble parameter today. Non-linear effects are important for . The current observations on large scales around are not so accurate but can be improved in future. The upper bound corresponds to , 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 ) for the mode , then the parameter today is constrained to be When the linear perturbations have been always in the GR regime by today, in which case the models are not distinguished from the CDM model. The bound (8.113) is relaxed for non-linear perturbations with , but the linear analysis is not valid in such cases.If the transition characterized by the condition (8.111) occurs during the deep matter era (), we can estimate the critical redshift at the transition point. In the following let us consider the models (4.83) and (4.84). In addition to the approximations and during the matter dominance, we use the the asymptotic forms and with . Since the dark energy density today can be approximated as , it follows that . Then the condition (8.111) translates into the critical redshift [589]

For , , , and the numerical value of the critical redshift is , which is in good agreement with the analytic value estimated by (8.114).The estimation (8.114) shows that, for larger , the transition occurs earlier. The time at the transition has a -dependence: . For the matter perturbation evolves as by the time corresponding to the onset of cosmic acceleration (). The matter power spectrum at the time shows a difference compared to the case of the CDM model [568]:

We caution that, when is close to (the redshift at ), the estimation (8.115) begins to lose its accuracy. The ratio of the two power spectra today, i.e., is in general different from Eq. (8.115). However, numerical simulations in [587] show that the difference is small for of the order of unity.The modified evolution (8.110) of the effective gravitational potential for leads to the integrated Sachs–Wolfe (ISW) effect in CMB anisotropies [544, 382, 545]. 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. (8.115) there should be a difference between the spectral indices of the CMB spectrum and the galaxy power spectrum on the scale (8.112) [568]:

Observationally we do not find any strong signature for the difference of slopes of the two spectra. If we take the mild bound , we obtain the constraint . Note that in this case the local gravity constraint (5.60) is also satisfied.In order to estimate the growth rate of matter perturbations, we introduce the growth index defined by [484]

where . This choice of comes from writing Eq. (4.59) in the form , where and we have ignored the contribution of radiation. Since the viable f (R) models are close to the CDM model in the region of high density, the quantity approaches 1 in the asymptotic past. Defining and in the above way, the Friedmann equation can be cast in the usual GR form with non-relativistic matter and dark energy [568, 270, 589].The growth index in the CDM model corresponds to [612, 395], which is nearly constant for . In f (R) gravity, if the perturbations are in the GR regime () today, is close to the GR value. Meanwhile, if the transition to the scalar-tensor regime occurred at the redshift larger than 1, the growth index becomes smaller than 0.55 [270]. Since , the smaller implies a larger growth rate.

In Figure 4 we plot the evolution of the growth index in the model (4.83) with and for a number of different wavenumbers. In this case the present value of is degenerate around independent of the scales of our interest. For the wavenumbers and the transition redshifts correspond to and , respectively. Hence these modes have already entered the scalar-tensor regime by today.

From Eq. (8.114) we find that gets smaller for larger and . If the mode crossed the transition point at and the mode 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 and one can expect that the transition to the regime has not occurred by today. The following three cases appear depending on the values of and [589]:

- (i)
- All modes have the values of close to the CDM value: , i.e., .
- (ii)
- All modes have the values of close to the value in the range .
- (iii)
- The values of are dispersed in the range .

The region (i) corresponds to the opposite of the inequality (8.113), i.e., , in which case and take large values. The border between (i) and (iii) is characterized by the condition . The region (ii) corresponds to small values of and (as in the numerical simulation of Figure 4), in which case the mode entered the scalar-tensor regime for .

The regions (i), (ii), (iii) can be found numerically by solving the perturbation equations. In Figure 5 we plot those regions for the model (4.84) 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 (4.83) is also similar to that in the model (4.84). The parameter space for and is dominated by either the region (ii) or the region (iii). While the present observational constraint on is quite weak, the unusual converged or dispersed spectra found above can be useful to distinguish metric f (R) gravity from the CDM model in future observations. We also note that for other viable f (R) models such as (4.89) the growth index today can be as small as [589]. If future observations detect such unusually small values of , this can be a smoking gun for f (R) models.

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