The perturbation equations for the action (10.10) are given in Eqs. (6.11) – (6.18) with , , and . We use the unit , but we restore when it is necessary. In the Longitudinal gauge one has , , , and in these equations. Since we are interested in sub-horizon modes, we use the approximation that the terms containing , , , and are the dominant contributions in Eqs. (6.11) – (6.19). We shall neglect the contribution of the time-derivative terms of in Eq. (6.16). 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 is smaller than the matter-induced mode . In Fourier space Eq. (6.16) gives[596, 547] (see also [84, 632, 631])
Substituting Eq. (10.39) into Eq. (8.93), we obtain the equation of matter perturbations on sub-horizon scales [with the neglect of the r.h.s. of Eq. (8.93)]
In the regime (“scalar-tensor regime”) we havef (R) gravity, it follows that . Note that the result (10.42) agrees with the effective Newtonian gravitational coupling between two test masses [93, 175]. The evolution of and during the matter dominance in the regime is
As an example, let us consider the potential (10.23). During the matter era the field mass squared around the potential minimum (induced by the matter coupling) is approximately given by
When and the matter perturbation evolves as and , respectively (apart from the epoch of the late-time cosmic acceleration). The matter power spectrum at time (at which ) shows a difference compared to the CDM model, which is given by
The CMB power spectrum is also modified by the non-standard evolution of the effective gravitational potential for . 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 () :f (R) gravity ( and ) as a special case. Under the criterion we obtain the bounds for and for . As long as is close to 1, the model can be consistent with both cosmological and local gravity constraints. The allowed region coming from the bounds and (10.35) are illustrated in Figure 9.
The growth rate of for is given by . As we mentioned in Section 8, the observational bound on is still weak in current observations. If we use the criterion for the analytic estimation , we obtain the bound (see Figure 9). The current observational data on the growth rate as well as its growth index is not enough to place tight bounds on and , but this will be improved in future observations.
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