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

Using this relation together with Eqs. (6.13) and (6.18), it follows that Combing this equation with Eqs. (6.11) and (6.13), we obtain [596, 547] (see also [84, 632, 631]) where we have recovered . Defining the effective gravitational potential , we find that satisfies the same form of equation as (8.99).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)]

where the effective gravitational coupling is In the regime (“GR regime”) this reduces to , so that the evolution of and during the matter domination () is standard: and .In the regime (“scalar-tensor regime”) we have

where we used the relation (10.11) between the coupling and the BD parameter . Since in f (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 Hence the growth rate of for is larger than that for .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

which decreases with time. The perturbations cross the point at time , which depends on the wavenumber . Since the evolution of the mass during the matter domination is given by , the time has a scale-dependence: . More precisely the critical redshift at time can be estimated as [596] where the subscript “0” represents present quantities. For the scales , which correspond to the linear regime of the matter power spectrum, the critical redshift can be in the region . Note that, for larger , decreases.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 () [596]:

Note that this covers the result (8.116) in 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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