From Eq. (9.28) the perturbation can be expressed by the matter perturbation , as

where . This equation clearly shows that the perturbation is sourced by the matter perturbation only, unlike metric f (R) gravity in which the oscillating mode of is present. The matter perturbation and the velocity potential obey the same equations as given in Eqs. (8.86) and (8.87), which results in Eq. (8.89) in Fourier space.Let us consider the perturbation equations in Fourier space. We choose the Longitudinal gauge () with and . In this case Eq. (9.26) gives

Under the quasi-static approximation on sub-horizon scales used in Section 8.1, Eqs. (9.24) and (8.89) reduce to Combining Eq. (9.30) with Eq. (9.31), we obtain where Then the matter perturbation satisfies the following Eq. [597] The effective gravitational potential defined in Eq. (8.98) obeys In the above approximation we do not need to worry about the dominance of the oscillating mode of perturbations in the past. Note also that the same approximate equation of as Eq. (9.35) can be derived for different gauge choices [597].The parameter is a crucial quantity to characterize the evolution of perturbations. This quantity can be estimated as , which is much larger than for sub-horizon modes (). In the regime the matter perturbation evolves as . Meanwhile the evolution of in the regime is completely different from that in GR. If the transition characterized by occurs before today, this gives rise to the modification to the matter spectrum compared to the GR case.

In the regime , let us study the evolution of matter perturbations during the matter dominance. We shall consider the case in which the parameter (with evolves as

where is a constant. For the model () the power corresponds to , whereas for the models (4.83) and (4.84) with one has . During the matter dominance the parameter evolves as , where the subscript “” denotes the value at which the perturbation crosses . Here and signs correspond to the cases and , respectively. Then the matter perturbation equation (9.35) reduces toWhen , the growing mode solution to Eq. (9.38) is given by

This shows that the perturbations exhibit violent growth for , which is not compatible with observations of large-scale structure. In metric f (R) gravity the growth of matter perturbations is much milder.When , the perturbations show a damped oscillation:

where , and is a constant. The averaged value of the growth rate is given by , but it shows a divergence every time changes by . These negative values of are also difficult to be compatible with observations.The f (R) models can be consistent with observations of large-scale structure if the universe does not enter the regime by today. This translates into the condition [597]

Let us consider the wavenumbers that corresponds to the linear regime of the matter power spectrum. Since the wavenumber corresponds to (where “0” represents present quantities), the condition (9.41) gives the bound .If we use the observational constraint of the growth rate, [418, 605, 211], then the deviation parameter today is constrained to be - for the model () as well as for the models (4.83) and (4.84) [597]. Recall that, in metric f (R) gravity, the deviation parameter can grow to the order of 0.1 by today. Meanwhile f (R) dark energy models based on the Palatini formalism are hardly distinguishable from the CDM model [356, 386, 385, 597]. Note that the bound on becomes even severer by considering perturbations in non-linear regime. The above peculiar evolution of matter perturbations is associated with the fact that the coupling between non-relativistic matter and a scalar-field degree of freedom is very strong (as we will see in Section 10.1).

The above results are based on the fact that dark matter is described by a cold and perfect fluid with no pressure. In [358] it was suggested that the tight bound on the parameter can be relaxed by considering imperfect dark matter with a shear stress. Although the approach taken in [358] did not aim to explain the origin of a dark matter stress that cancels the -dependent term in Eq. (9.35), it will be of interest to further study whether some theoretically motivated choice of really allows the possibility that Palatini f (R) dark energy models can be distinguished from the CDM model.

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