From Eq. (9.28) the perturbation can be expressed by the matter perturbation , asf (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 .
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
When , the growing mode solution to Eq. (9.38) is given byf (R) gravity the growth of matter perturbations is much milder.
When , the perturbations show a damped oscillation:
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 
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) . 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  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  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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