In the following we consider the evolution of perturbations in f (R) gravity in the Longitudinal gauge (6.33). Since , , , and in this case, Eqs. (6.11), (6.13), (6.15), and (8.89) give
Let us consider the wavenumber deep inside the Hubble radius (). In order to derive the equation of matter perturbations approximately, we use the quasi-static approximation under which the dominant terms in Eqs. (8.90) – (8.93) correspond to those including , (or ) and . In General Relativity this approximation was first used by Starobinsky in the presence of a minimally coupled scalar field , which was numerically confirmed in . This was further extended to scalar-tensor theories [93, 171, 586] and f (R) gravity [586, 597]. Precisely speaking, in f (R) gravity, this approximation corresponds to and weak lensing observations . From Eq. (8.97) we have
From Eq. (6.12) the term is of the order of provided that the deviation from the CDM model is not significant. Using Eq. (8.97) we find that the ratio is of the order of , which is much smaller than unity for sub-horizon modes. Then the gauge-invariant perturbation given in Eq. (8.88) can be approximated as . Neglecting the r.h.s. of Eq. (8.93) relative to the l.h.s. and using Eq. (8.97) with , we get the equation for matter perturbations:[586, 597]
We recall that viable f (R) dark energy models are constructed to have a large mass in the region of high density (). During the radiation and deep matter eras the deviation parameter is much smaller than 1, so that the mass squared satisfiesf (R) models (4.83) and (4.84) the transition from the former regime to the latter regime, which is characterized by the condition , can occur during the matter domination for the wavenumbers relevant to the matter power spectrum [306, 568, 587, 270, 589].
In order to derive Eq. (8.100) we used the approximation that the time-derivative terms of on the l.h.s. of Eq. (8.92) is neglected. In the regime , however, the large mass can induce rapid oscillations of . In the following we shall study the evolution of the oscillating mode . For sub-horizon perturbations Eq. (8.92) is approximately given by
As long as the frequency satisfies the adiabatic condition , we obtain the solution of Eq. (8.104) under the WKB approximation:[568, 587]
For viable f (R) models, the scale factor and the background Ricci scalar evolve as and during the matter era. Then the amplitude of relative to has the time-dependencef (R) models (4.83) and (4.84) behave as with in the regime . During the matter-dominated epoch the mass evolves as . In the regime one has and hence the amplitude of the oscillating mode decreases faster than . However the contribution of the oscillating mode tends to be more important as we go back to the past. In fact, this behavior was confirmed in the numerical simulations of [587, 36]. This property persists in the radiation-dominated epoch as well. If the condition is violated, then can be negative such that the condition or is violated for the models (4.83) and (4.84). Thus we require that is smaller than at the beginning of the radiation era. This can be achieved by choosing the constant in Eq. (8.106) to be sufficiently small, which amounts to a fine tuning for these models.
For the models (4.83) and (4.84) one has in the regime . Then the field defined in Eq. (2.31) rapidly approaches as we go back to the past. Recall that in the Einstein frame the effective potential of the field has a potential minimum around because of the presence of the matter coupling. Unless the oscillating mode of the field perturbation is strongly suppressed relative to the background field , the system can access the curvature singularity at . This is associated with the condition discussed above. This curvature singularity appears in the past, which is not related to the future singularities studied in [461, 54]. The past singularity can be cured by taking into account the term , as we will see in Section 13.3. We note that the f (R) models proposed in  [e.g., ] to cure the singularity problem satisfy neither the local gravity constraints  nor observational constraints of large-scale structure .
As long as the oscillating mode is negligible relative to the matter-induced mode , we can estimate the evolution of matter perturbations as well as the effective gravitational potential . Note that in [192, 434] the perturbation equations have been derived without neglecting the oscillating mode. As long as the condition is satisfied initially, the approximate equation (8.100) is accurate to reproduce the numerical solutions [192, 589]. Equation (8.100) can be written as , so the oscillating mode tends to be unimportant with time.
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