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 [567], which was numerically confirmed in [403]. 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
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:
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 satisfies
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 [568
]. 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:
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-dependence
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
[266
]. 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 [37
], as we will see in Section 13.3. We note that the
f (R) models proposed in [427] [e.g.,
] to cure the singularity problem
satisfy neither the local gravity constraints [580] nor observational constraints of large-scale
structure [194].
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
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