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
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
)]
In the regime (“scalar-tensor regime”) we have
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
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]:
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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