We are interested in the wavenumbers relevant to the linear regime of the galaxy power
spectrum [577, 578]:
If the transition characterized by the condition (8.111) occurs during the deep matter era (
), we
can estimate the critical redshift
at the transition point. In the following let us consider the models
(4.83
) and (4.84
). In addition to the approximations
and
during the
matter dominance, we use the the asymptotic forms
and
with
. Since the dark energy density today can be approximated as
, it
follows that
. Then the condition (8.111
) translates into the critical redshift [589
]
The estimation (8.114) shows that, for larger
, the transition occurs earlier. The time
at the
transition has a
-dependence:
. For
the matter perturbation evolves as
by the time
corresponding to the onset of cosmic acceleration (
). The
matter power spectrum
at the time
shows a difference compared to the case of the
CDM model [568
]:
The modified evolution (8.110) of the effective gravitational potential for
leads to the
integrated Sachs–Wolfe (ISW) effect in CMB anisotropies [544
, 382
, 545
]. However this is limited to very
large scales (low multipoles) in the CMB spectrum. Meanwhile the galaxy power spectrum is directly
affected by the non-standard evolution of matter perturbations. From Eq. (8.115
) there should be a
difference between the spectral indices of the CMB spectrum and the galaxy power spectrum on the scale
(8.112
) [568
]:
In order to estimate the growth rate of matter perturbations, we introduce the growth index defined
by [484
]
The growth index in the CDM model corresponds to
[612, 395
], which is nearly constant
for
. In f (R) gravity, if the perturbations are in the GR regime (
) today,
is
close to the GR value. Meanwhile, if the transition to the scalar-tensor regime occurred at the redshift
larger than 1, the growth index becomes smaller than 0.55 [270]. Since
, the smaller
implies a larger growth rate.
In Figure 4 we plot the evolution of the growth index
in the model (4.83
) with
and
for a number of different wavenumbers. In this case the present value of
is
degenerate around
independent of the scales of our interest. For the wavenumbers
and
the transition redshifts correspond to
and
, respectively. Hence these modes have already entered the scalar-tensor regime by
today.
From Eq. (8.114) we find that
gets smaller for larger
and
. If the mode
crossed the transition point at
and the mode
has marginally entered (or
has not entered) the scalar-tensor regime by today, then the growth indices should be strongly dispersed.
For sufficiently large values of
and
one can expect that the transition to the regime
has not occurred by today. The following three cases appear depending on the values of
and
[589
]:
The region (i) corresponds to the opposite of the inequality (8.113), i.e.,
, in which case
and
take large values. The border between (i) and (iii) is characterized by the condition
. The region (ii) corresponds to small values of
and
(as in the numerical
simulation of Figure 4
), in which case the mode
entered the scalar-tensor regime for
.
The regions (i), (ii), (iii) can be found numerically by solving the perturbation equations. In Figure 5 we
plot those regions for the model (4.84
) together with the bounds coming from the local gravity constraints
as well as the stability of the late-time de Sitter point. Note that the result in the model (4.83
) is also
similar to that in the model (4.84
). The parameter space for
and
is dominated by
either the region (ii) or the region (iii). While the present observational constraint on
is quite weak,
the unusual converged or dispersed spectra found above can be useful to distinguish metric
f (R) gravity from the
CDM model in future observations. We also note that for other viable
f (R) models such as (4.89
) the growth index today can be as small as
[589]. If
future observations detect such unusually small values of
, this can be a smoking gun for
f (R) models.
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