In the CDM model the effective gravitational potential is constant during the matter dominance, but
it begins to decay after the Universe enters the epoch of cosmic acceleration (see the left panel of Figure 7
).
This late-time variation of
leads to the contribution to
, which works as the ISW
effect.
For viable f (R) dark energy models the evolution of during the early stage of the matter era is
constant as in the
CDM model. After the transition to the scalar-tensor regime, the effective
gravitational potential evolves as
during the matter dominance [as we have shown in
Eq. (8.110
)]. The evolution of
during the accelerated epoch is also subject to change compared to the
CDM model. In the left panel of Figure 7
we show the evolution of
versus the scale factor
for
the wavenumber
in several different cases. In this simulation the background
cosmological evolution is fixed to be the same as that in the
CDM model. In order to quantify the
difference from the
CDM model at the level of perturbations, [628, 544
, 545
] defined the following
quantity
From the right panel of Figure 7 we find that, as
increases, the CMB spectrum for low multipoles
first decreases and then reaches the minimum around
. This comes from the reduction in the
decay rate of
relative to the
CDM model, see the left panel of Figure 7
. Around
the
effective gravitational potential is nearly constant, so that the ISW effect is almost absent (i.e.,
). For
the evolution of
turns into growth. This leads to the increase of the
large-scale CMB spectrum, as
increases. The spectrum in the case
is similar to that in the
CDM model. The WMAP 3-year data rule out
at the 95% confidence level because of the
excessive ISW effect [545
].
There is another observational constraint coming from the angular correlation between the
CMB temperature field and the galaxy number density field induced by the ISW effect [544].
The f (R) models predict that, for , the galaxies are anticorrelated with the CMB
because of the sign change of the ISW effect. Since the anticorrelation has not been observed
in the observational data of CMB and LSS, this places an upper bound of
[545].
This is tighter than the bound
coming from the CMB angular spectrum discussed
above.
Finally we briefly mention stochastic gravitational waves produced in the early
universe [421, 172, 122, 123
, 174, 173, 196, 20]. For the inflation model
the
primordial gravitational waves are generated with the tensor-to-scalar ratio
of the order of
, see
Eq. (7.73
). It is also possible to generate stochastic gravitational waves after inflation under the
modification of gravity. Capozziello et al. [122, 123] studied the evolution of tensor perturbations for a toy
model
in the FLRW universe with the power-law evolution of the scale factor. Since the
parameter
is constrained to be very small (
) [62, 160], it is very difficult to detect
the signature of f (R) gravity in the stochastic gravitational wave background. This property should hold
for viable f (R) dark energy models in general, because the deviation from GR during the radiation and the
deep matter era is very small.
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