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

where . If the effective equation of state defined in Eq. (4.69) is constant, it then follows that and hence . The stability of cosmological perturbations requires the condition [544, 526]. The left panel of Figure 7 shows that, as we increase the values of today (), the evolution of at late times tends to be significantly different from that in the CDM model. This comes from the fact that, for increasing , the transition to the scalar-tensor regime occurs earlier.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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