8.4 Cosmic Microwave Background

The effective gravitational potential (8.98View Equation) is directly related to the ISW effect in CMB anisotropies. This contributes to the temperature anisotropies today as an integral [308214Jump To The Next Citation Point]
∫ η0 −τd-Φeff ΘISW ≡ d ηe d η jℓ[k(η0 − η )], (8.122 ) 0
where τ is the optical depth, ∫ η = a −1dt is the conformal time with the present value η0, and jℓ[k(η0 − η )] is the spherical Bessel function for CMB multipoles ℓ and the wavenumber k. In the limit ℓ ≫ 1 (i.e., small-scale limit) the spherical Bessel function has a dependence ℓ−1∕2 jℓ(x) ≃ (1∕ℓ)(x∕ℓ), which is suppressed for large ℓ. Hence the dominant contribution to the ISW effect comes from the low ℓ modes (ℓ = 𝒪(1)).
View Image

Figure 7: (Left) Evolution of the effective gravitational potential Φeff (denoted as Φ− in the figure) versus the scale factor a (with the present value a = 1) on the scale −1 3 k = 10 Mpc for the ΛCDM model and f (R) models with B0 = 0.5, 1.5, 3.0, 5.0. As the parameter B0 increases, the decay of Φe ff decreases and then turns into growth for B0 ≳ 1.5. (Right) The CMB power spectrum ℓ(ℓ + 1)C ℓ∕(2 π) for the ΛCDM model and f (R) models with B0 = 0.5, 1.5, 3.0, 5.0. As B 0 increases, the ISW contributions to low multipoles decrease, reach the minimum around B0 = 1.5, and then increase. The black points correspond to the WMAP 3-year data [561Jump To The Next Citation Point]. From [545Jump To The Next Citation Point].

In the ΛCDM 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 7View Image). This late-time variation of Φeff leads to the contribution to ΘISW, which works as the ISW effect.

For viable f (R) dark energy models the evolution of Φeff during the early stage of the matter era is constant as in the ΛCDM model. After the transition to the scalar-tensor regime, the effective gravitational potential evolves as √-- Φeff ∝ t(33−5)∕6 during the matter dominance [as we have shown in Eq. (8.110View Equation)]. The evolution of Φe ff during the accelerated epoch is also subject to change compared to the ΛCDM model. In the left panel of Figure 7View Image we show the evolution of Φ eff versus the scale factor a for the wavenumber −3 −1 k = 10 Mpc in several different cases. In this simulation the background cosmological evolution is fixed to be the same as that in the ΛCDM model. In order to quantify the difference from the ΛCDM model at the level of perturbations, [628544Jump To The Next Citation Point545Jump To The Next Citation Point] defined the following quantity

R˙H-- B ≡ m R ˙, (8.123 ) H
where m = Rf,RR∕f,R. If the effective equation of state weff defined in Eq. (4.69View Equation) is constant, it then follows that R = 3H2 (1 − 3weff ) and hence B = 2m. The stability of cosmological perturbations requires the condition B > 0 [544Jump To The Next Citation Point526]. The left panel of Figure 7View Image shows that, as we increase the values of B today (= B 0), the evolution of Φ eff at late times tends to be significantly different from that in the ΛCDM model. This comes from the fact that, for increasing B, the transition to the scalar-tensor regime occurs earlier.

From the right panel of Figure 7View Image we find that, as B0 increases, the CMB spectrum for low multipoles first decreases and then reaches the minimum around B0 = 1.5. This comes from the reduction in the decay rate of Φeff relative to the ΛCDM model, see the left panel of Figure 7View Image. Around B0 = 1.5 the effective gravitational potential is nearly constant, so that the ISW effect is almost absent (i.e., ΘISW ≈ 0). For B0 ≳ 1.5 the evolution of Φe ff turns into growth. This leads to the increase of the large-scale CMB spectrum, as B0 increases. The spectrum in the case B0 = 3.0 is similar to that in the ΛCDM model. The WMAP 3-year data rule out B0 > 4.3 at the 95% confidence level because of the excessive ISW effect [545Jump To The Next Citation Point].

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 B0 ≳ 1, 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 B0 ≲ 1 [545]. This is tighter than the bound B0 < 4.3 coming from the CMB angular spectrum discussed above.

Finally we briefly mention stochastic gravitational waves produced in the early universe [421172122Jump To The Next Citation Point123Jump To The Next Citation Point17417319620]. For the inflation model f(R ) = R + R2 ∕(6M 2) the primordial gravitational waves are generated with the tensor-to-scalar ratio r of the order of 10− 3, see Eq. (7.73View Equation). It is also possible to generate stochastic gravitational waves after inflation under the modification of gravity. Capozziello et al. [122123] studied the evolution of tensor perturbations for a toy model 1+𝜖 f = R in the FLRW universe with the power-law evolution of the scale factor. Since the parameter 𝜖 is constrained to be very small (|𝜖| < 7.2 × 10− 19[62160], 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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