Figure 6. (Top) Evolution of the time derivative \( \dot { \Psi } + \dot { \Phi } \) for \( \Lambda ^ { \prime } \) CDM and the models listed in Table I, computed at \( k \)=0.01 \( \mathrm { M p c } ^ { - 1 } \). (Bottom) Relative difference of \( \dot { \Psi } \! + \! \dot { \Phi } \), computed with respect to \( \Lambda ^ { \prime } \) CDM. See the discussion after Eq. (4.11) for the physical interpretation of this figure.

that, apart from BH1 in which \( \Sigma \) is close to 1, the lensing power spectra in other three cases are subject to the en-hancement with respect to \( \Lambda ^ { \prime } \) CDM. Since today’s values of \( \mu \) and \( \Sigma \) increase for larger \( x _ { 3 } ^ { ( 0 ) } \), the deviation from \( \Lambda \) CDM tends to be more significant with the order of GGC, BH2, and BH3.

Let us proceed to the discussion of the impact of BH and GGC models on the CMB temperature anisotropies. The CMB temperature-temperature (TT) angular spec-trum can be expressed as [100]

\[ C _ { \ell } ^ { \mathrm { T T } } = ( 4 \pi ) ^ { 2 } \int \frac { \mathrm { d } k } { k } \; \mathcal { P } ( k ) \Big | \Delta _ { \ell } ^ { \mathrm { T } } ( k ) \Big | ^ { 2 } \, , \qquad \mathrm { ( 4 . 9 ) } \]

where

\[ \Delta _ { \ell } ^ { \mathrm { T } } ( k ) = \int _ { 0 } ^ { \tau _ { 0 } } \mathrm { d } \tau \, e ^ { i k \tilde { \mu } ( \tau - \tau _ { 0 } ) } S _ { \mathrm { T } } ( k , \tau ) j _ { \ell } [ k ( \tau _ { 0 } - \tau ) ] \, , \, \, ( 4 . 1 0 ) \]

with \( \tilde { \mu } \) being the angular separation, and \( S _ { \mathrm { T } } ( k , \tau ) \) is the radiation transfer function. The contribution to \( S _ { \mathrm { T } } ( k , \tau ) \) arising from the integrated-Sachs-Wolfe (ISW) effect is of the form

\[ S _ { \mathrm { T } } ( k , \tau ) \sim \left( \frac { \mathrm { d } \Psi } { \mathrm { d } \tau } + \frac { \mathrm { d } \Phi } { \mathrm { d } \tau } \right) e ^ { - \kappa } \, , \qquad \qquad ( 4 . 1 1 ) \]

where \( \kappa \) is the optical depth. Besides the early ISW effect which occurs during the transition from the radiation to matter eras by the time variation of \( \Psi + \Phi \), the pres-ence of dark energy induces the late-time ISW effect. In the \( \Lambda \) CDM model, the gravitational potential −(Ψ + Φ),which is positive, decreases by today with at least more than 30 % relative to its initial value (see Fig. 3). As we observe in Fig. 6 we have \( \dot { \Psi } + \dot { \Phi } > \) 0 in this case, so the ISW effect gives rise to the positive contribution to Eq. (4.9). In Fig. 7, we plot the CMB TT power spectra \( D _ { \rho } ^ { \mathrm { T T } } = \ell ( \ell + 1 ) C _ { \rho } ^ { \mathrm { T T } } / ( 2 \pi ) \) for the models listed in Table I and \( \Lambda \) CDM. In BH1 the parameter \( \Sigma \) is close to 1 at low redshifts due to the smallness of \( x _ { 3 } ^ { ( 0 ) } \), so the late-time ISW effect works in the similar way to the GR case. Hence the TT power spectrum in BH1 for the multipoles \( \ell \lesssim \) 30 is similar to that in \( \Lambda \) CDM.

In the GGC model of Fig. 7, we observe that the large-scale ISW tail is suppressed relative to that in \( \Lambda ^ { \prime } \) CDM. This reflects the fact that the larger deviation of \( \Sigma \) from 1 leads to the time derivative \( \dot { \Psi } + \dot { \Phi } \) closer to 0, see Fig. 6. Hence the late-time ISW effect is not significant, which results in the suppression of \( D _ { \ell } ^ { \mathrm { T T } } \) with respect to \( \Lambda \) CDM. In Ref. [72] this fact was first recognized in the GGC model, which exhibits a better fit to the Planck CMB data. As the deviation of \( \Sigma \) from 1 increases further, the sign of \( \dot { \Psi } + \dot { \Phi } \) changes to be negative (see Fig. 6). The BH2 model can be regarded as such a marginal case in which the large-scale ISW tail is nearly flat. In BH3, the increase of \( \Sigma \) at low redshifts is so significant that the largely negative ISW contribution to Eq. (4.9) leads to the enhanced low-\( \ell \) TT power spectrum relative to \( \Lambda \) CDM.

The modified evolution of the Hubble expansion rate from \( \Lambda \) CDM generally leads to the shift of CMB acoustic peaks at high-\( \ell \). In Fig. 8, we observe that the largest deviation of H(a) at high redshifts occurs for BH1 by the dominance of \( x _ { 4 } \) over \( _ { x _ { 1 , 2 , 3 } } \). This leads to the shift of acoustic peaks toward lower multipoles (see Fig. 7). We also find that BH3 is subject to non-negligible shifts of high-\( \ell \) peaks due to the large modification of H(a) at low redshifts, in which case the peaks shift toward higher multipoles. Moreover, there is the large enhancement of ISW tails for BH3, so it should be tightly constrained from the CMB data. We note that the shift of CMB acoustic peaks is further constrained by the datasets of BAO and SN Ia. For BH2 and GGC the changes of peak positions are small in comparion to BH1 and BH3, but still they are in the range testable by the CMB data. Moreover, the large-scale ISW tail is subject to the sup-pression relative to \( \Lambda \) CDM in BH2 and GGC.

In BH1, we also notice a change in the amplitude of acoustic peaks occurring dominantly at high \( \ell \). This is known to be present in models with early-time modifi-cations of gravity [101, 102]. The modification of gravi-tational potentials affects the evolution of radiation per-turbations (monopole and dipole) through the radiation driving effect [101, 103], thus resulting in the changes in amplitude and phase of acoustic peaks at high \( \ell \).

The modified time variations of \( \Psi \) and \( \Phi \) around the recombination epoch also give a contribution to the early ISW effect. This is important on scales around the first acoustic peak, corresponding to the wavenumber \( k \simeq \) 0.016 \( \mathrm { M p c ^ { - 1 } } \) for our choice of model parameters. To have a more qualitative feeling of this contribution, we have estimated the impact of the early ISW effect on \( D _ { \ell } ^ { \mathrm { T T } } \) by using the approximate ISW integral presented