Figure 3. (Top) Evolution of the gravitational potential \( \Psi \) normalized by its initial value \( \Psi ^ { ( s ) } \) for the wavenumber \( k \) =0.01 \( \mathrm { M p c } ^ { - 1 } \). We show the evolution of \( \Psi / \Psi ^ { ( s ) } \) for four models listed in Table I and also for \( \Lambda ^ { \prime } \) CDM (black line). (Bottom) Percentage relative difference of \( \Psi \) relative to that in \( \Lambda ^ { \prime } \) CDM. The cosmological parameters used for this plot are the Planck 2015 best-fit values for \( \Lambda \) CDM [105] (which is also the case for plots in Figs. 5 and 6). The physical interpretation of this figure is discussed in Sec. IV.

\( \Lambda \) CDM at high redshifts (\( a \lesssim 1 0 ^ { - 2 } \)). This property arises from the dominance of \( x _ { 4 } \) over \( x _ { 1 , 2 , 3 } \) at early times, in which case the relative density abundances between dark energy and matter fluids are modified. Besides this effect, the non-negligible early-time contribution of \( x _ { 4 } \) to scalar perturbations gives rise to a scale-dependent evolution of gravitational potentials, which manifests itself in the k-dependent variation of \( \mu ( a , k ) \) and \( \Sigma ( a , k ) \). In Fig. 4, we plot the evolution of \( \Psi \) in BH1 for three different values of k. For perturbations on smaller scales, the deviation from \( \Lambda \) CDM tends to be more significant. In models BH2, BH3, and GGC, the early-time evolution of \( \Psi \) is similar to that in \( \Lambda ^ { \prime } \) CDM, but they exhibit large deviations from \( \Lambda \) CDM at late times.

At low redshifts, the lensing gravitational potential \( \phi _ { \mathrm { l e n } } = ( \Psi + \Phi ) / 2 \) evolves in a similar way to \( \Psi \), by re-flecting the property \( \mu \simeq \Sigma \) for \( x _ { A } ^ { ( 0 ) } \ll 1 \). The lensing angular power spectrum can be computed by using the line of sight integration method, with the convention [99]

\[ C _ { \ell } ^ { \phi \phi } = 4 \pi \int \frac { \mathrm { d } k } { k } \mathcal { P } ( k ) \left[ \int _ { 0 } ^ { \chi _ { * } } \mathrm { d } \chi \, S _ { \phi } ( k ; \tau _ { 0 } - \chi ) j _ { \ell } ( k \chi ) \right] _ { ( 4 . 7 ) } ^ { 2 } \, , \]

where \( \mathcal { P } ( k ) = \Delta _ { \mathcal { R } } ^ { 2 } ( k ) \) is the primordial power spectrum of curvature perturbations, and \( j _ { \ell } \) is the spherical Bessel function. The source \( S _ { \phi } \) is expressed in terms of the transfer function

\[ S _ { \phi } ( k ; \tau _ { 0 } - \chi ) = 2 T _ { \phi } ( k ; \tau _ { 0 } - \chi ) \left( \frac { \chi _ { * } - \chi } { \chi _ { * } \chi } \right) \, , \qquad ( 4 . 8 ) \]

with \( T _ { \phi } ( k , \tau ) \, = \, k \phi _ { \mathrm { l e n } } \), \( \chi \) is the comoving distance with \( \chi _ { * } \) corresponding to that to the last scattering surface, \( \tau _ { 0 } \) is today’s conformal time \( \textstyle \tau = \int a ^ { - 1 } \mathrm { d } t \) satisfying the relation \( \chi = \tau _ { 0 } - \tau \) . In Fig. 5, we show the lensing power spectra \( \bar { D } _ { \ell } ^ { \phi \phi } = \ell ( \ell + 1 ) C _ { \ell } ^ { \bar { \phi } \phi } / ( 2 \pi ) \) and relative differences in units of the cosmic variance for four models listed in Table I. Since \( \Sigma _ { i } > \) 1 at low redshifts in BH and GGC models, this works to enhance \( D _ { \ell } ^ { \phi \phi } \) compared to \( \Lambda \) CDM. We note that the amplitude of matter density contrast \( \delta _ { m } \) in these models also gets larger than that in \( \Lambda \) CDM by reflecting the fact that \( \mu > \) 1. In Fig. 5, we observe

Figure 4. (Top) Evolution of the gravitational potential \( \Psi \) normalized by its initial value \( \Psi ^ { ( s ) } \) for BH1 and \( \Lambda ^ { \prime } \) CDM with three different wavenumbers: \( k \, = \, 0 . 0 1 , 0 . 1 , 0 . 5 \, \, \, \mathrm { M p c } ^ { - 1 } \). In Table I, we list the starting values of parameters \( x _ { i } \) at the initial redshift \( z _ { s } = 1 . 5 \times 1 0 ^ { 5 } \) for the BH1 model. (Bottom) Percentage relative difference of \( \Psi \) relative to that in \( \Lambda ^ { \prime } \) CDM for the same values of k in the top panel.

Figure 5. (Top) Lensing angular power spectra \( D _ { \ell } ^ { \phi \phi } ~ = ~ \)\( \ell ( \ell + 1 ) C _ { \ell } ^ { \phi \phi } / ( 2 \pi ) \) for \( \Lambda \) CDM and the models listed in Ta-ble I, where \( C _ { \ell } \) is defined by Eq. (4.7). (Bottom) Rela-tive difference of the lensing angular power spectra, com-puted with respect to \( \Lambda \) CDM, in units of the cosmic variance \( \bar { \sigma } _ { \ell } = \sqrt { 2 / ( 2 \ell + 1 ) } C _ { \ell } ^ { \Lambda \mathrm { C D M } } \).

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