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 \( \bar { \sigma } _ { \ell } = \sqrt { 2 / ( 2 \ell + 1 ) } C _ { \ell } ^ { \Lambda \mathrm { C D M } } \).