Figure 1

Left panel: $f(a,k)$ as a function of $z(=1/a-1)$ for the model $n=1$. The solid curves are obtained by solving the differential equation (11) numerically, for $K(=k/k_C)={10}^2$, 10, and 1, respectively, from the top to the bottom. The long dashed curve, the dotted curve, and the short dashed curve adopt the approximate formula, for $K(=k/k_C)={10}^2$, 10, and 1, respectively, from the top to the bottom. Right panel: The growth index $\gamma (a,k)$ as a function of $z(=1/a-1)$, corresponding to the left panel. The parameter of the curves is the same as that of the left panel. The curves correspond to $K(=k/k_C)={10}^2$ and 10, 1, respectively, from the bottom to the top.Reuse & Permissions

Figure 2

Left panel: Redshift $z_x$ when the difference of the growth rate becomes $f^{(\mathrm{appr})}-f^{(\mathrm{exac})}=0.03$, as a function of $K(=k/k_C)$. The curves are $n=1/2$, 1, 2, 3, 4, and 5, respectively, from the top to the bottom. Right panel: Transition redshift $z_c$ as a function of $K(=k/k_C)$. From the top to the bottom, curves are $n=1/2$, 1, 2, 3, 4, and 5, respectively. Here we adopted the background expansion of the Universe is the $\Lambda \mathrm{CDM}$ model with ${\Omega }_0=0.28$.Reuse & Permissions

Figure 3

The relative difference of the growth factor between the exact solution $f$, which is obtained by solving Eq. (11) numerically, and the approximate solution in the form ${\Omega }_m^{\gamma }$, as a function of $z$. The four panels correspond to $N=1$, 2, 3, and 4, respectively. In each panel, the curves correspond to $K=0.01$, 0.1, 1, respectively, from the bottom to the top.Reuse & Permissions

Figure 4

Left panel: The 1-sigma contour in the ($n-\lambda$) plane. The linear modeling for $P_{\mathrm{mass}}(k,z)$ in the range of $10\le l\le {10}^3$ is used. The target modes are $(n,\lambda )=(3,4)$, (2, 2), (5, 2) and (2, 8), respectively. The other target parameters are $w_0=-1$, $w_a=0$, ${\Omega }_0=0.28$, ${\Omega }_b=0.044$, $h=0.7$, ${\sigma }_8=0.8$, and $n_s=0.95$. The solid curve corresponds to $k_C=0.2h{\mathrm{Mpc}}^{-1}$. Right panel: Same as the left panel, but with the nonlinear modeling for $P_{\mathrm{mass}}(k,z)$ of the range of $10\le l\le 3\times {10}^3$.Reuse & Permissions

Figure 5

Left panel: The 1-sigma contour in the ($k_C-n$) plane. The linear modeling for $P_{\mathrm{mass}}(k,z)$ in the range of $10\le l\le {10}^3$ is used. We assume the target modes $k_C=0.1h{\mathrm{Mpc}}^{-1}$ and $n=5$, 4, 3, 2, 1, and $1/2$, respectively, from the larger circle to smaller one. The other parameters are the same as those of Fig. 4. Right panel: Same as the left panel, but with the nonlinear modeling for $P_{\mathrm{mass}}(k,z)$ of the range of $10\le l\le 3\times {10}^3$.Reuse & Permissions

Figure 6

The 1-sigma error on $k_C$ as a function of the target value of $k_C$, where the other parameters are marginalized over. The left (right) panels use the linear (nonlinear) modeling for $P_{\mathrm{mass}}(k,z)$ of the range of $10\le l\le {10}^3$ ($10\le l\le 3\times {10}^3$). In each panel, the curves assume the target parameter $n=5$, 4, 3, 2, 1, and $1/2$, from the top to the bottom, respectively. The other target parameters are the same as those of Fig. 4. The upper panels are the result of the 9 parameters, $k_C$, $n$, $w_0$, $w_a$, ${\Omega }_0$, ${\Omega }_b$, $h$, $A$, and $n_s$. The lower panels are the result of the 7 parameters, $k_C$, $n$, ${\Omega }_0$, ${\Omega }_b$, $h$, $A$, and $n_s$, where the background expansion is fixed as that of the $\Lambda \mathrm{CDM}$ model.Reuse & Permissions

Figure 7

The additional spectral index $\Delta n_s$ as a function of wave number $k(h{\mathrm{Mpc}}^{-1})$. Here we adopted $k_C=0.1h{\mathrm{Mpc}}^{-1}$. The curves assume $n=1/2$, 1, 2, 3, 4, and 5, from the top to the bottom, respectively.Reuse & Permissions