Results of fitting the dependence of A on ${\overline{x}}_{\mathrm{HI}}$ using Eq. (4). Top panel shows the best-fit values of the parameters A_⋆, ${\overline{x}}_{\mathrm{HI},\star }$ and α_A when we fit different curves of the left panel of Fig. 5 using Eq. (4). Here, A represents the ratio ${\mathrm{\Delta }}_{\delta T_{\text{b}}}^2/{\mathrm{\Delta }}_{\delta \delta }^2$ at k ∼ 0.05 h Mpc⁻¹. The bottom panel shows the difference between the input/true A values and predicted (or fitted) A_f values with the best-fit parameters as a function of average ionization fraction ${\overline{x}}_{\mathrm{HI}}$.