we can focus on the constant mode $c_{52}$, and thus the power spectrum in this case becomes $$\begin{aligned}
 \nonumber
&P_{\zeta} \sim c_{52}^2 k^{3} =\left\{
  \frac{c_{11}}{(1-2q_2)(1-2q_1)2(1-2q_1)}  \right. \\ \nonumber &
  \left.  \times \left[ -4 (1 + q_1 - q_2) (-2 + q_1 + q_2)
    \epsilon^{1 - 2 q_2}-4q_1\delta^{2q_1-1}  (-2 + q_1 + q_2)^2
    \epsilon^{-2 (-1 + q_1 + q_2)} \right. \right. \nonumber \\   &
    \left. \left. -4(q_1-1)\delta^{-2q_1+1} (1 + q_1 - q_2)^2
    \epsilon^{2 (q_1 - q_2) } \right] \right\}^2 k^{3}  .\end{aligned}$$

The initial power spectrum is $$\begin{aligned}
P_i = P_{\zeta} (-\tau_B-\epsilon) = \zeta^2 k^{3}  = c_{11}^2
\epsilon^{2-4q_2}  k^{3}\end{aligned}$$ and, thus, we can relate the final to the initial power spectrum as $$\begin{aligned}
\label{powerspectrum}
P_{\zeta} = \left(A_1  + A_2   \delta^{2q_1-1} \epsilon^{1-2 q_1 } +
A_3\delta^{-2q_1+1} \epsilon^{2q_1-1} \right)^2 P_i \, ,\end{aligned}$$ where $A_1$, $A_2$ and $A_3$ are constants that do not depend on $k$. Their explicit forms are $$\begin{aligned}
&A_1 = \frac{-2(1+q_1-q_2)(-2+q_1+q_2)}{(1-2q_2)(1-2q_1)^2},\\  &A_2 =
  \frac{-2q_1(-2+q_1+q_2)^2}{(1-2q_2)(1-2q_1)^2}, \\ &A_3 =
  \frac{-2(q_1-1)(1+q_1-q_2)^2}{(1-2q_2)(1-2q_1)^2}.\end{aligned}$$

For very large scale modes $k^{-1} \gg  k_*^{-1}$, $\delta \to
\tau_B$ and $\delta$ can be approximated as a constant time interval. Thus, the power spectrum in this case becomes

$$\begin{aligned}
\label{powerspectrum1}
(A_1  + A_2  \tau_B^{2q_1-1} \epsilon^{1-2 q_1 } + A_3
  \tau_B^{-2q_1+1} \epsilon^{2q_1-1} )^2 P_i 