two different modes:

$$\begin{aligned}
\begin{pmatrix}
\epsilon^{1-q_2} & \epsilon^{q_2} \\ -(1-q_2)  \epsilon^{-q_2} &
-q_2\epsilon^{q_2-1}
\end{pmatrix} \begin{pmatrix}
c_{11} \\ c_{12}
\end{pmatrix} = \begin{pmatrix}
\epsilon^{1-q_1} & \epsilon^{q_1} \\ (1-q_1) \epsilon^{-q_1} & q_1
\epsilon^{q_1-1}
\end{pmatrix} \begin{pmatrix}
c_{21} \\ c_{22}
\end{pmatrix} .\end{aligned}$$

### Between region II and region III

To be explicit, we focus in this case on the large scale modes for which the time duration of Region III is $-\Delta<\tau<\Delta$. For the small scale modes which enter the Hubble radius before $- \Delta$, we just make the substitution $\Delta\rightarrow\tau_H(k)$. Apart from that the discussion is the same. For the next subsection the convention will be the same. The matching conditions in this case are $$\begin{aligned}
v_2(-\Delta) = v_3 (-\Delta),\quad  v'_2(-\Delta) = v'_3 (-\Delta),\end{aligned}$$ which in matrix form can be expressed as

$$\begin{aligned}
\begin{pmatrix}
\delta^{1-q_1} & \delta^{q_1} \\ (1-q_1) \delta^{-q_1} & q_1
\delta^{q_1-1} 
\end{pmatrix} \begin{pmatrix}
c_{21} \\ c_{22}
\end{pmatrix} = \begin{pmatrix}
e^{-ik \Delta} & e^{i k \Delta} \\ i k e^{-ik\Delta} & - i k e^{i k
  \Delta}
\end{pmatrix} \begin{pmatrix}
c_{31} \\ c_{32}
\end{pmatrix},\end{aligned}$$

and where in the above equation we have defined $\delta$ as

$$\delta = \tau_B - \Delta .
\label{delta}$$

Note that for the small scale modes the definition of $\delta$ should 