In this subsection, we would like to analyze in detail these two cases.

First we would like to calculate the critical comoving momentum $k_*^{-1}$. We start by analyzing the Hubble parameter $H$. The corresponding comoving Hubble parameter in region II is

$$\begin{aligned}
aH =q_1 (\tau+\tau_B)^{-1}.\end{aligned}$$

The critical scale $k_{*}$, which is obtained by $k_*=aH(\tau=-\Delta)$, is therefore $$\begin{aligned}
k_{*} =  q_1 (\tau_B - \Delta)^{-1}.\end{aligned}$$ The analysis here is similar to the instantaneous matching case of the previous subsection and we can obtain the power spectrum as $$\begin{aligned}
&P_{\zeta} \sim c_{52}^2 k^{3} = \left\{ \frac{ 1
  }{(1-2q_2)(1-2q_1)2k(1-2q_1)}  \right.  \nonumber \\ & \left. \times
  \left[  2 a_{11} (1 + q_1 - q_2) (-2 + q_1 + q_2)   + a_{12} (-2 +
    q_1 + q_2)^2 \epsilon^{1 -2 q_1 } \right. \right.  \nonumber \\ &
    \left. \left. + a_{21} (1 + q_1 - q_2)^2 \epsilon^{2 q_1 -1}
    \right] \right\}^2 P_i ,
\label{powerspectrumforinstantaneous}\end{aligned}$$ where $a_{11}$, $a_{12}$ and $a_{21}$ were already defined by Eqs. , and , respectively.

Model with no Region III
------------------------

Let us here consider the model with no Region III. We expect that the result we obtained in the previous Subsection will approach the result derived here in the limit when $\Delta\rightarrow 0$. 