that $k \tau \ll 1$, the higher order terms of $v(\tau)$ given by $m> 0$ are subleading. Thus, in the following, we can just focus on the $m=0$ terms in Eq. (\[vtaum\]).

The scale factors and the solutions to the mode functions of the five regions can be obtained by shifting the time coordinate. Thus, they are given, respectively, by

-   Region I (contracting), where $\tau<-\tau_B$, we have that

    $$\begin{aligned}
    a \sim  (-\tau-\tau_B)^{q_2} \sim(-t-t_B)^{p_2}, \quad v_{1} = c_{11}
    (-\tau_B-\tau)^{1-q_2} + c_{12} (-\tau_B-\tau)^{q_2},\end{aligned}$$

-   Region II (expanding), where $ -\tau_B<\tau<-\Delta\,\,\, {\rm
      for}\,\,\, k^{-1}>k_*^{-1},   \,\,\, {\rm
      and}\,\,\,-\tau_B<\tau<-\tau_H \,\,\, {\rm for}\,\,\,
    k^{-1}<k_*^{-1}$, we have that

    $$\begin{aligned}
    a\sim (\tau+\tau_B)^{q_1}\sim(t+t_B)^{p_1},\quad v_2 =
    c_{21}(\tau+\tau_B)^{1-q_1} + c_{22} (\tau+\tau_B)^{q_1},\end{aligned}$$

-   Region III (intermediate), where $-\Delta<\tau<\Delta\,\,\, {\rm
      for}\,\,\, k^{-1}>k_*^{-1}, \,\,\, {\rm and}\,\,\,
    -\tau_H<\tau<\tau_H \,\,\, {\rm for}\,\,\, k^{-1}<k_*^{-1}$, we have that

    $$\begin{aligned}
    a\sim {\rm constant\,\,\,in\,\,\,-\Delta<\tau<\Delta} , \quad v_3 =
    c_{31} e^{i k \tau} + c_{32} e^{-i k \tau},\end{aligned}$$

-   Region IV (contracting), where $\Delta<\tau<\tau_B\,\,\, {\rm
      for}\,\,\, k^{-1}>k_*^{-1},   \,\,\, {\rm and}\,\,\,
    \tau_H<\tau<\tau_B \,\,\, {\rm for}\,\,\, k^{-1}<k_*^{-1}$, we have that

    $$\begin{aligned}
    a \sim  (-\tau+\tau_B)^{q_1} \sim (-t+t_B)^{p_1},\quad v_4 = c_{41}
    (-\tau+\tau_B)^{1-q_1} +c_{42} (-\tau+\tau_B)^{q_1},\end{aligned}$$

-   Region V (expanding), where $\tau>\tau_B$, we have that

    $$\begin{aligned}
    a\sim  (\tau-\tau_B)^{q_2} \sim (t-t_B)^{p_2},\quad v_5 = c_{51} (\tau
    - \tau_B)^{1-q_2} 