the result given by Eq. (\[finalPzeta\]).

Limiting case of Instantaneous matching
---------------------------------------

We first consider the limit as the duration of the plateau region of $a(t)$ goes to zero, corresponding to what we have denoted by Model 2 in Sec. \[sec2\]. This is the limit $\Delta\rightarrow 0$. In this case, large scale modes $k^{-1} > k_*^{-1}$ do not enter the Hubble radius in the region near $t = 0$, and we can set $\Delta = 0$ in the matching condition equations, i.e.,

$$\begin{aligned}
\sin(2k\Delta) \rightarrow 0, \quad \cos(2 k \Delta) \rightarrow 1 \,
.\end{aligned}$$

On the other hand, small scale modes $k^{-1} < k_*^{-1}$ will enter the Hubble radius at a time given by $-\tau_H(k)$, and in the matching condition equations we must replace $\Delta$ by $\tau_H(k)$.

### Large scale modes $k^{-1} > k_*^{-1}$

Let us first consider the case for large scale modes $k^{-1} >
k_*^{-1}$. In this case we have that $$\begin{aligned}
a_{11}\rightarrow -2k, \quad a_{12}\rightarrow -4k q_1\delta^{2q_1-1},
\quad a_{21}\rightarrow -4k (q_1-1)\delta^{-2q_1+1}.\end{aligned}$$ Because we are interested in the parameter region $1/3<p<1$, then, written in terms of $q$, we have $q>1/2$. So the $c_{51}$ mode in the expression for the power spectrum Eq. (\[finalPzeta\]) is a decaying solution. Hence, 