easy to read off when various modes enter and exit the Hubble radius.

All scales re-enter and re-exit the Hubble radius several times since at the extrema of $a(t)$ the Hubble radius is infinite. We will treat the transitions at $-\tau_B$ and $\tau_B$ as instantaneous[^2]. More specifically, we will cut out a time interval $$-\tau_B - \epsilon < \tau < - \tau_B + \epsilon,$$ (with $\epsilon \ll \tau_B$) and correspondingly another time interval of the same length about $\tau_B$ and we will match the solutions between the neighboring phases making use of the matching conditions given in Refs. [@HV; @DM], which are the cosmological version of the Israel [@Israel] ones[^3]. Hence, the only Hubble re-entry which is important to us is the one which occurs between $-
\tau_B$ and $+ \tau_B$.

In the first model, given by the plots on the left shown in Fig. \[bounceplot\], there are two characteristic comoving length scales. The first is $k_*^{-1}$ which is defined as the length which re-enters the Hubble radius at time $- \Delta$. The second one, $k_{\Delta}^{-1}$, is the mode which undergoes one oscillation between $\tau = - \Delta$ and $\tau = + \Delta$ (we are assuming $k_{\Delta}^{-1} >  k_*^{-1}$ if 