[@RHBbounceReview]).

### Small scale modes $k^{-1} < k_*^{-1}$

For small scale modes $k^{-1}<k_{*}^{-1}$, we set $\delta$ equal to the Hubble crossing time. Thus, we can use the Hubble crossing condition $a H= k$, which from $a\sim \tau^q$ gives $$\begin{aligned}
\delta =q_1  k^{-1} .\end{aligned}$$ But we need to have $k \tau_H = k\tau_B - k\delta= k\tau_B - q_1 $. As a consequence of the oscillations in the coefficients $a_{ij}$, Eqs. (\[a11\]), (\[a12\]) and (\[a21\]), the final power spectrum of fluctuations will oscillate for small wavelengths. This is explicitly manifested when we show a numerical example for the power spectrum in Fig. \[bounceplot1\], where we chose an initial pre-bounce spectrum which is scale-invariant. We see that the scale-invariance of the spectrum is maintained on large scales, but that on small length scales there is both a change in the slope of the spectrum, and superimposed oscillations.

In the following we discuss in what range we can reproduce the results of Ref. [@RHBcyclic], which hold for a cyclic cosmology. In that work, it was found that for modes which re-enter the Hubble radius during the bounce phase, the index of the spectrum of cosmological perturbations changes during each cycle. For a matter-dominated contracting 