phase the change in the index $n_s$ of the power spectrum was determined to be $\Delta n_s = - 2$.

The results of Ref. [@RHBcyclic] are applicable when $k^{-1}<k_*^{-1}$, but for quite large scales such that $k^{-1}\rightarrow k_*^{-1}$. In this range we have $k\tau_H\rightarrow 0$. Using this in Eq. (\[powerspectrum\]), we obtain that $$\begin{aligned}
\label{powerspectrum_cyclic}
P_f \, = \,  \left[ A_1  + A_2 (k \epsilon/q_1)^{1-2q_1} +  A_3
  (k\epsilon/q_1)^{2q_1-1} \right]^2 P_i.\end{aligned}$$ Since $2q_1 - 1 > 0$ it is the second term in Eq. (\[powerspectrum\_cyclic\]) which dominates. Hence, we conclude that there is a change in the index of the power spectrum by $$\Delta n_s = 2-4 q_1= - 2 \frac{3 p_1 - 1}{1 - p_1} ,
\label{tiltns}$$ which coincides with the results of Ref. [@RHBcyclic]. This is as expected because the case studied in Ref. [@RHBcyclic] corresponds to a big bounce where $\delta$ is (cosmologically) large.

Case with a flat plateau
------------------------

In the case with a flat plateau and when $\Delta$ is very small, we have just one characteristic comoving mometum. However, when $\Delta$ is big, we have two key comoving momenta which are characterized by the mode which cross the Hubble radius at $\Delta$ and $\tau_B-
\Delta$, respectively. 