the details of the bounce phase, we here consider a toy model where the scale factor undergoes small amplitude oscillations during the bounce phase. Such a behavior may emerge from certain models motivated by ideas from loop quantum gravity [@Alesci]. Heuristically, one would argue that those small bounces will not influence the large scale modes provided that the wavelengths of these modes are so large scale that they would not feel the small scale fluctuations of the scale factor. On the other hand, smaller scale modes whose wavelength is comparable or smaller to the total duration of the bounce phase should be sensitive to the details of the dynamics during the bounce. In this work, we would like to give a careful treatment to see if this is really the case.

In the case of a cyclic cosmology, when the time interval between cycles is larger than the wavelength of the modes being considered, it is generally sufficient [@RHBcyclic] to consider only the dominant modes in each phase (except during the bounce phase and when mode matching conditions are applied [@Durrer]). In our case however, the time scale between the small bounces is very small compared to the length scales 