Perturbations in Bouncing and Cyclic Models, a General Study

Being able to reliably track perturbations across bounces and turnarounds in cyclic and bouncing cosmology lies at the heart of being able to compare the predictions of these models with the Cosmic Microwave Background observations. This has been a challenging task due to the unknown nature of the physics involved during the bounce as well as the technical challenge of matching perturbations precisely between the expansion and contraction phases. In this paper, we will present general techniques (analytical and numerical) that can be applied to understand the physics of the fluctuations, especially those with "long" wavelengths, and test its validity in some simple bouncing/cyclic toy models where the physics is well understood. We will then apply our techniques to more interesting cosmological models such as the bounce inflation and cyclic inflation.