Nonperturbative Phenomenons of the Very Early Universe: Resonances in Primordial Fluctuations and Non-Gaussian Tails

In the era of precision cosmology, multi-messenger, high-resolution as well as small-scale cosmological observations have provided profound insights into the early universe, while also presenting new challenges to cosmological perturbation theory based on linear approximation and perturbative expansions. In recent years, the search for primordial gravitational waves and primordial black holes has generated significant research interest in the non-linear and non-perturbative processes in the early universe on small scales. This review paper summarizes the progress in understanding the generation of primordial black holes and gravitational waves at cosmological small scales. Specifically, we focus on the application of the resonance effect depicted by Mathieu equation to study small-scale power spectrum amplification and the induction of observable primordial gravitational waves. Furthermore, we attempt to explore the influence of non-Gaussian tails on primordial black hole formation. Our findings indicate that the resonance effect of the Mathieu equation provides an effective approach to describe non-perturbative and non-linear processes on small scales in the early universe, leading to a better understanding of the mechanisms behind primordial black hole formation and gravitational wave generation. Additionally, the non-perturbative non-Gaussianity may have a significant impact on the formation of primordial black holes.