Fully nonlinear Jeans instabilities for expanding Newtonian universes under homogeneous and isotropic perturbations

Based on mathematically rigorous analysis of nonlinear differential equations studied in our companion article [1], we construct a model which describes the \textit{nonlinear} gravitational instability on a local portion of the universe characterized by the expanding Newtonian universe. In this portion, the perturbations are homogeneous and isotropic. This result, to some extent, can be viewed as a nonlinear version of the Jeans instability. The growth rate of the relative density due to the nonlinear effects is much faster (at least $\sim \exp(t^{\frac{2}{3}})$ or blowup at a finite time according to the data) than the one predicted by the classical linear version of the Jeans instability ($\sim t^{\frac{2}{3}}$), and it leads to a better, or potentially substantial impacts on, understanding of the formation of the nonlinear structures in the universe and stellar systems. This article associated with [1] provides a new window into the rigorously mathematical and robust method, instead of the most used approximations and numerical calculations, of the fully nonlinear analysis of the Jeans instability for general cases.Comment: 15 page