Fully nonlinear gravitational instabilities for expanding Newtonian universes with inhomogeneous pressure and entropy: Beyond the Tolman's solution

Nonlinear gravitational instability is a crucial way to comprehend the clustering of matter and the formation of nonlinear structures in both the Universe and stellar systems. However, with the exception of a few exact particular solutions for pressureless matter, there are only some approximations and numerical and phenomenological approaches to study the nonlinear gravitational instability instead of mathematically rigorous analysis. We construct a family of particular solutions of the Euler-Poisson system that exhibits the nonlinear gravitational instability of matter with inhomogeneous pressure and entropy (i.e., the cold center and hot rim) in the expanding Newtonian universe. Despite the density perturbations being homogeneous, the pressure is not, resulting in significant nonlinear effects. By making use of our prior work on nonlinear analysis of a class of differential equations \cite{Liu2022b}, we estimate that the growth rate of the density contrast is approximately $\sim \exp(t^{\frac{2}{3}})$, much faster than the growth rate anticipated by classical linear Jeans instability ($\sim t^{\frac{2}{3}}$). Our main motivation for constructing this family of solutions is to provide a family of reference solutions for conducting a fully nonlinear analysis of inhomogeneous perturbations of density contrast. We will present the general results in a mathematical article \cite{Liu2023b} separately. Additionally, we emphasize that our model does not feature any shell-crossing singularities before mass accretion singularities since we are specifically interested in analyzing the mathematical mechanics of a pure mass accretion model, which poses limitations on the applicability of our model for understanding the realistic nonlinear structure formation.