Turbulence, Norm Inflation, and the Rendall Instability for the Einstein-Euler System with Positive Cosmological Constant

Abstract

We numerically study perturbations of spatially homogeneous and orthogonal solutions to the Einstein-Euler equations with positive cosmological constant and linear equation of state $p=Kρ$ for $K \in (\frac{1}{3},1)$. Recent numerical and analytic work has demonstrated that the fluid density gradient in these perturbed spacetimes can form steep gradients and blow up as future timelike infinity is approached, a phenomenon now known as the `Rendall instability'. In this article, we study the onset of this instability outside of one spatial dimension for the first time. Our results provide compelling evidence that the Rendall instability causes a forward turbulent cascade and $H^{s}$ norm inflation in the energy density of the fluid from arbitrarily small $U(1)$-symmetric perturbations. This is, to the best of our knowledge, the first example of small data turbulence for solutions of the field equations with a positive cosmological constant. Moreover, we find that this instability can drive structure formation in the late universe.