The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

In this paper, we study cosmological solutions to the Einstein-Euler equations. We frst establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fxed linearly expanding cosmological spacetimes with a linear equation of state p = Kρ for the parameter values K (0,1/3). This removes the restriction to irrotational perturbations in earlier work [15] and relies on a novel transformation of the fuid variables that is well-adapted to Fuchsian methods. We then apply this new transformation to show the global regularity and stability of the Milne spacetime under the coupled Einstein-Euler equations, again with a linear equation of state p = Kρ, K (0,1/3). Our proof requires a correction mechanism to account for the spatially curved geometry. In total, this is indicative that structure formation in cosmological fuid-flled spacetimes requires an epoch of decelerated expansion.