The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant

Abstract

ABSTRACT. We study small perturbations of the well-known family of Friedman-Lemaître-RobertsonWalker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the spacelike Cauchy hypersurfaces are diffeomorphic to T 3 . These solutions model a quiet pressureless fluid in a dynamic spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our analysis takes place relative to a harmonic-type coordinate system, in which the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends the results of [40,46,32], where analogous results were proved for the Euler-Einstein system under the equations of state p = c 2 s ρ, 0 < c 2 s ≤ 1/3. The dust-Einstein system is the Euler-Einstein system with c s = 0. The main difficulty that we overcome is that the energy density of the dust loses one degree of differentiability compared to the cases 0 < c 2 s ≤ 1/3. Because the dustEinstein equations are coupled, this loss of differentiability introduces new obstacles for deriving estimates for the top-order derivatives of all solution variables. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive a collection of elliptic estimates that complement the energy estimates of [40,46]. An important feature of our analysis is that we are able to close our estimates even though the top-order derivatives of all solution variables can grow much more rapidly than in the cases 0 < c 2 s ≤ 1/3. Our results apply in particular to small compact perturbations of the vanishing dust state.

Newtonian Limits of Isolated Cosmological Systems on Long Time Scales

“…The future nonlinear stability of the FLRW fluid solutions for a linear equation of state p = Kρ was first established under the condition 0 < K < 1/3 and the assumption of zero fluid vorticity by Rodnianski and Speck [58] using a generalization of a wave-based method developed by Ringström in [56]. Subsequently, it has been shown [19,24,40,61] that this future nonlinear stability result remains true for fluids with nonzero vorticity and also for the equation of state parameter values K = 0 and K = 1/3, which correspond to dust and pure radiation, respectively. It is worth noting that the stability results established in [40] and [19] for K = 1/3 and K = 0, respectively, rely on Friedrich's conformal method [17,18], which is completely different from the techniques used in [24,58,61] for the parameter values 0 ≤ K < 1/3.…”

The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

“…The Commuted Equations. As discussed in the introduction, the Ricci coefficients and curvature components will be estimated in L 2 using the null structure and Bianchi equations respectively 19 . In order to deal with the nonlinearities some of the error terms are estimated in L ∞ on the spheres.…”

“…There are more global stability results for the Einstein equations with a positive cosmological constant, for example the works of Friedrich [17], Ringström [30] and Rodnianski-Speck [31]. A more comprehensive list can be found in the introduction to the work of Hadzić-Speck [19]. the particles have zero mass and hence travel through spacetime along null curves, the decay properties of the function describing the matter are compatible in a nice way with those of the spacetime metric.…”

Nonlinear Stability of the Milne Model with Matter

“…Further stability results for cosmological spacetimes with matter models exist but to our knowledge consider the case of a positive cosmological constant. We refer here to the works of Rodnianski-Speck and Speck on the Einstein-Euler system [RS13,S12], Hadžić-Speck on the Einstein-dust system [HS15], Friedrich on the Einstein-dust system [Fr17] and Olyniyk on the Einstein-fluid system [Ol16].…”