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

Abstract

We study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 𝕋3. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state [mathematical equation]. The dust-Einstein system is the case c[subscript s] = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases [mathematical equation] which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions.