The nonlinear stability of n+1 dimensional FLRW spacetimes

Abstract

We prove nonlinear Lyapunov stability of a family of 'n + 1'-dimensional cosmological models of general relativity locally isometric to the Friedman Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein-Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant n = 3 case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein-Euler field equations in constant mean extrinsic curvature spatial harmonic gauge (CMCSH). In order to handle Euler's equations, we construct energy from a current that is similar to the one derived by Christodoulou [1] (and which coincides with Christodoulou's current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein-Euler field equations into a coupled elliptichyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant Λ is included, which suggests that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the non-linearities in the case of small data. A few physical consequences of this stability result are discussed.

Decay of solutions of the wave equation in cosmological spacetimes—a numerical analysis

“…We will restrict our analysis to the case Λ = 0, due to the existence of closed-form expressions for the solutions to the Friedmann equations. Recent developments on the problem can be found in [2,[5][6][7][8][9][10][11][12][13][14][15][16] and references therein, both for the case Λ = 0 and for a non-zero cosmological constant. It has been settled, in particular, that the decay of the energy associated to waves, caused by the redshift effect, is only one of the many factors that influence the propagation, together for instance with the non-compactness of the space sections and the dispersive properties of the background spacetime.…”

The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

“…In the context of Einstein-Euler, stability of homogeneous solutions on exponentially expanding cosmological spacetimes was first studied in [32], with several later works [17,18,21,22,23,24,26,28,33]. For various other results on fluid stabilization in the regime of accelerated expansion, we refer to [20,25,37]. We mention also earlier work [31] concerning the stability of solutions to the Einstein equations undergoing accelerated expansion.…”