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We prove nonlinear Lyapunov stability of a family of `$n+1$'-dimensionalcosmological models of general relativity locally isometric to the FriedmanLema\^itre Robertson Walker (FLRW) spacetimes including a positive cosmologicalconstant. In particular, we show that the perturbed solutions to theEinstein-Euler field equations around a class of spatially compact FLRW metrics(for which the spatial slices are compact negative Einstein spaces in generaland hyperbolic for the physically relevant $n=3$ case) arising from regularCauchy data remain uniformly bounded and decay to a family of metrics withconstant negative spatial scalar curvature. To accomplish this result, weemploy an energy method for the coupled Einstein-Euler field equations inconstant mean extrinsic curvature spatial harmonic gauge (CMCSH). In order tohandle Euler's equations, we construct energy from a current that is similar tothe one derived by Christodoulou \cite{christodoulou} (and which coincides withChristodoulou's current on the Minkowski space) and show that this energycontrols the desired norm of the fluid degrees of freedom. The use of a fluidenergy current together with the CMCSH gauge condition casts the Einstein-Eulerfield equations into a coupled elliptic-hyperbolic system. Utilizing theestimates derived from the elliptic equations, we first show that thegravity-fluid energy functional remains uniformly bounded in the expandingdirection. Using this uniform boundedness property, we later obtain sharp decayestimates if a positive cosmological constant $\Lambda$ is included, whichsuggests that the accelerated expansion of the physical universe that isinduced by the positive cosmological constant is sufficient to control thenon-linearities in the case of small data. A few physical consequences of thisstability result are discussed.

The nonlinear stability of n+1 dimensional FLRW spacetimes