The linear stability of the n + 1 dimensional FLRW spacetimes

Abstract

Here we prove the linear stability of a family of ‘n + 1’-dimensional cosmological models of general relativity locally isometric to the Friedmann Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. We show that the solutions to the linearized Einstein–Euler field equations around a class of FLRW metrics with compact spatial topology (negative Einstein spaces and in particular hyperbolic for n = 3) arising from regular initial data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish the result, we express the Einstein–Euler system in constant mean extrinsic curvature spatial harmonic gauge and linearize about the chosen FLRW background. Utilizing a Hodge decomposition of the fluid’s n-velocity one-form, the linearized system becomes elliptic–hyperbolic (and non-autonomous) in the CMCSH gauge facilitating an application of an energy type argument. Utilizing the estimates derived from the associated elliptic equations, we first prove the uniform boundedness of an energy functional (controlling an appropriate norm of the data) in the expanding direction. Utilizing the uniform boundedness, we later obtain a sharp decay estimate which suggests accelerated expansion (for Λ > 0) of this particular Universe model may be sufficient to control the non-linearities (including possible shock formation) of the Einstein–Euler system in a potential future proof of the fully non-linear stability. In addition, the rotational and harmonic parts of the fluid’s n-velocity field only couple to the remaining degrees of freedom in higher orders, which once again indicates a straightforward extension of current analysis to the fully non-linear setting in the sufficiently small data limit. In addition, our results require a certain integrability condition on the expansion factor and a suitable range of the adiabatic index γ ad ( i.e. in the physically relevant ‘3 + 1’ Universe) if the barotropic equation of state p = (γ ad − 1)ρ is chosen.