The linear stability of the Einstein-Euler system on negative Einstein spaces

Here we prove the linear stability of a family of `$n+1$'-dimensional Friedmann Lema\^{i}tre Robertson Walker (FLRW) cosmological models of general relativity. 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. Utilizing a Hodge decomposition of the fluid's $n-$velocity 1-form, the linearized Einstein-Euler 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 a Lyapunov functional (controlling appropriate norm of the data) in the expanding direction. Utilizing the uniform boundedness, we later obtain a sharp decay estimate which suggests that expansion 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 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 $\gamma_{a}$ ($(1,\frac{n+1}{n})$ i.e., $(1,\frac{4}{3})$ in the physically relevant `$3+1$' universe) if the equation of state $p=(\gamma_{a}-1)\rho$ is chosen.