2020/10/28 by Puskar Mondal, Mondal, Puskar
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2010.15108
openalex publication_date 2020/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Here we prove the linear stability of a family of `n+1'-dimensional Friedmann Lemaître 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 γa ((1,(n+1)/(n)) i.e., (1,(4)/(3)) in the physically relevant `3+1' universe) if the equation of state p=(γa-1)ρ is chosen.