Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry

Abstract

We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state $p=-\frac{A^2}{{\rho }^{\alpha }}$ relating the pressure to the mass-energy density, in which $A>0$ and $\alpha \in (0,1]$ are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the $(3+1)$-spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.