To understand to which extent cosmological models make sense as an approximation of our universe, it is of particular interest to study their stability within the Einstein equations. In this talk, I will present a recent result based on joint work with David Fajman, in which we prove that FLRW spacetimes with negative spatial sectional curvature and (non-trivial) spatially homogeneous scalar field are nonlinearly stable within the Einstein scalar-field system. This also verifies the Strong Cosmic Censorship conjecture (in a C^2-sense) near these FLRW spacetimes in both the collapsing and the expanding direction, which we analyse separately: Toward the Big Bang, one observes stable curvature blow-up that drives geodesic incompleteness. Crucially, this is shown with a covariant approach using Bel-Robinson variables that is largely independent of spatial geometry and may prove to be very robust when translated to other settings. Toward the future, near-FLRW solutions are future complete and asymptotically approach the vacuum Milne solution. This boils down to proving future stability of the Milne solution itself, which we then establish independently.
