  The Friedmann--Lema\^{\i}tre--Robertson--Walker (FLRW) solution to the
Einstein-scalar field system with spatial topology $\mathbb{S}^3$ models a
universe that emanates from a singular spacelike hypersurface (the Big Bang),
along which various spacetime curvature invariants blow up, only to re-collapse
in a symmetric fashion in the future (the Big Crunch). In this article, we give
a complete description of the maximal developments of perturbations of the FLRW
data at the chronological midpoint of its evolution. We show that the perturbed
solutions also exhibit curvature blowup along a pair of spacelike
hypersurfaces, signifying the stability of the Big Bang and the Big Crunch.
Moreover, we provide a sharp description of the asymptotic behavior of the
solution up to the singularities, showing in particular that various
time-rescaled solution variables converge to regular tensorfields on the
singular hypersurfaces that are close to the corresponding FLRW tensorfields.
Our proof crucially relies on $L^2$-type approximate monotonicity identities in
the spirit of the ones we used in our joint works with Rodnianski, in which we
proved similar results for nearly spatially flat solutions with spatial
topology $\mathbb{T}^3$. In the present article, we rely on new ingredients to
handle nearly round spatial metrics on $\mathbb{S}^3$, whose curvatures are
order-unity near the initial data hypersurface. In particular, our proof relies
on i) the construction of a globally defined spatial vectorfield frame adapted
to the symmetries of a round metric on $\mathbb{S}^3$; ii) estimates for the
Lie derivatives of various geometric quantities with respect to the elements of
the frame; and iii) sharp estimates for the asymptotic behavior of the FLRW
solution's scale factor near the singular hypersurfaces.
