  We consider near-FLRW solutions to the Einstein-massless-scalar field system
with a positive cosmological constant. We establish global existence in the
future direction, for all sufficiently regular initial data, and derive their
precise asymptotic behavior at infinity. As a corollary, we infer that, unlike
the FLRW background, the perturbed solutions do not describe a regular
irrotational stiff fluid with linear equation of state $p=\rho$, for general
asymptotic data at infinity. Our method of proof is inspired by recent works on
stable Big Bang formation and our results hold for open sets of initial data in
Sobolev spaces without symmetries.
