  We consider a system representing self-gravitating balls of dust in an
expanding Universe. It is demonstrated that one can prescribe data for such a
system at infinity and evolve it backward in time without the development of
shocks or singularities. The resulting solution to the Einstein-lambda-dust
equations exists for an infinite amount of time in the asymptotic region of the
spacetime. Furthermore, we find that if the density is small compared to the
Cosmological constant, then it is possible to construct Cosmological solutions
to the Einstein constraint equations on a standard Cauchy hypersurface
representing self-gravitating balls of dust. If, in addition, the density is
assumed to be sufficiently small, then this initial data gives rise to a future
geodesically complete solution to the Einstein-lambda-dust equations admitting
a smooth conformal extension at infinity which can be regarded as a
perturbation of de Sitter spacetime. The main technical tool in this analysis
are Friedrichs conformal Einstein field equations for the Einstein-lambda-dust
written in terms of a gauge in which the flow lines of the dust are recast as
conformal geodesics.
