Applications of the Junction Conditions Connecting the Robertson--Walker Metric and the Metric of a Local System on Our Universe

We investigate how the local and global metrics are connected in an ideal model of spacetime where the local system is assumed to be highly symmetric and the cosmological matter is kept away from the local system and does not disturbed by the local system. A boundary condition arising from the junction conditions is obtained and its implication in our universe is studied. We know that the total mass of a sufficiently large system must be that of the cosmological matter within the region of that size. This requirement is satisfied since it is just a consequence of the boundary condition. The analysis shows that at the very late epoch of the universe, there exists a particular time when the largest symmetric local systems stop growing and the observation of this time can be used to check the cosmological parameters. Adopting the popular values (Ω_M ,Ω_Λ)=(0.28,0.72), we find that particular time would be associated with z=0.726, the effect of dark matter on the clustering of objects would be insignificant, and the Virgo cluster would be gravitationally bound even if dark matter is ignored.