We study the general relativistic spacetime metrics surrounding massive cosmological objects, such as suns, stars, galaxies or galaxy clusters. The question addressed here is the transition of local, object-related spacetime metrics into the global, cosmological Robertson-Walker metrics. We demonstrate that the answer often quoted for this problem from the literature, the so-called Einstein-Straus vacuole, which connects a static outer Schwarzschild solution with the time-dependent Robertson-Walker universe, is inadequate to describe the local spacetime of a gravitationally bound system. Thus, we derive here an alternative model describing such bound systems by a metrics more closely tied to the fundamental problem of structure formation in the early universe and obtain a multitude of solutions characterising the time-dependence of a local scale parameter. As we can show, a specific solution out of this multitude is able to, as a by-product, surprisingly enough, explain the presently much discussed phenomenon of the PIONEER anomaly.