Solving Einstein Field Equations in Observational Coordinates with Cosmological Data Functions: Spherically Symmetric Universes with Cosmological Constant

摘要

Extending the approach developed by Ara\'ujo and Stoeger [1] and improved inAra\'ujo {\it et al} [2], we have shown how to construct dust-filled $\Lambda\neq 0$ Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmological modelsfrom FLRW cosmological data on our past light cone. Apart from being ofinterest in its own right -- demonstrating how such data fully determines themodels -- it is also illustrated in the flat case how the more generalspherically symmetric (SS) Einstein field equations can be integrated inobservational coordinates with data fit to FLRW forms arrayed on our past lightcone, thus showing how such data determines a FLRW universe -- which is not{\it a priori} obvious. It is also shown how to integrate these exact SSequations, in cases where the data are not FLRW, and the space-time is notknown to be flat. It is essential for both flat and non-flat cases to have datagiving the maximum of the observer area (angular-diameter) distance, and theredshift $z_{max}$ at which that occurs. This enables the determination of thevacuum-energy density $\mu_{\Lambda}$, which would otherwise remainundetermined.