Fate of the Universe, Age of the Universe, Dark Matter, and the Decaying Vacuum Energy

It is shown that in the cosmological models based on a vacuum energy decayingas a^{-2}, where a is the scale factor of the universe, the fate of theuniverse in regard to whether it will collapse in future or expand forever isdetermined not by the curvature constant k but by an effective curvatureconstant k_{eff}. It is argued that a closed universe with k=1 may expandforever, in other words simulate the expansion dynamics of a flat or an openuniverse because of the possibility that k_{eff}=0 or -1, respectively. Twosuch models, in one of which the vacuum does not interact with matter and inanother of which it does, are studied. It is shown that the vacuum equation ofstate p_{vac}= -\rho_{vac} may be realized in a decaying vacuum cosmologyprovided the vacuum interacts wuth matter. The optical depths for gravitationallensing as a function of the matter density and other parameters in the modelsare calculated at a source redshift of 2. The age of the universe is discussedand shown to be compatible with the new Hipparcos lower limit of 11Gyr. Thepossibility that a time-varying vacuum energy may serve as dark matter issuggested.Comment: AAS LaTex, 29 pages, published in the Astrophysical Journal, 520, 45, 199