Fate of the universe, age of the universe, dark matter, and the decaying vacuum energy

It is shown that in cosmological models based on a vacuum energy decaying as a(-2), where a is the scale factor of the universe, the fate of the universe in regard to whether it will collapse in the future or expand forever is determined not by the curvature constant k but by an effective curvature constant k(eff). It is argued that a closed universe with k = 1 may expand forever, in other words, simulate the expansion dynamics of a hat or an open universe because of the possibility that k(eff) = 0 or -1, respectively. Two such models, in one of which the vacuum does not interact with matter and in another of which it does, are studied. It is shown that the vacuum equation of state p(vac) = -rho(vac) may be realized in a decaying vacuum cosmology provided the vacuum interacts with matter. The optical depths for gravitational lensing as a function of the matter density and other parameters in the models are calculated at a source redshift of 2. The age of the universe is discussed and shown to be compatible with the new Hipparcos lower limit of 11 Gyr. The possibility that a time-varying vacuum energy may serve as dark matter is suggested.