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 a s a(-2), where a is the scale factor of the universe, the fate of the unive rse in regard to whether it will collapse in the future or expand forever i s 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 ope n 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 a nd in another of which it does, are studied. It is shown that the vacuum eq uation of state p(vac) = -rho(vac) may be realized in a decaying vacuum cos mology provided the vacuum interacts with matter. The optical depths for gr avitational lensing as a function of the matter density and other parameter s in the models are calculated at a source redshift of 2. The age of the un iverse 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 ser ve as dark matter is suggested.