Setting limits on q0 from gravitational lensing
Gravitational lensing by galaxies in a wide variety of cosmological models is considered. For closed models, the lensing depends on the parameter beta(crit). If beta(crit) is greater than zero, a normal lensing case can be obtained with two bright images separated by an angle twice beta(crit) and a third, arbitrarily dim image between them coincident with the position of the lensing galaxy nucleus. As the QSO approaches the antipodal redshift, which can occur in models with large values of the cosmological constant, the cross sections for lensing blow up. An overfocused case where beta(crit) is less than zero can be obtained for a QSO beyond the antipodal redshift. In this case, when a lensing event occurs, only one arbitrarily dim image coincident with the position of the lensing galaxy nucleus is seen. If galaxy rotation curves are always flat or slowly rising, the overfocused case always produces one image.