Setting limits on q0 from gravitational lensing

In this paper we consider gravitational lensing by galaxies in a wide variety of cosmological models. We model galaxies (with their heavy halos) as singular isothermal spheres. For closed models we find that the lensing depends on the parameter β_crit_ = 4π(σ_parallel_/c)^2^sin (X_Q_ - X_L_)/sin X_Q_, where σ_parallel_ is the line-of-sight velocity dispersion in the lensing galaxy and X_Q_ and X_L_ are the comoving distances to the QSO and lensing galaxy, respectively. If β_crit_ > 0, then we can obtain a normal lensing case with two bright images separated by {DELTA}θ = 2β_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 (X_Q) approaches π), which can occur in models with large values of the cosmological constant {LAMBDA}, the cross sections for lensing blow up. For a QSO beyond the antipodal redshift (X_Q_ > π) we can obtain an overfocused case where β_crit_ < 0. 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, as appears to be the case, the overfocused case always produces one image. This remains true even if the galaxy has an elliptical potential or if there is an additional quadrupole tidal part without excessive shear (γ < 1, where γ is dimensionless tidal shear). If a galaxy has a rotation curve which falls rapidly enough, the overfocused case can in principle produce three images which all lie to one side of the galaxy in the sky. So if we observe a normal lensed QSO consisting of a bright double with the lensing galaxy located roughly between them, we are looking at a normally lensed case where sin(X_Q_ - X_L_)/sin X_Q_ > 0. The existence of the apparently normally lensed case QSO 2016 with z_Q_ = 3.27 and z_L_ = 1.01 is difficult to understand if the antipodal redshift z_p_ is less than 3.27 (which would make X_Q_ > π). Indeed, the seven observed lensing cases taken together, as well as the fact that no large separation galactic lenses have been found, suggest that z_p_ > 3.27, thus implying that q_0_ > -2.3.