Systematic uncertainties in gravitational lensing models: a semi-analytical study of PG1115+080

While the Hubble constant can be derived from observable time delays between images of lensed quasars, the result is often highly sensitive to assumptions and systematic uncertainties in the lensing model. Unlike most previous authors, we put minimal restrictions on the radial profile of the lens and allow for non-elliptical lens potentials. We explore these effects using a broad class of models with a lens potential psi [rf(theta)], which has an unrestricted radial profile but self-similar iso-potential contours defined by rf(theta) = constant. For these potentials, the lens equations can be solved semi-analytically. The axis ratio and position angle of the lens can be determined from the image positions of quadruple gravitational lensed systems directly, independent of the radial profile. We give simple equations for estimating the power-law slope of the lens density directly from the image positions and for estimating the time delay ratios. Our method greatly simplifies the numerics for fitting observations and is fast in exploring the model parameter space. As an illustration, we apply the model to PG1115+080. An entire one-parameter sequence of models fits the observations exactly. We show that the measured image positions and time delays do not uniquely determine the Hubble constant.