Degeneracies and scaling relations in general power-law models for gravitational lenses

Abstract

The time delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. The most important uncertainty is related to the mass model used. We discuss a family of models with a separable radial power-law and an arbitrary angular dependence for the potential psi = r^beta * F(theta). Isothermal potentials are a special case of these models with beta=1. An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H0 as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to H0 proportional to (2-beta)/beta. The effect of external shear is quantified by introducing a `critical shear' gamma_c as a measure for the amount of shear that changes the result significantly. The analysis shows, that in the general case H0 and gamma_c do not depend on the position of the lens galaxy. We discuss these results and compare with numerical models for a number of real lens systems.