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. For systems with accurately measured positions and time-delays, the most important uncertainty is related to the mass model used. Simple parametric models like isothermal ellipsoidal mass distributions seem to provide consistent results with a reasonably small scatter when applied to several lens systems. We discuss a family of models with a separable radial power law and an arbitrary angular dependence for the potential ψ = rβF(θ). Isothermal potentials are a special case of these models with β= 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 ∝ (2 - β)/β. The effect of external shear is quantified by introducing a 'critical shear' γc as a measure for the amount of shear that changes the result significantly. The analysis shows that in the general case H0 and γc do not depend on the position of the lens galaxy. Spherical lens models with images close to the Einstein radius with fitted external shear differ by a factor of β/2 from shearless models, leading to H0 ∝ 2 - β in this case. We discuss these results and compare them with numerical models for a number of real lens systems.