Classification of the multiple deflection two point-mass gravitational lens models and application of catastrophe theory in lensing

We investigate the properties of a gravitational lens system consisting of two point masses, where the two lenses are allowed to be at different distances from the observer. Since even this simplest model for a multiple deflection lens system contains three essential parameters, corresponding to the transverse separation, the separation along the line-of-sight, and the mass ratio, a classification of the properties of this lens system with respect to the model parameters is essential. We obtain such a classification according to the topological properties of the critical curves and the caustics of the model. This is done by investigating the equation describing the critical curves. The transitions between different topologies of critical curves and caustics occur at those points in parameter space (bifurcation points) where the behaviour of solutions of this equation changes. For the case of a single lens plane, this method yields a closed expression for these bifurcation values as a function of transverse separation of the lenses and their mass ratio, whereas for the general case, a high order polynomial must be solved analytically. This explicit determination of the classification of lens models within the parameter space, which is obtained by methods from the theory of algebraic curves, is supplemented with methods and results from catastrophe theory. It turns out that all transitions between caustic topologies (metamorphoses) are either beak-to-beak singularities or elliptic umbilics. We argue that catastrophe theory provides an adequate tool to investigate the structure of parametrized lens models, and consider this paper as a first application of this much more general method. For the specific lens model under investigation, we present plots of critical curves and caustics together with their classification diagrams, and show examples of the mapping of extended sources by our lens. Differences between the properties of the multiple deflection lens system and that of a single lens plane situation are pointed out.