Bounds on number of cusps due to point mass gravitational lenses
Generic caustics in gravitational lensing occur locally either as folds or cusps. This paper rigorously proves that the total number of cusps, Ncusps, due to g point masses on a single plane having non-normalized external shear γ>0 and continuous matter with constant density σc, is bounded as follows: 0≤Ncusps≤12g2. For vanishing shear γ=0 we obtain the result 0≤Ncusps≤12g(g-1). Consequences of these bounds for the global geometry of caustics are discussed. It is also shown that if γ≥0 and σcis sufficiently large, then all cusps can be eliminated, that is, Ncusps=0. The paper also includes equations for calculating all the bi-caustics (i.e., curves yielding the positions of cusps during a one-parameter evolution) of a single point-mass lens with continuous matter and shear. The methods of the paper are based on a new approach to point-mass gravitational lensing using complex quantities and the theory of resultants. © 1996 American Institute of Physics.
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