Reflection Symmetry of Cusps in Gravitational Lensing

Criticality in graviational microlensing is an everyday issue because that is what generates microlensing signals which may be of photon-challenged compact objects such as black holes or planetary systems ET calls home. The criticality of these quasi-analytic lenses is intrinsically quadratic, and the critical curve behaves as a mirror generating two mirror images along the image line (‖ ±E−) at the same distances from the critical curve in the opposite sides. At the (pre)cusps where the caustic curve “reflects” and develops cusps, however, “would-be” two pairs of quadratic images “superpose” to produce three mirror images because of the degenerate criticality. The critical curve behaves as a parabolic mirror, and the image inside the parabola is indeed a superposed image having the sum of the magnifications of the other two that are outside the parabola. All three images lie on a parabolic image curve shaped by a property (∇J) of the (pre)cusp, and two distances of the system determine the position of the image curve and positions of the three images on the image curve. The triplet images satisfy ∑ J = 0, and the function ∑ J is discontinuous at a caustic crossing where a pair of quadratic images disappear into a critical point. The reflection symmetry of the image curve is a manifestation of the symmetry of the cusp which is also respected by a trio of parabolic curves that are tangnet at the (pre)cusp and define the image domains. The symmetry is guaranteed when J+− vanishes or can be ignored, and the cusps on the lens axis of the binary lenses are strongly symmetric having J+− = 0 because of the global reflection symmetry of the binary lenses. “E±-algebra” is laid out for users’ convenience. Subject headings: gravitational lensing binary stars, planets 0. “ Forget About This Universe” Q: Quazi-What lenses? Too special! What do you think is the probability that one of the baby universes will evolve into one just like this? Measure zero. A: Well, sir, one resolution is to preach a paradox. The other is to admit that you have not found the proper measure yet. – “Modo de Notre Dame”