Theory of Lensing Invariants I : The Power - Law ModelsC

The multiple images of lensed quasars provide evidence on the mass distribution of the lensing galaxy. The lensing invariants are constructed from the positions of the images, their parities and their uxes. They depend only on the structure of the lensing potential. The simplest is the magniication invariant, which is the sum of the signed magniications of the images. Higher order connguration invariants are the sums of products of the signed magniications with positive or negative powers of the position coordinates of the images. We consider the case of the four and ve image systems produced by elliptical power-law galaxies with / (x 2 + y 2 q ?2) =2. This paper provides simple contour integrals for evaluating all their lensing invariants. For practical evaluation, this ooers considerable advantages over the algebraic methods used previously. The magniication invariant is exactly B = 2=(2 ?) for the special cases = 0; 1 and 4=3; for other values of , this remains an approximation, but an excellent one at small source ooset. Similarly, the sums of the rst and second powers of the image positions (or their reciprocals), when weighted with the signed magniications, are just proportional to the same powers of the source ooset, with a constant of proportionality B. To illustrate the power of the contour integral method, we calculate full expansions in the source ooset for all lensing invariants in the presence of arbitrary external shear. As an example, we use the elliptical power-law galaxies to t to the data on the four images of the Einstein Cross (G2237+030). The lensing invariants play a role by reducing the dimensionality of the parameter space in which the 2 minimisation proceeds with consequent gains in accuracy and speed.