An analytical approach to gravitational lensing by an ensemble of axisymmetric lenses

Abstract

We consider the problem of gravitational lensing by an ensemble of identical axisymmetric lenses randomly distributed on a single lens plane and derive a formal expression for the joint probability density of finding shear (γ1, γ2) and convergence κ at a random point on the plane. The amplification probability for a source can be accurately estimated from the distribution in shear and convergence. We apply this method to two cases: lensing by an ensemble of point masses and by an ensemble of objects with Gaussian surface mass density. There is no convergence for point masses whereas shear is negligible for wide Gaussian lenses. We find excellent agreement with results from recent Monte Carlo simulations with σav = 0.2.