Measuring the three-dimensional shear from simulation data, with applications to weak gravitational lensing

We have developed a new three-dimensional algorithm, based on the standard (PM)-M-3 method, for computing deflections resulting from weak gravitationa l lensing. We compare the results of this method with those of the two-dime nsional planar approach, and rigorously outline the conditions under which the two approaches are equivalent. Our new algorithm uses a Fast Fourier Tr ansform convolution method for speed, and has a variable softening feature to provide a realistic interpretation of the large-scale structure in a sim ulation. The output values of the code are compared with those from the Ewa ld summation method, which we describe and develop in detail. With an optim al choice of the high-frequency filtering in the Fourier convolution, the m aximum errors, when using only a single particle, are about 7 per cent, wit h an rms error less than 2 per cent. For ensembles of particles, used in ty pical N-body simulations, the rms errors are typically 0.3 per cent. We des cribe how the output from the algorithm can be used to generate distributio ns of magnification source ellipticity, shear and convergence for large-sca le structure.