Weak gravitational lensing in the standard cold dark matter model, using an algorithm for three-dimensional shear

We investigate the effects of weak gravitational lensing in the standard co ld dark matter cosmology, using an algorithm that evaluates the shear in th ree dimensions. The algorithm has the advantage of variable softening for t he particles, and our method allows the appropriate angular diameter distan ces to be applied to every evaluation location within each three-dimensiona l simulation box. We investigate the importance of shear in the distance-re dshift relation, and find it to be very small. We also establish clearly de fined values for the smoothness parameter in the relation, finding its valu e to be at least 0.83 at all redshifts in our simulations. From our results , obtained by linking the simulation boxes back to source redshifts of 4, w e are able to observe the formation of structure in terms of the computed s hear, and also note that the major contributions to the shear come from a v ery broad range of redshifts. We show the probability distributions for the magnification, source ellipticity and convergence, and also describe the r elationships amongst these quantities for a range of source redshifts. We f ind a broad range of magnifications and ellipticities; for sources at a red shift of 4, 97.5 per cent of all lines of sight show magnifications up to 1 .39 and ellipticities up to 0.23. There is clear evidence that the magnific ation is not linear in the convergence, as might be expected for weak lensi ng, but contains contributions from higher order terms in both the converge nce and the shear. Our results for the one-point distribution functions are generally different from those obtained by other authors using two-dimensi onal (planar) approaches, and we suggest reasons for the differences. Our m agnification distributions for sources at redshifts of 1 and 0.5 are also v ery different from the results used by other authors to assess the effect o n the perceived value of the deceleration parameter, and we briefly address this question.