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 cold dark matter cosmology, using an algorithm that evaluates the shear in three dimensions. The algorithm has the advantage of variable softening for the particles, and our method allows the appropriate angular diameter distances to be applied to every evaluation location within each three-dimensional simulation box. We investigate the importance of shear in the distance-redshift relation, and find it to be very small. We also establish clearly defined values for the smoothness parameter in the relation, finding its value 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, we are able to observe the formation of structure in terms of the computed shear, and also note that the major contributions to the shear come from a very broad range of redshifts. We show the probability distributions for the magnification, source ellipticity and convergence, and also describe the relationships amongst these quantities for a range of source redshifts. We find a broad range of magnifications and ellipticities; for sources at a redshift 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 magnification is not linear in the convergence, as might be expected for weak lensing, but contains contributions from higher order terms in both the convergence and the shear. Our results for the one-point distribution functions are generally different from those obtained by other authors using two-dimensional (planar) approaches, and we suggest reasons for the differences. Our magnification distributions for sources at redshifts of 1 and 0.5 are also very different from the results used by other authors to assess the effect on the perceived value of the deceleration parameter, and we briefly address this question.