Madhura Killedar & Geraint Lewis

1. Statistical Lens Properties The large-scale distribution of mass, mostly dark matter, has associated lens properties: the strong lensing cross-section, the magnification probability distribution (MPD) and the shear probability distribution (SPD). Different cosmological models predict different matter distributions and their lens properties. These properties can be observed with weak lensing and supernovae Ia surveys. Comparisons between predictions and observations, are used to place constraints on the relevant cosmological parameters such as the matter density Ω0, the cosmological constant ΩΛ & the linear amplitude of mass fluctuations σ8. Backward Ray Tracing N-body simulations determine the non-linear evolution of large-scale matter distribution for a given cosmological model. By ray-tracing through these we can predict the lens properties of large-scale structure.The ray-shooting method1,2,3 sends light rays backwards from the observer; they travel through the lensing mass and are then mapped to the source plane. The more rays that land in a particular region of the source plane, the greater the magnification of a source if positioned there. The ray-bundle method4 sends rays in infinitesimal polygonal bundles; these maintain their association back to the source redshift. The magnification of a bundle is given by the ratio of its areas in the image and source planes. Both methods require that the 3D mass distribution is projected onto a series of lens planes. Light rays travel straight until they reach a lens plane; here they are deflected. That the net deflection due to masses projected onto a plane is equivalent to that due to their 3D distribution is the thin lens approximation. What’s new here? The accuracy of the thin lens approximation has been assessed through convergence of statistics by increasing the number of lens planes5. For this work, the 3D mass distributions need not be projected onto a lens plane → the thin lens approximation is not invoked. We directly solve the null geodesic equation by numerically integrating through a series of N-body simulations which are stacked end-to-end from z=0 to an initial source redshift. Integration across an expanding weak-field metric requires the gradients Ø(x) of the gravitational potential. These are computed at each grid point with the aid of 3D Fast Fourier Transforms, and the values at each point along the ray are found with a trilinear interpolation scheme. The scale factor a(t) is allowed to vary throughout the depth of each simulation box. Magnification, shear & convergence will be calculated as in the ray-bundle method. The resulting lens properties (MPD, SPD etc.) will be compared to those computed by the standard methods, and we can also compare the lens properties predicted by different cosmological models. Why quantify the thin lens approximation? When there exists large-scale structure along the line-of-sight such as filaments, their projection results in a 2D lens with a relatively high density; the implied deflection and shear is higher than the true value. The importance of multiple lenses along the line of sight (each with subcritical density, but together contributing to image splitting) has been recognised6. Furthermore, when there is no dominant lens, matter distributed along the line of sight would play a significant role. The thin lens approximation has been seen to produce a ‘bump’ in the MPD7 that is not seen when lensing through 3D mass distributions. Thus, we can expect errors in the strong lensing cross-section and the MPD (lower magnification events are interpreted as higher magnification events), especially for high redshift sources. Quantifying the Thin-Lens Approximation We are investigating the thin lens approximation when considering light propagation through 3D matter distributions. We will make statistical comparisons to standard multiple-lens-plane approaches. Multiple lens planes: Previous methods send light rays from the observer (left) to a source plane (right) traversing many lens planes (three shown here). The rays travel straight until they reach a lens plane, where they are deflected. Illustration Credit: Wambsganss, Cen & Ostriker 1998, ApJ. 494, 29 Madhura Killedar & Geraint Lewis Institute of Astronomy, School of Physics, The University Of Sydney m.killedar@physics.usyd.edu.au Our new model does not invoke the thin lens approximation. Light propagates through large-scale 3D structure (stacked N-body simulation boxes, one shown here) until it reaches the source plane. Circular images become sheared, rotated and (de)magnified. ∆ Illustration Credit: hetdex.org/dark_energy/how_find_it/gravitational_lensing.php (modified by author) References • Schneider P. & Weiss A., 1988, ApJ, 330,1 • Jaroszyński M., Park C., Paczyński B. & Gott J. R., 1990, ApJ, 365, 22 • Wambsganss J., 1990, Ph.D. Thesis, Munich University, MPA Rep. #550 • Fluke C. J., Webster R. L., & Mortlock D. J., 1999, MNRAS, 306, 567 • Wambsganss J., Cen R. & Ostriker J. P., 1998, ApJ, 494, 29 • Wambsganss J., Bode P. & Ostriker J. P., 2005, ApJ, 635, 1 • Rauch K.P., Mao S., Wambsganss J., Paczyński B., 1992, ApJ, 386, 30 Background Image: Abell 2218 – Andrew Fruchter (STScI) et al., WFPC2, HST, NASA