Semi-Analytical Models for Lensing by Dark Halos: I. Splitting Angles
  We use the semi-analytical approach to analyze gravitational lensing of quasars by dark halos in various cold dark matter (CDM) cosmologies, in order to determine the sensitivity of the prediction probabilities of images separations to the input assumptions regarding halos and cosmologies. The mass function of dark halos is assumed to be given by the Press-Schechter function. The mass density profile of dark halos is alternatively taken to be the singular isothermal sphere (SIS), the Navarro-Frenk-White (NFW) profile, or the generalized NFW profile. The cosmologies include: the Einstein-de Sitter model (SCDM), the open model (OCDM), and the flat \Lambda-model (LCDM). As expected, we find that the lensing probability is extremely sensitive to the mass density profile of dark halos, and somewhat less so to the mean mass density in the universe, and the amplitude of primordial fluctuations. NFW halos are very much less effective in producing multiple images than SIS halos. However, none of these models can completely explain the current observations: the SIS models predict too many large splitting lenses, while the NFW models predict too few small splitting lenses. This indicates that there must be at least two populations of halos in the universe. A combination of SIS and NFW halos can reasonably reproduce the current observations if we choose the mass for the transition from SIS to NFW to be ~ 10^{13} solar masses. Additionally, there is a tendency for CDM models to have too much power on small scales, i.e. too much mass concentration; and it appears that the cures proposed for other apparent difficulties of CDM would help here as well, an example being the warm dark matter (WDM) variant which is shown to produce large splitting lenses fewer than the corresponding CDM model by one order of magnitude. 
Li-Xin Li, Jeremiah P. Ostriker (Princeton University)