Semianalytical Models for Lensing by Dark Halos. I. Splitting Angles

Abstract

We use the semianalytical approach to analyze gravitational lensing of remote quasars by foreground dark halos in various cold dark matter (CDM) cosmologies in order to determine the sensitivity of the predictions for probabilities of image separations to the input assumptions regarding the properties of halos and cosmological models. The power spectrum of primordial fluctuations is normalized by the cluster abundance constraints. 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 being considered include the Einstein-de Sitter model (SCDM), the open model (OCDM), and the flat Λ-model (LCDM). As expected, we find that the lensing probability is extremely sensitive to the mass density profile of lenses (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. For NFW lenses, the SCDM model produces fewer lensing events than the OCDM/LCDM models by 2 orders of magnitude. For SIS lenses, the SCDM model produces more lensing events with small splitting angles and fewer lensing events with large splitting angles than the OCDM/LCDM models, which is due to the fact that for large-mass halos, the Press-Schechter function is very sensitive to the amplitude of primordial fluctuations. In all cases the difference between the OCDM model and the LCDM model is not dramatic. None of these models are consistent with current observations: the SIS models predict too many large splitting lenses, while the NFW models predict too few small splitting lenses. Essentially, the observed high ratio of small splitting to large splitting lenses is not predicted correctly. This indicates that there must be at least two populations of halos in the universe: small-mass halos with a steep inner density slope and large-mass halos with a shallow inner density slope. 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 ~1013 M☉, as might plausibly occur because of baryonic cooling and contraction in lower mass systems. Additionally, there is a tendency for CDM models to have too much power on small scales, i.e., too much mass concentration. From our sensitivity studies it appears that the cures proposed for other apparent difficulties of CDM would help here as well, an example being the warm dark matter variant, which is shown to produce fewer large splitting lenses than the corresponding CDM model by 1 order of magnitude.