Semi-Analytical Models for Lensing by Dark Halos: I. Splitting Angles

In this paper we use the semi-analytical approach to analyze gravitational lensing of remote objects (quasars) by dark halos in various cosmologies in order to determine the sensitivity of the predictions for probabilities of splitting-angles of multiple images to the input assumptions regarding halo properties and cosmological model. All models are variants of the cold dark matter scenario and all fail in that they predict too many occurrences of large splitting lenses. 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, the NFW (Navarro-Frenk-White) density profile, and the generalized NFW density profile which has a different slope at small radii. The cosmologies being considered include: the Einstein-de Sitter model (i.e. the standard cold dark matter model), which is a flat universe without a cosmological constant; the open model, which is a universe with negative spatial curvature but without a cosmological constant; and the Λ-model, which is a flat universe with a positive cosmological constant. As expected, we find that, the lensing probability is very sensitive to the mass density profile of the lenses (dark halos), the mass density of matter in the universe, and the amplitude of primordial fluctuations. The more concentrated the density distribution is, the higher the lensing probability it produces. The bigger the mass density in the universe and the amplitude of the primordial fluctuations are, the higher the lensing probability is. With suitable normalization of the amplitude of the primordial fluctuations, we find that the Ωm = 1 standard cold dark matter model produces the least lensing, smaller than that for the open model and the Λ model by two orders of magnitudes. But the difference in lensing probability in the open model and the Λ-model is not big, due to the fact that the mass density of matter and the