Gravitational Lensing and Dark Structures
  We examine whether a cosmologically significant distribution of dark galaxy group or cluster-sized objects can have an optical depth for multiple imaging of distant background sources which is comparable to that from known galaxies while at the same time producing angular splittings of the same order of magnitude. Our purpose is to explore whether such objects could realistically account for some of the observed lenses. Modeling such systems as isothermal spheres with core radii, and assuming a Schechter-type distribution function, we find that independent of the cosmology (open, flat matter dominated, or flat cosmological constant dominated) an allowed parameter range exists which is comparable in velocity dispersion to that for known compact groups of galaxies, although the preferred core radii are somewhat smaller than that normally assumed for compact groups. Dark cluster-sized objects, on the other hand, cannot reproduce the observed lensing characteristics. If the one known Dark cluster were a good representative of such a distribution, most such objects would not produce multiple images. We also present a result for the angular splitting due to an isothermal sphere lens with non-zero core radius, extending earlier work of Hinshaw and Krauss (1987). Our results are expressed as contour plots for fixed lensing probabilities, and angular splittings. 
Yu-Chung N. Cheng and Lawrence M. Krauss (Case Western Reserve Univ.)