Gravitational lensing magnification without multiple imaging
  Gravitational lensing can amplify the apparent brightness of distant sources. Images that are highly magnified are often part of multiply-imaged systems, but we consider the possibility of having large magnifications without additional detectable images. In rare but non-negligible situations, lensing can produce a singly highly magnified image; this phenomenon is mainly associated with massive cluster-scale halos (>~1e13.5 Msun). Alternatively, lensing can produce multiply-imaged systems in which the extra images are either unresolved or too faint to be detectable. This phenomenon is dominated by galaxies and lower-mass halos (<~1e12 Msun), and is very sensitive to the inner density profile of the halos. Although we study the general problem, we customize our calculations to four quasars at redshift z~6 in the Sloan Digital Sky Survey (SDSS), for which Richards et al. (2004) have ruled out the presence of extra images down to an image splitting of 0.3" and a flux ratio of f=0.01. We predict that 9-29% of all z~6 quasars that are magnified by a factor of mu>10 would lack detectable extra images, with 5-10% being true singly-imaged systems. The maximum of 29% is reached only in the unlikely event that all low-mass (<~1e10 Msun) halos have highly concentrated (isothermal) profiles. In more realistic models where dwarf halos have flatter (NFW) inner profiles, the maximum probability is ~10%. We conclude that the probability that all four SDSS quasars are magnified by a factor of 10 is <~1e-4. The only escape from this conclusion is if there are many (>10) multiply-imaged z~6 quasars in the SDSS database that have not yet been identified, which seems unlikely. In other words, lensing cannot explain the brightnesses of the z~6 quasars, and models that invoke lensing to avoid having billion-Msun black holes in the young universe are not viable. 
Charles R. Keeton (Chicago/Rutgers), Michael Kuhlen (Santa Cruz),   Zoltan Haiman (Columbia)