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 nonnegligible situations, lensing can produce a single highly magnified image; this phenomenon is mainly associated with massive cluster-scale halos (≳1013.5 M☉). 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 (≲1012 M☉) 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 Δθ = 0farcs3 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 μ > 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 (≲1010 M☉) halos have highly concentrated (isothermal) profiles. In more realistic models in which dwarf halos have flatter (Navarro-Frenk-White) 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 ≲10-4. 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-solar-mass black holes in the young universe are not viable.