Distribution of image separations at z = 6.51, obtained assuming a Press–Schechter distribution of galaxies, following Barkana & Loeb (2000). The shaded area indicates the 1σ uncertainty due to the distribution of velocity dispersion in galaxies σ_v. The angular separation for J0439+1634 (∼0.″2), the HST angular resolution (∼0.″1), and the typical angular size of lens galaxies at z ∼ 1 (∼1″) are also shown for reference.