Cosmological Constant and Statistical Lensing of Giant Arcs

Using a singular isothermal sphere model for the matter distribution of foreground clusters of galaxies, we study the statistics of giant arcs in flat cosmologies with and without a cosmological constant. We find that the relative number of arcs predicted within $z=1$ in a universe with ${\mathrm{\Omega }}_0=0.3$ and ${\lambda }_0\equiv \mathrm{\Lambda }\mathrm{/}(3H_0^2)=0.7$ is a factor of $\sim 2$ larger than the one in the Einstein-de Sitter universe (${\mathrm{\Omega }}_0=1,{\lambda }_0=0$). For a luminosity-dependent evolution model of the number density of background galaxies that accounts for the over-density of faint blue galaxies at $z_s\approx 0.4$, the Einstein- de Sitter cosmological model predicts that about $5\mathrm{\%}$ of clusters of galaxies with an X-ray luminosity $L_x>2\times {10}^{44}$ erg s${}^{-1}$ should have giant arcs with length-to-width ratio larger than 10. This is a factor of $\sim 4$ lower than the observed fraction in the gravitational lensing survey of distant X-ray selected EMSS clusters of galaxies, indicating that the matter distribution of clusters of galaxies deviates significantly from simple isothermal spheres or/and the presence of a significant cosmological constant. It is profitable to further study the constraint on the cosmological constant from giant arcs using more realistic cluster models.