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 Ω_0 = 0.3 and λ_0 ≡ Λ/(3H_0~2) = 0.7 is a factor of ～2 greater than that predicted in the Einstein-de Sitter universe (Ω_0 = 1, λ_0 = 0). For a luminosity-dependent evolution model of the number density of background galaxies that accounts for the overdensity of faint blue galaxies at z_s ≈ 0.4, the Einstein-de Sitter cosmological model predicts that ～5% of clusters of galaxies with X-ray luminosity L_x > 2 x 10~(44) ergs s~(-1) should have giant arcs, with length-to-width ratio greater than 10. This is a factor of ～4 lower than the observed fraction in the gravitational-lensing survey of distant X-ray-selected Einstein Extended Medium-Sensitivity Survey clusters of galaxies, indicating a significant deviation of the matter distribution of clusters of galaxies from simple isothermal spheres and/or the presence of a significant cosmological constant. It will be profitable to further study the constraints on the cosmological constant from giant arcs with more realistic cluster models.
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