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Cosmology with Galaxy Clusters: III. Gravitationally Lensed Arc Statistics as a Cosmological Probe

We calculate the expected number of gravitationally lensed optical, radio and sub-mm lensed sources on the whole sky due to foreground galaxy clusters for different cosmological models. We improve previous calculations of lensed arc statistics by including redshift information for background sources and accounting for the redshift evolution of the foreground lensing clusters. The background sources are described based on the redshift and optical magnitude or flux distribution for sources in the Hubble Deep Field (HDF). Using the HDF luminosity function, we also account for the magnification bias in magnitude-limited observational programs to find lensed optical arcs. Based on the results from optical arc surveys, we find that the observed number of arcs can easily be explained in a flat universe with low values for cosmological mass density of the universe. At radio wavelengths (1.4 GHz), we predict \sim 1500 lensed radio sources with flux densities greater than 10 muJy, and with amplifications due to lensing greater than 2, in a flat cosmology with Omega=0.3 and Lambda=0.7. At sub-mm wavelengths (850 mum), the number of lensed sources expected towards the same foreground lens population and cosmology is \sim 3 x 10^4. We briefly consider the possibility of using the South Pole 10-m sub-mm telescope and the Planck surveyor to identify lensed sub-mm sources. A catalog of around 100 gravitationally lensed sources at 353 GHz may be a useful by-product of Planck. (Abridged)