1999A&A...341..653C

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. The foreground lensing clusters are modeled as singular isothermal spheres, and their number density and redshift distribution is calculated based on the Press-Schechter theory with normalizations based on the local cluster temperature function. Based on the results from optical arc surveys, we find that the observed number of arcs can easily be explained in a flat universe (Ω_m+Ω_Λ=1) with low values for cosmological mass density of the universe (Ω_m≲0.5). However, given the large systematic and statistical uncertainties involved with both the observed and predicted number of lensed arcs, more reliable estimates of the cosmological parameters are not currently possible. We comment on the possibility of obtaining a much tighter constraint based on statistics from large area optical surveys. At radio wavelengths (1.4GHz), we predict ∼1500 lensed radio sources with flux densities greater than 10µJy, and with amplifications due to lensing greater than 2, in a flat cosmology with Ω_m=0.3 and Ω_Λ=0.7. Given the recent detection of a sub-mm selected lensed µJy radio source towards A370, it is suggested that deep radio observations of clusters should contain such lensed sources. At sub-mm wavelengths (850µm), the number of lensed sources expected towards the same foreground lens population and cosmology is ∼3x10⁴. 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 353GHz may be a useful by-product of Planck.