Cosmological Parameters from Statistics of Strongly Lensed Radio Sources

We calculate the expected number of strongly lensed radio sources in a sample of ∼ 6500 sources observed with the Very Large Array as part of the Cosmic Lens All Sky Survey (CLASS) during the first two sessions of its observations. A comparison between the predicted and the observed number of lensed radio sources allows a determination of the current value of Ωm − ΩΛ, where Ωm is the cosmological mass density of the universe and ΩΛ is the normalized cosmological constant. If there are six strongly lensed sources in this sample, our 95% confidence lower limit on Ωm − ΩΛ is -0.58. For a flat universe with Ωm + ΩΛ = 1, then, ΩΛ < 0.79 (95% C.L.). If there are ten strongly lensed sources, the 95% confidence lower limit on Ωm − ΩΛ is -0.90. These lower limits are consistent with estimates based on high redshift supernovae and with previous limits based on gravitational lensing. Instead of considering a simple cosmological constant, we also consider the possibility of a quintessence scalar field responsible for the additional energy density of the universe, with an equation of state of the form w = Px/ρx, where Px and ρx are the pressure and energy density of the field. We present our constraints on the Ωx − w plane, where Ωx is the present day normalized energy density of the scalar-field component, assuming a flat universe such that Ωm +Ωx = 1. If there are 6 strongly lensed sources in the present CLASS sample, gravitational lensing statistics allow us to rule out the region with Ωx >∼ (1.2 − 0.5w ) ± 0.05 (95% C.L.). We discuss the region allowed by combined gravitational lensing statistics, high redshift Type Ia supernovae distances, and globular cluster ages. Instead of a cosmological model, we can constrain the redshift distribution of faint radio sources based on the observed gravitational lensing rate and an assumed cosmological model. If there are six strongly lensed sources, the 68% confidence upper limit on the average redshift 〈z〉 of radio sources with flux densities less than 150 mJy at 8.4 GHz is 〈z〉 < 1.4 + (Ωm −ΩΛ)± 0.1. In order to obtain a much tighter estimate on the cosmological parameters, it is essential that the redshift distribution for radio sources at the faint flux density levels be observationally determined. We strongly recommend that statistically complete optical spectroscopic programs be carried out to obtain redshifts for a representative subsample of faint background radio sources. Until such redshifts are obtained, it is unlikely that a major improvement could be made with respect to lensed radio source constraints on cosmological parameters.