Cosmological parameters from statistics of strongly lensed radio sources

We calculate the expected number of strongly lensed radio sources in a samp le of similar to 6500 sources observed with the Very Large Array as part of the Cosmic Lens All Sky Survey (CLASS) during the first two sessions of it s observations. A comparison between the predicted and the observed number of lensed radio sources allows a determination of the current value of Omeg a(m) - Omega(Lambda), where Omega(m) is the cosmological mass density of th e universe and Omega(Lambda) is the normalized cosmological constant. If th ere are six strongly lensed sources in this sample, our 95% confidence lowe r limit on Omega(m) - Omega(Lambda) is -0.58. For a flat universe with Omeg a(m) + Omega(Lambda) = 1, then, Omega(Lambda) < 0.79 (95% C.L.). If there a re ten strongly lensed sources, the 95% confidence lower limit on Omega(m) - Omega(Lambda) is -0.90. These lower limits are consistent with estimates based on high redshift supernovae and with previous limits based on gravita tional lensing. Instead of considering a simple cosmological constant, we a lso consider the possibility of a quintessence scalar field responsible for the additional energy density of the universe, with an equation of state o f the form w = P-x/rho(x), where P-x and rho(x) are the pressure and energy density of the field. We present our constraints on the Omega(x) - w plane , where Omega(x) is the present day normalized energy density of the scalar -field component, assuming a flat universe such that Omega(m) + Omega(x) = 1. If there are 6 strongly lensed sources in the present CLASS sample, grav itational lensing statistics allow us to rule out the region with Omega(x) greater than or similar to (1.2-0.5w(2)) +/- 0.05 (95% C.L.). We discuss th e region allowed by combined gravitational lensing statistics, high redshif t 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 an d 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 + (Omega(m) - Omega(Lambda)) +/- 0.1. In order to obtain a much tighter estimate on the cosmological parameters, it is essential that the redshift distribution fo r radio sources at the faint flux density levels be observationally determi ned. We strongly recommend that statistically complete optical spectroscopi c programs be carried out to obtain redshifts for a representative subsampl e of faint background radio sources. Until such redshifts are obtained, it is unlikely that a major improvement could be made with respect to lensed r adio source constraints on cosmological parameters.