Statistical lensing of faint QSOs by galaxy clusters

We investigate the anti-correlation between faint high redshift QSOs and low-redshift galaxy groups found by Boyle, Fong & Shanks (1988), on the assumption that it is caused by gravitational lensing of a flat QSO number count, rather than by dust in the galaxy groups, or any other systematic effect. Using an isothermal sphere lens model, the required velocity dispersion is σ = 1286 −91 km s −1. With an isothermal sphere plus uniform density plane, the velocity dispersion is σ = 1143 −153 km s −1, while the plane density is Σc = 0.081±0.032h g cm−2. Both these values for the velocity dispersion are considerably larger than the ∼ 400− 600 km s−1 expected for poor clusters and groups and imply that the mass associated with such groups is ∼4× larger than inferred from virial analyses. If due to lensing, this measurement clearly tends to favour high values of Ω0. We demonstrate how an estimate of Ω0 may be obtained, finding the relation Ω0 = 1.3(n/3×10−4h3Mpc−3)(r/1 h−1 Mpc)(σ/1286 km s−1)2 where r is the extent of the anti-correlation and n is the space density of groups. In the current data systematic errors in the determination of n and r may dominate this measurement, but this will be a potential route to estimating Ω0 in improved galaxy-QSO datasets where these systematics can be better controlled. We have compared our result with that of Williams & Irwin (1998) who find a positive correlation between bright LBQS QSOs and APM galaxies. Because the QSO number counts are steeper at bright magnitudes, there is no contradiction between this result and our own. Indeed, adapting the lensing analysis of Williams and Irwin to our use of groups rather than galaxies, we find that there is good agreement between the amplitude of the positive cross-correlation found for the bright QSOs and the amplitude of the negative cross-correlation found for the faint QSOs. This analysis leads to a common estimate of Ω0σ8 ∼ 3 − 4. This, however, is significantly higher than indicated from several other analyses. Further tests of the accuracy of the galaxy-QSO cross-correlation results and thus their implications for Ω0 and σ8 will soon be available from the new 2dF QSO catalogue.