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, 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 With an isothermal sphere plus uniform density plane, the velocity dispersion is while the plane density is Σ c =0.081±0.032 h  g cm −2 . Both of 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 times larger than inferred from virial analyses. If it is a result of lensing, this measurement clearly tends to favour high values of Ω 0 . We demonstrate how an estimate of Ω 0 may be obtained, finding the relation Ω 0 =1.3( n /3×10 −4   h 3  Mpc −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 Ω 0 in improved galaxy–QSO data sets where these systematics can be better controlled. We have compared our result with that of Williams & Irwin who find a positive correlation between bright Large Bright Quasar Survey (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 & 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 Ω 0 σ 8 ∼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 Ω 0 and σ 8 will soon be available from the new 2dF QSO catalogue.