Statistical lensing of faint QSOs by galaxy clusters

We investigate the anti-correlation between faint high-redshift QSOs and lo w-redshift galaxy groups found by Boyle, Fong & Shanks, on the assumption t hat it is caused by gravitational lensing of a flat QSO number count, rathe r than by dust in the galaxy groups, or any other systematic effect. Using an isothermal sphere lens model, the required velocity dispersion is sigma = 1286(-91)(+72) km s(-1). With an isothermal sphere plus uniform density p lane, the velocity dispersion is sigma = 1143(-153)(+109) km s(-1), while t he plane density is Sigma(c) = 0.081 +/- 0.032h g cm(-2). Both of these val ues for the velocity dispersion are considerably larger than the similar to 400-600 km s(-1) expected for poor clusters and groups and imply that the mass associated with such groups is similar to 4 times larger than inferred from virial analyses. If it is a result of lensing, this measurement clear ly tends to favour high values of Omega(0). We demonstrate how an estimate of Omega(0) may be obtained, finding the relation Omega(0) = 1.3(n/3 x 10(- 4) h(3) Mpc(-3))(r/1 h(-1) Mpc)(sigma/1286 km s(-1))(2) where r is the exte nt of the anti-correlation and n is the space density of groups. In the cur rent data systematic errors in the determination of n and r may dominate th is measurement, but this will be a potential route to estimating Omega(0) i n improved galaxy-QSO data sets where these systematics can be better contr olled. We have compared our result with that of Williams & Irwin who find a positi ve correlation between bright Large Bright Quasar Survey (LBQS) QSOs and AP M galaxies. Because the QSO number counts are steeper at bright magnitudes, there is no contradiction between this result and our own. Indeed, adaptin g 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 th e positive cross-correlation found for the bright QSOs and the amplitude of the negative cross-correlation found for the faint QSOs. This analysis lea ds to a common estimate of Omega(0)sigma(8) similar to 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 Omega(0) and sigma(8) will soon be available from t he new 2dF QSO catalogue.