Abstract
Magnification bias is a gravitational lensing effect that is normally overlooked because it is considered sub-optimal in comparison with the lensing shear. Thanks to the demonstrated optimal characteristics of the sub-millimetre galaxies (SMGs) for lensing analysis, in this work we were able to measure the magnification bias produced by a sample of QSOs acting as lenses, 0.2<z<1.0, on the SMGs observed by Herschel at 1.2<z<4.0. Two different methodologies were successfully applied: the traditional cross-correlation function approach and the Davis-Peebles estimator through stacking technique. The second one was found to be more robust for analysing the strong lensing regime (<20−30 arcsec in our case) and provides the possibility to take into account the positional errors of the sources in our samples. From the halo modelling of the cross-correlation function, the halo mass where the QSOs acting as lenses are located was estimated to be greater than log 10(Mmin/M⊙) > 13.6−0.4+0.9, also confirmed by the mass density profile analysis (M200c∼ 1014 M⊙). These mass values indicate that we are observing the lensing effect of a cluster size halo signposted by the QSOs, as in previous studies of the magnification bias. Moreover, we were able to estimate the lensing convergence, κ(θ), for our magnification bias measurements down to a few kpcs. The derived mass density profile is in good agreement with a Navarro-Frank-White (NFW) profile. We also attempt an estimation of the halo mass and the concentration parameters, obtaining MNFW=1.0+0.4−0.2×1014 M⊙ and C=3.5−0.3+0.5. This concentration value is rather low and it would indicate that the cluster halos around these QSOs are unrelaxed. However, higher concentration values still provides a compatible fit to the data.