QSOs sigposting cluster size halos as gravitational lenses: halo mass,
  projected mass density profile and concentration at $z\sim0.7$
  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 $\log_{10}{(M_{min}/M_\odot)} >
13.6_{-0.4}^{+0.9}$, also confirmed by the mass density profile analysis
($M_{200c}\sim 10^{14} M_\odot$). 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, $\kappa(\theta)$, 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
$M_{NFW}=1.0^{+0.4}_{-0.2}\times10^{14} M_\odot$ 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.
