# QSOs sigposting cluster size halos as gravitational lenses: halo mass, projected mass density profile and concentration at $z\sim0.7$

## TL;DR

This study measures the magnification bias caused by QSOs acting as gravitational lenses on background sub-millimetre galaxies, revealing that these QSOs are associated with massive, cluster-sized halos at redshift around 0.7, and analyzing their density profiles.

## Contribution

It introduces a robust methodology using stacking and cross-correlation to analyze magnification bias from QSOs, estimating halo masses and profiles at intermediate redshift.

## Key findings

- QSOs act as lenses for cluster-sized halos with masses > 10^13.6 M_6.
- The mass density profile aligns with an NFW profile, indicating unrelaxed cluster halos.
- Lensing convergence measured down to a few kiloparsecs, providing detailed halo structure insights.

## 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 $\log_{10}{(M_{min}/M_\odot)} > 13.6_{-0.4}^{+0.9}$, 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.