Lensing in the Darkness: A Bayesian Analysis of 22 Chandra Sources at $z \gtrsim 6$ Shows No Evidence of Lensing

More than $200$ quasars have been detected so far at $z > 6$, with only one showing clear signs of strong gravitational lensing. Some studies call for a missing population of lensed high-$z$ quasars, but their existence is still in doubt. A large fraction of high-$z$ quasars being lensed would have a significant effect on the shape of the intrinsic quasar luminosity function (QLF). Here, we perform the first systematic search for lensed X-ray-detected quasars at $z \gtrsim 6$ employing a Bayesian analysis, with the code BAYMAX, to look for morphological evidence of multiple images that may escape a visual inspection. We analyzed a sample of 22 quasars at $z > 5.8$ imaged by the Chandra X-ray observatory and found none with statistically significant multiple images. In the sub-sample of the 8 sources with photon counts $>20$ we exclude multiple images with separations $r>1''$ and count ratios $f>0.4$, or with separations as small as $0.''7$ and $f>0.7$ at $95\%$ confidence level. Comparing this non-detection with predictions from theoretical models suggesting a high and a low lensed fraction, we placed upper limits on the bright-end slope, $β$, of the QLF. Using only the sub-sample with 8 sources, we obtain, in the high-lensing model, a limit $β< 3.38$. Assuming no multiple source is present in the full sample of 22 sources, we obtain $β< 2.89$ and $β< 3.53$ in the high and low lensing models, respectively. These constraints strongly disfavor steep QLF shapes previously proposed in the literature.
Accepted for publication in MNRAS. This is the final version of the manuscript. 10 pages, 5 figures