Constraining the Gravitational Lensing of $z\gtrsim6$ Quasars from their Proximity Zones

Since their discovery twenty years ago, the observed luminosity function of $z\gtrsim6$ quasars has been suspected to be biased by gravitational lensing. Apart from the recent discovery of UHS J0439+1634 at $z\approx6.52$, no other strongly lensed $z\gtrsim6$ quasar has been conclusively identified. The hyperluminous $z\approx6.33$ quasar SDSS J0100+2802, believed to host a supermassive black hole of $\sim10^{10} M_\odot$, has recently been claimed to be lensed by a factor of $\sim450$, which would negate both its extreme luminosity and black hole mass. However, its Ly$\alpha$-transparent proximity zone is the largest known at $z>6$, suggesting an intrinsically extreme ionizing luminosity. Here we show that the lensing hypothesis of $z\gtrsim6$ quasars can be quantitatively constrained by their proximity zones. We first show that our proximity zone analysis can recover the strongly lensed nature of UHS J0439+1634, with an estimated magnification $\mu=28.0^{+18.4}_{-11.7}(^{+44.9}_{-18.3})$ at 68% (95%) credibility that is consistent with previously published lensing models. We then show that the large proximity zone of SDSS J0100+2802 rules out lensing magnifications of $\mu>4.9$ at 95% probability, and conclusively rule out the proposed $\mu>100$ scenario. Future proximity zone analyses of existing $z\gtrsim6$ quasar samples have the potential to identify promising strongly lensed candidates, constrain the distribution of $z\gtrsim6$ quasar lensing, and improve our knowledge of the shape of the intrinsic quasar luminosity function.