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Revisiting the Lensed Fraction of High-Redshift Quasars

The observed lensed fraction of high-redshift quasars $(\sim0.2\%)$ is significantly lower than previous theoretical predictions $(\gtrsim4\%)$. We revisit the lensed fraction of high-redshift quasars predicted by theoretical models, where we adopt recent measurements of galaxy velocity dispersion functions (VDFs) and explore a wide range of quasar luminosity function (QLF) parameters. We use both analytical methods and mock catalogs which give consistent results. For ordinary QLF parameters and the depth of current high-redshift quasar surveys $(m_z\lesssim22)$, our model suggests a multiply-imaged fraction of $F_\text{multi}\sim 0.4\%-0.8\%$. The predicted lensed fraction is $\sim1\%-6\%$ for the brightest $z_s\sim6$ quasars $(m_z\lesssim19)$, depending on the QLF. The systematic uncertainties of the predicted lensed fraction in previous models can be as large as $2-4$ times and are dominated by the VDF. Applying VDFs from recent measurements decreases the predicted lensed fraction and relieves the tension between observations and theoretical models. Given the depth of current imaging surveys, there are $\sim15$ lensed quasars at $z_s>5.5$ detectable over the sky. Upcoming sky surveys like the LSST survey and the {\em Euclid} survey will find several tens of lensed quasars at this redshift range.