A strategy for astrometric quasar selection with Gaia DR3 and its application to discover new luminous $z>2$ quasars

We aim to provide a cookbook for purely astrometric selection of quasars with Gaia DR3 and to provide an example of its use for the identification of very luminous $z>2$ quasars. We analyse the Gaia DR3 astrometry of confirmed quasars and stationary sources. We show that the proper motion signal to noise $\mu \mathrm{/}{\sigma }_{\mu }$ follows a Rayleigh-like distribution. We also show the importance of properly taking into account the correlation between proper motions in right ascension and declination, as well as the non-linear error propagation. We adopt the Mahalanobis distance to the null proper-motion hypothesis. A scaled ${\chi }^2$ distribution with $k=2$ and scale $s=1.1485$ is fitted to Gaia-CRF3; the corresponding $\pm 2\sigma$ Gaussian-equivalent threshold is $D_{M(\mu )}^2<7.097$. The final criteria combine this with the parallax condition and $\mathrm{r}\mathrm{u}\mathrm{w}\mathrm{e}<1.4$. Applying the new criteria to the SDSS DR16Q confirmed quasars, the recovery rate is $90.25\mathrm{\%}$. The suggested purely astrometric selection significantly improves both completeness and purity compared to other strategies used in the literature, reaching a quasar purity of approximately $95\mathrm{\%}$ ($\sim 2\sigma$ Gaussian-equivalent). It can be used to improve the selection purity and completeness for future quasar surveys. As an example, we use the suggested strategi to locate 24 very bright $z>2$ quasars without redshifts in the literature. Our proposed astrometric selection strategy recovers 95% of known quasars. Its potential use for new discoveries is demonstrated by the identification of 24 very luminous $z>2$ quasars. Ten of these are new discoveries and for an additional seven we correct the redshifts determined by other methods.}