Quasar clustering and duty cycle measurements at $0\leq z\leq 4$ with the Gaia-unWISE Catalog

Abstract: We measure the two-point correlation function of a uniformly selected, all-sky sample of $\sim$1.3 million quasars with magnitudes $G\leq20.5$ from the Gaia--unWISE Quasar Catalog (Quaia) over the redshift range $0 \leq z \leq 4$ to trace the evolution of the quasar clustering strength across cosmic time. We find a steady increase in the correlation length $r_0$ with redshift, i.e. $r_0 = 6.8 \pm 0.2\,h<sup>{-1}\mathrm{Mpc}$ at $0 \leq z &lt; 1$, $r_0=8.0 \pm 0.2\,h<sup>{-1}\mathrm{Mpc}$ at $1 \leq z &lt; 2$, $r_0=10.8 \pm 0.2\,h<sup>{-1}\mathrm{Mpc}$ at $2 \leq z &lt; 3$, and $r_0=13.9 \pm 1.2\,h<sup>{-1}\mathrm{Mpc}$ at $3 \leq z &lt; 4$, and slopes consistent with $γ\approx 2$. Our measurements suggest a slightly weaker clustering signal at $z&gt;3$ than previous studies, and thus we find a smooth, monotonic rise in clustering strength. Using a bias-halo mass relation and a step-function for the halo occupation distribution, we infer characteristic minimum halo masses of quasar hosts of $\log_{10}(M_{\mathrm{min}}/M_\odot) \approx 12.8$ across all redshifts. Combining these with the observed quasar number densities yields duty cycles that rise from $f_{\mathrm{duty}} \approx 2\%$ to $\approx 7\%$ with increasing redshift, corresponding to integrated quasar lifetimes of $t_{\rm QSO}\sim10<sup>8$~years. These results suggest that both the characteristic halo mass of active quasars and their typical lifetimes have remained remarkably stable over more than $12 \sim$ Gyr of cosmic time, implying a self-regulated growth process largely independent of epoch.