Trinity VII. Predictions for the Observable Correlation Functions of Accreting Black Holes

The quasar correlation function assesses the occurrence of quasar pairs as a function of separation, which is strongly influenced by quasar host halo masses. The empirical Trinity model recently inferred the redshift-dependent relationship between supermassive black hole (SMBH) mass, galaxy mass, and halo mass, using constraints other than correlation functions (e.g., quasar luminosity functions, active galactic nuclei occupation fractions, and SMBH mass-bulge mass relations). Hence, comparing the predicted quasar correlation functions from Trinity to real observations is an important test of Trinity's inferred SMBH -- halo relation. In this work, we use a compilation of observed two-point projected and redshift-space correlation functions from $0 \leq z \leq 3.5$. We find that Trinity accurately predicts quasar correlation functions within observed error bars, although observations do not have much constraining power at lower redshifts due to smaller observable volumes and lower quasar number densities. This finding is consistent with Trinity having the correct placement of quasars within their host galaxies and dark matter halos, without requiring quasar clustering constraints during model fitting. Using Trinity, we also predict the clustering as a function of quasar bolometric luminosity, finding that existing survey uncertainties are too large to show measurable differences ($\lesssim 0.3$ dex change in bias for $10^{42}$ erg s$^{-1}$ compared to $10^{46}$ erg s$^{-1}$ SMBHs across redshifts). This fact arises because most SMBH growth (and hence quasar luminosity) occurs in halos in a similar mass range ($10^{12}-10^{13} M_\odot$).