Erratum: TRINITY I: self-consistently modeling the dark matter halo-galaxy-supermassive black hole connection from z = 0−10 (MNRAS (2019) 488 (3143) DOI: 10.1093/mnras/stac2633)

PARAMETRIZATION OF TRINITY In the original version of TRINITY, we parametrized the correlation coefficient between supermassive black hole (SMBH) accretion rate and SMBH mass at fixed dark matter halo mass, ρ_BH, as a function of redshift: 1 ρ_BH (z) = ρ_BH,0 + ρ_BH,a^（_{1 + z} − 1 ^） + ρ_BH,z z. (1) We believed that at z ≾ 2, ρ_BH was constrained by active black hole mass functions (ABHMFs), i.e. the mass functions of quasars with Eddington ratios η ≥ 0.01. This is because any change in ρBH will change the fraction of active SMBHs at fixed SMBH mass, and thus change the ABHMFs. At high redshifts, the constraint on ρBH comes from the fact that z ∼ 6 quasars tend to have overmassive SMBHs compared to the intrinsic SMBH mass-galaxy mass (M_•-M_∗) relation predicted by TRINITY. Only a high ρ_BH can produce such a strong luminosity-dependent bias of the M_•-M_∗ relation. However, we later found that the algorithm to calculate ABHMFs was not updated to account for the parametrization of ρBH. Additionally, ρBH was not used correctly, such that the effective ρ_BH actually increased from 1 to 2, from z ≿ 8 to z ∼ 0, instead of decreasing from 1 to 0 towards low redshifts (see fig. 24 of Zhang et al. 2023). We also found that the z ∼ 0.2 active black hole mass function (ABHMF) from Schulze & Wisotzki (2010) was underestimated, because we incorrectly applied the correction for obscured AGN. After fixing these three bugs, we reran the original TRINITY model. We found that the current datasets are unable to meaningfully constrain ρBH. In light of this, we chose to fix ρ_BH ≡ 1 across cosmic time, i.e. we assume that all SMBHs in a given halo mass bin share the same Eddington ratio distribution. In Zhang et al. (2023), we applied an SMBH mass shift, ∆M_• ≡ 0.2 dex, when predicting the z ∼ 0.2 ABHMF. This is motivated by the systematic difference in the virial estimates adopted by Schulze & Wisotzki (2010) and Schulze et al. (2015). But with the corrected TRINITY code, we tend to overproduce the corrected z ∼ 0.2 ABHMF with ∆M_• ≡ 0.2 dex, leading to one of the parameterization updates in Section 2. 2 U P D AT E D T R I N I T Y PA R A M E T R I Z AT I O N As mentioned in Section 1, ρ_BH actually varies between 1 − 2 in the original TRINITY model. At z ≾ 3, ρ_BH ≿ 1.7. This makes overmassive SMBHs have even higher Eddington ratios to account for the bright end of QPDFs, without invoking a top-heavy Eddington ratio distribution. Therefore, the bright end of the Eddington ratio distribution is steep across cosmic time, which naturally reproduces the lack of super-Eddington quasars at z ∼ 6. After fixing the problems in Section 1, we found that the redshift evolution of the bright-end slope of Eddington ratio distribution is not flexible enough to simultaneously reproduce: 1) the QPDFs at z ≾ 3, which implies a shallower bright end of the Eddington ratio distribution; and 2) the lack of super-Eddington quasars at z ∼ 6, which requires the opposite. Therefore, we add a term that scales with z to the bright-end slope of the Eddington ratio distribution, c₂ (equation 51 of Zhang et al. 2023): 1 c₂ = c_2,0 + c_2,a^（_{1 + z} − 1 ^） + c_2,zz. (2) This allows the Eddington ratio distribution to steepen at high redshifts, which naturally decreases the number of super-Eddington AGNs. Finally, to reproduce the z ∼ 0.2 ABHMF, we chose to allow ∆M_• to vary in the Markov Chain Monte Carlo (MCMC) exploration. We chose not to change other parameterizations to avoid overfitting the potential uncertainties in virial estimates. We also apply a Gaussian prior centered at 0.2 dex, with an uncertainty of 0.1 dex. 2.1 Induced changes to TRINITY results We have verified that the vast majority of the main TRINITY results still hold with the fixes and these changes in parametrization, including: 3 FUTURE PROSPECTS ON CONSTRAINING ρBH Our current result shows that active SMBH mass functions (ABHMFs) are not able to constrain ρ_BH while also providing constraints for other model parameters in TRINITY. This is likely because ABHMFs only specify the abundance of active SMBHs at given SMBH masses, but not whether the SMBHs are overmassive/undermassive compared to their host haloes/galaxies. In principle, with the scatter around the M_•-M_∗ relation constrained by the shape of ABHMFs (as in TRINITY, see section 4.2 of Zhang et al. 2023), the SMBH mass distribution of AGNs as a function of AGN luminosity and host galaxy mass (e.g. Ding et al. 2020 and Li et al. 2021) would provide more information to constrain ρBH. However, it is much more practical to calculate such statistics and compare with real observations in a model where individual galaxies and SMBHs are traced, e.g. the UNIVERSEMACHINE (Behroozi et al. 2019). We therefore defer the incorporation of this observable to our future study of extending the UNIVERSEMACHINE to include individual SMBHs.