TRINITY VI: connection between galaxy star formation rates and supermassive black hole accretion rates from z = 0 - 10

Haowen Zhang, Peter Behroozi, Marta Volonteri, Joseph Silk, Xiaohui Fan, James Aird, Jinyi Yang, Feige Wang, Philip F. Hopkins

► To cite this version:

Haowen Zhang, Peter Behroozi, Marta Volonteri, Joseph Silk, Xiaohui Fan, et al. TRINITY VI: connection between galaxy star formation rates and supermassive black hole accretion rates from z = 0 - 10. Monthly Notices of the Royal Astronomical Society, 2025, 538, pp.503-517. 10.1093/mn-ras/staf260. insu-05101679

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# TRINITY VI: connection between galaxy star formation rates and supermassive black hole accretion rates from $z = 0 – 10$

Haowen Zhang (张昊文),<fnref n="1" />* Peter Behroozi<fnref n="1" />,<fnref n="1" />,<fnref n="2" /> Marta Volonteri,<fnref n="3" /> Joseph Silk,<fnref n="3" />,<fnref n="4" />,<fnref n="5" /> Xiaohui Fan,<fnref n="1" />

James Aird,<fnref n="6" />,<fnref n="7" /> Jinyi Yang (杨锦怡),<fnref n="1" />,<fnref n="8" /> Feige Wang (王飞格)<fnref n="1" />,<fnref n="8" /> and Philip F. Hopkins,<fnref n="9" />

Accepted 2025 February 7. Received 2025 February 4; in original form 2024 September 18

## ABSTRACT

We infer supermassive black hole (SMBH) accretion rates and Eddington ratios as a function of SMBH/host galaxy mass and redshift with the empirical TRINITY model of dark matter halo–galaxy–SMBH connection. The galaxy–SMBH mass and growth rate connection from TRINITY match galaxy observables from $0 < z < 13$ and SMBH observables from $0 < z < 6.5$. Key findings include: (1) the ratio between cosmic SMBH accretion rate and galaxy star formation rate stays constant at $\sim 2 \times 10^{-3}$ from $z = 0–4$, and decreases by 2 orders of magnitude from $z = 4–10$; (2) the average SMBH Eddington ratio $\bar{\eta}$ increases towards higher redshifts, nearly reaching $\bar{\eta} = 1$ at $z \sim 10$; (3) at fixed redshift for $z < 3$, SMBHs/galaxies with higher masses have lower $\bar{\eta}$, consistent with active galactic nucleus (AGN) downsizing; (4) the average ratio of specific SMBH accretion rate (SBHAR) to average specific star formation rate (SSFR) is nearly mass-independent, with a value SBHAR/SSFR $\sim 1$, which decreases slightly from $z = 10$ to $z = 0$; (5) similar to galaxies, SMBHs reach their peak efficiency to convert baryons into mass when host haloes reach $10^{12} M_{\odot}$; (6) given galaxy and SMBH growth histories from TRINITY, the local descendants of $1 < z < 11$ overmassive JWST AGNs will remain outliers from the local SMBH mass–galaxy mass relation. These findings combine to give a simple explanation for massive ($10^9 – 10^{10} M_{\odot}$) quasars at $z > 6$: at these redshifts, dark matter haloes grow with an $e$-folding time of $\sim 45$ Myr, driving similar growth in both galaxies and SMBHs.

**Key words:** galaxies: evolution – galaxies: haloes – quasars: supermassive black holes.

## 1 INTRODUCTION

Supermassive black holes (SMBHs) are believed to exist in the centres of most galaxies (Kormendy & Richstone 1995; Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Ho 2008; Gültekin et al. 2009; Kormendy & Ho 2013; Heckman & Best 2014). When accreting, SMBHs release huge amounts of energy, and are called active galactic nuclei (AGNs) because the resulting radiation often lights up galaxy centres. Given such high-energy output, SMBHs are a prime candidate for exerting feedback on host galaxies and regulating their star formation in theoretical models (Silk & Rees 1998; Bower et al. 2006; Somerville et al. 2008; Sijacki et al. 2015). At the same time, galaxies may also affect black hole growth by controlling the amount of available gas fuel through various physical mechanisms, such as stellar feedback and

galaxy mergers. Hence, elucidating the assembly histories of SMBHs in different galaxies is critical for understanding host galaxy–SMBH interactions (see, e.g. Hopkins et al. 2007; Alexander & Hickox 2012; Brandt & Alexander 2015).

Many types of SMBH observations exist. First, relatively tight scaling relations ($\sim 0.3$ dex scatter) are found between SMBH masses and host galaxy dynamical properties (e.g. velocity dispersion or bulge mass at $z \sim 0$, see Häring & Rix 2004; Gültekin et al. 2009; Kormendy & Ho 2013; McConnell & Ma 2013; Savorgnan et al. 2016). If SMBH growth tracks host galaxy growth, such relations will naturally occur. Further, the cosmic SMBH accretion rate (CBHAR) density is found to track the cosmic star formation rate (CSFR) density over $0 < z < 4$, with a roughly constant CBHAR/CSFR ratio between $10^{-4}$ and $10^{-3}$ (Merloni, Rudnick & Di Matteo 2004; Silverman et al. 2008; Shankar, Weinberg & Miralda-Escudé 2009; Aird et al. 2010; Delvecchio et al. 2014; Yang et al. 2018). In other words, the collective growth of SMBHs and galaxies proceed on similar time-scales. Towards even higher redshifts such as $z \gtrsim 6$, the

<footnote n="1">¹Department of Astronomy, University of Arizona, 933 N Cherry Ave., Tucson, AZ 85721, USA</footnote>

<footnote n="2">²Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan</footnote>

<footnote n="3">³Institut d'Astrophysique de Paris (UMR 7095: CNRS & Sorbonne Universite), 98 bis Bd. Arago, F-75014, Paris, France</footnote>

<footnote n="4">⁴Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA</footnote>

<footnote n="5">⁵BIPAC, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK</footnote>

<footnote n="6">⁶Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK</footnote>

<footnote n="7">⁷Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK</footnote>

<footnote n="8">⁸Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="9">⁹Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA</footnote>

<footnote>* E-mail: hwzhang0595@arizona.edu</footnote>
number of detectable AGNs drop significantly due to detection limits. Before JWST, 275 AGNs were found at $z > 6$, most of which are bright quasars with bolometric luminosities $L_{bol} > 10^{46}$ erg s$^{-1}$ (Fan, Bañados & Simcoe 2023). Fortunately, JWST has pushed the AGN detection limit down to $L_{bol} \sim 10^{44}$ erg s$^{-1}$ with its unique near-infrared capabilities (Harikane et al. 2023; Maiolino et al. 2023). As of the writing of this paper, JWST has found hundreds of AGN candidates between $4 \lesssim z \lesssim 11$ with colour, morphology, and/or the detection of broad emission lines. Many of these AGNs are found to be compact and red (hence named as 'Little Red Dots'), which seem to be less common at lower redshifts (Matthee et al. 2024).

Besides observational efforts, SMBH growth is also widely investigated in hydrodynamical simulations and semi-analytic models (SAMs) of galaxy evolution (see, e.g. Croton et al. 2006; Somerville et al. 2008; Dubois et al. 2012; Schaye et al. 2015; Sijacki et al. 2015; Weinberger et al. 2017). With detailed snapshots produced by these simulations, the assembly histories of individual galaxies and SMBHs are reconstructed and examined. However, resolution limits entail many assumptions on how SMBHs interact with host galaxies, which varies among different simulations. In addition, simulations and SAMs are usually calibrated against a small subset of available galaxy and SMBH data at low redshifts, and thus often do not match higher redshift observations such as quasar luminosity functions (see, e.g. Amarantidis et al. 2019; Habouzit et al. 2022). Different assumptions lead to divergent predictions of early SMBH properties like the black hole mass–galaxy mass ($M_{\bullet}-M_{*}$) relation (Habouzit et al. 2021). Although these different predictions provide many potential scenarios of galaxy–SMBH interaction for testing with data, they also show that little consensus has been reached on how the $M_{\bullet}-M_{*}$ relation evolves with time.

Designed to bridge observations and other theoretical models, empirical models are a complementary tool to elucidate SMBH evolution in different galaxies. Empirical models use observations to self-consistently and empirically characterize the properties of SMBHs and/or their connection with host galaxies. These results are then compared with different theoretical models to identify plausible physical mechanisms that drive SMBH formation and evolution. By matching multiple kinds of galaxy and AGN observable data, empirical models such as Yang et al. (2018), Shankar et al. (2020), and TRINITY (Zhang et al. 2023) have inferred the average SMBH accretion rates as functions of host galaxy mass and redshift. TRINITY, in particular, has reconstructed average SMBH growth histories in different halo and galaxy mass bins, which allows us to statistically connect observed AGN populations detected at different redshifts. By comparing TRINITY’s best-fitting $M_{\bullet}-M_{*}$ relation with those from different hydrodynamical simulations, Zhang et al. (2023) found (1) that stronger supernova feedback at higher redshifts may have delayed the growth of SMBHs in early low-mass galaxies, and (2) that the slow AGN accretion may have been maintaining the quiescence of low-redshift high-mass galaxies.

In this paper, we continue to present TRINITY’s predictions on the galaxy–SMBH connection, with a focus on SMBHs’ average accretion rates, average Eddington ratios, and Eddington ratio distributions, all as functions of host galaxy mass and redshift. With these predictions, we will further carry out a case study on JWST AGNs and AGN candidates above $z \sim 1$, to reconstruct their $M_{\bullet}$ and $M_{*}$ evolution histories. In this case study, we only include AGN (candidates) with broad emission lines given the stronger certainty about their AGN nature and $M_{\bullet}$ estimates.

The paper is organized as follows. Section 2 introduces the TRINITY model framework. In Section 3, we describe the dark matter simulation and real observations used in TRINITY. Section 4 presents

TRINITY’s predictions for average SMBH accretion rates, average Eddington ratios, and Eddington ratio distributions as functions of host galaxy mass and redshift. In Section 5, we compare TRINITY predictions with other models, and carry out a case study on JWST AGNs. Finally, we present the conclusions in Section 6. In this work, we adopt a flat Lambda cold dark matter (ΛCDM) cosmology with parameters ($\Omega_m = 0.307, \Omega_\Lambda = 0.693, h = 0.678, \sigma_8 = 0.823, n_s = 0.96$) consistent with Planck results (Planck Collaboration XIII 2016). We use data sets that adopt the Chabrier stellar initial mass function (IMF; Chabrier 2003), the Bruzual & Charlot (2003) stellar population synthesis model, and the Calzetti dust attenuation law (Calzetti et al. 2000). Halo masses are calculated following the virial overdensity definition from Bryan & Norman (1998).

## 2 METHODOLOGY

We start this section by reviewing TRINITY’s predictions for the galaxy–SMBH mass and growth rate connection, and explain which input galaxy and SMBH data (listed in Section 3.2) drove these predictions (Section 2.1). We then give a brief overview of TRINITY implementation in Section 2.2. For full details, we refer readers to Zhang et al. (2023).

### 2.1 Inferring galaxy–SMBH mass and growth rate connection with TRINITY

In the best-fitting TRINITY model, the $M_{\bullet}-M_{*}$ relation experiences mild evolution at above $\sim 3 \times 10^{10} M_{\odot}$ from $z = 0-10$, but the evolution is stronger at $\lesssim 5 \times 10^9 M_{\odot}$, where the median $M_{\bullet}$ at a fixed $M_{*}$ decreases by $\gtrsim 1$ dex from $z = 0-10$. The redshift evolution of the $M_{\bullet}-M_{*}$ relation in TRINITY is jointly constrained by quasar luminosity functions (QLFs), quasar probability distribution functions (QPDFs), the local black hole mass–galaxy mass ($M_{\bullet}-M_{*}$) relation, as well as the local galaxy stellar mass functions (SMFs). First, the AGN radiative efficiency, $\epsilon_{rad}$, is constrained by the Soltan argument (Soltan 1982). Specifically, TRINITY compares the total amount of AGN radiation across cosmic time (i.e. integral of QLFs) with the total amount of SMBH mass growth (i.e. the convolution of the local galaxy SMF with the local $M_{\bullet}-M_{*}$ relation), and $\epsilon_{rad}$ is simply the ratio between the total radiation and total SMBH mass growth. Note that although SMBH mergers contribute to the growth of individual SMBHs, the cosmic SMBH mass density can only increase via accretion. Therefore, the constraint of $\epsilon_{rad}$ is not affected by the continuing uncertainties in SMBH mergers.

Every redshift evolution of the $M_{\bullet}-M_{*}$ relation gives a different average SMBH accretion rate as a function of galaxy mass and redshift. With a constrained $\epsilon_{rad}$, TRINITY converts the resulting average BHARs into AGN luminosity distributions with parametrized Eddington ratio distribution shapes and AGN duty cycles. By comparing modelled luminosities with observed QPDFs for different galaxy populations and redshifts, TRINITY finds the best redshift-dependent $M_{\bullet}-M_{*}$ relation that matches all input QLFs and QPDFs.

The best-fitting TRINITY model predicts that towards higher redshifts, AGN Eddington ratio distributions become narrower and SMBHs have higher and higher typical Eddington ratios when accreting. This is mainly constrained by active black hole mass functions, i.e. AGN SMBH mass functions from $z = 0.3-5$. Specifically, with a constrained redshift evolution of the $M_{\bullet}-M_{*}$ relation, Eddington ratio distributions with broader shapes and/or lower typical Eddington ratios cannot produce enough active SMBHs (either defined with Eddington ratio or luminosity thresholds) to match the observed active black hole mass functions.
Finally, TRINITY also predicts that AGN duty cycle, defined as the fractions of halo/galaxy hosting SMBHs that are accreting at an Eddington ratio of $\eta > 10^{-4}$, is a strong increasing function of halo/galaxy mass. This is driven by the QPDFs as a function of galaxy mass, which yields much larger probabilities for massive galaxies to host AGNs at detectable luminosities.

Of note, our input galaxy and SMBH data cover redshift ranges of $0 < z < 13$ and $0 < z < 6.5$, respectively. Beyond these redshift ranges, TRINITY’s predictions of galaxies and/or SMBHs are effectively extrapolations of observed data with our chosen parametrizations. These predictions can be regarded as the observational consequences of our redshift-dependent $M_{\bullet}-M_{*}$, Eddington ratio distribution, and AGN duty cycle. By comparing future observations from, e.g. JWST, Euclid, Roman Space Telescope etc., with our predictions, we can determine if TRINITY gives a realistic picture of high-redshift galaxy–SMBH connection, and if not, how much information can be extracted from new data.

## 2.2 Implementation overview

TRINITY<fnref n="1" /> aims to find an optimal recipe to grow galaxies and SMBHs in a series of halo mass bins, which matches all the input galaxy and SMBH statistics. In TRINITY, these recipes are parametrized as a series of empirical scaling relations, instead of being built on any specific physical mechanism(s) in halo–galaxy–SMBH interaction. Given the complicated dependence of predicted galaxy and SMBH properties on model parameters, it is prohibitive to find an analytical likelihood function for our parametrization. Instead, we adopt a simulation-based inference framework. Specifically, TRINITY creates mock universes with halo populations, galaxies, and SMBHs. Each unique model parameter set specifies a different mock universe. For each universe, we forward model observable galaxy and SMBH properties and compare them with observed data. Such a comparison yields a posterior likelihood that the proposed recipe creates a realistic mock universe. This likelihood is used in a custom Markov chain Monte Carlo (MCMC) algorithm to find the posterior probability distribution of TRINITY model parameters. Based on this posterior distribution, we identify the best recipe to connect dark matter haloes, galaxies, and SMBHs from $z = 0-6.5$, as well as the uncertainties in this recipe given the error bars in input data constraints. In other words, these uncertainties in the recipe quantify how much the recipe can change without significantly violating our data constraints. This best recipe of the halo–galaxy–SMBH connection is then extrapolated out to $z \sim 15$ to make higher redshift predictions.

The statistical halo–galaxy–SMBH connection in TRINITY is comprised of the following components:

(i) **Halo–galaxy connection:** to connect galaxies with dark matter haloes, we parametrize a redshift-dependent scaling relation between galaxy star formation rate (SFR) and halo mass ($M_{\text{peak}}$). Galaxy masses ($M_{*}$) in different halo mass bins are then obtained by integrating the galaxy star formation and merger histories along the halo assembly histories, where stellar mass loss due to stellar evolution is accounted for (Behroozi, Wechsler & Conroy 2013b). We also parametrize the redshift-independent, log-normal scatter around the $M_{*}-M_{\text{peak}}$ relation as $\sigma_{*}$.

(ii) **Galaxy–SMBH connection:** TRINITY converts $M_{*}$ into galaxy bulge mass $M_{\text{bulge}}$ using a fixed fitted $M_{\text{bulge}}-M_{*}$ relation based on Sloan Digital Sky Survey (SDSS) and Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) data (Lang et al. 2014; Mendel et al. 2014). A parametrized redshift-dependent $M_{\bullet}-M_{\text{bulge}}$ relation is then used to specify the median $M_{\bullet}$ for each halo mass bin, based on the galaxy mass. To account for the random scatter around the $M_{\bullet}-M_{\text{bulge}}$ relations, we parametrize the redshift-independent, log-normal scatter as $\sigma_{\text{BH}}$.

**Modelling the radiative versus kinetic energy outputs in SMBH accretion:** at low Eddington ratios, accreting SMBH become radiatively inefficient, and release kinetic energy in the form of kinetic jets and/or outflows (Narayan & Yi 1994; Ho 2002; Nagar, Falcke & Wilson 2005). Inspired by Merloni & Heinz (2008), we adopted a fixed, empirical scaling relation between the radiative and kinetic energy efficiencies from accreting AGNs:

$$ \epsilon_{\text{kin}} = \begin{cases} \sqrt{\frac{\eta_{\text{crit}} L_{\text{Edd}} \epsilon_{\text{rad}}}{M_{\bullet} c^2}} - \epsilon_{\text{rad}}, & \eta_{\text{rad}} \le \eta_{\text{crit}} \\ 0, & \eta_{\text{rad}} > \eta_{\text{crit}} \end{cases}, \quad (1) $$

where $L_{\text{Edd}} = 4\pi GM_{\bullet}c$ is the AGN Eddington luminosity, $\eta_{\text{rad}} = \epsilon_{\text{rad}} M_{\bullet}c^2/L_{\text{Edd}}$ is the radiative Eddington ratio, and $\eta_{\text{crit}} = 0.03$ is the critical Eddington ratio below which AGNs become radiatively inefficient. The kinetic AGN luminosity is then calculated as $L_{\text{kin}} = \epsilon_{\text{kin}} M_{\bullet}c^2$. According to this scaling relation, AGN energy output is more and more dominated by kinetic energy ($\epsilon_{\text{kin}}/\epsilon_{\text{rad}} \to \infty$) as $M_{\bullet}$ decreases.

(iii) **Forward modelling observable SMBH luminosities:** as mentioned in Section 2.1, the redshift evolution of the $M_{\bullet}-M_{*}$ relation coupled with a well-constrained halo–galaxy connection only gives average BHARs as functions of halo/galaxy mass and redshifts, which cannot be directly compared to measurements of SMBH luminosity distributions. Hence, we parametrize the AGN radiative efficiency $\epsilon_{\text{rad}}$, duty cycle, and Eddington ratio distribution shapes that are flexible enough to match all the data without overfitting. $\epsilon_{\text{rad}}$ is constrained with the Soltan argument (see Section 2.1), whereas AGN duty cycles and Eddington ratio distribution shapes are constrained by the integrals and shapes of QPDFs, respectively. Given these AGN properties, we convert average BHARs into bolometric luminosity distributions as a function of halo/galaxy mass and redshift.

(iv) **Forward modelling systematic effects on TRINITY predictions:** to account for various systematic effects in measuring observed galaxy and AGN properties, we take a two-prong strategy: for systematic effects that can be modelled with one or two nuisance parameters, such as a systematic offset in $M_{*}$ in spectral energy distribution (SED) fitting, we parametrize them in the model, and constrain them with the residual inconsistencies in the input data constraints; other systematics are more complicated, such as the redshift- and luminosity-dependent AGN obscuration fraction. For these systematics, we opt to use empirical correction formulae from, e.g. Ueda et al. (2014) and Merloni et al. (2014), to correct them. As a result, TRINITY cannot predict AGN obscured fractions by itself.

(v) **Determining the best-fitting TRINITY predictions and their uncertainties:** starting with the model parameter yielding the highest posterior probability in MCMC, we run a gradient descent algorithm to obtain the best-fitting TRINITY model parameters. We then generate our fiducial predictions with these best-fitting parameters. To determine the uncertainties in these predictions, we randomly sample the MCMC chain with equal probabilities, and generate ensembles of predictions using sampled MCMC steps. Finally, we take the 16–

<footnote n="1">¹The C implementation of TRINITY can be found at https://github.com/HaowenZhang/TRINITY/tree/main/. The best-fitting model parameters, down-sampled MCMC chain, and correlations between model parameters can be found at https://github.com/HaowenZhang/TRINITY/tree/main/stats.</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Top Panel: the cosmic black hole accretion rate (CBHAR) as functions of redshift from TRINITY and previous studies (scale on the left), as well as the cosmic star formation rate density (CSFR) from TRINITY (scale on the right). Bottom Panel: the CBHAR/CSFR ratio as a function of redshift. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.1.</figcaption></figure>

84th percentile range of each prediction as its uncertainty. Such uncertainties thus reflect how much our predictions will change when the underlying model changes within the error bars of input observational constraints.

# 3 SIMULATIONS AND DATA CONSTRAINTS

## 3.1 Dark matter halo statistics

As noted in Section 2.2, TRINITY requires only halo population statistics from dark matter simulations, as opposed to individual halo merger trees. We use the peak historical mass ($M_{\text{peak}}$) halo mass functions from Behroozi et al. (2013b), for the cosmology specified in the introduction. These mass functions are based on central halo mass functions from Tinker et al. (2008), with adjustments to include satellite halo number densities as well as to use $M_{\text{peak}}$ instead of the present day mass, as detailed in Behroozi et al. (2013b) and Zhang et al. (2023).

In addition to smooth accretion, haloes also grow via halo mergers. For average halo mass accretion histories, we use the fitting formulae in appendix H of Behroozi et al. (2013b). For halo mergers, we fit

merger rates from the UNIVERSE MACHINE (Behroozi et al. 2019), with full details and formulae in appendix B of Zhang et al. (2023). The calibration and fitting of these halo statistics, including halo mass functions, accretion rates, and merger rates, are suitable for studying haloes from $10^{10} M_{\odot}$ to $10^{15} M_{\odot}$.

## 3.2 Observational data constraints

In TRINITY, we use the following galaxy data to constrain the halo-galaxy connection: galaxy SMFs from $z=0-8$, galaxy SSFRs as functions of galaxy mass from $z=0-8$, galaxy cosmic star formation rate from $z=0-10$, quiescent fractions of galaxies as functions of galaxy mass from $z=0-4$, and galaxy UV luminosity functions (UVLFs) from $z=9-13$. The references of these data can be found in tables 4–8 of Zhang et al. (2023) except for the $z=9-13$ UVLFs from JWSST (Harikane et al. 2023), which was introduced in section 3.2 of Zhang et al. (2024a).

To constrain the galaxy–SMBH connection as well as AGN luminosity distributions in TRINITY, we include the following SMBH statistics from observations: the $z=0 M_{\bullet}-M_{\text{bulge}}$ relation, QLFs from $z=0-5$, QPDFs from $z=0-2.5$, active black hole mass functions from $z=0-5$, and observed SMBH mass distribution of bright quasars from $z=5.8-6.5$. The characteristics and references of these data are summarized in tables 9 and 10 of Zhang et al. (2023).

Although these data sets come with different qualities, depths, redshift ranges and areas, we use them down to the mass/luminosity ranges where the completeness is high enough to make robust constraints. That being said, the redshift–mass/luminosity distribution of the data points do have effects on our constraints and resulting predictions: (1) due to the general constraints in observation depths, we inevitably have fewer data points in the low-mass and/or high-redshift regime. As a result, we can see that the uncertainties of our predictions generally increase with decreasing mass (e.g. Fig. 4) and increasing redshift (e.g. Figs 2 and 15). (2) An exception of this trend is $z=0$, where, e.g. average BHAR seems more uncertain than higher redshifts (Fig. 2). This is because compared to higher redshifts (e.g. $0.1 < z < 0.5$), the Universe has a much smaller volume at $z < 0.1$ (by a factor of $\sim 100$), and thus we do not have as good constraints on SMBH activity as at moderate redshifts.

# 4 RESULTS

In this section, we present the ratio between CBHAR and CSFR as a function of redshift in Section 4.1. Average black hole accretion rates (BHARs) and Eddington ratios as functions of galaxy mass ($M_*$), SMBH mass ($M_\bullet$), and redshift are presented in Section 4.2, the evolution of SFR/BHAR ratio in Section 4.3, and finally, black holes’ efficiency to convert baryonic matter into energy in Section 4.4.

## 4.1 The redshift evolution of the cosmic SFR/BHAR ratio

The top panel of Fig. 1 shows the CBHAR from TRINITY and previous studies as functions of redshift. We also show the CSFR from TRINITY. All these curves show broad peaks at $z \sim 2$ and decrease toward lower and higher redshifts. This trend is driven by the redshift evolution of the input QLF, whose normalization reaches maximum at $z \sim 2$.

Below $z \sim 1$, the CBHAR drops more slowly with cosmic time in TRINITY than in other models. This is likely because in TRINITY,
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** The average black hole accretion rate as a function of SMBH mass ($M_{\bullet}$, top panel) and galaxy mass ($M_{*}$, bottom panel). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.2.</figcaption></figure>

SMBHs release both radiation and kinetic energy during accretion (see Section 2.2), whereas in other models the energy output from SMBH accretion is purely radiative. As a result, TRINITY needs more SMBH accretion to account for the same observed AGN radiative luminosities. We also notice a significant offset in the CBHAR curves between TRINITY and Yang et al. (2018). This is likely due to our different input assumptions: TRINITY adopts the X-ray bolometric correction from Ueda et al. (2014), which is systematically larger than the one adopted by Yang et al. (2018). With the same X-ray QLFs from Ueda et al. (2014), TRINITY needs more accretion to account for larger bolometric quasar luminosities (also see Section 5.2).

The bottom panel shows the CBHAR/CSFR ratio as a function of redshift from TRINITY. The CBHAR/CSFR ratio stays constant at CBHAR/CSFR ~ $2 \times 10^{-3}$ until $z \sim 4$, and decreases by ~2 dex from $z = 4 - 10$. This is because most SMBH growth at $z < 4$ occurs in galaxies with masses $log(M_*/M_{\odot}) > 10.5$, where the $M_{\bullet}-M_*$ relation evolves mildly with redshift (see Section 2.1). At $z > 4$, more SMBH and galaxy growth occurs in lower mass galaxies. Due to the lower normalization of the $M_{\bullet}-M_*$ relation at such high redshifts, the absolute amount of cosmic SMBH growth is smaller, which lowers the CBHAR/CSFR ratio.

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** The black hole Eddington ratio ($\bar{\eta}$) as a function of SMBH mass ($M_{\bullet}$, top panel) and galaxy mass ($M_{*}$, bottom panel). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.2.</figcaption></figure>

## 4.2 The average black hole accretion rates, average Eddington ratios, and Eddington ratio distributions as functions of $M_{\bullet}$, $M_{\odot}$, and redshift

Fig. 2 shows the average BHARs as a function of $M_{\bullet}$ (top panel) and $M_{*}$ (bottom panel). Unless otherwise specified, these average values are calculated for all SMBHs, including active and inactive objects. At each given redshift, BHAR increases with $M_{*}$ or $M_{\bullet}$, with a mild flattening at the massive end between $z \sim 1-2$. At fixed $M_{*}$ or $M_{\bullet}$, BHAR is higher at higher redshift. This is because SMBHs have higher Eddington ratios at higher redshifts (see Section 2.1), which is shown in Fig. 3. At $z \sim 10$, black holes in all galaxies are accreting at a universal and slightly sub-Eddington rate. Below $z \sim 4$, the average Eddington ratio decreases with $M_{*}$ or $M_{\bullet}$ at the massive end. This is also known as the ‘AGN downsizing’ phenomenon, where more massive black holes experience the decline in activity level earlier than smaller black holes. That said, at $z \sim 0.1$, average Eddington ratio does increase slightly with mass above $log M_{\bullet} \sim 8.5$ or $log M_{*} \sim 11$. This is required by the observed QPDFs from Aird et al. (2018), in which massive SMBHs have higher duty cycles than lower mass SMBHs (also see Section 2.1). Physically, this higher AGN duty cycle in low-redshift massive galaxies could be attributed to the dominance of giant elliptical galaxies, where more cool gas is available due to smaller angular momentum compared to in lower
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Black hole Eddington ratio ($\eta$) distributions in different galaxy galaxy mass bins ($M_*$) at $z=0$, $z=3$, $z=6$, $z=10$. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.2.</figcaption></figure>

mass, discy galaxies (see, e.g. Gaspari, Brighenti & Temi 2015 and McDonald et al. 2021). We also note that at above $M_* \sim 10^{11} M_\odot$, the best-fitting average Eddington ratio lies outside of the 16–84th range. This may be caused by the irregular shape of the representative volume in our high-dimensional parameter space (54 parameters in total). In this case, the best-fitting parameters may lie at the edge of the representative volume in MCMC. When transformed into various predictions, such best-fitting parameters thus give predictions that lie at the edge of or even beyond the confidence intervals. Given the fact that the best-fitting parameter gives the highest posterior probability by definition, and that the 16–84th percentiles are more representative of the whole MCMC chain, we always show both for readers with different preferences.

Fig. 4 shows the SMBH Eddington ratio distribution functions (ERDFs) in different galaxy mass bins as functions of redshift. The kink in the ERDFs at $\eta = 0.03$ results from the non-linear scaling relation between Eddington ratio and mass-scaled SMBH accretion rate adopted in TRINITY. For full details of this scaling relation, we refer readers to section 2.7 of Zhang et al. (2023) and appendix A of Zhang et al. (2024b). Due to the lack of $z \gtrsim 3$ QPDFs in our data constraints, our $z \gtrsim 3$ Eddington ratio distributions are predictive extrapolations based on the input $z \lesssim 3$ QPDFs and other data. Overall, we see that the low-Eddington-end slope of ERDFs becomes steeper higher redshifts. This is constrained by (and extrapolated from) the shape evolution of the QPDFs from Aird et al. (2018).

AGN ERDFs also shift to the higher end towards the early universe. This demonstrates that unlike the CBHAR that peaks at $z \sim 2$, typical AGN Eddington ratios increase monotonically towards higher redshifts. At lower Eddington ratios, AGNs likely become radiatively inefficient and convert more accreted matter into kinetic energy than radiation (see Section 2.2). Thus, this redshift evolution in the ERDF suggests that the dominant AGN feedback mode transitions from predominantly radiative at higher redshifts to largely kinetic in the local universe (see, e.g. Somerville et al. 2008; Weinberger et al. 2017). At $z > 0$, the ERDF normalization increases towards higher galaxy masses, reflecting the strong mass-dependence of AGN duty cycles in the input QPDFs (see Section 2.1).

At $z \gtrsim 6$, the average Eddington ratio among all, i.e. active + in-active$^2$ SMBHs is a weak function of host galaxy mass. Therefore, the ERDFs of AGNs in low-mass galaxies are skewed higher to compensate for their low duty cycles. For more detailed discussions on the mass dependence of AGN duty cycles, we refer readers to Zhang et al. (2024b). But very briefly, such low duty cycles imply that early SMBHs in low-mass galaxies may have experienced intermittent accretion events. Hydrodynamical simulations and SAMs have shown that such low AGN duty cycles can be caused by supernova

<footnote n="2">²Unless otherwise noted, ‘active’ means accreting at any non-zero Eddington ratios.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** The redshift evolution of the ratio between average black hole accretion rate (BHAR) and average star formation rate (SSFR) as a function of $M_\bullet$ (top panel) and $M_*$ (bottom panel). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.3.</figcaption></figure>

feedback that removes gas fuel for SMBH growth (e.g. Dubois et al. 2015; Anglés-Alcázar et al. 2017; Tillman et al. 2022).

Beyond $z=0$, the ERDFs of $8 < \log M_* < 9$ galaxies peak at a strongly super-Eddington rate of $\eta > 10$. We would like to point out that such high characteristic Eddington ratios are largely extrapolations towards the low-mass, high-redshift, and high-Eddington ratio regimes based on the QPDFs from Aird et al. (2018). Therefore, we would refrain from taking these typical Eddington ratios at a face value.

## 4.3 The redshift evolution of the SFR/BHAR ratio as a function of $M_\bullet$ and $M_*$

Fig. 5 shows $\overline{\text{BHAR}}/\text{SFR}$ at different redshifts as functions of $M_\bullet$ (top panel) and $M_*$ (bottom panel). In both panels, $\overline{\text{BHAR}}/\text{SFR}$ increases with mass. As we will see in Fig. 6, this is because the ratio of specific BHAR (SBHAR) and specific SFR (SSFR) are nearly mass-independent except for the massive galaxies/SMBHs at $z \sim 0$. In other words, SMBH and galaxy specific growth are in lock step at a given redshift. In addition, the $M_\bullet-M_*$ relation predicted by TRINITY is always superlinear (the slope is $\gamma_{BH} \sim 1.1$ at $z=0$ and $\gamma_{BH} \sim 1.7$ at $z=10$). As a result, more massive SMBHs have

higher $\overline{\text{BHAR}}/\text{SFR}$ due to the bigger $M_\bullet/M_*$ ratios. This is also (expectedly) consistent with how $\overline{\text{BHAR}}/\text{SFR}$ changes with halo mass and redshift, which was presented in Zhang et al. (2023). We also notice that the normalization of $\overline{\text{BHAR}}/\text{SFR}$ decreases more strongly towards higher redshift at the low-$M_*$ end. This is because the $M_\bullet-M_*$ relation decreases more significantly in this mass and redshift regime. At a fixed specific accretion rate ratio (Fig. 6), smaller SMBHs would have lower absolute accretion rates compared to their galaxies.

In Fig. 6, we show the $\overline{\text{SBHAR}}/\text{SSFR}$ ratio as a function of $M_\bullet$, $M_*$, and redshift, since $\overline{\text{SBHAR}}/\overline{\text{SSFR}} = (\overline{\text{BHAR}}/\overline{\text{SFR}})/(M_\bullet/M_*)$, $\overline{\text{SBHAR}}/\overline{\text{SSFR}} = 1$ means that SMBHs and galaxies experience the same amount of fractional growth, and share the same mass doubling (or e-folding) time-scale. On the other hand, $\overline{\text{SBHAR}}/\overline{\text{SSFR}} > 1$ means that SMBHs gain more fractional growth than galaxies, which is the case above $z \sim 1$. This means that from $z \sim 10$ to $z \sim 1$, SMBHs have slightly higher specific growth rates than host galaxies. This results from the superlinear slope and increasing normalization of the $M_\bullet-M_*$ relation predicted by TRINITY, which is jointly constrained by the $z=0$ $M_\bullet-M_*$ relation, QLFs, and QPDFs (see Section 2.1). At $z \sim 0$, we see a strong increase in $\overline{\text{SBHAR}}/\text{SSFR}$ with mass at the massive end. This is constrained by the combination of galaxy and SMBH data at low redshifts. Specifically, the observed SSFRs of massive galaxies drop significantly, whereas SBHARs (or equivalently, SMBH Eddington ratios) are required to be higher than SSFRs by the QPDFs from Aird et al. (2018). The high $\overline{\text{BHAR}}/\text{SFR}$ and $\overline{\text{SBHAR}}/\text{SSFR}$ values for $M_* > 10^{11} M_\odot$ at $z \sim 0$ are in line with the measurements by McDonald et al. (2021) using X-ray cavities and $H\alpha$ luminosities in giant elliptical galaxies, which are not used to constrain TRINITY. We also note that Eddington ratios of these massive SMBHs still lie below $\eta \lesssim 1$ per cent, where kinetic energy makes up a significant portion of total AGN energy output (e.g. Narayan & Yi 1994; Nagar et al. 2005). Physically, this is consistent with the scenario where the kinetic AGN feedback causes and/or keeps the quiescence of massive galaxies (e.g. Somerville et al. 2008; Weinberger et al. 2017).

The top panel of Fig. 7 shows $\overline{\text{SBHAR}}/\text{SSFR}$ as a function of $M_{\text{peak}}$ and $z$. Similar to in Fig. 6, $\overline{\text{SBHAR}}/\overline{\text{SSFR}}$ is roughly constant except for the most massive haloes at low redshifts. This increase starts earlier in more massive objects. The bottom panel of Fig. 7 shows the $\overline{\text{SBHAR}}/\text{SSFR}$ ratio histories of different halo populations. Quantitatively, $\overline{\text{SBHAR}}/\overline{\text{SSFR}}$ can reach $\sim 1000$ for the most massive haloes at $z \sim 0$. This is much higher than the highest values seen in Fig. 6, i.e. $\sim 30$. This apparent discrepancy is due to the Eddington bias (Eddington 1913): for galaxies or SMBHs at a given mass, their host halo mass distributions are dominated by smaller haloes with overmassive galaxies/SMBHs rather than bigger haloes with typical galaxies/SMBHs. Hence, $\overline{\text{SBHAR}}/\overline{\text{SSFR}}$ at fixed galaxy/SMBH masses will reflect the properties of less-massive haloes, which have lower $\overline{\text{SBHAR}}/\overline{\text{SSFR}}$ values.

## 4.4 Baryon conversion efficiencies of SMBHs

The top panel of Fig. 8 shows the ratios between the average star formation rate (SFR) and the baryon accretion rate ($f_b \dot{M}_{\text{peak}}$) as a function of $M_{\text{peak}}$ and $z$. The baryon accretion rate is assumed to be the average halo mass accretion rate multiplied by the universal baryon fraction, $f_b = 0.157$. The halo mass accretion rate $\dot{M}_{\text{peak}}$ is calculated with the fitting formula from appendix H of Behroozi et al. (2013b). This SFR/($f_b \dot{M}_h$) ratio represents galaxies’ instantaneous efficiency for converting accreted baryonic matter into stars. At
all redshifts, this efficiency peaks broadly at $M_{\text{peak}} \sim 10^{12} M_{\odot}$, suggesting that the maximal baryon conversion efficiency is a weak function of redshift (Behroozi et al. 2013a). The bottom panel of Fig. 8 shows the histories of this ratio in different halo populations. All the haloes have increasing baryon conversion efficiencies before reaching $M_{\text{peak}} \sim 10^{12} M_{\odot}$, and keep decreasing consistently afterwards.

The top panel of Fig. 9 shows the ratios between the average black hole accretion rate (BHAR) and $f_b M_h$ as a function of $M_{\text{peak}}$ and $z$. Given the similar $M_{\text{peak}}$ and $z$ dependency between the BHAR and SFR, the baryon conversion efficiency of black holes also peaks broadly at $M_{\text{peak}} \sim 10^{12} M_{\odot}$ across cosmic time. The bottom panel of Fig. 9 shows the histories of this efficiency in different halo populations. The overall trend is the same as shown in the bottom panel of Fig. 8, but the average efficiencies stay constant or even increase slightly below $z \sim 1$. This behaviour is, again, driven by the QPDFs from Aird et al. (2018), which require significant residual AGN activity in low-redshift massive galaxies. Figs 8 and 9 show that the growth of SMBHs closely follows that of host galaxies in the high-redshift and/or low-mass regime. Below $z \sim 1$, massive black holes become more and more efficient in converting baryonic mass compared to their host galaxies. Overall, this change in the galaxy-SMBH growth correlation is consistent with the scenario where host galaxies and SMBHs regulate each other's growth, and/or they grow via the same source of fuel except for low-redshift massive objects. In the low-redshift high-mass regime, negative AGN feedback suppresses star formation.

## 4.5 Specific growth rates of SMBHs versus haloes

Fig. 10 shows the ratio between average specific BHAR and average specific halo mass accretion rate, SBHAR/SMAR, as a function of halo mass and redshift. Overall, SBHAR/SMAR decreases towards the massive end at a fixed redshift, except for $M_{\text{peak}} \gtrsim 10^{14} M_{\odot}$ haloes at $z \lesssim 2$. In this mass and redshift regime, the SBHAR/SMAR stays flat or increases with halo mass due to the residual AGN activities required by the QPDFs from Aird et al. (2018). At $z \lesssim 1$, SBHAR/SMAR is significantly larger than 5 for $M_{\text{peak}} \lesssim 10^{12} M_{\odot}$ haloes, indicating that SMBHs double their masses much faster than their host haloes. This is mainly due to the strong decline in halo mass accretion rate at a fixed halo mass towards lower redshifts, especially when $M_{\text{peak}} \lesssim 10^{12} M_{\odot}$. At $z \gtrsim 5$, SBHAR/SMAR $\sim 2$ for all halo populations, indicating similar paces for fractional SMBH and halo growth. The slightly faster growing pace of SMBHs at $z \gtrsim 5$ than haloes and galaxies provides a potential explanation for the overmassiveness of many quasars at such redshifts: In ΛCDM, early galaxies and SMBHs grow along with their haloes by converting accreted baryonic matters without strong feedback. Some quasars become overmassive relative to the $z = 0 M_{\odot} - M_{\star}$ relation by $z \sim 5$, because they doubled their masses slightly faster than their host haloes and galaxies (also see Section 5.4.1).

## 5 COMPARISON WITH PREVIOUS STUDIES AND DISCUSSION

Section 5.1 compares the specific black hole accretion rate from TRINITY and Merloni & Heinz (2008); Section 5.2 compares the relative growth rates of SMBHs and galaxies from TRINITY and previous studies; in Section 5.4, we carry out a case study on the $M_{\bullet}$ and $M_{\star}$ evolution of overmassive JWST AGNs, to ascertain how the descendants of these AGNs compare to the local $M_{\bullet}-M_{\star}$ relation.

<figure><img src="image_9.png" /><figcaption>**Figure 6.** The ratios between average specific black hole accretion rate (SBHAR) and average specific star formation rate (SSFR) as a function of $M_{\bullet}$ (top panel) and $M_{\star}$ (bottom panel). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.3.</figcaption></figure>

### 5.1 Specific black hole accretion rates

Fig. 11 shows specific black hole accretion rates (SBHARs) as functions of redshift and black hole mass from TRINITY and Merloni & Heinz (2008, hereafter MH08). Overall, we find broad agreement between both studies for black holes with $M_{\bullet} \sim 10^8 M_{\odot}$ at $0 \lesssim z \lesssim 3$, but also slightly stronger redshift evolution of SBHAR in TRINITY between $3 \lesssim z \lesssim 5$. This demonstrates that black holes experience a strong decline in activity level in TRINITY. This is not seen in MH08. A potential reason is that they adopted systematically lower quasar luminosity functions from Silverman et al. (2008), and did not have to produce as highly active black holes to account for the observed luminosities.

### 5.2 BHAR/SFR ratios as a function of stellar mass and redshift

As shown in Section 4.3, the BHAR/SFR ratio bears important insights into the potential galaxy-SMBH co-evolution scenarios. This is especially true when BHAR/SFR is shown as a function of stellar mass and redshift. Here, we compare the results from TRINITY, Aird et al. (2019) and Yang et al. (2018).

Aird et al. (2019) calculated BHAR/SFR ratios as a function of stellar mass and redshift. Their BHARs are converted from AGN X-
<figure><img src="image_10.png" /></figure>

**Figure 7.** Top Panel: the ratios between average specific black hole accretion rate (SBHAR = BHAR/$M_*$) and average specific star formation rate (SSFR) as a function of redshift and $M_{\text{peak}}$ for our best-fitting model. The white solid lines are the average mass growth curves of haloes with $M_{\text{peak}} = 10^{12}$, $10^{13}$, $10^{14}$, and $10^{15} M_{\odot}$ at $z=0$. Bottom Panel: SBHAR/SSFR ratio histories as a function of halo mass at $z=0$. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.3.

ray luminosities assuming a certain radiative efficiency and that all the accretion energy is converted into radiation. Given the systematic differences in these assumptions between TRINITY and Aird et al. (2019), we opt to compare the average $L_X$ to average SFR ratios from both studies, which are shown in Fig. 12. Since Aird et al. (2019) used the same data (from Aird et al. 2018) to calculate the quasar probability distribution functions in this work, this is effectively a sanity check. From Fig. 12 $M_* = 10^{10} M_{\odot}$, there is a decent agreement between TRINITY and Aird et al. (2019). None the less, discrepancy exists below this mass. This is mostly due to the inability of TRINITY to fully capture the exact shape of the QPDFs provided by Aird et al. (2018) (see fig. 9 of Zhang et al. 2023). Specifically, below $M_* = 10^{10} M_{\odot}$, TRINITY overpredicts more high-SBHAR AGNs and underpredicts low-SBHAR ones, which directly translates into this excess of average X-ray luminosity among low-mass galaxies. We note that we opt not to add more model parameters to to further improve the fit to the QPDFs, because it is not straightforward to ascertain whether the QPDF shapes are driven by prior assumptions in Aird et al. (2018).

<figure><img src="image_11.png" /></figure>

**Figure 8.** Top Panel: the ratios between the average star formation rate (SFR) and the baryon accretion rate ($f_b \dot{M}_h$) as a function of $M_{\text{peak}}$ and $z$. The white solid lines are the average mass growth curves of haloes with $M_{\text{peak}} = 10^{12}$, $10^{13}$, $10^{14}$, and $10^{15} M_{\odot}$ at $z=0$. Bottom Panel: SFR/($f_b \dot{M}_h$) ratio histories as a function of $M_{\text{peak}}$ at $z=0$. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.4.

Fig. 13 shows the BHAR/SFR ratios from TRINITY (solid lines) and Yang et al. (2018, dashed lines) as functions of redshift and stellar mass. A significant discrepancy exists between TRINITY and Yang et al. (2018): (a) BHAR/SFR is overall larger in TRINITY and (b) the BHAR/SFR increases strongly with stellar mass in Yang et al. (2018) at all redshifts. But in TRINITY, the mass dependence is much weaker at higher redshifts. As we have pointed out in Section 4.1, this discrepancy likely results from different input assumptions. Specifically, Yang et al. (2018) adopted a smaller X-ray bolometric correction than is adopted in Zhang et al. (2023). With the same input X-ray QLFs from Ueda et al. (2014), TRINITY would predict more total AGN accretion energy and needs more accretion to account for such energy.

Given this difference in our input assumption, we also directly compare the ratios of the average X-ray luminosity to average SFR ($L_X$/SFR) in Fig. 14. Compared to Fig. 13, the curves from the two studies have more similar normalizations. However, there is still some residual discrepancy between TRINITY and Yang et al. (2018) at the massive end. There are several reasons that could
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Top Panel: the ratios between the average black hole accretion rate (BHAR) and the baryon accretion rate as a function of $M_{\text{peak}}$ and $z$. The white solid lines are the average mass growth curves of haloes with $M_{\text{peak}} = 10^{12}$, $10^{13}$, $10^{14}$, and $10^{15}$ M$_{\odot}$ at $z=0$. Bottom Panel: BHAR/($f_b M_h$) ratio histories as a function of $M_{\text{peak}}$ at $z=0$. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.4.</figcaption></figure>

explain the disagreement in Fig. 14: (a) TRINITY was constrained by the QPDFs from Aird et al. (2018), which provide qualitatively the same kind of information as the AGN catalogues used by Yang et al. However, the exact mathematical procedures do differ between the model used by Aird et al. (2018) and Yang et al. (2018), which could lead to part of the discrepancy; (b) in TRINITY, we excluded the QPDFs at the brightest end, where the exact probability values may be driven by the smooth prior imposed by Aird et al. (2018). While this choice avoids fitting to prior assumptions rather than real data, it also decreases the amount of constraint from the QPDFs, which may affect TRINITY's accuracy; (c) In the observational data, the AGN sample size decreases strongly with redshift, mass, and AGN luminosity for both Yang et al. (2018) and TRINITY, which might cause less stringent constraints on the models' high-redshift, high-mass, and/or bright end behaviours; (d) in Yang et al. (2018), there was no explicit requirement that the black hole populations should be evolved self-consistently, which is present in TRINITY.

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Top Panel: average specific SMBH accretion rate (SBHAR) and the average specific halo mass accretion rate (SMAR) as a function of $M_{\text{peak}}$ and $z$. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. See Section 4.5.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** The specific black hole accretion rates (SBHAR) from TRINITY and MH08. The solid lines and the surrounding shaded regions are from TRINITY, representing the best-fitting model and the statistical uncertainties from MCMC, respectively. The lighter shaded regions within the dotted curves represent the range that different model span in MH08. See Section 5.1.</figcaption></figure>

## 5.3 Early SMBH growth into $z \sim 6$ quasars

Among the 275 quasars found at $z \gtrsim 6$ before JWST, many have elevated $M_{\bullet}$ compared to the $z = 0 M_{\bullet}-M_{*}$ relation. Therefore, these SMBHs must have either higher seed-to-galaxy mass ratio and/or had outgrown their host galaxies by $z \sim 6$. However, it remains unclear when and how these SMBHs have outgrown their host galaxies. In Section 4.3, we have shown that at a fixed $M_{\bullet}$ or $M_{*}$, the specific growth rates of SMBHs are higher than their host galaxies at $z \gtrsim 6$. Therefore, it is plausible that these quasars did not start out as overmassive, but instead outgrew their host galaxies between SMBH seeding and $z \sim 6$. To fully demonstrate this, we show the specific growth rates of haloes, galaxies, and SMBHs of the halo population with $M_{\text{peak}} = 10^{12} M_{\odot}$ at $z = 6$ in Fig. 15. Again, we see that haloes, galaxies, and SMBHs grow at similar specific rates between $6 < z < 15$, but SMBHs slightly outpace galaxies and haloes. This leads to the overmassiveness of the bright $z \gtrsim 6$ quasars lying well above the local $M_{\bullet}-M_{*}$ relation. In ΛCDM, both haloes
<figure><img src="image_15.png" /><figcaption>**Figure 12.** $L_X/SFR$ ratios as a function of stellar mass and redshift. The individual data points come from Aird, Coil & Georgakakis (2018, 2019). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. For visual clarity, an incremental offset of -0.25 dex is applied to the data and prediction each redshift bin except for $0.1 < z < 0.5$. See Section 5.2.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 13.** BHAR/SFR ratios as a function of stellar mass and redshift. The dashed lines come from Yang et al. (2018). The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. For visual clarity, an incremental offset of -0.45 dex is applied to the data and prediction each redshift bin except for $z = 0.5$. See Section 5.2.</figcaption></figure>

and galaxies grow at around the Eddington rate. The slightly higher specific SMBH accretion rate in Fig. 15 shows that certain physical mechanisms are at play for efficient removal of gas angular momenta and feeding central SMBHs. The same argument also applies to new, fainter AGNs detected by JWST between $1 < z < 11$, as shown in Section 5.4.

## 5.4 Case study: the redshift evolution of $M_\bullet/M_*$ ratios for JWST AGNs

With latest data from JWST, many AGNs and AGN candidates have been shown to have elevated $M_\bullet/M_*$ ratios compared to the $z=0$

<figure><img src="image_17.png" /><figcaption>**Figure 14.** The $L_X/SFR$ ratios as a function of stellar mass and redshift. The dashed lines come from Yang et al. (2018). The systematic offset between TRINITY and Yang et al. (2018) is largely reduced compared to Fig. 13. This is due to systematic differences in the conversion from X-ray AGN luminosities to black hole accretion rates: Yang et al. assumed that all the energy released from black hole accretion is converted into radiation, but in TRINITY, part of this energy is released in the form of kinetic jets or outflows. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. For visual clarity, an incremental offset of -0.45 dex is applied to the data and prediction each redshift bin except for $z = 0.5$. See Section 5.2.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 15.** The average specific growth rates of $M_{\text{peak}} = 10^{12} M_{\odot}$ haloes (fixed mass across redshifts), the galaxies, and the SMBHs hosted by these haloes as functions of time. The dashed horizontal line represents the Eddington rate, which is based on the best-fitting energy efficiency from TRINITY, i.e. $\epsilon = 0.067$. See Section 5.3.</figcaption></figure>

scaling relation (e.g. Kormendy & Ho 2013; Reines & Volonteri 2015). Although these AGNs may not be fully representative of the underlying SMBH populations due to selection effects (see, e.g. Li et al. 2024), the mere existence of these $M_\bullet$ calls for an examination of how these SMBHs evolve along with their host galaxies and investigation into their possible origins. In this section, we take the JWST AGNs and AGN candidates from Bogdan et al. (2023), Harikane et al. (2023), Kokorev et al. (2023), Larson et al. (2023), Maiolino et al. (2023), Mezcua et al. (2024), and Übler et al. (2023) to examine the redshift evolution of their $M_\bullet/M_*$ ratios to answer
<figure><img src="image_19.png" /><figcaption>**Figure 16.** The predicted high-redshift evolution of the $M_*/M_*$ ratios for JWST AGNs and AGN candidates above $z \sim 1$. Each AGN or AGN candidate is labelled with solid markers, colour-coded by galaxy mass ($M_*$) at their respective redshifts. TRINITY's predictions of the average $M_*/M_*$ histories are shown in curves. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. We only show uncertainties for two AGNs in shaded regions with solid edges, representing the upper ($\sim 4.82$ dex) and lower ($\sim 0.82$ dex) limits in these uncertainties at $z = 15$. See Section 5.4.1.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 17.** Similar to Fig. 16. We also only show uncertainties for two AGNs (shaded regions with solid edges), representing the upper ($\sim 1.32$ dex) and lower ($\sim 0.14$ dex) limits in $M_*$ uncertainties at $z = 15$. See Section 5.4.1.</figcaption></figure>

the following questions: (1) When did the SMBHs gain so much mass compared to the host galaxies? (2) Will they remain outliers from the $z = 0 M_* - M_*$ relation if they evolve according to TRINITY's predictions? Specifically, we take the average SMBH Eddington ratios (see Section 4.2) and SSFRs as functions of galaxy mass ($M_*$) and redshift from the fiducial TRINITY model, and evolve these JWST AGNs from their respective redshifts up to $z \sim 15$ and down to $z = 0$, respectively, to answer each of these two questions.

### 5.4.1 Early evolution from $z \sim 15$

Fig. 16 shows the predicted $M_*/M_*$ evolution of JWST AGNs/AGN candidates from $z \sim 15$ down to their respective redshifts. The

<figure><img src="image_21.png" /><figcaption>**Figure 18.** The predicted low-redshift evolution of the $M_*/M_*$ ratios for JWST AGNs and AGN candidates above $z \sim 1$. Each AGN or AGN candidate is labelled with solid markers, colour-coded by $M_*$ at $z = 0$. TRINITY's predictions of the average $M_*/M_*$ histories are shown in solid curves. The shaded regions show the 68 per cent confidence intervals inferred from the model posterior distribution. These intervals reflect the ranges that the model predictions are allowed to vary, while keeping the underlying SMBH properties matching input data constraints within error bars. We only show uncertainties for two AGNs (shaded regions with solid edges), representing the upper ($\sim 1.19$ dex) and lower ($\sim 0.14$ dex) limits in these uncertainties at $z = 0$. See Section 5.4.2.</figcaption></figure>

individual symbols, curves, and shaded regions are colour-coded by $M_*$ at these AGNs' respective redshifts. According to TRINITY's prediction, the $z=10.3$ SMBH called UHZ1 from Bogdan et al. (2023) and the $z=8.5$ SMBH called UNCOVER 20466 from Kokorev et al. (2023) are too overmassive and thus too rare to be found in the universe. As a result, TRINITY predictions for these two AGNs are likely inaccurate. For completeness, we still show them in our figures, but in dotted lines to remind readers of the potential inaccuracy. Since average specific BHAR is consistently larger than SSFR at $z \gtrsim 1$, the $M_*/M_*$ of all JWST AGNs decrease towards higher redshifts. In other words, these SMBHs may not be always overmassive since their birth. Instead, they experience faster exponential growth than their host galaxies by the time of detection, and have become overmassive along the way. To maintain visual clarity, we only show the $M_*/M_*$ uncertainties for two AGNs, representing the upper and lower limits of such uncertainties (i.e. 4.82 and 0.82 dex at $z \sim 15$, respectively).

Fig. 17 shows the $M_*$ evolution of JWST AGNs from $z \sim 15$ down to their respective redshifts. Given the high SMBH Eddington ratios at higher redshifts (also see Section 4.2), these JWST AGNs experience swift increase in $M_*$ between $8 < z < 15$. Towards $z \gtrsim 10$, $M_*$ of the $z \lesssim 6$ AGNs drop below $10^2 M_\odot$, which is the Population III star remnant regime for seed SMBHs. At face value, these predictions mean that new mechanisms are needed to overcome various physical hurdles against fast growth of light SMBH seeds, such as erratic motion, etc. On the flip side, the direct collapse SMBH seeding scenario is disfavoured for most of these JWST AGNs: out of 36 AGNs, only 3 had 99th percentile values of $M_* > 10^4 M_\odot$, the largest of which is $M_* \sim 5.5 \times 10^4 M_\odot$.

The increase in $M_*/M_*$ from $z \sim 15$ to $z \sim 8$ is at odds with the conclusions from Natarajan et al. (2024) and Scoggins & Haiman (2024): Natarajan et al. (2024) compared the multiwavelength properties of UHZ1 with theoretical templates from Natarajan et al. (2017), and found that the UHZ1 is likely seeded with direct collapse SMBHs with $M_* > 10^4 M_\odot$; Scoggins & Haiman (2024) built $z \gtrsim 10$
star formation histories (SFHs) and on halo merger trees generated by the extended Press–Schechter (EPS) theory (Press & Schechter 1974). By adding constant-Eddington-ratio SMBH growth on these SFHs, Scoggins & Haiman (2024) found that $M_\bullet/M_\star$ decreases or stay constant over time at $z \gtrsim 10$. TRINITY disfavours such $M_\bullet/M_\star$ evolution and seeding scenario, because SMBHs are predicted to accrete at super-Eddington rates at $z \gtrsim 12$. With $4 < z < 11$ AGNs as boundary conditions, the $M_\bullet/M_\star$ must have increased from $z \sim 15$ to $z \sim 8$, leading to smaller seed masses. On the contrary, Scoggins & Haiman (2024) capped SMBH growth at the Eddington rate, which led to slower SMBH growth. Taking Natarajan et al. (2024) and Scoggins & Haiman (2024) at face value, this discrepancy is likely due to the lack of input constraints at $z \sim 10$ in TRINITY, which entails extrapolating low-redshift Eddington ratio evolution towards $z \gtrsim 10$. In the current version of TRINITY, multiwavelength properties of SMBHs are not forward modelled, so it is difficult for us to use UHZ1 data like Natarajan et al. (2024) did. In the future, we will implement such models in TRINITY to enable direct constraints from AGN SEDs.

### 5.4.2 Later evolution down to $z=0$

In addition to the evolution to higher redshifts, JWST AGNs' $M_\bullet$ and $M_\star$ evolution down to $z=0$ is also informative for ascertaining: (1) whether these AGNs' descendants stay overmassive compared to the local $M_\bullet-M_\star$ relation, and (2) whether evolutional connections exist between these AGNs at different redshifts. In Fig. 18, we show the $M_\bullet/M_\star$ evolution of JWST AGNs from their respective redshifts down to $z=0$. The individual symbols and evolution curves are colour-coded by the predicted $M_\star$ at $z=0$. At a fixed redshift, the $M_\bullet/M_\star$ uncertainty is positively correlated with the initial $M_\bullet/M_\star$. This is because a higher $M_\bullet/M_\star$ corresponds to a stronger outlier from the $M_\bullet-M_\star$ relation predicted by TRINITY, and it takes more extrapolations to predict their evolutions. In addition, AGNs with higher initial redshifts also tend to accumulate larger $M_\bullet/M_\star$ error bars due to longer times for evolution. For visual clarity, we only show the widest and narrowest $M_\bullet/M_\star$ ranges at $z=0$, which are $\sim 1.19$ and $0.14$ dex, respectively.

According to TRINITY, the $M_\bullet/M_\star$ ratios of all $z \gtrsim 4$ JWST AGNs have increased by $z=0$. This indicates that these AGNs will remain overmassive outliers compared to the local $M_\bullet-M_\star$ relation. In particular, the two AGNs with highest initial $M_\bullet/M_\star$ would have $M_\bullet/M_\star > 1$ at $z=0$. Mathematically, this result comes from the assumption that all SMBHs with the same host galaxy mass share the same average Eddington ratio distribution. Physically, however, this assumption may not be true for such overmassive outliers, since the amount of gas fuel available in their (relatively) small host galaxies may not be sufficient to maintain the same Eddington ratios. In light of this, such high $M_\bullet/M_\star$ values are likely inaccurate and we show them in dotted lines only for completeness.

$z > 8$ JWST AGNs would experience a significant increase in $M_\bullet/M_\star$ before flattening at lower redshifts. The same trend is also seen in lower redshift AGNs with lower $M_\star$. This trend is due to the strong increase in the typical $M_\bullet$ at a fixed $M_\star$ for low-mass galaxies from $z=10$ to $z \sim 2$ (see Section 2.1). With higher redshifts, the $z > 8$ AGNs left the low-mass regime earlier than those at $4 < z < 8$, so they also reached the plateau in $M_\bullet/M_\star$ before the latter did. The other AGNs at $4 < z < 8$ are already in the massive regime, where the $M_\bullet-M_\star$ relation evolves more mildly. As a result, there is not as much initial increase in $M_\bullet/M_\star$ compared to their less massive counterparts. Below $z \sim 1$, $M_\bullet/M_\star$ increases in galaxies above $\sim 10^{11} M_\odot$ ($z=0$) at $z=0$, and slightly decreases for those below the same mass (e.g.

<figure><img src="image_22.png" /><figcaption>**Figure 19.** Similar to Fig. 18. We also only show uncertainties for two AGNs (shaded regions with black solid edges, representing the upper ($\sim 1.32$ dex) and lower ($\sim 0.14$ dex) limits in $M_\bullet$ uncertainties at $z=0$. See Section 5.4.2.</figcaption></figure>

those from Mezcua et al. 2024). This prediction is driven by the QPDFs from Aird et al. (2018), which requires higher average mass growth for SMBHs in massive galaxies than less-massive ones. But this later decrease in $M_\bullet/M_\star$ is very small, so that their descendants will remain outliers compared to the local $M_\bullet-M_\star$ relation.

Despite the similar $M_\bullet/M_\star$ ratios, it is difficult to connect the $1 < z < 3$ JWST AGNs to their higher redshift analogues only with observational data. But as shown in Fig. 18, such connections can be made via the $M_\bullet/M_\star$ evolution tracks from TRINITY. TRINITY suggests that these two AGN groups may be the same population detected at different life stages. To further demonstrate this, we show the $M_\bullet$ evolution tracks of these JWST AGNs in Fig. 19. Similar to Fig. 18, we only show the widest and narrowest uncertainty regions ($1.32$ and $0.14$ dex at $z=0$, respectively) for visual clarity. Many $4 < z < 6$ and $1 < z < 3$ JWST AGNs share not only the $M_\bullet/M_\star$, but also $M_\bullet$ evolution tracks. This means that these two AGN groups do have very similar average $M_\star$ and $M_\bullet$ growth histories, and are likely progenitors/descendants of one another.

None the less, there are some caveats behind such connections. First, the average $M_\bullet$ and $M_\star$ evolution tracks in Figs 16–19 are calculated using the populational average SMBH and galaxy growth rates and then applied to individual systems. In doing so, we implicitly assume that the ergodic hypothesis holds for SMBH growth. But in the real Universe, individual SMBH growth histories may deviate significantly from populational average.

Secondly, we also ignore the possibility that these galaxies will have merged into larger galaxies, which will likely lower the $M_\bullet/M_\star$ ratio. In light of this, it is not straightforward to predict the number density of local outliers from the $M_\bullet-M_\star$ relation by evolving overmassive JWST AGNs down to $z=0$. As a result, here, we instead estimate the number densities of $M_\bullet > 10^{10} M_\odot$ SMBHs at $z \sim 0$. By integrating over the $z=0$ SMBH mass function from TRINITY (Zhang et al. 2023), this number density is $1.3_{-0.4}^{+1.2} \times 10^{-7}$ Mpc$^{-3}$. We thus expect to see $1.9_{-0.6}^{+1.8}$ SMBHs with $M_\bullet > 10^{10} M_\odot$ out to 150 Mpc, which is consistent with the compilation by Greene et al. (2016).

## 6 CONCLUSIONS

In this work, we continue to use the empirical TRINITY model of halo–galaxy–SMBH connection to study the SMBH mass evolution in different galaxies. Compared to previous studies that are typically
focused on one or two kinds of observables, TRINITY features
in the ability to self-consistently match a comprehensive set of
observational data for galaxies from $z=0-13$ and SMBHs from
$z=0-6.5$. As a result, TRINITY has extracted information that is
only accessible with the joint constraints by multiple data sets. **Key
results are as follows:**

(i) Like other previous empirical models, TRINITY reproduces the trend that the CBHAR peaks at $z \sim 2$, and decreases towards lower and higher redshifts. Such results are driven from the redshift evolution of quasar luminosity functions in our input data constraints (Fig. 1, Section 4.1).

(ii) Compared to previous studies, TRINITY further expands the prediction to $z=10$, showing that although the CBHAR/CSFR ratio remains constant at $\sim 2 \times 10^{-3}$ over $z=0-4$, it does decrease by $\sim 2$ orders of magnitude from $z=4-10$. This is because in the early universe, SMBH growth happen mostly in low-mass galaxies, where the typical SMBH mass and accretion rate decreases strongly towards higher redshifts (Fig. 1, Section 4.1).

(iii) AGNs experience downsizing, in the sense that average Eddington ratios start to decrease earlier for more massive SMBHs. This *does not* hold for average black hole accretion rates, which do not decrease towards higher masses at low redshifts (Figs 2 and 3, Section 4.2).

(iv) At $z > 6$, haloes, galaxies, and SMBHs all have specific growth rates near the Eddington rate. This provides a natural explanation for the emergence of massive black holes at these redshifts. That is, ΛCDM causes haloes to assemble at around the Eddington rate between their formation and $z \sim 6$, and the galaxies and black holes within them are fed with gas at the same specific growth rates. As a result, it is no surprise that massive black holes at $z=6$ would have needed to grow at around the Eddington rate to reach their observed masses. Below $z=6$, specific halo growth rates continue declining, resulting in fewer and fewer massive black holes that can continue growing at the Eddington rate (Section 4.2).

(v) In general, the low-Eddington ratio end slope of AGN Eddington ratio distributions increases towards higher redshifts. This prediction is driven by (and extrapolated from) the shape evolution of the QPDFs from Aird et al. (2018). AGNs in lower mass galaxies generally have lower duty cycles, which is constrained by the mass-dependence of input QPDF normalizations. Given the weak mass dependence of average Eddington ratio among all (i.e. active + inactive) SMBHs, lower mass AGNs have higher typical Eddington ratios (Fig. 4, Section 4.2).

(vi) The ratio between the average BHAR and SFR ($\overline{\text{BHAR}}$/$\overline{\text{SFR}}$) increases with SMBH mass $M_\bullet$, but depends weakly on redshift. Due to the strong decrease in the typical $M_\bullet$ at a fixed galaxy mass $M_*$, $\overline{\text{BHAR}}$/$\overline{\text{SFR}}$ at a fixed low $M_*$ also decreases significantly with increasing redshift (Fig. 5, Section 4.3).

(vii) At each redshift, the ratio between the average specific BHAR and SFR ($\overline{\text{SBHAR}}$/$\overline{\text{SSFR}}$) is a weak function of $M_\bullet$, $M_*$, and halo mass $M_{\text{peak}}$, except for massive objects in the local universe. The increase of $\overline{\text{SBHAR}}$/$\overline{\text{SSFR}}$ in massive SMBHs/galaxies is jointly constrained by the following observations: (1) the QPDFs from Aird et al. (2018), which requires higher AGN activity level for massive objects, and (2) the low galaxy SFRs among massive galaxies (Figs 6 and 7, Section 4.3). Physically, the higher AGN duty cycles in massive galaxies could be due to the availability of cool gas with low angular momentum in elliptical galaxies that dominate high $M_*$ bins (Gaspari et al. 2015; McDonald et al. 2021). The elevated $\overline{\text{SBHAR}}$/$\overline{\text{SSFR}}$ in massive SMBHs/galaxies is also consistent with

the scenario that kinetic AGN feedback maintains the quiescence of
$z \lesssim 1$ massive galaxies.

(viii) The galaxy and SMBH baryonic conversion efficiencies show similar halo mass and redshift dependencies, except for low-redshift massive haloes/galaxies/SMBHs. This means that SMBH growth closely follows galaxy growth for high-redshift and/or low-mass objects. In low-redshift massive galaxies, SMBHs are more efficient than galaxies at converting baryon into their masses. This is consistent with the scenario where negative AGN feedback keep low-redshift massive galaxies quenched (Figs 8 and 9, Section 4.4).

(ix) SMBHs in $z \sim 6$ quasars with $M_\bullet \sim 10^9 - 10^{10} M_\odot$ have grown at around the Eddington rate, which is slightly higher than the specific growth rates of host haloes and galaxies. This is consistent with a simple explanation for how these quasars have grown to such masses: at these redshifts, dark matter haloes experience ~Eddington specific growth rates, driving ~Eddington specific growth rates in both galaxies and SMBHs (Figs 10 and 15, Section 4.5 and 5.3).

With the average SMBH and galaxy assembly histories from TRINITY, we carry out a case study on JWST AGNs between $1 \lesssim z \lesssim 11$. Our main conclusions are:

(i) Overmassive JWST AGNs are not overmassive from their birth. Instead, their $M_\bullet/M_*$ mass ratios are below $\sim 10^{-4}$ above $z \sim 10$. Significant growth in $M_\bullet/M_*$ took place between $6 \lesssim z \lesssim 10$, making most of the JWST AGNs overmassive by $z \sim 8$ (Figs 16 and 17, Section 5.4.1).

(ii) Overmassive JWST AGNs at $1 < z < 11$ will experience either mild increase or very slight decrease in $M_\bullet/M_*$ down to $z=0$, and will thus remain outliers compared to the local $M_\bullet$-$M_*$ relation. We also found that many $4 < z < 6$ and $1 < z < 3$ JWST AGNs have very similar $M_\bullet$ and $M_*$ growth histories, so they are likely progenitors/descendants of each other (Figs 18 and 19, Section 5.4.2).

## ACKNOWLEDGEMENTS

We thank the referee, Darren Croton, for the constructive comments. We also thank Alison Coil, Sandy Faber, Jenny Greene, Melanie Habouzit, David Koo, Andrey Kravtsov, Junyao Li, Joel Primack, George Rieke, Marcia Rieke, Xuejian Shen, Yue Shen, Rachel Somerville, Fengwu Sun, Wei-Leong Tee, and Minghao Yue for very valuable discussions.

Support for this research came partially via National Aeronautics and Space Administration (NASA) Astrophysics Theory Program (ATP) grant, 23-ATP23-0095. PB was partially funded by a Packard Fellowship, Grant #2019-69646.

Data compilations from many studies used in this paper were made much more accurate and efficient by the online WEBPLOTDIGITIZER code.<fnref n="3" /> This research has made extensive use of the arXiv and NASA's Astrophysics Data System.

This research used the Ocelote supercomputer of the University of Arizona. The allocation of computer time from the UA Research Computing High Performance Computing (HPC) at the University of Arizona is gratefully acknowledged. The Bolshoi-Planck simulation was performed by Anatoly Klypin within the Bolshoi project of the University of California High-Performance AstroComputing Center (UC-HiPACC; PI Joel Primack).

<footnote n="3">³https://apps.automeris.io/wpd/</footnote>
DATA AVAILABILITY

The parallel implementation of TRINITY, the compiled data sets
(Section 3.2), and the data for reproducing all the plots in this paper
are available at https://github.com/HaowenZhang/TRINITY.

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