The New Numerical Galaxy Catalogue (ν²GC): properties of active galactic nuclei and their host galaxies

ABSTRACT

1 INTRODUCTION

Galaxies are one of the main components of the Universe. Understanding galaxy formation and evolution is thus one of the main goals of astrophysics. Almost all galaxies have a supermassive black hole (SMBH) at their centre and the mass of SMBHs correlates with properties of their host galaxies, such as the mass and velocity dispersion of the bulges (e.g. Magorrian et al. 1998; Ferrarese & Merritt 2000; Häring & Rix 2004; McConnell & Ma 2013). SMBHs and their host galaxies would thus have co-evolved with each other. The gas can lose its momentum via the growing processes of the bulge and/or galactic bars, and a part of the gas would get accreted on to the SMBH (e.g. Mihos & Hernquist 1994; Wada & Habe 1995), which could be observed as active galactic nuclei (AGNs). After that, AGN radiation, jets, and outflow inject the energy and/or angular momentum to the surrounding gas, which would cause the increase/decrease of the star formation rate (SFR) of their host galaxies (e.g. Wagner et al. 2016, for a review). This ‘co-evolution’ is a standing question in astrophysics, and has been the subject of theoretical and observational studies over three decades. Such work has focused on the mechanism of black hole (BH) feeding and the energetic feedback related with BH growth in the context of galaxy formation (see, however, Jahnke & Macciò 2011).

Understanding the growth mechanisms and evolution of SMBHs is challenging because they cannot be directly observed. AGNs are the main sources to obtain information on SMBHs observationally, which emit light when material is accreted on to the SMBHs. To overcome the difficulty in investigating growth mechanisms and evolution of SMBHs, we need close comparisons between model predictions and observations of both galaxies and AGNs.

Semi-analytic models of galaxy formation (hereafter SA models) are powerful tools for making theoretical predictions that can be directly compared with observations. In SA models, merging histories of dark matter (DM) haloes are obtained from N-body simulations (e.g. Roukema et al. 1997; Okamoto & Nagashima 2003; Nagashima et al. 2005; De Lucia et al. 2010; Guo et al. 2016; Makiya et al. 2016) or analytic algorithms based on the extended Press–Schecter formalism (e.g. Press & Schechter 1974; Lacey & Cole 1993; Nagashima & Yoshii 2004; Menci et al. 2005; Valiante et al. 2011). The evolution of baryonic components such as gaseous haloes, galaxies, and SMBHs is followed by phenomenological modellings to diminish the computational cost and to enlarge the sample size. Therefore, SA models are an excellent approach for statistical studies of galaxies and SMBHs and particularly useful for theoretical studies of rare objects, such as AGNs.

There are a large number of previous studies using SA models aimed at revealing the evolution of SMBHs within their host galaxies. The evolution of the AGN luminosity function (LF), which has been well known to imply the ‘anti-hierarchical trend’ of SMBH growth, is regarded as one of the main observational constraints on the models. In the earlier studies, they tested the ‘merger-driven AGN scenario’ by comparing their QSO LFs with observational ones in optical bands (e.g. Kauffmann & Haehnelt 2000; Enoki, Nagashima & Gouda 2003). Other triggering mechanisms of AGN activities such as disc instabilities (e.g. Lagos, Cora & Padilla 2008; Fanidakis et al. 2011; Hirschmann et al. 2012), direct accretion from its hot gas halo (e.g. Fanidakis et al. 2012; Griffin et al. 2018), and galaxy–galaxy interactions (e.g. Menci et al. 2014) are also studied. Some authors investigated the SMBH and AGN growth over cosmic time (e.g. Fontanot et al. 2006; Monaco, Fontanot & Taffoni 2007; Marulli et al. 2008; Fanidakis et al. 2012; Enoki et al. 2014; Menci et al. 2014), the effect of BH spins (e.g. Lagos, Padilla & Cora 2009; Fanidakis et al. 2011; Griffin et al. 2018), the effect of the seed BH mass (e.g. Volonteri & Natarajan 2009; Shirakata et al. 2016), and AGN clustering (e.g. Fanidakis et al. 2013; Oogi et al. 2016, 2017).

There are, however, uncertainties related with phenomenological modellings of the SMBH evolution, e.g. the triggers and the duration of gas accretion, the relation between the accretion rate and the AGN luminosity, the dust attenuation, Compton absorption, BH seeding, and AGN feedback. Unfortunately, several physical processes degenerate. Different combinations of phenomenological modellings and free parameters in a model could equally well explain observational properties of AGNs. Therefore, it is important to understand the effect of each phenomenological modelling on properties of SMBHs and AGNs.

In this paper, we focus on the accretion time-scale on to SMBHs. Estimation of this time-scale is important as it reveals the co-evolution between SMBHs and their host galaxies. If all galaxies have undergone the AGN phase, the duration of this phase should be short to explain the observed AGN LFs. In contrast, AGNs should be long-lived if a small fraction of galaxies have experienced this phase (e.g. Soltan 1982).

There are some constraints on the accretion time-scales obtained from previous studies (see Martini 2004, for more details). Yu & Tremaine (2002) estimate the time-scale by comparing present-day mass density of BHs with the integrated accreted mass density in luminous AGN phases obtained from optical AGN LFs at various redshifts. They suggest that the average ‘AGN lifetime’ is 3–13 × 10⁷ yr for |$10^{8-9} \, \mathrm{M}_\odot$| BHs if the radiation efficiency, ε, is 0.1–0.3. On the theoretical side, Kauffmann & Haehnelt (2000, hereafter KH00) estimate the AGN lifetime by using an SA model. They assume a constant radiation efficiency for AGNs, which are triggered only by major mergers of galaxies. They derive the average AGN lifetime to explain observed AGN LFs with M_B ≲ −23 (where M_B is the B-band absolute magnitude). They suggest that the lifetime is ∼3 × 10⁷ yr at z = 0 and that the time-scale would scale with the dynamical time of the halo; ∝ (1 + z)^−1.5.

In these studies, the AGN lifetime is assumed to be the time-scale within which SMBHs are observed as optical AGNs. This time-scale is not necessarily equal to the accretion time-scale on to SMBHs. Hopkins et al. (2005) estimate not the AGN lifetime but the ‘total’ accretion time-scale considering the obscured accretion phases by using hydrodynamic simulations. They suggest that the accretion on to an SMBH is not visible at first because gas and dust components are surrounding the nuclear region. After blowing out these components by AGN winds, AGNs can be observed as optical sources. The AGN lifetime is then ∼20 Myr and the total accretion time-scale is ∼ 100 Myr for AGNs with M_B < −22.

There are still two uncertainties about the accretion time-scale. One is the physical processes that govern the time-scale. Several authors have proposed different mechanisms that determine the accretion time-scale. KH00 suggest it is proportional to the dynamical time of the host halo. Norman & Scoville (1988) propose that the gas accretion continues during a starburst in its host galaxy, because they assume that the gas fuelling to an SMBH is promoted by the mass-loss from large star clusters. Granato et al. (2004) and Fontanot et al. (2006) assume the accretion rate to be determined by the viscosity of the accretion disc. The effect of these different assumptions on statistical properties of AGNs and SMBHs remains unclear.

It is also unclear whether the time-scale of less luminous AGNs is the same order as that of luminous ones. Previous work has focused on the time-scale of optical AGNs with M_B < −22 (hard X-ray (2–10 keV) luminosity, L_X, corresponds to ∼5 × 10⁴³ erg s⁻¹), whose SMBH mass is larger than |${\sim } 10^8 \, \mathrm{M}_\odot$|⁠. Less luminous AGNs with L_X ≲ 10⁴⁴ erg s⁻¹ would have wide range of SMBH masses. The accretion time-scale of such less luminous AGNs is not necessarily in the same order as luminous AGNs.

In this paper, we test a new phenomenological and physically motivated model of the accretion time-scale and investigate whether the effect of the accretion time-scale can explain the cosmic evolution of the AGN number density. We employ a revised version of an SA model, ‘New Numerical Galaxy Catalogue’ (hereafter ν²GC; Makiya et al. 2016, hereafter M16). The model more accurately explains statistical properties of galaxies and AGNs at various redshifts than the model of M16. We show the statistical properties of AGNs and SMBHs obtained by the model. This paper is organized as follows. In Section 2, we present the details of the modellings which is relevant to the growth of SMBHs and their host bulges. In Section 3, we present the statistical properties of model SMBHs and AGNs. We mainly focus on the effect of the accretion time-scale on AGN properties. Finally, in Section 4, we discuss and summarize the results.

2 MODEL DESCRIPTIONS

We create merging histories of DM haloes from large cosmological N-body simulations (Ishiyama et al. 2015),² which have higher mass resolution and larger volume compared with previous simulations (e.g. four times better mass resolution compared with Millennium simulations, Springel et al. 2005). Table 1 summarizes basic properties of the simulations. The ν²GC-M, and -SS simulations have the same mass resolution with different box sizes (L = 560 and 70 h⁻¹ Mpc, respectively). The ν²GC-H2 simulation is one of the high-resolution simulations for our SA model which has ∼64 times higher mass resolution than the ν²GC-SS simulation with the same box size. Throughout this paper, we assume a ΛCDM Universe with the following parameters: Ω₀ = 0.31, λ₀ = 0.69, Ω_b = 0.048, σ₈ = 0.83, n_s = 0.96, and a Hubble constant of H₀ = 100 h km s⁻¹ Mpc⁻¹, where h = 0.68 (Planck Collaboration I 2014; Planck Collaboration XIII 2016).

The model used in this work is based upon the original ν²GC model of M16, although it has undergone numerous improvements. This model originates from Nagashima & Yoshii (2004) and Nagashima et al. (2005) (νGC model). Both νGC and ν²GC models have been used for a variety of astrophysical studies including gravitational waves, Lyα emitters, star formation, and AGN clustering (Enoki et al. 2003; Enoki & Nagashima 2007; Kobayashi, Totani & Nagashima 2007; Makiya et al. 2014; Oogi et al. 2016, 2017). The SMBH growth and AGN properties in M16 are based on Enoki et al. (2003, 2014), and Shirakata et al. (2015). In this section, we describe the processes which relate to the growth of SMBHs (and their host bulges). Other modellings are shown in Appendix  A. Schematics of the model are shown in Fig. 1. In our model, the values of adjustive parameters are determined by a Markov Chain Monte Carlo (MCMC) method. The details of fitting procedures, its result, and the resulting statistical properties of galaxies with the fiducial model are shown in Appendix  B.

Figure 1.

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Schematics of the model showing the determination of observable properties of galaxies and AGNs.

2.1 Bulge growth by mergers and disc instability

We assume that the bulge (spheroid) component within a galaxy grows via starbursts and the migration of disc stars, both of which are triggered by mergers of galaxies and disc instabilities. Our model for these processes is based on Shirakata et al. (2016).

2.1.1 Mergers of galaxies

When DM haloes merge with each other, the newly formed halo should contain several galaxies which are classified as satellite galaxies and a single central galaxy. All members of this galaxy group would eventually merge under the gravitational attraction of the resultant halo. Mergers of galaxies occur via dynamical friction (central–satellite merger) and random collision (satellite–satellite merger). We estimate the time-scales of dynamical friction and random collision in the same manner as M16. For the dynamical friction, we set the merger time-scale, τ_mrg, as τ_mrg = f_mrg τ_fric, where f_mrg is an adjustable parameter (in this paper, f_mrg = 0.81) and τ_fric is the time-scale of dynamical friction, for which we adopt the formula by Jiang et al. (2008) and Jiang, Jing & Lin (2010).³

These types of mergers induce bulge formation and growth within a galaxy. We introduce the model of the merger-driven bulge growth proposed by Hopkins et al. (2009a) based on hydrodynamic simulations. When galaxies merge, stars and gas lose their angular momentum through bar instabilities induced by the merger.

We define a primary galaxy as the galaxy with a larger baryon mass, M₁ (cold gas + stars + a central BH), between the merging pair, and secondary galaxy as the one with smaller baryon mass, M₂. We assume that the secondary is absorbed in the bulge of the primary. The bulge also obtains the cold gas and stars from the primary’s disc. The migrated stellar mass, ΔM_1ds, is determined as MIN(f_*M₂, M_1,ds), where f_* = G(μ) = 2μ/(1 + μ) is the mass fraction of the disc that is destroyed as a function of μ = M₂/M₁ (Hopkins et al. 2009a). This results in the bulge of the primary gaining the stellar mass of M₂ + ΔM_1ds ≲ 2M₂ per a merger.

As shown in equations (2) and (3), ΔM_1dg is smaller than M_1dg even when μ = 1 (i.e. an equal-mass merger). In this case, we cannot form pure bulge galaxies. We thus assume that the disc of the primary galaxy is completely destroyed when μ > f_major, where f_major is a free parameter (f_major = 0.89). We then set ΔM_1ds = M_1ds and ΔM_1dg = M_1dg.

2.1.2 Disc instability

We also consider bulge growths via disc instabilities. When a galactic disc becomes gravitationally unstable, a small fraction, f_bar, of the galactic disc is assumed to migrate to the bulge.

The critical value for disc stabilities, ε_DI,crit (equation 5), depends on the gas fraction and density profile of a galactic disc (e.g. Efstathiou et al. 1982; Christodoulou, Shlosman & Tohline 1995). If the velocity dispersion of galactic discs is neglected, the value of ε_DI,crit is ∼1.1 for the exponential stellar disc (Efstathiou et al. 1982) and ∼0.9 for the gaseous disc (Christodoulou et al. 1995). We, however, treat ε_DI,crit as an adjustable parameter, whose value should be ≤1.1 since the disc actually has the velocity dispersion and becomes more stable. We set ε_DI,crit = 0.75 to explain the observed cosmic SFR density. If we set ε_DI,crit = 1.1, the cosmic SFR density becomes constant at 4 < z < 6, which is inconsistent with the previous suggestions and such model cannot explain the observed stellar mass–SFR relation.

We note that some other SA models (e.g. Cole et al. 2000; Lacey et al. 2016) use the circular velocity and the half-mass radius of the disc instead of V_max and r_ds. The circular velocity would change by the effect of the supernovae (SNe) explosions. We thus use V_max following original prescription by Efstathiou et al. (1982). If we assume an exponential disc, the effective radius is only ∼1.67 times larger than the scale length.

The spheroids formed through this process might be so-called pseudo-bulges, although we do not differentiate between bulges formed by these instabilities and those formed by mergers. Starbursts triggered by these instabilities are also treated in the same way as those by mergers.

2.2 Growth of SMBHs and properties of AGNs

2.2.1 BH seeding

A seed BH is immediately placed within a newly formed galaxy. We use a mass of the seed BHs, |$M_\mathrm{BH,seed} = 10^{3} \, \mathrm{M}_\odot$|⁠, for all galaxies independent from the redshift. The minimum mass of the halo in which the gas cools and possibly forms a galaxy depends on redshift and the mass resolution of N-body simulations (see fig. 2 in M16). A seed BH is therefore placed in a halo with different mass with different mass resolution and/or at different redshift. The seed BH mass, however, does not affect the main results of this paper, focusing mainly on AGNs at z ≲ 6, since the seed mass is negligible compared with the total amount of the accreted gas on to a BH (see Shirakata et al. 2016). Shirakata et al. (2016) suggest that the mass of the seed BHs should be dominated by |${\sim }\,10^3\, \mathrm{M}_\odot$| to reproduce the M_BH−M_bulge relation at z ∼ 0, including galaxies with |$M_\mathrm{bulge}\,\lt \,10^{10} \, \mathrm{M}_\odot$|⁠.

Figure 2.

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Examples of the growth history of model SMBHs with the initial SMBH mass of |$10^6 \, \mathrm{M}_\odot$|⁠. We assume t_acc = 10⁷ yr and ΔM_acc = 10⁶ and |$10^7 \, \mathrm{M}_\odot$| in top and bottom panels, respectively. In this figure, we show the evolution of |$\skew4\dot{M}_\mathrm{BH}, L_\mathrm{bol}, \lambda _\mathrm{Edd},$| and M_BH from left-hand to right-hand panels.

2.2.2 Mass accreted by SMBHs

We note that equations (12) and (13) are valid for SMBH growth via both galaxy mergers and disc instabilities. Practically, the value of f_BH is not necessarily the same for both galaxy mergers and disc instabilities. There are, however, almost no suggestions about the difference of the fraction of the cold gas mass which gets accreted on to an SMBH with different triggering mechanisms. We, thus, employ the common f_BH, for diminishing the degree of freedom.

SMBHs also increase their mass via SMBH–SMBH coalescence following mergers of galaxies. As in M16, we simply assume that SMBHs merge instantaneously after the merger of their host galaxies.

2.2.3 The accretion time-scale for SMBHs

In this paper, we test three types of the accretion time-scale summarized in Table 2. The KH00model, t_acc = 3 × 10⁷ (1 + z)^−1.5 yr, means that the accretion time-scale is proportional to the dynamical time of the host halo (originally introduced by KH00).

Some SA models (e.g. Fanidakis et al. 2012; Pezzulli et al. 2017) instead use the GalModel, t_acc = α_bulge t_dyn,bulge by assuming the accretion continues until the gas supply from the host galaxy continues. The accretion time-scale is proportional to the dynamical time of the host bulge, t_dyn,bulge = r_b/V_b (where r_b and V_b are the size and 3D velocity dispersion of the bulge, respectively), and the coefficient, α_bulge, is a free parameter. We choose the value of α_bulge so that the bright-end of the model AGN LFs are consistent with observed AGN LFs. In this paper, we set α_bulge = 0.58.

When we use this model, we find that there are SMBHs whose accretion time-scale exceeds the age of the Universe. In this case, we set |$\skew4\dot{M}_\mathrm{BH} = 0$| implicitly assuming that accreted gas becomes gravitationally stable in a circumnuclear torus and/or a accretion disc, which cannot be accreted on to an SMBH. This treatment does not affect the shape of the AGN LFs since the accretion rates of such SMBHs are negligibly small.

There are some analytical estimates for the time-scale of the angular momentum loss in a circumnuclear torus (e.g. Kawakatu & Umemura 2002; Kawakatu & Wada 2008), which have been employed by some SA models (e.g. Antonini, Barausse & Silk 2015; Bromley, Somerville & Fabian 2004; Granato et al. 2004). We note that there are large uncertainties as to whether a circumnuclear torus with some common properties exists for all types of AGNs.

We do not consider an obscured phase (e.g. Hopkins et al. 2005), in which SMBHs do not appear as luminous AGNs at optical bands despite sufficiently large accretion rates on to SMBHs. To avoid this uncertainty, we compare the model results with observations by using AGN LFs in hard X-ray (2–10 keV) (see also Section 2.2.5).

2.2.4 AGN luminosity

2.2.5 ‘Observable fraction’ of AGNs

To compare the calculated AGN LFs with observed UV AGN LFs, we need to define ‘observable fraction’ in UV band, f_obs,UV, because we can only obtain the intrinsic luminosity of AGNs from our model. Since AGN obscuration and absorption processes are very complicated, we derive an empirical formula by the following procedures. Recent work (e.g. Ueda et al. 2014; Aird et al. 2015) has estimated the hydrogen column density distribution around AGNs by a compilation of available samples obtained by Swift/BAT, MAXI, ASCA, XMM–Newton, Chandra, and ROSAT. Therefore, one can estimate the ‘intrinsic’ luminosity in hard X-ray of observed AGNs by utilizing the hydrogen column density distribution. We thus use the observed hard X-ray LFs (Aird et al. 2015, table 9) to obtain the ‘observable fraction’. The procedures are as follows.

Hopkins, Richards & Hernquist (2007) propose an alternative formula for the ‘observable fraction’. They employ an observed distribution of hydrogen column density and assume a dust attenuation curve, then they derive intrinsic AGN LFs in hard X-ray (2–10 keV), soft X-ray (0.5–2 keV), optical B, and mid-IR (15 |$\mu\mathrm{ m}$|⁠). We show the difference between observable fractions obtained from Hopkins et al. (2007) and this paper in Appendix  E.

Ricci et al. (2017) suggest that observed UV LFs of AGNs are well explained by their hard X-ray LFs, whose hydrogen column densities are less than 10^21–22cm⁻². Since the modelling of the distribution of gas around an SMBH is difficult for SA models, we estimate the observable fraction by an empirical formulation.

2.3 ‘Radio-mode’ AGN feedback

3 STATISTICAL PROPERTIES OF AGNS AND SMBHs

We present statistical properties of model AGNs and SMBHs, and show their dependence on the models of the accretion time-scale on to SMBHs. We first present the local SMBH MF in Fig. 3 and the M_BH–M_bulge relation (including both AGNs and quiescent BHs) in Fig. 4. We show the results with the ν²GC-SS and ν²GC-H2 simulations in both figures for checking the effect of the mass resolution. The model SMBH MF at z ∼ 0 are shown as the grey dashed and black solid lines in Fig. 3. The SMBH MF is roughly consistent with the observational estimate (Shankar et al. 2004) (grey shaded region). The M_BH–M_bulge relation at z ∼ 0 is consistent with observations at |$M_\mathrm{BH} \gt 10^{9.5} \, \mathrm{M}_\odot$| (Fig. 4) since we adjust the parameter, f_BH, to reproduce this relation. We, however, find that the median value of the M_BH–M_bulge relation obtained by the fiducial model deviates from the observational estimates for |$M_\mathrm{bulge} \lt 10^{9.5} \, \mathrm{M}_\odot$|⁠. We do not use such low-mass galaxies for the model calibration since the observed sample is too small. Most observational data for less massive galaxies with |$M_\mathrm{bulge} \lt 10^{9.5} \, \mathrm{M}_\odot$| are AGN data. It is unclear whether the quiescent BHs with |$M_\mathrm{bulge} \lt 10^{9.5} \, \mathrm{M}_\odot$| have the same relation as the AGNs. In addition, the bulge mass of less massive galaxies is difficult to estimate by observations since the bulge is more rotational-support.

Figure 3.

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SMBH MF at z ∼ 0. The model result obtained with the ν²GC-SS and ν²GC-H2 simulations appear in grey dashed and black solid lines with analytical fit to the observational data obtained from Shankar et al. (2004) in grey shaded region.

Figure 4.

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The relation between bulge mass and SMBH mass at z ∼ 0. The colour contour and black solid line show the distribution and the median value of mock galaxies obtained from the fiducial model with the ν²GC-H2 simulation, respectively. We overplot the result with the ν²GC-SS simulation, for checking the effect of the mass resolution. Blue filled triangles, circles, and squares are observational results for quiescent BH systems (Kormendy & Ho 2013; McConnell & Ma 2013; Scott, Graham & Schombert 2013, respectively). Cyan filled triangles, squares, diamonds, stars, and pluses are observational results for AGNs (Jiang, Greene & Ho 2011; Mathur et al. 2012; Reines, Greene & Geha 2013; Busch et al. 2014; Graham, Ciambur & Soria 2016, respectively).

3.1 The effect of the accretion time-scale on AGN LFs

We show the AGN properties obtained with ν²GC. We present the luminosity of AGNs in the hard X-ray (2–10 keV) band because the effect of obscuration and absorption is small. We show how AGN LFs change when we use three different models of the accretion time-scale in Fig. 5. Black lines show the model hard X-ray LFs with different accretion time-scales. We also show the fitting function of the LFs from Aird et al. (2015) with grey dotted lines and observed data from Aird et al. (2015), Ueda et al. (2014), and La Franca et al. (2005). We have confirmed that the results have no statistical differences when we employ the high-resolution N-body simulations.

Figure 5.

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AGN LFs in hard X-ray (2–10 keV) at z < 0.5, z ∼ 0.7, z ∼ 1.3, z ∼ 2.0, z ∼ 3.25, and z ∼ 4.25. The model LFs are obtained with the ν²GC-M simulation. Black dashed, dot–dashed, and solid lines are the model LFs with different models of accretion time-scale; the KH00model, Galmodel, and GalADmodel, respectively. Observational results are obtained from red circles, blue triangles, and green squares are the data taken from Ueda et al. (2014), Aird et al. (2015), and La Franca et al. (2005), respectively. Grey dotted lines show the fitting LFs of observed data (Aird et al. 2015).

Black dashed lines show the hard X-ray (2–10 keV) AGN LFs with the KH00model, which is the time-scale proportional to the dynamical time of the host halo. The model is consistent with observational results at log (L_X/erg s⁻¹) > 43.5 within the dispersion of the observed data. We, however, find that the model underestimates the number density of AGNs at z < 1.0 with log (L_X/erg s⁻¹) < 43.5 (i.e. nuclei of Seyfert galaxies), whose UV (1450 Å) magnitude, M_UV, corresponds to ∼−20.6. Such less luminous AGNs are not considered in the estimation of the AGN lifetimes in KH00 and their lifetimes could significantly differ for luminous AGNs.

Black dot–dashed lines show hard X-ray AGN LFs by the model in which the Galmodel. This modelling is similar to previous SA models (e.g. Fanidakis et al. 2012; Shirakata et al. 2016; Pezzulli et al. 2017). The accretion time-scale does not cause a big difference in the faint-end slope of AGN LFs compared with that with the KH00model, since the Galmodel has the accretion time-scale with the same order as the KH00model as shown later in Fig. 6.

Figure 6.

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The redshift evolution of the accretion time-scale with KH00model, Galmodel, and t_loss. The black solid line shows the KH00model, which corresponds to the dynamical time of haloes. The red circles and blue squares with error bars show the median value of t_dyn,bulge and t_loss of AGNs with log (L_X/erg s⁻¹) > 41.0 obtained by the GalADmodel. The errorbars are 25th and 75th percentiles. We also show the value of t_loss of AGNs with log (L_X/erg s⁻¹) > 44.0 by green triangles.

Black solid lines show the hard X-ray AGN LFs with GalADmodel, implicitly considering the time-scale of angular momentum loss in the circumnuclear torus and the accretion disc. The model enables us to reproduce not only bright-ends of the LFs but also the faint-ends, especially at z < 1.5. When this model of the accretion time-scale is employed, a significant fraction of low-luminosity AGNs sustain their activity for a long time as we will show later. The model thus reproduces the both the bright- and faint-ends of AGN LFs much better than the other models.

Next, Fig. 6 shows the redshift evolution of the accretion time-scale of KH00model and Galmodel, and t_loss. We select AGNs with log (L_X/erg s⁻¹) > 41.0. The red circles and blue squares with error bars show the median value of α_bulget_dyn,bulge and t_loss with 25th and 75th percentiles. The redshift evolution of the dynamical time of the bulge (red circles) and the halo (black solid line) are similar although the difference becomes larger at higher redshift. This explains why the AGN LFs with the KH00model and Galmodel are similar. While t_loss distributes broadly, it is longer especially at lower redshift. This results in the increase of the number density of AGNs at log (L_X/erg s⁻¹) < 43.5 and z < 1.5. We also plot t_loss only for luminous AGNs with log (L_X/erg s⁻¹) > 43.5 as green triangles. The time-scale is more than one order of magnitude shorter than that of AGNs with log (L_X/erg s⁻¹) > 41.0 at all redshifts.

The GalADmodel predicts the longer accretion time-scales for the less luminous AGNs due to the effect of t_loss as shown in Fig. 7. This figure shows the relation between hard X-ray luminosity and time-scales (t_loss and α_bulget_dyn,bulge) at z ∼ 0, 2, and 4. We find that the time-scale is almost constant (∼2 × 10⁷ yr) for AGNs with log (L_X/erg s⁻¹) > 44.0 (corresponds to M_UV < −22.3), which is consistent with the constraints obtained by previous studies (Yu & Tremaine 2002; Kauffmann & Haehnelt 2000; Hopkins et al. 2005). Less luminous AGNs, in contrast, have negative correlations between the time-scale and L_X. We also find that the total accretion time-scale becomes longer at lower redshift for all AGNs.

Figure 7.

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The relation between hard X-ray luminosity and two different time-scales at z ∼ 0, 2, and 4 (blue, green, and red lines, respectively) obtained with the GalADmodel. Solid and dashed lines describe t_loss and α_bulget_dyn,bulge, respectively.

The results obtained with the GalADmodel naturally explains the evolution of the AGN number density, which is sometimes called as ‘anti-hierarchical trend’ of SMBH growth. Fig. 8 shows the number density of AGNs obtained with the GalADmodel, and those obtained from observations (Ueda et al. 2014; Aird et al. 2015). The reason why the model result shows mild anti-hierarchical trends would be partially because we consider the obscured fraction in hard X-ray (2–10 keV) is 0 at all redshift. We will show a more detailed analysis in future.

Figure 8.

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The redshift evolution of the AGN number density. Colour describes the luminosity bins (log (L_X/ergs⁻¹) = [42, 43], red; (log (L_X/ergs⁻¹) = [43, 44], yellow; (log (L_X/ergs⁻¹) = [44, 45], green; and (log (L_X/ergs⁻¹) = [45, 46], blue). The results obtained with GalADmodel are shown with lines. Filled squares with error bars and triangles are observational results (Ueda et al. 2014; Aird et al. 2015, respectively).

3.2 The effect of the time-scale on other properties of AGNs

To see dependencies of the accretion time-scale on M_BH and ΔM_acc, we show the relation between AGN bolometric luminosity and BH mass, M_BH (top panels), and accreted gas mass on to an SMBH, ΔM_acc (bottom panels) at z ∼ 0, in Figs 9 and 10. In Fig. 9, x-axes are the AGN bolometric luminosity at t = t_start, L_bol(t_start), while these are AGN bolometric luminosity at the output time, L_bol(t_out), in Fig. 10. The left-hand panels show the result with the Galmodel and the right-hand panels show that obtained by the GalADmodel. We note that the model AGNs have a weak correlation between M_BH and ΔM_acc, of the form |$M_\mathrm{BH} \propto \Delta M_\mathrm{acc}^{1.1}$|⁠, with a large dispersion. This positive correlation comes from the fact that the host galaxy of the heavier SMBH is more massive and has large amount of the cold gas.

Figure 9.

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The relation between the AGN bolometric luminosity at t = t_start, L_bol(t_start), and BH mass, M_BH (top) and accreted gas mass on to an SMBH, ΔM_acc (bottom) at z ∼ 0. The distributions are described as density contours, whose value is normalized by the number of total AGNs with GalADmodel. Left-hand and right-hand panels show the results obtained with the Galmodel and GalADmodel, respectively. Black solid, dashed, dot–dashed, and dashed lines show λ_Edd = 1, 0.1, 0.01, and 0.001, respectively.

Figure 10.

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The same figure as 9 although the x-axis shows the AGN bolometric luminosity at output time, L_bol instead of L_bol,peak.

Fig. 9 shows the clear correlation between L_bol(t_start) and ΔM_acc with the Galmodel (bottom left panel). Since t_dyn,bulge is similar for galaxies at the same redshift (see Fig. 6), the peak accretion rate, |$\skew4\dot{M}_\mathrm{peak} \equiv \Delta M_\mathrm{acc} / t_\mathrm{acc}$|⁠, is mainly determined by ΔM_acc. The higher peak bolometric luminosity therefore implies a larger amount of the accreted gas. The relation between L_bol(t_start) and M_BH with the same model (top left panel) comes from the correlation, |$M_\mathrm{BH} \propto \Delta M_\mathrm{acc}^{1.1}$|⁠.

The correlations obtained by the GalADmodel (right-hand panels) show bimodal distributions, which are quite different from the model with the Galmodel. The peak accretion rate is proportional to |$M_\mathrm{BH}^{-\gamma _\mathrm{BH}} \Delta M_\mathrm{acc}^{1 - \gamma _\mathrm{gas}}$| if α_bulget_dyn,bulge is smaller than t_loss. Since γ_BH = 3.5 and γ_gas = −4.0, |$\skew4\dot{M}_\mathrm{peak} \propto M_\mathrm{BH}^{-3.5} \Delta M_\mathrm{acc}^{5.0}$|⁠. The peak accretion rate, thus, can be written as |$\skew4\dot{M}_\mathrm{peak} \propto \Delta M_\mathrm{acc}^{1.15}$| (or |$\propto M_\mathrm{BH}^{1.05}$|⁠). These positive correlations appear as contour peaks at log (L_bol(t_start)/erg s⁻¹) < 44.0.

Fig 10 shows the same relations as shown in Fig. 9, but instead plotting bolometric luminosity estimated at an output time. Since L_bol(t_start) has positive correlations with M_BH and ΔM_acc when the Galmodel is employed, the dispersions of the correlation between L_bol and M_BH and ΔM_acc (left-hand panels) reflect the elapsed time from their AGN activity.

The relation between AGN luminosity and SMBH mass allows us to compare theoretical models with observations and to potentially place a stronger constraint on the accretion time-scale. There are numerous previous studies which present the relation between AGN luminosities and the SMBH mass at various redshifts (e.g. Schulze & Wisotzki 2010; Nobuta et al. 2012; Ikeda et al. 2017). Schulze & Wisotzki (2010) and Steinhardt & Elvis (2010) show the relation between the bolometric luminosity and the SMBH mass for broad line AGNs at z < 0.3 and 0.2 < z < 2.0, respectively. Since their sample are limited at λ_Edd > 0.01, we cannot distinguish the two models of the accretion time-scale. If complete AGN sample with λ_Edd > 0.001 are obtained, we could put a stronger constraint on the accretion time-scale.

In Fig. 11, we present AGN LFs in UV band (1450 Å) from z ∼ 6.0 to 0.0 obtained by the GalADmodel. The observable fraction is defined by equation (21). The results are roughly consistent with observed UV AGN LFs (Croom et al. 2001, 2009; Fan et al. 2001; Richards et al. 2005, 2006; Fontanot et al. 2007; Siana et al. 2008; Glikman et al. 2011; Fiore et al. 2012; Ikeda et al. 2012; Palanque-Delabrouille et al. 2013; Ricci et al. 2017; Akiyama et al. 2018), especially at z > 1.5. We, however, overproduce UV LFs at lower redshift. In such redshift range, we also overproduce hard X-ray LFs (see Fig. 5) compared with the fitting LFs of Aird et al. (2015) although the model LFs are consistent with observed data points within the range of a dispersion. We need to take the dispersion of observed hard X-ray LFs into account for estimating the observable fraction although we leave it for future studies. The UV LFs do not place a strong constraint on the accretion time-scale since the observed UV LFs are well determined only at M_UV < −20.8 [corresponds to log (L_X/erg s⁻¹) > 44.6] because of the contamination of galaxies’ emission (Parsa et al. 2016). The hard X-ray LFs obtained from models with the different assumption of the accretion time-scale show little difference at log (L_X/erg s⁻¹) > 44.6.

Figure 11.

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AGN LFs in UV band (1450 Å) at z < 0.5, z ∼ 0.75, z ∼ 1.25, z ∼ 1.75, z ∼ 2.25, z ∼ 3.00, z ∼ 4.00, z ∼ 5.00, and z ∼ 6.00. The model LFs (volume-weighted) obtained with the ν²GC-M simulation appear in black solid lines. Observational results are obtained from Croom et al. (2001, 2009), Fan et al. (2001), Richards et al. (2005, 2006), Fontanot et al. (2007), Siana et al. (2008), Glikman et al. (2011), Fiore et al. (2012), Ikeda et al. (2012), Palanque-Delabrouille et al. (2013), Ricci et al. (2017), and Akiyama et al. (2018).

We show Fig. 12 to show the effect of the time-scale on the Eddington ratio distribution function. The black solid and dashed lines are results obtained with GalADmodel and Galmodel, respectively. We select all AGNs with |$M_\mathrm{BH} \gt 10^6 \, \mathrm{M}_\odot$| and L_bol > 10^43.5 erg s⁻¹ at z ∼ 0. The results at log (λ_Edd) > −1.5 are roughly consistent with that obtained by the observation (Schulze & Wisotzki 2010) at z ∼ 0.3. We, however, note that it is difficult to compare model Eddington ratio distribution functions with observations since (1) optical observational sample is limited in type-1 AGNs with well-estimated SMBH mass, (2) SMBH masses of X-ray AGNs are simply estimated from e.g. the BH mass–stellar mass relation, (3) observational sample seems to be incomplete for less massive SMBHs, and (4) the obscured fraction of AGNs would depend on both their luminosity and Eddington ratio (e.g. Oh et al. 2015; Khim & Yi 2017). Also, if there is an obscured growing phase before visible AGN phase suggested by, e.g. Hopkins et al. (2005), then the super-Eddington accreting phase should be preferentially missed.

Figure 12.

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The Eddington ratio distribution functions at z ∼ 0 obtained with GalADmodel and Galmodel (black solid and dashed lines, respectively). In both models, AGNs with |$M_\mathrm{BH} \gt 10^6 \, \mathrm{M}_\odot$| and L_X > 10⁴³ erg s⁻¹ are selected. Also, we compare the results with that obtained by Schulze & Wisotzki (2010) at z ∼ 0.3 (blue filled circles with error bars).

Fig. 12 clearly shows the difference caused by the implementation of the accretion time-scale. The GalADmodel increases the number of AGNs with log (λ_Edd) < −1.5 and the difference between the two models becomes larger at smaller Eddington ratio. We find that the GalADmodel and Galmodel have no difference for active BHMF with AGNs |$M_\mathrm{BH} \gt 10^6 \, \mathrm{M}_\odot$|⁠, L_bol > 10^43.5 erg s⁻¹, and λ_Edd > 0.03 (roughly similar selection as that of Schulze & Wisotzki 2010). As we can expected from AGN LFs (Fig. 5), the Eddington ratio distribution functions at z > 1.0 also have little difference between the GalADmodel and Galmodel. The evolution of the Eddington ratio will appear in a future paper.

3.3 Triggers of the gas supply from host galaxies

Fig. 13 shows the fraction of AGN host galaxies at 0.0 < z < 7.0 in each luminosity bin, divided by triggering situations. We classify the galaxies by the mass ratio of the merging galaxies; major (mass ratio >0.7 = f_major; blue dash dotted line), intermediate (0.4–0.7; green dotted line), and minor (<0.4; red solid line). The grey dashed line shows the fraction of AGNs triggered only by a disc instability. For merger-driven AGN activities, the typical merging mass ratio becomes larger for more luminous AGNs. Interestingly, we find that the primary trigger of AGNs at z < 4.0 is mergers of galaxies, although, at higher redshift, disc instabilities become essential for less luminous AGNs. This result is inconsistent with Fanidakis et al. (2012) and Griffin et al. (2018), who suggest that disc instabilities and ‘hot halo mode accretion’ are dominant triggering mechanisms of AGNs even at z < 4.0. As we described in Section 2.1, we employ the smaller ε_DI,crit for reproducing the properties of star formation galaxies at z > 4. Also, we consider the effect of the bulge potential on the stability of galactic discs. With this effect, the number of disc-unstable galaxies becomes 60 per cent smaller at z ∼ 1 with ε_DI,crit = 0.75. These are why our model suggests such low efficiency of disc instabilities as a triggering mechanism. The critical point is that the observed number density of AGNs can be sufficiently reproduced at z < 4 only by mergers of galaxies, and the importance of disc instabilities and other processes should be investigated in more detail. Our model predicts disc instabilities drive only less than 20 per cent of AGNs at z ∼ 0. We will come back this topic in Section 4.

Figure 13.

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Fraction of the AGN host galaxies whose AGN activity is triggered by mergers of galaxies or disc instabilities. We pick out AGNs (in ν²GC-M box) with log (L_X/ergs⁻¹) = [41.5, 42.5], [42.5, 43.5], [43.5, 44.5], and >44.5. Mergers are classified according to the mass ratio of merging galaxies: >0.70 (major, blue dash–dotted), between 0.4 and 0.7 (middle, green dotted), and <0.4 (minor, red solid). We also show the fraction of AGNs triggered only by the disc instability (grey dashed).

4 DISCUSSION AND CONCLUSIONS

We have presented the latest results of an updated version of an SA model, ν²GC. The most important changes are related to the bulge and SMBH growth model. We assume that the gas accretion on to the SMBH and the bulge growth are triggered by mergers of galaxies and disc instabilities. For bulge and SMBH growths by mergers of galaxies, we employ a phenomenological model proposed by Hopkins et al. (2009a), whose model is based on results of hydrodynamic simulations. Along with this revision, we have also updated the way of calculating the velocity dispersion and size of bulges when bulges grow via minor mergers. For bulge and SMBH growths by disc instabilities, we employ a classical model originally proposed by Efstathiou et al. (1982). We consider the effect of the bulge potential on the gravitational stability of the disc.

We have investigated the effect of the accretion time-scale on statistical properties of AGNs, such as their LFs. We stress that the impact of the accretion time-scale especially for low-luminosity (L_X < 10⁴⁴ erg s⁻¹) AGNs has been almost neglected in previous SA models. When we assume that the accretion time-scale is proportional to the dynamical time of the host halo or the host bulge, as in the previous SA models, the number density of the low-luminosity AGNs is one order of magnitude smaller than observational estimates. We have found that the number density of such less luminous AGNs becomes consistent with the observational data when we take a phenomenological and physically motivated model for the time-scale of the angular momentum loss in the circumnuclear torus and/or the accretion disc into account. The GalADmodel predicts that low-luminosity AGNs at z < 1.0, such as local Seyfert-like AGNs, are mainly triggered by minor mergers. The contribution of disc instabilities is only less than 20 per cent.

Previous studies with SA models solve the inconsistent number density of less luminous AGNs by considering other AGN triggering mechanisms such as ‘efficient’ disc instabilities (e.g. Hirschmann et al. 2012), fly-by interactions of galaxies (e.g. Menci et al. 2014), and the direct gas accretion from the hot halo (e.g. Fanidakis et al. 2012, ‘hot halo mode’). Hirschmann et al. (2012) suggest the importance of disc instabilities as a triggering mechanism of less luminous AGNs. We, however, have to note that the phenomenological modelling of disc instabilities in SA models would be too simple and is not supported by numerical simulations (see Athanassoula 2008). We have tried to make more physically reasonable modelling of disc instabilities in this paper. For the first step, we include the stabilizing effect by the bulge component and take smaller ε_DI (Section 2.1.2). We then find that disc instabilities are not the main contributor to AGN triggering mechanisms. As another point, some SA models (e.g. Fanidakis et al. 2012; Griffin et al. 2018) assume that a disc instability destroys a galactic disc entirely and all the gas is exhausted by a starburst forming a spheroidal galaxy just as major mergers. By these two effects (ignoring bulge potential and the complete destruction of a disc), some SA models are likely to overproduce the number density of AGNs induced by disc instabilities. Further updates are necessary, and we leave it for future studies. Menci et al. (2014) suggest that fly-by interactions are important instead of disc instabilities. Although we do not introduce fly-by interactions, the random collision of galaxies may have similar effects. The ‘hot halo mode’ (Fanidakis et al. 2012; Griffin et al. 2018) is the same as our ‘radio-mode’ AGN feedback model, both of which are based on Bower et al. (2006). In our fiducial models, we do not calculate the AGN luminosity with this mode because the bolometric correction and the radiative efficiency are unclear. When we assume the same bolometric correction as that of QSOs, and the radiative efficiency is 0.1, the contribution of the radio-mode AGN to the AGN LFs becomes the same order as that of AGNs induced by mergers of galaxies and disc instabilities at L_X ∼ 10⁴¹ erg s⁻¹ at z ∼ 0. The contribution becomes smaller at more luminous regime and at higher redshift. Our results based on the time-scales show that observed AGN LFs can be reproduced without ‘radio-mode’ or ‘hot halo mode’ accretions. Even without the ‘radio-mode’ AGN feedback, GalADmodel produces a large number of AGNs with low Eddington ratios, which would be AGN jet and outflow sources. Considering the injected energy and momentum from the low Eddington ratio AGNs, they may have non-negligible impact on the star formation quenching of massive galaxies. We will examine which explanation is more plausible in a future study.

Marulli et al. (2008) suggest the importance of AGN light curve for determining the shape of AGN LFs. They assume three types of the Eddington ratio evolution models based on observations and hydrodynamical simulations. The faint end slope of AGN LFs at z < 1.0 are well fitted when they assume the constant Eddington ratio, namely, =0.3[(1 + z)/4]^1.4 at z < 3, and =1 at z > 3. By using this Eddington ratio, the accretion time-scale should be ∼0.17 Gyr at z ∼ 0, which is larger than the dynamical time of bulges (Fig. 6) and is qualitatively consistent with our suggestion. However, the model with this assumption of the constant Eddington ratio underestimates the number density of luminous AGNs at z > 1. They also introduce AGN light curve with two stages; rapid, Eddington-limited growth phase, and longer quiescent phase with lower Eddington ratios. By using this light curve, the accretion time-scale should be longer when the SMBH mass is smaller or the accreted gas mass is larger, which is the opposite to that suggested in the GalADmodel. The resulting faint end slope of AGN LFs at z < 1 is shallower than observations. They cannot explain the shape of the AGN LFs by changing just the Eddington ratio distribution. Finally, they introduce SMBH mass dependence to the f_BH and successfully reproduce AGN LFs at z < 5.

Hydrodynamic simulations (e.g. Hirschmann et al. 2014; Khandai et al. 2015; Sijacki et al. 2015) do explain AGN LFs well, assuming Bondi–Hoyle–Littleton (BHL) accretion for all SMBH growths. Generally, hydrodynamic simulations assume that the ‘effective’ accretion rate on to SMBHs is roughly 200 times larger than the BHL accretion rate, which is too small compared to that of observed AGNs (e.g. Ho 2009). The assumption of the accretion rate with ∼200 times larger than the BHL accretion, independent of any properties of galaxies and SMBHs, might be a too simplified assumption. Besides, we must care about another uncertainty; different AGN feedback models are employed in different cosmological simulations, which reproduce AGN LFs at the same extent.

As we have shown, there are several prescriptions to explain the faint end slopes of AGN LFs at z < 1. For discriminating the models, comparisons of model results with observed properties of AGNs and their host galaxies are necessary. We have shown the relation between M_BH and L_X (Figs 9 and 10), the Eddington ratio distribution function at z ∼ 0.3 (Fig. 12), and the fraction of AGNs with different triggering mechanisms (Fig. 13). Since the difference between the Galmodel and GalADmodel is clear for low-luminosity AGNs with the smaller SMBH masses, the comparisons with observations are challenging. The other possible way would be comparing the clustering properties with observations. Fanidakis et al. (2013) suggest that the host halo mass of luminous AGNs like QSOs and low-luminosity ones is different. In their model, luminous AGNs are triggered by starbursts induced by mainly disc instabilities (and mergers of galaxies) and their typical host halo mass is |${\sim } 10^{12} \, \mathrm{M}_\odot$|⁠. Low-luminosity AGNs, on the other hand, are triggered mainly ‘hot halo mode’ and their halo mass is larger than those of luminous AGNs, namely |${\sim } 10^{13} \, \mathrm{M}_\odot$|⁠. The ‘hot halo mode’ is efficient for cluster galaxies whose host halo is cooling inefficient. On the other hand, Oogi et al. (2016) suggest that when they assume AGNs are mainly triggered by mergers of galaxies, the host halo mass weakly depends on the AGN luminosities at 1 < z < 4. The GalADmodel also shows the same trend as Oogi et al. (2016) at 1 < z < 4. We, thus, can discriminate effects of the accretion time-scale and AGN triggering mechanisms by detailed comparisons with observational results.

One might think that the underproduction of less luminous AGNs results from the underestimation of the velocity dispersion of the bulge and/or the underestimation of the cold gas mass in galaxies. As shown in Fig. B5, the velocity dispersion of the bulge tends to be smaller than those obtained from observations, although the bulge size is broadly consistent with the observational data at z ∼ 0 (Fig. B6). The dynamical time of the bulge evaluated in the fiducial model is statistically longer than the value estimated from the observed velocity dispersion and bulge size. We thus underestimate the gas accretion rate on to SMBHs since the peak accretion rate is proportional to |$t_\mathrm{dyn,bulge}^{-1}$|⁠. In addition, low-mass galaxies in the model seem to have smaller gas masses than observed galaxies (Fig. B2) due to the insufficient resolution, which could also cause the underestimation of the gas accretion rate. In Fig. 14, we check these effects and find that both are insufficient to compensate the underproduction of the less luminous AGNs. We compare hard X-ray LFs at z ∼ 0 obtained by the following three models: (1) the Galmodel with the ν²GC-SS simulation (black solid line), (2) the Galmodel with the ν²GC-H2 simulation (black dotted line), and (3) the model with t_acc = 0.2 × α_bulget_dyn,bulge (black dashed line). The number density of AGNs obtained by the model (3) becomes smaller than that obtained by the model (1) since t_dyn,bulge is set to be smaller, and the AGN activity shut off sooner. Also, we find no effect of the gas deficiency by comparing (1) and (2), while the number of galaxies with |$M_{HI}\lt 10^8 \, \mathrm{M}_\odot$| increases when we employ the ν²GC-H2 simulation. The comparison (1) and (2) therefore suggests the gas deficiency is not the cause of the underestimation of the abundance of the less luminous AGNs. Even at z ∼ 1, the model (3) does not solve the inconsistency of the faint-end slope since the shorter accretion time-scale causes the shallower slope. We have confirmed that the faint-end slope of the AGN LF at z ∼ 1 also does not change with model (3). We conclude the underestimation of the gas mass of galaxies is not a primary cause of the underestimation of the number density of faint AGNs.

Figure 14.

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AGN LFs at z ∼ 0. To check the effect of the determination of t_dyn,bulge and the accreted gas mass, we compare three models: (1) the Galmodel with the ν²GC-SS simulation (black solid line), (2) the Galmodel with the ν²GC-H2 simulation (black dotted line), and (3) the model in which t_acc = 0.2 × α_bulget_dyn,bulge (black dashed line). Observational results are the same as the top left panel of Fig. 5.

Another problem of the AGN LFs obtained with the ν²GC is that there are no AGNs with log (L_X/erg s⁻¹) > 45.3 at z > 2.6. Such luminous AGNs do not appear even when we employ N-body simulations with larger volumes. The modelling of the radio-mode AGN feedback is likely to be responsible for this, which was originally proposed by Bower et al. (2006) and is similar to other SA models. Host halo masses of AGNs with log (L_X/erg s⁻¹) ∼ 45.0 at z ∼ 4 in the fiducial AGN model are |$10^{12-13} \, \mathrm{M}_\odot$|⁠. Such massive haloes could satisfy conditions of equations (22) and (23) and the gas cooling is quenched even at high redshifts. This is shown in Fig. 15, which shows the fraction of galaxies whose gas cooling is quenched by the radio-mode AGN feedback. We find that about the half of galaxies are quenched when |$M_\mathrm{halo} \gt 10^{12.5} \, \mathrm{M}_\odot$| at z ∼ 4. We will address this problem in future studies.

Figure 15.

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The fraction of central galaxies whose gas cooling is shut off by the radio-mode AGN feedback at z ∼ 4. The x-axis is the host halo mass of the galaxies. The number means [Quenchedhalo]/[Totalhalo].

ACKNOWLEDGEMENTS

We appreciate the detailed review and useful suggestions by the anonymous referee, which have improved our paper. We would like to express the deepest gratitude to A. R. Pettitt for thorough English proofreading that drastically improves the paper. We thank J. Aird to give the fitting function of hard X-ray luminosity functions of AGNs. We appreciate the fruitful comments from the observational side by T. Izumi, M. Onoue, Y. Ueda, D. Zhao, T. Nagao, Y. Matsuoka, Y. Kimura, and M. Akiyama. We also thank K. Wada for theoretical comments. HS has been supported by the Sasakawa Scientific Research Grant from The Japan Science Society (29-214) and Japan Society for the Promotion of Science (JSPS) KAKENHI (18J12081). TO has been financially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) KAKENHI (16H01085). TK was supported in part by an University Research Support Grant from the National Astronomical Observatory of Jpan (NAOJ) and JSPS KAKENHI (17K05389). MN has been supported by the Grant-in-Aid (25287041 and 17H02867) from the MEXT of Japan. TI and TO have been supported by MEXT as ‘Priority Issue on Post-K computer’ (Elucidation of the Fundamental Laws and Evolution of the Universe) JICFuS. TI has been supported JSPS KAKENHI Grant Number 15K12031. RM was supported in part by MEXT KAKENHI (15H05896). TO was supported by World Premier International Research Center Initiative (WPI). KO has been supported by JSPS KAKENHI (16K05299).

Footnotes

REFERENCES