1. INTRODUCTION

Studies of AGN host galaxies are fundamental to understand the physical mechanisms that trigger AGN activity and that govern the fuelling rate of the central black hole (e.g., ^{Gilmour et al. 2009}; ^{Alexander & Hickox 2012}; ^{Beckmann & Shrader 2012}).

The observed correlation between the mass of the central BH and the velocity dispersion (σ) of the bulge of the host galaxy suggest a strong connection between galaxy evolution and BH activity (e.g., ^{Gebhardt et al. 2000}; ^{Merritt & Ferrarese 2001}; ^{Tremaine et al. 2002}; ^{Kormendy & Ho 2013}). This activity is related to the accretion of material onto the central engine triggered by, for example: the merger of gas-rich galaxies (e.g., ^{Silk & Rees 1998}; ^{Springel et al. 2005b}; ^{Hopkins et al. 2008}), bar-driven inflows (e.g., ^{Jogee 2006}), disk instabilities (^{Bournaud et al. 2011}), collisions with molecular clouds (^{Hopkins & Hernquist 2006}), stellar winds from evolved stars (^{Ciotti & Ostriker 2007}), and the transportation of gas to galactic centers by supernova explosions (^{Chen et al. 2009}) or a combination of these effects.

Mergers and strong interactions can induce substantial gravitational torques on the gas content of a galaxy, depriving it of its angular momentum, and leading to gas inflows and the buildup of huge reservoirs of gas at its center (e.g., ^{Hernquist 1989}; ^{Barnes & Hernquist 1991}, ¹⁹⁹⁶; ^{Mihos & Hernquist 1996}; ^{Springel et al. 2005b}; ^{Cox et al. 2006}, ²⁰⁰⁸; ^{Di Matteo et al. 2007}). Major galaxy mergers are very efficient in moving gas to galaxy centers due to the generation of large torques. Visual inspections of host galaxies of AGNs (^{Treister et al. 2012}) have found that the more luminous AGNs show recent merger features, while such features are not commonly seen in the less luminous AGNs, which appear to be driven by another process. Low luminosity AGNs could also be triggered in non-merger scenarios (e.g., ^{Milosavljević et al. 2006}; ^{Hopkins & Hernquist 2006}, ²⁰⁰⁹), and probably also by the interaction with very small satellites (total mass ratio of about 1:100), as recently suggested by ^{Ramón-Fox & Aceves (2014)}.

The main process triggering AGN activity could be a function of redshift and/or halo mass. The anti-hierarchical evolution of AGNs (or AGN downsizing), in which the number density of low luminosity AGNs comes later in the universe than high-luminosity ones (^{Ueda et al. 2003}; ^{Hasinger et al. 2005}; ^{Ueda et al. 2014}), is probably not consistent with the theoretical predictions of a major merger AGN triggering scenario such as the one suggested by ^{Wyithe & Loeb (2003)}. Some recent theoretical studies based on cosmological simulations suggest the necessity of a combination of merger and secular processes (e.g. ^{Draper & Ballantyne 2012}), or of hot-halo accretion and star-burst induced triggering (^{Fanidakis et al., 2012}) to explain the evolution of the luminosity function of AGNs and their clustering properties (^{Fanidakis et al. 2013}). These results suggest that one or more mechanisms other than major mergers are at least partially responsible for triggering AGN activity.

Bias measurements of large scale AGN show that mXAGNs are on average associated with more massive DMHs than more luminous QSOs (e.g., ^{Miyaji et al. 2007}; ^{Krumpe et al. 2010}; ^{Allevato et al. 2011}). While the typical masses of DMHs associated with QSOs [ M_DMH ≈ 10 ^12-13 h ^-1M_ʘ; ^{Porciani et al. (2004}); ^{Croom et al. (2005)}; ^{Hopkins et al. (2007)}; ^{Coil et al. (2007)}; ^{da Angela et al. (2008)}; ^{Mountrichas et al. (2009)} ] are consistent with a major merger triggering scenario (e.g., ^{Shen 2009}), those associated with mXAGNs are typically more massive, with M_DMH ≈ 10^13-14 h ^-1 M_ʘ (i.e., the mass scale of rich groups and poor clusters), and results from ^{Allevato et al. (2011)} suggest that secular processes could trigger mXAGNs.

Cosmological simulations are an important tool to understand the distribution of dark matter in the universe, and the co-evolution and growth of BHs with respect to their host galaxies (e.g., ^{Sijacki et al.. 2007}; ^{Di Matteo et al., 2008}; ^{Thacker et al., 2006}). In N-body simulations, the requirement of associating the dark matter with galaxy distributions has to be satisfied (e.g., ^{Pujol & Gaztañaga, 2014}). Studies of AGN clustering using cosmological simulations have been carried out using the halo model (e.g., ^{Thacker et al., 2009}; ^{Degraf et al., 2011}) or the BH continuity equation approach (e.g., ^{Lidz et al., 2006}; ^{Bonoli et al., 2009}; ^{Shankar et al., 2010}). The HOD method allows to distinguish between AGN evolution models (e.g., ^{Chatterjee et al., 2012}). It has been used by several authors to interpret AGN and quasar clustering measurements from direct counts of AGNs within groups of galaxies (e.g., ^{Wake et al.. 2008}; ^{Shen et al., 2010}; ^{Miyaji et al., 2011}; ^{Starikova et al., 2011}; ^{Krumpe et al., 2012}; ^{Allevato et al.. 2012}; ^{Richardson et al., 2012}; ^{Kayo & Oguri, 2012}; ^{Chatterjee et al., 2013}; ^{Krumpe et al., 2014}).

The environment of AGN and, in particular, the mass of the typical DMHs in which they reside, is a powerful diagnostic of the physics that drive the formation of super massive black holes (SMBHs) and their host galaxies (e.g., ^{Mountrichas & Georgakakis, 2012}). By modeling the mean AGN occupation at z = 0.5, ^{Allevato et al. (2012)} found that the host halos of these AGNs have a DMH with a mass M_DMH ≥10 ^12.75 h ^-1M_ʘ, that is, a DMH mass corresponding to galaxy groups (^{Eke et al., 2004}). This result agrees with studies by ^{Georgakakis et al. (2008)} and ^{Arnold et al. (2009)}, which present evidence that AGNs at z ≈ 1 are frequently found in groups. Moreover, ^{Miyaji et al. (2011)} estimated a shape of the HOD of X-ray selected AGN that suggests that the AGN satellite fraction increases slowly with Mdmh, in contrast with the satellite HOD of low luminosity limited samples of galaxies. For these galaxies, ^{Allevato et al. (2012)} found that the slope a of the HOD distribution of satellite AGNs had a value of α _s ≤ 0.6, suggesting a picture in which the average number of satellite AGNs per halo mass decreases with the halo mass.

Considering that α ≈ 1 is the value inferred for galaxies in general (e.g., ^{Coil et al. 2009}; ^{Zehavi et al. 2011}), the study of the HOD of AGNs suggests that the AGN fraction in galaxies decreases with in creasing DMH mass. The reason for this may be that the cross-section of the merger between two galaxies decreases with increasing relative velocity and, thus, that the merging frequency is suppressed in a group/cluster environment with high velocity dispersion (^{Makino & Hut 1997}). In addition, gas processes such as ram-pressure stripping of cold gas in galaxies by hot intragroup/intracluster gas sup presses star formation activities that may feed AGN (^{Gunn & Gott 1972}).

In this work we investigated a scenario where satellite subhalos, from N-body cosmological simulations harbor mXAGNs within a group/cluster-sized parent halo of M_DMH ≥10 ^12.75 h ^-1M_ʘ, and were triggered by either a major or minor merger. We estimated the HODs of the simulated mXAGNs and compared them with the inferred HOD obtained by ^{Allevato et al. (2012)} and ^{Miyaji et al. (2015)} in order to see if we could reproduce such results by using this approach. In brief, we used a simple approach for coupling cosmological simulations of ΛCDM with semi-analytical results to determine the HOD of our numerical AGNs.

The outline of the paper is as follows. In § 2 we describe the method used in this work to determine AGN candidates in cosmological simulations to esimate their HOD. In § 3 we show our results; we discuss them in § 4, and, finally, in § 5 we indicate our main conclusions. Throughout this paper we assume a matter density Ω _m = 0.266, dark energy density Ω _Λ = 0.734, H _O = 72kms^-1Mpc^-1 and mass rms fluctuation σ₈ = 0.816 consistent with the W_MAP7 results of ^{Larson et al. (2011)}.

2. MODEL

In this section, we describe the cosmological simulations and the semi-analytical procedure used to establish an association between satellite subhalos and the observational properties of mXAGNs; we then describe the criteria used to determine when a merger occurs, as well as the actual computation of the HOD.

2.2. Halo finder algorithm

We identified DMHs and subhalos using the Amiga Halo Finder (AHF) code, which locates halo centers using an adaptive mesh refinement (AMR) approach. In brief, this code finds prospective halo centers, collects particles possibly bound to the center, removes unbound particles and calculates halo properties (^{Knollmann & Knebe 2009}; ^{Knebe et al. 2011}) ¹. Virial masses are defined at an overdensity of 200 p _c , where p _c is the critical density of the universe. We used a minimum number of particles N _p = 100 to define a bound halo.

As mentioned in § 1, we are interested in DMH with a virial mass of M_DMH ≥ 10 ^12.75 h ^-1 M _ʘ = M_TH at a z = 0.5 snapshot, in which mXAGNs preferentially reside (^{Allevato et al. 2012}; ^{Padmanabhan et al. 2009}). This is consistent with the HOD modeling of the cross-correlation function between ROSAT all-sky Survey AGNs and low-luminosity red galaxies (^{Miyaji et al. 2011}). We focused our attention on those DMH that reside at a non-central location of the host halo.

Halos with M_DMH > M_th at redshift z = 0.5 are called host-halos (HHs hereafter). We identified subhalos belonging to these HHs, and selected the subhalos that are satellites (subhalo-host, SH); see Figure 1.

Fig. 1 This schematic diagram illustrates the host-halo (HH of mass M_th) as a circle with virial radius; R _v. Several subhalos (SHs) are depicted inside R _v. The central subhalo is identified as the closest to the center of the host-halo (dark filled circle) while the others are considered satellite subhalos (empty circles). 

2.3. Connection of subhalos to mXAGN

Several approximate methods to assign AGN or quasar activity to halos are described in the litera ture. For example, ^{Croton (2009)} uses the M_bh ^-σ relation by requiring the reproduction of the observed luminosity function of quasars, using the abundance matching technique to "turn-on" halos in the Millennium simulation (^{Springel et al. 2005a}). ^{Conroy & White (2013)} invoked an empirical model of the halo population and M _BH -M _gal in which quasars are treated as light bulbs to match the luminosity function of quasars.

In this section we explain the different scaling relations used in our analysis to establish an asso ciation between the subhalos and the observational properties of AGNs.

2.3.1. Black hole mass

The central velocity dispersion σ of each SH is associated with the black hole mass M_BH (^{Kormendy & Ho 2013}) by:

1

where M₀ = 10⁹ h ^-1M_ʘ and σ ₀ = 200 km s^-1.

2.3.2. Assigning the Eddington ratio

Given a M_bh, an Eddington ratio (λ_Edd) can be defined as:

2

where L _bol is the bolometric luminosity and L_Edd(M _BH) is the Eddington luminosity which is proportional to M_BH

Combining equation 2 with the data provided in Table 2 from ^{Lusso et al. (2012)}, we can obtain the X-ray luminosity as:

3

where L_band corresponds to the 0.5-2 keV band luminosity, x = log L_bol-12; ɑ₁, ɑ_2, ɑ₃ and b are bolo metric correction coefficients.

We need to mimic a population of AGNs representing mXAGNs within our simulations and construct the HOD to compare it with the observed mXAGNs HOD by ^{Allevato et al. (2012)}. To do this, we used a representative value of λ_Edd =0.1, ɑ₁ = 0.248, ɑ₂ = 0.061, ɑ₃ = -0.041 and b = 1.431, restricting the calculations only to subhalos with L_x ≥10^42.4 h ^-2ergs^-1.

Using equations (1-3), we obtained the black hole mass threshold M_•, which we used to obtain the number density of subhalos with active or dormant black holes, n(≥M_•) We also considered a z = 0.5 snapshot, which is the median redshift of the sample used by ^{Allevato et al. (2012)}. We designated the black holes as active or dormant depending on whether they had undergone a galaxy merger within the AGN lifetime τ_agn in the past.

2.4. Determining the AGN lifetime

Instead of using the abundance matching technique, we selected the duty cycle (e.g., ^{Cappelluti et al. 2012}) as an indicator that the subhalo was turned on. In the following, we took a simple approach and assumed that all AGNs observed at z = 0.5 above the luminosity threshold, are shining at λ_Edd = 0.1 during their lifetime, τ_agn

Using the X-ray [2-10 keV] band luminosity function (XLF) of AGNs by ^{Miyaji et al. (2015)}, and the model of the distribution function of absorbing column density by ^{Ueda et al. (2014)}, in combination with the results of ^{Allevato et al. (2012)}, we estimated the number density of AGNs, including absorbed (within the Compton-thin range, i.e., N _H < 10²⁴[cm^-2]) and unabsorbed ones, above the intrinsic (i.e., before absorption) [0.5-2 keV] B-band luminosity of L _x ≥ 10^42.4 h ^-2ergs^-1. We obtained:

4

The idea behind using the 2-10 keV luminosity function is that the 0.5-2 keV sample used by ^{Alle vato et al. (2012)} is highly biased against absorbed AGNs. In order to estimate more accurately the lifetime of AGNs, we require the number density of both absorbed and unabsorbed AGNs. For simplicity, we here assume that the satellite HODs of absorbed and unabsorved mXAGNs have the same shape.

To constrain the number density of AGNs that can be observed at z = 0.5, we used the timescale τ_AGN, which indicates when the mXAGNs were activated. We estimated the AGN lifetime τ_AGN as follows:

5

where n _AGN is the observed number density of X ray AGNs, n (_≥M.) is the simulated number of SHs with BH mass threshold (active or dormant), and τ_age(z=o.5) is the age of the universe at z = 0.5.

2.6. Simulated HOD

The following formula was used in order to estimate the HOD of mXAGNs:

6

where n _HHagn is the number density of HHs that have a SH that has undergone a major or minor merger and that has a M_BH ≥ M _•, and n _HH is the total number density of HH (defined in § 2.2) in simulations. The mass bin size used was ΔlogM_th 0.4.

3. RESULTS

Before presenting our results, we should note the following: the HOD measured by ^{Allevato et al. (2012)} was evaluated at z = 0; their mea surement was made over a sample extending up to z ≈ 1 that was corrected for the 0.5-2 keV XLF and its luminosity-dependent evolution, as indicated by ^{Ebrero et al. (2009)}. Their sample is more represen tative of z ≈ 0.5 than of z ≈ 0.

To make a comparison with our results, we back-corrected their HOD to z ≈ 0.5 in the following way: the luminosity-dependent density evolution (LDDE) model, describing the 0.5-2 keV XLF derived by ^{Ebrero et al. (2009)}, indicates that the number den sity grows as ∞ (1 + z)^3.38 up to z ≈ 0.8 at all lu minosities. Thus, we converted the HOD at z = 0 to z = 0.5. The HOD results from multiplying by (1 +0.5)^3.38 = 3.9 at all DMH masses. Furthermore, since the ^{Allevato et al. (2012)} sample in the 0.5 2 keV band is highly biased against obscured AGNs, we further multiplied the HOD by a factor of 2 to ac count for the obscured AGN contribution using the recent 2-10 keV XLF model by ^{Miyaji et al. (2015)}. Figure 2 shows our results for the HOD of major and minor mergers and the corrected form of the HOD at a redshift of z = 0.5.

Fig. 2.  The HOD of the number of hosts that harbor a mXAGN triggered by either a major (blue dots) or minor merger (black triangles), and the inferred HOD of satel lites mXAGNs by ^{Allevato et al. (2012)} (red asterisks). The latter was corrected by an evolution factor and by addition of the unabsorbed AGN estimation by ^{Miyaji et al. (2015)}. Error bands were calculated as indicated in the text. The color figure can be viewed online. 

The slope of the HOD shows the same trend for both minor and major mergers at masses 5 x 10¹³ h ^-1M_ʘ. However, at larger masses, the minor and major HODs show a somewhat different behavior, with the major merger HOD increasing and the minor one tending to be flat (Figure 2). To make a comparison with the slope (α _s )found by ^{Al levato et al. (2012)}, we used the same occupation function model, which is described by:

7

where ƒ׳_ɑ is a normalization, M₁ is the halo mass at which the number of central AGN is equal to that of satellite AGNs (log M₁ = 13.8M_ʘ), and M_cut is a cut-off mass scale (logM_cut = 13.4M_ʘ).

The fitted slope for the minor merger case was α _s = 0.10 ± 0.09 and α _s = 0.20 ± 0.18 for the ma jor merger; both can be compared with the slope observed by ^{Allevato et al. (2012)}α _s ≤ 0.6 (Ta ble 1). The slope of the minor merger HOD is closer to that of the mXAGN HOD than to that of the ma jor merger case. However, both slopes are consistent with the observations.

The errors in Figure 2 were derived as follows: if nHH_agn was less than 15, we estimated 1σ errors using the equations (7) & (12) of ^{Gehrels (1986)}. If nHH_agn ≥ 15, we estimated the 1σ errors by nHHagn .

4.DISCUSSION

Different numerical studies have addressed the triggering of AGNs, in particular by mergers between galaxies, since it is a naturally expected contributing process (e.g., ^{Sanders et al. 1988}; ^{Hopkins et al. 2006}). Major mergers are considered to activate QSO's, a situation that has been studied through hydrodynamical cosmological simulations, measurements of their clustering properties and of the properties of AGNs (^{Sijacki et al. 2007}; ^{Di Matteo et al. 2008}; ^{Marulli et al. 2009}; ^{Ciotti et al. 2010}; ^{Degraf et al. 2011}; ^{Chatterjee et al. 2012}; ^{Van Wassenhove et al. 2012}; ^{Krumpe et al. 2015}).

Table 1.Number densities and hod slopes 

^{1Allevato et al. (2012)}.

Table 2 λ _edd and α _s  

However, ^{Schawinski et al. (2011)} found that most of the quasars in their sample have disk-like morphologies, suggesting that a secular evolution mechanism could drive the activity to this type of AGNs. Moreover, if only major mergers turn out to be important at high-redshifts, the AGNs should probably reside in more elliptical-shaped galaxies (^{Cisternas et al. 2011}). In contrast, ^{Lee et al. (2012)} and ^{Cisternas et al. (2013)} found that the bar-driven gas and the gas that could trigger nuclear activity do not correlate; a mechanism discussed in particular by ^{Wyse (2004)}.

Stochastic accretion models have been thought to trigger the mechanism of low/moderate AGNs (^{Hopkins & Hernquist 2006}; ^{Hopkins et al. 2014}; ^{Kocevski et al. 2012}). Thus, secular evolution (e.g., ^{Ehlert et al. 2015}) has become an important scenario, as well as minor mergers (^{De Robertis et al. 1998}; ^{Hernquist & Mihos 1995}; ^{Taniguchi 1999}; ^{Kendall et al. 2003}; ^{Hopkins & Hernquist 2009}; ^{Karouzos et al. 2014}), owing to the fact that injection of gas is recurrent and keeps the accretion going, which can explain the non-disk AGNs and can in some cases influence the growth of SMBH (e.g., ^{Kaviraj 2014}). As intermediate mergers are more common than major mergers (e.g., ^{Tapia et al. 2014}), they may also play a role. Recognizing that the main triggering mechanism is not obvious in all types of AGNs, we conducted a study in which mXAGNs are triggered by either major or minor mergers using the HOD formalism and a simplified model.

It is difficult to estimate the precise time of the merger, and, therefore, the time when the BH will be activated. Using the dynamical friction time (e.g., ^{Hopkins et al. 2010}) to estimate when the progenitors merge, causes an important overestimation compared to following as closely as possible the evolution in time of the subhalos in our cosmological simulations. Furthermore, results by ^{Jiang et al. (2014}) show that it is not adequate to use a single time scale to infer when the merging takes place.

In spite of the wide range of probable environments in which AGNs reside (^{Villarroel & Korn 2014}: ^{Karouzos et al. 2014}; ^{Leauthaud et al. 2015}), in this work we assumed that the host/environment of the mXAGN has a group-like halo mass, as indicated by some observations (e.g., ^{Allevato et al. 2012}; ^{Silverman et al. 2014}). We concentrated our attention on the mXAGNs that reside in non-central subhalos. The connection between galaxies and DMHs has been established in different ways: assigning the stellar mass of the galaxies to DMHs (^{Degraf et al. 2011}; ^{Behroozi et al. 2013}), introducing a gas fraction of galaxies into the DMHs (^{Hopkins et al. 2010}; ^{Zavala et al. 2012}) and using the luminosity function (^{Croton 2009}). Here we used a combination of semi-empirical and semi-analytic models to seed a black hole in a subhalo/satellite employing the relation σ-M _BH.

The number densities of AGNs in groups and clusters can help to establish on firmer ground whether there is any relation between environment density and AGN luminosity (^{Karouzos et al. 2014}), or even the possible two-phase evolution of X-ray AGNs (^{Miyaji et al. 2015}). Figure 2 shows that the satellite subhalos selected have a similar distribution in the two merger scenarios tested, i.e., the shape of the HOD can be described to a good extent by both types of mergers. These processes also approximately reproduce the observed number densities of mXAGNs. The results shown in Figure 2 indicate that minor mergers definitely play a role in establishing the HOD of these AGNs, and point to the necessity of further research on the igniting of AGNs.

While we chose to use λ _Edd =0.1 based on the median value of the λ _Edd distribution from ^{Lusso et al. (2012)}, it is instructive to show how the results change with λ _Edd . Thus, we also estimated simulated HODs for λ _Edd = 0.01,0.03 & 0.3. The results are shown in Figure 3, while α _s values are shown in Table 2. Figure 3 and Table 2 show that the slopes are consistently flat for the minor merger case (α_s ≈ 0.1 -0.2) for all λ _Edd values, and are consistent with the slopes estimated by ^{Allevato et al. (2012)}. For the major mergers, the slope becomes steeper (α_s ≈ 1 for λ _Edd = 0.01), while it is flat for larger λ _Edd values. The global normalization of the minor merger HOD seems to agree better with the observations for λ _Edd = 0.01 than with higher λ _Edd values. However, it is worth noting that the normalization is directly proportional to the AGN lifetime τagn estimated from Eq. 5, which is a rough approximation. Thus, the agreement or disagreement of the HOD normalization between the models and a few observations should not be used to choose one model over other.

Fig. 3 The HOD of different values of λ _Edd, the number of hosts that harbor a mXAGN triggered by either a major (blue dots) or minor merger (black triangles), and the inferred HOD of satellite mXAGNs according to ^{Allevato et al. (2012)} (red asterisks). The color figure can be viewed online. 

5. CONCLUSIONS

In this work, we used cosmological simulations and semi-analytical methods to assign activity to satellite subhalos within halos with mass M _th ≥ 10^12.75 h ^-1M_ʘ, and used the merger-driven scenario of BH triggering to obtain an estimate of the contribution of major and minor mergers to the inferred shape of the HOD of mXAGNs.

Testing different models can help to constrain the connection between the AGN and the host galaxy as well as the mechanism that triggers its BH. We used an estimate of the duty cycle to associate the number density obtained from the simulated mXAGNs with the observed ones. Our results support the hypothesis that minor mergers, typically not considered as triggering events, can be an important factor in activating mXAGNs.

Our work shows that the HODs of mXAGNs activated by major and minor mergers are very similar and are both consistent, in our approximate treatment, with the flat slope (α _s < 1) shape of HOD inferred from observations. The minor merger model reproduces the slope of the satellite HOD of mXAGNs even at λ_Edd=0.01, while that of the major merger model has a steeper slope at this low λ_Edd.

Moreover, since our simulations do not take into account any baryonic processes, other mechanisms such as the ram-pressure stripping of cold gas, are not excluded as a significant cause of the flat slope of the HOD. Our results, however, show that non-baryonic processes such as the decrease of the merging cross-section in the high velocity encounters (^{Makino & Hut 1997}), are at least able to produce a flat slope in satellite HODs.

With the help of large X-ray surveys like eROSITA and larger samples of mXAGNs in groups and clusters, we will be able to constrain the properties of the host of these mXAGNs (e.g., mass); more sophisticated models could lead us to a better un derstanding of the co-evolution of AGNs and their distribution.