The XMM-Newton Wide Field Survey in the COSMOS Field: Clustering Dependence of X-ray Selected AGN on Host Galaxy Properties

TL;DR

This study investigates how the clustering of X-ray selected AGNs relates to host galaxy properties, revealing that low-mass AGNs tend to reside in more massive halos than non-active galaxies, with no dependence on accretion rate.

Contribution

It provides the first detailed analysis of the dependence of AGN clustering on host galaxy stellar mass and accretion rate using XMM-COSMOS data.

Findings

High-mass AGNs reside in halos similar to non-active galaxies.

Low-mass AGNs are found in more massive halos than non-active galaxies.

No significant dependence of clustering on accretion rate.

Abstract

We study the spatial clustering through the projected two-point correlation function of $632$ $(1130)$ XMM-COSMOS Active Galactic Nuclei (AGNs) with known spectroscopic (spectroscopic or photometric) redshifts in the range $z = [0.1 - 2.5]$ in order to measure the AGN bias and estimate the typical mass of the hosting dark matter (DM) halo as a function of AGN host galaxy properties. We create AGN subsamples in terms of stellar mass $M_*$ and specific black hole accretion rate $L_X/M_*$, to probe how AGN environment depends on these quantities. For the full spectroscopic AGN sample, we measure a typical DM halo mass of $\log (M_\mathrm{halo} / h^{-1}\mathrm{M}_\odot)= 12.79_{-0.43}^{+0.26}$, similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. We find no significant dependence on $L_X/M_*$, with $\log (M_\mathrm{halo} / h^{-1}\mathrm{M}_\odot) = 13.06_{-0.38}^{+0.23}$ ($12.97_{-1.26}^{+0.39}$) for the low (high) $L_X/M_*$ subsample. We also find no difference in the hosting halos in terms of $M_*$ with $\log (M_\mathrm{halo} / h^{-1}\mathrm{M}_\odot) = 12.93_{-0.62}^{+0.31}$ ($12.90_{-0.62}^{+0.30}$) for the low (high) $M_*$ subsample. By comparing the $M_*-M_\mathrm{halo}$ relation derived for XMM-COSMOS AGN subsamples with what is expected for normal non-active galaxies by abundance matching and clustering results, we find that the typical DM halo mass of our high $M_*$ AGN subsample is similar to that of non-active galaxies. However, AGNs in our low $M_*$ subsample are found in more massive halos than non-active galaxies. By excluding AGNs in galaxy groups from the clustering analysis, we find evidence that the result for low $M_*$ may be due a larger fraction of AGNs as satellites in massive halos.

Tables

Table 1. Table 1: XMM -COSMOS AGN subsamples.

	${\sum }_i{\mathrm{pdf}}_i$	$⟨z⟩$	$⟨\log \left(M_{\ast }/M_{\odot }\right)⟩$	$⟨\log \frac{L_X/M_{\ast }}{\mathrm{erg}{s}^{-1}{M}_{\odot }}⟩$	type1/type2	${\chi }_{\min }^2$	dof	$b$	$\log \frac{M_{\mathrm{halo}}}{h^{-1}{M}_{\odot }}$
Specz
All	632	1.19	10.72	33.02	351/281	15.14	7	${2.20}_{-0.45}^{+0.37}$	${12.79}_{-0.43}^{+0.26}$
Low $L_X/M_{\ast }$	309	0.88	10.73	32.53	88/221	6.91	6	${2.14}_{-0.41}^{+0.35}$	${13.06}_{-0.38}^{+0.23}$
High $L_X/M_{\ast }$	309	1.50	10.73	33.49	253/ 56	1.88	4	${2.95}_{-1.42}^{+0.93}$	${12.97}_{-1.26}^{+0.39}$
Low $M_{\ast }$	309	0.97	10.39	33.03	134/175	7.96	6	${2.11}_{-0.58}^{+0.45}$	${12.93}_{-0.62}^{+0.31}$
High $M_{\ast }$	309	1.41	11.05	33.02	211/ 98	10.25	5	${2.69}_{-0.79}^{+0.61}$	${12.90}_{-0.62}^{+0.30}$
Specz + Photz Pdfs
All	664	1.20	10.72	33.03	372/292	16.28	7	${2.21}_{-0.40}^{+0.34}$	${12.77}_{-0.37}^{+0.23}$
Low $L_X/M_{\ast }$	325	0.88	10.73	32.53	95/230	8.45	5	${2.12}_{-0.45}^{+0.37}$	${13.03}_{-0.43}^{+0.25}$
High $L_X/M_{\ast }$	325	1.52	10.73	33.51	268/57	1.22	4	${2.91}_{-1.03}^{+0.75}$	${12.93}_{-0.77}^{+0.33}$
Low $M_{\ast }$	323	0.98	10.39	33.03	139/184	9.76	6	${2.25}_{-0.48}^{+0.40}$	${13.03}_{-0.43}^{+0.25}$
High $M_{\ast }$	323	1.42	11.05	33.03	224/99	13.36	5	${2.18}_{-0.78}^{+0.57}$	${12.53}_{-0.98}^{+0.39}$
Specz no groups
Low $M_{\ast }$	287	0.99	10.37	33.06	130/157	6.20	5	${1.69}_{-0.72}^{+0.49}$	${12.50}_{-1.67}^{+0.47}$
High $M_{\ast }$	292	1.45	11.05	33.05	207/85	9.12	4	${2.48}_{-0.71}^{+0.55}$	${12.73}_{-0.64}^{+0.32}$

Equations

$${\lambda }_{\mathrm{Edd}}=\frac{A\times k_{\mathrm{bol}}}{1.3\times 10^{38}}\times \frac{L_X}{M_{\ast }}.$$

$$\mathrm{d}P=n\left[1+\xi (r)\right]\mathrm{d}V,$$

$$\xi (r)=\frac{D{D}^{\prime }-2D{R}^{\prime }+R{R}^{\prime }}{R{R}^{\prime }},$$

$$D{D}^{\prime }$$

$$D{R}^{\prime }$$

$$R{R}^{\prime }$$

$$w_p(r_p)=2{\int }_0^{\infty }\xi (r_p,\pi )\mathrm{d}\pi .$$

$$w_p^{2-\mathrm{halo}}(r_p)=b^2{w}_{\mathrm{DM}}^{2-\mathrm{halo}}(r_p),$$

$$w_{\mathrm{DM}}^{2-\mathrm{halo}}(r_p)=2{\int }_{r_p}^{\infty }\frac{{\xi }_{\mathrm{DM}}^{2-\mathrm{halo}}(r)r\mathrm{d}r}{\sqrt{r^2-r_p^2}},$$

$${\xi }_{\mathrm{DM}}^{2-\mathrm{halo}}(r)=\frac{1}{2{\pi }^2}\int P^{2-\mathrm{halo}}(k)k^2\left[\frac{\sin kr}{kr}\right]\mathrm{d}k.$$

$$C_{ij}=\frac{1}{N_{\mathrm{rs}}}\sum _{k=1}^{N_{\mathrm{rs}}}\left[w_{p,k}(r_{p,i})-⟨w_p⟩(r_{p,i})\right]\left[w_{p,k}(r_{p,j})-⟨w_p⟩(r_{p,j})\right],$$

Full text

*Methods. *We measure the projected two-point correlation function $w_{p}(r_{p})$ using both the classic and the generalized clustering estimator based on photometric redshifts as probability distribution functions in addition to any available spectroscopic redshifts. We measure the large-scale ($r_{p}\gtrsim 1\,{h}^{-1}\mathrm{Mpc}$) linear bias $b$ by comparing the clustering signal to that expected of the underlying DM distribution. The bias is then related to the typical mass of the hosting halo $M_{\mathrm{halo}}$ of our AGN subsamples. Since $M_{*}$ and $L_{X}/M_{*}$ are correlated, we match the distribution in terms of one quantity, while split the distribution in the other.

*Results. *For the full spectroscopic AGN sample, we measure a typical DM halo mass of $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=12.79_{-0.43}^{+0.26}$, similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. We find no significant dependence on specific accretion rate $L_{X}/M_{*}$, with $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=13.06_{-0.38}^{+0.23}$ and $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=12.97_{-1.26}^{+0.39}$ for low and high $L_{X}/M_{*}$ subsamples, respectively. We also find no difference in the hosting halos in terms of $M_{*}$ with $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=12.93_{-0.62}^{+0.31}$ (low) and $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=12.90_{-0.62}^{+0.30}$ (high). By comparing the $M_{*}-M_{\mathrm{halo}}$ relation derived for XMM-COSMOS AGN subsamples with what is expected for normal non-active galaxies by abundance matching and clustering results, we find that the typical DM halo mass of our high $M_{*}$ AGN subsample is similar to that of non-active galaxies. However, AGNs in our low $M_{*}$ subsample are found in more massive halos than non-active galaxies. By excluding AGNs in galaxy groups from the clustering analysis, we find evidence that the result for low $M_{*}$ may be due a larger fraction of AGNs as satellites in massive halos.

1 Introduction

Supermassive black holes (SMBH) with $M\sim 10^{6-9}\,\mathrm{M}_{\odot}$ reside at the centers of virtually every massive galaxy. SMBHs reach these masses by growing via matter accretion and simultaneously shine luminously as an active galactic nucleus (AGN). Interestingly, BHs and their host galaxies seem to co-evolve, as suggested by the correlation between the SMBH and the host galaxy properties (velocity dispersion, luminosity, stellar mass). However, the co-evolution scenario, AGN feedback and accretion mechanisms are still poorly known (e.g. Alexander & Hickox, 2012).

AGNs and their host galaxies reside in collapsed dark matter (DM) structures i.e. halos. In the concordance $\Lambda$CDM cosmology these halos form hierarchially ’bottom up’ from the smallest structures (density fluctuations in the CMB) that grow via gravitational instability to the largest (galaxy groups and clusters). AGNs and DM halos they reside in are both biased tracers of the underlying DM distribution. By measuring the clustering of AGN, and comparing that to the underlying DM distribution, the AGNs may be linked to their hosting DM halos (e.g. Cappelluti et al., 2012; Krumpe et al., 2014). Recent AGN clustering measurements have not been able to paint a coherent picture of the complex interplay of AGN and their environment. It seems that optically selected luminous quasars prefer to live in halos $\mathrm{few}\times 10^{12}\,{h}^{-1}\mathrm{M}_{\odot}$ over a wide range in redshift (Croom et al., 2005; da Ângela et al., 2008; Ross et al., 2009) while moderate luminosity X-ray selected AGN prefer larger halos $10^{12.5-13}\,{h}^{-1}\mathrm{M}_{\odot}$ at similar redshifts (Coil et al., 2009; Allevato et al., 2011; Koutoulidis et al., 2013).

Mendez et al. (2016) suggest that the clustering of AGN could be understood as the clustering of galaxies with matched properties in terms of stellar mass and star-formation rate and redshift, and AGN selection effects. This would indicate that instead of the properties of the AGN itself, the properties of the host galaxy, such as, stellar mass $M_{*}$ or specific black hole accretion rate $L_{X}/M_{*}$ have a more significant role in driving the clustering of AGN.

Many authors have investigated the relation between the stellar mass and the DM halo mass, the so-called $M_{*}-M_{\mathrm{halo}}$ relation, for normal non-active galaxies via abundance matching (Moster et al., 2013; Behroozi et al., 2013), clustering measurements and HOD modeling (Zheng et al., 2007; Wake et al., 2011) or weak lensing (Coupon et al., 2015). For X-ray selected AGNs, the $M_{*}-M_{\mathrm{halo}}$ relation has only recently been studied observationally. Georgakakis et al. (2014) argue that AGN environment is closely related to $M_{*}$. However, they do not measure $M_{*}$ directly, but use the rest frame absolute magnitude in the J band as a proxy for $M_{*}$. Very recently, Mountrichas et al. (2019) measured the AGN clustering dependence directly in terms of $M_{*}$ and found that the environments of X-ray AGN at $z=0.6-1.4$ are similar to normal galaxies with matched SFR and redshift.

In this study, we wish to build upon the previous X-ray selected AGN clustering measurements in XMM-COSMOS (Miyaji et al., 2007; Gilli et al., 2009; Allevato et al., 2011), to investigate the clustering dependence on host galaxy properties ($M_{*}$, $L_{X}/M_{*}$). We compare this to the $M_{*}-M_{\mathrm{halo}}$ relation for normal non-active galaxies. In our clustering measurements, we also investigate the new generalized estimator which has been introduced (Georgakakis et al., 2014; Allevato et al., 2016), where photometric redshifts are included in the clustering analysis as probability density functions. Clustering measurements using photometric redshifts will be important in future X-ray AGN surveys, where spectroscopic redshifts are not available either due to AGN being optically faint, or because no extensive spectroscopic follow-up campaigns are available. In eROSITA, for example, spectroscopic redshifts will be available only for a certain portion of the sky, and only at later stages of the survey (Merloni et al., 2019).

We adopt a flat $\Lambda$CDM cosmology with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $\sigma_{8}=0.8$ and $h=0.7$. Distances reported are comoving distances and the dependence in $h$ is shown explicitly. The symbol ‘$\log$‘ signifies base 10 logarithm. DM halo masses are defined as the enclosed mass within the Virial radius, within which the mean density is 200 times more than the background density. DM halo masses scale as $h^{-1}$, while $M_{*}$ scales as $h^{-2}$.

2 XMM-COSMOS Multiwavelength Data Set

To study the dependence of AGN clustering in terms of host galaxy properties, we use the Cosmic Evolution Survey (COSMOS, Scoville et al., 2007). COSMOS is a multiwavelength survey over $1.4\times 1.4\,\mathrm{deg}^{2}$ field designed to study the evolution of galaxies and AGNs up to redshift $z\sim 6$. To date the field has been covered by a wide variety of instruments from radio to X-ray bands. XMM-Newton surveyed $2.13\,\mathrm{deg}^{2}$ of the sky in the COSMOS field in the $0.5-10\,\mathrm{keV}$ band for a total of $1.55\,\mathrm{Ms}$ (Hasinger et al., 2007; Cappelluti et al., 2007, 2009), providing an unprecedented large sample of point-like X-ray sources (1822).

Brusa et al. (2010) carried out the optical identification and presented the multiwavelength properties ($24\mu\mathrm{m}$ to UV) of ${\sim}1800$ sources with a spectroscopic completeness of ${\sim}50\%$ (e.g. Hasinger et al., 2018). Salvato et al. (2009, 2011) derived accurate photometric redshifts with $\sigma_{\Delta z/(1+z_{\mathrm{spec}})}\sim 0.015$. Bongiorno et al. (2012) used a Spectral Energy Distribution (SED) fitting technique based on AGN+Galaxy template SEDs and estimated the host galaxy properties, i.e. stellar mass $M_{*}$ and star-formation rate (SFR) of ${\sim}1700$ AGN in COSMOS up to $z\lesssim 3$. The quantity $L_{X}/M_{*}$ corresponds to the rate of accretion onto the central SMBH scaled relative to the stellar mass of the host galaxy. Assuming a $M_{*}-M_{\mathrm{BH}}$ relation and a constant bolometric correction to convert from $L_{X}$ to $L_{\mathrm{bol}}$, then Eddington ratio ($\lambda_{\mathrm{Edd}}\equiv L_{\mathrm{bol}}/L_{\mathrm{Edd}}$) can be expressed as:

[TABLE]

With $A=500$ and $k_{\mathrm{bol}}=25$, $L_{X}/M_{*}=10^{34}\,\mathrm{erg}\,\mathrm{s}^{-1}\,\mathrm{M}_{\sun}^{-1}$ corresponds to accretion at Eddington luminosity i.e. $\lambda_{\mathrm{Edd}}=1$ (Bongiorno et al., 2012).

In this paper we use the catalog presented in Bongiorno et al. (2012), and we focus on $1130$ AGN in the redshift range $0.1<z<2.5$, with mean $z\sim 1.2$. The redshifts are either spectroscopic (632) or high quality photometric (498) ones. The 2-10 keV luminosity $L_{X}$ spans $\log(L_{X}/\,\mathrm{erg}\,\mathrm{s}^{-1})=42.3-45.5$ with a mean $\log(L_{X}/\,\mathrm{erg}\,\mathrm{s}^{-1})=43.7$. The typical host galaxy of our AGN is a red and massive galaxy with mean $\log\left(M_{*}/\,\mathrm{M}_{\sun}\right)=10.7$. However, the host galaxies also span a wide range of stellar masses with $\log\left(M_{*}/\,\mathrm{M}_{\sun}\right)={7.6-12.3}$. The $L_{X}$ and $M_{*}$ distributions for our sample of $1130$ XMM-COSMOS AGN are shown in Figure 2. It would be of interest to also study the clustering as a function of host galaxy SFR or specific SFR (SFR$/M_{*}$) as recently done by Mountrichas et al. (2019). However, Bongiorno et al. (2012) conclude for XMM-COSMOS that while stellar masses from SED fitting are relatively robust for both type 1 and type 2 AGNs, SFRs are more sensitive to AGN contamination from type 1 AGN and are unreliable. Thus in order to increase statistics in our clustering analysis, we will not consider the host galaxy SFR, available only for type 2 AGN in XMM-COSMOS.

The recent Chandra COSMOS Legacy Survey (CCLS; Civano et al., 2016; Marchesi et al., 2016) contains the largest sample of X-ray selected AGNs to date. However, for CCLS AGN, host galaxy properties have only been estimated for type 2 AGNs, while Bongiorno et al. (2012) provide the estimates for both type 1 and 2 AGNs. Further, the clustering of XMM-COSMOS AGNs is well studied (Miyaji et al., 2007; Gilli et al., 2009; Allevato et al., 2011, 2012, 2014), but not in terms of host galaxy properties as in this work. For CCLS AGN, Allevato et al. (2016) measured the clustering at $2.9\leq z\leq 5.5$, and Koutoulidis et al. (2018) used multiple fields including COSMOS to measure the clustering. Thus, there are no clustering measurements for CCLS AGN at the redshift of interest ($z<2.5$).

2.1 AGN Subsamples

The full AGN sample with known spectroscopic redshifts consists of $N=632$ AGNs with mean $z=1.19$. For AGNs with only known photometric redshifts, we take into account the full probability distribution function $\mathrm{Pdf}(z)$. In this picture, the total weight of an AGN is the integral over $z$. We limit ourselves to $z<2.5$ and the combined weighted number of AGNs with photometric redshifts is $N=488.64$ with weighted mean $z=1.44$

To study the dependence on host galaxy properties, we divide our AGN sample effectively in two bins of $M_{*}$ and $L_{X}/M_{*}$ which we refer to as the low and high subsamples. In detail, first we bin the distribution of host galaxy stellar mass $\log M_{*}$ of the sample with binsize $0.1$ dex. Then, each bin is split individually exactly in half based on the logarithm of the specific BH accretion rate $\log L_{X}/M_{*}$ to create the low and high $L_{X}/M_{*}$ subsamples. The low and high $L_{X}/M_{*}$ subsamples consist of $309$ objects each. We find the average values for the low (high) $L_{X}/M_{*}$ subsamples to be mean $\log L_{X}/M_{*}=32.53\ (33.49)$, while the difference in mean $\log M_{*}$ is $\lesssim 0.01$. We then repeat this process by binning the $\log L_{X}/M_{*}$ and splitting in terms of $\log M_{*}$. The number of objects in the low and high $M_{*}$ subsamples is $309$. The average values for the low (high) $M_{*}$ subsamples are mean $\log M_{*}=10.39\ (11.05)$ and the difference in mean $\log(L_{X}/M_{*})$ is $\lesssim 0.01$.

COSMOS is known to be affected by cosmic variance that influences the clustering measurements (e.g. Gilli et al., 2009; Mendez et al., 2016). This means that it is also important to take into account how our low and high $M_{*}$ AGN subsamples relate to the large structures in the field. To this end, as an additional test, we associate the AGN sample with known spectroscopic redshifts with the co-added COSMOS galaxy group catalog (see Finoguenov et al., 2007; Leauthaud et al., 2010; George et al., 2011). An AGN is taken to belong to a galaxy group if the AGN-group angular separation on the sky is $<R_{200,\mathrm{deg}}$ (radius of the group in degrees enclosing 200 times the critical density), and the radial comoving distance separation is $<\pi_{\mathrm{max}}$ (see Section 3). We find $22$ ($17$) AGNs in our low (high) $M_{*}$ AGN subsamples with spectroscopic redshifts in galaxy groups with a total number of $39$ AGNs.

We summarize the properties of the different AGN subsamples in Table 1, and the $L_{X}$ and $M_{*}$ distributions are shown in Figures 2 and 3.

3 Methods

3.1 Two-Point Statistics

In clustering studies, a widely used measure to quantify clustering is the two-point correlation function $\xi(r)$ which is defined as the excess probability above random of finding a pair of AGNs in a volume element $\mathrm{d}V$ at physical separation $r$, so that

[TABLE]

where $n$ is the mean number density of AGNs. To estimate $\xi(r)$, we use the Landy & Szalay (1993) estimator

[TABLE]

where

[TABLE]

and $DD$, $DR$ and $RR$ are the number of data-data, data-random and random-random pairs with physical separation $r$, respectively. $N_{d}$ and $N_{r}$ are the total number of sources in the data and random catalogs. This estimator requires the creation of a random catalog to act as an unclustered distribution of AGNs with the same selection effects in terms of RA, Dec, and redshift, as present in the data catalog (see Section 3.4).

As the distances between AGN are inferred from their redshifts, the estimates are affected by distortions due to peculiar motions of AGNs. To avoid this effect, we express pair separations in terms of distance parallel ($\pi$) and perpendicular ($r_{p}$) to the line-of-sight of the observer, defined with respect to the mean distance to the pair. Then, the projected 2PCF, which is insensitive to redshift space distortions, is defined as (Davis & Peebles, 1983)

[TABLE]

In practice, the integration is not carried out to infinity, but to finite value $\pi_{\mathrm{max}}$. The estimation of the $\pi_{\mathrm{max}}$ is a balance between including all of the correlated pairs and not including noise to the signal by uncorrelated pairs. For the estimation of the 2PCFs, we use CosmoBolognaLib 111https://github.com/federicomarulli/CosmoBolognaLib(Marulli et al., 2016), which is a free (as in freedom) software library for numerical cosmological calculations.

We note that another common way to measure the clustering is to use the cross-correlation function where positions of both an AGN sample and a complete galaxy sample are used to decrease statistical uncertainties (e.g. Coil et al., 2009; Krumpe et al., 2015; Powell et al., 2018; Mountrichas et al., 2019). At our redshift of interest in COSMOS, especially at $1\lesssim z\lesssim 2.5$, it is difficult to build a complete galaxy sample with known spectroscopic redshifts (see Sec. 3.2 for discussion on the effect of photometric redshift in clustering measurements) to measure the clustering with, and thus we are limited to the AGN auto-correlation function.

3.2 Generalized Estimator

Motivated by recent progress in utilizing photometric redshifts in AGN clustering studies (Georgakakis et al., 2014; Allevato et al., 2016), we use the full probability distribution function $\mathrm{Pdf}(z)$ for AGNs with no known spectroscopic redshifts. In this approach, the classic Landy & Szalay (1993) estimator is replaced by a generalized one, where pairs are weighted based on $\mathrm{Pdf}(z)$ of the two objects. For the details, we refer the reader to Georgakakis et al. (2014, Section 3).

For the $498$ AGNs with photometric redshifts, we discretize the $\mathrm{Pdf}(z)$ by integrating the Pdfs in terms of $z$ with an accuracy of $\delta z=0.01$, and normalize the Pdfs to unity. Further, we only consider the part of the Pdf with $\mathrm{Pdf}(z)>10^{-5}$. Using our redshift limit, we only use the part of the Pdfs with $z<2.5$. This means that the AGNs with Pdfs that span over this redshift limit are cut, and for these AGNs, the Pdf does not necessarily sum to unity i.e. $\sum_{i}\mathrm{Pdf}(z_{i})\leq 1$.

Large uncertainties in photometric redshifts may lead to loss of not only accuracy, but also not being able to recover the full clustering signal. This is highlighted by the use of large values of $\pi_{\mathrm{max}}\gtrsim 200\,{h}^{-1}\mathrm{Mpc}$ (Georgakakis et al., 2014; Allevato et al., 2016) versus studies with only spectroscopic redshifts with $\pi_{\mathrm{max}}\lesssim 100\,{h}^{-1}\mathrm{Mpc}$ (e.g. Coil et al., 2009; Allevato et al., 2011; Mountrichas et al., 2016). Therefore, we select only Pdfs based on the following quality criteria: the comoving distance separation between the $z_{\mathrm{min}}$ and $z_{\mathrm{max}}$ may not exceed a critical value of $\Delta d=100\,{h}^{-1}\mathrm{Mpc}$. We define $z_{\mathrm{min}}$ and $z_{\mathrm{max}}$ separately for each AGN so that $\mathrm{Pdf}(z)<10^{-5}$ for $z<z_{\mathrm{min}}$ and $z>z_{\mathrm{max}}$.

In detail, from the total of 498 AGN with photometric redshifts, 32 AGN pass the quality criterion and are included in the subsample including spectroscopic and photometric redshifts. In terms of our $L_{X}/M_{*}$ ($M_{*}$) AGN subsamples, a total of 32 (28) AGN with photometric redshifts are kept and divided equally between the low and high subsamples in both cases. The number of AGN in each of our subsamples including photometric redshifts are shown in Table 1.

This quality cut is suggested by the fact that including all phot-z Pdfs will lead to large uncertainties in the measured clustering signal for all the AGN subsamples. The investigation of quality criteria for studies including phot-z Pdfs is beyond the scope of this work. However, given the importance of photometric redshifts in future large surveys such as eROSITA, we will explore clustering photz Pdfs in a future study (Viitanen et al., in prep.).

3.3 Halo model

In the halo model (e.g. Cooray & Sheth, 2002), the AGN clustering signal is the sum of the 1-halo and 2-halo terms, which arise from the clustering of AGN that occupy the same halo, and two distinct halos, respectively. On large scales ($r_{p}\gtrsim 1\,{h}^{-1}\mathrm{Mpc}$), the 2-halo term is the dominant term, and the AGN projected 2PCF may be related to the underlying DM projected 2PCF $w_{\mathrm{DM}}^{\mathrm{2-halo}}$ via the linear bias $b$

[TABLE]

where $w_{\mathrm{DM}}^{\mathrm{2-halo}}$ is estimated at the mean redshift of the corresponding AGN subsample and integrated to the same value of $\pi_{\mathrm{max}}$. The DM projected 2PCF is related to the DM one-dimensional 2PCF $\xi_{\mathrm{DM}}^{\mathrm{2-halo}}$

[TABLE]

where $\xi_{\mathrm{DM}}^{\mathrm{2-halo}}(r)$ is in turn estimated using the linear power spectrum $P^{\mathrm{2-halo}}(k)$:

[TABLE]

We base our estimation of the linear power spectrum on Eisenstein & Hu (1999), which is also implemented in CosmoBolognaLib.

The 1-halo term ($r_{p}\lesssim 1\,{h}^{-1}\,\mathrm{Mpc}$) also contains important information on the AGN halo occupation and could be contributing towards the clustering signal up to scales $r_{p}\sim 3\,{h}^{-1}\,\mathrm{Mpc}$. However, due to low number counts of pairs especially at small scales $r_{p}\lesssim 3\,{h}^{-1}\,\mathrm{Mpc}$ in our XMM-COSMOS subsamples (see Fig. 5), we are not able to constrain the AGN 1-halo term and excluding the 1-halo term from the modeling does not affect our results significantly at large scales.

3.4 Random catalog and error estimation

The random catalog consists of an unclustered set of AGNs with the same selection effects and observational biases. To this end, we follow Miyaji et al. (2007). In detail, for each random object, we draw right ascension and declination at random in the COSMOS field. In detail, right ascension is drawn uniformly, while for declination we draw $\sin(\mathrm{Dec})$ uniformly. Then, we draw a $0.5-2\,\mathrm{keV}$ flux from the data catalog, and if the drawn flux is above the limit given by the sensitivity map (Cappelluti et al., 2009, see also Figure 1), we keep the object. Otherwise we discard it. Each kept random object is given a redshift drawn from the smoothed redshift distribution of the data catalog with gaussian smoothing using $\sigma_{z}=0.3$. For each of the data catalogs, we create a random catalog with $N_{r}=100N_{d}$. We show the redshift distribution of the data and random catalogs for our AGN subsamples in Figure 4.

Poissonian errors are readily assigned to the projected 2PCF, but are known to underestimate the errors. For this reason we adopt a Bootstrap resampling technique by dividing the XMM-COSMOS survey into $N_{\mathrm{region}}=18$ subregions ($3\times 3\times 2$ for RA, Dec, and comoving distance, respectively) of roughly equal comoving volumes. We resample the regions $N_{\mathrm{rs}}=100$ times. In each of the resamples, the regions are assigned different weights based on the number of times they are selected (Norberg et al., 2009). The elements of the covariance matrix $C$ are then defined as

[TABLE]

where $i$ and $j$ refer to the $i$th and $j$th $r_{p}$ bins and the bar denotes the mean over $N_{\mathrm{region}}$ resamples. The $1\sigma$ error for $w_{p}(r_{p,i})$ is the square root of the corresponding diagonal element i.e. $\sqrt{C_{ii}}$.

4 Results

For each of the AGN subsamples, we estimate the projected 2PCF $w_{p}(r_{p})$ with $r_{p}=1.0-100\,{h}^{-1}\mathrm{Mpc}$ using $12$ logarithmic bins. We use one bin in the $\pi$ direction, where the upper limit of this bin is dictated by $\pi_{\mathrm{max}}$. In order to set $\pi_{\mathrm{max}}$, we try out all the values in the range $\pi_{\mathrm{max}}=20-75\,{h}^{-1}\mathrm{Mpc}$ with an accuracy of $\Delta\pi_{\mathrm{max}}=5\,{h}^{-1}\mathrm{Mpc}$. For the full spectroscopic AGN sample, we found that the signal converges at $\pi_{\mathrm{max}}=40\,{h}^{-1}\mathrm{Mpc}$, which is adopted for all the subsamples. This value is similar to previous clustering studies involving XMM-COSMOS AGNs (Gilli et al., 2009; Allevato et al., 2011). The AGN projected 2PCF $w_{p}(r_{p})$ is then estimated using Eq. 7 and the $1\sigma$ bootstrap errors are estimated using Eq. 11. We show the estimated projected 2PCF for our subsamples in Figure 5. Comparison between the spectroscopic subsamples and the specz+photz subsamples are shown in Figures 5 (full) and 6 ($M_{*}$ and $L_{X}/M_{*}$ subsamples).

We derived the best-fit large-scale bias (Eq. 8) using $\chi^{2}$ minimization for $r_{p}=1-30\,{h}^{-1}\mathrm{Mpc}$. In detail, we utilize the inverse of the full covariance matrix $C^{-1}$ and minimize $\chi^{2}=\Delta^{T}C^{-1}\Delta$, where $\Delta$ is a with the same number of elements as the number of $r_{p}$ bins used in the fit. $\Delta$ is defined explicitly as $\Delta=w_{p,\mathrm{AGN}}^{\mathrm{2-halo}}-b^{2}w_{\mathrm{DM}}^{\mathrm{2-halo}}$. With one free parameter, we estimate the $1\sigma$ errors on the best-fit bias, given by the lower and upper bounds of the region $(\chi^{2}-\chi_{\mathrm{min}}^{2})/\nu\leq 1.0$, where $\nu=N-1$ is the number of degrees of freedom. To exclude noisy bins in the fit, we require that the number of pairs in each bin is $>16$. The large-scale bias derived for all the XMM-COSMOS AGN subsamples are summarized in Table 1 and shown in Figure 7.

For the full spectroscopic AGN subsample (Figure 5), we find a best-fit bias of $b=2.20_{-0.45}^{+0.37}$. Following the bias-mass relation described in van den Bosch (2002) and Sheth et al. (2001), this corresponds to a typical mass of the hosting halo of $\log(M_{\mathrm{halo}}/h^{-1}\mathrm{M}_{\odot})=12.79_{-0.43}^{+0.26}$. Notice that in this work we define the typical mass explicitly as the DM halo mass which satisfies $b=b(M_{\mathrm{halo}})$ (e.g. Hickox et al., 2009; Allevato et al., 2016; Mountrichas et al., 2019). Albeit with large uncertainties, we find a small $\lesssim 1\sigma$ difference in the biases of the spectroscopic AGN subsamples split in terms of stellar mass (Figure 5). The biases are $b=2.11_{-0.58}^{+0.45}$ for the low stellar mass and $b=2.69_{-0.79}^{+0.61}$ for the high stellar mass. However, it is worth noting that the two subsamples peak at different redshifts ($z\sim 1.0$ versus $z\sim 1.4$). In terms of the typical masses of the hosting halos, we find no difference. For the $M_{*}$ subsamples, we find that excluding AGNs that are associated with groups has a greater effect on the measured best-fit bias of the low $M_{*}$ subsample. We measure $b=1.69_{-0.72}^{+0.49}$ ($b=2.48_{-0.71}^{+0.55}$) for the low (high) $M_{*}$ AGN subsample. This lower value for the bias could be an indication that AGNs in galaxies with lower stellar mass are more preferably satellites in their DM halos.

Moreover, we derive an AGN bias $b=2.14_{-0.41}^{+0.35}$ (at $z\sim 0.9$) and $b=2.95_{-1.42}^{+0.93}$ ($z\sim 1.5$) for the low and high $L_{X}/M_{*}$ subsamples, respectively (Figure 5). No significant difference is observed in the typical masses of the hosting halos.

Similar results in terms of bias dependence on $M_{*}$ and $L_{X}/M_{*}$ are found when using phot-z Pdfs in addition to any available spectroscopic redshifts. In particular, in our full AGN subsamples, an increase of ${\sim}5\%$ in the weighted number of AGNs introduces no systematic error in the estimation of the bias, but decreases the $1\sigma$ error of the bias by $(\delta b_{1}-\delta b_{2})/\delta b_{1}\sim 10\%$, where $\delta b_{i}$ is the average error derived from the lower and upper limits of the bias (see Table 1). However, since including photometric redshifts do not change the conclusions drawn from our clustering measurements, in the following sections we focus on the results from the AGN subsamples with known spectroscopic redshifts.

5 Discussion

We have performed clustering measurements of 1130 X-ray selected AGN in XMM-COSMOS at $0.1<z<2.5$ (mean $z\sim 1.2$) in order to study AGN clustering dependence on host galaxy stellar mass and specific BH accretion rate $L_{X}/M_{*}$. For our AGN subsamples we find a typical DM halo mass $\sim 10^{13}\,{h}^{-1}\,\mathrm{M}_{\sun}$ that roughly correspond to group-sized environments. This is in agreement with similar studies using X-ray selected AGNs at similar redshifts (Coil et al., 2009; Allevato et al., 2011; Fanidakis et al., 2013; Koutoulidis et al., 2013), as well as at lower redshifts $z<0.1$ (e.g. Krumpe et al., 2018; Powell et al., 2018). We have also investigated including photometric redshifts as Pdfs in the analysis in addition to any available spectroscopic redshifts.

In COSMOS, Leauthaud et al. (2015) use weak lensing measurements on X-ray COSMOS AGN at $z<1$ with $\log L_{X}/\mathrm{erg}\,\mathrm{s}^{-1}=[41.5-43.5]$ and $\log M_{*}/\mathrm{M}_{\odot}=[10.5-12]$. They infer that $50$ per cent of AGN reside in halos with $\log M_{\mathrm{halo}}/\mathrm{M}_{\odot}<12.5$ in tension with the claim that X-ray AGN inhabit group-sized environments with masses $\sim 10^{13}\,\mathrm{M}_{\odot}$. However, they also underline that due to the skewed tail in the halo mass distribution, the effective/typical halo mass derived from clustering measurements may be markedly different from the median of the distribution.

In fact, they found an effective mass of $M_{\mathrm{eff}}\sim 10^{12.7}\,\mathrm{M}_{\odot}$, which is close to the typical halo masses derived in this work. It is worth noticing that they derived the effective halo mass from modelling the AGN halo occupation (see Eq. (4) in Leauthaud et al., 2015), which may differ from the typical halo mass inferred from the 2-halo term as in this work.

Also, they found that the effective DM halo mass of their AGN sample lies between the median and the mean values of the DM halo mass distribution, which are lower and higher than the effective DM halo mass, respectively. Given the statistics in our XMM-COSMOS AGN sample, we are not able to constrain the median or the mean of the DM halo mass distribution. In the future this could be done through HOD modelling, provided the 1-halo term is constrained.

Moreover, different cuts in luminosity and host galaxy mass may reflect in different hosting DM halo mass distributions. For instance, our sample of XMM-COSMOS AGN spans a range of host galaxy stellar masses $\log M_{*}/\mathrm{M}_{\odot}=[8-12]$, including also low-mass systems with masses $<10^{10.5}\,\mathrm{M}_{\odot}$ (that are likely satellite galaxies in galaxy groups), and probes higher redshifts up to $z=2.5$.

5.1 Clustering in terms of specific BH accretion rate

We divided the full sample in low and high specific BH accretion rate subsamples with the same $M_{*}$ distributions and find no significant clustering dependence on $L_{X}/M_{*}$, and thus Eddington ratio. Krumpe et al. (2015) also found no dependence on $\lambda_{\mathrm{Edd}}$ for their sample of local ($0.16<z<0.36$) X-ray and optically selected AGN in the Rosat All-Sky Survey. They concluded that high accretion rates in AGN are not necessarily linked to high density environments where galaxy interactions would be frequent. Our result provides further evidence that this is also true for non-local AGN at intermediate redshifts $z\sim 1$. Mendez et al. (2016) studied the clustering of AGN in the PRIMUS and DEEP2 surveys (including the COSMOS field) at $z\sim 0.7$ based on multiple selection criteria. In their X-ray selected AGN sample, they did not find a significant dependence on clustering in terms of specific BH accretion rate, in line with our results.

5.2 Clustering in terms of host galaxy stellar mass

We also studied the AGN clustering dependence on host galaxy stellar mass, probing the $M_{*}-M_{\mathrm{halo}}$ relation for active galaxies. In Figure 8, we compare our results for XMM-COSMOS AGN with recent studies in literature using normal (non-active) galaxies. For our comparison, we convert the results to our adopted $h=0.7$ cosmology. DM halo masses defined with respect to 200 times $\rho_{\mathrm{crit}}$ have been re-defined to be with respect to mean density of the background. The blue curve shows the Moster et al. (2013) $M_{*}-M_{\mathrm{halo}}$ relation for central galaxies estimated using a multi-epoch abundance matching method which we have calculated at the mean redshift $z\sim 1.2$ of our AGN sample. The orange curve shows the galaxy $M_{*}-M_{\mathrm{halo}}$ relation of Behroozi et al. (2013) at $z\sim 1.2$. Coupon et al. (2015) estimated the $M_{*}-M_{\mathrm{halo}}$ relation in the CFHTLenS/VIPERS field at $z\sim 0.8$ using constraints from several different methods including galaxy clustering. Compared to our AGN sample, their sample has a similar range in stellar mass and a slightly lower redshift. Results from HOD modeling of galaxy clustering in DEEP2 (Zheng et al., 2007) and the NMBS (Wake et al., 2011) at comparable redshifts ($z\sim 1.0-1.1$) are shown as well. Using weak lensing methods, Leauthaud et al. (2015) studied a sample of moderate-luminosity AGN in COSMOS at a lower redshift $z\sim 0.66$ than our sample. At $M_{*}>10^{10.5}\,\mathrm{M}_{\sun}$, they suggest that AGN populate similar DM halos as normal galaxies. Similarly, we found that high $M_{*}$ ($\gtrsim 10^{10.5}\,\mathrm{M}_{\odot}$) XMM-COSMOS AGN follow the same $M_{*}-M_{\mathrm{halo}}$ relation as normal non-active galaxies. On the contrary, we estimated that low $M_{*}$ ($\lesssim 10^{10.5}\,\mathrm{M}_{\odot}$) AGN are more clustered than normal galaxies. Mountrichas et al. (2019) measured clustering of AGN from the XMM-XXL survey in terms of host galaxy properties ($M_{*}$, SFR, sSFR) at $z\sim 0.8$ and find a positive dependence on the environment with respect to $M_{*}$. Within errors, our results at slightly higher redshift are in agreement with their measurements (see Figure 8).

The $M_{*}-M_{\mathrm{halo}}$ relation obtained from our clustering analysis of XMM-COSMOS AGN is not consistent with results inferred for normal galaxies at similar redshifts, at least for the low $M_{*}$ bin. In fact, we found that AGN host galaxies with low $M_{*}$ reside in slightly more massive halos than normal galaxies of similar stellar mass. On the other hand, at high $M_{*}$, our results are in good agreement with the $M_{*}-M_{\mathrm{halo}}$ relation of normal galaxies. Following Figure 8, we do not expect the observed discrepancy at low $M_{*}$ to be due to the different mean redshift of the two subsamples ($z\sim 1$ and $z\sim 1.4$). If we exclude AGN that are associated with galaxy groups from our $M_{*}$ subsamples, we see that this affects our low $M_{*}$ bin more, while leaving the high $M_{*}$ bin relatively unchanged. This could indicate that XMM-COSMOS AGN with higher $M_{*}$ are more preferably found in central galaxies of their respective halos. For lower $M_{*}$, the fraction of AGNs as satellites would be higher. Nevertheless, excluding the galaxy groups from the analysis brings our result for the low $M_{*}$ closer to the $M_{*}-M_{\mathrm{halo}}$ of normal non-active galaxies.

It is important to note that our results for the $M_{*}$ subsamples include both type1 and type2 AGNs i.e. AGNs regardless of obscuration are considered in the same subsample. With the limited sample size of XMM-COSMOS, we are not able to further divide the subsamples and examine the $M_{*}-M_{\mathrm{halo}}$ relation for type1 and type2 AGNs separately, to see whether there are any differences between these to populations. However, this issue can be revisited with Chandra COSMOS Legacy Survey AGNs.

6 Conclusions

We have measured the clustering of XMM-COSMOS AGN in terms of host galaxy stellar mass $M_{*}$ and specific BH accretion rate $L_{X}/M_{*}$. Using these two quantities, we created AGN subsamples by splitting the full sample in terms of one quantity, while matching the distribution in the other. In addition, we investigated including AGNs with photometric redshifts as Pdfs in addition to AGNs with known spectroscopic redshifts. From our analysis, we make the following conclusions:

XMM-COSMOS AGNs are highly biased with a typical DM halo mass of $M_{\mathrm{halo}}\sim 10^{13}\,{h}^{-1}\mathrm{M}_{\odot}$, characteristic to group-sized environments and in broad agreement with previous results for moderate-luminosity X-ray selected AGN. 2. 2.

We find no significant clustering dependence in terms of specific BH accretion rate, consistent with a picture that higher accretion rates in AGNs do not necessarily correspond to more dense environments. 3. 3.

Also we find no significant clustering dependence in terms of host galaxy stellar mass. By comparing our results with various $M_{*}-M_{\mathrm{halo}}$ relations found for normal non-active galaxies, we find that our low $M_{*}$ AGN subsample is more clustered than what is expected of normal galaxies at similar $M_{*}$. We investigate this further by excluding AGNs that are associated with galaxy groups. We find that excluding objects in galaxy groups results in a lower AGN bias for the low $M_{*}$ AGN subsamples, but does not affect high $M_{*}$. This could be due to a higher fraction of satellites for the lower stellar mass systems. 4. 4.

Our selected quality criterion for including additional photometric redshifts as Pdfs decreases the errors on the measured best-fit bias and does not introduce a bias to the clustering signal. Optimal quality cuts for including photometric redshifts will be studied in a future work.

Acknowledgements.

We thank the referee for helpful comments that have improved this paper. AV is grateful to the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. VA acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 749348. TM is supported by UNAM-DGAPA PAPIIT IN111379 and CONACyT 252531. RG acknowledges support from the agreement ASI-INAF n.2017-14-H.O.