Clustering of X-Ray-Selected AGN

The study of the angular and spatial structure of the X-ray sky has been under investigation since the times of the Einstein X-ray Observatory. This topic has fascinated more than two generations of scientists and slowly unveiled an unexpected scenario regarding the consequences of the angular and spatial distribution of X-ray sources. It was first established from the clustering of sources making the CXB that the source spatial distribution resembles that of optical QSO. It then it became evident that the distribution of X-ray AGN in the Universe was strongly reflecting that of Dark Matter. In particular one of the key result is that X-ray AGN are hosted by Dark Matter Halos of mass similar to that of galaxy groups. This result, together with model predictions, has lead to the hypothesis that galaxy mergers may constitute the main AGN triggering mechanism. However detailed analysis of observational data, acquired with modern telescopes, and the use of the new Halo Occupation formalism has revealed that the triggering of an AGN could also be attributed to phenomena like tidal disruption or disk instability, and to galaxy evolution. This paper reviews results from 1988 to 2011 in the field of X-ray selected AGN clustering.


INTRODUCTION
After about 50 years from the opening of the X-ray window on the Universe with the discovery of Sco-X1 and the Cosmic X-ray background (CXB, Giacconi et al. 1962), our knowledge of high energy processes in the Universe has dramatically improved. One of the leading mechanism for the production of X-ray in the Universe is accretion onto compact objects. For this reason the study of astrophysical X-ray sources is a powerful tool for studying matter under the effects of extreme gravity. As the efficiency of converting matter into energy in accretion processes is proportional to the "compactness" of the object, (i.e. ∝ M/R), it is clear that the strongest sources powered by accretion are Super-Massive Black Holes (SMBH). It also became a cornerstone of astrophysics that every galaxy with a bulge-like component hosts a SMBH at its centre and that the BH mass and the bulge velocity dispersion are strictly related (Magorrian et al. 1998). It is also believed that black holes reach those high masses via one or more phases of intense accretion activity and therefore shining as Active Galactic Nuclei (AGN). It is believed that an AGN is basically shine mostly from the power emitted by a thin, viscous, accretion disk orbiting the central SMBH Shakura & Sunyaev (1976). Such a disk produces a high amount X-rays both from its hot inner regions (as far as the soft X-ray emission is concerned) and from a non thermal source which is supposed to be the primary source of X-rays (both soft and hard).
Since its discovery, the nature of the CXB has been strongly debated, but soon the community converged into interpreting most of the CXB as the integrated emission of AGN across the cosmic time.
While the discrete nature of the CXB has been proposed (Bergamini Londrillo & Setti 1967) and rapidly unveiled by experiments like Einstein (Giacconi et al. 1979) and ROSAT (see e.g. Hasinger et al. 1993), little cosmological information has been obtained from samples of AGN because of the scarce number of detected sources in the X-ray band. Structure formation models and numerical simulations have shown that structures in the Universe have undergone a hierarchical growth starting from the denser peaks in the primordial gaussian matter distribution. The Large Scale Structures (LSS) of the Universe are gravitationally dominated by Dark Matter (DM) and we can consider it as the responsible and one of the main driver of the Cosmological structures evolution. Dark matter is believed to clump in large scale halos (DMH Navarro, Frenk & White 1997) which are populated by galaxies. Thus galaxies can be considered as tracers of the DM distribution in the Universe and the study of their spatial clustering led us to a most comprehensive view of the LSS. On the other hand AGN/Quasar, as phase of the galactic evolution, is a quite rare phenomenon in the Universe as their space density of these objects is about 1/100-1/1000 lower than that of galaxies. This means that AGN/Quasar survey require large field of view and/or deep exposure to provide statistically significant samples. The study of their clustering and its evolution is a powerful tool to understand, from a statistical point of view, what kind of environment is more likely to host AGN. This is not just an academic question but, this is strictly related to the mechanism of AGN activation. We know that one of the candidate mechanism for triggering an AGN is galaxy merger (see e.g. Hopkins et al. 2007Hopkins & Henquist 2009;Silverman et al. 2011). The probability of such an event is definitely dependent on the environment inhabited by the host galaxy. Even if the mean distance between galaxies is relatively small, in high density (mass) environments, they have a high velocity dispersion and therefore, the likelihood of a major merger is very low. On the contrary, in the field the likelihood of galaxy mergers is low because of the large average distance between galaxies. The most favorable place to detect a merger is therefore a moderately low density (mass) environment like a group (see e.g McIntosh et al. 2009). In fact, merger-driven models (see e.g. Hopkins et al. 2007) accurately predict the observed large-scale clustering of quasars as a function of redshift up to z∼ 4. The clustering is precisely that predicted for small group halos in which major mergers of gas-rich galaxies should proceed most efficiently. Thus it is well established empirically and with theoretical predictions that quasar clustering traces a characteristic host halo mass ∼ 4 × 10 12 h −1 M ⊙ , supporting the scenario in which major mergers dominate the bright quasar populations.
In addition other phenomena like secular processes may become dominant at lower luminosities as suggested by Milosavljević et al. (2006); Hopkins et al. (2006); Hopkins & Henquist (2009). Low-luminosity AGN could be triggered in more common nonmerger events, like stochastic encounters of the black holes and molecular clouds, tidal disruption or disk instability. This leads to the expectation of a characteristic transition to mergerinduced fueling around the traditional quasar-Seyfert luminosity division (growth of BH masses above/below ∼ 10 7 M ⊙ ). However the triggering mechanism of the SMBH growth must be compliant with M BH -σ relation, that links the growth of the SMBH with growth of the bulge of the host galaxy (Magorrian et al. 1998). As shown in , the predicted largescale bias of quasars triggered by secular processes is, at all redshifts, lower than the bias estimated for quasars fueled by major mergers. This implies that low-luminosity Seyfert galaxies live in DMHs that never rich the characteristic mass associated with small group scales. On the other hand, the majority of the results on the clustering of X-ray selected AGN, suggest a picture where moderate-luminosity AGN live in massive DMHs (12.5< logM DMH [h −1 M ⊙ ] <13.5) up to z ∼ 2, i.e. X-ray selected AGN samples appear to cluster more strongly than bright quasars. The reason for this is not completely clear but several studies argued that these large bias and DMH masses could suggest a different AGN triggering mechanism respect to bright quasars characterized by galaxy merger-induced fueling.
This paper reviews results of clustering of X-ray selected AGN from the first Einstein to the most recent Chandra and XMM-Newton surveys. We give a detailed description of the methods used in this kind of analysis from simple power-law to halo models. In addition we discuss the results of X-ray AGN clustering in the framework of AGN evolution and triggering. We adopt a ΛCDM Cosmology with Ω Λ =0.7, Ω m =0.3, H 0 =100 h −1 km/s/Mpc with h=0.7 and σ 8 =0.8 (Larson et al. 2011, WMAP-7).

AMPLITUDE
As far as the X-ray source clustering results are concerned, the development of the field has always be driven by with the performance of the telescopes. In particular while first results studied the angular distribution of the unresolved CXB under the assumption that Quasars were its main contributors, recent Chandra and XMM-Newton surveys sample clustering of AGN with a precision comparable to that achievable with redshift galaxy surveys. In the following section we will use the following convention for reporting results of clustering analysis in the case of power-law representation of the auto(cross)correlation function: if the clustering is measured in the angular space, we will use: where θ 0 is the angular correlation length. If the measurements has been performed in the real (redshift) space this becomes: where γ is the 3D correlation slope and r 0 or s 0 are the correlation lengths. Barcons & Fabian (1988) measured with Einstein a clustering signal of the CXB on scales ≤ 5 ′ corresponding to an angular correlation length θ 0 ∼ 4 ′ . They have shown the importance of studying the angular structure of the CXB by pointing out that a large fraction of the CXB could have been attributed to sources with a redshift distribution similar to optical QSOs. In additon, the first prediction was not consistent with the hypothesis that the CXB was also partly produced by a diffuse hot Intergalactic Medium (IGM) component. It was also proposed that these sources were actually clustered on comoving scales of the order of ∼10 h −1 Mpc. Carrera & Barcons (1992), Georgantopoulos et al. (1993) and Soltan & Hasinger (1994) observed that the CXB was highly isotropic on scales of the order of 2 • -25 • . The first attempt of measuring the clustering of X-ray selected AGN was performed by Boyle & Mo (1993), that measured a barely significant signal by using a sample of 183 EMSS sources, mostly local AGN (z<0.2). These evidences have brought the attention to the study of the clustering of the CXB down to the arcminute scale. The first significant upward turn for the measurement of AGN clustering in the X-ray band has been brought to light by ROSAT. By using a set of ROSAT-PSPC pointing on an area of ∼40 deg 2 , Vikhlinin & Forman (1995) measured, for the first time an angular correlation signal of faint (ROSAT) X-ray sources on scales <10 ′ . By using the Limber equation (see Appendix and Peebles 1980) they have de-projected their angular correlation function into a real space correlation function and found that, under the assumption that the the redshift distribution of the sources was the same as that of optical QSOs, the spatial correlation length was in the range 6-10 h −1 Mpc. With such a result, they confirmed the hypothesis that the CXB was mostly produced by sources with a redshift distribution comparable to that of optically selected QSO, though with almost double source density. By using the results of Vikhlinin & Forman (1995) and Akylas et al. (2000, who obtained similar results), Barcons et al. (2001) has shown for the first time that X-ray selected AGN are highly biased tracers of the underlying LSS at z<1 by showing a redshift evolving bias factor as large as b∼2.
However, it is worth to consider that the deprojection of the angular correlation function into a 3D correlation relies on several assumptions, like the model dependent expected redshift distribution, which may lead to a biased estimate of the real space clustering. It is however worth noticing that angular correlation can be very useful to provide a first overview in the early phase of surveys, when optical identifications are not available, especially sampling new part of the parameter space of sources, like i.e. new unexplored luminosity/flux limits and therefore source classes. Detailed physical models are however much better investigated by more sophisticated techniques as shown in the following parts.
The first firm detection of 3-D spatial clustering of X-ray selected AGN has been claimed by Mullis et al. (2004) by using data of the ROSAT-NEP survey. They detected on an area of ∼81 deg 2 a 3σ significant signal in the redshift space auto-correlation function of soft Xray selected sources at < z >∼ 0.22. They have shown that at that redshift AGN cluster with a typical correlation length r 0 =7.4±1.9 h −1 Mpc. Their results suggest that the population of AGN in such a sample is consistent with an unbiased population with respect to the underlying matter . Their result suggested that at that redshift AGN were hosted in DMHs of mass of the order of 10 13 h −1 M ⊙ .
With the development of Chandra and XMM-Newton surveys and thanks to the high source surface densities (i.e. >400-1000 deg −2 ) our capabilities in tracing the LSS has dramatically increased. One of the first evidences that AGN are highly correlated with the underlying LSS has been pointed out by Cappi et al. (2001) and Cappelluti et al. (2005) and references therein, who showed that around massive high-z galaxy clusters the source surface density of Chandra point sources is significantly, up to two times, higher than that of the background. More recently, Koulouridis & Plionis (2010) showed that although the X-ray source surface density of AGN around galaxy clusters is larger than in the background, the amplitude of their overdensities is about 4 times lower than that of galaxies in the same fields. This has been interpreted as a clear indication of an environmental influence on the AGN activity. Silverman et al. (2011) in the COSMOS field and Koss et al. (2010) in the Swift-BAT all-sky survey have shown that the AGN fraction in galaxy pairs is higher relative to isolated galaxies of similar stellar mass providing an additional evidence of the influence of the environment on AGN activity.
Chandra and XMM-Newton performed several blanck sky extragalactic surveys, and most of them dedicated part of their efforts in the study of the LSS traced by AGN to unveil their co-evolution. Basilakos et al. (2004Basilakos et al. ( , 2005 by using data of the XMM-Newton 2dF-survey have measured an unexpected high correlation length both in the angular (θ 0 ∼10 ′′ ) and, by projection, in the real space (r 0 ∼16 h −1 Mpc). Such an high correlation length has been detected in this field only, thus one can explain such a measurement as a statistical fluctuation. With the same technique, Gandhi et al. (2006) obtained a marginal 2-3σ detection of angular clustering in the XMM-LSS survey and obtained θ 0 =6.3(42)±3( +7 −13 ) in the 0.5-2 (2-10) keV bands and a slope γ ∼2.2. Puccetti et al. (2006) measured the clustering of X-ray sources in the XMM-Newton ELAIS-S1 survey in the soft and hard energy bands with a sample of 448 sources. They obtained θ 0 = 5.2±3.8 4 ′′ and θ 0 =12.8±7.8 4 ′′ in the two bands respectively. These measurements have been deprojected with the Limber's inversion in the real space and obtained r 0 =9.8-12.8 h −1 Mpc and r 0 =13.4-17.9 h −1 Mpc in the two bands, respectively.
In the Chandra era, Gilli et al. (2005) measured the real space auto-correlation function of point sources in the CDFS-CDFN. They have measured in the CDFS r 0 =8.6±1.2 h −1 Mpc at z=0.73, while in the CDFN they obtained r 0 =4.2±0.4 h −1 Mpc. The discrepancy of these measurements has been explained with variance introduced by the relatively small field of view and the consequent random sampling of LSSs in the field. In the CLASXS survey Yang et al. (2006) obtained a measurement of the clustering at z=0.94 with r 0 =8.1 +1.2 −2.2 h −1 Mpc which proposes that AGN are hosted by DMH of mass of 10 12.1 h −1 M ⊙ (see next Section). In addition they proposed that AGN clustering evolves with luminosity and they found that the bias factor evolves with the redshift. Such a behavior is similar to that found in optically selected quasars. The XMM-Newton (Hasinger et al. 2007;Cappelluti et al. 2007Cappelluti et al. , 2009 and Chandra (Elvis et al. 2007;Puccetti et al. 2009) survey of the COSMOS field have provided a leap forward to the field of X-ray AGN clustering by surveying a 2 deg 2 field of view. The key of the success of this project is a redshift survey zCOSMOS (Lilly et al. 2007) performed simultaneously with the X-ray survey, together with observations in more than 30 energy bands from radio to X-ray, that allowed to measure either the spectroscopic or the photometric redshift of every source. In the X-ray band, the survey covers 2 deg 2 with XMM-Newton with a depth of ∼60 ks with the addition of a central 0.9 deg 2 observed by Chandra with ∼150 ks exposure. The first sample of ∼1500 X-ray sources  has been used by Miyaji et al. (2007) to determine their angular correlation function, without knowing their distance, and just assuming a theoretical redshift distribution for the purpose of Limber's deprojection. Significant positive signals have been detected in the 0.5-2 band, in the angular range of 0.5 ′ -24 ′ , while the positive signals were at the ∼ 2σ and 3σ levels in the 2-4.5 and 4.5-10 keV bands, respectively. With power-law fits to the ACFs without the integral constraint term, they have found correlation lengths of θ 0 =1.9 ±0.3 ′′ , 0.8 +0.5 −0.4 ′′ , and 6

±2
′′ for the three bands, respectively, for a fixed slope γ=1.8. The inferred comoving correlation lengths were r 0 =9.8±0.7, 5.8 +1.4 −1.7 , and 12±2 h −1 Mpc at the effective redshifts of z=1.1, 0.9, and 0.6, respectively. Comparing the inferred rms fluctuations of the spatial distribution of AGNs σ 8,AGN (see Appendix) with those of the underlying dark matter, the bias parameters of the X-ray source clustering at these effective redshifts were found in the range b = 1.5-4. Such a result lead to the conclusion that the typical mass of the DMH hosting an AGN is of the order M DMH ∼10 13 M ⊙ h −1 . Similar results have been found by Ebrero et al. (2009) using the angular correlation function of 30000 X-ray sources in the AXIS survey.
In the XMM-LSS survey Elyiv et al. (2011) measured the clustering of ∼5000 AGN and computed via Limber's deprojection the obtained r 0 = 7.2±0.8 Mpc/h and r 0 = 10.1±0.8 Mpc/h and γ ∼2 in the 0.5-2 keV and 2-10 keV energy bands, respectively. In the XMM-COSMOS field Gilli et al. (2009) measured the clustering of 562 Xray selected and spectroscopically confirmed AGN. They have obtained that the correlation length of these source, r 0 =8.6±0.5 h −1 Mpc and slope of γ=1.88±0.07. They also found that if source in redshift spikes are removed the correlation length decreases to about 5-6 h −1 Mpc . Even if not conclusively, they also showed that narrow line AGN and broad line AGN cluster in the same way, indicating that both class of sources share the same environment, an argument in favor of the unified AGN model which predicts that obscuration, and therefore the Type-I/Type II dichotomy is simply a geometrical problem. However it is worth noticing that such a procedure may artificially reduce the clustering signal and the effects of such a cut in the sample, may lead to an unreliable estimate of the clustering signal.
Even if the results of Gilli et al. (2009) provide a quite complete overview of the environments of the AGN in the COSMOS field, Allevato et al. (2011) analyzed the same field by using the halo model formalism (see Section 3). Their results show that AGN selected in the X-ray band are more biased than the more luminous optically selected QSO. This observation significantly deviates from the prediction of models of merger driven AGN activity (Hopkins et al. 2006;Bonoli et al. 2009), indicating that other mechanisms like disk/bar instability of tidal disruptions may trigger an AGN. They also found that Type 1 AGN are more biased than Type 2 AGN up to redshift of ∼ 1.5. In the Böotes field Hickox et al. (2009) explored the connection between different classes of AGN and the evolution of their host galaxies, by deriving host galaxy properties, clustering, and Eddington ratios of AGN selected in the radio, X-ray, and infrared (IR) wavebands from the wide-field (9 deg 2 ) Böotes survey. They noticed that radio and X-ray AGNs reside in relatively large DMHs (M DMH ∼3×10 13 and 10 13 M ⊙ h −1 , respectively) and are found in galaxies with red and green colors. In contrast, IR AGN are in less luminous galaxies, have higher Eddington ratios, and reside in halos with M DMH <10 12 On the same line, Coil et al. (2009) measured the clustering of non-quasar X-ray active galactic nuclei at z = 0.7-1.4 in the AEGIS field. Using the cross-correlation of Chandra-selected AGN with 5000 DEEP2 galaxies they measured a correlation length of r 0 = 5.95±0.90 h −1 Mpc and slope γ= 1.66±0.22. They also concluded that X-ray AGN have a similar clustering amplitude as red, quiescent and "green" transition galaxies at z ∼1 and are significantly more clustered than blue, star-forming galaxies. In addition they proposed a "sequence" of X-ray AGN clustering, where its strength is primarily determined by the host galaxy color; AGN in red host galaxies are significantly more clustered than AGN in blue host galaxies, with a relative bias that is similar to that of red to blue DEEP2 galaxies. They did not observe any dependence of clustering on optical brightness, X-ray luminosity, or hardness ratio. In addition they obtained evidence that galaxies hosting X-ray AGN are more likely to reside in groups and more massive DMHs than galaxies of the same color and luminosity without an X-ray AGN. Allevato et al. (2011; Mountrichas & Georgakakis (2011) concluded that DEEP2 X-ray AGN at z ∼ 1 are more clustered than optically selected quasars (with a 2.6σ significance) and therefore may reside in more massive DMHs. In an evolutionary picture their results are consistent with galaxies undergoing a quasar phase while in the blue cloud before settling on the red sequence with a lower-luminosity X-ray AGN, if they are similar objects at different evolutionary stages (Hickox et al. 2009). At lower redshift, Krumpe et al. (2010) confirmed the results of Coil et al. (2009). Various recent works have been presented indications and/or evidences, of varying significance, regarding a correlation between the X-ray Luminosity and the AGN clustering amplitude, based either on the spatial (Yang et al. 2006;Gilli et al. 2009;Coil et al. 2009;Cappelluti et al. 2010;Krumpe et al. 2010Krumpe et al. , 2011, or the angular ) correlation function. Note that luminosity dependent clustering is one of the key features of merger triggered AGN activity and is one of the prime motivations for AGN clustering analyses. Low L X AGN have been found to cluster in a similar way as blue star forming galaxies while high L X AGN cluster like red passive galaxies. Such a result has been confirmed by Cappelluti et al. (2010) using the Swift-BAT all sky survey at z∼0. They detected both a L X dependence of AGN clustering amplitude and a larger clustering of Type I AGN than that of Type II AGN. Krumpe et al. (2010Krumpe et al. ( , 2011 confirm the weak dependence of the clustering strength on AGN X-ray luminosity at a 2σ level for z < 0.5. Table D summarizes all the discussed results on the clustering of AGN in X-ray surveys with bias factors converted to a common cosmology (Ω Λ = 0.7,Ω m =0.3, σ 8 =0.8) in the EMSS, Boyle & Mo (1993) ; RASS, Vikhlinin & Forman (1995); Akylas et al. (2000); ROSAT-NEP, Mullis et al. (2004); AXIS, Ebrero et al. (2009); ELAIS-S1, Puccetti et al. (2006); CDFS, Gilli et al. (2005)

Techniques of investigation
The continuously increasing volume and quality of data, allowed a parallel improvement of the techniques of investigation. The first surveys of Einstein (see e.g. Barcons & Fabian 1988), used the autocorrelation function of the unresolved CXB and linked it to the clustering properties of the clustering of X-ray source that produced it.
Modern surveys have mostly estimated correlation function with estimators that use of random samples and real data pairs and then estimating physical clustering properties by fitting the correlation function functions with simple power-law models in the form of eq. 2. A detailed description of the method to estimate correlation functions is given in the appendix. Considering its power, here we give a detailed description of halo modeling which is by far the most reliable formalism to describe clustering of AGN/Galaxies and to determine the environment of a specific DMH tracer.

HALO MODEL
In the hierarchical model of cosmological structure formation, galaxies, group of galaxies, clusters an so on are built from the same initial perturbation in the underlying dark matter density field. Regions of dark matter denserthan-average collapse to form halos in which structures form. Galaxies and AGN, as well as, groups and clusters are believed to populate the collapsed DMHs.
The theoretical understanding of galaxy clustering has been greatly enhanced through the framework of the halo model (Kauffmann, Nusser & Steinmetz 1997;Peacock & Smith 2000;Cooray & Sheth 2002;Tinker et al. 2005;Zheng et al. 2005). One can fill DMHs with objects based on a statistical halo occupation distribution (HOD), allowing one to model the clustering of galaxies within halos (and thus at non-linear scales) while providing a self-consistent determination of the bias at linear scales. Similarly the problem of discussing the abundance and spatial distribution of AGN can be reduced to studying how they populate their host halos. The HOD analysis recasts AGN clustering measurements into a form that is more physically informative and conducive for testing galaxy/AGN formation theories.
Thus, one can use measurements of AGN two-point correlation functions to constrain the HOD of different sets of AGN and gain information on the nature of DMH in which they live. In fact, the power of the HOD modeling is the capability to transform data on AGN pair counts at small-scales into a physical relation between AGN and DMH at the level of individual halos.
The key ingredient needed to describe the clustering properties of AGN is their halo occupation distribution function P N (M h ), which gives the probability of finding N AGN within a single halo as a function of the halo mass, M h . In the most general case, P N (M h ) is entirely specified by all its moments which, in principle, could be observationally determined by studying AGN clustering at any order. Regrettably AGN are so rare that their two-point function is already poorly determined, so that it is not possible to accurately measure higher-order statistics. One overcomes this problem by assuming a predefined functional form for the lowest-order moments of P N (M h ), defining the halo occupation number N (M h ) which is the mean value of the halo occupation distri- terms of a few parameters whose values will then be constrained by the data.
An accurate description of matter clustering on the basis of the halo approach requires three major ingredients: the halo mass function n(M h ) (the number of DMHs per unit mass and volume), the mass-dependent biasing factor b(M h ) and the density profile of halos. These terms, along with a parametrization of N (M h ), allow us to calculate some useful quantities; the number density of AGN: and the average mass of the host dark halo: The number density and clustering properties of the DMHs can be easily computed, at any redshift, by means of a set of analytical tools which have been tested and calibrated against numerical simulations (Mo & White 1996;Sheth & Tormen 1999;Sheth et al. 2001;Tinker et al. 2005;Basilakos et al. 2008;Tinker et al. 2010;Pillepich et al. 2010;Ma et al. 2011). Popular choices for both n(M h ) and b(M h ) are the analytical spherical collapse (Sheth & Tormen 1999) or an ellipsoidal collapse model , see §4 for more details). A detailed description of HOD mathematical formalism is given in Appendix B.

Occupation Number
In the past ten years, a very successful framework for modelling the nonlinear clustering properties of galaxies has been developed and a number of halo models have been presented in the literature. These have been successfully used to describe the abundance and clustering properties of galaxies at both low (Peacock & Smith 2000 Zehavi et al. 2004;Tinker et al. 2005;Phleps et al. 2006;Zheng et al. 2009) and high (Bullock, Wechsler & Somerville 2002;Moustakas & Somerville 2002;Hamana et al. 2004;Zheng 2004;Zheng et al. 2007) redshifts, as well as whether these galaxies occupy the centers of the DMH or are satellite galaxies (Kravtsov et al. 2004;Zheng et al. 2005).
Partially due to the low number density of AGN, there have been few results in the literature interpreting AGN correlation function using HOD modelling, where the small-scale clustering measurements are essential. Porciani, Magliocchetti & Norberg (2004) studied the clustering of 2QZ QSO with the halo model to infer the mean number of optically selected quasars which are harboured by a virialized halo of given mass and the characteristic quasar lifetime. Padmanabhan et al. (2009) discussed qualitative HOD constraints on their LRG-optical QSO Cross-Correlation Function (CCF) and Shen et al. (2010) modelled with the HOD the observed two-point correlation function of 15 binary quasar at z > 2.9.
The standard halo approach used for quasars and galaxies is based on the idea that the elements of HOD can be effectively decomposed into two components, separately describing the properties of central and satellite galaxies within the DMH. A simple parametric form used to describe the galaxy HOD is to model the mean occupation number for central galaxies as a step function, i.e., N cen = 1 for halos with mass M ≥ M min and N cen = 0 for M < M min , while the distribution of satellite objects can be well approximated by a Poisson distribution with the mean following a power law, N sat = (M/M 1 ) α . Previously derived HOD of galaxies show α values ∼ 1 − 1.2 which implies a number of satellite galaxies approximately proportional to M h .
The clustering properties of X-ray selected AGN have been modelled with the HOD in two previous works for sources in the Bootes field Starikova et al. (2010) and in the ROSAT All-Sky Survey Miyaji et al. (2011). Starikova et al. (2010) used the the projections of the two-points correlation function both on the sky plane and in the line of sight to show that Chandra/Bootes AGN are located at the center of DM halos with M > M min = 4 × 10 12 h −1 M ⊙ , assuming a halo occupation described by a step function (zero AGN per halo/subhalo below M min and one above it). They also showed that Chandra/Boötes AGNs are located at the centers of DMHs, limiting the fraction of AGN in non-central galaxies to be < 0.09 at the 95% CL. The central locations of the AGN host galaxies are expected in the merger trigger model because mergers of equally-sized galaxies preferentially occur at the centers of DMH . Miyaji et al. (2011) modelled the AGN HOD testing the effects of having or not AGN in central galaxies by using the RASS AGN-LRG cross-correlation. In the first scenario they assumed that all the AGN are satellites and they visualized the HOD of the LRG as a step function with a step at logM h [h −1 M ⊙ = 13.5]. While formally they assumed that all AGN are not in central galaxies, the HOD constraints obtained from this assumption can be applied to satellite and central AGN if the AGN activity in central galaxies of high-mass halos (logM h [h −1 M ⊙ > 13.5]) is suppressed. In particular, they used a truncated power-law satellite HOD, with two parameters: the critical DMH mass below which the AGN HOD is zero and the slope α of the HOD for M h > M cr . They also investigated a model where the central HOD is constant and the satellite HOD has a power-law form, both at masses above M min . In all the cases they rejected α ∼ 1, finding a marginal preference for an AGN fraction among satellite galaxies which decreases with increasing M h . They argued that this result might be explained by a decrease of the cross-section for galaxy merging in the environment of richer groups or clusters. In fact previous observations infer that the AGN fraction is smaller in clusters than in groups (Arnold et al. 2009;Martini et al. 2009;Silverman et al. 2009;Koulouridis & Plionis 2010).
It is important to stress that the small number statistics has so far limited the accuracy of correlation function of X-ray AGN at small-scales, especially through the auto-correlation function of the AGN themselves. The situation can be improved by measuring the crosscorrelation function of AGN with a galaxy sample that has a much higher space density, with common sky and redshift coverage as the AGN redshift surveys. The AGN clustering through cross-correlation function with galaxies is emerging in the last years (Li et al. 2006;Coil et al. 2007;Coil et al. 2009;Hickox et al. 2009 can be used to improve our understanding of how AGN populate DMH Krumpe et al. 2011).

BIAS AND DMH MASS
In the literature, the bias parameter is often calculated with the power-law fits (Mullis et al. 2004;Yang et al. 2006;Miyaji et al. 2007;Coil et al. 2009;Krumpe et al. 2010;Hickox et al. 2011) over scales of 0.1-0.3 < r p < 10-20 h −1 Mpc. The power-law models of the ACF are usually converted to the rms fluctuation over 8 h −1 Mpc spheres or are averaged up to the distance of 20 h −1 Mpc. While some authors use only large scales (r p > 1 − 2 h −1 Mpc) to ensure that the linear regime is used, others include smaller scales to have better statistics. As an example, Hickox et al. (2009) fitted their data with a biased DMH projected correlation function. In the HOD analysis the bias factor only comes from the 2-halo term (r p >1-2 h −1 Mpc). Miyaji et al. (2011) compared the bias of RASS-AGN from the full HOD model (Eq. D4) with the one estimated using the powerlaw best fits parameters, finding that the bias estimates are consistent within 1σ. Moreover, using Eq. D1 one introduces large statistical errors. Allevato et al. (2011) found a similar results in comparing the bias of X-ray AGN in COSMOS field from the 2-halo term with Eq. D3 and the one estimated from the power-law best fits parameters. In Appendix C, we describe the mathematical procedures for the bias parameter calculation commonly used in the literature.
Most of the authors (Hickox et al. 2009;Krumpe et al. 2010;Cappelluti et al. 2010) used an analytical expression (as the one described in Sheth & Tormen 1999;Sheth et al. 2001;Tinker et al. 2005;Basilakos et al. 2008) to assign a characteristic DMH mass to the hosting halos. The large-scale bias is directly related to the mass function of halos, so that the mass of a halo dictates the halo clustering and the number of such halos. The halo mass can be quantified in terms of the peak height ν = δ c /σ(M h , z), which characterizes the amplitude of density fluctuations from which a halo of mass M h form at a given redshift. In general one assumes δ c = 1.686 and σ(M h , z) is the linear overdensity variance in spheres enclosing a mean mass M h . The traditional choice of the mass function and then of the bias has been that of Press & Schechter (1974): A commonly-used prescription was derived by Sheth & Tormen (1999): where a = 0.707 and p = 0.3 or the ellipsoidal collapse formula of Sheth et al. (2001):  2011) argued that this approach reveals an incongruity due to the fact that the AGN bias used in the formulas above, is the average bias of a given AGN sample at a given redshift. In fact, following this approach one can not take into account that the the average bias is sensitive to the entirety of the mass distribution; different mass distributions with different average masses can give rise to the same average bias.
On the contrary by using the halo model, the average bias and the average mass of the sample, Eq. D4 and Eq. 5 properly account for the shape of the mass distribution: the average bias depends on the halo number density and on the AGN HOD, integrated over the mass range of the particular AGN sample. They introduced a new method that uses the 2-halo term in estimating the AGN bias factor assuming an AGN HOD described by a δ-function. Following this approach they properly took into account for the sample variance and the growth of the structures over time associated with the use of large redshift interval of the AGN sample.
On the other hand, Miyaji et al. (2011) and Krumpe et al. (2011) applied the HOD modeling technique to the RASS AGN-LRG CCF in order to move beyond determining the typical DMH mass based on the clustering signal strength and instead constrain the full distribution of AGN as a function of DMH mass. Along with a parametrization of N (M h ) they estimated the large-scale bias and the typical mass of hosting DM halos using Eq. D4 and Eq. 5. This method improves the clustering analysis because it properly uses the nonlinear growth of matter in the 1-halo term through the formation and growth of DMHs. These results are significant improvements with respect to the standard method of fitting the signal with a phenomenological power law or using the 2-halo term (see Appendix C).
At high redshift, Gilli et al. (2005) measured the clustering of X-ray AGN with z=0-4 in both the ∼ 0.1 deg 2 CDFs, finding b = 1.87 +0.14 −0.16 for 240 sources in the northern field and b = 2.64 +0.29 −0.30 for 124 sources in the southern field. At z ∼ 1, Yang et al. (2006) measured the clustering of 233 spectroscopic sources in the 0.4 deg 2 Chandra CLASXS area and of 252 spectroscopic sources from the CDFN, both at z=0.1-3. They found b = 3.58 +2.49 −1.38 for the CLASXS AGN and b = 1.77 +0.80 −0.15 for the CDFN field. Gilli et al. (2009) studied 538 XMM-COSMOS AGN with 0.1 < z < 3 and they found a bias factor b = 3.08 +0.14 −0.14 at z ∼ 1. Using the Millennium simulations they suggested that XMM-COSMOS AGN reside in DMH with mass M DMH > 2.5 × 10 12 h −1 M ⊙ . Coil et al. (2009) measured the clustering of X-ray AGN at z=0.7-1.4 in the AEGIS field and they estimated b = 1.85 +0.28 −0.28 . Following Zheng et al. (2007) they infer from the bias factor that at z = 0.94 the minimum DM halo mass of the X-ray AGN is > 10 12 M ⊙ h −1 . These results combined with Mountrichas & Georgakakis (2011) show that moderate luminosity X-ray selected AGN live in DMHs with masses M h ∼ 10 13 h 1 M ⊙ at all redshifts since z ∼ 1. At lower redshift Hickox et al. (2009) analysed 362 AGES X-ray AGN at < z >=0.51. The bias factor equal to b = 1.40 ± 0.16 indicates that X-ray AGN inhabit DM halos of typical mass ∼ 10 13 M ⊙ h −1 .
In  Using the HOD model, Starikova et al. (2010) suggested that X-ray Chandra/Bootes AGN are located at the center of DM halos with M > M min = 4×10 12 h −1 M ⊙ while Miyaji et al. (2011) estimated for RASS AGN at z=0.25 b = 1.32 ± 0.08 and a typical mass of the host halos of 13.09 ± 0.08.
The redshift evolution of the clustering of X-ray selected AGN has been first studied by Yang et al. (2006) in the CLAXS+CDFN fields. They measured an increase of the bias factor with redshift, from b = 0.95 ± 0.15 at z=0.45 to b = 3.03 ± 0.83 at z=2.07, corresponding to an average halo mass of ∼12.11 h −1 M ⊙ . Allevato et al. (2011) studied the redshift evolution of the bias for a sample of XMM-COSMOS AGN at z < 2. They found a bias evolution with time from b(z = 0.92) = 1.80 ± 0.19 to b(z = 1.94) = 2.63 ± 0.21 with a DM halo mass consistent with being constant at logM [h −1 M ⊙ ] ∼ 13.1 at all redshifts z < 2. They also found evidence of a redshift evolution of the bias factor of XMM-COSMOS Type 1 AGN and Type 2. The bias evolves with redshift at constant average halo mass logM 0 [h −1 M ⊙ ] ∼ 13.3 for Type 1 AGN and logM 0 [h −1 M ⊙ ] ∼ 13 for Type 2 AGN at z < 2.25 and z < 1.5, respectively. In particular Allevato et al. (2011) argued that X-ray selected Type 1 AGN reside in more massive DMHs compared to X-ray selected Type 2 AGN at all redshifts at ∼ 2.5σ level, suggesting that the AGN activity is a mass triggered phenomenon and that different AGN classes are associated with the DM halo mass, irrespective of redshift z. Krumpe et al. (2011) measured the clustering amplitudes of both X-ray RASS and optically-selected SDSS broad-line AGNs, as well as for X-ray selected narrowline RASS/SDSS AGNs through cross-correlation func-tions with SDSS galaxies and derive the bias by applying the HOD model directly to the CCFs. They estimated typical DMH masses of broad-line AGNs in the range log(M h /[h 1 M ⊙ ]) = 12.4 − 13.4, consistent with the halo mass range of typical non-AGN galaxies at low redshifts and they found no significant difference between the clustering of X-ray selected narrow-line AGNs and broad-line AGNs up to z ∼ 0.5. Fig. 2 shows the bias parameter (Left Panel ) and the mass of the AGN hosting halos (Right Panel ) as a function of redshift for X-ray selected AGN (black data points), X-ray selected Type 1 AGN (blue data points) and X-ray selected Type 2 AGN (red data points) as estimated for different surveys (see the legend). The dashed lines show the expected b(z) of typical DM halo masses M DMH based on Sheth et al. (2001). The masses are given in logM DMH in units of h −1 M ⊙ .
There have been several studies of the bias evolution of optical quasar with the redshift as shown in fig. 2 (grey data points), based on large survey samples such as 2QZ, 2SLAQ and SDSS (Croom et al. 2005;Shen et al. 2009;Ross et al. 2009;daÂngela et al. 2008). These previous studies infer the picture that X-ray selected AGN which are moderate luminosity AGN compared to bright quasars, inhabit more massive DMHs than optically selected quasars in the range z = 0.5 − 2.25. Recently, Krumpe et al. (2011) verified that the clustering properties between X-ray and optically-selected AGN samples are not significantly different in three redshift bins below z = 0.5 (the differences are 1.5σ, 0.1σ and 2.0σ).The reason for the fact that X-ray selected AGN samples appear to cluster more strongly than optically-selected AGNs is still unclear. Allevato et al. (2011);Mountrichas & Georgakakis (2011) suggested that the difference in the bias and then in the host DMH masses is due to the different fueling mode of those sources from that of the X-ray selected moderate luminosity AGN. On the contrary, Krumpe et al. (2011) suggested that some of the X-ray clustering studies significantly underestimate their systematic uncertainties and then it may turn out that these measurements are consistent with optical AGN clustering measurements. More high-z AGN clustering measurements based on larger samples are needed to gain a clearer picture.

AGN Life Time
One of the most important tests for studying the evolution models of AGN is understanding their lifetime. It is widely accepted that AGN is phase of the galaxy life necessary to explain the coevolution of the bulge and the black hole. After a triggering event, of which we do not know the nature, yet, the central black holes begins its accretion phase and it is believed that undergoes several regimes of Eddington rates and bolometric luminosity. Martini & Weinberg (2001) proposed a method to derive the AGN life time by knowing their space density and their DMH host mass. By knowing the AGN and DMH halo space density at a given luminosity and mass (n AGN , n DMH ), one can estimate the duty cycle of the AGN, τ AGN (z) = nAGN(L,z) nDMH(M,z) τ H (z), where τ (H(z)) is the Hubble time at a given redshift. Actually this method provides only an upper limit since it assumes that the life of an halo of a given mass is similar to the Hubble time. A more exhaustive formulation would τ AGN (z) = nAGN(L,z) nDMH(M,z) τ DMH (z), where τ DMH (z) is the age of a DMH at given redshift. Unfortunately this quantity cannot be estimated analytically but could be estimate in a statistical way by using hidrodynamic simulations. Several results can be mentioned for this quantities but their dispersion is very large, therefore we report only some example. At z=1, Gilli et al. (2009) obtains that the typical duty cycle of AGN is <1 Gyr. At z=0, Cappelluti et al. (2010) has measured a duty cycle in the range 0.2 Gyr-5 Gyr with an expectation value of 0.7 Gyr. Both the measurements are fairly larger than the 40 million years determined by Martini & Weinberg (2001) at z=2-3. These differences however are not surprising if we assume that the different populations of AGN, grow with a different Eddington rate as function of their typical luminosities and/or redshifts (Fabian et al. 2009).

DISCUSSION
In this paper we reviewed the results in the field of X-ray AGN clustering, for energies between 0.1 keV to 55 keV over a period of more that 20 years. The literature has produced an increasingly convincing and consistent picture of the physical quantities derivable from this kind of study. Most of the advancements in the field have been achieved with the improvement of survey capabilities and instruments sensitivity. The availability of simultaneously wide and deep fields, coupled with multiwavelength information, has produced larger and larger samples of spectroscopically confirmed sources. This allowed several teams to refine the techniques needed to estimate the two point ACF and the quantities derived form it. In particular we are entering a phase where, at least at z< 2, AGN clustering studies won't probably provide any new result unless evaluated with the HOD formalism. Open questions as what is the AGN occupation number and the evolution of HOD define a new barrier which is necessary to break in order to understand the history of X-ray emission from accretion onto AGN. In this respect, samples of X-ray selected AGN always need a spectroscopical follow-up to provide a solid base to compute clustering in the real space rather than in the angular space.
Summarizing, the current picture is that X-ray selected AGN are highly biased objects with respect to the underlined matter distribution. Such an evidence is clearer when measuring the redshift dependence of AGN bias. At every redshift from z=0 to z=2, AGN cluster in way similar to DMH of mass of the order of log(M ⊙ h −1 )=13. The spread of such a value is of the order 0.3-0.5 dex at 1σ. This means that the determination of what kind of environment is inhabited by AGN, is relatively well constrained and identical at every redshift sampled by X-ray surveys. This allows us to formulate the hypothesis that every phase of AGN activity is mass triggered phenomenon (i.e. each AGN evolutionary phase is characterized by a critical halo mass).
It is believed that major mergers of galaxies is one of the dominant mechanisms for fueling quasars at high redshift and bright luminosities, while minor interactions, bar instabilities or tidal disruptions are important at low redshift (z 1) and low luminosities (L 10 44 erg s −1 ) (Hopkins et al. 2006Hasinger et al. 2008;Hopkins & Henquist 2009). In the local Universe, for example, the study of the environment of Swift BAT Seyfert galaxies (Koss et al. 2010) finds a larger fraction of BAT AGNs with disturbed morphologies or in close physical pairs (<30 kpc) compared to matched control galaxies or optically selected AGNs. The high rate of apparent mergers (25%) suggests that AGN activity and merging are critically linked for the moderate luminosity AGN in the BAT sample. Moreover models of major mergers appear to naturally produce many observed properties of quasars, as the quasar luminosity density, the shape and the evolution of the quasar luminosity function and the large-scale quasar clustering as a function of L and z (e.g., Shen 2009;Shankar et al. 2009Bonoli et al. 2009;Treister et al. 2010). Quasar clustering at all redshift is consistent with halo masses similar to group scales, where the combination of low velocity dispersion and moderate galaxy space density yields to the highest probability of a close encounter McIntosh et al. 2009). Moreover recent detections of an L X dependent clustering play in favor of major mergers being the dominant AGN triggering mechanism.
On the other hand it has became clear that many AGN are not fueled by major mergers and only a small fraction of AGN are associated with morphologically disturbed galaxies. Georgakakis et al. (2007) and Silverman et al. (2009) found that AGN span a broad range of environments, from the field to massive groups and thus major mergers of galaxies, possibly relevant for the more luminous quasar phenomenon, may not be the primary mechanism for fueling these moderate luminosity AGN. Georgakakis et al. (2009) suggest that bar instabilities and minor interactions are more efficient in producing luminous AGN at z 1 and not only Seyfert galaxies and low-luminosity AGN as the Hopkins & Henquist (2009) model predicts. Cisternas et al. (2010 analysed a sample of X-ray selected AGN host galaxies and a matched control sample of inactive galaxies in the COSMOS field. They found that mergers and interactions involving AGN hosts are not dominant and occur no more frequently than for inactive galaxies. Over 55% of the studied AGN sample that is characterized by L BOL ∼ 10 45 erg s −1 and by mass of the host galaxies M * 10 10 M ⊙ are hosted by disk-dominated galaxies, suggesting that secular fuelling mechanisms can be highly efficient. Moreover several works on the AGN host galaxies (Dunlop et al. 2003;Grogin et al. 2005;Pierce et al. 2007;Gabor et al. 2009;Reichard et al. 2009;Tal et al. 2009) show that the morphologies of the AGN host galaxies do not present a preference for merging systems.
At high redshift (z ∼ 2) recent findings of Schlegel et al. (2001) and Rosario et al. (2011), who examined a smaller sample of AGN in the ERS-II region of the GOODS-South field, inferred that late-type morphologies are prevalent among the AGN hosts. The role that major galaxy mergers play in triggering AGN activity at 1.5 < z < 2.5 was also studied in the CDF-S. At z=1.5-3 Schawinski et al. (2011) showed that for X-rayselected AGN in the Chandra Deep Field South and with typical luminosities of 10 42 erg s −1 <L X <10 44 erg s −1 the majority ( 80%) of the host galaxies of these AGNs have low Srsic indices indicative of disk-dominated light profiles, suggesting that secular processes govern a significant fraction of the cosmic growth of black holes. That is, many black holes in the present-day universe grew much of their mass in disk-dominated galaxies and not in early-type galaxies or major mergers.
Later, Kocevski et al. (2011) found that X-ray selected AGN at z ∼ 2 do not exhibit a significant excess of distorted morphologies while a large fraction reside in latetype galaxies. They also suggested that these late-type galaxies are fueled by the stochastic accretion of cold gas, possibly triggered by a disk instability or minor interaction. Allevato et al. (2011) argued that for moderate luminosity X-ray AGN secular processes such as tidal disruptions or disk instabilities might play a much larger role than major mergers up to z ∼ 2.2. It becomes important to study the clustering properties of AGN at high redshift when we assume the peak of the merger-driven accretion. Moreover given the complexity of AGN triggering, a proper selection of AGN samples, according to the luminosity or the mass of the host galaxies can help to test a particular model boosting the fraction of AGN host galaxies associated with morphologically disturbed galaxies.
From the evolutionary point of view the evidence of a bias segregation of optically and X-ray selected AGN might be a sufficient proof to claim that the two phenomena are sensitive to different environments and therefore likely driven by different triggering mechanisms. A more comprehensive picture will be available when the clustering of different phases of AGN activity will be studied and compared. Hickox et al. (2009) interpreted their clustering results  Yang et al. (2006); RASS, Akylas et al. (2000); ELAIS-S1, Puccetti et al. (2006)).
in terms of a general picture for AGN and galaxy evolution which is reproduced in Fig. 4. The picture consists of an evolutionary sequence that occurs at different redshifts for halos with different masses. In this scenario, luminous AGN accretion occurs preferentially (through a merger or some secular process) when a host DMH reaches a critical M DMH between 10 12 and 10 13 M ⊙ h −1 (this phase is indicated by the solid ovals). Once a large halo reaches this critical mass, it becomes visible as a ULIRG or SMG (owing to a burst of dusty star formation) or (perhaps subsequently) as a luminous, unobscured quasar. The ULIRG/quasar phase is associated with rapid growth of the SMBH and formation of a stellar spheroid, and is followed by the rapid quenching of star formation in the galaxy. Subsequently, the young stellar population in the galaxy ages (producing "green" host galaxy), and the galaxy experiences declining nuclear accretion that may be associated with an X-ray AGN. Eventually the aging of the young stars leaves a "red" and "dead" early-type galaxy, which experiences intermittent "radio-mode" AGN outbursts that heat the surrounding medium. For "medium" initial DMHs, the quasar phase and formation of the spheroid occurs later than for the systems with high halo mass, so that at z ∼ 0.5 we may observe the green X-ray AGN phase. Even smaller halos never reach the threshold mass for quasar triggering; these still contain star-forming disk galaxies at z 0.8, and we observe some of them as optical or IR-selected Seyfert galaxies. The dashed box indicates the AGN types (in their characteristic DMH) that would be observable in the redshift range 0.25 < z < 0.8.
Further steps in the field will require the study of clus-  tering of AGN from z=3 to z=6-7. This will likely lead to the determination of the mass of early DM spheroids who hosted primordial black holes seeds. However this is a very challenging task since it requires a very deep and wide survey with an almost complete optical follow-up.
BOSS (Eisenstein et al. 2011) and BigBOSS (Schlegel et al. 2001) will detect high redshift AGNs at z ∼ 2.2, which will improve AGN clustering measurements at higher redshifts. The only approved mission that at the moment will allow to study the z=3-5 X-ray Universe is eROSITA (Predehl et al. 2007, launch Dec. 2013 for which an estimate of the completeness of the typical follow up is still unavailable. Additionally, the Large Synoptic Survey Telescope (Ivezic et al. 2008, LSST) is expected to identify ∼2 million AGNs in optical bands. eROSITA and LSST have the potential to significantly improve AGN clustering measurements at low and high redshifts, though only if there are dedicated large spectroscopic follow-up programs. Another strong contribution will come form either Nustar that will likely provide a better view of AGN clustering without the selection biases introduced by photoelectric absorption. Athena the proposed ESA new generation telescope that will mount a wide field imager on a very large collecting area telescope, will provide a further view on the deep X-ray sky and likely push our knowledge of the high-z X-ray Universe.
In addition to better model the evolution of SMBH environments a fundamental point to start is to establish the nature of BH seeds at z=10. Such a determination will likely come with the new generation of telescope like JWST and ESO-ELT.
NC thanks the INAF-Fellowship program for support. NC thanks the Della Riccia and Blanceflor-Lodovisi-Boncompagni foundation for partial support. VA is supported by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de). NC, VA and AF thank the referees, Ryan Hickox and Manolis Plionis, for valuable suggestions for improving the paper.

DERIVING THE TWO POINTS AUTO-CORRELATION FUNCTION
The two-point auto-correlation function (ξ(r), ACF) describes the excess probability over random of finding a pair with an object in the volume dV 1 and another in the volume dV 2 , separated by a distance r so that dP = n 2 [1 + ξ(r)]dV 1 dV 2 , where n is the mean space density. A known effect when measuring pairs separations is that the peculiar velocities combined with the Hubble flow may cause a biased estimate of the distance when using the spectroscopic redshift. To avoid this effect it is usually computed the projected ACF (Davis & Peebles 1983): w(r p ) = 2 πmax 0 ξ(r p , π)dπ. Where r p is the distance component perpendicular to the line of sight and π parallel to the line of sight (Fisher et al. 1994). It can be demonstrated that, if the ACF is expressed as ξ(r) = (r/r 0 ) −γ , then where A(γ) = Γ(1/2)Γ[(γ − 1)/2]/Γ(γ/2) (Peebles 1980). The ACF is mostly estimated by using the minimum variance estimator described by Landy & Szalay (1993): where DD, DR and RR are the normalized number of data-data, data-random, and random-random source pairs, respectively. Equation A2 indicates that an accurate estimate of the distribution function of the random samples is crucial in order to obtain a reliable estimate of ξ(r p , π). Note that other estimator have been proposed in the literature, but the Landy & Szalay (1993) one has been shown to provide the smallest statistical variance. Such a formalism can be easily adopted when computing the angular or the redshift space correlation function, with the only difference that the evaluation is made on a single dimension. Several observational biases must be taken into account when generating a random sample of objects in a X-ray flux limited survey. In particular, in order to reproduce the selection function of the survey, one has to carefully reproduce the space and flux distributions of the sources, since the sensitivity in X-ray surveys is not homogeneous on the detector and therefore on the sky. This points out the necessity of create a random sample which includes as many selection effects as possible since the estimate of ξ(r) (or w(θ)) is strongly dependent on RR (see eq. A2). Moreover in several case optical follow-up of the X-ray source is not 100% complete, therefore one must carefully reproduce the mask effect. What is usually done is that to create random samples in 3D, sources are placed at the same angular position of the real sources and redshift are randomly drawn from a smoothed redshift distribution of the real sources. If instead the spectral completeness is close to 100% then the right procedures is to occupy the survey volume with random sources drawn from a L-z dependent luminosity function and accept check if they would be observable using a sensitivity map. An important choice for obtaining a reliable estimate of w(r p ), is to set π max in the calculation of the integral above. One should avoid values of π max too large since they would add noise to the estimate of w(r p ). If, instead, π max is too small one could not recover all the signal. Uncertainties in the ACF are usually evaluated with a bootstrap resampling technique but it is worth noting that in the literature, several methods are adopted for errors estimates in two-point statistics Norberg et al. (2009, for a detailed description). It is known that Poisson estimators generally underestimate the variance because they do consider that points in ACF are not statistically independent. Jackknife resampling method, where one divides the survey area in many sub fields and iteratively re-computes correlation functions by excluding one sub-field at a time, generally gives a good estimates of errors. But it requires that sufficient number of almost statistically independent sub-fields, this is not the case for most of X-ray surveys where the source statistics is moderately low. Coil et al. (2009) estimated the error bars on the two-point correlation function including both Poisson and cosmic variance errors estimated, using DEEP2 mock catalogs derived from the Millenium Run simulations.

LIMBER'S DEPROJECTION
The 2D Angular Correlation Function (ACF) is a projection of the real-space 3D ACF of the sources along the line of sight. In the following discussions and thereafter, r is in comoving coordinates. The relation between the 2D (angular) ACF and the 3D ACF is expressed by the Limber equation (e.g., Peebles 1980). Under the assumption that the scale length of the clustering is much smaller than the distance to the object, this reduces to where dA(z) is the angular distance, N is the total number of sources, and dN/dz is the redshift distribution (per z) of the sources. The redshift evolution of the 3D correlation function is customarily expressed by where ǫ=-3 and ǫ=γ-3 correspond to the case where the correlation length is constant in physical and comoving coordinates, respectively. In these notations, the zero-redshift 3-D correlation length r 0 can be related to the angular correlation length θ 0 by where τ (z) is the look-back time. We also define the comoving correlation length at the effective redshiftz ef f , which is the median redshift of the contribution to the angular correlation (the integrand of the second term). An essential ingredient of the deprojection process is the redshift distribution of the sources and when individual redshifts are not available this is derived from integration of the luminosity function.

1-HALO AND 2-HALO TERMS IN THE HOD FORMALISM
In the halo model approach, the two-point correlation function of AGN is the sum of two contributions: the first term (1-halo term) is due to the correlation between objects in the same halo and the second term (2-halo term) arises because of the correlation between two distinct halos: w p (r p ) = w p,1h (r p ) + w p,2h (r p ) Recent articles prefer to express w = (1 + w 1h ) + w 2h (Tinker et al. 2005;Zheng et al. 2005;Blake et al. 2008), instead of w = w 1h + w 2h , as used in older articles. This is because 1 + ξ represents a quantity that is proportional to the number of pairs ∝ [1 + ξ 1h ] + [1 + ξ 2h ]. In this new convention, the projected correlation function w p,1h represents the projection of 1 + ξ 1h rather than ξ 1h . Similarly, one express the power spectrum of the distribution of the AGN in terms of the 1-and 2-halo term contributions: P (k) = P 1h (k) + P 2h (k) and then the projected correlation function as: w p,1h (r p ) = k 2π P 1h (k)J 0 (kr p )dk (C3) w p,2h (r p ) = k 2π P 2h (k)J 0 (kr p )dk (C4) where J 0 (x) is the zeroth-order Bessel function of the first kind. Several parameterizations exist in literature for representing the DMH profile (Cooray & Sheth 2002;Knollmann et al. 2008;Stadel et al. 2009) and the Navarro, Frenk & White (1997) (NFW) profile is a popular choice. If y(k, M h ) expresses the Fourier transform of the NFW profile of the DMH with mass M h , normalized such that volume integral up to the virial radius is unity, then the one-halo term of the power spectrum can be written as: Assuming the linear halo bias model (Mo & White 1996), the two-halo term of the power spectrum reduces to: Since the clustering on large scales is dominated by the two-halo term, it is fairly insensitive to the assumption of AGN distribution inside the hosting halo (Berlind & Weinberg 2002). It should be noted that since y ∼ 1 on large scales (e.g. scales much larger than the virial radius of halos), on such scales the two-halo term can be rewritten as: ) or in terms of projected correlation function: w p,2h (r p ) = b 2 w m,2h (r p ) where b is the bias parameter of the sample and w m,2h is the DM projected correlation function. For the matter power spectrum, P m (k), one can use the primordial power spectrum with a fixed n s and a transfer function calculated using the fitting formula of Eisenstein & Hu (1998) or the nonlinear form given by Smith et al. (2003), Tinker et al. (2005).

BIAS PARAMETER CALCULATION
In the majority of works on clustering of X-ray AGN (Mullis et al. 2004;Yang et al. 2006;Gilli et al. 2005;Coil et al. 2009;Krumpe et al. 2010;Cappelluti et al. 2010) the standard approaches used to estimate the bias are based on the power-law fit parameters of the AGN correlation function. This method assumes that the projected correlation function is well fitted by a power-law and the bias factors are derived from the best fit parameters r 0 and γ of the clustering signal at large scale. Using the power-law fit one can estimate the AGN bias factor using the power-law best fit parameters: b P L = σ 8,AGN (z)/σ DM (z) where σ 8,AGN (z) is the rms fluctuations of the density distribution over the sphere with a comoving radius of 8 Mpc h −1 , σ DM (z) is the dark matter correlation function evaluated at 8 Mpc h −1 , normalized to a value of σ DM (z = 0) = 0.8. For a power-law correlation function this value can be calculated by (Peebles 1980): where J 2 (γ) = 72/[(3 − γ)(4 − γ)(6 − γ)2 γ ].
Differently in the halo model approach, the 2-halo term of the projected correlation function, which dominates at large scales, can be considered in the regime of linear density fluctuations. In the linear regime, AGN are biased tracers of the dark matter distribution and the bias factor is described by: b = (w p,1h (r p )/w m,2h (r p )) 1/2 (D3) HOD modeling is currently the optimal method to establish the large-scale bias parameter, provided the parametrization of N (M h ), by using: assuming the halo mass function n(M h )and the halo bias factor b(M h ). In fact, power law fit bias measurements commonly use smaller scales (< 1 − 2h −1 Mpc) that are in the 1-halo term in order to increase the statistical significance. If power law fits are restricted only to larger scales, the method suffers from the problem that the lowest scale, where the linear biasing scheme can still be applied, varies from sample to sample and remains ambiguous. HOD modeling allows, in principle, the use of the full range of scales since the method first determines the 1 and 2-halo terms and then constrains the linear using data down to the smallest r p values that are dominated by the 2-halo term for each individual sample. Krumpe et al. (2011) estimated the RASS-AGN bias following the power-law (Eq. D1) and the HOD (Eq. D4) approach, pointing out that using the first method the errors on the bias are much larger, but the values are statistically consistent which those derived from the HOD model fits. Allevato et al. (2011) found similar results in estimating the COSMOS-AGN bias following Eq. D1 and D3.
In order to derive a reliable picture of AGN clustering, bias parameters should be inferred from HOD modeling, or at least from the comparison of the correlation function with that of the DM only in the linear regime, because systematic errors based on power-law bias parameters will be larger than the statistical uncertainties of the clustering measurement.