THE SPATIAL CLUSTERING OF ROSAT ALL-SKY SURVEY AGNs. II. HALO OCCUPATION DISTRIBUTION MODELING OF THE CROSS-CORRELATION FUNCTION

In performing the HOD modeling, we consider that galaxies and AGNs are associated with DMHs, the mass function of which (per comoving volume per dM_h) is denoted by ϕ(M_h) dM_h, where M_h is the DMH mass. We use ϕ(M_h) based on Sheth & Tormen (1999), which is in good agreement with later analyses by Sheth et al. (2001) and Jenkins et al. (2001). A DMH may contain one or more galaxies and/or AGNs that are included in our samples. We then model the two-point projected CF as the sum of two terms,

$$\begin{equation} w_{\rm p}(r_{\rm p})=w_{\rm p,1h}(r_{\rm p}) + w_{\rm p,2h}(r_{\rm p}), \end{equation} \tag{ 4 }$$

where the terms are:

• 1.  
  the 1-halo term (denoted by a subscript "1h"), where both of the objects occupy the same DMH, and
• 2.  
  the 2-halo term (denoted by a subscript "2h"), where each of the pair of objects occupies a different DMH.

Recent articles prefer to define ξ_1h such that the total 3D CF is expressed by ξ = (1 + ξ_1h) + ξ_2h (Tinker et al. 2005; Zheng et al. 2005; Blake et al. 2008), instead of ξ = ξ_1h + ξ_2h, as used in older articles. This is because 1 + ξ represents a quantity that is proportional to the number of pairs and allows one to express each of the 1- and 2-halo terms consistently. In this case, the number of pairs is ∝1 + ξ = [1 + ξ_1h] + [1 + ξ_2h]. In this new convention, our w_p,1h represents the projection of 1 + ξ_1h rather than ξ_1h, i.e.,

$$\begin{equation} w_{\rm p,1h}(r_{\rm p}) = \int _{-\infty }^\infty [1+\xi _{\rm 1h}(r_{\rm p},\pi)] d\pi. \end{equation} \tag{ 5 }$$

Similarly, we express the power spectrum of the distribution of the objects in terms of the 1- and 2-halo term contributions:

$$\begin{equation} P(k) = P_{\rm 1h}(k)+P_{\rm 2h}(k), \end{equation} \tag{ 6 }$$

with

$$\begin{equation} \begin{array}{l} \dsty w_{\rm p,1h}(r_{\rm p})=\int \frac{k}{2\pi }P_{\rm 1h}(k) J_0(kr_{\rm p})dk \\ \ms \dsty w_{\rm p,2h}(r_{\rm p})=\int \frac{k}{2\pi }P_{\rm 2h}(k) J_0(kr_{\rm p})dk, \end{array} \end{equation} \tag{ 7 }$$

where J₀(x) is the zeroth-order Bessel function of the first kind.

In the case of modeling the ACF of a sample, the 2-halo term of the power spectrum can be approximated by (Cooray & Sheth 2002)

$$\begin{equation} P_{\rm 2h}(k) \approx b^2 P_{\rm lin}(k,z), \end{equation} \tag{ 8 }$$

where b is the bias parameter of the sample, which we model as

$$\begin{equation} b = \frac{\int b_{\rm h}(M_{\rm h})\langle N\rangle (M_{\rm h}) \phi (M_{\rm h})dM_{\rm h}}{\int \langle N\rangle (M_{\rm h}) \phi (M_{\rm h})dM_{\rm h}}. \end{equation} \tag{ 9 }$$

For the linear power spectrum, P_lin(k, z), we use the primordial power spectrum with n_s = 1 and a transfer function calculated using the fitting formula of Eisenstein & Hu (1998) under our assumed cosmology (Section 1). For the mass-dependent bias parameter of DMHs, b_h(M_h), we use Equation (8) of Sheth et al. (2001) with recalibrated parameter values by Tinker et al. (2005) (see their Appendix A).

In calculating the 1-halo term, the usual assumption is that the radial distribution of the involved objects that are not at the center of the halo—objects that are satellites—follows the mass profile of the DMH itself, i.e., there is no internal biasing. Following Zheng et al. (2009), whose results will be used later in this work, we use the Navarro et al. (1997) (NFW) profile for the DMH density distribution, which is still a popular choice for representing DMH profiles, although other parameterizations exist in the literature (e.g., Knollmann et al. 2008; Stadel et al. 2009). We express 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, by y(k, M_h). Again, in the case of the ACF,

$$\begin{eqnarray} P_{\rm 1h}(k)&=&\frac{1}{(2\pi)^3n^2}\int \phi (M_{\rm h})[\langle N_{\rm c}N_{\rm s}\rangle (M_{\rm h}) y(k,M_{\rm h})\nonumber \\ & &+\langle N_{\rm s}(N_{\rm s}-1)\rangle (M_{\rm h})|y(k,M_{\rm h})|^2]dM_{\rm h}, \end{eqnarray} \tag{ 10 }$$

where N_c and N_s are the numbers of the objects in the sample per DMH as a function of M_h for those that are at the center of the halos (central) and those that are off center (satellites), respectively, while n = ∫〈N〉(M_h)ϕ(M_h)dM_h is their overall space density (Hamana et al. 2004; Seljak 2000).

The HOD model is calculated using a set of software we have developed, based partially on a code developed by J. Peacock for the use in Peacock & Smith (2000) and Phleps et al. (2006). Our software includes a number of improvements over their code, most notably by including the application of the HOD modeling to a CCF. We note, however, that in our analysis we do not include some recent improvements in modeling the 2-halo term (Zheng et al. 2005, 2009; Tinker et al. 2005). The neglected factors include the convolution of the DMH profile to the 2-halo term and the scale-dependent bias. In addition, if a pair of objects are closer than the sum of the virial radii of their parent halos, the pair should not be counted in the 2-halo term; this effect is neglected. These improvements have enabled those authors to make accurate modelings of, e.g., the ACF of SDSS galaxies, where the CFs can be measured to a few percent level at r_p ≈ 1 h⁻¹ Mpc. However, as AGNs have a much smaller number density, the errors on our CCF measurements are ≳20% (Paper I) at this r_p, where these improvements are important. Therefore, the effects of these recent improvements by other authors are estimated to be within the 1σ measurement errors of the CCF.

In this subsection, we explain the application of the HOD modeling to the CCF between AGNs and LRGs (or more generally, between any two different populations). Using the HOD of the LRGs as a template, which has been accurately constrained by Z09 using the LRG ACF measured by Zehavi et al. (2005b), we constrain the HOD of the AGNs by fitting to the AGN–LRG CCF. A similar approach has been made by Z09 to model the CCF between LRGs and L_* galaxies, where they assumed that the LRGs are at the halo centers and the L_* galaxies are satellites in the 1-halo term calculation. Since LRGs occupy the centers of almost all massive DMHs (log M_h ≳ 14 [h⁻¹ M_☉], Z09), this is a good approximation for their LRG–L_* galaxy CCF as well as for our AGN–LRG CCF. However, for completeness, we develop a formulation of the HOD modeling that includes more general cases. Hereafter, the quantities for AGNs are represented by a subscript "A," LRGs by "G" (representing galaxies), and CCF between the two by "AG." We denote the LRG HODs at the halo center and of satellites by 〈N_G,c〉(M_h) and 〈N_G,s〉(M_h), respectively. Now 〈N_G〉(M_h) = 〈N_G,c〉(M_h) + 〈N_G,s〉(M_h). Likewise, the HODs of the AGNs at the halo centers, of satellites, and the sum of the two are denoted by 〈N_A,c〉(M_h), 〈N_A,s〉(M_h) and 〈N_A〉(M_h).

In our approximate treatment, the 2-halo term can be expressed as follows. Let the linear bias parameters of the LRGs and AGNs be b_G and b_A, respectively:

$$\begin{equation} b_{\rm G}=\frac{\int b_{\rm h}(M_{\rm h})\langle N_{\rm G}\rangle (M_{\rm h})\phi (M_{\rm h})dM_{\rm h}}{\int \langle N_{\rm G}\rangle (M_{\rm h})\phi (M_{\rm h})dM_{\rm h}} \end{equation} \tag{ 11 }$$

$$\begin{equation} b_{\rm A}=\frac{\int b_{\rm h}(M_{\rm h})\langle N_{\rm A}\rangle (M_{\rm h})\phi (M_{\rm h})dM_{\rm h}}{\int \langle N_{\rm A}\rangle (M_{\rm h})\phi (M_{\rm h})dM_{\rm h}}. \end{equation} \tag{ 12 }$$

Then, the 2-halo term of the power spectrum corresponding to the CCF (cross power spectrum) can be expressed by

$$\begin{equation} P_{\rm AG, 2h}(k) \approx b_{\rm A}b_{\rm G} P_{\rm lin}(k). \end{equation} \tag{ 13 }$$

The 1-halo term is composed of three terms:

$$\begin{eqnarray} P_{\rm AG,1h}(k)&=&\frac{1}{(2\pi)^3n_{\rm A}n_{\rm G}}\int \phi (M_{\rm h}) \nonumber \\ & &\times[\langle N_{\rm A,c}N_{\rm G,s}+N_{\rm A,s}N_{\rm G,c}\rangle (M_{\rm h})y(k,M_{\rm h})\nonumber \\ & &+\langle N_{\rm A,s}N_{\rm G,s}\rangle (M_{\rm h})\,\,|y(k,M_{\rm h})|^2]\, dM_{\rm h}, \end{eqnarray} \tag{ 14 }$$

provided that no object is in common between the AGN and LRG samples (which is the case in our CCF). In this case, the relation

$$\begin{equation} \langle N_{\rm A,x}N_{\rm G,y}\rangle (M_{\rm h})= \langle N_{\rm A,x}\rangle (M_{\rm h})\langle N_{\rm G,y}\rangle (M_{\rm h}) \end{equation} \tag{ 15 }$$

holds, where the subscripts x and y represent any combination of the subscripts s and c, except for the case both are c's. This relation is exact even if 〈N〉 ≲ 1, where one has to use the exact Poisson distribution for the probability distribution of having exactly N objects in a halo (sometimes called a sub-Poissonian case). The relation 〈N_A,cN_G,s + N_A,sN_G,c〉 = 〈N_A,cN_G,s〉 + 〈N_A,sN_G,c〉 also holds.

Even in cases where there are objects in common between the two samples, the above discussion does not change for the central–satellite pairs, because the two objects are surely different. However, the existence of the common objects makes assumptions behind the third term in the integrand of Equation (14) invalid. A detailed discussion of this case is beyond the scope of the present paper and will be made in a future paper.

3.2.1. The LRG HOD

The HOD of the LRGs has been extensively studied by Z09 based on the ACF measurements of the LRGs by Zehavi et al. (2005b) using SDSS Data Release (DR) 3. Their 〈N_G,s〉(M_h) model has essentially five parameters, which are the values at five M_h points, and they have spline-interpolation values between these points. They take an elaborate parameterization of 〈N_G,c〉(M_h), which is an integration of a luminosity-dependent smoothed step-function, where the value increases from 0 (at low M_h) to 1 (at high M_h), with a transition following a luminosity-dependent error function.

In Paper I, we re-calculate the LRG ACF for the 0.16 < z < 0.36, −23.2 < M_g < −21.2 sample based on DR4+, using ≈46,000 LRGs, instead of ≈30,000 LRGs from DR3 used by Zehavi et al. (2005b). We obtain practically the identical ACF to that calculated by Zehavi et al. (2005b). Since we want to use exactly the same model and the same LRG sample between the LRG ACF and the LRG–AGN CCF, we derive the best-fit LRG HOD to our DR4+ sample using our software. However, instead of newly exploring the full parameter space, we take advantage of the analysis in Z09 to find the HODs that give the best fit to our data by tweaking their LRG HODs as follows. Figure 1(b) of Z09 shows the Δχ² = 4 upper and lower bounds of the HODs separately for the central and satellite LRGs. For the satellites, we start with their Δχ² = 4 upper and lower bounds, denoted by N^u_Z,s and N^l_Z,s, respectively. We then interpolate between these curves:

$$\begin{equation} \langle N_{\rm G,s}\rangle (M_{\rm h}) = (1-f)=\big\langle N_{\rm Z,s}^{\rm l}\big\rangle (M_{\rm h})+f\big\langle N_{\rm Z,s}^{\rm u}\big\rangle (M_{\rm h}), \end{equation} \tag{ 16 }$$

where f is a weight parameter in the linear interpolating procedure.

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For the central LRGs, we shift their central HOD (N_Z,c) horizontally by d in log M_h:

$$\begin{equation} \langle N_{\rm G,c}\rangle (M_{\rm h}) = \langle N_{\rm Z,c}\rangle (M_{\rm h}10^d). \end{equation} \tag{ 17 }$$

We tweak N_G,c horizontally because of the 〈N_G,c〉(M_h) ⩽ 1 constraint. In calculating 〈N_s(N_s − 1)〉(M_h) of the LRG ACF based on the above HODs, we calculate the mean value using the exact Poisson statistics in the small number case (〈N_s〉 ⩽ 10.0), while we use 〈N_s〉(〈N_s〉 − 1) for larger values. As a whole, our model has two "tweak" parameters, f and d.

We calculate a series of model ACFs at z = 0.28 (the average redshift of our sample, see Table 1) in a parameter space grid exhaustively over a rectangular area in the (f, d) space that is large enough to include all the region with Δχ² ≲ 5. The grid spacings are 0.03 and 0.003 for f and d, respectively.

Then, we fit the model to the data by minimizing χ², taking the correlation of errors and the total number density of LRGs (n_LRG) into account:

$$\begin{eqnarray} \chi ^2 &=& \sum _{ij}\big\lbrace \big[w_{\rm p}(r_{{\rm p},i})-w_{\rm p}^{\rm mdl}(r_{{\rm p},i})\big]M^{-1}_{ij} \nonumber \\ & &\times\big [w_{\rm p}(r_{{\rm p},j})-w_{\rm p}^{\rm mdl}(r_{{\rm p},j})\big]\big\rbrace \nonumber \\ & &+ \big(n_{\rm LRG}-n_{\rm LRG}^{\rm mdl}\big)^2\big/\sigma _{n_{\rm LRG}}^2, \end{eqnarray} \tag{ 18 }$$

where the quantities from the model are indicated by a superscript "mdl," M is the covariance matrix, and $\sigma _{n_{\rm LRG}}$ is the 1σ error of the LRG number density. The covariance matrix and $\sigma _{n_{\rm LRG}}$ are estimated using the jackknife resampling method (see Paper I). During the minimization process, the models w^mdl_p(r_p,i) are calculated by interpolating from the four nearest grid points in the parameter space computed above. The minimization is performed using the MINUIT package⁷ distributed as a part of the CERN program library. Including the number density term in χ² gives important parameter constraints, not only on the overall normalization, but also the shape. This is because N_G,c, by definition, has the absolute maximum value of unity (as there cannot be more than one central galaxy in a halo), and for the LRGs, the value saturates at 1 for M_h ≳ 10¹⁴h⁻¹ M_☉. With this constraint, n_LRG becomes sensitive to the combination of the relative normalizations between N_G,c and N_G,s and the "cut-off" mass of N_G,c. For the LRG sample defined in Paper I, n_LRG = (9.56 ± 0.13) × 10⁻⁵ h⁻³ Mpc⁻³. In performing the fits, we use only the data points that are dominated by either the 1-halo or 2-halo terms, excluding the transition region (0.46 < r_p[h⁻¹ Mpc] < 2.8). In this transition region, our model ACF becomes inaccurate due to our approximations, especially in neglecting the halo-exclusion effect in the 2-halo term, as described above in Section 3.1. We use the data points down to r_p = 0.2 [h⁻¹ Mpc], which is slightly below the lower r_p limit used in Paper I. Our measurements to this scale are consistent with those measured by Masjedi et al. (2006).

Our best-fit model has "tweak" parameter values of f = 1.21(1.12; 1.29) and d = −0.087(−0.095; − 0.083), where the 68% confidence ranges for two interesting parameters (Δχ² < 2.3) are given in parentheses. Our HODs are slightly higher than those measured by Z09, probably because we include a data point at r_p = 0.2 [h⁻¹ Mpc] and we use a slightly different linear power spectrum. The best-fit model w_p(r_p) is compared with the observation as well as the Z09 model in Figure 1(a). The corresponding HODs (〈N_G,s(M_h)〉, 〈N_G,c(M_h)〉, and the total) are shown in Figure 1(b), and confidence contours in the (d,f) space are shown in Figure 1(c).

3.2.2. The AGN HOD

With the LRG HOD in hand, we come to our main purpose of obtaining constraints on the HOD of the AGNs in the RASS–SDSS survey.

As a parameterized form of the AGN HOD, we first try a simple truncated power-law form, assuming that all the AGNs are satellites (Model A):

$$\begin{equation} \langle N_{\rm A,s}\rangle \propto M_{\rm h}^\alpha \Theta (M_{\rm h}-M_{\rm cr}),\,\,\langle N_{\rm A,c}\rangle =0, \end{equation} \tag{ 19 }$$

where Θ(x) is the step function (=1 at x ⩾ 0; =0 at x < 0), M_cr is a critical DMH mass below which the HOD is zero, and α is a power-law slope of the HOD above M_cr. While formally we are assuming that all AGNs are satellites, the HOD constraints obtained from this assumption can be applied fairly to the sum of the central and satellite AGNs in cases where the AGN activity in the central galaxy of high-mass log  M_h[h⁻¹ M_☉] ≳ 13.5 DMHs is suppressed. For simplicity of explanation, we visualize the HOD of the central LRG as a step function with a step at log  M_h[h⁻¹ M_☉] = 13.5, and refer the DMH mass roughly above (below) this cut as higher (lower) mass. There is no contribution to the 1-halo term at lower mass DMHs, since the 1-halo term requires both AGNs and LRGs to reside at the same DMH mass. Since there is in practice no distinction between central and satellite AGNs in the 2-halo term (Equation (13)), whether the AGNs in low-mass DMHs are satellites or central makes no difference to the CCF, and indeed this model can fairly accommodate a case where a low-mass halo represents one galaxy (halo) at the center. At higher masses, among the three terms in Equation (14), the satellite AGN-central LRG pairs dominate the 1-halo term, because, as we show later, the present HOD analysis is sensitive at log  M_h[h⁻¹ M_☉] < 14.0, where there are in practice no satellite LRGs (Figure 1), and therefore the central AGN-satellite LRG term as well as the satellite AGN-satellite LRG term are negligible. Thus, Equation (19) represents the sum of the central and satellite AGN HOD if there is no central AGN at higher masses.

The underlying assumption in Model A is partially motivated by observations of AGN "downsizing" (Ueda et al. 2003; Hasinger 2008; Ebrero et al. 2009b; Yencho et al. 2009), where the number density of high-luminosity AGNs peaks earlier in the history of the universe and drops rapidly toward low redshift, while lower luminosity AGN activity peaks later. (We note, however, that a recent work by Aird et al. (2010) reports that this trend might be weaker than those reported previously.) One possible implication of this trend is that BHs at the centers of massive galaxies occupying the centers of (higher mass) DMHs stopped accreting long before z ∼ 0.3, while accretion occurs more frequently in lower mass galaxies in satellites or at the centers of lower mass halos in the redshift range of our sample. Thus, model A can be a demonstrative case of this scenario. Additionally, central galaxies of massive halos are mostly early-type luminous galaxies, where AGN activity is reported to be suppressed (Schawinski et al. 2010).

In Section 4.3, we consider in detail other models (Models B and C) which use well-studied galaxy HODs (central plus satellites) as templates and treat the separate central and satellite HODs explicitly.

Using a fixed set of 〈N_G,c〉(M_h) and 〈N_G,s〉(M_h) derived in the previous subsection and the formulations developed earlier in this section, we calculate the expected w_p,AG(r_p) in the two parameter (M_cr,α) model of 〈N_A〉(M_h). Due to much smaller errors of the LRG ACF compared with those in the AGN–LRG CCF, we fixed the tweak parameter f and d at the best-fit values during the fits to the CCF. Shifting the tweak parameters to any point on the Δχ² = 2.3 contour in Figure 1(c) causes a shift of only Δχ² < 0.05 to the same AGN HOD model. This justifies the use of fixed LRG HODs during the fit to the CCF.

We calculate a series of model CCFs in a parameter grid at z = 0.28 with the spacings of 0.1 and 0.05 for log  M_cr and α, respectively, over the rectangular region that we are interested in (see confidence contours in the next section). While the mean redshift of the AGN samples range from z = 0.24 to z = 0.28, the 1-halo and 2-halo terms vary by only ≈2% and ≈0.1%, respectively between these redshifts, justifying our single redshift calculations. We search for the best-fit model by minimizing the correlated χ²:

$$\begin{eqnarray} \chi ^2 &=& \sum _{ij}\big\lbrace \big[w_{\rm p,AG}(r_{{\rm p},i})-w_{\rm p,AG}^{\rm mdl}(r_{{\rm p},i)}\big]M^{-1}_{ij} \nonumber \\ & &\times\big [w_{\rm p,AG}(r_{{\rm p},j})-w_{\rm p,AG}^{\rm mdl}(r_{{\rm p},j})\big]\big\rbrace, \end{eqnarray} \tag{ 20 }$$

where the errors and covariance matrix are estimated using the jackknife resampling method (Paper I). Unlike in the case of LRGs, here we do not include the number density term in determining χ², because the CCF depends only on α and M_cr in Equation (19) but not the normalization. Thus, the CCF constraints on α and M_cr and the density constraints on the normalization can be separated. After the constraints of these two parameters are made, the normalization can be determined by matching to the number density of the AGNs. We use only data points with r_p>0.3 h⁻¹ Mpc, as smaller scale bins contain only a small number of AGN–LRG pairs (<15 pairs for the total and much less in the high- and low-L_X RASS-AGN samples), which would not warrant the applicability of the χ² statistic. Unlike in the case of the LRG ACF, we use all the data points in 0.3 ≲ r_p[h⁻¹ Mpc] ≲ 40. Due to the smaller mass for the DMH occupied by the AGN compared to satellite LRGs, the transition between 1-halo and 2-halo dominated regimes is very narrow for the AGN–LRG CCFs, as will be seen in the next section. Even within this small transition region, the effects of the major source of inaccuracy in our approximation, i.e., halo exclusion in the 2-halo term, is estimated be a few times smaller than our CCF measurement errors, based on the comparison of our LRG 2-halo term and that calculated by Z09.

Using the methods described above, we obtain constraints on the AGN HOD parameters log  M_cr and α in Equation (19). For each point in this two parameter space, we can also calculate the mean DMH mass occupied by the AGN sample,

$$\begin{equation} \langle M_{\rm h}\rangle =\frac{\int M_{\rm h} \langle N_{\rm A}\rangle (M_{\rm h}) \phi (M_{\rm h})dM_{\rm h}}{\int \langle N_{\rm A}\rangle (M_{\rm h}) \phi (M_{\rm h})dM_{\rm h}}, \end{equation} \tag{ 21 }$$

as well as the effective bias parameter of AGNs b_A (Equation (12)).

Figure 2(a) compares our AGN–LRG CCF for the total RASS-AGN sample and our best-fit model.

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Figure 2(b) shows the confidence contours in the log  M_cr–α space, overlaid on underlying thin (green) contours showing the mean halo mass 〈M_h〉. A line of constant bias, b_A, roughly follows a constant 〈M_h〉 contour, with small deviations caused by the nonlinearity of the b_h(M_h) function. The degree of the deviation is such that log 〈M_h〉 for a given b decreases typically by ≈0.4 when α is decreased from ≈1 to ≈−3. Figures 3 and 4 show our HOD fits and confidence contours, respectively, for the high-L_X and-low L_X RASS-AGN samples, with an emphasis on the comparison between the two.

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Figures 2(b) and 4 show that the parameter pair (log  M_cr, α) is tightly constrained roughly along the constant 〈M_h〉 line. This primary constraint comes from the amplitude of the 2-halo term, which is proportional to the bias parameter of the AGN sample (for the fixed bias of the LRG sample). In the HOD analysis, however, the inclusion of the 1-halo term adds additional constraints in the two-parameter space.

In any of the total, high-L_X and low-L_X RASS-AGN samples, models where the number occupation of AGNs is proportional to M_h above M_cr (α = 1) are excluded, while a constant number of AGNs per halo above M_cr is preferred. Figures 2(b) and 4 put constraints on the values of M_cr for each sample. Using the Δχ² = 2.3 contour (68% confidence level for two parameters), the minimum mass is constrained to be log  M_cr[h⁻¹ M_☉]≳ 11.9, 11.9, and 10.7 for the total, high-L_X and low-L_X RASS-AGN samples, respectively. For the total RASS-AGN sample, the contour is constrained to have α> − 3.0, while for the high-L_X and low-L_X RASS-AGN samples, the contour allows the smallest α that we have explored (α = −3.25). This essentially gives constraints on the width of the HOD, and at least for the total RASS-AGN sample, a delta-function type HOD could be marginally excluded.

The best-fit HOD parameters, the mean DMH mass occupied by the AGNs, and the linear bias from the fits are summarized in Table 2. For the fitting parameters log  M_cr and α, we show the full range corresponding to 68% confidence range in the 2D parameter space (Δχ² = 2.3). As shown in Figures 2 and 4, these two parameters are highly correlated and the confidence contours are skewed, thus one should be cautious in interpreting these ranges. On the other hand, if we project the probability distribution on the variable 〈M_h〉 or the variable b (bias), both of which are unique functions of our two fitting parameters, the projected probability distribution becomes roughly Gaussian. Thus, we take the Δχ² < 1 range (68% error for one parameter) to estimate the 1σ errors of these derived quantities, 〈M_h〉 and b_A, in Table 2.

We note that because of the exponential drop of the DMH mass function the HOD at very high M_h contributes little to the CCF. In order to check the maximum M_h that our analysis can reasonably constrain, we recalculate the CCF with the truncated power-law model with an upper mass cutoff

$$\begin{equation} \langle N_{\rm A}\rangle \propto M_{\rm h}^\alpha \;\Theta (M_{\rm h}-M_{\rm cr})\Theta (M_{\rm cut}-M_{\rm h}).\nonumber \end{equation} \tag{ 22 }$$

We recalculate χ² with varying M_cut while fixing (M_cr,α) at the best-fit value above. The change of χ² from the M_cut = ∞ is larger than unity at log M_h[h⁻¹ M_☉] < 14.0. Thus, our HOD analysis is sensitive up to M_h ≈ 10¹⁴h⁻¹M_☉, i.e., the mass of a poor cluster.

In order to illustrate the range of acceptable HODs, we plot a number of representative AGN HOD models accepted by our fits (〈N_A〉(M_h)) as function of DMH mass in Figure 5(a). In Figure 5(b), we also show the same sets of models in units of the spatial density per comoving volume per log of the DMH mass:

$$\begin{equation} \langle N_{\rm A} \rangle (M_{\rm h}) \frac{d \phi (M_{\rm h})}{d \log \,M_{\rm h}} \equiv \langle N_{\rm A} \rangle (M_{\rm h}) \cdot \ln (10)M_{\rm h}\phi (M_{\rm h}). \end{equation} \tag{ 23 }$$

These models have been normalized to the observed number densities (see Table 1). In each panel, three curves are plotted, representing the best-fit case and two extreme cases. These are (1) for the best-fit values (solid lines), (2) the point on the Δχ² = 2.3 contour that has the smallest M_cr value (i.e., the upper left tip of the two-parameter 68% confidence area in Figure 2(b) or 4; dotted lines), and (3) the smallest α point on the Δχ² = 2.3 (the bottom of the confidence region and narrowest possible HOD distribution, dashed lines). The best-fit model with α = −3.25 (the smallest α in our search grid) is used for the high-L_X and low-L_X RASS-AGN samples, for which the Δχ² = 2.3 contour continues below α = −3.25. These three models have been plotted to illustrate almost the full range of possible HODs statistically accepted by our analysis. In order to illustrate the DHM mass range that can be occupied by both LRGs and AGNs, and therefore contributes to the 1-halo term of the CCF, we also show the HODs of the LRGs for reference. Of course, these plots are for our rather restrictive truncated power-law model and by no means are intended to illustrate an exhaustive set of possible HODs. However, this illustrates that these three models have roughly the same average M_h, with varied widths of the HOD. An important constraint is that the AGN HOD can be no wider than the dotted line in Figures 5(a) and (b), which represents the point corresponding to the lowest extreme of M_cr and the roughly highest extreme of α, among acceptable models with Δχ² < 2.3. The meaning of this constraint is discussed in Section 5.2.

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In this section, we explore models in which central AGNs are explicitly included, using galaxy HODs as a template. Galaxy HODs have been extensively investigated using, e.g., SDSS data, with a high statistical accuracy. A simple model for luminosity–threshold galaxy samples (i.e., more luminous than a given limit), introduced by Zehavi et al. (2005a) is a three-parameter model including a step function for the central HOD and a truncated power-law satellite HOD. We express the AGN HODs here with the same form, except that the global normalization is also a free parameter (Model B):

$$\begin{equation} \begin{array}{l} \langle N_{\rm A,c}\rangle = f_{\rm A}\Theta (M_{\rm h}-M_{\rm min}), \\ \ms \langle N_{\rm A,s}\rangle = f_{\rm A}\Theta (M_{\rm h}-M_{\rm min})(M_{\rm h}/M_1)^{\alpha _{\rm s}}, \end{array} \end{equation} \tag{ 24 }$$

where f_A is the AGN fraction (duty cycle) among central galaxies at M_h ≳ M_min. We use the symbol α_s to emphasize that it represents the HOD slope for satellites. M₁ is the DMH mass at which the number of central AGNs is equal to that of satellite AGNs. For luminosity–threshold galaxy samples, it is usually assumed that the central galaxy HOD saturates at unity at the high-M_h end (i.e., f_A = 1 in Equation (24)), because central galaxies in the highest mass DMHs are expected to be luminous enough to be included in the sample (e.g., Section 3.2.1). Since only a small fraction of galaxies contain an AGN, this has to be multiplied by a factor f_A(≪1), the value of which does not affect the CCF as discussed above. The value of f_A can be determined by normalizing the HOD to n_AGN.

It is important to note the underlying physical assumption of Equation (24), which is that the AGN duty cycle among central galaxies does not depend on the DMH mass. We explore the consequences of this model as an illustrative contrast to Model A, which assumes a suppression of the AGN duty cycle in central galaxies of high-mass DMHs.

We investigate the behavior of χ² (Equation (20)) in the space of M_min, M₁, and α_s. Due to the low signal-to-noise ratio of the CCFs for our luminosity-divided subsamples, we limit our discussion to the total sample. Even with the total sample, our statistics do not allow us to constrain all three parameters; therefore we investigate the constraints on two parameters by fixing the remaining parameter. The HOD analysis of luminosity–threshold samples with −21.5 ⩽ M^max_r ⩽ −21.0, studied by Zehavi et al. (2005a) found M₁/M_min ≈ 23 and α_s ≈ 1.2. We consider the typical HOD of SDSS galaxy samples in this luminosity threshold range as a template because their M_min range (12.0 ≲ log M_min[h⁻¹ M_☉] ≲ 12.7) roughly coincides with the minimum DMH range for our AGNs, as we see from the range of M_cr in the confidence contour in Figure 2(b), which approximately coincide with the range of M_min as we see below. We therefore (1) fix M₁/M_min = 23 and obtain constraints in the (log  M_min, α_s)-space, and (2) fix α = 1.2 and obtain constraints in the (log  M_min, M₁/M_min)-space.

The resulting confidence contours for (1) are shown in Figure 6(a). The best-fit parameters and χ² values are summarized in Table 3 for both (1) and (2), where we also show the α_s = 1.0 case for (2), in which the number of satellites is proportional to the DMH mass.

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For the M₁/M_min = 23 case, the best-fit slope for the satellite is below unity (α_s = 0.55) and χ² = 6.9 is identical to that of Model A. The best-fit model with α_s = 1.0 is marginally rejected (Δχ² = 3.6) and the model with α_s = 1.2 is significantly rejected (Δχ² = 18). Models that are consistent with α_s = 1.2 can be found at M₁/M_min ⩾ 30, but they are not consistent with the HOD of luminosity–threshold galaxy samples in the −21.5 ⩽ M^max_r ⩽ −21.0 range.

We also explore a model (Model C) where only lower mass DMHs contain a central AGN, where the physical picture is explained at the introduction of Model A (Section 3.2.2). In this model, the central HOD in Model B is replaced by

$$\begin{equation} \langle N_{\rm A,c}\rangle = f_{\rm A}\Theta (M_{\rm h}-M_{\rm min})\Theta (M_{\rm cc}-M_{\rm h}), \end{equation} \tag{ 25 }$$

while the form of 〈N_A,c〉 stays the same. As a representative case, we choose log M_cc[h⁻¹ M_☉] = 13.0, below which the DMH center is more frequently occupied by blue galaxies than red galaxies in the HOD analysis of color-separated samples by Zehavi et al. (2005a), while we keep M₁/M_min = 23 (but see also Zehavi et al. (2010), which use a different parameterization on color-separated samples). The confidence contours are shown in Figure 6(b) and the best-fit parameters and χ² values are shown in Table 3. In this scenario, the best-fit slope for the satellite is α_s = 0, while α_s = 1.0 is rejected at the Δχ² = 4.8 level.

Table 4 compares our results with those from Paper I. The meanings of the table columns are: (1) sample, (2) the bias parameter from the HOD analysis (Equation (12)), (3) the bias parameter from Paper I, i.e., for the power-law fit in 0.3 < r_p[h⁻¹ Mpc] < 15, (4) the bias parameter for the power-law fit in 1.5 < r_p[h⁻¹ Mpc] < 15 calculated in the same way as Paper I (see below), (5) the "mean" DMH mass from Equation (21), (6) the "typical" DMH mass calculated from b_A from the HOD model, defined by b_h(M_typ) = b_A calculated with the b_h(M_h) by Tinker et al. (2005) (see Section 3.1), (7) the "typical" DMH mass from Paper I (i.e., from power-law fits) calculated with the b_h(M_h) by Sheth et al. (2001), and (8) the "typical" DMH mass from the Paper I bias parameter re-calculated using the b_h(M_h) relation by Tinker et al. (2005). While the results from the the HOD analysis presented here and the simple analysis of Paper I agree well for the high-L_X RASS-AGN sample, the HOD analysis gives larger bias parameters than those derived in Paper I for the total and low-L_X RASS-AGN samples at the level of 1.5σ. Since these bias parameters have been derived from the same data set with different methods, the discrepancies are systematic rather than statistical. One of the major findings of Paper I is the detection of the X-ray luminosity dependence of the CF. The difference in correlation lengths (r₀), between the high-L_X and low-L_X RASS-AGN samples measured in Paper I is ≈2.5σ, based on power-law fits with a fixed slope of γ = 1.9. The differences of the AGN bias parameters b_A and associated typical DMH mass M_h,typ derived from power-law fits are ≈1.8σ. In the HOD analysis, the significance of the difference in b_A decreases to 1.1σ, because it is constrained mainly by measurements at large scales where the 2-halo term dominates (see below). However, in terms of M_h,typ derived from the HOD analysis, the difference is ≈1.8σ because of the additional constraints from the 1-halo term.

There are a number of differences in the processes used to derive the bias parameter between Paper I and this work. In Paper I, the bias parameters are based on the power-law fits to Equation (3) over scales of 0.3 < r_p[h⁻¹ Mpc] < 15, assuming a linear bias over the entire scale range. Then, the power-law models are converted into the rms fluctuations over 8 h⁻¹ Mpc spheres (σ_8,AGN), and the bias parameter is calculated using σ_8,AGN/[σ₀D(z)], where D(z) is the linear growth factor. The main source of discrepancy is that, while Paper I also uses data points down to a scale of r_p = 0.3 h⁻¹ Mpc to obtain b, in the HOD analysis presented here the constraints on b mainly come from the 2-halo-term-dominated region (r_p ≳ 1.5 h⁻¹), while smaller scale data constrain other variables. Figure 3 shows that the difference between the high-L_X and low-L_X RASS-AGN samples is more prominent in the 1-halo-term-dominated regime, at r_p ≲ 1 h⁻¹, than in the 2-halo-term-dominated regime. This means that, for the AGN–LRG CCF, the 1-halo term is more sensitive to differences in the AGN HOD. The reason for this is that while b_h(M_h) increases only weakly with increasing with M_h, the 1-halo term is directly proportional to the number of AGNs that are in the same DMHs as those occupied by LRGs, i.e., very massive ones (log M_h ≳ 13.5[h⁻¹ M_☉]), i.e., the 1-halo term is more sensitive to the existence of AGNs in high-mass halos.

In the literature, relative or absolute bias parameters are often calculated based on results from power-law fits (e.g., Paper I; Mullis et al. 2004; Yang et al. 2006; Miyaji et al. 2007; Coil et al. 2009). The power-law models of ξ(r) 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 ($\bar{\xi }({<}20\, h^{-1}\;{\rm Mpc})$). While some authors use only large scales (r_p>1–2 h⁻¹ Mpc) to ensure that only the linear regime is used, others include smaller scales. Usually the rationale for including smaller scales is to have better statistics, combined with the empirical observation that the CFs are well fit with a power-law model over the scale of 0.1–0.3 ≲ r_p ≲ 10–20 h⁻¹ Mpc (0.3 < r_p < 15 h⁻¹ Mpc in the case of Paper I). As a check, we have re-calculated the bias parameter for power-law fits to the w_p(AGN|AGN) used in Paper I on scales of 1.5 < r_p < 15 h⁻¹ Mpc, to limit ourselves to the linear regime. These are shown in Column 4 of Table 4. The bias parameters obtained in this manner carry large statistical errors, but the values are consistent with those derived from the HOD analysis within 1σ in all three samples, while they are more than ≈1σ above the bias parameters derived in Paper I for the total and low-L_X RASS-AGN samples. The AGN bias parameters in our total and high-L_x samples from the HOD analysis, as well as the power-law fit 1.5 < r_p < 15 h⁻¹ Mpc, are consistent within combined 1σ errors with those measured by Padmanabhan et al. (2009) for their QSO samples ALL0 and LSTAR0, which cover a similar redshift range as ours (0.25 < z < 0.35). However, their bias parameters for their higher redshift QSO samples, ALL1 and LSTAR1 (0.33 < z < 0.50) as well as ALL2 and LSTAR2 (0.45 < z < 0.60) are lower.

Using the power-law fit results including the nonlinear regime is useful to detect the different clustering properties between, e.g., different source populations, luminosities, or types. In our case, this gives the most statistically significant difference between the clustering properties of the high-L_X and low-L_X RASS-AGN samples. However, one has to be careful in interpreting the results, e.g., in terms of the DMH mass based on the bias parameter. In principle, the HOD analysis overcomes this limitation, because we are able to derive the pure linear bias parameter, while at the same time utilizing the data in the nonlinear regime.

With the HOD analysis, we can also obtain constraints on at least one additional variable which is independent from the "mean" DMH mass, thanks to the constraints from the 1-halo term. The importance of the 1-halo term constraints has also been noted by Padmanabhan et al. (2009) using a similar CCF analysis between photometric-redshift LRG samples and optically selected QSOs in SDSS at z ∼ 0.5. They qualitatively concluded that at least some QSOs must be satellites and also that models of the form of our Model B (Equation (24)) with M₁/M_min = 20 and α_s = 0.9 are also acceptable, while lower α values are also possible. Our analysis here gives qualitatively similar results.

The acceptable parameter space for Model A is always in the α < 1 regime for all of our three AGN samples: the total, high-L_X, and low-L_X RASS-AGN samples. This means that the average number of AGNs in a DMH does not increase proportionally with mass. Figures 2(b) and 4 show that models with α ≲ 0 are preferred. It is interesting to compare this result with HOD analyses of galaxies, that find α_s ≈ 1–1.2 for a wide range of absolute magnitudes and redshifts at least up to z ≈ 1.2 (Zehavi et al. 2005a; Zheng et al. 2007; Zehavi et al. 2010). In principle, neither the results nor the assumptions of Model A say anything about the central fraction among AGNs at the lower mass DMHs (see Section 3.2.2), and thus they do not directly constrain the slope of the satellite AGN HOD α_s. However, if we take the extreme case where all the AGNs are satellites, α_s < 0.4, this would imply that AGN fraction among satellite galaxies strongly decrease with M_h. In Model B, where the central galaxy component is included explicitly under the assumption that the AGN duty cycle in central galaxies is constant against varying M_h above M_min, the best-fit slope is still less than unity (α_s = 0.55), and models with α_s ≈ 1 are only marginally rejected for M₁/M_min = 23 (a representative value of luminosity–threshold galaxy samples). In case of Model C, where only log  M_h[h⁻¹ M_☉] < 13.0 DMHs can contain a central AGN, the best-fit solution is α_s = 0, while α_s ≈ 1 is rejected at a ∼2σ level for M₁/M_min = 23. On the whole, our results tend to favor models in which the AGN fraction among satellite galaxies decreases with increasing DMH mass (or the richness of groups and clusters), in the mass range 12–13 ≲ log  M_h[h⁻¹ M_☉] ≲ 14.0.

Since our AGN sample is selected from X-ray point sources in RASS, and since clusters of galaxies (which represent the most massive DMHs) are also X-ray sources, an observational bias that might affect our results is that AGNs in the clusters are selected against in the RASS-AGN sample due to confusion, especially considering that the mean point-spread function (PSF) has an FWHM ≈2 arcmin (La Barbera et al. 2009), which is not negligible when compared to the extent of the cluster X-ray emission. We have estimated the significance of this effect as follows. The detection limit of the cluster diffuse X-ray emission is higher than that of point sources. A typical flux limit for clusters in RASS is S_X ≈ 2 × 10⁻¹² erg s⁻¹ cm⁻², (0.1–2.4 keV)(Böhringer et al. 2001; Popesso et al. 2004), corresponding to log  L_X ≳ 44.3 [h⁻²₇₀erg s^-1] at the lower bound of our redshift range of z ≈ 0.16. This scales to a cluster virial mass of log  M_h ≳ 14.5 h⁻¹ M_☉ (Reiprich & Böhringer 2002). As we found at the end of Section 4.1, our CCF is only sensitive to the HOD behavior at log  M_h ≲ 14.0 h⁻¹ M_☉, and thus this selection effect should play negligible role in our results. Additionally, a confusion due to two or more X-ray AGN point sources being artificially blended into a single source by the RASS PSF is not important either, as 〈N_A〉 is much less than unity over all masses (see Figure 5). Thus, we can state with a good degree of confidence that our limits on α (Model A) or α_s (Model B/C) are not an artifact of X-ray source confusion.

Our results of α_s < 1, found in our extensive explorations in the parameter space of the models that we consider, may be interpreted in view of the long-suggested deficiency of emission line diagnostic-selected AGNs in rich clusters (e.g., Gisler 1978; Dressler et al. 1985). While our CCF analysis is not sensitive to rich clusters (Section 4.1), a more recent work by Popesso & Biviano (2006) verified that this trend extends to the group scale by finding an anti-correlation between AGN fraction and the galaxy velocity dispersion in nearby (z < 0.08) groups/clusters of galaxies down to the velocity dispersion of σ_v ∼ 200(km s^-1). Arnold et al. (2009) also found that the X-ray-selected AGN (log  L_X ≳ 41  h⁻¹₇₀ erg s⁻¹, representing a much lower luminosity population than our AGNs) fraction is larger in groups than in clusters by a factor of two at low redshifts (0.02 < z < 0.06). Various physical mechanisms have been suggested to explain the deficiency of star formation galaxies in clusters of galaxies, and similar processes might be responsible for the lack of AGNs as well. These include ram pressure stripping (Gunn & Gott 1972) or evaporation (Cowie & Songaila 1977) of a galaxy's interstellar medium, which is a required ingredient of the AGN activity in galaxies, by the hot intracluster medium. Also, if AGN activity is triggered by mergers between two gas-rich galaxies, another possible mechanism for this trend is the decreased cross-section for galaxy mergers in the galaxy–galaxy close encounters at high relative velocities (Makino & Hut 1997). In high-mass DMHs, representing richer groups/clusters, the velocity dispersion is higher and thus one may expect that there are fewer mergers than in groups of galaxies (Popesso & Biviano 2006), although the density of galaxies increase at the center. Other mechanisms such as the accretion of small gas-rich dwarf galaxies (minor mergers) may play a role in fueling the AGN host, but it is not clear how this would affect the slope α_s.

In order to fully understand the growth history of SMBHs as well as the physical processes responsible for the AGN activity in groups and clusters, we need to explore the statistics at different redshifts, luminosities, and AGN types. The results mentioned above that investigate AGNs in groups/clusters are concerned with AGN at lower redshifts and with lower luminosities than the AGNs in our CCF analysis. In the AEGIS field, Georgakakis et al. (2008) found that AGNs at z ≈ 1 are more frequently found in the group environment than the overall galaxy population. However, deep field surveys such as AEGIS do not contain a large enough volume to explore the richness dependence of the AGN fraction in groups/clusters. Martini et al. (2009) investigate Chandra images of a sample of 33 clusters at 0.05 < z < 1.3, finding that the luminous AGN (log  L_X ≳ 43  h⁻¹₇₀ erg s⁻¹) fraction increases with redshift. They explore the redshift dependence of the AGN fraction, but not the richness dependence. At z < 0.4, where the luminosity range of their AGNs is closer to ours, they have only two AGNs in 17 clusters in the redshift range that is similar to our sample. A much larger sample of groups/clusters is needed to explore the richness dependence of the AGN fraction in a sufficiently narrow redshift range to avoid any degeneracy with the redshift dependence.

In general, directly investigating AGN prevalence in individual groups and clusters is observationally and analytically intensive. The HOD analysis of galaxy-AGN CCF provides a strong statistical tool with which to probe the AGN population in groups and clusters, which can be made without identifying individual groups and clusters. Thus, it is important to extend the HOD analysis to AGN clustering measurements at higher redshift, especially in view that the fraction of star-forming galaxies in clusters increase with redshift (Butcher & Oemler 1984). It is also important to have larger numbers of AGNs for the CCF measurements, which will enable measurements of small-scale clustering and produce stronger constraints on α_s, as well as to break the model degeneracy. Using non-LRG galaxies for the CCF to obtain the 1-halo term constraints in the lower M_h regime would also improve the analysis.

Linked to our constraints on α in Model A, we also have weak constraints on the range of M_cr. The range of M_cr is similar to that of M_min in Model B. Using the scaling relations and the luminosity range of our RASS-AGN samples, we can make an interesting comparison. Figure 1 of Paper I shows that the minimum 0.1–2.4 keV luminosities of the total and high-L_X RASS-AGN samples are log  L_X [h⁻²₇₀ erg s^-1] ≈ 43.7 and 44.3, respectively, while the low-L_X sample has the same minimum luminosity as the total RASS-AGN sample. Converting the 0.1–2.4 keV luminosity to a 0.5–2 keV luminosity, assuming a power-law spectrum with photon index Γ = 2.5 and applying the luminosity-dependent bolometric correction from Equation (2) of Hopkins et al. (2007), the corresponding bolometric luminosities are log  L_bol [h⁻²₇₀ erg s^-1]≈ 44.5 and 45.4 for the total and high-L_X RASS-AGN samples. The mass of the SMBH (M_•) corresponding to the minimum luminosity is then M_• ≈ 2 × 10⁶/λ and 2 × 10⁷/λ [M_☉], where λ ≡ L_bol/L_Edd and L_Edd is the Eddington Luminosity.

The relation between the BH mass M_• and the DMH of its host galaxy is explored by Ferrarese (2002) by comparing the rotation curves of spiral galaxies. While their M_•–DMH mass relation is subject to uncertainties due to different assumptions of the circular velocity and virial velocity by a factor of a few, using their Equation (6) with Bullock et al. (2001)'s recipe, the above BH mass corresponds to a galaxy halo mass of 4 × 10¹¹ (1.5 × 10¹²) · λ^−0.6[M_☉] (for h = 0.7). These masses are comparable or slightly lower than the lower bounds of M_cr derived above for those AGNs emitting at the Eddington luminosity and are an order of magnitude lower than our best-fit values. In general, the concept of the DMH of a galaxy explored by Ferrarese (2002) is different from the DMH derived in our CCF/HOD analysis, since the large-scale CF reflects the DMH mass as the largest virialized structure the object belongs to. If the largest virialized structure represents a group or cluster of galaxies, then there are multiple objects in a DMH, and the DMH that is related to Ferrarese (2002)'s scaling relation represents a sub-halo in a larger mass halo. While our statistics do not allow for strong constraints, if M_cr is close to our nominal value, then the lowest luminosity AGNs among our samples should be accreting at a sub-Eddington rate (λ < 0.1) or reside in groups.