The Spatial Clustering of ROSAT All-Sky Survey Active Galactic Nuclei. V. The Evolution of Broad-line AGN Clustering Properties in the Last 6 Gyr

This is the fifth paper in a series of investigations of the clustering properties of luminous, broad-emission-line active galactic nuclei (AGNs) identified in the ROSAT All-Sky Survey (RASS) and Sloan Digital Sky Survey (SDSS). In this work we measure the cross-correlation function (CCF) between RASS/SDSS Data Release 14 AGNs with the SDSS CMASS galaxy sample at 0.44 < z < 0.64. We apply halo occupation distribution (HOD) modeling to the CCF along with the autocorrelation function of the CMASS galaxies. We find that X-ray-selected and optically selected AGNs at 0.44 < z < 0.64 reside in statistically identical halos with a typical dark matter halo (DMH) mass of MDMHtyp,AGN∼1012.7h−1M⊙ . The acceptable HOD parameter space for these two broad-line AGN samples have only statistically marginal differences caused by small deviations of the CCFs in the one-halo-dominated regime on small scales. In contrast to optically selected AGNs, the X-ray AGN sample may contain a larger population of satellites at M DMH ∼ 1013 h −1 M ⊙. We compare our measurements in this work with our earlier studies at lower independent redshift ranges, spanning a lookback time of 6 Gyr. The comparison over this wider redshift range of 0.07 < z < 0.64 reveals (i) no significant difference between the typical DMH masses of X-ray-selected and optically selected AGNs, (ii) weak positive clustering dependencies of MDMHtyp,AGN with L X and M BH, (iii) no significant dependence of MDMHtyp,AGN on Eddington ratio, and (iv) the same DMH masses host more-massive accreting black holes at high redshift than at low redshifts.

DMHs (e.g., White & Frenk 1991).Processes that are still poorly understood can lead to a flow of matter onto the supermassive black holes (SMBHs) at the centers of the galaxies.Such an object would be observed as an active galactic nucleus (AGN) until the mass flow onto the SMBH stops.The physical processes leading to the formation and evolution of galaxies (and their AGN phases) are closely connected with their host DMHs.However, these DMHs cannot be observed directly.One approach to determine the properties of the DMHs is to measure the clustering of extragalactic objects hosted by these DMHs.This is commonly done by measuring the twopoint correlation function (2PCF; e.g., Peebles 1980).Since AGN can be detected at a variety of wavelength ranges in sufficiently large samples, observational and theoretical studies have investigated the relationship between AGN and their DMHs.AGN clustering measurements (see review by Krumpe et al. 2014) reveal important physical results on AGN/galaxy co-evolution, typical DMHs and the full distribution of DMH masses hosting AGN (e.g., Porciani et al. 2004;Gilli et al. 2005Gilli et al. , 2009;;Yang et al. 2006;Coil et al. 2009;Ross et al. 2009;Krumpe et al. 2010Krumpe et al. , 2018;;Cappelluti et al. 2010;Miyaji et al. 2011;Allevato et al. 2011;Mountrichas & Georgakakis 2012;Koutoulidis et al. 2013;Melnyk et al. 2018;Powell et al. 2018Powell et al. , 2020;;Plionis et al. 2018;Moun-trichas et al. 2019;Viitanen et al. 2019;Krishnan et al. 2020).Simulations and theoretical work (e.g., Springel 2005;Booth & Schaye 2010;Comparat et al. 2019;Georgakakis et al. 2019;Shankar et al. 2020) can use these observational constraints, as well as observed clustering dependences with AGN parameters, to improve models of AGN and galaxy evolution.
As AGN samples are substantially smaller than galaxy samples at low (z ≲ 0.7) and moderate redshifts (0.7 ≲ z ≲ 1.5), the AGN autocorrelations function (ACF; measuring the distances between AGN only) has a limited signal-to-noise ratio (S/N), particularly at small separations.To overcome this observational challenge, the AGN cross-correlation function (CCF) with a dense galaxy sample in the same volume can be used; Coil et al. (2009) demonstrate the potential of this approach.Due to the substantial increase in the number of AGN-galaxy pairs, the uncertainty in the clustering measurement is reduced compared to the measurement of the ACF in the same AGN sample.The AGN ACF can then be inferred from the AGN-galaxy CCF.Halo occupation distribution (HOD) modeling can then be used to interpret the observed clustering signal (e.g., Cooray & Sheth 2002;Rodríguez-Torres et al. 2017).
In Krumpe et al. (2010, hereafter paper I), we use the CCF technique to measure the clustering between ROSAT All-Sky Survey (RASS) AGN identified in the Sloan Digital Sky Survey (SDSS) and a large sample of SDSS luminous red galaxies (LRGs) at 0.16 < z < 0.36.The samples are drawn from a series of SDSS data releases, starting with data release 4 (Adelman- McCarthy et al. 2006).The high S/N of the measurement allows us to split the sample into low and high X-ray luminosity subsamples.We find a weak X-ray luminosity dependence for luminous, broad-line AGN in which higherluminosity (median intrinsic L 0.1−2.4keV = 3.0 × 10 44 erg s −1 ) AGN cluster more strongly than their lowerluminosity (9.8 × 10 43 erg s −1 ) counterparts.
In the second paper of this series (Miyaji et al. 2011, hereafter paper II), we develop a novel method of applying the HOD modeling technique directly to the measured CCF between RASS/SDSS AGN and SDSS LRGs.We constrain the distribution of AGN as a function of DMH mass instead of quoting only typical DMH masses.The major advantage of this method is that it does not use a phenomenological power-law fit, as is often done to derive typical DMHs.We find that models where the AGN fraction among satellite galaxies decreases with DMH mass beyond M DMH ∼ 10 12 h −1 M ⊙ are preferred for luminous, broad-line AGN.This is in contrast to HOD modeling of galaxy samples (Zheng et al. 2009;Zehavi et al. 2011).
In the third paper (Krumpe et al. 2012, hereafter paper III), we extend the cross-correlation measurements to lower and higher redshifts, covering a redshift range of z = 0.07 − 0.50 and apply the HOD modeling to all CCFs directly.We show that the weak X-ray luminosity dependence of broad-line AGN clustering is also found if radio-detected AGN are excluded, and that optically and X-ray selected AGN samples in SDSS show no significant difference in their clustering properties.
In the fourth paper (Krumpe et al. 2015, hereafter paper IV), we explore the physical origin of the weak X-ray luminosity.We find that the clustering strength of luminous broad-line AGN depends on M BH and not on L/L EDD .Thus, more-massive SMBHs are hosted by more-massive DMHs than lower-mass SMBHs.
In this paper we extend the redshift range for our RASS/SDSS AGN clustering measurements further.We compute the CCF between AGN and SDSS CMASS galaxies in the redshift range 0.44 < z < 0.64.We investigate the clustering dependencies for optically and X-ray selected AGN in this redshift range as well as for AGN parameters such as L X , M i , M BH , L/L EDD , and L Bol .Combining these new measurements with the previous independent measurements at lower redshift allows us to constrain the evolution of the AGN clustering properties over a look-back time of 6 Gyr.
This paper is organized as follows.In Section 2 we describe the properties of the CMASS galaxy tracer set and the X-ray and optical AGN samples.Section 3 provides details on how we fit the Hα line profile in the optical SDSS AGN spectra, derive the M BH , estimate L/L EDD , and define our AGN subsamples.In Section 4 we briefly summarize the cross-correlation technique, how the AGN ACF is inferred from this, and how we derive the clustering parameters using HOD modeling.Section 5 provides the results of our clustering measurements.Our results are discussed in Section 6, and we present our conclusions in Section 7.
Throughout the paper, all distances are measured in comoving coordinates and given in units of h −1 Mpc, where h = H 0 /100 km s −1 Mpc −1 , unless otherwise stated.We use a cosmology of Ω m = 0.3, Ω Λ = 0.7, and σ 8 (z = 0) = 0.8, which is consistent with the Wilkinson Microwave Anisotropy Probe data release 7 (Table 3 of Larson et al. 2011).The same cosmology is used in papers I-IV.Luminosities and absolute magnitudes are calculated for h = 0.7.We use AB magnitudes throughout the paper.The symbol "log" represents a base-10 logarithm.All uncertainties represent 1σ (68.3%) confidence intervals unless otherwise stated.

DATA
The data sets used in this study are drawn from the SDSS data releases 12 (Alam et al. 2015) and 14 (Abolfathi et al. 2018).The data were obtained as part of the SDSS-III (Eisenstein et al. 2011) Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013) using the BOSS spectrograph (Smee et al. 2013) on the 2.5 m SDSS telescope (Gunn et al. 2006).In the following subsections, we explain the sample selection of the CMASS (for "constant mass") SDSS sample (DR12), which serves as a tracer set of underlying dark matter density.We also give a description of the X-ray selected RASS/SDSS AGN (DR14) and optical SDSS AGN (DR14) samples.
The original spectroscopic CMASS sample can be downloaded from: https://data.sdss.org/sas/dr12/boss/lss/.For this work, we focus only on data from the SDSS North Cap (100 ≤ R.A. ≤ 270), exclude the South Cap, and use the galaxy catalog "galaxy DR12v5 CMASSLOWZTOT North.fits".
To minimize possible effects of redshift evolution in the clustering signal, we limit the sample in redshift range by using only spectroscopically identified CMASS galaxies with 0.44 < z < 0.64.To minimize systematic uncertainties and facilitate interpretation, we require the galaxy tracer set (CMASS) to exhibit a constant clustering strength over the full redshift range of interest.Galaxy clustering studies (e.g., Meneux et al. 2009, Law-Smith & Eisenstein 2017) show that the clustering strength depends on stellar mass in particular for red galaxies.Thus, restricting the stellar mass range will lead to a nearly constant clustering strength in a narrow redshift range.Comparat et al. (2017) performed spectral fitting of stellar population models on all SDSS spectra classified as galaxies in DR14 (including CMASS and LRGs) and derived galaxy physical properties; stellar masses in particular are relevant here.Fits are performed with the full spectral fitting code Firefly (Wilkinson et al. 2017; https://www.icg.port.ac.uk/firefly/) and the stellar population models of Maraston & Strömbäck (2011) (hereafter M11), which are provided for several choices of stellar initial mass functions and input stellar library.We use their results for the M11-ELODIE population model, calculated assuming the ELODIE stellar library (Prugniel et al. 2007) for a Salpeter stellar initial mass function (Salpeter et al. 1955).The CMASS stellar mass (M stellar ) as a function of redshift is shown in Fig. 1 (black and red data points).Initially, we limit the stellar mass to 11.25 < log(M stellar /M ⊙ ) < 11.43, which is a compromise between a narrow stellar mass range and maximizing the number of sources.The median stellar mass of this sample is ⟨log(M stellar /M ⊙ )⟩ = 11.33.
When we split the CMASS sample into low (0.44 < z < 0.54) and high (0.54 ≤ z < 0.64) redshift subsamples, the high-z CMASS sample shows on average a slightly higher clustering strengths on scales ≳1 h −1 Mpc.This is because the median value of the stellar mass increases slightly from z = 0.44 to z = 0.64.This results from an incompleteness in the CMASS sample stellar mass as a function of redshift as reported in, e.g., Leauthaud et al. (2016) and Rodríguez-Torres et al. (2016) (see their Fig. 3).To remove this bias, we compute the median stellar mass in bins of ∆z=0.01 over the redshift range of z = 0.44 − 0.64 and add objects either below or above the primary chosen stellar mass range in each individual bin until each redshift bin has an identical median stellar mass of ⟨log(M stellar /M ⊙ )⟩ = 11.33.This results in the selection of 148,686 spectroscopic CMASS galaxies, as show in red in Fig. 1.
As described in Appendix A we correct for fiber collisions in the spectroscopic sample when calculating the correlation function.This increases the number of objects in the final CMASS sample to 150,898 galaxies.We consider only the CMASS galaxy sample from the SDSS North Cap (∼6940 deg 2 ).The median redshift is ⟨z⟩ = 0.53 and the comoving number density is (8.00 ± 0.13) × 10 −5 h 3 Mpc −3 .Uncertainties on the number densities are determined using jackknife resampling, the details of which are given in Sect.4.2 below.

Random Catalogs
The computation of correlation functions requires a random catalog.
We use the random catalogs provided by the SDSS consortium (e.g., Reid et al. 2016).
The random catalogs for the CMASS sample in the North Cap can be found at https://data.sdss.org/sas/dr12/boss/lss/.As the data catalogs for the CMASS sample use weights to account for all observational biases (see Appendix A), the random catalogs can be used as provided.
For the random catalogs, we use 250 times as many objects as in the observed samples.This ensures that we minimize uncertainties due to statistical effects of the random catalog, in particular for pairs at the smallest scales measured.We verify that the randomly distributed objects fall within the same SDSS DR12 footprint as the observed CMASS galaxies.
As in papers I-IV, the corresponding redshifts for the random objects are drawn from the smoothed redshift distribution of the observed CMASS sample by applying a least-squares (Savitzky & Golay 1964) low-pass filter.The same jackknife subarea definition is used as that for the observed sample (see Sect. 4.2).

AGN Samples
The goal of this paper is to compare the clustering properties of X-ray and optically selected broad-line AGN samples.SDSS is perfectly suited for this purpose, as it included follow-up spectroscopy of extragalactic Xray sources and an optical AGN identification program based primarily on color selection.In addition, we aim to study AGN clustering properties as a function of various AGN parameters.The following subsections provide more details on the X-ray and optically selected AGN samples and how we create the different broad-line AGN subsamples.

X-ray RASS/SDSS AGN
The X-ray selected AGN sample is based on SDSS-IV DR14 SPIDERS (SPectroscopic IDentification of eROSITA Sources; Blanton et al. 2017).The X-ray telescope eROSITA (Predehl et al. 2021) on board of the SRG observatory (Sunyaev et al. 2021) was successfully launched in 2019 July.However, due to the delay of the launch, SDSS DR14 SPIDERS targets could not be selected using eROSITA.Instead the precursor ROSAT mission (Trümper et al. 1982) was used as the main input for target selection.The ROSAT All-Sky Survey (RASS, Voges et al. 1999) is an all-sky survey in the soft (0.1-2.4 keV) X-ray regime.Using this survey, Voges et al. (1999) and Voges et al. (2000) presented the RASS bright and faint source catalog, respectively.A goal of the ongoing SPIDERS project is to obtain highly complete and reliable optical identifications for these sources with the SDSS telescope.
We make use of SDSS DR14 and consider only X-ray sources that are listed in the revised version of the RASS catalog presented by Boller et al. (2016) (2RXS).Several major improvements and bug fixes compared to the previous version of RASS resulted in the deepest and cleanest X-ray all-sky survey collected by ROSAT.In particular, 2RXS aims for a significant reduction of the number of spurious sources.
We supplement the DR14 SPIDERS 2RXS sample (Coffey et al. 2019;Comparat et al. 2020) with RASS sources lying outside the eBOSS/DR14 footprint but within the footprint of earlier SDSS spectroscopic coverage.We cross-match the AllWISE counterparts to 2RXS sources (Salvato et al. 2018) to the SDSS-DR13 photometric catalog, choosing the brightest counterpart (in the r band) within a 3 arcsec radius of the AllWISE position.The optical counterparts are then matched to the SDSS DR14 optical spectroscopic catalog (Abolfathi et al. 2018) using a search radius of 1 arcsec.
The reported luminosities are 0.1-2.4keV k-corrected rest-frame luminosities, assuming a photon index of Γ=2.4.They are corrected for Galactic absorption.We restrict the X-ray selected AGN sample to the same redshift range as our CMASS galaxy sample (0.44 < z < 0.64) and to the same SDSS footprint.This yields a total of ∼2130 in the North Cap (100 ≤ R.A. ≤ 270).We apply an additional selection to this sample in Sect.3.1 below and obtain a final sample of 1701 broad-line X-ray AGN.

Optical SDSS AGN
The optical AGN sample is drawn from the SDSS Quasar Catalog DR14 (Pâris et al. 2018), which contains all AGN observed as part of SDSS I-IV.The newly discovered SDSS-IV/eBOSS AGN arise from the target selection presented by Myers et al. (2015).Pâris et al. (2018) defined an AGN as an object with a luminosity of M i [z = 2] < −20.5 that also has at least one emission line with FWHM > 500 km s −1 or has interesting or complex absorption features.Given the large number (526,356) of objects in the catalog, visual inspection of the optical spectra of these sources is not feasible.The automatic classification pipeline determines the redshift based on the maximum peak of the Mg II emission line, which is observable in eBOSS spectra from z = 0.3 − 2.5.
As with the X-ray selected AGN sample, we limit the optical AGN sample to the same SDSS footprint and redshift range (0.44 < z < 0.64) as our CMASS galaxy sample.This results in a total of 11,298 optical AGN in the North Cap.Note that we further down-select this sample in Sect.3.1 and obtain 10,994 broad-line optical

AGN.
There is a significant overlap between the X-ray and optical AGN samples: 95.7% of the X-ray selected AGN are also selected by the optical AGN selection.However, the two samples do reflect different AGN selection criteria and AGN populations.The RASS sample is essentially flux-limited.The optical AGN sample is based on a heterogeneous AGN selection based on a combination of nonstellar optical color selection and radio point-source detection.As the host galaxy can contribute significantly to the total optical emission, there is a selection bias against detecting low-luminosity AGN in galaxies where the optical host light is substantial.
Figure 2 shows the overlap between and differences in the two selection methods.The X-ray detected AGN have on average higher M i (z = 2) values than the average of the optical broad-line AGN (BLAGN) sample.However, there are also a substantial number of optical BLAGN with very high M i (z = 2) values that are not detected in X-rays: 30% percent of optical BLAGN brighter than M i (z = 2) = −24 mag are not detected in X-rays.For BLAGN brighter than M i (z = 2) = −25 mag, this fraction increases to almost 50%.Given the substantial differences in these samples, we aim to test whether the different BLAGN selections lead to differences in the clustering properties of the samples.

CREATING BROAD-LINE AGN SAMPLES
For the X-ray selected and optical AGN samples, we fit the individual SDSS optical spectra.In the redshift range of 0.44 < z < 0.64, the Hβ line is always in the available SDSS spectral wavelength range, except for objects in which this wavelength range is masked out based as a result of quality checks.We use the Hβ line properties to include or exclude an object in our broad-line AGN samples.In Krumpe et al. (2015) we used the Hα line to make this distinction as the AGN sample originated from a lower redshift range (0.16 < z < 0.36).Using the Hβ line properties not only allows for the generation of a broad-line AGN sample but is also used to estimate the M BH and L/L EDD .The details of the procedure are given in the following subsections.We show the individual model components, including a power law (black dashed line), the iron template (cyan line), the broad (blue) and narrow (magenta) Hβ lines, and the [O III] lines (green).

Hβ Spectral Line Measurements
We model the spectral region around the Hβ line for all SDSS optical spectra.For the X-ray sample, we visually inspect each fit to verify its robustness and improve the fit if necessary.For the larger optical AGN sample, we visually inspect a subset, focusing on potential problematic cases.
In particular, the X-ray sample includes not only broad-line AGN but narrow-line AGN, inactive galaxies, and stars as well.We use the spectral class given by the SDSS pipeline (Bolton et al. 2012) as an initial estimate of the nature of the X-ray source.For objects classified as QSO or BLAGN, we fit a broad-line model to the SDSS spectrum.In other cases, we first test if a potential line with a peak S/N> 3 per pixel is present at the location of Hβ.If so, we fit the spectrum with a narrow-line-only and a narrow+broad-line model.We use the latter model only if it leads to an improvement in the reduced χ 2 by at least 25%.This threshold is empirical and based on our visual quality checks of the fits.Otherwise we classify the object as a narrow-line object.
For the optical AGN sample, we initially fit every object with a broad-line model.For cases with a low amplitude of broad Hβ or cases identified during the visual inspection, we refit the spectrum with both a narrow-line and narrow+broad-line model and evaluate if the addition of a broad component is justified as described above.Based on this procedure, we identify 1701 and 10,994 objects that require a broad-line component in the fit in the X-ray and the optical AGN DR14 sample, respectively.
Our procedure for continuum and emission line fitting uses code originally presented in Schulze et al. (2017Schulze et al. ( , 2018)).It builds on a Levenberg-Marquardt least-squares minimization as implemented in MPFIT (Markwardt 2009).We first correct the spectra for galactic extinction using the extinction map from Schlegel et al. (1998) and the reddening curve from Cardelli et al. (1989) and shift the spectra to their rest frame.We then fit and subtract a local pseudo-continuum over the wavelength windows 4435 − 4680 Å and 5100 − 5535 Å.The model consists of a power-law continuum and an optical iron template (Boroson & Green 1992), broadened by a Gaussian.The pseudo-continuum subtracted spectrum is then fit with an emission line model over the range 4700 − 5100 Å .For the narrow-line model, we fit the narrow Hβ and the narrow [O III] λλ4959, 5007 lines, each fit by a single Gaussian.The [O III] λλ4959, 5007 doublet is fit with two Gaussians for each line, one for the line core and one for the blue wing often present in this line (e.g., Mullaney et al. 2013).The two [O III] lines are coupled in shape and their line ratio is fixed to 3.0.The line width and velocity centroids of the core components are tied together for Hβ and the [O III] lines.In the broad-line model we add a broad Hβ line, which we model with up to three Gaussians with FWHM> 1000 km s −1 each.We stress that we do use these to capture the often non-Gaussian profile shape of the broad Hβ line and do not assign any physical nature to the individual components.
We measure the FWHM of the broad Hβ line (corrected for the instrumental resolution) and the continuum flux at 5100 Å from the best-fit model.In Figure 3 we show spectra and the best fit models for six representative objects with typical S/N.For spectra with low S/N the uncertainties and systematic errors on the line width measurements can become substantial, compared to the systematic uncertainty of ∼ 0.3 dex of the virial method to estimate black hole masses.Therefore, we use a lower S/N threshold for the sample, similar to paper IV.We define the S/N over the Hβ range as the median S/N per pixel over the range 4750 − 4950 Å.In Appendix B we show that Hβ FWHM measurements with S/N>5 should be used to determine robust results and allow for a meaningful estimate of M BH .In Appendix C we compare our Hβ measurements with those from other studies and find good agreement.The broad-line X-ray and optically selected AGN samples used here (Hβ-S/N>5) have median S/N values of 14.1 and 10.8, respectively.

Estimating Black Hole Masses and Eddington
Ratios For broad-line AGN, black hole masses can be estimated using the viral method (e.g.McLure & Jarvis 2002;Vestergaard & Peterson 2006).This technique builds on the assumption of virialized motions of the broad-line region (BLR) gas (Peterson & Wandel 2000), where the broad-line width serves as an estimate of the gas velocity, and uses an empirical scaling relation between BLR size and continuum luminosity (Kaspi et al. 2000;Bentz et al. 2009).The typical uncertainty on M BH using the virial method for individual objects is ∼0.3-0.4 dex.The method is particularly powerful when studying the statistical properties of large samples of broad-line AGN (e.g.McLure & Dunlop 2004;Kelly & Shen 2013;Schulze et al. 2015).
The virial method is most directly calibrated for the Hβ line; thus, this line typically provides the most reliable black hole mass estimate.The Mg II line, which is covered by the SDSS spectra in our sample, is also known to be a reliable black hole mass estimator (Trakhtenbrot & Netzer 2012;Mejía-Restrepo et al. 2016).However, the Mg II mass estimator is calibrated to Hβ, so when both lines are available (as in our sample), Hβ is the preferred choice.We therefore do not fit the Mg II line for this study.We estimate black hole masses from Hβ using the formula given by Vestergaard & Peterson (2006): L 5100 10 44 erg s −1 For the broad luminosity range of our AGN sample, the host galaxy contamination to the continuum luminosity L 5100 is not negligible.Shen et al. (2011) showed that host galaxy contamination becomes significant at L 5100 < 10 45 erg s −1 .We account for the host contribution in an average sense by applying the formula for the average host contamination given by Shen et al. (2011).
We estimate the bolometric luminosity for our sample from the (host-corrected) continuum luminosity L 5100 .Specifically, we use a constant bolometric correction factor BC 5100 = 7.0 (Netzer & Trakhtenbrot 2007), which is consistent on average with the luminosity-dependent bolometric correction of Marconi et al. (2004) and excludes reprocessed emission from the mid-IR.Schulze et al. (2018) demonstrated the consistency of this bolometric correction factor with L bol obtained by direct integration of the spectral energy distribution (SED) and other common bolometric luminosity indicators.Combining our estimates of M BH and L bol provides the Eddington ratio L/L EDD for AGN in our sample, where We also compute the absolute magnitude as the z = 2 k-corrected i-band magnitude for both the X-ray and the optical AGN samples following Richards et al. (2006b) using the k-correction from their Table 4.This allows us to better compare the optical and X-ray AGN samples, as for the latter, no M i (z = 2) is provided.For the optical AGN we compare our M i (z = 2) values with those in Pâris et al. (2018) and find excellent agreement.In 68% of the sample, the magnitudes differ by less then 0.056 mag, and in 95% of the sample, they differ by less then 0.085 mag.

Defining Broad-line X-ray AGN Subsamples
Due to the relatively low number of sources, we decide to split the X-ray selected broad-line AGN sample into only two subsamples as a function of various AGN parameters.First we divide the sample into low and high X-ray luminosity subsamples at the median log (L X /[erg s −1 ]) = 44.82 of the full sample.The properties of all X-ray AGN samples as well as subsamples are given in Table 2.
The observed X-ray luminosity is a combination of the physical parameter M BH and L/L EDD .However, as in other redshift ranges (see paper IV), the RASS/SDSS AGN do not uniformly populate the L/L EDD -M BH plane (Fig. 4).Higher L/L EDD are usually found in AGN with lower M BH .The absence of objects in the lower-left corner of the plane is an observational bias.To remove the correlation between M BH and L/L EDD and test the clustering properties of each AGN parameter indepen- dently, we create subsamples that depend on one parameter only, while the distribution of the second parameter in both subsamples is identical.This "matching" of the subsamples is a commonly used method in clustering measurements (e.g., Coil et al. 2009) and was used in Paper IV. Thus we split the AGN sample into low and high M BH samples with identical distributions in L/L EDD and vice versa.To do so, we first remove extreme objects by considering only objects with −2.3 < log(L/L EDD ) < 0.0 for the M BH split and 7.4 < log(M BH /M ⊙ ) < 9.6 for the L/L EDD split.We then determine the number of objects in each bin in the parameter we are matching on, using a bin width of 0.1 for both parameters.Then in each bin, we split the sample at the median value of this bin and allow multiple draws of objects, to result in an identical number of objects in the low and high subsamples in each bin.As an example, we show the matched M BH division in Fig. 4. The L/L EDD distribution of the low and high M BH samples are identical.
We also split the sample at the median log (L Bol /[erg s −1 ]) = 45.43 and create low and high L Bol subsamples.Finally, we split the X-ray broad-line AGN sample into faint and bright M i (z = 2) subsamples using the median M i (z = 2) = −24.07mag of the full AGN sample with reliable M BH estimates.selected subsamples with similar redshift distributions, we limit the sample first at M i (z = 2) = −22.7 mag and M i (z = 2) = −25.4mag. Figure 5 shows the four resulting subsamples, each of which have a similar number of objects.We keep the objects with M i (z = 2) > −22.7 mag as an additional fifth M i subsample.The properties of all optical AGN subsamples are listed in Table 2.
When comparing the X-ray and optical broad-line AGN samples (see Fig. 6) in the space of the observed parameters L Hβ vs. FWHM (left panel) and the derived parameters L/L EDD vs. M BH (right panel), the two samples cover a similar though not identical parameter space.Unlike our previous lower-redshift samples (0.16 ≤ z ≤ 0.36; paper IV), here the X-ray selected AGN sample does not extend to lower L Hβ .Instead the optical selection is able to detect more objects with lower Hβ luminosities.ROSAT's flux sensitivity is limited, and so with increasing redshift, even luminous AGN will drop below the RASS flux limit.This is reflected in the total number of selected objects, as well as when directly comparing the number of objects with log (L Hβ /[erg −1 ]) ≥ 43.5, where we have 86 X-ray AGN and 141 optical AGN.
Differences in the observed parameter space naturally translate into differences in the derived M BH -L/L EDD space (Fig. 6, right panel).The X-ray AGN sample tends to select objects with somewhat higher M BH and higher L/L EDD .This is not surprising as the X-ray luminosity is increasing from the lower-left to the upper-right corner in this parameter space.The optical sample extends to lower M BH and lower L/L EDD .

Clustering Measurements
We measure the two-point correlation function ξ(r) (Peebles 1980), which measures the excess probability dP above a Poisson distribution.The auto-correlation function (ACF) measures the spatial clustering of objects in the same sample, while the cross-correlation function (CCF) measures the clustering of objects in two different samples.We use the same approach as described in detail in papers I-IV .Here we only repeat the most essential elements of our method.
We use the correlation estimator of Davis & Peebles (1983) in the form where DD(r) is the number of data-data pairs with a separation r, and DR(r) is the number data-random pairs.Both pair counts have been normalized by the number density of data and random points.For our purposes, the use of this simple estimator has a number of major advantages and results in only a marginal loss in the S/N when compared to more advanced estimators (e.g., Landy & Szalay 1993).The estimator in Equation ( 2) requires the generation of a random catalog for the tracer set only.
To separate the effects of redshift distortions, the correlation function is measured as a function of two components of the separation vector between two objects, one perpendicular to (r p ) and the other along (π) the line of sight.ξ(r p , π) is thus extracted by counting pairs on a two-dimensional grid of separations r p and π.We obtain the projected correlation function w p (r p ) by integrating ξ(r p , π) along the π-direction.
We measure r p in the range of 0.05-40 h −1 Mpc in 15 logarithmic bins, identical to those used in papers III and IV.We compute π in bins of 5 h −1 Mpc in the range π = 0 − 200 h −1 Mpc.To derive π max , we compute w p (r p ) for a set of π max ranging from 10-160 h −1 Mpc in steps of 10 h −1 Mpc.We then fit w p (r p ) over a r p range of 0.35-30 h −1 Mpc with a fixed γ = 1.9 (based on Krumpe et al. 2010) and determine the correlation length r 0 for the individual π max measurements.With increasing π max more clustering signal is included ,though at some π max value, the estimated w p (r p ) stops increasing and the uncertainty of w p (r p ) increases afterwards.For the CMASS galaxy ACF and all AGN-CMASS CCFs, the correlation signal saturates at π max,ACF = 80 h −1 Mpc and π max,CCF = 40 h −1 Mpc, respectively.Previous CMASS ACF studies also use π max,ACF = 80 − 100 h −1 Mpc (e.g., White et al. 2011;Guo et al. 2013;Nuza et al. 2013;Guo et al. 2015).Spectroscopic broad-line AGN redshifts have on average larger uncertainties than spectroscopic redshifts of galaxies.Typically the redshift uncertainties are δz ∼ 0.01.Since we are integrating over π max,CCF = 40 h −1 Mpc, these small redshift uncertainties have a negligible impact on the clustering signal.

Error Analysis
The error analysis is identical to that used in papers I-IV.We use the jackknife resampling technique to estimate the measurement errors based on the covariance matrix M ij , which reflects the degree to which bin i is correlated with bin j.
We divide the survey area into N T = 100 subsections, which have roughly an equal area of ∼70 square degrees.At the median redshift of the CMASS sample (⟨z⟩ = 0.55), each subarea spans a physical scale at least four times larger than our largest scales studied in this paper.
The N T jackknife-resampled correlation functions de- fine the covariance matrix: We calculate w p (r p ) N T times, where each jackknife sample excludes one section; w k (r p,i ) and w k (r p,j ) are from the kth jackknife samples of the CMASS ACF and AGN CCFs, respectively, and < w(r p,i ) > and < w(r p,j ) > are the averages over all of the jackknife samples.The uncertainties represent 1σ (68.3%) confidence intervals.
4.3.HOD Modeling HOD modeling is a popular method for interpreting correlation function results.In this model the sample objects are assumed to be in DMHs where the average number of objects per halo follows a parameterized function of halo mass separately for the central halo and satellite halos.The correlation function can be modeled as a sum of the contribution of pairs from the same DMH (one-halo term) and different DMHs (two-halo term).We interpret our results using the HOD modeling following an approach similar to that presented in papers II and IV as well as in Krumpe et al. (2018).We use the HOD approach to obtain linear bias parameters, as well as to investigate differences in the measured CCFs among various AGN subsamples with CMASS galaxies, probing differences beyond using the bias parameters alone.
As we do not intend to present full constraints of the AGN HODs in this paper, the HOD model of the AGN is deliberately designed to be simple.In order to enable historical comparisons with the results from our previous work, we use the same expressions for the b-M DMH relation (Tinker et al. 2005) and the halo mass function (Sheth et al. 2001), as in our previous work.In Krumpe et al. (2018), we use an improved version of our code that takes into account the effects of halo-halo collisions and scale-dependent bias (Tinker et al. 2005).Practically, this removes the need for excluding the transition range between one-halo and two-term-dominated regimes in the HOD modelings (see papers II-IV).
The outline of our approach is as follows: 1. We first apply the HOD modeling technique to the ACF of the CMASS galaxies and determine central and satellite HODs using a correlated χ 2 -fit with a model with four free parameters.The number density constraint is also included.The best-fitting parameter search and the associated confidence regions are generated using a Markov Chain Monte Carlo (MCMC) method.
2. In order to model the CCF between the CMASS galaxies and AGN samples, two HODs are required: the HOD of the CMASS galaxies derived in the previous step and that of the AGN.As the CMASS galaxy sample is much larger than the AGN samples (and therefore the statistical significance of the CMASS galaxy ACF is much higher than that of the CMASS galaxy-AGN CCF), we use the bestfit CMASS galaxy HOD derived above for calculating the uncertainties of the AGN HOD.Due to the lower S/N of the CCF measurements between AGN and CMASS galaxies, some parameters remain unconstrained when applying our HOD approach.We therefore fix these parameters to reasonable values.We use an MCMC parameter search when there are three free parameters to fit.If we fit only two free parameters, we apply a simple grid search.Details are given below.

HOD of CMASS Galaxies
For the HOD modeling of the CMASS galaxy sample, we use the five-parameter model of Zheng et al. (2007): where the model fit parameters are log M min , σ log M , log(M ′ 1 /M min ), and α s (satellite slope).Since M 0 is poorly constrained for our sample, we fix this to M 0 = 0, reducing the number of free parameters to four.For M 0 = 0 the convention is to use the symbol M 1 (which is the halo mass at which ⟨N s (M DMH )⟩ = 1) instead of M ′ 1 .We therefore use the variable M 1 hereafter.The MCMC points and confidence contour matrix of the CMASS HOD model parameters are shown, along with the marginal probability distribution of each parameter in the diagonal panels.The best fit value in a two parameter space is indicated with a red cross, while the best fit value for one parameter is marked with a red vertical line.Individual dots indicate the MCMC chain points in the corresponding two parameter space.The three contours in each panel correspond to ∆χ 2 = χ 2 − χ 2 min of 4.8, 7.9, and 12.1, below which 68, 90, and 98% of the chain points fall respectively.In these figures parameters name have been shortened for labeling purposes by omitting spaces and expressing log as lg.Best-fit parameters are searched in the range of 0.1 < r p [h −1 Mpc] < 40 and by minimizing the correlated χ 2 : where the quantities from the model are indicated by a superscript "mdl", M ij is the covariance matrix (Eq.3), n G is the number density of CMASS galaxies, and σ nG is the 1σ error.We find n G = (8.00±0.18)×10−5 h 3 Mpc −3 over 0.44 < z < 0.64, where the 1σ error is estimated using jackknife resampling.
The best-fit parameter search and determination of the confidence contours are made using an MCMC method with the MCMC-F90 library by Marko Laine 12 , which we have modified and linked to our HOD model calculation software.Table 1 provides the best-fit parameters and 12 http://helios.fmi.fi/lainema/mcmcf90/ their 90% confidence errors (or 95% upper limits).

HOD of AGN Samples
We fit HOD models to the CCFs between the CMASS galaxy and the AGN (sub)samples.To calculate the expected CCF w p (r p ), we use the best-fit HOD of the CMASS galaxies derived above and a model of the AGN HOD.For the AGN samples, we use the same HOD form as for the galaxy HOD, with an additional global normalization factor: where f A is the AGN fraction (duty cycle) among central galaxies at M DMH ≫ M min ; this global normalization factor scales both the central and satellite galaxies.For the AGN samples σ log M can be poorly constrained, 13 and therefore we fix this parameter to 0. In this case, the factor 1 2 1 + erf log MDMH−log Mmin σ log M becomes Θ(log M DMH − log M min ) where Θ(x) is a step function that has the value of 0 at x < 0 and 1 at x ≥ 0, respectively.Thus there are three free parameters for the AGN HODs: log M min , log(M 1 /M min ) and α.We note that the central HOD does not have to saturate at a constant value for AGN samples, unlike in the case of HODs for mass-or luminosity-thresholding galaxy samples, where it is usually assumed that the centers of the most-massive DMHs are occupied by a massive or luminous galaxy.Thus, in general, Eq. 6 may be too restrictive for an AGN HOD, and it would be ideal to introduce separate high-mass slopes for central and satellite AGN.A cosmological simulation-based forward model by Aird & Coil 2021 predicts that the central AGN HOD decreases at large halo masses in many AGN samples.Such cases cannot be fully represented by Eq. 6.If the central HOD does decrease with increasing halo mass, forcing to fit using Eq.6 could cause a misestimation of α s .However, using the simple form here provides a guide to the satellite HOD behavior and can highlight differences between different samples.
We do not use AGN density constraints for the χ 2fits and parameter searches, as the various observational biases in the AGN selection make the number density estimation very uncertain.The value of f A , which sets the global normalization of the AGN HOD, does not affect the resulting CCF.f A can be determined following the CCF fit using the AGN density constraint.
We apply the MCMC method for the AGN HOD parameter search to the CCFs between CMASS galaxies and the total broad-line X-ray and optical AGN samples.We repeat the analysis for objects with S/N at Hβ ≥ 5. We fit the range of 0.2 < r p [h −1 Mpc] < 25 and 0.1 < r p [h −1 Mpc] < 25 for the X-ray and optical AGN samples, respectively.The lower bounds are determined by the requirement that both samples have 15 pairs or more in the smallest r p bin.At r p [h −1 Mpc] > 25 redshift space distortions (RSD) substantially affect the measured w p (r p ) values.Measurements at r p [h −1 Mpc] > 25 have minimal contribution to the power-law fits used to determine π max as discussed above; therefore, we do not include them in the HOD model fits.The latest version of our HOD code accounts for RSD to the two-halo term based on the linear theory (Kaiser 1987) following the recipe by van den Bosch et al. (2013).Using it is computationally prohibitive for our MCMC chain searches.Instead we calculate the RSD corrected w p (r p ) model for the best-fit parameters to determine the potential level of impact (Sect.5.) 4.3.3.Luminosity, MBH, and L/LEDD AGN Subsamples One of the main goals of this work is to investigate the differences in AGN HOD properties as a function of AGN luminosity, M BH , L/L EDD , and M i , for which we create AGN subsamples.The constraints on M ′ 1 /M min are poor when fitting the full X-ray selected AGN sample.For the subsamples we therefore fix M ′ 1 /M min to be equal to 5, and we check the sensitivity of the results to different values of this parameter.This reduces the number of fit parameters to the most fundamental ones of interest, namely log M min and α.Fixing M ′ 1 /M min to 10 or 30 only modifies the large-scale bias in most cases by less than ±0.02.In the most extreme case, the change in bias is ±0.05.Even in this extreme scenario, this systematic error is well within the 1σ uncertainty range of the bias.
The purpose of reducing the number of fit parameters is two-fold.First, for these two parameters we do not need to use a computationally intensive MCMC for each subsample; we can use a two-dimensional grid of models in a tabular form to explore the confidence range.Second, with fewer free parameters we can highlight the differences in the HODs of the subsamples, as fixing the extra parameter may reduce the confidence regions in the two-parameter confidence contour plots by removing projection effects.As our purpose is to investigate the difference between samples, rather than obtaining full HOD constraints, comparing the confidence ranges in the plane of the two essential parameters, while holding other parameters fixed, is more demonstrative than comparing full confidence contour plot matrices in the full parameter space.

RESULTS
We compute the CMASS galaxy ACF within 0.44 ≤ z < 0.64 and at a median stellar mass of ⟨log(M stellar /M ⊙ )⟩ = 11.33 (which is constant across the full redshift range).The best-fit HOD model parameters, including the large-scale bias, are given in Table 1.The measured ACF and the HOD results are shown in Fig. 7.
We measure the CCFs between different X-ray and optical AGN samples and CMASS galaxies.As described above, we run the computationally intensive MCMC fitting for the full X-ray and optical AGN samples.For the AGN subsamples we determine the best-fit based on a search in a two-dimensional grid.Figure 8 shows the resulting CCFs (w p (r p )) and best fit HOD models, as well as residuals from the fits, for the SN H β > 5 samples.In each panel, the best-fit model corrected for the RSD effect is also shown, which demonstrates that the deviation of w p (r p ) at the largest r p can be accounted for by the RSD effect, though this effect is not included in the MCMC procedure due to the computational requirements.Confidence contour matrices for these results are shown in Fig. 9, with dots representing the corresponding MCMC chain.We present the best-fit values in Table 3.
The confidence contours can differ from the concentration of the MCMC chain points, as the contours are determined from the minimum χ 2 -values along the projections and not from the probabilities based on the marginal distributions of the MCMC points in the projected space.Results are also shown for the satellite fraction where dϕ/d log M DMH is the halo mass function.The larger uncertainties derived for the X-ray selected sample compared to the optical AGN sample are due to the smaller size of the X-ray sample.
Figure 10 (upper panel) shows the central and satellite HODs with 90% confidence ranges for the same samples as those in Fig. 8.The sum of central and satellite AGN HODs are shown with their 90% confidence ranges in hatches.The lower panel shows the sum of the central and satellite HODs multiplied by the halo mass function, again with 90% confidence ranges shown in hatches.
• As the DMH mass distribution for AGN is very wide, the best-fit typical DMH should be interpreted carefully.It is useful to compare the typical The best fit HOD model is shown with the one-halo (red dashed) term, two-halo (green dashed) term, and the sum of these terms (blue solid lines).The best fit two-halo term model with the RSD correction is shown with a green short-dashed line.

X-ray Selected AGN Sample
Optically Selected AGN Sample Fig. 9.-MCMC results with confidence contours for the parameters in Eq. 6 from the CCF HOD analysis of the broad-line X-ray (left) and optically selected AGN (right) samples.Constraints on the satellite fraction fsat are also derived from the model (see Eq. 7).In both cases we use the sample with a S/N ratio at Hβ of >5.The nomenclatures are the same as those in Fig. 7, except that ∆χ 2 -levels are 3.6, 6.2, and 10.4 for the left panel and 4.5, 7.7, and 10.9 for the right panel, respectively.These correspond to the 68, 90, and 98% of the chain points fall because of the non-Gaussian probability distributions.Note.-All samples span a redshift range of 0.44 < z < 0.64.For the HOD analysis of all (sub)samples presented in this table, we use a common, conservative fitting range of rp ≥ 0.9 h −1 Mpc.All bias values are computed at z = 0.53.Note.
-The quoted best-fit value is the 50th percentile (median) of the MCMC parameter distribution, and the uncertainties are 68% confidence values for one parameter, derived from the distribution of the MCMC chain.The fitting ranges for the full samples are 0.2 < rp < 25 and 0.1 < rp < 25 [h −1 Mpc] for the X-ray and optical AGN samples, respectively.An asterisk (*) for the error value indicates that it is pegged at a parameter search limit.Fig. 10.-Comparison of the HODs determined for the Xray (blue) and optical (red) AGN samples.Dotted, dashed, and solid lines show the 90% confidence ranges of the models in the MCMC chain evaluated at each log M DMH for ⟨N A,c ⟩(M DMH ), ⟨N A,s ⟩(M DMH ), and their sum, respectively.The total HOD constraint ranges are shown with hatches.Due to the functional form assumed for ⟨N A,c ⟩(M DMH ), the 5% and 95% percentile lines overlap with each other at high M DMH .Furthermore, the central, satellite, and total 90% limit lines also overlap at M DMH = M min (vertical part).The y-axis in the upper panel shows N (M DMH )/f A , such that large values may reflect a small f A .In the bottom panel the product of the HOD and the DMH mass function dϕ/d log M DMH for the X-ray (blue) and optical (red) AGN samples (central+satellite) is shown.The large difference in the satellite term between these samples at M DMH ∼ 10 13 h −1 M ⊙ is due to very different αs constraints and is discussed in Sect.5.1.
DMH masses of different AGN samples to look for clustering trends; however, it should not be interpreted to imply that most AGN reside in halos of this single mass.
• A positive value of α s > 0 implies that the number of AGN in satellite galaxies increases with increasing M DMH , as might be expected, as this is the behavior of the galaxy HOD.On the other hand, α s < 0 implies that AGN in satellite galaxies are preferentially found in the lower-mass halos (i.e., modest richness groups) and are rare in high-mass halos (i.e., in rich groups and clusters).
• By construction in the model log M 1 /M min becomes unconstrained for α s = 0.
• The two-halo term of the galaxy-AGN CCF is ∝ b Gal b AGN , where the bias value is an indicator of some weighted mean host DMH mass.
• There are two major constraints from the one-halo term: the amplitude of the clustering signal and the spatial extent of the one-halo-term-dominated region.The amplitude reflects the satellite-central and satellite-satellite pair counts within the same halos.The spatial extent reflects the maximum halo mass (i.e., the virial radius) that hosts a substantial satellite population.
• The value of f sat should not be over-interpreted, as it is highly model dependent, particularly the behavior of the model near M DMH ∼ M min where the halo mass function is large.

Findings from the AGN Grid Fitting Procedure
In addition to the MCMC method, we use the grid fitting procedure for all AGN (sub)samples.The lower r p value in each sample is determined such that there are at least 15 pairs in a bin.While for the full X-ray and optical AGN samples and most optical AGN subsamples, this is the case for r p ≥ 0.1 − 0.3 h −1 Mpc, for the X-ray AGN subsamples, this criterion is met for r p ≥ 0.9 h −1 Mpc.Thus we choose a conservative lower limit of r p = 0.9 h −1 Mpc for all AGN samples where we use the grid fitting method and provide the results in Table 2.In this table we list the number of objects per AGN (sub)sample, the median redshift of the AGN sample, median absolute i-band magnitude (k-corrected for z = 2), median X-ray luminosity (rest-frame 0.1-2.4keV, corrected for Galactic absorption), median black hole mass, median accretion ratio relative to Eddington, median bolometric luminosity, linear bias, typical DMH mass of the sample, and mean DMH mass.The constraints on the last three quantities are derived from the HOD modeling approach and use the above mentioned fitting ranges.All typical DMH masses of the sample and mean DMH masses are computed for z = 0.53.The constraints on the bias, the typical DMH mass (derived from the bias), and mean DMH mass (which mainly depends on the bias) all effectively result from the two-halo term.Thus, we can obtain constraints without using data in the one-halo-dominated regime (r p < 0.9 h −1 Mpc).
The results from the MCMC method show that the full broad-line X-ray and optical AGN samples have similar large-scale biases and HOD results.The largest difference is found for log(M 1 /M min ), though it is only 1.1σ when considering the combined uncertainties.In particular the exclusion of the AGN for which the M BH estimates are not as reliable does not significantly change the clustering results.
When comparing results from the MCMC and grid fitting approaches for the full X-ray and optical AGN samples, we obtain similar results (within the 1σ uncertainties).Differences arise due to having three free parameters in the MCMC method, while in the grid fitting method, two parameters are fit while the third is fixed at the same value for all samples.The grid fitting then results in tighter constraints on log M min and α s than the MCMC approach.
Using the grid fitting procedure, we find differences in the large-scale clustering properties of some X-ray AGN subsamples in excess of the combined 2σ uncertainties.There is a weak L X clustering dependence with the mean DMH masses, at a significance of 2.1σ, in that more X-ray luminous AGN cluster more strongly than their lower-luminosity counterparts.We also find a 2.1σ positive M BH clustering dependence in the X-ray AGN samples when considering the mean DMH masses.Among the optical AGN subsamples, we find a >2σ difference in the clustering properties with respect to L Bol (for details, see Sect.6.7).For all other parameters (e.g, L/L EDD , M i ), the clustering properties in the optical and X-ray subsamples agree within <2σ.In general, the mean DMH mass is a similar quantity as the typical DMH, but the mean has contributions from the one-and two-halo terms while the typical DHM reflects the large-scale bias value, which has contributions from the two-halo term only.

DISCUSSION
In discussing our findings we focus both on comparing our results with other studies as well as our previous work (in particular papers III and IV) to draw conclusions on the evolution of broad-line AGN clustering properties across a redshift range of z = 0.07 − 0.64.The upper value of this redshift range corresponds to a look-back time of 6 Gyr, almost half of the lifetime of the universe.Our studies provide four independent measurements across this redshift range.Since the X-ray and optical AGN samples are generated by the same instruments (ROSAT and SDSS) and the methodology used (cross-correlation approach & HOD modeling) is identical, systematic effects are expected to be minimized compared to considering other studies that use different samples and methods.However, there are some differences across our studies, including (i) our HOD analysis improved over the years, (ii) different galaxy tracer sets are used, and (iii) the AGN samples from ROSAT/SDSS differ in some properties across this wide redshift range (e.g., lower luminosities at low redshift compared to high redshift).

X-ray versus Optical AGN Clustering Properties
We first compare the clustering properties of the X-ray and optical AGN samples.Table 3 shows that the HOD parameters of the total X-ray and optical AGN samples agree well within the uncertainties, including for the samples with Hβ-S/N>5.Figure 9 shows that due to differences in the sample sizes, the optical AGN sample has much tighter constraints than the X-ray AGN sample.The confidence ranges of these two samples have some overlap in the covered parameter spaces.An illustration of the overlap of the X-ray and optical AGN HODs is given in Fig. 10.
Although the HOD results are not highly constraining due to the limited statistics (resulting from the relatively small AGN sample sizes), we are able to determine some salient characteristics.As shown in Fig. 6, the Xray AGN sample has on average higher-luminosity AGNs than the optical AGN sample, and the number density of the X-ray AGN sample is ∼ 1/5 of that of the optical AGN sample.As the two-halo terms for the two samples are consistent with each other, the "typical" DMH masses are also consistent.However, there are differences in the one-halo term clustering properties, as seen in Fig. 8.These differences are marginal, in that the 90% confidence ranges in the HOD parameter spaces as well as in HODs themselves overlap (Figs. 9 and 10).However, for the X-ray sample, the preferred χ 2 -values are in the α s < 0 region, while α s < 0 is almost excluded in the optical sample.As shown in Fig. 8, the one-halo term of the X-ray selected sample has a higher amplitude on small scales, and the scale at which the two-halo term dominates is lower.This points to higher M min and lower α s for X-ray AGN, where the AGN in satellites are more common in halos near M min and the occupation decreases for more-massive halos.
As discussed in Miyaji et al. (2011), possible mechanisms for low α s include the decrease of the crosssection of satellite-satellite mergers triggering AGN activity (Makino & Hut 1997;Altamirano-Dévora et al. 2016) in high-velocity encounters in massive halos and ram pressure stripping of cold gas in the intracluster medium.
One caveat in interpreting α s is that an underlying assumption of our HOD model is that the radial distribution of satellite AGN follows the Navarro-Frenk-White (NFW) halo mass density profile (Navarro et al. 1997) with the halo mass dependent concentration parameter of Zheng et al. (2007).It is possible that this assumption may not be valid for all AGN samples.An HOD model fit to the same data that assumes a different satellite radial profile would change the resulting α s .However, such a model is beyond the scope of this paper and may be addressed when larger AGN samples such as those based on eROSITA become available in the future.
As mentioned above, our results indicate that X-ray selected AGN, in contrast to optically selected AGN, contain a large population of satellites at 10 13 h −1 M ⊙ .This is reflected in the f sat distribution having higher values for the X-ray AGN sample.The marginal difference in the one-halo term behavior may be explained by X-ray selection identifying somewhat higher-luminosity AGN than optical selection.However, the difference is statistically marginal and therefore we conclude that X-ray and optically selected BLAGN samples show very similar clustering properties in the redshift range 0.44 ≤ z < 0.64.

X-ray versus Optical AGN Clustering as a
Function of Redshift In paper III we found no statistically significant difference in the clustering properties of X-ray and optical AGN at median redshifts of z ∼ 0.13 (cross-correlation with SDSS main galaxies), z ∼ 0.27 (cross-correlation with SDSS LRGs), and z ∼ 0.42 (cross-correlation with the most-luminous SDSS LRGs).The amplitude and spatial extent of the one-halo term region are virtually identical in these lower-redshift ranges for X-ray and optically selected AGN (see Figure 8 in paper III).Taking the results of all independent redshift measurements into consideration, it appears that there are very similar bias values for X-ray and optical BLAGN from very low redshift up to the redshift range studied of this work, with median z ∼ 0.53.
In each redshift range, the RASS X-ray selection and SDSS optical selection trace the full BLAGN population in different ways (see Fig. 2). Figure 11 shows that the X-ray and optical luminosities of AGN are correlated, though with substantial scatter.Thus, measuring clustering of optical AGN alone does not necessarily allow one to make conclusions about the clustering of X-ray selected AGN, and vice versa.
In this work, we find statistically marginal differences in the HOD parameters between the X-ray and optical AGN samples due to differences in the one-halo term clustering properties.This is reminiscent of similar differences seen between low-and high-L X AGN as well as type I and type II AGN in our previous work with the Swift/ BAT+INTEGRAL/IBIS nearby AGN sample (Krumpe et al. 2018).
Despite possible differences in the one-halo term for X-ray and optical AGN samples at low redshift, their two-halo term clustering (and therefore large-scale bias) agrees very well.This is in contrast with previous X-ray and optical AGN clustering studies at higher redshifts (z > 0.8), which generally find that X-ray AGN samples are hosted by more-massive DMHs than optical AGN samples at the same redshift (see Porciani et al. 2004;Coil et al. 2007;Ross et al. 2009;Gilli et al. 2005Gilli et al. , 2009;;Coil et al. 2009;Allevato et al. 2011;Eftekharzadeh et al. 2015;Laurent et al. 2017).The differences in these results are likely due to a combination of different methodologies used in the conversion from b DMH to M typ DMH in different studies, as well as differences in properties of the AGN samples (i.e., luminosity and BH mass) and in the properties of the galaxies hosting those AGN.Aird & Coil (2021) argued that AGN clustering results are most easily interpreted in terms of the relative bias of AGN to well-characterized galaxy samples and are more challenging to interpret in terms of the absolute bias value alone.Comparing the clustering properties of (inactive) galaxy samples with matched properties as observed in the AGN samples (e.g., in stellar mass, redshift range, and star formation rate) reveals whether AGN clustering can be explained purely by the known galaxy-halo connection or whether the presence of AGN in the center of a galaxy alters the clustering signal.Current measurements using this relative bias approach do find that the AGN clustering signal can be produced by inactive galaxies closely matched to the AGN in spectral class, stellar mass, and redshift (e.g., Mendez et al. 2016;Mountrichas et al. 2019;Krishnan et al. 2020).Measuring host galaxies properties such as stellar mass in the most-luminous BLAGN can be very challenging, making a relative bias measurement difficult.Alternatively, galaxy-galaxy lensing measurements can provide constraints on the M DMH of the individual AGN host galaxies.Semiempirical modeling suggests a halo mass distribution for X-ray selected AGN that peaks at M DMH ∼ 10 12 h −1 M ⊙ with a tail extending to higher halo masses (Leauthaud et al. 2015;Georgakakis et al. 2019).This tail shifts the average halo mass to higher values than M DMH ∼ 10 12 h −1 M ⊙ .In addition, the HOD modeling of the one-halo term should be improved to allow for distributions beyond a single power law.Equation 6 could be adjusted to allow for a more complex halo mass distribution of AGN.Ideally the model would allow for different slopes at different halo masses.However, the challenge for such models would be the limited S/N of current AGN clustering measurements, such that a more flexible model would result in poor constraints on additional HOD parameters.
6.3.L X Clustering Dependence of X-ray Selected AGN We find a weak and marginal clustering dependence on luminosity for X-ray selected BLAGN at a 2.1σ confidence level (0.44 < z < 0.64), in that higher-luminosity X-ray AGN have a higher mean DMH mass than lowerluminosity AGN.Interestingly, we previously discovered weak, positive, and marginally significant clustering dependences with L X in two lower-redshift ranges (0.07 < z < 0.16, 0.16 < z < 0.36).Considering the detection of a weak and tentative L X clustering dependence in three independent redshift ranges provides stronger evidence for the existence of this positive but weak correlation.We note that our X-ray selected AGN sample is drawn from a soft (0.1-2.4 keV) X-ray band, such that moderately X-ray absorbed AGN will be missed by our selection.In paper IV, we show with cosmological simulations that a weak L X dependence of the clustering is only expected for the most-luminous AGN while the more moderate or weak L X AGN are not expected to exhibit a L X dependence of the clustering properties.Aird & Coil (2021) also presented constraints on DMH mass distribution as a function of L X and have results for very luminous AGN that are consistent with our findings.
Figure 12 shows that there is not a simple continuous function of bias with L X that is independent of redshift; what is observed is that AGN with the same L X show different clustering strengths at different redshifts.Within any redshift range, the most X-ray luminous AGN have a similar high typical DMH mass of M DMH ∼ 10 13 h −1 M ⊙ .Within each individual redshift range, the lower L X AGN, being ∼5 times less luminous, have a typical M DMH ∼ 10 12.7 h −1 M ⊙ .There may be a tentative trend of lower DMH mass with increasing redshift for AGN at a given X-ray luminosity.
To further investigate this observation, we compare the number density of AGN at a given luminosity with that of DMHs at the corresponding typical halo mass.The ratio of these densities represents a "duty cycle", or the fraction of halos that contain an AGN.In principal, if a halo contains more than, e.g, two AGN at the same time, these count as two.However, as seen below, the duty cycle is much less than one, and such cases are rare.Here we compare the AGN duty cycles between the highestand lowest-redshift bins shown in Fig. 12.We estimate the duty cycle by: i.e., the ratio between the X-ray luminosity function (XLF) and the corresponding DMH mass function.The XLF (d ϕ X /d log L X ) is obtained from the 0.5-2 keV XLF (Miyaji et al. 2001;Hasinger, Miyaji, & Schmidt 2005) converted to our 0.1-2.4keV band assuming a photon index of Γ = 2.4.The DMH mass function (d ϕ DMH /d log M DMH ) is from Sheth et al. (2001) and is implemented in our HOD code.The factor |d log L X /d log M DMH | is estimated from the slope for the given redshift bin in Fig. 12.For our z ∼ 0.53 sample with M DMH ∼ 10 12.8 h −1 M ⊙ , the estimated duty cycle is ∼ 1 × 10 −3 , i.e., 0.1% of the DMHs at this halo mass could contain a BLAGN.Using the DMH mass growth rate of Fakhouri, Ma, & Boylan-Kolchin (2010), the descendants of halos of mass M DMH ∼ 10 12.8 h −1 M ⊙ at z ∼ 0.53 have M DMH ∼ 10 13.0 h −1 M ⊙ at z ∼ 0.13.The duty cycle for these descendants (at z ∼ 0.13) is estimated to be ∼ 2%.Thus, roughly speaking, the duty cycle represented by our high-L X z ∼ 0.13 sample at M DMH ∼ 10 13.2 h −1 M ⊙ DMHs has increased by an order of magnitude compared to their progenitor DMHs at z ∼ 0.53.
There are caveats in interpreting our results in this manner.We assume that the observed slope of the log M DMH -log L X relation at each redshift range can be extended to unprobed luminosities.The error on the slope is large, and it is not guaranteed that the observed relation applies beyond the observed luminosity range.In particular, at z ∼ 0.53, AGN with a luminosity of L 0.1−2 keV ≲ 10 44 h −2 70 erg s −1 are below the RASS flux limit and thus not included in our sample.Those AGN may well reside in ≳ 10 12.5 h −1 M ⊙ DMHs.This would result in a smaller |d ϕ DMH /d log M DMH | and thus a larger duty cycle.Future analysis of AGN samples collected with eROSITA will allow us to explore the typical DMH masses of luminosity-and redshift-defined AGN samples to compare the differences in duty cycles in this space with unprecedented precision.6.4.M i Clustering Dependence of Optically selected AGN We do not find a clear clustering dependence of the optically selected AGN with M i (see Table 2).The subsample with −23.25 < M i < −23.70 deviates the most from other subsamples but at a difference of only ∼1.4σ.Other than the M i > −22.7 mag subsample, all samples have similar redshift distributions.To remove differences in the redshift distributions completely, we create redshift-matched subsamples (except for the subsample M i > −22.7 mag) and analyze their clustering properties.The bias values for these samples vary within their 1σ uncertainties, such that no significant correlation is found.Considering the relatively small uncertainties of the optical sample bias measurements, there is little room for even a weak clustering dependence with M i .Shen et al. (2013) also explored the M i clustering dependence of quasars using cross-correlation measurements between DR7 (optical) quasar catalog and DR10 CMASS galaxies.Their data are also consistent with no significant luminosity (M i ) dependence.One difference is that they analyze a much wider redshift range from 0.3 < z < 0.9, while we focus on a much narrower redshift range to exclude redshift evolution of the clustering properties as well as having better control of the systematics (e.g., constant stellar masses for CMASS galaxies).In addition, they also select their M i subsamples differently.The advantage of these slightly different approaches, while finding the same (non)result, strengthens the robustness that there is no M i clustering dependence for optical AGN.
We also did not find an M i clustering dependence in our previous work (paper III) in the redshift ranges 0.16 < z < 0.36 and 0.36 < z < 0.50.In the lowest redshift range (0.07 < z < 0.16) we were not able to test for such a dependence due to the low number of objects.Thus, we conclude that from low to intermediate redshifts, there is no M i clustering dependence for optically selected AGN.This is in contrast to the result that in all of these redshift ranges we do find evidence for a weak L X dependence in the X-ray selected samples.In paper IV, we gave as a possible explanation the complex sample selection for the optical SDSS sample (e.g., host galaxy/AGN separation).However, in the redshift range studied here this effect should be minor.In the logical chain of correlations between M DMH → M stellar → M bulge → M BH → M i there is a scatter at each stage, which sums across the chain.It is possible that the last step in the chain has a higher scatter than that of M BH → L X , as for L X we consistently detect a positive but weak clustering dependence and we find no dependence with M i .Another potential explanation is that there are different dependencies of M stellar -L X (e.g., Carraro et al. 2022) and M stellar -M i .

M BH Clustering Dependence and Its Evolution with Redshift
Fig. 13.-Left:Typical DMH mass (derived from HOD modeling and grid fitting approach) as a function of M BH at several independent redshifts (shown with different symbols) for X-ray (red symbols) and optical (blue symbols) AGN samples.Right: Similar to the left panel, here as a function of the Eddington ratio.We show the results in the redshift range 0.16 < z < 0.36 with dotted lines to allow for a clear distinction of the different samples.
In the redshift range of 0.44 < z < 0.64 studied here, we find only a tentative positive correlation of M BH with the clustering strength for the X-ray AGN sample (1.3σ when comparing b values, 2.1σ confidence when comparing mean DMH masses).Despite the much larger sample size of the optical AGN, the largest difference between the M BH subsamples is only 1σ when comparing their bias values and 1.4σ when comparing mean DMH masses.This result is not sensitive to the specific cuts used in creating these subsamples or to reducing the number of subsamples (e.g., two subsamples only).Figure 6 (right) and Table 2 show that the X-ray and optical AGN samples span a similar parameter space in M BH .The samples differ only slightly in L Hβ in that X-ray AGN have on average higher L Hβ than optical AGN (Fig. 6,left).Thus if a very weak M BH dependence of the clustering strength exists, it remains unclear why the optical AGN samples do not reveal this correlation as the X-ray selected AGN samples do.
In Fig. 13 (left), we summarize the DMH mass dependence on M BH measured in different redshift ranges for X-ray and optical BLAGN samples.At 0.07 < z < 0.16, there are too few optical AGN to compute a robust clustering signal.In this redshift range, there is a clear signal for X-ray AGN; however, we are not able to create subsamples due to the relatively low number of objects.Thus in Fig. 13 we show only the measurement for the total X-ray AGN sample in the redshift range 0.07 < z < 0.16.
There is not a single M BH versus DMH mass correlation across all redshifts; i.e., increasing M BH does not necessarily result in higher DMH mass.Instead, we find that within each redshift range the DMH mass ranges are very similar across a range of M BH , consistent with a typical M DMH ∼ 10 12.7−13.3h −1 M ⊙ .For the redshift ranges in which M BH subsamples can be created (0.16 < z < 0.36 and 0.44 < z < 0.64), in both cases there is a weak positive correlation between M BH and the typical DMH mass for X-ray selected AGN.For the optical AGN, no clear trend is found.Within a given redshift range, measurements between the X-ray and optical selections with similar M BH agree well, within their uncertainties.
6.6.L/L EDD Clustering Dependence and Evolution with Redshift We do not find a correlation between L/L EDD and typical DMH mass in the redshift range 0.44 < z < 0.64; all subsamples, for both X-ray and optical AGN, agree in the typical and mean DMH masses within their combined uncertainties at less than 1.3σ.In Fig. 13 (right) we show the measurements for all redshift ranges.The results show that across all redshift ranges studied, there is no clear clustering dependence with L/L EDD and that at a given L/L EDD the measurements from different redshifts agree well.Determining any L/L EDD clustering dependence and its evolution with redshift for a hard X-ray selected sample would provide a more complete picture than the current sample, as it would mitigate many observational biases (e.g., soft X-ray selection for our AGN sample).

L Bol Clustering Dependence
We do not find a correlation between L Bol and typical DMH mass in the redshift range 0.44 < z < 0.64 for the two X-ray selected AGN subsamples.However, in the same redshift range we find a > 2σ L Bol dependence of the clustering strength in three of the four optical AGN subsamples.However, the highest L Bol subsample (median log (L Bol /[erg s −1 ]) = 45.67) has almost identical clustering properties as the lowest L Bol subsample (median log (L Bol /[erg s −1 ]) = 44.80).Although the X-ray selected AGN sample has a smaller dynamic range in L Bol , there is overlap with the L Bol range of the optical selected AGN sample.Within these individual subsamples we do not find a statistically significant discrepancy between the optical and X-ray selected AGN.As the L Bol results taken together do not present a consistent picture of a clear dependence, we refrain from interpreting a possible trend to the data.
Unfortunately no observational constraints on the L Bol dependence of AGN clustering exists from crosscorrelation measurements at other redshifts.In the future, larger samples will be needed to more fully study the L Bol clustering dependence at different redshifts.

CONCLUSIONS
In this paper, we extend our clustering studies of X-ray (ROSAT /SDSS) and optically (SDSS) selected broadline AGN to the redshift range of z = 0.44 − 0.64.As in our previous work, we use a cross-correlation approach with a large set of tracer galaxies (here the CMASS galaxy sample) to substantially improve the S/N of the AGN clustering measurement.We define a CMASS galaxy sample to (i) have an identical average stellar mass (and thus clustering strength) over the entire redshift range used in this paper, and (ii) explore the AGN clustering properties down to scales of r p = 0.1 h −1 Mpc.We estimate for the X-ray and optically selected broadline AGN samples the supermassive black hole mass, accretion ratio relative to Eddington, and the bolometric luminosity based on spectral fits to the Hβ line.We split the AGN samples by the physical properties of luminosity in the selection bandpass, M BH , L/L EDD , and bolometric luminosity.We apply HOD modeling directly to the cross-correlation measurements to infer the AGN distribution as a function of DMH mass.We also determine the HOD parameters for the CMASS galaxy sample.For the full X-ray and optical AGN samples, we apply an MCMC HOD parameter search, while for the AGN subsamples, we apply a grid search to estimate the large-scale bias and typical and mean DMH mass of each sample.
The HOD constraints on the optical BLAGN sample are tighter than those for the X-ray BLAGN sample, due to a larger sample size.X-ray and optical BLAGN have the same large-scale bias and thus extremely similar typical DMH masses (logarithmic DMH masses of 12.87 +0.11  −0.19 h −1 M ⊙ for the X-ray AGN sample and 12.72 +0.06 −0.06 h −1 M ⊙ for the optical AGN sample).While this reflects the typical DHM mass, the full range of halo masses occupied by AGN is very broad (e.g., Georgakakis et al. 2019;Aird & Coil 2021).There is marginal statistical evidence that the one-halo clustering properties between the X-ray and optical AGN samples are different.In contrast to optically selected AGN, X-ray selected AGN may have a larger population of satellite galaxies at M DMH ∼ 10 13 h −1 M ⊙ , indicated by the higher satellite fraction and lower α s (satellite slope) found for the X-ray AGN HOD compared to the optical AGN HOD.
When we compare results as a function of AGN properties, we do not find statistically significant correlations with L/L EDD and M i .For the X-ray selected AGN sample, we find a positive correlation between mean DMH mass and L X as well as with M BH at a > 2σ confidence level.The optical AGN sample does not show a significant correlation between mean DMH mass and M BH , but the measured values for the different optical subsamples also do not contradict the M BH versus mean and typical DMH mass correlations found for the X-ray se-lected AGN sample.Within the optical broad-line AGN sample, we find some dependencies at > 2σ confidence level in the clustering properties as a function of L Bol .However, the lowest L Bol and highest L Bol samples have very similar clustering properties, so a clear trend is not present.
We compare our results with those obtained in our paper III, in which we evaluate the clustering and HOD properties of X-ray (RASS/SDSS) and optically selected (SDSS) BLAGN in three lower-redshift ranges using different SDSS galaxy tracer sets.The full redshift range of z = 0.07 − 0.64 covered by all of these studies spans a cosmic time interval of 5 Gyr.In all redshift ranges we use the same cross-correlation technique to determine the AGN clustering properties, create samples from the same large area surveys (RASS and SDSS), and evaluate the results with similar HOD modeling.We find that X-ray and optically selected BLAGN, despite their different optical and X-ray luminosities at different redshifts, occupy DMHs with M typ DMH ∼ 10 12.5−13.0h −1 M ⊙ across this full redshift range.In other words, we find no statistically significant difference in the typical DMH masses between X-ray and optically selected BLAGN samples.However, at higher redshift, the same DMH mass hosts more X-ray luminous AGN than at lower redshift.
Semianalytic cosmological simulations find that an L X and M BH dependence of the AGN clustering exists, but only for the most-luminous AGN and not for lowor moderate-luminosity AGN.Since our X-ray selected BLAGN sample contains the most-luminous objects in independent redshift bins, we can test this prediction.From z = 0.07 to z = 0.64, we also find weak positive correlations between the typical DMH mass and L X for the X-ray BLAGN samples.In the redshift ranges z = 0.16−0.36and z = 0.44−0.64we can estimate M BH , and we have enough objects to create subsamples in M BH and L/L EDD .We find that the weak positive L X dependence of the AGN clustering appears to be due to a weak positive M BH dependence of the clustering signal, in that more X-ray luminous (and larger M BH ) BLAGN reside in more-massive DMHs than their lower X-ray luminosity (lower M BH ) counterparts.The observed clustering strength across the full redshift range does not depend on L/L EDD .Consequently, higher accretion rate AGN do not reside in more dense environments.
The number of RASS-detected AGN decreases sharply below z ≲ 0.07 and above z ≳ 0.6, due to limited volumes at low redshift and ROSAT's flux sensitivity.Consequently, the redshift range for high-S/N clustering studies using RASS-selected AGN cannot be extended further.However, the X-ray telescope eROSITA (Predehl et al. 2021) is a game changer for AGN studies.eROSITA is far more sensitive than ROSAT and scans the entire sky several times at X-ray wavelengths.The resulting stacked X-ray data will reach a flux limit ∼30 times more sensitive than RASS.This will provide unprecedented large AGN samples for clustering studies to improve constraints on dependences with redshift, L X , and M BH , as well as to test if the one-halo properties of X-ray and optically selected AGN differ.Recently the German eROSITA team published its first AGN clustering paper (Comparat et al. 2023) using several thousand AGN detected in the ∼140 square degree eROSITA Final Equatorial Depth Survey field (Brunner et al. 2022;Liu et al. 2022).The spectroscopic AGN samples from the eROSITA all-sky survey scans will contain hundreds of thousands of objects and will thus bring AGN clustering studies to a new level.fiber and why this target was selected (i.e., which object class).As described above, not all of these targets can be spectroscopically observed due to fiber collisions.The target catalog encodes information about why an object was targeted for spectroscopic follow-up observation using target IDs 14 .These targets IDs are bitmask values.We identify all photometric target IDs in the North Cap that lead to spectroscopically confirmed CMASS objects.Of those, we consider only IDs that lead to at least 100 spectroscopically confirmed CMASS galaxies.For the resulting target IDs (001, 134, 135), we compute the fraction c of spectroscopic CMASS targets with each of these IDs over photometric targets with these IDs (c 001 = 0.253, c 134 = 0.872, c 135 = 0.008).The meaning of these number is, e.g., that 25.3% of photometric targets with ID 001 led to spectroscopically confirmed CMASS objects.
Then we calculate the redshift distribution (f ID target (z)) of spectroscopic CMASS galaxies with these IDs in the range 0.44 < z < 0.64 (bin size ∆z = 0.01).Figure 15 (left) shows that spectroscopic CMASS galaxies with ID 134 are by far the dominant population.In each redshift bin, the sum over all three target IDs is 1.Thus this provides the normalized relative contribution of each target ID as a function of redshift.
Next we determine from Fig. 1 the fraction f selected/all (z) of spectroscopic CMASS objects that fulfill our final selection criteria of 0.44 ≤ z ≤ 0.64 and stellar mass selection (red data points) relative to the full spectroscopic CMASS sample (red+black data points) in as a function of redshift.The result is shown in Fig. 15 (right).With this information in hand, we are able to perform a statistical correction for the fiber collisions based on the photometric target catalog.To recover the small-scale clustering signal, we compute, for each photometric galaxy that has a fiber collision with a spectroscopic CMASS object, the probability that the photometric source (based on its photometric target ID and the redshift) should be assigned the redshift of the spectroscopic CMASS object and include it in the final sample as follows: P (ID, z) = c ID target × f ID target (z) × f selected/all (z) − P spur (A1) In practice, we search for all photometric targets that have IDs with the values 001, 134, or 135 within a radius of 62 arcsecs of each selected spectroscopic CMASS galaxy.Photometric targets within the search radius that already have spectroscopic data are rejected.If the spectroscopic CMASS galaxy has w CP > 1, we use the redshift of the spectroscopic CMASS object that the object collides with and calculate the probability P (ID, z) according Eq.A1.We then draw a random value n random between 0 and 1 for each colliding target.If n random ≤ P (ID, z), we consider this object for our collision correction further; otherwise, we reject it.The term P spur corrects for spurious matches.The number density per sky area for the photometric targets with the CMASS IDs is 8.78 × 10 −6 arcsec −2 .We calculate the distance between the spectroscopically confirmed CMASS object and the collided object.Using a simple circular area calculation (distance = radius) and considering the number density provides the chance that the collided object will spuriously fall at its observed position.
If n random ≤ P (ID, z), we adopt the R.A. and decl. of the photometric object and assign it the same redshift as the nearby spectroscopic CMASS galaxy.Finally, we recompute w CP for the spectroscopic CMASS galaxy by w CP,NEW = w CP,OLD − 1 if the collided object ends up being in our sample.This approach is repeated for the same spectroscopic CMASS galaxy if it collides with more than one object until w CP > 1 is not fulfilled anymore.If more than one object collides with a spectroscopic CMASS galaxy, we sort all objects by distance from the spectroscopic object and start by calculating (thus potentially correcting) the closest photometric object first.This approach accounts for the fact that close neighbors have a substantially higher probability of having similar redshifts.
Our approach for correcting the fiber-collision issue in the CMASS sample has several advantages: (i) it does not affect the large-scale clustering above ∼0.5 h −1 Mpc, because for each collision-corrected object we include in our final sample, we reduce the weight (w CP ) by 1, (ii) the resulting clustering signal is very similar in shape and amplitude to that measured for the full CMASS sample from SDSS DR10 by Guo et al. (2013) (0.43 < z < 0.7), and most importantly for our cross-correlation method (iii) when split into lower-and higher-redshift subsamples (0.44 < z < 0.54 and 0.54 < z < 0.64, respectively), the clustering signal has minimal differences, well within the expected statistical uncertainties (Fig. 14, right).We additionally tested various approaches using photometric redshifts to correct the small-scale clustering signal.However, these approaches resulted in strong discrepancies between the low-and high-redshift CMASS samples at small scales.

B -RELIABILITY OF THE Hβ BROAD-LINE FIT
In order to assess the reliability of our broad-line fits, we carry out Monte Carlo simulations using 21 objects in the Xray catalog with S/N> 50, stepwise degrading their S/N by adding random Gaussian noise, fitting the degraded spectra with our fitting routine, and measuring the FWHM from the best fit to the degraded spectrum.For each AGN, we perform 100 simulations per S/N bin.Following paper IV, we require an uncertainty on the Hβ FWHM measurement of less than 40% as above that value the uncertainties exceed the commonly assumed systematic uncertainties for viral M BH estimates of ∼0.3 dex.As shown in the upper panel in Fig. 16 at S/N=5, roughly 90% of our targets fulfill this requirement on the recovered FWHM.We therefore use S/N= 5 as our default threshold for spectra for which we determine M BH , and we verify that our results do not depend on the precise S/N value chosen.Our results agree with previous work, suggesting that at an S/N below ∼ 5 the uncertainties on FWHM dominate the M BH measurement (e.g., Denney et al. 2009;Shen et al. 2011).The typical FWHM uncertainty at S/N= 5 is ∼ 0.06 dex, as shown in the bottom panel of Fig. 16.After applying the S/N> 5 threshold, we have a sample size of 1632 for the X-ray AGN sample and 8889 for the optical AGN sample.

C -COMPARING OUR Hβ FIT RESULTS TO LITERATURE MEASUREMENTS
Our sample overlaps with previous studies of the spectral properties of AGN in the SDSS.We compare our Hβ measurements with other studies, to investigate the consistency of our results with previous work.In particular we cross-match our optical AGN sample (DR14) with the SDSS DR7 quasar catalog of Shen et al. (2011) and our SPIDERS X-ray selected AGN sample with the recent study by Coffey et al. (2019).The SDSS DR7 quasar catalog is a subset of the DR14 quasar catalog by Pâris et al. (2018) with 5764 objects in common with our optical DR14 sample.In the upper panels of Figure 16 we compare the FWHM and L 5100 measurements of Shen et al. (2011) to ours.We find an excellent agreement overall, with median differences in log FWHM = 0.02 and log L 5100 = 0.00 and a standard deviation of 0.12 dex and 0.04 dex, respectively.Coffey et al. (2019) present an M BH catalog for SPIDER AGN from the 2RXS catalog with 1124 AGN in common with our X-ray sample.We show this comparison in the lower panels of Figure 16.For the FWHM measurement, we find a similar agreement as in the comparison with Shen et al. (2011), providing an estimate of the systematic uncertainties of the line width measurements.For the continuum luminosity, L 5100 , we find a consistent mode, but there is a tail with lower L 5100 in the Coffey et al. (2019) catalog.This is caused by the inclusion of a host galaxy template in the fitting procedure of the continuum in Coffey et al. (2019).We do not explicitly account for the host galaxy contribution in our fitting routine, since for our sample it is   (2011).Lower panels: Comparison of our X-ray sample with the SDSS/SPIDERS (2RXS) catalog by Coffey et al. (2019).
often difficult to break the degeneracy between host galaxy spectrum and AGN continuum emission.We account for the effect of host galaxy contamination to L 5100 in a statistical fashion, as detailed in Section 3.2.
The differences between our method and those of Shen et al. (2011) and Coffey et al. (2019) lead to uncertainties in the M BH estimates that are lower than the systematic uncertainties of ∼0.3 dex; thus, we conclude that our line fitting routine provides robust measurements of the Hβ region spectral properties and enables the estimation of black hole masses based on these measurements.

Fig. 1 .
Fig. 1.-Stellar mass vs. redshift for the full spectroscopic CMASS sample (black data points) and the CMASS subsample (red) used for cross-correlation measurements with AGN.

Fig. 4 .
Fig. 4.-M BH vs. L/L EDD for broad-line X-ray RASS/SDSS AGN with S/N at Hβ ≥ 5. Blue symbols show the high M BH sample, while red symbols show the low M BH sample (the two samples have identical distributions in L/L EDD ).Vertical lines mark the restriction of the parameter space to remove extreme objects.

Fig. 5 .
Fig. 5.-Absolute i-band magnitude (k-corrected to z = 2) vs. redshift for the broad-line optical AGN with S/N at Hβ ≥ 5.The different colors represent the different M i (z = 2) subsamples.The few brightest objects with M i (z = 2) < −25.4 mag are not considered for the M i (z = 2) subsamples.

Fig. 6 .
Fig.6.-Comparison between broad-line X-ray selected RASS/SDSS AGN (black lines) and broad-line optically selected SDSS AGN (red lines) with S/N at Hβ > 5.The contours show the location of 10% (dashed lines), 50% (solid lines), and 90% (dotted line) of the full sample, respectively.The left figure shows the observed parameters of the luminosity and FWHM of the Hβ line, while the right figure shows the derived parameters L/L EDD and M BH .In both figures a boxcar smoothing with a width of two has been applied.

Fig. 7 .
Fig. 7.-Left: The CMASS galaxy ACF (black circles) with 1σ error bars is shown, along with the best fit HOD model (blue solid line).The one-halo and two-halo terms of the HOD model are shown with red short dashed and green long dashed lines, respectively.The lower panel shows the fit residuals in terms of the data/model ratio.Right: The MCMC points and confidence contour matrix of the CMASS HOD model parameters are shown, along with the marginal probability distribution of each parameter in the diagonal panels.The best fit value in a two parameter space is indicated with a red cross, while the best fit value for one parameter is marked with a red vertical line.Individual dots indicate the MCMC chain points in the corresponding two parameter space.The three contours in each panel correspond to ∆χ 2 = χ 2 − χ 2 min of 4.8, 7.9, and 12.1, below which 68, 90, and 98% of the chain points fall respectively.In these figures parameters name have been shortened for labeling purposes by omitting spaces and expressing log as lg.

Fig. 8 .
Fig.8.-The measured cross-correlation function between the X-ray selected (left) and the optically-selected AGN sample (right) with the CMASS galaxy sample.In both cases we show the AGN samples corresponding to "BL, Hβ-S/N>5" in Table2.The data points indicated by open circles and x-points represent the measurements included and excluded from the HOD model fitting process, respectively.The best fit HOD model is shown with the one-halo (red dashed) term, two-halo (green dashed) term, and the sum of these terms (blue solid lines).The best fit two-halo term model with the RSD correction is shown with a green short-dashed line.

Fig. 12 .
Fig.12.-TypicalDMH mass (derived from HOD modeling and grid fitting approach) as a function of 0.1-2.4keV luminosity in independent redshift ranges (shown with different colors).Data points with the solid error bars show the full samples, while the data points with dashed error bars displays the respective subsamples split by X-ray luminosity.

Fig. 14
between the ACFs of a CMASS sample with 0.44 < z < 0.64 and 11.25 < log(M stellar /M ⊙ ) < 11.43 when split into low-and high-redshift subsamples, before applying our correction algorithm.In addition we show the fullGuo et al. (2013) CMASS sample which utilizes a different approach to correct the small-scale clustering.Right: similar to the left panel showing the lowand high-redshift CMASS subsamples, here after our correction algorithm is applied.

Fig. 15 .
Fig. 15.-Left: fraction of different target IDs (bitmask values) to the full spectroscopic CMASS sample as a function of redshift (SDSS North Cap).Each line represents a different target ID whose tag is stated directly next to the corresponding line.Right: fraction of spectroscopic CMASS sources that fulfill our final selection criteria to the full spectroscopic CMASS sample as a function of redshift.

Fig. 16 .
Fig. 16.-Left: Upper panel: Simulation of the recovery rate of the intrinsic FWHM of Hβ within a 40% uncertainty as a function of spectral S/N, based on Monte Carlo simulations for 21 AGN spectra.Lower panel: Dependence of the mean uncertainty on the FWHM measurement as a function of S/N.The gray areas show the 68% (1σ) uncertainty range based on the simulated spectra with degraded S/N properties.Right: Comparison of the FWHM (left panels) and L 5100 measurements (right panels) between our samples and those from the literature.Upper panels: Comparison between our optical DR14 AGN catalog and the SDSS DR7 quasar catalog by Shen et al.(2011).Lower panels: Comparison of our X-ray sample with the SDSS/SPIDERS (2RXS) catalog byCoffey et al. (2019).

TABLE 2
Properties of the AGN Samples and Derived HOD Model Quantities Using Grid Fitting