11institutetext:  Max-Planck-Institut für extraterrestrische Physik (MPE),
Giessenbachstrasse 1, D-85748 Garching bei München, Germany 22institutetext:
Key Laboratory for Research in Galaxies and Cosmology, School of Astronomy and
Space Science, University of Science and Technology of China, Hefei, Anhui
230026, China 33institutetext: Department of Astronomy, School of Physical
Sciences, University of Science and Technology of China, Hefei, Anhui 230026,
China 44institutetext: The Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411007, India
55institutetext: Kavli Institute for the Physics and Mathematics of the
Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8583, Japan
66institutetext: Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte
16, 14482 Potsdam, Germany 77institutetext: Instituto de Astronomía sede
Ensenada, Universidad Nacional Autónoma de México, Km 107 Carretera Tijuana-
Ensenada, 22860, Ensenada, Mexico 88institutetext: Department of Astronomy and
Astrophysics, 525 Davey Lab, The Pennsylvania State University, University
Park, PA 16802, USA 99institutetext: Institute for Gravitation and the Cosmos,
The Pennsylvania State University, University Park, PA 16802, USA
1010institutetext: Department of Physics, 104 Davey Laboratory, The
Pennsylvania State University, University Park, PA 16802, USA
1111institutetext: Institute for Astronomy and Astrophysics, National
Observatory of Athens, V. Paulou & I. Metaxa, 11532, Greece 1212institutetext:
Astronomical Institute, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai, Japan
1313institutetext: Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI)
1414institutetext: Department of Astronomy, School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan 1515institutetext:
Research Center for Space and Cosmic Evolution, Ehime University, 2-5 Bunkyo-
cho, Matsuyama, Ehime 790-8577, Japan 1616institutetext: National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
1717institutetext: Department of Astronomy, Kyoto University, Kitashirakawa-
Oiwake-cho, Sakyo-ku, Kyoto, Kyoto 606-8502, Japan 1818institutetext: Academia
Sinica Institute of Astronomy and Astrophysics, 11F Astronomy-Mathematics
Building, AS/NTU, No.1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
1919institutetext: Department of Economics, Management and Information
Science, Onomichi City University, Hisayamada 1600-2, Onomichi, Hiroshima
722-8506, Japan 2020institutetext: Astronomy Department, University of
Washington, Box 351580, Seattle, WA 98195, USA 2121institutetext: The Carnegie
Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA

# The cosmic web of X-ray active galactic nuclei

seen through the eROSITA Final Equatorial Depth Survey (eFEDS)

Johan Comparat E-mail<EMAIL_ADDRESS>Wentao Luo 2233 Andrea Merloni 11
Surhud More 4455 Mara Salvato 11 Mirko Krumpe 66 Takamitsu Miyaji 77 William
Brandt 88991010 Antonis Georgakakis 1111 Masayuki Akiyama 1212 Johannes
Buchner 11 Tom Dwelly 11 Toshihiro Kawaguchi 1919 Teng Liu 11 Tohru Nagao 1515
Kirpal Nandra 11 John Silverman 13131414 Yoshiki Toba 1616 17 17 18 18 15 15
Scott F. Anderson 2020 Juna Kollmeier 2121

Which galaxies in the general population turn into active galactic nuclei
(AGN) is a keystone of galaxy formation and evolution. Thanks to SRG/eROSITA’s
contiguous 140 square degrees pilot survey field, we constructed a large,
complete, and unbiased soft X-ray flux-limited AGN sample at low redshift
$0.05<z<0.55$. Two summary statistics, the clustering using spectra from
SDSS-V and galaxy-galaxy lensing with imaging from HSC, are measured and
interpreted with halo occupation distribution and abundance matching models.
Both models successfully account for the observations. We obtain an
exceptional complete view of the AGN halo occupation distribution. The
population of AGN is broadly distributed among halos with a mean mass of
$3.9_{-2.4}^{+2.0}\times 10^{12}{\rm M}_{\odot}$. This corresponds to a large-
scale halo bias of $b(z=0.34)=0.99^{+0.08}_{-0.10}$. The central occupation
has a large transition parameter $\sigma_{\log_{10}(M)}=1.28\pm 0.2$. The
satellite occupation distribution is characterized by a shallow slope
$\alpha_{{\rm sat}}=0.73\pm 0.38$. We find that AGNs in satellites are rare,
with $f_{{\rm sat}}<20\%$. Most soft X-ray-selected AGNs are hosted by central
galaxies in their dark matter halo. A weak correlation between soft X-ray
luminosity and large-scale halo bias is confirmed (3.3$\sigma$). We discuss
the implications of environmental-dependent AGN triggering. This study paves
the way towards fully charting, in the coming decade, the co-evolution of
X-ray AGN, their host galaxies, and dark matter haloes by combining eROSITA
with SDSS-V, 4MOST, DESI, LSST, and Euclid.

###### Key Words.:

X-ray, active galactic nuclei

##  1 Introduction

Active galactic nuclei (AGN) are a keystone in galaxy evolution. How they are
triggered and are fueled are essential questions. Answering them will deepen
our understanding of the co-evolution between galaxies, the gas surrounding
them, and their central supermassive black holes (SMBH; see reviews from
Padovani et al. 2017; Eckert et al. 2021). This article focuses on the large-
scale environment of X-ray-selected AGN, namely the population of dark matter
haloes hosting them. X-ray selection provides AGN samples with higher
completeness and purity than selections at different wavelengths (Hickox et
al. 2009). As devised in simulations, this population is diverse (Georgakakis
et al. 2018; Comparat et al. 2019). To infer the population of dark matter
haloes hosting a sample of galaxies, the best technique to date consists of
interpreting the complementary signals from clustering and weak gravitational
lensing (see for example Comparat et al. 2013; More et al. 2015; Coupon et al.
2015; Favole et al. 2016; Zhang et al. 2021).

Previous studies of the clustering of X-ray-selected AGN were limited by the
total number of X-ray AGN or the small survey area. They typically measured
the large-scale halo bias of AGN selected in different fashions (Gilli et al.
2009; Cappelluti et al. 2010; Starikova et al. 2011; Koutoulidis et al. 2013,
2018; Leauthaud et al. 2015; Viitanen et al. 2019; Allevato et al. 2019). The
auto-correlation of X-ray-selected AGN was studied locally ($z\sim 0.045$)
with 199 AGN in the Swift-BAT all-sky survey (Cappelluti et al. 2010). They
found these bright low redshift AGN to be hosted on average by dark matter
haloes of mass $1.6-2.5\times 10^{13}h^{-1}{\rm M}_{\odot}$ corresponding to a
large-scale halo bias of 1.2$\pm$0.1. At higher redshift ($z\sim 1$) with deep
pencil beam surveys (COSMOS observed with XMM and Chandra, Bootes, and Chandra
compilations) and larger numbers of AGN (ranging from 500 to 1,500), Gilli et
al. (2009); Starikova et al. (2011); Koutoulidis et al. (2013); Viitanen et
al. (2019); Allevato et al. (2019) inferred a large-scale halo bias of $\sim
2\pm 0.2$ corresponding to halo masses $4-9\times 10^{12}h^{-1}{\rm
M}_{\odot}$. In these studies, further splitting of the samples as a function
of AGN type, luminosity, and host-galaxy properties, is not very conclusive
due to small statistics. There are hints of correlation with X-ray luminosity
and an indication of a low satellite fraction. The study of the angular auto-
correlation of photometrically selected AGN, so with much larger samples, led
to similar large-scale halo bias and typical dark matter halo masses (Myers et
al. 2007; Donoso et al. 2014; Koutoulidis et al. 2018). Finally, Leauthaud et
al. (2015) studied the galaxy-galaxy lensing signal around 382 X-ray-selected
AGNs in the COSMOS field. They find that AGN host occupation is no different
from that of galaxies. They explain the issue of quoting a mean for the halo
mass when, instead, complete halo occupation distributions should be
discussed; see also Georgakakis et al. (2018) for an extended discussion.
Also, after controlling for stellar mass, Yang et al. (2018a) found no clear
dependence between the environment and the sample-averaged SMBH accretion rate
or the AGN fraction, which indicates that environment-related physical
mechanisms might not significantly affect SMBH growth.

To circumvent the low signal-to-noise ratio in the auto-correlation functions,
the cross-correlation with a controlled galaxy population has been recently
fruitful. Such studies relate AGN populations to their host dark matter haloes
(Krumpe et al. 2010, 2012, 2015, 2018; Mendez et al. 2016; Mountrichas et al.
2019; Zhang et al. 2021). They cross-correlated a similar number of X-ray-
selected AGN (between 300 and 1500) with spectroscopic galaxy surveys: 2MASS,
SDSS, VIPERS, COSMOS (Skrutskie et al. 2006; York et al. 2000; Guzzo et al.
2014; Scoville et al. 2007). They obtain similar large-scale halo bias values
as the auto-correlation studies and investigate the correlation with host-
galaxy properties hinting at possible correlations with stellar mass. This
powerful technique works only with access to a well-studied galaxy sample
(Zehavi et al. 2011; Marulli et al. 2013). The limited signal-to-noise impedes
establishing a clear definitive picture of how X-ray AGNs populate the cosmic
web.

With the advent of eROSITA (Predehl et al. 2021), the number density of X-ray
AGN increased to more than a hundred per square degree in the eROSITA Final
Equatorial Depth Survey (eFEDS, 140 deg2, $\sim$1,400 ks Brunner et al. 2022;
Salvato et al. 2022). Accurate redshifts are required for precise clustering
and lensing analysis. The dedicated spectroscopic observations of the X-ray
sources detected in eFEDS (SDSS-IV, SDSS-V Abdurro’uf et al. 2022; Kollmeier
et al. 2017, Merloni et al. in preparation) enabled the accurate measurement
of redshifts for about eleven thousand X-ray point sources in eFEDS (i.e., for
$\sim 50\%$ of the sources). This number of X-ray AGN with spectra is already
comparable to its predecessor follow-up of ROSAT point sources (Comparat et
al. 2020b).

Outstanding weak-lensing data products are now available over wide areas
thanks to the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) (Aihara et
al. 2019). They measured accurate galaxy shapes for more than 20 source
galaxies per square arc minute over vast areas (1,400 deg2), which almost
completely cover the eFEDS field (Mandelbaum et al. 2018b).

With these two outstanding observational advances, we measure the auto-
correlation function and the galaxy-galaxy lensing signal of X-ray-selected
AGN to study their underlying dark matter halo distribution. We detail, in
Sect. 2, the construction of the X-ray AGN sample, and the weak-lensing data
products used. We describe the method to measure the clustering and galaxy-
galaxy lensing in Sect. 3. The halo occupation distribution and sub-halo
abundance matching models used are detailed in Sect. 4. Results are discussed
in Sect. 5, 6. Throughout, we assume a Flat LCDM cosmology with $H_{0}=67.74$
km s-1 Mpc-1 and $\Omega_{m}(z=0)$=0.3089 (Planck Collaboration et al. 2020).
The uncertainties are $1\sigma$ unless stated otherwise. Magnitudes are in the
AB system (Oke & Gunn 1983). Throughout the article, we use AGN to designate
X-ray-selected AGN.

##  2 Data

In this section, we describe the X-ray observations in Sect. 2.1 and the weak-
lensing data products in Sect. 2.2.

###  2.1 eROSITA eFEDS

Figure 1: Slice of the light cone sampled by the X-ray-selected eFEDS AGN
sample in the redshift range $0.05<z<0.55$ (blue crosses). The surrounding
large-scale structure is sampled by GAMA galaxies (grey) and GAMA galaxy
groups (purple) (Driver et al. 2022) as well as by eROSITA eFEDS clusters
(red) (Liu et al. 2022a).

We use the public Early Data Release eROSITA point source catalog of the eFEDS
Performance Verification survey (Brunner et al. 2022). The catalog contains
20,191 primary sources, over 140 deg2, detected with a likelihood greater than
8 ($\textrm{ERO}\\_\textrm{DET}\\_\textrm{LIKE}>8$) and with reliable
counterpart ($\textrm{CTP}_{\textrm{quality}}\geq 2$) determined as described
in Salvato et al. (2022). Simulations are only at X-ray wavelength and not in
the optical, so the impact of the determination of the counterpart is studied
empirically in Sect. 4 of Salvato et al. (2022). The trade-off between purity
and completeness study shows that counterparts with a threshold of
$\textrm{CTP}_{\textrm{quality}}\geq 2$ ($\textrm{p}\\_\textrm{any}>0.035$)
have a purity and completeness both equal to 95%. We follow the recommendation
of Salvato et al. (2022) and use the set of reliable counterparts above
threshold with $\textrm{CTP}_{\textrm{quality}}\geq 2$. There are 1,219 extra
sources
$\textrm{ERO}\\_\textrm{DET}\\_\textrm{LIKE}>8\;\&\;\textrm{CTP}_{\textrm{quality}}<2$
that are then discarded. These sources are on the faint end of the X-ray flux
distribution, inducing a 5% incompleteness at the faint end. Given the large
distribution of fluxes (luminosities) considered here, we neglect this bias.

2,160 sources are classified as stars either via astrometry, spectroscopy,
X-ray, and opt/IR colors or via a dedicated analysis as described in Schneider
et al. (2022) and removed from the rest of the study.

As shown by simulations (Liu et al. 2022c; Seppi et al. 2022), faint clusters
are contaminants of the point source catalog. In eFEDS, 129 clusters are
present in the point source catalog (Bulbul et al. 2022). They are identified
in Salvato et al. (2022) with the flag $\textrm{CLUSTER}\\_\textrm{CLASS}\geq
3$ and are masked.

After these cuts on the eFEDS point source catalog, we obtain 17,902 AGN
candidates over 140 deg2 (density of 127.9 per square degree). Fig. 1
illustrates the light cone considered in this analysis.

####  2.1.1 Masks

We must propagate the masks applied to the source catalog to the random
catalog to estimate clustering. The random catalog is a set of un-clustered
data points that cover the same sky area as the observations, see description
in Sect. 2.1.4. As the masking radius for each detected source, we use its
radius of maximum signal-to-noise augmented by 40 percent. This radius is
determined while extracting the X-ray spectrum of each source (Liu et al.
2022b), see Appendix A.1 for complete details.

The edges of the survey have a lower exposure time. We find that trimming the
survey edges by requiring a minimum exposure time of 830 seconds minimizes the
KS test values (between random and data vectors) with a minimal area loss, see
Sec § 2.1.4. After applying the minimum exposure time cut, we are left with
16,308 AGN candidates over 128 deg2, resulting in a density of $\sim$127.4
deg-2.

####  2.1.2 Photometric redshifts

Photometric redshift estimation for galaxies hosting active galactic nuclei is
complex (e.g. Salvato et al. 2018). In the eROSITA/eFEDs case, Salvato et al.
(2022) measured photometric redshifts to have
$\sigma_{NMAD}=1.48\times\textrm{median}\left(\frac{|z_{spec}-z_{phot}|}{1+z_{spec}}\right)\sim
0.05$ and a fraction of outliers, with
$\frac{|z_{spec}-z_{phot}|}{1+z_{spec}}>0.15$, of the order of 20%. At the
bright end (r¡21.5), we find that $\sigma_{NMAD}$ decreases to $\sim 0.03$
while the outlier fraction remains the same, 20%.

With the help of the simulation from Comparat et al. (2019), we find that the
measured clustering using photometric redshift with such dispersion and
fraction of outliers would result in losing between one-third and one-half of
the amplitude of the clustering signal. So we will not use the photometric
redshift to measure clustering statistics; instead, we focus on the sub-sample
of 10,680 AGNs with spectroscopic observations, see the following Sect. 2.1.3.

####  2.1.3 Spectroscopic redshifts

The eFEDS field was observed with the SDSS infrastructure (Gunn et al. 2006;
Smee et al. 2013) in March-April 2020 with both BOSS spectrographs (1000
fibers per plate, SDSS-IV, Blanton et al. 2017) and March-April 2021 with a
single BOSS spectrograph (500 fibers per plate, SDSS-V, Kollmeier et al. 2017,
Merloni et al. in preparation). A total of 31 plates were observed, see
Section ‘SPIDERS’ of Abdurro’uf et al. (2022) and the spectra are part of the
SDSS DR18 (SDSS collaboration in preparation). The total area covered by SDSS-
IV and V spectroscopic observations is 133.77 deg2 (95% of the eFEDS area).
The obtained spectroscopic redshift completeness depends on (i) the position
in the sky; (ii) the optical magnitude of the source. We consider the z-band
AB magnitude measured as in the legacy survey DR8 (Dey et al. 2019) and based
on observations made with DECam (Flaugher et al. 2015). Although photometric
redshifts are not accurate enough for clustering studies, they are of
sufficient quality to compare the distribution of magnitudes and fluxes in
broad redshift bins. Overall, we find that at a z-band magnitude of 21.25
(19.0), the completeness is 50% (90%). We find that, up to redshift
$\sim$0.55, the spectroscopic sample is a fair sub-sample (as a function of
optical magnitude and X-ray flux) of the entire population. SDSS-V
observations being limited to z-band magnitudes brighter than 21.5, beyond a
redshift of 0.55, we are missing a significant fraction of the AGN that are
optically faint X-ray-selected AGN, see Fig. 2.

Figure 2: Spectroscopic completeness as a function of r-band magnitude vs.
redshift (top) and soft X-ray luminosity (0.5-2 keV) vs. redshift (bottom).
The completeness coverage is homogeneous below the redshift of 0.6. At higher
redshift, completeness at the faint end impacts the sample significantly.

We estimate the spectroscopic completeness in $\sim$3.5 deg2 equal area pixels
(half the size of an SDSS plate $\sim$7 deg2). The minimum (maximum)
completeness measured in a pixel is 13% (69%). The relative variations of the
spectroscopic redshift distribution as a function of completeness are within
the expected fluctuations for pixels with completeness levels above 40%. So,
we discard areas with completeness lower than 40%. It removes about 20 deg2 of
the area located at the edge of the eFEDs field (most of it overlaps with the
low-exposure regions).

To summarize, to measure the clustering, we create an AGN sample covering
redshifts between 0.05 and 0.55, where the spectroscopic sample is not biased
compared to the parent photo-z sample. We obtain a sample of 1,992 AGN with
spectroscopic redshift covering 122.3 deg2. Figure 1 illustrates how the AGN
considered in this analysis sample the large-scale structure observed with
galaxies and groups from the GAMA survey and eROSITA eFEDS clusters. Table 1
summarizes the main properties of the considered sample. The mean redshift of
the sample is 0.34, with a standard deviation of 0.13. The sample’s mean X-ray
luminosity in the soft 0.5–2 keV band is 42.91. The distribution around the
mean is broad and has a standard deviation of 0.65.

We verified that the edges of the selection do not impact the clustering and
lensing summary statistics: by moving the redshift cut from 0.05 to 0.1 and
from 0.55 to 0.5 and by adding a minimum luminosity threshold of 41.5. Further
splitting the sample in soft X-ray luminosity-limited samples (or following a
visual inspection of the optical spectra) significantly decreases the signal-
to-noise in the measurements, and HOD model parameters become unconstrained.
Larger numbers of AGN with spectroscopic redshifts are required to investigate
trends with parameters defining the sample.

Among the 1992 AGN studied here, 1648 (82.7%) have their spectroscopic
redshift coming from SDSS observations. 270 (13.5%) come from GAMA
observations (Liske et al. 2015). Then in smaller numbers, spectroscopic
redshifts originate from: 46 from WiggleZ (Drinkwater et al. 2018), 8 from
LAMOST DR5 v3 (Luo et al. 2015), 7 from 2SLAQ (Cannon et al. 2006), 5 from
2MASS (Skrutskie et al. 2006), 3 from HCSC (Oguri et al. 2018), 2 from 6dFGS
(Jones et al. 2009), 1 from HYPERLEDA (Paturel et al. 2003), 1 from (Véron-
Cetty & Véron 2010), and 1 from RCSEDv2 (Chilingarian et al. 2021).

Table 1: Properties of the sample. Minimum, mean, maximum, and standard deviation of the redshift, soft band X-ray luminosity, and g, r, z magnitudes from the legacy survey (Dey et al. 2019). property | min | mean | max | std
---|---|---|---|---
redshift | 0.05 | 0.34 | 0.55 | 0.13
soft LX | 40.49 | 42.91 | 45.12 | 0.65
gAB | 15.16 | 20.1 | 23.13 | 1.41
rAB | 14.26 | 19.23 | 22.39 | 1.35
zAB | 13.62 | 18.57 | 21.82 | 1.32

####  2.1.4 Random catalogue

To measure clustering, one compares the set of observed points to a group of
points with no clustering but all other aspects equal (window function,
redshift distribution). In this section, we explain how the set of random
points is constructed.

We draw a set of random points with a large uniform density of $\sim$81,000
deg-2 on the sky (about 11.5 million points on eFEDS). We first trim it to
follow precisely the edges of the survey. We then follow the methodology of
Georgakakis et al. (2008) to downsample the uniform random catalog with the
sensitivity map, see details in Appendix A.2. Heuristically, this step applies
the X-ray flux limit and its variations across the field to the set of random
points. The total number of random points remaining after downsampling,
masking (see the previous section), and trimming (low exposure time region) is
3,713,726. The density of random points, $\sim 30,000$ deg-2, is more than 200
times larger than that of the data points (127.4 deg-2), which is largely
sufficient.

We downsample the random catalog to follow the spectroscopic redshift
completeness map and its dependency on R.A. and Dec. We cut the areas where
spectroscopic completeness is lower than 40 %. As the relative variation in
the redshift distribution is independent of the completeness (see the previous
section), we shuffle the set of observed redshifts and assign them to the
random points, regardless of the completeness level.

###  2.2 HSC-SSP weak-lensing data

We use the HSC S19A weak-lensing products based on the HSC-SSP accumulated
$i$-band imaging data from 2014 to 2019. The original HSC-SSP S19A wide-layer
data covers about 512 deg2, and it reduces to 433.48deg2 after the full-color
full-depth (FCFD) selection. With the $i$-band magnitude cut of 24.5, the
observed number density reaches up to 22.9 arcmin-2. The deep imaging data
enable a comprehensive redshift coverage ranging from 0 to 3, and the
calibrated bias residual shows no dependence on the redshift.

There are several major updates in the shape catalog version from hscPipe4 to
hscPipe7 (used here). These improvements include PSF modeling, image co-
addition, bright star masking, and background subtraction.

Due to the bad weather, volcanic activity, and other telescope downtimes, the
observation time has significantly decreased. To compensate for the loss,
besides the 30 nights additional observation time, the survey strategy has
been modified by (i) reducing to 80% the time in the Deep/Ultra Deep fields,
(ii) lowering the seeing conditions in $i$-band, (iii) changing the dither
pattern from 6 to 5 in the $i$, $z$ and $y$ bands. The latter results in a 0.1
magnitude difference in the $i$-band depth. Altogether, the expected coverage
should still reach up to 1200 deg2. We detail several essential aspects of the
S19A imaging data below.

####  2.2.1 Photometry

The most important updates to the pipeline and data processing are (i)
improved sky background subtraction; (ii) improved image co-addition warping
kernel from Lanczos3 to Lanczos5; (iii) the addition of two new filters $r2$
and $i2$ that substitute the original $r$ and $i$\- band filters; (iv)
improved Point Spread Function (PSF) based on PSFEx; (v) new masks around
bright stars composed of ghosts, blooming, halo. About the sky subtraction
algorithm, we apply the newest version based on Aihara et al. (2022) rather
than the global-global algorithm Aihara et al. (2019). The latter leads to
about 10% loss of extended source near the shape catalog cut at $i$-band
magnitude equals 24.5.

We use the Forward Global Calibration Method (FGCM; Burke et al. 2018) that
was firstly developed for the Dark Energy Survey (DES; Sevilla-Noarbe et al.
2021) and now has been merged into the LSST/HSC pipeline. It starts with
modeling the instrumental throughput measurements such as mirrors, filters,
and detectors. Then, the atmospheric model (MODTRAN4 Berk et al. 1999) carries
out detailed modeling of atmospheric throughput as a function of zenith
distance at the location of the SUBARU telescope on Mauna Kea.

The performance of the photometry is tested in two ways, an internal test by
comparing the PSF magnitude to the Kron magnitude for a bright star sample
($i$¡21.5) in the Wide XMM-LSS field. The standard deviation of the difference
(PsfMag-KronMag) achieved is better than 1%, independently of the filters and
fields. One exception is the $y$-band, which shows a slightly larger scatter
of 1.5%. In addition, the difference between the CModel magnitude and PSF
magnitude is below 0.2%. For the external test, PanSTARR1 stars brighter than
$r$-band 20 mag are used (PS1; Chambers et al. 2016). The scatter level is
also at about the 1% level indicating good photometric performance.
Observations are mixing the $i$ ($r$) and the $i2$ ($r2$) filters. It results
in a small offset in the $i$ ($r$) photometry in the Deep+UltraDeep fields and
small regions of the Wide fields. Good photometric performance is a pre-
condition for the redshift calibration, described below.

####  2.2.2 Photometric redshifts for sources

The overlapping photometric and public spectroscopic surveys with HSC-SSP
provide a wealth of data for photometric redshift calibration. These data sets
include zCOSMOS DR3 (Lilly et al. 2009), UDSz (Bradshaw et al. 2013; McLure et
al. 2013), 3D-HST (Skelton et al. 2014; Momcheva et al. 2016), FMOS-COSMOS
(Silverman et al. 2015) and so forth. There are about 170k spectroscopic
redshifts (spec-$z$) and 37k g/prism-$z$ with high-quality.

To cover the wide range of photo-$z$ methods (Tanaka et al. 2018; Nishizawa et
al. 2020) used different techniques on the HSC-SSP data: template-fitting,
empirical-fitting, and machine learning. These include the Mizuki template-
fitting method Tanaka (2015), MLK Self-Organise Map (SOM; More et al. in
prep), NNPZ Nearest Neighbors P(z) (Cunha et al. 2009), FRANKEN-Z Flexible
Regression over Associated Neighbors with Kernel dEnsity estimatioN for
Redshifts, DEmP Direct Empirical Photometric code (Hsieh & Yee 2014) and Ephor
Extended Photometric Redshift (EPHOR). There is a newly developed machine
learning photometric method for HSC-SSP Y3 shape catalog, i.e., dNNz (A. J.
Nishizawa et al. in preparation).

The metrics to quantify the performance of each method are the bias defined as
$\Delta z=(z_{phot}-z_{ref})/(1+z_{ref})$, the dispersion
$\sigma_{zphot}=1.48\times MAD(\delta z)$ (MAD is the median absolute
deviation), the outlier rate $f_{outlier}=N(\Delta z)>0.15/N_{total}$ and the
loss function $L(\Delta z)=1-1/(1+(\Delta z/\gamma)^{2})$ with $\gamma=0.15$.
The photometric redshift used in this work is based on dNNz method, which
achieves an accuracy with a $\Delta z=10^{-4}$ bias, a dispersion
$\sigma_{zphot}=3\%$, and an outlier rate smaller than $f_{outlier}\leq 10\%$.

####  2.2.3 Shape catalog

The HSC galaxy sample is selected following a series of basic flag cuts, such
as i_detect_isprimary, i_extendedness_value and i_sdsscentroid_flag. The
detailed descriptions are listed in Table 2 of Li et al. (2022). The shapes of
galaxies are measured with re-Gaussianization method (Hirata & Seljak 2003)
(reGauss which has been merged to GalSim Rowe et al. (2015)), the PSF effects
are corrected during the measurement process. HSC covers six discrete fields
named after overlapping regions from previous surveys: XMM, HECTOMAP, WIDE12H,
GAMA09H, GAMA15H, and VVDS. The eFEDS region overlaps with GAMA09H. Note that
the HSC region GAMA09H covers a larger area than the original GAMA09H field
and completely encompasses the eFEDS field.

The final shape catalog contains the two components of the ellipticity:

$\rm(e_{1},e_{2})=\frac{1-(b/a)^{2}}{1+(b/a)^{2}}(cos2\phi,sin2\phi),$ (1)

where $b/a$ is the ratio between the minor axis and major axis, and $\phi$ is
the position angle of the major axis with respect to the sky coordinates. The
shear distortion, $\gamma_{i}$, is then related to the $e_{i}$ (i=1,2) such
that

$\rm\gamma_{i}=\frac{1}{2\mathcal{R}}\langle e_{i}\rangle(i=1,2),$ (2)

where $\mathcal{R}$ is the response of the galaxy ellipticity to a small
distortion defined in Kaiser et al. (1995); Bernstein & Jarvis (2002). The
response is calculated from the calibrated parameters $e_{rms}$ and
$\sigma_{e}$ based on simulations (Mandelbaum et al. 2018a; Li et al. 2022) as
follows

$\rm\mathcal{R}=1-\frac{\sum_{i}w_{i}e_{rms,i}^{2}}{\sum_{i}w_{i}}.$ (3)

The weighting term $\rm w_{i}$ in Eq.3 is composed of the per-component error
from simulation due to photon noise $\sigma_{e;i}$, and the rms of galaxy
shape distribution $e_{rms;i}$, $\rm
w_{i}=1/(\sigma^{2}_{e;i}+e_{rms;i}^{2})$.

The reGauss algorithm suffers from several estimation biases, e.g., model
bias, noise bias, and selection bias which can be classified into
multiplicative bias $\rm m_{i}$ and additive bias $\rm c_{i}$ (i=1,2), so that

$\rm\gamma_{i}=(1+m_{i})\gamma_{i}^{true}+c_{i}.$ (4)

The final shear estimator is obtained with Eq. 5. It does not incorporate the
geometry factor $\Sigma_{crit}$ described in Sec.3.2.

$\rm\langle\gamma_{i}\rangle=\frac{\sum_{j}w_{i}e_{i;j}}{2\mathcal{R}(1+\langle
m_{i}\rangle)\sum_{j}w_{j}}-\frac{\langle c_{i}\rangle}{1+\langle m\rangle}.$
(5)

Both multiplicative and additive biases are calibrated based on the
simulations mentioned above. The two biases are then assigned to each galaxy
as a function of SNR and resolution $R_{2}$. Additionally, there is selection
bias as well as weight bias. The overall bias is quantified as the residuals
for both multiplicative $\delta m$ and additive bias $\delta a$, both of which
can reach below 1% level for HSC-SSP Y3 shape catalog Li et al. (2022).

##  3 Summary statistics

Galaxy clustering and gravitational lensing probe the galaxy and matter over-
density field’s auto and cross-correlations as a function of scale via a
biasing function (Tegmark & Peebles 1998; Tegmark & Bromley 1999; Dekel &
Lahav 1999). These measurements are well suited to constrain the biasing
function, also named more generically the galaxy-halo connection (Sheth &
Tormen 1999; Wechsler & Tinker 2018). With the data described above, we
compute two summary statistics: the AGN-AGN auto-correlation (clustering,
Sect. 3.1) and galaxy-galaxy lensing with the AGN population being the lenses
(Sect. 3.2).

###  3.1 Clustering measurement

We use the Landy & Szalay (1993) estimator to measure the projected two-point
correlation function, labeled $w_{p}(r_{p})$ (for detailed definition, see
e.g., Davis & Peebles 1983). To count pairs and integrate along the line of
sight, we use the corrfunc software (Sinha & Garrison 2020). For the
integration, we use $\pi_{max}=40\;h^{-1}$Mpc. We carried out measurements
with shorter and longer $\pi_{max}$ and found that with 40, we would obtain
the largest signal-to-noise in the clustering measurement.

We randomly down-sample the catalog of random points for the clustering
measurement to have twenty times the number of AGN. To have consistent 3D
positions between the optical spectra and the X-ray sources, we compute the
clustering using the position on the sky of the optical counterparts (Salvato
et al. 2022); we do not use the positions of the X-ray sources. The projected
correlation function obtained is shown in Fig. 3 (black error bars). The
clustering measurement’s uncertainty is estimated using the diagonal component
of the covariance matrix obtained with 18 eFEDS simulated catalogs (Liu et al.
2022c). These simulated eFEDS observations are based on the empirical models
of the X-ray cosmic web from Comparat et al. (2019, 2020a). The yellow shaded
area in Fig. 3 shows the prediction from the 18 mocks. We find that the
forecast is faithful to the observations.

Following Driver & Robotham (2010), we estimate the cosmic variance in this
field to be 1%, which we add as a constant systematic uncertainty at all
scales to the clustering measurement. Note that it is small compared to
statistical uncertainties. Using the eFEDS simulations, we find that
clustering summary statistics are significantly biased low for separations
$r_{p}>$40 $h^{-1}$Mpc. This is due to the finite volume observed. So, we
exclude from the fitting procedure clustering measurements with a separation
larger than 40 $h^{-1}$Mpc. The total signal-to-noise in the clustering
measurement is 17.7, split into 11 radial bins. We sample the separation range
with five bins per decade evenly log-spaced (0.2 dex steps) between 0.25
($10^{-0.6}$) and 39.8 ($10^{0.6}$) $h^{-1}$Mpc.

The fiber collision radius in SDSS is 62 arc seconds ($\sim$0.25$h^{-1}$Mpc at
the mean redshift of the sample). The eROSITA PSF is 30 arc seconds, so X-ray-
selected AGN pairs with a separation smaller than one arc minute are hardly
detected. Moreover, since the AGN sample considered here is sparse (120
deg${}^{-2}\sim$0.03 arc minutes-2) and spread over a long line of sight, AGN
close pairs are small in numbers. Using the mock catalogs limited to redshifts
$0.05<z<0.55$, we estimate the number of expected pairs with an angular
separation smaller than 62 arc seconds to be typically a handful: less than
10. Only half have physical separation smaller than 40 $h^{-1}$Mpc. So, the
number of missed pairs due to fiber collisions is negligible. So in our case,
we consider that fiber collisions are not an issue, and we define our lowest
separation bin at 0.25 $h^{-1}$Mpc.

Figure 3: Projected clustering measurement of the X-ray flux-limited eFEDS AGN
sample in the redshift range $0.05<z<0.55$ (black). The prediction from the 18
eFEDS simulations appears in yellow. The best-fit model (jointly with weak-
lensing observations) is in red.

###  3.2 Galaxy-galaxy lensing measurement

The galaxy-galaxy lensing measurement is a cross-correlation between positions
of foreground lenses (AGN in our case) and shapes of background galaxies
acting as sources (HSC galaxies), see reviews from Bartelmann & Schneider
(2001); Refregier (2003). This measurement directly traces the galaxy halo
connection (e.g. Mandelbaum et al. 2005; Seljak et al. 2005). Numerous studies
have used galaxy-galaxy lensing (sometimes combined with galaxy clustering) to
trace the galaxy-halo connection in general (Leauthaud et al. 2011; Coupon et
al. 2015; Zu & Mandelbaum 2015; Dvornik et al. 2018; Zacharegkas et al. 2022).

We combine the X-ray point sources from the eFEDS region and the HSC shape
catalog to compute the galaxy-galaxy lensing using each source galaxy and its
probability distribution function as a function of redshift ($p(z)$). The
physical interpretation of the galaxy-galaxy lensing signal is the difference
between the average density inside a certain projected radius $R$ and the
average density at that same radius, so an excess surface density (ESD,
$\Delta\Sigma$), that is

$\rm\Delta\Sigma(R)=\bar{\Sigma}(\leq R)-\Sigma(R).$ (6)

We follow the measurement procedure described in Miyatake et al. (2019); Luo
et al. (2022), that

$\rm\Delta\Sigma(R)=\frac{1}{2\mathcal{R}(R)}\frac{\sum_{l}^{N_{l}}w_{l}\sum_{s}^{N_{s}}w_{ls}e_{t,ls}[\langle\Sigma_{cr}^{-1}\rangle]^{-1}}{[1+K(R)]\sum_{l}^{N_{l}}w_{l}\sum_{s}^{N_{s}}w_{ls}}.$
(7)

In the above ESD estimator, $\mathcal{R}(R)$ is the response of the shape
estimator, which, for this work, takes a value of 0.84. $\rm w_{ls}$ is the
weight for each lens-source galaxy pair. $\rm w_{l}$ is a weight assigned to
each lens galaxy. Here we use $\rm w_{l}=1$ in that there are no particular
requirements on redshift, stellar mass or other properties for the lens
catalog. $e_{t,ls}$ is the tangential component of the source galaxy shape
with respect to the lens. The factor $\rm K(R)$ accounts for the
multiplicative bias calibrated based on a suite of simulations developed in
Mandelbaum et al. (2018a); Li et al. (2022).

Two blinding schemes are provided for systematic sanity tests, the low and the
high-accuracy blinding scheme. We use the former where a value of $\delta m$
is added to the original calibrated additive bias $m$ where
$e_{i}=(1+m_{i})e_{i}+c_{i}$ (i=1,2). In the low accuracy scheme, only $\delta
m_{1}$ is added and encrypted for each user of the shape catalog. It is then
decrypted and removed by subtracting $\delta m_{1}$ term.

An extra selection function for each lens-source pair is applied following
Medezinski et al. (2018) so that the accumulated probability of the P(z)
satisfies

$\rm P(z_{s}\geq z_{l}+0.2)=\int_{z_{l}+0.2}^{\infty}p(z)dz\geq 0.98.$ (8)

Fig. 4 shows the galaxy-galaxy lensing measurements obtained (black) as well
as the best-fit HOD model (red). Measurements with a separation smaller than
20 $h^{-1}$Mpc are included in the fitting procedure. The total signal-to-
noise in the lensing measurement is 46, split into 15 radial bins. We sample
the separation range with 5.5 bins per decade evenly log-spaced (0.18 dex)
steps between 0.025 ($10^{-1.6}$) and 8.3 ($10^{0.9}$) $h^{-1}$Mpc. Compared
to the clustering measurement, the measurement extends to ten times smaller
separations. We measured the same using the public HSC one-year S16A (KIDS
DR4) lensing products. They are consistent but have a lower total signal-to-
noise of 21.4 (6.5) compared to 46.

Figure 4: Galaxy-galaxy lensing measurement. Excess surface mass density
measurement with HSC S19A data using as lenses the X-ray flux-limited eFEDS
AGN sample in the redshift range $0.05<z<0.55$ (black). The best-fit model
(jointly with clustering measurements) is shown in red.

##  4 Models

We interpret the clustering and lensing summary statistics using models of
halo occupation statistics (Cooray & Sheth 2002; Guo et al. 2010). There are
three flavours of models: Halo Occupation Distribution (HOD; Berlind &
Weinberg 2002; Kravtsov et al. 2004; Zheng et al. 2005, 2007), Sub Halo
Abundance Matching (SHAM; Conroy et al. 2006; Trujillo-Gomez et al. 2011;
Klypin et al. 2013) and emulators of the large-scale structure of the Universe
(e.g. DarkEmulator Nishimichi et al. 2019; Nishizawa et al. 2020). HOD and
SHAM reproduce measurements of the galaxy clustering as a function of
luminosity and color (Zehavi et al. 2011; Marulli et al. 2013). The
exploration and parametrization of the assembly bias in such models is still a
matter of debate (Contreras et al. 2021; Xu et al. 2021). Emulators enable a
precise prediction of the cross-correlation between haloes and the dark
matter, though due to the finite resolution of simulations they build upon,
they are currently limited to predict statistics at the high mass end
(Nishimichi et al. 2019). So, first, we adjust the HOD model parameters to the
measurements obtained in the previous section 4.1. Then, we compare
measurements to the prediction of the SHAM model from Comparat et al. (2019)
in Sect. 4.2.

###  4.1 Halo occupation distribution model

Table 2:  HOD parameters obtained (median of the posterior) by jointly fitting the auto-correlation function and the galaxy-galaxy lensing of the X-ray flux-limited AGN sample. The uncertainties quoted are $1\sigma$ (15.9–84.1 percentiles). Priors are flat in linear space. parameters | min | max | $0.05<z<0.55$
---|---|---|---
$M_{{\rm min}}$ | 8.0 | 15.0 | 13.06 $\pm$ 0.44
$\sigma_{\log_{10}M}$ | 0.05 | 1.5 | 1.28 $\pm$ 0.2
$\alpha_{{\rm sat}}$ | 0.1 | 1.5 | 0.73 $\pm$ 0.38
$M_{{\rm sat}}-M_{{\rm min}}$ | -3.0 | 2.45 | 1.46 $\pm$ 0.52
$M^{*}_{12}$ | -4.0 | 0.1 | -0.96 $\pm$ 0.45
evidence (logZ) |  |  | -39.88 $\pm$ 0.14
deduced parameters
$b(z=\bar{z})=0.991^{+0.078}_{-0.096}$
$b(z=0.1)=0.915^{+0.065}_{-0.08}$
$f_{{\rm sat}}<20.6\%$
4-parameter fit, $\sigma_{\log_{10}M}=1.3$
$M_{{\rm min}}$ |  |  | 13.09 $\pm$ 0.19
$\alpha_{{\rm sat}}$ |  |  | 0.75 $\pm$ 0.39
$M_{{\rm sat}}-M_{{\rm min}}$ |  |  | 1.56 $\pm$ 0.46
$M^{*}_{12}$ |  |  | -0.97 $\pm$ 0.46
evidence (logZ) |  |  | -38.6 $\pm$ 0.12
deduced parameters
$b(z=\bar{z})=1.001^{+0.075}_{-0.094}$
$b(z=0.1)=0.918^{+0.065}_{-0.076}$
$f_{{\rm sat}}<16.8\%$
4-parameter fit, $\sigma_{\log_{10}M}=1.0$
$M_{{\rm min}}$ |  |  | 12.47 $\pm$ 0.26
$\alpha_{{\rm sat}}$ |  |  | 0.75 $\pm$ 0.39
$M_{{\rm sat}}-M_{{\rm min}}$ |  |  | 1.84 $\pm$ 0.53
$M^{*}_{12}$ |  |  | -1.0 $\pm$ 0.5
evidence (logZ) |  |  | -40.34 $\pm$ 0.11
deduced parameters
$b(z=\bar{z})=0.996^{+0.067}_{-0.097}$
$b(z=0.1)=0.919^{+0.059}_{-0.081}$
$f_{{\rm sat}}<66.4\%$

As a baseline, we use the HOD model formulated by More et al. (2015). It is
described by the two equations below. $\langle N_{C}\rangle$ ($\langle
N_{S}\rangle$) gives the average occupation of a dark matter halo of mass $M$
by a central (satellite) galaxy. The model has 5 parameters
$\@vec{\theta}=(M_{{\rm min}},\sigma_{\log_{10}M},\alpha_{{\rm sat}},M_{{\rm
sat}},M^{*}_{12}$).

$\displaystyle\langle N_{C}\rangle$
$\displaystyle(M,\@vec{\theta})=\frac{f_{A}}{2}\left(1+\operatorname{erf}\left(\frac{(log_{10}(M)-M_{{\rm
min}})}{\sigma_{\log_{10}M}}\right)\right)$ (9) $\langle
N_{S}\rangle(M,\@vec{\theta})=\langle
N_{C}\rangle(M,\@vec{\theta})\left(\frac{M-10^{M_{{\rm sat}}-1}}{10^{M_{{\rm
sat}}}}\right)^{\alpha_{{\rm sat}}}$ (10)

Only a fraction of distinct haloes hosts a central galaxy with an AGN. The
$f_{A}$ parameter, as introduced by Miyaji et al. (2011, Eq. 24), can be
interpreted as the duty cycle of halo centers being an AGN. In this study,
since the correlation function measurements do not depend on the normalization
of the occupation distribution, we arbitrarily set $f_{A}$ to 1.

To avoid sampling un-physical values of $M_{{\rm sat}}$, the parameter passed
to the fitting routine is $M_{{\rm sat}}-M_{{\rm min}}$ with boundaries
specified in Table 2.

To fit for the $\Delta\Sigma$ measurement at small separations and benefit
from the signal present, we need to add the prediction for a point-like mass
term that represents the baryonic lensing mass of the AGN host galaxies. It
adds the parameter $M^{*}_{12}$ as follows:

$\Delta\Sigma^{*}(r)=\frac{10^{M^{*}_{12}+12}}{\pi r^{2}}.$ (11)

The posterior of this parameter represents the mean baryonic lensing mass of
the galaxies hosting AGN. This mass is related to stellar mass (inferred with
stellar population synthesis models) but will also encompass gas in and around
the galaxy. This baryonic lensing mass can be considered the upper limit of
the mean stellar mass of galaxies hosting AGN.

In total, we fit for five parameters on the two measurements $\Delta\Sigma$
($w_{p}(r_{p})$) which have S/N=46 (17.7) in 15 (11) radial bins. The
parameters are sampled with a flat prior (in linear space) within broad
boundaries as specified in Table 2.

###  4.2 Sub halo abundance matching models

The Comparat et al. (2019, 2020b) empirical AGN model statistically links the
dark matter haloes to the probability of hosting an AGN and its spectral
energy distribution. By construction, it follows the X-ray luminosity function
from Aird et al. (2015). Importantly for interpretation, the assignment is
done regardless of the environment in which haloes live. The model has two
parameters: the fraction of AGN in satellites sub haloes and the scatter in
the abundance matching relation between stellar mass and hard X-ray
luminosity.

We show the direct $w_{r}(r_{p})$ prediction from the mock catalogues of Liu
et al. (2022c) in Fig. 3. It is consistent with observations. It was obtained
with $f_{{\rm sat}}=10\%$ (fraction of AGN being satellites) and $\sigma=1$
(scatter in the abundance matching procedure between hard X-ray luminosity and
stellar mass). These parameters were chosen by hand by Comparat et al. (2019,
2020a). At that time (before this study), such parameters resulted in
reasonable predictions.

Creating a complete SHAM-base mock catalog is long (order of a few CPU hours)
and thus impractical for fitting purposes. Furthermore, with current light
cones constructed with replications, predicting the galaxy-galaxy lensing
signal is tedious as the dark matter particles are not kept. So, instead of
predicting summary statistics as measured as a function of the SHAM
parameters, we directly predict the halo occupation distribution curves as a
function of $f_{{\rm sat}}$ and $\sigma$. Thus, we sample a small and finite
number of ($f_{{\rm sat}}$, $\sigma$) combinations and create individual mock
catalogs to predict the HOD curves.

Figure 5: Constraints were obtained on the HOD parameters when fitting only
the clustering measurement (yellow), the lensing measurement (purple), or both
jointly (blue). Contours show 1 and 2 $\sigma$ constraints. Most of the
constraining power comes from galaxy-galaxy lensing.

##  5 Results

We discuss here the results of the fitting procedure and the comparison
between models. In Sec. 5.1 (5.2), we discuss the results obtained with the
HOD (SHAM) model. For the first time, we measure with relatively small
uncertainties the complete halo occupation distribution of a low redshift
flux-limited X-ray-selected sample of AGN. We obtain a global view of the
distribution of haloes hosting X-ray AGN; see Fig. 6.

###  5.1 HOD results

Figure 6: Inferred halo occupation distribution (solid) split into central
(dashes) and satellite (dots) for the 4 and 5 parameter HOD fits
(yellow/orange and black). The 4-parameter best-fit model is within the
$1\sigma$ uncertainty of the 5-parameter best-fit model. Due to the degeneracy
between $M_{{\rm min}}$ and $\sigma_{\log_{10}M}$, the 4-parameter fit HOD
with $\sigma_{\log_{10}M}=1$ is skewed towards lower masses compared to the
5-parameter fit. The direct predictions from mock catalogs from Leauthaud et
al. (2015); Georgakakis et al. (2018); Comparat et al. (2019) are shown on the
right panel. They are within the fitted contours obtained. The mocks from
Leauthaud et al. (2015); Georgakakis et al. (2018) have a lower
$\sigma_{\log_{10}M}$ value (sharper transition) and are thus more in line
with the 4-parameter fit. The mock from Comparat et al. (2019) has a higher
$\sigma_{\log_{10}M}$ value and is comparable to the 5-parameter fit.

We fit the parameters of the HOD model with a nested sampling method ultranest
(Buchner 2021). The resulting parameters are given in Table 2. The constraints
on the HOD parameters (when fitting individually each summary statistic or
both jointly) are shown in Fig. 5. It illustrates the complementary nature of
the two measurements. The comparison between the joint best-fit model and the
clustering measurements and lensing measurements are shown in Figs. 3, 4. The
models are sensible. They account for the observations.

The five parameters are meaningful, although not precisely constrained by the
joint fit of both summary statistics. For the central haloes, $M_{{\rm min}}$
takes a median posterior value of 13.06$\pm$0.44, the width of the error
function is found at $\sigma_{\log_{10}M}=$ 1.3 $\pm$ 0.2. There is a low
1$\sigma$-level tension between the constraints on these parameters obtained
by each summary statistic; see Fig. 5. Due to the higher signal-to-noise on
the lensing statistic, the combined best-fit values are closer to the
individual best-fit value on the lensing statistic. For the satellites, the
slope is best-fit at $\alpha_{{\rm sat}}=$0.73$\pm$0.38 and the transition
occurs in haloes 10 to 100 times more massive than the typical halo : $M_{{\rm
sat}}-M_{{\rm min}}=$ 1.46 $\pm$ 0.52. Both summary statistics point to these
parameter values (Fig. 5). The typical baryonic lensing mass of galaxies
hosting these AGN is $M^{*}_{12}=$ -0.96 $\pm$ 0.45. It sets an upper limit to
the mean stellar mass of galaxies hosting AGN of $\sim 10^{11}{\rm
M}_{\odot}$. The 1$\sigma$ boundaries encompass $3.9\times 10^{10}$ and
$3.1\times 10^{11}{\rm M}_{\odot}$, which is in fair agreement with
expectations from AGN host stellar mass function (Bongiorno et al. 2016; Yang
et al. 2018b). This parameter is not degenerate with others and is constrained
only by the lensing measurements at small separations.

We note a degeneracy between $M_{{\rm min}}$ and $\sigma_{\log_{10}M}$, which
we investigate. We decrease the number of parameters by fixing
$\sigma_{\log_{10}M}$ to 1.3 (similar to the best-fit value) and 1 (to force a
less broad halo distribution, a sharper transition). We obtain a set of best-
fit parameters (see Table 2) that are compatible with the 5-parameter fit.
When fixing $\sigma_{\log_{10}M}$ to 1.3, we obtain $M_{{\rm min}}13.09\pm
0.19$, similar value to the 5-parameter fit but with half the uncertainty.
Results for other parameters remain unchanged. When fixing
$\sigma_{\log_{10}M}$ to 1, $M_{{\rm min}}$ is logically forced to lower
values to $\sim 12.5$ to be able to fit the overall signal. The halo
occupation distribution posterior is close to that of the five parameter fit,
see Fig. 6 black and yellow/orange contours.

We obtain a global view of the distribution of haloes hosting X-ray AGN; see
Fig. 6. The 4-parameter best-fit model is within the $1\sigma$ uncertainty of
the 5-parameter best-fit model. Due to the degeneracy between $M_{{\rm min}}$
and $\sigma_{\log_{10}M}$, the 4-parameter fit HOD with
$\sigma_{\log_{10}M}=1$ is skewed towards lower masses compared to the
5-parameter fit. With the HOD model, we derive the average halo mass hosting
central (central or satellite) AGN is $3.93_{-2.44}^{+2.03}\times 10^{12}{\rm
M}_{\odot}$ ($4.95_{-1.99}^{+2.63}\times 10^{12}{\rm M}_{\odot}$). These
values are comparable to the findings of Rodríguez-Torres et al. (2017). We
measure that the distribution of halo masses is broad. We thus confirm that
quoting a typical halo mass will be extremely sensitive to the definition of
what ‘typical’ means; see discussion in Leauthaud et al. (2015).

The direct HOD predictions from mock catalogs from Leauthaud et al. (2015);
Georgakakis et al. (2018); Comparat et al. (2019) are shown on the right panel
of Fig. 6. They are within the fitted contours obtained. The mocks from
Leauthaud et al. (2015); Georgakakis et al. (2018) have a lower
$\sigma_{\log_{10}M}$ value (sharper transition) and are thus more in line
with the 4-parameter fit. The mock from Comparat et al. (2019) has a higher
$\sigma_{\log_{10}M}$ value and is comparable to the 5-parameter fit (see more
discussion in the SHAM section below). The normalization of $\langle
N(M)\rangle$ can be added as a parameter and possibly constrained by jointly
fitting the clustering and lensing summary statistics with the stellar mass
(or luminosity) function of galaxies hosting X-ray AGN to have a handle on the
fraction of galaxies hosting an AGN. Though measuring reliable host-galaxy
stellar masses in the case of type 1 AGN is complex (Ciesla et al. 2015; Zou
et al. 2022), and is left for future studies.

Figure 7: Inferred large-scale halo bias as a function of redshift (top panel)
and luminosity (bottom panel) compared to (Krumpe et al. 2015) (redshift range
0.16-0.36). We confirm the trend with soft X-ray luminosity and obtain a best-
fit of $y=(0.482\pm 0.143)x+(-19.684\pm 6.173)$ between the soft X-ray
luminosity and the large-scale halo bias.

The large-scale halo bias inferred is given in Table 2 and shown in Figs. 7.
At the mean redshift (0.34), it takes a value of
$b(\bar{z}=0.34)=0.99^{+0.08}_{-0.10}$, which extrapolated to redshift $z=0.1$
becomes $b(z=0.1)=0.92^{+0.07}_{-0.08}$. The deduced large-scale halo bias is
the same if we fit the HOD model with 4 or 5 parameters, see Table 2. Krumpe
et al. (2015) measured the bias of X-ray-selected AGN in a similar redshift
range but for intrinsically more luminous AGN. With this analysis, we add a
new measurement of the bias at lower soft X-ray luminosity: $8.1\times
10^{42}$ erg s-1. We confirm the weak positive correlation between bias and
soft X-ray luminosity found by Krumpe et al. (2015), see Fig. 7 bottom panel.
We fit a linear relationship between the quantities and obtain $b=(0.48\pm
0.14)L_{X}+(-19.68\pm 6.17)$. The slope value obtained is $3.3\sigma$
(0.48/0.14=3.3) away from 0.

Other X-ray-selected AGN clustering studies were either at lower redshift
(Cappelluti et al. 2010) or higher redshift (Gilli et al. 2009; Starikova et
al. 2011; Koutoulidis et al. 2013; Viitanen et al. 2019; Allevato et al. 2019)
and always covered higher luminosities. This new study is complementary to
them.

###  5.2 Halo abundance matching results

The model has two parameters: the fraction of satellite AGNs and the scatter
in the relation between stellar mass and X-ray luminosity. The predicted
curves extend to halo masses of $10^{11.5}{\rm M}_{\odot}$ (and not lower) due
to the resolution of the simulation used. Both impact the shape and amplitude
of the clustering signal and the HOD. Figure 8 shows the predicted HOD curves
for a subset of the parameter space explored. In the top left panel, the
satellite fraction is fixed to 10%, and the $\sigma$ parameter varies. The
lower the $\sigma$, the sharper the transition is. The $\sigma$ parameter from
SHAM is related to the $\sigma_{\log_{10}M}$ parameter from the HOD model.
Mock catalogs with higher $\sigma$ have a distribution of dark matter haloes
more extended towards lower masses. In the top right panel, $\sigma$ is fixed
to 0.8, and the $f_{{\rm sat}}$ varies. The higher the $f_{{\rm sat}}$, the
steeper the slope of the satellite occupation curve (the larger the $\alpha$
parameter). The $f_{{\rm sat}}$ parameter is related to both the $\alpha_{{\rm
sat}}$ and the $M_{{\rm sat}}$ HOD parameters.

We compute a distance, denoted $d$, between each predicted HOD curve
($N^{SHAM}(M)$) and the 50th precentile of the inferred HOD model as follows

$d=\Sigma^{M=15.5}_{M=11.5}\frac{\left[N^{SHAM}(M,f_{{\rm
sat}},\sigma)-N^{50\%}_{HOD}(M)\right]^{2}}{\left[N^{84.1\%}_{HOD}(M)-N^{15.9\%}_{HOD}(M)\right]^{2}}.$
(12)

Figure 8 (bottom panels) shows the distances as a function of $\sigma$ and
$f_{{\rm sat}}$. We find mock catalogs constructed with parameters satisfying
$\sigma<2-f_{{\rm sat}}/10$ predict halo occupation distributions well within
the contours of the 5-parameter best-fit HOD inferred from the observations
(Fig. 8 bottom left panel). Parameter combinations such as $\sigma>2-f_{{\rm
sat}}/10$ are less preferred (top right corner of the bottom left panel). $d$
is minimized for $\sigma=0.8$ and $f_{{\rm sat}}=4\%$, see the star in the
Figure. The six smallest distances are pointed out with empty circles. When
comparing to the 4-parameter best-fit HOD contours (Fig. 8 bottom right
panel), parameters within $\sigma<2.4-f_{{\rm sat}}/10$ and $\sigma>0.5$ are
acceptable. Here we find that solutions with low $\sigma$ are less preferred.
In that case, $d$ is minimized for $\sigma=1.2$, and $f_{{\rm sat}}=4$, see
the star in the Figure. The six smallest distances are pointed out with empty
circles. In both cases (comparing either to the 4-parameter of the 5-parameter
HOD fit), the best solutions point towards low $f_{{\rm sat}}$ values.

Mocks with both high $\sigma$ and high $f_{{\rm sat}}$ are far from the
observations and are ruled out.

Figure 8: SHAM predictions plotted with the fitted HOD model 1 $\sigma$
contour of the 4 and 5 parameters HOD fit results. On the top left panel, the
satellite fraction is fixed to 10%, and the $\sigma$ parameter is varied. On
the top right panel, $\sigma$ is fixed to 0.8, and the $f_{{\rm sat}}$ is
varied. On the bottom left (right) panel is shown, as a function of $\sigma$
and $f_{{\rm sat}}$, the distance between the HOD predicted by SHAM models and
the 5-parameter (4-parameter) HOD inferred from the observations. The black
dashed line on the bottom left panel corresponds to $\sigma=2-f_{{\rm
sat}}/10$. When comparing to the 5-parameter HOD fit, the bottom left half,
below the $\sigma<2-f_{{\rm sat}}/10$ line of the parameter space, is
preferred. Compared to the 4-parameter HOD fit (bottom right panel), the
bottom left half is also preferred. The dashed line represents
$\sigma=2.4-f_{{\rm sat}}/10$. The star identifies the lowest distance model.
Empty circles identify the six lowest distance models.

##  6 Summary and discussion

This article provides a complete picture of how soft X-ray AGNs populate the
cosmic web (Fig. 1). This achievement is possible thanks to two factors: (i)
the combination of the eROSITA eFEDS X-ray survey with its dedicated SDSS
spectroscopic follow-up and with the HSC S19A lensing products (ii) the
complementary nature of the two summary statistics fitted (Figs. 3, 4). We
obtain meaningful HOD constraints for an X-ray-selected AGN sample (Fig. 5,
6). We interpret the summary statistics with state-of-the-art HOD and SHAM
models (Sect. 5). We set upon firm footage the fact that the mass distribution
of haloes hosting X-ray-selected AGN is broad, as hinted by previous studies.
Both models point to a shallower satellite slope than for galaxy surveys
meaning that the satellite fraction for X-ray-selected AGN is low, similar to
the findings of Miyaji et al. (2011). Interestingly, we find a relatively
large $\sigma_{\log_{10}M}$ that is likely related to the width of the
specific accretion rate distribution. Contrasting our results with those of
Krumpe et al. (2015), the large-scale halo bias of X-ray-selected AGN appears
to correlate (3.3$\sigma$ significance) with soft band X-ray luminosity (Fig.
7). We compare the results with predictions from SHAM models and can rule out
a portion of the parameter space (Fig. 8).

###  6.1 On the $\sigma$ and $\sigma_{\log_{10}M}$ parameters

The $\sigma$ parameter in the SHAM model is the scatter in the abundance
matching relation between the stellar mass of the galaxy hosting the AGN and
the AGN hard X-ray luminosity (2–10 keV). The probability distribution
function of specific accretion rate resulting from a broad range of galaxy
stellar masses (hosting AGN) is close to a power-law (with slope -1) in the
range $31.5<\log_{10}(\lambda_{SAR})<33.5$ (see Fig. 5 of Comparat et al.
(2019)). The distribution obtained deviates from the power law at high
accretion rates. Indeed the scatter induces an exponential cut-off. The
distribution at the faint end of the function would require higher-resolution
simulations to be populated. The HOD results obtained here are compatible with
SHAM models if $\sigma\in[0.8,1.2]$ and incompatible for low values of
$\sigma<0.5$ (for the four parameters HOD fit) or high values (for the
5-parameter HOD fit). Indeed low values of $\sigma$ induce a steeper
probability distribution function of specific accretion rate, which are
excluded by observations (Georgakakis et al. 2017). In the opposite regime,
large values of $\sigma$ induce a shallow (tending to become flat) probability
distribution function of specific accretion rate when considering the entire
population. In a sense, the $\sigma$ SHAM parameter is related to how broad
the distribution of specific accretion rate and its slope is.

The $\sigma_{\log_{10}M}$ parameter characterizes how broad the host halo
distribution is. It is related to the diversity of host galaxies and their
stellar mass via the stellar-to-halo mass relation (Moster et al. 2013;
Behroozi et al. 2013). The relatively high $\sigma_{\log_{10}M}\sim 1.3$
parameter obtained indicates that the host halo mass distribution and, thus,
the host stellar mass distribution are both broad. This is consistent with
studies of the AGN host-galaxy stellar mass function (Bongiorno et al. 2016).

So, it seems that both models point to the same general interpretation: the
distribution of host-galaxy stellar mass and that of specific accretion rate
are ‘broad’, which strengthens direct observations of these distributions
(Bongiorno et al. 2016; Georgakakis et al. 2017), even if these might be
subject to systematic effects in the measurement of the stellar mass of type 1
AGN (Ciesla et al. 2015).

With the innermost lensing measurements, we measure the baryonic lensing mass
for this sample ($M^{*}_{12}$) to be within $4\times 10^{10}$ and $3\times
10^{11}{\rm M}_{\odot}$. From the mock catalog, we predict a broad stellar
mass distribution of AGN host galaxies with a median $4\times 10^{10}{\rm
M}_{\odot}$ and a large standard deviation of 0.6 dex. So, the SHAM predicted
stellar mass being smaller than the HOD inferred baryonic lensing mass; we
find that interpretations from the two models are consistent.

###  6.2 On the satellite occupation

As suggested in Leauthaud et al. (2015), the combination of clustering and
lensing best constraints satellite occupation statistics. Compared to previous
studies, we take here a significant step forward. Indeed satellite fractions
inferred from clustering studies are limited by the precision of redshift in
the presence of broad line AGNs, and for example Shen et al. (2013) or
Rodríguez-Torres et al. (2017) could not constrain it. Lensing studies were
limited by small numbers of X-ray-selected AGN (Leauthaud et al. 2015) and
showed large uncertainties in the satellite occupation statistics. By
combining eFEDS with HSC, we find a preference for low satellite fractions
(HOD upper limit is $f_{{\rm sat}}<20\%$ and SHAM best fits are with $f_{{\rm
sat}}<12\%$). The HOD result shows a preference for a shallow satellite slope
($\sim 0.75$) that is smaller than measured for galaxy samples ($\alpha\sim
1-1.1$ and $f_{{\rm sat}}$ of 40% for galaxies with a stellar mass of 3$\times
10^{10}{\rm M}_{\odot}$) (Zehavi et al. 2011; Zu & Mandelbaum 2015).

The low satellite fraction could, in part, be due to the soft X-ray selection
of the AGN. Indeed, satellite AGN could be obscured and only detectable in
hard X-ray or the infrared (Kocevski et al. 2015; Krumpe et al. 2018). Krumpe
et al. (2018) compared the cross-correlation functions (CCF) of Swift BAT AGNs
with 2MASS redshift survey galaxies and their HODs. Since Swift BAT AGN sample
is hard (14-195 keV) X-ray-selected, it contains a larger fraction of type 2
obscured AGN than eROSITA-based samples. They found clear suppression of the
1-halo term in type 1 AGN CCF compared to type 2. The HOD analysis shows
$\alpha\sim 1$ for the type 2 AGN HOD while that of the type 1 AGN was
$\alpha\la 0.6$. Powell et al. (2018) obtained similar results. A possible
scenario causing the low $\alpha$ is the suppression of sub-halo mergers in
high velocity encounters in high mass halos (Altamirano-Dévora et al. 2016;
Oogi et al. 2020). An alternative interpretation of the apparent shallow slope
is that the satellite HOD slope is not shallow, but the satellite distribution
profile within the dark matter halo does not follow the mass density profile
assumed in the HOD modeling. If the satellite distribution is suppressed
towards the outer part of the halo, the ordinary HOD modeling would result in
low fitted $\alpha$. Indeed, it would appear as if the satellites were
suppressed in high-mass halos with large virial radii. However, one should be
cautious as such interpretations are still a debate.

###  6.3 Triggering mechanism for soft X-ray AGN in the cosmic web

The general SHAM scheme applied to populate mock catalogs with AGN accounts
for the observations satisfactorily. One important assumption made in the SHAM
model is that the assignment of an AGN to a galaxy is independent of the
environment: it ignores the properties of the neighboring haloes. It implies
that, to the first order, the larger scale environment, beyond the galaxy host
halo, is not the primary driver to turn on the AGN. Instead, the local
environment (within the virial radius) i.e., the circumgalactic medium, the
interstellar medium, and the stellar populations, are likely more decisive
parameters. It agrees with the findings of Yang et al. (2018a); Allevato et
al. (2019); Siudek et al. (2023). It emphasizes internal processes and their
role as AGN triggers, for example, disc instabilities (e.g. Bournaud et al.
2011) or the presence of bars (e.g. Ohta et al. 2007).

The fact that the satellite slope is shallower than that of galaxies with
equivalent stellar mass means that the in-fall of a satellite on a larger
structure makes it less likely to host an AGN, even more, when structures are
larger. It likely illustrates that the gas stripping from satellite galaxies
in deep potential wells suppresses AGN. It is compatible with the environment
quenching mechanism described by Peng et al. (2010, 2012).

###  6.4 Outlook

The eROSITA eFEDs observations constitute about 1% of the full eROSITA All-Sky
Survey (eRASS). This study paves the way towards charting the co-evolution of
X-ray AGN and their host galaxies and dark matter haloes.

In the coming decade, by combining eROSITA with SDSS-V, 4MOST, and DESI
spectroscopic redshifts (Kollmeier et al. 2017; Merloni et al. 2019; DESI
Collaboration et al. 2016) and with LSST and Euclid lensing products, one will
be able to carry out the similar analysis over a larger area and on an
extended redshift range, up to z=1. Between eFEDS (120 deg2, $z<0.55$) and
future analysis (13,000 deg2, $z<1$), the comoving volume will increase by a
factor 450. HOD parameters should be inferred to the percent level. We will
accurately measure the halo occupation distributions as a function of host-
galaxy properties and AGN properties towards characterizing possible
correlations between HOD parameters and host-galaxy, AGN, and environmental
properties. With that, one should unravel the role of AGN in shaping the
galaxy population and its hot circumgalactic medium (Hopkins et al. 2006;
Comparat et al. 2022).

Complementary to HOD analysis is the direct or partial correlations with host
galaxy properties; see reviews from Brandt & Yang (2021); Brandt & Alexander
(2015). In recent years, spectral energy distribution fitting has dramatically
improved in retrieving unbiased galaxy stellar parameters of galaxies hosting
AGN (e.g., Mountrichas et al. 2021; Yang et al. 2022, Buchner et al. in prep).
The upcoming Rubin observatory LSST survey111https://www.lsst.org (Ivezić et
al. 2019) will provide deep multi-band imaging to be used to determine host
galaxy properties. In addition, the future
Euclid222https://www.cosmos.esa.int/web/euclid imaging space mission (Laureijs
et al. 2011) will enable accurate morphological measurements of AGN hosts on a
significant fraction of the extra-galactic sky. Together these will allow
charting of the physics of the connection between AGN, host galaxy morphology,
and stellar properties (e.g. Yang et al. 2019; Ni et al. 2019) and give
further insight on the ecology of the cosmic web of X-ray AGN.

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###### Acknowledgements.

MK is supported by the DFG grant KR 3338/4-1. T. M. is supported by UNAM-DGAPA
PAPIIT 111319 and CONACyT Ciencias Básica 252531.
This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE).
The development and construction of the eROSITA X-ray instrument was led by
MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP
(FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA.
The eROSITA data shown here were processed using the eSASS/NRTA software
system developed by the German eROSITA consortium.
The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University.
This paper makes use of software developed for Vera C. Rubin Observatory. We
thank the Rubin Observatory for making their code available as free software
at http://pipelines.lsst.io/.
This paper is based on data collected at the Subaru Telescope and retrieved
from the HSC data archive system, which is operated by the Subaru Telescope
and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out
with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ. We
are honored and grateful for the opportunity of observing the Universe from
Maunakea, which has the cultural, historical and natural significance in
Hawaii.
Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P.
Sloan Foundation, the Heising-Simons Foundation, the National Science
Foundation, and the Participating Institutions. SDSS acknowledges support and
resources from the Center for High-Performance Computing at the University of
Utah. The SDSS web site is www.sdss5.org.
SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration, including the Carnegie Institution for
Science, Chilean National Time Allocation Committee (CNTAC) ratified
researchers, the Gotham Participation Group, Harvard University, Heidelberg
University, The Johns Hopkins University, L’Ecole polytechnique fédérale de
Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Extraterrestrische Physik (MPE), Nanjing University, National Astronomical
Observatories of China (NAOC), New Mexico State University, The Ohio State
University, Pennsylvania State University, Smithsonian Astrophysical
Observatory, Space Telescope Science Institute (STScI), the Stellar
Astrophysics Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Illinois
at Urbana-Champaign, University of Toronto, University of Utah, University of
Virginia, Yale University, and Yunnan University.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah. The SDSS web
site is www.sdss.org.
SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration including the Brazilian Participation
Group, the Carnegie Institution for Science, Carnegie Mellon University,
Center for Astrophysics — Harvard & Smithsonian (CfA), the Chilean
Participation Group, the French Participation Group, Instituto de Astrofísica
de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatório Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

##  Appendix A X-ray data analysis

###  A.1 X-ray mask

We use the region files created by eSASS/srctool to create X-ray masks for
point sources (PS) and extended sources (EXT) (Liu et al. 2022b, a). Each
source has its signal-to-noise ratio measured as a function of radius
(circular apertures). An optimal radius for source extraction is found by
maximizing the signal-to-noise ratio given the local background surface
brightness. It is clipped to a minimum radius of 10″(MINIMUM_SOURCE_RADIUS
parameter) and a maximum radius of the 99% energy enclosed fraction radius of
the point spread function. We use this maximum signal-to-noise radius as a
starting point to determine the area to be masked around sources.

We measure the cross-correlation as a function of scale between events
(0.2-2.3 keV) and sources in the catalog. We measure it for bins of the number
of counts measured per source in the detection band. The cross-correlation
becomes constant above a particular angular scale, which corresponds to a
conservative masking radius of a source (with a given number of counts), i.e.,
its average imprint on the sky, see Fig. 9 top panels. For each cross-
correlation curve, we measure the radius at which its value is between 1.25
and two times that of the constant values measured at large separation. This
brackets the masking radius within the black vertical error bars shown in Fig.
9 bottom panels. We find that this cross-correlation masking radius for point
sources is, on average, 40 percent (20 for extended sources) larger than the
eSASS/srctool radius of maximum signal-to-noise, see Fig. 9 bottom panels. The
srctool mask is likely not conservative enough for our purpose. For instance,
the detection of a point source just beyond the eSASS/srctool masking radius
of another point source will be subject to biases due to the residual events
measured via the cross-correlation. Though, if we followed the average masking
radius suggested by the cross-correlation (Fig. 9 bottom panels), the large
scatter in the relation between the maximum signal-to-noise radius and the
total number of counts would be missed. So to have a conservative mask that
closely follows the data, we multiply the masking radii from eSASS/srctool by
a factor of 1.4 (this is more conservative than required for the extended
sources, but it simplifies the procedure). In that way, the masking radius
will reach, on average, the line obtained from the cross-correlation. Doing so
ensures no remaining correlation between the set of events outside the mask
and the source catalog. We are conservatively masking both point sources and
extended sources individually.

After applying the mask, we are left with 17,523 AGN candidates. We are using
the fraction of random points (see § 2.1.4) that fall in the masks, we
estimate the area of the observed X-ray sky effectively occupied by sources.
In all, sources occupy 9.805 deg2 out of 141.97 deg2. AGN occupy 6.914 deg2,
stars 0.988 deg2, and extended sources 2.057 deg2.

Figure 9: Cross-correlation between sources and events as a function of
angular separation for extended (EXT, top left) and point-like (PS, top right)
X-ray sources. Bottom. Masking radius v.s. log10 of the counts measured
(bottom). Average masking radius obtained with the cross-correlation (black
line). The masking radius obtained with eSASS/srctool for individual sources
is systematically lower than the black line. Its best-fit polynomial (blue
dashed) is multiplied by 1.2 (extended sources) or 1.4 (point sources) to
align with the black line.

###  A.2 Random catalogue

We use the sensitivity map produced by the eROSITA pipeline Brunner et al.
(eSASS, apetool 2022) with a parameter $P_{thres}=e^{-8}=0.00033$ (the Poisson
probability threshold below which an excess of counts is considered a source),
corresponding to a detection likelihood of 8. It is a pixelated fits image
with size $[0,9000)\times[0,18000)$ containing the sensitivity limit (in
counts). Each random point falls in a pixel of this map, and we attach the
corresponding count limit $C^{lim}_{X}$ to the random number. We draw a large
set of redshift and X-ray fluxes ($f_{x}$, $z$) from the AGN X-ray luminosity
function projection to assign to each random point Aird et al. (2015);
Comparat et al. (2019). It is sampled down to $2\times 10^{-15}$erg cm-2 s-1,
a flux value at which the area curve is smaller than 0.5 deg2. We convert the
flux into an expected number of counts

$CT^{expected}=f_{x}\times ECF\times EEF\times t_{exp}+CT^{background}.$ (13)

where the energy conversion factor is $ECF=1.164\times 10^{12}$. The encircled
energy fraction is set to $EEF=0.65$. The exposure time, $t_{exp}$, is
obtained with the exposure map. $CT^{background}$ is obtained from the
background map. We draw a random Poisson variable $R_{v}$ for each
$CT^{expected}$. If this value exceeds the count limit, $R_{v}>C^{lim}_{X}$,
the point is accepted in the random sample. We remove the shallower areas at
the edge of the field through a minimum exposure time threshold to minimize
the maximum offset between the normalized cumulative distribution of the data
sample and the random sample. We find that an 830 seconds threshold minimizes
the KS-test values at 0.19% (0.81%) for R.A. (Dec.). It removes $\sim 10$
deg2. It is sufficiently accurate to estimate clustering on the photometric
sample. After masking extended sources and stars and trimming the low exposure
time region, the total number of random points remaining is 3,713,726.