The cosmic web of X-ray active galactic nuclei seen through the eROSITA Final Equatorial Depth Survey (eFEDS)

Which galaxies in the general population turn into active galactic nuclei (AGNs) is a keystone of galaxy formation and evolution. Thanks to SRG/eROSITA's contiguous 140 square degree pilot survey field, we constructed a large, complete, and unbiased soft X-ray flux-limited ($F_X>6.5\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$) 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 exceptionally complete view of the AGN halo occupation distribution. The population of AGNs is broadly distributed among halos with a mean mass of $3.9 _{- 2.4 }^{+ 2.0 }\times10^{12}M_\odot$. 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, $\sigma_{\log_{10}(M)}=1.28\pm0.2$. The satellite occupation distribution is characterized by a shallow slope, $\alpha_{{\rm sat}}=0.73\pm0.38$. 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 toward fully charting, in the coming decade, the coevolution of X-ray AGNs, their host galaxies, and dark matter halos by combining eROSITA with SDSS-V, 4MOST, DESI, LSST, and \textit{Euclid} data.


Introduction
Active galactic nuclei (AGNs) are a keystone in galaxy evolution.How they are triggered and fueled are essential questions, and answering them will deepen our understanding of the coevolution between galaxies, the gas surrounding them, and their central supermassive black holes (SMBHs; see reviews from Padovani et al. 2017;Eckert et al. 2021).This article focuses on the large-scale environment of X-ray-selected AGNs, namely the population of the dark matter halos that host 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 halos that host 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 AGNs were limited by the total number of X-ray AGNs or a small survey area.They typically measured the large-scale halo bias of AGNs selected in different fashions (Gilli et al. 2009;Cappelluti et al. 2010;Starikova et al. 2011;Koutoulidis et al. 2013Koutoulidis et al. , 2018;;Leauthaud et al. 2015;Viitanen et al. 2019;Allevato et al. 2019).The autocorrelation of X-ray-selected AGNs was studied locally (z ∼ 0.045) with 199 AGNs in the Swift-BAT (Burst Alert Telescope) all-sky survey by Cappelluti et al. (2010).They found these bright low redshift AGNs to be hosted, on average, by dark matter halos of mass 1.6 − 2.5 × 10 13 h −1 M , corresponding to a large-scale halo bias of 1.2±0.1.At higher redshifts (z ∼ 1) with deep pencil beam surveys (COSMOS observed with XMM and Chandra, Bootes, and Chandra compilations) and larger numbers of AGNs (ranging from 500 to 3,100), Gilli et al. (2009), Starikova et al. (2011), Koutoulidis et al. (2013), Plionis et al. (2018), Viitanen et al. (2019), and Allevato et al. (2019) inferred a large-scale halo bias of ∼ 2±0.2, corresponding to halo masses of 4 − 10 × 10 12 h −1 M .In these studies, further splitting the samples as a function of AGN type, luminosity, or host-galaxy properties is not very conclusive due to small statistics.There are hints of a correlation with X-ray luminosity and an indication of a low satellite fraction.The study of the angular autocorrelation of photometrically selected AGNs, 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 (Cosmic Evolution Survey) field (Scoville et al. 2007).They find that the 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 (HODs) 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.
Cross-correlations with a controlled galaxy population, done to circumvent the low signal-to-noise ratio in the autocorrelation functions, have recently been fruitful.Such studies relate AGN populations to their host dark matter halos (Krumpe et al. 2010(Krumpe et al. , 2012(Krumpe et al. , 2015(Krumpe et al. , 2018(Krumpe et al. , 2023;;Mendez et al. 2016;Mountrichas et al. 2019;Zhang et al. 2021).They cross-correlated a similar number of X-ray-selected AGNs (between 300 and 1500) with spectroscopic galaxy surveys: 2MASS (Two Micron All Sky Survey), SDSS (Sloan Digital Sky Survey), VIPERS (VIMOS Public Extragalactic Redshift Survey), and 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 autocorrelation 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 (extended ROentgen Survey with an Imaging Telescope Array) (Predehl et al. 2021), the number density of X-ray AGNs increased to more than a hundred per square degree in the eROSITA Final Equatorial Depth Survey (eFEDS; 140 deg 2 , ∼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 and SDSS-V; Abdurro'uf et al. 2022;Kollmeier et al. 2017, Merloni et al. in preparation) enabled the accurate measurement of redshifts for about 11 thousand X-ray point sources in eFEDS (i.e., for ∼ 50% of the sources).This number of X-ray AGNs with spectra is already comparable to its predecessor follow-up of ROSAT (ROentgen SATellite) 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).It measured accurate galaxy shapes for more than 20 source galaxies per square arcminute over vast areas (1,400 deg 2 ), which almost completely cover the eFEDS field (Mandelbaum et al. 2018b).
With these two outstanding observational advances, we measure the autocorrelation function and the galaxy-galaxy lensing signal of X-ray-selected AGNs 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 for measuring the clustering and galaxy-galaxy lensing in Sect.3. The HOD and sub-halo abundance matching (SHAM) models used are detailed in Sect. 4. Results are discussed in Sects.5 and 6.Throughout, we assume a flat Lambda cold dark matter cosmology with H 0 = 67.74km s −1 Mpc −1 and Ω m (z = 0)=0.3089(Planck Collaboration et al. 2020).The halo mass is defined as 200 times the background density.The halo mass function used is that of Tinker et al. (2010).The uncertainties are 1σ unless stated otherwise.Magnitudes are in the AB system (Oke & Gunn 1983).Throughout the article, we use the shorthand "AGNs" to designate "X-rayselected AGNs."

Data
In this section we describe the X-ray observations (Sect.2.1) and the weak-lensing data products (Sect.2.2).

eROSITA eFEDS
We used 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 deg 2 , detected with a likelihood greater than 8 (ERO_DET_LIKE > 8) and with a reliable counterpart (CTP quality ≥ 2) determined as described in Salvato et al. (2022).The catalog is flux-limited to F X > 6.5 × 10 −15 erg s −1 cm −2 in the soft X-ray (0.5-2 keV).Simulations are only at X-ray wavelengths 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).A study of the trade-off between purity and completeness shows that counterparts with a threshold of CTP quality ≥ 2 (p_any > 0.035) have a purity and completeness both equal to 95%.A total of 2,160 sources are classified as stars either via astrometry, spectroscopy, X-ray, and optical/IR colors or via a dedicated analysis as described in Schneider et al. (2022) and were 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 CLUSTER_CLASS ≥ 3 and are masked here.
After these cuts from the eFEDS point source catalog, we are left with 17,902 AGN candidates over 140 deg 2 (density of 127.9 per square degree).Figure 1 illustrates the light cone considered in this analysis.

Masks
We had to 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 the description in Sect.2.1.5).As the masking radius for each detected source, we used its radius of maximum signal-to-noise augmented by 40 percent.This radius was determined while extracting the X-ray spectrum of each source (Liu et al. 2022b); the details of this process are provided in Appendix A.1.
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 Kolmogorov-Smirnov test values (between random and data vectors) with a minimal area loss (see Sect. 2.1.5).After applying the minimum exposure time cut, we are left with 16,308 AGN candidates over 128 deg 2 , resulting in a density of ∼127.4 deg −2 .

Photometric redshifts
Photometric redshift estimation for galaxies that host AGNs is complex (e.g., Salvato et al. 2018).In the eROSITA/eFEDS case, Salvato et al. (2022) 2019), we find that the clustering measured using a photometric redshift with such a dispersion and fraction of outliers would result in losing between one-third and one-half of the amplitude of the clustering signal.So we did not use the photometric redshift to measure clustering statistics; instead, we focused on the subsample of 10,680 AGNs with spectroscopic observations (see Sect. 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 (Baryon Oscillation Spectroscopic Survey) spectrographs (1000 fibers per plate; SDSS-IV; Blanton et al. 2017) and in 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 the section entitled "SPIDERS" of Abdurro'uf et al. 2022), and the spectra are part of the SDSS Data Release (DR)18 (Almeida et al. 2023).The total area covered by SDSS-IV and V spectroscopic observations is 133.77deg 2 (95% of the eFEDS area).The obtained spectroscopic redshift completeness depends on (i) the position in the sky and (ii) the optical magnitude of the source.We considered 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 ∼0.55, the spectroscopic sample is a fair subsample (as a function of optical magnitude and X-ray flux) of the entire population.Since SDSS-V observations are limited to z-band magnitudes brighter than 21.5, beyond a redshift of 0.55 we are missing a significant fraction of the AGNs that are optically faint X-ray-selected AGNs (see Fig. 2).
We estimated the spectroscopic completeness in ∼3.5 deg 2 equal area pixels (half the size of an SDSS plate, ∼7 deg 2 ).The minimum (maximum) completeness measured in a pixel is 13% (69%).The relative variations in the spectroscopic redshift dis-Fig.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 a redshift of 0.6.At higher redshifts, completeness at the faint end impacts the sample significantly.
tribution as a function of completeness are within the expected fluctuations for pixels with completeness levels above 40%.So, we discarded areas with completeness lower than 40%.This removed about 20 deg 2 of the area located at the edge of the eFEDS field (most of it overlaps with the low-exposure regions).

AGN sample
In summary, to measure the clustering, we created an AGN sample that covered redshifts between 0.05 and 0.55, where the spectroscopic sample is not biased compared to the parent photoz sample.We obtained a sample of 1,992 AGNs with spectroscopic redshift covering 122.3 deg 2 .Figure 1 illustrates how the AGNs considered in this analysis sample the large-scale structure observed with galaxies and groups from the GAMA (Galaxy and Mass Assembly) survey and eROSITA eFEDS clusters.Table 1 summarizes the main properties of the considered sample.
Table 1.Properties of the sample.The minimum, mean, maximum, and standard deviation of the redshift, the soft-band X-ray luminosity, and the g, r, and z magnitudes are from the legacy survey (Dey et al. 2019 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.Among the 1992 AGNs studied here, 1648 (82.7%) have their spectroscopic redshifts measurements from SDSS observations, and 270 (13.5%) from GAMA observations (Liske et al. 2015).The other spectroscopic redshifts originate from: Wig-gleZ Dark Energy Survey ( 46 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.Given the negligible impact of the verification on the summary statistics, we did not cut the sample further. Additional splitting the sample in soft X-ray luminositylimited 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 AGNs with spectroscopic redshifts are required to investigate the trends with parameters that define the sample.

Random catalog
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 drew a set of random points with a large uniform density of ∼81,000 deg −2 on the sky (about 11.5 million points on eFEDS).We first trimmed it to precisely follow the edges of the survey.We then followed the methodology of Georgakakis et al. (2008) to down-sample 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 down-sampling, masking (see the previous section), and trimming (low exposure time region) is 3,713,726.The density of random points, ∼ 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 down-sampled the random catalog to follow the spectroscopic redshift completeness map and its dependency on right ascension and declination.We cut 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 shuffled the set of observed redshifts and assigned them to the random points, regardless of the completeness level.

HSC-SSP weak-lensing data
We used 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 cover about 512 deg 2 , and it reduces to 433.48deg 2 after the full-color full-depth selection.With the i-band magnitude cut of 24.5, the observed number density reached 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.We refer the readers to the year 3 shape catalog paper by Li et al. (2022) for more details.
We briefly describe the photometric redshift accuracy, the year 3 shape catalog, and the measurements of the shear estimator in the following subsections.

Photometry and photometric redshifts
Good photometry is a precondition for the measurement of photometric redshifts.The performance of the HSC photometry was tested in two ways.The first was an internal test that compared the point spread function (PSF) magnitude to the Kron magnitude for a bright star sample (i<21.5) in the Wide XMM-LSS (X-ray Multi-Mirror satellite Large-Scale Structure Survey) field.The standard deviation of the difference (PsfMag-KronMag) achieved is better than 1%, independently of the filters and fields.In addition, the difference between the CModel magnitude and PSF magnitude is below 0.2%.For the external test, Pan-STARRS-1 (PS1) stars brighter than r-band 20 mag are used (Chambers et al. 2016).The scatter level is also at about the 1% level, indicating good photometric performance.

Shape catalog
The HSC galaxy sample was 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 were measured via a re-Gaussianization method (Hirata & Seljak 2003) (reGauss, which has been merged with GalSim, Rowe et al. 2015), and the PSF effects were 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.We 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: where b/a is the ratio between the minor axis and major axis, and φ is the position angle of the major axis with respect to the sky coordinates.The shear distortion, γ i , is then related to the e i (i=1,2) such that where R is the response of the galaxy ellipticity to a small distortion defined in Kaiser et al. (1995) and Bernstein & Jarvis (2002).The response was calculated from the calibrated parameters e rms and σ e based on simulations (Mandelbaum et al. 2018a;Li et al. 2022) as follows The weighting term in Eq. 3, w i , is composed of the percomponent error from the simulation due to the photon noise, σ e;i , and the rms of galaxy shape distribution, e rms;i , w i = 1/(σ 2 e;i + e 2 rms;i ).The reGauss algorithm suffers from several estimation biases, for example model bias, noise bias, and selection bias, which can be classified into multiplicative bias, m i , and additive bias, c i (i=1,2), such that The final shear estimator was obtained with Eq. 5.It does not incorporate the geometry factor Σ crit described in Sect.3.2: Both multiplicative and additive biases are calibrated based on the simulations mentioned above.The two biases were then assigned to each galaxy as a function of S/N and resolution R 2 .Additionally, there is selection bias as well as weight bias.The overall bias was quantified as the residuals for both multiplicative bias, δm, and additive bias, δa, both of which can reach below the 1% level for the HSC-SSP Y3 shape catalog (Li et al. 2022).

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 more generically named the galaxy-halo connection (Sheth & Tormen 1999;Wechsler & Tinker 2018).With the data described above, we computed two summary statistics: the AGN-AGN autocorrelation (clustering; Sect.3.1) and galaxy-galaxy lensing with the AGN population as the lenses (Sect.3.2).

Clustering measurement
We used the Landy & Szalay (1993) estimator to measure the projected two-point correlation function, labeled w p (r p ) (for a detailed definition, see, e.g., Davis & Peebles 1983).To count pairs and integrate along the line of sight, we used the corrfunc software (Sinha & Garrison 2020).For the integration, we used π max = 40 h −1 Mpc.We carried out measurements with shorter and longer π max and found that with 40, we would obtain the largest signal-to-noise in the clustering measurement.
We randomly down-sampled the catalog of random points for the clustering measurement to have 20 times the number of AGNs.To have consistent 3D positions between the optical spectra and the X-ray sources, we computed the clustering using the position on the sky of the optical counterparts (Salvato et al. 2022); we did 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 was 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. (2019Comparat et al. ( , 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 estimated the cosmic variance in this field to be 1%, which we added as a constant systematic uncertainty at all scales to the clustering measurement.We note that it is small compared to the 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 excluded from the fitting procedure clustering measurements with a separation larger than 40 h −1 Mpc.The total signal-tonoise in the clustering measurement is 17.7, split into 11 radial bins.We sampled 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 arcseconds (∼0.25h −1 Mpc at the mean redshift of the sample).The eROSITA PSF is 30 arcseconds, so X-ray-selected AGN pairs with a separation smaller than one arcminute are hardly detected.Moreover, since the AGN sample considered here is sparse (120 deg −2 ∼0.03 arc minutes −2 ) and spread over a long line of sight, AGN close pairs are small in number.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 arcseconds to typically be a handful (i.e., fewer than ten).Only half have physical separations smaller than 40 h −1 Mpc.So, the number of missed pairs due to fiber collisions is negligible.In our case, we thus consider that fiber collisions are not an issue, and we defined our lowest separation bin at 0.25 h −1 Mpc.

Galaxy-galaxy lensing measurement
The galaxy-galaxy lensing measurement is a cross-correlation between positions of foreground lenses (AGNs in our case) and shapes of background galaxies acting as sources (HSC galaxies; see the reviews Bartelmann &Schneider 2001 andRefregier 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 combined 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; ∆Σ), that is We followed the measurement procedure described in Miyatake et al. (2019) and Luo et al. (2022), that In the above ESD estimator, R(R) is the response of the shape estimator, which, for this work, takes a value of 0.84.The w ls is the weight for each lens-source galaxy pair, and w l is a weight assigned to each lens galaxy.We used w l = 1, meaning there are no particular requirements on redshift, stellar mass, or other properties for the lens catalog.The e t,ls is the tangential component of the source galaxy shape with respect to the lens.The factor K(R) accounts for the multiplicative bias calibrated based on a suite of simulations developed in Mandelbaum et al. (2018a) and Li et al. (2022).
Two blinding schemes are provided for systematic sanity tests, the low-and the high-accuracy blinding scheme.We used the former, in which a value of δm 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 δm 1 is added and encrypted for each user of the shape catalog.It is then decrypted and removed by subtracting the δm 1 term.
We applied an extra selection function for each lens-source pair following Medezinski et al. (2018) so that the accumulated probability of the P(z) satisfies Figure 4 shows the obtained galaxy-galaxy lensing measurements (black) as well as the best-fit HOD model (red).Measurements with a separation smaller than 20 h −1 Mpc were included in the fitting procedure.The total signal-to-noise in the lensing measurement is 46, split into 15 radial bins.We sampled the separation range with 5.5 bins per decade in evenly log-spaced (0.18 dex) steps between 0.025 (10 −1.6 ) and 8.3 (10 0.9 ) h −1 Mpc.The measurement extends to separations that are one-tenth of the size of the clustering measurement.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.

Models
We interpreted the clustering and lensing summary statistics using models of halo occupation statistics (Cooray & Sheth 2002;Guo et al. 2010).There are three flavors of models: HOD (Berlind & Weinberg 2002;Kravtsov et al. 2004;Zheng et al. 2005Zheng et al. , 2007)), 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).The HOD and SHAM models reproduce measurements of the galaxy clustering as a function of luminosity and color (Zehavi et al. 2011;Marulli et al. 2013).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 halos and the dark matter.However, due to the finite resolution of the simulations they build upon, they are currently limited to predicting statistics at the high mass end only (Nishimichi et al. 2019).So, we first adjusted the HOD model parameters to the measurements obtained in the previous section (see Sect. 4.1).Then, we compared measurements to the prediction of the SHAM model from Comparat et al. (2019, see our Sect.4.2).

Halo occupation distribution model
As a baseline, we used the HOD model formulated by More et al. (2015).It is described by the two equations below, in which N C ( N S ) gives the average occupation of a dark matter halo of mass M by a central (satellite) galaxy: The model has five parameters θ = (M min , σ log 10 M , α sat , M sat , M * 12 ).Only a fraction of distinct halos host 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 sat , the parameter passed to the fitting routine is M sat − M min with the boundaries specified in Table 2. To fit for the ∆Σ measurement at small separations and benefit from the signal present, we needed to add a prediction for a point-like mass term that represents the baryonic lensing mass of the AGN host galaxies.We added the parameter M * 12 as follows: The posterior of this parameter represents the mean baryonic lensing mass of the galaxies that host AGNs.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 that host AGNs.
In total, we fit for five parameters on the two measurements, ∆Σ (w p (r p )), which have S/N=46 (17.7) in 15 (11) radial bins.The parameters were sampled with a flat prior (in linear space) within broad boundaries, as specified in Table 2.

Sub-halo abundance matching models
The Comparat et al. (2019Comparat et al. ( , 2020b) ) empirical AGN model statistically links the dark matter halos 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 the halos exist.The model has two parameters: the fraction of AGNs in satellites sub-halos 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 catalogs of Liu et al. (2022c) in Fig. 3.It is consistent with observations.It was obtained with f sat = 10% (fraction of AGNs that are satellites) and σ = 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. (2019Comparat et al. ( , 2020a)).At that time (before this study), such parameters resulted in reasonable predictions.
Creating a complete SHAM-base mock catalog is timeconsuming (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 predicted the HOD curves as a function of f sat and σ.Thus, we sampled a small and finite number of ( f sat , σ) combinations and created individual mock catalogs to predict the HOD curves.

Results
We discuss here the results of the fitting procedure and the comparison between models.In Sects.5.1 and 5.2 we discuss the results obtained with the HOD and SHAM models, respectively.For the first time, we measure with relatively small uncertainties the complete HOD of a low redshift flux-limited X-ray-selected sample of AGNs.We obtain a global view of the distribution of halos that host X-ray AGNs (see Fig. 6).

HOD results
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 each summary statistic individually 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 and 4. The models are meaningful, 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 halos, M min takes a median posterior value of 13.06±0.44,and the width of the error function is found at σ log 10 M = 1.3 ± 0.2.There is a low 1σ 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 fit best at α sat =0.73±0.38,and the transition occurs in halos 10 to 100 times more massive than the typical halo : M sat − M min = 1.46 ± 0.52.Both summary statistics point to these parameter values (Fig. 5).The typical baryonic lensing mass of galaxies that host these AGNs is M * 12 = -0.96± 0.45.This sets an upper limit to the mean stellar mass of galaxies that host AGNs of ∼ 10 11 M .The 1σ boundaries encompass 3.9 × 10 10 and 3.1 × 10 11 M , which is in fair agreement with expectations from the 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 noted a degeneracy between M min and σ log 10 M , which we investigated.We decreased the number of parameters by fixing σ log 10 M to 1.3 (similar to the best-fit value) and 1 (to force a less broad halo distribution and a sharper transition).We obtained a set of best-fit parameters (see Table 2) that are compatible with the five-parameter fit.When fixing σ log 10 M to 1.3, we obtained M min 13.09 ± 0.19, a value similar to the five-parameter fit but with half the uncertainty.Results for other parameters remain unchanged.When fixing σ log 10 M to 1, M min is logically forced to lower values, to ∼ 12.5, to be able to fit the overall signal.The HOD posterior is close to that of the five-parameter fit (see the black and orange contours in Fig. 6 ).
We obtain a global view of the distribution of halos that X-ray AGNs (see Fig. 6).The four-parameter best-fit model is within the 1σ uncertainty of the five-parameter best-fit model.Due to the degeneracy between M min and σ log 10 M , the fourparameter fit HOD with σ log 10 M = 1 is skewed toward lower masses compared to the five-parameter fit.With the HOD model, we derive an average halo mass hosting a central (central or satellite) AGN of 3.93 +2.03 −2.44 ×10 12 M (4.95 +2.63 −1.99 ×10 12 M ).These values are comparable to the findings of Rodríguez-Torres et al. (2017).We find the distribution of halo masses to be broad.We thus confirm that quoting a typical halo mass will be extremely sensitive to the definition of what "typical" means (see the discussion in Leauthaud et al. 2015).
The direct HOD predictions from mock catalogs from Leauthaud et al. ( 2015 2019) has a higher σ log 10 M value and is comparable to the five-parameter fit (see more discussion in the SHAM section below).The normalization of N(M) 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 that host X-ray AGNs.However, measuring reliable host-galaxy stellar masses in the case of type 1 AGNs is complex (Ciesla et al. 2015;Zou et al. 2022) and is left for future studies.
The large-scale halo bias inferred is given in Table 2 and shown in Fig. 7.At the mean redshift (0.34), it takes a value of b(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 four or five parameters (see Table 2).Krumpe et al. (2015) measured the bias of X-ray-selected AGNs in a similar redshift range but for intrinsically more luminous AGNs.With our analysis, we add a new bias measurement at lower soft X-ray luminosity: 8.1 × 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 the bottom panel of our Fig.7 ).We fit a linear relationship between the quantities and obtain b = (0.48 ± 0.14)L X + (−19.68 ± 6.17).The slope value obtained is 3.3σ (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

Halo abundance matching results
The model has two parameters: the fraction of satellite AGNs and the scatter in the relation between stellar mass and Xray luminosity.The predicted curves extend to halo masses of 10 11.5 M (and not lower) due to the resolution of the simulation used.Both parameters 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 σ parameter varies.The lower the σ, the sharper the transition is.The σ parameter from SHAM is related to the σ log 10 M parameter from the HOD model.Mock catalogs with higher σ have a distribution of dark matter halos more extended toward lower masses.In the top-right panel, σ is fixed to 0.8, and the f sat varies.The higher the f sat , the steeper the slope of the satellite occupation curve (the larger the α parameter).The f sat parameter is related to both the α sat and the M sat HOD parameters.
We computed a distance, denoted d, between each predicted HOD curve (N S HAM (M)) and the 50th percentile of the inferred HOD model as follows, Mocks with both high σ and high f sat differ significantly from the observations and are ruled out.

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, and (ii) the complementary nature of the two fitted summary statistics (Figs. 3 and 4).We obtain meaningful HOD constraints for an X-rayselected AGN sample (Fig. 5, 6).We interpreted the summary statistics with state-of-the-art HOD and SHAM models (Sect.5).We provide evidence supporting the idea that the mass distribution of halos that host X-ray-selected AGN is broad, as hinted at 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 σ 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 AGNs appears to correlate (3.3σ significance) with soft-band X-ray luminosity (Fig. 7).We compared the results with predictions from SHAM models and can rule out a portion of the parameter space (Fig. 8).
6.1.On the σ and σ log 10 M parameters The σ 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 the specific accretion rate resulting from a broad range of stellar masses of galaxies (that host AGNs) is close to a power law (with slope -1) in the range 31.5 < log 10 (λ S AR ) < 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 cutoff.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 those of SHAM models if σ ∈ [0.8, 1.2] and incompatible for low values of σ < 0.5 (for the four-parameter HOD fit) or high values (for the five-parameter HOD fit).Indeed, low values of σ induce a steeper probability distribution function of the specific accretion rate and are excluded by observations (Georgakakis et al. 2017).In the opposite regime, large values of σ induce a shallow (tending to flat) probability distribution function of the specific accretion rate when considering the entire population.In a sense, the σ SHAM parameter is related to how broad the distribution of the specific accretion rate and its slope is.
The σ 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 σ log 10 M ∼ 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 the specific accretion rate are "broad," which strengthens the validity of direct observations of these distributions (Bongiorno et al. 2016;Georgakakis et al. 2017), even if they might be subject to systematic effects in the measurement of the stellar mass of type 1 AGNs (Ciesla et al. 2015).
With the innermost lensing measurements, we measure the baryonic lensing mass for this sample (M * 12 ) to be within 4×10 10 and 3 × 10 11 M .From the mock catalog, we predict a broad stellar mass distribution of AGN host galaxies with a median of 4 × 10 10 M and a large standard deviation of 0.6 dex.Therefore, the SHAM method predicted a stellar mass smaller than the HOD-inferred baryonic lensing mass; we find that interpretations from the two models are consistent.

On the satellite occupation
As suggested in Leauthaud et al. (2015), the combination of clustering and lensing best constrains satellite occupation statistics.Compared to previous studies, here we take a significant step forward.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 AGNs (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 sat < 20% and SHAM best fits are with f sat < 12%).The HOD result shows a preference for a shallow satellite slope (∼ 0.75) that is smaller than that measured for galaxy samples (α ∼ 1 − 1.1 and f sat of 40% for galaxies with a stellar mass of 3×10 10 M ; 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 AGNs 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 of Swift BAT AGNs with 2MASS redshift survey galaxies and their HODs.Since the Swift BAT AGN sample is hard (14-195 keV) X-ray-selected, it contains a larger fraction of type 2 obscured AGNs than Fig. 8. SHAM predictions plotted with the fitted HOD model 1 σ contour of the four-and five-parameter HOD fit results.In the top-left panel, the satellite fraction is fixed to 10%, and the σ parameter is varied.In the top-right panel, σ is fixed to 0.8, and the f sat is varied.In the bottom-left (bottom-right) panel, shown as a function of σ and f sat , is the distance between the HOD predicted by SHAM models and the five-parameter (four-parameter) HOD inferred from the observations.The dashed black line in the bottom-left panel corresponds to σ = 2 − f sat /10.When compared to the five-parameter HOD fit, the bottom-left half, below the σ < 2 − f sat /10 line of the parameter space, is preferred.Compared to the four-parameter HOD fit (bottom-right panel), the bottom-left half is also preferred.The dashed line represents σ = 2.4 − f sat /10.The star identifies the lowest distance model.Empty circles identify the six lowest distance models.
eROSITA-based samples.They find clear suppression of the one-halo term in type 1 AGN cross-correlation functions compared to type 2. The HOD analysis shows α ∼ 1 for the type 2 AGN HOD, while that of the type 1 AGN was α < ∼ 0.6.Powell et al. ( 2018) obtained similar results.A possible scenario causing the low α is the suppression of sub-halo mergers in high velocity encounters in high mass halos (Altamirano-Dévora et al. 2016;Oogi et al. 2020).
At low redshift, previous dynamical analyses of clusters (M halo > 10 14 M ) have shown that luminous X-ray-selected AGNs are preferentially in-falling onto clusters and that the cluster environment suppresses their X-ray activity (e.g., Haines et al. 2012;Ehlert et al. 2013Ehlert et al. , 2014)).More recent studies of AGN abundance as a function of cluster-centric radius involving larger numbers of clusters confirmed this picture up to a redshift of 1 (Koulouridis et al. 2018;Koulouridis & Bartalucci 2019;Mishra & Dai 2020).This activity may in part be due to ram pressure feeding and stripping (Poggianti et al. 2017a,b;Peluso et al. 2022) or to close pair interaction of galaxy mergers (Koulouridis et al. 2013;Ehlert et al. 2015).Pimbblet et al. (2013) reached similar conclusions using optically selected AGNs.An alternative interpretation of the apparent shallow slope is therefore that the satellite HOD slope is not shallow but rather the satellite distribution profile within the dark matter halo does not follow the mass density (Navarro-Frenk-White) profile assumed in the HOD modeling.If the satellite distribution is suppressed toward the outer (inner) part of the halo, the ordinary HOD modeling would result in low (high) fitted α.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 matter of debate.Petter et al. (2023) also find a low satellite fraction (between 5 and 20%) at redshift 1.4.But the luminous AGN fraction in clusters was found to evolve (increase) rapidly with redshift (Martini et al. 2013;Mishra & Dai 2020).We still lack a clear picture of how AGNs are satellites of galaxy clusters.Upcoming large samples of X-ray-selected clusters and AGNs with spectroscopic redshifts are key to settling this issue (Merloni et al. 2019;Finoguenov et al. 2019).

Triggering mechanism for soft X-ray AGNs in the cosmic web
The general SHAM scheme applied to populate mock catalogs with AGNs satisfactorily accounts for the observations.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 halos.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) -that is, the circumgalactic medium, the interstellar medium, and the stellar populations -are likely more decisive parameters.This is in agreement with the findings of Yang et al. (2018a), Allevato et al. (2019), and 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 masses means that the in-fall of a satellite on a larger structure makes it less likely to host an AGN; it is increasingly unlikely the larger the structure is.This likely illustrates that the gas stripping from satellite galaxies in deep potential wells suppresses AGNs.It is compatible with the environment quenching mechanism described by Peng et al. (2010Peng et al. ( , 2012)).

Outlook
The eROSITA eFEDS observations constitute about 1% of the full eROSITA All-Sky Survey (eRASS).This study paves the way toward charting the coevolution of X-ray AGNs and their host galaxies and dark matter halos.
In the coming decade, by combining eROSITA with SDSS-V, 4MOST, and DESI (Dark Energy Spectroscopic Instrument) spectroscopic redshifts (Kollmeier et al. 2017;Merloni et al. 2019;DESI Collaboration et al. 2016) and with LSST (Vera C. Rubin Observatory 10-year Legacy Survey of Space and Time) and Euclid lensing products, one will be able to carry out a 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 analyses (13,000 deg 2 , z < 1), the comoving volume will increase by a factor of 450, and HOD parameters should be inferred to the percent level.We will accurately measure the HODs as a function of host-galaxy properties and AGN properties with the aim of characterizing possible correlations between HOD parameters and host-galaxy, AGN, and environmental properties.With that, one should be able to unravel the role of AGNs in shaping the galaxy population and its hot circumgalactic medium (Hopkins et al. 2006;Comparat et al. 2022).
Complementary to HOD analysis are 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 become dramatically better at retrieving unbiased galaxy stellar parameters of galaxies that host AGNs (e.g., Mountrichas et al. 2021;Yang et al. 2022;Zou et al. 2022, Buchner et al. in prep).The upcoming Rubin Observatory LSST survey1 (Ivezić et al. 2019) will provide deep multi-band imaging to be used to determine host galaxy properties.In addition, the future Euclid 2 imaging space mission (Laureijs et al. 2011) will enable accurate morphological measurements of AGN hosts on a significant fraction of the extragalactic sky.Together they will allow the physics of the connection between AGNs, host galaxy morphology, and stellar properties to be charted (e.g., Yang et al. 2019;Ni et al. 2019) and provide further insight into the ecology of the cosmic web of X-ray AGNs.

Fig. 1 .
Fig. 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; Salvato et al. 2022).The surrounding large-scale structure is sampled by GAMA galaxies (gray points; Driver et al. 2022) and GAMA galaxy groups (green squares; Robotham et al. 2011) as well as by eROSITA eFEDS clusters (red circles; Liu et al. 2022a).

Fig. 3 .
Fig. 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.

Fig. 4 .
Fig. 4. Galaxy-galaxy lensing measurement.The excess surface mass density measurement with HSC S19A data uses as lenses the X-ray fluxlimited 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.

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

Fig. 6 .
Fig. 6.Inferred HOD (solid) split into central (dashes) and satellite (dots) for the four-and five-parameter HOD fits (orange and black).The four-parameter best-fit model is within the 1σ uncertainty of the five-parameter best-fit model.Due to the degeneracy between M min and σ log 10 M , the four-parameter fit HOD with σ log 10 M = 1 is skewed toward lower masses compared to the five-parameter fit.The direct predictions from the Leauthaud et al. (2015), Georgakakis et al. (2018), and Comparat et al. (2019) mock catalogs are shown in the right panel.They are within the fitted contours obtained.The mocks from Leauthaud et al. (2015) and Georgakakis et al. (2018) have a lower σ log 10 M value (sharper transition) and are thus more in line with the four-parameter fit.The mock from Comparat et al. (2019) has a higher σ log 10 M value and is comparable to the five-parameter fit.

Figure 8 (
Figure8(bottom panels) shows the distances as a function of σ and f sat .We find that mock catalogs constructed with parameters satisfying σ < 2 − f sat /10 predict HODs well within the contours of the five-parameter best-fit HOD inferred from the observations (Fig.8bottom-left panel).Parameter combinations

Fig. 7 .
Fig. 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±0.143)x+(−19.684±6.173)between the soft X-ray luminosity and the large-scale halo bias.

Fig
Fig. A.1.Cross-correlation between sources and events as a function of angular separation for extended (EXT, top-left panel) and point-like (PS, top-right panel) X-ray sources.The lower panels show the masking radius vs. log10 of the counts measured.The average masking radius obtained with the cross-correlation is shown with the black line.The masking radius obtained with eSASS/srctool for individual sources is systematically lower than the black line.Its best-fit polynomial (dashed blue line) is multiplied by 1.2 (extended sources) or 1.4 (point sources) to align with the black line.
measured photometric redshifts to have σ N MAD = 1.48 × median , we find that σ N MAD decreases to ∼ 0.03, while the outlier fraction remains the same, 20%.With the help of the simulation from Comparat et al. ( ).