11institutetext: Max Planck Institute for Extraterrestrial Physics (MPE),
Giessenbachstrasse 1, 85748 Garching bei München, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
33institutetext: Dipartimento di Fisica e Astronomia ”Augusto Righi”, Alma
Mater Studiorum Università di Bologna, via Gobetti 93/2, 40129 Bologna, Italy
44institutetext: INAF-Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, via Gobetti 93/3, 40129 Bologna, Italy 55institutetext: Institute for
Astronomy and Astrophysics, National Observatory of Athens, V. Paulou and I.
Metaxa, 11532, Greece 66institutetext: Dr. Karl Remeis-Sternwarte and Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander Universität Erlangen-
Nürnberg, Sternwartstraße 7, 96049 Bamberg, Germany 77institutetext: Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam,
Germany 88institutetext: Instituto de Astronomía y Ciencias Planetarias,
Universidad de Atacama, Copayapu 485, Copiapó, Chile 99institutetext:
Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile.
1010institutetext: Instituto de Astrofísica, Facultad de Física, Pontificia
Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860,
Macul Santiago, Chile, 7820436 1111institutetext: Centro de Astroingeniería,
Facultad de Física, Pontificia Universidad Católica de Chile, Campus San
Joaquín, Av. Vicuña Mackenna 4860, Macul Santiago, Chile, 7820436
1212institutetext: Millennium Institute of Astrophysics, Nuncio Monseñor
Sótero Sanz 100, Of 104, Providencia, Santiago, Chile 1313institutetext: Space
Science Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado 80301
1414institutetext: Department of Astronomy & Astrophysics, 525 Davey Lab, The
Pennsylvania State University, University Park, PA 16802, USA
1515institutetext: Institute for Gravitation and the Cosmos, The Pennsylvania
State University, University Park, PA 16802, USA 1616institutetext: Department
of Physics, 104 Davey Laboratory, The Pennsylvania State University,
University Park, PA 16802, USA 1717institutetext: Max-Planck-Institut für
Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany

# The eROSITA Final Equatorial Depth Survey (eFEDS):

The hard X-ray selected sample

K. Nandra 11 S. G. H. Waddell 11 T. Liu 11 J. Buchner 11 T. Dwelly 11 M.
Salvato 11 Y. Shen 22 Q. Wu 22 R. Arcodia 11 Th. Boller 11 H. Brunner 11 M.
Brusa 3344 W. Collmar 11 J. Comparat 11 A. Georgakakis 55 M. Grau 11 S.
Hämmerich 66 H. Ibarra-Medel 88 Z. Igo 11 M. Krumpe 77 G. Lamer 77 A. Merloni
11 B. Musiimenta 3344 J. Wolf 11 R.J. Assef 99 F.E. Bauer 1010111112121313
W.N. Brandt 141415151616 H.-W. Rix 1717

(Received XXX; accepted XXX)

###### Abstract

Context. During its calibration and performance verification phase, the
eROSITA instrument aboard the SRG satellite performed a uniform wide–area
X-ray survey of approximately 140 deg2 in a region of the sky known as the
eROSITA Final Equatorial Depth Survey (eFEDS).

Aims. The primary aim of eFEDS is to demonstrate the scientific performance to
be expected at the end of the 8-pass eROSITA all sky survey. This will provide
the first focussed image of the whole sky in the hard X-ray ($>2$ keV)
bandpass. The expected source population in this energy range is thus of great
interest, particularly for AGN studies.

Methods. We use the 2.3–5 keV selection presented by Brunner et al. (2022) to
construct a sample of 246 point-like hard X-ray sources for further study and
characterization. These are classified as either extragalactic ($\sim 90$ %)
or Galactic ($\sim 10$ %), with the former consisting overwhelmingly of AGN
and the latter active stars. We concentrate our further analysis on the
extragalactic/AGN sample, describing their X-ray and multiwavelength
properties and comparing them to the eFEDS main AGN sample selected in the
softer 0.2-2.3 keV band.

Results. The eROSITA hard band selects a subsample of sources that is a factor
$>10$ brighter than the eFEDS main sample. The AGN within the hard population
reach up to $z=3.2$ but on the whole are relatively nearby, with median
$z$=0.34 compared to $z$=0.94 for the main sample. The hard survey probes
typical luminosities in the range $\log L_{\rm X}=43-46$. X-ray spectral
analysis shows significant intrinsic absorption (with $\log N_{\rm H}>21$) in
$\sim 20$ % of the sources, with a hard X-ray power law continuum with mean
$<\Gamma>=1.83\pm 0.04$, typical of AGN, but slightly harder than the soft-
selected eROSITA sample. Around $\sim 10$ % of the hard sample show a
significant ”soft excess” component. The sampled black hole mass distribution
in the eFEDS broad line AGN population is consistent with that of the deeper
COSMOS survey that probes a higher redshift populations. On the other hand,
the Eddington ratios appear systematically lower, consistent with the idea
that the decline in SMBH activity since $z\sim 1$ is due to a reduction in the
typical accretion rate, rather than a shift towards activity in lower mass
black holes.

Conclusions. The eFEDS hard sample provides a preview of what can be expected
from the eRASS final survey in terms of data quality. This pilot survey
indicates the power of eROSITA to shed new light on the demographics and
evolution of AGN, and the potential for discovery of new and rare populations.

###### Key Words.:

Surveys – Catalogs – X-rays: galaxies – X-rays: stars

##  1 Introduction

The eROSITA instrument (Predehl et al., 2021) aboard the Spectrum-RG satellite
(Sunyaev et al., 2021) consists of seven Wolter-1 type telescopes focussing
X-rays onto seven focal plane X-ray cameras. It is designed to survey the sky
in the X-ray band, with the driving science being to constrain the
cosmological parameters via the evolution of clusters of galaxies (Merloni et
al., 2012). A key aim is to detect all clusters in the Universe more massive
than about $3\times 10^{14}M_{\odot}$. This requires all-sky coverage with
high surface-brightness sensitivity, combined with angular resolution
sufficient to resolve clusters from the much larger population of X-ray point
sources in the sky, dominated by active galactic nuclei (AGN) and coronally
active stars. To do this efficiently requires an imaging telescope with large
field of view and effective area in the soft X-ray band.

Figure 1: The 2.3–5 keV eFEDS image, smoothed by a Gaussian kernel of
$\sigma=16\arcsec$, with each point source overlaid as a red annulus with
radii of $60\arcsec$ and $300\arcsec$. Representations of the regions of the
XMM-XXL-north (orange) and the COSMOS (yellow) fields, which do not overlap
with eFEDS, are placed at the lower right corner for size comparison.

While the primary driving science of eROSITA is cluster cosmology, the high
sensitivity of the instrument and mission profile of SRG enable a vast array
of additional science (Merloni et al., 2012). Unlike its predecessor ROSAT
(Truemper, 1982), the eROSITA telescope system features significant effective
area in the harder X-ray band above 2 keV. Together with its sister instrument
Mikhail Pavlinsky ART-XC (Pavlinsky et al., 2021), which covers an even harder
X-ray band (4–30 keV), it is the first focussing telescope to have performed
an all-sky survey at energies $>2$ keV. eROSITA therefore opens up a new
parameter space for large area hard X-ray surveys.

These build on a considerable body of survey work in this energy range, which
have covered various regions of area-depth parameter space. One early example
was the HEAO-1/A2 survey, which covered the whole sky. Despite its high flux
limit this has nonetheless been very influential as it established a reference
set of X-ray bright extragalactic objects (Piccinotti et al., 1982) which have
been the subject of increasingly intense study by subsequent X-ray missions.
Numerically, the extragalactic hard X-ray population found by HEAO-1 was
dominated by Active Galactic Nuclei (AGN) hosting accreting supermassive black
holes, with the next most numerous class being massive clusters of galaxies.

While soft X-ray surveys, such as those performed by ROSAT and now eROSITA,
are arguably more efficient for the selection of galaxy clusters, harder X-ray
surveys are of particular interest for AGN studies. One reason for this is
that hard X-ray emission can be used to uncover heavily obscured AGN and
measure their level of obscuration (e.g. Awaki et al., 1991; Turner et al.,
1997), as well as their contribution to the X-ray background (e.g. Comastri et
al., 1995; Gilli et al., 2007; Ueda et al., 2014). These factors have
motivated numerous surveys with X-ray imaging telescopes with sensitivity
above 2 keV, for example with ASCA (Ueda et al., 1999), BeppoSAX (Fiore et
al., 2001), Chandra (e.g., Nandra et al., 2015; Civano et al., 2016; Luo et
al., 2017) and XMM-Newton (e.g. Fiore et al., 2003; Hasinger et al., 2007;
Ranalli et al., 2013; Pierre et al., 2016). Even harder bandpasses with
greater sensitivity to the most heavily obscured AGN have now also been
covered by Swift (Oh et al., 2018) and NuSTAR (Alexander et al., 2013). These
and other similar surveys have formed the backbone for AGN demographic and
evolutionary studies over the past two decades (e.g., Ueda et al., 2003; La
Franca et al., 2005; Aird et al., 2015; Buchner et al., 2015). Despite the
great depth of surveys in particular with Chandra and XMM-Newton, they can
necessarily only cover limited areas. For example, the largest area surveys
with each facility XBootes (Murray et al., 2005; Kenter et al., 2005) and XMM-
XXL (Pierre et al., 2016) reach flux limits of around $2\times 10^{-14}$ erg
cm-2 s-1 in the 2-10 keV band but cover only 9 and (two) 25 deg2 contiguous
regions respectively. A significant region of parameter space exists in
between these Chandra/XMM surveys and the with HEAO-1/A2 (Piccinotti et al.,
1982) and the similar-depth RXTE All-Sky Survey (Revnivtsev et al., 2004),
which cover the whole sky but at a flux limit around 3 orders of magnitude
brighter. This gap in depth-area coverage can neatly be filled by a wide-field
imaging X-ray telescope like eROSITA.

As a precursor to the eROSITA all-sky survey (eRASS), eROSITA performed a
field scan of an equatorial region of approximately 140 deg2 during its
calibration and performance verification phase. Aside from all-sky surveys
such as ROSAT and the eRASS itself, this survey, known as the eROSITA Final
Equatorial Depth Survey (eFEDS) currently represents the largest contiguous
dedicated X-ray imaging survey performed thus far, despite taking only
approximately 5 days. Occupying a new part of area-depth parameter space for
contiguous surveys (Brunner et al., 2022), and with excellent supporting data
in numerous other wavebands, eFEDS is capable of yielding new insights,
particularly into the properties and evolution of clusters and AGN. It also
provides a verification and foretaste of what can be expected ultimately from
the full 4-years exposure of the entire sky (eRASS:8).

In this paper, we present an analysis of the X-ray and optical properties of
the hard X-ray (2.3–5 keV) point sources detected in eFEDS. In Section 2 we
briefly describe the sources of the data and the methodology from which the
catalogue products are derived. Section 3 presents the basic properties of the
sample. In Section 4 we focus on the subsample of extragalactic sources,
predominantly AGN. A discussion and outlook for the eventual 8-pass, 4 year
eRASS:8 all-sky survey is presented in Section 5.

Throughout this paper, we adopt a flat $\Lambda$CDM cosmology with
$\Omega_{\Lambda}=0.7$, $\Omega_{m}=0.3$, and $H_{0}=70$ km s-1 Mpc-1.

##  2 Data and Methods

###  2.1 The eROSITA Final Equatorial Depth Survey

A full description of the eFEDS X-ray survey is presented in Brunner et al.
(2022), which also contains the X-ray catalogue used to construct the sample
presented in this paper. The eFEDS field covers approximately 140 deg2 in an
equatorial field chosen to have excellent multi-wavelength supporting data. Of
particular note for the current study are: (i) high quality optical imaging
and photometry provided by the HSC Wide Area Survey (Aihara et al., 2018);
(ii) multi-band photometry from the DESI Legacy Imaging survey (Dey et al.,
2019) data release 8 (LS8), which also incorporates mid infrared data from the
Wide-Field Infrared Spectroscopic Explorer (WISE; Wright et al. (2010)) and
(iii) extensive optical spectroscopic coverage mainly from the fourth and
fifth incarnations of the Sloan Digital Sky Survey (SDSS; Blanton et al.,
2017; Kollmeier et al., 2017, Kollmeier et al., in preparation). A fuller
description of the multi-wavelength data in this field can be found in Salvato
et al. (2022a).

The X-ray data were acquired in a field-scanning mode providing an unusually
uniform exposure over the field compared to the ”point and stare” observing
strategy commonly used for X-ray surveys. The nominal exposure at a given
position is typically 2.2ks, making the eFEDS survey representative of, but
slightly deeper than, the expectation for the typical equatorial sky position
in the full 8-pass eRASS:8.

The X-ray data processing and reduction was performed using the eROSITA
Science Analysis Software system (eSASS) using the eROSITA early data release
(EDR) version (eSASSusers_201009). The eSASS procesing chain is described in
full in Brunner et al. (2022). Source detection involves the generation of an
initial seed source catalogue using a sliding box detection algorithm, and
subsequent PSF-fitting to determine source counts and significances using a
likelihood ratio approach. An advantage to this method is that the detection
can be performed using event files/images in multiple energy bands
simultaneously. In practice, the eFEDS main sample is assembled using a soft
X-ray selection in the 0.2-2.3 keV band, where the instrument is most
sensitive. This main eFEDS source catalogue contains a total of 27,910 sources
above a detection likelihood threshold DET_LIKE$>6$ in the detection band,
where DET_LIKE$=-\ln P$ where $P$ is the chance probability of the source
being a background fluctuation based on a PSF-fitting detection algorithm (see
Brunner et al., 2022). The detection algorithm allows for the sources to be
extended, also providing a likelihood estimation for this, the quantity
EXT_LIKE. Sources with EXT_LIKE$>=6$ are considered candidate galaxy clusters
whose properties are presented and analysed in detail by Liu et al. (2022a).
Sources with EXT_LIKE$<6$ are considered point-like and have their EXT_LIKE
fixed to zero in the catalogue of Brunner et al. (2022). The point sources can
have a variety of classifications, and are characterised by a dedicated
algorithm (see below and Salvato et al., 2022b).

###  2.2 The eFEDS hard X-ray catalogue

Figure 2: Histograms of the 2-10 keV X-ray flux (left) and detection
likelihood DET_LIKE (right) for our sample. Sources detected only in the hard
band are shown separately.

To supplement the main, soft X-ray source catalogue, Brunner et al. (2022)
also present a hard X-ray selected catalogue. This is constructed using a
source–detection run performed simultaneously in three energy bands: 0.2–0.6,
0.6–2.3 and 2.3–5 keV (see Brunner et al., 2022, for details), which results
in a likelihood in each of the these individual bands, as well as a combined
likelihood for all three bands. To construct the catalogue presented here, a
threshold of the three-band summary likelihood DET_LIKE_0$>5$ was adopted. We
then selected sources with a detection likelihood in the hardest of those
three bands (2.3–5 keV) of DET_LIKE_3$>10$. The former threshold is
sufficiently loose that it has no impact on the latter, in other words, the
DET_LIKE_0$>5$ threshold does not remove any potential source with
DET_LIKE_3$>10$. There are eight extended sources with DET_LIKE_3$>10$ and
EXT_LIKE$>=6$ which are candidate galaxy clusters (Liu et al., 2022a) and thus
excluded from this work. The multi-band detection is performed because, as we
show below, the vast majority of the hard-selected sources are also detected
in the soft band. Inclusion of the soft photons in these cases can
substantially improve the determination of the X-ray source position, which
will lead to a more accurate determination of the true hard source counts and
hence the likelihood of the hard detection. In addition, the better positional
accuracy should aid considerably in the counterpart identification. According
to detailed simulations by Liu et al. (2022c), the selection threshold imposed
for this catalog ensures a low fraction (2.5%) of spurious sources.

The resulting hard X-ray sample consists of 246 point-like sources, more than
two orders of magnitude fewer than the main sample. The sources’ positions
within the eFEDS field are shown in Fig. 1, superimposed on the 2.3–5 keV
smoothed image. Just 20 of the 246 sources are not also listed in the main
catalogue i.e. they are detected in the hard band but not the softer 0.2-2.3
keV band which is used to construct the main catalogue. The lower sensitivity
in the hard band can be attributed to two main factors. First, as can be seen
in (Predehl et al., 2021, ,see their Fig. 9) there is a pronounced drop in the
effective area of eROSITA around $\sim 2.3$ keV due to M-absorption edges in
the gold mirror coating. Above this energy the effective area continues to
decline due to the relatively short eROSITA focal length of 1.6m (compared,
e.g., to Chandra or XMM-Newton). Second, at harder X-ray energies, the
instrumental background starts to become dominant. This is larger than the
pre-flight predictions, partly related to the solar cycle during the first few
months after launch (Freyberg et al., 2020; Predehl et al., 2021), and reduces
the source detection sensitivity accordingly.

Figure 3: Normalised histogram of the soft X-ray (0.5-2 keV) flux comparing
the main sample (selected in the 0.2–2.3 keV band) with the hard sample
(selected in the 2.3–5 keV band). Note that the hard-band only sources are
included in the blue histogram, and will have large uncertainties in their
soft X-ray fluxes.

###  2.3 Optical identifications, classifications and redshifts

Optical counterparts for the hard X-ray sample have been identified by Salvato
et al. (2022b). A total of three different matching algorithms were applied:
the Bayesian NWAY code (Salvato et al., 2018), a maximum likelihood approach
(e.g., Sutherland & Saunders, 1992; Brusa et al., 2007) and HamStar, a
dedicated matching algorithm designed to identify X-ray emitting stars
(Schneider et al., 2022). The key datasets to which the X-ray source positions
are matched are LS8, which includes forced photometry from unWISE (Lang,
2014), and Gaia EDR3. The latter also provides parallax and proper motion
information that is crucial in distinguishing between Galactic and
extragalactic counterparts to the X-ray sources.

Figure 4: Legacy survey $r$-band (uppers panels) and Wise W1 3.4$\mu m$ (lower
panels) magnitude versus the 0.5–2 keV (left panels) and 2-10 keV (right
panels) X-ray flux for the eFEDS hard sample. Source classified as being
extragalactic (green symbols) and Galactic (purple symbols) are shown
separately. As a comparison the data for the AEGIS-XD survey (Nandra et al.,
2015) are shown (small black symbols), which cover slightly different bands in
the optical/IR, specifically Subaru Rc and Spitzer/IRAC 3.6$\mu$m.

A well-defined decision tree is applied to find the most likely counterpart,
or identify cases where the counterpart is unlikely or ambiguous. This results
in the allocation of a counterpart quality flag between 0-4, described in
detail in Salvato et al. (2022b). In short, however, counterpart qualities 0
and 1 are considered to be insecure, 3 and 4 to be secure. Counterpart quality
2 represents cases where the counterpart is ambiguous, and/or where more than
one source could contribute to the observed X-ray flux. The results for the
eFEDS hard sample are given in Section 3.1.

In addition to the counterparts, Salvato et al. (2022a) present a compilation
of optical spectroscopy of the most likely counterparts to our hard sources,
along with photometric–redshift determinations. The vast majority of the
optical spectra come from the Sloan Digital Sky Survey (SDSS) telescope (Gunn
et al., 2006) using the BOSS spectrographs (Smee et al., 2013). Of these, most
are from a dedicated program for X-ray followup within SDSS-IV (Blanton et
al., 2017), the SPIDERS program (Dwelly et al., 2017; Comparat et al., 2020).
X-ray sources in the eFEDS field have also been targeted explicitly as part of
the SDSS-V (Kollmeier et al., 2017) special plates program. The SDSS-V program
greatly increases both the spectroscopic completeness and the quality of the
data, given that the exposures for the SDSS-V special plates were generally
signigicantly longer. Including a smattering of redshifts from additional
programs, a total of 197/246 of our hard sample objects have a spectroscopic
redshift as detailed in Table 1. The redshifts used here were derived from the
compilation of eFEDS spectroscopic redshifts released as part of SDSS DR18
(Almeida et al., 2023) 111see also
https://www.sdss.org/dr18/data_access/value-added-catalogs/?vac_id=10001,
which have been verified by visual inspection, and superseded those presented
by Salvato et al. (2022b). The vast majority of spectroscopic objects are
extragalactic, with just 12 being spectroscopic stars.

Table 1: Optical Spectroscopy. Where multiple spectra exist of the same object, the table lists the surveys in preference order: (1) Blanton et al. (2017); (2) Almeida et al. (2023); (3) Baldry et al. (2018); (4) Jones et al. (2009); (5) Wenger et al. (2000); (6) Huchra et al. (2012);  Origin | Redshifts | Stellar | Ref
---|---|---|---
| $z$ | ($z<0.001$) |
SDSS | 180 | 2 | (1,2)
Gaia-RVS | 8 | 8 | –
GAMA | 2 | – | (3)
6dFGS | 2 | – | (4)
LAMOST | 2 | 1 | –
Simbad | 2 | 1 | (5)
2mrs | 1 | – | (6)
Total | 197 | 12 | –

Most of the spectroscopy in the field yields high–confidence redshifts. Three
objects have spectra from which no redshift could be determined, but are
detected in the radio in the FIRST survey. They are considered candidate BL
Lac type objects (see Section 4.6 for more discussion), with the lack of a
redshift determination being attributed to a lack of spectral features.
Reflecting the the fact that the hard sources are significantly brighter, the
hard sample has very high spectroscopic completeness (80%) compared to the
main sample(24%). Several previous hard X-ray surveys have also benefitted
from high spectral completeness for the same reason (e.g. Akiyama et al.,
2003; Della Ceca et al., 2004; Eckart et al., 2006; Cocchia et al., 2007).

For the relatively small but significant fraction of our sample for which
optical spectroscopy is not available, other information must be used to
classify the objects. Salvato et al. (2022b) present a classification scheme
primarily designed to differentiate between Galactic and extragalactic
objects, which is particularly important for the current study which focusses
on the properties of the latter. Aside from the spectroscopic stars (see
above), X-ray source counterparts which have a significant Gaia parallax
measurement are considered to be securely Galactic, as are all other objects
using the ”Hamstar” scheme designed to identify X-ray emitting stars
(Schneider et al., 2022). Conversely, objects which are extended in their
optical images are assumed to be securely extragalactic. For the remainder,
when possible a separation scheme based on optical/mid-IR colors is applied
(Salvato et al., 2022a) or otherwise an X-ray/mid-IR criterion (Salvato et
al., 2018).

Once secure or likely Galactic sources have been identified, we assign
photometric redshift estimates to the extragalactic sources without
spectroscopic redshifts. These are again discussed and presented in Salvato et
al. (2022b), and are based on template fitting of the IR-optical-UV Spectral
Energy Distribution (SED) using the LePhare code (Ilbert et al., 2006). The
photometric redshifts are assigned a “grade”, with a higher grade indicating
greater reliability of the photo-z estimate and in particular a lower outlier
fraction. Here we adopt REDSHIFT_GRADE $\geq 4$ in the Salvato et al.
catalogue, selecting only objects with either spectroscopic or the most
reliable photometric redshift measurements. The latter are selected by
comparing the SED-fitting photo-z with an independent measurement using
machine learning techniques, specifically a multi-layer perceptron (DNNZ;
Nishizawa et al., 2020), and adopting only those measurements where both
methods agree. For such cases the photometric redshift show a low scatter of
$\sigma_{\rm NMAD}=0.043$ and outlier fraction $5.3$ per cent (see Salvato et
al., 2022b, for details). The redshift information for the eFEDS hard sample
is provided in Section 4.1.

###  2.4 X-ray spectral fitting

We characterize the spectral properties of our sources both by hardness ratios
and full X-ray spectral fitting. We adopt a standard hardness ratio (HR)
definition:

$HR=(H-S)/(H+S)$

where $H$ is the flux in the 2.3–5 keV band and $S$ that in the 0.6–2.3 keV
band.

A systematic analysis of the X-ray spectra of the eFEDS main sample has been
presented by Liu et al. (2022b), who fit a variety of spectral models to
determine fluxes and luminosities, as well as physical parameters such as the
photon index and $N_{\rm H}$. Although most of the hard sources are included
in the main sample, the photon-level data can be different. For example the
source positions in the multi-band detected catalogue will in general be
subtly different to those in the soft band. We therefore perform an
independent spectral analysis for the hard sample separately from the main
sample, but using the same method.

Using the eSASS task srctool, for each source we calculate a signal-to-noise-
ratio optimized source extraction circular region and a source-free background
extraction annular region, adopting the same settings as used for the main
sample (Liu et al., 2022b). The source and background spectra and the response
files in the source regions are then extracted using srctool. Before fitting
the source spectra, each background spectrum is first fit using a model
composed of components extracted from a Principal Component Analysis (PCA) of
all the background spectra (described in the appendix of Simmonds et al.,
2018). This defines a background model specific to the given source. Then we
fit the source and background spectra simultaneously in the 0.2-8 keV energy
range (unless otherwise noted) using a source spectral model and the dedicated
background spectral model. Few source photons are detected above 8 keV due to
the decreasing effective area and consequent dominance of the instrumental
background at higher energies. A Bayesian spectral fit is performed with
BXA222https://github.com/JohannesBuchner/BXA (Buchner et al., 2014), which
connects XSPEC with the
UltraNest333https://github.com/JohannesBuchner/UltraNest/ nested sampling
package (Buchner, 2021). The robust MLFriends nested sampling algorithm
(Buchner, 2016, 2019) implemented in UltraNest explores the model parameter
space in a global fashion.

We adopt an absorbed power-law as the baseline model, and modify it according
to the requirements of sources with various signal-to-noise ratio, either
enriching it with additional components or decreasing the complexity by fixing
certain parameters. The models are listed in Table 2 and described in more
detail by Liu et al. (2022b). Two double-component models (m2: power-law plus
a softer power-law and m3: power-law plus a blackbody component) are applied
to describe the potential soft excess. Both of these models are
phenomenological in describing the soft excess, which may have different
origins and thus spectral shapes in different sources. The soft–excess
properties of this sample are discussed in further detail by Waddell et al.
(2023). Without the soft–excess component (m1), the power-law slope $\Gamma$
should be considered as a general description of the broad-band spectral
shape. With the soft excess component (m2 and m3), the primary power-law slope
can be taken as an intrinsic property of the X-ray emitting corona. The
$\Gamma$-fixed-powerlaw model (m4) is the same as the single-powerlaw model
(m1) but with $\Gamma$ fixed at 1.8. Model 5 (m5) is also the same as model 1
but fitting the 2.3–6 keV spectrum instead of the default 0.2–8 keV band. In
addition to the power-law-based models, a hot plasma model (APEC; Smith et
al., 2001) (m0), which is appropriate for stars, is used to calculate fluxes
of Galactic sources in this sample. In all the models, the Galactic absorption
is considered, adopting the total Galactic column density measured from the
HI4PI survey (HI4PI Collaboration et al., 2016) and the Schlegel et al. (1998)
dust map using the empirical correlation presented by Willingale et al. (2013)
(see Liu et al., 2022b, for details).

All the models in Table 2 are fitted to all the spectra. With each model, the
observed and absorption-corrected fluxes in the 0.5–2 and 2.3–5 keV band and
the intrinsic (absorption-corrected, rest-frame) fluxes and luminosities are
calculated in the 0.5–2, 2–10, and a narrow band around 2 keV (1.999–2.001
keV) to provide a monochromatic luminosity at that energy. In the case of the
hard-band fitting (m5), we only calculate the fluxes and luminosities in the
hard band ($>2$keV). In the case of the APEC stellar model (m0), we only
calculate the observed fluxes. Then we choose the most appropriate luminosity
measurements as described in Liu et al. (2022b). Unlike the main eFEDS sample
that contains many extremely faint sources, almost all the sources in this
hard sample have at least moderate S/N. The selection of luminosity
measurements can therefore be simplified by omitting special treatment of the
faintest sources with NHclass of ”uninformative”, since there are only two
such cases. For faint sources with unconstrained $\Gamma$ (selected based on
Kullback-Leibler divergence as described in Liu et al., 2022b), we adopt the
”$\Gamma$-fixed-powerlaw” model (m4). For the other, brighter sources, we
adopt the ”powerlaw+blackbody” model (m3) for the unobscured ones and the
”single-powerlaw” model (m1) for the obscured ones. The selection is discussed
in detail in Liu et al. (2022b). It provides reasonable but not-uniformly-
defined uncertainty measurements. One could also adopt the luminosity
measurement using one particular model (e.g., m3) if desired. These are
provided in the published catalogue (see Appendix A).

Name | Energy Range | Description
---|---|---
| (keV) |
m0 | 0.2-8 | single-temperature APEC model with redshift$=0$, free kT
m1 | 0.2-8 | single-powerlaw, free $\Gamma$, $N_{\rm H}$
m2 | 0.2-8 | double-powerlaw, free $\Gamma$s, $N_{\rm H}$
m3 | 0.2-8 | powerlaw plus blackbody, free $\Gamma$, kT, $N_{\rm H}$
m4 | 0.2-8 | same as m1 but fixed $\Gamma=1.8$
m5 | 2.3-6 | same as m1 but in a narrower band
Table 2: X-ray spectral models applied to the hard sample. In m1, m2, and m3,
a log-uniform prior is adopted for $N_{\textrm{H}}$ in the range $4\times
10^{19}\sim 4\times 10^{24}$ cm-2, and a Gaussian prior centered at 1.8 with
$\sigma=0.5$ is adopted for $\Gamma$ in the range $-2\sim 6$. The appropriate
Galactic absorption is applied to all models. More details including other
parameter prior ranges are described in Liu et al. (2022b).

###  2.5 Line Measurements and Black Hole Mass estimates

For objects with broad optical/UV emission lines, the central black hole mass
can be estimated (e.g., Kaspi et al., 2000). The high spectroscopic
completeness means such estimates are possible for a large fraction of our
sample.

We achieved this using the PyQSOFit code (Guo et al., 2018; Wu & Shen, 2022),
a python package for $\chi^{2}$ fiting of optical spectra of quasars. The
model consists of several components: a continuum which comprises a power-law,
a polynomial component and Balmer continuum; FeII emission in optical and
ultraviolet regime; and emission lines that are fitted as single or multiple
gaussians.

After host-galaxy decomposition via the PCA method of Yip et al. (2004) and
transformation into the rest-frame, the continuum model is fitted to
predefined regions that do not contain strong emission lines. Accurate
modeling of the FeII emission is of particular importance, and this was fitted
using the following templates in the relevant wavelength range:

  * •

$1000-2200$ Å: Vestergaard & Wilkes (2001)

  * •

$2200-3090$ Å: Salviander et al. (2007)

  * •

$3090-3500$ Å: Tsuzuki et al. (2006)

  * •

optical: Boroson & Green (1992)

The continuum and FeII model is subtracted from the data, leaving only the
contribution from emission lines. The emission lines are then fitted using
Gaussian functions; the maximum number of Gaussians was predefined for each
line component (e.g. 3 Gaussians for broad H$\beta$, 1 gaussian for narrow
H$\beta$). The output of the code consists of all parameters used to fit the
spectrum and an error estimate for each of them based on Monte Carlo
simulation.

Using the results, the black hole mass was calculated via:

$\log\left(\frac{M_{BH}}{M_{\odot}}\right)=A+B\log\left(\frac{\lambda
L_{\lambda}}{10^{44}\text{erg
s\textsuperscript{-1}}}\right)+C\log\left(\frac{\text{FWHM}}{\text{km
s\textsuperscript{-1}}}\right)$ (1)

with the monochromatic luminosity $L_{\lambda}$ at wavelength $\lambda$, the
full width at half maximum FWHM of the broad component of the emission line
and the constants A, B and C. The black hole mass was calculated using the
H$\beta$ and the MgII emission lines. For the H$\beta$ line, the constants of
Vestergaard & Peterson (2006) are used with the monochromatic luminosity at
5100Å; for MgII, the calibration by Shen & Liu (2012) was selected together
with the luminosity at 3000Å. The error of the black–hole mass was calculated
via error propagation of the Monte Carlo errors from the fit. These single-
epoch black hole mass estimates also carry a systematic uncertainty of around
0.4 dex (Shen, 2013). Bolometric luminosities were calculated from the 5100 or
3000Å monochromatic luminosities adopting the bolometric corrections from the
SDSS quasar sample (Richards et al., 2006).

##  3 Basic properties of the hard sample

While eROSITA is significantly more sensitive in the soft band, the eFEDS data
show that the instrument is also able to open up new parameter space for
large-area, hard X-ray science compared to previous surveys. Histograms of the
2–10 keV flux and DET_LIKE of our sample are shown in Fig. 2. The typical flux
of the hard band sources peaks at around $(1-2)\times 10^{-13}$ erg cm-2 s-1.
This is more than 2 orders of magnitude fainter than the HEAO-1 A2 all-sky
hard X-ray sample of Piccinotti et al. (1982) and the RXTE all-sky survey of
Revnivtsev et al. (2004), which have similar sensitivites below 10 keV.

A comparison of the soft X-ray (0.5–2) keV fluxes between the hard and main
sample is shown in Fig. 3. The former samples typically much brighter X-ray
fluxes, by more than an order of magnitude, given the higher flux limit
overall. There is nonetheless a tail to faint soft X-ray fluxes, mainly due to
the sources which are detected primarily in the hard band, some of which may
be heavily obscured and therefore have weak soft X-ray emission.

Table 3: Optical/IR counterpart quality (see text) comparing the hard and main samples, as well as the sources detected only in the hard band  Sample | Total | CTP$\\_$quality | Secure
---|---|---|---
|  | 0 | 1 | 2 | 3 | 4 | %
Hard | 246 | 4 | 11 | 3 | 0 | 228 | 92
Hard only | 20 | 4 | 7 | 1 | 0 | 8 | 40
Main | 27,369 | 1225 | 1370 | 2552 | 1379 | 20837 | 81

###  3.1 Optical properties

Turning to the optical identifications, the numbers of counterparts of various
types are shown in Table 3. For our hard sample, we find a very high fraction
of good counterparts, with 232 sources ($\sim 94$%) having a counterpart
quality $\geq 2$, significantly higher than for the main (soft) sample (81%).
This is most likely due to the fact that the typical main sample source is
fainter and hence has a fainter counterpart and additional ambiguity in the
association.

Table 4: Source classifications for the 232 hard-band sources with ${\tt CTP\\_quality}$ 2 or higher with comparison to classifications of the main sample. Sample | Extragalactic | Galactic
---|---|---
| Secure | Likely | All | Secure | Likely | All
Hard | 181 | 36 | 217 | 14 | 1 | 15
Hard (%) | 78.0 | 15.5 | 93.5 | 6.0 | 0.4 | 6.4
Main (%) | 48.4 | 46.6 | 95.0 | 3.3 | 1.7 | 5.0
Figure 5: Histogram of the hardness ratios split between sources classified as
Galactic and extragalactic, respectively. Note that sources without a high
confidence counterpart are not assigned a class in this plot. The
extragalactic sources show a tail to large hardness ratio which is suggestive
of a population of obscured AGN. Several of these are hard-band only
detections with consequently high values of hardness ratio. The vertical
dotted line indicates the hardness ratio expected for an absorbed power law
with photon index $\Gamma=1.8$ and Galactic column density typical of eFEDS of
$N_{\rm H}=3\times 10^{20}$ cm-2.

The assigned source classifications as described in Section 2.3 are shown in
Table 4. The hard sample has a higher fraction of secure classifications
compared to the main sample, reflecting the higher spectroscopic completeness.
The fraction of Galactic counterparts is also higher, most likely because on
average the hard sample sources are brighter, and the stellar content of hard
X-ray samples typically increases with flux.

Fig. 4 shows a comparison between the X-ray flux and optical $R$ and mid-IR W1
magnitudes for the sample, compared to the equivalent data for a
representative Chandra deep survey, AEGIS-XD (Nandra et al., 2015). The full
optical magnitude range for our sample is extremely broad, spanning the range
$R_{AB}=9-24$. The bulk of the AGN population in eFEDS covers a narrower range
from $R_{AB}=15-22$. As is typical of X-ray surveys, a locus is traced by the
AGN with a rough anti-correlation between X-ray flux and optical magnitude,
with large scatter. X-ray emitting stars are relatively optically-bright and
can be distinguished by their low X-ray/optical ratios which stand out from
the AGN locus. Interestingly, a few of the objects identified as Galactic show
ratios more typical of AGN. These could be accretion-powered binaries with
enhanced X-ray emission compared to normal stars. Conversely there are some
extragalactic objects which are optically as bright as stars.

###  3.2 Hard-band only sources and catalogue integrity

As noted earlier, of the 246 sources in the hard sample, 226 are also detected
and listed in the main, soft-selected X-ray catalogue of Brunner et al.
(2022). Thus 20 of our sources have no significant detection in the softer
bands, hereafter referred to as hard-band only (HBO) sources. If real, these
sources must be quite unusual and interesting spectrally, most likely being
very heavily obscured, with their soft X-ray emission suppressed by
photoelectric absorption hence evading detection in the highly sensitive
eROSITA soft bandpass.

On the other hand, the purity of our sample is not expected to be 100%.
Specifically, at our chosen likelihood threshold of DET_LIKE_3$>10$, the
simulations of Liu et al. (2022c) suggest a spurious source fraction of
approximately 2.5%. Based on this we would expect around 6 of the 246 sources
in the hard band not to reflect astrophysical X-ray emission and instead could
be random background flucuations. These spurious sources are more likely to be
found in the HBO subsample, as the detection in the softer band(s) for the
others increases the probability that the X-ray emission has an astrophysical
origin. It can be seen from Fig. 2 that the HBO sources tend to be clustered
at low detection likelihood (albeit not at particularly faint hard X-ray flux)
compared to the remainder of the sample.

The suspicion that some of the HBO sources may be spurious is reinforced by
the identification statistics. As can be seen in Table 3 the fraction of the
HBO sources without a valid counterpart is much higher than for the whole
sample. Indeed, considering the 226 sources which are detected in both the
hard and softer bands, $>98\%$ of them have a valid counterpart
(CTP_QUALITY$\geq 2$) while only $\sim 40\%$ of the HBO sources do. Indeed
these statistics are entirely consistent with all of the predicted $\sim 6$
spurious sources in our sample being HBO sources. On the other hand, the 9 HBO
sources with valid optical counterparts seem likely to be real, with the
association of the optical source with the X-ray emission lending additional
confidence to the X-ray detection itself. We note further that these sources
all have an extragalactic classification, which is reassuring given that
coronally-emitting stars are typically very strong soft X-ray emitters.

Figure 6: (left panel) Redshift distribution for the extragalactic sample.
Sources with spectroscopic redshift and high reliability photometric redshift
are shown separately. (right panel) Normalised redshift histograms for
spectroscopic and highly reliable photo-z comparing the hard and soft samples.
The hard X-ray selection yields a relatively local AGN sample, with a median
redshift $<z>=0.34$, compared to the main sample which has $<z>=0.94$.

###  3.3 Basic spectral properties

Fig. 5 shows the hardness ratio distribution of our sample, split into
Galactic and extragalactic sources for sources with reliable classifications.
Both span a reasonably wide range of hardness ratio, which in part will
reflect the uncertainties in the fluxes from which the HR is calculated. The
extragalactic sources show a peak hardness ratio corresponding to an
unobscured power law with $\Gamma\sim 1.7-1.8$, typical of AGN, but it can be
seen clearly that the extragalactic sources show a tail of hard X-ray sources
which is not present in the stellar sample. This is entirely expected because
the extragalactic sample will contain some fraction of obscured AGN (see
Section 4.3.2), while the stars will generally show soft emission. A very hard
X-ray source with a stellar identification would tend to be indicative of an
absorbed binary system, where the ID is of a star, and the X-ray emission from
accretion onto a compact companion.

##  4 The AGN sub-sample

In the remainder of this paper we restrict our discussion to a subsample of
the sources which meet all the following criteria:

  * •

Are contained within the 90%-area region444This exposure depth thresholding
results in 90% of the total area of eFEDS excluding the edges of the field,
which have a strong exposure gradient (Brunner et al., 2022), and provides a
relatively clean selection function for subsequent statistical analysis such
as the X-ray luminosity function (Buchner et al., in preparation), where the
exposure is at least 500s (241/246 sources).

  * •

Have counterpart quality 2 or higher (228/241 sources).

  * •

Are considered secure or likely extragalactic sources (213/228 sources)

This sample is likely to consist overwhelmingly of AGN and we examine their
properties in further detail below.

###  4.1 Redshifts and Luminosities

The compilation of spectroscopic redshifts in this field is summarised for the
hard sample in Table 1. A total of 179 out of 213 candidate extragalactic
sources have a secure spec-z. On top of this spectroscopic sample, an
additional 21 sources have a highly reliable photometric redshift
determination. We focus our further attention on this subsample of 200
sources, excluding the 13 where the redshift determination is less certain.
Fig. 6 (left panel) shows a histogram of the redshift distribution separating
the sources into these three categories. The source with the highest reliable
redshift measurement in the hard X-ray sample is at $z=3.2$, spectroscopically
confirmed and likely a blazar (See Section 4.6). Two objects show high
photometric redshift estimates ($z>3$), but come from the subsample with less
reliable redshifts, and we do not consider them to be secure. Henceforth,
unless otherwise noted, we plot and consider only the 200/213 (94%) sources
which have a spectroscopic or highly reliable photometric redshift.

The right panel of Fig. 6 shows the hard sample redshift distribution compared
to that of the main sample. The hard selection clearly samples a significantly
lower redshift range than the main sample, with a median redshift $<z>=0.34$
for the former compared to $<z>=0.94$ for the latter. The eROSITA hard
selection thus provides an interesting low-$z$ AGN sample for comparison to
the higher redshift populations seen in deeper surveys (see Section 4.4).
Together with the Swift-BAT survey (Koss et al., 2022b) eFEDS thus provides
information about the low redshift X-ray selected AGN population, which is
poorly sampled by deep X-ray surveys due to their limited area and hence
sampled cosmological volume at low-z.

Luminosities for our sample have been derived from the X-ray spectral fits, as
described above and in Liu et al. (2022b) and are thus absorption-corrected.
The luminosity-redshift relation in Fig. 7 is shown in the soft X-ray (0.5–2
keV) band to facilitate comparison to the main sample. The vast majority of
the sample covers the range $\log L_{\rm X}=42-45$ with a few lower luminosity
sources at very low redshift and the “standout” $z=3.2$ object at $\log L_{\rm
X}=46$ being amongst the most luminous even in comparison to the main sample.
As expected, the hard selection samples higher luminosities at a given
redshift, given the higher flux limit compared to the main sample.

Figure 7: Soft X-ray (0.5–2 keV) luminosity-redshift relation for the eFEDS
hard sample compared to the main sample. The luminosities are corrected for
absorption and show the subsamples with either spectroscopic redshift or high
quality photometric redshift (see text).

###  4.2 Optical spectroscopic classifications

For the sub-sample of our objects which have optical spectroscopy, we are able
to provide a basic optical spectral classification. This is most easily
achieved by fitting objects with SDSS spectra for which we have access the
spectra themselves. We have visually inspected the 172 spectra with the
primary intention of identifying objects with broad lines, predominantly
$H\alpha$, $H\beta$, MgII or CIV. A total of 159 objects ($\sim$92% of the
sample) show evidence for broad emission in at least one of these lines, with
the majority of these having prominent broad $H\beta$, MgII and/or CIV. A
subset of these objects (25 in total), however, show only broad wings to
H$\alpha$, sometimes marginally visible, and with no evidence for broad
H$\beta$ which we henceforth consider them as optical type 1.9, following the
definition thereof by Koss et al. (2022b). Of the 13 objects that show no
visual evidence for broad emission in any line, 12 show narrow emission lines
and one is an absorption-line galaxy. These objects will be either type 2 AGN
or host-galaxy dominated. Thus, despite the hard X-ray selection being
sensitive to obscured objects, more than 90% of our sample appear to be
optical broad-lined AGN of some kind.

###  4.3 X-ray spectral properties

Figure 8: Hardness ratio distribution for the extragalactic/AGN subsample as a
function of spectroscopic classification from SDSS. The vertical dotted line
indicates the hardness ratio expected for an absorbed power law with photon
index $\Gamma=1.8$ and Galactic column density typical of eFEDS of $N_{\rm
H}=3\times 10^{20}$ cm-2. Objects with broad optical lines (Type 1
classification) dominate the sample, but some of these show large hardness
ratios indicative of X-ray absorption. AGN without broad lines or without
spectroscopic classifications show systematically larger values of the
hardness ratio, indicating that they are obscured.

The hard X-ray selection and relative brightness of the current sample enable
analysis of the spectra with relatively complex models. In particular, the
existence of significant hard X-ray counts makes it much easier to infer
whether there is line-of-sight absorption of the soft X-rays, or if there is
excess emission at low energies above the extrapolation of the power law
continuum in the hard X-ray band (a.k.a the “soft excess”). As a first test,
we plot in Fig. 8 the distribution of hardness ratios for three different
classes based on the SDSS spectrum. We separate the objects with a measured
broad optical line and those without, and plot in addition those sources
without optical spectroscopy.

The sources without an optical broad line detection are more likely to exhibit
large hardness ratios indicative of absorption, as are those objects without a
spectroscopic redshift. Broad-line objects can also nonetheless exhibit large
hardness ratios, and are discussed in Section 4.5. We note that the HR is a
relatively crude measure of the spectral properties and turn our further
attention to the physical properties based on spectral fitting.

####  4.3.1 Continuum properties

Figure 9: Distributions of the primary-powerlaw slope in the ”power-law plus
blackbody” model (m3) for the hard (blue) and main (orange) AGN sample,
obtained using the HBM method adopting histogram (points with 1-$\sigma$ error
bar) and Gaussian model (lines with 1-$\sigma$ range).

Figure 10: Distributions of the obscuring column density in the ”single-
powerlaw” model (m1) for the hard AGN sample (blue) and the main AGN sample
(orange), obtained using the HBM method. The 1-$\sigma$ error bar is plotted
on the histogram.

The photon index distribution of the hard sample was determined based on a fit
to the spectrum in the full energy range with the ”powerlaw+blackbody” model
(m3 in Table 2). The posterior distribution of $\Gamma$ is obtained for each
source. We combine the posteriors of all the ($209$) AGN with good redshifts
(redshift measurement quality CTP_REDSHIFT_GRADE$\geqslant 3$) with a
Hierarchical Bayesian Model (HBM) to obtain a sample distribution. Following
the approach described in Baronchelli et al. (2020) and Liu et al. (2022b), we
adopt both a Gaussian model with a mean $\mu$ and a standard deviation
$\sigma$ and a non-parametric, histogram model for the sample distribution.
The posterior distribution of the parameters of the sample distribution can be
inferred using the PosteriorStacker
tool555https://github.com/JohannesBuchner/PosteriorStacker, which uses the
nested sampling code UltraNest. The hierarchical model implicitly allows
poorly constrained posteriors to be informed by better-constrained posteriors,
as the same model is assumed to describe the distribution for all objects.
Fig. 10 shows the photon index distribution comparing the main and hard
samples.

The main sample, selected in the soft band, has inferred Gaussian distribution
centered at $\Gamma=2.07\pm 0.006$ with a intrinsic dispersion (accounting for
measurement errors) of $0.24\pm 0.006$ (Liu et al., 2022b). For the hard
sample, the Gaussian model is found to be centered at $1.83\pm 0.04$,
significantly harder than the main sample, with a larger dispersion of
$0.47\pm 0.04$. This is easily understood as sources with very steep intrinsic
continua are likely to be missed by the hard selection and vice versa. The
histogram model shows that the $\Gamma$ distribution peaks at $1.9-2$ and has
a tail at lower $\Gamma$ values. This low-$\Gamma$ tail may not reflect the
true intrinsic distribution and instead be due to additional spectral
complexity (e.g., complex or ionized absorption) that is not considered in our
modelling (but see Waddell et al., 2023). Hence the mean intrinsic index in
the hard sample may be somewhat softer than that indicated by the mean of the
Gaussian HBM.

####  4.3.2 Obscuration

We use the same HBM method as described above to infer the sample distribution
of AGN column density $N_{\textrm{H}}$ from the posteriors measured with the
”single-powerlaw” model (m1) of all the AGN with good redshifts. We adopt only
the histogram model and not the Gaussian model because in practice the shape
of the distribution is very different from a Gaussian. We use the ”single-
powerlaw” model rather than the double-component models because in the
obscured cases a soft–excess component can be strongly degenerate with the AGN
obscuration, and our data quality does not allow constraints on both
simultaneously. Figure 10 compares the $N_{\textrm{H}}$ distribution for the
main and hard samples. In both cases, the distributions show a dominant bin at
the lower boundary $\sim 10^{20}\mathrm{cm}^{-2}$, i.e. the majority of
sources in both hard and main samples are unobscured.

Both the main and the hard sample $N_{\textrm{H}}$ distribution show a bi-
modality. This can be quantified by drawing a line between the fractions in
the $\sim 10^{20}\mathrm{cm}^{-2}$ bin and the $\sim 10^{22}\mathrm{cm}^{-2}$
bin, and noticing that both the $\sim 10^{20.5}\mathrm{cm}^{-2}$ and the $\sim
10^{21}\mathrm{cm}^{-2}$ bins are $10\sigma$ lower, indicating a dip and thus
multi-modality (Hartigan & Hartigan, 1985). Therefore, we have evidence that
there is a low-column density mode $<10^{20.5}\mathrm{cm}^{-2}$ and a high-
column density mode $\sim 10^{22}\mathrm{cm}^{-2}$. The latter can be
associated with nuclear or host galaxy obscuration (Buchner et al., 2017;
Buchner & Bauer, 2017).

The distributions diverge at high $N_{\textrm{H}}$. The main sample, selected
in the soft band (0.2–2.3 keV), has a very low fraction of obscured sources
with $N_{\textrm{H}}>10^{22}$ cm-2 (4%; Liu et al., 2022b) in the HBM-inferred
distribution, reflecting the strong selection bias against obscured sources.
For the hard sample, the fraction above $10^{22}$ cm-2 is much higher (13%).
Even more highly obscured sources with $N_{\textrm{H}}>10^{23}$ cm-2 are
almost absent from the main sample, while in the hard sample the numbers are
significant, with a probability of around 4% of having such a high
$N_{\textrm{H}}$. This is roughly in line with expectations. At the median
redshift of $z=0.34$ a column density of $10^{22}$ cm-2 only suppresses the
2.3–5 keV flux by around 5 per cent, whereas the flux is reduced by 40 per
cent at a column of $N_{\rm H}=10^{23}$ cm-2, which will start impacting
significantly on source detectability.

To define an “obscured” subsample, we adopt the same definition as Liu et al.
(2022b). Based on the posterior PDF for N${}_{\textrm{H}}$ obtained with the
single-powerlaw model (m1), Liu et al. (2022b), we divide the
N${}_{\textrm{H}}$ measurements into four classes (NHclass): 1) uninformative,
2) unobscured, 3) mildly-measured, and 4) well-measured. The posterior median
N${}_{\textrm{H}}$ value (lognH_Med_m1) is invalid in the first two cases.
Following Liu et al. (2022b), we adopt the criterion NHclass$\geqslant 3$ and
lognH_Med_m1$>21.5$ and select $36$ obscured AGN, which comprises 18% of the
hard AGN sample ($200$). The largest measured value us lognH_Med_m1$=24.0$

####  4.3.3 Soft excesses

The hard-selected sample has by selection a defined hard X-ray continuum and,
combined with the high soft X-ray sensitivity of eROSITA in the soft X-ray
band, is ideal to search for so-called ”soft excess” emission. The origin of
this component is still debated, as it can arise e.g., from a second, cooler
coronal component, ionized disk reflection, or even be an artefact of complex
soft X-ray absorption. In the present work we restrict ourselves to relatively
simple modelling of the soft excess emission.

The standard fits shown in Table 2 include both a double power law (model 2)
and a power law plus black body. We can therefore identify sources with a soft
excess by comparing the Bayesian evidence of these models to a single power
law. Specifically we use the criterion $logZ_{m2}-logZ_{m1}>1$, which selects
17 sources better fit with a double power law, or $logZ_{m3}-logZ_{m1}>1$
which yields 7 sources better fit with a powerlaw plus blackbody. The 7
sources with apparent ”blackbody” soft excesses are a subset of the 17 power
law soft excesses, indicating that the latter is a better model and more
sensitive for soft excess detection. Sources with a soft excess comprise about
10% of the AGN/extragalactic sample.

Figure 11: Distribution of the redshift (top left), bolometric luminosity (top
right), black hole mass (bottom left) and Eddington ratio (bottom right) for
spectroscopically confirmed type 1 AGN in the eFEDS hard sample, shown in
blue. The dotted black histogram shows a comparison sample from the Chandra
deep survey in the COSMOS field (Suh et al., 2020).

###  4.4 Black hole masses and Eddington ratios

As discussed above, black hole masses have been estimated using the single-
epoch virial method for the objects in our sample exhibiting a broad H$\beta$,
MgII or Civ line, in other words the spectroscopic type-1 AGN. For the 134 AGN
which show broad emission at H$\beta$ or blueward thereof, we calculate the
black hole masses and other derived parameters with optical spectral fitting
using pyQSOFit (see Section 4.4. We do not consider H$\alpha$ based masses
(e.g., Greene & Ho, 2005) because there are indications that these may be
biased and/or unreliable in Seyfert 1 type AGN.

Of the 200 AGN in the sample with reliable redshift, 179 are spectroscopic. We
restrict our black holes analysis to the 172 objects with spectra originating
from SDSS, to which we have full access and for which we have optical spectral
fitting results based on PyQSOFit as described in Wu & Shen (2022) and Section
2.5.

As described there, for type 1 objects the black hole masses and bolometric
luminosities can be estimated using the line widths and optical/UV continuum,
and the accretion rate relative to Eddington can also be estimated from these
quantities. As we are interested in the latter derived quantity, we follow
Waddell et al. (2023) in sub-selecting only objects with a constrained
Eddington ratio. This results in a subsample of 154 sources. The distributions
of redshift, black hole mass, bolometric luminosity and Eddington ratio for
this BH mass subsample are shown in Fig. 11. This subsample has a median
redshift of $z_{\rm med}=0.39$ and median luminosity $\log L_{\rm
bol,med}=45.3$.

The black hole masses cover $\log M_{\rm BH}=7-10$, peaking in the middle of
this range with a median $\log M_{\rm BH,med}=8.6$. The objects with Mg II-
derived masses are at the top end of the mass range. This is unsurprising, as
they are the highest redshift and hence amongst the highest luminosity sources
in this flux-limited sample. The Eddington ratios cover a very broad range,
from $\sim 10^{-3}$ up to the Eddington limit. Certainly, eFEDS samples
objects at the high ends of both the mass and accretion rate distributions. We
note, however, that there are significant uncertainties on our estimated
quantities, so the most extreme cases should be treated cautiously. In
addition, the objects with the highest bolometric luminosities in the eFEDS
sample are the handful of blazars (see Section 4.6 and, e.g., Fig. 11). We
caution that in these objects, the optical/UV continuum may be contaminated by
jet emission which could lead to an overestimate of the bolometric luminosity
and quantities derived therefrom (mass and accretion rate). As there are
relatively few of these in our sample, however, we do not consider this a
major effect.

To place the eFEDS results in context, we show in Fig 11 the corresponding
distributions from the work of Suh et al. (2020), from the Chandra COSMOS
legacy survey. The latter samples a significantly higher redshift range, with
$z_{\rm med}=1.58$, but due to the higher sensitivity of the deep Chandra
observations in this region the two surveys sample a similar limiting
luminosity at their median redshift, around $\log L_{\rm 2-10}=44$ in the rest
frame. It is therefore of interest to compare the two samples in relation to
possible evolution of the AGN population.

From Fig 11 we can see that the black hole mass range in the two samples is
rather similar, covering the same range and with a remarkably similar median,
with $\log M_{\rm BH,med}=8.55$ in COSMOS compared to 8.63 for non-blazars in
eFEDS. A K-S test reveals no significant difference between the distributions,
with false probability $p=0.08$. In contrast, the Eddington ratio
distributions are dramatically different with the K-S probability of them
being the same $<4\times 10^{-5}$ (once again excluding blazars). The median
Eddington ratio for eFEDS ($\lambda_{\rm Edd,med}=0.03$) is about three times
lower than that in COSMOS ($\lambda_{\rm Edd,med}=0.09$).

###  4.5 X-ray absorbed type 1 AGN

Figure 12: Black hole mass versus Eddington ratio for the type 1 AGN sample.
Type 1 AGN that show evidence for significant absorption, having an NHclass of
3 or 4 and a column density greater than 1021.5 cm-2, are indicated with black
squares.

With a combination of relatively deep and sensitive soft X-ray data, as well
as comprehensive follow-up optical spectroscopy, we are able simultaneously to
classify the objects in our sample optically and measure their absorption in
the X-ray. A particularly interesting population in this regard are X-ray
obscured type 1 AGN (e.g., Page et al., 2001; Brusa et al., 2003; Page et al.,
2011), i.e., objects which show broad optical lines but exhibit soft X-ray
absorption. The demographics of this population have been explored in, e.g.,
Merloni et al. (2014).

In the very simplest unification scenarios, this is not expected, because the
same gas which absorbs the X-rays should contain dust which extinguishes the
optical broad lines. On the other hand, there is evidence that the obscuring
material is clumpy (e.g., Nenkova et al., 2008; Markowitz et al., 2014), that
the broad line clouds can have a complex geometry (Maiolino et al., 2010) and
that there may be substantial departures from normal gas-to-dust ratios
(Maiolino et al., 2001). The broad lines and AGN continuum may also be seen
only in scattered light (Antonucci & Miller, 1985; Alexandroff et al., 2018;
Assef et al., 2020). An interesting possibility is that the X-ray absorbing
gas is dust-free, which would be the case if it is close to the central engine
within the dust sublimation radius. If so, the gas may be significantly
ionized and in the form of “warm absorbers” (Halpern, 1984; Yaqoob et al.,
1989) which are, in turn, associated with ionized winds (e.g., Kaastra et al.,
2000). Evidence for significant X-ray absorption in type 1 AGN has also been
found in broad absorption line quasars (e.g. Gallagher et al., 2006; Gibson et
al., 2009), which have direct evidence for strong outflows.

Our hard X-ray sample is dominated by broad-line AGN, but nonetheless contains
a number of objects with significant soft X-ray absorption. Considering the
154 objects in the ”black hole mass” sample as defined in Section 4.4 \- which
by definition must have a visible and constrained broad optical line - we find
that 4 also satisfy the criterion of being obscured by a column of $\log
N_{\rm H}>21.5$ as defined in Section 4.3.2. These obscured type 1 AGN are
identified in black hole mass-accretion rate parameter space in Fig. 12.
Naively, one might expect these absorbers to be seen in AGN with high
accretion rates where, close to the Eddington ratio, radiation pressure drives
an outflow (e.g., Fabian et al., 2008; Ricci et al., 2017). In practice they
show a wide range of Eddington ratios. None of the objects shows BAL
signatures in the Sloan spectra, but but only one (at $z$=2.25) is within the
redsfhit range where classical UV BAL signatures would be evident.

In addition to the black hole mass subsample, as noted in Section 4.2, Seyfert
1.9 galaxies comprise a substantial subset of our sample. Several of these
(6/25) also show significant soft X-ray absorption but this is unsurprising
given the evidence for line-of-sight obscuration based on the suppression of
optical broad lines blueward of H$\alpha$.

###  4.6 Blazars

Due to the large area of the eFEDS survey it is sensitive to rare AGN such as
those at the highest luminosities. In some fraction of these, the luminosity
is enhanced due to the presence of a relativistic jet whose emission is
Doppler boosted for sightlines close to the jet axis. Such blazars can be
identified via strong radio and/or Gamma-ray emission. A more detailed study
of the blazars in eFEDS is given in Collmar et al. 2024 (in preparation), but
here we highlight the sources in the hard X-ray sample which are likely to
host strong emission from a relativistic jet. Confirmed blazars and blazar
candidates have been identified by cross-correlating the hard sample with
catalogues of blazars identified via their radio and/or gamma-ray emission.

As input catalogs we used the following: the Fermi Large Area Telescope Fourth
Source Catalog (4GL; Abdollahi et al., 2020); the ROMA-BZCAT Multifrequency
Catalog of Blazars (Massaro et al., 2015); the 3HSP catalog of extreme and
high-synchrotron peaked blazars (Chang et al., 2019);the Swift-XRT catalog of
blazars (Giommi et al., 2019); the Blazar Radio and Optical Survey (BROS; Itoh
et al., 2020) and the WISE Blazar-like Radio-Loud Sources catalog (D’Abrusco
et al., 2019).

To select blazars or blazar candidates we searched for a positional
coincidence of $<15$ arcseconds and cross-checked the identifications with the
eFEDS counterpart catalog of Salvato et al. (2022b).

There are 15 such objects in our sample, as detailed in Table 5, of which two
are outside the 90% area region, both BL Lac type objects. Of the remaining
13, 7 are flat spectrum radio quasars, all spectroscopically confirmed, two
are BL Lac type objects and four are candidate blazars or of unknown type. Our
own visual examination of the SDSS spectra largely confirms the literature
classifications. The four BL Lac type objects show largely featureless blue
continua, with only one (4FGL J0831.8+0429) showing weak lines sufficient for
a redshift determination. The Flat Spectrum radio quasars generally exhibit
QSO broad emission lines, but two show optical properties indicating
obscuration, with 5BZQ J0924+0309 having the optical spectrum of a Seyfert 1.9
and WIBRaLS2 J092203.20-004443.3, listed as a FSRQ candidate, having narrow
lines indicative of a type-II QSO classification. A final noteworthy object
spectroscopically is 3HSPJ093303.0+045235, which presents as an absorption-
line galaxy with no AGN emission lines, but a possibly enhanced blue
continuum. This is plausibly a BL-Lac type object in which the non-thermal jet
emission is subdominant in the optical compared to the host galaxy.

Table 5: Blazars and blazar candidates identified in the hard sample. a Outside Area90. b bll= BL Lac; fsrq = Flat Spectrum Radio Quasar; fsrqc = fsrq candidate; bcu = Unclassified blazar; bcuc = Unclassified blazar candidate.  Name | SRCID | DET_LIKE_3 | Name | z | z_orig | Classb
---|---|---|---|---|---|---
eFEDS J083148.9+042939a | 65 | 29.8 | 4FGL J0831.8+0429 | 0.174 | SDSS | bll
eFEDS J083949.5+010427 | 112 | 57.1 | 4FGL J0839.8+0105 | 1.124 | SDSS | fsrq
eFEDS J085301.2-015049 | 148 | 41.7 | 5BZQ J0853-0150 | 1.498 | Simbad | fsrq
eFEDS J085920.5+004711 | 19 | 45.4 | 4FGL J0859.2+0047 | 1.47 | photz | bll
eFEDS J090111.8+044900 | 1116 | 10.8 | 5BZQ J0901+0448 | 1.862 | SDSS | fsrq
eFEDS J090910.2+012135 | 25 | 68.4 | 4FGL J0909.1+0121 | 1.024 | SDSS | fsrq
eFEDS J090915.8+035442 | 78 | 75.1 | BROS J0909.2+0354 | 3.262 | SDSS | bcuc
eFEDS J090939.9+020005 | 354 | 15.4 | 4FGL J0909.6+0159 | 0.15 | photz | bll
eFEDS J091408.2-015944 | 10 | 61.9 | 4FGL J0914.1-0202 | 1.15 | photz | bcu
eFEDS J091437.8+024558 | 55 | 41.9 | 4FGL J0914.4+0249 | 0.427 | SDSS | fsrq
eFEDS J092203.5-004442 | 16185 | 11.3 | WIBRaLS2 J092203.20-004443.3 | 0.576 | SDSS | fsrqc
eFEDS J092400.9+053345a | 20 | 50.4 | 4FGL J0924.0+0534 | 1.99 | photz | bll
eFEDS J092414.7+030859 | 23 | 114.1 | 5BZQ J0924+0309 | 0.128 | SDSS | fsrq
eFEDS J092507.8+001913 | 457 | 19.9 | 5BZQ J0925+0019 | 1.721 | SDSS | fsrq
eFEDS J093303.3+045235 | 41 | 27.9 | 3HSPJ093303.0+045235 | 0.378 | SDSS | bcu

A particularly extreme object amongst these and indeed the eFEDS hard sample
as a whole is eFEDS J090915.8+035442, identified with the blazar candidate
BROS J0909.2+0354 (Itoh et al., 2020). This is both the highest redshift and
the highest luminosity source in the sample. It has optical broad lines, but a
measured absorption column of $\log N_{\rm H}=10^{21.4}$ cm-2, almost
satisfying our criterion for being obscured. This absorption may not be
intrinsic to the quasar, but instead may originate in the intergalactic medium
(Arcodia et al., 2018).

##  5 Discussion

In this paper we have presented the sample of hard X-ray (2.3–5 keV) selected
objects detected in the eFEDS survey. At the time the data were acquired, the
140 deg2 of coverage provided by eFEDS constituted the largest contiguous
X-ray survey above 2 keV, now surpassed only by the all-sky eRASSs (Merloni et
al. 2023, in preparation). The large area and cosmological volume probed,
together with comprehensive supporting data particularly in terms of
spectroscopy, have yielded new insights into the hard X-ray source populations
and this hard X-ray parent sample can also be used to select objects or
subsamples or particular interest for further detalied study (e.g. Brusa et
al., 2022; Waddell et al., 2023).

###  5.1 The hard X-ray sky seen by eROSITA

Due to the short focal length and hence soft response of eROSITA, the hard
X-ray sensitivity is considerably less than in the main (0.2-2.3 keV) eROSITA
detection band, with the sample size being around 100 times smaller. The 246
hard X-ray sources in our sample have been identified optically and classified
according to whether they are Galactic or extragalactic. The vast majority (90
%) are in the latter category and are overwhelmingly AGN. Extensive
spectroscopic coverage in the field thanks mainly to the SDSS-IV and SDSS-V
surveys yields a high spectroscopic redshift completeness for our sample of
around 80%, with the remaining sources having photometric redshift estimates.
This high redshift completeness at bright fluxes is in part what distinguishes
eFEDS from larger, non-contiguous wide field surveys such as Champ, Chandra
Source Catalogue and 4XMM (Green et al., 2004; Evans et al., 2010; Webb et
al., 2020). The redshift distribution shows that the eFEDS hard survey yields
a relatively nearby population of extragalactic sources, with median redshift
$<z>=0.34$, compared to the eFEDS soft X-ray sample or deeper pencil-beam
surveys with Chandra and XMM-Newton. The hard survey therefore provides an
interesting lower-redshift baseline for evolutionary studies of AGN. The
extensive optical spectroscopy in the field shows that the overwhelming
majority of the objects in the eFEDS hard X-ray sample are type 1 AGN, despite
the hard X-ray selection being sensitive to at least moderately obscured
objects. Similar effects have been found also in previous hard X-ray surveys
at relatively bright fluxes (La Franca et al., 2002; Della Ceca et al., 2008,
e.g.)

An interesting sub-class of objects are the optically bright extragalactic
sources which stand out somewhat from the general population in Fig. 4. For
example our extragalactic/AGN subsample contains six objects with $r<15$.
Optical spectroscopy show that these are all very low redshift galaxies with
$z<0.05$. In principle these may be normal galaxies whose X-ray emission is
dominated by binary populations and hot gas emission (see Vulic et al., 2022,
for a discussion of the normal galaxies in eFEDS) but in reality most of these
hard X-ray selected sources show AGN signatures including broad H$\alpha$ and
their X-ray luminosities are also indicative of AGN activity, with $\log
L_{\rm X}>41$. One such object, eFEDS J085547.9+004747, is a hard-band only
detected object and one of the most obscured objects in the entire sample,
with $\log N_{\rm H}=23.3$ for an absorbed power law fit. Despite our concern
that many hard-band only sources could be spurious, this object is very well
detected with DET$\\_$LIKE$\\_$3=30.6 and a confirmatory detection in the
Swift 105-month catalogue (Oh et al., 2018). There is a hint in Fig 4 that the
optical magnitude distribution of the hard-band selected objects is bimodal,
and it will be interesting to see if this is borne out in the eRASS.

###  5.2 X-ray spectral properties of the hard sample

X-ray spectral analysis of the hard sources shows that a significant fraction
exhibit soft X-ray absorption, with the survey having some sensitivity to
column densities as high as $N_{\rm H}=10^{24}\mathrm{cm}^{-2}$. eROSITA hard
samples can therefore be used to explore the demographics of the (Compton-
thin) obscured populations. We observe a dip in the column density
distribution near $N_{\rm H}\sim 10^{20.5}\mathrm{cm}^{-2}$. This may indicate
that the medium around AGN is not a single, continous density distribution
from $10^{20}$ to $10^{23}\mathrm{cm}^{-2}$ and above, but composed of
discrete clumps of a significant, typical column density $N_{\rm H}\sim
10^{22}\mathrm{cm}^{-2}$ (see, for example Buchner & Bauer, 2017; Buchner,
2019), with unobscured sight-lines in between. A bi-modality has also been
seen in the optical obscuration (Assef et al., 2013).

At column densities of $N_{\rm H}>10^{22}$ $cm^{-2}$, the selection function
needs to be taken into account carefully, given that the survey will become
increasingly incomplete for higher absorbing columns. After accounting for
absorption, the hard sample exhibits similar intrinsic photon indices as the
eROSITA soft sample, but with a tail to flat photon indices that might be
indicative of unmodelled soft X-ray spectral complexity, including e.g.
ionized absorbers which may not be correctly matched by the cold absorption
model employed here.

This conclusion is supported by the fact that a small but significant fraction
of our sources exhibit soft X-ray absorption, while being optical type-1s
(see, e.g. Brusa et al., 2003; Page et al., 2011; Merloni et al., 2014). One
explanation for this is that the absorber is ionized and dust-free, presumably
being located close to the origin of the nuclear ionizing radiation and well
within the dust sublimation radius. Ionized or ”warm” absorbers have been
shown via high resolution X-ray spectroscopy to take the form of outflows
(e.g. Kaastra et al., 2000; Kaspi et al., 2002) and such outflows could be an
important component of the accreting system as well as having wider influence.
Naively, one might expect these winds to be launched at the highest accretion
rates, but as shown in Fig. 12 the absorbed type 1 AGN are preferentially seen
at more modest accretion rates. This is in concurrence with the conclusions of
Liu et al. (2018) based on the XMM-XXL field and the more detailed analysis of
the eFEDS hard sample data presented in Waddell et al. (2023). It should also
be noted that the exact relationship between accretion rate and the launching
of winds could be complicated by factors such as geometry and the presence or
absence of dust (e.g., Fabian et al., 2008; Ricci et al., 2017)

Soft X-ray spectral complexity of the opposite kind is also evident in the
form of a soft excess, detected in around 10% of the extragalactic sources.
Our modeling indicates that this has a power law, rather than blackbody form.
The origin of the soft excess emission in AGN is still debated, with
possibilities including a secondary corona, complex absorption or relativistic
X-ray reflection from an ionized accretion disk. The spectral modeling favours
the warm corona interpretation. Again, the above results confirm the more
sophisticated analysis of the soft X-ray complexity of the eFEDS hard X-ray
sample excesses presented in Waddell et al. (2023). The most likely overall
explanation for this is that at the highest accretion rates, the warm absorber
becomes over-ionized decreasing the efficiency of radiative line-driving
(Proga et al., 2000).

###  5.3 AGN evolution and downsizing

The relatively local nature of the eFEDS hard sample offers an interesting
baseline to be compared to deeper pencil beam surveys, with the large area of
eFEDS offering sufficient cosmological volume to provide a fair sample of the
low redshift Universe. In this work, in particular, we have compared the black
hole mass and accretion rate distributions between eFEDS and the deeper COSMOS
survey. Fortuitously, the luminosity limit of eFEDS and COSMOS observations at
their respective median redshift is similar, as is their rest frame selection
bandpass, and they also yield similar sample sizes.

We find a remarkable similarity between the black hole mass distributions, but
with a clear difference in the accretion rate distributions, in the sense that
the typical black hole growth rate relative to Eddington is about a factor 3
lower in eFEDS than in COSMOS. This result is of particular interest given
that the global accretion rate density has declined substantially between the
two cosmic epochs sampled by the COSMOS and eFEDS samples. Under the
assumption that the populations sampled by these surveys are representative of
the overall AGN populations and hence the global accretion rate it suggests
that the decline in accretion rate is due primarily to a reduction in the
typical fuelling rate of black holes, rather than a ”downsizing” effect where
the sites of active accretion occur in objects with lower typical black holes
mass.

This general effect is similar to that seen in work which analyses the
evolution of the specific accretion rate distribution of AGN over cosmic time,
as traced by the ratio of the X-ray luminosity to host stellar mass (e.g. Aird
et al., 2012; Bongiorno et al., 2016; Georgakakis et al., 2017). Based on the
extensive optical spectroscopy in our field, it has been possible here to
calculate the accretion rate more directly via the line width and black hole
mass estimates. Our finding of a lack of ”downsizing” effect in the overall
black hole populations contrasts somewhat with some previous studies of
optical quasars, which do see such an effect (e.g. Kelly & Shen, 2013; Schulze
et al., 2015). The difference may be due to the selection band, but full
understanding requires careful modeling of the selection effects (Trump et
al., 2015; Jones et al., 2016).

Figure 13: Luminosity-redshift relation for the eFEDS hard X-ray sample
compared to the Swift-BAT sample (Ricci et al., 2017) and approximate flux
limit for the HEAO-1 A2 all-sky survey (Piccinotti et al., 1982). The eFEDS
luminosities are absorption-corrected and dervied from the spectra (see
Secttion 4.1). eFEDS probes a similar redshift range to Swift-BAT but samples
luminosities around an order of magnitude fainter.

##  6 Conclusions and outlook

We have presented the properties of a sample of X-ray sources selected in the
relatively hard 2.3–5 keV band from the eROSITA eFEDS survey. The sample of
246 sources is around two orders of magnitude smaller than the main, soft
X-ray selected eROSITA catalogue in this region, reflecting the lower
effective area and higher background of the instrument in this band. The hard
sample consists overwhelmingly of extragalactic sources and more specifically
AGN, with a small fraction of stars. Being relatively bright and benefiting
from extensive dedicated spectroscopy from the SDSS-IV and SDSS-V surveys, the
sample has very high spectroscopic completeness (80%) which increases to
$\sim$94% when only the extragalctic/AGN subsample is considered, and enables
a number of properties to be explored in detail:

  1. 1.

The hard-selected AGN cover a wide luminosity and redshift range up to z$\sim
3$, but the selection predominantly results in a relatively nearby sample with
$<z>\sim 0.3$, which is interesting to compare to higher redshift objects in
the main sample and in deeper surveys.

  2. 2.

The intrinsic power law index appears similar, if perhaps slightly harder,
than the main eFEDS sample.

  3. 3.

The hard selection yields a small but significant subsample of highly obscured
AGN, with $\log N_{\rm H}>23$. We characterize the column density distribution
from $\log N_{\rm H}=20-22$ in fine detail.

  4. 4.

The hardest sources in the sample tend to be optically unclassified
spectroscopically and are probably obscured, type 2 AGN. On the other hand a
significant subsample ($\sim$10%) of the optical broad-line AGN are also X-ray
obscured. These may be ”warm absorbers” with an outflowing nuclear wind, and
are observed preferentially at low Eddington ratios (see also Waddell et al.,
2023).

  5. 5.

Several of the brightest and most luminous sources in our sample are blazars,
predominantly flat-spectrum radio quasars many of which are also $\gamma$-ray
emitters.

  6. 6.

For the objects with optical broad lines, we are able to estimate black hole
masses and accretion rates. Comparing to a similar sample of high redshift
objects in the COSMOS field, we find a very similar distribution of black hole
masses, but significantly lower Eddington ratios (by a factor $\sim 3$) in the
lower redshift eFEDS sample. This provides important clues as to the origin of
the decrease in the overall accretion luminosity of the Universe since $z\sim
1-2$.

As mentioned earlier, eFEDS was designed as a precursor to the 8-pass eRASS:8
survey, intended to demonstrate the power and potential of the full survey.
With the exception of the XMM-Slew survey (Saxton et al., 2008), previous all-
sky or near all-sky surveys in the hard band have had to rely on non-imaging
instruments. Notable examples include the HEAO-1 A2 all sky survey catalogue
(Rothschild et al., 1979), the RXTE all-sky survey (Revnivtsev et al., 2004)
and the Swift-BAT survey (BASS; Oh et al., 2018; Koss et al., 2022b). With
resulting AGN catalogues presented by e.g. Piccinotti et al. (1982) for HEAO-1
A2, Sazonov & Revnivtsev (2004) for RXTE and, for BASS,Ricci et al. (2017) and
Koss et al. (2022a). The luminosity-redshift relation of eFEDS is compared to
some of these surveys in Fig. 13. This clearly shows the superior sensitivity
even of the single-pass eRASS1 survey Merloni et al. (2024) compared to
previous hard X-ray all-sky surveys, being effectively around 2 orders of
magnitude deeper than HEAO-1 (or the similar RXTE all-sky survey) in the
standard 2-10 keV bandpass.

Arguably the most interesting comparison survey from the point of view of AGN
science is the Swift-BAT survey (Oh et al., 2018) which is highly
spectroscopically complete (Koss et al., 2017, 2022b) and has given important
new insights into the local AGN population. The harder bandpass of Swift BAT
means it is much more complete in terms of obscured AGN, but Fig. 13 shows
that the eFEDS hard sample, and by extension the eRASS, clearly probes to
significantly fainter fluxes, being a true imaging survey, and hence
luminosities an order of magnitude lower at any given redshift. A combination
of these surveys will undoubtedly yield new insights into the nature of the
AGN population at low redshift.

Overall the eFEDS survey illustrates the high potential for the full eROSITA
all sky survey for studies of AGN physics and evolution, with the all-sky
survey yielding unprecedented samples of hard X-ray AGN. A first taste of this
is presented in Waddell et al. (2024), who present the hard-selected sample
from the first eROSITA sky survey eRASS1 (Merloni et al. 2024). Based on eFEDS
and applying approximate corrections for the sky area and the slightly deeper
exposure for compared to the expectation for the eRASS:8 survey, we would
expected approximately 50,000 hard X-ray selected AGN all-sky. While these
will be mostly unobscured, the eFEDS analysis shows that objects with
absorbing columns up to N${}_{\rm H}\sim 10^{24}$ cm-2 are sampled in the
survey, so provided obscuration bias can be accurately characterised via the
selection function, the eRASS has the potential to add significantly to AGN
demographic studies. The information content of the resulting dataset will be
very substantial, given the excellent eROSITA spectral response, especially
when augmented by massive optical ground-based spectroscopy from SDSS and,
eventually, 4MOST (de Jong et al., 2019; Merloni et al., 2019), both of which
have dedicated eROSITA follow-up programs.

###### Acknowledgements.

FEB acknowledges support from ANID-Chile BASAL CATA FB210003, FONDECYT Regular
1200495, and Millennium Science Initiative Program – ICN12_009. RJA was
supported by FONDECYT grant number 123171 and by the ANID BASAL project
FB210003. MB, AG and BM are supported by the European Union’s Innovative
Training Network (ITN) ”BiD4BEST”, funded by the Marie Sklodowska-Curie
Actions in Horizon 2020 (GA N. 860744). MK acknowledges support from DLR grant
FKZ 50 OR 2307. 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 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. 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.
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.sdss.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.

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##  Appendix A The eFEDS hard source catalogue

Description of catalogue contents are listed in Table LABEL:table:cat. In the
column names, the band flags indicate a few energy bands as listed in Table.
7. The suffix flags in the flux-related columns (FluxObsv, FluxCorr, FluxIntr,
LumiIntr) include “Med” (posterior median), “Lo1”, “Lo2”
(1-$\sigma$/2-$\sigma$ percentile lower limit), “Up1”, “Up2”
(1-$\sigma$/2-$\sigma$ percentile upper limit), and “BF” (best-fit). For other
X-ray spectral parameters, the suffix flags include “Med” (posterior median),
“Mean” (posterior mean), “Std” (standard deviation), “Lo” (1-$\sigma$ lower
limit), “Up” (1-$\sigma$ upper limit). For a few spectral shape parameters,
e.g., $\Gamma$ and NH, we also provide “HLo” (1-$\sigma$ HDI lower limit),
“HUp” (1-$\sigma$ HDI upper limit), and “KL” (KL divergence). These X-ray
spectral parameters are named and calculated in the same way as described in
Liu et al. (2022b).

Table 6: Columns in the catalog Column name |  Description
---|---
X-ray source properties from Brunner et al. (2022)
NAME |  Source name
ID_SRC |  Source ID in the eFEDS hard X-ray catalog
ID_main |  ID of the corresponding source in the eFEDS main X-ray catalog
RA_CORR |  X-ray right ascension (deg; J2000), astrometrically corrected
DEC_CORR |  X-ray declination (deg; J2000), astrometrically corrected
RADEC_ERROR_CORR |  combined positional uncertainty (arcsec)
DET_LIKE_band |  source detection likelihood in band band; band 1,2,3 for 0.2–0.6, 0.6–2.3, and 2.3–5 keV, band 0 for combined 3-band likelihood
inArea90 |  Whether located inside the inner 90%-area region of eFEDS
ML_RATE_band |  source count rate (cts/s) in band 1,2,3,u,b1,b2,b3,b4
ML_CTS_band |  source net counts in band 1,2,3,u,b1,b2,b3,b4
ML_FLUX_band |  source flux (erg cm-2 s-1) in band 1,2,3,u,b1,b2,b3,b4
Multiband properties, mainly from Salvato et al. (2022b)
CTP_quality |  Counterpart quality
CTP_CLASS |  Classification of the optical counterpart . For AGN it can be 2: “likely extraGalactic” or 3: “secure extraGalactic”.
Redshift |  redshift
RedshiftOrig |  origin of redshift
RedshiftGrade |  Redshift Grade . Grade 4 indicates high-quality photo-z. The highest value 5 indicates spec-z.
CTP_LS8_UNIQUE_OBJID |  ID of the best LS8 counterpart
CTP_LS8_RA |  Right ascension (deg; J2000) of the best LS8 counterpart
CTP_LS8_DEC |  Declination (deg; J2000) of the best LS8 counterpart
W1 |  LS8 Wise W1 magnitude (AB)
W1_ERR |  LS8 Wise W1 magnitude error
W2 |  LS8 Wise W2 magnitude (AB)
W2_ERR |  LS8 Wise W2 magnitude error
LS8_g |  LS8 g-band magnitude (AB)
LS8_g_ERR |  LS8 g-band magnitude error
LS8_r |  LS8 r-band magnitude (AB)
LS8_r_ERR |  LS8 r-band magnitude error
LS8_z |  LS8 z-band magnitude
LS8_z_ERR |  LS8 z-band magnitude error
in_KiDS |  Whether located inside the region of the KiDS survey
galNH |  Total Galactic absorption column density (cm-2)
AGN |  Boolean flag for the AGN subsample
X-ray spectral properties
LxModel |  Index of the selected model for X-ray luminosity measurement. 1: single-powerlaw; 3: powerlaw+blackbody; 4: powerlaw with Gamma fixed at $1.8$. 0 means no luminosity measurement.
NHclass |  Class of measurement of AGN $N_{\rm H}$ based on model 1, which can be 1: uninformative, 2: unobscured, 3: mildly-measured, and 4: well-measured.
Exposure |  Exposure time from the X-ray spectral file (s)
SrcCts |  Source net counts in the 0.2–5 keV band measured from the spectra
Rate_band |  Net count rate in the 0.2–2.3, 0.2–0.5, 0.5–1, 1–2, 2–4.5, 2.3–5, and 5–8 keV bands (with band suffixes of d2_2d3, d2_d5, d5_1, 1_2, 2_4d5, 2d3_5, and 5_8) measured from the spectra
RateErr_band |  Net count rate error in the corresponding energy band
BkgCts_band |  Background counts in the 0.2–0.6, 0.6–2.3, 2.3–5, and 5–8 keV bands (with band suffixes of d2_d6, d6_2d3, 2d3_5, and 5_8)
BkgCtsErr_band |  Background counts error in the corresponding energy band
FluxObsv_suffix_band_m$n$ |  Observed energy flux (erg cm2 s-1) in an observed-frame energy band s (0.5–2 keV) or t (2.3–5 keV); for all the models (m0–m5)
FluxCorr_suffix_band_m$n$ |  Absorption corrected energy flux (erg cm2 s-1) in an observed-frame energy band s (0.5–2 keV) or t (2.3–5 keV); for m1–m5
FluxIntr_suffix_band_m$n$ |  Absorption corrected energy flux (erg cm2 s-1)) in a rest-frame energy band s (0.5–2 keV), h (2–10 keV), or 2keV (1.999–2.001 keV); for m1–m5
LumiIntr_suffix_band_m$n$ |  Intrinsic (absorption corrected) luminosity (erg/s) in a rest-frame energy band s (0.5–2 keV), h (2–10 keV), or 2keV (1.999–2.001 keV); for m1–m5. Having no model index suffix m$n$ indicates the selected luminosity measurement using model LxModel.
logZ_m$n$ |  log Bayesian evidence for each mode (m0– m5)
$\Gamma$_suffix_m$n$ |  Powerlaw slope; for m1, m2, m3, m5
logPowNorm_suffix_m$n$ |  Power-law normalization for AGN models (m1$\sim$m5)
lognH_suffix_m$n$ |  AGN absorption column density NH (cm-2) for AGN models, or Galactic NH for m0
logBkgNorm_suffix_m$n$ |  Background normalization for all the models
logApecNorm_suffix |  APEC normalization; only for m0
logBBNorm_suffix |  Blackbody normalization; only for m3
logkT_suffix_m$n$ |  Temperature (keV) of blackbody (m3) or APEC (m0)
logAbundanc_suffix |  Abundance of the APEC model; only for m0
dGm_suffix |  Slope of the additional soft power law minus slope of the primary power law; only for m2
logFrac_suffix |  Ratio of the additional power-law to the primary power law at 1 keV; only for m2
SDSS optical spectral properties
SDSS_DR |  SDSS data release version
PLATE |  plate ID
MJD |  MJD
FIBERID |  Fiber ID
CATALOGID |  catalog ID
SDSS_RA |  SDSS target RA [deg]
SDSS_DEC |  SDSS target DEC [deg]
SDSS_Z |  SDSS redshift
SNR_conti |  continum signal-to-noise ratio
LogL1350 |  1350Åluminosity [erg/s]
LogL1350_err |  1350Åluminosity [erg/s]
LogL3000 |  3000Åluminosity [erg/s]
LogL3000_err |  3000Åluminosity [erg/s]
LogL5100 |  5100Åluminosity [erg/s]
LogL5100_err |  5100Åluminosity [erg/s]
logMbh_Hbeta |  Hbeta black hole mass
logMbh_Hbeta_err |  Hbeta black hole mass 1-$\sigma$ error
logMBH_MgII |  MgII black hole mass
logMBH_MgII_err |  MgII black hole mass 1-$\sigma$ error
logMBH_CIV |  CIV black hole mass
logMBH_CIV_err |  CIV black hole mass 1-$\sigma$ error
Ha_br_flux |  H$\alpha$ flux
Ha_br_flux_err |  H$\alpha$ flux uncertainty
Ha_br_LogL |  H$\alpha$ luminosity
Ha_br_LogL_err |  H$\alpha$ luminosity uncertainty
Ha_br_FWHM |  H$\alpha$ FWHM
Ha_br_FWHM_err |  H$\alpha$ FWHM uncertainty
Hb_br_flux |  H$\beta$ flux
Hb_br_flux_err |  H$\beta$ flux uncertainty
Hb_br_LogL |  H$\beta$ luminosity
Hb_br_LogL_err |  H$\beta$ luminosity uncertainty
Hb_br_FWHM |  H$\beta$ FWHM
Hb_br_FWHM_err |  H$\beta$ FWHM uncertainty
MgII_br_flux |  MgII flux
MgII_br_flux_err |  MgII flux uncertainty
MgII_br_LogL |  MgII luminosity
MgII_br_LogL_err |  MgII luminosity uncertainty
MgII_br_FWHM |  MgII FWHM
MgII_br_FWHM_err |  MgII FWHM uncertainty
CIV_br_flux |  CIV flux
CIV_br_flux_err |  CIV flux uncertainty
CIV_br_LogL |  CIV luminosity
CIV_br_LogL_err |  CIV luminosity uncertainty
CIV_br_FWHM |  CIV FWHM
CIV_br_FWHM_err |  CIV FWHM uncertainty
Lbol |  bolometric luminosity
Lbol_err |  bolometric luminosity error
logMBH |  adopted black hole mass
logMBH_err |  adopted black hole mass error
Ledd |  Eddington luminosity
Ledd_err |  Eddington luminosity error
GoodMBH |  whether the black hole mass is good
Table 7: Dictionary of energy band suffixes in the column names Band |  Energy range
---|---
1 |  0.2–0.6 keV
2 |  0.6–2.3 keV
3 |  2.3–5 keV
s |  0.5–2 keV
t |  2.3–5 keV
u |  5–8 keV
b1 |  0.2–0.5 keV
b2 |  0.5–1 keV
b3 |  1–2 keV
b4 |  2–4.5 keV
2keV |  1.999–2.001 keV