11institutetext:  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, E-20118 Donostia-San Sebastián, Spain
11email<EMAIL_ADDRESS>Max-Planck-Institut für
Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany33institutetext:
Institute of Astronomy, Pontificia Universidad Católica de Chile, Avenida
Vicuña Mackena 4690, Santiago, Chile44institutetext: Universität Heidelberg,
Seminarstrasse 2, D-69117 Heidelberg, Germany55institutetext: Max-Planck-
Institut für extraterrestrische Physik (MPE), Giessenbachstrasse 1, D-85748
Garching, Germany66institutetext: MIT Kavli Institute for Astrophysics and
Space Research, 70 Vassar Street, Cambridge, MA 02139, USA77institutetext:
Ikerbasque, Basque Foundation for Science, E-48013 Bilbao, Spain

# Massive black holes in nuclear star clusters:

Investigation with SRG/eROSITA X-ray data

N. Hoyer 11223344 R. Arcodia NASA Einstein fellow5566 S. Bonoli 1177 A.
Merloni 55 N. Neumayer 22 Y. Zhang 55 J. Comparat 55

(Received xx yy, 2023; accepted xx yy, 2023)

###### Abstract

Context. Massive black holes (MBHs) are typically hosted in the centres of
massive galaxies but they appear to become rarer in lower mass galaxies, where
nuclear star clusters (NSCs) frequently appear instead. The transition region,
where both an MBH and NSC can co-exist, has been poorly studied to date and
only a few dozen galaxies are known to host them. One avenue for detecting new
galaxies with both an MBH and NSC is to look for accretion signatures of MBHs.

Aims. Here, we use new SRG/eROSITA all-sky survey _eRASS:4_ data to search for
X-ray signatures of accreting MBHs in NSCs, while also investigating their
combined occupation fraction.

Methods. We collected more than $200$ galaxies containing an NSC, spanning
multiple orders in terms of galaxy stellar mass and morphological type, within
the footprint of the German eROSITA Consortium survey. We determined the
expected X-ray contamination from binary stellar systems using the galaxy
stellar mass and star formation rate as estimated from far-ultraviolet and
mid-infrared emission.

Results. We find significant detections for $18$ galaxies ($\sim
8.3\text{\,}\mathrm{\char 37\relax}$), including one ultra-luminous X-ray
source; however, only three galaxies (NGC 2903, 4212, and 4639) have X-ray
luminosities that are higher than the expected value from X-ray binaries,
indicative of the presence of an MBH. In addition, the X-ray luminosity of six
galaxies (NGC 2903, 3384, 4321, 4365, 4639, and 4701) differs from previous
studies and could indicate the presence of a variable active galactic nucleus.
For NGC 4701 specifically, we find a variation of X-ray flux within the
_eRASS:4_ data set. Stacking X-ray non-detected galaxies in the dwarf regime
($M_{\star}^{\mathrm{gal}}\leq${10}^{9}\text{\,}\mathrm{M_{\odot}}$$) results
in luminosity upper limits of a few times
${10}^{38}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$. The combined
occupation fraction of accreting MBHs and NSCs becomes non-zero for galaxy
masses above $\sim 10^{7.5}\,$\mathrm{M_{\odot}}$$ and this result is slightly
elevated as compared to the literature data.

Conclusions. Our data extend, for the first time, towards the dwarf elliptical
galaxy regime and identify promising MBH candidates for higher resolution
follow-up observations. At most galaxy masses (and with the exception of three
cases), the X-ray constraints are consistent with the expected emission from
binary systems or an Eddington fraction of at most $0.01\text{\,}\mathrm{\char
37\relax}\mathrm{,}$ assuming a black holes mass of
$10^{6.5}\,$\mathrm{M_{\odot}}$$. This work confirms the known complexities in
similar-type of studies, while providing the appealing alternative of using
X-ray survey data of in-depth observations of individual targets with higher
resolution instruments.

###### Key Words.:

Galaxies: star clusters: general – X-rays: galaxies – Galaxies: nuclei

##  1 Introduction

Since the first detections of massive compact objects in nearby galaxy centres
almost forty years ago (Tonry, 1984), it has become evident that massive black
holes (MBHs) occupy many nearby galaxy centres (e.g. Kormendy & Richstone,
1995; Maggorian et al., 1998; Tremaine et al., 2002; Kormendy & Ho, 2013).
This insight was made possible by significant advancements in the performance
and capabilities of many ground-based facilities, including NIRC on Keck (Ghez
et al., 1998; Filippenko & Ho, 2003; Walsh et al., 2012), SHARP on NTT (Genzel
et al., 2000; Gillessen et al., 2009), GRAVITY (Abuter et al., 2017, 2021),
SAURON (Bacon et al., 2001a; van den Bosch & de Zeeuw, 2010), CFHT (Bender et
al., 1996; Kormendy et al., 1997), SINFONI (Nowak et al., 2008; Rusli et al.,
2011; Saglia et al., 2016), VLBI (Kuo et al., 2011), GEMINI/NIFS (Nguyen et
al., 2018; Merrell et al., 2023), VLT (Marconi et al., 2001), and with the
Hubble Space Telescope (HST; e.g. Gebhardt et al., 2003; Devereux et al.,
2003; Atkinson et al., 2005; Gültekin et al., 2009; Walsh et al., 2010; Nguyen
et al., 2019), as well as improvements in dynamical models of galaxy centres
(e.g. Cappellari & Emsellem, 2004; Cappellari, 2020; Thater et al., 2019,
2022a, but see also Thater et al., 2022b and references therein). These
measurements were only performed on massive galaxies as secure detections of
MBHs towards the lowest galaxy masses become rare both because of weaker
observational signatures and an apparent decline in the MBH occupation
fraction, as suggested by observational (e.g. Miller et al., 2015a; Trump et
al., 2015; Nguyen et al., 2018) and theoretical (e.g. Volonteri et al., 2003;
Bellovary et al., 2011; Habouzit et al., 2017; Haidar et al., 2022)
studies.111See Bustamente-Rosell et al. (2021) and Regan et al. (2023) for a
discussion on a ${10}^{6}\text{\,}\mathrm{M_{\odot}}$ MBH in the nearby Leo I
($M_{\star}^{\mathrm{gal}}\sim${10}^{7}\text{\,}\mathrm{M_{\odot}}$$) dwarf
galaxy. Despite numerous investigations (see e.g. Sharma et al., 2022;
Beckmann et al., 2023; Spinoso et al., 2023, for recent studies), the
functional shape and value of the decline of the occupation fraction from
unity as a function of galaxy stellar (or halo) mass remain only loosely
constrained.

Galaxy centres can also host dense stellar systems, known as nuclear star
clusters (NSCs), which are more commonly found in the dwarf galaxy regime,
occupying about $80\text{\,}\mathrm{\char 37\relax}$ of
$M_{\star}^{\mathrm{gal}}\sim${10}^{9}\text{\,}\mathrm{M_{\odot}}$$ galaxies
in the local universe (Sánchez-Janssen et al., 2019; Neumayer et al., 2020;
Hoyer et al., 2021; Ashok et al., 2023). Contrary to MBHs, their occupation
fraction rapidly declines in the most massive galaxies, where MBHs are most
common, potentially due to interactions between the two objects (e.g. Antonini
et al., 2015; Arca-Sedda & Capuzzo-Dolcetta, 2017) or tidal evaporation of
progenitor clusters (Leaman & van de Ven, 2022). Nevertheless, a transition
region where both types of nuclei are present exists and includes, for
example, the Milky Way (e.g. Genzel et al., 2010). As the functional shape of
the MBH occupation fraction with respect to the host galaxy stellar mass is
currently unclear, the extent of this transition region is unclear as well.

Due to observational constraints all firm MBH detections within NSCs are
confined to relatively nearby galaxies (see e.g. Figure $2$ in Greene et al.,
2020 and the compilation of Neumayer et al., 2020) and are located in the
NSC’s centre with the exception of M 31 (e.g. Lauer et al., 1993; Bacon et
al., 1994, 2001b; Bender et al., 2005). Consequently, the total number of
these systems is limited to a few dozen (e.g. Neumayer et al., 2020; Nguyen et
al., 2022; Thater et al., 2023), including ultra-compact dwarfs as previous
NSCs of accreted galaxies (e.g. Seth et al., 2014; Pfeffer et al., 2016; Ahn
et al., 2017; Pechetti et al., 2022). As we are now aware of more than 1000
nucleated galaxies (Muñoz et al., 2015; Venhola et al., 2018; Sánchez-Janssen
et al., 2019; Carlsten et al., 2020; Habas et al., 2020; Poulain et al., 2021;
Su et al., 2021; Hoyer et al., 2023) and given the significant overlap between
the NSC and MBH occupation fractions, we should expect a significantly higher
number of galaxies with both an MBH and NSC. While dynamical measurements are
important to obtain reliable mass measurements for MBHs within NSCs, focusing
on accretion signatures can help to identify larger samples out to higher
distances, including dwarf galaxies where NSCs are most common (e.g. Kauffmann
et al., 2003; Baldassare et al., 2018; Birchall et al., 2020; Mezcua &
Domínguez Sánchez, 2020; Mezcua et al., 2023; Cann et al., 2023).

Accretion events onto MBHs from gas or stars via tidal disruption events
(Rees, 1988) leads to bright X-ray emission (e.g. Komossa & Bade, 1999; Esquej
et al., 2008; Maksym et al., 2010) which can be used to study the mass of the
black hole (e.g. Mockler et al., 2019) and potentially that of black hole
binaries (Mockler et al., 2023). Additionally, data from large-scale surveys
was previously used to trace MBHs (e.g. Miller et al., 2015b) and to constrain
their occupation fraction (Miller et al., 2015a). One avenue to detect more
MBHs in NSCs is to combine optical and X-ray data to detect and characterise
the NSC and MBH, respectively, requiring an active galactic nucleus (AGN) that
does not outshine the NSC in the optical regime.

Previous works have already taken advantage of combining various wavelength
regimes (Seth et al., 2008; Baldassare et al., 2022), using, among other
instruments, Chandra for X-rays. Another approach compared to using high-
resolution Chandra data is to perform a shallower wide-area survey, allowing
us to study a greater number of NSCs in galaxies of various masses and
morphologies. The extended ROentgen Survey with an Imaging Telescope Array
instrument (eROSITA; Predehl et al., 2021) aboard the Spectrum-Roentgen-Gamma
(SRG; Sunyaev et al., 2021) takes this approach and serves as an ideal
laboratory for such a study. The poorer resolution of eROSITA operating in its
survey mode (half-energy width of
$26\text{\,}\mathrm{\SIUnitSymbolArcsecond}$; Predehl et al., 2021) does not
allow us to distinguish clearly between nuclear and off-nuclear emission as
securely as Chandra but can still be used to detect MBH candidates for follow-
up studies and to potentially probe MBH signatures in a large number of NSCs
directly.

Figure 1: Cutouts of eROSITA _eRASS:4_ images (_top panels_) and of the DESI
Legacy Imaging Surveys (_bottom panels_) Data Release 10 [Legacy Surveys / D.
Lang (Perimeter Institute)] of three galaxies in our sample: NGC 4651, IC 3602
and LEDA 40679. Both images are centered at the input optical coordinates of
the NSC (white cross), with the $30\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
aperture circle used for X-ray photometry highlighted in white. In case of an
X-ray detection (_left column_) X-ray contours are also overlayed to the
optical image (red).

In this paper, we explore these possibilities using the cumulative data from
eROSITA’s already completed four all-sky surveys (dubbed _eRASS:4_ ; Predehl
et al., 2021) to locate X-ray emission in a large sample of NSCs. We introduce
the eROSITA, galaxy and literature data sets in Section 2 and we analyse their
properties in Section 3. Section 4 presents a discussion of the results and
Section 5 gives the conclusions.

##  2 Data

###  2.1 Sample of nucleated galaxies

To generate an all-sky catalogue of nucleated galaxies, we first consider all
galaxies up to a distance of $100\text{\,}\mathrm{Mpc}$, which are part of the
HyperLEDA222https://leda.univ-lyon1.fr/ data base (Makarov et al., 2014),
containing approximately $63\,000$ objects. Based on this catalogue, we search
the Hubble Legacy Archive333https://hla.stsci.edu/ for available high-
resolution imaging data (Advanced Camera for Surveys, Wide Field and Planetary
Camera 2, and Wide Field Camera 3). Based on these data, we assigned a nuclear
classification to all galaxies, not taking into account previous
classifications in the literature. The HyperLEDA data base becomes incomplete
towards the dwarf galaxy regime, which is why we add to the classified galaxy
sample the data of den Brok et al. (2014); Muñoz et al. (2015); Sánchez-
Janssen et al. (2019); Zanatta et al. (2021); Su et al. (2022) for members of
the Fornax, Virgo, and Coma galaxy clusters. The combined catalogue contains
$888$ nucleated galaxies across the whole sky, which we used to cross-match
with the German footprint of eROSITA.

###  2.2 eROSITA observations: _eRASS:4_

We systematically extracted X-ray photometry at the input coordinates of the
nucleated galaxies in the cumulative _eRASS:4_ images within the footprint of
the German eROSITA Consortium (i.e. Galactic longitudes between $179.944$ and
$359.944$). This led to a starting sample of $239$/$888$ galaxies, with mean
exposure of $\sim$418\text{\,}\mathrm{s}$$ and standard deviation of
$\sim$58\text{\,}\mathrm{s}$$ (see Figure 1) for three examples. A detailed
description of the methodology is presented in Arcodia et al. (2023) and we
only outline here the basic steps.

X-ray counts were extracted between $0.2$-$2.0\text{\,}\mathrm{keV}$ within a
circular aperture of $30\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,
corresponding to $\sim$75\text{\,}\mathrm{\char 37\relax}$$ of the encircled
energy fraction of eROSITA’s point spread function in the adopted energy band.
The background contribution was estimated from an annulus with inner and outer
radii of $120\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ and
$360\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, respectively. Contaminating
X-ray sources were masked following the prescription from Comparat et al.
(2023) (Appendix A). For a small number of cases ($21$/$239$), more than 70
percent of the source aperture was masked out and the NSC was therefore
excluded from the analysis. Consequentially, the sample size of nucleated
galaxies with extracted X-ray properties from the automated pipeline reduced
down to $$239$-$21$=$218$$.

From _eRASS:4_ X-ray spectra were extracted from the same aperture using the
srctool task in the eROSITA Science Analysis Software System (eSASS, Brunner
et al., 2022), with products version 020. The spectral analysis was performed
with the Bayesian X-ray Analysis software (BXA) version 4.0.5. (Buchner et
al., 2014), which connects the nested sampling algorithm UltraNest (Buchner,
2019, 2021) with the fitting environment XSPEC version 12.12.0. (Arnaud et
al., 1996), in its Python version PyXspec.444The documentation for PyXspec can
be found here. We adopted a simple power-law model (zpowerlw) with absorption
fixed at the Galactic column density from HI4PI (HI4PI Collaboration et al.,
2016) and redshifted to rest-frame. We quoted median, first, and 99th
percentiles from the fit posteriors for fluxes and luminosity, unless
otherwise stated. In our approach, we followed the method from Arcodia et al.
(2023) and adopted $P_{\mathrm{binom}}=$3\text{\times}{10}^{-4}$$ as threshold
for a significant detection, which corresponds to a spurious fraction of
$\sim$1\text{\,}\mathrm{\char 37\relax}$$. For non-detections
($P_{\mathrm{binom}}>$3\text{\times}{10}^{-4}$$), we quoted upper limits using
the 99th percentiles of the fit posteriors.

Potential individual sources of contamination from within the source aperture
were treated a posteriori after visual inspection and were considered on a
case-by-case basis. For instance, we cross-matched our sample with the
catalogue from Walton et al. (2022), which compiled ultraluminous X-ray source
(ULXs) candidates from _XMM–Newton_ , _Swift-XRT_ , and _Chandra_ data. We
manually masked out a handful of apertures with known ULX candidates and other
obvious off-nuclear X-ray sources, whose centroid lies within the source
aperture. In some cases, this resulted in the NSC being non-detected after the
masking: the galaxy NGC 4559 contains a known ULX (Walton et al., 2022) and
after its masking, the whole source aperture is masked-out and no products
from the NSC can be analysed. Therefore, after a visual inspection, the number
of galaxies with extracted X-ray properties from the automated pipeline was
$217$/$218$. We provide in Table 1 the properties of all $217$ galaxies,
derived as explained in the next subsections, with their measured eRASS:4
X-ray luminosities.

From this sample, we obtain that $18$/$217$ targets are detected. After
computing 1$\sigma$ binomial uncertainties on this fraction from Cameron
(2011), this results in a detection fraction of
$8.3$${}^{+2.3}_{-1.5}\,$\mathrm{\char 37\relax}$$.

Finally, we performed a stacking analysis of non-detections following the
methodology outlined in Comparat et al. (2022) and Arcodia et al. (2023).
Around each galaxy, we retrieved a photon cube with the angular position,
physical distance to the associated galaxy, exposure time, observed and
emitted energy (shifted to the rest-frame of the galaxy), and the effective
area for each photon. Detected sources in the field were masked. The cubes
were merged for the desired sub-sample of non-detected galaxies. We took a
weighted mean of all events within $10\text{\,}\mathrm{kpc}$ using the weight
described in Equ. 3 of Comparat et al. (2022) to obtain the surface
brightness. We estimated the background surface brightness by repeating the
procedure with events located at a distance between $15$ and
$50\text{\,}\mathrm{kpc}$. Finally, we subtracted the background from the mean
surface brightness (within a $10\text{\,}\mathrm{kpc}$ radius) and converted
it to a luminosity.

Table 1: Literature and eROSITA _eRASS:4_ data for our sample of nucleated
galaxies. The complete data table is available for download online.

Galaxy | RA | DEC | $m-M$ | $T$ | $\log_{10}\,M_{\star}^{\mathrm{gal}}$ |  $\log_{10}\mathrm{SFR}$ (a) | $L_{X,\,\mathrm{median}}^{\mathrm{2-10}\,$\mathrm{keV}$}$ | $L_{X,\,\mathrm{1}^{\mathrm{st}}}^{\mathrm{2-10}\,$\mathrm{keV}$}$ | $L_{X,\,\mathrm{99}^{\mathrm{th}}}^{\mathrm{2-10}\,$\mathrm{keV}$}$ |  $L_{X}^{\mathrm{cont}}$ (b) | $t_{\mathrm{exp}}$ | $P_{\mathrm{binom}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [mag] |  |  [$\mathrm{M_{\odot}}$] |  [$\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}$] |  [$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$] |  [$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$] |  [$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$] |  [$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$] |  [$\mathrm{s}$] |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
BTS 76 | $179.68375$ | $27.58500$ | $30.50$ | $10.0\pm 2.0$ | $7.53\pm 0.16$ | $-3.47$ | – | – | $3.97\times 10^{38}$ | $(6.2\pm 2.3)\times 10^{36}$ | $326$ | 0.06078
DDO 084 | $160.67458$ | $34.44889$ | $29.99$ | $9.8\pm 0.6$ | $7.65\pm 0.35$ | $-3.84$ | – | – | $1.07\times 10^{38}$ | $(8.1\pm 6.5)\times 10^{36}$ | $325$ | 0.39909
DDO 088 | $161.84292$ | $14.07028$ | $29.44$ | $8.9\pm 0.3$ | $7.88\pm 0.15$ | N/A | – | – | $6.54\times 10^{38}$ | $(1.37\pm 0.48)\times 10^{37}$ | $282$ | $0.04318$
dw 1048+13 | $162.14917$ | $13.06000$ | $30.13\pm 0.12$ | $-2.0\pm 1.0$ | $6.71\pm 0.43$ | N/A | – | – | $4.25\times 10^{38}$ | $(9.3\pm 9.2)\times 10^{35}$ | $297$ | $0.40543$
dw 1049+12b | $162.35833$ | $12.55250$ | $30.17\pm 0.06$ | $-2.0\pm 1.0$ | $7.55\pm 0.06$ | N/A | – | – | $1.84\times 10^{38}$ | $(6.37\pm 0.81)\times 10^{36}$ | $275$ | $0.76436$
… | … | … | … | … | … | … | … | … | … | … | … | …

  * Columns: (1) galaxy name; (2) Right ascension; (3) Declination; (4) Distance modulus; (5) Hubble type; (6) Galaxy stellar mass; (7) Star formation rate; (8) Median X-ray luminosity in the $2$-$10\text{\,}\mathrm{keV}$ band; (9) & (10) 1st and 99th percentiles of the X-ray luminosity; (11) Expected X-ray luminosity due to binary systems; (12) Exposure time; (13) Binomial probability that the detection is a background fluctuation.

  * (a)

The star formation rate is set to zero for dwarf elliptical galaxies in the
core of the Virgo cluster.

  * (b)

If no star formation rate could be determined, the estimated X-ray luminosity
due to binaries is a lower limit.

###  2.3 NSC and galaxy parameters

The imaging data from HST used to classify galaxies are inhomogeneous with
different spatial resolutions and available filters. Instead of deriving new
NSC parameter estimates from these data, we looked for available values from
the literature. As a consequence, not all nucleated galaxies of our sample
have available NSC properties (seen in Figures 4 and 5 below). More
specifically, we searched for available NSC parameters for nucleated galaxies
within the Local Volume ($d\lesssim$11\text{\,}\mathrm{Mpc}$$; Seth et al.,
2006; Georgiev et al., 2009; Graham & Spitler, 2009; Baldassare et al., 2014;
Schödel et al., 2014; Calzetti et al., 2015; Carson et al., 2015; Crnojević et
al., 2016; Nguyen et al., 2017; Baumgardt & Hilker, 2018; Nguyen et al., 2018;
Bellazzini et al., 2020; Pechetti et al., 2020; Hoyer et al., 2023), the field
environment outside the Local Volume (Georgiev & Böker, 2014; Georgiev et al.,
2016), and the Virgo galaxy cluster (Sánchez-Janssen et al., 2019). Nucleated
galaxies in the Coma and Fornax galaxy clusters are not part of our data set.

Galaxy stellar masses were determined as presented in the next subsection.
Morphological type values are adopted from HyperLEDA and for the data sample
of Sánchez-Janssen et al. (2019), we assumed a value of $-5$, corresponding to
elliptical galaxies.

####  2.3.1 Galaxy stellar masses

To compute galaxy stellar masses, we used three different tracers: (1) the
$B-V$ colour, (2) the $g-r$ colour, and (3) the $K$-band luminosity. First, we
obtained the photometric parameters and distance estimates from both the
HyperLEDA and NED data bases. We then computed the aforementioned colours and
$B$-, $g$-, and $K$-band luminosities using the absolute magnitude of the
Sun555URL: http://mips.as.arizona.edu/~cnaw/sun.html (Willmer, 2018),
accounting for Galactic extinction via the re-calibrated version of the
Schlegel et al. (1998) extinction maps (Schlafly & Finkbeiner, 2011), assuming
$R_{V}=3.1$ (Fitzpatrick, 1999). Internal extinction was not taken into
account. The mass-to-light ratios for the different colours and $K$-band
luminosity were taken from McGaugh & Schombert (2014) and Du et al. (2020),
which give re-calibrated versions of the original relations by Bell et al.
(2003); Portinari et al. (2004); Zibetti et al. (2009); Into & Portinari
(2013); Roediger & Courteau (2015). The uncertainties on the final stellar
mass estimates were based on those of the photometric parameters, the distance
estimate, the absolute magnitude of the Sun (assumed to be
$0.04\text{\,}\mathrm{mag}$), and the stellar mass-to-light relation (assumed
to be $0.3\text{\,}\mathrm{dex}$). Usually, the latter one dominates over all
other uncertainties.

For the dwarf galaxies in the Virgo cluster, we directly took the mass
estimates from Sánchez-Janssen et al. (2019), which are based on fits to the
spectral energy distributions and overall match to the other three approaches
outlined above (Hoyer et al., 2021). Their stellar mass estimates lack an
uncertainty which is why we assumed a value of $0.3\text{\,}\mathrm{dex}$.

###  2.4 Literature X-ray data

As pointed out in the Introduction, previous work investigated the X-ray
emissivity of NSCs in search for MBHs. Based on optical spectroscopy as well
as radio and X-ray data (via Chandra, ROSAT, and XMM-Newton), Seth et al.
(2008) found X-ray emission indicative of the presence of MBHs consistent with
the position of NSCs in a sample of $176$ early- and late-type galaxies. Most
recently, Baldassare et al. (2022) used data from the Chandra X-ray
observatory to search for such signatures in $108$ nearby
($d\lesssim$40\text{\,}\mathrm{Mpc}$$) nuclei from the galaxy sample of
Georgiev & Böker (2014), which is composed of massive late-type galaxies. They
classified $29$ targets as having significant X-ray emission and, thus,
harbouring AGN.

Some other studies investigated the X-ray luminosity of the central region of
galaxies without taking into account their nuclear classification. Here, we
took into account data from She et al. (2017) and Ohlson et al. (2023), which
used archival Chandra data.

We extracted fluxes and luminosities in the $0.5$-$7\text{\,}\mathrm{keV}$
range to compare with Ohlson et al. (2023) and in the
$2$-$10\text{\,}\mathrm{keV}$ range to compare with Baldassare et al. (2022).
We note that compared to Chandra, eROSITA is most sensitive in the
$0.2$-$2.3\text{\,}\mathrm{keV}$ energy band (Predehl et al., 2021). From
Ohlson et al. (2023) we used their luminosity values for a circular aperture
with a radius of $3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ to better match
the PSF of eROSITA. Baldassare et al. (2022) gives luminosities in both bands
and we confirmed that the results (given in Section 3.4) remain unchanged when
changing to the $0.5$-$7\text{\,}\mathrm{keV}$ band.

##  3 Analysis

###  3.1 X-ray contamination from binaries

Both low- and high-mass X-ray binaries, namely, binary stellar systems
composed of a donor and either a neutron star or stellar mass black hole, can
significantly contribute to a galaxy’s total X-ray luminosity (e.g. Iwasawa et
al., 2009), sometimes rivalling that of the AGN (e.g. Lehmer et al., 2010).
This contribution is especially important for our analysis given the size of
eROSITA’s PSF (half-energy width of approximately
$30\text{\,}\mathrm{arcsec}$; Predehl et al., 2021). The formation of low-mass
X-ray binaries (LMXBs) typically takes $1$-$10\text{\,}\mathrm{Gyr}$ (Verbunt
& van den Heuvel, 1995) as we require stellar evolution to first produce a
neutron star, which then has to dynamically enter into a binary system with a
donor. The collective X-ray luminosity of these systems in older disks and
bulges is related to the stellar mass of the galaxy (Gilfanov, 2004) via
$L_{X}^{\textrm{LMXB}}=\alpha\times M_{\star}^{\mathrm{gal}}$ (e.g. Colbert et
al., 2004; Lehmer et al., 2010), where
$\log_{10}\,\alpha=29.15^{+0.07}_{-0.06}\;$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{M_{\odot}}}^{-1}$$
(Lehmer et al., 2019).

In contrast, high-mass X-ray binaries (HMXBs) require a stellar mass black
hole and their X-ray emission is related to the stellar evolution timescale of
the massive donor star, resulting in a luminous phase about
$100\text{\,}\mathrm{Myr}$ after formation of the binary (Verbunt & van den
Heuvel, 1995). Due to the high-mass of the donor and its short lifetime, the
X-ray luminosity is related to the star formation rate ($\mathrm{SFR}$) of the
host galaxy via $L_{X}^{\textrm{HMXB}}=\beta\times\mathrm{SFR}$ (e.g. Grimm et
al., 2003), where
$\log_{10}\beta=39.73^{+0.15}_{-0.10}\;$\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$\,($\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}$)^{-1}$
(Lehmer et al., 2019).

To estimate the current star formation rate of our sample, which we assume to
be constant over the last $100\text{\,}\mathrm{Myr}$ (i.e. no star bursts or
quenching effects from tidal interactions or bright AGN), we used the
correlation from Hao et al. (2011, but see also from , , then) relating the
emission in the far-ultraviolet ($L_{\mathrm{FUV}}$) with the mid-infrared
($L_{\mathrm{MIR}}$) and star formation rate via:

$\log_{10}\,\mathrm{SFR}=\log_{10}\big{(}L_{\mathrm{FUV}}+3.89\times
L_{\mathrm{MIR}}\big{)}-$43.35$\,.$ (1)

We took this approach, contrary to, for instance, an estimation via X-rays
(Colbert et al., 2004; Symeonidis et al., 2011; Riccio et al., 2023), due to
the availability and homogeneity of the available luminosities: to estimate
the far-ultraviolet luminosity, we used the publicly available data from Galex
(Morrissey et al., 2007). For the mid-infrared luminosity, we used the AllWISE
W4 (Wright et al., 2010; Cutri et al., 2013) or Spitzer MIPS (Rieke et al.,
2004) magnitudes. For dwarf elliptical galaxies in the Virgo cluster, we
assumed that no star formation occured over the last few hundred
$\mathrm{Myr}$ and that the expected X-ray binary contamination is solely
produced by LMXBs.

After computing the galaxy stellar mass and star formation rates, we
determined the luminosities of both classes of binary systems. The expected
contamination by X-ray binary systems is the sum of the two components,
$L_{X}^{\mathrm{bin}}=L_{X}^{\mathrm{LMXB}}+L_{X}^{\mathrm{HMXB}}$. Objects
that were detected above this expected X-ray binary emission could indicate
the presence of an MBH (however, see Section 4.2 for caveats).

###  3.2 Galaxy properties of X-ray detected sources

We compared the expected X-ray luminosity from the binary populations in the
$2$-$10\text{\,}\mathrm{keV}$ range with the _eRASS:4_ data in Figure 2,
distinguishing between X-ray detected sources
($P_{\mathrm{binom}}\leq$3\text{\times}{10}^{-4}$$) and undetected sources
($P_{\mathrm{binom}}>$3\text{\times}{10}^{-4}$$, shown as upper limits). We
also distinguished between objects only detected with eROSITA and the ones
also detected with other instruments taken from Seth et al. (2008); She et al.
(2017); Baldassare et al. (2022) or Ohlson et al. (2023). All significant
detections in the _eRASS:4_ data have measured X-ray luminosities of
$L_{X,\,\mathrm{obs}}^{\mathrm{2-10}\,$\mathrm{keV}$}>${10}^{38}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$
and similarly high expected luminosities from the galaxies LMXBs and HMXBs.
Only three galaxies in our sample (NGC 2903, NGC 4212, and NGC 4639) have
measured luminosities greater than the expected values from binaries at
$3\sigma$ confidence. Some of the X-ray detected sources are also measured
below the expected value which may be related to uncertainties in the
estimates of the galaxy-only predictions. We will discuss this observation
further in Section 4.2. Below
$L_{X,\,\mathrm{obs}}^{\mathrm{2-10}\,$\mathrm{keV}$}\sim$5\text{\times}{10}^{38}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$
we find no significant emission and can only determine upper limits.

Figure 2: Expected X-ray luminosity from the combined high- and low-mass
binary population in the $2$-$10\text{\,}\mathrm{keV}$ range
($L_{X,\,\mathrm{bin}}^{2\mathrm{-}10\,\mathrm{keV}}$) versus the measurements
$\big{(}L_{X,\,\mathrm{obs}}^{2\mathrm{-}10\,\mathrm{keV}}\big{)}$. We show
the detected galaxies ($P_{\mathrm{binom}}\leq$3\text{\times}{10}^{-4}$$) in
the _eRASS:4_ footprint with blue squares. For non-detections, we plot upper
limits using the 99th percentile of the X-ray luminosity (blue arrows).
Additionally, we highlight the _eRASS:4_ luminosities of galaxies with
available literature X-ray data (see Section 2.4; orange circles). These
include data from Seth et al. (2008); She et al. (2017); Baldassare et al.
(2022); Ohlson et al. (2023). Some of the galaxies are not detected in the
_eRASS:4_ data but have secure measurements from other instruments available
in the literature (orange arrows). A more detailed comparison between the
_eRASS:4_ and literature data is presented in Section 3.4.

Some of the NSCs with X-ray upper limits in Figure 2 reside in dwarf
elliptical galaxies in the core of the Virgo cluster ($29$/$117$ galaxies at
$M_{\star}^{\mathrm{gal}}\leq${10}^{9}\text{\,}\mathrm{M_{\odot}}$$). A lack
of photometric data in the literature makes it challenging to determine star
formation rates and we assumed that no star formation takes place for these
objects. While this assumption may be justified for the dwarf galaxy sample,
the presented X-ray luminosity from binaries remains only a lower limit.

Figure 3 shows the distribution of measured X-ray luminosity versus galaxy
stellar mass. To further constrain the emission in the dwarf galaxies, we
stack non-detected galaxies within bins of stellar mass of
$1\text{\,}\mathrm{dex}$ starting at
$M_{\star}^{\mathrm{gal}}=10^{5.5}\,$\mathrm{M_{\odot}}$$ until
$10^{10.5}\,$\mathrm{M_{\odot}}$$ (see Section 2.2). None of the stacked X-ray
images contains signals above background level, with upper limits of the order
of $2\text{\times}{10}^{38}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$
found in each stellar mass bin. Estimating the expected X-ray luminosity from
LMXBs with these galaxy stellar masses reveals that these upper limits are
either matching or higher than the expected values; so, while we cannot rule
out the presence of MBHs with low Eddington fractions (see Section 4.1 for
more discussion below), these galaxies do not contain X-ray bright AGN.

From the low-mass towards the high-mass end in Figure 3 the X-ray luminosity
of the significant detections appears to increase, starting around
$M_{\star}^{\mathrm{gal}}\sim${10}^{10}\text{\,}\mathrm{M_{\odot}}$$. While
this increase could be related to the increasing contribution from the nuclear
emission, as seen for a few sources in Figure 2, most of the detected sources
feature the expected luminosities from X-ray binary systems. Therefore, it
appears plausible that this increase is mostly related to the increasing
strength of LMXBs and not due to the presence of AGN.

Figure 3: Measured X-ray luminosity in the $2$-$10\text{\,}\mathrm{keV}$ range
($L_{X,\,\mathrm{obs}}^{\mathrm{2-10}\,$\mathrm{keV}$}$) versus galaxy stellar
mass ($\log_{10}\,M_{\star}^{\mathrm{gal}}$). The markers, colour-coding and
literature references are all the same as in Figure 2. We stack non-detections
(upper limits, blue and orange arrows) in mass bins of
$1\text{\,}\mathrm{dex}$ width starting at a galaxy mass of
$10^{5.5}\,$\mathrm{M_{\odot}}$$ to set tighter constraints on the X-ray
emission in dwarf galaxies. Only galaxies above galaxy masses of approximately
$10^{9.3}\,$\mathrm{M_{\odot}}$$ are detected in the _eRASS:4_ data.

###  3.3 NSC properties of X-ray detected sources

Regarding NSC properties, we show the NSC versus host galaxy stellar mass
relation (e.g. Georgiev et al., 2016; Neumayer et al., 2020; Ashok et al.,
2023) in Figure 4. We complement our data with the sample of Baldassare et al.
(2022), which is based on the data of Georgiev & Böker (2014). Similar to the
previous observation for galaxy masses in Figure 3, only the most massive NSCs
are X-ray detected, above
$M_{\star}^{\mathrm{nsc}}\sim${10}^{7}\text{\,}\mathrm{M_{\odot}}$$. Other
lower-mass NSCs are not detected, including objects, which were detected
previously with Chandra down to
$M_{\star}^{\mathrm{nsc}}\gtrsim${10}^{5}\text{\,}\mathrm{M_{\odot}}$$.

Figure 5 shows the NSC effective radius versus stellar mass plane. The
_eRASS:4_ detected NSCs have both high stellar mass and large radii, as is
expected from scaling relations (Georgiev et al., 2016; Neumayer et al., 2020;
Ashok et al., 2023). The average half-mass density of these objects,
$\overline{\rho}$, is consistent with other X-ray detected but lower-mass NSCs
in the literature, falling between
$${10}^{4}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-3}$\lesssim\bar{\rho}\lesssim${10}^{6}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-3}$$.

Baldassare et al. (2022) also investigated the properties of their NSC sample
distinguishing between X-ray luminosities likely originating from a massive
black hole and X-ray binaries. Due to our limited sample of new detections, we
refer to their study for further discussion with NSC properties.

Figure 4: Nuclear star cluster ($\log_{10}\,M_{\star}^{\mathrm{nsc}}$) versus
host galaxy stellar mass ($\log_{10}\,M_{\star}^{\mathrm{gal}}$). We show
X-ray detected NSCs with full colour, distinguishing between sources only
detected with eROSITA (blue) and sources also detected with other instruments
(orange). A fainter shade is used for non-detection in the _eRASS:4_ data. In
addition, we show NSCs analysed by Seth et al. (2008, grey hexagons) and
Baldassare et al. (2022) (green triangles) for NSCs with X-ray emission
outside the _eRASS:4_ footprint (or with significant contamination). A lack of
literature data for NSC properties limits the included data set.  Figure 5:
Nuclear star cluster effective radius ($r_{\mathrm{eff}}$) versus stellar mass
($\log_{10}\,M_{\star}^{\mathrm{nsc}}$). The colour-coding and symbols are the
same as in Figure 4. Note that compared to Figure 4 fewer NSCs are shown
because of a lack of measured effective radii for the dwarf galaxy sample in
the core of the Virgo cluster (Sánchez-Janssen et al., 2019; Ferrarese et al.,
2020). Uncertainties are omitted for clarity.

###  3.4 X-ray variable sources

We compare in Figure 6 the X-ray luminosity of nucleated galaxies in the
German footprint of the _eRASS:4_ data with matches in Chandra, ROSAT, or XMM-
Newton data from Seth et al. (2008), She et al. (2017), Baldassare et al.
(2022), and Ohlson et al. (2023), as introduced in Section 2.4. Given their
upper limits, most of the _eRASS:4_ values are in agreement with the
literature. However, six other galaxies (NGC 2903, 3384, 4321, 4365, 4639, and
4701) have values not in agreement with the literature.

For one galaxy, NGC 2903, there exist literature data from both Baldassare et
al. (2022) and Ohlson et al. (2023). Although the same data were analysed,
they quote fluxes differing by a factor $\sim 100$, which is likely due to the
difference between catalogue fluxes (Evans et al., 2010; Ohlson et al., 2023)
and those estimated through aperture photometry (Baldassare et al., 2022). We
computed the X-ray fluxes through spectral analysis and obtained that the
_eRASS:4_ spectrum is sufficiently well described by a simple power-law with
photon index $2.08\pm 0.20$, although a more complex spectral model would be
most likely required with higher count statistics in the
$2$-$10\text{\,}\mathrm{keV}$ band. Based on this, we are not able to infer
whether the difference between eROSITA and Chandra values is due to intrinsic
variability or differences in the flux estimate methods.

We investigate this object further by looking at the X-ray luminosity in each
eROSITA survey to find that it was detected in eRASS2 (with
$L_{X,\,\mathrm{obs}}^{\mathrm{0.2}-\mathrm{2.0}\,$\mathrm{keV}$}\sim$1.1\text{\times}{10}^{40}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$)
but not in any other individual image. This indicates that NGC 2903 likely
hosts an AGN, which is variable on time scales shorter than six months (i.e.
the time between all-sky scans by eROSITA).

For the other five galaxies, we find no significant signs of X-ray variability
within the _eRASS:4_ data. This could indicate that the inconsistency detected
here, if not caused by any differences in analysis strategy between our
approach and the one of Ohlson et al. (2023), occurs on time scales longer
than six months but shorter than the time difference between the Chandra and
eROSITA observations of a few years.

Figure 6: Comparison of literature Chandra, XMM-Newton, or ROSAT X-ray
luminosity versus _eRASS:4_ values. Full colour-coded symbols show X-ray
detections in _eRASS:4_. The dashed line gives the one-to-one values. We
compare to Seth et al. (2008), She et al. (2017) and Baldassare et al. (2022)
using the $2$ to $10\text{\,}\mathrm{keV}$ band. Ohlson et al. (2023) give
Chandra X-ray data in the $0.5$ to $7\text{\,}\mathrm{keV}$ band. Galaxies,
which likely show some X-ray variability, are specifically named.

##  4 Discussion

###  4.1 Evaluating the presence of massive black holes

Most of the X-ray detected NSCs in _eRASS:4_ are in agreement with the
expected luminosity from binary system, therefore, we are not able to
unambiguously associate it with an AGN. Three NSCs (NGC 2903, NGC 4212, and
NGC 4639) have emission above the expected value and this supports the
presence of an AGN. For NGC 4212 the NSC properties are not known and no
secure black hole measurement exists. For the other two galaxies, She et al.
(2017) estimates the black hole mass using the $M_{\mathrm{BH}}-\sigma$
relation from Kormendy & Ho (2013) to find
$\log_{10}\,M_{\mathrm{BH}}^{2903}\,/\,\mathrm{M}_{\odot}=6.48^{+0.10}_{-0.10}$
and
$\log_{10}\,M_{\mathrm{BH}}^{4639}\,/\,\mathrm{M}_{\odot}=6.65^{+0.09}_{-0.09}$,
resulting in $\lambda_{\mathrm{Edd}}^{2903}\sim$7\text{\times}{10}^{-5}$$ and
$\lambda_{\mathrm{Edd}}^{4639}\sim$4\text{\times}{10}^{-4}$$, respectively,
after taking into account a bolometric correction from Duras et al. (2020).
The NSC masses are
$\log_{10}\,M_{\star}^{\mathrm{nsc},\,2903}\,/\,\mathrm{M}_{\odot}\sim 7.71$
(Pechetti et al., 2020) and
$\log_{10}\,M_{\star}^{\mathrm{nsc},\,4639}\,/\,\mathrm{M}_{\odot}\sim 7.05$
(Georgiev et al., 2016), resulting in mass fractions of
$\log_{10}(M_{\mathrm{bh}}\,/\,M_{\star}^{\mathrm{nsc}})\sim-1.23$ and $-0.4$,
respectively. These values compare well to other literature values, as we show
in Figure 7.

Figure 7: Mass ratio of the SMBH and NSC stellar mass
($\log_{10}\,M_{\mathrm{BH}}/M_{\star}^{\mathrm{nsc}}$) versus the host galaxy
stellar mass. We show the compiled data of Neumayer et al. (2020) and add to
that data for NGC 1336 (Saulder et al., 2016; Fahrion et al., 2019; Thater et
al., 2023), NGC 3593 (Bertola et al., 1996; Nguyen et al., 2022), and four
galaxies from Ashok et al. (2023), all shown with blue squares. The sample of
ultra-compact dwarf galaxies (UCDs, purple pentagons) include the compilation
of Neumayer et al. (2020) and the additional B023-G078, M 31’s most massive
globular cluster, from Pechetti et al. (2022). We estimate the most likely
previous UCD’s host galaxy mass using the
$M_{\star}^{\mathrm{nsc}}$-$M_{\star}^{\mathrm{gal}}$ relation of Neumayer et
al. (2020) for massive galaxies (their Equation $2$). Additionally, we add
nucleated galaxies with significant X-ray excess, hinting at the existence of
a massive black hole, by using a black hole mass estimate from She et al.
(2017) and NSC mass estimates from Georgiev et al. (2016), shown with red
hexagons. We add to this last group the data points of NGC 2903 and NGC 4639,
which could host massive black holes based on their X-ray variability (see
Figure 6) with an orange and green triangle, respectively.

In comparison to previous studies, our investigation also takes into account a
large sample of $111$ early-type galaxies of various stellar masses. We show
in Figure 8, the galaxy stellar mass versus Hubble morphological type plane,
with additions of Seth et al. (2008) and Baldassare et al. (2022) for NSCs
without observational data in _eRASS:4_. As previously explored, only the most
massive galaxies
($M_{\star}^{\mathrm{gal}}\gtrsim${10}^{10}\text{\,}\mathrm{M_{\odot}}$$) have
significant X-ray emission, irrespective of the host galaxy’s morphology. The
focus of Baldassare et al. (2022) on the late-type sample of Georgiev & Böker
(2014) results in detections down to galaxy masses of
$\sim{10}^{8}\text{\,}\mathrm{M_{\odot}}$, as explored previously. In
contrast, with the exception of one NSC (NGC 4467) from Seth et al. (2008), no
early-type galaxy in the same mass range is X-ray detected within _eRASS:4_.
This could imply several points:

  1. 1.

The accreting MBHs fall below the sensitivity of the cumulative data of
eROSITA.

  2. 2.

The black hole occupation fraction is different for galaxies of different
Hubble type resulting in fewer X-ray detections at the same host galaxy
stellar mass.

  3. 3.

Assuming that the massive black hole occupation fraction does not depend on
environment, this could indicate smaller black hole masses in these elliptical
galaxies, assuming the same Eddington fraction.

  4. 4.

While assuming the same massive black hole occupation fraction and typical
black hole masses, our results could indicate that the Eddington fraction is
different between different morphologies, likely caused by a lack of gas
available for accretion in the centres of early-type galaxies.

Figure 8: Galaxy stellar mass ($\log_{10}\,M_{\star}^{\mathrm{gal}}$) versus
Hubble morphological type ($T$) of the X-ray detected (solid blue) and non-
detected (fainter blue points) _eRASS:4_ data. Galaxies, which are part of
_eRASS:4_ but were also detected by previous work (Seth et al., 2008; She et
al., 2017; Baldassare et al., 2022; Ohlson et al., 2023) are shown in orange
colour. In addition, we show the data by Seth et al. (2008) and Baldassare et
al. (2022) with grey hexagons and green triangles, respectively, for galaxies,
which are not part of the _eRASS:4_ sample. The single X-ray detected early-
type galaxies at a galaxy stellar mass of approximately
${10}^{9}\text{\,}\mathrm{M_{\odot}}$ is NGC 4467, whose X-ray properties were
analysed by Seth et al. (2008) and Graham & Soria (2019).

Regarding the first and last items, we can estimate an upper limit to the
Eddington fraction of these objects. Assuming that we can ignore the
contributions of HMXBs, the expected X-ray luminosity from binaries for a
galaxy with stellar mass $M_{\star}^{\mathrm{gal}}\sim
10^{8.5}\,$\mathrm{M_{\odot}}$$ is
$L_{X,\,\mathrm{bin.}}\sim${10}^{37}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$,
which is about a factor ten below the upper limits of _eRASS:4_ (see Figure
3). Using a bolometric correction factor of about ten (Duras et al., 2020),
the upper limit on the luminosity of a massive black hole would be
$L_{\mathrm{bol},\,\mathrm{max}}\sim${10}^{39}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$.
In galaxies of this mass, we would expect to find MBHs with
$M_{\mathrm{BH}}\sim${10}^{5}\text{\,}\mathrm{M_{\odot}}$$ from observational
data in early-type galaxies (Erwin & Gadotti, 2012; Reines & Volonteri, 2015;
Capuzzo-Dolcetta & Tosta e Melo, 2017; Greene et al., 2020) and
$M_{\mathrm{BH}}\sim 10^{6.5}$\mathrm{M_{\odot}}$$ from simulations (e.g.
Spinoso et al., 2023). For these MBH masses
(${10}^{5}\text{\,}\mathrm{M_{\odot}}$, $10^{6.5}\,$\mathrm{M_{\odot}}$$), a
Bolometric luminosity as quoted above
($L_{\mathrm{bol},\,\mathrm{max}}\sim${10}^{39}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$)
would imply an Eddington fraction of at most $0.01\text{\,}\mathrm{\char
37\relax}$. This value roughly matches and is one dex higher than he values
determined for NGC 2903 and NGC 4639 above, yielding an explanation for why we
most likely do not detect these low-luminosity AGN in X-rays, if present. This
also sets an upper limit to the hot gas accretion of such systems.

Regarding the other items, current observational data indicate that MBHs in
early-type galaxies are more massive than their counterparts in late-types
(see the compilation of Greene et al., 2020) but the scaling relations are
solely based on measurements in massive galaxies and were extrapolated to the
dwarf galaxy regime. The occupation fraction of MBHs appears to be similar,
according to recent X-ray and dynamical results (see, again, the compilation
of Greene et al., 2020), making the first and last items of the above list
most likely.

####  4.1.1 MBH occupation fraction from X-rays

Assuming that all significantly X-ray detected NSCs host an AGN, we can infer
the combined occupation fraction of NSCs and AGN. To gain statistical
significance, we add to the _eRASS:4_ data the sample of Baldassare et al.
(2022). For their sample we assume that all galaxies classified as having
“diffuse” emission are non-detections.

We show the fraction of detected over the total sample as a function of galaxy
and NSC stellar mass in Figure 9. For comparison, we also add the sample
occupation fraction of Seth et al. (2008) for NSCs and the AGN occupation
fractions (without information of whether an NSC is present) from Miller et
al. (2015a) and Ohlson et al. (2023) from observations and Tremmel et al.
(2023) from a simulation. We find that above
$M_{\star}^{\mathrm{gal}}\sim${10}^{7}\text{\,}\mathrm{M_{\odot}}$$ the
combined AGN & NSC fraction increases and reaches $100\text{\,}\mathrm{\char
37\relax}$ around ${10}^{10}\text{\,}\mathrm{M_{\odot}}$. Our data are
slightly elevated compared to the data of Seth et al. (2008) and Ohlson et al.
(2023), which could be related different selection effects (Seth et al., 2008
use optical spectroscopy, radio, and X-ray data to find evidences of the
presence of an AGN) of the samples or instrument-related response functions.
Additionally, because of the big half-energy width of eROSITA’s PSF (half-
energy width of about $30\text{\,}\mathrm{arcsec}$; Predehl et al., 2021),
off-nuclear star forming region can contaminate the measurements and result in
a too-high estimate of the NSC and AGN fraction. Given the uncertainties of
the data it remains unclear whether the presence of an NSC can enhance the
occupation fraction of active galactic nuclei, not taking into account an
enhanced rate of tidal disruption events (Pfister et al., 2020).

Figure 9: Occupation fraction of nuclear star clusters and active galactic
nuclei versus host galaxy stellar mass
($\log_{10}\,M_{\star}^{\mathrm{gal}}$). We combine the _eRASS:4_ data with
the data from Baldassare et al. (2022) to gain statistical significance (green
line). Literature data for NSCs & AGN come from Seth et al. (2008, gray line).
In addition, we show the occupation fractions of AGN from the observational
studies of Miller et al. (2015a) and Ohlson et al. (2023, blue and purple
lines, respectively), and the computational results of Tremmel et al. (2023,
red line).

###  4.2 Caveats

There exist several caveats in the analysis both related to the measured and
expected X-ray luminosities. The spatial resolution of eROSITA results in an
uncertainty on both the central position of the emission (roughly
$4\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) and allows for contamination by
off-nuclear sources, thus not guaranteeing that the emission stems from the
NSC. Instead, HMXBs or ULXs may mimic the emission of an accreting massive
black hole in NSCs. Such contaminating sources may still be present in our
sample, despite matching it with the ULX sample of Walton et al. (2022)
(Section 2.2), thus requiring follow-up observations with higher spatial
resolution facilities like Chandra.

In low-mass galaxies, the above argument is not problematic because the
circular aperture typically contains the galaxy’s stellar body out to at least
one effective radius (cf. middle and right columns in Figure 1). However,
there are several other challenges in this mass range. As noted by Lehmer et
al. (2019), the scaling relations to track the contribution by HMXBs and LMXBs
contain uncertainty in the low-mass regime because of a poorly sampled X-ray
luminosity function. This makes it unclear how to interpret measured X-ray
emission in dwarf galaxies in future eROSITA data releases in case dwarf
galaxies (or stacks of dwarf galaxies) become X-ray detected. Additionally,
the scaling relations of Lehmer et al. (2019) apply for the expected X-ray
luminosity of the whole galaxy. However, in most cases, the aperture used to
extract the X-ray photometry only covers part of the galaxy (see Section 2.2),
thus overestimating the contamination from binaries.

Additionally, the influence of globular clusters to the X-ray binary
contamination remains unclear. It is well-known that globular clusters
efficiently produce LMXBs (e.g. Clark, 1975; Sivakoff et al., 2007; Cheng et
al., 2018) and that they can heavily influence the X-ray properties of
elliptical galaxies (e.g. Irwin, 2005; Lehmer et al., 2014, 2020) and, to some
degree, late-types as well (Pfahl et al., 2003; Peacock et al., 2009; Hunt &
Reffert, 2023). This effect is especially important in the dwarf galaxy regime
where the importance of globular clusters towards the total mass budget of the
galaxy increases, as probed by the specific globular cluster frequency666The
specific globular cluster frequency is often calculated as the total number of
globular clusters divided by the galaxy’s stellar mass,
$S_{N}=N_{\mathrm{GC}}\,/\,M_{\star}^{\mathrm{gal}}$. (e.g. Miller & Lotz,
2007; Liu et al., 2019; Carlsten et al., 2022). There also exists some scatter
in the specific frequency of dwarf galaxies (see e.g. the environmental
dependence discussed in Carlsten et al. 2022) requiring a detailed
investigation of the X-ray luminosity from globular clusters in dwarf
galaxies.

Furthermore, what is not taken into account here is the X-ray contribution
from binaries within the NSC itself. This contribution may be similar to
globular clusters, especially in dwarf galaxies where the properties of both
systems become similar (e.g. Fahrion et al., 2022; Hoyer et al., 2023), but it
remains somewhat unclear in higher mass NSCs. Several works have found that
denser globular clusters have a higher probability of hosting X-ray binaries
(e.g. Kundu et al., 2002; Jordán et al., 2007; Sivakoff et al., 2007; Riccio
et al., 2022; Hunt & Reffert, 2023) and this probability should increase
further for NSCs, which are the densest stellar systems known (Neumayer et
al., 2020), especially in massive galaxies (Pechetti et al., 2020). The
expected LMXB contribution to the X-ray budget of NSCs is currently unknown
and distinguishing them from low-luminosity AGN requires future work.

In summary, there exist several caveats related to both the measured X-ray
luminosity and the expected value from X-ray binaries. Further studies
disentangling the contributions from binaries are required for the targets
with significant detections, which are high-mass galaxies at present. The
sample size may increase and extend towards the dwarf galaxy regime if future
eROSITA or Athena (Nandra et al., 2013) data are added.

##  5 Conclusions

We combined a compilation of galaxies containing a nuclear star cluster (NSC)
with eROSITA _eRASS:4_ data to probe X-ray signatures of an accreting massive
black hole (MBH) within them. Using a sample of more than $200$ nucleated
galaxies with overlapping _eRASS:4_ data within the footprint of the German
eROSITA Consortium, we find $18$ significant detections of which one is
related to the presence of an off-nuclear ultra-luminous X-ray source.
However, compared to the expected X-ray contamination from both low- and high-
mass X-ray binaries, only three galaxies (NGC 2903, 4212, and 4639) have
measured luminosities indicative of the presence of an MBH. Another six
galaxies (NGC 2903, 3384, 4321, 4365, 4639, and 4701) have significantly
different X-ray luminosities compared to previous archival measurements, which
we interpret as indicative of a variable X-ray AGN. For NGC 4701, we find
variability within the _eRASS:4_ data set, which could be related to an
intrinsic variability or changes in obscuration. To confirm the nature of
these objects, follow-up observations are necessary.

The MBH to NSC stellar mass fraction versus host galaxy stellar mass of the
newly identified AGN compares well to other known systems. By adding X-ray-
based black hole mass estimates, we can significantly expand this parameter
space towards lower galaxy stellar masses, apparently confirming a drop in the
mass ratio around galaxy stellar masses of
${10}^{10}\text{\,}\mathrm{M_{\odot}}$.

Assuming that all X-ray detected NSCs above the expected luminosity from X-ray
binaries host AGN, we construct an NSC + AGN occupation fraction by adding
data from Baldassare et al. (2022) to gain statistical significance. The
resulting curve has higher occupation fraction than the one of Seth et al.
(2008) and the AGN only fractions of Miller et al. (2015a); Tremmel et al.
(2023) and Ohlson et al. (2023). The differences may be related to instrument-
related response functions and the different half-energy widths of the
instruments used.

Large-scale surveys, as those carried out by eROSITA or by Athena in the
future, offer a unique view on X-ray emission in dwarf galaxies, covering low-
mass early-type galaxies as well, whose X-ray properties had not been
investigated previously, with respect to the presence of NSCs.

###### Acknowledgements.

The authors thank the editor and anonymous referee for their constructive
feedback. N.H. is a fellow of the International Max Planck Research School for
Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD). N.H.
received financial support from the European Union’s HORIZON-MSCA-2021-SE-01
Research and Innovation programme under the Marie Sklodowska-Curie grant
agreement number 101086388 - Project acronym: LACEGAL. R.A. received support
for this work by NASA through the NASA Einstein Fellowship grant No HF2-51499
awarded by the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555. 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 Tuebingen, with
the support of DLR and the Max Planck Society. The Argelander Institute for
Astronomy of the University of Bonn and the Ludwig Maximilians Universitaet
Munich also participated in the science preparation for eROSITA. The eROSITA
data shown here were processed using the eSASS software system developed by
the German eROSITA consortium. This work is based in part on observations made
with the Spitzer Space Telescope, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology under a contract with NASA.
This work acknowledges funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 865637). This work used the following software: Astropy (The
Astropy Collaboration et al., 2013, 2018), dustmaps (Green, 2018), Matplotlib
(Hunter, 2007), NumPy (Harris et al., 2020), Pandas (McKinney, 2010), eSASS
(Brunner et al., 2022), and BXA (Buchner et al., 2014), PyXspec (Arnaud et
al., 1996, ,and Craig Gordon and Keith Arnaud). The data underlying this
article are available in the article and in its online supplementary material.

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