# A link to the past: characterizing wandering black holes in Milky Way-type
galaxies

J.Untzaga1,2 , S. Bonoli2,3 , D. Izquierdo-Villalba4,5 , M. Mezcua1,6 D.
Spinoso7
1Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Magrans, 08193
Barcelona, Spain
2Donostia International Physics Center (DIPC), Manuel Lardizabal Ibilbidea, 4,
San Sebastián, Spain
3IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain
4Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-
Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
5INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
6Institut d’Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC,
08860 Castelldefels (Barcelona), Spain
7Department of Astronomy, 6th floor, MongManWai Building, Tsinghua University,
Beijing 100084, People’s Republic of China E-mail<EMAIL_ADDRESS>
###### Abstract

A population of non-stellar black holes ($\gtrsim$100 M⊙) has been long
predicted to wander the Milky Way. We aim to characterize this population by
using the L-Galaxies semi-analytical model applied on top of the high
resolution Millennium-II merger trees. Our results predict $\sim$10 wandering
black holes with masses $\sim$2 $\times$ 103 M⊙ in a typical $z$ = 0 Milky Way
galaxy, accounting for $\sim$2$\%$ of the total non-stellar black hole mass
budget of the galaxy. We find that the locations of these wanderers correlate
with their formation scenario. While the ones concentrated at $\lesssim$1 kpc
from the galactic nucleus on the disk come from past galactic mergers, the
ones formed as a consequence of ejections due to gravitational recoils or the
disruption of satellite galaxies are typically located at $\gtrsim$100 kpc.
Such small and large distances might explain the absence of strong
observational evidence for wandering black holes in the Milky Way. Our results
also indicate that $\sim$67$\%$ of the wandering population is conformed by
the leftovers of black hole seeds that had little to no growth since their
formation. We find that wandering black holes that are leftover seeds become
wanderers at an earlier time with respect to grown seeds, and also come from
more metal-poor galaxies. Finally, we show that the number of wandering black
holes in a Milky Way-type galaxy depends on the seeding efficiency.

###### keywords:

methods: numerical – software: simulations – Galaxy: evolution – black hole
physics

††pubyear: 2023††pagerange: A link to the past: characterizing wandering black
holes in Milky Way-type galaxies–A link to the past: characterizing wandering
black holes in Milky Way-type galaxies

##  1 Introduction

While the formation mechanism of stellar-mass black holes (BHs) is well
established, the origin of supermassive BHs ($\geq$106 M⊙) is unclear (see,
e.g., Inayoshi et al. 2020; Volonteri et al. 2021 for a full review).
Observational evidence of more than 300 supermassive BHs at z $\sim$ 7 (Fan et
al., 2001, 2003; Willott et al., 2007, 2010; Mortlock et al., 2011; Venemans
et al., 2013; Wu et al., 2015; Banados et al., 2018; Decarli et al., 2018;
Matsuoka et al., 2019; Wang et al., 2021; Naidu et al., 2022; Castellano et
al., 2022; Adams et al., 2023) and even z $\sim$ 11 (Maiolino et al., 2023)
requires the formation of BH “seeds” at even higher redshift from which these
supermassive BHs grew via accretion and mergers.

Possible seed BH formation channels include (1) the direct collapse of low-
metallicity Population III stars of a few hundred solar masses (Fryer et al.,
2001; Madau & Rees, 2001; Heger & Woosley, 2002; Heger et al., 2003; Spera &
Mapelli, 2017), (2) the direct collapse of rapidly inflowing dense gas in the
first protogalaxies (Loeb & Rasio, 1994; Eisenstein & Loeb, 1994; Bromm &
Loeb, 2003; Lodato & Natarajan, 2006), (3) the direct collapse induced by the
major merger of massive gas-rich galaxies (Mayer & Bonoli, 2018) and (4)
runaway collisions of stars in dense stellar clusters (Coleman Miller &
Hamilton, 2002; O’leary et al., 2006; Devecchi & Volonteri, 2009; Giersz et
al., 2015; Mapelli, 2016). Theoretical models support that a fraction of these
seeds did not grow into supermassive BHs, and instead remained with a similar
mass up until current redshift (e.g., Greene 2012; Reines & Comastri 2016;
Spinoso et al. 2023). This means that characterizing the properties and
formation mechanisms of leftover seeds found in the local Universe is crucial
for a thoroughgoing understanding of the non-stellar BH population all along
its history. But where can we find these seeds?

One of the potential locations for leftover seeds could be the core of dwarf
galaxies, where seed BHs might not have had the chance to grow substantially
(Volonteri, 2010; Greene, 2012; Reines & Comastri, 2016). Indeed, black holes
in the intermediate mass regime (102 M⊙ – 106 M⊙, IMBH onwards) could be
leftover seeds. The first observational evidence for IMBHs traces back to the
late 80s, with the elliptical galaxy Pox 52 (Kunth et al., 1987) and the
spiral galaxy NGC 4395 (Filippenko & Sargent, 1989). To detect BHs through
kinematics, one must be able to resolve the stellar motions that take place
within their gravitational sphere of influence. Since this type of observation
is greatly dependant on the spatial resolution of the sample, the limiting
distance for kinematic observations is set to $\sim$4 Mpc (Nguyen et al.,
2018; Greene et al., 2020). For this reason, a variety of methods can be
employed in the absence of kinematic signatures or to complement them. For
example, IMBHs in dwarf galaxies are often found in active galactic nuclei
(AGNs) through optical spectroscopy (Greene & Ho, 2004, 2007; Dong et al.,
2007, 2012; Izotov et al., 2007; Reines et al., 2013; Chilingarian et al.,
2018; Mezcua & Sánchez, 2020; Salehirad et al., 2022; Mezcua & Sánchez, 2024),
through the detection of hard X-ray emission or radio emission (Reines et al.,
2011; Reines et al., 2014; Reines et al., 2020; Mezcua et al., 2016; Mezcua et
al., 2018, 2019) or even by utilizing near-infrared photometry (Fischer et
al., 2006; Giallongo et al., 2015; Bohn et al., 2021).

IMBHs can also be found as the result of in-situ formation in nearby globular
clusters (Kızıltan et al., 2017). The presence of seeds in globular clusters
was first suggested by Silk & Arons (1975) when studying their X-ray emission.
Since then, many studies have aimed at detecting the radiative accretion
signature of IMBHs within globular clusters, but no conclusive results have
been obtained (e.g., Maccarone et al. 2005; Bash et al. 2007; Maccarone &
Servillat 2008; Cseh et al. 2010; Strader et al. 2012; Haggard et al. 2013;
Wrobel et al. 2015; Mezcua 2017; Tremou et al. 2018; Baumgardt et al. 2019;
Wrobel & Nyland 2020). While several candidates have been found from dynamical
models, there has been no direct X-ray nor radio detections of the sources. A
few minor exceptions arise, like globular cluster G1 in M31 (Pooley &
Rappaport, 2006; Kong, 2007; Ulvestad et al., 2007) and more recently the
outbursting source 3XMM J215022.4-055108 (Lin et al., 2018, 2020), both of
them detected in the X-ray regime. In any case, since globular clusters have
scarce amounts of gas and dust, any signatures of accretion from a putative
leftover seed in the X-ray and radio regimes are expected to be very low.

There is also observational evidence of non-nuclear IMBHs in the form of of
ultraluminous and hyperluminous X-ray sources (ULXs and HLXs, respectively;
e.g., Matsumoto et al. 2001; Farrell et al. 2009; Jonker et al. 2010; Mezcua &
Lobanov 2011; Mezcua et al. 2013a, b; Kim et al. 2015; Mezcua et al. 2015;
Comerford et al. 2015; Lin et al. 2016; Mezcua et al. 2018; Lin et al. 2020).
ULXs are extragalactic and off-nuclear X-ray sources with luminosities LX
$\geq$ 1039 erg/s that are less luminous than AGNs but more luminous than any
other stellar process with intrinsic X-ray luminosities. Since most of this LX
$\sim$ 1039 erg/s sources can be explained by stellar-mass BHs undergoing
super-Eddington accretion, HLXs, characterized by their even higher X-ray
luminosities (LX $\geq$ 1041 erg/s) remain the best seed BH candidates
(Barrows et al., 2019). Some examples include HLX-1 (e.g., Farrell et al.,
2009; Davis et al., 2011; Webb et al., 2012), M82-X1 (Kaaret et al., 2001;
Pasham et al., 2014) and NGC 2276-3c (Sutton et al., 2012; Mezcua et al.,
2013b, 2015), three ULXs that would have belonged to the nucleus of a dwarf
galaxy before being stripped during a minor merger with the current host
galaxy (King & Dehnen, 2005; Soria et al., 2013; Mezcua et al., 2015).

Plenty of observational evidence has been found regarding IMBHs, both central
and non-nuclear, both at local Universe and up to redshift z $\sim$ 3.4
(Mezcua et al., 2019); and since most of these detections are carried out by
examining radiative signatures, it is safe to assume that an even greater
population of IMBHs is yet to be found when also accounting for non-active
ones. This population –a fraction of which is theorized to correspond to
leftover seeds– would have been accreted via hierarchical merging into the
Milky Way (MW) throughout its history (Volonteri & Perna, 2005; Bellovary et
al., 2010). At $z$ = 0, these leftover seeds should be wandering within the
MW, thus contributing to its population of wandering black holes (WBHs).

The study of WBHs has recently gained significant attention thanks in part to
the advancements in numerical simulations of galaxy formation and evolution
(Volonteri et al., 2003; Volonteri & Perna, 2005; Bellovary et al., 2010;
González & Guzmán, 2018; Izquierdo-Villalba et al., 2020; Greene et al., 2021;
Ricarte et al., 2021b). In terms of the total expected amount of WBHs in MW-
type galaxies, Ricarte et al. (2021a) used the ROMULUSC zoom-in cosmological
simulation on a 1014 M⊙ galaxy cluster to predict over one thousand WBHs and
found a correlation between the expected number of wanderers in a galaxy and
its halo mass. Re-scaling to the MW halo mass, $\sim$10 WBHs are predicted to
wander our galaxy. This prediction is in agreement with a previous work by
Tremmel et al. (2018) where they studied a sample of 26 MW-mass halos
extracted from the ROMULUS25 cosmological simulation. They found $\sim$12
wanderers within 1 virial radius of a MW-mass galaxy, $\sim$5 of them within
10 kpc of the galactic center. In agreement with a higher concentration of
WBHs in the central regions, Ricarte et al. (2021a) found that 50$\%$ of all
wanderers are within 0.1 virial radii. Following a similar trend, Weller et
al. (2022) used the illustris TNG50 simulation to study the kinematics and
dynamics of star clusters acting as IMBH proxies in MW analogues, and they
found that over 4$\%$ of the total population of wanderers was predicted
within 1 kpc of the galactic center.

We find a more optimistic prediction of the overall number of WBHs in a MW
analog in the work of Greene et al. (2021), who used the semi-analytic model
SatGen (Jiang et al., 2021) to estimate $\sim$100 expected WBHs in the MW
halo. However, this prediction assumes that every infalling dwarf satellite
hosts a central BH that will later become a WBH of the MW-type galaxy. O’Leary
& Loeb (2009, 2012) used $\gtrsim$1000 merger tree histories of the MW to
calculate the number and mass distribution of those WBHs that may have been
ejected from their small host galaxies during the MW formation history. They
found that $\gtrsim$100 WBHs with masses $\gtrsim$104 M⊙ should be in the MW
halo today, surrounded by compact star clusters that make up for $\sim$1$\%$
of their WBH’s mass.

Kruijssen et al. (2019) compared the age-metallicity of 96 MW globular
clusters to the globular cluster population of the E-MOSAICS project,
consisting of 25 cosmological zoom-in simulations of MW-mass galaxies based on
the EAGLE galaxy formation simulations. The study concluded that the MW has
experienced $\sim$15 accretion events from $\gtrsim$4.5 $\times$ 106 M⊙
satellites. Of these $\sim$15 accretion events, the MW experienced $\sim$9
mergers at $z$ > 2 and $\sim$6 mergers at $z$ < 2\. On the other hand, Weller
et al. (2022) used the Illustris TNG50 cosmological simulation to determine
that $\sim$87$\%$ of WBHs in a MW-type galaxy exhibit a tendency to drift
towards the galactic center, with sinking timescales $\sim$10 Gyr. Combining
the findings of Kruijssen et al. (2019) (who observed mergers at $z$ > 2 and
$z$ < 2) and Weller et al. (2022) (who established an upper bound limit of
sinking timescales $z$ $\sim$ 2), it seems plausible that both a population of
non-disrupted infalling satellites hosting non-stellar BHs and a population of
recoiled WBHs should be expected in MW-type galaxies at the current redshift.

As for the spatial location of these wanderers, Tremmel et al. (2018)
estimated with a >4$\sigma$ significance that WBHs are preferentially found
outside of the Galactic disk. This could incentivize a search for small,
compact star clusters around these wanderers in the low-density environment of
the halo. However, most of the WBHs seem not to have retained a stellar
counterpart or to be located at larger fractions of the virial radius (Ricarte
et al. 2021a).

The aim of this research is to characterize both the WBH and the leftover seed
populations in the MW at the current redshift, and to improve our
understanding of their merger history. We expect to obtain useful information
that will help us in detecting these objects through observational means in
future works. This characterization has been carried out by using the semi-
analytic simulation for galaxy evolution L-Galaxies (Henriques et al. 2015;
see §2). In §3 we study the populations of WBHs and leftover seeds at $z$ = 0
and the time of their formation, and we also review the effects of varying the
initial seeding prescription in the model. We summarize and discuss our
results in §4. A Lambda Cold Dark Matter $(\Lambda$CDM) cosmology with
parameters $\Omega_{\rm m}\,{=}\,0.315$, $\Omega_{\rm\Lambda}\,{=}\,0.685$,
$\Omega_{\rm b}\,{=}\,0.045$, $\sigma_{8}\,{=}\,0.9$ and $\rm
H_{0}\,{=}\,67.3\,\rm km\,s^{-1}\,Mpc^{-1}$ is adopted throughout the paper
(Planck Collaboration et al., 2014).

##  2 The L-Galaxies semi-analytical model

In this work, we make use of the L-Galaxies semi-analytic code (Guo et al.
2011; Henriques et al. 2015; see §2.2), which was designed to model galaxy
formation and evolution on the merger trees of the Millennium (MR, Springel et
al. 2005) and Millennium-II (MR-II, Boylan-Kolchin et al. 2009) cosmological,
N-body simulations. Since we are particularly interested in studying the WBH
population and its link with the leftover seeds from the high-$z$ Universe, we
make use of the MR-II merger trees, whose mass resolution allows us to trace
the formation and evolution of WBHs in dark matter (DM) halos of 108 – 1014
M⊙. In the following subsections, we briefly describe the properties of the
MR-II simulation and the baryonic physics included in L-Galaxies to trace the
formation and evolution of galaxies and their WBHs.

###  2.1 The DM component: The Millenium-II simulation

Millenium-II is a large DM N-body simulation that follows 21603 particles of
6.89 $\times$ 106 h-1 M⊙ within a 100 h-1 Mpc comoving box. Structures formed
in the simulation (i.e., the so-called DM halos) are identified and stored at
68 different epochs or snapshots by making use of the SUBFIND algorithm. DM
halos hosting at least more than 20 particles are stored and arranged
according to their evolutionary path in the so-called merger trees (Springel
et al., 2005). Taking into account the particle mass resolution and the box
size, the minimum and maximum mass of a DM halo inside the MR-II merger trees
corresponds to 1.38 $\times$ 108 M⊙ and 8.22 $\times$ 1014 M⊙, respectively.
This makes MR-II the perfect simulation to trace not only the hosts of MW-type
galaxies but also the ones of the dwarf satellites that at some point might
deposit their WBH population in them. Even though the finite time resolution
offered by the MR-II merger trees (${\sim}\,300\,\rm Myr$) is adequate to
study the evolution of DM halos, it hinders the possibility of tracing
accurately the baryonic physics involved in galaxy evolution (with dynamical
times that can be up as short as few Myrs). To overcome this limitation,
L-Galaxies does an internal time discretization between the Millenium-II
outputs with a time resolution that varies between $5\,{-}\,20\,\rm Myr$,
depending on redshift.

###  2.2 The treatment of the galaxies

L-Galaxies is a state-of-the-art semi-analytical model that couples many
different astrophysical processes with DM merger trees extracted from N-body
simulations. In this work, we use as a foundation the version presented in
Henriques et al. (2015) with the modifications of Izquierdo-Villalba et al.
(2019), Izquierdo-Villalba et al. (2020) and Izquierdo-Villalba et al. (2022),
included to improve the predictions for galaxy morphology, extend the physics
of WBHs and introduce the formation and evolution of BH binaries.

The model assumes that as soon as a DM halo is formed, a fraction of the
baryonic component of the universe collapses within it. This baryonic
component has a mass that corresponds to 15.5$\%$ of the DM halo mass, and is
given by the cosmic mean baryon fraction for the Planck cosmology (see
Henriques et al. 2015). The baryon component is heated and distributed within
the halo in the form of a diffuse, spherical and quasi-static hot gas
atmosphere. The hot gas atmosphere undergoes a cooling process regulated by
the functions of Sutherland & Dopita (1993), and, as a result, a fraction of
the gas condensates and migrates towards the center of the DM halo. The
continuous cooling results in the cold gas component becoming massive enough
for star formation to take place, leading to the formation of a stellar disc
(White & Rees, 1978; White & Frenk, 1991; Springel et al., 2001).

Right after the star formation events, massive and short-lived stars explode
and inject energy into the surrounding medium via supernovae feedback, warming
up and expelling part of the galaxy’s cold gas. Besides supernovae feedback,
the model also invokes the so-called radio-mode feedback from central
supermassive BHs, which efficiently suppresses gas cooling in massive
galaxies. On top of the secular evolution, galaxies interact among themselves
thanks to the hierarchical assembly of DM halos. These interactions are
classified by L-Galaxies as major and minor depending on the mass ratio of the
two merging systems, and their outcomes are completely different. While major
interactions are assumed to be able to destroy the disc component of central
galaxies and lead to the formation of pure elliptical galaxies, minor
interactions do not have an effect on the disc of central galaxies and instead
trigger the build-up of a stellar bulge due to the absorption of satellite
galaxies. Finally, environmental processes are also accounted for by
L-Galaxies. Specifically, the infalling of galaxies in more massive halos
causes the loss of the entire galaxy’s hot gas atmosphere and the removal of
its cold gas and stellar component through tidal forces (ram pressure and
galaxy tidal disruption processes, respectively).

###  2.3 The treatment of BHs and the formation of WBHs

On top of the galaxy physics, the version of L-Galaxies used in this work
includes a sophisticated model to treat the formation and evolution of BHs,
the dynamical evolution of BH binaries and the formation and evolution of
WBHs. In the following section, we outline the relevant physics for this work.

####  2.3.1 Formation and growth of BHs

The formation of the first BHs in L-Galaxies has been extensively studied in
Spinoso et al. (2023). Specifically, the authors included the genesis of light
seeds (Population III remnants) and massive seeds (i.e. intermediate-mass and
heavy BH-seeds) by using sub-grid approaches and accounting for the local
spatial variations of the intergalactic metallicity and the UV-background
produced by star formation events. In this work, we address the genesis of the
first BHs in a much more simplified way, taking into account the analytical
seeding probability presented in Spinoso et al. (2023) and modified by
Izquierdo-Villalba et al. (2023). In brief, at any redshift, the probability
that a newly resolved halo inside the MR-II merger trees is seeded with a BH
is similar to the one predicted by the complete model of Spinoso et al.
(2023). In this way, every time a halo is resolved at $z$ $\geq$ 7 (i.e.
redshift at which galaxies tend to be pristine in L-Galaxies), it has a
probability $\mathcal{P}$ of being seeded with a central BH with mass randomly
chosen from [102 – 104] M⊙ (see Figure 3 of Spinoso et al. 2023):

$\mathcal{P}\ =\ \mathcal{A}\ (1\ +\ z)^{\gamma}\
\left(\frac{M_{\text{halo}}}{M^{\text{th}}_{\text{halo}}}\right)$ (1)

where $M_{\rm halo}$ is the virial mass of the newly resolved halo, $z$ its
redshift and $\mathcal{A}$, $\gamma$, $M^{\text{th}}_{\text{halo}}$ three free
parameters set to 0.015, 7/2 and 7 $\times$ 1010 M⊙, respectively. To guide
the reader, Fig. 1 shows how the seeding probability varies with redshift in
the case of two representative DM halo masses, 109 M⊙ and the DM halo mass
resolution of Millenium-II, which corresponds to 1.38 $\times$ 108 M⊙. To
quantify how the uncertainties related to BH formation impact the population
of WBHs in MW-type galaxies, we will assume different values for the parameter
$\mathcal{A}$ in Eq.(1). Specifically, we will explore (1) a boosted seeding
scenario ($\mathcal{A}$${}_{\text{boosted}}$ = 2.0$\mathcal{A}$) where the
galaxies have a larger probability of being seeded with respect to the
fiducial model, and (2) a weaker seeding scenario
($\mathcal{A}$${}_{\text{weaker}}$ = 0.5$\mathcal{A}$) where galaxies are less
frequently seeded than in the fiducial case (see Fig. 1 for the behaviour of
the boosted and weaker models compared to the fiducial one).

Figure 1: Probability used by L-Galaxies to seed a central BH in function of
redshift in a DM halo of 109 M⊙ (blue) and in a DM halo with the minimum mass
allowed of 1.38 $\times$ 108 M⊙ (green). The solid lines indicate the
probability function followed by the standard seeding prescription (see Eq.1).
The dashed lines represent the probability function of the boosted seeding
while the dotted lines indicate the probability function of the weaker
seeding.

Once a BH is formed, it can grow via three different channels: hot gas
accretion, cold gas accretion, and mergers with other BHs. The first growth
mechanism is triggered by the continuous accretion of hot gas that surrounds
the galaxy (Croton et al., 2006). However, since the rate of accretion is
orders of magnitude below the Eddington limit, the contribution of this hot
gas accretion to the BH mass growth is minimal. On the other hand, cold gas
accretion is the main channel driving BH growth. This mechanism is triggered
by galaxy mergers (where the central BH accumulates a fraction of cold gas
from the merged galaxy) and disc instability events (where the BH accretes an
amount of cold gas proportional to the mass of stars that have triggered the
stellar instability). In both cases, the cold gas available for accretion is
assumed to settle in a reservoir around the BH and is progressively consumed
according to a two-phase model (as suggested by Hopkins & Hernquist 2009 and
first implemented in L-Galaxies by Bonoli et al. 2009; Marulli et al. 2009).
In the first phase, the central BH consumes a fraction (set to 70%) of the
available gas reservoir following an Eddington-limited growth. In the second
phase, the central BH consumes the remaining gas following a quiescent growth
regime characterized by progressively smaller accretion rates (for further
details on the most recent implementation see Izquierdo-Villalba et al.,
2020).

Finally, BHs in the model can also increase their masses through BH
coalescences. These do not happen instantaneously, but L-Galaxies includes a
detailed model for the dynamical evolution of massive black hole binaries. The
evolutionary pathway of these systems can be described by three different
stages, as suggested by Begelman et al. (1980): pairing, hardening and
gravitational wave phases. The former takes place just after a galaxy merger
when the two BHs are separated at ${\sim}\,\rm kpc$ scales. In particular, the
BH hosted by the satellite galaxy experiences dynamical friction that causes
its sinking towards the galactic nucleus. In the model, to estimate the time
spent by a black hole in the pairing phase, we use the expression by Binney &
Tremaine (2008) to take into account, among others variables, the mass of the
pairing WBH, the velocity dispersion of the satellite galaxy and the position
at which the BH is deposited by its satellite galaxy just after the galactic
merger (which is considered to be when the satellite galaxy has lost 85$\%$ of
its total mass by tidal stripping). Once the dynamical friction phase ends,
the satellite BH reaches the galactic nucleus of the primary galaxy and binds
with its central BH ($\rm{\sim}\,pc$ scales). As a result, a BH binary is
formed. This is when the hardening phase starts and the orbital separation of
the BH binary begins to shrink. This process is modeled differently according
to whether the environment that drives the two BHs to coalescence is a gas-
rich or gas-poor environment. Mergers in gas-rich environments require the
binary to be surrounded by a gas reservoir with a mass larger than the mass of
the binary. For mergers in gas-poor environments, it is assumed that the gas
reservoir around the BHs is smaller than the total mass of the binary. In
these cases, the hardening is caused by the extraction of binary energy and
angular momentum through 3-body interactions with background stars that cross
the binary orbit. Independently of the environment, the BH binary eventually
reaches ${\sim}\,\rm sub\,{-}\,pc$ scales, and the emission of gravitational
waves brings the two BHs to the final coalescence in a few Myrs. In some
cases, the lifetime of a BH binary can be long enough that a third BH which
finishes the pairing phase can reach the galaxy nucleus and interact with the
binary system. When this happens, the model refers to the tabulated values of
Bonetti et al. (2018) to determine if the triple interaction leads to the
prompt merger of the BH binary or to the ejection of the lightest BH from the
system (hereafter tagged as slingshot scenario). In this latter case, the
separation of the BH binary that remains in the galactic core is computed
following Volonteri et al. (2003) and the final eccentricity is select as a
random value between [0 – 1].

####  2.3.2 The WBH population

Although all the BHs formed in L-Galaxies start their lives inside a galactic
nucleus, many different mechanisms can move them away from the heart of the
host galaxy. BHs displaced outside their galaxies nuclei but in bound orbits
within the DM halo are tagged as WBHs. L-Galaxies distinguishes and tracks the
orbits of three different classes of WBHs according to the physical mechanism
that displaced them: disrupted, pairing and recoiled. In the following
paragraphs, we outline these types of WBHs and briefly describe their
formation mechanism. For a full detailed description of the wandering
population we refer the reader to Izquierdo-Villalba et al. (2020) and
Izquierdo-Villalba et al. (2022).

  * •

Disrupted WBHs: The lifetime of these WBHs starts in the nucleus of small
satellite galaxies that at some point in their lives start infalling into a
more massive DM halo. During this process, the satellite galaxy is not able to
survive the tidal field exerted by the the massive DM halo, and its cold gas
(stellar) component is stripped and deposited in the hot gas atmosphere
(inter-cluster medium) that surrounds the primary galaxy. The BHs from the
satellite galaxy (including the central and the WBHs) are also left behind in
the inter-cluster medium in the form of WBHs. A schematic view of the
formation of these WBHs is presented in the first row of Fig. 2. Once a
“disrupted” WBH is formed, L-Galaxies follows its orbit by solving self-
consistently its equation of motion. To do so, the code uses the final
position and velocity of the galaxy before being disrupted as the starting
point of integration of the disrupted WBH population. Then, the code follows
their orbit by taking into account the gravity acceleration and the dynamical
friction exerted by any of the inter-cluster components (i.e dark matter,
stars and hot gas, see Eq.32 of Izquierdo-Villalba et al. 2020).

  * •

Pairing WBHs: Similarly to disrupted WBHs, pairing WBHs are BHs that
originally were hosted in the nuclei of a satellite galaxy. Unlike disrupted
WBHs, however, the host galaxies of pairing WBHs survive the tidal forces
exerted by the primary galaxy, managing to merge with it. During this process,
the central BH of the satellite galaxy is deposited in the outskirts of the
newly formed galaxy and progressively sinks towards the nucleus of the merger
remnant thanks to the dynamical friction force exerted by the stellar
component. A schematic view of the formation of these WBHs is presented in the
second row of Fig. 2. Unlike disrupted WBHs, we do not follow the orbit of
pairing WBHs. Instead, we assume that they will reach the galactic centre
after spending a time $t_{\rm dyn}$ in the paring phase. This time is computed
according to the dynamical friction equation of Binney & Tremaine (2008) which
assumes, for simplicity, a spherical isothermal galaxy. Regarding the
evolution of the pairing WBHs, we assume a simple model in which the BHs
spiral towards the galactic nucleus inside an isothermal host on a near-
circular orbit. This assumption predicts that the radius of the orbit should
decay linearly in time. Consequently, the radial position of the BH inside the
galaxy is updated at each time step of L-Galaxies by multiplying the previous
distance by the factor $(1\,{-}\,\Delta t/t_{\rm dyn})^{1/2}$, where $\Delta
t$ is the time since the BH started the pairing phase (see similar approach
used in Guo et al., 2011, to follow the orbit of satellite galaxies inside DM
halos). Finally, in this work, the minimum allowed distance with respect to
the galactic centre to consider a pairing WBH as a wandering object is
$100\,\rm pc$. Pairing WBHs at distances below this threshold are assumed to
no longer wander since they will form in the next time step of L-Galaxies (a
few Myr) a binary system with the central BH of the central galaxy.

  * •

Recoiled WBHs: This type of WBHs follows the same formation pathway of pairing
WBHs; a satellite galaxy merges with a central galaxy after surviving the
tidal forces applied by a DM halo and the BH of the satellite becomes a
pairing WBH. After a certain time (from mega to giga years), the pairing WBH
reaches the nucleus of the galaxy and forms a BH binary that evolves through
the hardening and gravitational wave phase. When these two phases are over,
the two BHs in the binary system coalesce. Due to the merger and conservation
of linear momentum, the remnant BH receives a kick (determined according to
Lousto et al. 2012). If the magnitude of the velocity characterizing such a
kick is large enough, the BH escapes from the center of the galaxy and is
deposited inside the DM halo that hosts the galaxy. A schematic view of the
formation of these WBHs is presented in the last row of Fig. 2. Once the BH is
ejected from the galaxy, L-Galaxies follows its orbit in the same way as
described for the disrupted WBHs. Notice that the slingshot scenario presented
in the §2.3.1 forms wandering BHs similar to the recoiled ones. In these
cases, the BH kick that ejects the BH is not caused by the merger of two BHs
but by scattering after three-body interaction (the two BHs in the binary and
the intruder pairing WBH). For these cases, the orbits are followed in the
same way as presented in the recoiled case and, for convenience, they are
tagged as recoiled as well, since they only represent $\ll$1$\%$ of the total
WBH population.

Figure 2: Schematic representation of the three different WBH formation
mechanisms accounted in L-Galaxies. Each row represents a type of WBH, and the
columns showcase, from left to right, the steps that lead them to become
wanderers. First row: Disrupted WBHs originate from the complete disruption of
their host galaxy during their infall towards a central galaxy. Once the BH is
deprived of its host galaxy, it is assumed that its accretion disk (in case it
was in an accreting phase), is also removed. Second row: Pairing WBHs are
infalling BHs that have arrived to the galactic center of the larger galaxy
and are considered by L-Galaxies as being a few mega to giga years to form a
binary system with the central BH of the larger galaxy. Third row: Recoiled
WBHs are BHs that have also arrived to the galactic center of the larger
galaxy, but then they have continued their sinking process until reaching a
distance of $<$100 pc to the central BH. As a result, they have undergone a
merging process with its subsequent gravitational recoil. The kick velocity
depends on the properties of the two progenitors. If the modulus of the recoil
velocity is larger than the escape velocity of the galaxy, the kicked BH is
ejected from the host galaxy, starting its wandering phase. The WBH has been
depicted bigger in this case to highlight that it is the result of adding the
masses of the infalling wanderer and the previous central BH. However, since
the central BH mass of high-$z$ galaxies is usually $<$106M⊙, recoiled WBHs
rarely fit the supermassive BH category. Also, even though a recoil event
removes the central BH from the original host galaxy from its galactic
nucleus, due to the active merging history of most galaxies and the fact that
a pairing WBHs eventually fill this absence, the majority ($>$90$\%$) of
recoiled WBHs end up in galaxies that showcase a central BH.

We stress that as the redshift gets lower and the merger history of galaxies
gets more complex, satellite galaxies deposit not only their central BHs in
the DM halos of larger galaxies but also the WBH population that is orbiting
them as a consequence of the three main wandering events we just mentioned.
Consequently, it is expected that the majority of the WBHs orbiting low
redshift galaxies are not generated in situ but inherited from other galaxies
(see e.g Fig. 8 of Izquierdo-Villalba et al. 2020). For instance, we have
checked that $\sim$70$\%$ of the WBHs detected in a MW-type galaxy at $z$ = 0
are actually inherited from the wandering populations of previous galaxies.

###  2.4 Milky Way-type galaxies

Similarly to previous works (see, e.g, Licquia & Newman, 2013; Murphy et al.,
2022), we have defined MW-type galaxies as those fulfilling the following
constraints:

  * •

Have a stellar mass within [2$\times$1010 – 8$\times$1010] M⊙

  * •

Have a DM halo with mass [6$\times$1011 – 3$\times$1012] M⊙

  * •

Have a star formation rate in the range [0.5 – 2.5] M⊙yr-1

  * •

Have a bulge to total stellar mass ratio within [0.1 – 0.5]

  * •

Are central galaxies of their subhalos

###  2.5 Leftover seeds

Although plenty of literature has discussed the characterization of leftover
seeds (e.g, Rashkov & Madau 2013; Ferrara et al. 2014; Greene et al. 2019;
Mezcua 2019, 2020), no definitive consensus has been reached regarding their
properties. Typically, they are theorized as BHs that have had little to no
growth in their mass since their time of formation. In this work we will tag
as a leftover seed any BH that has not outgrown more than twice its original
mass. Since the range of masses for newly seeded BHs is within [102 – 104] M⊙,
the mass range for leftover seeds at $z$ = 0 will be [102 – 2 $\times$ 104]
M⊙.

##  3 Results

In this section, we present the results obtained from utilizing L-Galaxies to
characterize the WBH population in MW-type galaxies at the current redshift
(see §3.1). Additionally, we delve into the merger history of the WBH
population, establishing a connection between their current properties and the
redshift at which they became wanderers (see §3.2). In §3.3 we focus on the
leftover seed population of WBHs, characterizing it in the same way as the
previous subsections and comparing it with WBHs that instead managed to
substantially grow in mass. Finally, we conduct a comprehensive review of the
L-Galaxies seeding prescription and its effects on the stellar mass of the
galaxies that hosted the WBHs before they became wanderers (see §3.4).

###  3.1 The population of WBHs in MW-type galaxies

The L-Galaxies fiducial model run on the MR-II offers information on
$\sim$27.000 galaxies at $z$ = 0. After applying our constraints for MW-type
galaxies (see §2.4), we end up with a sample of 117 galaxies. All of these MW-
type galaxies host at least one WBH, with the predicted number of WBHs per
galaxy being 10${}_{-3}^{+5}$111All predicted numbers are computed as the
median of the distributions, and the errors as the 20th and 80th-percentiles,
respectively. (see the yellow histogram in the top panel of Fig. 3).

We also study the statistics of WBHs depending on their formation channel (see
§2.3.2). The results are presented in the bottom panel of Fig 3. Of the 117
MW-type galaxies, $\sim$97$\%$ of them host at least one disrupted WBH, with
the expected number being 7${}_{-3}^{+5}$ for this type of wanderer;
$\sim$85$\%$ of MW-type galaxies host one or more pairing WBH, with the
expected number being 3${}_{-2}^{+1}$; and $\sim$42$\%$ of MW-type galaxies
host one or more recoiled WBH, with the expected number being 1${}_{-0}^{+1}$.
The higher expected number of disrupted WBHs compared to the other two
families might be explained by the fact that pairing and recoiled WBHs result
from their host satellite galaxies sinking into a massive DM halo and merging
with a larger galaxy. Since this entire process takes more time than a
disruption event, which can occur as soon as a satellite galaxy loses most of
its mass, disrupted WBHs end up being the most abundant population.

Fig. 4 presents a view of the mass distribution of the WBH population (points
and diamonds) as a function of its distance to the galactic center, color-
coded according to each of the three WBH types. We find that pairing WBHs are
expected to be found at 1.1${}_{-0.8}^{+2.1}$ kpc from the galactic center and
to have a mass of $\bigl{(}$ 2.0${}_{-1.6}^{+9.5}$ $\times$ 103 $\bigr{)}$ M⊙
(Fig. 4, green square); disrupted WBHs are expected to be found at
174${}_{-78}^{+84}$ kpc from the galactic center and to have a mass of
$\bigl{(}$ 2.0${}_{-1.7}^{+7.0}$ $\times$ 103 $\bigr{)}$ M⊙ (Fig. 4, red
square); and recoiled WBHs are expected to be found at 182${}_{-92}^{+174}$
kpc from the galactic center and to have a mass of $\bigl{(}$
1.6${}_{-11}^{+79}$ $\times$ 104 $\bigr{)}$ M⊙ (Fig. 4, blue square). This
picture is coherent with the evolutionary stage of our population of WBHs.
Pairing WBHs are in the process of sinking towards their galactic nucleus
after the galaxy interaction whereas disrupted and recoiled WBHs are embedded
in the DM halos in orbits that can linger in the halo outskirts thanks to
large gravitational wave recoils and galaxy disruption processes occurring
primarily at the halo outskirts. Delving into the pairing population, these
type of WBHs are most likely found in the galactic disk, where the stellar
density is higher and the dynamical friction allows for a more efficient
sinking. This, in turn, exacerbates the detection of potential compact stellar
clusters bound to them, as stellar overdensities in the disk are easily
confused with open clusters.

On the other hand, we see that recoiled WBHs showcase higher mass values than
the other WBH families. This is expected since recoil events are the result of
a coalescence that includes a central BH, and central BHs have generally had
some time to grow before the merger. A small fraction (<0.1$\%$) of the WBH
population in MW-type galaxies does not experience disruption, pairing or
recoiled events. This is the case for the slingshot wanderers. Slingshot
wanderers are the result of a triple BH interaction at the center of a galaxy,
which leads to the ejection of the lightest BH (see §2.3.2). Of the 1310 WBHs
scattered among 117 MW-type galaxies, only one WBH was tagged as having
undergone a slingshot event. When compared to the overall sample of galaxies,
slingshot events are slightly more probable, occurring in $\sim$0.3$\%$ of all
WBHs.

In Fig. 5 we compare our results with the ROMULUS25 hydrodynamic simulation
from Tremmel et al. (2018). We find a good agreement in the total number of
WBHs expected in a MW-type galaxy ($\sim$10), and especially in the number of
WBHs within 5 kpc of the galactic center ($\sim$2), which in our case
corresponds to the pairing WBH population. However, we find the radial
distribution of the non-pairing WBH population to be far more extended than
predicted in Tremmel et al. (2018), with $\sim$60$\%$ of all WBHs being
farther than 100 kpc from their galactic center (see Fig. 5). This is probably
due to the fact that the initial seeding mass for BHs in ROMULUS25 is 106 M⊙,
while L-Galaxies uses a lighter distribution (100 – 104) M⊙. Since the WBH
population in ROMULUS25 has larger masses, the sinking process takes place
more efficiently, and therefore WBHs end up closer to the galactic center. It
is worth mentioning that ROMULUS25 does not consider multi-body BH
interactions nor gravitational wave recoil from BH mergers as wanderer-
triggering events for their BHs. In ROMULUS25, a BH is tagged as a wanderer if
it is further than 700 pc from the center of its host halo, but no recoiled
WBH population is contemplated.

In Fig. 6 we compute the ratio between the total mass of all WBHs in each MW-
type galaxy with respect to the mass of their respective central BH, and
compare our results with the analysis of the ROMULUS25 simulation from Ricarte
et al. 2021a. We find that in L-Galaxies WBHs account for
2${}_{-1.8}^{+14}$$\%$ of the local non-stellar BH mass budget of a MW-type
galaxy, which is lower than the $\sim$10$\%$ found by ROMULUS25. The
disagreement between these two values is most likely due to the minimum WBH
mass in ROMULUS25 being 106 M⊙. In comparison, the minimum WBH mass accounted
for in L-Galaxies is $\sim$100 M⊙, which makes for a more centralized
distribution of the BH mass budget in the galaxy.

Note that some of the galaxies in our sample have a central BH with a mass
significantly lower than the ones predicted by the scaling relations derived
from massive galaxies. This is in part because the inclusion of recoil events
causes central BHs to be off-center (thus not growing) for even extended
periods of time, and partially because in some cases, the galaxy incorporated
as central BH is one of the infalling satellites (see Izquierdo-Villalba et
al. 2020).

(a)

(b)

Figure 3: Distribution of MW-type galaxies as a function of the number of WBHs
that they are hosting. The dashed lines indicate the expected number of WBHs
of a given population. Top. Fraction of MW galaxies hosting any kind of WBH
(yellow) compared to the fraction of MW galaxies hosting leftover seeds
(violet). Bottom. Fraction of MW galaxies hosting WBHs that have undergone a
process of disruption (magenta), an ejection (blue), or that are in the
process of forming a binary system with their central BH (green). See more
details regarding the formation mechanisms of the WBHs in §2.3.2. Figure 4:
Mass distribution of the WBHs in MW-type galaxies as a function of their
distance to the galactic center. The WBHs are color-coded according to whether
they are in the process of forming a binary system with the central BH of
their galaxy (green), whether they have undergone a recoil event (blue) or if
they have undergone a disruption event (magenta). The diamond-shaped points
represent the population of leftover seeds while the dots their non-leftover
counterpart. The big symbols on top of the points indicate the median values
of WBH mass and distance to galactic center for the total sample of WBHs
(square), the leftover seed subsample (diamond) and the non-leftover seed
subsample (circle). The errors represent the 20th and 80th percentile of each
distribution. Figure 5: Expected number of WBHs that MW-type galaxies host
within a certain distance to their galactic center, with the magenta shaded
region indicating the 80th and 20th percentiles. The green dashed line
represents the median cumulative number of wandering SMBHs as a function of
distance from halo center for all MW-Mass halos in the ROMULUS25 hydrodynamic
simulation by Tremmel et al. (2018). Figure 6: The magenta dots show the
relation between the total WBH mass of the MW galaxies as a function of the
mass of their central BHs. Only those MW-type galaxies hosting a central BH
are shown (103 out of 117). The magenta line indicates the median of the ratio
for MW-type galaxies in our simulation, while the shaded area indicates the
20th and 80th percentiles. We find that WBHs in a MW-type galaxy have a
combined total mass that represents a $\sim$2$\%$ with respect to the mass of
their central BH. The green line indicates the same ratio in the hydrodynamic
simulation ROMULUS25, as extracted from Ricarte et al. (2021a), where they
find a ratio of $\sim$10$\%$. Grey dotted lines have been added for visual
reference, showing ratios of mass separated by two orders of magnitude.

###  3.2 The history of WBHs in MW-type galaxies

The study of the merger history of the WBH population is crucial to improve
our understanding of these objects in the current Universe. Fig. 7 shows some
of the properties that BHs (points and diamonds) showcase at the time they
become wanderers. We find the earliest event of disruption to occur at $z$
$\sim$ 14, while in the case of recoil events this happens at $z$ $\sim$ 7,
and in the case of pairing at $z$ $\sim$ 6\. Until $z$ $\sim$ 5, the formation
channel for WBHs is heavily dominated by disruption events. These early
disruption events result in wanderers that do not continue to grow beyond the
upper mass limit of the initial BH seeding given by L-Galaxies, hence being
all below 104 M⊙. When focusing exclusively on disruption events, early ones
($z$ $\geq$ 5) take place in galaxies with lower metallicities (-1.47 compared
to -1.12 for disruption events at $z$ $<$ 5) and are also linked to wanderers
that end up wandering in farther orbits at current redshift (193 kpc compared
to 169 kpc). This is relevant when trying to detect WBHs in the MW through
observational means. As discussed by Greene et al. (2021), $\sim$100 BHs with
masses 103 – 105 M⊙ are expected to be wandering in the MW surrounded by
$\sim$1 pc size stellar clusters containing 500 – 5000 stars. Even though in
this work we find a smaller number of putative WBHs, the values found for
galactic metallicity in the galaxies of origin can be used to tighten the
search for stellar clusters that might still be bound to the wanderers, and
that would have once belonged to that same galaxy of origin. Constraining the
metallicity values is also useful when proceeding with the isochrone fitting
of the stellar population’s color magnitude diagrams (e.g., to verify the old
nature of these kind of clusters).

Figure 7: Redshift at which WBHs first became wanderers versus their current
mass (left), distance to galactic center (middle), and the metallicity of the
galaxy where the wandering event takes place (right). The colors and symbols
used are the same as in Fig. 4. Note that in the middle Figure, the blue and
red squares representing the whole sample medians of the recoiled and
disrupted populations respectively, overlap in the same space.

.

###  3.3 The population of leftover seeds in MW-type galaxies

We study the subsample of WBHs that have not outgrown more than two times
their original seeding mass (see 2.5). We find that $\sim$67$\%$ of the WBH
population in a MW-type galaxy at $z$ = 0 satisfies this constraint. We also
find that all of the MW-type galaxies host at least one or more leftover seed,
with the median number per galaxy being 6${}_{-2}^{+4}$ (see Fig. 3, Top).

Of the 117 MW-type galaxies, $\sim$93$\%$ of them host at least one disrupted
leftover seed, with the expected number being 5${}_{-2}^{+3}$ for this type of
seed; $\sim$72$\%$ of MW-type galaxies host one or more pairing leftover
seeds, with the expected number being 1${}_{-0}^{+2}$; and $\sim$12$\%$ of MW-
type galaxies host one or more recoiled leftover seeds, with the expected
number being 1${}_{-0}^{+0}$. When compared with the overall sample of WBHs,
it seems that leftover seeds are less affected by recoiled events. This can be
explained by the fact that recoiled WBHs are generally the most massive of the
three families, and since leftover seeds are defined as having a limit in
growth, it is expected that fewer of them will be tagged as recoiled leftover
seeds.

The diamond-shaped points in Fig. 4 showcase the mass distribution of the
leftover seed wandering population as a function of its distance to the
galactic center, color-coded according to each of the three formation
mechanisms. We find that pairing seeds are expected to be found at
0.6${}_{-0.4}^{+1.3}$ kpc from the galactic center and to have a mass of
$\bigl{(}$ 0.7${}_{-0.5}^{+2.4}$ $\times$ 103 $\bigr{)}$ M⊙; disrupted seeds
are expected to be found at 177${}_{-76}^{+81}$ kpc from the galactic center
and to have a mass of $\bigl{(}$ 1.1${}_{-0.9}^{+3.1}$ $\times$ 103 $\bigr{)}$
M⊙; and recoiled seeds are expected to be found at 138${}_{-50}^{+175}$ kpc
from the galactic center and to have a mass of $\bigl{(}$
6.1${}_{-2.5}^{+5.5}$ $\times$ 103 $\bigr{)}$ M⊙. When compared to the sample
of non-leftover seeds, disrupted seeds are $\sim$13$\%$ farther from the
galactic center; recoiled seeds are $\sim$25$\%$ closer to the galactic
center; and pairing seeds are $\sim$70$\%$ closer to the galactic center.

As per the merger history, we find that the majority of WBHs in MW-type
galaxies (both leftover seeds and non-leftover seeds) that became wanderers at
high redshift are most likely tagged as disrupted (see Fig. 7). This is
probably due to pairing and recoiled WBHs being the result of a galaxy merger
that requires a satellite galaxy to sink into a massive DM halo, therefore
occurring at a later time in the simulation.

We also compare the expected wandering event redshift of a leftover seed to
its non-leftover counterpart. We find that disrupted seeds become wanderers at
a higher redshift ($\sim$3.4 vs $\sim$1.4); pairing seeds become wanderers at
a lower redshift ($\sim$0.4 vs $\sim$0.8); and recoiled seeds and non-seeds
have very similar redshifts of formation. In the case of the disrupted
population, this is most likely due to BHs drastically slowing their growth
rate once becoming wanderers. That would explain why the majority of leftover
seeds experience their disruption event at a higher redshift compared to their
non-leftover counterparts, which have had more time to continue their growth.
In the case of the pairing population, leftover seeds are tagged as pairing at
a lower redshift probably due to being hosted by galaxies with a lower mass
compared to their non-leftover counterparts. This means that pairing seeds
would sink slower into the galactic core due to experiencing less dynamical
friction. Finally, in the case of the recoiled population, especially the
recoiled seeds, we find very low numbers, as expected.

Finally, We compare the leftover and non-leftover populations in terms of the
metallicity of the galaxy that originally hosted them (Fig. 7, right). We find
that disrupted and pairing leftover seeds come from more metal-poor galaxies
than non-leftover seeds (–1.3 vs –0.9 in the case of disrupted ones and –1.4
vs –0.7 in the case of pairing ones). This is most likely due to old, low-
mass, low-metallicity galaxies being the ones to host BHs that have little to
no growth during their lives. In the case of the recoiled WBHs (both leftover
seeds and non-leftover seeds), the metallicity Z does not correspond to the
galaxy that originally hosted the BH, but rather the one where the coalescence
process took place. We find that, in contrast to disrupted and pairing
leftover seeds, recoiled seeds come from more metal-rich environments (–0.4 vs
–0.6). Despite $\sim$70$\%$ of the recoiled seed population being related to
galactic environments with metallicity Z > –0.5, only $\sim$30$\%$ of their
non-leftover counterparts are found in that regime.

###  3.4 The impact of BH seeding efficiency on the population of WBHs

We study the effects of varying the initial seeding prescription of L-Galaxies
on the merger history of the WBH population. We do so by running the model on
two new seeding prescriptions: a boosted one where the seeding probability
$\mathcal{P}$ is doubled with respect to the fiducial model, and a weaker one
where the probability is halved (see §2.3 for more details).

####  3.4.1 The global population of WBHs in MW-type galaxies

Figure 8: Number of disrupted (left), pairing (middle), and recoiled (right)
WBHs as a function of the stellar mass of the galaxy that hosted them
originally. In the case of the recoiled WBHs (right), the x-axis does not
represent the stellar mass of the galaxy where the WBHs were first seeded, but
rather the galaxy with which the BH-BH fusion and the subsequent recoil took
place. The fiducial model is shown in red, while a version with boosted BH
seeding is shown in olive and a weaker one in blue (see §2.3 for more
details).

.

We do not show the plots here to not overcrowd the paper, but we compared the
WBH population in the boosted and weaker models to the fiducial one. In the
boosted model, a typical MW galaxy hosts 50$\%$ more WBHs, with masses
$\sim$10$\%$ smaller for the non-pairing WBHs and $\sim$25$\%$ smaller for the
pairing ones. In the weaker model, a typical MW galaxy hosts 50$\%$ fewer
WBHs, with masses $\sim$20$\%$ greater for the non-pairing WBHs and
$\sim$85$\%$ greater for the pairing ones. As per the locations of the WBHs
within the galaxy, we find that the distance to the galactic center of all the
wandering populations is invariant to the seeding probability of the model.

We also study the seeding probability’s effect in the WBH population’s merger
history. Fig. 8 showcases the distribution of the number of WBHs per bin of
stellar mass of the galaxy that hosted them when becoming wanderers. As
expected, a higher seeding probability in the early Universe results in a
greater amount of galaxies hosting the type of BHs that will later become
wanderers in MW-type galaxies. However, we see an inverted correlation between
the expected stellar mass of the host galaxy and the seeding probability of
its model. This correlation can be observed in the disrupted WBH population,
which is the largest of the three samples, where host galaxies in the boosted
model have an expected stellar mass of (5.8 $\times$ 105) M⊙, while the
fiducial and weaker models showcase bigger ones (8.0 $\times$ 105 M⊙ and 1.2
$\times$ 106 M⊙, respectively). The pairing WBH population mirrors this
behavior with the boosted, fiducial and weaker models having an expected
stellar mass for the host galaxies of 8.9 $\times$ 105 M⊙, 2.6 $\times$ 106 M⊙
and 3.8 $\times$ 106 M⊙ respectively. The same applies for the recoiled WBH
population, with the boosted, fiducial and weaker models being, in this case,
1.8 $\times$ 107 M⊙, 5.5 $\times$ 107 M⊙ and 6.2 $\times$ 107 M⊙ respectively.
However, the overall higher expected mass of the three models in the recoiled
population (at least when compared to the other two populations) is due to
L-Galaxies not considering the stellar mass of the galaxy where the WBH was
originally seeded (as it was the case for the disrupted and pairing
populations), but rather the central galaxy where the coalescence and
subsequent recoil took place.

To explain the inverse correlation between the seeding probability and the
expected stellar mass of galaxies during the merger history, we need to take
into account that low-mass galaxies are the most abundant type of galaxies at
high-$z$. This means that an increase in the initial seeding probability will
have a greater effect on them, making them much more likely to host a central
BH. Hence, we can argue that the larger the number of low-mass galaxies seeded
with central BHs, the larger the number of small galaxies undergoing a merger
event where a BH is deposited, and therefore the smaller the average stellar
mass of that type of galaxy. Also, a boosted seeding model is related to a
highly active merger history that forces an earlier transition of central BHs
into wanderers. In such cases, the galaxies hosting them would not have as
much time to grow as those affected by a fiducial seeding. On the other hand,
a weaker seeding is related to an early Universe characterized by a merger-
deprived galactic evolution, where galaxies that contain central BHs have more
time to merge with each other until they find one that indeed hosts a central
BH and triggers a wandering event.

####  3.4.2 The population of leftover seeds in MW-type galaxies

We study the effects of varying the BH seeding prescription in the leftover
seed population. In the boosted model, a typical MW galaxy hosts $\sim$80$\%$
more leftover seeds, with masses $\sim$5$\%$ smaller for the non-pairing seeds
and $\sim$10$\%$ smaller for the pairing ones. In the weaker model, a typical
MW galaxy hosts 50$\%$ fewer leftover seeds, with masses $\sim$10$\%$ smaller
for the non-pairing seeds and $\sim$40$\%$ greater for the pairing ones. As
per the locations of the leftover seeds within the galaxy, we find again
invariability except for two of the WBH populations of the boosted model which
are located farther from the galactic center compard to the fiducial model;
$\sim$35$\%$ in the case of the pairing population, and $\sim$45$\%$ in the
case of the recoiled one.

As per the effect of the seeding prescription in the leftover seeds’ merger
history, Fig. 9 displays the distribution of the number of leftover seeds per
bin of stellar mass of the galaxy that hosted them when becoming wanderers.
The same inverse correlation between seeding probability and stellar mass can
be observed also for the leftover seeds. In the disrupted population, the
stellar mass of the host galaxy increased from 2.8 $\times$ 105 M⊙ in the
boosted model, to 3.1 $\times$ 105 M⊙ in the fiducial and 3.6 $\times$ 105 M⊙
in the weaker one. The same applies for the pairing population, where the
stellar mass increases from 2.3 $\times$ 105 M⊙ in the boosted model, to 2.4
$\times$ 105 M⊙ in the fiducial and 3.9 $\times$ 105 M⊙ in the weaker one.
Even though the recoiled population has too little data compared to the other
two, it is clear that the inverse correlation also applies for the leftover
seed population.

Figure 9: Number of disrupted (left), pairing (middle), and recoiled (right)
leftover seeds as a function of the stellar mass of the galaxy that hosted
them originally. In the case of the recoiled seeds (right), the x-axis does
not represent the stellar mass of the galaxy where the leftover seeds were
first formed, but rather the galaxy with which the BH-BH fusion and the
subsequent recoil took place. The fiducial model is shown in red, while a
version with boosted BH seeding is shown in olive and a weaker one in blue
(see §2.3 for more details).

.

##  4 Conclusions

We have utilized the semi-analytical model L-Galaxies to characterize the WBH
population of MW-type galaxies at $z$ = 0. We divide the wandering population
in three categories: disrupted WBHs (BHs from satellite galaxies that got
disrupted by tidal forces upon entering a massive DM halo), pairing WBHs (BHs
hosted by satellite galaxies that managed to reach and merge with a central
galaxy and are now sinking towards the galactic core) and recoiled WBHs (BHs
that got ejected into a bound orbit as the result of a pairing WBH merging
with the central BH of a galaxy). We also study the population of leftover
seeds, which in this work are defined as WBHs that have not grown more than
twice their original seeding mass. We extract the following conclusions:

  * •

A typical MW galaxy is expected to host $\sim$10 WBHs, $\sim$70$\%$ of which
have undergone a disruption event, $\sim$25$\%$ are in the process of forming
a binary system with the central BH and $\sim$5$\%$ have undergone a recoil
event. They account for $\sim$2$\%$ of the total BH mass budget of the galaxy.

  * •

The locations of WBHs within the galaxy correlate with their formation
scenario. While pairing WBHs are concentrated at $\lesssim$1 kpc from the
galactic nucleus on the stellar disk, disrupted and recoiled are found
$\gtrsim$100 kpc away from it. Such small and large distances might hinder
their observational detection and explain the absence of strong observational
evidence of WBHs in the MW.

  * •

A typical WBH has an expected mass of $\sim$2 $\times$ 103 M⊙, but recoiled
WBHs, which have undergone a coalescence event, are predicted to have a mass
one order of magnitude higher.

  * •

While the total number of expected WBHs in a MW-type galaxy is in agreement
with the prediction by Tremmel et al. (2018) using the hydrodynamic simulation
ROMULUS25, we find a higher spread in the radial distribution. This could be
explained by ROMULUS25 having higher seed masses and thus WBHs probably
infalling in shorter timescales. Also, L-Galaxies takes into account the
recoil scenario when studying the WBH populations.

  * •

A typical MW galaxy is expected to host $\sim$6 leftover seed BHs, which we
define in this work as WBHs that have not outgrown more than twice their
original seeding mass. Non-pairing leftover seeds are located $\sim$14$\%$
farther from the galactic center compared to their non-leftover counterparts.
Pairing seeds, however, are expected $\sim$70$\%$ closer to the galactic
center.

  * •

Leftover seeds come from more metal-poor galaxies than non-leftover seeds
(–1.4 compared to –0.8). This difference in the metallicity of the stellar
environment surrounding them could be used when searching for hyper-compact
clusters (Greene et al., 2021).

  * •

We have also analyzed the impact of varying the BH seeding prescription of
L-Galaxies. We find that a higher seeding prescription leads to an increase of
low-mass galaxies hosting central BHs at high-$z$. This in turn provokes that
more satellite galaxies are likely to leave their central BH as wanderers of
the new DM host.

Overall we find that WBHs are either located within 1 kpc of the galactic
center —most likely on the stellar disk— or more than 100 kpc away from it,
making them difficult to detect. The wide spread in metallicity found in the
galaxies that hosted the WBHs before they became wanderers could be an
indicator of the range of metallicities of the compact stellar populations
that are predicted to be bound around WBHs. In a future work we will explore
this in more depth by modeling clusters of stars around the WBHs.

##  5 Data Availability

This work is based on an extension of the 2015 public version of the Munich
model of galaxy formation and evolution L-Galaxies, available at
http://galformod.mpagarching.mpg.de/public/LGalaxies/.

##  6 Acknowledgements

S.B. acknowledges support from the Spanish Ministerio de Ciencia e Innovación
through project PID2021-124243NB-C21. D.I.V. acknowledges the financial
support provided under the European Union’s H2020 ERC Consolidator Grant
“Binary Massive Black Hole Astrophysics” (B Massive, Grant Agreement: 818691).
M.M. acknowledges support from the Spanish Ministry of Science and Innovation
through the project PID2021-124243NB-C22. This project was also partially
supported by the program Unidad de Excelencia María de Maeztu
CEX2020-001058-M. D.S. acknowledges support from the National Key R$\&$D
Program of China (grant No. 2018YFA0404503), the National Science Foundation
of China (grant No. 12073014), the science research grants from the China
Manned Space Project with No. CMS-CSST-2021-A05, and the Tsinghua University
Initiative Scientific Research Program (No. 20223080023).

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