††thanks<EMAIL_ADDRESS>
# Massive Black Hole Seeds

John A. Regan1 Marta Volonteri2 1Centre for Astrophysics and Space Science
Maynooth, Department of Physics, Maynooth University, Maynooth, Ireland
2Institut d’Astrophysique de Paris, Sorbonne Université, CNRS, UMR 7095, 98
bis bd Arago, 75014 Paris, France

###### Abstract

The pathway(s) to seeding the massive black holes (MBHs) that exist at the
heart of galaxies in the present and distant Universe remains an unsolved
problem. Here we categorise, describe and quantitatively discuss the formation
pathways of both light and heavy seeds. We emphasise that the most recent
computational models suggest that rather than a bimodal-like mass spectrum
between light and heavy seeds with light at one end and heavy at the other
that instead a continuum exists. Light seeds being more ubiquitous and the
heavier seeds becoming less and less abundant due the rarer environmental
conditions required for their formation. We therefore examine the different
mechanisms that give rise to different seed mass spectrums. We show how and
why the mechanisms that produce the heaviest seeds are also among the rarest
events in the Universe and are hence extremely unlikely to be the seeds for
the vast majority of the MBH population. We quantify, within the limits of the
current large uncertainties in the seeding processes, the expected number
densities of the seed mass spectrum. We argue that light seeds must be at
least $10^{3}$ to $10^{5}$ times more numerous than heavy seeds to explain the
MBH population as a whole. Based on our current understanding of the seed
population this makes heavy seeds ($\rm{M_{seed}}>10^{3}$ $\rm{M_{\odot}}$) a
significantly more likely pathway given that heavy seeds have an abundance
pattern than is close to and likely in excess of $10^{-4}$ compared to light
seeds. Finally, we examine the current state-of-the-art in numerical
calculations and recent observations and plot a path forward for near-future
advances in both domains.

Figure 1: Schematic showing the Number Density (rarity, y-axis) versus Time
and Redshift. Light seeds can form in ’normal’ galaxies with relative ubiquity
in the early universe with number densities of the order of 1 cMpc-3 at
redshifts between z = 20 and z = 10. However, their (initially) small masses
make them unlikely to be the progenitors for all of the ‘JWST’ AGN or the
high-z quasar population. On the other hand forming heavier seeds requires
rarer environmental conditions and number densities varying from $10^{-1}$ to
$10^{-5}$ cMpc-3 (perhaps lower) but their initially heavier masses can
facilitate more rapid growth potentially explaining the early MBH population
as well as the high-z quasar population.

##  1 Introduction

Massive black holes (MBHs) with masses up to $10^{10}$
$\rm{M_{\odot}}~{}$exist at the centre of most, perhaps all, massive galaxies.
The ongoing challenge in the field is to understand the origin of the central
MBHs. Up until recently the most distant MBHs discovered were quasars with
masses up to $10^{9}$ $\rm{M_{\odot}}~{}$observed at redshifts just beyond
seven (approximately 700 Myr after the Big Bang) (Fan, Carilli & Keating,
2006; Mortlock et al., 2011; Venemans et al., 2013, 2015; Bañados et al.,
2018; Yang et al., 2021; Wang et al., 2021). However, while these observations
strongly indicate that the seeds for MBHs must form in the early Universe they
offer little insight into their origin, as the initial conditions within which
the seed black hole formed have long since been erased and the black hole
itself contains no signature of that event.
More recently, observations with JWST have uncovered MBHs with masses in the
range $10^{6}$ $\rm{M_{\odot}}~{}$to $10^{8}$ $\rm{M_{\odot}}~{}$(Kovacs et
al., 2024; Maiolino et al., 2024b; Larson et al., 2023; Bogdán et al., 2023;
Matthee et al., 2023; Greene et al., 2024) at even higher redshifts. These MBH
masses are lower than the high-z quasar masses previously detected but the
galaxies within which these “JWST-quasars” have been detected likely represent
the tip of the iceberg at this time in terms of both halo and stellar masses
(Boylan-Kolchin, 2023; Juodžbalis et al., 2024; Li et al., 2024). Nonetheless,
with inferred masses down as low as $5\times 10^{6}$
$\rm{M_{\odot}}~{}$(Maiolino et al., 2024b) these observations set an upper
limit to the seed masses of this population of MBHs at the very least. The
number density of these MBH host galaxies is currently unclear with
potentially significant evolution of the central black hole following seeding.
If we assume that current candidate AGN being observed by JWST are indeed true
AGN then the number densities could be of the order $10^{-4}$ to $10^{-3}$
cMpc-3 (Matthee et al., 2023; Greene et al., 2024; Scholtz et al., 2023).The
goal of this paper is to highlight what the current state-of-the-art in MBH
seeding is, to identify challenges in the current models and to plot a path
forward in terms of reconciling current theoretical models with observational
data.
The models for the astrophysical formation of MBH seeds can be loosely broken
down into light and heavy seed models. Here, we define a light seed as any
seed black hole with a mass less than 1000 $\rm{M_{\odot}}~{}$and a heavy seed
as any seed black hole with an initial mass in excess of 1000
$\rm{M_{\odot}}$. A key challenge is then to understand what the mass spectrum
and number densities of MBH seeds is and to understand the growth prospects of
the seeds. We illustrate this challenge to the field in Figure 1.
The crucial question for the black hole population is “Does the entire black
hole mass spectrum originate from stellar mass black holes with characteristic
masses of approximately 40 $\rm{M_{\odot}}~{}$and a tail to higher and lower
masses, which grow through accretion and mergers”? This would imply a
continuum from stellar black holes to MBHs, with a single origin for all of
them. Or is another channel required to populate the MBH mass spectrum?
Depending on the mass distribution of this channel then this channel will, if
it exists, give rise to an additional peak in the mass spectrum of black
holes. Tantalisingly this peak could, in theory, be detected by future
gravitational wave observations (Sesana, Volonteri & Haardt, 2007, 2009).
The simplest case is to assume that the seeds are indeed light and that they
originated from the remnants of the first stars (e.g. Madau & Rees, 2001).
However, significant challenges to light seed growth exist and we will address
this question in detail in §2.
Alternative routes to MBH formation may be through either a dynamical runaway
pathway where either stellar collisions and/or black hole mergers result in a
heavy seed. Another possibility is a near monolithic collapse, or at least
reduced fragmentation of a massive gas cloud, resulting in the formation of a
supermassive star (SMS). Both of these mechanisms create a heavy seed with
masses in excess of approximately $10^{3}$ $\rm{M_{\odot}}~{}$and in some
cases possibly up to $10^{5}$ $\rm{M_{\odot}}$. We will address these
questions and gauge their possible number density in §3. We begin by first
briefly reviewing the light seed model and then the heavy seed model. More in-
depth reviews of the mechanisms can be found in reviews by Latif & Ferrara
(2016); Woods et al. (2017); Inayoshi, Visbal & Haiman (2020); Volonteri,
Habouzit & Colpi (2021). Additional mechanisms, which we will not consider in
this paper, include supermassive stars fuelled by dark matter annihilation
(e.g., Spolyar, Freese & Gondolo, 2008; Singh, Monaco & Tan, 2023) or
primordial black holes (e.g., Carr & Silk, 2018; Hasinger, 2020).

##  2 Light Seeds as the origin of all MBHs

While this is indeed the simplest explanation there are significant challenges
in growing light seeds in order to explain the entire MBH mass spectrum. The
problem is particularly acute when attempting to use light seed growth to
explain the appearance of MBHs at $z\gtrsim 7$. In order to achieve such
growth the seed black holes would need to grow, uninterrupted, at the
Eddington rate (under the assumption that light seeds form from the first or
second generation of stars). The Eddington accretion rate is given by

$\rm{M(t)=M_{0}}\
\rm{exp}\Big{(}\frac{1-\epsilon_{r}}{\epsilon_{r}}\frac{t}{t_{Edd}}\Big{)},$
(1)

where $\rm{t_{Edd}}=0.45$ Gyr and $\epsilon_{r}$ is the radiative efficiency.
For a “standard” radiative efficiency of $\epsilon_{r}\sim 0.1$ and a black
hole seed mass of $\rm{M_{0}}=10^{2}$ $\rm{M_{\odot}}$, it takes nearly 1 Gyr
to grow from $\rm{M(0)}$ up to $\sim 10^{9}$ $\rm{M_{\odot}}$. While therefore
in principle marginally allowable with current observational constraints the
growth must continue at the Eddington rate all the way up to the maximum mass
(i.e. the duty cycle must be unity). Such a scenario has been shown to be
exceedingly unlikely with numerous models and cosmological simulations showing
that light seeds do not grow efficiently and tend to remain at their seed
masses across cosmic time (Alvarez, Wise & Abel, 2009; Milosavljević, Couch &
Bromm, 2009; Smith et al., 2018; Spinoso et al., 2022).
The reasons for this are complex and multifaceted but are primarily driven by
the fact that the light seed black holes are unable to sink to, or remain in,
the galactic centre where gas densities are higher and also because they must
compete for accretion with the star formation processes evolving nearby while
local ionising radiation can further hamper accretion. However, in rare
circumstances rapid accretion onto a light seed may be possible (e.g.
Alexander & Natarajan, 2014; Zubovas & King, 2021; Shi, Kremer & Hopkins,
2024). We will return to this topic in §7.

##  3 Heavy Seeds as a Sub-population

The need for heavy seed black holes has largely been driven by observations of
MBHs with masses up to and in excess of $10^{9}$ $\rm{M_{\odot}}~{}$at
redshifts beyond seven (e.g. Fan, Bañados & Simcoe, 2023). With continued
observations putting tight constraints on the light seed pathway the strong
possibility of heavy seeds being the progenitors for the MBH which inhabit
galactic centres strengthens (e.g. Bogdán et al., 2023; Natarajan et al.,
2024).
A number of different avenues remain open to achieving heavy seed black holes,
including dynamical runaway processes, the direct collapse of a SMS into a MBH
and the direct formation of a MBH via galactic mergers. We now briefly discuss
the environmental conditions required to drive each of these pathways but
refer the interested reader to a review on the subject for a more in-depth
analysis (e.g. Inayoshi, Visbal & Haiman, 2020; Volonteri, Habouzit & Colpi,
2021).
Seminal work by Portegies Zwart et al. (2004) showed that MBHs can be formed
through runaway collisions of stars in dense young star clusters. This work
has been built on by several groups in more recent years with at times
contrasting results (Devecchi & Volonteri, 2009; Katz, Sijacki & Haehnelt,
2015; Rizzuto et al., 2021; González et al., 2021). The differences can often
be ascribed to different initial conditions and environmental factors. For
example Arca Sedda et al. (2023) has shown using detailed numerical models
using the Nbody6+ +GPU code that in clusters of up to $10^{6}$ stars that
black hole growth via stellar collisions tops out at a few hundred
$\rm{M_{\odot}}$. While on the other hand simulations by Reinoso et al. (2023)
in smaller clusters using the AMUSE (Portegies Zwart et al., 2009; Pelupessy
et al., 2013) framework have demonstrated that stellar collisions can grow
stars up to several tens of thousands of solar masses when accounting also for
gas accretion. In any case, while convergence of results has perhaps not yet
been achieved it does appear that initial black hole seed growth up to masses
greater than 1000 $\rm{M_{\odot}}~{}$does appear viable and hence this route
to heavy seeds appears plausible (although sensitive to cluster initial
conditions).
Another related pathway is hierarchical BH mergers within a star cluster
undergoing core collapse. In this case repeated black hole mergers can in
principle grow a black hole up to masses in excess of 1000
$\rm{M_{\odot}}~{}$(Fragione & Leigh, 2018; Fragione et al., 2022; Antonini,
Gieles & Gualandris, 2019; Mapelli et al., 2021; Arca Sedda, Amaro Seoane &
Chen, 2021). As before however, the final black hole mass is sensitive to the
conditions found in the cluster. Nonetheless, again a pathway to heavy seed
MBH does exist.
In addition to the dynamical pathway heavy seeds can form via the formation of
a very massive or super-massive star (SMS). In contrast to the formation of a
dense cluster and subsequent dynamical runaway inside the cluster a (near-)
monolithic collapse is allowed to develop. In this case where fragmentation is
suppressed, either through low metallicity or the lack of other efficient
coolants, then the possibility of forming very massive or SMSs arises. SMSs
are defined by their extremely high accretion rate onto the stellar surface
($\rm{\dot{M}}\gtrsim 0.001$ $\rm{M_{\odot}/yr^{-1}}$) which, through the
accretion of high entropy gas, causes the photosphere of the star to expand
(so long as the accretion rate is maintained, Haemmerlé et al., 2018). Such
stars are expected to be very red, in contrast to massive PopIII stars whose
spectrum would be blue, and cool with effective temperatures of approximately
$\rm{T_{eff}}\sim 5000$ K (Hosokawa, Omukai & Yorke, 2013; Hosokawa et al.,
2013; Haemmerlé et al., 2018; Woods et al., 2021). With maximum theoretical
masses of greater than $10^{5}$ $\rm{M_{\odot}}~{}$(Woods et al., 2017) these
SMS stars would be ideal progenitors for MBHs and among the heaviest seeds
possible.

Finally, Mayer et al. (2010, 2015, 2023) have postulated a "direct" mechanism
for forming extremely massive black holes. In their model the merger of
massive galaxies ($\rm{M_{Gal}\gtrsim 10^{11}}$ $\rm{M_{\odot}}$) can result
in the direct creation of a supermassive black hole with a mass up to $10^{8}$
$\rm{M_{\odot}}$. The formation of the MBH directly bypasses both the stellar
and seed formation stages and may explain the existence of the quasar
population seen at $z\sim 6$. However, it cannot explain the existence of less
massive black holes at high redshift as the mechanism relies on the merger of
massive haloes ($\rm{M_{Halo}}\sim 10^{12}$ $\rm{M_{\odot}}~{}$at z $\sim$ 10
corresponding to 4$\sigma$-5$\sigma$ peaks) to drive large gas inflows
creating the central MBH.

Figure 2: Gas Number Density projection plots from each of the three halos
selected for our study. In the top panel we show the projection from the 1000J
halo, in the middle panel the projection from the Rapid halo and in the bottom
panel the PopIII halo. Contours in each panel demarcate density levels in
logspace. The 1000J halo (top panel) shows a very symmetric density profile,
while both the Rapid (in particular) and PopIII haloes show much more diverse
structure. Labelled in each panel is the halo mass, the LW flux impacting onto
the halo, the redshift and time of the collapse.

##  4 Seed Number Densities

A crucial question is what are the expected number densities of both light and
heavy seeds in the high-z Universe. Light seeds will be ubiquitous in the
early universe - forming from the end points of massive stars. The exact
number density of light seeds is non-trivial to compute and will depend on the
initial mass function of the first stars (which is unknown). However, a
reasonable estimate would be between $10^{-1}$ cMpc-3 and up to $10$ cMpc-3
(at z $\sim 10$) (Xu, Wise & Norman, 2013; Smith et al., 2018; Piana et al.,
2020; Trinca et al., 2022). The number density of heavy seeds will of course
be significantly less.
By definition certain, perhaps rare, environmental conditions must occur for
heavy seed formation pathways to be activated. On the contrary were such
conditions common place, we would likely see them in action in the present day
universe (at least for cases where gas/stellar metallicity does not need to be
very small). Additionally, the rarity and perhaps uniqueness of this pathway
to the early Universe has a significant impact on the expected number
densities of heavy seeds. See Woods et al. (2017); Inayoshi, Visbal & Haiman
(2020) for a review of heavy seed formation environments.
Dijkstra, Ferrara & Mesinger (2014) calculated the number density of heavy
seeds forming through the so-called Lyman-Werner (LW) channel. In this case an
atomic halo is irradiated with a super-critical LW by a nearby galaxy
(Dijkstra et al., 2008; Agarwal et al., 2012, 2013; Visbal, Haiman & Bryan,
2014; Agarwal et al., 2015; Regan et al., 2017). In principle this can lead to
the formation of an extremely massive seed (see §5). However, the calculated
number densities are small with expected number densities in the range
$10^{-10}$ cMpc-3 to $10^{-5}$ cMpc-3 (at $z\sim 10$). It should be noted here
that the higher number densities quoted here (i.e. $10^{-5}$ cMpc-3) should be
treated as strong upper limits as they rely on very optimistic formation
criteria, namely, low values of the LW radiation needed to keep $\rm{H_{2}}$
dissociated or allowing for metal pollution. Other authors have drawn similar
conclusions on the expected number densities from this channel (Inayoshi &
Tanaka, 2015; Habouzit et al., 2016; Agarwal et al., 2016; Trinca et al.,
2022).
Moving to channels which potentially produce heavy seeds below the theoretical
maximum mass (of SMSs) but well in excess of light seeds, the so-called rapid
assembly channel (Yoshida et al., 2003; Fernandez et al., 2014; Wise et al.,
2019; Regan et al., 2020c; Lupi, Haiman & Volonteri, 2021; Latif et al., 2022)
can produce heavy seeds with number densities in the range 10-5 to $10^{-1}$
cMpc-3 (at $z\sim 10$) (McCaffrey et al. in prep) depending on environmental
factors. Number densities achieved through other channels like baryonic
streaming velocities result in number densities close to approximately 10-5
cMpc-3 (at $z\sim 10$) as well (Hirano et al., 2017).
Overall the number densities of heavy seeds range over several orders of
magnitude (varying according to the expected mass of the seed and the
environmental conditions required) from $10^{-10}$ cMpc-3 up to $10^{-1}$
cMpc-3 (at $z\sim 10$). The variance of these estimates spanning almost 10
orders of magnitude speaks to the complexity of the problem. However, what
should be kept in mind here is that while producing the heaviest seeds may be
exceedingly rare - producing more moderate mass seeds becomes progressively
more straightforward and the higher number densities. This leads to the
conclusion that even in the case of a heavy seed channel that the seed mass
function could well be a continuum with a peak at some characteristic mass. We
now move on to discuss the types of haloes expected to form both light and
heavy seeds.

Model Name |  Halo Mass [$\rm{M_{\odot}}$] |  Redshift |  Mass of Most Massive Star [$\rm{M_{\odot}}$] |  Halo Number Density [cMpc-3] at $z\sim 10$
---|---|---|---|---
1000J Halo |  $2.59\times 10^{7}$ |  23.70 |  76123 |  $10^{-10}-10^{-5}$
Rapid Halo |  $9.3\times 10^{7}$ |  15.05 |  6127 |  $10^{-5}-10^{-1}$
PopIII Halo |  $3.7\times 10^{6}$ |  17.21 |  173 |  $10^{-1}-10^{+1}$
Table 1: Characteristic of the three different haloes used to illustrate our
arguments. The three haloes are the 1000J halo, the Rapid halo and the PopIII
halo. For each halo we give the halo mass (second column), the redshift at
which star formation first occurs in each halo (third column), the mass of the
most massive star in each halo (fourth column) and in the last column the
expected number density of each halo type.

Figure 3: Similar to Figure 2 except using the temperature field as the
projection variable. In the top panel we show the projection from the 1000J
halo, in the middle panel the projection from the Rapid halo and in the bottom
panel the PopIII halo. The 1000J shows very little temperature differences
reflecting the fact that the strong LW field ($\rm{J_{LW}=1000\ J_{21}}$)
largely dissociates the $\rm{H_{2}}$ ensuring the collapse is isothermal. The
middle panel, with only a very mild LW field ($\rm{J_{LW}\sim 5\ J_{21}}$),
shows significantly more temperature structure (which drives more structure in
the density field). Likewise the bottom panel shows a strong temperature
gradient as $\rm{H_{2}}$ is able to cool the gas to below 1000 K in the halo
centre.

##  5 Model Haloes

To illustrate the different environmental conditions that have been
investigated in the literature over the last decade or more we have selected
three haloes that broadly reflect the mainstream pathways of investigation.
Additionally, the environmental conditions required feed directly into the
expected number densities of these seeds. We are not aiming for complete
sampling of all scenarios and environments investigated but rather the small
sample of haloes selected here reflects in broad terms the emergence of light
and heavy seed pathways. The haloes we select we name the Rapid halo, the
1000J halo and the PopIII halo. The characteristics of each halo are given in
Table 1.
The 1000J halo is the prototypical halo used to test the intense LW field
model (see §3 for details). The Rapid halo is the result of the rapid assembly
paradigm and results in the formation of heavy seed black holes via the rapid
assembly (or dynamical heating) mechanism. Finally, the PopIII halo is a
standard mini-halo whose environment is not influenced by either (extreme) LW
radiation or rapid assembly. The PopIII halo gives rise to light seeds while
the 1000J halo and Rapid halo give rise to heavy seeds albeit almost certainly
of different initial masses. The expected number densities of seeds as well as
their potential mass ranges are given in Table 1. We now discuss each of the
haloes in detail comparing and contrasting the different environmental
conditions and the reasons why different halo configurations lead to light or
heavy seed formation and whether they offer the potential for significant
future growth.

###  5.1 The 1000J Halo

The 1000J is taken from Regan & Downes (2018) and consists of a single halo
irradiated by a LW background field with flux 1000 J21111J21 is in units of
$10^{-21}$ $\rm{erg\ cm^{-2}\ s^{-1}\ Hz^{-1}\ sr^{-1}}$ (an intensity of 1000
J21 is generally accepted in the literature as required for substantial
$\rm{H_{2}}$ suppression (e.g. Shang, Bryan & Haiman, 2010)). The simulation
evolves with this background field strength until one halo exceeds the atomic
cooling threshold. At this point cooling due to Lyman-alpha line emission
dominates the cooling and allows the gas to collapse almost isothermally at
approximately 8000 K. The most massive star formed in the halo has a mass of
approximately 76,000 $\rm{M_{\odot}}~{}$and there were only four stars in the
halo at the end of the simulations ($\sim 250$ kyr after the formation of the
first star). This is the proto-typical isothermal collapse seen by most
studies in which collapse in the presence of a large LW field is studied
(Latif et al., 2013; Visbal et al., 2014; Latif et al., 2016; Dunn et al.,
2018; Schauer et al., 2021). Even larger LW fields may result in less
fragmentation at the scales studied while lower LW fields will likely result
in more fragmentation (taking into account halo stochasticity).
In the top panel of Figure 2 we show a gas number density projection centred
on the 1000J halo. The halo is highly symmetric indicating a near spherical
collapse. The symmetry of the halo is also evidenced by the (density-weighted)
temperature projection plot shown in the top panel of Figure 3. The strong LW
field homogenises the temperature field leading to a near isothermal collapse
(at $\sim 8000\rm{K}$). The homogenisation of the temperature field has a
knock on effect on the density - making it more homogeneous as well due to the
removal of temperature gradients which can lead to cooling instabilities. This
in turn leads to less fragmentation. This is an ideal scenario for creating
heavy seeds.
In this specific case most of the stellar mass ends up in two SMSs with four
SMSs forming in total. The mass of each of the stars in shown in Figure 4. The
accretion rate onto the most massive star in the 1000J halo is shown in the
top panel of Figure 5 (green line). The SMS accretes at super-critical rates
(Nandal et al., 2023) for more than 200 kyr before the gas supply becomes
restricted. The final mass of each of the stars are shown in Figure 4 (green
bars). In Figure 5 we show the mass evolution and accretion rate onto the most
massive star in the simulation (green lines). The most massive star achieves a
mass of more than 76,000 $\rm{M_{\odot}}~{}$after approximately 250 kyr. At
this point the accretion rate falls off as most of the locally available gas
is consumed and the star contracts to the main sequence.
The number density of haloes subjected to a super-critical LW flux cannot be
obtained from the simulations discussed above due to their idealistic nature.
However, both Visbal, Haiman & Bryan (2014) and Regan et al. (2020b) (using
the Renaissance simulation suite) computed the abundance of so-called
synchronised pairs which can lead to the emergence of haloes similar to the
1000J halo (Regan et al., 2017). Both authors found the abundance of
synchronised haloes to be less than $10^{-2}$ cMpc-3 (at z $\sim 10$).
However, this number should be seen as an upper limit. For the "Normal" region
of the Renaissance simulation suite Regan et al. (2020b) found exactly zero
synchronised pairs while finding five, in the clustered, "Rarepeak" region.
Taking into account the relative rarity of the Rarepeak region, the number
densities obtained should be decreased by a factor of approximately 1000. This
leads to number densities of less than $10^{-5}$ cMpc-3 (at z $\sim 10$) for a
1000J like object.

###  5.2 The Rapid Halo

The Rapid halo is taken from Regan et al. (2020c) which were themselves zoom-
in simulations of the Renaissance simulation suite. The halo was originally
discovered as part of Wise et al. (2019). In this paradigm, as described in
§3, the halo experiences periods of extremely rapid growth, which dynamically
heats the halo, overcoming the ability of $\rm{H_{2}}$ to cool the gas. As a
result the gas stays ‘hot’ increasing the Jeans mass of any fragmenting clumps
(and reducing the number of fragmenting clumps) - see also Latif et al. (2022)
who explore a similar case to that of Wise et al. (2019) albeit in a rarer
environment.
The gas number density distribution of the halo is shown in the middle panel
of Figure 2. The morphology is clearly very different to that of the 1000J
shown in the panel above. In the case of the Rapid halo there is a distinct
lack of symmetry. The (density-weighted) temperature projection shows a
similar diversity compared to the 1000J halo. The temperature distribution
with cooler gas towards the centre clearly contains far more structure. The
Rapid halo forms 99 stars over the course of 2 million years with the most
massive star having a mass of 6127 $\rm{M_{\odot}}~{}$at the end of the
simulation. The mass of each star at the end of the simulation is shown in
Figure 4 (blue bars). The halo has a peak star mass of approximately 1000
$\rm{M_{\odot}}$. The mass of the most massive star and the accretion rate
onto the most massive star in the Rapid halo is shown, as a function of time,
in Figure 5 (blue lines). Similarly, to the 1000J the most massive star stops
accreting after a few tens of kyr as the locally available gas supply runs
out. At this point the star contracts to the main sequence and becomes a
massive (and extremely luminous) PopIII star.
The number density of rapidly growing haloes from the Renaissance simulations
is approximately $5\times 10^{-3}$ cMpc-3 at z $\sim 10$ (Regan et al.,
2020a)222This value is taken from the ’Normal’ simulation which runs to z =
11.6. This is at least two orders of magnitude higher than the probablility of
forming a 1000J like object. The criteria used to define these haloes from the
Renaissance simulations was all haloes with mass inflow rates exceeding 0.1
$\rm{M_{\odot}/yr^{-1}}~{}$in the halo centre and with metallicities
$Z\lesssim 10^{-3}$ Z⊙.

Figure 4: The masses of all of the stars formed in each of the three
characteristic haloes. Only four stars form in total in the 1000J halo after
250 kyr, approximately 100 in the Rapid halo after 2 Myr while the PopIII halo
forms approximately 20 stars after 2 Myr. The masses of the stars formed in
the 1000J are up to an order of magnitude more massive than those in the Rapid
halo and more than two orders of magnitude more massive than those formed in
the PopIII halo.

###  5.3 The PopIII Halo

As with the Rapid halo the PopIII Halo is taken from Regan et al. (2020c). The
halo represents a canonical mini-halo with no LW field flux or rapid assembly.
The PopIII halo forms PopIII stars out of primordial Hydrogen and Helium. The
bottom panel of Figure 2 shows the gas number density projection of the
collapsing PopIII halo. The collapse is somewhat symmetric while also showing
clear evidence of an extended morphology. Similarly, the (density-weighted)
temperature projection shown in the bottom panel of Figure 3 shows a clear
temperature gradient with cooler gas forming deeper inside the halo. The
PopIII halo forms 21 stars by the end of the simulation with the most massive
star having a mass of 173 $\rm{M_{\odot}}$(see Figure 4, red lines). The top
panel of Figure 5 shows the cumulative mass as a function of time for the most
massive star (red line) while the bottom panel shows the accretion rate onto
the most massive star in the PopIII halo.
The number density of haloes, at z $\sim 10$, in the Renaissance simulation
suite hosting light seeds (i.e. PopIII remnant black holes) is approximately
0.3 cMpc-3. This is approximately $10^{3}$ times higher than the number
density of rapidly accreting haloes from the same simulation volume and at
least $10^{5}$ times higher than the number density of LW haloes expected in a
similar volume.

##  6 Model Halo Comparison

Differences are clearly apparent across the three characteristic haloes
chosen. The 1000J has a near homogeneous density distribution as a result of
the super-critical LW flux it is exposed to. This density (and temperature)
homogeneity promotes a strong suppression of fragmentation resulting in two
extremely massive objects forming (masses of 76000 $\rm{M_{\odot}}~{}$and
44000 $\rm{M_{\odot}}$) along with two less massive stars. The other two
haloes (Rapid and PopIII) show a more inhomogeneous density (and temperature)
distribution (due to the presence of higher $\rm{H_{2}}$ fractions) and result
in lower mass stars. The distribution in stellar masses across each halo is
shown in Figure 4 with the 1000J showing fewer but significantly more massive
stars compared to either the Rapid halo or the PopIII halo.
There are two primary reasons for the elevated stellar masses found in the
1000J compared to the Rapid and PopIII halo. On the one hand and as described
above the LW flux dissociates $\rm{H_{2}}$ and leads to larger Jeans masses
and hence larger protostars (Prole et al., 2023) but also these protostars can
accrete more readily and for longer (see Figure 5). In the Rapid halo and
PopIII halo the structures are far more asymmetric making continued accretion
much more difficult (at least for the stellar masses probed here). For the
cases of stars in both the Rapid and PopIII haloes the stars can initially
accrete gas readily but the gas supply quickly gets consumed and the stars
contract to the main sequence (Regan et al., 2020c) with masses significantly
below that seen in more symmetric and homogeneous halos.
It should be noted that in Figure 5 we show only accretion over a relatively
short period (hundreds of kilo-years). The subsequent growth of the seed
(light or heavy) will depend fundamentally on the evolution of the host galaxy
over several tens and hundreds of megayears. Presently, simulations able to
track the model the formation of light and heavy seeds and to subsequently
follow their growth are extremely computationally challenging and essentially
intractable. We nonetheless describe progress in this direction in the
following section bearing in mind that this remains an extremely challenging
modelling problem.

Figure 5: The evolution of the most massive star that forms in each of the
three haloes (1000J, Rapid and PopIII). In the top panel we show the mass as a
function of time. In the bottom panel we show the accretion rate onto the most
massive star in each halo as a function of time. The 1000J halo forms the most
massive object followed by the Rapid halo and the PopIII halo. The dashed line
shows the critical accretion that must be maintained onto the stellar surface
to meet the criteria for being a super-massive star (Nandal et al., 2023).

##  7 Black Hole Growth

The sustained growth of black holes is clearly critical for their evolution
into the MBHs we observe as both high-z quasars and as the "JWST-quasars" at z
$\gtrsim 7$. The question then becomes which seeds are more likely to
experience significant growth?

###  7.1 Growth of Light Seeds

Light seeds can suffer from initially severely stuttered growth as a result of
their formation pathway. Depending on their final mass the seeds are often
born "starving" into an environment which is hostile to initial growth (e.g.
Whalen, Abel & Norman, 2004; Alvarez, Wise & Abel, 2009; Milosavljević, Couch
& Bromm, 2009). This is because the ionising feedback from the stellar phase
plus the energy injected into the surrounding gas results in a light seed born
into a relatively underdense local environment.
Light seeds can nonetheless grow in two ways, firstly via gas accretion and
secondly via mergers with other compact objects. Due to their smaller masses
light seeds have a significantly smaller Bondi radius (which scales as
M${}^{2}_{BH}$) compared to heavy seeds. Furthermore, their mass relative to
that of their host galaxy and indeed the surrounding stellar and gas
distribution is negligible. This means that light seed black holes will tend
to move easily within their environment - a movement which can appear random.
A number of authors have investigated this property of light seeds
numerically.
Smith et al. (2018) et al. used the Renaissance suite to track, in post-
processing, the evolution of light seed black holes formed from PopIII stars.
They tracked the evolution of over 20,000 light seeds for up to 200 Myrs and
found negligible growth across the entire light seed population. The reason
being is that the light seeds experience extremely low duty cycles due to the
surrounding (negative) feedback processes of star formation and stellar
feedback. Moreover, higher resolution studies of (PopIII) star forming
environments show that PopIII stars form in multiples (e.g. Turk, Abel &
O’Shea, 2009; Clark et al., 2011; Prole et al., 2023) setting off a
competitive accretion scenario further mitigating against efficient accretion.
Other authors (e.g. Pfister et al., 2019; Ma et al., 2021) find that only
seeds with masses in the heavy seed (up to $10^{8}$ $\rm{M_{\odot}}~{}$in the
case of the Ma et al. (2021) study) range can effectively sink to the centre
of the potential well where efficient accretion may be sustainable. None of
the above studies rule out light seed accretion but instead find that the
probability of efficient light seed growth is extremely low.
Building on these challenging conclusions from studies of light seed growth a
number of groups have investigated how growth can be accelerated when the
conditions are favourable. In the majority of these cases the light seed black
hole is modelled to either sit at the centre of a converging flow (e.g.
Volonteri & Rees, 2005) or is "trapped" within the confines of a nuclear star
cluster (Alexander & Natarajan, 2014; Natarajan, 2020) or within a
circumnuclear disk (e.g. Lupi et al., 2016). More recent work by Shi, Kremer &
Hopkins (2024) shows that a small fraction ($\sim$ 1%) of light seeds can
achieve bursts of rapid growth when placed in environments that are favourable
(i.e. are gas rich and support growth). In each of these cases the growth of
the initially light seed proceeds via bursts of super-Eddington accretion. The
physical process of super-Eddington accretion has been demonstrated
extensively via extremely high resolution simulations (e.g. Sa̧dowski, 2009;
Sa̧dowski & Narayan, 2016; Sa̧dowski et al., 2016; Jiang, Stone & Davis, 2017;
Jiang et al., 2020). However, what is lacking are robust numerical simulations
which show self-consistently the growth of light seeds in a cosmological
setting. The reason for this is that a combination of high resolution, with
resolved dynamics, and a realisation containing a rare dynamical occurrence is
required. This has, thus far, not been realised. However, such a setup should
be tractable with current simulation codes and techniques at least for the
early evolutionary phase.
In the above discussion we have focused on growth through gas accretion.
However, a light seed can grow also via mergers. The classic case here is a
dense stellar cluster where stars undergo collisions resulting in light or
heavy seed formation and the seed subsequently merges with other stars and BHs
resulting in a final seed mass. The final seed mass attainable here is unclear
and detailed state-of-the-art n-body simulations have not yet converged on a
final result (e.g. Arca Sedda et al., 2023; Rantala, Naab & Lahén, 2024) due
to the complex parameter space that must be explored. Additionally, the impact
of gas on both the dynamics and accretion must be incorporated into this
scenario and this could potentially result in much larger seed masses (e.g.
Chon & Omukai, 2020; Reinoso et al., 2023).
Conversely, light seed mergers in the aftermath of galaxy mergers (the classic
case considered, for instance, to estimate merger rates for gravitational wave
detectors such as LISA) is unlikely. This is because of the same reasons why
sinking and retaining light seeds in galaxy centres is difficult, as already
discussed. In a galaxy merger a light seed will have mass not much larger than
that of stars and therefore dynamical friction is inefficient: the probability
that two light seeds, which have very small cross section, find each other
within this sea of stars is exceedingly small.

###  7.2 Heavy Seed Growth

Given that the growth of light seeds appears very inefficient in the main, it
is worth exploring whether heavy seeds can avoid this bottleneck. As discussed
above this will likely depend sensitively on the mass of the heavy seed.
Pfister et al. (2017) showed that growth is extremely inefficient below at
least $10^{5}$ $\rm{M_{\odot}}~{}$and perhaps as high as an initial mass of
$10^{8}$ $\rm{M_{\odot}}~{}$in the case of Ma et al. (2021). These seminal
works appear to be borne out by more recent simulations. For example Lupi et
al. (2023) model the progenitor of a high-z quasar using an initial black hole
seed mass of $10^{5}$ $\rm{M_{\odot}}~{}$(the model sets the initial dynamical
mass of the heavy seed to be $10^{6}$ $\rm{M_{\odot}}$). This initial mass
helps to stabilise the seed prior to rapid growth - nonetheless the initial
growth is stunted by supernova feedback and requires significant halo growth
to facilitate the black hole growth. Although the initial barrier to growth
depends on the strength of supernova feedback, what this shows is that growth
can occur onto a heavy seed but only if the seed is massive and if the
conditions are favourable (i.e. the halo is sufficiently massive). Latif &
Khochfar (2020) reached similar conclusions, albeit from somewhat more
idealised conditions - following the growth of a heavy seed
(M$\rm{{}_{seed}}=10^{5}$ $\rm{M_{\odot}}$) starting from a halo similar to
the 1000J halo described above.
On larger scales the Astrid (e.g. Bird et al., 2022; Ni et al., 2022),
NewHorizon (Dubois et al., 2021; Beckmann et al., 2022) and Romulus (e.g.
Tremmel et al., 2017; Ricarte et al., 2019) simulation suites all seed black
holes with masses well inside the heavy seed regime as well as implementing
dynamical friction subgrid modelling. For higher mass galaxies the results for
the simulatiion suites are boardly consistent. However, for smaller mass
galaxies (M${}_{*}\lesssim 10^{9}$ $\rm{M_{\odot}}$), the results are not
converged. Ricarte et al. (2019) show that MBHs can grow in galaxies
independent of the stellar mass (and hence show little or no effect from
supernova feedback). This is in contrast to other results (e.g. Dubois et al.,
2014; Habouzit, Volonteri & Dubois, 2017; Anglés-Alcázar et al., 2017). The
reasons for this difference are not obvious and require further investigation
(Ricarte et al., 2019).
So overall where does the extensive work on black hole growth leave us? Across
the various simulation setups there is convergence that black holes can grow
once the host halo becomes massive enough with some differences arising for
black hole growth in lower mass galaxies which needs to be fully understood.
Additionally, the black hole must achieve a mass of at least $10^{5}$
$\rm{M_{\odot}}~{}$before growth can be achieved - this is particularly
evident from the Astrid simulations which have the lowest seed masses
($\rm{M_{seed}\gtrsim 10^{4}}$ $\rm{M_{\odot}}$) and see very limited growth
for the lightest seeds. Similar results are obtained in semi-analytical models
(Trinca et al., 2022; Spinoso et al., 2022). The main result therefore is that
any MBH seeds below approximately $10^{5}$ $\rm{M_{\odot}}~{}$grow extremely
inefficiently due to dynamics. In addition to this, in low mass galaxies
($\rm{M_{*}\lesssim 10^{9}}$ $\rm{M_{\odot}}~{}$) growth may by hampered by
supernova feedback but this result suffers from different interpretations in
the literature and requires further study.

##  8 Summary and Conclusions

The goal of this paper is to weave together recent developments in modeling
MBH seeding and pathways to their growth. The current paradigm is broken into
light and heavy seeds with light seeds typically defined as those with initial
masses less than 1000 $\rm{M_{\odot}}~{}$and heavy seeds as those with initial
masses in excess of 1000 $\rm{M_{\odot}}$. Scenarios considered in the past
gave rise to a bimodal initial mass function: common light seeds related to
PopIII stars with mass $\sim 100$ $\rm{M_{\odot}}~{}$and rarer heavy seeds
related to SMSs with mass $\sim 10^{5}$ $\rm{M_{\odot}}$. In order to grow
these seeds into the massive and supermassive regime, two main pathways were
considered: super-Eddington on light seeds and sub-Eddington on heavy seeds,
although of course super-Eddington accretion can occur, and indeed it may be
more likely to occur, for heavy seeds. What the most recent simulations are
instead suggesting is that these two regimes are likely to be the extreme
cases of a broad distribution of initial properties.

To characterise the pathways to achieving each seed (light or heavy) we select
three representative haloes which we use to illustrate the different
environmental conditions in which they form. Our findings regarding seed
formation are as follows:

  1. 1.

The heaviest seeds: Haloes subjected to a strong or super-critical LW flux
will most likely result in the largest single stellar object forming, up to
$\sim 10^{5}$ $\rm{M_{\odot}}~{}$(1000J halo). This is because the absence of
any coolant, apart from Hydrogen, results in a halo whose temperature is
nearly uniform across the halo inducing a near spherical collapse. Such
conditions are very rare (seed number densities $\sim 10^{-10}-10^{-5}\
\rm{cMpc}^{-3}$ at z $\sim$ 10) and cannot explain the number densities of
either the high-z AGN population or the local MBH population.

  2. 2.

Heavy seeds: Other mechanisms that lead to environments conducive to heavy
seed formation (e.g. a rapid assembly process, baryonic streaming velocities
or mechanisms that lead to the emergence of dense stellar systems) but not
necessarily a monolithic collapse will generate a spectrum of heavy seed
masses. In this, more fragmented, environment heavy seed multiplicity will be
higher with the mass therefore spread across more fragments. For the specific
examples shown here the largest stellar masses are up to $\sim 10^{4}$
$\rm{M_{\odot}}~{}$(e.g. in the Rapid halo). However, the number densities are
up to $\sim 10^{5}$ times higher than in the monolithic case making it
possible to explain the number density of the whole massive BH population.

  3. 3.

Light Seed: Smaller haloes or those enriched with metals will produce
relatively less massive stars. These stars will be distributed inhomogeneously
around the halo centre and will accrete very inefficiently. The largest
stellar masses shown here have masses up to a few times $\sim 10^{2}$
$\rm{M_{\odot}}~{}$(PopIII halo). The number densities of these halos are
between $10^{-1}$ cMpc-3 and up to $10$ cMpc-3 (z $\sim$ 10).

Based on these findings and results of other investigations, we speculate on
seed growth as follows:

  1. 1.

Need for gas replenishment: In all three halos considered the accretion rate
on the stellar progenitors of the seeds drops after $\sim 10^{5}$ years as the
readily accretable gas in the parent halo is consumed. Gas replenishment is
needed (Johnson & Bromm, 2007; Lupi et al., 2023) through either mergers with
gas-rich halos or accretion from the cosmic web.

  2. 2.

Light Seed Accretion: Light seeds accrete inefficiently because of a
combination of a small Bondi radius (Pacucci et al., 2017) and their
collective dynamics that leads them to move randomly in the inhomogenous
stellar+dark matter distribution (Smith et al., 2018; Pfister et al., 2019).
In order to overcome this barrier, some mechanism is necessary to allow highly
efficient and sustained accretion to take place: for instance if they are
embedded in a star cluster (Alexander & Natarajan, 2014; Natarajan, 2020), are
captured/capture gas clumps (Lupi et al., 2016) or a later merger brings them
into an extremely gas rich region (Johnson & Bromm, 2007; Pezzulli, Valiante &
Schneider, 2016). The question is for how long the BHs are able to remain
within this dense region?

  3. 3.

Heavy Seed Accretion: Heavy seeds experience the same bottlenecks as light
seeds, but their effect is mitigated in relation to the seed masses: Bondi
accretion scales as $M_{BH}^{2}$ and gravity scales as $M_{BH}$. A 10-times
more massive seed has better chances of remaining in dense regions and
accreting from them (Pfister et al., 2019; Lupi et al., 2023).

  4. 4.

Light Seed Mergers: The more erratic dynamics and small cross-section of light
seeds will hamper their growth by BH-BH mergers except when they are located
in dense stellar environments (Reinoso et al., 2023; Arca Sedda et al., 2023;
Rantala, Naab & Lahén, 2024). After a galaxy merger existing seeds with masses
below $\sim 10^{5}$ $\rm{M_{\odot}}~{}$are highly unlikely to bind in binaries
(Pfister et al., 2019; Ma et al., 2021).

  5. 5.

Heavy Seed Mergers: similarly to what was described for accretion, heavy seeds
are favored proportionally to their mass, since gravity-related process are
eased as the BH mass increases, but even heavy seeds have a hard time reaching
the “center” of the shallow and granular potential of high-redshift galaxies
(Ma et al., 2021).

In summary heavy seeds production is more likely the more symmetric the halo
is. The advance of more sophisticated models and simulations over the last two
decades in particular tracks this idea well. Initially the relatively coarse
simulations which also lacked advanced cooling algorithms showed the formation
of central massive stars (e.g. Abel, Bryan & Norman, 2000; Bromm, Coppi &
Larson, 1999). The inclusion of more sophisticated simulations at increased
resolution has in the vast majority of cases led to more fragmentation and a
more complex morphology within collapsing haloes, for both light and heavy
seed scenarios.

While heavy seed formation has still been shown to be viable in these more
sophisticated simulations the most massive objects formed tend to be lighter
than the maximum mass (theoretically) possible. Moreover, while light seeds
will be abundant with characteristic masses in the range of a few tens of
solar masses they will be scattered around a morphologically complex halo.
Rapid growth in such an environment will be extremely unlikely. Indeed the
growth of heavy seeds in such an environment is still challenging albeit
mitigated by the larger Bondi radius which scales as $\rm{M_{BH}^{2}}$.

So then the question becomes - what seeds will grow and within what
environment? Rapid growth will only become viable if a seed (light or heavy)
can sink towards and remain in the galaxy centre where the gas densities are
highest. The heavier the initial mass the more likely this is to occur. Light
seeds, in isolation, are not likely to do this, but there are regions of the
parameter space that allow for their growth. Smith et al. (2018) showed in
their study that out of 20,000 light seeds none grew: this suggests that the
fraction of light seeds able to grow is less than $10^{-4}$. More
optimistically Shi, Kremer & Hopkins (2024) find that the fraction of light
seeds that grow, under favourable and somewhat idealised assumptions, is $\sim
10^{-2}$. If the light seed that grows is able to reach the typical mass of a
heavy seed, and the abundance of light seeds is at least $10^{3}-10^{4}$ times
that of heavy seeds then light seeds become equally likely to be the source of
MBHs. A point of note here is that it is currently impossible to constrain
seed models based on current observational data of MBH masses by extrapolating
the growth backwards. The available parameter space, which allows both
Eddington and super-Eddington phases makes this an impossible challenge. All
that can be garnered from current observational data is an upper limit on the
seed masses.

##  9 Outlook

The most recent generation of large scale hydrodynamical simulations are
beginning to achieve mass and spatial resolutions at a level where self-
consistently tracking light and heavy seed formation and growth becomes
tractable. As discussed in §7 the Astrid, NewHorizon and Romulus simulation
suites already model heavy seed-like objects and study their growth over
cosmic timescales. While the formation of the heavy seeds from either a
stellar or collisionally driven pathway is not followed self-consistently the
subsequent growth through accretion and mergers of the MBH is followed
reasonably accurately. Moreover, the simulations include subgrid models for
dynamical friction which model the dynamics of these heavy seed black holes
within growing proto-galaxies.
The next generation of simulations are likely to push this mass resolution
limit even further slowing encroaching into the regime of the light seed
(simulations by Bhowmick et al. (2024) already incorporate some of this
thinking, albeit with optimistic dynamics as BHs are pinned to halo centres,
increasing the chances of growth by both accretion and mergers). By including
a self-consistent treatment of both light and heavy seeds and their dynamics,
the growth and relative number densities of each seed population can in
principle be determined. However, expectations must be tempered. For the
foreseeable future a subgrid model for both light and heavy seed formation
will exist particularly as the formation pathways for heavy seeds via
collisional runaway (e.g. Arca Sedda et al., 2023; Rantala, Naab & Lahén,
2024) and via SMSs remain unconverged in detailed high resolution simulations
(e.g. Prole et al., 2023).
From the observational point of view, JWST has detected a new population of
AGN, selected spectroscopically via broad (Maiolino et al., 2024b; Matthee et
al., 2023) or narrow emission lines (Scholtz et al., 2023), or photometrically
via color-color selection and then followed up spectroscopically to identify
broad emission lines (Greene et al., 2024). This population has a number
density above the extrapolation of the previously-known bright end of the AGN
luminosity function: this is because these AGN have little or no X-ray
emission (Furtak et al., 2023; Yue et al., 2024b) and they are UV-faint with
respect to the host galaxies (Matthee et al., 2023; Scholtz et al., 2023). The
physical reason for this is still unclear, but previous selections were based
on X-ray and restframe-UV, and therefore this population was missed.
Interestingly, reanalysis of HST images using variability has uncovered a
similarly sized population that is UV bright (Hayes et al., 2024) and some of
the high-z AGN are X-ray bright (Bogdán et al., 2023). However, as a
population, the JWST AGN, or perhaps candidates, do not show X-ray emission
(Yue et al., 2024a; Maiolino et al., 2024a; Ananna et al., 2024). This lack of
X-rays has led to different hypotheses: the sources are not AGN, they’re AGN
but the masses of MBHs have been overestimated, or they are intrinsically
X-ray weak, or they are surrounded by dust-free Compton-thick gas.

These sometimes conflicting results make it difficult to anchor theoretical
models to observations, and in these early days of these new JWST results we
may consider taking an agnostic attitude and perhaps consider the AGN as
candidates, while waiting for additional in-depth analyses. No doubt, however,
this influx of data to previously inaccessible wavelengths has opened new
pathways to test and constrain theoretical models of MBH formation and growth
by accretion. In the not too far future, LISA and ET will probe MBH growth by
mergers for massive and intermediate-mass BHs up to very high redshift (eLISA
Consortium et al., 2013; Maggiore et al., 2020).

## References

  * Abel, Bryan & Norman (2000) Abel T., Bryan G. L., Norman M. L., 2000, ApJ, 540, 39
  * Agarwal et al. (2013) Agarwal B., Davis A. J., Khochfar S., Natarajan P., Dunlop J. S., 2013, MNRAS, 432, 3438
  * Agarwal et al. (2015) Agarwal B., Johnson J. L., Zackrisson E., Labbe I., van den Bosch F. C., Natarajan P., Khochfar S., 2015, ArXiv e-prints:1510.01733
  * Agarwal et al. (2012) Agarwal B., Khochfar S., Johnson J. L., Neistein E., Dalla Vecchia C., Livio M., 2012, MNRAS, 425, 2854
  * Agarwal et al. (2016) Agarwal B., Smith B., Glover S., Natarajan P., Khochfar S., 2016, MNRAS, 459, 4209
  * Alexander & Natarajan (2014) Alexander T., Natarajan P., 2014, Science, 345, 1330
  * Alvarez, Wise & Abel (2009) Alvarez M. A., Wise J. H., Abel T., 2009, ApJ, 701, L133
  * Ananna et al. (2024) Ananna T. T., Bogdán Á., Kovács O. E., Natarajan P., Hickox R. C., 2024, arXiv e-prints, arXiv:2404.19010
  * Anglés-Alcázar et al. (2017) Anglés-Alcázar D., Faucher-Giguère C.-A., Quataert E., Hopkins P. F., Feldmann R., Torrey P., Wetzel A., Kereš D., 2017, MNRAS, 472, L109
  * Antonini, Gieles & Gualandris (2019) Antonini F., Gieles M., Gualandris A., 2019, MNRAS, 486, 5008
  * Arca Sedda, Amaro Seoane & Chen (2021) Arca Sedda M., Amaro Seoane P., Chen X., 2021, A&A, 652, A54
  * Arca Sedda et al. (2023) Arca Sedda M., Kamlah A. W. H., Spurzem R., Rizzuto F. P., Naab T., Giersz M., Berczik P., 2023, MNRAS, 526, 429
  * Bañados et al. (2018) Bañados E. et al., 2018, Nature, 553, 473
  * Beckmann et al. (2022) Beckmann R. S. et al., 2022, arXiv e-prints, arXiv:2211.13301
  * Bhowmick et al. (2024) Bhowmick A. K., Blecha L., Torrey P., Weinberger R., Kelley L. Z., Vogelsberger M., Hernquist L., Somerville R. S., 2024, MNRAS, 529, 3768
  * Bird et al. (2022) Bird S., Ni Y., Di Matteo T., Croft R., Feng Y., Chen N., 2022, MNRAS, 512, 3703
  * Bogdán et al. (2023) Bogdán Á. et al., 2023, Nature Astronomy
  * Boylan-Kolchin (2023) Boylan-Kolchin M., 2023, Nature Astronomy, 7, 731
  * Bromm, Coppi & Larson (1999) Bromm V., Coppi P. S., Larson R. B., 1999, ApJ, 527, L5
  * Carr & Silk (2018) Carr B., Silk J., 2018, MNRAS, 478, 3756
  * Chon & Omukai (2020) Chon S., Omukai K., 2020, MNRAS
  * Clark et al. (2011) Clark P. C., Glover S. C. O., Smith R. J., Greif T. H., Klessen R. S., Bromm V., 2011, Science, 331, 1040
  * Devecchi & Volonteri (2009) Devecchi B., Volonteri M., 2009, ApJ, 694, 302
  * Dijkstra, Ferrara & Mesinger (2014) Dijkstra M., Ferrara A., Mesinger A., 2014, MNRAS, 442, 2036
  * Dijkstra et al. (2008) Dijkstra M., Haiman Z., Mesinger A., Wyithe J. S. B., 2008, MNRAS, 391, 1961
  * Dubois et al. (2021) Dubois Y. et al., 2021, A&A, 651, A109
  * Dubois et al. (2014) Dubois Y. et al., 2014, MNRAS, 444, 1453
  * Dunn et al. (2018) Dunn G., Bellovary J., Holley-Bockelmann K., Christensen C., Quinn T., 2018, ApJ, 861, 39
  * eLISA Consortium et al. (2013) eLISA Consortium et al., 2013, ArXiv e-prints 1305.5720
  * Fan, Bañados & Simcoe (2023) Fan X., Bañados E., Simcoe R. A., 2023, ARA&A, 61, 373
  * Fan, Carilli & Keating (2006) Fan X., Carilli C. L., Keating B., 2006, ARA&A, 44, 415
  * Fernandez et al. (2014) Fernandez R., Bryan G. L., Haiman Z., Li M., 2014, MNRAS, 439, 3798
  * Fragione et al. (2022) Fragione G., Kocsis B., Rasio F. A., Silk J., 2022, ApJ, 927, 231
  * Fragione & Leigh (2018) Fragione G., Leigh N., 2018, Monthly Notices of the Royal Astronomical Society, 480, 5160
  * Furtak et al. (2023) Furtak L. J. et al., 2023, arXiv e-prints, arXiv:2308.05735
  * González et al. (2021) González E., Kremer K., Chatterjee S., Fragione G., Rodriguez C. L., Weatherford N. C., Ye C. S., Rasio F. A., 2021, ApJ, 908, L29
  * Greene et al. (2024) Greene J. E. et al., 2024, ApJ, 964, 39
  * Habouzit, Volonteri & Dubois (2017) Habouzit M., Volonteri M., Dubois Y., 2017, MNRAS, 468, 3935
  * Habouzit et al. (2016) Habouzit M., Volonteri M., Latif M., Dubois Y., Peirani S., 2016, MNRAS, 463, 529
  * Haemmerlé et al. (2018) Haemmerlé L., Woods T. E., Klessen R. S., Heger A., Whalen D. J., 2018, MNRAS, 474, 2757
  * Hasinger (2020) Hasinger G., 2020, J. Cosmology Astropart. Phys, 2020, 022
  * Hayes et al. (2024) Hayes M. J. et al., 2024, arXiv e-prints, arXiv:2403.16138
  * Hirano et al. (2017) Hirano S., Hosokawa T., Yoshida N., Kuiper R., 2017, Science, 357, 1375
  * Hosokawa, Omukai & Yorke (2013) Hosokawa T., Omukai K., Yorke H. W., 2013, ApJ, 778, 178
  * Hosokawa et al. (2013) Hosokawa T., Yorke H. W., Inayoshi K., Omukai K., Yoshida N., 2013, ApJ, 778, 178
  * Inayoshi & Tanaka (2015) Inayoshi K., Tanaka T. L., 2015, MNRAS, 450, 4350
  * Inayoshi, Visbal & Haiman (2020) Inayoshi K., Visbal E., Haiman Z., 2020, ARA&A, in press; e-print arXiv:1911.05791, arXiv:1911.05791
  * Jiang et al. (2020) Jiang L. et al., 2020, Nature Astronomy
  * Jiang, Stone & Davis (2017) Jiang Y.-F., Stone J., Davis S. W., 2017, ArXiv e-prints
  * Johnson & Bromm (2007) Johnson J. L., Bromm V., 2007, MNRAS, 374, 1557
  * Juodžbalis et al. (2024) Juodžbalis I. et al., 2024, arXiv e-prints, arXiv:2403.03872
  * Katz, Sijacki & Haehnelt (2015) Katz H., Sijacki D., Haehnelt M. G., 2015, MNRAS, 451, 2352
  * Kovacs et al. (2024) Kovacs O. E. et al., 2024, arXiv e-prints, arXiv:2403.14745
  * Larson et al. (2023) Larson R. L. et al., 2023, ApJ, 953, L29
  * Latif & Ferrara (2016) Latif M. A., Ferrara A., 2016, PASA, 33, e051
  * Latif & Khochfar (2020) Latif M. A., Khochfar S., 2020, MNRAS, 497, 3761
  * Latif et al. (2016) Latif M. A., Omukai K., Habouzit M., Schleicher D. R. G., Volonteri M., 2016, ApJ, 823, 40
  * Latif et al. (2013) Latif M. A., Schleicher D. R. G., Schmidt W., Niemeyer J. C., 2013, MNRAS, 436, 2989
  * Latif et al. (2022) Latif M. A., Whalen D. J., Khochfar S., Herrington N. P., Woods T. E., 2022, Nature, 607, 48
  * Li et al. (2024) Li J. et al., 2024, arXiv e-prints, arXiv:2403.00074
  * Lupi et al. (2016) Lupi A., Haardt F., Dotti M., Fiacconi D., Mayer L., Madau P., 2016, MNRAS, 456, 2993
  * Lupi, Haiman & Volonteri (2021) Lupi A., Haiman Z., Volonteri M., 2021, MNRAS, 503, 5046
  * Lupi et al. (2023) Lupi A., Quadri G., Volonteri M., Colpi M., Regan J. A., 2023, arXiv e-prints, arXiv:2312.08422
  * Ma et al. (2021) Ma L., Hopkins P. F., Ma X., Anglés-Alcázar D., Faucher-Giguère C.-A., Kelley L. Z., 2021, MNRAS, 508, 1973
  * Madau & Rees (2001) Madau P., Rees M. J., 2001, ApJ, 551, L27
  * Maggiore et al. (2020) Maggiore M. et al., 2020, J. Cosmology Astropart. Phys, 2020, 050
  * Maiolino et al. (2024a) Maiolino R. et al., 2024a, arXiv e-prints, arXiv:2405.00504
  * Maiolino et al. (2024b) Maiolino R. et al., 2024b, Nature, 627, 59
  * Mapelli et al. (2021) Mapelli M. et al., 2021, MNRAS, 505, 339
  * Matthee et al. (2023) Matthee J. et al., 2023, arXiv e-prints, arXiv:2306.05448
  * Mayer et al. (2023) Mayer L., Capelo P. R., Zwick L., Di Matteo T., 2023, arXiv e-prints, arXiv:2304.02066
  * Mayer et al. (2015) Mayer L., Fiacconi D., Bonoli S., Quinn T., Roškar R., Shen S., Wadsley J., 2015, ApJ, 810, 51
  * Mayer et al. (2010) Mayer L., Kazantzidis S., Escala A., Callegari S., 2010, Nature, 466, 1082
  * Milosavljević, Couch & Bromm (2009) Milosavljević M., Couch S. M., Bromm V., 2009, ApJ, 696, L146
  * Mortlock et al. (2011) Mortlock D. J. et al., 2011, Nature, 474, 616
  * Nandal et al. (2023) Nandal D., Regan J. A., Woods T. E., Farrell E., Ekström S., Meynet G., 2023, A&A, 677, A155
  * Natarajan (2020) Natarajan P., 2020, MNRAS
  * Natarajan et al. (2024) Natarajan P., Pacucci F., Ricarte A., Bogdán Á., Goulding A. D., Cappelluti N., 2024, ApJ, 960, L1
  * Ni et al. (2022) Ni Y. et al., 2022, MNRAS, 513, 670
  * Pacucci et al. (2017) Pacucci F., Natarajan P., Volonteri M., Cappelluti N., Urry C. M., 2017, ApJ, 850, L42
  * Pelupessy et al. (2013) Pelupessy F. I., van Elteren A., de Vries N., McMillan S. L. W., Drost N., Portegies Zwart S. F., 2013, A&A, 557, A84
  * Pezzulli, Valiante & Schneider (2016) Pezzulli E., Valiante R., Schneider R., 2016, MNRAS, 458, 3047
  * Pfister et al. (2017) Pfister H., Lupi A., Capelo P. R., Volonteri M., Bellovary J. M., Dotti M., 2017, MNRAS, 471, 3646
  * Pfister et al. (2019) Pfister H., Volonteri M., Dubois Y., Dotti M., Colpi M., 2019, MNRAS, 486, 101
  * Piana et al. (2020) Piana O., Dayal P., Volonteri M., Choudhury T. R., 2020, MNRAS
  * Portegies Zwart et al. (2009) Portegies Zwart S. et al., 2009, New A, 14, 369
  * Portegies Zwart et al. (2004) Portegies Zwart S. F., Baumgardt H., Hut P., Makino J., McMillan S. L. W., 2004, Nature, 428, 724
  * Prole et al. (2023) Prole L. R., Regan J. A., Glover S. C. O., Klessen R. S., Priestley F. D., Clark P. C., 2023, arXiv e-prints, arXiv:2312.06769
  * Rantala, Naab & Lahén (2024) Rantala A., Naab T., Lahén N., 2024, arXiv e-prints, arXiv:2403.10602
  * Regan & Downes (2018) Regan J. A., Downes T. P., 2018, MNRAS, 478, 5037
  * Regan et al. (2020a) Regan J. A., Haiman Z., Wise J. H., O’Shea B. W., Norman M. L., 2020a, arXiv e-prints, arXiv:2006.14625
  * Regan et al. (2017) Regan J. A., Visbal E., Wise J. H., , Haiman Z., Johansson P. H., Bryan G. L., 2017, Nature Astronomy, 1, 0075
  * Regan et al. (2020b) Regan J. A., Wise J. H., O’Shea B. W., Norman M. L., 2020b, MNRAS, 492, 3021
  * Regan et al. (2020c) Regan J. A., Wise J. H., Woods T. E., Downes T. P., O’Shea B. W., Norman M. L., 2020c, The Open Journal of Astrophysics, 3, 15
  * Reinoso et al. (2023) Reinoso B., Klessen R. S., Schleicher D., Glover S. C. O., Solar P., 2023, MNRAS
  * Ricarte et al. (2019) Ricarte A., Tremmel M., Natarajan P., Quinn T., 2019, MNRAS, 489, 802
  * Rizzuto et al. (2021) Rizzuto F. P. et al., 2021, MNRAS, 501, 5257
  * Sa̧dowski (2009) Sa̧dowski A., 2009, ApJS, 183, 171
  * Sa̧dowski et al. (2016) Sa̧dowski A., Lasota J.-P., Abramowicz M. A., Narayan R., 2016, MNRAS, 456, 3915
  * Sa̧dowski & Narayan (2016) Sa̧dowski A., Narayan R., 2016, MNRAS, 456, 3929
  * Schauer et al. (2021) Schauer A. T. P., Glover S. C. O., Klessen R. S., Clark P., 2021, MNRAS, 507, 1775
  * Scholtz et al. (2023) Scholtz J. et al., 2023, arXiv e-prints, arXiv:2311.18731
  * Sesana, Volonteri & Haardt (2007) Sesana A., Volonteri M., Haardt F., 2007, MNRAS, 377, 1711
  * Sesana, Volonteri & Haardt (2009) Sesana A., Volonteri M., Haardt F., 2009, Classical and Quantum Gravity, 26, 094033
  * Shang, Bryan & Haiman (2010) Shang C., Bryan G. L., Haiman Z., 2010, MNRAS, 402, 1249
  * Shi, Kremer & Hopkins (2024) Shi Y., Kremer K., Hopkins P. F., 2024
  * Singh, Monaco & Tan (2023) Singh J., Monaco P., Tan J. C., 2023, MNRAS, 525, 969
  * Smith et al. (2018) Smith B. D., Regan J. A., Downes T. P., Norman M. L., O’Shea B. W., Wise J. H., 2018, MNRAS, 480, 3762
  * Spinoso et al. (2022) Spinoso D., Bonoli S., Valiante R., Schneider R., Izquierdo-Villalba D., 2022, arXiv e-prints, arXiv:2203.13846
  * Spolyar, Freese & Gondolo (2008) Spolyar D., Freese K., Gondolo P., 2008, Phys. Rev. Lett., 100, 051101
  * Tremmel et al. (2017) Tremmel M., Karcher M., Governato F., Volonteri M., Quinn T. R., Pontzen A., Anderson L., Bellovary J., 2017, MNRAS, 470, 1121
  * Trinca et al. (2022) Trinca A., Schneider R., Valiante R., Graziani L., Zappacosta L., Shankar F., 2022, MNRAS, 511, 616
  * Turk, Abel & O’Shea (2009) Turk M. J., Abel T., O’Shea B., 2009, Science, 325, 601
  * Venemans et al. (2015) Venemans B. P. et al., 2015, ApJ, 801, L11
  * Venemans et al. (2013) Venemans B. P. et al., 2013, ApJ, 779, 24
  * Visbal, Haiman & Bryan (2014) Visbal E., Haiman Z., Bryan G. L., 2014, MNRAS, 445, 1056
  * Visbal et al. (2014) Visbal E., Haiman Z., Terrazas B., Bryan G. L., Barkana R., 2014, ArXiv e-prints
  * Volonteri, Habouzit & Colpi (2021) Volonteri M., Habouzit M., Colpi M., 2021, Nature Reviews Physics, 3, 732
  * Volonteri & Rees (2005) Volonteri M., Rees M. J., 2005, ApJ, 633, 624
  * Wang et al. (2021) Wang F. et al., 2021, arXiv e-prints, arXiv:2101.03179
  * Whalen, Abel & Norman (2004) Whalen D., Abel T., Norman M. L., 2004, ApJ, 610, 14
  * Wise et al. (2019) Wise J. H., Regan J. A., O’Shea B. W., Norman M. L., Downes T. P., Xu H., 2019, Nature, 566, 85
  * Woods et al. (2017) Woods T. E., Heger A., Whalen D. J., Haemmerlé L., Klessen R. S., 2017, ApJ, 842, L6
  * Woods et al. (2021) Woods T. E., Willott C. J., Regan J. A., Wise J. H., Downes T. P., Norman M. L., O’Shea B. W., 2021, ApJ, 920, L22
  * Xu, Wise & Norman (2013) Xu H., Wise J. H., Norman M. L., 2013, ApJ, 773, 83
  * Yang et al. (2021) Yang J. et al., 2021, ApJ, 923, 262
  * Yoshida et al. (2003) Yoshida N., Abel T., Hernquist L., Sugiyama N., 2003, ApJ, 592, 645
  * Yue et al. (2024a) Yue M., Eilers A.-C., Ananna T. T., Panagiotou C., Kara E., Miyaji T., 2024a, arXiv e-prints, arXiv:2404.13290
  * Yue et al. (2024b) Yue M., Eilers A.-C., Annana T. T., Panagiotou C., Kara E., Miyaji T., 2024b, arXiv e-prints, arXiv:2404.13290
  * Zubovas & King (2021) Zubovas K., King A., 2021, MNRAS, 501, 4289