The pathways 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 bimodallike 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 103 to 105 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   a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 104 compared to light seeds. Finally we examine the current stateoftheart in numerical calculations and recent observations and plot a path forward for nearfuture advances in both domains.
['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']
Massive Black Hole Seeds
{'MASSIVE BLACK HOLE SEEDS': "John A. Regan 1 and Marta Volonteri 2  \n1 Centre for Astrophysics and Space Science Maynooth, Department of Physics, Maynooth University, Maynooth, Ireland 2 Institut d'Astrophysique de Paris, Sorbonne Université, CNRS, UMR 7095, 98 bis bd Arago, 75014 Paris, France Version September 4, 2024", '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 ( M seed > 10 3 M ⊙ ) 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.', '1. INTRODUCTION': 'Massive black holes (MBHs) with masses up to 10 10 M ⊙ 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 M ⊙ 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.  \nMore recently, observations with JWST have uncovered MBHs with masses in the range 10 6 M ⊙ to 10 8 M ⊙ (Kovács et al. 2024; Maiolino et al. 2024b; Larson et al. 2023; Bogdán et al. 2023; Matthee et al. 2024; 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 × 10 6 M ⊙ (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  \npotentially significant evolution of the central black hole following seeding. If we assume that current candidate AGNbeing 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. 2024; 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 .  \nThe 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 M ⊙ and a heavy seed as any seed black hole with an initial mass in excess of 1000 M ⊙ . 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.  \nThe 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 M ⊙ 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).  \nFig. 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.  \n<!-- image -->  \nThe 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.  \nAlternative 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 M ⊙ and in some cases possibly up to 10 5 M ⊙ . 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; Feng, Yu & Zhong 2021; Singh, Monaco & Tan 2023) or primordial black holes (e.g., Carr & Silk 2018; Hasinger 2020). We do not consider these mechanisms here as there is still considerable uncertainty in their formation pathway and their possible number densities. However, continued research into these mechanisms is warranted given the outstanding questions over the mainstream pathways, as we now discuss, as well as the ability of "non-standard"  \ndark matter to potentially solve other dynamical hurdles (e.g Alonso-Álvarez, Cline & Dewar 2024).', '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 ≳ 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  \nM(t) = M 0 exp ( 1 -ϵ r ϵ r t t Edd ) , (1)  \nwhere t Edd = 0 . 45 Gyr and ϵ r is the radiative efficiency. For a 'standard' radiative efficiency of ϵ r ∼ 0 . 1 and a black hole seed mass of M 0 = 10 2 M ⊙ , it takes nearly 1 Gyr to grow from M(0) up to ∼ 10 9 M ⊙ . 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. 2023).  \nThe 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 M ⊙ 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).  \nA 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).  \nSeminal 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 M ⊙ . 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 M ⊙ does appear viable and hence this route to heavy seeds appears plausible (although sensitive to cluster initial conditions).  \nAnother 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 M ⊙ (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.  \nIn addition to the dynamical pathway heavy seeds can form via the formation of a very massive or supermassive 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 ( ˙ M ≳ 0 . 001 M ⊙ / 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 T eff ∼ 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 M ⊙ (Woods et al. 2017) these SMS stars would be ideal progenitors for MBHs and among the heaviest seeds possible.  \nFinally, Mayer et al. (2010, 2015, 2024) have postulated a "direct" mechanism for forming extremely massive black holes. In their model the merger of massive galaxies ( M Gal ≳ 10 11 M ⊙ ) can result in the direct creation of a supermassive black hole with a mass up to 10 8 M ⊙ . 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 ∼ 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 ( M Halo ∼ 10 12 M ⊙ at z ∼ 10 corresponding to 4 σ -5 σ peaks) to drive large gas inflows creating the central MBH.  \nWe note here that high resolution modelling including state-of-the-art physical models is crucial here. When attempting to model the collapse of gas creating the central MBH. resolution and/or physical models can result in fragmenting objects which are unrealistically massive. Early simulations of PopIII formation in particular suffered from this issue (e.g. Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2000) while more recent approaches (when modelling heavy seed formation) have shown, for example, the importance of resolving dynamical heating (e.g. Wise et al. 2019) and ISM turbulence (e.g. Latif et al. 2022).', '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 ∼ 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.  \nBy definition certain, perhaps rare, environmental conditions must occur for heavy seed formation pathways to  \nFig. 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.  \n<!-- image -->  \nbe 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.  \nDijkstra, Ferrara & Mesinger (2014) calculated the number density of heavy seeds forming through the socalled 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. 2016; 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 ∼ 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 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).  \nMoving now to channels which potentially produce heavy seeds below the theoretical maximum mass (of SMSs) but well in excess of light seeds masses. 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 ∼ 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 ∼ 10 ) as well (Hirano et al. 2017).  \nOverall 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 ∼ 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.', '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  \nTABLE 1  \nCharacteristic 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.  \nof 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.  \nThe 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 J 21 1 (an intensity of 1000 J 21 is generally accepted in the literature as required for substantial 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 M ⊙ and there were only four stars in the halo at the end of the simulations ( ∼ 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).  \nIn 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  \nthe top panel of Figure 3. The strong LW field homogenises the temperature field leading to a near isothermal collapse (at ∼ 8000K ). 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.  \nIn 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 M ⊙ 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.  \nThe number density of haloes subjected to a supercritical 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 ∼ 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 ∼ 10 ) for a 1000J like object. An important additional point here is that in-situ LW radiation can potentially ease the thresholds somewhat but remains in delicate balance with other factors like metalpollution which can suppress heavy seed formation (e.g. Dunn et al. 2018; Bhowmick et al. 2022b; Chiaki et al. 2023).', '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,  \n<!-- image -->  \n<!-- image -->  \nFig. 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 ( J LW = 1000 J 21 ) largely dissociates the H 2 ensuring the collapse is isothermal. The middle panel, with only a very mild LW field ( J LW ∼ 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 H 2 is able to cool the gas to below 1000 K in the halo centre.  \n<!-- image -->  \nFig. 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.  \n<!-- image -->  \nas described in §3, the halo experiences periods of extremely rapid growth, which dynamically heats the halo, overcoming the ability of 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.  \nThe 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 M ⊙ 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 M ⊙ . 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.  \nThe number density of rapidly growing haloes from the Renaissance simulations is approximately 5 × 10 -3 cMpc -3 at z ∼ 10 (Regan et al. 2020a) 2 . 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 M ⊙ / yr -1 in the halo centre and with metallicities Z ≲ 10 -3 Z ⊙ .", '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 M ⊙ (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.  \nThe number density of haloes, at z ∼ 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 M ⊙ and 44000 M ⊙ ) 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 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.  \nThere 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 H 2 and leads to larger Jeans masses and hence larger protostars (Prole et al. 2024) 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.  \nIt should be noted that in Figure 5 we show only accretion over a relatively short period (hundreds of kiloyears). The time-resolution needed for these simulations is very short - of the order of years compared to large  \n<!-- image -->  \nFig. 5.The evolution of the most massive star that forms in each of the three haloes (1000J, Rapid and PopIII). Stars are modelled as accreting sink particles. The spatial resolution of each simulation is approximately 10 -3 pc (physical). 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).  \n<!-- image -->  \nscale cosmological simulations with timesteps many orders of magnitude higher. 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.', '7. BLACK HOLE GROWTH': 'The sustained growth of black holes is clearly critical for their evolution into the MBHs we observe as both high-  \nz quasars and as the "JWST-quasars" at z ≳ 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.  \nLight 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.  \nSmith 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. 2024) 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 M ⊙ 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.  \nBuilding 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 ( ∼ 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  \ndemonstrated extensively via extremely high resolution simulations (e.g. S¸adowski 2009; S¸adowski & Narayan 2016; S¸adowski et al. 2016; Jiang, Stone & Davis 2019; 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.  \nIn 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).  \nConversely, 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 M ⊙ and perhaps as high as an initial mass of 10 8 M ⊙ 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. (2024) model the progenitor of a high-z quasar using an initial black hole seed mass of 10 5 M ⊙ (the model sets the initial dynamical mass of the heavy seed to be 10 6 M ⊙ ). 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 (see also Bhowmick et al. 2022a). 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  \nmore idealised conditions - following the growth of a heavy seed (M seed = 10 5 M ⊙ ) starting from a halo similar to the 1000J halo described above.  \nOn 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 ∗ ≲ 10 9 M ⊙ ), 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).  \nSo 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 M ⊙ before growth can be achieved - this is particularly evident from the Astrid simulations which have the lowest seed masses ( M seed ≳ 10 4 M ⊙ ) and see very limited growth for the lightest seeds. Similar results are obtained in semi-analytical models (Trinca et al. 2022; Spinoso et al. 2023). The main result therefore is that any MBH seeds below approximately 10 5 M ⊙ grow extremely inefficiently due to dynamics. In addition to this, in low mass galaxies ( M ∗ ≲ 10 9 M ⊙ ) 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 M ⊙ and heavy seeds as those with initial masses in excess of 1000 M ⊙ (e.g. Sassano et al. 2021; Evans, Blecha & Bhowmick 2023; Jeon et al. 2024). Scenarios considered in the past gave rise to a bimodal initial mass function: common light seeds related to PopIII stars with mass ∼ 100 M ⊙ and rarer heavy seeds related to SMSs with mass ∼ 10 5 M ⊙ . 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.  \nTo characterise the pathways to achieving each seed ( light or heavy ) we select three representative haloes  \nwhich we use to illustrate the different environmental conditions in which they form. Our findings regarding seed formation are as follows:  \n- 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 ∼ 10 5 M ⊙ ( 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 ∼ 10 -10 -10 -5 cMpc -3 at z ∼ 10) and cannot explain the number densities of either the high-z AGN population or the local MBH population.\n- 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 ∼ 10 4 M ⊙ (e.g. in the Rapid halo). However, the number densities are up to ∼ 10 5 times higher than in the monolithic case making it possible to explain the number density of the whole massive BH population.\n- 3. Light Seeds: 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 ∼ 10 2 M ⊙ (PopIII halo). The number densities of these halos are between 10 -1 cMpc -3 and up to 10 cMpc -3 (z ∼ 10).  \nBased on these findings and results of other investigations, we speculate on seed growth as follows:  \n- 1. Need for gas replenishment: In all three halos considered the accretion rate on the stellar progenitors of the seeds drops after ∼ 10 5 years as the readily accretable gas in the parent halo is consumed. Gas replenishment is needed (Johnson & Bromm 2007; Lupi et al. 2024) through either mergers with gas-rich halos or accretion from the cosmic web.\n- 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  \nextremely 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?  \n- 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 2 BH 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. 2024).\n- 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 ∼ 10 5 M ⊙ are highly unlikely to bind in binaries (Pfister et al. 2019; Ma et al. 2021).\n- 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 highredshift galaxies (Ma et al. 2021).  \nIn 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.  \nWhile 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 M 2 BH .  \nSo 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  \noptimistically Shi, Kremer & Hopkins (2024) find that the fraction of light seeds that grow, under favourable and somewhat idealised assumptions, is ∼ 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.  \nSemi Analytic Models (SAMs) rely on a backbone of dark matter merger trees to model galaxy and MBH evolution over long timescales. Their relative simplicity, compared to hydrodynamic simulations make them ideal for rapidly traversing the large parameter space of galaxy and black hole formation (e.g. Schneider et al. 2006; Dayal et al. 2019; Trinca et al. 2022). However, this power comes at the cost of small scale details. SAMs inherently do not have spatially-resolved structures and the detailed physical modelling capacity necessary to capture emergent physics like turbulence, detailed ISM physics as well as the likely episodic nature of MBH accretion.  \nIn Figure 6 we show a schematic outlining the range of spatial and temporal scales that must be bridged by our models. No one simulation suite, technique or code can cross the domains and multiple suites and techniques (SAMs + hydrodynamics) must be used. To model MBH evolution different models utilise different seeding and accretion prescriptions - usually driven by the numerical resources available to them at a given scale. For example the large scale simulations (top right) must seed galaxies with relatively large black holes which represent black holes not necessarily at their seed masses but at some time after their physical formation.  \nSAMs based on high-resolution parent DM halo trees can in principle seed MBHs in arbitrarily low-mass galaxies/halos. However, SAMs have to seed BHs based on global quantities rather than local, spatially-resolved properties. Common to both the large scale hydrodynamic simulations and the SAMs is a difficulty in resolving the dynamics and detailed accretion physics of the MBHs. As we negotiate the parameter space of the  \nFig. 6.Schematic comparing Semi Analytic Models (SAMs) against hydrodynamic simulations. The computational complexity increases from the top right to the bottom left. Models in the top right run extremely quickly but can lack the physical complexity to resolve small scale features like MBH dynamics and seed formation. On the other hand models occupying the bottom left panel focus on scales close to the event horizon and can only be evolved for relatively short physical times given the extreme computational expenses. An array of simulations volumes and resolutions must therefore be used to "join" the scales. The simulation images shown include Illustris (e.g. Vogelsberger et al. 2014), Romulus (e.g. Tremmel et al. 2017), Renaissance (e.g. Xu, Wise & Norman 2013) and results from small scale GRMHD simulations (Chatterjee et al. 2019).  \n<!-- image -->  \nschematic moving from top right to bottom left the models can becomes increasingly sophisticated in how, and at what mass, black holes are seeded. At smaller scales and higher resolution emergent physics gives rise to a better handle on the physics thought necessary for MBH formation (be they light or heavy seeds). Conversely, these simulations are unable, to follow over long timescale, the evolution of the seeds.  \nThe 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. (2024a,b) 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 selfconsistent 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  \n(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. 2024).  \nFrom the observational point of view, JWST has detected a new population of AGN, selected spectroscopically via broad (Maiolino et al. 2024b; Matthee et al. 2024) 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. 2024; Yue et al. 2024b) and they are UV-faint with respect to the host galaxies (Matthee et al. 2024; 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  \npopulation, 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.  \nThese 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  \ntoo 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).', 'ACKNOWLEDGEMENTS': 'We thank the referee for providing highly constructive feedback which greatly improved the manuscript. JR acknowledges support from the Royal Society and Science Foundation Ireland under grant number URF\\R1\\191132. JR also acknowledges support from the Irish Research Council Laureate programme under grant number IRCLA/2022/1165. MV acknowledges funding from the French National Research Agency (grant ANR-21-CE31-0026, project MBH\\_waves).', 'REFERENCES': "- Abel T., Bryan G. L., Norman M. L., 2000, ApJ, 540, 39\n- Agarwal B., Davis A. J., Khochfar S., Natarajan P., Dunlop J. S., 2013, MNRAS, 432, 3438 Agarwal B., Khochfar S., Johnson J. L., Neistein E., Dalla Vecchia C., Livio M., 2012, MNRAS, 425, 2854 Agarwal B., Smith B., Glover S., Natarajan P., Khochfar S., 2016, MNRAS, 459, 4209 Alexander T., Natarajan P., 2014, Science, 345, 1330 Alonso-Álvarez G., Cline J. M., Dewar C., 2024, arXiv e-prints, arXiv:2401.14450 Alvarez M. A., Wise J. H., Abel T., 2009, ApJ, 701, L133 Ananna T. T., Bogdán Á., Kovács O. E., Natarajan P., Hickox R. C., 2024, ApJ, 969, L18 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 F., Gieles M., Gualandris A., 2019, MNRAS, 486, 5008 Arca Sedda M., Amaro Seoane P., Chen X., 2021, A&A, 652, A54 Arca Sedda M., Kamlah A. W. H., Spurzem R., Rizzuto F. P., Naab T., Giersz M., Berczik P., 2023, MNRAS, 526, 429 Bañados E. et al., 2018, Nature, 553, 473 Beckmann R. S. et al., 2022, arXiv e-prints, arXiv:2211.13301 Bhowmick A. K. et al., 2022a, MNRAS, 516, 138 Bhowmick A. K., Blecha L., Torrey P., Kelley L. Z., Vogelsberger M., Nelson D., Weinberger R., Hernquist L., 2022b, MNRAS, 510, 177 Bhowmick A. K., Blecha L., Torrey P., Weinberger R., Kelley L. Z., Vogelsberger M., Hernquist L., Somerville R. S., 2024a, MNRAS, 529, 3768 Bhowmick A. K., Blecha L., Torrey P., Weinberger R., Kelley L. Z., Vogelsberger M., Hernquist L., Somerville R. S., 2024b, MNRAS, 529, 3768 Bird S., Ni Y., Di Matteo T., Croft R., Feng Y., Chen N., 2022, MNRAS, 512, 3703 Bogdán Á. et al., 2023, Nature Astronomy Boylan-Kolchin M., 2023, Nature Astronomy, 7, 731 Bromm V., Coppi P. S., Larson R. B., 1999, ApJ, 527, L5 Carr B., Silk J., 2018, MNRAS, 478, 3756 Chatterjee K., Liska M., Tchekhovskoy A., Markoff S. B., 2019, MNRAS, 490, 2200 Chiaki G., Chon S., Omukai K., Trinca A., Schneider R., Valiante R., 2023, MNRAS, 521, 2845 Chon S., Omukai K., 2020, MNRAS Clark P. C., Glover S. C. O., Smith R. J., Greif T. H., Klessen R. S., Bromm V., 2011, Science, 331, 1040 Dayal P., Rossi E. M., Shiralilou B., Piana O., Choudhury T. R., Volonteri M., 2019, MNRAS, 486, 2336 Devecchi B., Volonteri M., 2009, ApJ, 694, 302 Dijkstra M., Ferrara A., Mesinger A., 2014, MNRAS, 442, 2036 Dijkstra M., Haiman Z., Mesinger A., Wyithe J. S. B., 2008, MNRAS, 391, 1961 Dubois Y. et al., 2021, A&A, 651, A109 Dubois Y. et al., 2014, MNRAS, 444, 1453 Dunn G., Bellovary J., Holley-Bockelmann K., Christensen C.,\n- Quinn T., 2018, ApJ, 861, 39\n- eLISA Consortium et al., 2013, ArXiv e-prints 1305.5720 Evans A. E., Blecha L., Bhowmick A. K., 2023, arXiv e-prints, arXiv:2309.11324\n- Fan X., Bañados E., Simcoe R. A., 2023, ARA&A, 61, 373 Fan X., Carilli C. L., Keating B., 2006, ARA&A, 44, 415 Feng W.-X., Yu H.-B., Zhong Y.-M., 2021, ApJ, 914, L26 Fernandez R., Bryan G. L., Haiman Z., Li M., 2014, MNRAS, 439, 3798\n- Fragione G., Kocsis B., Rasio F. A., Silk J., 2022, ApJ, 927, 231 Fragione G., Leigh N., 2018, Monthly Notices of the Royal Astronomical Society, 480, 5160\n- Furtak L. J. et al., 2024, Nature, 628, 57\n- González E., Kremer K., Chatterjee S., Fragione G., Rodriguez C. L., Weatherford N. C., Ye C. S., Rasio F. A., 2021, ApJ, 908, L29\n- Greene J. E. et al., 2024, ApJ, 964, 39\n- Habouzit M., Volonteri M., Dubois Y., 2017, MNRAS, 468, 3935\n- Habouzit M., Volonteri M., Latif M., Dubois Y., Peirani S., 2016, MNRAS, 463, 529\n- Haemmerlé L., Woods T. E., Klessen R. S., Heger A., Whalen D. J., 2018, MNRAS, 474, 2757\n- Hasinger G., 2020, J. Cosmology Astropart. Phys., 2020, 022 Hayes M. J. et al., 2024, ApJ, 971, L16\n- Hirano S., Hosokawa T., Yoshida N., Kuiper R., 2017, Science, 357, 1375\n- Hosokawa T., Omukai K., Yorke H. W., 2013, ApJ, 778, 178\n- Hosokawa T., Yorke H. W., Inayoshi K., Omukai K., Yoshida N., 2013, ApJ, 778, 178\n- Inayoshi K., Tanaka T. L., 2015, MNRAS, 450, 4350 Inayoshi K., Visbal E., Haiman Z., 2020, ARA&A, 58, 27\n- Jeon J., Bromm V., Liu B., Finkelstein S. L., 2024, arXiv e-prints, arXiv:2402.18773\n- Jiang L. et al., 2020, Nature Astronomy\n- Jiang Y.-F., Stone J. M., Davis S. W., 2019, ApJ, 880, 67 Johnson J. L., Bromm V., 2007, MNRAS, 374, 1557\n- Juodžbalis I. et al., 2024, arXiv e-prints, arXiv:2403.03872\n- Katz H., Sijacki D., Haehnelt M. G., 2015, MNRAS, 451, 2352\n- Kovács O. E. et al., 2024, ApJ, 965, L21\n- Larson R. L. et al., 2023, ApJ, 953, L29\n- Latif M. A., Ferrara A., 2016, PASA, 33, e051\n- Latif M. A., Khochfar S., 2020, MNRAS, 497, 3761\n- Latif M. A., Omukai K., Habouzit M., Schleicher D. R. G., Volonteri M., 2016, ApJ, 823, 40\n- Latif M. A., Schleicher D. R. G., Schmidt W., Niemeyer J. C., 2013, MNRAS, 436, 2989\n- Latif M. A., Whalen D. J., Khochfar S., Herrington N. P., Woods T. E., 2022, Nature, 607, 48\n- Li J. et al., 2024, arXiv e-prints, arXiv:2403.00074\n- Lupi A., Haardt F., Dotti M., Fiacconi D., Mayer L., Madau P., 2016, MNRAS, 456, 2993\n- Lupi A., Haiman Z., Volonteri M., 2021, MNRAS, 503, 5046\n- Lupi A., Quadri G., Volonteri M., Colpi M., Regan J. A., 2024, A&A, 686, A256\n- Ma L., Hopkins P. F., Ma X., Anglés-Alcázar D., Faucher-Giguère C.-A., Kelley L. Z., 2021, MNRAS, 508, 1973  \nMadau P., Rees M. J., 2001, ApJ, 551, L27  \n- Maggiore M. et al., 2020, J. Cosmology Astropart. Phys., 2020, 050 Maiolino R. et al., 2024a, arXiv e-prints, arXiv:2405.00504 Maiolino R. et al., 2024b, Nature, 627, 59 Mapelli M. et al., 2021, MNRAS, 505, 339 Matthee J. et al., 2024, ApJ, 963, 129 Mayer L., Capelo P. R., Zwick L., Di Matteo T., 2024, ApJ, 961, 76 Mayer L., Fiacconi D., Bonoli S., Quinn T., Roškar R., Shen S., Wadsley J., 2015, ApJ, 810, 51 Mayer L., Kazantzidis S., Escala A., Callegari S., 2010, Nature, 466, 1082 Milosavljević M., Couch S. M., Bromm V., 2009, ApJ, 696, L146 Mortlock D. J. et al., 2011, Nature, 474, 616 Nandal D., Regan J. A., Woods T. E., Farrell E., Ekström S., Meynet G., 2023, A&A, 677, A155 Natarajan P., 2020, MNRAS Natarajan P., Pacucci F., Ricarte A., Bogdán Á., Goulding A. D., Cappelluti N., 2024, ApJ, 960, L1 Ni Y. et al., 2022, MNRAS, 513, 670 Pacucci F., Natarajan P., Volonteri M., Cappelluti N., Urry C. M., 2017, ApJ, 850, L42 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 E., Valiante R., Schneider R., 2016, MNRAS, 458, 3047 Pfister H., Lupi A., Capelo P. R., Volonteri M., Bellovary J. M., Dotti M., 2017, MNRAS, 471, 3646 Pfister H., Volonteri M., Dubois Y., Dotti M., Colpi M., 2019, MNRAS, 486, 101 Piana O., Dayal P., Volonteri M., Choudhury T. R., 2020, MNRAS Portegies Zwart S. et al., 2009, New A, 14, 369 Portegies Zwart S. F., Baumgardt H., Hut P., Makino J., McMillan S. L. W., 2004, Nature, 428, 724 Prole L. R., Regan J. A., Glover S. C. O., Klessen R. S., Priestley F. D., Clark P. C., 2024, A&A, 685, A31 Rantala A., Naab T., Lahén N., 2024, MNRAS, 531, 3770 Regan J. A., Downes T. P., 2018, MNRAS, 478, 5037 Regan J. A., Haiman Z., Wise J. H., O'Shea B. W., Norman M. L., 2020a, The Open Journal of Astrophysics, 3, E9 Regan J. A., Visbal E., Wise J. H., , Haiman Z., Johansson P. H., Bryan G. L., 2017, Nature Astronomy, 1, 0075 Regan J. A., Wise J. H., O'Shea B. W., Norman M. L., 2020b, MNRAS, 492, 3021 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 B., Klessen R. S., Schleicher D., Glover S. C. O., Solar P., 2023, MNRAS Ricarte A., Tremmel M., Natarajan P., Quinn T., 2019, MNRAS, 489, 802 Rizzuto F. P. et al., 2021, MNRAS, 501, 5257  \nThis paper was built using the Open Journal of Astrophysics L A T E X template. The OJA is a journal which  \n- Sassano F., Schneider R., Valiante R., Inayoshi K., Chon S., Omukai K., Mayer L., Capelo P. R., 2021, MNRAS, 506, 613 S¸adowski A., 2009, ApJS, 183, 171\n- S¸adowski A., Lasota J.-P., Abramowicz M. A., Narayan R., 2016, MNRAS, 456, 3915\n- S¸adowski A., Narayan R., 2016, MNRAS, 456, 3929\n- Schauer A. T. P., Glover S. C. O., Klessen R. S., Clark P., 2021, MNRAS, 507, 1775\n- Schneider R., Salvaterra R., Ferrara A., Ciardi B., 2006, MNRAS, 369, 825\n- Scholtz J. et al., 2023, arXiv e-prints, arXiv:2311.18731\n- Sesana A., Volonteri M., Haardt F., 2007, MNRAS, 377, 1711\n- Sesana A., Volonteri M., Haardt F., 2009, Classical and Quantum Gravity, 26, 094033\n- Shang C., Bryan G. L., Haiman Z., 2010, MNRAS, 402, 1249 Shi Y., Kremer K., Hopkins P. F., 2024\n- Singh J., Monaco P., Tan J. C., 2023, MNRAS, 525, 969\n- Smith B. D., Regan J. A., Downes T. P., Norman M. L., O'Shea B. W., Wise J. H., 2018, MNRAS, 480, 3762\n- Spinoso D., Bonoli S., Valiante R., Schneider R.,\n- Izquierdo-Villalba D., 2023, MNRAS, 518, 4672\n- Spolyar D., Freese K., Gondolo P., 2008, Phys. Rev. Lett., 100, 051101\n- Tremmel M., Karcher M., Governato F., Volonteri M., Quinn T. R., Pontzen A., Anderson L., Bellovary J., 2017, MNRAS, 470, 1121\n- Trinca A., Schneider R., Valiante R., Graziani L., Zappacosta L., Shankar F., 2022, MNRAS, 511, 616\n- Turk M. J., Abel T., O'Shea B., 2009, Science, 325, 601\n- Venemans B. P. et al., 2015, ApJ, 801, L11\n- Venemans B. P. et al., 2013, ApJ, 779, 24\n- Visbal E., Haiman Z., Bryan G. L., 2014, MNRAS, 445, 1056\n- Visbal E., Haiman Z., Terrazas B., Bryan G. L., Barkana R., 2014, MNRAS, 445, 107\n- Vogelsberger M. et al., 2014, MNRAS, 444, 1518\n- Volonteri M., Habouzit M., Colpi M., 2021, Nature Reviews Physics, 3, 732\n- Volonteri M., Rees M. J., 2005, ApJ, 633, 624\n- Wang F. et al., 2021, ApJ, 907, L1\n- Whalen D., Abel T., Norman M. L., 2004, ApJ, 610, 14\n- Wise J. H., Regan J. A., O'Shea B. W., Norman M. L., Downes T. P., Xu H., 2019, Nature, 566, 85\n- Woods T. E., Heger A., Whalen D. J., Haemmerlé L., Klessen R. S., 2017, ApJ, 842, L6\n- 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 H., Wise J. H., Norman M. L., 2013, ApJ, 773, 83 Yang J. et al., 2021, ApJ, 923, 262\n- Yoshida N., Abel T., Hernquist L., Sugiyama N., 2003, ApJ, 592, 645\n- Yue M., Eilers A.-C., Ananna T. T., Panagiotou C., Kara E., Miyaji T., 2024a, arXiv e-prints, arXiv:2404.13290  \nYue M., Eilers A.-C., Annana T. T., Panagiotou C., Kara E.,  \nMiyaji T., 2024b, arXiv e-prints, arXiv:2404.13290  \n- Zubovas K., King A., 2021, MNRAS, 501, 4289  \nprovides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike. Learn more at http://astro.theoj.org ."}