3.1. Pop III initial mass function

Understanding the IMF is a key challenge in the study of primordial star formation and the masses of stellar BHs are strongly associated with it. The first numerical simulations (Abel et al. Reference Abel, Bryan and Norman2002; Bromm et al. Reference Bromm, Coppi and Larson2002) suggested that primordial stars were massive with typical masses of a few hundred solar. The latter studies also showed that Pop III stars were born in isolation with typical mass accretion rates of 10⁻⁴-10⁻² M_⊙yr⁻¹ (Yoshida et al. Reference Yoshida, Abel, Hernquist and Sugiyama2003, Reference Yoshida, Omukai, Hernquist and Abel2006; O’Shea & Norman Reference O’Shea and Norman2007; Yoshida, Omukai, & Hernquist Reference Yoshida, Omukai and Hernquist2008). However, during the past few years, high resolution simulations including a detailed treatment of physical processes found that a protostellar disk formed as a consequence of gravitational collapse becomes unstable and fragments into multiple clumps, see Figure 1. This may lead to the formation of multiple stars per halo (Turk, Abel, & O’Shea Reference Turk, Abel and O’Shea2009; Stacy, Greif, & Bromm Reference Stacy, Greif and Bromm2010; Clark et al. Reference Clark, Glover, Smith, Greif, Klessen and Bromm2011; Greif et al. Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013e). These simulations were evolved only up to a few tens to 5 000 yr, about a factor of 100 lower than the time required for a protostar to reach the main sequence and also did not include the stellar UV feedback.

Figure 1.

The evolution of a protosetllar system in four different minihalos. Density projections of hydrogen nuclei are shown for the central 10 AU. Each row represents the minihalo whilst each column shows the time evolution after the formation of a central star. Adopted from Greif et al. (Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012).

However, simulations taking into account the UV feedback from the protostar show that it shuts accretion onto the protostar by photoevaporating the protostellar disk and the central star cannot grow beyond 40 M_⊙ (Hosokawa et al. Reference Hosokawa, Omukai, Yoshida and Yorke2011; Stacy et al. Reference Stacy, Greif and Bromm2012). These simulations either employed an approximate treatment for the radiative feedback or performed only two dimensional simulations. Hirano et al. (Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014) have derived the stellar mass distribution for one hundred minihalos under the assumption that only single star forms per halo by including the radiative feedback from a star as well as stellar evolution. They found that stellar masses range from 10–1 000 M_⊙ and depend on the properties of their natal halos, also see Susa et al. (Reference Susa, Hasegawa and Tominaga2014). Latif & Schleicher (Reference Latif and Schleicher2015a) employed an analytical model to study the properties of a protostellar disk around a primordial star and argue that although the disk is susceptible to fragmentation but the clump migration time is shorter than the KH timescale and therefore clumps may be able to migrate inwards. In fact, 3D radiation hydrodynamical simulations by Hosokawa et al. (Reference Hosokawa, Hirano, Kuiper, Yorke, Omukai and Yoshida2015) show that clump migration leads to intermittent accretion (also see Vorobyov & Basu Reference Vorobyov and Basu2010; Vorobyov, DeSouza, & Basu Reference Vorobyov, DeSouza and Basu2013) and disk fragmentation does not halt the formation of massive stars. Therefore, stellar masses may range from a few tens to a few hundred solar.

The most recent simulations of Stacy et al. (Reference Stacy, Bromm and Lee2016) found that disk fragmentation leads to the formation of a stellar cluster with a top heavy IMF. In their simulations, the most massive star reaches 20 solar masses in 5 000 yr after its formation and some of the sinks get ejected from the disk before ionisation front breaks out. However, these simulations were evolved only for 5 000 yr after the formation of the first sink and therefore final masses of Pop III stars are still uncertain. In Figure 2, we show the stellar mass distribution from Hirano et al. (Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014) which gives an upper limit on the expected stellar masses as their calculations do not take into account the multiplicity of stars per halo. These results suggest that the typical mass of Pop III stars is about 100 M_⊙ with the exception of a few cases of a 1 000 M_⊙.

Figure 2.

The stellar mass distribution of 110 first stars assuming that single star forms in each minihalo. Each colour represents different stellar evolution path, see Hirano et al. (Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014) for details. Adopted from Hirano et al. (Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014).

3.2. Effects of metallicity and rotation

Both the metallicity and the rotation speed of stars play a vital role in defining their fates and have important implications for the formation of stellar mass BHs. In the presence of trace amounts of metals, dust cooling becomes important at high densities, and triggers the formation of multiple low mass stars (Ferrara, Pettini, & Shchekinov Reference Ferrara, Pettini and Shchekinov2000; Schneider et al. Reference Schneider, Ferrara, Salvaterra, Omukai and Bromm2003; Omukai et al. Reference Omukai, Tsuribe, Schneider and Ferrara2005; Schneider et al. Reference Schneider, Salvaterra, Ferrara and Ciardi2006). Numerical simulations show that for Z/Z_⊙ ⩾ 10⁻⁵ dust cooling induces fragmentation and fosters low mass star formation (Dopcke et al. Reference Dopcke, Glover, Clark and Klessen2011, Reference Dopcke, Glover, Clark and Klessen2013; Smith et al. Reference Smith, Wise, O’Shea, Norman and Khochfar2015). Similarly, the dark matter halos with higher spin have a longer collapse timescale which results in enhanced fragmentation. Moreover, higher rotation decreases the mass accretion onto a protostar and consequently the final stellar mass gets reduced (Hirano et al. Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014; Dutta Reference Dutta2016).

The presence of metals and rotation does not only influence the formation of the first/second generation of stars by reducing their final masses but also strongly affects their evolution. Stellar evolution models show that mass loss from the stellar winds is metallicity dependent, scales with $\dot{m}_{{\rm loss}} \propto {\it Z}^{0.5}$ and consequently metal rich stars show much higher mass loss compared to the metal poor stars (Nugis & Lamers Reference Nugis and Lamers2000; Kudritzki & Puls Reference Kudritzki and Puls2000; Baraffe, Heger, & Woosley Reference Baraffe, Heger and Woosley2001; Heger et al. Reference Heger, Fryer, Woosley, Langer and Hartmann2003; Meynet & Maeder Reference Meynet and Maeder2005). Similarly, fast rotation enhances the surface enrichment of CNO cycle elements which in turn derive the mass loss by stellar winds from metal poor stars. For example, a fast rotating 60 M_⊙ star can lose 30–55% of its initial mass for Z/Z_⊙ = 10⁻⁸-10⁻⁵ (Heger, Langer, & Woosley Reference Heger, Langer and Woosley2000; Meynet & Maeder Reference Meynet and Maeder2000; Meynet, Ekström, & Maeder Reference Meynet, Ekström and Maeder2006b; Meynet et al. Reference Meynet, Hirschi, Ekström, Maeder, Lamers, Langer, Nugis and Annuk2006a; Chiappini et al. Reference Chiappini, Frischknecht, Meynet, Hirschi, Barbuy, Pignatari, Decressin and Maeder2011). Furthermore, an enhanced rotation modifies the Eddington limit known as the ΩΓ limit (Langer Reference Langer1998; Meynet & Maeder Reference Meynet and Maeder2000) and prevents the star from growing beyond 20–40 M_⊙ by making it more compact (Lee & Yoon Reference Lee and Yoon2016). Such rapidly rotating low metallicity stars may also produce gamma-ray bursts (Yoon & Langer Reference Yoon and Langer2005).

3.3. Stellar tracks leading to BH formation

Stellar evolution calculations suggest that the growth of primordial stars is significantly different from the present-day counterparts. Larger mass accretion rates, about two-three orders of magnitude higher than normal stars, and the absence of heavier elements makes the evolution of Pop III stars significantly different from ordinary stars (Omukai & Palla Reference Omukai and Palla2001; Schaerer Reference Schaerer2002; Omukai & Palla Reference Omukai and Palla2003; Schaerer Reference Schaerer2003). Stellar evolution also strongly depends on the time evolution of the mass accretion rates. Deuterium burning and pp cycle do not generate enough energy to counteract the KH contraction. The hydrogen burning from the CN cycle significantly increases the stellar luminosity and a protostar quickly reaches the ZAMS. Therefore, more massive stars of the order of a hundred solar masses are expected to form from a zero metallicity gas whilst for Z/Z_⊙ ⩾ 10⁻² radiation pressure on dust grains becomes important and leads to the formation of low mass stars (Omukai & Palla Reference Omukai and Palla2003; Hosokawa et al. Reference Hosokawa, Yoshida, Omukai and Yorke2012b).

The mass loss from stars become significant even in the presence of trace amount of metals whilst for primordial stars such losses are minimal. Stellar evolutionary tracks leading to the formation of BHs are indicated in Figure 3. The primordial stars with masses between 40–140 M_⊙ and above 260 M_⊙ are expected to directly collapse into BHs of similar masses (Heger & Woosley Reference Heger and Woosley2002; Heger et al. Reference Heger, Fryer, Woosley, Langer and Hartmann2003). For a trace amount of metals, all stars above 40 M_⊙ directly collapse in a BH but with increasing metallicity the mass loss starts to increase. Moreover, the fraction of massive stars collapsing into a BH depends on the shape of the IMF and is about twice for top-heavy IMF compared to the Salpeter IMF (Heger et al. Reference Heger, Fryer, Woosley, Langer and Hartmann2003). Although numerical simulations suggest that some of the Pop III stars might be rotating at ⩾ 1 000 kms⁻¹, i.e. close to their break up limit (Stacy et al. Reference Stacy, Bromm and Loeb2011), stellar evolution calculations including rotation (Ekström et al. Reference Ekström, Meynet, Chiappini, Hirschi and Maeder2008) indicate that the mass loss from the fast rotating Pop III stars is still very low. In a nutshell, metal free stars with low rotation speeds are favoured for the formation of stellar mass BHs as they retain most of their mass until they collapse into a BH.

Figure 3.

The fate of single stars as a function of their initial mass and initial metallicity. The tracks for the formation of direct BHs from the stars are highlighted by the black colour whilst the white region in the bottom right indicates the range for a pair instability supernova. Adopted from Heger et al. (Reference Heger, Fryer, Woosley, Langer and Hartmann2003).

3.4. Expected properties of stellar mass BHs

The current numerical simulations indicate that the formation of massive primordial stars up to a 1 000 M_⊙ is possible with the characteristic mass scale between 10–100 M_⊙. This suggests that BH seeds of a few hundred solar masses can be formed at z = 20–30. In order to reach a few billion solar masses by z = 7, they must continuously grow at the Eddington limit. The three-dimensional numerical simulations studying the growth of stellar mass BHs show that feedback from a BH photo-evaporates the gas the in its hosting halo and consequently accretion onto a BH gets halted (Johnson & Bromm Reference Johnson and Bromm2007; Alvarez, Wise, & Abel Reference Alvarez, Wise and Abel2009). They further found that the BH accretion rate is about 10⁻¹⁰ M_⊙yr⁻¹, i.e. several orders of magnitude below the Eddington limit which makes their growth extremely difficult. Moreover, the progenitor star creates an HII region, evacuates the gas from a minihalo and accretion remains halted for about 10⁸ yr. Park et al. (Reference Park, Ricotti, Natarajan, Bogdanović and Wise2016) found that stellar mass black holes cannot coevally grow with budge via accretion until it reaches the critical mass of 10⁶ M_⊙. There is a growing consensus that the stellar mass BHs require various episodes of super Eddington accretion to reach a billion solar masses by z ⩾ 6 (Madau, Haardt, & Dotti Reference Madau, Haardt and Dotti2014; Volonteri, Silk, & Dubus Reference Volonteri, Silk and Dubus2015; Pacucci, Volonteri, & Ferrara Reference Pacucci, Volonteri and Ferrara2015; Inayoshi, Haiman, & Ostriker Reference Inayoshi, Haiman and Ostriker2016a).

4.1. Binarity of early star formation

Observations of contemporary star formation suggest that about 50% of the stars are born in binaries or multiple systems (Sana, Gosset, & Evans Reference Sana, Gosset and Evans2009; Sana, James, & Gosset Reference Sana, James and Gosset2011). In the context of primordial stars, recent numerical simulations show that the first generation of stars may also have formed in multiple systems due to the fragmentation of protostellar disk (Turk et al. Reference Turk, Abel and O’Shea2009; Stacy et al. Reference Stacy, Greif and Bromm2010; Clark et al. Reference Clark, Glover, Smith, Greif, Klessen and Bromm2011; Greif et al. Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013e; Latif & Schleicher Reference Latif and Schleicher2015a; Stacy et al. Reference Stacy, Bromm and Lee2016). Some of the clumps from disk fragmentation migrate inward and merge with the central protostar whilst the rest survive to form a multiple system (Greif et al. Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012; Stacy et al. Reference Stacy, Bromm and Lee2016). In fact, numerical simulations suggest that up to 35% of Pop III stars may have formed in binaries; thus, the probability for a PopIII star to have a companion is about 50% (Stacy et al. Reference Stacy, Greif and Bromm2010; Stacy & Bromm Reference Stacy and Bromm2013). This may also suggest the formation of X-ray binaries at earlier cosmic times (Mirabel et al. Reference Mirabel, Dijkstra, Laurent, Loeb and Pritchard2011; Ryu, Tanaka, & Perna Reference Ryu, Tanaka and Perna2016).

Simulations of the first galaxies show that supernovae from Pop III stars quickly enrich halos with metals where gas can be cooled down to the cosmic microwave background temperature (T _CMB) by dust and metal line cooling at z = 15 even in the presence of UV flux (Tornatore et al. Reference Tornatore, Ferrara and Schneider2007; Smith et al. Reference Smith, Turk, Sigurdsson, O’Shea and Norman2009; Greif et al. Reference Greif, Glover, Bromm and Klessen2010; Wise et al. Reference Wise, Turk, Norman and Abel2012; Whalen et al. Reference Whalen2013; Johnson, Dalla Vecchia, & Khochfar Reference Johnson, Dalla Vecchia and Khochfar2013a; Safranek-Shrader, Milosavljević, & Bromm Reference Safranek-Shrader, Milosavljević and Bromm2014a; Bovino et al. Reference Bovino, Grassi, Schleicher and Latif2014; Pallottini et al. Reference Pallottini, Ferrara, Gallerani, Salvadori and D’Odorico2014). Stellar clusters of Pop II stars are expected to form above the critical value of the metallicity, i.e. Z/Z_⊙ ~ 5 × 10⁻⁴(Schneider et al. Reference Schneider, Ferrara, Salvaterra, Omukai and Bromm2003; Omukai et al. Reference Omukai, Tsuribe, Schneider and Ferrara2005; Cazaux & Spaans Reference Cazaux and Spaans2009; Latif, Schleicher, & Spaans Reference Latif, Schleicher and Spaans2012). The recent three-dimensional cosmological simulations suggest that the first bona fide stellar clusters form at z = 15 where cooling is triggered by the metal lines (such as CII and OI) and stellar masses range from 0.1-10 M_⊙ (Safranek-Shrader, Milosavljević, & Bromm Reference Safranek-Shrader, Milosavljević and Bromm2014b; Safranek-Shrader et al. Reference Safranek-Shrader, Montgomery, Milosavljević and Bromm2016). Although current cosmological simulations are unable to constrain the binary fraction in the first stellar clusters due to the numerical constraints but the binary fraction is expected to be higher for metal poor stars (Komiya et al. Reference Komiya, Suda, Minaguchi, Shigeyama, Aoki and Fujimoto2007; Machida Reference Machida2008).

4.2. Compact stellar clusters

The compactness of a stellar cluster is the key property for the stellar dynamical processes to occur as it determines whether a seed BH can form or not. This requirement arises from the fact that the time for the core collapse should be shorter than the time for massive stars to go off supernova (~3 Myr) otherwise latter may halt the core collapse, also see Yajima & Khochfar (Reference Yajima and Khochfar2016). The more compact the cluster shorter the core collapse timescale. For a given cluster mass, the number of collisions and the increase in mass per collision strongly depend on the half-mass radius of a cluster (Portegies Zwart & McMillan Reference Portegies Zwart and McMillan2002). The mass of a VMS almost linearly increases with the initial central density of a cluster (Katz, Sijacki, & Haehnelt Reference Katz, Sijacki and Haehnelt2015).

In the context of BHs, the first compact nuclear cluster are expected to form in massive halos of 10⁸ M_⊙ at z > 10 with Z/Z_⊙ ~ 10⁻⁵-10⁻³. In the presence of a LW flux, the formation of H₂ at low densities remains suppressed and gas initially collapses isothermally to form a self-gravitating accretion disk at the centre of a halo (Devecchi & Volonteri Reference Devecchi and Volonteri2009). Fragmentation occurs only in the nucleus of the disk above densities of 10³ cm⁻³ due to the metal line cooling and forms a compact cluster of 10⁵ M_⊙ with half-mass radius of about 1 pc. On the other hand, for higher metallicities, fragmentation already occurs at densities < 10³ cm⁻³ and results in the core collapse time longer than 3 Myr (Omukai et al. Reference Omukai, Schneider and Haiman2008; Devecchi & Volonteri Reference Devecchi and Volonteri2009; Devecchi et al. Reference Devecchi, Volonteri, Rossi, Colpi and Portegies Zwart2012).

Semi-analytical models suggest that large accretion rates and efficient star formation are required for the formation of a such compact nuclear stellar and conditions for their formation are feasible in the first massive halos polluted by a trace amount of metals at z > 10 (Devecchi & Volonteri Reference Devecchi and Volonteri2009). Latif et al. (Reference Latif, Omukai, Habouzit, Schleicher and Volonteri2016) have performed 3D cosmological simulations to study the impact of trace amount of dust and metals in massive halos of 10⁸ M_⊙ and found that a dense cluster may also form for Z/Z_⊙ ⩾ 10⁻⁴ in the presence of a strong UV flux. The study of first nuclear clusters in still in infancy as their formation cannot be resolved in numerical simulations due to the enormous range of spatial scales. In future, more work is required to asses how compact and massive the clusters can be formed at high redshift.

4.3. Dynamical evolution

The dynamical evolution of dense stellar clusters has been studied via direct N-body simulations which show that the mergers between stars destroy binaries and avoid three body binary heating (Portegies Zwart et al. Reference Portegies Zwart, Makino, McMillan and Hut1999). Moreover, in contrary to the previous studies, stellar collisions are fostered by the dynamically formed binaries which increase the star collision rate (Portegies Zwart et al. Reference Portegies Zwart, Makino, McMillan and Hut1999; Portegies Zwart & McMillan Reference Portegies Zwart and McMillan2002). Consequently, stellar collisions can occur even in a low mass system with 12 000 stars and may lead to the formation of a BH of up to ~ 1 000 M_⊙. This was further confirmed from the Monte Carlo simulations by Gürkan, Freitag, & Rasio (Reference Gürkan, Freitag and Rasio2004) where they explored a range of IMFs and cluster structural parameters. Their followed-up work including stellar dynamics showed that core collapse occurred on ~ 3 Myr timescale irrespective of the cluster size, mass, concentration, and a VMS reached up to 1 000 M_⊙ (Freitag, Gürkan, & Rasio Reference Freitag, Gürkan and Rasio2006). Moreover, the mass of a VMS is about 10⁻³ times the cluster mass and is mainly contributed by the stars in the mass range of 60–120 M_⊙ (Goswami et al. Reference Goswami, Umbreit, Bierbaum and Rasio2012).

Metallicity also plays an important role in the dynamical evolution of a stellar cluster as mass loss from stars can deeply affect the core collapse. Schulman, Glebbeek, & Sills (Reference Schulman, Glebbeek and Sills2012) and Downing (Reference Downing2012) found that the size of cluster depends on the metallicity and metal rich clusters expand more rapidly. For more massive clusters, Sippel et al. (Reference Sippel, Hurley, Madrid and Harris2012) found that both metal poor and metal rich clusters are structurally similar and no significant differences in their half-mass radii were observed. The effect of metallicity seems to be more important in intermediate mass young clusters and leads to the differences in the core radius as well as in the half-mass radius (Mapelli & Bressan Reference Mapelli and Bressan2013). These differences arise from an interplay between the mass loss from stellar winds, dynamical heating, and three-body interactions. Therefore, low metallicity young clusters forming at z > 10 are preferred for the formation of a massive seed black hole.

4.4. Expected properties of seed BHs

The BH mass resulting from the core collapse of a dense stellar cluster depends on its initial stellar mass, initial stellar density, compactness, cluster geometry, fraction of primordial binaries, and the IMF of a cluster. The N-body simulations exploring the range of above mentioned parameters show that clusters with different geometries, fractal distributions, and density profiles converge within 1 Myr and do not significantly influence the mass of a VMS. However, the final mass of a VMS strongly depends on the compactness for a given cluster mass and almost linearly increases with initial central density. Recently, Katz et al. (Reference Katz, Sijacki and Haehnelt2015) used a combination of hydrodynamical cosmological simulations and direct N-body simulations to estimate the mass of a VMS forming via stellar run-away collisions in a metal poor stellar cluster. They found that a VMS of ⩾ 400 M_⊙ can be formed in a stellar cluster of 10⁴ M_⊙ at z = 15 and may reach up to a 1 000 M_⊙ for a cluster mass of a few times 10⁵ M_⊙.

The IMF of a stellar cluster influences the number of collisions between stars and the mass of a VMS tends to increase for a top heavy IMF but the probability of producing a VMS remains the same. Similarly, the introduction of primordial binaries and the initial mass segregation significantly change the evolution of a cluster but the mass of a VMS differs only by a factor of 3 which is less significant than the changes in the central density and the mass of a cluster (Katz et al. Reference Katz, Sijacki and Haehnelt2015). The mass of a seed black hole forming from the core collapse of a cluster can be estimated as follows (Portegies Zwart & McMillan Reference Portegies Zwart and McMillan2002):

(1) $$\begin{equation} M_{{\rm BH}} = m_{*} + 4 \times 10^{-3} f_{{\rm c}} M_{{\rm c}0} \gamma \ln \Lambda _{\rm C}. \end{equation}$$

Here, m _* is the mass of a massive star in the cluster, f _c is the fraction of dynamically formed binaries, M _c0 is the birth mass of the cluster, ln Λ_c is the Coulomb logarithm, and γ ~ 1, is the ratio of timescales, see Portegies Zwart & McMillan (Reference Portegies Zwart and McMillan2002) for details. For a M _c0 = 10⁴ M_⊙, m _* = 100 M_⊙, f _c = 0.2, ln Λ_C = 10, M _BH ~ 180 M_⊙ whilst for M _c0 = 10⁵ M_⊙, the expected black hole mass is ~ 900 M_⊙.

Devecchi et al. (Reference Devecchi, Volonteri, Colpi and Haardt2010, Reference Devecchi, Volonteri, Rossi, Colpi and Portegies Zwart2012) employed an analytical model to estimate the expected mass range of seed BHs forming via dynamical processes in nuclear stellar clusters at z ⩾ 10. Their estimates for BH masses are shown in Figure 4 and indicate that BHs of up to a 1 000 M_⊙ can be formed. Simulations self-consistently modelling the formation and evolution of a nuclear stellar cluster are necessary to better understand the dynamics of the first dense nuclear clusters.

Figure 4.

The mass function of BHs formed via stellar dynamical process in the first nuclear cluster at z ~ 15. Adopted from Devecchi et al. (Reference Devecchi, Volonteri, Rossi, Colpi and Portegies Zwart2012).

5.1. Birthplaces of DCBHs

The potential embryos for the formation of DCBHs via isothermal collapse are the first massive metal-free halos with T _vir ⩾ 10⁴ K and masses of > 10⁷ M_⊙ at z ~ 15. They have sufficient gas reservoir to feed a massive object and their potential wells are deep enough to foster a rapid collapse required for the formation of a DCBH. Moreover, their virial temperature is high enough for the atomic line cooling to operate. The formation of a DCBH mandates that they should be of a primordial composition and the formation of H₂ remains suppressed (Shang et al. Reference Shang, Bryan and Haiman2010; Petri, Ferrara, & Salvaterra Reference Petri, Ferrara and Salvaterra2012; Latif et al. Reference Latif, Bovino, Van Borm, Grassi, Schleicher and Spaans2014c; Yue et al. Reference Yue, Ferrara, Salvaterra, Xu and Chen2014). In the absence of H₂ cooling, collapse proceeds isothermally with T ~ 8000 K and warm gas flows towards the centre at the rate of $\rm \dot{\it m} \sim {c_s^3}/{G} \sim 0.1\;M_{\odot } yr^{-1} \left( {\it T}/{8\,000\;K}\right)^{3/2}$ , where c _s is the thermal sound speed.

The suppression of molecular hydrogen requires the presence of a strong LW flux (Omukai Reference Omukai2001; Omukai et al. Reference Omukai, Schneider and Haiman2008; Shang et al. Reference Shang, Bryan and Haiman2010; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013a). As we discuss in the next subsections that constraint of a strong LW fluxes requires that DCBH hosting halo should form in the vicinity (about few kpc) of a massive star forming galaxy (Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Habouzit et al. Reference Habouzit2016b). In mean time, it has also to avoid the metal pollution for the above mentioned reasons. Under these conditions, an isothermal monolithic collapse is expected to occur which later may lead to the formation of a DCBH.

5.2. UV flux constraints

One of the main constraints for the formation of DCBHs via isothermal direct collapse is the suppression of molecular hydrogen formation which requires the presence of a strong LW flux (Omukai Reference Omukai2001; Omukai et al. Reference Omukai, Schneider and Haiman2008; Shang et al. Reference Shang, Bryan and Haiman2010; Petri et al. Reference Petri, Ferrara and Salvaterra2012; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013a). The gas phase reactions in primordial gas can lead to the formation of a trace amount of H₂. The main pathway for H₂ formation is

(2) $$\begin{equation} \mathrm{H + e^{-} \rightarrow H^{-} +} \gamma, \end{equation}$$

(3) $$\begin{equation} \mathrm{ H + H^{-} \rightarrow H_{2} + e^{-}.} \end{equation}$$

The formation of H₂ can be suppressed either by directly dissociating H₂ or indirectly via the photo-detachment of H⁻. The destruction channel for H₂ depends on the stellar spectra. The photons with energy between 11.2–13.6 eV can be absorbed in the LW bands of H₂ and photo-dissociate it shortly after putting it into an excited state, this is known as the Solomon process. Whilst the low energy photons with energy above 0.76 eV can photo-detach H⁻. The reactions for the both processes are the following:

(4) $$\begin{equation} \mathrm{H_{2}} + \gamma _{{\rm LW}} \mathrm{\rightarrow H + H}, \end{equation}$$

(5) $$\begin{equation} \mathrm{H^{-}} + \gamma _{0.76} \mathrm{\rightarrow H + e^{-}}. \end{equation}$$

The stars with hard spectrum of T _rad = 10⁵ K are more efficient for the direct dissociation of H₂ whilst the stars characterised with T _rad = 10⁴ K are more effective in the photo-detachment of H⁻. The complete quenching of H₂ requires a critical value of UV flux (J ^crit ₂₁) which depends on the shape of a radiation spectrum. The previous studies used idealised spectra to compute the J ^crit ₂₁ and found that J ^crit ₂₁ = 30–1 200 for T _rad = 10⁴ K and J ^crit ₂₁ = 1 000 for T _rad = 10⁵ K (Omukai Reference Omukai2001; Shang et al. Reference Shang, Bryan and Haiman2010; Van Borm & Spaans Reference Van Borm and Spaans2013; Latif et al. Reference Latif, Bovino, Van Borm, Grassi, Schleicher and Spaans2014c; Johnson et al. Reference Johnson, Whalen, Agarwal, Paardekooper and Khochfar2014). It has been found that the strength of J ^crit ₂₁ is about an order of magnitude above the background UV flux and can only be achieved in the close vicinity of a star forming galaxy (Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Habouzit et al. Reference Habouzit2016b). Moreover, such flux is mainly provided by Pop II stars due to their long lives and higher abundance. Recently, realistic spectra of the first galaxies was computed using the stellar synthesis code STARBURST (Leitherer et al. Reference Leitherer1999) and it was found that J ^crit ₂₁ depends on the mode of a star formation (either bursty or constant) as well as on the age and metallicity of stars (Sugimura et al. Reference Sugimura, Omukai and Inoue2014; Agarwal & Khochfar Reference Agarwal and Khochfar2015; Agarwal et al. Reference Agarwal, Smith, Glover, Natarajan and Khochfar2016a). Sugimura et al. (Reference Sugimura, Omukai and Inoue2014) show that such spectra can be mimicked with T _rad= 2 × 10⁴ − 10⁵ K.

Recent estimates of J ^crit ₂₁ from three-dimensional cosmological simulations (Latif et al. Reference Latif, Bovino, Grassi, Schleicher and Spaans2015)Footnote ² for a realistic Pop II spectra vary between 20 000–50 000 considering a uniform isotropic background UV flux and are shown in Figure 5. Similar values of J ^crit ₂₁ have been obtained for anisotropic source (Regan, Johansson, & Wise Reference Regan, Johansson and Wise2014b, Reference Regan, Johansson and Wise2016a). Moreover, the presence of X-rays may further influence the J ^crit ₂₁ depending on its strength (Latif et al. Reference Latif, Bovino, Grassi, Schleicher and Spaans2015; Inayoshi & Tanaka Reference Inayoshi and Tanaka2015; Regan, Johansson, & Wise Reference Regan, Johansson and Wise2016b) and also an accurate modelling of H₂ self-shielding is required (Wolcott-Green, Haiman, & Bryan Reference Wolcott-Green, Haiman and Bryan2011; Hartwig et al. Reference Hartwig, Glover, Klessen, Latif and Volonteri2015).

Figure 5.

The estimates of critical value of UV flux (J ^crit ₂₁) both from one zone models and three-dimensional simulations including variations from halo to halo, dependence on the radiation spectra, and the impact of X-ray ionisation. Adopted from Latif et al. (Reference Latif, Bovino, Grassi, Schleicher and Spaans2015).

5.3. Metal pollution

One of the key requirements for the formation of DCBHs via isothermal collapse is that the hosting halo should be metal free. The metal enrichment is expected to be patchy in the early universe and therefore some halos may remain unpolluted even down to z = 6 (Tornatore et al. Reference Tornatore, Ferrara and Schneider2007; Trenti et al. Reference Trenti, Stiavelli and Michael Shull2009; Maio et al. Reference Maio, Khochfar, Johnson and Ciardi2011; Ritter et al. Reference Ritter, Sluder, Safranek-Shrader, Milosavljević and Bromm2015; Pallottini et al. Reference Pallottini, Ferrara, Gallerani, Salvadori and D’Odorico2014; Habouzit et al. Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016a). In fact, observations show that pockets of extremely metal poor gas (Z/Z_⊙ ~ 10⁻⁵) can exist down to z = 7 (Simcoe et al. Reference Simcoe, Sullivan, Cooksey, Kao, Matejek and Burgasser2012). So, it is conceivable that primordial halos where star formation is suppressed in the presence of LW background may stay metal free at z ⩾ 10.

However, the DCBH host halo is expected to form in the surrounding of a star-forming galaxy to receive a strong LW flux and therefore has to avoid the possible pollution by the supernova winds. Dijkstra, Ferrara, & Mesinger (Reference Dijkstra, Ferrara and Mesinger2014) employed an analytical model to study the impact of metal pollution by the supernova winds and found that it can significantly affect the expected abundance of DCBHs. Visbal, Haiman, & Bryan (Reference Visbal, Haiman and Bryan2014b) propose that metal pollution can also be avoided in a synchronised pair of halos. In such a case, star-forming halo forms first whilst the DCBH host halo forms later and rapidly collapses before it gets enriched by the supernova winds. Moreover, such pair of halos is expected to be in a clustered environment where metals might be ejected in a preferential direction with low density and metal pollution may be avoided (Ritter et al. Reference Ritter, Sluder, Safranek-Shrader, Milosavljević and Bromm2015; Pallottini et al. Reference Pallottini, Ferrara, Gallerani, Salvadori and D’Odorico2014). It has been recently found that some of the DCBH host halos may get tidally disrupted in such a scenario (Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016).

Cosmological hydrodynamical simulations including both star formation and supernova feedback show that a significant fraction of halos remains metal free down to z = 10 with mass range between 2 × 10⁷ − 10⁸ M_⊙ (Latif et al. Reference Latif, Omukai, Habouzit, Schleicher and Volonteri2016; Habouzit et al. Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016a), see Figure 6. These results provide an upper limit on the fraction of metal free halos in the above mentioned range as simulations are unable to resolve halos below 10⁷ M_⊙. In future, cosmological hydrodynamical simulations self-consistently taking into account the metal pollution as well as radiations from the star forming galaxies are required to better understand the metal pollution in the DCBH host halos.

Figure 6.

Fraction of halos with metallicity below the given value in the figure legged and masses between 2 × 10⁷-10⁸ M_⊙. Adopted from Latif et al. (Reference Latif, Omukai, Habouzit, Schleicher and Volonteri2016).

5.4. Formation of a supermassive/quasi star

SMSsFootnote ³ are considered as potential cradles for the formation of DCBHs (Begelman Reference Begelman2010; Volonteri & Begelman Reference Volonteri and Begelman2010; Ball et al. Reference Ball, Tout, Żytkow and Eldridge2011; Hosokawa, Omukai, & Yorke Reference Hosokawa, Omukai and Yorke2012a; Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Schleicher et al. Reference Schleicher, Palla, Ferrara, Galli and Latif2013; Johnson et al. Reference Johnson, Whalen, Li and Holz2013b; Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014). They may collapse via GR instabilities into a DCBH whilst retaining ⩾ 50% of their initial mass (Baumgarte & Shapiro Reference Baumgarte and Shapiro1999; Shibata & Shapiro Reference Shibata and Shapiro2002; Montero, Janka, & Müller Reference Montero, Janka and Müller2012; Reisswig et al. Reference Reisswig, Ott, Abdikamalov, Haas, Mösta and Schnetter2013) or may evolve towards a ZAMS and later collapse into a massive BH (Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014). The stellar evolution calculations suggest that the formation of such objects requires rapid accretion with $\dot{\it m} \ge 0.1\; {\rm M}_{\odot } {\rm yr}^{-1}$ (Begelman Reference Begelman2010; Ball et al. Reference Ball, Tout, Żytkow and Eldridge2011; Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Schleicher et al. Reference Schleicher, Palla, Ferrara, Galli and Latif2013). For such high accretion rates, the radius of star monotonically increases with mass due to the shorter accretion time in comparison with the KH contraction timescale (Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Sakurai et al. Reference Sakurai, Hosokawa, Yoshida and Yorke2015). Consequently, their surface temperatures remain as low as 5 000 K, they produce weak UV stellar feedback and can grow up to $\rm 3.6 \times 10^8 \; \dot{\it m} \;M_{\odot }$ . The energy released by the nuclear burning remains subdominant compared to the energy produced by the stellar contraction.

Schleicher et al. (Reference Schleicher, Palla, Ferrara, Galli and Latif2013) found that for $\rm \dot{\it m} \ge 0.14\; M_{\odot } yr^{-1}$ the core of a SMS collapses into a BH and forms a so-called quasi star, also see Begelman, Rossi, & Armitage (Reference Begelman, Rossi and Armitage2008). It comes from the fact that accretion timescale is considerably shorter than the nuclear burning timescale and therefore the core collapses into a black hole (Begelman et al. Reference Begelman, Volonteri and Rees2006; Begelman Reference Begelman2010). A possible advantage of such composition is that a central BH may accrete at the super-Eddington rate but overall accretion is limited by the Eddington rate of a quasi star. Ball et al. (Reference Ball, Tout, Żytkow and Eldridge2011) have found that about 10% of the stellar mass is accreted onto the BH before the hydrostatic equilibrium breaks down and results are sensitive to the choice of boundary conditions. Dotan, Rossi, & Shaviv (Reference Dotan, Rossi and Shaviv2011) quantified the potential impact of radiation-driven winds from the envelope of a quasi star and found that winds can be so strong that they may blow away the envelope before the BH doubles its mass. The effect of winds from the radiation dominated objects was reconsidered by Fiacconi & Rossi (Reference Fiacconi and Rossi2016b) by solving the equations of motion and including the previously neglected advection energy term. They found that the super Eddington accretion onto the newly born BH within quasi stars is likely responsible for vigorous mass loss and limits the BH growth. In the follow-up study, Fiacconi & Rossi (Reference Fiacconi and Rossi2016a) explored the impact of rotation in quasi-star scenario via simplified analytical model and found that for ⩾ 10⁵ M_⊙ massive SMSs the quasi-star phase may be skipped whilst the growth of less massive objects may get prevented. However, 3D radiation hydrodynamics simulations are necessary to validate these findings.

The numerical experiments employing a Jeans resolution of four cells did not observe any fragmentation (Bromm & Loeb Reference Bromm and Loeb2003; Wise et al. Reference Wise, Turk and Abel2008; Regan & Haehnelt Reference Regan and Haehnelt2009; Prieto, Jimenez, & Haiman Reference Prieto, Jimenez and Haiman2013; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013a) whilst the simulations employing both higher Jeans (64 cells per Jeans length) as well as spatial resolution show that fragmentation occasionally occurs (Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013c, Reference Latif, Schleicher, Schmidt and Niemeyer2013d; Regan, Johansson, & Haehnelt Reference Regan, Johansson and Haehnelt2014a; Becerra et al. Reference Becerra, Greif, Springel and Hernquist2015). They further found that a self-gravitating accretion disk forms in the centre of the halo (see Figure 7) which becomes marginally unstable and fragments into multiple clumps but most of the clumps get merged with the central clump. Large accretion rates of 0.1 − 1 M_⊙yr⁻¹ are observed in these simulations and seem to be sufficient for forming a massive central object. Moreover, subgrid scale turbulence and strong magnetic fields amplified via the small scale dynamo help in suppressing fragmentation, see Latif et al. (Reference Latif, Schleicher, Schmidt and Niemeyer2013c, Reference Latif, Schleicher, Schmidt and Niemeyer2013b), Latif, Schleicher, & Schmidt (Reference Latif, Schleicher and Schmidt2014a). These simulations did not include H⁻ cooling which becomes important at densities of 10⁸–10¹⁶ cm⁻³ and also cooling was artificially switched off to mimic the formation of a protostar. Both idealised (Van Borm et al. Reference Van Borm, Bovino, Latif, Schleicher, Spaans and Grassi2014; Inayoshi, Omukai, & Tasker Reference Inayoshi, Omukai and Tasker2014) and cosmological (Latif, Schleicher, & Hartwig Reference Latif, Schleicher and Hartwig2016) simulations employing a detailed chemical model have been performed. Particularly, they included the H⁻ cooling and opacities for H⁻ bound-free emission, free–free absorption as well as the Raleigh scattering of hydrogen atoms. These simulations show that H⁻ cooling does not halt the formation of a SMS and may lead to a binary formation in some cases but realistic opacities help in stabilising the collapse on small scales (Latif, Schleicher, & Hartwig Reference Latif, Schleicher and Hartwig2016).

Figure 7.

Self-gravitating accretion disks formed at the centre of a massive primordial halos illuminated by a strong LW flux as a consequence of isothermal collapse. Each panel represents a halo of above 10⁷ M_⊙ forming at z = 10–15 and shows the density projection in the central 300 AU. Adopted from Latif et al. (Reference Latif, Schleicher, Schmidt and Niemeyer2013c).

Cosmological simulations using sink particles (also without sinks) confirm the formation of a SMS of 10⁵ M_⊙ within a 1 Myr after its birth (Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013d; Shlosman et al. Reference Shlosman, Choi, Begelman and Nagamine2016). Moreover, stellar evolution simulations employing both constant (⩾ 0.1 M_⊙yr⁻¹) and time-dependent mass accretion rates show that a SMS can be formed and resulting UV feedback is too weak to hinder the mass accretion (Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Sakurai et al. Reference Sakurai, Hosokawa, Yoshida and Yorke2015, Reference Sakurai, Vorobyov, Hosokawa, Yoshida, Omukai and Yorke2016). The present studies suggest the formation of a SMS as a potential outcome (Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Sakurai et al. Reference Sakurai, Vorobyov, Hosokawa, Yoshida, Omukai and Yorke2016) but do not consider the effect of rotation. However, it is not yet clear if the final outcome of an isothermal collapse is always a SMS or a quasi star or even a massive BH. In future three-dimensional cosmological hydrodynamical simulations including the UV feedback from a SMS and coupled stellar evolution will be required to asses it.

5.5. Birth mass function

The studies investigating the formation of a SMS could not evolve their simulations long enough to assess its final fate due to the numerical constraints (Hosokawa et al. Reference Hosokawa, Yoshida, Omukai and Yorke2012b, Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013). So, it is not clear whether a SMS directly collapses into a BH via GR instabilities provided that accretion continues with ⩾ 0.1 M_⊙yr⁻¹ or it evolves to a ZAMS (if the mass accretion onto the SMS stops) and later collapses into a massive BH. It is also not well understood what is the birth mass function of DCBHs. Moreover, previous works ignored the effect of rotation on the evolution of a SMS which can play important role in determining its final fate. Ferrara et al. (Reference Ferrara, Salvadori, Yue and Schleicher2014) used an analytical model to investigate the fate of an accreting SMS and found that it becomes GR unstable when its equation of state drops below a critical value (Chandrasekhar Reference Chandrasekhar1964; Montero et al. Reference Montero, Janka and Müller2012) whilst the presence of rotation halts the collapse. The mass of a SMS can be estimated as (Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014)

(6) $$\begin{equation} {M_{*}} \le 8.48 \times 10^{5} \left( \frac{\dot{m}}{{\rm M}_{\odot }\, {\rm yr}^{-1}} \right)^{2/3} No \;Rotation \end{equation}$$

(7) $$\begin{equation} {M_{*}} \le 6.01 \times 10^{5} \left( \frac{\dot{m}}{{\rm M}_{\odot }\, {\rm yr}^{-1}} \right) Rotation \end{equation}$$

SMSs more massive than these limits are expected to directly collapse into a BH.

Ferrara et al. (Reference Ferrara, Salvadori, Yue and Schleicher2014) used merger tree simulations to compute the birth mass function of DCBHs by following the growth of a SMS. Depending on the mass accretion onto the proto-SMS they determined whether it forms a DCBH via GR instabilities or a protostar contracts and evolves into a ZAMS SMS. The mass distribution of DCBHs is shown in Figure 8 and depicts that typical masses of DCBHs are ~ 2 × 10⁵ M_⊙ whilst the masses of SMSs vary from 2 × 10⁴-10⁵ M_⊙. The final mass further depends on whether the merged minihalo/accreted gas was metal free (dubbed as the sterile case) or metal enriched (fertile case). Overall, the masses of DCBHs are in the range of 10^{4 − 5} M_⊙.

Figure 8.

The mass distribution of DCBH seeds (dotted histogram) and SMS (yellow histogram). The upper panel shows the case of fertile mini halos whilst the bottom panel sterile mini halos. The results are computed from merger tree simulations and averaged over 50 milky-way merger histories with ± σ error bars. Adopted from Ferrara et al. (Reference Ferrara, Salvadori, Yue and Schleicher2014).