2.1. How and when did the z > 6 SMBHs form and the nature of their progenitors

The formation mechanism and properties of the first seed BHs are the subject of several studies which focus on three distinct scenarios (see e.g. Volonteri et al. Reference Volonteri2010; Natarajan Reference Natarajan2011; Latif & Ferrara Reference Latif and Ferrara2016 for complete reviews).

The first scenario relies on low-mass seeds, namely BHs of few tens to few hudreds solar masses, formed as remnants of Population III (Pop III) stars in the mass range [40–140] and >260 M_⊙ (e.g. Madau & Rees Reference Madau and Rees2001; Abel, Bryan, & Norman Reference Abel, Bryan and Norman2002; Heger et al. Reference Heger, Fryer, Woosley, Langer and Hartmann2003; Volonteri, Haardt, & Madau Reference Volonteri, Haardt and Madau2003; Yoshida, Omukai, & Hernquist Reference Yoshida, Omukai and Hernquist2008; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013b) up to ~1 000 M_⊙ stars that may form at z > 20 (Hirano et al. Reference Hirano, Hosokawa, Yoshida, Omukai and Yorke2015).

On the other hand, intermediate mass, 10³–10⁴ M_⊙, BHs may arise from from stars and stellar-mass BHs collisions in dense clusters (e.g. Omukai, Schneider, & Haiman Reference Omukai, Schneider and Haiman2008; Devecchi & Volonteri Reference Devecchi and Volonteri2009; Regan & Haehnelt Reference Regan and Haehnelt2009b; Katz, Sijacki, & Haehnelt Reference Katz, Sijacki and Haehnelt2015; Davies, Miller, & Bellovary Reference Davies, Miller and Bellovary2011; Lupi et al. Reference Lupi, Colpi, Devecchi, Galanti and Volonteri2014; Yajima & Khochfar Reference Yajima and Khochfar2016).

Finally, a third SMBH formation channel has been proposed: high-mass seeds, forming in T _vir ⩾ 10⁴K halos, exposed to an intense H₂ photo-dissociating ultra-violet (UV) flux (but see e.g. Spaans & Silk Reference Spaans and Silk2006, for a different scenario), via direct collapse (DC) of low metallicity gas clouds into 10⁴–10⁶ M_⊙ BHs. Such a scenario has been explored in details by means of both analytic works (e.g. Loeb & Rasio Reference Loeb and Rasio1994; Bromm & Loeb Reference Bromm and Loeb2003; Eisenstein & Loeb Reference Eisenstein and Loeb1995; Volonteri & Rees Reference Volonteri and Rees2005a; Begelman, Volonteri, & Rees Reference Begelman, Volonteri and Rees2006; Lodato & Natarajan Reference Lodato and Natarajan2006; Spaans & Silk Reference Spaans and Silk2006; Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014) and simulations (e.g. Wise, Turk, & Abel Reference Wise, Turk and Abel2008; Regan & Haehnelt Reference Regan and Haehnelt2009a, Reference Regan and Haehnelt2009b; Shang, Bryan, & Haiman Reference Shang, Bryan and Haiman2010; Inayoshi & Omukai Reference Inayoshi and Omukai2012; Regan, Johansson, & Wise Reference Regan, Johansson and Wise2014; Inayoshi, Omukai, & Tasker Reference Inayoshi, Omukai and Tasker2014; Becerra et al. Reference Becerra, Greif, Springel and Hernquist2015).

Another debated issue is related to the seed BH growth mechanism that is needed in order to explain z > 6 SMBHs.

Alvarez, Wise, & Abel (Reference Alvarez, Wise and Abel2009) pointed out that Pop III star remnants forming in minihalos at z > 15 do not grow efficiently in mass to become miniquasars (BHs with mass ~10⁶ M_⊙). However, after merging with atomic cooling halos (i.e. halos with virial temperatures of ⩾10⁴ K), the BH feedback may be able to inhibit star formation, thus leading to efficient accretion and growth of the BH.

In addition, if Pop III stars are less massive than expected, i.e. not exceeding 100 M_⊙ (e.g. O’Shea & Norman Reference O’Shea and Norman2007; Hosokawa et al. Reference Hosokawa, Omukai, Yoshida and Yorke2011; Greif et al. Reference Greif, Springel, White, Glover, Clark, Smith, Klessen and Bromm2011; Stacy, Greif, & Bromm Reference Stacy, Greif and Bromm2012; Hirano et al. Reference Hirano, Hosokawa, Yoshida, Omukai and Yorke2015), the resulting BHs of ~20–60 M_⊙ may receive a kick during their formation, ejecting them out of their host halos and thus preventing their subsequent growth (Whalen & Fryer Reference Whalen and Fryer2012). Moreover, because of their low mass, such BHs are not expected to settle in the galaxy centre. They would rather wander in the halo, without accreting gas (see e.g. Volonteri et al. Reference Volonteri2010, for a discussion).

Various studies suggest that BHs may evolve via uninterrupted gas accretion at the Eddington rate and/or episodic super-Eddington accretion phases, to grow up to billion solar masses, especially in the case of low-mass seeds (Haiman Reference Haiman2004; Yoo & Miralda-Escudé Reference Yoo and Miralda-Escudé2004; Shapiro Reference Shapiro2005; Volonteri & Rees Reference Volonteri and Rees2005a, Reference Volonteri and Rees2006a; Pelupessy, Di Matteo, & Ciardi Reference Pelupessy, Di Matteo and Ciardi2007; Tanaka & Haiman Reference Tanaka and Haiman2009; Johnson et al. Reference Johnson, Whalen, Li and Holz2013; Madau, Haardt, & Dotti Reference Madau, Haardt and Dotti2014; Volonteri, Silk, & Dubus Reference Volonteri, Silk and Dubus2015).

We refer the interested reader to reviews by Volonteri et al. (Reference Volonteri2010), Natarajan (Reference Natarajan2011), Volonteri & Bellovary (Reference Volonteri and Bellovary2012), Volonteri et al. (Reference Volonteri, Bogdanović, Dotti and Colpi2016a), Latif & Ferrara (Reference Latif and Ferrara2016), and Johnson & Haardt (Reference Johnson and Haardt2016), and references therein for details on the first seed BHs formation and feeding mechanisms.

The seeds of the first SMBHs are still elusive even to the most sensitive instruments that exist today, thus preventing us from putting observational constraints on their nature. A good example is the bright Lyα emitter CR7 observed at z ~ 6.6 (Matthee et al. Reference Matthee, Sobral, Santos, Röttgering, Darvish and Mobasher2015; Sobral et al. Reference Sobral, Matthee, Darvish, Schaerer, Mobasher, Röttgering, Santos and Hemmati2015; Bowler et al. Reference Bowler, McLure, Dunlop, McLeod, Stanway, Eldridge and Jarvis2016) where either Pop III stars (Sobral et al. Reference Sobral, Matthee, Darvish, Schaerer, Mobasher, Röttgering, Santos and Hemmati2015; Visbal, Haiman, & Bryan Reference Visbal, Haiman and Bryan2016; Dijkstra et al. Reference Dijkstra, Gronke and Sobral2016) or an accreting DCBH (Pallottini et al. Reference Pallottini2015; Hartwig et al. Reference Hartwig2016; Agarwal et al. Reference Agarwal, Johnson, Zackrisson, Labbe, van den Bosch, Natarajan and Khochfar2016b; Smith, Bromm, & Loeb Reference Smith, Bromm and Loeb2016; Smidt, Wiggins, & Johnson Reference Smidt, Wiggins and Johnson2016; Agarwal et al. Reference Agarwal, Johnson, Khochfar, Pellegrini, Rydberg, Klessen and Oesch2017a; Pacucci et al. Reference Pacucci, Pallottini, Ferrara and Gallerani2017) has been suggested as the primary constituent of its metal-poor component.

Although the observational signatures of seed BHs still remain unexplored, Pacucci et al. (Reference Pacucci, Ferrara, Grazian, Fiore, Giallongo and Puccetti2016) suggest a promising method to search for DCBH candidates in deep multi-wavelength surveys, based on photometric observations. By modelling the spectral energy distribution (SED) and colours of objects selected from the CANDELS/GOODS-S field catalogues (Guo et al. Reference Guo2013), they identify two X-ray-detected faint active galactic nuclei (AGN), 33160 and 29323 (Giallongo et al. Reference Giallongo2015) (but see also Weigel et al. Reference Weigel, Schawinski, Treister, Urry, Koss and Trakhtenbrot2015; Cappelluti et al. Reference Cappelluti2016; Vito et al. Reference Vito2016) as DCBHs prototypes at z ~ 6 and ~9.7, respectively.

The existence of such low-luminosity AGN at very high redshift, together with the recent reduction in the optical depth due to free electrons, τ _e reported by the Planck Collaboration et al. (2016) has renewed the interest in the role of the first quasars in cosmological reionisation. Although the idea of quasars substantially contributing to, or even being the main responsible for, reionisation (e.g. Madau & Haardt Reference Madau and Haardt2015) is still highly debated (see e.g. D’Aloisio et al. Reference D’Aloisio, Sanderbeck, McQuinn, Trac and Shapiro2016) the recent discoveries strengthen the motivation for a better understanding of their demographics and origin.

2.2. What are the properties of high-z SMBHs hosts?

High-z quasars are predicted to be hosted in the most massive dark matter (DM) halos residing in over-dense environments (e.g. Overzier et al. Reference Overzier, Guo, Kauffmann, De Lucia, Bouwens and Lemson2009; Di Matteo et al. Reference Di Matteo, Khandai, DeGraf, Feng, Croft, Lopez and Springel2012; Angulo, Hahn, & Abel Reference Angulo, Hahn and Abel2013) However, clear observational evidences of such a scenario are still missing, as observations provide controversial results (e.g. Stiavelli et al. Reference Stiavelli2005a; Willott et al. Reference Willott, Crampton, Hutchings, Sawicki, Simard, Jarvis, McLure, Percival, Merloni, Nayakshin and Sunyaev2005; Wang, Malhotra, & Rhoads Reference Wang, Malhotra and Rhoads2005; Zheng et al. Reference Zheng2006; Kim et al. Reference Kim2009a; Utsumi et al. Reference Utsumi, Goto, Kashikawa, Miyazaki, Komiyama, Furusawa and Overzier2010; Husband et al. Reference Husband, Bremer, Stanway, Davies, Lehnert and Douglas2013; Simpson et al. Reference Simpson, Mortlock, Warren, Cantalupo, Hewett, McLure, McMahon and Venemans2014; Morselli et al. Reference Morselli2014; McGreer et al. Reference McGreer, Eftekharzadeh, Myers and Fan2016; Mazzucchelli et al. Reference Mazzucchelli, Bañados, Decarli, Farina, Venemans, Walter and Overzier2017; Balmaverde et al. Reference Balmaverde2017).

The quasar hosts are chemically evolved, metal and dust-rich galaxies. Although their metallicity is quite difficult to trace, constraints on the gas-phase elemental abundances in the interstellar medium (ISM) come from the detection of emission line ratios in broad- and the narrow-line regions (BLRs and NLRs, respectively).

Although BLRs are representative of a small fraction of the gas content, concentrated within the central region (10⁴ M_⊙ on parsec scales, close to the AGN), the observed emission line ratios, such as Feii/Mgii (e.g. Barth et al. Reference Barth, Martini, Nelson and Ho2003), NV/CIV (e.g. Pentericci et al. Reference Pentericci2002), (Si IV+OIV)/CIV (Nagao, Marconi, & Maiolino Reference Nagao, Marconi and Maiolino2006; Juarez et al. Reference Juarez, Maiolino, Mujica, Pedani, Marinoni, Nagao, Marconi and Oliva2009), and metal lines like Cii and Oi (e.g. Maiolino et al. Reference Maiolino2005; Becker et al. Reference Becker, Sargent, Rauch and Simcoe2006) trace up to ~7 Z_⊙ metallicities (Nagao et al. Reference Nagao, Marconi and Maiolino2006; Juarez et al. Reference Juarez, Maiolino, Mujica, Pedani, Marinoni, Nagao, Marconi and Oliva2009) suggesting a fast evolution of the ISM chemical properties. By using emission line ratios as tracers, Jiang et al. (Reference Jiang, Fan, Vestergaard, Kurk, Walter, Kelly and Strauss2007) estimated gas metallicity of a sample of 5.8 < z < 6.3 quasars, powered by 10⁹–10¹⁰ M_⊙ SMBHs, finding values as high as ~4~Z_⊙.

A better proxy of the host galaxy ISM metallicity, on larger scales (comparable to the host galaxy size), is provided by NLRs. A mean gas-phase metallicity $Z_{\text{NLR}}=1.32^{+0.25 }_{-0.22} \text{ Z}_\odot$ is inferred from Civ/Heii and Ciii/Civ flux ratios in quasar, with no significant evolution up to z ~ 4 (Nagao et al. Reference Nagao, Marconi and Maiolino2006; Matsuoka et al. Reference Matsuoka, Nagao, Maiolino, Marconi and Taniguchi2009). Such super-solar metallicities are reminiscent of the star formation history (SFH) of the system (see e.g. Matsuoka et al. Reference Matsuoka, Nagao, Maiolino, Marconi and Taniguchi2009 and references therein) and can serve as a lower limit for the z ~ 6 quasar host galaxies.

Constraints on the cool/warm dust content come from the observations of far-infrared (FIR) and sub-millimetre (sub-mm) continuum radiation, while NIR and MIR observations may provide indications of the hot dust component (e.g. Jiang et al. Reference Jiang, Fan, Vestergaard, Kurk, Walter, Kelly and Strauss2007).

The observed ⩾10¹³ L_⊙ quasar FIR luminosities are consistent with emission from dust with temperatures of the order of 30–60 K and masses >10⁸ M_⊙ (Bertoldi et al. Reference Bertoldi, Carilli, Cox, Fan, Strauss, Beelen, Omont and Zylka2003; Priddey et al. Reference Priddey, Isaak, McMahon, Robson and Pearson2003; Robson et al. Reference Robson, Priddey, Isaak and McMahon2004; Beelen et al. Reference Beelen, Cox, Benford, Dowell, Kovács, Bertoldi, Omont and Carilli2006; Wang et al. Reference Wang2008; Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014; Michałowski et al. Reference Michałowski, Murphy, Hjorth, Watson, Gall and Dunlop2010). From the same FIR luminosities, high star-formation rates (SFRs), ⩾1 000 M_⊙ yr ⁻¹, can be inferred, suggesting that a large fraction of these systems has ongoing, highly efficient, star-formation activity (see e.g. Table 1 in Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014 and references therein).Footnote ¹

2.3. Is there a stellar mass crisis?

The rapid enrichment by metals and dust at very high redshift discussed above suggests that quasar host galaxies could have undergone intense episodes of star formation. Similar chemical abundances are typically found in local galaxies which, however, evolved on longer time scales.

The estimated mean BH-stellar bulge mass ratio, M _BH/M _star, of z ~ 6 quasars is about 10 times higher than the one observed in the local Universe (e.g. Wang et al. Reference Wang2010, Reference Wang2013), suggesting that high-redshift BHs may have formed or assembled earlier than their host galaxies (e.g. Lamastra et al. Reference Lamastra, Menci, Maiolino, Fiore and Merloni2010; Venemans et al. Reference Venemans, Walter, Zschaechner, Decarli, De Rosa, Findlay, McMahon and Sutherland2016). Although this result could be strongly affected by observational selection effects (Lauer et al. Reference Lauer, Tremaine, Richstone and Faber2007; Volonteri & Stark Reference Volonteri and Stark2011) and large uncertainties in the estimation of the mass and size of the stellar bulge (Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014; Pezzulli, Valiante, & Schneider Reference Pezzulli, Valiante and Schneider2016), it is difficult to explain how the ISM has been enriched to chemical abundances similar to that of local galaxies, albeit with ≲ 10% of the stars (Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011; Calura et al. Reference Calura, Gilli, Vignali, Pozzi, Pipino and Matteucci2014; Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014).

2.4. What is the role of BH feedback?

It is expected that galaxy-scale winds, triggered by the large amount of energy released in the BH accretion process, play a crucial role in regulating the BH-host galaxy co-evolution, shaping the SFH and BH accretion history itself (e.g. Silk & Rees Reference Silk and Rees1998; Granato et al. Reference Granato, De Zotti, Silva, Bressan and Danese2004; Di Matteo, Springel, & Hernquist Reference Di Matteo, Springel and Hernquist2005; Springel, Di Matteo, & Hernquist Reference Springel, Di Matteo and Hernquist2005a; Ciotti, Ostriker, & Proga Reference Ciotti, Ostriker and Proga2009, Reference Ciotti, Ostriker and Proga2010; Hopkins & Elvis Reference Hopkins and Elvis2010; Zubovas & King Reference Zubovas and King2012).

Indeed, massive and fast large scale gas outflows, associated to quasar activity, have been observed in local and high-redshift quasars (Feruglio et al.; Reference Feruglio, Maiolino, Piconcelli, Menci, Aussel, Lamastra and Fiore2010, Reference Feruglio2015; Alatalo et al. Reference Alatalo2011; Aalto et al. Reference Aalto, Garcia-Burillo, Muller, Winters, van der Werf, Henkel, Costagliola and Neri2012; Alexander et al. Reference Alexander, Swinbank, Smail, McDermid and Nesvadba2010; Nesvadba et al. Reference Nesvadba2010, Reference Nesvadba, Polletta, Lehnert, Bergeron, De Breuck, Lagache and Omont2011, Maiolino et al. Reference Maiolino2012; Cano-Díaz et al. Reference Cano-Díaz, Maiolino, Marconi, Netzer, Shemmer and Cresci2012; Farrah et al. Reference Farrah2012; Trichas et al. Reference Trichas2012; Carniani et al. Reference Carniani2016). At z > 6, a massive gas outflow has been inferred from observations of [Cii] emission line in J1148, revealing an outflow rate ⩾2 000–3 000 M_⊙ yr⁻¹ (Maiolino et al. Reference Maiolino2012; Cicone et al. Reference Cicone2015).

However, there are still open issues like what is the outflow powering mechanism, what are the effects of BH feedback on the host galaxy, how can the observed strong outflows and starbursts be simultaneously sustained? Although there are hints of star formation being quenched by quasar feedback at high redshift (Cano-Díaz et al. Reference Cano-Díaz, Maiolino, Marconi, Netzer, Shemmer and Cresci2012; Farrah et al. Reference Farrah2012; Trichas et al. Reference Trichas2012; Carniani et al. Reference Carniani2016), it is unclear if such feedback is able to completely suppress star formation in galaxies (Peng, Maiolino, & Cochrane Reference Peng, Maiolino and Cochrane2015). On the other hand, it has been pointed out that AGN-driven positive feedback (Zinn et al. Reference Zinn, Middelberg, Norris and Dettmar2013; Cresci et al. Reference Cresci2015) which triggers or enhances star formation, may be as important as quenching mechanisms in galaxy formation (e.g. Gaibler et al. Reference Gaibler, Khochfar, Krause and Silk2012; Wagner, Umemura, & Bicknell Reference Wagner, Umemura and Bicknell2013; Silk Reference Silk2013; Bieri et al. Reference Bieri, Dubois, Silk and Mamon2015).

4.1. Seeds formation sites

As they are the end products of massive Pop III stars, low-mass seed formation is enabled by nearly primordial conditions: metal and dust poor gas fragmenting into one or few massive stars at redshift z ~ 20 (e.g. Abel et al. Reference Abel, Bryan and Norman2002; Heger et al. Reference Heger, Fryer, Woosley, Langer and Hartmann2003; Madau & Rees Reference Madau and Rees2001; Yoshida et al. Reference Yoshida, Omukai and Hernquist2008; Hosokawa et al. Reference Hosokawa, Omukai, Yoshida and Yorke2011; Latif et al. Reference Latif, Schleicher, Schmidt and Niemeyer2013b; Hirano et al. Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014, Reference Hirano, Hosokawa, Yoshida, Omukai and Yorke2015). Gas enriched up to metallicity Z _cr ⩾ 10⁻⁴~Z_⊙, or dust-to-gas ratios $\cal {D}$ >4 × 10⁻⁹, fragments more efficiently (thanks to metal lines cooling and dust continuum radiation), to form instead lower mass, population II (Pop II) stars (Schneider et al. Reference Schneider, Ferrara, Natarajan and Omukai2002, Reference Schneider, Ferrara, Salvaterra, Omukai and Bromm2003; Omukai et al. Reference Omukai, Tsuribe, Schneider and Ferrara2005; Schneider et al. Reference Schneider, Omukai, Bianchi and Valiante2012). Such conditions are expected to be easily met in the first virialised structures at early times, the so-called minihalos, characterise by virial temperatures of 1.2 × 10³ < T _vir < 10⁴ K and masses M _h ~ 10^{5 − 6} M_⊙ (see e.g. Bromm Reference Bromm2013 for a review).

Although early studies suggest that Pop III star formation in these halos is characterised by high-mass stars (⩾100 M_⊙, e.g. Abel et al. Reference Abel, Bryan and Norman2002; Bromm, Coppi, & Larson Reference Bromm, Coppi and Larson2002; Bromm & Loeb Reference Bromm and Loeb2004; Yoshida et al. Reference Yoshida, Omukai and Hernquist2008), more recent simulations have shown that Pop III stars forming under different minihalo environmental conditions (e.g. determined by the presence or absence of photo-dissociating and ionising feedback) may span a wider range of masses, from few tens up to ~1 000 M_⊙ (e.g. Hirano et al. Reference Hirano, Hosokawa, Yoshida, Umeda, Omukai, Chiaki and Yorke2014, Reference Hirano, Hosokawa, Yoshida, Omukai and Yorke2015; Hosokawa et al. Reference Hosokawa, Hirano, Kuiper, Yorke, Omukai and Yoshida2016). In these works, only one star per halo is formed. However, a number of studies, resolving protostellar scales ( ~ 100R_⊙), show that fragmentation of protostellar disks may lead to the formation of multiple stars, with a wide mass spectrum (down to few solar masses), in small clusters (e.g. Clark, Glover, & Klessen Reference Clark, Glover and Klessen2008, Reference Clark, Glover, Smith, Greif, Klessen and Bromm2011; Turk, Abel, & O’Shea Reference Turk, Abel and O’Shea2009; Stacy, Greif, & Bromm Reference Stacy, Greif and Bromm2010, Reference Stacy, Bromm and Lee2016, Greif et al. Reference Greif, Springel, White, Glover, Clark, Smith, Klessen and Bromm2011, Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012; Susa, Hasegawa, & Tominaga Reference Susa, Hasegawa and Tominaga2014).

Pop III stars also represent the first sources of light and heavy elements (including dust, e.g. Nozawa et al. Reference Nozawa, Kozasa, Habe, Dwek, Umeda, Tominaga, Maeda and Nomoto2007; Heger & Woosley Reference Heger and Woosley2010; Marassi et al. Reference Marassi, Schneider, Limongi, Chieffi, Bocchio and Bianchi2015), setting the stage for all subsequent structure formation in their neighbourhood. Therefore, it is imperative that their formation is captured in the models for a consistent identification of the seed BH hosts. Resolving minihalos, in which these stars form, is thus crucial for models, at least at z > 20. Unfortunately, the mass/size resolution limit in both hSAMs (i.e. the box size and DM particle mass) and the pSAMs (i.e. the minimum DM halo mass) is often determined by the inherent computational costs.

Depending on the aim of the model, different scale/mass resolutions are suited for different studies. Resolving arbitrarily small halos is computationally prohibitive even for analytic binary Monte Carlo algorithms. In pSAMs, the resolution of the merger tree is thus defined by the minimum halo mass, which, together with the adaptive redshift interval (Δz) are chosen to maintain manageable computational times, simultaneously matching the EPS predictions at different redshifts (e.g. Volonteri et al. Reference Volonteri, Haardt and Madau2003; Tanaka & Haiman Reference Tanaka and Haiman2009).

In N-body simulations, the need to resolve a minihalo sets an upper limit on the box size that can be simulated in a reasonable time frame. N-body simulations with volumes ~ 100 cMpc³ allow one to resolve minihalos, capturing the small-scale sub-grid physics. These simulations offer insights on the formation sites of the first stars and seed BHs but lack statistical significance in terms of SMBH abundance for which larger volumes are required as discussed in section 3.

The formation of a DCBH requires the absence of star formation and of efficient coolants (metals and dust) in order to maintain isothermal collapse of gas clouds in Lyman-α-cooling halos (Lyα, T _vir ~ 10⁴ K), leading to a Jeans halo mass (which scales as T ^3/2) which is high enough to avoid fragmentation. Thus, high-mass seed BHs are expected to form out of poorly enriched gas (Z < Z _cr) if star formation is somehow inhibited. Colliding cold accretion flows (e.g. Inayoshi & Omukai Reference Inayoshi and Omukai2012) or high relative velocity galaxy mergers ( ⩾ 200 km s⁻¹) can shock heat the gas in the dense central regions of galaxies, collisionally dissociating the H₂ molecules (e.g. Inayoshi, Visbal, & Kashiyama Reference Inayoshi, Visbal and Kashiyama2015), thus preventing the gas from forming stars. Alternatively, the presence of H₂ photo-dissociating flux, i.e. photons in the Lyman Werner (LW) band (11.2–13.6 eV) emitted by nearby external sources, may suppress star formation in Lyα cooling halos (e.g. Bromm & Loeb Reference Bromm and Loeb2003; Begelman et al. Reference Begelman, Volonteri and Rees2006; Spaans & Silk Reference Spaans and Silk2006; Inayoshi et al. Reference Inayoshi, Omukai and Tasker2014; Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014). These conditions indeed enable the formation of a supermassive star (SMS) of 10^{4 − 5} M_⊙ that may eventually lead to a massive seed BH by accreting the surrounding material (e.g. Bromm & Loeb Reference Bromm and Loeb2003; Begelman et al. Reference Begelman, Volonteri and Rees2006; Lodato & Natarajan Reference Lodato and Natarajan2006, Reference Lodato and Natarajan2007; Inayoshi & Omukai Reference Inayoshi and Omukai2012; Inayoshi et al. Reference Inayoshi, Omukai and Tasker2014; Ferrara et al. Reference Ferrara, Salvadori, Yue and Schleicher2014; Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013; Sakurai et al. Reference Sakurai, Hosokawa, Yoshida and Yorke2015; Umeda et al. Reference Umeda, Hosokawa, Omukai and Yoshida2016; Woods et al. Reference Woods, Heger, Whalen, Haemmerle and Klessen2017; Haemmerlé et al. Reference Haemmerlé, Woods, Klessen, Heger and Whalen2017). Another pathway to create massive BHs in the presence of an external LW radiation field is via a quasi-star system. A massive star rapidly forms a 10–100 M_⊙ BH embedded in a radiation pressure supported dense gas cloud which then experiences high gas infall (and therefore accretion) rates ~1 M_⊙ yr⁻¹, eventually resulting in a more massive 10^4–5 M_⊙ DCBH seed (Spaans & Silk Reference Spaans and Silk2006; Begelman, Rossi & Armitage Reference Begelman, Rossi and Armitage2008; Volonteri & Begelman Reference Volonteri and Begelman2010). This peculiarity of the environmental conditions, and the frequency of their occurrence is still under debate (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012, Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014; Habouzit et al. Reference Habouzit2016b; Dijkstra et al. Reference Dijkstra, Ferrara and Mesinger2014; Yue et al. Reference Yue, Ferrara, Salvaterra, Xu and Chen2014; Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016). The conditions are sensitive to galaxies’ assembly histories and on the interplay between the effect of chemical, radiative, and mechanical feedback, driven by star formation and BH growth itself.

4.2. Forming the first stars

In star-forming halos, both Pop III or Pop II stars form depending on the chemical enrichment (metallicity) of the gas. Pop III stars form out of metal-free/poor gas (Z < Z _cr) while metal/dust-rich gas clouds instead lead to Pop II star formation.

The metallicity of a galaxy is usually the result of the interplay between in-situ and external metal pollution, i.e. stellar nucleosynthetic products injected in the galaxy ISM, and in-falling metal-rich (and dusty) gas ejected from nearby galaxies via supernovae (SNe) and AGN-driven winds.

Most hSAMs allow Pop III stars to form in metal-free halos, i.e. the ones that have never hosted a star in their past and/or pass the critical mass threshold (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012). The mass threshold can be understood as a negative feedback effect of LW photons that delay Pop III SF by a fraction of dissociating H₂ molecules in a minihalo. While exposed to LW radiation, J _LW,Footnote ² the halo must grow (or accrete more gas) in order to replenish the H₂ content, thereby becoming suitable for Pop III SF (e.g. Machacek, Bryan, & Abel Reference Machacek, Bryan and Abel2001; Wise & Abel Reference Wise and Abel2007; O’Shea & Norman Reference O’Shea and Norman2008). We show this M _crit–J _LW curve expressed as Equation (1) (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012), in Figure 2 (from O’Shea & Norman Reference O’Shea and Norman2008) where

(1) $$\begin{eqnarray} M_{\rm crit} \approx 4 \left(1.25 \times 10^5 + 8.7 \times 10^5 \left({ 4 \pi J_{\rm LW} } \right)^{0.47}\right). \ \end{eqnarray}$$

Figure 2. The M _crit − J _LW relation from O’Shea & Norman (Reference O’Shea and Norman2008), Figure 3. The squares represent their updated calculations while the Machacek et al. (Reference Machacek, Bryan and Abel2001) relation is depicted by the dashed line. The empty square represent the case with J _LW = 0. If the mass of a pristine minihalo exposed to a given J _LW, lies above the curve formed by the squares, it is considered Pop III star forming.

In their recent pSAMs, Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) and de Bennassuti et al. (Reference de Bennassuti, Salvadori, Schneider, Valiante and Omukai2017) compute the fraction of gas that can cool down and form stars in minihalos as a function of halo virial temperature, redshift, gas metallicity, and level of LW flux J _LW at which the halo is exposed. At a given redshift, the halo mass threshold increases with J _LW. Progressively more massive minihalos are expected to form stars at lower redshifts, at a fixed J _LW. A value J _LW ⩽ 0.1 is already high enough to suppress star formation in the less massive minihalos (<(3–4) × 10⁶ M_⊙) at z > 20. In good agreement with the gas collapse simulations of O’Shea & Norman (Reference O’Shea and Norman2008), Pop III star formation is inhibited in ⩽10⁷ M_⊙ pristine (Z = 0) minihalos exposed to an LW flux J _LW ⩾ 1, at redshift z < 17. Stronger J _LW levels (e.g. >10) sterilise all pristine minihalos already at redshift z = 20.Footnote ³

To date, observations do not provide strong enough constraints on the Pop III IMF. On the other hand, theoretical studies provide predictions on the mass distribution of these stars, that varies among different study (see e.g. the reviews by Bromm Reference Bromm2013; Glover Reference Glover, Wiklind, Mobasher and Bromm2013).

The most commonly adopted scenario in hSAMs (e.g. Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012, Reference Agarwal, Davis, Khochfar, Natarajan and Dunlop2013; Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016) is to form one Pop III star in a minihalo, randomly picked from a top-heavy IMF that ranges from 100to1 000 M_⊙. For atomic cooling pristine halos, where molecular hydrogen is still present in the central region, generally a cluster of 10–100 Pop III stars are allowed to form (e.g. Greif & Bromm Reference Greif and Bromm2006; Greif et al. Reference Greif, Springel, White, Glover, Clark, Smith, Klessen and Bromm2011, Reference Greif, Bromm, Clark, Glover, Smith, Klessen, Yoshida and Springel2012; Clark et al. Reference Clark, Glover, Smith, Greif, Klessen and Bromm2011), following the same IMF.

Regardless of the DM halo mass, massive Pop III stars with an average mass of ~100–200 M_⊙ are allowed to form in high-z pSAMs (e.g. Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014; Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016). The number of stars depends on the total stellar mass formed in each star formation episode, and thus on the star formation efficiency and available gas mass. An alternative scenario for Pop III star formation in pSAMs has been proposed by Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016): Pop III stars form with an intrinsic top-heavy IMF in the mass range [10–300] M_⊙. Then, this IMF is stochastically sampled, on the fly, according to the time-dependent total mass of newly formed stars. We will discuss the effect of these two different assumptions for Pop III stars formation on the low-mass seed BHs distribution later.

In metal-rich halos, Pop II star formation is generally accounted for by converting a fixed fraction of the available gas into stars. The time/redshift evolution of the gas content is modelled either by scaling the DM halo mass with the universal baryon fraction (e.g. Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008, Reference Dijkstra, Ferrara and Mesinger2014; Habouzit et al. Reference Habouzit2016b) or solving a set of differential equations (e.g. Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014, Reference Valiante, Schneider, Volonteri and Omukai2016; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016). In hSAMs, the star formation recipes are usually calibrated to reproduce the cosmic star-formation rate density (CSFRD) observed at z > 6 (Hopkins Reference Hopkins2004; Mannucci et al. Reference Mannucci, Buttery, Maiolino, Marconi and Pozzetti2007; Bouwens et al. Reference Bouwens, Illingworth, Franx and Ford2008; Laporte et al. Reference Laporte2012). Since pSAMs are generally targeted to explain the existence of a single quasar, the models are designed to match the observables of the quasar in question.

4.3. Conditions for direct collapse

The treatment of the DC scenario is now taking advantage of hybrid models where instead of Press–Schechter merger trees, one uses a fully cosmological N-body simulation as a playground for the various recipes critical to DCBH formation. One of the main advantages of using hSAMs to study the formation of SMBHs at early times is the spatial information that enables one to study the dependence of various processes on the halos’ physical distribution within the simulated volume. Nearby star-forming halos emit LW photons that are able to photo-dissociate H₂ (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), and thus the spatial distance between halos is a crucial ingredient as it controls the strength of the irradiation flux (e.g. Agarwal et al. Reference Agarwal, Smith, Glover, Natarajan and Khochfar2016a). Anisotropies (fluctuations) in the LW background , due to source clustering and/or proximity to the DCBH host candidate, are indeed the key of the radiation-driven DCBH formation scenario (e.g. Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008, Reference Dijkstra, Ferrara and Mesinger2014; Sugimura, Omukai, & Inoue Reference Sugimura, Omukai and Inoue2014; Agarwal et al. Reference Agarwal, Smith, Glover, Natarajan and Khochfar2016a; Regan, Johansson, & Wise Reference Regan, Johansson and Wise2016, Reference Regan, Johansson and Wise2014). When a proto-galaxy is located nearby an emitting source, spatial correlation makes the difference. Far from the emitting source, the LW photons flux seen by the target halo is too low to affect the fraction of molecular gas which remains high. On the hand, the halo is photo-evaporated, by ionising radiation, if it is too close to the illuminating source (e.g. Regan et al. Reference Regan, Johansson and Wise2016). Time synchronisation matters too. The time elapsed between the starburst onset in the primary halo and the gas collapse in the companions must be short in order to avoid halo photo-evaporation or pollution by heavy elements (e.g. Visbal, Haiman, & Bryan Reference Visbal, Haiman and Bryan2014; Regan et al. Reference Regan, Visbal, Wise, Haiman, Johansson and Bryan2017; Agarwal et al. Reference Agarwal, Regan, Klessen, Downes and Zackrisson2017b).

We provide here an overview of the large-scale feasibility of the DC model, i.e. we do not consider studies related to the formation of individual DCBHs (see e.g. Latif & Ferrara Reference Latif and Ferrara2016 for a review), and rather discuss studies which aim at deriving statistical properties, such as the number density of DCBH sites that form in the early Universe and the conditions leading to them.

In order to identify a DCBH formation site within an average volume of the Universe, one must account for the entire LW and metal pollution history of the atomic cooling halo in question, especially taking into consideration the effects of the local environment. This is one of the biggest strengths of hSAMs as painting galaxies on N-body simulations allows us to compute spatial locations.

4.3.1. Critical LW flux

We have discussed above how (low level) LW flux can delay Pop III star formation in pristine minihalos. Once the halo becomes atomic cooling, i.e. when it attains a virital temperature of T _vir > 10⁴ K and the primary coolant becomes atomic hydrogen (Omukai Reference Omukai2000), an extremely high level of flux can completely shut down H₂ cooling by dissociating these molecules in the most dense (thus efficiently self-shielded) regions (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).

The critical level J _cr, above which DC of gas clouds into massive seeds is enabled, is still a matter of debate and remains a free parameter for models. Assuming that Pop III stellar populations mimic a T = 10⁵ K and Pop II stellar populations a T = 10⁴ K blackbody, Omukai (Reference Omukai2000) computed the critical value of J _crit using their 1D spherically symmetric gas collapse model. Since the shape of the blackbody spectrum depends on its temperature, J _crit depends on the type of the stellar population externally irradiating the pristine atomic cooling halo. They found J ^III _cr ≈ 10⁴–10⁵ and J ^II _cr ≈ 10²–10³ is needed from Pop III and Pop II populations to cause DCBH formation in a neighbouring pristine atomic cooling halo. Revisions in this estimate followed by employing high-resolution 3D hydrodynamical simulations and better recipes for H₂ self-shielding, leading to an estimate of J ^III _cr ~ 1 000 and J ^II _crit ≈ 10 − 100 (Shang et al. Reference Shang, Bryan and Haiman2010; Wolcott-Green, Haiman, & Bryan Reference Wolcott-Green, Haiman and Bryan2011; Latif et al. Reference Latif, Bovino, Van Borm, Grassi, Schleicher and Spaans2014; Hartwig et al. Reference Hartwig, Glover, Klessen, Latif and Volonteri2015).

In addition, ionising photons and X-rays can both increase the free electron fraction promoting H₂ formation (Inayoshi & Omukai Reference Inayoshi and Omukai2011; Yue et al. Reference Yue, Ferrara, Salvaterra, Xu and Chen2014; Johnson et al. Reference Johnson, Whalen, Agarwal, Paardekooper and Khochfar2014; Aykutalp et al. Reference Aykutalp, Wise, Spaans and Meijerink2014; Inayoshi & Tanaka Reference Inayoshi and Tanaka2015). As a result, a higher critical LW level, up to J _cr ~ 10⁴–10⁵, is required (Latif et al. Reference Latif, Bovino, Van Borm, Grassi, Schleicher and Spaans2014; Regan et al. Reference Regan, Johansson and Wise2014; Latif & Volonteri Reference Latif and Volonteri2015).

Besides H₂ molecules, H^- ions play a critical role in pristine gas collapse as they regulate H₂ formation at densities n ≲ 10³ cm⁻³ via the reactions

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

(3) $$\begin{eqnarray} {\rm H}^{-} +\ {\rm H} \rightarrow {\rm H}_{2} + e^{-}. \end{eqnarray}$$

The importance of this network is further understood by their corresponding photo-destruction channels

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

(5) $$\begin{eqnarray} {\rm H}^{-} +\ \gamma _{0.76} \rightarrow {\rm H} + e^{-}, \end{eqnarray}$$

where γ_LW and γ_0.76 represent the photons in the LW band and photons with energy greater than 0.76 eV, respectively. Ignoring the role of 1 eV photons can lead to a gross overestimation in the value of LW flux required to suppress H₂ cooling, as demonstrated by Wolcott-Green et al. (Reference Wolcott-Green, Haiman and Bryan2011) and Haiman (Reference Haiman2012). Furthermore, Glover (Reference Glover2015a, Reference Glover2015b) showed that inconsistencies in the chemical networks and reaction rate coefficients can lead to a factor ~3 difference in the determination of J _cr.

The assumption of representing Pop III and Pop II SEDs as blackbodies was questioned by Sugimura et al. (Reference Sugimura, Omukai and Inoue2014), Agarwal & Khochfar (Reference Agarwal and Khochfar2015), and Agarwal et al. (Reference Agarwal, Smith, Glover, Natarajan and Khochfar2016a), who showed that using realistic SEDs to represent stellar populations instead drastically alters the paradigm. This is because the change in the slope of a SED with the age of a stellar population alters the rate of production of LW photons (e.g. Schaerer Reference Schaerer2002) with respect to 1 eV photons. Agarwal et al. (Reference Agarwal, Smith, Glover, Natarajan and Khochfar2016a) and Wolcott-Green, Haiman, & Bryan (Reference Wolcott-Green, Haiman and Bryan2017) demonstrated that indeed, one cannot expect a single value of J _cr from a given stellar population, but that it is a value dependent on the underlying stellar population’s SFH and varies from 0.1–1 000 in their 1D models. Needless to say, given that these studies are very recent, this variation in the nature of J _cr needs to be further explored.

In Figure 3, we show the global and spatial LW intensities from the Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) hybrid fiducial model, and compare them to other studies. The averaged background LW intensity,J _bg, at a given redshift is computed as a function of the stellar mass density at that redshift.

$$\begin{equation*} J_{\rm bg}(z) = \frac{hc}{4\pi m_{\rm H}}\eta _{LW}\rho _\star (1+z)^3\ , \nonumber \end{equation*}$$

where η_LW is the number of LW photons emitted per stellar baryon, and ρ_⋆ is the stellar mass density at a given redshift, z. Both quantities are linked to the stellar population, so that J _bg = J ^III _bg + J ^II _bg (see Greif & Bromm Reference Greif and Bromm2006 and Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012 for more details). The green dotted line is instead the specific intensity J _bg given by Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014). The yellow dotted line in Figure 3 shows the average LW emission computed in the pSAM of (Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016) (similar values are also shown by Petri et al. (Reference Petri, Ferrara and Salvaterra2012)).

Figure 3. From Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012): The background and local level of LW radiation plotted for each redshift. ‘The red triangles (J ^II _local) and blue crosses (J ^III _local) indicate the maximum value of LW radiation to which a pristine halo is exposed at each redshift in their volume. The red and blue dashed lines represent J ^II _crit and J ^III _crit respectively. It is interesting to see that the maximum value of J ^III _local (blue crosses) falls short of J ^III _crit (blue dashed line). However, in the case of Pop II sources, the maximum value of J ^II _local (red triangles) is several orders of magnitude higher than the J ^II _crit (red dashed line). The green dotted line is the specific intensity J _bg given by Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014). Finally, the yellow dotted line shows the average LW emission from Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016).

As it can be seen from the figure, the global LW background radiation, J _bg is always far below the critical value for DC, J _cr (horizontal dashed red and blue lines). Thus, the study of the spatial variation of the photo-dissociating emission is fundamental to identify potential DCBH formation sites.

Ahn et al. (Reference Ahn, Shapiro, Iliev, Mellema and Pen2009) presented the first study of the evolution of the inhomogeneous LW background, in which the local LW flux intensity is self-consistently computed in a cosmological N-body simulation, explaining its importance. Their study is based on a suite of runs that were originally aimed at understanding reionisation (Iliev et al. Reference Iliev, Mellema, Shapiro and Pen2007), but was modified to include a radiative-transfer module for LW photons. Ahn et al. (Reference Ahn, Shapiro, Iliev, Mellema and Pen2009) find that the average intensity of the LW radiation exceedes the threshold value for H₂-cooling and star-formation suppression in minihalos well before the reionisation process is complete. In their scenario, both the average and local LW flux can be ⩾10⁻² already at z < 20 (see e.g. Ahn et al. Reference Ahn, Shapiro, Iliev, Mellema and Pen2009: Figure 10). As a result, Lyα-cooling halos are the dominant sources of reionsation while minihalos are sterilised before they can significantly contribute to the ionising and LW background radiation. Following this study, several other models (pSAMs and hSAMs) pointed out the importance of LW flux fluctuations due to sources clustering in the formation of DCBHs (e.g. Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008, Reference Dijkstra, Ferrara and Mesinger2014; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Habouzit et al. Reference Habouzit2016b; Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016; Pawlik, Bromm, & Milosavljević Reference Pawlik, Bromm and Milosavljević2014).

In Figure 3, we also show the values of the local LW flux, J _local, from single stellar populations as computed by Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) in their hSAM volume at each redshift. They show that while Pop III stars are never able to produce the J ^III _crit in their vicinity, Pop II stars are able to produce J ^II _crit quite easily (see Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012 for details). This result was later confirmed by Agarwal et al. (Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014) and Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) in their suite of hydrodynamical runs, and by Chon et al. (Reference Chon, Hirano, Hosokawa and Yoshida2016).

Due to the lack of spatial information, pSAMs instead cannot capture the spatial variations of J _LW with respect to the background flux as hSAMs do. However, the LW emission from Pop III/II stars and accreting BHs is self-consistently computed, according to their SED, as a function of stellar age and metallicity and of BH accretion rate (e.g. Petri et al. Reference Petri, Ferrara and Salvaterra2012; Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016). An important difference with respect to hSAMs is that in pSAMs, the star formation and BH accretion efficiency are usually calibrated to match the observed SFR and BH mass of specific, single, obejcts (e.g. quasar J1148 in Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016). Within the biased region occupied by the progenitors of a 10¹³ DM halo, the computed LW flux can be interpreted as a mean value for the local fluctuations exceeding the background level, as expected by several models (e.g. Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008; Tanaka & Haiman Reference Tanaka and Haiman2009; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Dijkstra et al. Reference Dijkstra, Ferrara and Mesinger2014). In addition, Petri et al. (Reference Petri, Ferrara and Salvaterra2012) and Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) show that stellar emission provides the dominant contribution to the photo-dissociating flux with respect to accreting BHs. For example, the global LW emission from stellar populations in Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016), taken as a proxy of the local flux in their biased region (orange dotted line Figure 3), is in good agreement with the maximum local Pop II LW flux, at z < 11 (red triangles), and with the large scatter in the maximum local Pop III flux, at larger redshifts (blue crosses), from Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012).

4.4. DCBHs number density

Over the past few years, the question of the number density of DCBHs has become a topic of great interest, and has led to values that span several orders of magnitude, from ~10⁻¹ to 10⁻⁹ cMpc⁻³.

Here we compare the results of both hSAMs (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Habouzit et al. Reference Habouzit2016b) and pSAMs of Dijkstra et al. (Reference Dijkstra, Haiman, Mesinger and Wyithe2008, Reference Dijkstra, Ferrara and Mesinger2014) and Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016). We include DCBH number densities from the Agarwal et al. (Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014) and Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) hydrodynamical simulations as they offer a direct comparison of semi-analytic and hydrodynamic approaches.

Dijkstra et al. (Reference Dijkstra, Haiman, Mesinger and Wyithe2008) compute the probability distribution function of the LW flux at which DM halos are exposed to at z ~ 10 taking into account their clustering properties. They find that only a small fraction, <10⁻⁶, of all atomic cooling halos are exposed to a LW flux exceeding the assumed critical threshold, J _LW > 10³ and thus derive a number density of <10⁻⁶ cMpc⁻³ potential DCBHs hosts.

In contrast, using a semi-analytic model on top of a cosmological N-body simulation, Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) find a higher number density, in the range 10⁻²–10⁻¹ cMpc⁻³ for J _crit = 30 (their fiducial model), even accounting for in-situ metal pollution from previous star-formation events.

In their fiducial model, Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014) include star formation in atomic cooling halos (but do not include minihalos), metal pollution from progenitor halos, and galactic outflows and estimate n _DCBH ~ 10⁻⁹–10⁻⁶ cMpc⁻³ between z = 20 and 7. They explore the dependence of their predictions on model assumptions, such as the value of LW photons escape fraction and critical flux for DC, underlying the important effect of galactic winds decreasing the number density by several orders of magnitudes. The fraction of LW photons escaping from galaxies, and contributing to the photo-dissociating background radiation, indeed plays a crucial role in this scenario. However, the LW escape fraction is still highly uncertain (may increases from 0 to 1 depending on halo and stellar mass) and strongly dependent on the ionisation front propagation (e.g. Kitayama et al. Reference Kitayama, Yoshida, Susa and Umemura2004; Schauer et al. Reference Schauer, Whalen, Glover and Klessen2015, Reference Schauer2017b).

More recently, Habouzit et al. (Reference Habouzit2016b) find a number density of DCBH regions in the range 10⁻⁷–5 × 10⁻⁶ cMpc⁻³, consistent with what found by Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014). A factor of two higher number density can be found in cosmological N-body simulations in which primordial fluctuations are described by a non-Gaussian distribution. In addition, they also estimate the Pop III remnant BHs number density, being about two order of magnitude higher than that of DCBHs, although they do not resolve minihalos in their simulations. Similar values are found in hydrodynamical simulations by Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) for different box sizes and resolutions.

In their pSAM aimed to study the role of Pop III remnant BHs and DCBHs in the formation of a z ~ 6 SMBH, Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) predict an average number density of ~10⁻⁷ cMpc⁻³ DCBHs. These are the DCBHs expected to form in J1148 progenitor galaxies, along the hierarchical history of a 10¹³ M_⊙ DM halo. As we will discuss later, only a fraction of these high-mass seeds eventually end in the final SMBH, driving its fast growth.

In Figure 4, we show a collection of DCBH number densities derived from some of the studies discussed above. Symbols represent different radiation intensity thresholds: squares refer to J _{LW, crit} = 30, circles to J _{LW, crit} = 100, and triangles to J _{LW, crit} = 300. The figure is taken from Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) who compare the results of semi-analytic studies by Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014) (dark gray symbols) with hydrodynamical simulations: one of the FiBy simulations based on the smoothed particle hydrodynamics (SPH) code gadget (e.g. Springel et al. Reference Springel, Di Matteo and Hernquist2005a) presented by Agarwal et al. (Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014) (light grey crossed square at z = 10.5); two runs of the 10 cMpc box Chunky simulation with a collapse times scale equal to 10 Myr (purple symbols) and to the halo free fall time, t _ff (orange square); the large-scale (142 cMpc side box) cosmological simulation Horizon-noAGN (cyan symbols, Dubois et al. Reference Dubois2014b; Peirani et al. Reference Peirani2016). We refer the reader to the original paper Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) for a detailed discussion. We have included in this figure the predictions by Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) in the z = 7 − 10 redshift range (light gray squares) and Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) at z = 18 and 15 (black triangles). Finally, the horizontal blue solid line show the SMBH number density observed at z ~ 6 of 1 cGpc⁻³.

Figure 4. Co-moving number density of halos that can host a DCBH, at a given redshift. Symbols represent different radiation intensity thresholds. Squares: J _{LW, crit} = 30, circles: J _{LW, crit} = 100, triangles: J _{LW, crit} = 300. The horizontal solid blue line shows the co-moving number density of z ~ 6 SMBHs. The light gray crossed square at z = 10.5 is from the hydrodynamical simulation by Agarwal et al. (Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014), the light gray squares in the range z = 10–7 are from Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012, private communication), dark gray squares and black triangles are the results of Dijkstra et al. (Reference Dijkstra, Ferrara and Mesinger2014) and Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016), respectively. The orange square shows the number density for Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) (10 cMpc side box, t _ff, see text). The purple squares and circles show the number density for Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c) (10 cMpc side box, 10 Myr, see text). The cyan squares, circle, and triangle represent the large-scale cosmological simulation Horizon-noAGN (Dubois et al. Reference Dubois2014b; Habouzit et al. Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c, 142 cMpc side box).

4.4.1. Consensus between different studies

One of the most restrictive ingredient of the DC scenario is the absence of H₂ (through both H₂ destruction and prevention of H₂ formation) to keep the gas temperature and thus the Jeans mass high enough to avoid the fragmentation of gas clouds. This should favour the formation of only one massive object. As mentioned in section 2, the exposure to a strong LW radiation is one of the possible way to strongly depress H₂ abundances (Omukai Reference Omukai2000; Omukai et al. Reference Omukai, Schneider and Haiman2008; Shang et al. Reference Shang, Bryan and Haiman2010). From Ahn et al. (Reference Ahn, Shapiro, Iliev, Mellema and Pen2009), we have understood that the spatial variations of the radiation intensity, driven by LW photons able to photo-dissociate H₂, was certainly a key requirement of the scenario. Most of the models for the radiation intensity include now a spatial varying component based on local photo-dissociating sources. The radiation intensity is either computed directly from stellar particles according to their age, distance, and redshift (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012, Reference Agarwal, Dalla Vecchia, Johnson, Khochfar and Paardekooper2014; Habouzit et al. Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016c), or from the stellar mass painted on DM halos (Dijkstra et al. Reference Dijkstra, Ferrara and Mesinger2014; Habouzit et al. Reference Habouzit2016b; Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016).

Moreover, the critical radiation flux needed to destroy H₂, seems to be driven mainly by Pop II stars. This is supported by three main ideas. First of all, the LW radiation background created by Pop III stars emission, impacts their surrounding by photo-dissociating molecular hydrogen. Cooling rate decreases, which delays the gas collapse, and this vicious circle lowers and delays the formation of new Pop III stars at later time (O’Shea & Norman Reference O’Shea and Norman2008; Johnson et al. Reference Johnson, Whalen, Fryer and Li2012). The life time of Pop III stars is also thought to be short (~10 Myr), it could be too short for providing a high LW radiation intensity during the whole free-fall time of a DM halo. One can compute the redshift at which the free-fall time is approximately equal to ~10 Myr, and finds z ~ 45. This means that a halo illuminated only by Pop III radiation, could form a BH only at very early times, around z ~ 45. Finally, the intensity of Pop III radiation itself may be not enough to provide the critical radiation intensity commonly assumed for the DC model (O’Shea & Norman Reference O’Shea and Norman2008; Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012). In Figure 5 (reproduced from Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012), we show the distribution of the local varying radiation intensity seen by pristine halos at z = 16, before the formation of the first Pop II stars, and z ~ 9, after their formation. Radiation intensity from Pop III stars is shown in blue, and from Pop II stars in red. Dashed lines indicate the critical radiation intensity expected for Pop III stars (in blue) and Pop II stars (in red). Pop III stars radiation intensity appears to be almost always below the critical intensity (below the corresponding red dashed line), whereas a majority of pristine halos under Pop II stars radiation flux can meet the critical radiation intensity condition. The distribution of radiation intensity to which halos are exposed to, is in good agreement between various studies, using similar methods and LW radiation modellings (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012; Chon et al. Reference Chon, Hirano, Hosokawa and Yoshida2016), or different approaches (Dijkstra et al. Reference Dijkstra, Haiman, Mesinger and Wyithe2008).

Figure 5. Distribution of local radiation intensity (Agarwal et al. Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) seen by pristine halos at z = 16 (top panel), before the formation of Pop II begins, and later on at z ~ 9 (bottom panel) when Pop II is already in place. f _pris is the number fraction of pristine halos exposed to a given J _LW. Radiation intensity from Pop III stars is shown in red, and radiation intensity from Pop II stars in blue. Dashed lines indicate the critical radiation intensity expected for Pop III stars (in red) and Pop II stars (in blue). Pop III stars radiation intensity appears to be almost always below the critical intensity (below the corresponding dashed line), whereas a fraction of pristine halo illuminated by Pop II stars radiation flux can meet the critical radiation intensity condition.

Finally, all studies agree that metal pollution from both heritages, previous episodes of star formation in halo progenitors and galactic winds from nearby halos, could play a fundamental role. Galactic winds could sterilise potential DCBH regions by enriching them in metals, on a scale of ⩽ 10 kpc, thereby reducing their number density (Dijkstra et al. Reference Dijkstra, Ferrara and Mesinger2014). The process is a complex interplay of metals mixing in a gas medium of varying density, the propagation of metals in the IGM, and the winds launching out from their host halo (Cen & Riquelme Reference Cen and Riquelme2008; Smith et al. Reference Smith, Wise, O’Shea, Norman and Khochfar2015). Agarwal et al. (Reference Agarwal, Regan, Klessen, Downes and Zackrisson2017b) recently devised a semi-analytical model working under worst case assumptions for DCBH formation under the influence of metals originating from neighbouring galaxies that provide the necessary LW flux. Even after assuming an extremely short (300 pc) separation between their DCBH candidate halo and external LW sources, and instantaneous metal mixing, they find that the metal mixing is insufficient to shut down DCBH formation. This is because during the time window when the target halo can form a DCBH, its metallicity remains well below the critical threshold above which SF is expected (Omukai et al. Reference Omukai, Schneider and Haiman2008; Latif et al. Reference Latif, Omukai, Habouzit, Schleicher and Volonteri2016).

5.1. Low-mass vs. high-mass seeds

In Figure 6, we show the distribution of the number of seed BHs formed along the hierarchical build-up of a z ~ 6 quasar (i.e. in its progenitor galaxies) as a function of the host DM halo mass. In the left panel, we show the number of equal mass stellar BHs, low-mass seeds of 10² M_⊙, assumed to form in newly virialised halos, as long as they are metal poor, Z < Z _cr = 10^−3.8, i.e. at z ⩾ 20, as predicted by Pezzulli et al. (Reference Pezzulli, Valiante and Schneider2016). The other two panels instead are for a mixed-seed-based seeding prescription (Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016): (40–140) and (260–300) M_⊙ Pop III remnant BHs (middle panel) plus 10⁵ M_⊙ high-mass seeds (right panel), forming along the same merger history. In this scenario, the formation of low-mass and high-mass seeds is simultaneously explored thus, allowing to directly compare the role of the two channels in the formation of an SMBH. In all panels, histograms and data points are obtained by averaging over 29(10) different merger histories of the 10¹³ M_⊙ DM halo in the low-mass-seed(mixed-seed) case, with error bars showing the 1σ dispersion. Both prescriptions have been adopted to model the quasar, J1148 at z = 6.4, with a SMBH of (2–6) × 10⁹ M_⊙ (Barth et al. Reference Barth, Martini, Nelson and Ho2003; Willott, McLure, & Jarvis Reference Willott, McLure and Jarvis2003; De Rosa et al. Reference De Rosa, Decarli, Walter, Fan, Jiang, Kurk, Pasquali and Rix2011). As we discussed in the previous sections, the number, redshift, and typical host halo mass of both low-mass and high-mass seeds is determined by the interplay between the early chemical enrichment—due to metal-rich infalling gas from the external medium, polluted by SN- and AGN-driven winds from other galaxies—and the intensity of the LW radiation (from both stars and accreting BHs) to which the halos are exposed.

Figure 6. Distribution of the average number of seed BHs as function of the DM halo mass from different seeding prescriptions adopted in pSAMs: (i) equal-mass 100 M_⊙ low-mass seeds (left panel) and (ii) (10–140) and (260–300) M_⊙ Pop III remnant BHs (middle panel) plus 10⁵ M_⊙ high-mass seeds (right panel). Histograms and data points show the number of total (in lighter colours) and real SMBH progenitors (darker histograms, see text). Error bars account for the 1σ dispersion. The figures are adapted from Pezzulli et al. (Reference Pezzulli, Valiante and Schneider2016) and Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016). The average redshift range in which seeds form, according to these two models, is given in each panel.

The inclusion of radiative feedback effects results in a less efficient and slightly slower metal enrichment, enabling Pop III stars to form on average down to lower redshift, e.g. z ~ 16 in the model shown on Figure 6. As we see in the right panel of the figure, DCBH form in 10⁷–10⁸ M_⊙ progenitor halos [and in the narrow redshift range 15–18, see Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) for details], consistent with what is expected from their formation theory and the findings of Bellovary et al. (Reference Bellovary, Volonteri, Governato, Shen, Quinn and Wadsley2011), Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012), Habouzit, Volonteri, & Dubois (Reference Habouzit, Volonteri and Dubois2016a), and Chon et al. (Reference Chon, Hirano, Hosokawa and Yoshida2016).

In their pSAM, Petri et al. (Reference Petri, Ferrara and Salvaterra2012) combine both low-mass and high-mass seeds to investigate their relative role in the formation of SMBHs in a pSAM. They explore the dependence of the resulting SMBH evolutionary scenario on the fraction of halos (exposed to an LW flux J _cr > 10³) that can actually host DCBHs. A 10⁹–10¹⁰ M_⊙ BH is formed at z ~ 6 if at least (1–10)% of all the halos host a high-mass seed (see Petri et al. Reference Petri, Ferrara and Salvaterra2012: Figures 4 and 9).

For a critical LW threshold J _cr > 300 (Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016) predict an average high-mass seeds occurrence ratio (the number of galaxies with Z < Z _cr when J _LW > J _cr divided by the number of all the halos exposed to a flux J _LW > J _cr) of ~5% at z > 15. This suggests that chemical feedback plays a dominant role in determining of the birth environment.Footnote ⁴

Recently, Schauer et al. (Reference Schauer, Regan, Glover and Klessen2017a) explore the effects of baryonic streaming velocities on minihlaos, offering an alternative pathway to inhibit Pop III star formation before the pristine halo reaches the atomic cooling limit. Chon et al. (Reference Chon, Hirano, Hosokawa and Yoshida2016) combined a semi-analytic model for galaxy formation with halo merger trees extracted from N-body DM simulations to select possible DCBH hosts among atomic cooling halos. By means of zoom-in cosmological hydrodynamical simulations of the selected halos, they explore the evolution of gas collapse in the DCBH sites. They mostly follow the approach of Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) but bring a previously unexplored effect to light: tidal gravitational fields affecting gas collapse. They show that unless assembled via major mergers, their DCBH sites do not survive the tidal fields and get disrupted before an isothermal collapse can ensue at gas densities of n ⩾ 10 cm⁻³. A DCBH occupation fraction of ~5% (2 out of the selected 42) is found in this study, in good agreement with the pSAM of Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016).

5.2. The role of mergers and BH dynamics

Merger events can serve as an important physical process that drives the growth of BHs. However, binary (or multiple) BH interactions, driven by dynamical friction, are quite complex, multi-scale processes. The physical scales of interest span from sub-pc scales of the Schwarzschild radius (e.g. ~10⁻¹¹ pc for 100–300 M_⊙ BHs and ~10⁻⁸–10⁻⁷ pc for BHs of 10⁵–10⁶ M_⊙) up to the Mpc scale of the host galaxy mergers. In addition, the mechanism leading to BH–BH mergers, the time it takes for BHs to coalesce via gravitational wave (GW) emission, and the relation between the end state of the merger and the properties of the respective host galaxies, are still open questions.

However, SAMs aimed to study the formation and evolution of SMBHs through cosmic time usually adopt simple prescriptions to account for the contribution of mergers to the BH growth (see e.g. Tanaka & Haiman Reference Tanaka and Haiman2009 and references therein).

In major mergers,Footnote ⁵ BHs follow the fate of their host galaxies, coalescing to form a more massive BH. However, during this process, a large centre-of-mass recoil (kick) can be imparted to the newly formed BH as a consequence of asymmetric GW emission (e.g. Campanelli et al. Reference Campanelli, Lousto, Zlochower and Merritt2007; Schnittman et al. Reference Schnittman, Buonanno, van Meter, Baker, Boggs, Centrella, Kelly and McWilliams2008; Baker et al. Reference Baker, Boggs, Centrella, Kelly, McWilliams, Miller and van Meter2008). The acquired kick velocity can be as large as ~ 100 kms⁻¹, enough to eject the coalesced binary out of the host galaxy (see e.g. Yoo & Miralda-Escudé Reference Yoo and Miralda-Escudé2004; Volonteri & Rees Reference Volonteri and Rees2006b; Tanaka & Haiman Reference Tanaka and Haiman2009; Barausse Reference Barausse2012 and references therein for details). On the other hand, in minor mergers, one of the two merging BHs, usually the least massive one, is assumed to remain as a satellite in low-density regions, without accreting or contributing to the growth of the final BH.

The effective number of seed BHs from which an SMBH forms depends on these assumptions. Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) predict that only ~13% of the low-mass and high-mass seeds in their model (darker histograms in middle and right panels in Figure 6) contribute to the final mass of the SMBH of J1148, at z = 6.4, as a large fraction of BHs is lost due to minor mergers.

A similar fraction, ~15% (indicated by the darker histogram in the left panel) is left by taking into account the combined effect of minor mergers and gravitational recoil on growing low-mass seeds. On average, ~56% satellites BHs are lost along the entire merger tree in minor mergers while ~32% of the coalescing BHs, in major merger events, gain a recoil velocity large enough to exceed the retention speed, being kicked out of the galaxies (Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016; a much larger fraction, ~99% is found by Volonteri et al. Reference Volonteri, Haardt and Madau2003).

The effect of BH recoil due to GW emission during BH mergers has been also studied by Sijacki, Springel, & Haehnelt (Reference Sijacki, Springel and Haehnelt2009). They resimulate the most massive z = 6 DM halo extracted from the Millennium simulation in order to study the effect of BH mergers (Blecha et al. Reference Blecha2016) in the growth of high-redshift massive BHs. An SMBH of 10⁹–10¹⁰ M_⊙ is produced in an Eddington-limited scenario, by planting massive BH seeds of 10⁵ M_⊙, in DM halos with masses 10^9–10 M_⊙ at z = 15. They find that if the initial BH spin is high, the growth of mostly isolated (only a small number of mergers occur) massive BHs is hampered. However, BH kicks substantially expel low-mass BHs, and thus do not affect the overall growth of the SMBHs.

BH mergers are found to play a minor role in the formation of the first SMBHs (at relatively lower redshifts), in pSAMs (e.g. Figure 6 in Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016) and recently in hydrodynamical simulations like MassiveBlack and BlueTides (e.g. Feng et al. Reference Feng, Di Matteo, Croft and Khandai2014a and Di Matteo et al. Reference Di Matteo, Croft, Feng, Waters and Wilkins2016).

Mergers between BHs drive the BH mass assembly only at high redshifts (but see Petri et al. Reference Petri, Ferrara and Salvaterra2012). For example, although driving the BH growth process at z > 11, BH–BH coalescences contributes to less than 1% of the J1148 final BH mass at z = 6.4 (Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011). Similarly, in Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016), BH mergers (of mainly low-mass seeds) are predicted to drive the BH growth down to z ~ 15, before the gas accretion regime triggered by the formation of the first high-mass seeds, sets in.

Conversely, in the large-volume, cosmological hydrodynamical simulation Horizon-AGN (box size of 100 h ⁻¹ Mpc and resolution mass of 8 × 10⁷ M_⊙), Dubois et al. (Reference Dubois, Volonteri and Silk2014a) show an accretion-dominated BH growth at high redshift, while in the older Universe, the galactic centres tend to be less gas-rich, and, thus, the mass growth of the central BHs is mostly driven by mergers. In addition, a demographic study of BHs has been recently carried out by Volonteri et al. (Reference Volonteri, Dubois, Pichon and Devriendt2016b) within the same simulation. They show that the fraction of BH host galaxies is higher at higher stellar masses and that multiple BHs are hosted in the most massive halos as a consequence of merger events. A population of dual AGN, a central, and an off-centre accreting BH, is found in the simulated halos.

Recent ALMA observations presented by Trakhtenbrot et al. (Reference Trakhtenbrot, Lira, Netzer, Cicone, Maiolino and Shemmer2017a) have revealed a large number, ~50%, of massive star-forming galaxies interacting with quasar hosts (within <50 kpc scales). The authors argue that this may support the idea of major merger-driven growth playing an important role in the formation of SMBHs in high-redshift quasars, at least those showing sub-mm galaxy (SMG) companions. The z ~ 5 quasar in the sample shows similar properties in terms of BH mass and bolometric luminosity but varies in terms of host galaxy properties (see Netzer et al. Reference Netzer, Mor, Trakhtenbrot, Shemmer and Lira2014 and Trakhtenbrot et al. Reference Trakhtenbrot, Lira, Netzer, Cicone, Maiolino and Shemmer2017a for details on the sample), suggesting different accretion mechanisms may be operating in different environments.

5.3. The role of gas accretion

Semi-analytic techniques have been largely employed to study the role of different gas accretion regimes and/or the effect of dynamical processes in the early growth of SMBHs (e.g. Volonteri et al. Reference Volonteri, Haardt and Madau2003, Reference Volonteri and Rees2005a; Begelman et al. Reference Begelman, Volonteri and Rees2006; Volonteri & Rees Reference Volonteri and Rees2005b, Reference Volonteri and Rees2006b; Tanaka & Haiman Reference Tanaka and Haiman2009; Volonteri et al. Reference Volonteri, Silk and Dubus2015).

Volonteri & Rees (Reference Volonteri and Rees2006b) show that the observed high-z SMBH masses can be reproduced starting from low-mass seeds (~100 M_⊙) if they accrete gas at super-Eddington rates, at early stages. Super-Eddington accretion is a selective and biased process, occurring only for a small fraction of BH seeds if they form in metal-free atomic cooling (T _vir ⩾ 10⁴K) halos (e.g. Volonteri & Rees, Reference Volonteri and Rees2005b, Reference Volonteri and Rees2006b).

Gas accretion rates that are 10⁴ times higher than the Eddington rate can be reached by low-mass seeds in super-Eddington models (e.g. Volonteri & Rees Reference Volonteri and Rees2005b, Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016, and references therein). However, mildly super-Eddington intermittent accretion at ${\sim }3-4 \dot{M}_{\rm Edd}$ (or in general, $<20 \dot{M}_{\rm Edd}$ ) may be efficient enough to grow an SMBH in less than 800 Myr (at z ~ 7) starting from a single (e.g. Madau et al. Reference Madau, Haardt and Dotti2014: Figure 2) or a population (e.g. Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016: Figure 5) of 100 M_⊙ BH seeds.

In Figure 7, we show the plot presented by Volonteri & Rees (Reference Volonteri and Rees2006b) to illustrate the SMBH mass growth along the merger tree of a 10¹³ M_⊙ halo at z = 6. The figure depicts the effect of different accretion regimes and/or radiative efficiencies on the mass assembly of a ~100 M_⊙ seed that starts accreting at z = 24: Eddington-limited with a radiative efficiency ε = 0.1, 0.2, 0.4 (solid, short-dashed, and dot-dashed lines, respectively) and super-Eddington (long-dashed line). Radiatively efficient gas accretion disks (ε > 0.1) strongly limit the growth of their BH, even while accreting continuously at the Eddington rate.

Figure 7. The growth of a low-mass seed BH mass as a function of redshift in different regimes: Eddington-limited gas accretion with radiative efficiencies ε = 0.1, 0.2, 0.4 (solid, short-dashed and dot-dashed lines, respectively); super-critical accretion (long-dashed line). The figure is taken from Volonteri & Rees (Reference Volonteri and Rees2006b).

The requirement for episodic, radiatively inefficient, super-critical gas accretion onto stellar mass seed of 20–100 M_⊙ is supported by sub-pc resolution hydrodynamical simulations presented by Lupi et al. (Reference Lupi, Haardt, Dotti, Fiacconi, Mayer and Madau2016). They compare two different methods, the Adaptive Mesh Refinement technique used in the code ramses, and the Lagrangian Godunov-type method adopted in gizmo. In addition, 3D radiation magnetohydrodynamic simulations suggest that super-Eddington accretion flows can drive the rapid growth of low-mass BHs simultaneously, enabling high levels of both radiative and mechanical feedback (Jiang, Stone, & Davis Reference Jiang, Stone and Davis2014). On the other hand, super-critical accretion onto low-mass BH seeds is not supported by radiation hydrodynamic models for BH formation in Hii regions, which instead suggest rather low rates of accretion, below (or at most close to) the Eddington limit (e.g. Milosavljević, Couch, & Bromm Reference Milosavljević, Couch and Bromm2009a; Milosavljević et al. Reference Milosavljević, Bromm, Couch and Oh2009b; Park & Ricotti Reference Park and Ricotti2011, Reference Park and Ricotti2012, Reference Park and Ricotti2013).

Very recently, the analysis of a sample of 20 quasars, including ULAS J1120 and SDSS J0100 at z ⩾ 5.8 presented by Trakhtenbrot, Volonteri, & Natarajan (Reference Trakhtenbrot, Volonteri and Natarajan2017b) suggest that the inferred BH masses and luminosities can be naturally explained by means of a classical thin accretion disk model, with radiative efficiencies in the range [0.04–0.4] and sub-Eddington accretion rates. This support the idea that super-critical growth may have occurred at earlier cosmic epochs (z > 10, e.g. Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016).

Super-Eddington gas accretion regime is not only adopted for low-mass seeds growth. In their recent analytic model, Volonteri et al. (Reference Volonteri, Silk and Dubus2015) show that galactic inflow rates as high as 1–100 M_⊙ yr⁻¹ may trigger a sequence of fast (10⁴–10⁷ yr) episodes of super-critical accretion, onto both low-mass or high-mass seeds, at rates which are 10²–10⁴ times larger than in the Eddington-limited scenario (see Volonteri et al. Reference Volonteri, Silk and Dubus2015: Figure 2). As a result of these intermittent phases of short super-Eddington gas accretion, an SMBH can be produced.

In the super-Eddington scenarios, the radiatively inefficient slim disk model (Abramowicz et al. Reference Abramowicz, Czerny, Lasota and Szuszkiewicz1988) ensures that even in the presence of hyper-Eddington accretion ( ${>}>20 \dot{M}_{\rm Edd}$ ), the bolometric luminosity of the accreting BH is only mildly super-Eddington, L _bol/L _Edd ⩽ (2–4) (e.g. Mineshige et al. Reference Mineshige, Kawaguchi, Takeuchi and Hayashida2000; Volonteri & Rees Reference Volonteri and Rees2006b; Madau et al. Reference Madau, Haardt and Dotti2014; Volonteri et al. Reference Volonteri, Silk and Dubus2015; Pezzulli et al. Reference Pezzulli, Valiante and Schneider2016).

In Eddington-limited gas accretion scenarios, in which the BH can accrete at most at the Eddington rate, the formation of high-mass seeds, enabled by the LW radiative feedback is crucial to explain the fast growth of z ~ 6 SMBHs (see e.g. Johnson et al. Reference Johnson, Whalen, Li and Holz2013, the recent pSAMs of Petri et al. Reference Petri, Ferrara and Salvaterra2012; Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016, and the review by Johnson & Haardt Reference Johnson and Haardt2016). In their mixed-seed-based model, Valiante et al. (Reference Valiante, Schneider, Volonteri and Omukai2016) determine the relative contribution of low-mass and high-mass seeds to the final BH mass of J1148. They report that efficient Eddington-limited growth relies on the formation of ≈1–10 high-mass seeds in order to produce the expected SMBH mass at z = 6.4. If high-mass seed formation is prevented, the predicted final BH mass does not exceed ~10⁶ M_⊙, thus warranting the need for super-Eddington accretion in the low-mass seeds scenario.

Finally, a new cosmological semi-analytic model for galaxy formation, including the growth of SMBHs within a large box size (1.12 cGpc h ⁻¹) N-body simulation (hSAM), has been presented by Makiya et al. (Reference Makiya2016). Their model is currently tuned to reproduce the properties of local galaxies. Using this simulation, Shirakata et al. (Reference Shirakata2016) suggest that stringent constraints on the seed BH mass, may come from less massive bulges observed at z ~ 0, rather than the high-redshift BH-bulge mass relation. Their study suggests that the mass of BHs observed in ~10⁹ M_⊙ bulges is overpredicted if only seeding by high-mass seeds (10⁵ M_⊙) is considered. Such small stellar mass bulges instead favour seeding by smaller seed BHs (10³ M_⊙) or a mixed population of seed BHs randomly distributed in the mass range 10³–10⁵ M_⊙.

Numerical simulations of equal-mass protogalaxies encounters show that merger-driven gas inflows are able to trigger the formation (without requiring the suppression of star formation) and rapid growth of a massive BH (Mayer et al. Reference Mayer, Kazantzidis, Escala and Callegari2010) as well as of actively accreting SMBH binaries (Mayer et al. Reference Mayer, Kazantzidis, Madau, Colpi, Quinn and Wadsley2007). Recently, a suite of high spatial resolution simulations (~10 pc) have been devoted to study the effect of galaxy mergers on BH accretion, as a function of the initial merging galaxies’ mass ratio, orbital configuration, and gas fraction. These different stages of galactic encounters is described in Capelo et al. (Reference Capelo, Volonteri, Dotti, Bellovary, Mayer and Governato2015). They confirm that more efficient BH accretion is induced during galaxy mergers with the initial mass ratio being the most critical parameter affecting BH accretion and AGN activity.

In the simulations presented by Feng et al. (Reference Feng, Di Matteo, Croft and Khandai2014a) and Di Matteo et al. (Reference Di Matteo, Croft, Feng, Waters and Wilkins2016), the rapid growth of BHs, occurring in bulge dominated galaxies, is driven by large-scale filamentary cold gas accretion, rather than by major gas-rich mergers. Feng et al. (Reference Feng, Di Matteo, Croft and Khandai2014b) extract three DM halos from the cosmological hydrodynamical simulation MassiveBlack, hosting 10⁹ M_⊙ BHs and re-simulate them with zoom-in techniques. They find that dense cold gas is able to sustain accretion. During the accretion phase at the Eddington rate, the cold gas directly feeds the BH, while in the sub-Eddington phase (that they find for z ≲ 6), the accretion disc is disturbed and disrupted by feedback. A recent numerical simulation, including X-rays radiation transport, presented by Smidt et al. (Reference Smidt, Whalen, Johnson and Li2017) suggest that both the SMBH observed in ULAS J1120 and SDSS J0100 (at z ~ 7 and z = 6.3, respectively) can form from 10⁵ M_⊙ BH seeds (planted at z = 19.2) growing via cold accretion streams. The models reproduce the observed properties of the two quasars, such as the host galaxy mass, SFR, metallicity, luminosity, and ionised near zone, including the dynamical mass enclosed within the inner 1.5 kpc region of the ULAS J1120 host galaxy, inferred from recent ALMA observations (Venemans et al. Reference Venemans2017).

Although the numerous studies presented to date, we cannot yet draw firm conclusions on which growth mechanism (via super- or sub-critical accretion disks, cold accretion streams, mergers) and/or seed formation channel (low-mass vs. high-mass seeds) is to be preferred, or more viable than the others, for high-redshift SMBH formation.

5.4. BH feedback

As discussed in Section 2, the physical processes involved in quasar formation and evolution are expected to be regulated by AGN and stellar feedback. During the quasar-dominated regime (z ≲ 8, see Section 6.2), a strong, galaxy-scale wind is predicted to be driven by the energy released during both BH accretion and SN explosions. This feedback is expected to clear the ISM of gas and dust leaving a un-obscured line of sight towards the central emitting source. In addition, radiation emitted from the optically bright quasar J1148 may contribute to at least 30% of the observed FIR luminosity (>20 μm) heating the large amount of dust (~3 × 10⁸ M_⊙) in the host galaxy ISM, outside the un-obscured cone. Both stellar and quasar optical/UV emission are expected to be reprocessed by dust, thus contributing to the observed FIR luminosity (Schneider et al. Reference Schneider, Bianchi, Valiante, Risaliti and Salvadori2015).

Adopting an energy-driven wind prescription similar to that usually adopted by numerical simulations (e.g. Di Matteo et al. Reference Di Matteo, Springel and Hernquist2005), pSAMs show that the AGN feedback is the main driver of the massive observed gas outflow rates at z > 6. This is predicted to have a dominant effect with respect to stellar feedback (energy-driven winds from SN explosions) in shaping the high-z BH-host galaxy co-evolutionary path. For example, a powerful quasar-driven gas outflow is launched during the latest stages of the evolution (~100–200 Myr) in the best-fitting models of Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Maiolino, Salvadori and Bianchi2012 and Pezzulli et al. (Reference Pezzulli, Valiante and Schneider2016), for J1148. The predicted outflow rates are in good agreement with the observations, >1 000–3 000 M_⊙ yr⁻¹ (Maiolino et al. Reference Maiolino2012; Cicone et al. Reference Cicone2015) and ~10³ times more efficient than the sub-dominant SN-driven contribution.

However, it is worth noting that the prescription usually adopted in SAMs to describe the energy-driven wind effects cannot provide insights on the physical processes determining the observed properties of the outflowing gas and its complex dynamics.

Although described by sub-grid prescriptions, the response of the gas to the energy released by the accreting BH is now well described by hydrodynamical simulations. Costa, Sijacki, & Haehnelt (Reference Costa, Sijacki and Haehnelt2014) study AGN feedback using the moving-mesh code AREPO. They find that, despite the fact that momentum-driven outflows predict an M _BH − σ relation similar to that observed, the energy-driven scenario better reproduces the observed, large-scale anisotropic AGN-driven outflows. With the same code, Costa, Sijacki, & Haehnelt (Reference Costa, Sijacki and Haehnelt2015) re-simulate a zoom-in region around the six most massive halos at z ~ 6 to study the brightest quasars. They show that the high-velocity extended cold gas observed out to ~30 kpc (Maiolino et al. Reference Maiolino2012; Cicone et al. Reference Cicone2015) requires the combined effect of SN and AGN feedback. SN-driven winds are responsible for the pre-enrichment of the circumgalactic and intergalactic medium in which the massive, fast ( > 1 400 kms⁻¹) AGN-driven hot outflow is launched, ensuring efficient radiative cooling (see e.g. Figure 2 in Costa et al. Reference Costa, Sijacki and Haehnelt2015) to explain the presence of cold gas (see e.g. Cicone et al. Reference Cicone2015).

Finally, high velocity (10²–10³ km s⁻¹) energy-driven winds on large scales have been recently also studied by Bieri et al. (Reference Bieri, Dubois, Rosdahl, Wagner, Silk and Mamon2017) by means of radiation-hydrodynamic simulations of isolated galactic discs. They suggest that outflow rates as high as ~10³ M_⊙ yr⁻¹ are sustained by IR radiation, with scattering on dust grains enabling efficient momentum transfer to the gas.

6.1. The origin of high-z dust

Several theoretical models have been devoted to the study of the rapid enrichment of the ISM in z > 6 galaxies and quasars, and in particular to the origin of the huge amount of dust (>10⁸ M_⊙) inferred from the FIR and sub-mm observations (e.g. Hirashita & Ferrara Reference Hirashita and Ferrara2002; Morgan & Edmunds Reference Morgan and Edmunds2003; Dwek, Galliano, & Jones Reference Dwek, Galliano and Jones2007; Dwek & Cherchneff Reference Dwek and Cherchneff2011; Valiante et al., Reference Valiante, Schneider, Bianchi and Andersen2009, Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014; Gall et al. Reference Gall, Hjorth and Andersen2011a, Reference Gall, Andersen and Hjorth2011b; Mattsson Reference Mattsson2011, Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014; Calura et al. Reference Calura, Gilli, Vignali, Pozzi, Pipino and Matteucci2014).

An SN origin for the dust observed in the early Universe has often been advocated because of the shorter evolutionary time scale of core collapse SNe progenitors (10–40 M_⊙ stars, with an age <10 Myr) with respect to that of AGB stars (e.g. Morgan & Edmunds Reference Morgan and Edmunds2003; Dwek, Galliano, & Jones Reference Dwek, Galliano and Jones2007). This scenario was supported by the deviation of the dust extinction curves of z > 4 quasars and gamma ray bursts (GRB) from the Small Magellanic Cloud (SMC) extinction curve, typical of z < 2 quasars (Maiolino et al. Reference Maiolino, Schneider, Oliva, Bianchi, Ferrara, Mannucci, Pedani and Sogorb2004; Stratta et al. Reference Stratta, Maiolino, Fiore and D’Elia2007; Perley et al. Reference Perley2010; Gallerani et al. Reference Gallerani2010). This suggests either a different dust production mechanism or dust processing into the ISM at high redshift.

However, subsequent studies pointed out that stellar sources alone cannot account for the entire dust budget and grain growth in cold, dense gas clouds must also have a dominant role, even at z > 6 (e.g. Michałowski et al. Reference Michałowski, Murphy, Hjorth, Watson, Gall and Dunlop2010; Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011; Pipino et al. Reference Pipino, Fan, Matteucci, Calura, Silva, Granato and Maiolino2011; Rowlands et al. Reference Rowlands, Gomez, Dunne, Aragón-Salamanca, Dye, Maddox, da Cunha and van der Werf2014; but see Ferrara et al. Reference Ferrara, Viti and Ceccarelli2016).

Moreover, in contrast to what was previously thought, AGB stars are able to significantly contribute to dust production in high-redshift quasars, producing a dust mass at least similar to that of SNe, already at z ~ 8–10 depending on the host galaxies’ SFH and IMF (see Valiante et al. Reference Valiante, Schneider, Bianchi and Andersen2009 and Figure 8 in Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011).

Figure 8. The cosmic cycle of a typical quasars at z ~ 6. Models reproduce the properties of J1148 (see text). Left panel: the build-up of the M _BH–M _star relation through cosmic time as compared with data and empirical fit for local galaxies (Sani et al. Reference Sani, Marconi, Hunt and Risaliti2011). Middle panel: the predicted star formation history via quiescent and merger-driven bursts (see e.g. Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011). Left panel: the assembly of the dust mass into the ISM as a function of the stellar mass. In all panels, the solid lines show the average over 50 different DM halo merger trees with shades representing the 1σ dispersion. These figures are adapted from Valiante et al. (Reference Valiante, Schneider, Salvadori and Bianchi2011).

Modelling the properties, and in particular the evolution of dust, in quasar host galaxies at z > 6 is still a major challenge. Li et al. (Reference Li2007, Reference Li2008) carried out the first multi-scale simulation, using GADGET2 (Springel et al. Reference Springel, Di Matteo and Hernquist2005a), aimed to follow the formation of quasar J1148 in a hierarchical scenario, accounting for self-regulated BH growth (starting from Pop III seeds), AGN feedback and the host galaxy properties evolution. They showed that the metallicity and dust mass of J1148 are produced through a series of efficient bursts of star formation (see Figure 7 in Li et al. Reference Li2007) resulting in a final stellar mass of 10¹² M_⊙, similar to what is expected from the local M _BH–M _⋆ relation. To date, this is the only attempt to study the high-z dust properties made with numerical approaches (Li et al. Reference Li2008). However, only a single plausible hierarchical build-up of the J1148 DM halo, extracted (and re-simulated) from the 1h ⁻¹ Gpc³ volume is explored in these works and thus, the resulting SFH is unique. Semi-analytic models, which instead enable a statistical investigation of different SFHs, provide similar conclusions. The chemical properties of the host galaxy require an order of magnitude higher stellar mass with respect to the dynamical constraint, as discussed in the following sections.

6.2. The BH-host galaxy co-evolution

Observational campaigns at z > 5 show that quasars and their host galaxies are characterised by similar properties in terms of the BH, dynamical, dust, and molecular gas masses, suggesting a common evolutionary scenario.

In Figure 8, we show the best-fit evolutionary scenario for the BH and host galaxy properties of J1148 as predicted by Valiante et al. (Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014). Solid lines show the redshift evolution of the total massesFootnote ⁶ of BH and stars (on the left), the total SFR (in the middle), and dust and stars again (on the right) averaged over 50 different DM halo merger trees, with shaded areas representing the 1σ error.

As soon as efficient star formation starts, the BH grows in the buried AGN. At this stage, its optical emission is outshined by the ongoing strong star burst, SFRs from 100 up to >1 000 M_⊙ yr⁻¹, at z ~ 8 (middle panel). The mass of dust (right panel) rapidly grows, reaching values as high as 10⁹ M_⊙, when the bulk of the stellar mass, ~(2–4) × 10¹¹ M_⊙, is already in place. During this dust-obscured phase, the total nuclear BH mass reaches ~2 × 10⁸ M_⊙.

In this scenario, the progenitor galaxies of J1148 at z ~ 8–10 are predicted to have similar properties (e.g. BH, stellar, and dust mass) as the observed SMGs at lower redshifts (e.g. Santini et al. Reference Santini2010; Michałowski et al. Reference Michałowski, Murphy, Hjorth, Watson, Gall and Dunlop2010; Magnelli et al. Reference Magnelli2012). These sub-mm galaxies are suggested to be the evolutionary stage preceding the active quasar phase.

The transition between the starburst-dominated regime and the quasar-dominated evolution, at z < 8, is triggered by powerful energy-driven winds which clear up the ISM of dust and gas (see e.g. the down turn indicated by the black arrow in the right panel of Figure 8), un-obscuring the line of sight towards the quasar and damping the SFR (we will discuss the AGN feedback in the following section).

SMBH evolution models suggest a steeper evolution of the BH-stellar bulge mass relation at high redshift, with the SMBH forming before/faster than the stellar bulge (e.g. Petri et al. Reference Petri, Ferrara and Salvaterra2012). In addition, the observed deviation of high-redshift quasars from the local BH-stellar-mass ratio seems to be a natural outcome of SMBH growth driven by episodic super-Eddington accretion which leads to a BH accretion rate-to-SFR ratio of >10² (Volonteri et al. Reference Volonteri, Silk and Dubus2015).

Agarwal et al. (Reference Agarwal, Davis, Khochfar, Natarajan and Dunlop2013) track the subsequent growth of DCBH seeds by using a modified version of the Agarwal et al. (Reference Agarwal, Khochfar, Johnson, Neistein, Dalla Vecchia and Livio2012) hSAM. In their simulated volume, they find that the merger of a DCBH host satellite with the neighbouring galaxy (source of the LW radiation field), leads to the resultant system lying above the local M _BH–M _star relation, already at these early stages of the evolution. The authors term this phase as ‘obese black hole galaxies’ or OBGs as the DCBH is able to outshine the stellar component, leading to unique observables that distinguish them from normal galaxies. The OBGs are expected to transition onto the local M _BH–M _star relation via mergers. However, they do not account for the formation and evolution of metals and dust in the ISM, which represent a strong constraint on the host galaxy SFH and final stellar mass.

Chemical evolution models instead point out that SFR, gas, metals, and dust content of quasar host galaxies are well reproduced with standard assumptions of stellar IMF, star-formation efficiency, and dust grain growth, for galaxies with stellar masses ⩾10¹¹ M_⊙ (see left panel of Figure 8). These are about one order of magnitude higher than the stellar masses inferred from the observations of high-redshift quasars (e.g. Wang et al.; Reference Wang2010, Reference Wang2013) and would bring the predicted M _BH–M _star relation closer to the local value, suggesting that high-redshift dynamical (and thus stellar) masses may be underestimated (Valiante et al., Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014; Calura et al. Reference Calura, Gilli, Vignali, Pozzi, Pipino and Matteucci2014).

Although a top-heavy IMF scenario (i.e. biased to more massive stars) can reproduce the observed dust mass and the deviation of J1148 from the local M _BH–M _star relation, it requires a less-efficient SFH to do so. This results in an SFR at z = 6.4 that is more than 10 times smaller than the observed rate (Valiante et al. Reference Valiante, Schneider, Salvadori and Bianchi2011), too small to provide the observed FIR luminosity even if the AGN contribution to dust heating (Schneider et al. Reference Schneider, Bianchi, Valiante, Risaliti and Salvadori2015) is accounted for.

Instead, assuming a short evolutionary time scale does not solve the tension either. At the observed SFR ~1 000 M_⊙ yr⁻¹, the ~(3–4) × 10¹⁰ M_⊙ stellar mass estimated for quasars like J1148 would be produced in a quite short time interval, ~10–20 Myr. Such an evolutionary time scale is too short for stellar evolution to account for dust enrichment up to >10⁸ M_⊙, even with a maximally efficient mode of dust formation by SNe (see Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014 for a detailed discussion).

Following this discussion, it is important to note that, at z > 6, stellar masses cannot be convincingly obtained via SED fitting as in local and lower redshift systems. A lower limit to the stellar mass (dynamical bulge) is usually obtained as M _star = M _dyn − M _H2 where dynamical and molecular gas masses, M _dyn and M _H2 , respectively, are derived from CO observations.

Large uncertainties are introduced by the methods adopted to infer M _dyn and M _H2 . A large scatter (>60%) in the estimated molecular gas mass is due to the adopted CO line luminosity-to-H₂ mass conversion factor, α_CO = 0.8 ± 0.5 M_⊙/(K km s⁻¹ pc²). This is typical of ultra luminous infrared galaxies (ULIRGs, Solomon et al. Reference Solomon, Downes, Radford and Barrett1997; Downes & Solomon Reference Downes and Solomon1998) and usually adopted for high-redshift quasars too. In addition, M _dyn strongly depends on geometrical assumptions for gas distribution which is usually considered to be disk-like, with given inclination angle i and radius R, which are difficult to infer from observations at such high redshifts. An uncertainty of more than 50% must be associated to the inferred values, M _dynsin² i = (10¹⁰–10¹¹) M_⊙. A radius R = 2.5 kpc and an inclination angle i = 65 have been inferred for J1148, in which the CO emitting region is spatially resolved (Walter et al. Reference Walter, Carilli, Bertoldi, Menten, Cox, Lo, Fan and Strauss2004). For other quasars, a similar radius and a mean inclination angle of 40 are usually assumed (see e.g. Maiolino et al. Reference Maiolino2007; Wang et al. Reference Wang2010).

Theoretical studies suggest that lower inclination angle (i < 30) and/or larger disk radius (R ~ 5–30) kpc may solve the so-called stellar mass crisis (see e.g. Figure 9 and discussion in Valiante et al. Reference Valiante, Schneider, Salvadori and Gallerani2014).

Recent Atacama Large Millimeter and sub-mm Array (ALMA) observations of [Cii] emission in quasars have suggested that a large fraction of the CO may be still undetected (Wang et al. Reference Wang2013), supporting the idea that dynamical mass estimates could be missing some of the stars. Moreover, IRAM Plateau de Bure Interferometer (PdBI) follow-up observations of [Cii] 158μm emission line and FIR continuum in J1148 host galaxy have revealed the presence of an extended cold gas component out to ~30 kpc which may be an indication of star-formation activity on larger scales with respect to the size of the CO emission (Cicone et al. Reference Cicone2015).

Thus, stellar mass estimates from model predictions and observations may be reconciled by accounting for a more complex and/or more extended star and gas distribution, beyond the few kpc radius inferred from the CO emitting regions. Observations (Cicone et al. Reference Cicone2015), SAMs (Valiante et al., Reference Valiante, Schneider, Salvadori and Bianchi2011, Reference Valiante, Schneider, Salvadori and Gallerani2014; Calura et al. Reference Calura, Gilli, Vignali, Pozzi, Pipino and Matteucci2014) and numerical simulations (e.g. Khandai et al. Reference Khandai, Feng, DeGraf, Di Matteo and Croft2012) seem to agree with this scenario. Quasars at z ~ 5 resolved in the MassiveBlack simulation are predicted to be compact and gas-rich systems with intense burst of star formation occurring in both the innermost and outer regions, out to the DM halo virial radius (~200h ⁻¹ kpc, Khandai et al. Reference Khandai, Feng, DeGraf, Di Matteo and Croft2012).

In addition, Di Matteo et al. (Reference Di Matteo, Croft, Feng, Waters and Wilkins2016) show that the most massive BHs (>10⁸ M_⊙) at z ~ 8 reside in compact bulge-dominated galaxies (more than 80% of the stars are in the spheroidal component). The total stellar masses of these systems are already >10¹⁰ M_⊙ (see e.g. Figure 1 and Table 1 of Di Matteo et al. Reference Di Matteo, Croft, Feng, Waters and Wilkins2016), bringing them well within the scatter of the observed local M _BH–M _star relation. Pure SAMs provide very similar results.Footnote ⁷

Finally, Lyu, Rieke, & Alberts (Reference Lyu, Rieke and Alberts2016) derived a typical stellar mass of (3–5) × 10¹¹ M_⊙ on the basis of the IR SED analysis of about 100 quasars at z > 5, suggesting a BH-galaxy mass ratio of 10⁻³–10⁻², consistent with local relations.