3.1. Basic halo properties

To understand the basic properties of the halos that could be able to host DCBH events, we start our investigation by looking at their typical dark-matter masses, chemical content, thermal conditions, and star formation rate.

The halo samples at different redshifts contain objects with dark-matter masses between ~ 10⁵ M_⊙ and 10⁸ M_⊙. Their baryonic properties are affected by the assumed features of the primordial stellar populations, as well as of the emitting radiation spectrum.

In Figure 1, halo properties at z = 9 are shown for the run with top-heavy popIII IMF and black-body spectrum with T _eff = 10⁵ K (TH.1e5 model). At this epoch there are two star-forming sites with masses of about 2 × 10⁷ M_⊙ and 4 × 10⁷ M_⊙, respectively. The same stellar populations that provide UV photons are also responsible for enriching nearby regions up to metallicities of ~10⁻²Z_⊙ once they explode as supernovae. As a result, metal spreading involves both the halo hosting star formation and four additional halos in which star formation is not taking place. The powerful radiation emitted is responsible for destroying most of the molecular content in the simulated volume. The residual H₂ fraction (x _mol) usually lies below 10⁻¹⁰ and reaches values as low as 10⁻²⁰ in the smaller unshielded haloes. The gas mass-weighted temperatures vary correspondingly between 10³ K and a few times 10⁴ K due to radiative heating from primordial stars. The haloes mostly affected are the ones with small masses—below a few 10⁶ M_⊙—that are not dense enough to cool against photo-heating, are not able to efficiently self-shield, and suffer strong evaporation effects (Maio et al. Reference Maio, Petkova, De Lucia and Borgani2016). Most of the objects have been heated to temperatures around 10⁴ K, corresponding to average molecular fractions of x _mol ~ 10⁻¹³–10⁻¹⁵. Haloes with temperatures as low as ~10³ K still retain certain amounts of H₂, resulting in x _mol ~ 10⁻¹⁰–10⁻¹². The biggest halo undergoes star formation, has a mass of ~ 4 × 10⁷ M_⊙, an average fraction x _mol ≃ 10⁻¹¹, and a mass-weighted gas temperature T≃10⁴ K.

Figure 1. Mean H₂ content, x _mol, temperature, T, star formation rate, SFR, and metallicity, Z, as a function of the dark-matter mass, M _dm, of haloes at z = 9 for the run with a T _eff = 10⁵ K black body as popIII SED.

Different assumptions for the popIII IMF and SED have clear implications on the basic halo properties. In Figure 2 results corresponding to the run with Salpeter-like popIII IMF and T _eff = 4 × 10⁴ K black body (SL.4e4 model) are displayed. In this case, there are four star-forming haloes, that is, twice as much as the TH.1e5 case, although only the most massive halo with mass of ~4 × 10⁷M_⊙ is found to be enriched at Z≃10⁻²Z_⊙. The other three haloes (with masses between ~7 × 10⁶ M_⊙ and ~2 × 10⁷ M_⊙) do not feature signatures of metal enrichment, yet. This is not surprising, because, once compared to the previous TH.1e5 model, the SEDs of the SL.4e4 model are less powerful (up to 2 dex), hence radiative feedback is not able to rapidly shut off star formation in distant haloes and metal spreading is less efficient in enriching nearby haloes. The trend for the average molecular content in each halo, x _mol, shows that radiative feedback in the SL.4e4 scenario reduces the molecular fraction down to 10⁻⁸–10⁻¹⁵, but x _mol never reaches values of the order of 10⁻²⁰, as in the TH.1e5 case. This means that the gas is not heated up to very high values and stays confined below 10⁴ K, as shown by the trend for T as function of mass.Footnote ^c In particular, photon propagation seems to play a little role for the thermal behaviour of the haloes. There is a well-defined trend of increasing temperature for increasing mass, whose corresponding molecular content shows typical fractions x _mol ~ 10⁻⁸–10⁻¹². The few haloes that are affected by radiation deviate from the displayed increasing trend (low masses and gas temperatures of 2 × 10³–10³K) and suffer molecule destruction with x _mol values going down to ~10⁻¹³–10⁻¹⁵.

Figure 2. As Figure 1 for the run with a T _eff = 4 × 10⁴ K black body as popIII SED.

The most conservative scenario with a Salpeter-like popIII IMF and a T _eff = 10⁴ K black body as popIII SED (SL.1e4) is shown in Figure 3. The SL.1e4 model is the least powerful in terms of radiation emitted. As a consequence, it predicts more star-forming haloes (7) than in the TH.1e5 and SL.4e4 models, as well as localised metal enrichment in one single halo with Z ≃ 10⁻² Z_⊙, consistently with the previous considerations. In this scenario, radiative effects are negligible, as clearly visible from the trend of both x _mol and T. Molecules are not significantly dissociated and thus average molecular fractions never decline below 10⁻¹² at z = 9 and barely reach 10⁻¹³ at later times. The trend for mass-weighted gas temperatures is little affected by emitted photons, too. In this case, the chemical and thermal evolutions are mainly led by cosmological growth and mechanical feedback, rather than radiative feedback.

Figure 3. As Figure 1 for the run with a T _eff = 10⁴ K black body as popIII SED.

We mention that basic host properties at higher redshift do not show evident differences among the models considered, due to the limited structure evolution at early times. In all the cases, there is no or little star formation and metal enrichment; there is no relevant effect from radiative feedback; average molecular fractions are close to initial-condition values, i.e. x _mol ~ 10⁻⁴ –10⁻⁶; and the haloes are smaller, with mass-weighted gas temperatures between a few hundreds and a few thousands Kelvin.

We summarise mean halo properties for star-forming haloes in the different models in Table 2. Statistics are shown for three samples corresponding to the three different scenarios considered here, TH.1e5, SL.4e4, and SL.1e4. Each sample consists of all the haloes found at redshift z ≥ 9. The different columns refer to the mean values of stellar mass (M _⋆) in solar units, fraction of haloes hosting metal-enriched star formation (f _host), star formation efficiency (M _⋆/M _gas), gas fraction (f _gas), absolute UV magnitude in the AB system at 1 500Å (M_UV), bolometric luminosity (L _bol) in solar units, and number of ionising photons per second (Ṅ _ph,ion). AB magnitudes and luminosities are computed by employing the spectral templates for the emission at 1 500Å, as a function of stellar lifetimes and metallicities, from the stellar population synthesis code GALAXEV (Bruzual & Charlot Reference Bruzual and Charlot2003). We use the star particle properties as inputs, adopt the instantaneous burst model, assume a Salpeter IMF, and do not consider any nebular emission. Since GALAXEV is calibrated for enriched stellar populations with conventional low-mass IMFs (either Salpeter or Chabrier), the resulting magnitudes and luminosities lack the contribution of harder radiation from powerful popIII sources, that for a 10⁵ K black body accounts for roughly one dex increased emission at 1 500 Å, and that we consider for massive popIII stars. In the TH.1e5 scenario, where star formation is more inhibited by early powerful popIII stars, the resulting mean stellar mass is slightly smaller, although, due to the higher emitting power, the mean magnitude and bolometric-luminosity estimates are brighter than in SL.4e4 or SL.1e4. In these latter cases, star formation suppression due to radiative feedback is milder, hence mean stellar masses are ~0.3 dex larger, mean absolute AB magnitudes more than one unit fainter, and mean bolometric luminosities about one dex dimmer. The effects of radiative feedback in the different cases can be better revealed from star formation efficiencies and gas fractions. The TH.1e5 scenario has a lower mean star formation efficiency, M _⋆/M _gas, due to stronger gas heating and cooling suppression linked to powerful popIII sources. In the SL.4e4 and SL.1e4 scenarios, mean star formation efficiencies, M _⋆/M _gas, are a factor ~1.6 larger, because of the more limited impact of early solar-like stars on the surrounding gas. This is also reflected by their ~6% larger mean gas fractions that, in average, are less subject to photo-heating and/or photo-evaporation than in the TH.1e5 case. The fraction of haloes hosting metal-enriched star formation shows an opposite trend, highlighting the effects of metal spreading: the more powerful the source the higher its ability to enrich and contaminate the medium out to larger distances. For this reason, the mean f _host varies from unity in the TH.1e5 case down to 0.18 and 0.12 in the SL.4e4 and SL.1e4 cases, respectively. The average rate of ionising photons is estimated by Ṅ _ph,ion = f _escQ _iM _⋆, with f _esc escape fraction, Q_i ionisation parameter giving the number of ionising photons per second per unit mass of simple stellar population, and M _⋆ stellar mass. An escape fraction of f _esc = 0.5 is adopted,Footnote ^d consistently with expectations for star-forming haloes in the mass range around 10⁷ M_⊙. Q_i parameters are taken from the tabulated values of the evolutionary synthesis model by Schaerer (Reference Schaerer2003) for starbursts with given IMF and metallicity. For our SL.1e4, SL.4e4, and TH.1e5 models, we use Schaerer (Reference Schaerer2003)’s case A, case B, and case C, respectively. While the two scenarios for regular and massive stars SL.1e4 and SL.4e4 adopt a Salpeter IMF similar to Schaerer (Reference Schaerer2003)’s case A and case B, our extreme case TH.1e5 has no close equivalent in Schaerer (Reference Schaerer2003) and we have to rely on case C therein. TH.1e5 model results to be more powerful than SL.1e4 and SL.4e4 models by ~1 dex. We have to stress, though, that f _esc is a poorly known parameter in the literature and its value might span over a wide range. While Wise et al. (Reference Wise, Demchenko, Halicek, Norman, Turk, Abel and Smith2014) suggest average values of the order of 50%, other authors, such as Yoshida et al. (Reference Yoshida, Oh, Kitayama and Hernquist2007), suggest values closer to unity, that is, a factor of 2 larger, and dependent both on the particular stellar mass range considered and on the features of the environment. Because of these uncertainties, expected Ṅ _ph,ion values could vary sensibly.

Table 2. Mean halo properties at z ≥ 9 for the three different models adopted.

From left to right, different columns indicate: name of the model considered, mean stellar mass in solar units (1), mean fraction of haloes hosting metal-enriched star formation (2), mean star formation efficiency (3), mean gas fraction (4), mean absolute UV magnitude in the AB system at 1500Å (5), mean bolometric luminosity in solar units (6), mean number of ionising photons per second (7).

3.2. DCBH host candidates

To understand the properties of the haloes hosting DCBH events in the three models considered in this work, we select the simulated candidates by referring to the conditions listed and discussed in Section 2.2. The resulting trends and requirements for DCBH formation are shown in Figures 4 and 5 for z = 11.5 and z = 9, respectively. They refer to the baryon properties that need to be checked to investigate the possibility of a direct collapse of the gas. In each figure, the host gas mass is plotted against the molecular fraction for the TH.1e5 model (top), the SL.4e4 model (middle), and the SL.1e4 model (bottom). Since only gas (and not dark matter) would collapse directly into a black hole, the gas masses quoted in the figures are comparable to the expected mass of the DCBH which would be born from such process. Metal-enriched haloes are denoted by red triangles, star-forming haloes by blue diamonds, and haloes more massive than 2 × 10⁶ M_⊙ by green asterisks. Magenta squares refer to pristine non-star-forming haloes, while the resulting DCBH host candidates are highlighted by bullet points. The halo population at very early times (first hundreds of million years after the Big Bang) is dominated by pristine non-star-forming small haloes that retain their molecular content irrespectively from the radiative model considered for primordial stars. Larger amounts of molecules start to form only in halos with dark-matter mass higher than 2 × 10⁶ M_⊙ and gas content of about 3 × 10⁵ M_⊙. Because of the local metal pollution, ongoing star formation, and large molecular fraction, these primordial halos are not suitable candidates for hosting a DCBH, since local gas is going to cool below 8 000 K and fragment further (McCourt et al. Reference McCourt, Oh, O’Leary and Madigan2016). Most of the haloes are still H₂ rich and too small to induce a direct gas collapse and no DCBH candidates are found at these epochs.

Figure 4. Gas mass versus molecular fraction of the simulated haloes at z = 11.5 for runs with different popIII SEDs: TH.1e5 (top panel), SL.4e4 (middle panel), and SL.1e4 (bottom panel). Different symbols refer to different types of haloes: metal-enriched haloes (red triangles), star-forming haloes (blue diamonds), haloes with dark-matter mass larger than 2 × 10⁶ M_⊙ (green asterisks), pristine non-star-forming haloes (magenta squares), and DCBH host candidates (black bullets) that have no metals, no star formation, dark-matter mass larger than 2 × 10⁶ M_⊙, gas temperature larger than 8 × 10³ K, and molecular fraction lower than 10⁻¹³. No DCBH host candidates are present at this epoch in any of the radiative models.

Figure 5. At z = 9 there are three possible DCBH host candidates in the top panel. Gas mass versus molecular fraction of the simulated haloes at z = 9 for runs with different popIII SEDs: TH.1e5 (top panel), SL.4e4 (middle panel), and SL.1e4 (bottom panel). Different symbols refer to different types of haloes: metal-enriched haloes (red triangles), star-forming haloes (blue diamonds), haloes with dark-matter mass larger than 2 × 10⁶M_⊙ (green asterisks), pristine non-star-forming haloes (magenta squares), and DCBH host candidates (black bullets) that have no metals, no star formation, dark-matter mass larger than 2 × 10⁶M_⊙, gas temperature larger than 8 × 10³ K, and molecular fraction lower than 10⁻¹³.

Results at z = 11.5 are shown in Figure 4, about 380 million yrs after the Big Bang. The first stars have formed at z ≃ 14.5 and have evolved for about ~100 million yrs, while new stars are born in the meantime.

In the top panel (TH.1e5), we see that photon propagation dissociates molecules up to levels as low as x _mol ~ 10⁻⁸–10⁻¹⁴ and that, due to the powerful stellar feedback, there are only two star-forming haloes and four enriched haloes, two of which are quiescent non-star-forming objects polluted by nearby spreading events.

The SL.4e4 model (centre) produces similarly strong variations (with x _mol ~ 10⁻⁸–10⁻¹³), although mechanical and radiative feedback is not so extreme to push metals into other haloes (they rather remain confined in their birth place) and to shut off star formation in the larger ones. Consequently, there are several massive objects (green asterisks) with typical dark masses above 2 × 10⁶ M_⊙ and typical gas masses between a few times 10⁵ and 10⁶ M_⊙, but none of them is a good DCBH host candidate. In fact, these massive-enough haloes feature values of x _mol that are always higher than 10⁻¹³ and range around x _mol ≃ 10⁻¹²–10⁻¹⁰. Considering that lower-mass objects reach smaller x _mol values, it is clear that shielding effects in the denser regions of the bigger haloes play a crucial role to prevent DCBH formation (see also Sections 2.1 and 3.1).

The trends in the bottom panel follow from the weaker sources assumed for this case (SL.1e4). Thus, most of the haloes with dark-matter mass > 2 × 10⁶ M_⊙ undergo star formation and/or get enriched with metals, still keeping average molecular fractions of the order of x _mol ~10⁻⁶ – 10⁻⁴. Smaller haloes suffer minimal radiative effects and x _mol always stays above 10⁻¹⁰. The bulk of the haloes in this scenario is either ‘large’ and star forming or small and quiescent; therefore, they are unlikely to host DCBHs.

Environmental effects (such as mergers, feedback, and ongoing cosmological growth) contribute to spreading the trends in the plots.

We show the results at z = 9 in Figure 5. This redshift corresponds to a cosmic time of half Gyr, that is, slightly more than 100 million yrs after z = 11.5. At such epoch, the differences among the three models are striking.

In the TH.1e5 case (top panel), powerful radiative emissions from primordial stars have heavily dissociated H₂, so that most of the haloes have x _mol ~ 10⁻¹⁵ and the smaller ones get down to 10⁻²⁰. The two largest haloes (blue diamonds) with gas masses of about 2–4 × 10⁶ M_⊙ are forming stars and are metal enriched, as well as two other smaller nearby objects (red triangles). The quiescent pristine haloes with dark-matter mass larger than 2 × 10⁶ M_⊙, despite their low molecular fraction, are mostly cold, with gas temperatures below 8 × 10³ K (see Section 3.1). Thus, they cannot host DCBH formation. Only three objects with dark mass of roughly 2 × 10⁶ M_⊙ and gas mass of about 2–3 × 10⁵ M_⊙ are hotter and poor of molecules; hence they are good DCBH host candidates. We note that the minimum temperature threshold of 8 × 10³ K is mainly led by H collisional cooling ((e.g. Oh & Haiman Reference Oh and Haiman2002; Smith et al. Reference Smith, Bromm and Loeb2017)), active during cosmic structure formation, when diffuse cosmic gas falls into the growing potential wells of dark-matter haloes and gets shock-heated to typical temperatures of the order of 10⁴−10⁵ K. In absence of additional coolants, it is not possible to bring temperatures below ~8 × 10³ K. In the quiescent haloes mentioned above, gas is still at a diffuse stage, as demonstrated by their small masses. Thus, the hosted gas simply follows the thermal cosmic expansion; hence it is by no means able to collapse nor to survive nearby feedback events. Only haloes with dark-matter masses larger than ~ 2 × 10⁶ M_⊙ × are going to host collapsing events and survive nearby star formation or photo-evaporation feedback (Whalen et al. Reference Whalen, Abel and Norman2004; Jeon et al. Reference Jeon, Pawlik, Greif, Glover, Bromm, Milosavljević and Klessen2012; Wise et al. Reference Wise, Abel, Turk, Norman and Smith2012a,b, Reference Wise, Demchenko, Halicek, Norman, Turk, Abel and Smith2014; Maio et al. Reference Maio, Petkova, De Lucia and Borgani2016).

In the SL.4e4 case (middle panel), the behaviour of the gas is bimodal, with many pristine quiescent haloes having still an average molecular fraction x _mol > 10⁻¹³ and a few others lying at x _mol < 10⁻¹³. These latter are the ones that surround star-forming haloes (blue diamonds) and result affected by the nearby radiative feedback, although not by chemical feedback. These objects are too small to host DCBHs, though. Some of the largest haloes are undergoing star formation and/or metal enrichment; however, the remaining ones (green asterisks) are still cold or feature x _mol > 10⁻¹³. Thus, no eligible DCBH hosts are found for this model.

In the SL.1e4 case (bottom panel), the scenario is much less dramatic for the environment of star-forming halos. Indeed, radiation from weaker solar-like sources affects very marginally the thermal and chemical properties of the local gas and has little implications for external haloes. For this reason, star formation is not severely inhibited by radiation and all the objects with gas masses higher than ~ 10⁶ M_⊙ form stars and are locally enriched by metals. The entire population of primordial haloes has an average molecular content that is always above the threshold of 10⁻¹³, and therefore, it cannot host DCBH formation. In this case, gas molecular evolution is mainly led by mergers and feedback effects, while photon propagation acts as a minor character.

In general, cosmological evolution in the first-half Gyr is responsible for spreading the values displayed in the plots, mostly in the low-mass end that is very susceptible to environmental processes.

3.3. DCBH host candidate location

In Figure 6, we display the position where DCBH host candidates are found (bullet points) with respect to the dissociating radiative sources (blue diamond) at redshift z = 9 in the TH.1e5 model (top). These hosts have similar total masses slightly larger than 2 × 10⁶ M_⊙. The three candidates are found at comoving (physical) distances of about 72 (7.2) kpc/ h, 290 (29) kpc/ h, and 80 (8) kpc/ h from the central radiative source. For sake of clarity, they have been marked by letters A, B, and C, and we will refer to them in the following as candidates A, B, and C, respectively. Since temperature is the main driver of DCBH formation, the temperature map allows us to get hints about the thermal properties of cosmic gas in different environments.Footnote ^e Interestingly, DCBH candidates are all located near cosmic filaments that preserve mass-weighted temperatures of ~10⁴ K. Isolated haloes are found not to be suitable DCBH host candidates. This is not surprising, because early isolated objects are usually smaller (hence they do not satisfy the mass requirement for DCBH formation) and thinner (hence they can severely suffer radiative heating and photo-evaporation effects, instead of gas collapse). In clustered environments, such as filaments or filament intersections, there is a wider variety of halo masses, thermal conditions, and molecular content, so there are higher chances that DCBH formation requirements are fulfilled.

Figure 6. Top. Mass-weighted temperature map where we have marked the position of the DCBH host candidates at z = 9 for the run with top-heavy popIII SED (TH.1e5). The map is a projection of the simulated structures on the xy plane centred on the middle of the z-axis and within a slice of width equal to 1/20^th the box length. The DCBH host candidates are denoted by black bullets and the letters A, B, and C. Centre. Same as top panel, but for the run with OB-like popIII SED (SL.4e4). The positions of the haloes marked by the yellow bullets and the letters A, B, and C are the same as in the top; but in this case, they are not DCBH host candidates. Bottom. Same as top panel, but for the run with standard solar-like popIII SED (SL.1e4). Also in this case, the haloes marked by the yellow bullets and the letters A, B, and C are not DCBH host candidates.

As a comparison, in the figure we also check the thermal conditions of the corresponding haloes in the SL.4e4 (centre) and SL.1e4 (bottom) models. In both cases, the medium is typically colder because of more efficient molecular cooling and weaker radiative fluxes. In fact, haloes denoted by A, B, and C (yellow bullets) host colder gas with mass-weighted temperatures below 10⁴ K and ranging between roughly 10² and 10³ K. In the SL.4e4 scenario, haloes A and C have mass-weighted temperatures of ~10³ K, while halo B features values ≲10² K. In the weaker SL.1e4 scenario, instead, only halo A has a mass-weighted temperature of ~10³ K, while haloes B and C are both at ≲10² K. Given these typical thermal conditions, the gas in these haloes results too cold to experience direct collapse, irrespectively from the hosting dark-matter mass, chemical composition, and local star formation. The main reason why the temperatures of these haloes are lower than in the previous case lies in the adopted features of stellar sources. In these two cases, central sources are not powerful enough to reach haloes at large distances and to dissociate their molecular content. Therefore, the hosts A, B, and C evolve almost unaffected by them, can retain cold gas, and will probably fragment in the next epochs. With such thermal conditions they are unlikely to turn into DCBHs. These structures could be, instead, small diffuse cold objects that are just assembling and represent the theoretical counterparts of currently debated early damped Lyman-alpha systems or dwarf galaxies forming at the end of reionisation (for further discussions on these topics we refer the interested reader to available works in the literature, such as the ones by Simcoe et al. (Reference Simcoe, Sullivan, Cooksey, Kao, Matejek and Burgasser2012), Maio, Ciardi, & Müller (Reference Maio, Ciardi and Müller2013a), Keating et al. (Reference Keating, Haehnelt, Becker and Bolton2014), Bosman & Becker (Reference Bosman and Becker2015), Bosman et al. (Reference Bosman, Becker, Haehnelt, Hewett, McMahon, Mortlock, Simpson and Venemans2017), García et al. (Reference García, Tescari, Ryan-Weber and Wyithe2017).

3.4. DCBH host candidate radiative properties

The DCBH host candidates are exposed to external local radiation; therefore, it is interesting to estimate the average spectral intensity, J_v, they experience at different times and frequencies, v. As we are interested in the LW band, if not otherwise specified we will refer to this frequency range for the next calculations of J_v. To this aim, we focus on the locations of the three hosts identified at z = 9 at physical distances from the radiative sources of 7.2 kpc/ h (A), 29 kpc/ h (B), and 8 kpc/ h (C), respectively. Due to the cosine law for isotropic radiation, they observe a spectral intensity J_v = F_v / π, where F_v is the average monochromatic flux and π is the value of the solid angle under which each of the three host candidates ‘sees’ the radiation.Footnote ^f The average monochromatic flux emitted by radiative sources is given by luminosity divided by surface area and frequency bin. Under spherical approximation this leads to:

$$J_\nu = {{F_\nu } \over \pi } = {{\dot N_{{\rm{p}}h} h_p \nu } \over {4\pi ^2 r^2 \Delta \nu }},$$(1)

with Ṅ _ph number of ionising photons per second, h_p Planck constant, v central frequency of the considered frequency range, r distance, and Δv frequency bin. To have an estimate of the expected order of magnitude for v LW central frequency (corresponding to 12.4 eV) and Δv LW band (corresponding to [11.2, 13.6] eV), it is convenient to rewrite the previous expression as

$$J_{{\rm{L}}W} \simeq 9.1 \times 10^{ - 21} \left( {{{\dot N_{{\rm{p}}h} } \over {10^{50} {\kern 1pt} {\rm{s}}^{ - 1} }}} \right)\left( {{{{\rm{k}}pc} \over r}} \right)^2 {\rm{e}}rg/s/cm^2 /Hz/sr.$$(2)

The redshift evolution of the spectral intensity, in units of J ₂₁ = 10⁻²¹ erg/s/cm²/Hz/sr, to which the three halos, A, B, and C are exposed to, is displayed in Figure 7. We note that such commonly adopted reference unit, that is, J ₂₁, represents a quite large spectral intensity, being equal to 100 Jansky per steradiant.

Figure 7. Spectral intensity in the LW band, J _LW, in units of J ₂₁ at the locations of the three DCBH host candidates (A, B, and C) identified at z = 9 as function of redshift, z. For each candidate, as indicated by the legends, the figure shows the actual amount of LW radiation produced at each snapshot (dotted lines), the cumulative LW radiation resulting from the sum of all the LW radiation produced until any given redshift z (dashed lines), and the radiation entering the LW band estimated by including redshifted photons from earlier times (solid lines).

The actual production of LW photons (due to radiation emitted by sources at that specific redshift), considering the processes of stellar mass growth and loss, is described by the dotted lines, while their cumulative distribution is given by the dashed lines. In order to quantify the exact amounts of LW radiation influencing the DCBH candidates at z = 9, the solid lines include the integrated effect of line shift into the LW band of LW photons produced at earlier times. For the sake of simplicity, we treat the LW central line at 12.4 eV as representative of the whole LW band. This is not a crude approximation, because the range of Δv around the central frequency is rather small. Thus, we impose a redshift constrain for z _ref = 9 of |Δz / (1 + z _ref)|< Δv / v ≃ 0.2, or z ≲ 11. This means that, roughly speaking, LW photons produced until z ~11 result redshifted in the LW band itself at z = 9, hence increasing the total amount of radiative intensity experienced by the three involved structures. We additionally take into account harder UV energies that can be shifted into the LW band. We limit our calculations only to a few representative linesFootnote ^g at about 15, 15.4, 24.6, 44.5, 54.4 eV. They increase the resulting LW intensity according to their relative contribution to the adopted black-body shape, ∝ v ³[1 − exp(h _pv/kT)]⁻¹, rescaled by redshift as [(1+z)/ (1+z _ref)]⁻⁴. These lines are produced at earlier times, when they have higher frequency than the LW central frequency. During cosmological evolution they get redshifted into the LW range and become dimmer.

The values experienced by the three DCBH host candidates are quite heterogeneous. Candidates A and C feature actual values (dotted lines) that are always above J ₂₁, with an initial burst of J _LW ~ 10J ₂₁ and z = 9 values of 2–3J ₂₁. Candidate B instead is located further away and is exposed to radiation of spectral intensity around only ~ 0.1−1J ₂₁. In all the three cases the cumulative amount of LW photons emitted (dashed lines) is always higher than 2J ₂₁ at z = 9 and sums up to 20–30J ₂₁ for candidates A and C. When including line shift (solid lines), these latter ones result exposed to values of J _LW ≃ 50J ₂₁ and J _LW ≃ 40J ₂₁, respectively. In this case, the radiation entering the LW band from previous epochs is accounted for. In general, line shift into the LW band from earlier sources appears to have a certain relevance (a factor of ~ 2). Despite the large uncertainties on the exact critical level of LW radiation, we definitely find the three candidates in conditions where resolved molecular cooling is severely inhibited (i.e. when molecular gas is exposed to J _LW > J ₂₁) and that favour DCBH formation. We highlight that the presence of metal spreading inhibits DCBH formation where J _LW is larger, that is, closer to stellar sources, around or below kpc distance (see Equation 2). Differently from more idealised setups, these findings strengthen the role of metal pollution for assessing DCBH formation within numerical three-dimensional studies (see, however, Valiante et al. Reference Valiante, Schneider, Volonteri and Omukai2016). When checking similar radiative properties for the other two runs, we find J _LW values that are always smaller than J ₂₁ due to the 100-times lower adopted Ṅ _ph. This explains why the two runs with solar-like and OB-type sources fail in producing viable DCBH host candidates and is consistent with the thermodynamical trends presented in the previous section.

Finally, as a warning, we point out that literature works are very uncertain about J _LW critical values and different authors give different LW thresholds varying in the large range between 1 and 10⁵J ₂₁. Thus, there is no general consensus about the exact critical flux. For example, Shang et al. (Reference Shang, Bryan and Haiman2010) run three-dimensional simulations of pristine gas and manage to identify different candidates in the presence of LW fluxes of 1–1 000J ₂₁, for standard stars, and of 1–10⁵ J ₂₁, for primordial stars. For the production of the observed population of z ~ 6 black holes, large values of the order of ~10⁴ –10⁵J ₂₁ are considerably too high and it is by no means clear whether such fluxes are in fact required for the formation of massive objects. In addition, the stabilising impact of viscous heating alleviates the need for a strong UV background to keep the gas atomic and objects more massive than 10⁴ M_⊙ can form even with moderate values of J _LW ~ 100J ₂₁ (Latif & Schleicher Reference Latif and Schleicher2015). Analyses of simulations by, for example, Latif et al. (Reference Latif, Bovino, Van Borm, Grassi, Schleicher and Spaans2014b) conclude that the typical mean flux may vary from halo to halo, although values larger than ~ 1 500J ₂₁ are rare, while massive objects still form for radiative fluxes of 10–500J ₂₁ (Latif et al. Reference Latif, Schleicher, Bovino, Grassi and Spaans2014c). Habouzit et al. (Reference Habouzit, Volonteri, Latif, Dubois and Peirani2016b,a) find that, depending on the feedback scheme and the metal spreading implementation, the critical flux must be at least one or two orders of magnitude lower than predicted by pristine-chemistry models. These works practically set an upper limit of J _LW << 1000J ₂₁ (in clear contrast with e.g. Sugimura et al. Reference Sugimura, Omukai and Inoue2014) and favour smaller values around 30–300J ₂₁. Yoshida et al. (Reference Yoshida, Abel, Hernquist and Sugiyama2003) even suggest values of the order of J ₂₁ to prevent primordial gas runaway cooling (for a deeper discussion we refer the interested reader to the review by Valiante et al. Reference Valiante, Agarwal, Habouzit and Pezzulli2017) Furthermore, pristine-chemistry studies finding very large J _LW thresholds in the close vicinity of star-forming regions typically neglect stellar evolution and metal pollution from stellar sources. This can seriously affect the results about critical J _LW values. Given the large uncertainties, we have no reason to favour the results of some authors with respect to other authors and, in practice, the only general agreement among the various literature studies is that required critical J _LW values should be between J ₂₁ and 10³J ₂₁.

3.5. DCBH host candidate structure

To discuss the possibility that a halo candidate selected with the basic necessary requirements mentioned above turns in fact into a DCBH host, it is important to understand whether the local halo structural properties could have an impact on the actual gas direct collapse. Indeed, mergers and/or substructure formation alter the gas thermal state and lead to fragmentation, inhibiting a ‘pure’ direct collapse. For this reason we check the presence of substructures in the three halo candidates.

The biggest halo at z = 9 (candidate A) has no sub-haloes, while the other two candidates (candidate B and C) are composed by one major halo and one smaller sub-halo.Footnote ^h The shape of the three candidates can be retrieved from the left panels of Figure 8, where gas over-density maps, δ, of the DCBH host candidates at z = 9 are displayed. For each candidate, we show the projection on the xy plane containing the vertical coordinate of the centre of mass. From the δ distributions, three equally spaced contour levels are derived and overplotted in black solid lines. Candidate A is a quiescent halo featuring a fairly spherical structure, mostly on the xy plane. On the contrary, candidates B and C show more irregular shapes because of the presence of a main halo and a smaller sub-halo. The presence of a smaller halo (on the right) constituting candidate B is well visible from the central stream bridging it with the main halo (on the left). Candidate C, instead, is clearly constituted by two distinct bound objects. The projection highlights that active interactions between the two components (merging event) are taking place and are responsible for the asymmetric compression of the material surrounding them.

Figure 8. Left: Maps of the gas over-density with respect to the mean, δ, of three DCBH host candidates at z = 9 for the run with a T _eff = 10⁵ K black body as popIII SED. The maps are obtained via projection of each candidate on its xy plane containing the vertical coordinate of the centre of mass and smoothed over a grid of 128 pixels a side. Three equally spaced isocontour levels are overplotted in solid black lines. The colour scale is the Log₁₀ δ. Right: Over-density profiles, δ, of gas (dashed lines) and total-matter (solid lines) as function of the physical radial distance, r, for candidate A (top), B (centre), and C (bottom).

Despite their local molecular content being rather low and the typical gas temperature being around 10⁴ K, the multiple structure of candidates B and C suggests that the occurrence of a pure direct collapse is unlikely for them and that they are not favoured DCBH hosts. On the contrary, candidate A is composed by one single quiescent gaseous structure with mean temperature of about 10⁴ K and is not subject to evident fragmentation (x _mol << 1) nor merger activity.

The presence of satellites in two out of three cases implies that the formation of a pure DCBH is a rare event, even in those haloes where the basic criteria mentioned in Section 2.2 are met.

The right panels of Figure 8 show corresponding over-density profiles of the candidates A (top), B (centre), and C (bottom) as function of the radius normalised to the virial one. Gas profiles and total-matter profiles are displayed in the three panels by dashed and solid lines, respectively. The three profiles are computed over radial shells around the centre of mass of each halo; therefore, the differences in the shapes of the objects are smeared out. Over-densities reach values ≳10³ (consistently with the maps on the left panels) and in all cases the declining trend for both gas and total content is well visible. In particular, gas behaviour is very regular due to the lack of ongoing star formation and feedback effects. These results are consistent with quiescent haloes, since active star-forming structures are expected to have larger central over-densities, δ. For example, molecular cooling ignites catastrophic runaway and collapses at typical number densities of ~1 cm⁻³, that is, at values roughly 10⁵ times the critical density (with a modest z dependence).Footnote ⁱ

The profiles of the total-matter content are led by dark-matter, while the gas profiles show some divergence towards the centre of about a factor of 2 for candidates A and C, and of a factor of 3 for candidate B. Smaller deviations from the total trend can be observed even at larger distances, where δ ~ 10–10². Since the candidates B and C contain substructures, it is unlikely that they will collapse directly and form a DCBH, because, in order to do so, their substructures should merge during collapse. However, mergers are commonly accompanied by gas compression and shocks that cause H₂ (re)formation capable of stimulating violent thin-shell instabilities even in the presence of intense LW flux and for a significant range of shock velocities, as well as a boost of HD abundances determined by the increased H₂ fractions (Shapiro Kang 1987; Ahn & Shapiro Reference Ahn and Shapiro2007; Whalen & Norman Reference Whalen and Norman2008; Petkova & Maio Reference Petkova and Maio2012). More quantitatively, the cooling timescale, t _cool, for H-dominated monoatomic gas around 10⁴ K can be written as

$$t_{{\rm{c}}ool} = {3 \over 2}{{k_{\rm{B}} T} \over {\Lambda n_{\rm{H}} }} \simeq 10^3 {{\left( {T/10^4 {\rm{K}}} \right)} \over {\left( {\Lambda /10^{ - 22} {\rm{e}}rg{\kern 1pt} s^{ - 1} {\kern 1pt} cm^3 } \right)n_{\rm{H}} }} {\rm{y}}r,$$(3)

where k _B is the Boltzmann constant, T the gas temperature, Λ the cooling function, and n _H the gas H number density (see e.g. Figure 4 in Maio et al. Reference Maio, Dolag, Ciardi and Tornatore2007).

During merger events, gas compression, molecule reformation, and tidal effects are likely to make the gas fragment into several clumps, instead of leading the formation of a massive black-hole seed (mediated by the collapse of a single supermassive star; Hosokawa, Omukai, & Yorke Reference Hosokawa, Omukai and Yorke2012; Hosokawa et al. Reference Hosokawa, Yorke, Inayoshi, Omukai and Yoshida2013, Reference Hosokawa, Hirano, Kuiper, Yorke, Omukai and Yoshida2016).

3.6. DCBH host candidate thermal and chemical characteristics

In Figure 9 we show the main chemical and thermal characteristics of the three candidates.

Figure 9. Left: Radial profile of electron fraction (dotted lines), H₂ molecular fraction (solid lines), and mass-weighted temperature (dashed lines) for candidate A (top), B (middle), and C (bottom). The right scale (in red) refers to temperature values in Kelvin. Right: Evolution as a function of redshift, z, of the enclosed mass within the innermost 50 pc/ h (solid lines), of the H₂ molecular mass times 10⁵ (dotted lines) and of the expected inflow rate (dashed lines). The right scale (in blue) refers to the values of the inflow rate in solar masses per year.

On the left column we show radial profiles of electron fraction, H₂ molecular fraction and mass-weighted temperature for the candidate A (top raw), B (middle raw), and C (bottom raw). On the right column, the redshift evolution of the mass within the innermost 50 physical pc/h, M ₅₀, is displayed for the enclosed material, as well as for the corresponding H₂ molecular mass and inflow rates.

The radial profiles are computed under spherical approximation and by assuming that the centre of the halo corresponds to the position of its most bound particle.Footnote ^j The resulting shapes are very similar for all three candidates. Due to dissociating radiation, residual H₂ fractions are always tiny, with values about 10⁻¹⁴ (left column in Figure 9), while electron abundances vary between ~ 10⁻⁴ (in the neutral central regions with T ≲ 10⁴ K) and fractions of unity (around or beyond the virial radii, ≳ 0.3 kpc/h, where T ≳ 10⁴ K and H atoms start getting ionised). These trends are consistent with the profiles of gas temperature (dashed line, right scale in the panels) that give typical values of about or slightly lower than 10⁴ K in the innermost denser regions and slightly larger in the diffuse outskirts. This is in line with the corresponding gas density profiles of these haloes (dashed lines in previous Figure 8) and consistent with previous studies by, for example, Kitayama et al. (Reference Kitayama, Yoshida, Susa and Umemura2004), Whalen et al. (Reference Whalen, Abel and Norman2004) and Abel, Wise, & Bryan (Reference Abel, Wise and Bryan2007), who showed that ionising shocks from massive popIII sources can leave behind a warm ~3 × 10⁴ K diffuse medium. Ionisation and recombination processes around 10⁴ K are very fast and give origin to the patterns observed in the three profiles of the electron fraction. Given the negligible amount of available H₂ molecules, the gas of these pristine haloes cannot ignite metal-free cooling nor fragmentation below ~8 × 10³ K.

On the right column of Figure 9, the redshift evolution of the three candidates, A, B, and C, is computed by tracing in time their progenitors and by evaluating the mass in a sphere of 50 pc/h radius around the most bound particle, M ₅₀, at each snapshot, for both the enclosed halo mass and the H₂ content. We also compute the corresponding mass inflow rate, $$\dot M_{{\rm{in}}} \sim c_s^3 /G$$, with c_s sound speed and G gravitation constant (O’Shea & Norman Reference O’shea and Norman2007, Reference O’shea and Norman2008). Mass evolution appears to be quite smooth. The progenitors of the considered haloes feature M ₅₀ values increasing by one order of magnitude, from roughly 10⁴ M_⊙ at z ≃ 20 up to almost 10⁵ M_⊙ at z ≃ 9. In particular, candidate A reaches M ₅₀ ≃ 8 × 10⁴ M_⊙, candidate B M ₅₀ ≃ 7 × 10⁴ M_⊙, and candidate C M ₅₀ ≃ 4 × 10⁴ M_⊙. The evolution of candidate B is the most regular one, instead candidates A and C that are closer to the star-forming regions suffer more photo-heating at lower redshifts, as visible from the small decline of M ₅₀ at z ~ 10 and its subsequent stabilisation. Since the total mass of these candidate haloes is around 10⁶ M_⊙, the amount of material enclosed in the innermost 50 pc/h constitutes a kernel of almost 1/10^th the halo mass. The most striking event is definitely the dramatic decrease of H₂ molecules, that, during the star formation episodes at z ≳ 10, are destroyed down to fractions < 10⁻¹³ by the dissociating radiation coming from the newly established star-forming regions. As a comparison, mass profiles do not show such sudden variations. The expected inflow rates (right scale in the plots) evolve accordingly to the mass growth and are sensitive to the thermal state of the gas. The final value achieved by candidates A and B is almost 2M_⊙/yr, while candidates C features Ṁ_in ≃ 1M_⊙/yr. These values are reached only in the final part of their evolution (z ≃ 9 – 10), when the kernel mass M ₅₀ stabilises and Ṁ_in steepens. Before z ~ 10 typical inflow rates are small, Ṁ_in << 10⁻¹ M_⊙/yr. We warn the reader that exact estimates for the inflow rates bear dependences on assumptions and environment. Standard commonly used rates of $$0.975 c_s^3 /G$$ refer to spherical hydrostatic isothermal clouds accreting at constant rate (Shu Reference Shu1977; Hunter Reference Hunter1977). Violent inflow rates of $$46.84 c_s^3 /G$$ removed are expected for Larson (Reference Larson1969)–Penston (Reference Penston1969) self-similar solutions after a discontinuous jump from an initial value of $$29 c_s^3 /G$$ removed (Whitworth & Summers Reference Whitworth and Summers1985) Time-dependent accretion rates with initial central values of $$47 c_s^3 /G$$ removed were suggested by Foster & Chevalier (Reference Foster and Chevalier1993) who also found a rapid decline at later times to $$ \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} 10 c_s^3 /G$$. Modest inflow rates in more realistic axisymmetric MHD contracting cloud undergoing runaway collapse may vary up to maximum rates of $$40 c_s^3 /G$$, with late-phase lower limit $$ \sim 2.5 c_s^3 /G$$ and time-averaged value of $$ \sim 4 c_s^3 /G$$ (Galli & Shu Reference Galli and Shu1993a,b; Tomisaka Reference Tomisaka1996; Safier, McKee, Stahler 1997). These values are not dramatically different from more recent studies of magnetic and non-magnetic clouds (Mellon & Li Reference Mellon and Li2008; Machida & Doi Reference Machida and Doi2013; Susa, Doi, & Omukai Reference Susa, Doi and Omukai2015. With respect to these considerations, the values showed in the right plots of Figure 9 might represent a lower limit and actual rates can episodically reach values of a few to tens times higher.

These trends clearly suggest that the three candidates A, B, and C can remain metal free during their entire lifetimes, because they are sufficiently far from star-forming haloes and metal-free star formation is prevented by the fact that destroyed H₂ molecules in the halo progenitors are not able to form again by z ~ 9.

3.7. DCBH host candidate rotational patterns

Independently of the structure of the haloes, rotational patterns play an important role for the occurrence of gas collapse. In fact, gas rotational motions will halt direct collapse and prevent DCBH formation. For this reason, we investigate the angular momentum per unit mass of the material hosted by the DCBH host candidates, j = r × v, with r and v particle position and velocity vectors (Binney & Tremaine Reference Binney and Tremaine2008). For each particle, we compute the circularity ε = j _z/j _circ distributions, being j _z the component of the angular momentum perpendicular to the plane of rotation and j _circ the expected angular momentum of a circular orbit in the gravitational potential determined by the enclosed mass, M, at the same radial distance, r, that is, j _circ = r v _circ, with v _circ = [GM(r)/r]^1/2. Obviously, ε is sensitive to the rotational patterns of the constituting halo particles and can give us hints about the dynamics of the material inside the hosting halo. Values of ε≃0 represent test particles that have negligible angular momentum and that can eventually collapse. The closer is ε to unity, the closer is the motion of a test particle to a stable (i.e. non-collapsing) circular orbit. Larger ε values refer to unbound particles that move along escaping orbits.

In Figure 10 the distributions of the ε values for the gas of the three DCBH candidates are plotted. The trends reflect the structure of the haloes, and the spread of the distributions is consistent with the variety of statistical properties of rotational patterns expected at these early times (Biffi & Maio Reference Biffi and Maio2013; de Souza et al. Reference de Souza, Ciardi, Maio and Ferrara2013; Prieto et al. Reference Prieto, Jimenez, Haiman and González2015). In particular, candidate A is an isolated object with a quite regular distribution, while candidate B shows a double peak that is linked to its internal substructure. The candidate C shows a very irregular behaviour as a consequence of its disturbed state due to the ongoing internal interactions. Values of |ε| > 1 refer to particles which are not dynamically bound to the structure. Their amounts vary significantly in the three host candidates. For candidate A, a fraction ≲20% of gas parcels is going to escape, while the other two candidates feature larger amounts of escaping material, ranging from more than 25% for candidate B up to about 75% for candidate C. The unbound material has little implications for candidates A and B, which are going to preserve most of their gas mass. However, for candidate C the situation is more critical, as it is going to be left with a gas mass that is almost one order of magnitude smaller. This complicates the formation of any intermediate-mass black hole from this halo.

Figure 10. Distributions of the gas circularity, ε, for the three DCBH host candidates A (top), B (centre), and C (bottom) at z = 9 for the run with a T _eff = 10⁵ K black body as popIII SED.

A similar conclusion is reached in Figure 11, where the ratio between j_z and the absolute value of the mean halo angular momentum, |j _mean|, for both gas and dark-matter is considered. This ratio is linked to disk formation in the inner structure of the halo and provides a measure of the alignment of the angular momentum of every single particle with the mean angular momentum.

Figure 11. Distributions of the ratios between gas and dark-matter j _z values and the mean halo angular momentum for the three DCBH host candidates A (top), B (centre), and C (bottom) at z = 9 for the run with a T _eff = 10⁵ K black body as popIII SED.

Candidate A shows a gas ratio distribution that is peaked around zero and declines symmetrically as a consequence of its roughly spherical shape. Candidates B and C, instead, have distributions that decline in a non-symmetric way, with a more prominent high-j _z tail. This arises from the particles of the sub-halo that are located at distances larger than those of the main halo.

The trends of gas and dark-matter are quite similar, although for j _z/ |j _mean| ≃ 0 there is a higher gas peak in the gas distribution. This originates mostly from the central regions of the halo, where gas settles in a spherical shape, and also explains why the difference is more marked in candidate A, rather than in the more irregular candidates B and C.

The tails of the distributions for gas and dark-matter angular momenta have similar behaviours for values different from zero, meaning that the dynamical state of those gas particles is significantly influenced by the underlying dark-matter distribution.

3.8. DCBH candidate host turbulent motions

Turbulence is another important limiting factor for gas direct collapse, since it enhances gas fragmentation and hinders DCBH formation. Strictly speaking, turbulence arises from the nonlinear advection term, (u · ▽) u, and the dissipative term, v ▽²u, in the equations of motion for a fluid whose velocity field is u and kinematic viscosity is v. The (dimensionless) Reynolds number quantifies the effects of inertial forces with respect to viscosity forces and is defined as Re ≡||(u · ▽) u|| / ||(v ▽²)u|| ~ ul/v, where l is the typical scale of the turbulent motion. When Re tends to zero the system is viscosity dominated and turbulence decays. The scale at which Re = 1 is denominated dissipation scale. When Re >> 1 the advection term dominates and the influence of viscous forces is negligible. In most practical situations this usually happens for fluids with Re values above 4 000 that are commonly considered turbulent. As an example, in the cool ISM, it is well established since long time that the Reynolds number assumes values between 10⁵ and 10⁷ (Cordes, Weisberg, & Boriakoff Reference Cordes, Weisberg and Boriakoff1985; Armstrong, Rickett, & Spangler Reference Armstrong, Rickett and Spangler1995).

Quantitatively, the kinematic viscosity of the gas, v, can be estimated as v ~ c / nσ, with c sound speed, n mean number density, and σ ~ 10⁻¹⁵cm² typical interaction cross section (Elmegreen & Scalo Reference Elmegreen and Scalo2004). Consequently, Re depends on the Mach number of the gas and its thermal state. Despite these idealisations, in reality turbulence is not uniformly distributed but shows clear spatial and temporal intermittency effects: regions particularly active coexist with regions completely inactive. In general, it is possible to take into account these effects, but they will not be considered here, because the expected deviations are small in comparison to the order-of-magnitude estimates presented below.

Figure 12 shows results for Re as expected at z = 9. To have an idea of the global behaviour of the gas in each halo, we estimate Re by employing the (physical) quantities describing each object, as computed at post-processing time, both via friend-of-friend algorithm (velocities, radii, and masses at over-densities of 200 and 500 with respect to the background) and substructure finder (namely, velocity dispersion, maximum radial velocities and corresponding radii, as well as substructure masses). Given the different nature of all these quantities, Re results are not expected to coincide exactly, but should give us a broad overview of the orders of magnitudes reached case by case.

Figure 12. Reynolds number estimated with different approaches (Re _sub, Re _max, Re ₅₀₀, and Re _vir) as a function of substructure mass, M _sub, mass within an over-density of 500, M ₅₀₀, and virial mass, M _vir, at z = 9 for the run with a T _eff = 10⁵ K black body as popIII SED (see details in the text). Black bullet points highlight the values for DCBH host candidates (three friend-of-friend objects overplotted on the red diamonds and cyan asterisks). Main haloes and substructures/satellite haloes are overplotted, respectively, by black and magenta bullets on the sequences of blue triangles and green crosses.

We show them as function of the substructure mass, M _sub, the mass within an over-density of 500, M ₅₀₀, and the virial mass M _vir to get the corresponding Re _sub, Re _max, Re ₅₀₀, and Re _vir values. In particular, Re _sub is obtained by substructure velocity dispersion and maximum radial velocities (blue triangles); Re _max is obtained by maximum radial velocities and the radii at which such velocities are reached (green crosses); Re ₅₀₀ is obtained by velocity dispersions and radii within an over density of 500 (red diamonds); Re _vir is obtained by velocity dispersion and virial radii (cyan asterisks).

We warn that the smallest haloes, despite their virial values can be obtained, do not always have well-defined quantities for Re _sub and Re _max calculations, due to the limited number of their constituting particles.

Bullet points highlight the values for the DCBH host candidates identified in the previous sections. Consistently with what mentioned before, there are three friend-of-friend objects candidate to host a DCBH and they are over-plotted both on the sequences of red diamonds and on the sequences of cyan asterisks. Among these three halo candidates, two are composed by a main halo and a satellite, for resulting five substructures that are over-plotted by bullets on the green crosses and the blue triangles (black bullets refer to main haloes and magenta bullets to satellites).

Independently from the details about Re in the figure, the turbulent nature of the primordial haloes is striking. The values obtained for bigger objects vary between roughly 10⁵ and 10⁸, depending on the assumptions; however, they are in broad agreement with previous studies of early star-forming haloes (Maio et al. Reference Maio, Khochfar, Johnson and Ciardi2011b) as well as with the values expected in turbulent ISM environments. Moreover, in most cases Mach numbers are close to unity, which makes early turbulent motions nearly supersonic.

Figure 12 shows the effects of different ways of estimating Re; however, it does not give us information on the role of metallicity for DCBH candidates. For this reason, in Figure 13 the results for metal-free haloes (grey symbols) are directly compared to the results for haloes that are not metal-free (color symbols).

Figure 13. Reynolds numbers, estimated with the same approaches as Figure 12, are shown in grey for pristine haloes and with coloured symbols for metal-enriched haloes.

In this case, cyan asterisks refer to metal-enriched haloes whose Re numbers have been computed through virial quantities (i.e. Re _vir vs M _vir). They lie along the trend of the whole halo population, although their Re > 10⁵ values are higher than DCBH candidates (bullets) both at larger (~10⁷ M_⊙) and smaller (≲ 10⁶ M_⊙) masses. Similarly, also red diamonds (Re ₅₀₀ vs M ₅₀₀ for metal-enriched haloes) are above the bullet points, roughly following the general trend. In the high-mass end, the larger Re values are mainly due to the larger masses of haloes above ~ 10⁷ M_⊙. In the low-mass end (see red diamond and cyan asterisk at masses ≲ 10⁶ M_⊙), Re values are affected by the metal enrichment process.

The values denoted by green crosses (Re _max vs M _sub) and blue triangles (Re _sub vs M _sub) are computed by using substructure information and refer to both enriched haloes and satellites. Their distribution is quite sparse; however, most of the points lie above the trend inferred from the bullets by up to 1 dex. In this respect, DCBH candidates host lower chaotic motions than metal-enriched haloes, although they are in line with the expectations for the most quiescent pristine haloes (grey symbols).

The physical reason for such behaviour relies on the additional entropy injected during metal pollution from nearby star-forming regions. Enriched material spreading into the surrounding objects affects the velocity and thermodynamical structure of hosting haloes, possibly causing an increase in the resulting Re with respect to DCBH host candidates and pristine haloes. This is particularly clear from the deviations from the general trend of the smaller enriched structures (asterisks and diamonds). Re values of larger polluted objects, instead, seem to be less sensitive to metal spreading, due to the higher degree of stochasticity implied by their bigger masses.

The wide scatter in the relations denoted by crosses and triangles indicates a strong influence of cosmic environment and substructure formation, too, that will influence velocity and thermodynamical features depending on whether haloes are in clustered or isolated regions (in fact, the spread is wider at the low-mass end, that is dominated by small structures and satellites). The position of DCBH host candidates on the bottom edge of these two samples in the figure reveals their formation preferentially in weakly clustered regimes or in isolation, where shocks from structure formation Wise & Abel Reference Wise and Abel2007; Wise et al. Reference Wise, Abel, Turk, Norman and Smith2012b) are less common.

From these considerations it emerges that DCBH host candidates are on average less turbulent structures than metal-enriched haloes (diamonds and asterisks) and are among the lowest-turbulence structures with pristine composition in the corresponding mass range (crosses and triangles).

Figure 14. Distributions of the Reynolds number estimated at the hydro smoothing length scale for the three DCBH candidates A (solid line), B (dotted line), and C (dashed line). The distributions are normalised to their peak value.

To judge better the level of turbulence within the DCBH hosting structures we compute the Re values for their constituting gas particles, by employing the gas SPH smoothing length (hsml) and the gas velocity. This allows us to assess turbulent motion locally rather than globally. In Figure 14 we show the distribution of Re numbers at the SPH smoothing scales (hsml). We find a broad trend within Re ≳ 10³ and ~ 10⁵ for all the three candidates, with peak values at Re ~10⁴ –10^4.7 for candidate A, Re ~ 10⁴ for candidate B, and Re ~ 10^3.6 – 10^4.2 for candidate C. The distributions are very disperse, but all of them lie at Re ≳ 10^3.6 ~ 4 000, meaning that even at particle level, turbulent motions are relevant. The orders of magnitude of these local estimates are consistent with the orders of magnitude of the global estimates shown above, once taken into account the different typical scales (radius vs hsml) involved in the two cases. In practice, they show that turbulence plays an important role at all resolved scales.

We have checked that by adopting the mean square velocity, instead of the particle velocity, as typical velocity of the fluid, the results about Re remain unchanged.

3.9. DCBH host candidate fate

The presence of turbulence might affect direct collapse, providing additional non-thermal pressure support to the gas. Since the turbulent dissipation timescale can be written as l/σ ≈ 4 Myr (l/40 pc)(σ/10 km/s)⁻¹, with l turbulent scale and σ velocity dispersion (Semenov, Kravtsov, Gnedin 2016), the timescales during which direct-collapse events are possible lie between ~ 4 and 40 Myr. The former estimate is led by turbulence decay, while the latter by molecular-chemistry arguments. In practice, the conditions for DCBH formation hold less than 40 Myr and hence the direct collapse should be as rapid as that. As an example, the innermost isothermal core of the candidate A at z = 9 has a typical dimension of ~ 100 pc. Reasonably, this value is comparable or larger than the turbulent decay scale and, considering the typical σ ~ 10 km/s, the resulting dissipation timescale is ≲10 Myr.

Hydro- and magneto-hydrodynamical simulations generally show turbulence dissipation on times that are much shorter than crossing times, both in subsonic and supersonic regimes (Stone, Ostriker, & Gammie Reference Stone, Ostriker and Gammie1998; Mac Low et al. Reference Mac Low, Klessen, Burkert and Smith1998; Burkert Reference Burkert2006; Kim & Basu Reference Kim and Basu2013; Semenov et al. Reference Semenov, Kravtsov and Gnedin2016), albeit still of the order of a few Myr.

So, the occurrence of turbulent motions should delay DCBH formation until turbulence dissipates.

At the same time, two-third of the candidates show substructures, which means that a pure direct collapse is rather unlikely and the process must be eventually accompanied by a merger (Latif et al. Reference Latif, Bovino, Grassi, Schleicher and Spaans2015; Becerra et al. Reference Becerra, Greif, Springel and Hernquist2015).

Merger events and substructure evolution (as e.g. in candidates B and C) are likely to inhibit gas direct collapse, enhancing gas fragmentation and halting accretion via tidal forces (in agreement with Chon et al. Reference Chon, Hosokawa and Yoshida2018).

Local rotational patterns and photo-evaporation effects (see previous sections) have maybe less severe implications, as they could cause the loss of only a fraction of the gas hosted in the halo potential wells.

Thus, it turns out that the basic requirements necessary for DCBH formation, whenever met, do not guarantee the actual gas direct collapse. In practice, the required necessary conditions are not sufficient for pure DCBH formation in two-third of the cases, while DCBH is probably expected for the remaining candidate A.

Figure 15. DCBH host candidates at z = 8.5 for the run with a T _eff = 10⁵ K black body as popIII SED.

What happens next will strongly depend on the surroundings of the host halos. If gas photo-evaporation is stronger than the host potential wells (no matter whether they are generated by the whole halo or the newly formed DCBH) accretion will be inhibited. Otherwise, accretion is expected to be episodic. When averaged over several duty cycles, it should typically proceed at sub-Eddington rates (Johnson & Bromm Reference Johnson and Bromm2007; Milosavljević et al. Reference Milosavljević, Bromm, Couch and Oh2009; Maio et al. Reference Maio, Dotti, Petkova, Perego and Volonteri2013b; Ricci et al. Reference Ricci2017). In cases when the surrounding environment is very dense, it is possible to reach super-Eddington rates (Wyithe & Loeb Reference Wyithe and Loeb2012; Alexander & Natarajan Reference Alexander and Natarajan2014; Madau, Haardt, & Dotti Reference Madau, Haardt and Dotti2014; Inayoshi et al. Reference Inayoshi, Kashiyama, Visbal and Haiman2016; Pezzulli, Valiante, & Schneider Reference Pezzulli, Valiante and Schneider2016; Pezzulli et al. Reference Pezzulli, Volonteri, Schneider and Valiante2017). The existence of massive black holes at z ~ 7 is challenging to explain, because it requires a growth process taking place in less than a billion years. More likely, the growth mechanisms could be active for only a few 100 s Myr, since massive black-hole seeds are usually expected at redshifts of z ~ 6–20, after the first structures form. Currently, no observational detections of DCBHs are available. Studies of possible identifications of DCBH candidates in the CANDELS/GOODS-S survey have claimed only two candidates with predicted mass greater than 10⁵ M_⊙, with robust X-ray detection and with photometric redshift z > 6 (Pacucci et al. Reference Pacucci, Ferrara, Grazian, Fiore, Giallongo and Puccetti2016). In theory, ad hoc seeding with 10⁵M_⊙ black holes at z ≳ 10 of all the haloes with mass larger than 10⁹ M_⊙ (as commonly done in large numerical simulations) could lead to Eddington accretion rates at z = 9–6 and to black hole masses higher than 10⁹ M_⊙ at z ≲ 7. Nevertheless, early attainment of a minimum mass of at least 10⁶ M_⊙ remains crucial for a fast growth. These considerations suggest that DCBHs could be interesting massive seeds, since the 10⁵ M_⊙ black-hole seeds commonly adopted in large numerical simulations are easily justified with this channel, although it is quite unlikely that all the early haloes blindly seeded in simulations are actual hosts of DCBHs.

To conclude, we briefly investigate thermal and chemical properties of the halo populations identified at a time later than z = 9, specifically at z = 8.5 (when the Universe is about 0.58 Gyr old). In Figure 15, DCBH host candidates (bullet points) for the run with top-heavy IMF and a T _eff = 10⁵ K black body as popIII SED (TH.1e5) are shown for redshift z = 8.5. Once compared to the corresponding results of Figures 4 and 5 at z = 11.5 and z = 9, the changes in the molecular content are evident. They are a consequence of the strong dissociating photon field established by primordial massive stars. As mentioned before, at z = 9.5 no DCBH host candidate is found, while a few Myrs later, at z = 9, we have identified three haloes satisfying the basic requirements (previous Figure 5). Later on, at z = 8.5 we find three additional systems. Of the three z = 8.5 DCBH candidates (mapped in Figure A1 in Appendix), two are composed by two substructures. The third candidate is, instead, an isolated object (see host properties in Figure A2 in Appendix). Notwithstanding the similarities, the three DCBH host candidates identified at z = 8.5 do not coincide with the ones in Figure 5 at z = 9. Their comoving (physical) distances from the central star-forming region are about 348 (37) kpc/h 179 (19) kpc/h and 48 (5) kpc/h, respectively. In comparison to the situation at z = 9, these candidates are slightly farther from the radiation sources, but they are still found in proximity of denser filamentary regions. Given the presence of substructures, rotational patterns (Figure A2), photo-heating, and turbulent Re numbers (Figures A3, A4), the formation of DCBHs in the z = 8.5 candidates is as difficult as for the z = 9 candidates.