A Radial Dependence of Ly\alpha Trapping

# Enhanced direct collapse due to Lyman alpha feedback

###### Key Words.:
radiative transfer – cosmology: theory – black hole physics – dark ages, reionization, first stars – quasars: supermassive black holes – molecular processes

We assess the impact of trapped Lyman cooling radiation on the formation of direct collapse black holes (DCBHs). We apply a one-zone chemical and thermal evolution model, accounting for the photodetachment of H ions, precursors to the key coolant H, by Lyman photons produced during the collapse of a cloud of primordial gas in an atomic cooling halo at high redshift. We find that photodetachment of H by trapped Lyman photons may lower the level of the H-dissociating background radiation field required for DCBH formation substantially, dropping the critical flux by up to a factor of a few. This translates into a potentially large increase in the expected number density of DCBHs in the early Universe, and supports the view that DCBHs may be the seeds for the BHs residing in the centers of a significant fraction of galaxies today. We find that detachment of H by Lyman has the strongest impact on the critical flux for the relatively high background radiation temperatures expected to characterize the emission from young, hot stars in the early Universe. This lends support to the DCBH origin of the highest redshift quasars.

## 1 Introduction

The direct collapse scenario for black hole (BH) formation in the early Universe has received much attention in recent years, in particular for its ability to explain the formation of BHs with masses 10 M within the first billion years of cosmic history (e.g. Mortlock et al. 2011; Wu et al. 2015). The key ingredients for the formation of the massive ( 10 M) seed BHs in this theory are (1) primordial gas collapsing into an atomic cooling dark matter halo and (2) a sufficiently low fraction of H molecules in the gas to prevent cooling below the 10 K cooling limit of atomic hydrogen (for reviews see Volonteri 2012; Haiman 2013; Johnson & Haardt 2016; Latif & Ferrara 2016).

The main ways that are envisioned for keeping the primordial gas devoid of molecules is photodissociation of H due to so-called Lyman-Werner (LW) radiation at energies 11.2 - 13.6 eV and photodetachment of the H ion, which is an intermediary in the formation of H (e.g. Bromm & Larson 2004), by photons with energies 0.76 eV (e.g. Chuzhoy et al. 2007). The relative importance of each of these processes has been found to be strongly dependent on the spectrum of the incident radiation (e.g. Shang et al. 2010; Sugimura et al. 2014; Glover 2015; Agarwal et al. 2015; Latif et al. 2015; Wolcott-Green et al. 2016), presumably produced by a nearby star-forming galaxy (e.g. Dijkstra et al. 2008; Agarwal et al. 2012; Visbal et al. 2014; Regan et al. 2016a).

An additional source of radiation which contributes to the photodetachment of H and so limits the formation rate of H is the trapped Lyman cooling radiation that is emitted from the collapsing atomic gas in the halo itself (Spaans & Silk 2006; Schleicher et al. 2010). Here we explore the impact that this trapped radiation has on the production of H molecules in the gas and, in turn, on its ability to cool below the 10 K required for direct collapse black hole (DCBH) formation. In the next Section, we outline the one-zone chemical and cooling model that we employ for our study and we describe our approach to modeling the photodetachment of H by Lyman cooling radiation. In Section 3 we present the basic results of our calculations, and in Section 4 we explore the impact of Lyman feedback on the critical LW flux required for DCBH formation. Finally, we give our conclusions and provide a brief discussion of our results in Section 5.

## 2 Feedback from Lyman α Cooling Radiation

For our study, we begin with the same one-zone model for the collapse of the primordial gas as presented in Johnson & Bromm (2006), which is very similar to other one-zone models that have been routinely applied to DCBH formation (e.g. Omukai et al. 2005, 2008; Schleicher et al. 2010; Agarwal et al. 2016a). The model assumes that the density of the primordial gas increases on the free-fall timescale, and the non-equilibrium chemical and thermal evolution of the collapsing gas is calculated. All of the pertinent primordial chemical species are included, as are all of the pertinent radiative processes.

While the reader is referred to Johnson & Bromm (2006) for more details, here we describe the key ingredients in the model that we draw on for our study of the direct collapse scenario. One important update to this code has been the adoption of the H self-shielding prescription presented in Wolcott-Green et al. (2011; see also Hartwig et al. 2015, Wolcott-Green et al. 2016), which replaced the simpler prescription presented in Bromm & Loeb (2003). We have also updated the collisional dissociation rate of H to that presented in Martin et al. (1996), which is now the commonly adopted rate (e.g. Shang et al. 2010; Agarwal et al. 2016a). The model includes the main cooling processes that are relevant for the direct collapse scenario, which are atomic hydrogen line cooling and molecular (H) line cooling (e.g. Cen 1992; Abel et al. 1997). In addition, the model tracks the non-equilibrium chemistry of the primordial gas and the formation of H molecules, the main channel for which is the following two reactions:

 H+e−→H−+γ (1)
 H−+H→H2+e−\ , (2)

where e is a free electron and is a photon. Given that H is the main precursor to H, the photodetachment of H is a key reaction to track in order to accurately calculate the formation rate of H. Thus, we track the photodetachment of H as well as the photodissociation of H in our model, adopting the rates presented in Shang et al. (2010) as functions of the temperature of the radiation field.1

We solve additional equations in order to assess the impact of photodetachment of H by Lyman photons. To begin, we make the simple assumption that the luminosity of Lyman cooling emission in the cloud balances the rate of gravitational potential energy release during the collapse of the cloud (e.g. Dijkstra et al. 2016). This is a sound approximation, as it has been shown in numerous cosmological simulations that the collapse of primordial gas in atomic cooling halos is roughly isothermal and occurs on the free-fall timescale (e.g. Wise et al. 2008; Regan & Haehnelt 2009). Thus, we adopt the following simple expression for the Lyman luminosity:

 LLyα=GM2cloudrcloud1tff\ , (3)

where = (3 / 32) is the free-fall time, where is Newton’s constant and is the density of the collapsing gas. Here = 10 M is the typical mass of the central gas cloud collapsing in an atomic cooling halo (e.g. Wise et al. 2008; Johnson et al. 2011, 2014; Latif et al. 2013; Choi et al. 2013). Assuming a uniform cloud density, which is appropriate for our simplified one-zone calculations, this implies a cloud radius of where is the number density of hydrogen nuclei. As the gas cools, this is the characteristic length scale over which Lyman photons must diffuse in order to escape the cloud.

The diffusion of Ly photons out of the cloud enhances the energy density in Ly photons by an amount that depends on the cloud column density, . The total line center optical depth to Ly is given by , where is the temperature of the gas and the column density of hydrogen atoms is = (e.g. Osterbrock & Ferland 2006). Following Adams (1975, see also Smith et al. 2017 for an updated discussion), the pathlength traversed by the photons in escaping the cloud is enhanced by a factor ), where = 4.7 10 ( / 10 K) is the Voigt parameter. We estimate the total energy density in Ly radiation, , to be

 uα=MFLLyαrcloudVcloudc, (4)

where denotes the volume of the cloud, and equals

 MF∼60(n102cm−3)2/9(M106M⊙)1/9(T104K)−1/3. (5)

Assuming that the Lyman radiation field is isotropic within the cloud due to the large optical depth to scattering, we can then approximate the photodetachment rate of H ions by Lyman photons as

 Rdetach=σH−MFLLyαELyα34πr2cloudB\ , (6)

where the cross section for this process is = 5.9 10 cm and = 10.2 eV is the energy of a Lyman photon (e.g. de Jong 1972; Shapiro & Kang 1987). As the dependence on the optical depth to scattering shows, this rate is elevated due to the many scatterings that Lyman photons make in passing out of the cloud. Finally, accounts for the fact that spatial diffusion of Ly photons does not necessarily uniformly enhance the Ly intensity throughout the cloud, especially when Ly emission is concentrated more towards the center of the cloud (see Fig. A.1). In the Appendix we show that can be as large as toward the center of the cloud, which is where the DCBH forms. Throughout, we will investigate the impact of varying within the range .

Combining the above equations, we obtain the following expression for the photodetachment rate as a function of cloud temperature, mass and density:

 Rdetach≃10−8s−1(Mcloud106M⊙)109(T104K)−13(n102cm−3)3118(B2)\ . (7)

This is the equation that we include in our calculations in order to assess the role that Lyman feedback plays in the formation of DCBHs.

## 3 Basic Results

Here we show our results for two sets of calculations, one in which the effect of H detachment by Lyman photons is included and another in which it is neglected. In both cases, we also include the effect of a background LW radiation field, which is assumed to only contribute to the photodissociation of H molecules and not to the detachment of H ions. In the next Section, we explore how the inclusion of the photodetachment rate due to the background radiation impacts the evolution of the collapsing gas. Finally, here we only consider cases with = 1, corresponding to the simplest case of uniform Lyman emission from the collapsing cloud. We explore cases with higher values, corresponding to strongly centralized emission, in the next Section.

Figure 1 shows the evolution of the H fraction of the gas, as a function of density, both with and without the above equations for Lyman photodetachment included. The three sets of curves correspond to different values of the LW background radiation field , which is expressed in the standard units of 10 erg s cm Hz sr. As expected, the H fraction is steadily depressed as the level of the background radiation increases. The impact of Lyman photodetachment is also evident, resulting in the peak H abundances dropping by orders of magnitude in the cases with relatively high background radiation levels 100.

The thermal evolution of the gas in these same sets of calculations is shown in Figure 2.2 Due to the depressed H fraction, molecular cooling is less effective with higher levels of the background radiation field. However, in all cases shown here the gas is still able to cool to 10 K when H detachment is not included in the calculation. With this effect included, the cooling of the gas is suppressed at high density, resulting in much higher temperatures. Importantly, we find that with photodetachment included, the temperature remains high enough for DCBH formation in the case with = 10. Thus, H detachment by Lyman photons has the effect of lowering the critical value of the background radiation level required for the formation of DCBHs.

To more fully elucidate the impact of photodetachment, the photodetachment rates in our calculations, as a function of the cloud density, are shown in Figure 3. The density and time dependence of the photodetachment rate makes comparison with previous determinations of the critical rate of photodetachment for DCBH formation difficult (e.g. Sugimura et al. 2014; Agarwal et al. 2016a; Wolcott-Green et al. 2016), as constant photodetachment rates have typically been assumed. However, it is clear that the photodetachment rates we find rise well above the critical value of 10 s found, for instance, by Agarwal et al. (2016a) for the case of a weak H-dissociating radiation field. Thus, in this sense, our results are consistent with, and can be understood in the context of, previous work. Noting from Figure 2 that the gas evolves roughly adiabatically up to 10 cm such that , the scaling provides a good match to our calculations, as shown in Figure 3. At higher densities an isothermal scaling of provides a better fit, as is also shown in the Figure. We next turn to assessing the impact of Lyman feedback on the value of the critical LW flux required for DCBH formation.

## 4 The Impact on the Critical Lyman-Werner Flux

Here we consider how our results change when including the H photodetachment rate due to the background radiation. To do so, we carry out the same calculations as shown in Figure 2, but now including also the H detachment rate due to the background radiation field. We adopt the rates presented in Shang et al. (2010) assuming simple blackbody spectra at = 10 and 10 K, and we evaluate the critical LW flux that is required to maintain the gas at 10 K, leading to the formation of a DCBH.

Our results are presented in Figures 4 and 5, and are summarized in Table 1. As shown in the left panels of Figures 4 and 5, for a relatively low background radiation temperature of = 10 K the additional suppression of H cooling due to Lyman feedback is relatively small, as the LW flux required to maintain the gas at 10 K at a density of 10 cm is 26 neglecting the effect and 22 - 24, depending on the geometry of the Lyman emission (i.e. for = 1 - 10), when accounting for it. However, as shown in the right panels of Figures 4 and 5, for a larger background radiation temperature of = 10 K accounting for Lyman feedback results in a much larger drop in the critical flux from 1.1 10 to 200 - 900, depending on the geometry of the Lyman emission. Thus, for the spectra expected from hot, young stars in the early universe, the impact of Lyman feedback may be especially important. As shown in Figure 3 and in equation (7), it is detachment rates due to Lyman feedback at high densities, which are higher than the detachment rate due to the background radiation field, that result in a lower critical LW background flux.

It is important to note the reason for the much larger difference in the critical flux in the case of the higher background radiation temperature. This is ultimately due to the much lower rate of H photodetachment, relative to the H photodissociation rate, for the higher temperature background radiation field. Specifically, the photodetachment rate at a given value of is some four orders of magnitude lower for a background temperature of 10 K than it is for one of 10 K (Shang et al. 2010). This implies that the rate of photodetachment by Lyman photons, which is independent of the spectrum of the background radiation field, is much higher relative to the rate due to the background radiation for the hotter spectrum than it is for the colder one. This leads directly to the much larger drop in the critical value of due to Lyman feedback that we find for the hotter background spectrum than for the colder one.

The values we find for the critical LW flux () in the cases neglecting Lyman feedback are broadly consistent with the values found by previous authors (see e.g. Omukai et al. 2008; Sugimura et al. 2014; Latif et al. 2015; Hartwig et al. 2015; Agarwal et al. 2016a,b; Glover 2016), although they are different in detail due to differences in the models adopted in these studies (see also Glover 2015 on rate coefficient uncertainties). As shown in Table 1, however, we can conclude from our calculations that the impact of Lyman feedback can be strong and, importantly, results in a particularly large drop in the critical LW background flux required for DCBH formation for background radiation temperatures characteristic of young, hot stars in the early Universe (e.g. Tumlinson et al. 2001; Bromm et a. 2001; Oh et al. 2001; Schaerer 2002).

Recent analyses have pointed out that it is more accurate to go beyond , and quantify the requirements for direct collapse in terms of both the photodetachment rate of H and the photodissociation rate of H (e.g. Sugimura et al. 2014; Agarwal et al. 2016a; Wolcott-Green et al. 2016). These works show that once s, the photodissociation rate that is required for direct collapse decreases rapidly. Our calculations indicate that the constraint s is reached for 2.5, implying that the thermal evolution of the gas at these high densities becomes strongly impacted by Lyman feedback.

## 5 Discussion and Conclusions

We have applied a one-zone chemical and thermal evolution model to investigate the role that trapped Lyman cooling radiation, generated during the collapse of atomic cooling halos, has in suppressing molecular cooling. We find that, while this feedback from Lyman emission is not strong enough on its own to suppress H cooling, it does have the effect of lowering the intensity of the background LW radiation level that is required for the formation of DCBHs. While our modeling is simplified, the effect can be pronounced, potentially dropping the critical LW flux by up to a factor of a few for the background radiation temperatures expected to be produced by young, hot stars in the early Universe.

One implication of our results is that the number density of DCBHs may be higher than previously anticipated based on calculations neglecting H detachment by cooling radiation. Previous works have shown that the number density of DCBHs increases roughly as (Dijkstra et al. 2008, 2014; Inayoshi & Tanka 2015; Chon et al. 2016), which suggests that the impact of H detachment by Lyman photons results in a large increase of up to a factor of order 10 in the number density of DCBHs in regions of the early universe illuminated by bright, young stellar populations. This is important, as DCBH formation may have to occur relatively early in the epoch of galaxy formation, when stellar populations are still young, in order to be the seeds for the highest-redshift quasars. The lower values of implied by our results also mean that overall higher rates of DCBH formation may be realized, perhaps high enough for DCBHs to account for the BHs residing in the centers of a fraction of normal galaxies today (e.g. Habouzit et al. 2016). We do note, however, that perhaps the most likely sources of the LW radiation that leads to DCBH formation are metal-enriched stellar populations which are likely to emit radiation with characteristic temperatures intermediate between the 10 and 10 K that we have considered here (e.g. Agarwal et al. 2012; Johnson et al. 2013). The precise enhancement of the DCBH formation rate that is due to Lyman feedback will clearly depend on the spectra of the sources producing the LW radiation, and it is possible that if the spectra are sufficiently soft then the impact of this feedback may be limited.

The extremely bright Lyman emitter known as CR7 is an intriguing candidate for a high-redshift quasar that may be powered by accretion onto a DCBH (Sobral et al. 2015). Recent modeling efforts have demonstrated that a nearby galaxy may well have produced a high enough level of LW radiation to induce the formation of a DCBH in this galaxy and that the nebular emission could be explained by an accreting BH with a mass consistent with formation as a DCBH (e.g. Pallottini et al. 2015; Agarwal et al. 2016c; Hartwig et al. 2016; Smidt et al. 2016; Smith et al. 2016; Dijkstra et al. 2016a).3 In suggesting that the critical LW flux may be lower than previously thought, our results lend support to DCBH scenario for the origin of CR7. We note that this is also consistent with recent work suggesting that a massive cluster of Population III stars, an alternative explanation for the origin of CR7 (e.g. Sobral et al. 2015; Visbal et al. 2016; see also Johnson 2010), is dubious since it is unknown how a sufficiently high mass of Population III stars could be assembled rapidly enough to explain the observed extremely bright emission (e.g. Hartwig et al. 2016; Yajima & Khochfar 2016; Xu et al. 2016; Visbal et al. 2017).

We note that we have neglected the 2-photon and other hydrogen line emission that is produced at very high densities ( 10 cm) where Lyman photons can be destroyed before escaping the collapsing cloud (e.g. Schleicher et al. 2010; Dijkstra et al. 2016b). While not resonant emission lines, these photons are energetic enough to detach H and, in fact, the cross section for this process is greater at these photon energies than for Lyman photons (e.g. de Jong 1972). Thus, neglecting this emission may also lead to a slight overestimate of the critical LW flux.

We note also, though, that we have neglected the absorption of Lyman photons by H molecules, as described in Neufeld (1990; see also Dijkstra et al. 2016b). However, we estimate that this results in a reduction in the Lyman flux of, at most, a factor of two at the column densities ( 10 cm) and the low H fractions ( 10) that occur with an elevated background radiation field. In addition, the LW photons produced in the subsequent radiative decay of the H molecules are also able to detach H. Thus, we do not expect that accounting for this effect would strongly impact our conclusions.

Our results also carry implications for the impact of X-rays on the collapse of gas in atomic cooling halos, which numerous authors have shown is to produce free electrons which catalyze H formation, resulting in an increase in the critical flux (e.g. Inayoshi & Omukai 2011; Inayoshi & Tanaka 2015; Latif et al. 2015; Glover 2016; Regan et al. 2016b). We note, in particular, that our results for the critical LW flux for DCBH formation are in reasonable agreement with those of Glover (2016) for the case neglecting X-ray feedback. While X-rays may have the effect of raising the critical flux by up to two orders of magnitude in the absence of Lyman feedback for a hard spectrum (Glover 2016), another impact of X-rays is to enhance the Lyman emission within the halo (e.g. Dijkstra et al. 2016a). As we have shown, this should result in an enhanced rate of H photodetachment that will again lower the critical flux.

Finally, we note that atomic cooling halos which grow rapidly, due to mergers or due to growth in high density environments, likely produce Lyman cooling radiation at a higher rate than assumed in our calculations. This more intense emission leads, in turn, to larger photodetachment rates and lower values for the critical externally-produced LW flux required for DCBH formation. As the earliest supermassive black holes form in relatively rare, overdense regions, this implies that Lyman feedback may play an especially strong role in paving the way for the formation of the DCBH seeds of the earliest bright quasars (e.g. Mortlock et al. 2011; Wu et al. 2015). Future work incorporating the feedback effect of Lyman radiation on the chemical evolution of atomic cooling halos in 3D cosmological simulations will further elucidate the role that this process plays in determining the overall rate of DCBH formation.

## Acknowledgements

Work at LANL was done under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. JLJ would like to thank Aaron Smith, Bhaskar Agarwal, Zoltan Haiman, Marta Volonteri, Aycin Aykutalp, Nicole Lloyd-Ronning and Simon Glover for helpful discussions. JLJ also acknowledges support for this work from the LANL LDRD program.

## Appendix A Radial Dependence of Lyα Trapping

In one-zone models, the physical conditions of the collapsing gas cloud are described completely by its temperature and density. When we interpret one-zone models as clouds of uniform density in which Ly emission is produced uniformly (as we did when deriving equation (7), the energy density in Ly photons is enhanced almost uniformly throughout the cloud (see discussion below and Figure A.1). The spatial diffusion of Ly photons out of the cloud introduces only small gradients in the Ly energy density. However, larger gradients exist if Ly is not emitted uniformly throughout the cloud, as is generally the case in more realistic scenarios, in which we expect Ly cooling to increase towards the center of the cloud.

Here we compute the radial dependence of the Ly energy density in a suite of spherical gas clouds. We vary the HI column density of the cloud and where Ly is emitted, and compute by comparing this energy density to our estimate for given by equation (4). Ly transfer through static, spherical clouds of uniform density can generally be solved analytically for large line-center optical depths . Dijkstra et al. (2006) derive expressions for the (angle-averaged) Ly intensity as a function of radius and frequency in a spherical cloud of radius (see their equation C12). For a central Ly point source (at ) their expression for the total (integrated over frequency) intensity simplifies to

 J(r/R)=A∞∑n=1∫∞−∞dx(Rr)sinpnexp(−λn|σ(x)|κ0), (8)

where is a normalization constant, and

 pn =πn(rR)(1−23τ0ϕ(x)) (9) λnκ0 =πnτ0(1−23τ0ϕ(x)) (10) σ =√2π27x3av. (11)

We obtain by dividing the energy density to given by equation (4).

Figure A.1 shows as a function of for the analytic model (thick grey line). This line is independent of provided that . We overplot 3 lines with different colors, which we obtained from Monte-Carlo simulations of the Ly radiative transfer. The blue, red, and black lines represent the cloud when its density is , , and 10 cm, respectively. First, we note that (not shown here) the total average trapping time we found for Ly photons in the Monte-Carlo simulation agreed well with our estimate used for equation (4). Figure A.1 shows clearly that at , and that at . That is, in the case of a central Ly source, photodetachment of by Ly is signficantly more important for the inner of gas than in the exterior regions. Clearly, the case of a central point source represents an extreme case of centrally enhanced Ly emission, and we consider the values of that we obtain for these models to represent upper limits.

For completeness, the black line shows obtained from our Monte-Carlo simulations in which Ly photons are produced uniformly throughout the cloud. For clarity, we have only shown the case cm, but we have verified that the curve does not change for higher densities. The Ly energy density is enhanced close to uniformly throughout the cloud, and at a level that is in good agreement with equation (4).

### Footnotes

1. While H is also a precursor to H formation in the primordial gas, the rate of H formation via this channel is much lower than that through the H channel for the relatively hot radiation spectra ( 10 K) that are of interest here (see e.g. Sugimura et al. 2015). For this reason, we neglect the radiative destruction of H in our modeling.
2. Note that we recover the canonical cooling behavior of the gas for the case with no background radiation ( = 0) and no H photodetachment (e.g. Bromm & Larson 2004; Greif et al. 2015), as expected since we are employing effectively the same code as in previous studies of such processes (Johnson & Bromm 2006).
3. We note that recent observations of CR7 suggest that the bright Lyman source may be enriched to some degree with heavy elements (Bowler et al. 2016), suggesting that it is somewhat evolved if it did intially host the formation of a DCBH (see e.g. Aykutalp et al. 2014; Agarwal et al. 2017).

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