A UV flux constraint on the formation of direct collapse black holes
M. A. Latif, S. Bovino, C. Van Borm, T. Grassi, D. R. G. Schleicher, M. Spaans
aa r X i v : . [ a s t r o - ph . GA ] J un Mon. Not. R. Astron. Soc. , 1– ?? (2009) Printed 28 June 2018 (MN LaTEX style file v2.2) A UV flux constraint on the formation of direct collapseblack holes
M. A. Latif, ⋆ S. Bovino, C. Van Borm, , T. Grassi, , D. R. G. Schleicher, M. Spaans Institut f¨ur Astrophysik, Georg-August-Universit¨at, Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany Kapteyn Astronomical Institute, University of Groningen, The Netherlands Centre for Star and Planet Formation, Natural History Museum of Denmark, Øster Voldgade 5-7, DK-1350 Copenhagen, Denmark Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
ABSTRACT
The ability of metal free gas to cool by molecular hydrogen in primordial halos isstrongly associated with the strength of ultraviolet (UV) flux produced by the stellarpopulations in the first galaxies. Depending on the stellar spectrum, these UV photonscan either dissociate H molecules directly or indirectly by photo-detachment of H − as the latter provides the main pathway for H formation in the early universe. Inthis study, we aim to determine the critical strength of the UV flux above whichthe formation of molecular hydrogen remains suppressed for a sample of five distincthalos at z > by employing a higher order chemical solver and a Jeans resolution of32 cells. We presume that such flux is emitted by PopII stars implying atmospherictemperatures of K. We performed three-dimensional cosmological simulations andvaried the strength of the UV flux below the Lyman limit in units of J . Our findingsshow that the value of J crit21 varies from halo to halo and is sensitive to the local thermalconditions of the gas. For the simulated halos it varies from 400-700 with the exceptionof one halo where J crit21 ≥ . This has important implications for the formation ofdirect collapse black holes and their estimated population at z >
6. It reduces thenumber density of direct collapse black holes by almost three orders of magnitudecompared to the previous estimates.
Key words: methods: numerical – cosmology: theory – early Universe – galaxies:formation
Observations of quasars at z > reveal that supermas-sive black holes (SMBHs) of a few billion solar masseswere assembled within the first billion years after theBig Bang (Fan et al. 2003, 2006; Willott et al. 2010;Mortlock et al. 2011; Venemans et al. 2013). Their for-mation mechanisms in the juvenile Universe remainunknown. The potential progenitors of SMBHs includethe remnants of Pop III stars (Haiman & Loeb 2001;Haiman 2004; Tanaka & Haiman 2009; Whalen & Fryer2012; Hirano et al. 2014; Madau et al. 2014), dense stel-lar cluster (Portegies Zwart et al. 2004; Omukai et al.2008; Devecchi & Volonteri 2009) and direct collapseof a protogalactic gas cloud into so-called directcollapse black holes (DCBHs) (Oh & Haiman 2002;Bromm & Loeb 2003a; Spaans & Silk 2006; Begelman et al. ⋆ Corresponding author: [email protected] − M ⊙ whichformed in the early universe at z = − are the po-tential cradles for these DCBHs. It is imperative thattheir halos remain metal free and cooling is mainly regu-lated by atomic line radiation instead of H . These condi-tions may lead to a monolithic isothermal collapse wherefragmentation is suppressed and a supermassive star of − M ⊙ forms which later collapses into a blackhole (i.e., DCBH). This scenario is supported by numer-ical simulations which show that fragmentation remains c (cid:13) Latif et al. inhibited and massive objects may form (Bromm & Loeb2003a; Wise et al. 2008; Regan & Haehnelt 2009; Latif et al.2011, 2013c,b,a). Recently, the feasibility of this scenariohas been explored via high resolution numerical experimentsand it is found that ∼ M ⊙ objects can form (Latif et al.2013e; Regan et al. 2014). These results are consistent withtheoretical predictions (Begelman et al. 2008; Begelman2010; Ball et al. 2011; Hosokawa et al. 2012; Ball et al. 2012;Schleicher et al. 2013; Hosokawa et al. 2013; Whalen et al.2013). Depending on the mass accretion rates these stud-ies suggest the formation of supermassive stars or quasi-stars (stars with BH at their center) as potential embryos ofDCBHs (Schleicher et al. 2013).In primordial gas, trace amount of H can be formed viagas phase reactions in the early universe which then leadsto cooling and star formation. The main channel for theformation of H is: H + e − → H − + γ (1) H + H − → H + e − . (2)Once the first generation of stars, so-called PopIII stars,are formed they produce UV flux, pollute the intergalac-tic medium with metals via supernova explosions, and leadto a second generation of stars known as PopII stars.The UV flux produced by these stellar populations eitherphoto-dissociates the molecular hydrogen directly or photo-detaches electrons from H − which provides the main routefor the formation of H in primordial gas chemistry.The stellar spectra of PopIII stars are harder with acharacteristic temperature of K while PopII stars arecharacterized by soft spectra with temperatures of K .PopIII stars mainly contribute to the direct dissociationof H while PopII stars also photo-detach H − . UV photonswith energies between 11.2 and 13.6 eV are absorbed in theLyman-Werner bands of molecular hydrogen and put it intoan excited state. The H molecule later decays to the groundstate and gets dissociated as H + γ → H ∗ → H + H , (3)a process known as the Solomon process. On the other hand,H − photo-detachment occurs via low energy photons above0.76 eV. In this study, we focus on the background UVflux predominantly emitted by PopII stars as tiny amountsof metals can lead to fragmentation (Omukai et al. 2005;Dopcke et al. 2013).The critical value of the UV flux, hereafter called J crit21 ,above which H cooling remains suppressed, can be deter-mined by comparing the H formation and dissociation timescales. Omukai (2001) found from one-zone calculations that J crit21 = in units of J = − erg cm − s − Hz − sr − for T ∗ = K which was later confirmed by Bromm & Loeb(2003b) in 3D simulations for a single halo. These esti-mates were revised by Shang et al. (2010) (hereafter S10)through three dimensional simulations using the H self-shielding formula of Draine & Bertoldi (1996), finding that J crit21 = − . They attributed these differences to the choiceof a more accurate and higher H collisional dissociationrate, and focused on rather massive halos forming at z < self-shielding function of Draine & Bertoldi (1996) andanticipated that it may further reduce the value of J crit21 . Such values of J crit21 are much larger than the global backgroundflux but can be achieved in the close vicinity (about 10kpc) of nearby star forming galaxies (Dijkstra et al. 2008;Agarwal et al. 2012, 2014).In this article, we derive the values of J crit21 for a stellarspectrum of T ∗ = K employing the improved H self-shielding fitting function provided by Wolcott-Green et al.(2011). Major improvements compared to the previous stud-ies are the following: • Selection of a larger sample of halos with collapse red-shifts at z > • Employed higher order chemical solver DLSODES(Bovino et al. 2013). • Accurate determination of J crit21 for the individual halos. • Higher Jeans resolution of 32 cells. • Improved self-shielding function of WG11.We note that the importance of an accurate chemical solverin high resolution simulations was previously reported byBovino et al. (2013). The impact of higher Jeans resolutionhas also been shown by Latif et al. (2013b) and Turk et al.(2012). Our selected halos are collapsed at z >
10 in contrastto S10 where halos collapsed at z <
10. All these improve-ments distinguish the present work from S10.We perform three dimensional cosmological simulationsfor five different halos of a few times M ⊙ and vary thestrength of the background UV flux (hereafter J , i.e. UVflux below Lyman limit). We use the chemical network listedin table A1 of S10 which includes all the relevant process forthe formation and dissociation of molecular hydrogen. Wefurther employed a fixed Jeans resolution of 32 cells through-out the simulations for better resolving the shocks and tur-bulence. A particular goal of this paper is to provide a rathernarrow constraint on J crit21 for individual halos, and to pointat potential correlations with halo properties. This studyhas important implications for the formation of DCBHs asit provides stronger estimates for the value of J crit21 requiredfor dissociation of molecular hydrogen.The organization of this article is as follows. In section2, we provide the details of simulations setup and summaryof a chemical network. In the third section, we present ourfindings and discuss our conclusions in section 4. The simulations presented here are performed using theadaptive mesh refinement, grid based, cosmological hydro-dynamics code ENZO (O’Shea et al. 2004; Bryan et al.2014). The hydrodynamical equations are solved employingthe piece-wise parabolic method (PPM) and the dark mat-ter dynamics is computed using the particle-mesh technique.The code makes use of multigrid Poison solver for solving thegravity.Our simulations start at z=100 with cosmological initialconditions. We first run uniform grid simulations andselect the most massive halos of a few times M ⊙ in oursimulated periodic box by using a standard halo finder basedon the friends of friends algorithm (Turk et al. 2011). Our http://enzo-project.org/, changeset:48de94f882d8c (cid:13) , 1– ?? lack hole formation in the early universe computational volume has a size of 1 Mpc / h and is centeredon the most massive halo. We employ two nested refinementlevels each with a grid resolution of cells besides thetop grid resolution of . To solve the dark matter (DM)dynamics, 5767168 particles are used which yield an effec-tive DM resolution of about 600 M ⊙ . Further, additional 18levels of refinement were employed during the course of thesimulations with a fixed Jeans resolution of 32 cells. Ourrefinement strategy is exactly the same as mentioned in anumber of previous studies (Latif et al. 2013a,d, 2014). Forfurther details about the simulation setup the reader is re-ferred to the above mentioned articles. The simulations werestopped once they reached the maximum refinement level,and were performed for five distinct halos selected from cos-mological initial conditions. The masses of these halos andtheir collapse redshifts are listed in table 1 for various valuesof J .To self-consistently solve the evolution of the follow-ing chemical species H , H + , He , He + , He ++ , e − , H − , H , H + in cosmological simulations, we employed the publicly avail-able KROME package (Grassi et al. 2014). The KROMEpackage was previously employed for 3D simulations of pri-mordial star formation and halo mergers (Bovino et al. 2013,2014). The chemical network used in this study is the sameas listed in table A1 of S10 with only two modifications, theinclusion of an improved fitting formula for H self-shieldingby WG11 and an addition of the dissociative tunneling ef-fect which contributes to the collisional dissociation of H (Martin et al. 1996). The latter is the dominant factor in thetotal dissociation rate ( γ tot = γ CID + γ DT ) of H for tempera-tures up to 4500 K, see detailed discussion in section 3.3 ofMartin et al. (1996). We presume that the background UVflux is emitted by PopII stars with an atmospheric temper-ature of T ∗ = K .Here, we are interested in the UV flux below the Ly-man limit (i.e. < . ). As pointed out in the previoussection, the low energy photons with energy above 0.76 eVemitted by PopII stars photo-detach H − which provides themain pathway for the formation of H in primordial composi-tion of the gas. We have not included the photo-ionization ofhydrogen and helium species as they require energies abovethe Lyman limit. The H cooling function of Galli & Palla(1998) is used here as in S10 for direct comparison and allother relevant cooling processes for primordial gas are in-cluded i.e, cooling by collisional excitation, collisional ion-ization, recombination and Bremsstrahlung radiation (fordetails see Grassi et al. 2014). We checked that using thecooling function by Glover & Abel (2008) has no impact onthe results. To test our chemical model, we have performed one-zone cal-culations similar to S10. The thermal evolution and speciesfractions of H , H − and e − are shown in figure 1. We startwith an initial density of − , a gas temperature of 200K, electron and H fractions of − and − respectively (same as S10). It can be seen from figure 1 that for J < ,H formation takes place and leads to cooling down to a fewhundreds K. The H − fraction initially increases up to den-sities of 100 cm − , then gets depleted during the formationof H . For stronger fluxes, the formation of H remains in-hibited, cooling due to Lyman alpha photons kicks in andthe temperature remains about 8000 K. The thermal evo-lution and the abundances of the species are in agreementwith S10. The value of J crit21 found from one-zone calculationsis between 30-40 and is consistent with findings of S10.However, we note that the value of J crit21 differs in 3Dsimulations. The differences in J crit21 between one-zone andthree-dimensional simulations arise from the fact that theone-zone calculation do not capture shocks, collapse dynam-ics, hydrodynamical effects as well as the effects of the darkmatter potential which are crucial in determining the J crit21 . To determine J crit21 , we have performed more than 20 cos-mological simulations for five distinct halos with massesabove 10 M ⊙ including all the necessary processes for theformation and dissociation of molecular hydrogen by self-consistently solving the rate equations along with the hy-drodynamics. In figure 2, we present the spherically aver-aged profiles of temperature, density, H and e − abundancesfor all simulated halos. It is found that in all halos the for-mation of H takes place for J < . At radii > × au,the temperature in all cases for the various strengths of J is about 7500 K but the fraction of the molecular hydrogenremains higher for the lower values of J . This is because the H − photo-detachment rate scales linearly with the strengthof J , i.e. the stronger the flux the lower the fraction ofmolecular hydrogen. For fluxes below the critical values, thefraction of H gets further boosted during the collapse andreaches ∼ − . It cools the gas down to temperatures ofabout thousand K. Similarly, the electron fraction declinestowards the center. It is about − in the center and − atlarger radii. The impact of H cooling is also visible in thedensity profile which significantly deviates from the isother-mal profile. Small bumps in the density profile are typicalsignatures of fragmentation which is expected in these cases.This trend is observed in all the halos for J = .For some halos the fraction of H sufficient to inducecooling still occurs at later times for J > , at densitiesof about cm − . This lowers the halo central tempera-tures down to about a thousand K. For J above the criticalthreshold the fraction of H remains quite low, i.e. − . Sucha low fraction is not sufficient to trigger H cooling and thenthe halos collapse isothermally. We note that value of the J crit21 varies from 400-700 for four halos as listed in table 2. Anotable variation in the critical value is observed for halo Bwhere J crit21 = . Such variations from halo to halo are notsurprising as halos have different density distributions aftervirialization, different spins and accretion shocks at differentdensities. As shown in figure 6 & 7 of S10, collisional dissoci-ation is the main destruction channel for H above densitiesof cm − . The rate of collisional dissociation exponentiallydepends on the temperature (Martin et al. 1996), and smalllocal variations in the temperature can change J crit21 as ver-ified from our one-zone model by artificially changing thetemperature by few hundreds K. Additional fluctuations in c (cid:13) , 1– ?? Latif et al. T [ K ] J21=10J21=30J21=40J21=100 10 -8 -7 -6 -5 -4 -3 -2 f H -8 -7 -6 -5 -4 f e - n [cm -3 ] 10 -16 -15 -14 -13 -12 -11 -10 f H - n [cm -3 ] Figure 1.
The abundances of H , e − , H − and temperature are plotted against the number density for various strengths of backgroundUV flux. These results are one-zone calculations are performed using the H self-shielding function of Draine & Bertoldi 1996. Table 1.
Properties of the simulated halos are listed here.halo A halo B halo CJ Mass ( M ⊙ ) z T cent (K) J Mass( M ⊙ ) z T cent (K) J Mass( M ⊙ ) z T cent (K)300 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × the electron number density may further influence it andthere is a weak dependence of the collisional dissociation ondensity as well. As these quantities change simultaneouslywhen different halos are considered, the individual depen-dencies cannot be explored in isolation, but the functionalform of collisional dissociation rate suggests that tempera-ture dependence provides the dominant effect (also see dis-cussion in S10).In figure 3 we show the phase plots of temperature and H fraction against density. Initially, the gas is heated up tothe virial temperature of the halo, i.e. above K , and thencools down to about 8000 K by Lyman alpha radiation. Atdensities of about cm , cooling due to H becomes effec-tive for the weaker J case while for the stronger flux case itremains inhibited. It is found that temporarily different gasphases may coexist at the same density. The latter reflects lo- cal variations in gas density, temperature and self-shielding.It may also be noted that in the latter case, the amount ofgas in the cold phase is negligible compared to the top panelwhich has a significant amount of cold gas and remains inthe cold phase. In these simulations, we find that if the gasat densities ≥ cm has a temperature lower than 1000 K(as shown in top panel of 3) then cooling due to H becomesimportant and halos remains in cold phase. On the otherhand if gas at densities ≥ cm has a temperature higherthan 2000 K shown in the bottom panel of figure 3 then thehalo remains in the hot phase and collapses isothermally. Wehave also checked that the behavior remains the same if thesimulations are evolved to higher densities.To further understand the origin of J crit21 , we have plottedthe radially averaged density, temperature, abundances of H and e − , radial infall velocities as well as rotational velocities c (cid:13) , 1– ?? lack hole formation in the early universe Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=300J21=600J21=700 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=600J21=900J21=1000J21=1500J21=2000 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=600J21=700J21=900 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=300J21=400J21=500J21=600 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=300J21=500J21=600 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Figure 2.
Spherically averaged and radially binned profiles of temperature, density, H and e − fractions are plotted in this figure. Eachpanel represent a single halo. Each line style represents the value of J as mentioned in the legend. Top panels represent halo A & B(left to right), middle panels halos C & D (left to right) and bottom panel halo E. of all halos for J = in figure 4. For the given strengthof J two halos (D & E) are already in an isothermal statewith central temperature around 7000 K while the otherthree have a sufficient H fraction to reduce their centraltemperatures down to about 1000 K. It can be noted thatparticularly halos D & E have larger infall velocities alreadyat radii > au compared to the halos with lower centraltemperatures. Particularly, the halo B with the highest J crit21 in our sample has the lowest infall velocity. The differences inthe infall velocities arise due to the ambient sound speed ofgas cloud. We further note that the two most massive halos have a lower value of J crit21 than the lower-mass halos. S10 haveconsidered even more massive halos at lower redshift, findinga further decrease in J crit21 . The latter suggests a potentialdependence on the mass of the halo with some fluctuations. For a comparison of our results with S10, we have performeda couple of simulations where we employed exactly the samechemical and thermal processes as described by S10 with-out any modifications and using the H self-shielding fit- c (cid:13) , 1– ?? Latif et al. -26 -24 -22 -20 -18 -16 Density [g/cm ] T e m p e r a t u r e [ K ] -4 -3 -2 -1 C e ll M a ss M s un ( M ⊙ ) -26 -24 -22 -20 -18 -16 Density [g/cm ] -21 -19 -17 -15 -13 -11 -9 -7 -5 -3 f H -4 -3 -2 -1 C e ll M a ss M s un ( M ⊙ ) -26 -24 -22 -20 -18 -16 Density [g/cm ] T e m p e r a t u r e [ K ] -2 -1 C e ll M a ss M s un ( M ⊙ ) Figure 3.
Phase plots of temperature, density and H fraction for representative cases of halos E and D are shown here. The top panelshows J = for halo E, the middle panel J = for the halo E, and the bottom panel J = for the halo D. The latter case alsoillustrates the existence of two gas phases at the same density. c (cid:13) , 1– ?? lack hole formation in the early universe Radius [au] -20 -18 -16 ρ [ g/ c m ] -7 -6 -5 -4 f e T [ K ] ABCDE Radius [au] -8 -6 -4 -2 f H V r o t [ k m / s ] −15−10−50 V r a d [ k m / s ] Figure 4.
Spherically averaged and radially binned profiles of temperature, density, H fraction, e − fraction, radial infall velocity androtational velocities are plotted for all halos for J = . Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=300J21=600J21=1000 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Radius [au] -8 -7 -6 -5 -4 -3 -2 f e -24 -22 -20 -18 -16 D e n s i t y [ g/ c m ] J21=300J21=600J21=1000 Radius [au] -10 -8 -6 -4 -2 f H T e m p e r a t u r e [ K ] Figure 5.
Same as figure 2 for halo D & C but using the H self-shielding fitting function of Draine & Bertoldi 1996. Radius [au] M [ M ⊙ ] J21=300J21=400J21=500J21=600 Radius [au] -4 -3 -2 -1 ˙ M [ M ⊙ / y r ] J21=300J21=400J21=500J21=600
Figure 6.
Enclosed masses and mass accretion rates are shown for a representative case of ’halo D’. The left panel shows enclosed massprofiles while the right panel shows the mass accretion rates.c (cid:13) , 1– ?? Latif et al.
Table 2.
Properties of the simulated halos for J crit21 are listed here.Model Mass Redshift J crit21 spin parameterNo M ⊙ z in units of J λ A . × . × . × . × . × ting function of Draine & Bertoldi (1996). These simulationswere performed for halos C & D and are shown in figure5. It suggests that the value of J crit21 differs by a factor of afew from their estimates as they suggest J crit21 varies from 30-300. Our study brackets the value of J crit21 within a factor oftwo. The self-shielding fitting function of Draine & Bertoldi(1996) is known to overestimate the H shielding. Therefore,a more accurate determination of J crit21 is not necessary. In ad-dition, we found significant variations from halo to halo aswell. Given the strong dependence of J crit21 on the variationsin the local gas temperatures, halo merger histories, the oc-currence of shocks at various densities as well as the densityand temperature dependence of the H collisional dissocia-tion rates, the results may still be consistent with S10, eventhough we typically obtain higher values for the J crit21 . Vari-ations in the temperature and the occurrence of the shocksis evident from figure 3. As mentioned in previous section,we further verified it by artificially changing the tempera-ture in our one-zone model. This was already confirmed byS10 as well (see paragraph 2, section 3.5 of S10). It mayalso be noted that the halos in our sample are assembledat higher redshifts and have lower masses compared to theS10, and therefore presumably different structures as well asformation histories.We further noted that the value of J crit21 very weakly de-pends on the choice of self-shielding function. It is due tothe fact that for a stellar temperature of T ∗ = K themain dissociation channel is H − photo-detachment not thedirect dissociation of H . The value of J crit21 changes within afactor of two at-most by employing the self-shielding fittingfunction of Draine & Bertoldi (1996) which overestimatesthe shielding effect. S10 also found similar results (privatecommunication with Zoltan Haiman). The key requirements for the formation of DCBHs are thatthe gas in halos with T vir > K must be of primordialcomposition and the formation of molecular hydrogen re-mains suppressed to avoid fragmentation. To keep the gasfree from H requires the presence of a strong UV flux abovethe critical value. Such values of J crit21 can be achieved inthe surrounding of star-burst galaxies or even in the ha-los which are satellites of such galaxies (Dijkstra et al. 2008;Agarwal et al. 2012, 2014). We have computed the enclosedmass and mass accretion rates for halo “D” as a represen-tative case which is shown in figure 6. It is found that foran isothermal collapse, the gas in the halo remains hotter, leads to higher accretion rates and consequently the enclosedmass within au is about two orders of magnitude highercompared to the case with the weaker UV flux. The massaccretion rate peaks around 1 M ⊙ /yr for isothermal casesand overall is about an order of magnitude higher comparedto the H cooling cases. One may expect similar differencesin the mass of resulting objects. These enhanced accretionrates help in rapidly building up the DCBHs.Our estimates of J crit21 are highly relevant for comput-ing the number density of DCBHs and their comparisonwith observations. Some recent studies predict the num-ber density of DCBHs of few per comoving Mpc − com-pared to the observed SMBHs density of few per comov-ing Gpc − (Dijkstra et al. 2008; Agarwal et al. 2012). Our re-sults suggest that for the most of halos simulated in thesestudies J crit21 ≥ . The choice of such critical values changesthe number density of DCBHs and also helps these mod-els to have better agreement with observations. Based onthe prescription for estimating the fraction of halos ex-posed to super-critical UV flux by Dijkstra et al. (2008) andAgarwal et al. (2012), we estimate that fraction of the ha-los hosting DCBHs may be reduced by about three ordersof magnitude due to the increase in J crit21 by an order magni-tude, i.e. adapting J crit21 = . The expected number densityof BHs is per comoving Gpc − at z = . The latter isstill factor of 500 or so higher than observed number den-sity. These estimates do not include the metal enrichmentby supernova driven winds (see Dijkstra et al. (2008) andAgarwal et al. (2012)) which will further reduce the num-ber density of DCBHs. We also report here a potential massdependence of J crit21 , which tends to increase with decreasinghalo mass. In order to give black holes more time to accrete,the initial collapse should take place at higher redshift, at avirial temperature of K. The latter implies a decreasingmass for higher-redshift dark matter halos, corresponding toa higher value of J crit21 . One of the main obstacles for the formation of direct col-lapse black holes is to avoid fragmentation in massive pri-mordial halos which are the potential birthplaces of seedblack holes. This may only be possible in the absence ofmolecular hydrogen which may induce fragmentation andtrigger star formation. The ubiquity of background UV fluxcan photo-dissociate H molecules and may overcome thisobstruction. The prime objective of this work is to deter-mine the critical value of the background UV flux required tosuppress the molecular hydrogen formation in atomic cool-ing halos. As photo-dissociation of H depends on the typeof stellar spectrum, we have considered here UV photonsbelow 13.6 eV emitted by PopII stars. We have conductedthree-dimensional cosmological simulations for five distincthalos by including all relevant processes for the formationand dissociation of H . The halos studied here have typicalmasses of a few times 10 M ⊙ and were illuminated by vari-ous strengths of background UV flux. We here employed the H self-shielding fitting function provided by WG11.Our findings show that the value of J crit21 strongly dependson the properties of the halo and may vary from halo to halo.For the halos studied here, we found that the value of J crit21 c (cid:13) , 1– ?? lack hole formation in the early universe varies from 400-700 with the exception of one halo where it isabout 1500. It is also found that J crit21 may depend on the massof halo, as the two most massive halos have reduced values of J crit21 . This trend is consistent with results by S10, where moremassive halos at have even lower values of J crit21 . We note thatour one-zone calculations are in agreement with S10. Thehighly non-linear collapse dynamics leads to the occurrenceof shocks with Mach numbers of about 3 at various densitieswhich changes the local gas temperature and consequently J crit21 differs from halo to halo due to the strong dependence ofH collisional rate on temperature and density (Martin et al.1996). To build up the supermassive black holes at z 6-7,one should preferentially consider halos that collapse early,implying a lower mass and potentially higher J crit21 at the samevirial temperature. However, it will be desirable in the futureto verify it for a larger sample of halos with broader massrange. Our estimates for J crit21 are quite robust as we consider alarger sample of halos and higher Jeans resolution leading tobetter resolved shocks and employed the high order chemicalsolver DLSODES. We also find that the value of J crit21 weaklydepends on the choice of H self-shielding. Although, thefitting formula of Draine & Bertoldi (1996) overestimates H self-shielding compared to WG11, its impact is very low forthe adapted stellar spectra.The value of J crit21 is about an order of magnitude higherin 3D calculations compared to the one-zone results. Thisis because of the inability of one-zone calculations to modelshocks and hydrodynamical effects. Similar results have beenfound in the study of S10. We also included the effect of dis-sociative tunneling and found from the one-zone test thatit decreases J crit21 by a factor of three. The estimates of J crit21 determined in this work have important implications for theformation of DCBHs. Our results suggest that the value of J crit21 > should be employed in computing the numberdensity of DCBHs. The value of J crit21 used in previous stud-ies (Agarwal et al. 2012, 2014) seems rather low (i.e. 30)and may be one of the reasons for the high abundance ofDCBHs predicted from semi-analytical calculations. Fromour results, the expected BH number density is percomoving Gpc − at z= 6 for J crit21 = . This estimate is ob-tained by rescaling the values of (Agarwal et al. 2012) fora higher J crit21 . These estimates do not take into account themetal enrichment in the intergalactic medium by supernovadriven winds which may further reduce the number densityof DCBHs.In our previous studies (Latif et al. 2013a,e, 2014), wehave shown that the presence of strong UV flux leads toan isothermal collapse where conditions are fertile for theformation of DCBHs. In fact, under these conditions largeaccretion rates of > M ⊙ /yr are observed which result inthe formation of supermassive stars of M ⊙ , the potentialprogenitors of DCBHs.We have presumed here that these halos exposed tothe intense UV flux by PopII stars are metal-free and re-main pristine throughout their evolution. The transitionfrom PopIII to PopII stars may inject metals and pollutethese halos. Depending on the critical value of metallic-ity which can be as low as − Z/Z ⊙ (Cazaux & Spaans2009; Aykutalp & Spaans 2011; Latif et al. 2012), once themetal content in halos exceeds this value fragmentationbecomes inevitable (Omukai et al. 2008). Particularly, thecooling due to the dust even in the presence of a strong UV flux becomes effective at densities around − cm − and may lead to the formation of dense stellar clusters(Devecchi & Volonteri 2009). Nevertheless, metal enrich-ment in the universe is expected to be patchy and pristinehalos may exist down to z > .Given the strong dependence of J crit21 on the local ther-mal conditions as argued above, the local heating/coolingeffects such as heating by ambiploar diffusion, turbulencedissipation as well as cooling by HD molecules may changethe critical value of the flux by a factor of few. In fact theimpact of turbulence and magnetic field in the presence ofUV flux was explored by Van Borm & Spaans (2013) andthey found that in turbulent halos with stronger initial seedfields the value of J crit21 is reduced by an order of magnitude.Furthermore, the presence of cosmic rays/X-rays may signif-icantly enhance the critical value of J (Inayoshi & Omukai2011). It was recently pointed out by Richings et al. (2014)that including the effect of turbulence in Doppler broaden-ing reduces the H self-shielding. This should be explored infuture studies. ACKNOWLEDGMENTS
The simulations described in this work were performed usingthe Enzo code, developed by the Laboratory for Computa-tional Astrophysics at the University of California in SanDiego (http://lca.ucsd.edu). We thank Zoltan Haiman andGreg Bryan for useful discussions on the topic. We acknowl-edge research funding by Deutsche Forschungsgemeinschaft(DFG) under grant SFB / (project A12) and comput-ing time from HLRN under project nip00029. DRGS andSB thank the DFG for funding via the SchwerpunktprogramSPP 1573 “Physics of the Interstellar Medium” (grant SCHL / − ). The simulation results are analyzed using thevisualization toolkit for astrophysical data YT (Turk et al.2011). REFERENCES
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