Accretion onto Intermediate-mass Seed Black Holes in Primordial Galaxies
DDraft version November 15, 2018
Preprint typeset using L A TEX style emulateapj v. 8/13/10
ACCRETION ONTO INTERMEDIATE-MASS SEED BLACK HOLES IN PRIMORDIAL GALAXIES
Yuexing Li
Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA
Draft version November 15, 2018
ABSTRACTThe origin of the supermassive black holes that power the most distant quasars observed is largelyunknown. One hypothesis is that they grew rapidly from intermediate-mass seeds ( ∼
100 M (cid:12) ) leftby the first stars. However, some previous studies argued that accretion onto these black holes wastoo low to build up the mass due to strong suppression by radiative feedback. Here, we re-exam theaccretion process of such a black hole embedded in a primordial gas cloud, by considering a widerange of physical and numerical parameters not explored before. We find that, while radiative heatingand pressure indeed suppress accretion effectively, self-gravity of the gas eventually overcomes thefeedback effects and boosts the accretion to the Eddington rate after one free-fall timescale of thecloud. Moreover, for a given black hole mass, there exists a critical density above which the accretioncan reach Eddington limit. Furthermore, we find a universal correlation between black hole accretionrate and ambient gas density, which may serve as a realistic recipe for black hole growth in simulations.
Subject headings: accretion – black hole physics – radiative feedback – hydrodynamics – methods:numerical – quasars: high redshift INTRODUCTION
A major recent development in observational cosmol-ogy has been the discovery of dozens luminous quasarsat high redshifts ( z (cid:38)
6) when the universe was lessthan 7% of its current age (Fan et al. 2001, 2003; Willottet al. 2007, 2010b; Mortlock et al. 2011). These quasarsare believed to be powered by supermassive black holes(SMBHs) of ∼ − M (cid:12) , and the BHs appear to accreteat near Eddington rate (e.g., Fan et al. 2006; Willottet al. 2010a). Furthermore, intense star formation (Car-illi et al. 2004; Walter et al. 2009), abundant CO gas anddust (Walter et al. 2003; Jiang et al. 2006, 2010; Wanget al. 2010), and super-solar metallicity (Maiolino et al.2005) were detected in the quasar hosts, indicating a co-eval formation of the SMBHs and host galaxies.The seeds and growth of these SMBHs, however, areunsolved puzzles. It has long been proposed that theymight grow from remnants of the very first stars, the so-called PopIII stars (e.g., (Madau & Rees 2001; Haiman& Loeb 2001; Volonteri & Rees 2005; Li et al. 2007;Volonteri 2010)). Sophisticated cosmological simulationsover the past decade suggested that PopIII stars formedat early cosmic times z ∼ −
50 in minihalos withmass ∼ M (cid:12) , and were likely massive, with mass ∼ −
300 M (cid:12) (e.g., Abel et al. 2002; Bromm & Lar-son 2004; Tan & McKee 2004; Gao et al. 2007; Yoshidaet al. 2008; Bromm et al. 2009) (see, however, recentwork by Clark et al. 2011; Greif et al. 2011 for sugges-tions of a smaller mass range). Massive PopIII stars of100 −
140 M (cid:12) or above 260 M (cid:12) were predicted to rapidlycollapse to BHs of ∼
100 M (cid:12) (Heger et al. 2003). Thesubsequent growth of these stellar-mass BHs might playan important role in the formation of luminous quasars atlater times. If a 100 M (cid:12)
BH starts to accrete at the Ed-dington rate at z >
20, it will attain a final mass greaterthan 10 M (cid:12) by z ∼
6, implying a viable explanation forthe observations of bright quasars in this epoch. [email protected]
Black holes are thought to accrete through a diskshaped by the outward transfer of the angular momen-tum (Shakura & Sunyaev 1973; Rees 1984; Antonucci1993; Krolik 1999). However, in metal-free, primordialgalaxies with virial temperature above 10 K, conditionsmay exist for the formation of a thick disk of gas at tem-perature of 5000 - 10000 K due to insufficient cooling(Oh & Haiman 2002; Volonteri & Rees 2005). As a re-sult, accretion proceeds in a quasi-radial manner onto thecentral BH. Such spherical Bondi-Hoyle accretion model(Bondi & Hoyle 1944; Bondi 1952) has been widely em-ployed in cosmological simulations on BH growth in theearly universe (Li et al. 2007; Johnson & Bromm 2007; DiMatteo et al. 2008; Alvarez et al. 2009; Sijacki et al. 2009;Di Matteo et al. 2011). In particular, it was shown byLi et al. (2007) that 100 M (cid:12)
BH seeds from PopIII starsin gas-rich protogalaxies residing in highly overdense re-gions could grow to 10 M (cid:12) within 800 million years andproduce luminous quasars at z ∼ cm( ∼ − pc) for a 100 M (cid:12) BH. This clearly poses a signif-icant challenge for large-scale cosmological simulations. a r X i v : . [ a s t r o - ph . C O ] S e p Recent analytical (Milosavljevi´c et al. 2009a) and ide-alized simulations (Milosavljevi´c et al. 2009b; Park & Ri-cotti 2011) of accretion onto a 100 M (cid:12)
BH in a gas sphereargued that radiation feedback strongly suppressed theaccretion rate to be at most a small fraction of the Ed-dington limit. However, these studies have focused onlyon the regime where the ionization radius is much largerthan the sonic radius. Furthermore, self-gravity of thegas was ignored on the assumption that the BH’s grav-ity dominates in the ionized zone. But even if the initialdensity is relatively low, as long as the radius of the gassphere is much larger than the ionization region, self-gravity would build up the density just outside the ion-ization radius to a high level after one free-fall timescale.Here, we report new findings of accretion onto a stellar-mass BH ( M BH = 100 M (cid:12) ) embedded in a primordial gassphere from numerical simulations, which cover a widerrange of numerical and physical parameters than previ-ous studies. Our model includes not only the relevantfeedback processes, but also self-gravity of the gas. Weuse a modified version of the public, grid-based hydro-dynamics code VH-1 (Blondin & Lufkin 1993) to followthe accretion in a spherical geometry. VH-1 is based onthe piece-wise parabolic method (Colella & Woodward1984), and uses a Lagrange-Remap approach to solve theEuler equations.The modifications we made to VH-1 include two ma-jor aspects: a logarithmic grid to ensure sufficiently highresolution near the central hole while covering a large re-gion with a reasonable number of grid cells, and a uniquetreatment of radiative feedback by using a finely pre-computed two-parameter grid, the ionization parameterand gas temperature, to tabulate the cooling, heating,and radiation pressure. Our feedback algorithm is animprovement to those previously used (Ciotti & Ostriker2001; Sazonov et al. 2005; Ciotti & Ostriker 2007) in thatit calculates the cooling, heating, and radiation forcemore accurately. Moreover, it can handle high-densityclouds accurately and efficiently. These modifications al-low us to achieve an unprecedentedly high spatial reso-lution of 10 cm ( ∼ − pc), and simulate clouds ofa wide range of gas density in 10 − cm − . In par-ticular, the ability to reach densities several orders ofmagnitude higher than the previous limit of 10 cm − iscritical to our new results.The paper is organized as follows. In §
2, we describeour analytical considerations and numerical methods indetail. In §
3, we first demonstrate that including self-gravity enhances the gas density just outside the ioniza-tion radius by several orders of magnitude, which leadsto enhanced accretion rate. We then present various sim-ulations with higher but fixed ambient densities withoutincluding self-gravity, and show that beyond a certaincritical density, the accretion rate indeed reaches Edding-ton limit, and that there exists a correlation between BHaccretion rate and ambient gas density. We discuss theimplications of our results to large scale numerical simu-lations of SMBH formation, and summarize in § METHODOLOGY
Analytical Considerations
In the analytical work of Milosavljevi´c et al. (2009a)and numerical simulations of Milosavljevi´c et al. (2009b) and Park & Ricotti (2011), there is an implicit assump-tion that the ionization radius, r ion , is much larger thanthe sonic radius, r s . As we will show later, the conclusionthat the accretion rate is significantly suppressed by ra-diative feedback processes is in fact critically dependenton this assumption, more accurately, the assumption that r ion (cid:29) r B , where r B is the standard Bondi accretion ra-dius ignoring radiative feedback processes: r B = 5 × M T , cm , (1)where M is the BH mass in unit of 100 M (cid:12) , and T , isthe ambient gas temperature at infinity in unit of 10 K.According to Milosavljevi´c et al. (2009a), the ioniza-tion radius r ion takes the form: r ion = 4 . × l / f / ion M / T HII , . f / turb n / T / HI , . cm , (2)where f ion is the average fraction of energy of an ab-sorbed photon that goes to photo-ionization, and is 1 / .
5, as we willassume in the present work; l is the BH luminosity rela-tive to the Eddington luminosity for Thomson scattering;and n is the ambient gas density in unit of 10 cm − .By taking other numerical factors to be of order unity,we have r ion < r B , if n M > l / × . For a BHwith M = 1, and at near Eddington limit, this means n > × cm − . This density is much higher thanthose considered in the simulations of Milosavljevi´c et al.(2009b) and Park & Ricotti (2011). It is therefore notsurprising that they found strongly suppressed accre-tion rates. However, one can naively imagine that thereshould be qualitative differences in the properties of theaccretion flow when the r ion = r B boundary is crossed.The reason is that r ion primarily determines the regionwhere photon-heating and photo-ionization are impor-tant processes, while r B determines the region where theBH’s gravitational field starts to significantly modify thegas flow. When r ion < r B , the radiation feedback is notimportant at radius larger than r B except for Thomsonscattering. The gas flow can therefore achieve significantinward velocity of the order of local sound speed at r B .On the other hand, at locations immediately inside r ion ,radiation feedback may slow down the inward velocitysignificantly, creating a strong shocked region between r ion and r B . It is this region that may help to maintainthe accretion rate of the flow at near Eddington limit.One critical question is therefore what kind of gas den-sity can one expect to find near the accreting BHs. Inthis paper, we suggest that densities as high as, or higherthan 10 cm − might be quite common. One reason isthat the near isothermal collapse of the gas outside of r ion under self-gravity can naturally enhance the ambi-ent density to high values. The self-gravity of gas was ig-nored in the numerical simulations of Milosavljevi´c et al.(2009b) and Park & Ricotti (2011), on the ground thatthe BH’s gravity dominate at r < r ion . However, evenif we start at a relatively low density initially, as longas the radius of the gas sphere is much larger than r ion ,the densities just outside r ion may reach very high lev-els shortly after one free-fall time scale. In this workwe explore this possibility and the resulting accretionrates with one-dimensional hydrodynamic simulations inspherical geometry. Numerical Methods
As discussed above, there are two major feedback pro-cesses that affect the BH accretion, namely the photo-ionization heating and the radiation pressure. To studythese processes and their effects on the BH accretion, onemust resolve the spatial scale smaller than the sonic ra-dius, which is of the order of 10 cm for a 100 M (cid:12) BH.Moreover, in order to investigate the dependence of BHaccretion on gas properties, a variety of gas density mustbe considered.In order to achieve a sufficiently high spatial resolu-tion, and simulate a wide range of gas density efficiently,we use a modified version of the one-dimensional hydro-dynamics code VH-1 by Blondin & Lufkin (1993), whichis publicly available . VH-1 is based on the piece-wiseparabolic method (Colella & Woodward 1984), and usesa Lagrange-Remap approach to solve the Euler equa-tions: ∂ t ρ + ∇ · ( ρ u ) = 0 ∂ t ( ρ u ) + ∇ · ( ρ uu ) + ∇ P = ρ a ∂ t ( ρE ) + ∇ · ( ρE u + P u ) = ρ u · a + H − C, (3)where the primary variables are the mass density ρ , gaspressure P , and fluid velocity u . In our implementation,the acceleration field a includes both gravity and radia-tive pressure forces, and H and C are heating and coolingsource terms arising from radiative feedback.The total energy per unit mass E is the sum of thekinetic energy and the internal energy e : E = 12 u + e, (4)and the internal energy is related to the pressurethrough the equation of state for an ideal gas for a givenadiabatic index γ . P = ρe ( γ − . (5)The modifications we made to VH-1 include two mainaspects. The first one is the straightforward change of auniform radial grid in spherical geometry to a logarithmicone. This is to ensure sufficiently high resolution near thecentral BH while covering a large spherical region with areasonable number of grid cells. The second one involvesa detailed treatment of radiative feedback. Specifically,we added the heating and cooling terms in the energyconservation equation, and the radiative pressure forcesin the momentum and energy conservation equations.The radiative feedback is modeled by formulating thecooling, heating, and radiation pressure with two param-eters: the ionization parameter ξ and gas temperature T .We assume that the photo-ionized region around the BHcan be fully described by ξ and T . The ionization pa-rameter ξ is defined as: ξ = L ( r ) n ( r ) r , (6) http://wonka.physics.ncsu.edu/pub/VH-1/ where L ( r ) is the local luminosity, and n ( r ) is the hydro-gen number density at radius r . The BH luminosity L (0)is assumed to have a power law spectrum, L ν ∼ ν − / ,and the attenuation of the luminosity is described by dL ( r ) dr = − πr ρca rabs , (7)where a rabs is the radiative acceleration due to photo-absorption, and c is the speed of light. In solving Equa-tion 7, we make another simplifying assumption that thespectral shape of L ( r ) is the same as the original L (0),so that the heating, cooling, and radiation accelerationat any radius can be obtained from pre-computed tablesof the ξ - T grid, H = n ˜ H ( ξ, T ) C = n ˜ C ( ξ, T ) a rabs = n ˜ a rabs ( ξ, T ) a rline = n ˜ a rline ( ξ, T ) , (8)where the scaling on the local hydrogen density is explic-itly factored out, and a rline is the radiation accelerationmainly due to resonant Ly α line scattering.We compute these two dimensional tables using thephoto-ionization code CLOUDY version 08.00 (Ferlandet al. 1998), assuming a primordial elemental abundance,and a power-law input spectrum with index − . a rcont = − a g ( r ) L (0) /L edd ( r ), where a g ( r ) and L edd arethe gravitational acceleration and Eddington luminosityof the BH, respectively. We note that the unattenuatedBH luminosity, L (0), is used in calculating a rcont , as theabsorbed luminosity is assumed to be re-emitted at en-ergies below the ionization threshold, which also con-tributes to Thomson scattering.Our feedback method is an improvement to thosepreviously used (e.g., Ciotti & Ostriker 2001; Sazonovet al. 2005; Ciotti & Ostriker 2007) in that it com-putes the heating, cooling, and radiation pressure forcesmore accurately, and applies to a temperature range of1 ≤ T ≤ K, much larger than the previous range10 ≤ T ≤ × K. As demonstrated in the mainpage, our simulations produce similar results as those byMilosavljevi´c et al. (2009b) and Park & Ricotti (2011) us-ing more sophisticated treatments on the same problems.Moreover, this method allows us to simulate high-densityclouds accurately and efficiently, which is critical in ourinvestigation.Overall, these modifications allow us to achieve anunprecedentedly high spatial resolution of 10 cm(10 − pc), and model a large range of gas density10 − cm − . In particular, the ability to reach densi-ties several orders of magnitude higher than the limit of10 cm − in previous work by Milosavljevi´c et al. (2009b)and Park & Ricotti (2011) is the key to our new findings. Table 1
Important Parameters and Properties of Simulations PerformedRuns n (cm − ) [1] r min (pc) [2] r max (pc) [3] N grid [4] ˙ M min [5] ˙ M avg [6] τ cycle (yr) [7] r ion (10 − pc) [8] A 10 × − [9] × − · · · · · · · · · · · · C 10 − × − − × − . × × − × × − × − × − − × − [10] − − [1] Ambient density of the gas sphere. [2]
Inner radius of the simulation volume. [3]
Outer radius of the simulation volume. [4]
Number of points for the logarithmic radial grid. [5]
Asymptotic minimum accretion rate in unit of 10 − M (cid:12) yr − . [6] Asymptotic average accretion rate in unit of 10 − M (cid:12) yr − . [7] Asymptotic period of the oscillation cycle. [8]
Ionization radius where the ionization fraction is 0.5. [9]
Only run B includes self-gravity. All other runs are without self-gravity in order to exam the main dependence of the accretion processon the ambient density. [10] At n = 10 cm − , the outer radius is reduced to 0.01 pc to accommodate the much smaller inner radius, and to make itcomputationally feasible. Initial Conditions and Parameters of Simulations
In this study, we model the radial accretion onto acentral BH embedded in a gas sphere. We assume a BHmass M BH = 100 M (cid:12) , a radiative efficiency of 0.1, andthe gas clouds are assumed to be metal-free and uniformwith an ambient density n , and an initial temperature T = 10 K. Outflow and inflow boundary conditions areused in the inner and outer boundaries, respectively. Theinner and outer boundaries of the sphere are chosen insuch a way that both sonic radius and ionization radiusare located well within the simulation region in order tosufficiently resolve the accretion flow.A total of 13 simulations with different initial hydrogendensities were performed in the present work. The mostimportant parameters and properties of these models arelisted in Table 1. In this table, column 1 lists the name ofthe simulations. Columns 2-5 give the numerical param-eters: the ambient density of the gas cloud, the inner andouter radius of the simulated sphere, and number of gridpoints, respectively. Columns 6-9 give the properties ofthe accretion process: the minimum and average accre-tion rates, the oscillation period of the accretion process,and the ionization radius where the ionization fraction is0.5, respectively.Run A used the same initial conditions as in other pre-vious work by Milosavljevi´c et al. (2009b) and Park &Ricotti (2011). It produced similar results of many as-pects of the accretion process, including the general in-termittent pattern, accretion rates, and oscillation pe-riod. This confirms the previous findings that radiativefeedback strongly suppresses BH accretion. In Run B,self-gravity of the gas was included. We find that self-gravity greatly modifies the accretion flow. It helps tomaintain a large inflow rate and increase density buildup outside of the ionization radius r ion , which leads to theshrink of r ion . When the density reaches a critical thresh-old, the radiative feedback becomes less effective, so theaccretion rate can reach Eddington limit. Runs C – Mcovered a wide range of gas density of 10 − cm − inorder to explore the dependence of accretion on gas den-sity. In these simulations, the self-gravity is deliberatelyturned off to avoid the density enhancement due to thecollapse outside of r ion . Such controlled simulations arenot only more computationally tractable than simply al-lowing the gas to collapse infinitely under self-gravity,they also better illustrate the main dependence of theaccretion process on the ambient density. RESULTS
Enhanced Accretion by Self-gravity of Gas
The BH accretion rate is dictated by the interplay be-tween inflow driven by gravity and external pressure andoutflow driven by radiation pressure. Without gas self-gravity, the accretion exhibits a pattern of periodic os-cillations, as demonstrated in Figure 1 ( top panel ) foran ambient density of 10 cm − . The intermittency isdue to alternating expulsion and fallback of the gas flowwith an average period of approximately 200 years. Themaximum rates occasionally exceed the Eddington limit( ˙ M Edd ∼ × − M (cid:12) yr − ), but the mean remainsroughly constant at about 25% of the Eddington rateover a very long timescale. These results are in goodagreement with previous ones from 2-dimensional simu-lations (Milosavljevi´c et al. 2009b; Park & Ricotti 2011).Similar intermittent behavior were also seen in other sim-ulations of Bondi-like (e.g., Krumholz et al. 2005; Ciotti& Ostriker 2007), or rotating accretion (e.g., Proga et al.2008) on different scales. Figure 1.
Time evolution of accretion rates onto a 100 M (cid:12) blackhole embedded in a uniform, primordial gas sphere with ambientdensity of 10 cm − , without (top panel) and with (bottom panel)self-gravity of the gas in the simulation. The dotted line in eachpanel indicates the Eddington rate ˙ M Edd ∼ × − M (cid:12) yr − . Theaccretion behavior exhibits a periodical oscillation caused by thealternating inflow driven by gravity and external pressure and out-flow driven by radiative feedback from the BH accretion. Withoutgas self-gravity, the average accretion is about 25% of the Edding-ton rate, but with self-gravity, it reaches ∼
86% of the Eddingtonrate after one free-fall timescale of the gas sphere.
However, when self-gravity of the gas is included, theBH accretion pattern changes dramatically, as shown inFigure 1 ( bottom panel ). It is clear that the minimumaccretion rate within one oscillation cycle gradually in-creases. In about one free-fall timescale of the gas sphere( ∼ . × years), the mean accretion rate reaches aconstant at ∼ . × − M (cid:12) yr − , or about 86% of theEddington rate.The reason for such a significant enhancement is thatthe BH accretion rate critically depends on the relationbetween ionization radius r ion and sonic radius, or moreaccurately, the standard Bondi accretion radius ignor-ing radiative feedback r B ( ∼ × cm for a 100 M (cid:12) BH in a 10 K cloud): the accretion is strongly sup-pressed by radiative feedback if r ion (cid:29) r B , but is lessaffected if r ion < r B . The properties of the accretionflow change when the r ion = r B boundary is crossed. Atradius smaller than r ion , radiative heating and photo-ionization dominate, while the BH’s gravitational fieldstarts to substantially modify the gas flow at radius closeto r B . When r ion < r B , the radiation feedback is notimportant at radius larger than r B except for Thomsonscattering. The gas flow can therefore achieve a largeinward velocity of the order of local sound speed at r B .On the other hand, when r ion > r B , radiation feedbackmay reduce the local accretion rate significantly, creat-ing a strong shocked region between r ion and r B . It isthis region with enhanced density that helps to maintaina relatively large inflow rate only limited by Thomsonscattering. Self-gravity gradually overcomes the radia- Figure 2.
Time evolution of BH accretion rate for different am-bient gas density, n = 10 , 10 , 10 , 10 , and 10 cm − , re-spectively. Note the accretion curves of the last four densities areshifted vertically for easy comparison, and the red line representsthe mean accretion rate, ˙M avg , of each simulation with the actualvalue (in unit of 10 − M (cid:12) yr − ) given next to it. The horizontalaxis is the scaled time, where the scale factor for different densityis given in vertical orientation at the end of the corresponding ac-cretion curve. In these simulations, self-gravity of the gas is delib-erately turned off to avoid density enhancement due to the collapseoutside of r ion . Such controlled simulations not only better unveilthe important dependence of the accretion process on the ambientdensity, they are also more computationally tractable than simplyallowing the gas to collapse infinitely under self-gravity. tion force and increases density buildup outside of r ion ,which leads to the shrinking of r ion to within r B . In theregime where r ion < r B , the accretion rate can approachto Eddington limit when the density at r B reaches a crit-ical value. Dependence of Accretion on Ambient Gas Density
An important question is then what the critical den-sity would be. Figure 2 shows the time evolution of ac-cretion rate at several different densities. As the densityincreases, the accretion rate becomes constant and ap-proaches to the Eddington limit.The oscillation period also decreases as density in-creases, and scales as n − . , as illustrated in Figure 3( top panel ). When the density reaches ∼ cm − , theoscillation is significantly damped after an initial time pe-riod, and the ionization radius r ion drops to below r B , asshown in Figure 3 ( bottom panel ). The radiation feedbackbecomes less effective. As a result, the mean asymptoticaccretion rate rises to about 90% of the Eddington limit.A density of 10 cm − or higher may be common in col-lapsing clouds in massive dark matter halos in the earlyuniverse, as reported in high-resolution simulations of thefirst protostars (e.g., Yoshida et al. 2008). This densityis much higher than those considered in previous workof similar problem (e.g., Alvarez et al. 2009; Milosavl- Figure 3.
Dependence of the oscillation period of the BH accre-tion, τ cycle ( top panel ), and ionization radius r ion ( bottom panel ),on the ambient density, n , of the gas sphere. The filled circles aresimulation data, while the solid lines are fitting curves. Both τ cycle and r ion decrease with density: τ cycle = 3 × ( n / cm − ) − . years, and r ion = 18 . n / cm − ) − . pc. At n (cid:38) cm − , theionization radius becomes r ion < r B ( r B ∼ . × − pc), the radi-ation feedback is weakened, and the oscillation is strongly damped. jevi´c et al. 2009b; Park & Ricotti 2011). It is thereforenot surprising that they found severely reduced accretionrates. A Universal Correlation Between Black HoleAccretion Rate and Ambient Gas Density
Figure 4 shows the relationship between the BH accre-tion rate and ambient density, which can be describedas ˙ M BH = ˙ M Edd (cid:16) n /n c n /n c (cid:17) β , where n c is the character-istic density at which the asymptotic accretion rate ap-proaches to the Eddington limit. As shown in the Figure,the fitting of the simulation data gives n c = 2 × cm − ,and β = 0 .
48. At densities above 2 × cm − , radiativefeedback other than Thomson scattering becomes lesseffective, and the accretion reaches the Eddington limit,while below that, radiative heating and photo-ionizationfeedback dominates and strongly suppresses the accre-tion. This relationship would imply an accretion ratelarger than the Bondi rate at densities below ∼ cm − .However, because both ionization and heating timescalesdecrease as density becomes smaller, they eventually be-come much less than the dynamical timescale of the gas.Under such conditions, the radiative feedback plays anegligible role, and the accretion becomes Bondi-like.More intriguingly, we repeat the simulations for aBH mass range of 10 − M (cid:12) , and find that thiscorrelation is universal over a wide range of BH massand gas density, with a simple scaling relation betweenthe BH mass and the critical density, n crit M BH =(2 × / cm − )(10 / M (cid:12) ). This scaling relation is easyto understand, as the hydrodynamic equations governingthe accretion process without self-gravity can be shownto depend only on n M BH , under the assumption that Figure 4.
Correlation between the BH asymptotic accretion rateand the ambient gas density. The filled circles represent datafrom simulations with radiation feedback, while the solid line isthe least-square fitting, which takes the following form: ˙ M BH =˙ M Edd (cid:16) n /n c n /n c (cid:17) β , where n c = 2 × cm − , and β = 0 .
48. Thisplot shows that, at densities below 2 × cm − , radiative feedbackdominates and strongly reduces the accretion to be sub-Eddington,while above that, radiative feedback becomes less effective, and theaccretion reaches the Eddington rate. the density is sufficiently high so that the local gas tem-perature is close to the thermal equilibrium determinedby the heating and cooling functions. That means, fora given BH mass, there exists a critical density, abovewhich the accretion rate can reach the Eddington limit.This general relation between BH accretion and am-bient gas density has important implications and appli-cations in studies of BH growth. We note that Bondiaccretion has been commonly used in simulations of BHgrowth (e.g., Li et al. 2007; Johnson & Bromm 2007; Al-varez et al. 2009; Di Matteo et al. 2005; Springel et al.2005; Hopkins et al. 2006; Di Matteo et al. 2008; Sijackiet al. 2009; Di Matteo et al. 2011). This simplified pre-scription neglects the effects of radiation feedback andoverestimates the accretion rate by up to two orders ofmagnitude at some densities below 10 cm − . Our re-sults and the fitting formula above can serve as a morerealistic recipe for BH accretion, and can be implementeddirectly into numerical simulations. SUMMARY
To summarize, we have presented a set of one-dimensional hydrodynamic simulations of the accretionof a black hole embedded in a primordial gas cloud, usingthe modified grid-based VH-1 code. We include not onlyimportant feedback processes from the accreting blackhole, but also self-gravity of the gas. We achieved anunprecedentedly high spatial resolution of 10 cm, andcovered a wide range of gas density of 10 − cm − .These advantages allowed us to study the accretion pro-cess in regimes not explored by previous work, and unveilthe following new findings:1. The accretion behavior exhibits a periodical oscil-lation caused by the alternating inflow driven bygravity and external pressure and outflow drivenby radiative feedback from the BH accretion. Self-gravity of the gas can boost the accretion rate bybuilding up high gas density which weakens thefeedback effects. Without gas self-gravity, the av-erage accretion is about 25% of the Eddington rate,but with self-gravity, it reaches ∼
86% of the Ed-dington rate after one free-fall timescale of the gassphere.2. The accretion depends strongly on the ambient gasdensity. For a given black hole mass, there exists acritical density above which the accretion can reachEddington limit. For example, for a 100 M (cid:12)
BH,the critical density is ∼ cm − .3. There exists a universal correlation between blackhole accretion rate and ambient gas density:˙ M BH = ˙ M Edd (cid:16) n /n c n /n c (cid:17) β , where n c = 2 × cm − , and β = 0 .
48. This fitting formula mayserve as a realistic recipe for BH accretion, andcan be implemented directly into numerical simu-lations.In the rare, high-density peaks (5 − σ ) of the cosmicfilaments where the first most massive dark matter haloscollapsed, the first stars were born and might evolve intothe first stellar-mass BHs. The extremely deep gravita-tional potential in these regions retained an abundant gassupply, and induced vigorous interactions and mergers ofprotogalaxies. Strong gravitational torques removed an-gular momentum from the highly shocked and dense gasand transported it to fuel the central BHs, while trig-gered global starburst on a larger scale. Seed BHs insuch a gas-rich and dynamical environment may growrapidly and co-evally with host galaxies to become thefirst luminous quasars at the cosmic dawn. ACKNOWLEDGMENTS
I thank Tom Abel, Tiziana Di Matteo, Mike Era-cleous, Carlos Frenk, Alex Heger, Lars Hernquist, Pe-ter M´esz´aros, Peng Oh, Massimo Ricotti, and DanielSchaerer for stimulating discussions and helpful com-ments. Support from NSF grants AST-0965694 andAST-1009867 is gratefully acknowledged. I thank theInstitute for Theory and Computation (ITC) at Har-vard University where the project was started for warmhospitality, and the Research Computing and Cyberin-frastructure unit of Information Technology Services atThe Pennsylvania State University for providing com-putational resources and services that have contributedto the research results reported in this paper. URL:http://rcc.its.psu.edu.
REFERENCESAbel, T., Bryan, G. L., & Norman, M. L. 2002, The Formation ofthe First Star in the Universe, Science, 295, 93Alvarez, M. A., Wise, J. H., & Abel, T. 2009, Accretion onto theFirst Stellar-Mass Black Holes, ApJ, 701, L133Antonucci, R. 1993, Unified models for active galactic nuclei andquasars, ARA&A, 31, 473Blondin, J. M. & Lufkin, E. A. 1993, The piecewise-parabolicmethod in curvilinear coordinates, ApJS, 88, 589Bondi, H. 1952, On spherically symmetrical accretion, MNRAS,112, 195 Bondi, H. & Hoyle, F. 1944, On the mechanism of accretion bystars, MNRAS, 104, 273Bromm, V. & Larson, R. B. 2004, The First Stars, ARA&A, 42,79Bromm, V., Yoshida, N., Hernquist, L., & McKee, C. F. 2009,The formation of the first stars and galaxies, Nature, 459, 49Carilli, C. L., Walter, F., Bertoldi, F., Menten, K. M., Fan, X.,Lewis, G. F., Strauss, M. A., Cox, P., Beelen, A., Omont, A., &Mohan, N. 2004, Radio Continuum Imaging ofFar-Infrared-Luminous QSOs at z >
6, AJ, 128, 997Ciotti, L. & Ostriker, J. P. 2001, Cooling Flows and Quasars. II.Detailed Models of Feedback-modulated Accretion Flows, ApJ,551, 131—. 2007, Radiative Feedback from Massive Black Holes inElliptical Galaxies: AGN Flaring and Central Starburst Fueledby Recycled Gas, ApJ, 665, 1038Clark, P. C., Glover, S. C. O., Smith, R. J., Greif, T. H., Klessen,R. S., & Bromm, V. 2011, The Formation and Fragmentationof Disks Around Primordial Protostars, Science, 331, 1040Colella, P. & Woodward, P. R. 1984, The Piecewise ParabolicMethod (PPM) for Gas-Dynamical Simulations, Journal ofComputational Physics, 54, 174Di Matteo, T., Colberg, J., Springel, V., Hernquist, L., & Sijacki,D. 2008, Direct Cosmological Simulations of the Growth ofBlack Holes and Galaxies, ApJ, 676, 33Di Matteo, T., Khandai, N., DeGraf, C., Feng, Y., Croft, R.,Lopez, J., & Springel, V. 2011, Cold flows and the first quasars,astro-ph 1107.1253Di Matteo, T., Springel, V., & Hernquist, L. 2005, Energy inputfrom quasars regulates the growth and activity of black holesand their host galaxies, Nature, 433, 604Fan, X., Carilli, C. L., & Keating, B. 2006, Observationalconstraints on Cosmic Reionization, ARA&A, 44, 415Fan, X., Narayanan, V. K., Lupton, R. H., Strauss, M. A.,Knapp, G. R., Becker, R. H., White, R. L., Pentericci, L.,Leggett, S. K., Haiman, Z., Gunn, J. E., Ivezi´c, ˇZ., Schneider,D. P., Anderson, S. F., Brinkmann, J., Bahcall, N. A.,Connolly, A. J., Csabai, I., Doi, M., Fukugita, M., Geballe, T.,Grebel, E. K., Harbeck, D., Hennessy, G., Lamb, D. Q.,Miknaitis, G., Munn, J. A., Nichol, R., Okamura, S., Pier,J. R., Prada, F., Richards, G. T., Szalay, A., & York, D. G.2001, A Survey of z > > > Jiang, L., Fan, X., Brandt, W. N., Carilli, C. L., Egami, E.,Hines, D. C., Kurk, J. D., Richards, G. T., Shen, Y., Strauss,M. A., Vestergaard, M., & Walter, F. 2010, Dust-free quasarsin the early Universe, Nature, 464, 380Jiang, L., Fan, X., Hines, D. C., Shi, Y., Vestergaard, M.,Bertoldi, F., Brandt, W. N., Carilli, C. L., Cox, P., Le Floc’h,E., Pentericci, L., Richards, G. T., Rieke, G. H., Schneider,D. P., Strauss, M. A., Walter, F., & Brinkmann, J. 2006,Probing the Evolution of Infrared Properties of z ˜ 6 Quasars:Spitzer Observations, AJ, 132, 2127Johnson, J. L. & Bromm, V. 2007, The aftermath of the firststars: massive black holes, MNRAS, 374, 1557Krolik, J. H. 1999, Active galactic nuclei : from the central blackhole to the galactic environment, ed. Krolik, J. H., Activegalactic nuclei : from the central black hole to the galacticenvironment (Princeton University Press)Krumholz, M. R., McKee, C. F., & Klein, R. I. 2005, BondiAccretion in the Presence of Vorticity, ApJ, 618, 757Li, Y., Hernquist, L., Robertson, B., Cox, T. J., Hopkins, P. F.,Springel, V., Gao, L., Di Matteo, T., Zentner, A. R., Jenkins,A., & Yoshida, N. 2007, Formation of z˜6 Quasars fromHierarchical Galaxy Mergers, ApJ, 665, 187Li, Y., Hopkins, P. F., Hernquist, L., Finkbeiner, D. P., Cox,T. J., Springel, V., Jiang, L., Fan, X., & Yoshida, N. 2008,Modeling the Dust Properties of z ˜6 Quasars withARTˆ2-All-Wavelength Radiative Transfer with AdaptiveRefinement Tree, ApJ, 678, 41Madau, P. & Rees, M. J. 2001, Massive Black Holes as PopulationIII Remnants, ApJ, 551, L27Maiolino, R., Cox, P., Caselli, P., Beelen, A., Bertoldi, F., Carilli,C. L., Kaufman, M. J., Menten, K. M., Nagao, T., Omont, A.,Weiß, A., Walmsley, C. M., & Walter, F. 2005, First detectionof [CII]158 µ m at high redshift: vigorous star formation in theearly universe, A&A, 440, L51Milosavljevi´c, M., Bromm, V., Couch, S. M., & Oh, S. P. 2009a,Accretion onto ”Seed” Black Holes in the First Galaxies, ApJ,698, 766Milosavljevi´c, M., Couch, S. M., & Bromm, V. 2009b, AccretionOnto Intermediate-Mass Black Holes in Dense ProtogalacticClouds, ApJ, 696, L146Mortlock, D. J., Warren, S. J., Venemans, B. P., Patel, M.,Hewett, P. C., McMahon, R. G., Simpson, C., Theuns, T.,Gonz´ales-Solares, E. A., Adamson, A., Dye, S., Hambly, N. C.,Hirst, P., Irwin, M. J., Kuiper, E., Lawrence, A., & R¨ottgering,H. J. A. 2011, A luminous quasar at a redshift of z = 7.085,Nature, 474, 616Narayanan, D., Li, Y., Cox, T. J., Hernquist, L., Hopkins, P.,Chakrabarti, S., Dav´e, R., Di Matteo, T., Gao, L., Kulesa, C.,Robertson, B., & Walker, C. K. 2008, The Nature of COEmission from z˜6 Quasars, ApJS, 174, 13Oh, S. P. & Haiman, Z. 2002, Second-Generation Objects in theUniverse: Radiative Cooling and Collapse of Halos with VirialTemperatures above 104