The Growth of Black Holes from Population III Remnants in the Renaissance Simulations
Britton Smith, John Regan, Turlough Downes, Michael Norman, Brian O'Shea, John Wise
MMon. Not. R. Astron. Soc. , 1–12 (2016) Printed 2 August 2018 (MN L A TEX style file v2.2)
The Growth of Black Holes from Population III Remnants in theRenaissance Simulations
Britton D. Smith (cid:63) , John A. Regan † , Turlough P. Downes , Michael L. Norman , ,Brian W. O’Shea , , , & John H. Wise San Diego Supercomputer Center, University of California, San Diego, 10100 Hopkins Drive, La Jolla, CA 92093 Centre for Astrophysics & Relativity, School of Mathematical Sciences, Dublin City University, Glasnevin, Ireland Center for Astrophysics and Space Sciences, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093 National Superconducting Cyclotron Laboratory, Michigan State University, MI, 48823, USA Department of Physics and Astronomy, Michigan State University, MI, 48823, USA Department of Computational Mathematics, Science and Engineering, Michigan State University, MI, 48823, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA Center for Relativistic Astrophysics, Georgia Institute of Technology, 837 State Street, Atlanta, GA 30332, USA
ABSTRACT
The formation of stellar mass black holes from the remnants of Population III stars provides asource of initial black hole seeds with the potential to grow into intermediate or, in rare cases,possibly super-massive black holes. We use the
Renaissance simulation suite to follow thegrowth of over 15,000 black holes born into mini-haloes in the early Universe. We compute theevolution of the black holes by post-processing individual remnant Population III star particlesin the
Renaissance simulation snapshots. The black holes populate haloes from M (cid:12) upto M (cid:12) . We find that all of the black holes display very inefficient growth. On average,the black holes increase their initial mass by a factor − , with the most active black holesincreasing their mass by approximately 10%. Only a single black hole experiences any periodof super-Eddington accretion, but the duration is very short and not repeated. Furthermore,we find no correlation of black hole accretion with halo mass in the mass range sampled.Within most haloes, we identify clumps of cool, dense gas for which accretion rates wouldbe high, but instances of black holes encountering these clumps are rare and short-lived. Starformation competes with black hole growth by consuming available gas and driving downaccretion rates through feedback. We conclude that the black holes born from Population IIIremnants do not form a significant population of intermediate mass black holes in the earlyUniverse and will need to wait until later times to undergo significant accretion, if at all. Key words:
Cosmology: theory – large-scale structure – first stars, methods: numerical
The existence of supermassive black holes (SMBHs) in the firstbillion years of the Universe presents a significant challenge to ourunderstanding of the formation of the first compact objects in ourUniverse. The earliest SMBHs observed have masses upwards of abillion solar masses (e.g. Fan et al. 2006; Mortlock et al. 2011; Ven-emans et al. 2013; Wu et al. 2015; Ba˜nados et al. 2018). The meansby which black holes could grow to be so massive so quickly repre-sents a serious theoretical challenge. Beginning with the sphericalaccretion model, the maximum accretion rate for a black hole can (cid:63)
E-mail:[email protected] † Marie Skłodowska-Curie Fellow be expressed as M ( t ) = M exp (cid:16) − (cid:15) r (cid:15) r tt Edd (cid:17) , (1)where t Edd = 0 . Gyr and (cid:15) r is the radiative efficiency. For a“standard” radiative efficiency of (cid:15) r ∼ . and a black hole seedmass of M = 10 M (cid:12) , it takes nearly 1 Gyr to grow to M(t) ∼ M (cid:12) . At present, there exist two potential origin stories forthe SMBHs that inhabit the centres of massive galaxies and shineas bright quasars. The seeds of massive black holes may have been“light” ( ∼ − (cid:12) ), beginning as the remnants of the firststars (e.g. Madau & Rees 2001) or from the core collapse of densestellar clusters (G¨urkan et al. 2004, 2006; Devecchi & Volonteri2009; Katz et al. 2015). Alternatively, the seeds may have been“heavy” ( ∼ − M (cid:12) ), being the end product of supermassivestar formation (e.g., Hosokawa et al. 2013a,b; Woods et al. 2017).The first (Population III) stars in the Universe form out of c (cid:13) a r X i v : . [ a s t r o - ph . GA ] J u l B.D. Smith, J.A. Regan, T.P. Downes, M.L. Norman, B.W. O’Shea & J.H. Wise metal-free gas with H as the primary coolant. Within a few hun-dred million years after the Big Bang, Population (Pop) III starswill begin forming in 10 M (cid:12) haloes (Tegmark et al. 1997; Yoshidaet al. 2003). The inefficiency of H as the sole gas coolant re-sults in initial stellar masses that are tens to hundreds of timesmore massive than the sun (Bromm et al. 1999; Abel et al. 2000,2002a; Bromm et al. 2002; O’Shea & Norman 2007a; Turk et al.2009a; Clark et al. 2011; Hirano et al. 2014). This range of massesprovides multiple pathways for black hole formation, includingcore-collapse supernovae or hypernovae (11 M (cid:12) ≤ M ≤ M (cid:12) ;Woosley & Weaver 1995; Nomoto et al. 2006) and direct formation(40 M (cid:12) ≤ M ≤ M (cid:12) and M > M (cid:12) ; Heger & Woosley2002), leading to a population of light seeds that have been im-planted into the building blocks of galaxies. The expected largenumber density of Pop III stars in the early Universe (Trenti & Sti-avelli 2009; Crosby et al. 2013) makes them natural candidates forthe seeds of SMBHs.Accretion onto Pop III remnants has been investigated as apathway for forming SMBHs through both analytical (e.g. Madau& Rees 2001) and semi-analytical mechanisms (e.g. Tanaka &Haiman 2009; Pezzulli et al. 2016). Even if only a small fraction ofthe Pop III remnant black holes grow at the Eddington rate, it wouldbe enough to seed the entire population of SMBHs observed in theUniverse. Detailed numerical simulations have also been used tostudy the initial conditions surrounding the black hole which formsfrom a Pop III star. These simulations take into account the radi-ation field generated by the Pop III star, the formation of an HIIregion surrounding the star, and any associated supernova explo-sion. The stellar radiation and supernova successfully evacuate thegas from the host halo, resulting in black holes that are “ born starv-ing” (Whalen et al. 2004; O’Shea et al. 2005; Johnson & Bromm2007; Milosavljevi´c et al. 2009). Simulations following the evolu-tion for up to 200 Myr after initial formation find that these blackholes continue to experience no significant growth (Alvarez et al.2009). However, these simulations did not have sufficient dynamicrange to follow the subsequent mergers of remnant mini-haloes intolarger atomic cooling haloes in which the black holes may be ableto experience significant accretion events.Jeon et al. (2012) simulate the growth of 100 M (cid:12) black holesfrom Pop III remnants with and without feedback from the accret-ing black hole. In the case of no feedback, they find that growth isnegligible for ∼
80 Myr until the halo reaches the atomic coolinglimit, at which time growth increases significantly. When feedbackis included, Jeon et al. (2012) find that growth remains insignificantthrough the end of the simulation at z = 10. However, Volonteriet al. (2015) claim that the early bottleneck in growth might also bealleviated by short periods of super-Eddington growth that allowblack holes to grow by several orders of magnitude in only 10 Myr.If a black hole is able to migrate into an environment where super-critical accretion becomes possible, then growth to a SMBH masswithin the timescale of approximately 500 Myr becomes possible(Lupi et al. 2016; Valiante et al. 2016; Pezzulli et al. 2017; Pacucciet al. 2017).Heavy seeds emerge from a rarer, more exotic channel whereunusually high accretion rates lead to the formation of a super-massive star (Begelman et al. 2006, 2008; Schleicher et al. 2013;Hosokawa et al. 2013b; Woods et al. 2017; Haemmerl´e et al. 2018).These direct collapse black holes (DCBHs) are thought to form inpristine atomic cooling haloes where H formation has been sup-pressed preventing the formation of smaller Pop III stars (Wiseet al. 2008; Regan & Haehnelt 2009a,b; Agarwal et al. 2012, 2013;Becerra et al. 2015; Latif et al. 2013b,a; Regan et al. 2014; Agar- wal et al. 2016; Regan et al. 2016, 2017). DCBH scenarios havethe distinct advantages of starting from much larger masses thanlight seeds and also existing in environments where significantlymore fuel is likely to be present (Hosokawa et al. 2016; Nakauchiet al. 2017). However, the very existence of supermassive stars isdebated and furthermore it is not clear whether the DCBH scenariocan provide a sufficient number density of black holes to explainthe existence of all SMBHs.In this work, we seek to quantify the range of possibilities forthe Pop III light seed scenario in the early Universe. We follow thegrowth of 15,000 black holes in the Renaissance simulations (Xuet al. 2013a, 2014; O’Shea et al. 2015; Xu et al. 2016b,a) over ap-proximately 300 Myr, three orders of magnitude in halo mass, andthree different large-scale galactic environments. We supplementthis data set with 12 Pop III remnants from the
Pop2Prime simu-lations (Smith et al. 2015), which have superior mass, spatial, andtime resolution. Both of these sets of simulations follow the for-mation and evolution of individual Pop III stars using radiation-hydrodynamics in a cosmological context. We examine the growthrates of the total black hole population and identify the commonali-ties of those that grow the most. We then focus on the halo with themost black holes. Finally, we study how the black hole growth rateis regulated by star formation. The layout of the paper is as follows.In Section 2, we describe the simulations used in this work. In Sec-tion 3.1, we discuss the methods for modeling black hole growth.In Section 4, we present the results of the investigations describedabove. Finally, we conclude with a discussion and summary in Sec-tion 5.
All simulations analyzed in this work were performed withthe open-source, adaptive mesh-refinement + N-body code,
Enzo (Bryan et al. 2014).
Enzo has been used extensively tosimulate high-redshift structure, including the formation of PopIII stars (Abel et al. 2002b; O’Shea et al. 2005; O’Shea & Nor-man 2007b, 2008a; Turk et al. 2009b, 2010, 2011a, 2012), low-metallicity stars (Smith & Sigurdsson 2007; Smith et al. 2009a;Meece et al. 2014; Smith et al. 2015), and the first galaxies (Wiseet al. 2012; Wise et al. 2012a; Xu et al. 2013b; Wise et al. 2014;Chen et al. 2014a). The two suites of simulations used here are de-scribed below.
The Renaissance Simulations have been well detailed previously inthe literature (Xu et al. 2013a, 2014; Chen et al. 2014b; Ahn et al.2015; O’Shea et al. 2015; Xu et al. 2016a,b), and here we onlysummarize the simulation characteristics relevant to this study. Allof the Renaissance simulations were carried out in a comoving vol-ume of (40 Mpc) , created with the M USIC (Hahn & Abel 2011)initial conditions generator. The cosmological parameters were setusing the 7-year WMAP Λ CDM+SZ+LENS best fit (Komatsu et al.2011b): Ω m = 0.266, Ω Λ = 0.734, Ω b = 0.0449, h = 0 . , σ =0 . and n = 0 . . First, an exploratory simulation with particles ( . × M (cid:12) per dark matter particle) was run to z = 6 .Three regions of interest were then selected for re-simulation athigher resolution, namely a rare-peak region, a normal region anda void region. To generate the three regions the initial, lower res-olution, volume was smoothed on a physical scale of 5 comovingMpc, and regions of high ( (cid:104) δ (cid:105) ≡ (cid:104) ρ (cid:105) / (Ω M ρ C ) − ∼ . ) - c (cid:13)000
The Renaissance Simulations have been well detailed previously inthe literature (Xu et al. 2013a, 2014; Chen et al. 2014b; Ahn et al.2015; O’Shea et al. 2015; Xu et al. 2016a,b), and here we onlysummarize the simulation characteristics relevant to this study. Allof the Renaissance simulations were carried out in a comoving vol-ume of (40 Mpc) , created with the M USIC (Hahn & Abel 2011)initial conditions generator. The cosmological parameters were setusing the 7-year WMAP Λ CDM+SZ+LENS best fit (Komatsu et al.2011b): Ω m = 0.266, Ω Λ = 0.734, Ω b = 0.0449, h = 0 . , σ =0 . and n = 0 . . First, an exploratory simulation with particles ( . × M (cid:12) per dark matter particle) was run to z = 6 .Three regions of interest were then selected for re-simulation athigher resolution, namely a rare-peak region, a normal region anda void region. To generate the three regions the initial, lower res-olution, volume was smoothed on a physical scale of 5 comovingMpc, and regions of high ( (cid:104) δ (cid:105) ≡ (cid:104) ρ (cid:105) / (Ω M ρ C ) − ∼ . ) - c (cid:13)000 , 1–12 the rare peak; average ( (cid:104) δ (cid:105) ∼ . ) - the normal region; and low( (cid:104) δ (cid:105) ∼ − . ) - the void region. The comoving volumes of thethree regions were 133.6, 220.5 and 220.5 Mpc , respectively. Eachsimulated region was then re-initialized with a further three nestedgrids for an effective resolution of 4096 and a dark matter particleresolution of . × M (cid:12) within the high-resolution region. Dur-ing the simulation, further adaptive refinement was allowed up to amaximum 12 levels, leading to a maximum spatial resolution of 19comoving pc (1.2 proper parsecs at z = 15 ). The simulations wereevolved to a final redshift z = 15 , . and 9.9 for the Rare-Peak,Normal and Void realisation respectively. The halo mass function iswell-resolved down to × M (cid:12) (70 particles per halo), and at theending redshift, the three realisations contained a total of 822, 758,458 galaxies having at least 1,000 particles ( M vir (cid:39) . × M (cid:12) )and ∼ , Pop III remnant black holes (see Table 1).The simulations include both self consistent Pop III and metal-enriched star formation (Pop II) at the maximum refinement leveland capture star formation in haloes as small as × M (cid:12) (Xuet al. 2013a). Pop III star formation is selected if the metallic-ity is less than − of the solar fraction in the highest densitycell with metal-enriched star formation proceeding otherwise. Thefunctional form of the IMF is a power-law with a slope of -1.3 withan exponential cutoff above a characteristic mass of 40 M (cid:12) . Theoperational mass range of the IMF is 1 M (cid:12) ≤ M ≤ M (cid:12) (seeWise et al. (2012b) for additional details.) Stellar feedback usesthe M ORAY radiative transport framework (Wise & Abel 2011) forH ionizing photons. Lyman-Werner (LW) radiation that dissociates H is modeled using an optically thin, inverse square law profile,centered on all star particles. At the end of their main-sequencelifetimes, Pop III stars in the mass range, 11 M (cid:12) ≤ M ≤ M (cid:12) ,explode as core-collapse supernova with total energies and metal-yields calculated by Nomoto et al. (2006). Pop III stars in the massrange 140 M (cid:12) ≤ M ≤ M (cid:12) explode as pair-instability super-nova (PISN) with total energy of ∼ − × erg over thePISN mass range and metal yields calculated by Heger & Woosley(2002). For Pop III stars outside of the above mass ranges (40 M (cid:12) < M < M (cid:12) and M > M (cid:12) ), no feedback is added afterthe main-sequence lifetime. The ionization states of hydrogen andhelium are followed with a 9-species primordial non-equilibriumchemistry and cooling network (Abel et al. 1997), supplementedby metal-dependent cooling tables (Smith et al. 2009b). No H self-shielding in included in the simulations as the densities at which itbecomes relevant are not fully resolved by the simulations. A LWbackground radiation field is also included to model radiation fromstars which are not within the simulation volume (Wise et al. 2012).In the high density region of the rare-peak simulation, the LW ra-diation from stars dominates over the background. Although thesimulations cannot follow Pop III star formation in haloes below × M (cid:12) , star formation is suppressed by the LW backgroundin such haloes (Machacek et al. 2001; Wise & Abel 2007; O’Shea &Norman 2008b). Finally, we note the existence of multiple versionsof each of these three simulations in the literature. The variationswere run to different redshifts and used slightly different methodsfor calculating the global Lyman-Werner radiation field. For clarity,we list the simulations used here in Table 1 by the names given tothem in the upcoming public data release along with the referenceof their first appearance. We supplement the
Renaissance simulations with an extension ofthe simulation presented in Smith et al. (2015), referred to here as
Table 1.
Simulation SummarySimulation a V b hr [(Mpc com.) ] z c f N d black holes Ref. e Rare-Peak LWB 133.6 15.0 6518 1Normal BG1 220.5 11.6 6225 2Void BG1 220.5 9.9 2487 3Pop2Prime 0.004 10 ∗
12 4(a) the simulation name; (b) volume of the high resolution region; (c) thefinal redshift of the simulation; (d) the total number of Pop III remnantblack holes at the final redshift. * - after z = 11 . , this simulation wascontinued to z = 10 with star formation turned off; (e) publication offirst appearance. 1: O’Shea et al. (2015), 2: Xu et al. (2016b), 3: Xu et al.(2016a), 4: Smith et al. (2015). the Pop2Prime simulation. With significantly higher mass and spa-tial resolution, the
Pop2Prime simulation provides some constrainton the dependence of the results on resolution. The
Pop2Prime sim-ulation uses a 500 comoving kpc/h box, initialized at z = 180 with the M USIC initial conditions generator with the WMAP 7best-fit cosmological parameters, Ω m = 0 . , Ω λ = 0 . , Ω b = 0 . , H = 71 . km/s/Mpc, σ = 0 . , and n s = 0 . (Komatsu et al. 2011a), and using a Eisenstein & Hu (1999) trans-fer function and second-order Lagrangian perturbation theory. Thesimulation follows the region around a halo reaching a virial massof 1.7 × M (cid:12) at z = 10 . The initial conditions are gener-ated with 512 grid cells and dark matter particles on the rootgrid and two additional levels of nested refinement surroundingthe target halo, corresponding to a comoving spatial resolutionof 0.244 kpc/h, and a baryon (dark matter) mass resolution of0.259 M (cid:12) (1.274 M (cid:12) ).The Pop2Prime simulation includes the formation and feed-back from Pop III stars in a manner similar to the
Renaissance sim-ulations, with the addition of He ionizing radiation (only H ion-izing radiation was used in the
Renaissance simulations) usingthe M
ORAY adaptive ray-tracing method and treating LW radia-tion as optically thin with 1/ r attenuation. In contrast to the Re-naissance simulations, which adopt a power-law Pop III IMF, allPop III stars in
Pop2Prime are given a mass of 40 M (cid:12) and endtheir main-sequence lifetimes (3.86 Myr) in a core-collapse super-nova with total energy of 10 erg. Since the original goal of the Pop2Prime simulation was to study the collapse and fragmentationof metal-enriched gas, this simulation does not form any Pop IIstars. Instead, gas with metallicity greater than 10 − Z (cid:12) is allowedto collapse until a number density of ∼ cm − is reached,at which time the simulation stops. This occurs at z ∼ . af-ter a total of 12 Pop III stars have formed. The simulation is thencarried forward to z = 10 , roughly an additional 100 Myr, withstar formation turned off. This serves as an illuminating experi-ment of the effects of stellar feedback on black hole growth. The Pop2Prime simulation uses the same chemistry and cooling ma-chinery as the
Renaissance simulations, but with the additions ofthree deuterium species (D, D + , and HD), H formation on dustgrains (described in Meece et al. 2014), and self-shielding of LWradiation using the model of Wolcott-Green et al. (2011). The simulations discussed here do not contain a subgrid prescrip-tion for black hole formation. Regardless of their initial mass, Pop c (cid:13) , 1–12 B.D. Smith, J.A. Regan, T.P. Downes, M.L. Norman, B.W. O’Shea & J.H. Wise
III star particles are given a negligible mass at the end of theirmain-sequence lifetimes and do not accrete from their surround-ings. They then effectively act as extremely low mass dark matterparticles. The simulations continue to update the positions, veloci-ties, and accelerations of these particles due to gravity, so they willbe located in approximately the same positions as if they had beenevolved fully self-consistently as black holes. We rely on this fact tomodel the growth of black holes represented by these particle usingaccretion rates calculated from the local gas conditions within theavailable simulation snapshots. This is not self-consistent and com-pletely ignores the gravitational force of the black holes on the sur-rounding material, changes in momentum/trajectory of the particledue to accretion, and radiative feedback from accretion. Neverthe-less, this exercise can provide a rough estimate of the evolution ofstellar mass black holes in the early universe across different galac-tic environments. Within the uncertainties created by the first twocaveats, the lack of radiative feedback serves to provide an upperlimit on the overall black hole growth.
We compute growth histories for each Pop III star particle expectedto form a black hole based on its zero-age main-sequence (ZAMS)mass. Given the initial mass of the star particle, we calculate itsinitial black hole mass by linearly interpolating from the results ofWoosley & Weaver (1995, Table 3) for stars with M < M (cid:12) .For stars with 140 M (cid:12) < M ≤ M (cid:12) , the star has undergone aPISN and so we assume no compact remnant. For stars with 260 M (cid:12) < M ≤ M (cid:12) (the upper limit of the Pop III IMF in the Renaissance simulations), we set the initial black hole mass to bethe mass of the He core using the relation from Heger & Woosley(2002, Equation 1), given by M He (cid:39) M ∗ −
20 M (cid:12) ) . (2)We model black hole growth as spherical Bondi-Hoyle accre-tion (Hoyle & Lyttleton 1941; Bondi 1952), where the growth rateis given by ˙ m B − H (cid:39) απρG M BH max( | (cid:126)v | , c s ) , (3)where M is the mass of the black hole, ρ is the gas density sur-rounding the black hole, c s is the local sound speed, | (cid:126)v | is the mag-nitude of the velocity of the black hole relative to the surroundingmaterial, and α is a dimensionless boost factor. Krumholz et al.(2005) show that the accretion rate is decreased when the gas hasnon-zero vorticity, but we ignore this effect to consider the mostoptimistic growth scenario. The boost factor term was first addedto Equation 3 by Springel et al. (2005) to account for underestima-tion in the gas density in the vicinity of the black hole caused bylimited spatial resolution of the simulation. This scale is the Bondiradius, given by r b = 2 GM BH c s , (4)which we do not resolve in our simulations. Booth & Schaye (2009)excellently summarize the subsequent use of the boost factor inproceeding works, noting the commonly adopted constant values of α = 100 − (although see Pelupessy et al. 2007; Kim et al. 2011,for alternative approaches to the boost factor). However, Booth &Schaye (2009) argue that values of α > are unphysical when themedium is single-phase and its associated Jeans length is resolved by the simulation. In the Pop2Prime simulations, the Jeans lengthis resolved explicitly by a minimum of 64 grid cells. In the
Renais-sance simulations, Jeans length-based refinement is not used, butwe find that in practice the Jeans length is refined in the vicinity ofthe black hole particle by at least 4 cells (i.e., the grid cell contain-ing it) roughly 99.9% of the time. Booth & Schaye (2009) arguethat boost factors should be used when the medium is expectedto be multi-phase and the associated spatial scales are unresolved.The scale of the multi-phase medium is not set by the Jeans length,but instead by the cooling length (the cooling time multiplied bythe sound speed) as it forms through thermal instability (Voit et al.2017; McCourt et al. 2018). For the circumgalactic medium, Field-ing et al. (2017) find that gas is stable against going multi-phasefor halo masses below roughly 10 . M (cid:12) , relating the associatedvirial temperature to the point in the cooling curve where coolingtimes become long. In our case, we find that the black hole particlesspend the majority of their time in two regimes: hot ( T ∼ − K), underdense ( n < − cm − ) gas that is the product of stel-lar feedback; and cooler ( T ∼ K), denser (10 − cm − < n < cm − ) gas heated to the virial temperature, but unable to coolfurther due to its low metallicity. Due to the long cooling times, weexpect that in practice the thermal instability will have no impactand hence the gas will be single phase in both these regimes. There-fore, we choose to adopt a constant value of α = 1 (i.e., no boost)in our growth model. Finally, we do not cap the black hole growthrate at the Eddington limit, which is given by ˙ m Edd = 4 πGM BH m p (cid:15) r σ T c (cid:39) . × − (cid:18) . (cid:15) r (cid:19) (cid:18) M BH M (cid:12) (cid:19) [ M (cid:12) /yr ] , (5)where m p is the proton mass, (cid:15) r is the radiative efficiency, σ T isthe Thomson cross-section, and c is the speed of light. Throughoutthis work, we refer to the Eddington rate assuming (cid:15) r = 0 . , ap-propriate for a non-rotating Schwarzschild black hole (Shakura &Syunyaev 1973). As we show below, instances of near-Eddingtonaccretion are extremely rare. We allow for super-Eddington accre-tion only to highlight instances where the physical conditions createa situation where it could be possible.Starting with the first simulation snapshot after which a PopIII star particle has exceeded its main-sequence lifetime, we useEquation 3 to compute the particle’s instantaneous growth rate .Assuming the density of the grid cell decreases negligibly due toaccretion by the particle, Equation 3 can be solved analytically togive the black hole’s mass at snapshot i + 1 , given its mass at snap-shot i , M i , and the timestep between snapshots, ∆ t as M i +1 = M i − ˙ m B − H ∆ tM i . (6)We compute the final mass of each black hole particle by iteratingover all available snapshots for each simulation. For the Renais-sance simulations, the average time between snapshots is roughly4 Myr. For the
Pop2Prime simulation, the average time betweensnapshots is about 0.8 Myr.
Below, we present the results of growing the Pop III remnant blackholes for each simulation to its final snapshot, focusing primarily on All analysis codes used in this work, including figure-generating scripts,are available as an extension package for the yt analysis code (Turk et al.2011b) at https://github.com/brittonsmith/yt_p3bh .c (cid:13) , 1–12 N b l a c k h o l e s rare peakz = 15 10 f ( N > ) N b l a c k h o l e s normalz = 11.6 10 f ( N > ) M halo [M ]020406080 N b l a c k h o l e s voidz = 9.9 10 f ( N > ) Figure 1.
Number of black holes in a halo as a function of halo mass forthe rare peak (top), normal (middle), and void (bottom) runs of the
Renais-sance simulations. The black line indicates the median in bins of 0.5 dexand the red line shows the fraction of haloes with at least one black hole. the
Renaissance simulations. We note that the Rarepeak and Nor-mal simulations were run until a qualitatively similar amount ofstructure (number of haloes, stars, etc.) had formed, hence the simi-larity in the number of black holes formed. However, this is not truefor the Void simulation, whose final redshift was determined by aprior simulation that was used to create a LW background modelfor the Void simulation. Unless otherwise stated, the results shownrefer to the final output of each simulation. Table 1 lists the finalredshift and total number of black holes formed in each simulation.
In Figure 1, we show the number of black holes as a function ofhalo mass at the final snapshot of each of the
Renaissance simu-lations. Over the three simulations we find that the distribution ofblack holes is scattered from haloes as small as a few times M (cid:12) , roughly the resolution limit, up to approximately M (cid:12) ,the maximum halo mass. The void haloes (bottom panel) showthe smallest number of black holes per halo with on average lessthan one black hole per halo up to M halo ∼ M (cid:12) . The normaland rare haloes show a slightly larger scatter with a handful ofhaloes having up to 40 black holes per halo up to M halo ∼ M (cid:12) . In all cases, less than roughly 1% of haloes with M halo < M (cid:12) are populated with Pop III black holes. Above a mass of M halo ∼ M (cid:12) , the number of black holes shows a markedincrease in number, especially for the normal and rare peak haloes.This is because at this mass scale atomic haloes form throughthe merger and accretion of mini-haloes which previously hostedblack holes. Xu et al. (2013a) found in their investigation of PopIII stars in the Renaissance simulations that the number of PopIII stars and remnants peaks in haloes with masses of a few × M (cid:12) and that the growth in the number of remnants comes mainlyfrom mergers of mini-haloes. They find that Pop III stars formonly in haloes with masses between × and × M (cid:12) . PopIII stars found in higher mass haloes appear there via mergers. Itis the remnants of these Pop III stars that now populate the moremassive atomic cooling haloes. The median line (solid black linein Figure 1) shows a clear increase in the number of black holesin haloes more massive than approximately M (cid:12) due to theeffect of mergers. We therefore sample quite well the black holeoccupation fraction in haloes up to M halo ∼ M (cid:12) . In Figure 2, we show histograms of black hole properties, includ-ing formation redshift, relative growth, final mass, and maximuminstantaneous growth rate. As expected, the black hole formationrate is indicative of the large-scale overdensity associated with eachsimulation. The landscape of overall black hole growth is notablybleak. Not a single black hole is able to double in size, with the bestcases growing by roughly 13% in the Rarepeak. In the less densegalactic environments, the maximal mass growth is even lower,with the best case in the Normal run growing only by 2%, and thatof the Void run by just under 1%. In all three cases, the distributionof relative black hole growth is bimodal, with peaks at ∼ − to ∼ − and a broader peak from ∼ − to ∼ − . The distri-bution of maximum instantaneous growth rates closely resemblesthe overall relative growth. In all, only a single black hole in theRarepeak is able to achieve super-Eddington accretion. The over-whelming majority of black holes accrete maximally at less than10 − of their Eddington rates. Figure 3 shows a probability distri-bution function of all instantaneous growth rates for all black holesand all snapshots. Only 2-3% of all growth rates exceed 10 − ofthe Eddington rate the total number of super-Eddington events isjust one, i.e., the one black hole that experiences super-Eddingtongrowth does so only once.The Pop2Prime simulation shows a similar bimodal distribu-tion of individual growth rates, albeit with narrower peaks and anoverall much smaller range of total values. The two peaks corre-spond to two dinstinct physical conditions in which the black holestend to exist. The lower of the two peaks is from hot, underdensegas associated with stellar feedback. All black holes forming ina supernova event will live in this phase at least once, and likelymuch longer given the long associated cooling times and contin-ually occurring star formation. The higher peak comes from gasabout to form stars, where the medium is slighly denser and heatedto roughly the virial temperature. In
Pop2Prime , the lower peak oc-curs at a higher growth rate because of the relative weakness ofthe stellar feedback producing lower temperatures in the hot phase.Haloes in
Pop2Prime , with masses of only a few hundred thou-sand M (cid:12) , form only 1-2 stars total. These smaller haloes also havelower virial temperatures and central gas densities, thus moving thelocation of the second peak in Figure 3 to lower accretion rates.In Figure 4, we plot the relative growth of black holes as afunction of their age. For black holes in the bulk of the relative c (cid:13) , 1–12 B.D. Smith, J.A. Regan, T.P. Downes, M.L. Norman, B.W. O’Shea & J.H. Wise
30 25 20 15 10z form N b l a c k h o l e s M f / M i - 110 N b l a c k h o l e s ave. growth rate [M /yr]10 N b l a c k h o l e s max growth rate / Eddington10 N b l a c k h o l e s Figure 2.
Black hole population statistics for the final snapshot of the rare peak (blue, z = 15 ), normal (green, z = 11 . ), and void (orange, z = 9 . ) runs ofthe Renaissance simulations.
Top Left Panel : black hole formation redshift. The drop-off corresponds to the final redshift of each simulation.
Top Right Panel :relative overall black hole growth. The vast majority of black holes grow by a neglible amount. No black holes are able to increase their mass by more than10%.
Bottom Left Panel : average absolute growth rate.
Bottom Right Panel : maximum instantaneous growth rate, as a fraction of the Eddington rate, achievedat any point during the simulation. growth distribution (Figure 2, top-right panel), there is effectivelyno relation between overall growth and age apart from the lack ofolder black holes at the lowest values of relative growth. The blackholes showing the most growth are relatively young, with ages lessthan about 50 Myr. In Figure 5, we show the individual growthhistories for all black holes growing by at least 0.5%. In all butone case, these black holes reach > of their final mass in lessthan 10 Myr. There are 9 black holes in the Rarepeak realisationthat grow by at least 0.5%, 4 in the Normal, and only one in theVoid. Of the 14 black holes shown here, 6 reach accretion ratesof at least one quarter of Eddington, with one reaching 2.5 timesEddington. In all cases, this strong growth lasts for only a singlesnapshot, and is therefore likely overestimated. Not surprisingly, all14 of these black holes are in the mass windows where formationoccurs without a preceding supernova, i.e., 40 M (cid:12) < M < M (cid:12) and M > M (cid:12) . These stellar mass ranges correspond toinitial black hole masses of 16.6 M (cid:12) < M < M (cid:12) and M > M (cid:12) . Apart from this initial period of super critical growth, noblack holes are able to accrete at rates exceeding the Eddington limit, with most accreting at rates many orders of magnitude belowthe Eddington rate.Finally, in Figure 6 we plot the specific growth rate (averagegrowth rate divided by initial mass) for all black holes as a func-tion of host halo mass. Within the mass range tracked by the Re-naissance simulations (10 M (cid:12) < M halo < M (cid:12) ), we see noevidence of the larger gas reservoirs of more massive haloes aidingin black hole growth, except to the extent that higher mass haloesshow a scarcity of the most slowly growing black holes. This find-ing appears to be in agreement with the isolated galaxy simulationsof Pelupessy et al. (2007), who find no instances super-Eddingtongrowth in haloes up to 10 M (cid:12) . The cosmological simulations ofHabouzit et al. (2017), which include black hole growth with feed-back in a 10 Mpc comoving box, also find very limited accretionat high redshift. Similar to this work, the distribution of accretionrates in Habouzit et al. (2017) also show a peak around 10 − ofthe Eddington rate, although their larger box size and lower finalredshift (at the cost of lower resolution) are able to capture moreinstances of much higher accretion rates. c (cid:13)000
Bottom Right Panel : maximum instantaneous growth rate, as a fraction of the Eddington rate, achievedat any point during the simulation. growth distribution (Figure 2, top-right panel), there is effectivelyno relation between overall growth and age apart from the lack ofolder black holes at the lowest values of relative growth. The blackholes showing the most growth are relatively young, with ages lessthan about 50 Myr. In Figure 5, we show the individual growthhistories for all black holes growing by at least 0.5%. In all butone case, these black holes reach > of their final mass in lessthan 10 Myr. There are 9 black holes in the Rarepeak realisationthat grow by at least 0.5%, 4 in the Normal, and only one in theVoid. Of the 14 black holes shown here, 6 reach accretion ratesof at least one quarter of Eddington, with one reaching 2.5 timesEddington. In all cases, this strong growth lasts for only a singlesnapshot, and is therefore likely overestimated. Not surprisingly, all14 of these black holes are in the mass windows where formationoccurs without a preceding supernova, i.e., 40 M (cid:12) < M < M (cid:12) and M > M (cid:12) . These stellar mass ranges correspond toinitial black hole masses of 16.6 M (cid:12) < M < M (cid:12) and M > M (cid:12) . Apart from this initial period of super critical growth, noblack holes are able to accrete at rates exceeding the Eddington limit, with most accreting at rates many orders of magnitude belowthe Eddington rate.Finally, in Figure 6 we plot the specific growth rate (averagegrowth rate divided by initial mass) for all black holes as a func-tion of host halo mass. Within the mass range tracked by the Re-naissance simulations (10 M (cid:12) < M halo < M (cid:12) ), we see noevidence of the larger gas reservoirs of more massive haloes aidingin black hole growth, except to the extent that higher mass haloesshow a scarcity of the most slowly growing black holes. This find-ing appears to be in agreement with the isolated galaxy simulationsof Pelupessy et al. (2007), who find no instances super-Eddingtongrowth in haloes up to 10 M (cid:12) . The cosmological simulations ofHabouzit et al. (2017), which include black hole growth with feed-back in a 10 Mpc comoving box, also find very limited accretionat high redshift. Similar to this work, the distribution of accretionrates in Habouzit et al. (2017) also show a peak around 10 − ofthe Eddington rate, although their larger box size and lower finalredshift (at the cost of lower resolution) are able to capture moreinstances of much higher accretion rates. c (cid:13)000 , 1–12 M B H / M
Edd f rare peaknormalvoidp2p Figure 3.
Probability distribution function of all instantaneous growth ratesfor all black holes over all times for all simulations. Growth rates are shownas a fraction of the Eddington rate. For the
Pop2Prime simulation (p2pabove), the period of time where star formation is turned off is not shownas this is unphysical.
The global statistics shown in Figures 4, 5 and 6 support theconclusion that light seeds, born from Pop III remnant black holesdo not grow efficiently by accretion in haloes up to M halo ∼ M (cid:12) . Next, we examine a single halo in detail to understand whythese black holes are unable to grow. As we have seen the black hole accretion rate shows no markedincrease as a function of time or halo mass. To further understandthe evolution of Pop III remnant black holes we examine in de-tail the halo with the most black holes, coming from the Normalrun with a total of 77 black holes at the final output. In Figure7, we plot the halo’s large-scale gas distribution with the effective“accretability” of the central, dense gas shown in the bottom panel.Here, we define the accretability as the ratio of the Bondi-Hoylerate to the Eddington rate, divided by black hole mass. If we ignorethe relative motion term in Equation 3 and consider only the soundspeed of the gas, the above quantity is independent of black holemass and is simply a measure of the gas properties. Accretabilityhas units of M − (cid:12) , meaning that for an accretability of 0.1 M − (cid:12) ,a 1 M (cid:12) black hole would accrete at 0.1 of Eddington and a 10 M (cid:12) black hole would accrete at Eddington. Regions of high acc-retability are clearly associated with dense gas, but the converse ofthat statement is not necessarily true. Interestingly, high accretabil-ity clumps do not appear to be centrally concentrated.In the bottom panel of Figure 7, we overplot the locations ofall black holes in the inner halo, ( r < . r vir .) We use the yt clump finder (Smith et al. 2009b; Turk et al. 2011b) to identifyall topologically disconnected regions with accretability of at least − M − (cid:12) . We choose this value as it is the minimum value for afew hundred M (cid:12) black hole (the maximum mass considered in thiswork) to approach the Eddington limit. To remove projection ef-fects, the black holes shown in Figure 7 are colored by the distanceto the nearest gas clump with accretability of at least − M − (cid:12) .The closest encounter between a black hole and a highly accretableclump is roughly 30 pc, but on average, black holes are many hun- M f / M i - rare peak10 M f / M i - normal0 100 200 300black hole age [Myr]10 M f / M i - void Figure 4.
Relative black hole growth as a function of black hole age, whereM i and M f are the initial and final masses, respectively. No strong trend ofblack hole growth versus age exists. However, black holes with the mostand least growth are generally young ( (cid:46) Myr). dreds of pc away from these clumps. In this snapshot, no blackholes are within such a clump.In order for a black hole to experience high growth rates, itmust intersect with a high accretability clump at some time. Toquantify the frequency of interactions between black holes andclumps, we measure the distances from each black hole to theedge of the nearest clump over the history of this halo and all ofits progenitors. We construct a merger-tree of this halo using the consistent-trees merger-tree code (Behroozi et al. 2013b).We use the ytree code (Smith 2018) to walk the tree, interfacewith yt , and run the clump finding algorithm as described abovefor all progenitors of the halo in question. In Figure 8, we plot thedistribution of distances between black holes and nearest clumpsas a function of time, with the median value for black holes withinone quarter of the virial radius shown in black. Throughout mostof the halo’s history, black holes remain on average a few hundredpc away from clumps in which they could grow rapidly, with theclosest black holes still tens of pc away. In total, we note 10 occa-sions of black holes existing in highly accretable clumps, with theirdetails shown in Table 2. Half of these ten occurrences consist of c (cid:13) , 1–12 B.D. Smith, J.A. Regan, T.P. Downes, M.L. Norman, B.W. O’Shea & J.H. Wise black hole age [Myr]10 M ( t ) / M i - rare peaknormalvoid Figure 5.
Relative black hole mass as a function of age for all black holeswith total relative growth of at least 0.005. Stars indicate the time when theblack hole has accomplished 90% of its total growth and the circles denoteits final mass and age. Dashed lines indicate periods of black hole growthof at least 0.25 of the Eddington rate. M halo [M ]10 s p e c i f i c g r o w t h r a t e [ / y r ] rare peaknormalvoid Figure 6.
Specific black hole growth rate as a function of halo mass, wherespecific growth rate is defined as (M f - M i ) / (M i × age). Solid lines indi-cate median number of black holes per halo in mass bins of 0.25 dex. Wefind no correlation of black hole growth with halo mass up to M halo ∼ M (cid:12) . a new black hole in the direct formation mass range. Two othersin the same mass range are only 2.5 Myr old. The maximum timespent inside a clump was just over 3 Myr with an average accretionrate of − M (cid:12) /yr, and in all cases the accretion rate was sub-Eddington. Black holes appear to have a difficult time remaining inclumps, either because they migrate out or because those clumpsare consumed or destroyed. If the latter is true, then the most likelycause is star formation, as highly accretable gas is cold and dense.We use the Pop2Prime simulation to test the hypothesis thatblack hole growth is regulate by star formation and feedback. Asdescribed in § Pop2Prime simulation is an extremely highresolution simulation in a very small volume and hence only 12black holes exist by z ∼
12. At z (cid:39) . , we turn off star forma-tion, but allow the simulation to evolve in every other respect. The [ g / c m ]
500 pc 10 ( m B H / m E dd ) / m b h [ / M ]
100 400 700bh-clump distance [pc]
Figure 7.
Top Panel: projection of mass-weighted mean density for the halowith the most black holes (M halo = 1.6 × M (cid:12) , n BH = 77 ). The sizeof the projected region denotes the halo’s virial radius of 3.4 kpc. BottomPanel: mass-weighted projection of gas accretability, defined as the ratioof Bondi-Hoyle to Eddington, divided by black hole mass. For example, avalue of 0.1 M − (cid:12) would allow a 1 M (cid:12) black hole to accrete at 0.1 Edding-ton and a 10 M (cid:12) black hole to accrete at Eddington. The projected regionis 0.25 of the virial radius. Circles indicate the locations of black holes, withcolors denoting the distance to the nearest gas clump with an accretabilityof at least 10 − M − (cid:12) . growth rates for all 12 black holes are shown in Figure 9. After starformation is disabled, the mean growth rate increases by roughlytwo orders of magnitude in 100 Myr. However, more notably, thegrowth rates of the oldest black holes, whose haloes have had muchmore time to reassemble, have increased by a much greater degree.Jeon et al. (2014) find that ∼ M (cid:12) mini-haloes can take morethan 100 Myr to reassemble, so the low growth rates of the latestforming black holes are not surprising. The top-right panel of Fig-ure 9 indicates that it is the increase in density which drives theenhancement of black hole growth after star formation has ceased.The sound speed has also dropped considerably at this time, butthis turns out to be unimportant as the gas/particle relative veloc-ity remains roughly 10 km/s. With a self-consistent treatment of c (cid:13) , 1–12 Table 2.
Black holes interacting with clumps in Figure 8.Particle ID a z b m c bh [M (cid:12) ] age d [Myr] ˙m e bh [M (cid:12) / yr] ˙m f bh / ˙m Edd ∆ t g [Myr] ∆ m h [M (cid:12) ]618050081 15.2 46.053 0.00 1.167e-07 1.152e-01 2.547 2.973e-01618051131 15.2 48.921 0.00 1.878e-07 1.745e-01 2.547 4.784e-01618048385 15.2 28.510 0.00 8.338e-08 1.329e-01 2.547 2.124e-01” 15.1 28.722 2.54 5.292e-08 8.374e-02 2.587 1.369e-01618049168 15.2 27.457 0.00 5.915e-08 9.793e-02 2.547 1.507e-01” 15.1 27.607 2.54 2.032e-08 3.346e-02 2.587 5.258e-02618062820 15.1 17.272 0.00 4.052e-08 1.066e-01 2.587 1.048e-01617970154 14.8 8.547 33.88 1.957e-09 1.041e-02 2.712 5.307e-03617970240 13.8 42.303 67.36 5.976e-08 6.422e-02 3.195 1.910e-01617977875 13.8 17.271 63.07 7.217e-09 1.899e-02 3.195 2.306e-02(a) the particle id of the black hole; (b) redshift of interaction; (c) black hole mass; (d) black hole age; (e) Bondi-Hoyle accretion rate; (f) fraction ofEddington accretion rate; (g) time between current and next snapshots; (h) mass accreted in ∆ t.
200 250 300 350 400t [Myr]10 d i s t a n c e t o c l u m p [ p c ] %
18 17 16 15 14 13 12 z Figure 8.
Distribution of distances between a black hole and a high acc-retability gas clump (greater than 10 − M − (cid:12) ) for all progenitor haloes ofthe halo hosting the most black holes (shown in Figure 7.) Where the shadedregion extends to the bottom of the figure, some black holes exist withinhigh accretion rate clumps. See Table 2 for a list of all instances of blackholes within clumps. Values shown in red correspond to all black holeswithin a halo’s virial radius. The black, dashed line denotes the median sep-aration for black holes within 0.25 of the virial radius. black hole formation and evolution (i.e., not what we have donehere), black holes should eventually sink to the center of the halodue to dynamical friction, although this will be slow because oftheir low masses. Indeed, Sugimura et al. (2018) find that dynami-cal friction is unable to transport black holes formed in mini-halosinto the inner, gas-rich regions of atomic cooling halos. As galax-ies grow larger, they will be more able to co-locate black holes andaccretable gas. Regardless, this provides strong evidence that theability of stellar feedback to destroy cold, dense gas through ra-diation and supernovae is quite important in regulating black holegrowth. The goal of this paper was to use the large sample of Pop III rem-nant black holes provided by the
Renaissance simulations to studyboth their evolution and growth over the course in the early Uni-verse. The
Renaissance simulations sample three large-scale galac- tic environments, span three orders of magnitude in halo mass (10 M (cid:12) ≤ M halo ≤ M (cid:12) ), provide roughly 300 Myr of black holeevolution time, and form roughly 15,000 Pop III remnant blackholes. This mass range allows us to span the boundary between H -cooling mini-haloes and atomic cooling haloes. We supplementthis with 12 Pop III remnants from the extremely high-resolution Pop2Prime simulation, in which we disable star formation part ofthe way through to test the effect of stellar feedback on black holegrowth. Our results can be summarized as: • In all simulations, the vast majority of Pop III remnant blackholes grow by negligible amounts, with relative mass gains rangingfrom 10 − to 0.1. Less than 100 of the 15,000 total black holesgrew by more than 10 − of their initial masses, with most of thesecoming from the Rarepeak (densest region), followed by the Nor-mal (next densest), and just a few from the Void simulation. Theinstantaneous accretion rates only exceed the Eddington rate onetime for one black hole. • The black holes that grew the most did so within about 10Myr of their formation, after which time they grew negligibly. Allformed from Pop III stars in the mass range where no supernovaoccurs. These black holes show no preference in host halo mass. • Clumps of gas with high Bondi-Hoyle accretion rates existwithin galaxies, but the instances of black holes existing withinthem are rare and short-lived. On average, black holes are tensto hundreds of pc away from highly accretable (cold and dense)clumps. These clumps appear to be rapidly detroyed by star for-mation and feedback before black holes have a chance to accretesignificantly from them. In examining the halo hosting the mostblack holes in the
Renaissance simulations, most of the instancesof black holes located within accretable clumps were newly bornand formed without a supernova. • In the
Pop2Prime simulations, the average black hole growthincreased by more than two orders of magnitude within 100 Myrof turning off star formation. Black holes in haloes that had moretime to reassemble after the black hole-forming supernova showedsignificantly increased growth rates, up to five orders of magnitude.This is due primarily to the increase in gas density.Overall the
Renaissance simulations indicate that black holesborn from the remnants of Pop III stars never enter regions whererapid and sustained accretion is possible. The early bottleneckwhich has been previously shown to prevent early black holegrowth continues as black holes migrate into more massive haloes,although the halos studied here remain under the minimum masswhere Pelupessy et al. (2007) find super-Eddington growth may be c (cid:13) , 1–12 B.D. Smith, J.A. Regan, T.P. Downes, M.L. Norman, B.W. O’Shea & J.H. Wise M [ M / y r ]
24 20 17 15 13 12 11 10z 10 [ g / c m ]
24 20 17 15 13 12 11 10z200 300 400 500t [Myr]10 c s [ k m / s ]
200 300 400 500t [Myr]10 v r e l [ k m / s ] Figure 9.
Growth histories and associated gas physical conditions for all 12 black holes in the
Pop2Prime simulation, with median values shown by the thick,black line. The vertical, red dashed line denotes the time when star formation was turned off.
Top-left: accretion rate.
Top-right: gas density.
Bottom-left: localsound speed.
Bottom-right: relative velocity between gas and particle. possible. We conclude that black holes born from Pop III remnantsin mini-haloes are likely to experience very limited growth in theearly Universe. We see no appreciable growth in any black holes inhalo masses up to M (cid:12) . This provides further evidence of thedifficulty of Pop III remnant black holes reaching SMBH scales by z = 6 , although the possibility of doing so in rarer, more mas-sive haloes still exists. This has additional consequences for theformation of intermediate mass black holes (IMBHs). If the PopIII remnants seeds are unable to accrete as cosmic time proceedsin more massive haloes then they will not grow to form a popu-lation of IMBHs and may go someway to explaining the relativedearth of IMBHs observed thus far (Koliopanos 2018). It is alsopossible that over a longer period these black holes will sink to thecentre of larger haloes or undergo several interactions with clumpswhich enable the black hole to grow. However, Renaissance doesnot follow the evolution of structure sufficiently far for us to answerthat question. Alternatively, it may also be that periods of high ac-cretion (possibly super-Eddington) can be achieved in metal-freehaloes with reduced H fractions rather than in the minihaloes thatled to the creation of the black holes examined here. However, Re-naissance has no prescription to form stars in such environments,though the environments do exist in
Renaissance (Wise et al. in prep). Such regions with their deeper potential wells may be moreadvantageous for forming rapidly accreting black hole seeds whichcan then go on to become IMBH and/or SMBHs.A number of other simplifications made in this work or by thesimulations studied have the potential to alter the findings presentedhere. The use of post-processing limits the temporal resolution tothe time between snapshots, which here is generally a few Myr.Any events occuring between snaphots will naturally be lost. Inparticular, this likely leads to an underestimation of the early-timegrowth of black holes forming without a supernova, i.e., the situa-tion that we find to yield the most growth. As well, any other dy-namic effects requiring an accurate calculation of the black hole’smass evolution during the simulation, such as dynamical friction,cannot be accounted for. The influence of feedback from the accret-ing black hole on the survival of high accretability clouds is alsonot clear. Black hole feedback could act similarly to stellar feed-back and destroy these clouds, or it may simply delay star forma-tion while largely keeping the cloud in tact. This could potentiallygive any embedded black holes additional time to grow. Finally, theoptically-thin treatment of the LW radiation field may lead to arti-ficially low H fractions in some instances, thus reducing the cool-ing rate in ∼ K gas and lowering both the star formation and c (cid:13)000
Renaissance (Wise et al. in prep). Such regions with their deeper potential wells may be moreadvantageous for forming rapidly accreting black hole seeds whichcan then go on to become IMBH and/or SMBHs.A number of other simplifications made in this work or by thesimulations studied have the potential to alter the findings presentedhere. The use of post-processing limits the temporal resolution tothe time between snapshots, which here is generally a few Myr.Any events occuring between snaphots will naturally be lost. Inparticular, this likely leads to an underestimation of the early-timegrowth of black holes forming without a supernova, i.e., the situa-tion that we find to yield the most growth. As well, any other dy-namic effects requiring an accurate calculation of the black hole’smass evolution during the simulation, such as dynamical friction,cannot be accounted for. The influence of feedback from the accret-ing black hole on the survival of high accretability clouds is alsonot clear. Black hole feedback could act similarly to stellar feed-back and destroy these clouds, or it may simply delay star forma-tion while largely keeping the cloud in tact. This could potentiallygive any embedded black holes additional time to grow. Finally, theoptically-thin treatment of the LW radiation field may lead to arti-ficially low H fractions in some instances, thus reducing the cool-ing rate in ∼ K gas and lowering both the star formation and c (cid:13)000 , 1–12 black hole growth efficiencies. A significantly higher H fractioncould potentially even give rise to thermal instability, which couldrequire the use of a boost factor higher than one for the accretionrate. However, an increase in the cooling would also allow stars toform more readily, which will act to destroy high-accretability gas,making the precise balance unclear. As always, these and any othershortcomings provide ample avenues for further progress. ACKNOWLEDGEMENTS
We thank the anonymous referee for their useful comments onthe manuscript. BDS is supported by NSF AST-1615848. JARacknowledges the support of the EU Commission through theMarie Skłodowska-Curie Grant - “SMARTSTARS” - grant num-ber 699941. MLN was supported by NSF AST-1102943 andAST-1615848. BWO was supported in part by NSF grants PHY-1430152 and AST-1514700, by NASA grants NNX12AC98G,NNX15AP39G, and by Hubble Theory Grants HST-AR-13261.01-A and HST-AR-14315.001-A. JHW is supported by Na-tional Science Foundation grant AST-1614333, NASA grantNNX17AG23G, and Hubble theory grants HST-AR-13895 andHST-AR-14326. The
Renaissance and
Pop2Prime simulationswere performed on Blue Waters, operated by the National Centerfor Supercomputing Applications (NCSA) with PRAC allocationsupport by the NSF (award number ACI-0832662). The subsequentanalysis was performed using time awarded under award numberACI-1514580. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the NSF (awardnumber ACI-1238993 and ACI-1514580) and the state of Illinois.Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its NCSA. Finally, this research was made pos-sible by a mountain of open-source scientific software, including
Enzo (Bryan et al. 2014), yt (Turk et al. 2011b), Rockstar (Behroozi et al. 2013a), consistent-trees (Behroozi et al.2013b), and ytree (Smith 2018), sitting on a vast continentof open-source software packages and libraries like
Python , matplotlib , and NumPy , just to name a few. We are gratefulto everyone who contributed to these projects.
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