Initial mass function of intermediate mass black hole seeds
aa r X i v : . [ a s t r o - ph . GA ] J un Mon. Not. R. Astron. Soc. , 1–18 (2012) Printed 20 August 2018 (MN L A TEX style file v2.2)
Initial mass function of intermediate mass black hole seeds
A. Ferrara , , S. Salvadori , B. Yue , D. R. G. Schleicher Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands Institut f¨ur Astrophysik, Georg-August-Universit¨at G¨ottingen, Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, the University of Tokyo
20 August 2018
ABSTRACT
We study the Initial Mass Function (IMF) and hosting halo properties of IntermediateMass Black Holes (IMBH, 10 − M ⊙ ) formed inside metal-free, UV illuminated atomiccooling haloes (virial temperature T vir > K) either via the direct collapse of thegas or via an intermediate Super Massive Star (SMS) stage. These IMBHs have beenrecently advocated as the seeds of the supermassive black holes observed at z ≈
6. Weachieve this goal in three steps: (a) we derive the gas accretion rate for a proto-SMSto undergo General Relativity instability and produce a direct collapse black hole(DCBH) or to enter the ZAMS and later collapse into a IMBH; (b) we use merger-treesimulations to select atomic cooling halos in which either a DCBH or SMS can formand grow, accounting for metal enrichment and major mergers that halt the growth ofthe proto-SMS by gas fragmentation. We derive the properties of the hosting haloesand the mass distribution of black holes at this stage, and dub it the “Birth MassFunction”; (c) we follow the further growth of the DCBH by accreting the leftovergas in the parent halo and compute the final IMBH mass. We consider two extremecases in which minihalos ( T vir < K) can ( fertile ) or cannot ( sterile ) form starsand pollute their gas leading to a different IMBH IMF. In the (fiducial) fertile casethe IMF is bimodal extending over a broad range of masses, M ≈ (0 . − × M ⊙ ,and the DCBH accretion phase lasts from 10 to 100 Myr. If minihalos are sterile, theIMF spans the narrower mass range M ≈ (1 − . × M ⊙ , and the DCBH accretionphase is more extended (70 −
120 Myr). We conclude that a good seeding prescriptionis to populate halos (a) of mass 7 . < log( M h /M ⊙ ) <
8, (b) in the redshift range8 < z <
17, (c) with IMBH in the mass range 4 . < (log M • /M ⊙ ) < . Key words: cosmology — star formation — black hole physics — galaxies: high-redshift
Along with the formation of first stars (Bromm & Yoshida2011), the appearance of black holes (Volonteri & Bellovary2012) is one of the most remarkable events occur-ring well within the first cosmic billion year (red-shift z > ∼ molecules by UV ( > .
755 eV)and Lyman-Werner (LW, 11 . − . c (cid:13) Ferrara et al. sity of the LW flux raises above a critical threshold, J ⋆ν,c (Machacek et al. 2001; Fialkov et al. 2012) gas cannot cooland form stars, and consequently stellar-mass black holes,in minihalos that have virial temperature T vir < K.On the other hand, when larger, metal-free , atomic-cooling( T vir > ∼ K) halos are illuminated by a sufficiently strongLW flux J ν,c > J • ν,c (Loeb & Rasio 1994; Eisenstein & Loeb1995; Begelman et al. 2006; Lodato & Natarajan 2006;Shang et al. 2010; Johnson et al. 2012; Regan & Haehnelt2009; Agarwal et al. 2012; Latif et al. 2013a) a “direct col-lapse black hole” (DCBH) can form . The precise val-ues of J ⋆ν,c and J • ν,c depend on radiative transfer, chem-istry and spectral shape of the sources and they are onlyapproximately known; however there is a broad agree-ment that J ⋆ν,c ≪ J • ν,c = 30 − − ergs − cm − Hz − sr − .Recent numerical simulations and stellar evolution cal-culations have provided strong support in favor of the di-rect collapse model. In a cosmological framework, Latif et al.2013a have shown that strong accretion flows of ≈ M ⊙ yr − can occur in atomic cooling halos illuminated by strong ra-diation backgrounds. As stellar evolution calculations by(Hosokawa et al. 2013) and (Schleicher et al. 2013) suggestonly weak radiative feedback for the resulting protostars,these calculations were followed for even longer time, sug-gesting the formation of ≃ M ⊙ black holes (Latif et al.2013b). Their accretion can be potentially enhanced in thepresence of magnetic fields, which may suppress fragmenta-tion in the centers of these halos (Latif et al. 2014).Even if these conditions (metal-free, atomic cooling ha-los illuminated by a J ν,c > J • ν,c UV field) for the forma-tion of a DCBH are met, little is known on: (i) the ex-istence of an upper mass limit of DCBH host halos; (b)the duration of the DCBH formation/growth phase; (c) thefinal DCBH mass function. A fourth important question,not addressed here, concerns the final fate (e.g. inclusionin a super-massive black-hole, ejection from the host) ofthis intermediate ( M • = 10 − M ⊙ ) black hole population.These questions are at the core of a large number of cos-mological and galaxy formation problems and therefore thequest for solid answers is very strong. Additional motivationscome from a possible interpretation of the near-infrared cos-mic background fluctuations and its recently detected cross-correlation with the X-ray background (Cappelluti et al.2013), which might imply that an unknown faint populationof high- z black holes could exist (Yue et al. 2013a;Yue et al.2013b).As we will show, answering the above questions requiresa detailed description of the mass accretion and merger his-tory of the atomic halos that satisfy the conditions describedabove. The process starts with the growth of a proto-SMSstar inside metal-free atomic cooling halos embedded in astrong LW radiation field. The growth, fed by a high ac- Unless differently stated, we express the field intensity in theusually adopted units J = J × − erg s − cm − Hz − sr − . DCBH formation has however to pass through an interme-diate stellar-like phase, as discussed later. In addition, it hasbeen suggested (e.g. Omukai et al. 2008, Davies et al. (2011),Devecchi et al. (2012), Miller & Davies (2012)) that intermedi-ate mass black holes can also form via runaway stellar collisionsin nuclear clusters. Here we do not consider such a scenario. cretion rate, typically ≈ . M ⊙ yr − , can be blocked by atleast two type of events: the first is accretion of polluted gas,either brought by minor mergers or smooth accretion fromthe IGM.Metals would enhance the cooling rate driving thermalinstabilities finally fragmenting the gas into clumps whichcannot be accreted as their angular momentum is hard todissipate. The second stopping process could be a majormerger that generates vigorous turbulence, again disruptingthe smooth accretion flow onto the central proto-SMS star.Note that at the rates discussed above it only takes 10 yr tobuild a 10 M ⊙ SMS. If these events indeed occur, the starstops accreting and rapidly evolves toward very hot ZeroAge Main Sequence (ZAMS) SMS emitting copious amontsof UV photons clearing the remaining halo gas out of the po-tential well. After a very brief lifetime ( < < ∼ M ⊙ . Based on the same prescriptions,Volonteri et al. (2008) follow the mass assembly of SMBHresulting from such seeds up to present time using a MonteCarlo merger tree. Perhaps closer in spirit, but fundamen-tally different for the physical processes involved, is the pro-posal by Volonteri & Begelman (2010) that seed black holesmay form via the already mentioned “quasi-star” phase, inwhich an embedded black hole forms from the collapse of aPop III star and accretes gas at high ( ≈ M ⊙ yr − ) rates.The resulting IMBH IMF peaks at a few × M ⊙ , but raresupermassive seeds, with masses up to 10 M ⊙ , are possi-ble. This approach however, does not deal with complicat-ing effects, included here, as the quenching of accretion dueto metal pollution and feedback effects from the accretingblack hole.Throughout the paper, we assume a flat Universe c (cid:13) , 1–18 he IMF of black holes seeds with cosmological parameters given by the PLANCK13(Planck Collaboration et al. 2013) best-fit values: Ω m =0 . Λ = 1 − Ω m = 0 . b h = 0 . h = 0 . σ = 0 . n s = 0 . After the early pioneering studies (Iben 1963;Chandrasekhar 1964; Loeb & Rasio 1994;Shapiro & Shibata 2002), the interest in the evolutionof supermassive stars (SMS) has recently received renewedattention in the context of DCBH formation. In particular,(Hosokawa et al. 2012a; Latif et al. 2013a; Johnson et al.2013b; Hosokawa et al. 2013) research has focused on thepreviously unexplored cases of very rapid mass accretion,˙ M = 0 . − M ⊙ yr − . In the following we summarize thecurrent understanding of the aspects of SMS evolution thatare relevant to the present work.The rate at which the proto-SMS accretes gas fromthe surroundings plays a key role in its evolution and inparticular on the stellar radius-mass relation. This can beappreciated by comparing two characteristic evolutionarytimescales: the Kelvin-Helmholtz (KH), t KH ≡ GM ⋆ R ∗ L ∗ , (1)and the accretion, t acc ≡ M ⋆ ˙ M , (2)timescales. In the early phases of the evolution, even for verysmall values of ˙ M , the strong inequality t KH ≫ t acc holds,i.e. the time scale on which the star radiates its gravita-tional energy is much longer than the time on which mass isadded to the star by accretion. Hence, the star grows almostadiabatically. As free-free absorption, providing the neces-sary opacity to radiative losses, depends on temperature as κ ∝ ρT − . , while the stellar temperature increases withmass, at some point t KH ≈ t acc . After this stage, the sub-sequent evolution of the star depends on the accretion rate.For sufficiently low values, ˙ M < ∼ − M ⊙ yr − the star en-ters a contraction phase during which temperatures becomesufficiently high to ignite hydrogen burning and the star en-ters the ZAMS (for ˙ M < ∼ − M ⊙ yr − this occurs when thestar has reached M ⋆ ≈ M ⊙ ) on the standard metal-freeeffective temperature-mass relation (Bromm et al. 2001b) T eff = 1 . × K (cid:18) M ⋆ M ⊙ (cid:19) . . (3)Note that the weak mass dependence implies that massivemetal-free stars are very hot and therefore produce copi-ous amounts of ultraviolet radiation that rapidly ionizes andclears out the remaining surrounding envelope. Thus, radia-tive feedback effects rapidly quench the further growth ofstars once they reach the ZAMS.The situation is drastically different for accretionrates exceeding ˙ M ≃ − M ⊙ yr − , as pointed out byHosokawa et al. (2012a) and Schleicher et al. (2013): evenwhen t KH has become much shorter than t acc , the stellar radius continues to increase following very closely the mass-radius relation: R ⋆ = 2 . × R ⊙ (cid:18) M ⋆ M ⊙ (cid:19) / ≡ R (cid:18) M ⋆ M ⊙ (cid:19) / , (4)while the effective temperature is thermostated to rela-tively low values, T eff ≈ L E = 4 πGm p cσ T M = L (cid:18) MM ⊙ (cid:19) (5)where L = 1 . × erg s − , as inferred from L ⋆ = 4 πR ⋆ σT eff , (6)combined with eq. 4.Why does the radius continue to grow even when theKH time has become shorter than the accretion timescale?Schleicher et al. (2013) have analyzed this question in de-tail. The key point is that t KH ∝ R − ⋆ : this implies thateven if at the time of formation a shell of mass M mightbe initially characterized by t KH ≪ t acc , such inequal-ity will be reversed as soon as its contraction begins. Asa result, the accreting envelope phase can last consider-ably longer than t acc . Schleicher et al. (2013) preliminarlyfind that the accreting phase could continue until M ⋆ =3 . × ( ˙ M/ M ⊙ yr − ) M ⊙ . Beyond this point the systemwill evolve into a main-sequence SMS, eventually collapsinginto a black hole .If this phase can be prolonged up to such high massesis a question that needs further investigation. There arehints from ongoing calculations (Hosokawa et al. 2013; K.Omukai, private communication) that when M ⋆ approaches10 M ⊙ for accretion rates > ∼ . M ⊙ yr − , the star enters acontraction phase (i.e. similarly to what happens for loweraccretion rates at smaller masses and possibly due to a H − opacity drop); however, at this stage numerical difficultiesdo not allow to confirm this hypothesis. If this will turnout to be the case, the SMS final mass will be limited to M ⋆ ≈ M ⊙ as the surrounding gas will be prevented fromaccreting by radiative feedback connected with the increased T eff . As we will see later, the growth of proto-SMS is limitedanyway by either general relativistic instabilities or cosmo-logical effects (as for example the accretion of polluted gas,see Sec. 5) to masses < ∼ × M ⊙ , so the above difficultiesdo not represent a major source of uncertainty on the finalresults.It is important to note that although D and H nuclearburning can be ignited relatively early (e.g. at 600-700 M ⊙ for ˙ M = 0 . M ⊙ yr − ) during the evolution, the associatedenergy production is however always subdominant comparedto the luminosity released via Kelvin-Helmholtz contrac-tion. Therefore it does not sensibly affect the proto-SMSevolution and final mass. The same conclusion is reachedby Montero et al. (2012) who performed general relativistic The thermostat is provided by the strong temperature sensi-tivity of H − opacity; such effect is missed if electron scattering isconsidered as the only source of opacity, e.g. Begelman (2010). Obviously the star can grow only as long as there is sufficientgas to accrete in the host halo. This point will be considered inthe next Section.c (cid:13) , 1–18
Ferrara et al. simulations of collapsing proto-SMS with and without ro-tation, including thermonuclear energy release by hydrogenand helium burning.The radius-mass relation for an accreting supermassiveprotostar can be obtained in a simple manner following themethod outlined in Schleicher et al. (2013). As we have dis-cussed above, as long as ˙ M > ∼ − M ⊙ yr − during the ac-cretion phase, the star radiates at a luminosity ≈ L E . Byintroducing the mass, radius and time non-dimensional vari-ables m = M/M ⊙ , (7) r = R/R , (8) τ = t/t , (9)where t is defined starting from the KH time as t KH = GM ⊙ R L mr ≡ t mr = 0 .
316 yr mr , (10)a shell of accreted gas that forms at time τ i = m/ ˙ m withinitial mass m and radius r i = m / (see eq. 4), starts tocontract according to the local KH timescale. At a generictime t the stellar radius is given by drdτ = − r m , (11)or Z rr i drr = − Z ττ i dtm , (12)finally yielding the solution r ( m ) = mm / + [( τ ( m ⋆ ) − τ i ( m ))] . (13)The evolution or r as function of M for various values of ˙ M is shown in Fig. 1. The radius grows proportionally to m / for the inner shells, but flattens out in the envelope whosesize relative to the stellar radius containing 90% of the stellarmass increases with ˙ M , becoming as large as 10 R ⊙ in themost extreme case ( M ⋆ , ˙ M ) = (10 M ⊙ , M ⊙ yr − ).The previous results imply that in principle the SMSfinal mass could be extremely large, provided a sufficientlylarge halo gas reservoir is present to feed it. However, thegrowth might be hindered by at least three factors. Thefirst is the possible transition to the ZAMS (shown in Fig.1). In this case, as already discussed above, radiative feed-back of UV photons from the now hot stellar surface willquench accretion. Although not expected, our simple treat-ment cannot exclude that. Detailed numerical simulationsaccounting for the opacity evolution will be required to ad-dress this possibility. As today, though, there is no sign ofsuch transitions for stars that have masses up to 5 × M ⊙ .Second, the proto-SMS might instead become general rela-tivistically (GR) unstable, directly collapsing into a blackhole before reaching the ZAMS. Finally, an earlier stop tothe proto-SMS growth can be imposed when the two re-quired conditions (metal-free gas, strong UV background)for the direct collapse of the gas into a black hole cease tobe valid. Whether and for how long these conditions hold Note that time is simply related to mass according to t = M/ ˙ M . Figure 1.
Internal mass-radius structure of accreting super-massive protostars of different final mass M ⋆ = 10 − M ⊙ ; foreach mass three different values of the accretion rate ˙ M =10 , , . M ⊙ yr − are reported from the uppermost to the low-ermost set of curves. Also shown are the regions corresponding togeneral relativistic instability for (grey) a non-rotating ( T = 0)and (green) for a maximally rotating ( T = 0 . | W | ) proto-SMS,along with the zero-age main sequence (ZAMS) relation. No hy-drostatic equilibrium is possible beyond 10 M ⊙ . can only be ascertained from a cosmological analysis thatwe defer to the next Section. In the following we analyzethe constraints posed by the most stringent local condition,i.e. GR gravitational instability. A non-rotating proto-SMS becomes gravitationally unsta-ble (Chandrasekhar 1964, Montero et al. 2012) when thegas equation of state (EOS) cannot be made stiff enoughto compensate for the de-stabilizing general relativistic ef-fects. Mathematically, this happens when the adiabatic in-dex drops below the critical valueΓ c = 43 + 1 . R s R ⋆ , (14)where R s = 2 GM ⋆ /c is the Schwarzschild radius of thestar. The de-stabilizing role of the relativistic term is evi-dent. This condition is easily translated into a critical cen-tral density at which a spherical star becomes unstable toradial perturbations: ρ c = 1 . × (cid:16) µ . (cid:17) (cid:18) M ⋆ M ⊙ (cid:19) − / g cm − , (15) We assume a fully ionized H+He gas and adopt a He abundance Y = 0 . µ = 0 .
59. c (cid:13) , 1–18 he IMF of black holes seeds which corresponds to a mass-radius relation condition forGR instability expressed in the non-dimensional units intro-duced above r c < . × − m / . (16)The unstable region is shown as a grey area in Fig. 1. Asseen from there, depending on the accretion rate stars abovea certain mass can become GR unstable and collapse intoa black hole. By equating eq. 16 and eq. 13 (in the limit τ ≫ τ i ) we can determine the proto-SMS mass upper limit M ⋆ < ∼ . × (cid:18) ˙ M ⋆ M ⊙ yr − (cid:19) / M ⊙ . (17)Thus, for an accretion rate of 0.15 M ⊙ yr − (typical ofatomic cooling halos) a proto-SMS will collapse into a DCBHwhen its mass reaches ≈ . × M ⊙ . Interestingly, thismass limit is comparable to the mass possibly marking thetransition to ZAMS according to ongoing stellar evolution1D numerical calculations. If the star is rotating, this has a stabilizing effect and canhold up the collapse. In this case the expression for the adi-abatic index must be modified (Janka 2002) as follows:Γ c = 2(2 − η )3(1 − η ) + 1 . R s R ⋆ , (18)which of course gives the correct limit (eq. 14) if the ro-tational to gravitational energy ratio, η = T / | W | → n = 3 polytrope,Baumgarte & Shapiro (1999) find that η approaches the uni-versal value 0.009 and that the instability criterion simplybecomes: (cid:18) R ⋆ R s (cid:19) c = 321 , (19)where we have defined the stellar radius as the equatorialone, ≈ / r c < . × − m. (20)By equating eqs. 16 and 20 we find that rotation increasesthe stability of stars with masses M r⋆ > . × M ⊙ (greenregion in Fig. 1), while at lower masses thermal pressurealone is sufficient to stabilize the star.The results obtained in this Sec. can then be summa-rized by the following formulae giving the stability condi- Note that the dynamical time ( Gρ c ) − / ≪ t KH for M ⋆ > M ⊙ . Hence no hydrostatic equilibrium is possible beyond thismass. tions for a proto-SMS: M ⋆ < ∼ . × M ⊙ (cid:18) ˙ M ⋆ M ⊙ yr − (cid:19) / ( T = 0) , (21) M ⋆ < ∼ . × M ⊙ (cid:18) ˙ M ⋆ M ⊙ yr − (cid:19) ( T = 0 , M ⋆ > M r⋆ ) . (22)proto-SMS that are more massive than the above limits willinevitably collapse to form a DCBH.Likely, the entire proto-SMS mass will be finally lockedinto the DCBH. This is confirmed by the results presentedin Montero et al. (2012) who performed general relativisticsimulations of collapsing supermassive stars with and with-out rotation and including the effects of thermonuclear en-ergy released by hydrogen and helium burning. They findthat at the end of their collapse simulation ( t ≈ s)of a proto-SMS of mass 5 × M ⊙ , a black hole has al-ready formed and its apparent horizon contains a mass > ∼
50% of the total initial mass. Analytical arguments dis-cussed by Baumgarte & Shapiro (1999) and later refined byShapiro & Shibata (2002) reach a similar conclusion, indi-cating that ≈
90% of the proto-SMS mass actually endsup into the DCBH, leaving a bare 10% of matter in anouter region, possibly a circumstellar disk. This high col-lapse efficiency is essentially a result of the highly concen-trated density profile of n = 3 polytropes. Similar conclu-sions are reached by Reisswig et al. (2013) who studied thethree-dimensional general-relativistic collapse of rapidly ro-tating supermassive stars. In the following, therefore, we willmake the assumption that M • ≈ M ⋆ . So far we have built a physical framework for the for-mation of IMBH occurring either via GR instability of aproto-SMS followed by accretion, or as the end point ofthe evolution of a more standard SMS. We now aim atembedding such a framework in a cosmological scenarioto derive the global population properties of these poten-tial SMBH seeds. As the first stars form in minihalos atvery high redshifts (Naoz et al. 2006; Salvadori & Ferrara2009; Trenti & Stiavelli 2009) their cumulative UV radiationboosted the intensity of the LW background to values > ∼ J ⋆ν,c for which star formation in newly born minihalo is quenched.Precisely quantifying the level of suppression is difficult asit depends of several fine-grain details, although reasonableattempts have been made (Ahn et al. 2009; Xu et al. 2013).For these reasons, and to bracket such uncertainty, we willassume that (a) all minihalos with mass above a certainthreshold corresponding to a virial temperature T sf ≈ In principle these conditions should be complemented with theone expressing the ability of the SMS to maintain its extended ac-creting envelope during the growth. This relation has been derivedby Schleicher et al. (2013): M ⋆ < ∼ . × M ⊙ ( ˙ M ⋆ /M ⊙ yr − ) ,or M ⋆ = 1 . × M ⊙ for an accretion rate of 0.15 M ⊙ yr − .However, in practice, this condition is met only when the star isalready GR unstable for the accretion rates > . M ⊙ yr − con-sidered here, and therefore we will disregard it in the following.c (cid:13) , 1–18 Ferrara et al.
Figure 2.
Schematic view of the DCBH formation and growthscenario discussed in Sec. 3. is either moderate or extremely effective . We will refer tothese two possibilities as the “fertile” or “sterile” minihalocases, respectively.Following the growth of cosmic structures, atomic-cooling halos start to appear as a result of accretion andmerging of minihalos. In the fertile minihalo case a frac-tion of them are born polluted; if minihalos are insteadsterile, atomic halos are metal-free by construction as nostars/metals have been produced at earlier epochs. In bothcases a fraction of them will be located in regions in which J ν > J • ν,c and therefore they are candidate IMBH forma-tion sites. Recent studies (Ahn et al. 2009; Yue et al. 2013b)have shown that during cosmic dawn, large spatial UV fieldintensity fluctuations existed and persisted for long times.High illumination regions are then found near the peak ofthe field intensity.Our main goal here is to determine the Initial MassFunction of the IMBH once they are allowed to form by theenvironmental conditions discussed above. We will not at-tempt here to quantify (as done e.g. in Dijkstra et al. 2008)how many among the unpolluted, atomic-cooling halos re-side in J ν > J • ν,c regions: this multiplicative (to first-order)function is only required to determine the number densityof such objects. Such an assumption is equivalent to statethat newly-born T vir > K halos reside in highly biased Strictly speaking even in hypothesis (b) a small number of halosmust form anyway to provide the UV radiation field. regions where the field is sufficiently intense; moreover, if thefirst few among them manage actually to form IMBH, theradiation field of the latter will greatly amplify J ν , triggeringthe birth of additional IMBH (Yue et al. 2013b).The formation of IMBH in atomic halos starts with anisothermal, coherent collapse centrally accumulating the gasat rates comparable or larger than the thermal accretionrate, ˙ M i ≈ π G c s = 0 . (cid:18) T K (cid:19) / M ⊙ yr − . (23)As simple as it is, this formula is in remarkable agreementwith the results of most recent and complete simulations ofthe collapse of atomic halos. For example, by analyzing 9such halos extracted from a cosmological large-eddy simula-tion, Latif et al. (2013a) found a very similar accretion rateof 0 . − M ⊙ yr − , measured at z = 15, with little depen-dence on the galactocentric radius.As long as this high accretion rate can be maintained,the proto-SMS continues to grow. Because of its low effec-tive temperature, radiative feedback is unable to stop thehalo gas from accreting. Eventually, the proto-SMS hits theGR unstable boundary shown in Fig. 1 and collapses intoa DCBH in a very short time (about 10 s). Once formed,the DCBH will continue to grow increasing its birth mass byaccreting the gas leftover (if any) in their parent atomic halofinally becoming an IMBH. This feedback-regulated growthis a complex process and we will discuss it separately in Sec.6. However, if during this accretion phase the rate for anyreason drops below ≈ . M ⊙ yr − , then the proto-SMS be-gins to contract and evolves into the ZAMS SMS. Given thecorresponding high effective temperature (eq. 3), the SMSluminosity exerts a sufficient radiation pressure on the sur-rounding gas. Hence accretion is completely halted. As aconsequence, in this case the mass of the IMBH that formsat the end of the brief ( ≈ Myr) stellar lifetime is the same asthe SMS. The main difference between these two IMBH for-mation channels is that for the GR instability channel, thebirth mass function of DCBHs is modified during the sub-sequent feedback-regulated growth. This does not happen ifthe proto-SMS reaches the ZAMS, as already explained. Forthis reason we will compute the IMF of IMBH in two steps:first computing the DCBH IMF (Sec. 5), and then modifyingit to account for the additional feedback-regulated growth.What could cause the gas accretion rate to drop-offbefore the proto-SMS has become GR unstable? Thereare several potential show-stopper effects that could comeinto play. The first is that major galaxy mergers, in con-trast with the smoother accretion of small lumps of mat-ter, are likely to dramatically perturb the smooth accre-tion flow onto the proto-SMS. An additional effect of themerger could be that the shock-induced electron fraction en-hances the cooling. This mechanism was initially proposedby Shchekinov & Vasiliev (2006) and Prieto et al. (2014),and recently confirmed by a numerical simulation in the ab-sence of radiative backgrounds (Bovino et al. 2014).Secondly, the gas brought by the merging halos (or col-lected from the intergalactic medium) can be already pol-luted with heavy elements. As a result, clump formationfollowing metal-cooling fragmentation of the gas is likely todrastically quench the accretion rate onto the proto-SMS,limiting its growth. c (cid:13) , 1–18 he IMF of black holes seeds Finally, the halo could be very gas-poor as a result ofgas ejection by supernova explosions occurred in the pro-genitor halos. In conclusion, even if the sufficient conditionsfor IMBH formation in a give halo are met, the hidden andquiet growth of the proto-SMS finally leading to a DCBHvia GR instability is hindered by a number of effects. Allthese possible physical paths are graphically summarized inFig. 2.The main challenge of the problem is to consistently fol-low the growth of a proto-SMS inside an atomic halo withina cosmological context, i.e. following the history of the par-ent halo as it merges with other halos and accretes gas fromthe intergalactic medium. We accomplish this by using amerger tree approach as described in the following.
In order to quantitatively investigate the above scenario wefollow the merger and mass accretion history of dark matterhalos and their baryonic component. To this aim we usethe data-calibrated merger-tree code GAMETE (GAlaxyMErger Tree and Evolution, Salvadori et al. 2007), whichhas been developed to investigate the properties of present-day ancient metal-poor stars. The code successfully re-produces the metallicity-luminosity relation of Milky Way(MW) dwarf galaxies, the stellar Metallicity DistributionFunction (MDF) observed in the Galactic halo, in classicaland ultra-faint dwarfs (Salvadori & Ferrara 2009), and theproperties of very metal-poor Damped Lyman α Absorbers(Salvadori & Ferrara 2012). Here we only summarize themain features of the code, deferring the interested readerto the previous papers for details.GAMETE reconstructs the possible merger historiesof a MW-size dark matter halo via a Monte Carlo al-gorithm based on the Extended Press-Schechter theory(Salvadori et al. 2007), tracing at the same time the starformation (SF) along the hierarchical trees with the follow-ing prescriptions: (i) the SF rate is proportional to the massof cold gas in each galaxy, and to the SF efficiency ǫ ∗ ; (ii) inminihalos ǫ ∗ is reduced as ǫ H = 2 ǫ ∗ [1+( T vir / × K) − ] − due to the ineffective cooling by H molecules; (iii) Pop-ulation II stars form according to a Larson IMF if thegas metallicity exceeds the critical value, Z cr = 10 − ± Z ⊙ (Schneider et al. 2006), here assumed Z cr = 10 − Z ⊙ . Atlower metallicity, PopIII stars form with reference mass m ∗ = 25 M ⊙ and explosion energy E SN = 10 erg consis-tent with faint SNe (Salvadori & Ferrara 2012). The chem-ical evolution of the gas is simultaneously traced in haloesand in the surrounding MW environment by including theeffect of SN-driven outflows, which are controlled by the SNwind efficiency. The metal filling factor, Q Z = V totZ /V mw , iscomputed at each z by summing the volumes of the indi-vidual metal bubbles around star-forming haloes, V totZ , andwhere V mw ≈ z ) − Mpc , is the proper MW volume atthe turn-around radius (Salvadori et al. 2014). The proba-bility for newly formed halos to reside in a metal enriched re-gion is then computed as P ( z ) = [1 − exp( Q Z )] /Q δ>δ c , where Q δ>δ c ( z ) is the volume filling factor of fluctuations withoverdensity above the critical threshold, δ > δ c = 1 . Z vir = Z GM / [1 − exp( Q Z )] ( Z vir = 0), where Z GM is the average metallicityof the MW environment.As primordial composition ( Z < Z cr ) halos in themerger tree cross the T vir = 10 K threshold, we postu-late that a proto-SMS can form in each of them and followits growth in the following manner. We assume that theproto-SMS is fed by an accretion rate, ˙ M ∗ = max( ˙ M i , ˙ M e ),that is the maximum between the “internal” accretion ratefrom the host halo gas, ˙ M i , and the “external” accretionrate due to minor merger events, ˙ M e . ˙ M i is computed asthe thermal accretion rate (eq. 23). The external rate ˙ M e istaken as the ratio between the gas mass of the merging haloand the dynamical friction time-scale, t merge , between thetwo colliding objects. We compute t merge from the classicChandrasekhar formula (Mo et al. 2010): H ( z ) t merge = 0 . ζx ln(1 + x ) (24)where x = M /M is the ratio of the merging halo masses(with M > M ), and ζ is the circularity parameter encodingthe eccentricity of the orbit decay, which we randomly selectin the interval ζ = [0 . , .
25] following Petri et al. (2012).This prescription allows the occurrence of supra-thermalaccretion rates, consistently with the numerical simulationresults of Mayer et al. (2010) and Bonoli et al. (2014) (seehowever Ferrara et al. (2013) for a critical discussion) thatshow that following merger events the gas rapidly loses an-gular momentum and is efficiently funneled towards the nu-clear region. As long as the gas in the host halo (includingthat brought by mergers) remains metal-free, the proto-SMSgrows at a rate set by ˙ M ∗ until it eventually becomes GRunstable (eq. 17).However, GR instability is not necessarily the finalfate for the proto-SMS. In fact, the protostar growth canstop because of two distinct physical processes (a) a majormerger event, i.e. a collision with a halo of comparable mass, M /M = [0 . , For reasons explained later in this Section, we stop the proto-SMS growth after a major merger.c (cid:13) , 1–18
Ferrara et al.
Figure 3. Left:
As a function of their formation redshift we show the mass of: (i) halos with T vir > K (gray open squares), (ii) haloshosting a DCBH or SMS (green squares), (iii) DCBH (black circles), and (iv) SMS whose growth has been halted by a major mergerevent (red exagones) or by metal-pollution (yellow exagones). The results are shown for a single MW merger history
Right:
Mass of gasleft in the halos after a DCBH (gray) or SMS (yellow-orange) formation as a function of the DCBH/SMS mass. Contours refer to 68%,95%, 99.7% confidence levels (50 realizations).
We are now ready to derive what we call the “birth massfunction”. This is the mass distribution including the newlyformed DCBH originating from GR instability of a proto-SMS, and the black holes corresponding to the end point ofthe SMS evolution. The final IMF of IMBH seeds (Sec. 6)needs to additionally account for the subsequent feedback-regulated growth of DCBH.The birth mass function will be presented for the twolimiting cases of fertile (representing the fiducial case) andsterile minihalos. The fertile case assumes that all mini-halos with T vir > T sf = 2000 K can form stars when z >
10. At lower redshifts T sf slowly increases up to thevalue T sf ≈ × K reached at z ≈
6, when MW envi-ronment is reionized (Salvadori et al. 2014). This empiricalfunctional form catches the essence of the increasing abilityof the LW radiation to suppress star formation in halos asits intensity climbs and it is calibrated on a detailed com-parison with the dwarf galaxy population of the MW halo(Salvadori & Ferrara 2009). The sterile minihalo case, in-stead, assumes that minihalos never form stars. These twocases are meant to bracket the uncertain role of radiativefeedback in suppressing star formation via H destructionin these small systems. Note that in the first case atomichalos resulting from the merger of smaller progenitors canbe polluted with heavy elements when they form; if insteadminihalos are sterile, atomic halos are all born unpolluted. The main results for this case are depicted in Fig. 3. Inthe left panel we show the masses of DCBH and SMS, and of their hosting halos ( M h ) as a function of their for-mation redshift for a single realization of the merger tree.For comparison purposes, the mass of all atomic halos inthe merger tree at different redshifts are also shown (graypoints). The halos hosting DCBH or SMS span a well-defined and narrow range of masses M h ≈ (2 − × M ⊙ during the entire formation epoch, 8 < z <
17. They arelow-mass systems, which at any given redshift have roughlythe minimum virial temperature required for atomic cool-ing, T vir ≈ (1 − . × K. While the chances to remainunpolluted are relatively high for these small systems (30such halos out of a total of 36, i.e. ≈ T vir > . × K objects,which form via merging of smaller progenitors that have al-ready formed stars. These results are more quantitativelyillustrated in the upper panel of Fig. 4, which shows theprobability distribution function (PDF) of host halo masses.The mass distribution of DCBH hosts (gray histogram) hasan almost symmetric distribution with a pronounced peakaround M h ≈ (3 . − × M ⊙ and rapidly declines to-wards the tails of the distribution. Most of the halos host-ing DCBH ( >
80% of the total) are low-mass objects with M h ≈ (2 . − . × M ⊙ . Moreover, the external accretionoccurs at a subdominant rate with respect to internal one,˙ M e = [0 . − . ˙ M i . This implies that most of thetime the proto-SMS accretes gas at the thermal accretionrate set by the halo virial temperature (eq. 23), and hencecomparable for all of them, ˙ M ∗ ≈ (0 . − .
57) M ⊙ yr − .Since the final mass of DCBHs is entirely determined by ˙ M ∗ (see eq. 17), it follows that also the DCBHs span a very nar-row range of masses, 2 . × M ⊙ < ∼ M • < ∼ . × M ⊙ , ascan be appreciated by inspecting Fig. 3 (black points).In the same Figure we show the mass of SMS (filled c (cid:13) , 1–18 he IMF of black holes seeds Figure 4.
Mass probability distribution function of DCBH (dot-ted histogram) or SMS (yellow shaded histogram) host halos forthe fertile (upper panel) and sterile (lower panel) minihalo cases.The results are averaged over 50 MW merger histories, and the ± σ errors are shown. exagones) whose growth has been blocked before reachingthe GR instability because of: (i) a major merging event(red symbols), or (ii) a minor merger with a metal pollutedhalo (yellow symbols). It is clear from the Figure that metalpollution is the dominant process stopping the proto-SMSgrowth (22 out of 24 SMS share this origin). Moreover, thetypical masses of SMS are smaller than DCBH, althoughthey span a larger range, M SMS ≈ (3 − × M ⊙ . Thisis due to the stochastic nature of the merging/accretionprocesses, which can quench the growth of the proto-SMSat different stages. We can also note that the formationepoch of SMS is shifted towards lower redshifts with re-spect to DCBH, z ≈ (14 − z > Q Z . ≈ z
9, when the growth of all proto-SMS is stoppedbecause of metal pollution. Due to this delay the hostinghalos of SMS are typically more massive than DCBH hostsas seen in Fig. 4.All these findings can be better interpreted by inspect-ing Fig. 5, where the comoving number density of DCBHs(black points), SMS (yellow/red exagones), and halos withdifferent physical properties (all, T vir > K, unpolluted),are shown as a function of redshift. It is evident that thenumber density, n , of unpolluted atomic halos (blue trian-gles), differently from the other curves does not monotoni-cally increase with time. Instead n gently grows from z ≈ z ≈
13, reaches a maximum, and then slowly decreaseswhen metal pollution starts to dominate. The number den-sity of both DCBH and SMS are tightly connected with thisevolution. From z = 20 to z ≈
17 the amount of DCBHsincreases steeply, tightly following the rise of unpolluted ha-los, while SMS are very rare, n . − . At lower z the steepness of the curve progressively decreases, becoming flatat z ≈ .
5, while the SMS density progressively increases,gradually approaching the DCBH value. Below z ≈
8, alsothe formation of SMS is stopped, and n becomes constant.At this z , metals have already reached the high density re-gions in which halos form, Q Z ≈ . > Q δ>δ c , making theonset of proto-SMS formation impossible. The final num-ber density of DCBHs and SMS are expected to be roughlythe same, n ≈ − . We recall that this number hasbeen obtained assuming that all host halos reside in regionsin which J ν > J • ν,c , and therefore represent a strong upperlimit.In Fig. 6 we show the mass probability distribu-tion functions of DCBH (gray histogram) and SMS (yel-low histogram) normalized to the total number of objects(DCBH+SMS), i.e. what we call the ”Birth Mass Function”.The mass distribution of DCBH exhibits a peak roughlyat the low-mass end, M • ≈ . × M ⊙ , and monotoni-cally declines towards higher masses. The lower limit of thePDF is populated by objects accreting at the thermal rate,˙ M ∗ ≈ . T vir ≈ K halos, the mostcommon DCBH hosts. This sharp low-mass cut is set by GRinstability (eq. 17). On the other hand, more massive DCBHform in unpolluted halos with higher T vir , that therefore aremuch less common. This explaines the rapid downturn of thedistribution.The PDF of SMS has a very different, roughly sym-metric shape, displaying a wide plateau in the mass interval M ≈ (0 . − . × M ⊙ ; it then rapidly declines towardslower/higher masses. The decreasing number of SMS withmasses < × M ⊙ depends on the limited number of merg-ing/accretion (driving the proto-star evolution towards theZAMS) occurring on timescales equal to 8 × M ⊙ / ˙ M ∗ ≈ . M > . × M ⊙ is due tothe same processes decribed for DCBH.Another quantity that we can derive from the previousanalysis is the gas left in the halo after DCBH or SMS forma-tion. Such a quantity is shown as a function of DCBH/SMSmass in Fig. 3 (right panel). Although DCBH masses spana very small range the gas mass can vary by more one orderof magnitude M g ≈ (3 − × M ⊙ . This gas can be po-tentially accreted by DCBH. Thus, as we will discuss in thenext Section, the final mass of IMBH seeds will crucially de-pend on the subsequent accretion phase and feedback effects.SMS are instead able to evacuate the gas that is not quicklyturned into stars. Halos residing within 68% confidence levelhave M g = f b (Ω b / Ω M ) M h with f b ≈ .
2. Such a reduced gasfraction with respect to the cosmological value is the resultof the previous star-formation activity and SN feedback pro-cesses, occurred in their progenitors. Only a few halos thatform at z ≈
20, and correspond to rare high- σ density fluc-tuations, are found to have f b ≈
1. The halos hosting SMScover roughly the same M g range as DCBH hosts. The bulkof SMS hosts, however, are more gas rich than DCBH hosts.This is because SMS typically form in more massive halos(see Fig. 4), which therefore contain more gas. In the second case, we consider the other extreme possibil-ity in which the UV flux is sufficiently intense to completelysuppress star formation in minihalos. As we commented al- c (cid:13) , 1–18 Ferrara et al.
Figure 5.
Comoving number density evolution of: (a) all halos in the simulation (top curve), (b) halos with T vir > K (purple filledsquares), (c) unpolluted (
Z < Z crit = 10 − Z ⊙ ) halos with T vir > K (blue triangles) (d) DCBH hosts (black circles), (e) SMS hosts(exagons) formed after (f) a major merger (orange) or (g) metal-pollution event (yellow). The errors are only shown for the total numberof halos and represent the ± σ dispersion among different merger histories. Left : fertile minihalos case;
Right : sterile minihalos case. ready, strictly speaking this case corresponds to an unphys-ical situation as at least some stars must form in order toproduce the required radiation field (unless some other UVsource is present, as for example dark matter annihilation).The sterile minihalo scenario requires that the fraction ofbaryons converted into stars in these systems is negligible.With this hypothesis and caveat in mind we can an-alyze the results of the merger trees and the predictedproperties of DCBH/SMS and of their hosting halos. FromFig. 7 we highlight two major differences with respect tothe previous case. First, the formation era of DCBH/SMSstretches towards lower redshifts, z ≈
7. Second, the prob-ability to merge with an already polluted halo strongly de-creases. Both these effects are simply a consequence of thelack of an early metal enrichment, impying that all atomichalos are unpolluted at birth. In these objects therefore star-formation, and the subsequent metal-enrichment, is only ac-tivated if a major merger event induces a vigorous fragmen-tation of the gas, thus stopping the proto-SMS growth.An inspection of the right panel of Fig. 5 further il-lustrates these points. The number density of unpollutedatomic halos and T vir > K halos exactly overlap downto z ≈
15. At lower z , however, the two functions start toslowly deviate. At z ≈
10 the number density of unpol-luted atomic halos, i.e. the sites for SMS/DCBH formation,reaches the maximumand then rapidly declines since metalpollution start to dominate. The maximum value is almostone order of magnitude larger than found fertile minihalos.As a consequence, the final number density of DCBH is muchlarger, n ≈
65 Mpc − .Despite of the drastically different conditions betweenthe fertile and sterile cases, the properties of DCBH/SMSand of their hosting halos are surprisingly similar. This is Figure 6.
Mass probability distribution function of DCBH seeds(dotted histogram) and SMS (yellow shaded histogram) for thefertile (upper panel) and sterile (lower panel) minihalos case. Thesymbols are the same of Fig. 4. evident both from the mass probability distribution func-tions shown (Fig.4) and in the birth mass function (Fig. 6).Nevertheless there is a remarkable difference, that can benoted by comparing the right panels of Fig. 7 and Fig. 3. Ifminihalos are sterile the gas mass at the DCBH/SMS forma-tion is larger, M g ≈ (1 − × M ⊙ . At their formation,indeed, all T vir ≈ K haloes have a gas mass fraction c (cid:13) , 1–18 he IMF of black holes seeds Figure 7.
As Fig. 3 for the sterile mini-halo case. close to f b ≈
1, since no gas have been consumed withintheir sterile progenitor minihalos. This implies that the sub-sequent feedback-regulated DCBH accretion phase, will becrucial in setting the final IMF of IMBH for the two differentscenarios.
In order to determine the IMF of the IMBH the final step isto asses wheter they were able to accrete the gas eventuallyleft at the time of DCBH/SMS formation. This is the goalof this Section.
If the transition to ZAMS occurs before the onset of GRinstability (eq. 16) an SMS forms. Due to the large amountof UV photons emitted by the hot ( T eff ≈ K, see eq. 3)stellar surface the radiation pressure on the remaining gas isvery likely to evacuate it during its lifetime, expected to be t ⋆ ≈ . M ⋆ c /L E = 3 Myr, virtually independent on itsmass. Obviously we cannot exclude that some fraction of thisgas can be turned into stars before this happens. Irrespectiveof these details by the time the SMS collapses into an IMBHthere will be virtually no gas left to accrete and furthergrowth becomes impossible. In this case therefore, the IMFfor these type of IMBH is the birth mass function itself. As the DCBH of mass M i • emerges from the collapse ofthe proto-SMS, it will be surrounded by the remaning halogas mass, M g , that has not been previously included intothe DCBH. These two initial values are obtained from themerger tree outputs, along with the total halo mass M h (i.e. 3 and 7). We assume that during the DCBH growthphase M h ≈ const. given the short duration of such phase.In principle, all the remaining gas could be eventually incor-porated into the DCBH unless feedback from energetic ra-diation emitted during the accretion process is able to stopor reverse the accretion flow.The typical density structure resulting from the isother-mal collapse of the halo gas prior to DCBH formation is con-stituted by a central (adiabatic) core in which the collapseis stabilized, and an outer envelope where ρ ∝ r − : ρ ( r ) = ρ c r/r c ) . (25)The above density profile has been confirmed by simulationsby Latif et al. (2013a), who showed that the 9 candidateDCBH host halos (all of mass M ≈ M ⊙ ) remarkablyfollow the distribution given by eq. 25, independent on theirmass and formation redshift (note that both the mass andredshift range are rather narrow as we have shown in theprevious Section). The core radius, r c , is comparable to theJeans length of the gas ∝ c s t ff as in a King profile for which r c = 3 c s √ πGρ c = 65 . (cid:18) T K (cid:19) / (cid:18) ρ c − g cm − (cid:19) − / AU , (26)where the core density reference value is taken from Fig.1 of Latif et al. (2013a). The previous formula gives a coreradius in very good agreement with the simulated value.Finally, we require that the mass contained within the outerradius, r out , at which we truncate the distribution is equalto M g . This gives (in the reasonable limit r out ≫ r c ), r out ≈ M ig πρ c r c = GM ig c s = 14 . . (27)Note that, due to collapse, the gas concentration increases,i.e. r out is more than 10 times smaller than the halo virial c (cid:13) , 1–18 Ferrara et al. radius r vir = 583 (cid:18) T vir K (cid:19) / (cid:18) z (cid:19) − / pc . (28)Assuming that, to a first approximation, the DCBH isat rest and that the accretion flow is close to spherical ,the relevant scale for accretion is the Bondi radius, r B = 2 GM • c s, ∞ = 9 . (cid:18) M • M ⊙ (cid:19) (cid:18) T K (cid:19) − pc , (29)where we denote with the subscript ∞ quantities evaluatedat large distances from the DCBH. Note that r B is about 2%of the virial radius of a typical DCBH host halo, and r B ≈ r out , implying that the DCBH can easily drain gas fromthe entire volume in which gas is present. This fact has twoimportant consequences that we analyze in the following.The first implication of the approximate equality be-tween the Bondi and outer radius is that the initial gas den-sity distribution will be modified by the accreting DCBH.The rearrangement of the gas requires that the dynamicaltime is shorter than the Salpeter time, i.e. t ff = (cid:18) π Gρ (cid:19) / ≪ M • ˙ M • ≡ t S = 4 . × ǫ yr , (30)where we have conservatively assumed that accretion occursat the Eddington rate. The minimum density required to re-arrange the profile fast enough is ρ = 2 . × − g cm − ,having further assumed a standard radiative efficiency ǫ =0 .
1. As from eq. 25 we obtain that the gas density is alwayslarger than the previous value we can safely assume thatthis is the case.In order to obtain an explicit expression for the accre-tion flow density profile, let us proceed as follows. The one-dimensional mass and momentum conservation equations fora steady adiabatic accretion flow, i.e. the classical Bondiproblem, read 1 ρ dρdr = − r − v dvdr , (31) v dvdr + 1 ρ dpdr + GM • r = 0 . (32)Taking c s, ∞ = γp ∞ /ρ ∞ , where γ is the adiabatic index, asthe sound speed at large distances, and further assuming v ∞ = 0, we can integrate eq. 32 to get the Bernoulli equa-tion, 12 v + c s ( r ) γ − − GM • r = c s, ∞ γ − , (33)from which the classical Bondi accretion rate can be derivedby evaluating the previous expression at the sonic radius r s = GM • / c s ( r s ):˙ M B = 4 πr s ρ ( r s ) c s ( r s ) = πq s r B c s, ∞ ρ ∞ , (34) The spherical approximation holds if r B is larger than the cir-cularization radius r c = j /GM • , where j is the specific angularmomentum of the gas. This may hold only approximately if the gas accretion ontothe halo from the intergalactic medium is still occurring where q s ( γ ) = 14 (cid:18) − γ (cid:19) (5 − γ ) / (2 γ − . (35)The numerical value of q s ranges from q s = 1 / γ = 5 / q s = e / / ≈ .
12 when γ = 1 (isothermal); in a radiation-dominated fluid ( γ = 4 /
3) then q s = √ / r s = 1 / r B for γ = 4 /
3, theBernoulli equation reduces to (1 / v ≈ GM • /r , whichyields v ( r ) = c s, ∞ ( r/r B ) − / . To conserve the Bondi ratethen the radial density dependence can be easily shown tosatisfy ρ ( r ) = ρ B (cid:18) rr B (cid:19) − / , (36)where ρ B = 3 M g / πr B is a normalization constant obtainedby requiring that at each time the mass contained within r out is equal to the current gas mass M g ( t ).Two points are worth noting. First, the − / γ . More-over, although it has been obtained under a steady-state as-sumption, it has been shown to hold also for time-dependent(Sakashita 1974), and even optically thick (Tamazawa et al.1975) accretion flows. Thus we consider it as a robust fea-ture of our model. In addition, as the dynamical time atsmall radii is much shorter than at r out , we keep the latterfixed during the evolution and allow ρ B to decrease as gasis incorporated into the DCBH.From the density we can compute the optical depthto Thomson scattering (we will discuss later on when thissimple opacity prescription breaks down and adopt a moreprecise formulation): τ ( r ) = − Z rr out ρ ( r ) µm p σ T dr = 2 τ B (cid:16) r B r (cid:17) / (cid:12)(cid:12)(cid:12)(cid:12) rr out (37)where we have defined τ B = n B σ T r B . Towards the centerthe density increases to values that are large enough to effec-tively trap photons; within this region the energy is convec-tively rather than radiatively transported by diffusion. It iseasy to transform this condition (which is also equivalent tothe Schwarzschild criterion for stability against convection)into one on τ . The radiative luminosity can be written as L r = − πr c κρ aT dTdr , (38)where a is the radiation density constant, and κ = σ T /µm p .On the other hand, the convective luminosity can be writtenas L c = 16 πr pv ; (39)if pressure inside the trapping region is dominated by radia-tion, then p = aT /c . By equating the two luminosities andrecalling that dτ = κρdr we obtain the implicit definitionfor the trapping radius, r tr : τ ( r tr ) v ( r tr ) c = 1 . (40)Thus, for r < r tr radiation is convected inward faster than itcan diffuse out and therefore within this radius ( convective region) photons cannot escape; the flow is then almost per-fectly adiabatic. At larger radii, radiation can start diffuseand transport energy outwards: we refer to this region as the c (cid:13) , 1–18 he IMF of black holes seeds radiative layer. It can be easily shown that L c ( r tr ) = βL E ,with β = O (1). Using eqs. 37 and 34 to express τ and re-calling that v = ˙ M B,γ =4 / / πρr , we finally obtain r tr = √ τ B (cid:16) c s, ∞ c (cid:17) r B ≪ r B , (41)and L c = 48 β (cid:18) c s, ∞ cτ B (cid:19) ˙ M B . (42)The temperature at the trapping radius is then simply ob-tained from T tr = L/πacr tr . We will show later that theradiative region is very thin. The DCBH growth rate canthen be determined by equating its accretion luminosity η ˙ M • c / (1 − η ) to L c to obtain˙ M • = 48 β (cid:18) − ǫǫ (cid:19) (cid:18) c s, ∞ cτ B (cid:19) ˙ M B . (43)To determine the thermal structure of the radiative regionand compute the photospheric temperature of the accretingDCBH we need to improve our treatment of the opacity. Sofar we have assumed a constant electron scattering opac-ity, κ T = σ T /µm p . In a metal-free gas, this is a good ap-proximation as long as the temperature remains > ∼ × K. At lower temperatures, as the gas starts to recombine,additional processes increase the gas opacity: (a) free-free( κ ff ∝ ρT − / , known as the Kramers opacity); (b) bound-free and free-bound; (c) H − ( κ H − ∝ ρ / T ) which is mostlyeffective in the temperature range (0 . − × K. A fullcalculation of the opacity is given in Mayer & Duschl (2005);here we use a fit to their results suggested by Begelman et al.(2008): κ ( T ) = κ T T /T ∗ ) − s , (44)with T ∗ = 8000 K and s = 13. The above expressionfor the Rosseland mean opacity is independent on density.This turns out to be a very good approximation as long as ρ < ∼ − g cm − . As the use of the correct opacity becomesimportant outside the trapping radius where densities arecomparable or below the above validity threshold, we con-sider this approximation as a safe and handy one.Armed with these prescriptions, we can solve for thetemperature structure in the radiative region using the en-ergy transport equation in the diffusion approximation: Z TT tr T ′ " (cid:18) T ′ T ∗ (cid:19) − s dT ′ = − Z rr tr κ T ρ ( r ′ ) L πacr ′ dr ′ , (45)whose solution can be written, using the expression for thedensity eq. 36 and the definition of τ B (eq. 37) as " T + 4 T ∗ − s (cid:18) TT ∗ (cid:19) − s TT tr = 3 τ B L πacr B "(cid:16) r B r (cid:17) / − (cid:18) r B r tr (cid:19) / . (46)To get the photospheric radius, r ph , we solve numerically theabove equation together with the additional constrain that τ ( r ph ) = 2 / r ph is definedas the photospheric temperature of the system.The resulting structural properties of an accreting flowonto a DCBH of mass M • = 10 M ⊙ , located in a dark mat-ter halo with T vir = 10 K, f b = 1 formed at z = 14 are Figure 8.
Dependence of various characteristic scales of the prob-lem as a function of the core gas density, ρ c for a DCBH of mass M • = 10 M ⊙ , located in a dark matter halo with T vir = 10 K formed at z = 14. We have assumed ǫ = 0 . f b = 1 .
0, and aradiation-dominated equation of state corresponding to γ = 4 / r c , Bondi, r B , the outer gas distribution, r out , and virial, r vir radii, also shown are the Bondi and Edding-ton density regimes corresponding to the above ( T vir , M • ) pairand the electron scattering optical depth out to r B . shown, as an example, in Fig. 8. We find that the radiativeregion is extremely narrow, i.e. r ph ≈ r tr = 2 . × − pc.Both radii are considerably smaller than the Bondi (9.9 pc)and virial (559.5 pc) radii. Inside r ph the optical depth raisesto very large values. However, the effective temperature ofthe system remains relatively low, reaching in this case only15970 K. Because of this low temperature the ionizing ratefrom the accreting envelope has a relatively mild feedbackeffect onto the overlying atmosphere, making it difficult tostop the accretion of the leftover gas onto the DCBH.Fig. 9 gives a full view of the evolution of the system asthe DCBH mass increases due to accretion, self-consistentlycalculated using eq. 43. As the DCBH mass increases theconvective region shrinks due to the decreasing density asmatter is progressively swallowed by the DCBH. At the sametime, such contraction induces a temperature increase at theconvective/radiative layer boundary, paralleled by a similarincrease in the photospheric temperature. In particular, inthis specific case of a DCBH growing inside a dark matterhalo with T vir = 10 K formed at z = 14 and f b = 1, T ph initially increases slowly and remains below 5500 K up tothe point at which the DCBH mass crosses the value M • =10 . M ⊙ . Beyond that point the photospheric temperatureincreases more rapidly and reaches about 30,000 K once theDCBH has grown to M • = 10 M ⊙ .Thus it is only in these more advanced evolutionaryphases that copious amount of ionizing photons start to beproduced. As a result of radiative energy deposition the gascan be heated to a temperature far exceeding the virial tem-perature of the halo and therefore be evacuated from the c (cid:13) , 1–18 Ferrara et al.
Figure 9.
Dependence of several characteristic radii of the system(see text for definitions) on the DCBH mass along with the tem-perature at the trapping radius, T tr , (blue) and photospheric tem-perature, T ph (red). The DCBH of initial mass M • = 10 . M ⊙ , islocated in a dark matter halo with T vir = 10 K formed at z = 14.We have assumed ǫ = 0 . f b = 1 .
0, and a radiation-dominatedequation of state corresponding to γ = 4 / halo, preventing the accretion of gas located beyond r ph .The gas within the photosphere is eventually accreted andthe final state of the system is a naked IMBH embedded inthe parent dark matter halo. Thus we are left with the finalquestion of establishing when accretion, and hence DCBHgrowth, will come to a halt.From the detailed properties computed above we derivethe ionizing rate, Q ( M • ) = πφ ( T ph ) h hν i r ph acT ph (47)where h hν i ≈ φ is the fraction of the bolometric energy emitted bythe accreting DCBH, whose spectrum is assumed to be ablack-body, B ν ( T ph ): φ ( T ph ) = R ∞ ν L dνB ν ( T ph ) R ∞ dνB ν ( T ph ) , (48)where hν L = 1 Ryd. In order to ionize the entire atmosphere(i.e. the gas outside r ph ) and increase the gas temperatureabove T vir ≈ K, the ionization rate must exceed therecombination rate, R , of the gas within r out . The lattercan be written as R ( M • ) = 4 π Z r out r ph dr (cid:18) r t rec (cid:19) (cid:18) ρµm p (cid:19) (49)where t rec = ( nα (2) ) − is the recombination timescale and α (2) = 2 . × − ( T / K) − / is the Case B recombinationrate of hydrogen (Maselli et al. 2003). By substituting eq. 36and performing simple algebra we obtain the final expressionfor the recombination rate: R ( M • ) = 4 πα (2) ρ B ( µm p ) r B ln (cid:18) r out r ph (cid:19) . (50) Figure 10.
Initial Mass Function of IMBH seeds (shaded grayhistogram) averaged over 10 MW merger histories for the fertile(upper panel) and for the sterile (lower panel) minihalo cases.The errorbars correspond to ± σ errors. The birth mass functionof DCBH and SMS (see Fig. 6) is also shown (dotted histogram).
Figure 11.
Probability distribution functions of the accretionphase duration for the fertile (upper panel) and the sterile (lowerpanel) minihalo cases. The results are averaged over 10 MWmerger histories and the ± σ errors are shown. Once the condition Q > R is satisfied, we assume that theremaining gas has been heated and ejected by the accretingDCBH radiative feedback and its growth is quenched. Thissets the final mass of the DCBH, or the mass of the resultingIMBH.In order to determine the IMF of IMBH we followthe feedback-regulated growth of DCBH present in 10 re- c (cid:13) , 1–18 he IMF of black holes seeds alizations of the merger tree by assuming that the hostinghalo mass remain constant during this phase (no mergingprocesses). The final mass distribution of DCBH is thensummed with the birth mass function of SMS (as theseobjects also finally evolve into black holes, see Section 5),and normalized. The results of this calculation are shownin Fig. 10, where the final IMF of IMBH (gray shaded his-tograms) is compared with the birth mass function of DCBHand SMS (dotted histograms). The low-mass end of the IMFis identical to the birth mass function, while the peak isshifted towards higher masses. This is simply a consequenceof the feedback-regulated growth, only affecting DCBH. Asthe growth of DCBH is fed by the available halo gas, thedisplacement is larger when minihalos are sterile because inthis case DCBH hosts are more gas rich (Fig. 6 and Fig. 3,right panels).As a consequence, the IMBH IMF is very different in thetwo scenarios: in the fertile case, in particular, it exhibits abimodal distribution with two separate peaks at M ≈ (0 . − . × M ⊙ and M ≈ (5 − × M ⊙ . The distributionextends over a broad range of masses, from M ≈ (0 . − × M ⊙ . If minihalos are sterile, the IMF spans the narrowermass range, M ≈ (1 − . × M ⊙ , which contains > −
120 Myr, due to the largerreservoir of gas available for accretion in the halo) and moreconcentrated (atomic halos have similar gas content, f b ≈ z = 17; however by z = 8 theirformation is already quenched as a result of the accretion ofpolluted gas and/or a major merging disrupting the quietaccretion flow and inducing gas fragmentation. If minihalosare sterile, then the termination epoch is delayed by about2 redshift units, and DCBH become the dominant source ofproduction for IMBH seeds.We finally comment on the relation between the IMBHand their host halo mass. This relation is often necessary toformulate physical seeding prescriptions, e.g. in studies ofSMBH formation based on merger trees or numerical sim-ulations. Our results show that a very reasonable prescrip-tion is to populate a given fraction of halos (a) of mass7 . < log M h <
8, (b) in the redshift range 8 < z < . < log M • < . Figure 12.
Probability distribution function of the formationredshifts DCBH (dotted histogram) and SMS (yellow shaded his-togram) for the fertile (upper panel) and sterile (lower) minihalocases are shown. The results are averaged over 10 MW mergerhistories. The errorbars correspond to ± σ errors. . < log M h < .
75, (b) in the redshift range 6 < z < < log M • < .
25. Werecall once again that the above fraction of such halos to bepopulated cannot be obtained from our method as it wouldrequire a detailed knowledge of the LW UV background field.Therefore such information must be fixed from other physi-cal considerations or left as a free parameter.
In this paper we have derived for the first time the InitialMass Function of Intermediate Mass Black Holes (10 − M ⊙ )formed inside metal-free, UV illuminated atomic cooling(virial temperature T vir > K) halos either via directcollapse followed by GR instability or via an intermedi-ate Super Massive Star (SMS) stage. These objects havebeen recently advocated as the seeds of the supermassiveblack holes observed at z ≈
6. Assembling the SMBH mass( M • = 2 × M ⊙ ) deduced for the most distant quasarULAS J1120+0641 at z = 7 .
085 (Mortlock et al. 2011) when t ( z ) = 0 .
77 Gyr, requires a seed mass > M ⊙ . Such valueis uncomfortably large when compared to the most recentestimates of the mass of first stars, which now converge to-wards values ≪ M ⊙ (Greif et al. 2011; Hosokawa et al.2012b; Hirano et al. 2014). This is why IMBH seeds, withtheir larger masses, are now strongly preferred as the mostpromising seeds.We have obtained the IMBH IMF with a three-stepstrategy, as described below. • We have first derived the condition for a proto-SMS toundergo GR instability and directly collapse into a DCBHdepending on the gas accretion rate; we found that, for anon-rotating SMS, GR instability kick in when the stellar c (cid:13) , 1–18 Ferrara et al. mass reaches M ⋆ = 8 . × (cid:18) ˙ M ⋆ M ⊙ yr − (cid:19) / M ⊙ . (51)Thus, for an accretion rate of 0.15 M ⊙ yr − (typical ofatomic cooling halos) a proto-SMS will collapse into a DCBHwhen its mass reaches ≈ . × M ⊙ . A similar expressionhas been obtained for rotating SMS, and given by eq. 22.However, the SMS growth can come to an end before thestar crosses the above critical mass. This occurs if the hosthalo accretes polluted gas, either brought by minor merg-ers or smooth accretion from the IGM, or suffers a majormerger that generates vigorous turbulence, again disruptingthe smooth and quiet accretion flow onto the central proto-SMS star. • We followed these processes in a cosmological contextusing the merger tree code GAMETE, which allows us tospot metal-free atomic cooling halos in which either a DCBHor SMS can form and grow, accounting for their metal en-richment and major mergers that halt the growth of theproto-SMS by gas fragmentation. We derive the mass dis-tribution of black holes at this stage, and dub it the “BirthMass Function” (BMF). Most DCBH host halos ( >
80% ofthe total) have M h ≈ (2 . − . × M ⊙ . As a result of ac-cretion physics, DCBHs span a very narrow range of masses,2 . × M ⊙ < ∼ M • < ∼ . × M ⊙ . We find that the metalpollution is by far the dominant process stopping the proto-SMS growth. The resulting SMS are smaller than DCBH, al-though they span a larger range, M SMS ≈ (3 − × M ⊙ ,due to the stochastic nature of the merging/accretion pro-cesses. We can also note that the formation epoch of SMSis shifted towards lower redshifts with respect to DCBH,8 < z <
14 instead of 8 < z <
17. The previous results referto the fiducial ( fertile ) case in which minihalos ( T vir < K) can form stars and pollute their gas. Results are alsogiven and discussed for the sterile case in Sec. 5. • As a third and final step towards the IMBH IMF wehave followed the accretion of the halo gas leftover after theformation of the DCBH onto the DCBH itself. This is neces-sary because, contrary to the case of the SMS in which ioniz-ing radiation from the exposed hot photosphere ionized anddisperses the surrounding gas, the General Relativity (GR)instability induces a rapid, direct collapse into a DCBH, i.e.without passing through a genuine stellar phase. The twocases differ dramatically, as virtually no ionizing photonsare produced if a DCBH forms. Therefore the newly formedDCBH will find itself embedded in the gas reservoir of thehalo and start accrete again. This accretion phase, similar tothe quasi-stellar phase advocated by Begelman et al. (2008)(see also Ball et al. (2012)), remains highly obscured and itis only in the latest phases (several tens of Myr after theDCBH formation) that the DCBH will be able to clear theremaining gas when the photospheric temperature starts toclimbs from about 5000 K when DCBH mass crosses thevalue M • = 10 . M ⊙ . Beyond that point the photospherictemperature increases rapidly and reaches about 30,000 Konce the DCBH has grown to M • = 10 M ⊙ , thus allow-ing radiative feedback to clear the gas, stop accretion, anddetermine the final IMBH mass.The IMBH IMF is different in the two scenarios consid-ered: in the (fiducial) fertile case it is bimodal with two broad peaks at M ≈ (0 . − . × M ⊙ and M ≈ (5 − × M ⊙ .The distribution extends over a wide range of masses, from M ≈ (0 . − × M ⊙ and the DCBH accretion phaselasts from 10 to 100 Myr. If minihalos are sterile, the IMFspans the narrower mass range M ≈ (1 − . × M ⊙ con-taining >
90% of the IMBH population (the remaining partbeing represented by the SMS low mass tail, see Fig. 10).We conclude that a good seeding prescription is to populatehalos (a) of mass 7 . < log( M h /M ⊙ ) <
8, (b) in the red-shift range 8 < z <
17, (c) with IMBH in the mass range4 . < (log M • /M ⊙ ) < . J • ν,c during their formation epoch. Fortu-nately, given the very narrow mass range of the IMBH hosthalos (7 . < log M h <
8) the LW intensity can be factorizedsafely.Second, to follow the feedback regulated growth ofDCBH we have assumed that the total mass of their hostinghaloes remain constant during this short accretion phase.Using our merger tree model we found that the averagetime after which DCBH hosts experience a minor or ma-jor merger is respectively equal to ≈
60 Myr and 90 Myr.Hence the approximation is good for our fiducial fertileminihalo case, as on average the accretion phase lasts for h ∆ T accr i ≈
50 Myr. However, this assumption may affectthe IMBH IMF obtained for the sterile minihaloes as in thiscase ∆ T accr ≈
100 Myr.Third, a very interesting remaining question is the fi-nal fate of the population of IMBH that do not merge intoSMBHs. As we have discussed, the IMBH seeds at forma-tion are located inside dark matter halos that have lost all oftheir gas. Some of these systems will be able to re-accrete gasand turn it into stars (the raining gas is progressively morelikely to be polluted); others might be included in larger ha-los and their IMBH merge with other black holes. Finally,some of them could remain isolated and dead, thus becomingvirtually undetectable.As a last remark, we stress that during the feedback-regulated growth we have assumed spherically symmetricaccretion. Although we have given arguments in support ofthis assumption, it is unclear if accretion might go througha disk that could become thermally unstable (e.g. because ofH formation), form stars and SNe, thus stopping the IMBHgrowth. We plan to address these issues, which require dedi-cated high.resolution numerical simulation, in a forthcomingstudy.On the observational side, our scenario can have im-portant implications. If a prolonged, obscured phase ofDCBH growth exists, this might explain the puzzling near-infrared cosmic background fluctuation excess and its re-cently detected cross-correlation with the X-ray background(Cappelluti et al. 2013), which might imply that an un-known faint population of high- z black holes could exist(Yue et al. 2013a;Yue et al. 2013b). In addition, hints of apervasive presence of IMBH in the center of nearby dwarf c (cid:13) , 1–18 he IMF of black holes seeds galaxies have been convincingly collected by Reines et al.(2013). Rashkov & Madau (2014) also pointed out thatabout 70-2000 (depending on the assumptions made on theirdynamics) relic IMBHs should be present in the Galacticbulge and halo. These objects might be indirectly traced bythe clusters of tightly bound stars that should accompanythem. Thus, our results might be a solid starting point tomake more detailed predictions on these and other relatedissues, including of course the puzzling presence of super-massive black hole in the first billion year after the Big Bang. ACKNOWLEDGMENTS
AF acknowledges financial support from PRIN MIUR 2010-2011 project, prot. 2010LY5N2T. SS acknowledges sup-port from Netherlands Organization for Scientific Research(NWO), VENI grant 639.041.233. DRGS thanks the Ger-man Science Foundation (DFG) for financial support via theCollaborative Research Center (CRC) 963 on ”Astrophysi-cal Flow Instabilities and Turbulence” (project A12) and viathe Priority Program SPP 1573 ”Physics of the InterstellarMedium” (grant SCHL 1964/1-1).
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