arXiv:astro-ph/0311487v2 8 Jul 2004 o.Nt .Ato.Soc. Astron. R. Not. Mon. eesd20 xx XX Xxxxx 2002 Released avsM Koushiappas, M. Savvas momentum angular low from material seeds hole black Massive otalglx peod elpiasadble fspi- of masses al- with bulges that holes, and black the revealed (ellipticals super-massive near spheroids have host kinematics rals) galaxy galaxies stellar all present-day and most of gas of centers studies Detailed INTRODUCTION 1 (10 ⋆ exist quasar to known with engines associated quasar-powering the now-dormant not to the are counterparts represent they holes presumably 2001; today, black Although activity 1998). these Gebhardt al. & of et Kormendy Magorrian most 2000; (e.g. Merritt & galaxies Ferrarese parent their c 3 2 1 ubefellow Hubble aa nttt fPyis h erwUiest,Jerusal University, Hebrew The Physics, of Institute Racah avr mtsna etrfrAtohsc,6 adnSt. Garden 60 Astrophysics, for Center Smithsonian Harvard eateto hsc,TeOi tt nvriy 7 .18t W. 174 University, State Ohio The Physics, of Department 03RAS 2003 6 − 10 9 )M ⊙ or , ∼ 10 000 − 4 –5(03 rne 5Otbr21 M L (MN 2018 October 25 Printed (2003) 1–15 , ftettlselrms of mass stellar total the of ABSTRACT le hteeypooglx hudcnangsta nsu nahig of a threshold in critical up a ends than that massive gas ta more contain momentum haloes should angular proto-galaxy distr low cosm every in significant a that matter a plies has dark has the haloes distribution obs for This derived rare-peak as simulations. that early-forming, to properties, similar in spheroid momentum n gas angular and proto-ga leads the masses in this that gas hole assumptions, assume momentum black reasonable angular between under lowest relation that the show of We out redshift. form holes black otne ni h rtmsiesasdsutteds.Tese b of seed mass The characteristic disc. a the have disrupt manner stars th this massive assume We in first inflow. the mass visco until allows and continues Lin-Pringle outward efficient momentum an angular fers experience and unstable, gravitationally ihro o prpit diinllmnu rwhdrn ae q after later fr process a inferred merger during growth hierarchical that luminous The to additional phase. appropriate comparable for becomes room density with mass comoving the i eoiainat opeetgscoigi hs o-assystem low-mass these in gro sa cooling seeds and gas hole haloes, prevent mass black to critical acts in m of reionization black-hole mass production for mic of continuous bound density lower the comoving a tracking The as today. serves This centers formation. of redshift n h eaieyipratrl fmresi h ulu ftdysb today’s words: of Key buildup scenario. the proposed in a the mergers seeds in of massive elements role negligib key of important with formation relatively mass, The the spheroid and galaxies. stellar disc-dominated and in masses mass black-hole between lation epeetacsooial-oiae oe nwihteseso su of seeds the which in model cosmologically-motivated a present We 1 ae .Bullock, S. James omlg akmte aais omto aais structur galaxies: - formation galaxies: – matter dark – cosmology M bh m sal [email protected] Israel; em, = abig A018UA [email protected] USA; 02138 MA Cambridge v,Clmu,O 31 S;[email protected] USA; 43210 OH Columbus, Ave, h hsppr epoieamdlfrpouigrte mas- rather producing for In model unknown. but a remain holes, provide objects black we these local paper, of or this nature AGN and explain origin to the aims that model the light via off give luminosity. grow Eddington process, can its the hole to in black proportion and, in black the gas “seed” available time, a of early with accretion some that populated at is is 2003 hole galaxy expectation Weinberg primordial & standard a Steed The ob- once e.g. as therein). (see masses references of spheroids accretion and range local mass same the in modeling have served by that explained holes black be onto can Nu reasonable (AGN) Galactic under Active clei Indeed, of function 2001). luminosity al. the et assumptions, (Fan redshift high at 2 ⋆ h edpplto sa motn nrdeti any in ingredient important an is population seed The vsa Dekel, Avishai A T E tl l v2.2) file style X z ∼ ∼ ∼ 10 5ntrlylast iercorre- linear a to leads naturally 15 7 5 M × ⊙ 10 3 ogl needn fthe of independent roughly , 7 M ⊙ at z ∼ bto fspecific of ibution akhlscreated holes lack uae hncos- when turates trlyt cor- a to aturally -est ic In disc. h-density mobservations, om re oa.We today. erved se tgalactic at asses ihredshifts, high t 5 h ic are discs the 15, lgclN-body ological iyta trans- that sity aa accretion uasar tti process this at swt time, with ws akhls are holes, lack .By s. aisa high at laxies ebakhole black le l hc im- which il, permassive z ∼ e. 15, - 2 Koushiappas, Bullock & Dekel
4−5 sive ( 10 M⊙ ) seed black holes at early times from high dial gas (Haehnelt & Rees 1993; Loeb & Rasio 1994; ∼ density, low angular momentum gas in rare halos. We argue Eisenstein & Loeb 1995a; Haehnelt, Natarajan & Rees that massive, early forming seeds help explain the high red- 1998; Gnedin 2001; Bromm & Loeb 2003). Our model is shift AGN population, and, unlike lower mass seeds, may related most closely to this third type of scenario. give rise to the observed correlations between host galaxy The key obstacle in any model of black hole formation properties and black hole masses without the need for fine- is clearly the centrifugal barrier. In the standard cosmolog- tuned feedback. ical framework of hierarchical structure formation, proto- Consider, for example the analysis of Barth et al. haloes acquire angular momentum by tidal torques exerted (2003), who have estimated a mass of Mbh = (2 6) by the background fluctuation field (Hoyle 1949; Peebles 9 − × 10 M⊙ for a black hole powering a z = 6.4 quasar found 1969; Barnes & Efstathiou 1987; Porciani, Delel & Hoffman in the Sloan Digital Sky Survey. For the ΛCDM cosmology 2002a,b). Haloes obtain a specific angular momentum that adopted below, the age of the universe at this redshift is 840 is on average 10 100 times smaller than that required for − Myr. Even the remnant of a very massive, early-forming star centrifugal support at the virial halo size (with a spin pa- of 100M⊙ would have to accrete mass at the Eddington rameter of λ 0.01 0.1, see below). This is roughly inde- ∼ ∼ − limit for 17 e-folding times in order to grow to the observed pendent of mass and time. Thus, even after cooling, most of mass. For a standard, thin-disk accretion model with an ef- the gas can contract only to radii 10-100 times smaller than ficiency 0.1, the e-folding Salpeter time is 45 Myrs, so the initial halo radius before it is halted in circular orbits. ∼ ∼ an intermediate-mass seed would require 770 Myrs of ac- The implied scale (at z = 0) is kpc, many orders of mag- 1 ∼ cretion in order to reach to observed mass, corresponding nitude larger than the corresponding Schwarzschild radius, > −5 9 to a formation redshift for the seed of zf 40. This is an 10 (M/10 M⊙ )pc. The situation may get even worse ∼ ∼ extremely early redshift in the context of ΛCDM structure at earlier times because the typical collapsing mass M⋆(a) formation, since the minimum halo mass for H2 cooling at is smaller in proportion to a high power of the expansion this time represents a & 5σ fluctuation. While this does not factor a (1 + z)−1. This implies that the ratio of the cen- ≡ 1/3 completely rule out the possibility that the seed was small, trifugally supported size ( aM⋆(a) ) to the Schwarzschild 5 ∝ it does require an extremely rare event. A 10 M⊙ seed, on radius ( M⋆(a) if the black hole mass is a constant frac- ∝ the other hand, would require 10 e-folding times of accre- tion of the halo mass) increases dramatically at early times, ∼ > tion, and thus a more recent formation redshift zf 11. Of unless the sites of black hole formation are rare haloes much ∼ course these numbers are sensitive to certain assumptions, more massive than M⋆(a). This line of argument indicates and may be influenced by allowing low efficiency accretion that the seed black holes originate from material at the very or super-Eddington luminosities. Nevertheless, it is gener- lowest end of the angular-momentum distribution within ally easier to understand the population of z> 6 quasars if rare, massive haloes at high redshift. This material should massive seeds are present (see also Haiman &∼ Loeb 2001 for then perhaps lose all its angular momentum via an effective a related discussion). mechanism of angular momentum transfer, which would be Perhaps the most intriguing clues to the nature of black enhanced in such a high-density environment. hole formation and growth are the strong correlations be- Eisenstein & Loeb (1995a) considered the sites of black tween black hole masses and global properties of their host hole formation to be the very high density discs which form spheroids — scales that differ by many orders of magnitude. in those rare haloes that have extremely low total angular The black hole mass is tightly correlated with the central momentum. Assuming a standard CDM scenario, they used velocity dispersion of the spheroid, and shows a significant an analytic model to predict the distribution of halo spins correlation with the spheroid luminosity or mass. The theo- and estimated the abundance of haloes with low enough spin retical challenge is to connect the galaxy-formation process for efficient viscosity (Eisenstein & Loeb 1995b). They ar- originating on scales of megaparsecs with the black-hole for- gued that there may be enough low-spin haloes to explain mation processes on sub-parsec scales. In addition, the epoch the black hole abundance. They then adopted the common of spheroid formation is likely much later than the time when view that viscosity in a differentially rotating gaseous disc is the first quasar black holes begin to shine, so an additional the mechanism that transfers angular momentum outward, challenge is to connect these two components in time, in thus allowing gas to contract into a black hole. In their cal- a way that maintains, or even creates, the observed local culations, they considered hydrodynamic (or perhaps hydro- correlations. magnetic) turbulence to be the viscous mechanism respon- Several scenarios have been presented for the for- sible for the angular momentum loss, and appealed to the mation of massive black hole seeds. The general frame- α-disc prescription (Shakura & Sunyaev 1973). works range from those that rely on the coalescence Our approach is related, but qualitatively different. We of massive stellar remnants that sink to the galaxy propose a scenario where seed black holes form from mate- centers (Quinlan & Shapiro 1990; Hills & Bender 1995; rial with low angular momentum in all haloes that are mas- Lee 1995; Ebisuzaki et al. 2001) to one that models sive enough to host an unstable self-gravitating disc. The black hole growth via ballistic particle capture dur- required high-density discs can only occur in haloes that ing bulge collapse (Adams et al. 2001, 2003). Other collapse at high redshift, z & 10 20, when the angular mo- − models are based on the hydro-dynamical evolution of mentum distribution of the (unprocessed) gas should be the super-massive objects formed directly out of primor- most similar to that of the dark matter, and before reioniza- tion or some other feedback mechanism (e.g., Dekel & Silk 1986; Dekel & Woo 2003) prevents the growth of cold discs. 1 This timescale is much larger than the estimated lifetimes of The crucial point is that every halo, regardless of its to- quasars at lower redshift (see Martini 2003 for a recent review). tal angular momentum, is expected to have a distribution
c 2003 RAS, MNRAS 000, 1–15 Black holes from low-spin material 3 of specific angular momentum including a tail of low-spin efficient cooling mechanism, gas cools and condenses to a material. centrifugally supported disc. The haloes we consider are massive enough so that the Atomic line cooling can bring the gas to 104K, and ∼ baryons trapped in their potential wells can cool and con- further cooling down to 300K requires the presence of ∼ dense. In the absence of any other effect that may alter the molecular hydrogen. The minimum halo mass for atomic 7 −3/2 angular momentum content of baryons, the lowest spin bary- cooling is Ma 6 10 M⊙ [(1 + z)/18] , and for molec- ≃ × 6 −7 onic material in each system ends in a high-density central ular hydrogen survival it is MH2 5 10 M⊙ [(1+z)/18] ≃ × region forming a proto-galactic disc. As an illustrative fidu- in the redshift range z 13 30 (Tegmark et al. 1997; ≃ − cial model, we assume that the angular-momentum distri- Abel et al. 1998). The haloes we consider below are just bution in each halo matches the “universal” distribution of above the limit for atomic line cooling and significantly Bullock et al. (2001a) (B01 hereafter). If the angular mo- above the limit for molecular hydrogen cooling. We therefore mentum is approximately preserved during baryonic infall expect the gas to cool and settle in an angular momentum into the disk, the B01 angular momentum distribution im- supported disc on a timescale comparable to the dynamical plies that the inner disk becomes self gravitating with the time: surface density inversely proportional to the radius. We show −3/2 −1/2 −1/2 1+ z below that if this behavior extends below the inner 2% of tdyn (Gρvir) 77 Myr ∆178(z) , (1) ∼ ≡ ≃ 18 the disc radius, then enough mass may acquire a high enough density for viscosity to efficiently redistribute the angular where we have assumed our standard cosmological param- eters and ∆178(z) is the virial overdensity in units of 178 momentum of the disc. For the final contraction of disc mate- 2 rial into a black hole we evaluate the angular-momentum loss times the background density. . Hereafter, due to the small by the effective kinematic viscosity that arises from gravi- redshift dependence of ∆vir at the redshifts of interest we set ∆178(z) = 1. tational instabilities (Lin & Pringle 1987). This mechanism 3 is somewhat more efficient than that from a typical α-disc Consider a halo of virial mass Mv and spin parameter √ −1 model. This viscosity-driven mass inflow cannot continue in- λ J[ 2MvVvRv] , where J is the total halo angular mo- ≡ 1/3 −1 definitely; once the disc significantly fragments, or once it is mentum. The virial radius is Rv = 384 pc M7 [(1+z)/18] −1 1/3 1/2 heated or even disrupted by feedback, the inflow stops. and the virial velocity is Vv = 11 km s M7 [(1+z)/18] . 7 The semi-quantitative scenario presented here can be Here, M7 Mv/10 M⊙ . B01 found that in each halo the ≡ regarder as an illustration of a general model based on the cumulative mass with specific angular momentum less than assumption that proto-galactic gaseous discs develop high j is described well by the function density inner regions dominated by low angular momentum j material. Our current analysis is basically a feasibility study M(< j)= µ Mv , µ> 1. (2) j0 + j of such a model, and an attempt to recover its robust predic- tions and to constrain the model parameters by theoretical This profile has an implicit maximum specific angular mo- mentum jmax = j0/(µ 1), where j0 = √2VvRvλ/b(µ), with considerations as well as by the observations. −1 − b(µ)= µln(1 µ ) 1. The parameter µ characterizes the The outline of this paper is as follows. In 2 we discuss − − − § shape of the angular momentum distribution. When µ 1 the angular momentum distribution in proto-galaxies and → the implied surface density profile for the cold discs. In 3 the deviation from a power law is pronounced with a larger § we investigate the effect of kinetic viscosity on the evolution fraction of the mass having low specific angular momentum, of the disc. In 4 we calculate the resulting seed black hole while if µ 1 the distribution resembles a pure power law. § ≫ mass and mass function. In 5 we calculate the redshift The key property of this distribution is that at least half § evolution of the seed black hole mass density. In 6 we make of the halo mass is in the power-law regime, M(< j) j, § ∝ a preliminary attempt to address the correlation between and the simulations confirm that the power-law behavior ex- tends at least over two decades in mass, down to 1% of the masses of black holes and galactic stellar spheroids, to ∼ be pursued in a subsequent paper. In 7 we summarize the total halo mass. § our conclusions and discuss our results. When necessary, we The B01 profile was confirmed by Chen & Jing (2002) assume a ΛCDM cosmology, with a Hubble constant h = 0.7 and Col´ın et al. (2002) using N-body simulations, and by −1 −1 van den Bosch et al. (2002), Chen, Jing & Yoshikaw (2003) (H0 = 100h km s Mpc ), universal mass density ΩM = and Sharma & Steinmetz (2003) using adiabatic hydrody- 0.3, cosmological constant ΩΛ = 0.7, and rms fluctuation namic simulations. These studies reveal that the B01 angular amplitude σ8 = 0.9. momentum distribution remains a good fit to haloes at red- shifts as high as z = 3. In what follows we will assume that
2 SPIN DISTRIBUTION AND DISC PROFILE 2 If we use the approximation of spherical top-hat collapse given 2 We focus on the proto-galaxies forming in the redshift range in Bryan & Norman (1998) we get ∆vir(z) ≃ (18π + 82x − 2 3 z 10 30. In the standard picture of structure forma- 39x )/ΩM(z), where x +1 = ΩM(z) = ΩM(1 + z) /[ΩM(1 + ∼ − z)3 +Ω ] and Ω (z) is the ratio of the mean matter density to tion, galaxies form as a uniform mixture of dark matter and Λ M critical density at redshift z. In the ΛCDM cosmology that we baryons in small over-densities that grow by gravitational adopt, ∆vir(z = 0) ≃ 337 and ∆vir(z) → 178 at high redshift, instability. Until its expansion turns around into a collapse, approaching the standard CDM (ΩM = 1) value. each halo acquires angular momentum through tidal torques 3 This spin parameter has been proposed by B01 as a practical induced by the density fluctuations around it. The dissipa- modification of the conventional spin parameter, defined as λ = 5/2 tionless collapse brings the dark matter into virial equilib- J |E|/GMv , where E is the halo energy. The two definitions rium in an extended spheroidal halo. In the presence of an ofpλ coincide for a singular isothermal sphere truncated at Rv.
c 2003 RAS, MNRAS 000, 1–15 4 Koushiappas, Bullock & Dekel
total halo spin), and claimed that this fraction may be as large as 20%. 4 By using the B01 profile as is we ∼ actually adopt the conservative approach which minimizes the efficency of black-hole formation, as follows. If a fraction of the gas starts with negative spin, it will mix dissipatively with the positive-spin gas leading to a purely rotating disc with no negative component. The implications on the disc formation depends on the way by which this mixing occurs. In the limiting case where the mixing occurs after the gas has already condensed to the disc, the negative component will mix positive spin material of exactly the same distribution of spin magnitudes. (van den Bosch et al. 2002). This results in a fraction of mass with near-zero angular momentum, which would increase the efficiency of black-hole production. If angular momentum mixing occurs during the process of gas cooling and collapse before the disc forms, there is no a priori reason for the negative angular momentum material to mix with only positive angular momentum material of the same magnitude. If this mixing is uniform across the angular momentum profile, Figure 1. The mass per unit specific angular momentum for then the resulting distribution should be similar to the B01 different values of the parameter ξ. Note that in our fiducial case form, but with a slightly smaller overall spin parameter. of ξ = 1 (B01), this function approaches a constant at small j. The B01 form, ignoring the negative tail, is therefore the conservative case of minimum blak-hole formation. We assume that the specific angular-momentum distri- the distribution holds at much higher redshifts, z 15. Since bution of the baryons resembles that of the dark matter ∼ the ΛCDM universe is effectively Einstein-de Sitter beyond halo. This assumption has been found to hold reasonably z 1, and the power spectrum is effectively a power law in well in both radiative and non-radiative hydrodynamic simu- ∼ the relevant mass range, one expects self-similarity between lations. (van den Bosch et al. 2002; Chen, Jing & Yoshikaw the haloes at z 3 and at z 15. Therefore, the profile at 2003). This assumption should be valid especially in sys- ∼ ∼ z 15 can be assumed to resemble the B01 shape derived at tems that collapse early, as baryons would not have been ∼ z = 0. Simple theoretical considerations (B01, Dekel et al. affected by feedback associated with star formation. In ad- 2000; Maller et al. 2002) argue that this angular momentum dition we assume that the angular momentum of gas is con- distribution is a natural outcome of the hierarchical cluster- served as gas cools and falls to the center forming a centrifu- ing process. A qualitatively similar distribution is obtained gally supported disc. This assumption is backed by obser- when linear tidal torque theory is applied to haloes shell vations, where disc sizes seem to indicate that the angular by shell, or when the orbital angular momentum of merging momentum of the gas gained by tidal torques is retained. In haloes turns into spin of the product halo via tidal stripping addition, B01 found that the direction of angular momen- and dynamical friction. Other justifications follow from more tum is well aligned at different radii, suggesting that the detailed models (Maller & Dekel 2002; van den Bosch et al. formation of a disc is indeed possible. Therefore, if there are 2001). These analytic models provide addivtional support to no significant changes in the specific angular momentum of the assumption that a similar average halo profile is valid at the baryons during the formation of the disc, then the disc low and high redshifts. angular-momentum distribution is Md(< j) = f0M(< j), As mentioned above, the B01 distribution is a good where f0 is the final mass fraction in cold baryons. For the fit to simulation results for angular momentum as small as circular orbits in the disc one has j(r)= rV (r), so one can 1% the characteristic angular momentum of the halo (cor- map the Md(< j) distribution to a radial mass profile in the ∼ responding to 1% the characteristic disc radius. As dis- disc. Note that if Md(< j) j (as with the B01 distribu- ∼ ∝ cussed below (see Eq. 13), our calculation assumes that the tion at small j) and the disc has a self-gravitating rotation 2 B01 form holds on scales below 2% of the characteristic curve V (r) GMd(r)/r, then V (r) =const. and Md(r) r ∼ ≃ ∝ angular momentum, so only a mild extrapolation of the low regardless of the initial properties of the halo and the pos- angular momentum tail is required, and being a power law, sible effects of baryonic contraction on it. This implies a −1 this extrapolation seems plausible. Nevertheless, the angular disc surface density profile of Σ(r) r in the inner disc. ∝ momentum distribution of the gas may not exactly mirror Since we expect these general trends to hold regardless of that of the dark matter. In order to evaluate the qualita- our assumptions about the halo profile, we make the simple tive effects of this possibility and the explore alternative approximation that the final total mass profile of halo plus extrapolations, we will also condier deviations from the B01 disc follows that of an isothermal sphere, with M(r) r ∝ distribution. Unlike the analysis in B01 (but see Bullock et al. 2002), other works such as Chen & Jing (2002), van den Bosch et al. (2002) and Chen, Jing & Yoshikaw 4 See the extensive discussion in B01 on the pitfalls that may arise (2003) focused on estimating the fraction of dark mass when attempting to measure the negative fraction of material in with negative angular momentum (in the direction of the dark halos.
c 2003 RAS, MNRAS 000, 1–15 Black holes from low-spin material 5
with the normalization constant N = µ−1[(µ 1)ξ + 1]. − In Fig. 1 we show the differential angular momentum distributions for different values of ξ. The specific case of ξ = 1 is the B01 form, while larger (smaller) values of ξ correspond to distributions with less (more) low angular momentum material. In Fig. 2 we show the resulting disc surface density profiles for different values of ξ. Note the steep increase in density for ξ = 1 (B01), while for ξ = 2 the density profiles are similar to an exponential disc. Throughout this paper, we focus on the regime where r˜ 1, and the surface density is well-approximated by ≪ NξfµMv ξ−2 Σd(˜r) 2 r˜ . (7) ≃ 2πrd We assume the B01 limit for the most part of this paper (which corresponds to ξ = 1), but also investigate how our results change with varying the value of ξ.
3 ANGULAR MOMENTUM LOSS Figure 2. The disc surface density profile [eq. (6)] for different values of ξ. Our fiducial case, ξ = 1, shows a steep increase toward After baryons cool and settle into a dense inner disc, an- small radii, while the ξ = 2 case has a flat core, similar to an gular momentum is redistributed as a result of viscosity. exponential disc. We adopt the kinematic viscosity prescription proposed by Lin & Pringle (1987) (see Laughlin & Rozyczka 1996 for a detailed test of this picture). According to this mechanism, and V (r) = Vv. In this case, the disc mass profile implied global gravitational instabilities in the disc give rise to a by the distribution (2) is transfer of angular momentum outward, which can be mod- r˜ eled as a local kinematic viscosity. Md(˜r)= fµMv , r˜ r/rd , (3) 1 +r ˜ ≡ Consider a disc with local surface density Σ, angular frequency ω, and temperature T , expressed as a sound speed where rd is a characteristic disc radius, 1/2 −1 cs 2(T/300K) kms . As discussed by Toomre (1964), a ≃ −1 disc is gravitationally unstable as long as the minimum size √2λRv − 1/3 1+ z 1 −2 rd 22pc λ0.04b (µ) M7 (4) of shear-stabilizing disturbances Ls G Σω , is larger than ≡ b(µ) ≃ 18 ≃ the maximum size of dispersion-supported regions, Ld 2 −1 −1 ≃ and λ0.04 = λ/0.04. At r rd, we have again Md r. This c G Σ . This condition is expressed by evaluating the ≪ ∝ s profile is valid out to a maximum radiusr ˜max = 1/(µ 1). Toomre parameter Q, − The constant f is the fraction of cold baryonic mass in the 1/2 Ld csω halo that has fallen into the disc by redshift z (f 6 f0). Q . (8) ≡ Ls ≃ GΣ Recall that it takes a finite time, of order tdyn, for h i ∼ a centrifugally supported disc to completely form after the In discs with Q< 1, the interaction of the global instability baryons cool. Thus f in eq. (3) becomes equal to the total perturbations with the background shear field drives a mass cold mass fraction f0 only after a time t > tdyn. Before that inflow. The transfer of angular momentum occurs across re- > −1 time, we assume f = f0 (t/tdyn), such that the disc mass gions of maximum size Ls over a typical timescale ω ∼ grows linearly with time until t = tdyn, due to an effective viscosity ∼ L2 G2Σ2 t ν s . (9) Md(t)= fMv f0Mv , 0 6 t 6 tdyn . (5) ω−1 ω3 ≃ tdyn ≃ ≃ In what follows, we use ω2 = GM(r)r−3. Hereafter we assume that the total cold mass fraction is A typical viscous time for mass inflow at a radius r in f0 = 0.03, which is roughly 1/5 of the universal baryon 2 −1 ∼ the disc is tν r ν . As mass flows in, the central re- fraction of the standard ΛCDM Universe we assume in this ≃ gions of the disc become denser, so the value of Q decreases. paper (see also Klypin et al. 2002). This is a conservative If at any point Q became as small as Q2 < H/r, where estimate for the fraction of baryons that can form enough H2 H cs/ω is the height of the disc, then the instabilities to cool significantly (Tegmark et al. 1997; Abel et al. 1998). ∼ would have become fully dynamical, fragmenting the disc Since our approach relies on an uncertain extrapola- into self-gravitating bodies, which would have stopped the tion of the B01 distribution, we explore a more general form angular momentum transfer. However, this effect is likely to for the angular momentum distribution, of which B01 is a be unimportant over the timescales of concern here because special case. We consider the family of angular-momentum while Q is decreasing, the ratio H/r is actually decreasing distributions that correspond to the following disc surface at a faster rate (Mineshige & Umemura 1997), thus delaying density profiles, parameterized by ξ, the global fragmentation and allowing the viscosity to act NfµM ξr˜ξ−2 until some external process disrupts the disc. The underly- Σ (˜r)= v , r˜ < (µ 1) , (6) d 2 ξ 2 ing assumption is that while the global instabilities give rise 2πrd (1+˜r ) −
c 2003 RAS, MNRAS 000, 1–15 6 Koushiappas, Bullock & Dekel to density waves, which cause the transfer of angular mo- mentum, the disc does not fragment instantaneously. Star formation can occur concurrently with the viscous process until feedback from the first massive stars stops the gas in- flow either by heating the disc or by disrupting it altogether (if its virial velocity is small enough).
3.1 Critical halo mass for viscous transport Only haloes more massive than a critical minimum mass host unstable discs in which the angular-momentum loss is efficient and a seed black hole can form. This is because, from eq. (8), we expect Q M −1/3, implying a critical ∼ mass above which Q < 1. For the specific case of the B01 profile the scaling is Q M −1/2r1/2 M −1/3 (1 + z)−1/2, ∝ d d ∝ d independent of radius. If we assume Md 6 f0Mv, then set- ting ξ = 1 in eq. (7) yields a lower limit for host halo masses hosting Q< 1 discs:
3/2 crit 7 3/2 18 Mv > 1.2 10 M⊙ T300 Figure 3. Timescales as a function of halo mass. Shown are (a) × 1 + z 3/2 the time needed for the disc to become self-gravitating (tQ), (b) λ 0.03 1.25 . (10) the time between virialization and the following major merger × 0.04b (µ) f0 µ (tmm), and (c) the dynamical time (tdyn). Thick (red) lines cor- respond to z = 17 and thin (blue) lines to z = 13. The critical All the quantities in brackets are of order unity, and T300 ≡ mass for a halo that can form a seed black hole is determined by T/300 K is the typical temperature for gas that has cooled the intersection of the corresponding lines for tQ and tmm. via molecular Hydrogen. A slightly improved estimate of this critical halo mass can be obtained by considering potential disc perturbations eq. (10) is an underestimate of the critical halo mass needed by major mergers during the gradual growth of disc mass for disc instability that leads to efficient angular momentum until it reaches the baryon fraction f0 at t tdyn after viri- transfer. In the regime we consider, t is always less than ∼ mm alization. If we approximate the rate of mass growth in the tdyn. −1/3 − disc by eq. (5), then Q M t 1/3 implies that ini- Figure 3 also shows the characteristic timescale for ma- ∝ d ∝ tially self-gravity is negligible compared to gas pressure. As jor mergers tmm, in the extended Press-Schechter formalism more material is accreted, Q decreases, and it may eventu- (Press & Schechter 1974; Bond et al. 1991; Lacey & Cole ally reach Q 6 1 where angular momentum transfer begins. 1993). Specifically, we associate tmm with the peak of the We define the timescale tQ as the timescale needed for the probability distribution for the time it would take a halo to disc to become gravitationally unstable had the accretion double its mass (see equation 2.21 in Lacey & Cole 1993). been uninterrupted. To evaluate this time, we assume that The intersection of the tQ and tmm curves (for a given red- the disc is growing according to eq. (5), compute the surface shift) marks the more realistic critical minimum mass, which density from the B01 profile (eq. (7) with ξ = 1), and sub- turns out to be roughly a factor of 4 larger than the un- ∼ stitute in eq. (8) for Q. We then set Q = 1 for the stability interrupted estimate (eq. 10). crit threshold and obtain Figuer 4 shows Mv calculated in the way just de- − 18 scribed as a function of redshift. The thick solid line was t = 1.4 t T M 2/3 Q dyn 300 7 computed assuming the B01 profile for the gas density dis- 1+ z 0.03 1.25 λ tribution. The redshift scaling is similar to that estimated . (11) in eq. (10), confirming that the critical mass is varying only × f0 µ 0.04b(µ) slowly with redshift. The critical mass is about a factor of Figure 3 shows tQ as a function of host halo mass at 4 larger than the uninterrupted estimate throughout the ∼ z = 17 and at z = 13. We assumed the fiducial values whole redshift range of interest. Shown for reference is the ξ = 1, T = 300K, f0 = 0.03, µ = 1.25, and λ = 0.04. minimum halo mass for molecular hydrogen to survive as These curves are compared to the virial dynamical times an efficient coolant; it justifies our assumption that gas can tdyn at the two corresponding redshifts. The intersection of cool in critical-mass haloes. the tQ and tdyn curve marks the uninterrupted critical mass Also shown in Fig. 4 is our estimated critical mass at that redshift, as estimated in eq. (10). However, a ma- for a flatter surface density profile (or narrower angular- jor merger may destroy the disc and disrupt the gas in-fall. momentum distribution) with ξ = 1.3. As would be ex- crit If the typical time for a major merger to occur after the pected, Mv is higher here than it is for the ξ = 1 case. halo virializes is tmm, the disc reaches a maximum mass of This is because densities at small radii are lower for a fixed Md (tmm/tdyn)f0Mv < f0Mv. In order to obtain a given mass disc, so a more massive host is needed in order to ≃ critical disc mass where viscosity efficiently redistributes an- supply the extra disc mass required for Q < 1. Unfortu- gular momentum, we require tQ radius can be written as 3 3−ξ −4 4 f µ 0.04b(µ) r˜vis = 3.9 10 Nξ × 0.03 1.25 λ 2 1+ z 3 t (12) × 18 Myr wherer ˜vis = rvis/rd and f = f0(tQ/tdyn). For the special case of ξ = 1, we have 1/2 1/2 3/2 −2 f µ 0.04b(µ) r˜vis = 1.9 10 × 0.03 1.25 λ 1+ z 3/2 t (13) × 18 Myr We see that the inflow radius grows linearly with the time available for viscosity to efficiently transfer angular momen- tum from the inner parts of the disc to outer radii. According to the above equations, if the viscous angular-momentum transfer were to continue uninterrupted Figure 4. The minimum mass of a halo capable of forming a seed for a long enough time, then the whole disc would have even- black hole as a function of redshift. The thick solid curve marked tually lost its angular momentum and collapsed. However as B01 refers to our fiducial ξ = 1 distribution. The thin solid curve mentioned before, the major merger timescale sets an upper refers to ξ = 1.3, with a longer time of 20 Myr assumed for star limit on the time available for this process, and in fact the formation. The dotted lines correspond to 2-8 σ fluctuations in the random fluctuation field, illustrating that the minimum masses feedback from star formation is likely to terminate the pro- of concern are rare systems. The short dashed line represents the cess even earlier. minimum mass for atomic line cooling only, while the long dashed In our adopted viscosity prescription, the first stars be- line is an approximation for molecular hydrogen cooling . gin to form from the same instabilities that drive angular- momentum loss. Of course, if the disc completely fragments into stars then the angular momentum transfer stops, but the timescale for this is likely longer than the lifetimes of Q r˜(1−ξ)/2. Indeed, for ξ > 1 discs become unstable from the very first massive stars. In addition, the presence of ad- ∝ the outside in; however, once inflow begins, mass transport ditional feedback on star formation, for example from mag- always tends to increase the central density with time and netic fields, may play a role in setting this timescale (Gnedin therefore it is natural to assume that the central disc will 2001). Therefore, we assume that a more realistic estimate become unstable within a viscous timescale. We can crudely of the time available for viscosity to work is comparable estimate the critical mass in this case by evaluating Q and to the timescale for collapse and evolution of the first self- the viscous timescale at some characteristic radius. If we as- gravitating bodies in the disc. These “first stars” should have sociate the viscous timescale with the timescale over which very low metallicities because the haloes under consideration at z 20 had very few progenitors massive enough to have the disc would get disrupted (see next section), we can de- ∼ formed stars before z 20. Thus, the stellar evolution in fine the critical halo mass for black hole seed formation to ∼ these stars is quite rapid, with lifetimes between 1 30 be the mass scale at which Q < 1 for the given viscous ∼ − timescale. The ξ = 1.3 curve in Fig. 4 has been computed Myr, depending on many assumptions (Shapiro & Shibata assuming t = 20 Myr. Despite the fact that this result re- 2002; Barafee et al. 2001; Heger & Woosley 2002; Shaerer 2002). We explore the range t = 1 30 Myr for the time lies on very crude approximations, it does serve to illustrate − how narrower angular momentum distributions require more available for viscosity to transfer angular momentum away massive (and less abundant) haloes as the sites of black-hole from the inner disc. formation. We come back to the implications of this later in If angular momentum transfer is allowed to oper- the paper. ate for a timescale t, then a large fraction of mass withinr ˜vis would lose its angular momentum and form a central, massive object. This object would be pressure- supported and short-lived 5, and inevitably collapse to a 3.2 Viscous inflow 5 Consider now haloes more massive than the critical mass It is perhaps a concern that the presence of a pressure- for disc instability and assume that they have formed discs supported object that grows over the viscous timescale might in- fluence the temperature of the disc. We can estimate the heating with surface density profiles described by eq. (7). The vis- of the disc if we assume that this objects radiates at Eddington cous process transports mass inward from a radius r over 2 −1 luminosity with an efficiency of ∼ 0.1 and that the disc is heated the viscous time tν r ν . We make the assumption that ≃ due to Thomson scattering. We find that it will take a time of if viscosity has been acting for time t, then all of the disc order ∼ 5tvis to increase the temperature of the disc by 50% (this 1/2 material originally within rvis = (νt) has lost its angu- timescale is reduced to ∼ 3tvis if the object radiates at the Ed- lar momentum. Assuming a viscosity given by eq. (9), this dington limit). Since these timescales are greater than the viscous c 2003 RAS, MNRAS 000, 1–15 8 Koushiappas, Bullock & Dekel black hole owing to the post-Newtonian gravitational in- stability (Shapiro & Teukolsky 1983). The exact mass of the resulting black hole depends on the mass loss expe- rienced during the short lifetime of this “star”. This is an unknown quantity, but the mass loss is typically as- sumed to be in the range 10% 90% of the initial mass − of the object (Barafee et al. 2001; Heger & Woosley 2002; Shapiro & Shibata 2002). Hereafter we assume that the mass of the resulting black hole is some fraction κ of the total mass with zero angular momentum, and take κ = 0.5 as our fiducial value where necessary. 4 MASS FUNCTION OF SEED BLACK HOLES We define the mass of the black hole Mbh as the fraction κ of the mass withinr ˜vis(t), i.e., the mass that lost its angular momentum over time t. Using equations (13) and (7) we obtain for the generalized profile 2η 3η Figure 5. Probability distributions of seed black hole masses −4 κ ξ t 1+ z Mbh = 5 10 Mv (14) for the threshold halo mass at z = 17. The assumed angular- × 0.5 Myr 20 momentum distribution is ξ = 1, and the assumed viscosity 1/ξ 3η timescale is 5 Myr. (N f µ) b (µ) 2η−1 [20]3η−3/2 1.1 10−5 , × λ × where η ξ/(3 ξ). For the B01 case with ξ = 1 the result where x0 = 0.035, 1.25 and σx = 0.5, 0.3 for x = λ, β respec- ≡ − reduces to tively. Given these distributions, it is easy to compute P (γ), 3/2 and from this the probability distribution of the seed black κ f 3/2 µb(µ) 3 holes given for a halo of mass Mv at redshift z for a given Mbh = 3.8 10 M⊙ (15) × 0.5 0.03 1.25 viscous timescale t. The result is −3/2 3/2 λ Mv 1+ z t 1 1 P (Mbh Mv) dMbh = 7 0.04 10 M⊙ 18 Myr | √ Mbh × 2πσMbh 2 0 Thus, for typical values, the black hole masses are quite ln (Mbh/Mbh) 4 −3 exp 2 dMbh. (17) large, 10 M⊙ , and of order 10 the mass of the host × − 2σ ∼ ∼ Mbh halo. Notice however that the black hole mass is quite sen- 0 sitive to the value of the profile slope parameter ξ (e.g. the where Mbh is last term in eq. 14). For example, with ξ = 1.4, the black- 3/2 −4 0 4 κ f Mv hole mass is 10 of the halo mass. A similar calculation Mbh = 3.8 10 M⊙ ∼ × 0.5 0.03 107M⊙ using an exponential-like profile (with ξ = 2) gives negligible 1+ z 3/2 t seed black hole masses, unless the viscous timescale is much (18) larger than the Hubble time at that high redshift. × 18 10 Myr In general, for a halo of a given mass M forming a v and σ = 0.9. black hole at redshift z we expect a spread in the parame- Mbh Figure 5 shows the seed black hole probability distri- ters characterizing the distribution of angular momentum. bution normalized to a peak value of one, for a host halo We compute the conditional probability distribution of seed 7 of mass Mv = 5 10 M⊙ at z = 17 and a time available black hole masses given Mv at z for the ξ = 1 case, where the × for angular momentum transfer t = 5 Myr. In this case, distributions of the parameters λ and µ are well-known. Note 5 3/2 tdyn(z = 17) 73 Myr, tQ(M = 5 10 M⊙ , z = 17) 35 first that the black-hole mass scales as Mbh [µb (µ)/λ] , ≈ × ≈ ∝ Myr and therefore f 0.014 yielding a mean seed black where we define for convenience β µb (µ) and γ β/λ, ≈ 4 3/2 ≡ ≡ hole mass of Mbh 3 10 M⊙ with possible masses in the such that Mbh γ . The N-body simulations used by 3 6 ≈ × ∝ range 10 10 M⊙ . Such a distribution function of seed B01 show that the probability distributions of λ and β are − black hole masses should provide a useful starting point well-parameterized by log-normal distributions of the form, for studies aimed at modeling the quasar properties within 1 1 ln2(x/x ) a cosmological context (Bromley, Somerville & Fabian P (x)dx = exp 0 dx (16) 2 2003; Menou, Haiman & Narayanan 2001; √2πσx x − 2σx Kauffmann & Haehnelt 2000) and can be used in of the merger history of supermassive black holes (Madau & Rees timescales we consider here (set by the timescales for the forma- 2001; Volonteri, Haardt & Madau 2003). tion and evolution of the first self-gravitating bodies in the disc) Consider now black holes forming only in haloes of crit- crit crit we conclude that the presence of this central short-lived object ical mass Mv . If we approximate Mv with equation (10), does not affect the thermal evolution of the disc. then the seed black hole given in equation (14) is indepen- c 2003 RAS, MNRAS 000, 1–15 Black holes from low-spin material 9 dent of redshift of formation, host halo mass, and spin pa- rameter, and has a “universal” value of 3/2 4 3/2 κ f0 t Mbh 6 10 M⊙ T300 .(19) ≃ × 0.5 0.03 10 Myr crit However, it is expected that since Mv depends on the spin parameter λ and µb (µ) there will be a spread in the crit critical host halo masses. The critical mass Mv scales as M crit γ−3/2. If we use the probability distribution v ∝ for P (γ) calculated as explained above, it is straightfor- ward to calculate the distribution in the host halo masses. The result is a log-normal distribution with a mean of crit,0 7 3/2 3/2 M 5 10 M⊙ T [18/(1 + z)] and a standard ≃ × 300 crit deviation of σMv = 0.9. Thus, if we focus on black holes forming within critical mass haloes only, our model predicts 5 a near universal mass of 10 M⊙ , with a log-normal dis- ∼ tribution of the masses hosting these seeds. 5 TOTAL MASS IN SEED BLACK HOLES Figure 6. Total mass density in seed black holes as a function of In this section we calculate the comoving mass density of redshift. Thick lines represent our fiducial ξ = 1 (B01) angular- mass of seed black holes and make a comparison with the momentum distribution for various viscous timescales, while the thin line corresponds to ξ = 1.3 and a viscous timescale of 20 observed density of mass in black holes today in the local Myr. The shaded area represents the observed range of total mass Universe. In this calculation we do not include any addi- density in black holes at z = 0. The WMAP reionization range is tional black hole growth via luminous accretion (which must marked by the vertical dotted lines. occur at some level in order to match the observed quasar population). We therefore expect the calculated density to ∞ be smaller than the observed value. If, for example, most dρ(z) d dn(z) = Mbh(M,z)dM . (20) black holes experience a luminous accretion phase charac- dz dz crit dM ZMv (z) terized by a few e-folding times, we want our cumulative mass density to fall below the locally observed black hole Here [dn(z)/dM]dM is the number density of haloes with density by roughly an order of magnitude. mass in the interval between M and M + dM at redshift z, As halo masses grow with time, some haloes become given by crit more massive than the minimum mass threshold Mv and 2 dn(z) ρ0 dσ (M) become the sites of potential seed black hole formation. Since dM = f1[σ(M),z] dM (21) dM M dM the threshold mass decreases only slowly with redshift (see figure 3.1), the mass density due to seed black holes is dom- and σ(M) is the mean square fluctuation amplitude at a inated by the seeds formed at the particular redshift of in- mass scale M. Here f1 is the fraction of mass associated terest. Naturally, all haloes more massive than the critical with haloes of mass M corresponding to the given range host halo mass scale were able to form a seed black hole at in σ(M) at redshift z. We use the Press-Schechter (1974) some higher redshift. However, these mass scales correspond approximation for f1 (see, e.g. Lacey & Cole 1993). Since crit to higher-sigma fluctuations and therefore do not contribute the value of Mv (z) is associated with increasingly higher significantly to the mass density. In order to include only the sigma fluctuations at high redshifts, our results are not sen- contribution coming from seed black holes formed in haloes sitive to the choice of zi. We choose zi = 30, but our results that have just crossed the minimum mass threshold, we com- are unchanged for zi = 50. pare the typical growth rate of halo mass with the growth Figure 6 shows the comoving density of seed black holes crit rate of Mv . We find that the probability of a halo crossing obtained by integrating eq. (20). As expected, this function the threshold having previously been massive enough to host is strongly dependent on the time available for viscosity to − black hole forming progenitors is vanishingly small ( 10 6). redistribute angular momentum. At high z there is a rapid ∼ Therefore it is safe to assume that the contribution to the growth, which flattens off toward lower redshifts. The hor- seed black hole mass density at each redshift comes predom- izontal band marks the observationally-inferred density of inantly from those seeds formed in haloes which have just mass in black holes today (Soltan 1982; Chokshi & Turner crossed the minimum mass threshold near that particular 1992; Kormendy & Richstone 1995; Fabian & Iwasawa redshift. 1999; Salucci et al. 1999; Merritt & Ferrarese 2001; The comoving density of mass in black holes as a Yu & Tremaine 2002). The total mass in black holes function of redshift can be determined by integrating over increases with time, until the cool gas supply stops due to the formation rate of haloes at the threshold mass from some feedback mechanism that prevents and/or destroys some initial redshift zi to the redshift of interest z, i.e., the production of molecular hydrogen. Here we assume z ρbh(z) = (dρ/dz)dz. We approximate the evolution in zi that cosmological reionization at zre can be responsible for the co-movingR number density of seed black holes, as preventing the further cooling and collapse of haloes of c 2003 RAS, MNRAS 000, 1–15 10 Koushiappas, Bullock & Dekel crit mass Mv for redshifts z c 2003 RAS, MNRAS 000, 1–15 Black holes from low-spin material 11 in the definition of the major merger makes only a small difference to the final slope of the relation between black hole and spheroid mass, because the merger history tends to converge to the final mass after the last major merger. Among the subhaloes that reach the center before the redshift of the last major merger, we determine if they have a black hole, and if so, we assume it merges with the central black hole over a short timescale. In principle, we could com- pute the black hole mass contained in each merged halo by following its full merger tree back in time, until some early > redshift z zre. However, since our aim is simply to investi- gate the ramifications∼ of our scenario, we have chosen only to follow the merger tree of the most massive progenitor, and estimate the formation times of each merged subunit in a statistical fashion. This choice saves a significant amount of computational time. Specifically, since seed black holes stopped forming at z = zre, we can estimate the total black hole mass present in a subhalo of mass Ms that merges into the main host at a redshift zmer by the following approximation. If zmer < zre, then the black-hole mass populating the merging subhalo is determined by integrating over the mass function of its progenitors at zre, Figure 7. Today’s black-hole mass versus host-spheroid mass. ∞ The filled symbols are based on data from Tremaine et al. (2002), Ms Mbh = Mbh(M,zre) where the bulge mass is derived from the B absolute magni- crit M ZMv (zre) tude assuming the constant mass-to-light ratio advocated by van der Marel (1991). The model predictions are based on merger f2[σ(M),zre σ(Ms),zmer] × 2 | growth only, ignoring accretion. Top: for a fixed halo mass to- dσ (M) 12 dM. (25) day, Mv = 3.2 × 10 M⊙ , using 50 random realizations of the × | dM | merger tree. The variations in merger history by themselves tend to spread the masses along the observed relation. Bottom: for Here, M is the mass of progenitors. The function 11 13.5 a range of present halo masses, Mv = 10 − 10 M⊙ , with f2[σ(M),zre σ(Ms),zmer] represents the fraction of mass in | logarithmic half order of magnitude steps. The slope of he pre- haloes of mass Ms at zmer that was in haloes of mass M dicted relation is in agreement with the observed slope. In cases at redshift zre (see equation 2.15 of Lacey & Cole (1993)). where the black-hole mass is calculated to be zero, we have placed The mass Mbh(M,zre) is the mass of the seed black hole the symbol at the bottom of the panel for clarity. The predicted residing in a halo of mass M at zre according to eq. (15). If black-hole masses could potentially increase by about an order of the merger occurs before zre, then the black-hole residing in magnitude due to accretion in an AGN phase. the merging subhalo is assumed to have a mass according to eq. (15), given the halo mass Ms at zmer. We only need to follow the merger history back to the time when the major dom merger history is varied from point to point. Though crit progenitor is as small as Mv (z), because when a progenitor not shown, the associated disc masses for these haloes range crit 10 11 is smaller than M , no seed black holes will form. from 10 10 M⊙ with no clear correlation between the v − The integral above gives only an average black hole mass disk and black hole masses. for an ensemble of host haloes of mass Ms. So the approx- We find that the black hole mass correlates with the imation is valid if Mbh M0, where M0 is the seed mass mass of the spheroid component, as observed, even for a ≫ associated with critical-mass haloes at zre. However, for low- fixed-mass host halo. This result arises because subhaloes mass subhaloes, the approximation can break down because that are massive enough to sink to the center of the parent it allows solutions with black hole masses smaller than the halo and contribute to a bulge, are generally massive enough 4 initial seed masses, Mbh < M0 ( 6 10 M⊙ , see Eq. 18). to host black holes as well. Haloes which experienced many ∼ × This is unphysical, because we assume that each black hole mergers with small subhaloes tend to have a small spheroids is built up by a series of mergers with seeds of mass M0. We and small black holes. This would naturally explain why attempt to remedy this shortcoming in cases by randomly black hole mass is observed to correlate with spheroid mass re-assigning the value of Mbh in these cases. Therefore if and not disc mass(Gebhardt et al. 2001), since disc mass Mbh < M0 we choose a random uniform deviate r, and set does not correlate well with the merger history in our pic- Mbh = 0 if r > Mbh/M0 and Mbh = M0 otherwise. ture. Figure 7 shows the resulting black-hole mass as a func- The bottom panel of Fig. 7 shows several ensemble re- tion of the host spheroid mass from the merger tree realiza- alizations for a series of host halo masses with z = 0 values 11 13.5 tions described above (open symbols), in comparison with of 10 10 M⊙ . The computed relation follows closely − the observed data (solid points with errors, see caption). We Mbh Msp, and is qualitatively similar to the observed re- ∝ have adopted model parameters of t = 10 Myr and zre = 15. lation at z = 0. The normalization (set primarily in this The model predictions in the top panel refers to a given fixed case by setting t = 10Myr) allows for an additional growth 12 halo mass today, M = 3.2 10 M⊙ , where only the ran- by a few e-folds during an AGN accretion phase, as desired. × c 2003 RAS, MNRAS 000, 1–15 12 Koushiappas, Bullock & Dekel In addition, for the lowest-mass spheroids, we see in some to the way we assign mass M to a black hole which results cases the absence of any black hole mass (points along the from the merger of two black holes, of masses M1 and M2. In bottom edge represent Mbh = 0). This comes about because our fiducial calculation we have assumed that the black hole low mass host haloes are less likely to have had progeni- masses sum up linearly, M = M1 + M2. On the other hand, crit tors more massive than Mv at z = zre. Black holes cannot if entropy is conserved, the mass of the merger product is 4 2 2 be smaller than the characteristic seed mass 6 10 M⊙ M = M + M (Hawking 1971a,b). This should serve as ∼ × 1 2 while there is no fixed mass threshold for a progenitor to a lowerp bound to the possible mass of the merger product, contain gas that will eventually end up in a spheroid. This and should yield the flattest-possible Mbh Msp relation. − gives rise to a slightly steeper slope in the bottom left of the Consider a black hole in the center of the halo formed from 5 relation. The apparent increase in the number of low mass Nm mergers of seed black holes of mass Mbh 10 M⊙ . If ∼ spheroids with no black holes also suggests the presence of a entropy is conserved, and Nm 1 then the black hole mass ≫ 1/2 1/2 lower mass limit on spheroids hosting black holes, of roughly would be Mbh = Mseed(1 + Nm) Nm . In our model, 8 ∝ Msp 10 M⊙ . This is a potentially verifiable prediction of spheroids grow via a series of mergers with the gas from ∼ the model. nearly equal-mass haloes, roughly as Msp f0MsNm Nm, 1/2 ≃ ∝ It is of interest to investigate the robustness of the de- so we expect Mbh Msp in this limit. When we follow this ∝ rived relation between Mbh and Msp to specific assumptions recipe for mass adding using our full merger-tree method, of the model presented here. For example, the result is prac- we indeed obtain a result very close to our expectation: 0.55 tically independent of the finite duration of a merger due Mbh Msp . Thus, in this limit, the derived relation would ∝ to dynamical friction. Major mergers are quick and have be somewhat flatter than observed. In order to comply with an immediate effect on the growth rate of the spheroid as the observed relation, one would have to appeal to a sce- well as the black hole, while minor mergers take a long time nario where black-hole growth via accretion is more efficient to complete and therefore have a delayed effect that may in high-mass spheroids (e.g., Volonteri, Haardt & Madau not materialize in a Hubble time (Zentner & Bullock 2003; 2003). Taylor & Babul 2001, 2003). When we repeat the calcula- With the above tests in mind, we conclude that the de- tion under the assumption that all mergers are instanta- rived linear relation between black hole and spheroid mass neous, ignoring the finite duration of the spiral-in process, depends primarily on the fact that the seeds were massive, we find that the predicted relation between black holes and that they formed at high redshift, and that their masses add spheroids remains almost unchanged, with only a slight shift roughly linearly when they merge. In this case the build-up of typically less than 10% towards larger spheroid masses for of black holes and spheroids is occurring in concert through the same black hole mass. This small shift is to be expected mergers, thus setting the linear relationship. A similar linear since the less massive subhaloes that now contribute to the relation would hold for the final black-hole masses provided spheroid mass tend to be below the critical mass for seed that it has not been changed by the subsequent growth ex- formation and thus do not add to the black hole mass. perienced by black holes via accretion. If the accretion is We find the predicted slope of the relationship to be proportional to the mass of the seed, as would be the case crit for accretion at the Eddington rate, then we expect the slope insensitive to the actual value of Mv (as long as Mv crit ≫ of the relation to persist. Of course, the effect of accretion Mv ). The normalization of the relation, namely the black- hole mass for a given spheroid mass, is roughly inversely depends on what fraction of the black hole mass was ac- crit creted. If the increase in mass is only by a factor of a few, proportional to Mv . This is because for a lower value of crit then one could assume that the role of mergers in setting Mv more haloes are able to form a seed and contribute to the merged black-hole mass without affecting the spheroid the observed relationship is important. On the other hand, mass. if the growth by accretion is by more than an order of mag- nitude, then the accretion process would have the dominant We also find that the slope is insensitive to the value role in determining the slope of the relation. According to of zre (as long as zre > 6), while the normalization does de- our model, the normalization of the observed relationship is pend on it. For a lower zre, seed formation takes place in a determined additively by two free parameters: the viscous larger fraction of the haloes that eventually merge to form timescale and the growth via accretion. When adopting a spheroids. By lowering zre from 15 to 6 we obtain an or- physically motivated value for the viscous timescale, we find der of magnitude increase in black-hole mass. A comparable that growth via accretion is limited to be by an order of mag- decrease in black-hole mass is obtained when increasing zre nitude or less, thus suggesting that mergers do play an im- from 15 to 20. portant role in setting the relationship. We note that in mod- However, the slope of the relation is found to be mildly els where the seeds are of smaller masses, 100 M⊙ (e.g., ∼ sensitive to the original scaling between the seed black hole Volonteri, Haardt & Madau 2003), the growth by mergers mass and its host halo mass. If we replace the predicted cor- is negligible and the accretion process has to be tuned with relation of our model (Mbh Mv) with an ad hoc seed ∝ a specific feedback scenario in order to obtain the correct assignment to critical-mass haloes at high redshift, e.g., relation between black holes and spheroids. crit 3 Mbh (Mv ) , we find that the linear Mbh Msp relation Although we focused here on black hole and spheroid ∝ 3/2 ∝ steepens to roughly Mbh Msp . This is because the ra- ∝ growth, we do not expect all the seed black holes accreted tio Mbh/Mv is now monotonically increasing with Mv. This by a dark matter halo to be incorporated in the spheroid. mild sensitivity means that the predicted correlation of our A number of black holes could be orbiting in haloes of physical model for seeds is important, but it does not have big galaxies. They could have arisen from subhaloes that to be exactly Mbh Mv. ∝ were disrupted after entering the host halo. Such a scenario Finally, we point out that our result is quite sensitive has been investigated by Islam et al. (2003a) and the possi- c 2003 RAS, MNRAS 000, 1–15 Black holes from low-spin material 13 ble observational consequences are explored in Islam et al. for an order of magnitude growth in black hole masses via (2003b,c). quasar powering accretion. We find that a viscous timescale of t 5 10 Myr provides quite reasonable results. ∼ − We explored the development of a correlation between black-hole mass and spheroid mass in today’s galaxies by 7 CONCLUSION tracking our early forming massive seed population through We presented a physical model for the production of massive a halo merger tree. Under the assumption that spheroids seed black holes at high redshifts, with a characteristic mass form via major mergers, we showed that the merger process 5 10 M⊙ . We argued that this model provides a useful leads naturally to a linear relation between black-hole mass ∼ starting point for explaining the observed relation between and spheroid mass. This result is mainly due to the forma- the masses of today’s black holes and the spheroidal compo- tion of massive seeds only in rare, high-peak haloes at high nents of galaxies. Massive seeds may also help explain the redshifts, and is not too sensitive to the details of our seed existence of the supermassive black holes associated with formation model. Starting with 1 100M⊙ seeds would ∼ − luminous AGN at z> 6. not lead to a linear correlation via mergers alone; additional Our model for∼ seed formation is based on the general feedback associated with AGN accretion is needed. The seed idea that the mass in every early-collapsing halo is expected black holes in our model have masses that are proportional to have a distribution of specific angular momentum, with at to the virial masses of their host haloes at formation. We least some fraction of its gas having a much smaller specific demonstrated that this linear relation is preserved if black angular momentum than is characterized by the global spin holes merge in concert with the hierarchical growth of haloes parameter of the halo. After the low-spin gas cools, it can fall and spheroids. into the central region and form a dense disc, where certain One potentially interesting ramification of our scenario viscous processes become effective and may potentially lead is that a population of massive black holes at high redshift to black hole formation. might serve to power small quasars, fueled perhaps by low- In our exploration of this idea, we assumed that the angular-momentum gas liberated by merger events that oc- specific angular momentum of the gas is similar to that cur after the seeds are in place. Madau et al. (2003) dis- measured for the dark-matter in N-body simulations. In this cussed how a population of “mini-quasars” like this might case, if angular momentum is conserved during the gas infall, be responsible for reionizing the universe at zre 15. How- ∼ the central gas discs become self-gravitating and Toomre un- ever, in their model, the mini-quasar power-sources were stable in all the haloes above a critical mass. This critical intermediate-mass 100M⊙ black holes. Under similar as- 7 ∼ mass has a median value of 7 10 M⊙ at z 15, and it sumptions, the massive seed black holes of our model would ∼ × ∼ is expected to be scattered log-normally with a ln standard produce even more ionizing photons per site, and perhaps deviation of 0.9. The sites of seed production are thus lead more naturally to an early reionization. In our cur- ∼ much more massive than the typical collapsing mass at that rent scenario the reionization epoch was an input param- time, and are therefore associated with high-sigma peaks eter detached from the black-hole formation process, but, in the fluctuation distribution. Gravitational instabilities in in principle, the accretion onto the seeds themselves could the disc drive an effective viscosity that redistributes angu- be responsible for the reionization that eventually inhibits lar momentum. As a result, material from the low angular further black-hole formation. momentum tail of the distribution loses its angular momen- Our quantitative predictions should not be interpreted tum on a viscous timescale and contribute to the formation too literally. For example, our fiducial model assumes that of a black hole. We associate this viscous timescale with the the initial angular-momentum distribution of the gas mir- timescale for disruption and/or heating of the disc due to rors that of a typical CDM halo, and that it is preserved substantial star formation. We expect this timescale to be during disk formation. There is no evidence in local galax- on the order of 1 30 Myr, but its exact value is not well ies for discs with inner densities as high as invoked by our − determined. model for proto-galaxies. This might be explained by pro- We found that the typical seed black holes, which cesses that have altered the gas angular-momentum distri- form in critical mass haloes, have a characteristic mass of bution preferentially at late epochs, such as feedback effects 5 10 M⊙ , quite insensitive to the epoch of formation. This (e.g. Maller & Dekel 2002), but one should clearly assign ∼ predicts a lower bound to possible masses of massive black a substantial degree of uncertainty to our model assump- holes, which can be confronted with observations today. tion. In order to address this uncertainty, at least qualita- This scenario should operate continuously as additional tively, we explored the effect of varying the initial angular haloes grow to above the critical mass and create new poten- momentum distribution. We found that if the low angular tial sites for black-hole formation. The process is expected momentum tail is as minor as in an exponential disc of a to stop, however, when the Universe becomes sufficiently constant-density core (ξ = 2), the resulting seed black holes reionized that H2 molecules are destroyed and cannot pro- are too small to be of relevance. An intermediate case, with vide efficient cooling. We therefore expect seed black holes ξ = 1.3, can produce sensible black-hole seeds, compara- to form before zre 13 21. ble to our fiducial case (ξ = 1), provided that the viscosity ∼ − We showed that the comoving density of mass in seed timescale is slightly longer, 20 6 t 6 30 Myr. For ξ = 1.3, 8 black holes grows gradually with time, in association with the critical halo mass becomes 10 M⊙ at z = 15, and the ∼ the birthrate of critical mass haloes. This density becomes inner 10% of the disc mass manages to lose its angular ∼ comparable to the observed mass density of supermassive momentum by viscosity and produce a black hole of mass 6 black holes by z 15, near the redshift of reionization. 10 M⊙ . This seed mass is larger than in the fiducial case, ∼ ∼ Depending on the adopted viscous timescale, there is room but the seeds in this case are rarer events (see figure 3.1). c 2003 RAS, MNRAS 000, 1–15 14 Koushiappas, Bullock & Dekel Therefore, in order to recover a similar total mass density Klypin A. A., Porciani C., Primack J. R., 2001a, ApJ, in black holes, the viscosity must operate for a longer du- 555, 240 (B01) ration. The actual low angular-momentum tail of the gas Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. 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