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PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 IS THERE A MAXIMUM MASS FOR BLACK HOLES IN GALACTIC NUCLEI? Kohei Inayoshi and Zoltań Haiman DepartmentofAstronomy,ColumbiaUniversity,550West120thStreet, NewYork,NY10027, USA ABSTRACT The largest observed supermassive black holes (SMBHs) have a mass of MBH 1010 M⊙, nearly ≃ 6 independent of redshift, from the local (z 0) to the early (z > 6) Universe. We suggest that the ≃ 1 growthofSMBHsaboveafew 1010 M⊙ ispreventedbysmall-scaleaccretionphysics,independentof 0 thepropertiesoftheirhostgala×xiesorofcosmology. GrowingmoremassiveBHsrequiresagassupply 2 rate from galactic scales onto a nuclear region as high as > 103 M⊙ yr−1. At such a high accretion n rate, most of the gas converts to stars at large radii ( 1∼0 100 pc), well before reaching the BH. ∼ − u We adopt a simple model (Thompson et al. 2005) for a star-formingaccretion disk, and find that the J accretionrateinthesub-pcnuclearregionisreducedtothesmallervalueofatmostafew M⊙ yr−1. 3 This prevents SMBHs from growing above 1011 M⊙ in the age of the Universe. Furthe×rmore, once ≃ 1 aSMBHreachesasufficiently highmass,thisratefalls belowthe criticalvalueatwhichthe accretion flow becomes advection dominated. Once this transition occurs, BH feeding can be suppressed by ] strong outflows and jets from hot gas near the BH. We find that the maximum SMBH mass, given A by this transition, is between MBH,max (1 6) 1010 M⊙, depending primarily on the efficiency of ≃ − × G angular momentum transfer inside the galactic disk, and not on other properties of the host galaxy. . Subject headings: galaxies: active — quasars: supermassive black holes — black hole physics h p - 1. INTRODUCTION fixed comoving volume are determined by the physics o of cooling and galactic feedback processes, but in gen- r MostmassivegalaxiesinthelocalUniverseareinferred t eral, they should increase as galaxies are assembled over s to host supermassive black holes (SMBHs) with masses time. However, local surveys probe smaller comoving [a of 105−1010 M⊙ at their centers. The correlations ob- volumes than high-z surveys, and can miss the rarest, served between the masses (M ) of the SMBHs and 2 the velocity dispersion (σ) andBoHther bulk properties of mcidoesnttamllaysslievaedgtaolaaxiems.axiImnupmringcailpalxey, tmhaissscothualdt sctoaiyns- v their host galaxies suggest that they co-evolved during roughly constant with redshift. In practice, this ex- 1 theircosmichistory(e.g.Kormendy & Ho2013,andref- planation requires the M σ correlation to evolve 1 erencestherein). ThecorrelationscouldbecausedbyBH BH − (Netzer 2003; Natarajan& Treister 2009), and also the 6 feedback,whichcansuppressstarformationandgassup- quasar luminosity function to steepen at the bright end 2 plyongalacticscales(e.g.Silk & Rees1998;Fabian1999; (Natarajan & Treister 2009). 0 King2003;Murray et al.2005). Theseobservationshave Here we pursue a possible alternative interpretation. 1. also revealed a maximum SMBH mass of ∼ 1010 M⊙, Namely, the observations suggest that SMBHs stopped 0 in the largest elliptical galaxies (e.g. McConnell et al. growing at near-Eddington short after z 5, once they 6 2011). reached 1010 M⊙ (Trakhtenbrot et al.≃2011). On the 1 Observations of distant quasars, with redshift as high otherhan∼d, galaxiesdonotlikewise stoptheir growthat : as z 7, have found that the SMBH masses fu- iv eling t∼he brightest quasars are similarly ∼ 1010 M⊙ tthoihsaevaerlaysespemocbhl:edthaetmzost1ma2ss(iev.eg.elBlieprtnicaarldsiaerteable.l2ie0v0e3d; X (e.g. Fan et al. 2001; Willott et al. 2010; Mortlock et al. ≃ − Thomas et al. 2005). This motivates us to hypothesize 2011; Wu et al. 2015). Intriguingly, this apparent r that there is a limiting mass, determined by small-scale a maximum mass is nearly independent of redshift (e.g. physicalprocesses,independent ofgalaxyevolution,star Netzer 2003; Vestergaard 2004; Marconi et al. 2004; formation history, or background cosmology. In this pa- Trakhtenbrot & Netzer 2012). Since the e-folding time per,we discusssucha“microphysical”scenario,limiting for BH mass growth (at the fiducial Eddington-limited the growthofSMBHs to a few 1010 M⊙: diskswith the accretion rate, with a 10% radiative efficiency) is 40 × ∼ high accretion rates required to produce more massive Myr, much shorter than the cosmic age. Given suffi- SMBHs fragment into stars. The small residual fraction cientfuel,SMBHscouldthuscontinuetogrow,andreach ofgasthattricklestotheinnerregionisunabletoforma mnoatssseesewSeMllBabHosvesi∼gn1ifi0c1a0nMtly⊙abbyozve≃∼0.1H0o10weMve⊙r,iwnetdhoe sotnatnodtahredBgeHo,maentdricisalilnysttheaindaecxcpreeltlieodnidniswkianndds toorajectcsr.e1te local Universe (or indeed at intermediate redshift). The rest of this paper is organized as follows. In 2, Naively, the near-constant value of the maximum § we discuss the model for star-forming accretion disks, SMBH mass with redshift is therefore surprising. It and the implied maximum SMBH mass. In 3, we show is tempting to attribute this observation to the same § that our results can explain the observed M L galactic-scale feedback that ties SMBH masses to their BH − bol host galaxies. The maximum masses of galaxies in a 1 As this paper was being completed, we became aware of a recent preprint proposing a similar idea (King 2016). We discuss KI:[email protected]–SimonsSocietyofFellows thesimilaritiesanddifferencesbetweenthetwoworksin§4below. 2 Kohei Inayoshi & Zoltán Haiman relation for most AGN/QSOs, as well as the maximum Eq. (4) is the radiation pressure on dust grains in the SMBH mass. In 4, we discuss possible caveats, and opticallythicklimit(τ 1), andthe secondtermrepre- § ≫ in 5 we summarize our conclusions. Throughout this sentsstellarUVradiationpressureandturbulentsupport pap§er,wedefinetheEddingtonaccretionrateasM˙ by supernovae in optically thin limit (τ 1), which is 10 LEdd/c2 =230 M⊙ yr−1(MBH/1010 M⊙). Edd ≡ characterizedby the non-dimensionalvalu≪e of ξ. Energy balance between cooling and heating is given by 2. LIMITONSMBHGROWTHVIAANACCRETIONDISK 1 3 2.1. Star-forming accretion disks σSBTe4ff = 2ǫΣ˙∗c2+ 8πM˙ Ω2, (5) We here consider a model for a star-forming accretion disk around a SMBH with MBH 108−11 M⊙ where the effective temperature Teff is given by ∼ based on Thompson et al. (2005, hereafter TQM05). In 3 2 4 this model, the gas fueling rate from galactic scales T4 = T4 τ + + . (6) 4 eff(cid:18) 3τ 3(cid:19) (& 100 pc) to the nuclear region (. 1 pc) is estimated self-consistently, including gas depletion due to star- In this disk, a fraction of gas forms stars at a rate of formation. Because of star-formation, the central BH is Σ˙∗(r) and the gas accretionrate decreasesinward,given fed at a rate of <M˙Edd and thus the BH growth is lim- by ited. ThisisconsistentwithmostobservedAGNs/QSOs, r whose Eddington ratios are inferred to be modest (e.g. M˙ (r)=M˙out 2πrΣ˙∗dr. (7) −Z L/LEdd 0.16 for 0.35 < z < 2.25 and L/LEdd 0.43 Rout ∼ ∼ for z > 4; Shen et al. 2011; De Rosa et al. 2011). A Eqs. (1)–(7)determinetheradialprofilesofallphysical few exceptionally bright QSOs at higher redshift are be- quantities,oncethe fiveparametersM , σ, m, ǫ, ξ and BH lievedtoaccretemorerapidly,atorevensomewhatabove theouterboundaryconditionsoftheaccretionrateM˙ M˙ . In the high-rate case, fragmentation of a nu- out Edd at the radius R are chosen. As our fiducial model, ∼ out cleardisksuppressestheBHfeeding(seediscussionin 4 we set ǫ = 10−3, appropriate for a Salpeter initial mass § and King 2016). function(IMF)with1 100M⊙,andξ =1,appropriate The TQM05 model assumes that radiation pressure − when turbulent support by supernovae is negligible, as from stars forming in the disk supports the gas against inahigh-M˙ (orρ)disk. The velocitydispersionissetto gravity in the vertical direction, and keeps the disk σ = 400 km s−1, motivated by the empirical correlation marginally stable; the Toomre parameter is then between BH mass and σ of its host galaxy for M BH Q csΩ 1, (1) 1010 M⊙ (e.g. Tremaine et al. 2002; Gu¨ltekin et a∼l. ≃ πGΣ ≃ 2009). Note that the dependence of our results on the g choice of σ is very weak because the stellar gravita- wherecs isthesoundspeed,Σg isthegassurfacedensity, tional potential is subdominant at r . GMBH/(2σ2) and Ω is the orbital frequency, given by 140pc(MBH/1010M⊙)(σ/400km s−1)−2,wheretheB≃H GM 2σ2 1/2 feeding rate is determined. Ω= BH + . (2) We consider a very high accretion rate of M˙ = (cid:18) r3 r2 (cid:19) 103 M⊙ yr−1 at the boundary, in order tooutgive Hereσ isthevelocitydispersioncharacterizingthegravi- a conservative estimate on the maximum BH mass. tationalpotentialongalacticscales. Fromthecontinuity From cosmological simulations (e.g. Genel et al. 2009; equation, the surface density is given by Fakhouri et al. 2010), the maximum gas accretion rate onto a dark matter halo is estimated as . Σ = M˙ = M˙ (3) 103M⊙ yr−1(Mhalo/1012M⊙)1.1[(1+z)/7]5/2,whereMh g 2πrv 2πrmc is the halo mass. This could be exceeded only for brief r s periodsduringmajormergereventsintheearlyUniverse where M˙ is the gas accretion rate through a radius of (Mayer et al. 2015), and for Mh & 1012 M⊙, gas heat- r, vr is the radial velocity and m (=vr/cs) is the radial ing by a virial shock prevents cold gas supply because Mach number. Note that the viscosity in this model is of inefficient radiative cooling (Birnboim & Dekel 2003; specified by assuming a constantvalue of m (see below), Dekel & Birnboim 2006). insteadoftheα-prescription(Shakura & Sunyaev1973). The nature of the angular momentum transfer, allow- The disk is supported vertically by both thermal gas ing gas to flow from large scales to the inner sub-pc re- pressure (pgas = ρkBT/mp) and radiation pressure due gions remains uncertain. Following TQM05, we assume tostarsinthedisk,whereρ=Σg/(2h)isthegasdensity, thetransferisprovidedbyglobaldensitywaves,andalso h = cs/Ω is the pressure scale height and T is the gas that the radial Mach number m is constant (indepen- temperature. The radiation pressure is given by dent of radius). Since our results depend on the choice τ of m, we here consider three cases of m = 0.05, 0.1 and prad =ǫΣ˙∗c +ξ , (4) 0.2. Thesevaluesaremotivatedby analyticalarguments 2 (cid:16) (cid:17) yielding the limit m . 0.2 (Goodman 2003). Note that whereτ =κΣg/2istheopticaldepth,κisthedustopac- intermsofthestandardα-prescriptionasamodelforthe ity (Semenov et al. 2003), Σ˙∗ is the star-formation rate diskviscosity(Shakura & Sunyaev1973),theviscouspa- per unit disk surface area and ǫ is the matter-radiation rameterisrelatedtotheMachnumberasα=m(2r/3h). conversion efficiency, which depends on the mass func- Fig. 1 shows radial profiles of the gas accretion rate tion of stars. The first term on the right-hand side of (solid) and star formation rate (dashed) for three differ- A maximum mass of SMBHs 3 (a) 30 1000 MBH = 10 9 Msun 10 MB0H.3 yr) 100 yr) /n /n u u Ms Ms n rate ( 10 6×1010 ・M (BH 1 etio 1011 ccr 1 a 0.2 (b) MB-0H.7 m=0.2 10-1 m=0.1 0.1 1 10 100 m=0.05 radius (pc) d d (daFsihge.d1).—in aGsatsar-afcocrrmetiinognarcactreeti(osnoliddi)ska.nTdhsetacrurfvoersmcaotriroensproantde ・/ MH E 10-2 toSMBHmassesofMBH=109(red),6×1010(blue),and1011M⊙ ・MB 10-3 (black). Theaccretionrateattheouterboundary(Rout=200pc) is set to M˙out = 103 M⊙ yr−1. In each case, the accretion rate intheinnerregion(.1pc)approaches aconstantvalue,whichis 10-4 muchsmallerthanM˙out becauseofstar-formationatlargerradii. 109 1010 1011 BH mass (Msun) entBHmassesandform=0.1. ForthelowestBHmass Fig.2.— Gas accretion rateinto thenuclear region(<1pc) as (MBH = 109 M⊙; in red), star formation is inefficient atrafunnsfcetrioenffiocfieSnMciBesH; mma=ss,0f.o0r5t(hbrleuee)d,iff0e.r1en(rteadn)gaunladr0m.2om(belnatcukm). at r & 10 pc, and the accretion rate remains close to The other parameters are the same as in Fig. 1. The horizontal its value at the outer boundary. At r < 10 pc, vigorous lineinthebottom panel marksM˙BH/M˙Edd =10−2, belowwhich star-formationdepletesmostofthegas,∼andtheaccretion a thin disk changes to an ADAF (Narayan&McClintock 2008). rate rapidly decreases inward. In this domain, the disk TheverticallinesmarkthecriticalSMBHmassMtr,abovewhich theBHfeedingissuppressedbystrongoutflowsandjetsaccording temperature reaches the dust sublimation temperature toYuan&Narayan(2014). (T 103 K), above which the dust opacity drops dust,sub ≃ rapidly. Since Σ˙∗ Σg/κ in the optically thick limit ∝ (see Eq. A2 in the Appendix), a higher star formation rate from larger scales (&1 pc). Fig. 2 shows the accre- rate is required to maintain the marginally-stable disk tionrate into the nuclearregionfor three different Mach structure with Q 1 when the opacity decreases. Note ≃ numbersm=0.05(blue), 0.1(red)andm=0.2(black). that since dust is composed of multiple species and each Up to a turn-over at a critical M , the accretion rates has a different sublimation temperature (Semenov et al. BH in panels (a) and (b) are fit by the single power-laws, 2003), the jagged radial profile for the star formation oraptaeciattyra∼t t1h0e−co1r0r0esppconisdcinagusseudbbliymsamtiaolnl dtermoppseoraftduurset. M˙BH ≃4.2 m0.1MB0.H3,10 M⊙ yr−1, (8) Within r < 0.5 pc, the disk becomes stable, star for- or mation cea∼ses, and the gas accretion rate approaches a M˙BH 1.8 10−2 m M−0.7 , (9) constant value. This accretion rate in the nuclear re- M˙ ≃ × 0.1 BH,10 gion does not depend on the value of M˙ , as long as Edd out AM˙poputen>dixM)˙,craintd≃th2u8s0hMar⊙dlyyro−n1 t(hseeemEoqd.el Apa1r2aminettehres Twhheesreesmca0l.i1ng≡remla/ti0o.n1sacnadnbMeBeHx,p10lai≡nedMbByHa/n(1a0n1a0lyMti⊙ca).l of the star-forming disk except the radial Mach number argument in the Appendix. Assuming a constant radia- m (see 2.2). For higher SMBH masses, the accretion tiveefficiency η,wecanintegrateEq. (8)overthe ageof rate in t§he nuclear region gradually increases. However, the Universe and obtain M 7.4 m10/7(1 η)10/7. for MBH & 6 1010 M⊙, the accretion rate at < 1 This suggests that SMBHBsHw,1o0u≃ld not 0g.1row −above pc decreases ag×ain, because gas is depleted more effi- 1011 M⊙ within the finite age of the Universe. ∼ ciently due to star formation at larger radii (r 100 The above argument yields a maximum BH mass, ∼ pc), where evaporationof volatile organicsdecreases the which comes close to the largest observed masses. Here, dustopacitymoderately(T &400K)andthestarforma- we discuss further constraints on the maximum value, tion rate increases in the optically thick star-burst disk considering properties of accretion flows in the vicinity (Σ˙∗ ∝Σg/κ). o1f09thMe⊙BHis.MI˙nBpHa/nMe˙lE(dbd), t0h.e1.noFromratlhizisedvarlautee,faorstManBdHar≃d 2.2. Accretion in the sub-pc nuclear region geometrically thin, opti∼cally thick nuclear disk can form We next consider the stable nuclear (sub-pc) regionof (Shakura & Sunyaev 1973). Through the disk, the BH the accretion disk, which is embedded by the galactic grows via accretion at a rate given by Eq. (8). On the star-forming disk. The properties of this disk are deter- otherhand,the normalizedratedecreaseswithBHmass mined primarily by the BH mass and the gas accretion and reaches the critical value of M˙ /M˙ . 10−2, BH Edd 4 Kohei Inayoshi & Zoltán Haiman 1049 η=0.42, m=0.2 η=0.42, m=0.1 ) s / 1048 η=0.10, m=0.1 g r η=0.10, m=0.05 e ( y t si 1047 o n mi u c l 1046 ri t e L=LEdd m o ol 1045 b 0.1 0.01 1044 107 108 109 1010 1011 BH mass (Msun) Fig.3.—ComparisonofthepredictedLbol−MBHrelationwithobservationaldata. ThedataaretakenfromtheAGN/QSOssamplesin Shenetal.(2011)for0<z<5(graydots)andfromseveralotherstudiesforz>5(magenta;Willottetal.2010,DeRosaetal.2011and green;Mortlocketal.2011,Wuetal.2015). Theorangelinesshowsisodensitycontoursofthesesamples. Thefourthicklinescorrespond to the Lbol−MBH relation with different radiative efficiencies 0.1≤η ≤0.42 and Mach numbers 0.05≤m≤0.2. The diagonal dotted linesindicateconstantEddingtonratios(L/LEdd),withvaluesaslabeledinthebottom leftofthefigure. at which the nuclear disk can not remain thin, because 2004). Moreover, numerical simulations of ADAFs of inefficient radiative cooling (Narayan& McClintock suggest that the gas accretion rate decreases inward 2008). The inner disk would then likely be replaced by within the transition radius as M˙ (r/R )s BH tr aradiatively-inefficientADAF (advection-dominatedac- (Igumenshchev & Abramowicz 1999; Stone∝et al. 1999; cretion flow; Narayan& Yi 1994, 1995).2 Adopting the Hawley & Balbus 2002; McKinney & Gammie 2004, see critical rate to be M˙ /M˙ = 10−2, we find that the alsoBlandford & Begelman1999). The power-lawindex BH Edd transition occurs at MBH & Mtr = 2.3×1010m100.1/7 M⊙ iisndeesptiemndaetendt oafs t0h.4e s.tresng.th0o.6f vatisc2os.ityr/aRndSchma.gn1e0ti4c, for0.05.m.0.2. WenotethatthetransitionBHmass field (Yuan et al. 2012). For a conservative estimate, M does notdependonthemodelparametersofthestar- fortmr ing disk, except on the Mach number m, while the we set s = 0.3 and Rtr = 100 RSch for M˙BH/M˙Edd behavioroftheaccretionrateatM >M dependson 10−2. This reduces the BH feeding rate by a factor o≤f BH tr the choices of the other model parameters. (RSch/Rtr)0.3 0.25 from the original accretion rate at ≃ Once the transition to an ADAF occurs, the accre- R (Eq. 8). As a result, the BH growth time is roughly tr tion flow through the disk becomes hot because of inef- given by 16 [M˙ /(10−2M˙ )]−1 Gyr, and we con- BH Edd ficient cooling. The hot gas near the BH would launch clude tha∼t once an SMBH reaches the critical mass of strong outflows and jets, suppressing the feeding of the MBH,max Mtr (0.9 6.2) 1010 M⊙, it cannot gain BH (e.g. Blandford & Begelman 1999). The location of significant≃mass≃within−the Hu×bble time. the transition radius R , inside which a thin disk turns into a hot ADAF, has btreen discussed by severalauthors 3. COMPARISONTOOBSERVATIONS (e.g. Yuan & Narayan2014, and referencestherein). Al- In the TQM05 disk model, the BH feeding rate is a though this location is uncertain, the theoretical mod- function of SMBH mass (Eq. 8). We can compare the (eqls su>gge0s)t Rfotrr/RMS˙ch &/M3˙00−1.03 [M1˙0B−H2/.(10−T2hMi˙sEdvda)]l−ueq (cworhreersepoLnbdoilngisptrheedibctoiloonmseftorirctlhuemLinbooslit−y)M, wBHithreolabtsieorn- BH Edd vational data. For this comparison, we use AGN/QSO is consistent with results obtained from fitting the samples from Shen et al. (2011)3 for 0 < z < 5 spectra of observed BH accretion systems (R tr 100 R for M˙ /M˙ 10−2; Yuan & Naraya∼n and from Willott et al. (2010), De Rosa et al. (2011), Sch BH Edd ∼ Mortlock et al. (2011) and Wu et al. (2015) for z >5. For simplicity, we estimate the bolometric luminosity 2 Lietal. (2013) discussed a transition to a rotating accretion ofthe nuclearBH disk assuminga constantradiationef- flow. ForM˙/M˙Edd.10−1.5,therotatingflowresultsinasolution withanevenloweraccretionrateandconicalwindoutflows. 3 http://das.sdss.org/va/qso properties dr7/dr7.htm A maximum mass of SMBHs 5 ficiency (L =ηM˙ c2) as long as the disk is thin, i.e. M˙ /M˙ bol> 10−2B.H The radiative efficiency depends 105 BH Edd on the BH spin. Although we do not have any direct measurementsoftheSMBHspinevolution,applyingthe 104 Paczynski-Soltan(Soltan1982)argumenttothedifferen- pgas > prad tial quasar luminosity function, Yu & Tremaine (2002) have inferred typical radiative efficiencies of ǫ > 0.3 )Sch 103 prad > pgas R for the brightest quasars with the most massive SM∼BHs s ( (TMraBkHht>e∼nb1r0o9tM(⊙20)1,4c)oninsdisetpenentdweintthlyrasupgidgessptiend.tRhaetcehnitglhy-, Radiu 102 Rsg redshift SMBHs with 1010 M⊙ have rapid spin with Rsg-Rgap ∼ 10 a 1, based on the band luminosities in accretion disk ≃ models (e.g. Davis & Laor 2011). Semi-analytical mod- elsandnumericalsimulationshavepredictedthatahigh 1 value of the BH spin is maintained (a 1) for high- 106 107 108 109 1010 1011 1012 ≃ z SMBHs growing via sustained accretion of cold gas Mass (Msun) (Volonteri et al.2007;Dubois et al.2014). Here,wecon- sidertwooppositelimitsfortheefficiency;η =0.1thatis Fig.4.— Thefragmentation radius Rsg of astandard Shakura- oftenusedandη =0.42foranextremeKerrBH(a=1). Sunyaev disk in units of RSch (solid red curve) for α = 0.1 and In Fig. 3, we show the L M relation predicted M˙BH/M˙Edd=1. Thefilledcirclemarksthelocationwherepgas= bol− BH prad(withradiationpressuredominatingathigherBHmass). The for four different combinations of BH spin and Mach dashedcurveshowsthevalueof(Rsg−Rgap)forfH=1.5,where number. As explained in 2.2 above, once the BH mass Rgap istheradialsizeoftheannulargapclearedbyaccretiononto reaxtceeefdasllsthbeelcorwiticMa˙l va/lMu˙e §Mtr<, t1h0e−n2o,ramnadlitzhede aBcHcrefeteiodn- acircclulems)p,ainstcairbclueladrisokrbcaitnnatotResxgi.stA(ti.Me.BRHsg=−4.R5g×ap1≈010RMISC⊙O()oapnend BH Edd the BH feeding would be suppressed. We set the ISCO radius to ing drops. Within the range of model parameters shown RISCO=3RSch (horizontal solidline). in the figure, the maximum BH mass is in good agree- mentwiththeobservationaldata(MBH <few 109M⊙), × but favors high values of a and m. Th∼e bolometric lu- require a high radial Mach number (m & 1). However, minosities are predicted to be between 1045 1047 such a large m is unlikely to be realized by global spiral erg s−1, in good agreement with the va∼lues fou−nd in waves in a marginally stable disk Q 1. Instead, this ≃ the AGN/QSO samples. Moreover,the slope we predict rapidinflowcouldbetriggeredbyamajorgalaxymerger, (dlnL /dlnM = 0.3) agrees well with the upper and sustained for a few dynamical timescales of a few bol BH envelope of these samples. However, the model would 107 yr (Hopkins & Quataert2010, 2011). After a brief × requireahighermtoreachthe luminositiesofthe rarest burst phase, the BH feeding rate would decrease to the bright objects ( 1% of all sources) with & 2 1047 erg value given by Eq. (8). As long as these major-merger s−1 (e.g. J01001∼3.02+280225.8;Wu et al. 201×5). triggedinflows are sufficiently rare and brief, the SMBH Webrieflynoteuncertaintiesoftheinferredbolometric masses will remain limited by the physics of the star- luminosities. The bolometricluminosityistypicallyesti- forming disks, as discussed in 2.2. mated from the luminosity measured in a narrow wave- We next argue that BH gro§wth at MBH & 1010 M⊙ lengthrange,usingaconstantconversionfactorbasedon wouldalsobesuppressedbyfragmentationofthenuclear template spectra (e.g. Elvis et al. 1994; Richards et al. disk, even at the higher accretion rates of M˙ . In Edd ∼ 2006). However, overestimates of the conventional cor- thiscase,thediskbecomescoldandthininsteadofahot rectionfactorfromtheopticalluminositieshavebeendis- ADAF. Such a thin disk is better described by the stan- cussed (Trakhtenbrot & Netzer 2012), and several stud- dardα-viscosityprescription(Shakura & Sunyaev1973). ies have suggested that the bolometric correction fac- Theα-diskbecomesself-gravitatingandunstableatlarge tors depend on M (Kelly et al. 2008) and increase radii, where Q.1, BH with the Eddington ratio L/L (Vasudevan & Fabian 2007). Thus, this method to eEsdtdimate Lbol would have R 8.1 α104.1/27m˙ B−H8/27MB−H26,1/027 (pgas >prad), intrinsic uncertainties, especially for high-z QSOs. In sg = addition, beaming could be present, and produce over- RSch 85.3 α1/3m˙ 1/6M−1/2 (p >p ), estimates of L for QSOs with weak emission lines 0.1 BH BH,10 rad gas bol (e.g. Haiman & Cen 2002). Since the fraction of weak-  (10) line QSOs is higher at z 6 than at lower redshift where α0.1 α/0.1 and m˙ BH M˙BH/M˙Edd (Banãdos et al. 2014), the b≃olometric luminosities could (Goodman & T≡an 2004). The top (bot≡tom) expression be overestimated for these high-z sources. is valid when gas (radiation) pressure dominates. Fig. 4 shows the fragmentation radius R as a function of the sg 4. DISCUSSION BH mass (solid curve). The filled circle marks the loca- 4.1. Maximum BH mass of the brightest QSOs tionwherepgas =prad,insideofwhichradiationpressure Among observed SMBHs, the brightest QSOs with > dominates (MBH &4×107 M⊙). Gas clumps formed in the unstable region (r & R ) 2 1047 erg s−1, which are inferred to have Eddingto∼n subsequently grow via gas accretion from the ambiesngt × ratMio˙s ne.arInutnhiteyT(QLM∼05LmEdodd),elwwoeuladdgorpotwed,atthaisrwatoeulodf dpilsektedan,dwhtheeregRas niesarthteheclculmump’ps wHiitlhlirnad∼iufsHaRnHd ifs de- Edd H H ∼ ∼ 6 Kohei Inayoshi & Zoltán Haiman O(1). Assuming that the clump grows until a density gapis created,the mass reachesa substantialfractionof 1000 M・out = 10 3 Msun/yr the isolation mass (Goodman & Tan 2004), Mc,iso ≃1.3×109 α0−.11/2m˙ 5B/H4MB7/H4,10 fH3/2M⊙, (11) /yr)n 100 u where the clump location is set to r = Rsg. The Ms width of the gap is estimated as Rgap fHRH e ( fHRsg(Mc,iso/3MBH)1/3, and thus ≈ ≈ on rat 10 M・out = 400Msun/yr RRgsagp ≈0.36 fH3/2α0−.11/6m˙ 5B/H12MB1/H4,10. (12) accreti 1 Fig. 4 shows the value of (R R ) for f = 1.5 sg gap H − (dashed blue curve). A stable disk can exist only below this line, down to the inner-most stable circular or- 0.1 1 10 100 radius (pc) bit(ISCO),R 3R . Thesizeofthestableregion ISCO Sch ≃ shrinkswithincreasingBHmass,anddisappearsentirely Fig.5.— The same as Fig. 1 but M˙out =400 M⊙ yr−1 (blue) at MBH = 4.5×1010 M⊙ (i.e. Rsg −Rgap ≈ RISCO). and103 M⊙yr−1(red)(MBH=1010 M⊙andm=0.1). Forboth Subsequently, the BH could not be fed via a stable disk. thecases,theBHfeedingratewithin1pcisidentical. Instead, the BH could be fed stars from a nuclear star cluster, forming by the gravitational collapse of a mas- However, we argued that the large physical size of the siveclumpatR withM . The stellarfeeding occurs sg c,iso clumps in this case prevents a stable disk from forming on the timescale of t ln(2/θ ) (e.g. Frank & Rees relax lc ≃ allthe waydownto smallerradii,comparabletothe R 1976;Syer & Ulmer1999),wheret isthe(two-body) sg relax inthefiducialgas-dominatedcase(seedashedbluecurve relaxation timescale, estimated as in Fig. 4). As a result, our main conclusion agrees with 0.34 σ3 that of King (2016). t ∗ 6 Gyr, (13) relax ≃ G2M∗2ρ∗ln(MBH/M∗) ≃ 4.2. MBH M∗ relation for the most massive BHs − (Binney & Tremaine 2008; Kocsis & Tremaine 2011), In the star-forming disk model, a high accretion rate whereσ∗ =[G(MBH+Mc,iso)/Rsg]1/2 isthestellarveloc- is required to feed the central BH ( 2 and 3). This ity dispersion, M∗2 is the ratio of the mean-square stel- fact means that a large amount of §stars wo§uld form lar mass to the mean stellar mass of the stars, and ρ∗ = around the SMBHs. We briefly discuss the stellar mass 3Mc,iso/(4πRs3g) is the stellar density of the cluster. As- of massive galaxies hosting the most massive BHs with suming the Salpeter IMFwith Mmin(max) =1(100)M⊙, 1010 M⊙. ∼ we obtain M∗2 11 M⊙. Since the angular size of the Fig. 5 shows radial profiles of the gas accretion rate ≃ loss cone is estimated as θlc =√2RSchGMBH/(σ∗Rsg) and star formation rate for the two different values of 0.19 and ln(2/θlc) 2.4, the stellar feeding time fo∼r M˙out =400 and 103 M⊙ yr−1 (for MBH =1010 M⊙ and MBH > 4.5 1010 M∼⊙ exceeds the age of the Universe m = 0.1). We note that M˙out = 400 M⊙ yr−1(> M˙crit) × (at z = 0). Therefore, we expect disk fragmentation to is sufficient to maintain the universalfeeding rate in Eq. suppress BH growth above this mass (placing the corre- (8). Inthe casewithM˙out =400M⊙ yr−1,apopulation tshpeonludminignouspiptye)r.limit of L≃LEdd ≃6×1048 erg s−1 on opfcsitnartshewmithastsodtaolumblainsgs ∼tim1e01o2fMth⊙e 1fo0r1m0 sMw⊙itBhHin(∼ 220.05 We note that King (2016) recently proposed the exis- Gyr). Note that such a compact star forming regi∼on is tence of an upper limit on the masses of SMBHs, due to consistentwithobservedultra-luminousinfraredgalaxies fragmentation of the nuclear disk. King (2016) suggests where stars form in a few 100 pc nuclear disk at a rate thatthemaximummassistheoneforwhichthefragmen- up to several 100 M⊙ yr−1 (Medling et al. 2014). tationradiusis locatedatthe ISCO.This is verysimilar The most massive elliptical galaxies are as old as to our discussion of the case of the M˙ /M˙ ( 1) &8 Gyr (e.g. Bernardi et al. 2003; Thomas et al. 2005). BH Edd ≃ α-disk above. The main difference is that King (2016) Thus, we can observe stars with masses of < 1.1 M⊙, adopts a gas-pressure dominated disk, though the ra- whose lifetimes are longer than the age of the galaxies diation pressure in fact dominates at R for M & (at least 8 Gyr). Although the IMF of stars around sg BH 4 107 M⊙ (below dashed-dotted line in Fig. 4). King the most massive BHs is highly uncertain, many au- (2×016) arguesthat a largeradiation–pressuredominated thors have discussed the possibility that stars forming diskextendingallthewayouttoR wouldbethermally in SMBH disks, including those observed in the Galac- sg unstable and can not form at all. Then, the BH mass tic center, have a top-heavy IMF (e.g. Paumardet al. limit is estimated as 3 1010 M⊙ from Rsg RISCO 2006; Levin 2007; Nayakshin et al. 2007, and see also assuming p > p ≃. A×s the implications of≃this in- Goodman & Tan 2004). Assuming the Salpeter IMF gas rad stability are not yet understood, we here conservatively with Mmin(max) = 1 (100) M⊙, 4 % of the stars in assumed that a radiation-pressuredominated disk could mass live in longer lifetimes of > 8 Gyr and can be ob- still feed the central BH, as long as it is gravitation- served in the most massive elliptical galaxies. There- ally stable. This, in principle, would greatly increase fore, we can estimate the stellar mass surface density as the fragmentation radius (see red solve curve in Fig. 4). 3 1011 M⊙ kpc−2, which is consistent with a max- ∼ × A maximum mass of SMBHs 7 imum value of dense stellar systems within a factor of in the sub-pc nuclear region is reduced to the smaller three (Hopkins et al. 2010, see also Lauer et al. 2007). value of at most 4 M⊙ yr−1(MBH/1010 M⊙)0.3. This ∼ prevents SMBHs from growing above 1011 M⊙ in ≃ 4.3. Super(Hyper)-Eddington growth of intermediate the age of the Universe. Furthermore, at this low rate massive BHs (M˙BH/M˙Edd . 10−2), the nuclear BH disk can not maintain a thin structure and changes to a radiatively We briefly mention rapid growth of intermediate mas- inefficient ADAF. Once this transition occurs, the BH sive BHs. According to Eq. (9), the BH feeding feeding is further suppressed by strong outflows from rate in the Eddington units M˙ /M˙ ( M−0.7) ex- BH Edd ∝ BH hot gas near the BH. The maximum mass of SMBHs rceegedimseunoiftyM˙aBtHM&BHM˙.Ed3d.,2t×he10n7ucmle10a0.r1/7acMcr⊙e.tionIndtihske iMsgBiHv,emnaxby≃t(h0e.9cr−iti6c.a2l)m×a1s0s1w0hMer⊙e,tahnisdtrdaenpseintidosnporcicmuarrs-, transits to an optically thick ADAF solution, so-called ily on the angular momentum transfer efficiency in the slim disk, where super-Eddington accretion would be galacticdisk, and only weakly on other properties of the possible (e.g. Abramowicz et al. 1988; Jiang et al. 2014; host galaxy. Sa¸dowski et al. 2015). However, radiation heating sup- Althoughthismodelgivesacompellingexplanationfor presses gas supply from larger scales, which results in the observedmaximum SMBH masses, it underpredicts, a lower accretion rate M˙ (e.g. Ciotti & Ostriker bya factoroffew, the highestobservedquasarluminosi- ∼ Edd ties. These rare high-luminosity objects would require 2001; Novak et al. 2011; Park & Ricotti 2012). For in- a high (near-Eddington) accretion rate, but we have ar- termediate massive BHs with MBH . 104 M⊙, the BH gued that they do not significantly add to the SMBH feeding rate becomes M˙ & 3000 m L /c2, where BH 0.1 Edd masses, because these bursts may correspond to brief hyper-Eddington accretion could be realized unimpeded episodes following major mergers, and because we find by radiation feedback, and the massive BHs would grow that self-gravity prevents a stable accretion disk from rapidly (Inayoshi et al. 2016; Ryu et al. 2016). forming for MBH > 4.5 1010 M⊙ even in this high- M˙ /M˙ regime. × BH Edd Finally, if the explanation proposed here is correct, 5. SUMMARYANDCONCLUSIONS it requires that stars forming in disks around the most Observations of SMBHs have revealed an upper limit massive SMBHs have a top-heavy IMF, in order to of a few 1010 M⊙ on their mass, in both the local and avoidover-producingthemassesofcompactnuclearstar- the early×Universe, nearly independent of redshift. In clusters in massive elliptical galaxies. This is consistent this paper, we have interpreted this to imply that the with theoretical expectations. growth of SMBHs above this mass is stunted by small- ACKNOWLEDGEMENTS scalephysicalprocesses,independentofthepropertiesof their host galaxies or of cosmology. The growth of more We thank Jeremiah Ostriker, Yuri Levin, Nicholas massive SMBHs requires a high rate (> 103 M⊙ yr−1) Stone, Benny Trakhtenbrot, Kazumi Kashiyama, Shy of cold gas supply from galactic scales i∼nto a nuclear re- Genel, Kohei Ichikawa and Jia Liu for fruitful discus- gion. We have argued that even if gas is supplied to sions. This work is partially supported by Simons the galaxy at such high rates, most of the gas forms Foundation through the Simons Society of Fellows (KI), stars at larger radii ( 100 pc). Adopting the model andbyNASA grantsNNX11AE05GandNNX15AB19G by TQM05 for a star∼-forming disk, the accretion rate (ZH). APPENDIX ANALYTICAL DERIVATIONS OF THE SCALINGRELATIONS We here give derivations of the scaling relations of Σ˙∗ ∝ Σg/κ (§2.1) and M˙BH ∝ mMB1/H3 (§2.2), and an analytical expression of M˙ ( 2.1). These arguments are based on TQM05 (see their 2 and Appendix A). crit § § In a star-burst disk, we assume that the accretion disk is marginally stable against the self-gravity (Q 1 or ≃ Σ c Ω) and is a hydrostatic equilibrium state to the vertical direction, the total pressure is given by g s ∝ p=ρh2Ω2 =Σ c Ω Σ2. (A1) g s ∝ g For a radiation-pressure dominant (p p T4) and optically thick (τ 1) disk, the pressure is expressed as rad ≃ ∝ ≫ p τΣ˙∗ (see Eq. 4). Combining these relations with τ κΣg, we can obtain two relations ∝ ≃ Σ˙∗ Σg/κ, (A2) ∝ T Σ1/2. (A3) ∝ g As we discussed in 2.1, the star formation rate increases at radii, where dust opacity decrease by sublimation (e.g. § T 1000 K), to maintain the marginally-stable disk structure. dust,sub ≃ Next, we derive the relation of the BH feeding rate M˙ with the BH mass and the Mach number of the radial BH velocity(m=v /c ). AsthegastemperatureinthediskincreasesinwardandreachesT ( 1000K),theopacity r s dust,sub ≃ rapidly decreases because of dust sublimation (κ Tβ at T &T , where β < 20). In this opacity gap, higher dust,sub ∝ − star formation rates are required to support the disk in the vertical direction via radiation pressure (see Eq. A2). 8 Kohei Inayoshi & Zoltán Haiman Because of the gas consumption, the gas accretion rate decreases inward inside the opacity gap, where timescales of the star formation t∗ Σg/Σ˙∗ and the radial advection tadv r/vr are balanced. These timescales are estimated as ≡ ≡ t∗ κ Tβ, (A4) ∝ ∝ rΩ rΩ t , (A5) adv ∝ Σ m ∝ T2m g where we use Eqs. (A2) and (A3). Thus, the condition where t∗ tadv gives us a relation of ≃ rm2 −1/(4+2β) T , (A6) ∝(cid:18)M (cid:19) BH whichmeansthatT T iskeptinside theopacitygap. Sincethe accretionandthe starformationarebalanced dust,sub ≃ (M˙ r2Σ˙∗), we obtain a relation from Eqs. (A2) and (A6) ∼ M˙ r2T2−β r64++52ββM42+−2ββmβ2+−β2 r5/2M−1/2m for β . (A7) ∝ ∝ BH −→ BH →−∞ The accretion rate decreases approximately following M˙ r5/2 in the opacity gap, where the temperature does not ∝ change but the density increase toward the center. As a result, the gas pressure dominates the radiation pressure eventually, and thus star formation becomes less important as a energy source to support the disk structure. We estimate the characteristicradiusR withinwhichp >p . Fromthe equationofcontinuity (Eq. 3), weestimate gas gas rad M˙ Ω p Σ c Ω . (A8) gas g s ≃ ∼ rm Since p Ω2T and p T4, the condition of p p gives gas rad gas rad ∝ ∝ ≃ 4/3 M p , (A9) rad ∝(cid:18)r3(cid:19) and thus we obtain R M˙ −2/3M5/9m2/3. (A10) gas ∝ BH At r < R , the star formation rate becomes below the gas accretion rate and thus M˙ (r) const, which is the BH gas ≃ feeding rate M˙ . Substituting Eq. (A10) into Eq. (A7), we find the relation BH M˙ mM1/3, (A11) BH ∝ BH whichisingoodagreementwithEqs. (8)and(9). CombiningEqs. (A10)and(A11),weobtainR 1.4M7/9 pc. gas ≃ BH,10 Note that viscous heating is still subdominant at r = R , but stabilizes the disk at r < R , where the Toomre gas gas parameter exceeds unity (Q>1). Finally, we estimate the critical gas accretion rate M˙ at a large radius R . For M˙ >M˙ , the gas accretion crit out out crit rate is high enough to maintain the universal BH feeding rate (Eqs. 8 and A11). Otherwise, the gas in the disk is depletedduetoefficientstarformationat R andthustheBHfeedingratebecomesmuchlowerthantheuniversal out value. Since the dustopacity is givenby κ∼=κ T2 atthe largeradius,where the gastemperature is .100K,the star 0 formationtimescale is t∗ ǫκ0T2. Thus, the accretionrate at Rout requiredto feed the BH at the universalrate (Eq. ∝ 8) is given by t∗ &tadv(≃ΣgRo2ut/M˙out ∝T2Ro2ut/M˙out), that is, R 2 ǫ −1 κ −1 M˙out &M˙crit ≃280 M⊙ yr−1(cid:18)200ouptc(cid:19) (cid:16)10−3(cid:17) (cid:18)2.4 10−4 cm0 2 g−1 K−2(cid:19) , (A12) × (see also Eq. 44 in TQM05). REFERENCES Abramowicz,M.A.,Czerny,B.,Lasota,J.P.,&Szuszkiewicz, E. 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