The evolution of active galactic nuclei and their spins
Marta Volonteri, Marek Sikora, Jean-Pierre Lasota, Andrea Merloni
aa r X i v : . [ a s t r o - ph . H E ] A ug Draft version October 15, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE EVOLUTION OF ACTIVE GALACTIC NUCLEI AND THEIR SPINS
M. Volonteri , M. Sikora , J.-P. Lasota and A. Merloni Draft version October 15, 2018
ABSTRACTMassive black holes (MBHs) in contrast to stellar mass black holes are expected to substantiallychange their properties over their lifetime. MBH masses increase by several order of magnitude overthe Hubble time, as illustrated by So ltan’s argument. MBH spins also must evolve through the series ofaccretion and mergers events that grow the MBH’s masses. We present a simple model that traces thejoint evolution of MBH masses and spins across cosmic time. Our model includes MBH-MBH mergers,merger-driven gas accretion, stochastic fueling of MBHs through molecular cloud capture, and a basicimplementation of accretion of recycled gas. This approach aims at improving the modeling of low-redshift MBHs and AGN, whose properties can be more easily estimated observationally. Despite thesimplicity of the model, it captures well the global evolution of the MBH population from z ∼ z = 0 the spin distribution in gas-poor galaxies peaks atspins 0 . − .
8, and is not strongly mass dependent. MBHs in gas-rich galaxies have a more complexevolution, with low-mass MBHs at low redshift having low spins, and spins increasing at larger massesand redshifts. We also find that at z >
Subject headings:
Black hole physics — Galaxies: active, nuclei INTRODUCTIONAstrophysical black holes span a large range of masses,from the remnants of stellar evolution to monsters weigh-ing by themselves almost as much as a dwarf galaxy.Notwithstanding the several orders of magnitude differ-ence between the smallest and the largest black holeknown, all of them can be described by only two pa-rameters: mass and spin. So, besides their masses, M ,astrophysical black holes are completely characterizedby their dimensionless spin parameter, a ≡ J h /J max = c J h /G M , where J h is the angular momentum of theblack hole, and 0 ≤ a ≤ ǫ rad , which is al-most equal to the mass-to-energy conversion efficiency, Institut d’Astrophysique de Paris, 98bis Bd. Arago, 75014,Paris, France Astronomy Department, University of Michigan, 500 ChurchSt. , Ann Arbor, MI 48109 Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland Astronomical Observatory, Jagiellonian University, ul. Orla171, 30-244 Krak´ow, Poland Max-Planck-Institut f¨ur Extraterrestrische Physik, Giessen-bachstr., D-85741, Garching, Germany ǫ rad ≃ ǫ ≡ − E ISCO , where E ISCO = p − / (3 r ISCO )is the specific energy of the gas particle (in rest mass-energy units) in the innermost stable (= marginally sta-ble) circular orbit (ISCO), and r ISCO is the radius ofthis orbit in
GM/c units. This radius and thereforeradiation efficiency depend solely on the BH spin param-eter a . Maximal efficiency ( ǫ ≃ .
42) is achievable bydisks rotating around maximally spinning BHs; it dropsto ≃ .
06 for non-spinning BHs, and to ≃ .
05 for max-imally counter-rotating BHs. This entails a dependenceof the BH mass-growth rate on the spin value, implyinglongest growth time scales for larger positive spins. Moreprecisely, for a hole accreting at the Eddington rate, theblack hole mass increases with time as: M ( t ) = M (0) exp (cid:18) − ǫǫ tt Edd (cid:19) , (1)where t Edd = M BH c /L Edd = σ T c π G m p = 0 .
45 Gyr. Thehigher the spin, the higher ǫ , implying longer timescalesto grow the MBH mass by the same number of e–foldings.The radiative efficiency is also the fundamental freeparameter in the So ltan argument (Soltan 1982) and,more recently, in synthesis models (e.g., Merloni & Heinz2008) which relate the integrated MBH mass density tothe integrated emissivity of the AGN population, via theintegral of the luminosity function of quasars. If the aver-age efficiency of converting accreted mass into luminosityis ǫ = L/ ˙ Mc , then the MBH will increase its mass by˙ M = (1 − ǫ ) ˙ M in , accounting for the fraction of the in-flowing mass, ˙ M in , that is radiated away. Applying thisargument to the whole MBH population, the MBH massdensity, ρ BH , can be related to the integral of the lu-minosity function of quasar, Ψ( L, z ), with the radiativeefficiency being a free parameter: Volonteri et al. ρ BH ( z ) = Z ∞ z dtdz dz Z ∞ (1 − ǫ ) Lǫc Ψ( L, z ) dL. (2)Recent results suggest that this approach might betoo simplistic, as the radiative efficiency evolves alongthe cosmic time. Wang et al. (2009) for instance sug-gests that quasars at the peak of their activity ( z ∼ z < aligned with the orbital angularmomentum are expected to recoil with velocities below200 km s − . The recoil is much larger, up to thousandskm s − , for MBHs with large spins in non-aligned con-figurations (Campanelli et al. 2007; Gonz´alez et al. 2007;Herrmann et al. 2007).Finally, the spin of a hole might determine how muchenergy is extractable from the hole itself (Blandford &Znajek 1977; Tchekhovskoy et al. 2011; McKinney etal. 2012). The so-called “spin paradigm” asserts thatpowerful relativistic jets are produced in AGNs with fastrotating black holes (Blandford et al. 1990), implyingthat MBHs rotate slowly in radio-quiet quasars, whichrepresent the majority of quasars (Wilson & Colbert1995). Sikora et al. (2007) proposed a “spin-accretionparadigm”, suggesting that the production of powerfulrelativistic jets is conditioned by the presence of fast ro-tating holes, while it also depends on the accretion rateand on the presence of disk magneto-hydrodynamicalwinds required to provide the initial collimation of thecentral Poynting flux dominated outflow, as in, e.g., theBlandford-Znajek process. Recently Sikora & Begelman(2013) proposed that the magnetic flux threading theblack hole, rather than BH spin or Eddington ratio, isthe dominant factor in launching powerful jets.As described above, MBH spins determine directly themass-to-energy conversion efficiency of quasars. On theother hand, accretion determines the evolution of MBHspins. A hole that is initially non-rotating gets spun upto a maximally-rotating state ( a = 1) after increasingits mass by a factor √ ≃ .
4. A maximally-rotatinghole is spun down by retrograde accretion to a = 0 aftergrowing by a factor p / ≃ .
22. Different modes ofMBH feeding imply different spin histories. Spin-up isa natural consequence of prolonged disk-mode accretion:any hole that increases substantially its mass by cap-turing material with constant angular momentum axiswould ends up spinning rapidly (“coherent accretion”).Spin-down occurs when counter-rotating material is ac- creted, if the angular momentum of the accretion diskis strongly misaligned with respect to the direction ofthe MBH spin. It has been suggested that accretionmay proceed also via small (i.e., the accreted mass isa very small fraction of the MBH mass, ∼
1% or less)and short uncorrelated episodes (“chaotic accretion”,Moderski & Sikora 1996; King & Pringle 2006), whereaccretion of co-rotating (causing spin-up) and counter-rotating (causing spin-down) is equally probable. Asthe ISCO for a retrograde orbit is at larger radii thanfor a prograde orbit, the transfer of angular momen-tum is more efficient in the former case. Accretion ofcounter-rotating material therefore spins MBHs downmore efficiently than co-rotating material spins them up.King et al. (2008) considered a MBH evolution scenariowhere chaotic accretion very rapidly adjusts the hole’sspin parameter to average values a ∼ . − . > . only if MBHs start already withlarge spins and they do not experience many major merg-ers. Therefore, the common assumption that mergers be-tween MBHs of similar mass always lead to large spinsneeds to be revised.The focus of this paper will be on the cosmic evolu-tion of spins of massive black holes, M ∼ − M ⊙ (Richstone et al. 1998; Ferrarese & Ford 2005), specifi-cally on how accretion and MBH-MBH mergers deter-mine the magnitude of spins. Very few works to-datehave studied the joint MBH mass and spin coevolu-tion (Moderski et al. 1998; Volonteri et al. 2005; Shapiro2005; Lagos et al. 2009; Fanidakis et al. 2011; Barausse2012). In common with previous efforts we adopt a semi-analytical approach, in order to capture both the cosmicevolution of structures and the processes that occur nearMBHs. This approach allows us to model accretion pro-cesses using an analytical formalism, that in principle hasunlimited spatial resolution. This is particularly relevantas the physical processes that influence spin evolution oc-cur near the MBH, and unfortunately direct cosmologicalsimulations at sub-pc resolution are still unfeasible. Theother advantage of this approach is that each assump-tion is clearly described mathematically, making the cal-culation easily reproducible, or modifiable and testableunder different assumptions by scientists with differenttheoretical stances. Finally, one should appreciate thatour formalism does not have many more “cranks” thansub-grid prescriptions adopted in numerical simulations,while offering a clear framework that can be replicated,or modified, in a very economical way using a standarddesktop by any scientist who decides so. It is importantto notice that our model does reproduce a large numberof observational constraints (luminosity function of AGNand mass function of MBHs, relation between MBHs andhosts, mass density in MBHs at low and high-redshift).Since we are comparing our models to a large numberof observables, there is not much leverage for the modelparameters or assumption to be varied. In section 5 wediscuss how the model’s parameters can be changed (andccretion and spin 3which cannot be modified). Of course, since there are notmany observational constraints on MBH spins (but seesection 6 and 7) the possibilities to compare our mod-els with observations are rather limited. Within the as-sumptions made, and the observational constraints usedto anchor our calculation, the model is robust. Beingthis a theoretical investigation, we present a frameworkthat predicts a set of properties for the MBH popula-tion. In contrast with previous investigations that fo-cused on high-redshift quasars (e.g. Volonteri et al. 2005;Shapiro 2005) our main interest here is to study the pop-ulations of low-redshift MBHs and AGN whose spins maybe directly measured through X-ray spectroscopy, or in-directly estimated through their average radiative effi-ciency. In particular, our models aim at translating thetheoretical expectations in a framework that can be di-rectly applied to observational samples, for instance bycasting our results in terms of AGN luminosity ratherthan MBH mass as typically done in the literature (asonly a small subsample of AGN have mass measure-ments).The outline of the paper is as follows. In section 2 wedescribe the basic infrastructure that we use to model thecosmic evolution of structures. In section 3 we summa-rize how we model spin evolution in MBH-MBH mergers,while in section 4 we describe how different phases of ac-cretion, related to the cosmic evolution of galaxies andof the MBHs they host, influence MBH spins. In section5 we consider a series of observational constraints thatwe adopt to anchor our model. In section 6 we discussour results, and we present our conclusions in section 7. THE BACKBONE: DARK MATTER HALOS ANDGALAXIESWe investigate the evolution of MBHs via cosmologicalrealizations of the merger hierarchy of dark matter halosfrom early times to the present in a ΛCDM cosmology(WMAP5, Komatsu et al. 2009). We track the mergerhistory of 300 parent halos with present-day masses inthe range 10 < M h < M ⊙ with a Monte Carloalgorithm (Volonteri et al. 2003). The mass resolutionof our algorithm reaches 10 M ⊙ at z = 20, and the mostmassive halos are split into up to 600,000 progenitors.We wish to keep our models as simple as possible,while making sure that the properties of the MBHs westudy are correctly determined through the cosmic evo-lution of their hosts. We do not explicitly model theevolution of the baryonic component of the host galaxiesthrough cooling, star formation and various feedbacks(see Lagos et al. 2009; Fanidakis et al. 2011; Barausse2012, and references therein for models that treat in de-tail semi-analytically the baryonic component of galaxiesand its link to MBH evolution). In our models we useonly one parameter to link the host halo to the centralMBH, and it is the host’s central velocity dispersion. Welink the central stellar velocity dispersion of the host tothe asymptotic virial velocity ( V c ) assuming a spherical,isothermal halo, so that σ = V c / √
2. We calculate thecircular velocity from the mass of the host halo and itsredshift. A halo of mass M h collapsing at redshift z has a circular velocity: V c = 142 km s − (cid:20) M h M ⊙ (cid:21) / (cid:20) Ω m Ω zm ∆ c π (cid:21) / (1+ z ) / , (3)where ∆ c is the over–density at virialization relativeto the critical density. For a WMAP5 cosmology weadopt here the fitting formula ∆ c = 18 π + 82 d − d (Bryan & Norman 1998), where d ≡ Ω zm −
1, and Ω zm =Ω m (1 + z ) / (Ω m (1 + z ) + Ω Λ + Ω k (1 + z ) ).At high redshift we seed dark matter halos with MBHscreated by gas collapse. Specifically, we adopt herethe formation model detailed in Natarajan & Volonteri(2011) based on Toomre instabilities (Lodato & Natara-jan 2006). The Toomre parameter is defined as Q = c s κπG Σ ,where Σ is the surface mass density, c s is the sound speed, κ = √ V c /R is the epicyclic frequency, and V c is the cir-cular velocity of the disk. When Q approaches a criticalvalue, Q c , of order unity, the disk is subject to gravita-tional instabilities. If the destabilization of the system isnot too violent, instabilities lead to mass infall instead offragmentation into bound clumps and global star forma-tion in the entire disk (Lodato & Natarajan 2006). Thisprocess stops when the amount of mass transported tothe center is sufficient to make the disk marginally sta-ble. The mass that has to be accumulated in the centerto make the disk stable, M inf , is obtained by requiringthat Q = Q c . This condition can be computed fromthe Toomre stability criterion and from the disk prop-erties, determined from the dark matter halo mass, via T vir ∝ M / h , and angular momentum, via the spin pa-rameter, λ spin : M inf = f d M halo − s λ spin f d Q c (cid:18) j d f d (cid:19) (cid:18) T gas T vir (cid:19) / . (4)for λ spin < λ max = f d Q c / f d /j d )( T vir /T gas ) / . Here λ max is the maximum halo spin parameter for which thedisk is gravitationally unstable, f d = 0 .
05 is the gas frac-tion that participates in the infall, and j d = 0 .
05 is thefraction of the halo angular momentum retained by thecollapsing gas. We further assume T gas = 5000K and Q c = 2 (see Volonteri, Lodato & Natarajan 2008 for avalidation of the parameter choice). Given the mass andspin parameter of a halo, the mass that accretes to thecenter in order to make the disk stable, M inf , is an upperlimit to the mass that can go into MBH formation. Weassume here M seed = M inf . Please refer to Natarajan &Volonteri (2011) and references therein for details on theMBH formation process.Our model for MBH growth requires only to knowwhether a galaxy is gas-rich (we typically refer to gas-rich galaxies as “disks” in this paper) or gas-poor(“spheroid”). Since forming galaxy disks even in high-resolution cosmological simulations is extremely chal-lenging (Governato et al. 2007), we keep our model forgalaxy morphology as simple as possible. Morphologyis related to the merger history, using a three-parametermodel, where spheroid formation depends on both halomass ratio and the absolute halo mass, and a spheroidcan re-acquire a disk through cold flows and mergerswith gas-rich galaxies. Koda et al. (2007) show that Volonteri et al.the fraction of disk- vs spheroid-dominated galaxies iswell explained if the only merger events that lead tospheroid formation have mass ratio > >
55 km s − ; also, the merger timescale must beinferior to the time between when the merger starts andtoday, z = 0. We assume that spheroids form after amerger that meets these requirement. We additionallyallow a disk to reform after 5 Gyrs in galaxies with virialvelocity <
300 km s − where no major mergers occurredto include the effect of cold flows. In section 5 we discussthe sense of this approach. SPIN EVOLUTION DUE TO MBH-MBHMERGERSWe assume that, when two galaxies hosting MBHsmerge, the MBHs themselves merge within themerger timescale of the host halos (Sesana et al. 2007;Dotti et al. 2007, and references therein). We adoptthe relations suggested by Boylan-Kolchin et al. (2008)for the galaxy merger timescale. We model MBH spinchanges due to mergers adopting an analytical schemesimilar to that described in Berti & Volonteri (2008),based on simulations of black hole mergers in full gen-eral relativity (Rezzolla et al. 2008; Lousto et al. 2009).Kesden et al. (2010) has validated the consistency of dif-ferent fitting formulae for calculating the spin of MBHremnants, and we refer the reader to Lousto et al. (2010)for the most comprehensive fitting formulae. Due to com-putational constraints, we adopt here the fitting formulaeof Rezzolla et al. (2008) for their easy implementation: | a fin | = q ) h | a | + | a | q + 2 | a || a | q cos α +2 (cid:0) | a | cos β + | a | q cos γ (cid:1) | ℓ | q + | ℓ | q i / , (5)where q ≡ M /M α, β and γ are definedbycos α ≡ ˆ a · ˆ a , cos β ≡ ˆ a · ˆ ℓ , cos γ ≡ ˆ a · ˆ ℓ . (6)where | ℓ | is the magnitude of the orbital angular momen-tum, and | ℓ | = s (1 + q ) (cid:0) | a | + | a | q + 2 | a || a | q cos α (cid:1) + (cid:18) s ν + t + 21 + q (cid:19) (cid:0) | a | cos β + | a | q cos γ (cid:1) +2 √ t ν + t ν , (7)where ν is the symmetric mass ratio ν ≡ M M / ( M + M ) , and the coefficients take the values s = − . ± . s = − . ± . t = − . ± . t = − . ± . t = 2 . ± . align their spins, initially oriented at random, to the angular momentum ofthe nuclear disk (Liu 2004; Bogdanovi´c et al. 2007;Dotti et al. 2010): in response to the Bardeen–Petterson(Bardeen & Petterson 1975) warping of the small–scaleaccretion disks grown around each MBH, total angularmomentum conservation imposes fast ( ∼ < α = cos β = cos γ =1, and the MBH remnant retains the spin direction of theparent MBH spins, both oriented parallel to the angu-lar momentum of their orbit: the post-coalescence MBHmay thus acquire a large spin > . − . α , cos β , and cos γ areisotropically distributed. Berti & Volonteri (2008) showthat for isotropic configurations mergers tend to “spin-down” a fast-spinning hole (see also Hughes & Blandford2003). For intermediate-large mass ratios (mass ratio q = M BH , /M BH , ≃ . SPIN EVOLUTION DUE TO ACCRETIONWe discuss here the feeding of MBHs in the quasarphase and its aftermath and in the more quiescent Seyfertgalaxies. Simulations of galaxy mergers and MBH ac-tivity (Di Matteo et al. 2005; Hopkins et al. 2006) showthat for every accretion episode triggered by a galaxymerger a three phase picture can be drawn. At the be-ginning the MBH has an “healthy diet”, with f Edd ≡ L/L
Edd
1. When the MBH mass reaches the “M- σ ”relation, the MBHs feedback can be sufficient to unbindand “blowout” the gas feeding it (Hopkins & Hernquist2006), causing a final “starvation”, when f Edd rapidlydecreases, until no more gas is available to feed theMBH. We argue that during the healthy diet and blowoutphases MBHs gain a high spin while accreting efficientlyand coherently (Dotti et al. 2010), building the popula-tion of high- z quasars. Coherent accretion ensues becauseMBHs in merger remnants are expected to be surroundedby dense circum-nuclear disks (Sanders & Mirabel 1996).Maio et al. (2012) study the evolution of the angularmomentum of material feeding MBHs embedded in cir-cumnuclear disks, and they find that coherence of theccretion and spin 5accretion flow near each MBH reflects the large-scale co-herence of the disk’s rotation.After the “blowout” phase, starving MBHs are nolonger surrounded by a thick gas disk, that determinesthe angular momentum of the material ending up in theaccretion disk. During this last phase we do not expectaccretion to necessarily proceed coherently any longer.During the starvation phase the accretion rate decreasesrapidly. The depletion of gas in galaxies and the decreasein the galaxy interaction rate at late cosmic times causestherefore a widespread “famine” in low- z ellipticals (i.e.,the merger remnants), where AGN with low accretionrates dominate (“radio” mode, see, e.g., Croton et al.2006; Churazov et al. 2005).We investigate the evolution of MBH spins during thequasar phase expanding previous work (Volonteri et al.2005, 2007) to more realistic models. We model the jointmass and spin evolution by coupling the results on themass accretion rate as a function of time in simulations(Hopkins et al. 2005; Volonteri et al. 2006), with the spinevolution due to disk accretion (Volonteri et al. 2007),thus solving a system of two coupled differential equa-tions ( f Edd as a function of M and time; spin a as afunction of f Edd , M . The framework has been derived inVolonteri et al. 2005, Volonteri et al. 2006, Hopkins &Hernquist 2006, and Volonteri et al. 2007). We remindhere the relevant information.4.1. Quasar phase
After a halo merger with mass ratio larger that 3:10,in which at least one of the two is a disk galaxy (hence,with conspicuous cold gas content) we assume that amerger-driven accretion episode is triggered . After adynamical timescale is elapsed (roughly, after the firstpericentric passage, cf. Van Wassenhove et al. 2012) ac-cretion starts. If at that point the MBH mass lies belowthe M- σ relation, accretion occurs at the Eddington rate( f Edd = 1). Additionally, during this early phase theMBH is nested into a nuclear disk that feeds the MBHcoherently. The following scheme is applied to the jointevolution of mass, spin and radiative efficiency in thisphase. Let us define M and a as the black hole mass andspin parameter at the beginning of the timestep, and µ as the cosine of the angle between the MBH spin andthe inner accretion disk angular momentum. Irrespec-tive of the infalling material’s original angular momen-tum vector, Lense-Thirring precession imposes axisym-metry close in, with the gas accreting on either prograde( µ = 1) or retrograde equatorial orbits ( µ = − c = G = 1: r ISCO = 3 + Z ∓ p (3 − Z )(3 + Z + 2 Z ) , (8)is the radius of the ISCO, where Z and Z are functionsof a only (Bardeen, Press, & Teukolsky 1972), Z ≡ − a ) / [(1 + a ) / + (1 − a ) / ] , (9) Z ≡ [3 a + Z ] / , (10) Please note that this assumption is at variance with previ-ous models of MBH cosmic evolution within our framework (e.g.,Volonteri et al. 2003, 2008; Volonteri & Natarajan 2009), wherethe threshold was set to a lower value of 1:10. The reason forincreasing the threshold for QSO phase to occur is the addition ofavenues for MBH growth other than merger-driven accretion andMBH-MBH mergers. and the upper (lower) sign refers to prograde (retrograde)orbits. We calculate the accretion efficiency as: ǫ = 1 − E ISCO , (11) E ISCO = (cid:18) − r ISCO (cid:19) / , (12)which is is also a plausible assumption for the radia-tive efficiency ( ǫ , mass-to-energy conversion) for thin-disk accretion occurring at large fractions of the Edding-ton rate (see below for the case of radiatively inefficientflows). We calculate self-consistently the radiative effi-ciency from the MBH spin and the location of the ISCO(i.e., taking into consideration the direction of the rela-tive angular momentum of spin and disk, co- or counter-rotating).Assuming that during a timestep ∆ t ∼ − yr theradiative efficiency and Eddington rate remain constant,( ǫ =const and f Edd =const,) from the derivation shown inthe Appendix, one obtains that the MBH mass grows as: M ( t + ∆ t ) = M ( t ) exp (cid:18) f Edd ∆ tt Edd − ǫ ( t ) ǫ ( t ) (cid:19) (13)where t Edd = σ T c π G m p = 0 .
45 Gyr and f Edd representsthe Eddington fraction. We update the magnitude ofthe MBH spin through: a ( t + ∆ t ) = r ISCO ( t ) / M ( t ) M ( t + ∆ t ) (14) " − (cid:18) M ( t ) M ( t + ∆ t ) r ISCO ( t ) − (cid:19) / for M ( t + ∆ t ) M ( t ) ≤ r / ( t ) ,a ( t + ∆ t ) = 0 .
998 for M ( t + ∆ t ) M ( t ) ≥ r / ( t ) (15)(Bardeen 1970) . After updating the spin magnitude, wealso update the mass-to-energy conversion efficiency bydetermining the new ISCO corresponding to a ( t + ∆ t ),and therefore ǫ ( t + ∆ t ) to be used at the successivetimestep iteratively. We here assume fast alignment be-tween accretion disk and spin (see Natarajan & Pringle1998; Volonteri et al. 2005, 2007; Perego et al. 2009), asthe alignment timescale is ≃ Myr, so that accretion isprograde during the quasar phase.4.2.
Decline phase
When a MBH reaches a mass close to the valuecorresponding to the M- σ correlation ( M BH,σ ) forits host, we assume, following Hopkins et al. (2006); We limit the MBH spin to a = 0 .
998 following the calculationof Thorne (1974) that showed that the radiation emitted by thedisk and swallowed by the hole produces a counteracting torque,which prevents spin up beyond this value. We note that magneticfields connecting material in the disk and the plunging region mayfurther reduce the equilibrium spin by transporting angular mo-mentum outward in non-geometrically thin disks. Fully relativisticmagnetohydrodynamic simulations for a series of thick accretiondisk models show that spin equilibrium is reached at a ≈ . a = 0 . Volonteri et al.Hopkins & Hernquist (2006) that a self-regulation ensuesand the MBH feedback unbinds the gas closest to theMBH, thus reducing its feeding. For simplicity we fur-ther assume that the black hole– σ ( M – σ ) scaling is: M = 10 (cid:16) σ
200 km s − (cid:17) M ⊙ (16)(Tremaine et al. 2002). To match the luminosity func-tion of quasars, we start the decline phase when the MBHmass is 0 . × M BH,σ . The MBH continues accretingduring the decline of accretion and it typically reachesa value closer to the M BH,σ by the end of the accretionepisode. Note that our model does not necessarily implythat the M BH,σ relation is a tight correlation. We as-sume only that the feedback during a high-accretion ratequasar phase establishes at that time for that object an M – σ relation (cf. Silk & Rees 1998; Fabian 1999). Ad-ditional processes, such as MBH-MBH mergers (sec. 3),accretion during the “decline phase” (sec. 4.2), accre-tion of recycled gas (sec. 4.3), accretion of gas stolenfrom molecular clouds (sec. 4.4) do not have any limitimposed by the M BH,σ relation, and in fact they pro-duce scatter, by pushing the MBHs above or below therelationship, depending also on the galaxy history (seeVolonteri & Ciotti 2012).We model the decrease of the accretion rate throughthe analytical formula from Hopkins & Hernquist (2006): f Edd ( t ) = (cid:18) t + t f Edd t f Edd (cid:19) − η L , (17)where η L ≃ t f Edd ≃ . × ( M BH,σ / )M ⊙ yr,and t = 0 (where f Edd = 1) represents the time whenthe MBH reaches the threshold (0 . × M BH,σ ). Since˙ M in ( t ) = f Edd ( t ) M ( t ) /t Edd , depends on time, the ac-cretion rate must be integrated self-consistently, and themass now grows with time as: M ( t ) = M (0) exp η L − " t − η L f Edd − t + t f Edd ) − η L t η L f Edd t Edd − ǫ ( t ) ǫ ( t ) ! . (18)We use again Eq. 8–15 to model spin evolution, how-ever, we explore two possible scenarios. In our referencecase we assume that the “outflow” causes some stirringof the angular momentum of the gas within the centralregion. We therefore explore a “chaotic” case where inthe decline phase we pick a new random µ = 1 or µ = − ≃ − years) to mimic the lack ofcoherence in the accretion flow after MBH feedback hasblown away the surrounding gas. In a second model weassume instead that the “blow-out” of gas does not af-fect strongly the angular momentum of the material nearthe MBH, and persist with keeping µ = 1. Given thatthe timescale for decline is longer for larger MBHs (cf.Eq. 17), the larger the MBH the longer the phase at rel-atively high accretion rates, and the faster the decreaseof MBH spin, as more mass is accreted in a non-coherentfashion.When the accretion rates become very sub-Eddington,we assume that the accretion flow becomes optically thin and geometrically thick. In this state the radiativepower is strongly suppressed (e.g., Narayan & Yi 1994;Abramowicz et al. 1995), so that the radiative efficiencydiffers from the mass-to-energy conversion efficiency, ǫ ,that depends on the location of the ISCO only. Indeed,the radiative efficiency becomes very model dependentand uncertain. In order to estimate the effect of radia-tively inefficient accretion on the MBH population weadopt here a specific functional form for the radiative ef-ficiency. Following Merloni & Heinz (2008) we write theradiative efficiency, ǫ rad , as a combination of the mass-to-energy conversion, ǫ , and of a term that depends on theproperties of the accretion flow itself. Merloni & Heinz(2008) suggest that the transition in the disk proper-ties occurs at f Edd < f
Edd , cr = 3 × − , and that ǫ rad = ǫ ( f Edd /f Edd , cr ). This specific choice allows us toestimate qualitatively the impact of radiatively inefficientsources to the AGN populations, and on the inferencesthat one can (or not) make from observables.4.3. Quiescent elliptical phase
After the formation of an elliptical galaxy, the feed-ing of the MBH can be sustained by the recycled gas(primarily from red giant winds and planetary nebu-lae) of the evolving stellar population (Ciotti & Ostriker1997, 2001, 2007; Ciotti et al. 2010). As shown byCiotti & Ostriker (2011) the behaviour of the accretionrate is similar to what we describe above: at early timesthe evolution is characterized by major, albeit intermit-tent, accretion episodes, while at low redshift accretion issmooth and characterized by f Edd ≪
1. We have imple-mented this channel of MBH feeding only for quiescentellipticals, that is only after a spheroid is formed and hadtime to relax. When the MBH mass is ∼ − of thespheroid mass (Magorrian et al. 1998; Marconi & Hunt2003; H¨aring & Rix 2004), and the spheroid is mod-eled as a Hernquist profile, the geometrical model byVolonteri et al. (2011a) implies that the quiescent levelof accretion onto a central MBH due to recycled gas is f Edd ≃ − . We assume here that this mode of accretionhas a constant f Edd = 10 − , and we model spin evolutionassuming that that the accreted material is isotropicallydistributed (i.e., we pick a random µ = 1 or µ = − Molecular cloud accretion in disk galaxies
Several observations suggest that single accretionevents last ≃ years in Seyfert galaxies, while the to-tal activity lifetime (based on the fraction of disk galax-ies that are Seyfert) is 10 − years (e.g., Kharb et al.2006; Ho et al. 1997). This suggests that accretion eventsare very small and very ‘compact’. Smaller MBHs,powering low luminosity AGN, likely grow by accret-ing smaller packets of material, such as molecular clouds(Hopkins & Hernquist 2006). Compact self-gravitating Please note that Merloni & Heinz (2008) use a different no-tation and terminology. Their λ is our f Edd ≡ L/L
Edd ≡ ǫ rad ˙ Mc /L Edd , and their ˙ m = ǫ f Edd /ǫ rad . As long as the ac-cretion flow is optically thick and geometrically thin, i.e., beforethe transition to very sub-Eddington flows, ˙ m = f Edd . ccretion and spin 7cores of molecular clouds (MC) can occasionally reachsubparsec regions. In gas-rich, star-forming disk galaxiesthe MBH is likely to be fed by short, recurrent, uncorre-lated accretion episodes. The spin evolution of a MBHhosted by a quiescent disk galaxy would then resemblethe “chaotic accretion” scenario. This argument wasdiscussed only qualitatively by Volonteri et al. (2007).We now wish to provide quantitative statistical predic-tions for the distribution of MBH spins in different hosts.We follow here Sanders (1981) and Hopkins & Hernquist(2006) to determine the event rate, and Volonteri et al.2007 to couple accretion episodes to spin evolution.In a disk galaxy, at each timestep, ∆ t , we determinethe probability of a MC accretion event as: P = ∆ tt MC ≃ − σ ∆ tR cl (19)where R cl ≃
10 pc (Hopkins & Hernquist 2006). Asin Volonteri et al. (2007) we further assume a lognor-mal distribution (peaked at log( M MC / M ⊙ ) = 4, witha dispersion of 0.75) for the mass function of MC closeto galaxy centers (based on the Milky Way case, e.g.,Perets et al. 2007).We model accretion of MCs through a description in-spired by Bottema & Sanders (1986) and Wardle &Yusef-Zadeh (2008). We assume that the MBH capturesonly material passing within the Bondi radius, R B , andwe also assume that specific angular momentum is con-served, so that the outer edge of the disk that formsaround the MBH corresponds to the material originallyat the Bondi radius: R d = 2 λ R B = 8 . λ M M ⊙ (cid:16) σ
100 km s − (cid:17) − , (20)where λ is the fraction of angular momentum retainedby gas during circularization. The maximum capturedmass will be contained in a cylinder with radius R B andlength 2 × R cl , the MC diameter. If κ is the ratio of themass going into the disk with respect to the whole massin the cylinder, then: M d , max = κ R B R cl M MC = 4 . × κ (cid:18) M M ⊙ (cid:19) (cid:16) σ
100 km s − (cid:17) − M ⊙ . (21)The inflow time will be of order of the viscous timescalefor the disk, t visc = (cid:18) R d α v GM d (cid:19) / = 3 . × λ α v M M ⊙ (cid:16) σ
100 km s − (cid:17) − yr , (22)so that for the whole disk to be consumed we can cal-culate an upper limit to the mean accretion rate andluminosity:˙ M max = M d , max t visc = 0 . α v κλ M M ⊙ (cid:16) σ
100 km s − (cid:17) − N M ⊙ yr − , (23) Figure 1.
Properties of all MBHs fed by MCs at all redshifts:normalized distribution of the masses of MBHs (top) and theirhost halos (bottom). The probability of MC accretion increaseswith galaxy mass, on the other hand most massive galaxies arespheroids, and therefore they do not have a population of MCsavailable. where N is the column density in the MC in units of10 cm − , and α v = 0 . λ = 0 . κ = 1. If ˙ M max is less than the Eddington rate (assuminga radiative efficiency of 10%) we let the MBH accrete thewhole M d , max over a time t visc , otherwise we treat accre-tion similarly to the “decline” phase of quasars (Eq. 17and 18), as feedback from the high luminosity producedby accreting the cloud will limit the amount of materialthe MBH can effectively swallow.From Eq. 21, it is evident that the mass accreted in oneof these episodes is typically much less than the mass ofthe MBH (typically between 10 − and 10 − of the MBHmass), we therefore assume that no alignment betweenaccretion disk and MBH spin can occur, and that retro-and prograde accretion is equally probable, i.e., we as-sign µ = 1 or µ = − µ constant over the accretion phase, us-ing Eq. 15 to evolve the spin magnitude. As shown byVolonteri et al. (2007) these assumptions result is a spindown in a random walk fashion that depends on the massof the MBH and on the number of events. We note thatthis is an extreme condition of randomness in the orbitsand distribution of MCs, as any common sense of rota-tion caused by the presence of disk-like structures in thehost would decrease the degree of anisotropy (Dotti etal. 2012).Eq. 19, instead, shows that accretion of MCs is moreprobable in large galaxies, since the accretion probabil-ity is directly proportional to the velocity dispersion ofthe galaxy. Large galaxies, however, are more likely tobe gas-poor spheroids. Large galaxies therefore have alower probability of being gas-rich and host a populationof MCs, while at the same time their MBHs have a higher Volonteri et al.probability of capturing MCs, if clouds are present.Thus, this type of accretion events occur typically ingalaxies hosted in halos with mass ∼ − M ⊙ andfuel MBHs with mass ∼ − M ⊙ . In Fig. 1 we showthe distribution of the masses of MBHs and their hosthalos where accretion of MCs takes place according toour scheme. ANCHORING THE MODELHere we present the constraints that we use to “an-chor” our model to observed properties of galaxies andAGN. After validating our scheme for accretion and hostevolution, we will discuss what its implications are forthe, still unknown, distribution of MBH spins.First, Fig. 2 we compare our model to constraints at z = 0. In the bottom panel of Fig. 2 we show the morpho-logical fraction as a function of galaxy stellar mass. Wescale from halo mass to stellar mass through the data de-scribed in Fig. 1 of Hopkins et al. (2010), assuming thata fixed fraction 10-50% of the baryons is in stars (openred points and filled orange ones respectively). Whileagreement is far from perfect, our simple approach re-produces the correct trend. We recall here that we donot model the whole evolution of gas and stars, nor thedisk formation (cf. Barausse 2012, for a comprehensivemodel), but to derive all properties our scheme uses onlya single quantity, the halo mass.In the top panel of Fig. 2 we reproduce the re-lationship between MBH masses and circular velocity(Volonteri et al. 2011b). Circles are model MBHs at z = 0, while errorbars show the data (Kormendy &Bender 2011; Volonteri et al. 2011b). We note thatour model fails to reproduce some of the massive MBHsat z = 0 in spheroids. The reason for the suppressedgrowth is that no MC accretion can happen in our schemein gas-poor galaxies, and we have not implemented atime-dependent accretion of recycled gas. As noted byCiotti & Ostriker (2007) the behavior of a MBH fueledby stellar mass loss is self-regulated between “on” and“off” phases. Our model includes only the “off” (quies-cent) phase and therefore underestimates the growth dueto this fueling channel.Our second anchor is the luminosity function of AGNin the redshift range 0 . z
3. This is a strong con-straint for our accretion scheme, and it is shown in Fig. 3.Our scheme produces an AGN population in good agree-ment with the observations at z = 0 . z = 2 and z = 3,while we slightly underproduce AGN at z = 1. The lackof high luminosity quasars at z = 2 and z = 3 is dueto our merger trees not including halos massive enoughto host MBHs with masses above a few 10 M ⊙ at thoseredshifts. Overall, however, we obtain the correct trends.This figure shows that there is little difference in the twomodels we explore on the effect of feedback over the an-gular momentum of the nuclear gas (“chaotic” or “co-herent” decline phases). The difference between chaoticand coherent decline (the “quasar” phase is coherent inall cases), reflects only on spin and as a consequence onradiative efficiency, not on the Eddington ratio. In thefollowing we will distinguish the two models only whenthe results are significantly different, otherwise we showonly the reference case (“chaotic” decline phase).The main parameters influencing the performance ofthe model against the constraints are the mass ratio above which a mergers can trigger quasar activity and thefraction of M BH,σ when the decline phase starts. Thesetwo parameters are weakly degenerate. The former pa-rameter is set to > z = 0, as happens instead by choosing alower threshold. A much lower threshold would also bein disagreement with simulations of galaxy mergers thatstudy merger-drive AGN activity, and show that with amass ratio of 1:6 high level of AGN activity does not oc-cur (Van Wassenhove et al. in preparation). We testeda case where we instead increased the threshold to 1:2,and in this case the bright end of the luminosity func-tion would disappear at z > . × M BH,σ . We tested a case thatbrought the MBHs exactly on M BH,σ , but this leads tolargely overestimating the MBH masses at z = 0 at agiven V c . We also tested a case with 0 . × M BH,σ , andin that case we still overestimated MBH masses at z = 0at a given V c (the overestimate is a factor of 2 overallin this case, over the best fit relationship). Decreasingthe parameter value to 0 . × M BH,σ instead underes-timates MBH masses at z = 0 at a given V c , by a factor2.25 overall, over the best fit relationship. Finally, wetested a case where we decreased the mass ratio thresh-old for merger-driven AGN activity to 1:10, and at thesame time we decreased the mass limit to 0 . × M BH,σ to compensate. In this case the luminosity function issimilar to the case with 1:10 and 0 . × M BH,σ , but therelationship between MBH mass and V c is tilted, hav-ing a shallower slope that underestimates the real MBHmasses at the high V c end. If we were to choose 1:10 and0 . × M BH,σ , then the relationship between MBH massand V c is overall overestimated by more than a factor of2. In summary, we have run several tests to limit thespace of free parameters, and to disentangle weakly de-generate ones until we found the set that best matchesthe set of observational constraints.Accretion of molecular clouds affects the faint end ofthe luminosity function. One could in principle boost theprobability of MC accretion by assuming more compactclouds, however the mass gained through this process isconstrained by the faint end of the luminosity function ofAGN (Fig. 3), and only small variations can be toleratedby our model, as the current implementation gives a goodmatch with observations. We note that MC accretionaccounts for almost all sources up to L ≃ L ⊙ at z = 0 . − L ≃ L ⊙ at z = 2 − M h =2 × M ⊙ halos at z = 5 − z ∼
6, and that the mass density we obtain at z > × M ⊙ /Mpc , does not overproduce the X-raybackground (upper limit of 10 M ⊙ /Mpc , Salvaterra etal. 2012). Fig. 4 compares the theoretical mass functionof MBHs that power quasars with bolometric luminositylarger than 10 erg/s at z = 6 to the empirical massfunction derived by Willott et al. (2010) from a sampleof z = 6 quasars in the Canada–France High-z QuasarSurvey.As discussed above, our model, while far from beingccretion and spin 9 Figure 2.
Top panel: relationship between MBH masses and cir-cular velocity. Circles are model MBHs at z = 0, errorbars showthe datapoints collected in Kormendy & Bender (2011) and thebest fit derived in Volonteri et al. (2011b). Bottom panel: fractionof spheroids as a function of stellar mass. We scale from halo massto stellar mass through the data described in Fig. 1 of Hopkinsetal. 2010, assuming that a fixed fraction 10-50% of the baryons isin stars (open red points and filled orange ones respectively). Wecompare the fraction of spheroids to Conselice et al. (2006). able to explain every single detail of the MBH populationand its growth, qualitatively grasps most of the globalbehavior. We therefore consider our attempts to modelthe spin evolution also of qualitative nature. Regardlessof the simplified nature of our models, we can learn howdifferent patterns influence the evolution of MBH spins. ACCRETION AND SPIN EVOLUTION: RESULTSIn Figure 5 we show examples of spin evolution of anMBH hosted in a large spheroid today along its cosmichistory. Most of the accreted MBH mass is accumulatedduring episodes of efficient growth at early times (up un-til z ∼ a = 0 . a = 0 . z ≃ .
3. In the case of diskgalaxies, late phases of MC accretion at substantial ac-cretion rates ( > − in Eddington units ) contribute tosetting the final spin of the MBH (Fig. 6).Statistically, Fig. 7 shows the evolution of the loga-rithmic Eddington ratio as a function of redshift. Thetop panel shows a sample selected on the MBH mass( M > M ⊙ and M > M ⊙ ) showing classic “anti-hierarchical” behavior (Merloni 2004), with the the mostmassive MBHs being more active at earlier cosmic times.At a given luminosity threshold the typical Eddingtonratio decreases slightly at late times (bottom panel ofFig. 7), tracking instead the overall increase in the mass Figure 3.
Luminosity function at different redshifts. We showhere minimum and maximum values, considering both 1- σ sta-tistical uncertainties, using Poissonian statistics, and fraction ofabsorbed AGN (La Franca et al. 2005). Orange (45 ◦ hatching):coherent accretion during the decline of the quasar phase. Gray(horizontal hatching): chaotic accretion during the decline of thequasar phase. Figure 4.
Red histogram: theoretical mass function of MBHs at z = 6 that power quasars with bolometric luminosity larger than10 erg/s. Blue points: mass function derived by Willott et al.2010 from a sample of z = 6 quasars. of MBHs.In Fig. 8 we compare the spins of MBHs hosted in disksand spheroids, in different redshifts bins. At low MBHmass ( M ∼ M ⊙ ) MBHs hosted in gas-rich galaxiestend to have low spins. The spin distribution in gas-rich0 Volonteri et al. Figure 5.
Bottom panel: mass growth of a MBH in a galaxy thatbecomes a large spheroid by z = 0. The evolution is extractedfrom a merger tree describing a dark matter halo of mass 4 × M ⊙ at z = 0. The galaxy is the central galaxy in a group-sizedhalo. Middle panel: evolution of the Eddington rate vs redshift.Top panel: evolution of the spin parameter. After an early phaseof rapid accretion and growth the accretion rate declines and theMBH is fed by stellar winds only (quiescent phase) in the past ≃ q = 0 .
35 that occurred ∼ Figure 6.
Evolution of a MBH in a galaxy that becomes a MilkyWay-type disk by z = 0. The evolution is extracted from a mergertree describing a dark matter halo of mass 10 M ⊙ at z = 0. Thegalaxy is the central galaxy in a Local Group-sized halo. Note theoccurrence of MC accretion at z < .
5. Panels, lines and symbolsas in Fig. 5.
Figure 7.
Mean values of the Eddington ratio (logarithmic units)as a function of redshift. The top panel shows a sample selectedon the MBH mass (10 M ⊙ < M < M ⊙ and M > M ⊙ )showing classic “anti-hierarchical” behavior. The bottom panelfocuses instead on luminosity-selected AGN, for which the typicalaccretion rate is much higher. galaxies tends to move towards higher spins as mass in-creases. This is mostly related to MC accretion. For themost massive MBHs most of the growth occurs earlieron through merger-driven accretion that tends to spin-up MBHs. Accretion of MCs does not modify much thespins of these MBHs because the total angular momen-tum accreted through MCs is less than the total angularmomentum the MBH has (i.e., the total mass accretedby the MBH through MCs is much less than the massof the MBH). On the other hand, for low-mass MBHsthe mass accreted in MCs is of the same order as theMBH mass, therefore they have a stronger effect on thespin distribution, lowering the typical spin of low-massMBHs (we remind that we have assumed that MCs ac-crete isotropically on MBHs). The distribution of spins ofMBHs hosted in gas-poor galaxies has little dependenceon mass and redshift. In these galaxies, in general, mostMBHs have spin a ∼ . − .
8. Spins tend to slightlydecrease as MBH mass increases. We find no strong de-pendence on whether accretion occurs mostly chaoticallyor coherently after ‘feedback’ effects take place, exceptat the highest masses. We have run a test case wherewe have artificially “turned-off” spin evolution via MBH-MBH mergers (while keeping the mass increase throughmergers). In general, the effect of MBH-MBH mergers isto decrease the spins of the most massive MBHs in thecase of coherent post-feedback phase, while it increasestheir spins in the case of chaotic post-feedback phase.In Fig. 9 we focus on active MBHs. MBHs accret-ing at high rates, f Edd > .
1, have very large spinsat all z >
2. These are for the most part MBHs inthe “quasar” phase. At 1 < z < > M ⊙ )and reshift (0 . < z <
1) range is the most suitable toprobe how feedback affects the angular momentum ofnuclear gas. For low-mass BHs the spin distribution ismostly insensitive to the chaotic and coherent models(green triangles in Fig.9), while there is a stronger im-pact on high-mass holes (orange squares in Fig.9). There-fore the changes most affect the high-luminosity end ofthe luminosity function. Finally, at z < . DISCUSSION AND CONCLUSIONSWe developed a model for the evolution of MBHs thattakes into account several physical mechanisms of MBHgrowth: MBH-MBH mergers, merger-driven accretion,stochastic accretion, and accretion of recycled gas. Thismodel, however, does not include MBH feeding throughdisk instabilities, nor the burst phase of recycled gasfeeding in elliptical galaxies. Under a series of plausi-ble assumptions we have derived the growth of MBHs,the properties of the AGN population and the evolu-tion of MBH spins. Our approach produces a populationof MBHs and AGN consistent with the observed one interms of, e.g., luminosity function of AGN, relationshipbetween MBHs and their hosts, high-redshift quasars.The main results of our models of MBH evolution canbe summarized as follows: • At high-redshift MBHs grow mostly by merger-driven
Figure 8.
Spins in galaxies of different morphologies as a functionof MBH mass (mean and 1- σ dispersion). Filled circles: gas-poorgalaxies (spheroids). Stars: gas-rich galaxies (disks). Bottom: z < .
5. Top: 0 . < z <
1. Lower mass MBHs in gas-rich galaxies tendto spin less rapidly than higher mass ones, and also less rapidlythan MBHs in gas-poor galaxies.
Figure 9.
Spins in AGN, selected by mass and Eddington ratio.Triangles: MBH mass 10 M ⊙ < M < M ⊙ . Squares: MBHmass > M ⊙ . Top: QSOs accreting at high accretion rate (inEddington units) f Edd > .
1. Bottom: AGN accreting at all accre-tion rates f Edd > . Figure 10.
Top: radiative efficiency of AGN of different luminosi-ties. Diamonds: 10 erg s − < L bol < erg s − . Asterisks: L bol > erg s − . Bottom: spins of the same AGN. 30-40% oflow luminosity sources ( L bol ∼ erg s − ) are genuinely inef-ficient accretors. Sources with L bol > erg s − are poweredby efficient accretors, and spin solely determines their radiativeefficiency. Figure 11.
Spins of the primary MBHs in a binary prior to co-alescence (pink filled histogram) and spin of the newly mergedMBH post coalescence (violet hatched histogram) . Left: chaoticaccretion during the decline of the quasar phase. Right: coherentaccretion during the decline of the quasar phase. accretion, while at later times other channels becomemore important. In gas-rich galaxies, MC accretion dom-inates the growth of low-mass ( < M ⊙ ) MBHs at z <
2. In gas-poor galaxies, MBH-MBH mergers arethe main growth channel, especially at high MBH mass( > M ⊙ ); • The mass of most active black holes decreases with in-creasing cosmic time, in “anti-hierarchical” fashion. Sus-tained accretion grows the most massive black holes sinceearly times without overproducing the MBH populationas a whole; • MBH spins tend to be larger at redshifts z >
2, typ-ically a ≥ .
8. They result from massive, coherent ac-cretion events triggered by major mergers. This result isin general agreement with the trend found by Barausse(2012) using a complementary, more refined approach tomodeling galaxy evolution; • A significant drop in the average value of MBH spinstakes place at z <
2. This is caused by the increas-ing number of dry MBH-MBH mergers at lower red-shifts in the case of spheroids. Additionally, a dramaticdrop is predicted at z < .
5, for low-mass MBHs in gas-rich galaxies. This is due to low-mass, chaotic accretionevents involving capture of molecular clouds; • In general, in gas-rich galaxies at z < > M ⊙ ) the statisticsare poor in the case of disk galaxies (between a coupleand ∼
50 objects per bin), as the most massive amongMBHs tend to reside in gas-poor galaxies. • If outflows do not affect the angular momentum of nu-clear disks, and accretion proceeds coherently both in thequasar and decline phase, the spin distribution and itsevolution is not very differed for highly accreting vs lowaccreting MBHs. Differences are clearer in the popula-tion of the most massive black holes ( > M ⊙ ). • If quasar feedback disrupts the nuclear disk feeding theMBH and accretion proceeds chaotically in the declinephase, the spin distribution and its evolution shows astronger dependence on mass, but not on morphology,again, except at the highest masses ( > M ⊙ ) and z <
1. The same comment on statistical significance as aboveapplies here as well.Qualitatively similar results have been obtained by Li,Wang & Ho (2012) on observational grounds. They in-ferred the MBH spin evolution by tracing the evolution ofthe radiative efficiency of accretion flows, using the con-tinuity equation for the MBH number density. Both ourand Li et al. results seem to contradict the predictionsof the ‘spin paradigm’ scenario according to which thejet production efficiency - and therefore, radio-loudnessof AGN - should reflect the MBH spin distribution andits evolution (Wilson & Colbert 1995; Hughes & Bland-ford 2003). Applying such a scenario to our results,one should expect the radio-loud fraction of AGN to bemuch larger at high redshifts than in the present epoch.At least in the case of quasars, an opposite trend hasbeen inferred, i.e. such a fraction has been suggestedto decrease with redshift (Jiang et al. 2007), althoughVolonteri et al. (2011c) find that the radio-loud fractionis roughly constant with redshift for the most luminoussources (
L > erg/s). For high-redshift blazars pow-ccretion and spin 13ered by M > M ⊙ MBHs, also, activity seems to peakaround z ∼ ≈ . ÷ .
20, with some tension among thepublished results that can be traced back to the partic-ular choice of AGN LF and/or scaling relation assumedto derive the local mass density.Recently, Gilfanov and Merloni (2013), have summa- rized our current estimate of the (mass-weighted) aver-age radiative efficiency in just one formula, relating h ǫ i tovarious sources of systematic errors in the determinationof supermassive black hole mass density:1 − ξ i − ξ CT + ξ lost = 1 − h ǫ ih ǫ i R (24)where ξ = ρ BH , z=0 / . × M ⊙ Mpc − is the local( z = 0) MBH mass density in units of 4.2 × M ⊙ Mpc − (Marconi et al. 2004) and using the integrated bolo-metric luminosity function from Hopkins et al. (2007),they obtain R ∼ . /ξ . Here ξ i is the mass density ofblack holes at the highest redshift probed by the bolo-metric luminosity function, z ≈
6, in units of the lo-cal one, and encapsulates uncertainties on the process ofMBH formation; ξ CT is the fraction of SMBH mass den-sity (relative to the local one) grown in unseen, heavilyobscured, Compton Thick AGN, still missing from ourcensus; finally, ξ lost is the fraction black hole mass con-tained in “wandering” objects, that have been ejectedfrom a galaxy nucleus following, for example, a merg-ing event and the subsequent production of gravitationalwave, the net momentum of which could induce a kickcapable of ejecting the black hole form the host galaxy.The model presented in this paper allows us to provide anestimate of two of the unknowns in Equation 24: ξ i and ξ lost . In our models they are both of order 0.1 and theyroughly cancel each other. We can also directly estimate h ǫ i = ( P ǫ i ∆ M MV,i ) / P ∆ M MV,i = 0 . − .
18 (the sumis done for all accreting MBHs starting from z = 20 downto z = 0, so it is an average over mass and time), leadingto an estimate of ξ CT ∼ .
5. Current estimates of theCompton Thick AGN fraction (that are, however, not“mass weighted” as in the formalism of Equation 24, seealso the note in the Appendix) based on local Universeand models of the X-ray background range between 20and 50% (e.g., Akylas et al. 2012).We thank M. Dotti for helpful discussions and com-ments. MV acknowledges funding support from NASA,through award ATP NNX10AC84G; from SAO, throughaward TM1-12007X, from NSF, through award AST1107675, and from a Marie Curie Career Integrationgrant (PCIG10-GA-2011-303609). JPL was supportedin part by a grant from the French Space Agency CNES,by the Polish Ministry of Science and Higher Educa-tion within the project N N203 380336, and by thePolish National Science Centre through grant DEC-2012/04/A/ST9/00083.
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BLACK HOLE GROWTH
We here recall how one can describe the growth of a MBH as a function of a constant or variable Eddington rate.We start by defining f Edd = L/L
Edd , and L Edd = M c /t Edd , where t Edd = σ T c π G m p = 0 .
45 Gyr and f Edd representsthe Eddington fraction. Therefore, if the accretion rate is ˙ M in , and ˙ M is the mass that goes into increasing the MBHmass: L = ǫ ˙ M in c = f Edd L Edd c (A1)and dM = (1 − ǫ ) dM in and dMdt = 1 − ǫǫ L Edd f Edd c = 1 − ǫǫ f Edd
M c t Edd , (A2)one obtains dMM = 1 − ǫǫ f Edd t − dt. (A3)If ǫ and f Edd are constant over the time of integration, then: Z M − dM = Z − ǫǫ f Edd t − dt, (A4)and the MBH mass grows as M ( t + ∆ t ) = M ( t ) exp (cid:18) f Edd ∆ tt Edd − ǫ ( t ) ǫ ( t ) (cid:19) , (A5)while if, for instance f Edd is a function of time, one has to self-consistently integrate: Z M − dM = Z − ǫǫ t − f Edd ( t ) dt, (A6)as shown in section 4.2.Note that this mathematical formalism differs from the approximate form M BH = (1 − ǫ ) ˙ M in ∆ t (the two expressionsagree in the limit − ǫǫ f Edd ∆ t t − → ǫ , f Edd and ∆ t the approximate expressionunderestimates the mass growth, and therefore, statistically, So ltan’s argument tends to underestimate ǫ with respectto our formalism. ALIGNMENT OF BLACK HOLE SPINS IN ACCRETION DISCS
In this paper we have assumed that most MBHs evolve in thin accretion discs where the importance of jetsand magnetic fields is limited. In this case warp propagation occurs diffusively (Bardeen & Petterson 1975;Papaloizou & Pringle 1983). In thick accretion discs (
H/R > α ) warp propagation occurs instead through bend-ing waves (Nelson & Papaloizou 2000), while in magnetized discs with jets a “magneto-spin alignment” mechanismhas been recently discovered in numerical simulations (McKinney et al. 2012).We refer the reader to Nelson & Papaloizou (2000); Sorathia et al. (2013) and references therein for a full discussionof the mathematical treatment and the differences between diffusive and wave propagation, and we summarize herethe relevant information. Bardeen & Petterson (1975) showed that a viscous disc would be expected to relax to a formin which the inner regions become aligned with the equatorial plane of the black hole (Lense-Thirring precession) outto a transition radius, beyond which the disc remains aligned with the outer disc. This because the Lense-Thirringprecession rate drops off sharply as the radius increases. The transition radius ( r tr ) is expected to occur approximatelywhere the rate at which Lense-Thirring precession is balanced by the rate at which warps are diffused or propagatedaway.In the diffusive regime, the warping of the disc is counteracted by diffusion of the warp, which acts over the diffusiontimescale t diff ∼ r α/ ( H Ω). In the bending wave regime warps evolve on the sound crossing time t wave ∼ r/c ss