Towards an understanding of the evolution of the scaling relations for supermassive black holes

0.26 MB·English

by  C. M. Booth

Mon.Not.R.Astron.Soc.000,1–8(2010) Printed4January2011 (MNLATEXstylefilev2.2)
Towards an understanding of the evolution of the scaling
relations for supermassive black holes.
C. M. Booth1⋆ and Joop Schaye1
1Leiden Observatory,Leiden University,PO Box 9513, 2300 RA Leiden, the Netherlands
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ABSTRACT
J
The growth of the supermassive black holes (BHs) that reside at the centres of most
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galaxies is intertwined with the physical processes that drive the formation of the

galaxiesthemselves.Theevolutionofthe relationsbetweenthe massofthe BH, mBH,
]
O andthepropertiesofitshostthereforerepresentcrucialaspectsofthegalaxyformation
process.Weuseacosmologicalsimulation,aswellasananalyticalmodel,toinvestigate
C
how and why the scaling relations for BHs evolve with cosmic time. We find that a
.
h simulation that reproduces the observedredshift zero relations between mBH and the
p properties of its host galaxy, as well as the thermodynamic profiles of the intragroup
o- medium, also reproduces the observed evolution in the ratio mBH/m∗ for massive
r galaxies, although the evolution of the mBH/σ relation is in apparent conflict with
t observations.ThesimulationpredictsthattherelationsbetweenmBH andthebinding
s
a energies of both the galaxy and its dark matter halo do not evolve, while the ratio
[ mBH/mhalo increases with redshift. The simple, analytic model of Booth & Schaye

(2010), in which the mass of the BH is controlledby the gravitationalbinding energy
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ofitshosthalo,quantitativelyreproducesthelattertworesults.Finally,wecanexplain
v
4 the evolution in the relations between mBH and the mass and binding energy of the
4 stellar component of its host galaxy for massive galaxies (m∗ ∼ 1011M⊙) at low
8 redshift (z <1) if these galaxies grow primarily through dry mergers.
0
Keywords: Cosmology:Theory–Galaxies:Active–Galaxies:Evolution–Galaxies:
.
5 Formation – Hydrodynamics – Galaxies: Quasars:General
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0
1
:
v
1 INTRODUCTION mation. However, despite a wide variety of theoretical and
i
X observationalstudies,theoriginoftheserelationsisstillde-
Over the past decade it has become clear that the su-
bated.ThestudyoftheevolutionoftheBHscalingrelations
r permassive black holes (BHs) found at the centres of
a thereforerepresentsacrucialaspectofthegalaxyformation
virtually all galaxies with spheroidal components, have
process that may provide us with additional clues regard-
masses that are coupled to the properties of their host
ing the physical processes that give rise to the BH scaling
galaxies (Magorrian et al. 1998; Ferrarese & Merritt 2000;
relations.
Tremaine et al. 2002; H¨aring & Rix 2004; Hopkinset al.
2007). Additionally, there exists evidence that BH masses Addressing these questions observationally is challeng-
are coupled to the properties of the dark matter haloes ing.Duetotheirextremelyhighluminosities,brightquasars
in which they reside (Ferrarese 2002; Booth & Schaye provide a promising route to measuring BH masses at high
2010). Further correlations between quasar activity (e.g. redshift through the widths of low-ionization lines that are
Boyle & Terlevich 1998) and the evolution of the cosmic associated with the broad-line region close to the BH and
star formation rate (e.g. Madau et al. 1996) provide evi- usingtheassumption of virialequilibrium(e.g. Vestergaard
dencethatthereexistsalinkbetweengalacticstarformation 2002). It has, however, been claimed that this procedure
and accretion onto a central AGN. systematicallyunderestimatesBHmasses(Jarvis & McLure
It has long been recognised that the growth of BHs is 2002).Measuringgalaxymassesfortheseobjectsisverydif-
likelyself-regulated (Silk & Rees1998)andthatthesetight ficult as the BH outshines the galaxy by a large factor (see
correlations indicate that the growth of BHs is tightly in- e.g. the discussion in Merloni et al. 2010). Since AGN sur-
tertwined with thephysicalprocesses that drivegalaxy for- veysare biased towards more massive black holes, selection
effectsalsoneedtobetakenintoaccount(e.g.Shen & Kelly
2009;Bennert et al.2010),whichcanmakeitdifficulttodis-
⋆ E-mail:[email protected](CMB) tinguish between evolution in the normalization and in the
(cid:13)c 2010RAS
2 C. M. Booth & J. Schaye
scatterinthescalingrelations(Lauer et al.2007).Inspiteof spread in thepredictions for theevolution of themBH−σ∗
these difficulties, measurements of the BH scaling relations relationmayreflectthatitismoredifficulttopredictveloc-
havebeen made as far out as redshift three. ity dispersions, which depend on both mass and size, than
McLure et al. (2006) found that the BHs associated it is to predict masses.
with radio loud AGN residing in galaxies of a given stellar In Booth & Schaye (2009, hereafter BS09) we pre-
mass are a factor of four more massive at redshift two than sentedself-consistent,hydrodynamicalsimulationsoftheco-
in thelocal Universe.Decarli et al. (2010) studied theCIV evolution of theBH and galaxy populations that reproduce
line associated with thequasarbroad lineregion in R-band the redshift zero BH scaling relations. These same simula-
selected hosts at both redshifts zero and three and found tions also match group temperature, entropy and metallic-
that BHs are typically a factor of seven more massive at ity profiles, as well as the stellar masses and age distribu-
highredshiftforagivengalaxymass.Theseresultsarecon- tionsof brightest group galaxies (McCarthy et al. 2010).In
sistent with otherobservational studies(Walter et al. 2004; Booth & Schaye (2010) (hereafter BS10) we used the same
Peng et al.2006a,b;Merloni et al.2010;Greene et al.2010; simulations, as well as an analytic model, to demonstrate
Bennert et al. 2010). Taken together, these papers suggest that mBH is determined by the mass of the dark matter
an emerging consensus that at higher redshift BHs in hosts (DM) halo with a secondary dependence on the halo con-
of a given mass are systematically more massive than in centration, of the form that would be expected if the halo
thelocalUniverse,althoughseeJahnke et al.(2009)forone bindingenergywerethefundamentalpropertythatcontrols
study that findsnosignificant evolution. the mass of the BH. In the present work we use the same
The evolution of the relation between BH mass, m , modelstoinvestigatewhyandhowtheBHscalingrelations
BH
and stellar velocity dispersion, σ∗, has been studied util- evolvefor massive galaxies.
ising the width of the OIII line as a proxy for stellar This paper is organised as follows. In Sec. 2 we sum-
velocity dispersion (Nelson & Whittle 1996). These studies marise the numerical methods employed in this study and
suggest that the mBH −σ∗ relation either does not evolve thesimulationanalysed.InSec.3wepresentpredictionsfor
(Shields et al. 2003; Gaskell 2009), or does so weakly, with theevolutionoftheBHscalingrelationsandcomparethem
BHs∼0.1−0.3dexmoremassiveatz=1(Salviander et al. toobservations.WefindthattheevolutioninthemBH−m∗
2006;Gu et al. 2009; Woo et al. 2008; Treu et al. 2007). relation predicted by the simulations is in excellent agree-
Theevolution oftheBHscalingrelationshasalso been ment with the observations, while the measured weak evo-
studied using numerical simulations (e.g. Robertson et al. lution in themBH−σ relation is in apparent disagreement,
2006;Johansson et al. 2009) and semi-analytic models (e.g. and predict that while BH mass increases with redshift for
Malbon et al.2007;Lamastra et al.2010;Kisaka & Kojima fixedhalomass,therelationsbetweenmBH andthebinding
2010). Robertson et al. (2006) employed simulations of ide- energies of both the host galaxies and DM haloes do not
alised galaxy mergers, initialised to have properties typical evolve.Wedemonstratein 4that theaanalytic description
of merger progenitors at various redshifts, to construct the in which mBH is coupled to the DM halo binding energy
relation between galaxy stellar mass, m∗, and σ∗ as a func- canreproducetheevolutionoftherelationbetweenBHand
tion of redshift and found that, at a given value of σ∗, the halo mass. Furthermore, we show that the evolution in the
corresponding m decreases mildly with increasing red- relations between the BH and the stellar mass and binding
BH
shift. At z = 1 the simulations of Di Matteo et al. (2008) energycanbeunderstoodintermsofthemorefundamental
have BHs that lie slightly above the z = 0 normalization relation with the binding energy of the dark halo and the
of the m −σ relation. However, these simulations were growthofmassivegalaxiesthroughdrymergers.Finally,we
BH
stopped at z = 1 and so cannot inform us about the evo- summarise our main conclusions in Sec. 5.
lution of the mBH−σ∗ toward lower redshift. However, for
z>1theypredictaweakevolutioninthemBH−σ∗relation
such that at higher redshift galaxies of a given velocity dis-
2 NUMERICAL METHOD
persion contain slightly less massive BHs. Johansson et al.
(2009)employedsimilar numericaltechniquestoarguethat Wehavecarriedoutacosmologicalsimulationusingasignif-
it is unlikely that BHs are able to form significantly be- icantly extended version of the parallel PMTree-Smoothed
fore their host bulges. Semi-analytic models that reproduce Particle Hydrodynamics (SPH) code gadget iii (last de-
many redshift zero properties of galaxies also predict that, scribed in Springel 2005). The simulation and code are
at a fixed σ∗, BH masses decrease with increasing redshift described in detail in BS09, we provide only a brief sum-
(Malbon et al. 2007). These theoretical models thus pre- mary here. In addition to hydrodynamic forces, we treat
dict evolutionary trends that go in the opposite direction star formation (Schaye& Dalla Vecchia 2008), supernova
to those inferred from observations. Finally, the models of feedback (Dalla Vecchia & Schaye 2008), radiative cool-
Hopkinset al. (2009) predict that, at a fixed stellar veloc- ing(Wiersma et al.2009a),chemodynamics(Wiersma et al.
ity dispersion, BH masses at higher redshift are either the 2009b) and black hole accretion and feedback (BS09,
same (for mBH ∼ 108M⊙) or slightly more massive (for Springel et al. 2005). We summarise in Sec. 2.1 the essen-
mBH > 108M⊙) at fixed σ∗ than their redshift zero coun- tial features of the BH model.
terparts, in agreement with observation. The properties of central galaxies and DM haloes
On the other hand, the relation between BH mass and are calculated by first identifying the most gravitationally
galaxy bulge mass shows a positive evolution in both semi- boundparticleineachDMhalousingthealgorithmsubfind
analytic models (Malbon et al. 2007; Hopkinset al. 2009) (Springelet al. 2001; Dolag et al. 2009), which is then con-
andnumericalsimulations(Di Matteo et al.2008),themag- sideredthehalocentre.Allstarswithinaradiusof0.15r
halo
nitude of which is comparable to that observed. The larger are then assigned to the central galaxy. We note that our
(cid:13)c 2010RAS,MNRAS000,1–8
Evolution of the black hole scaling relations 3
conclusions are insensitive to the exact choice for this ra- ties gives very similar results (Booth & Schaye 2009). This
dius, and whether we use a fixed physical value or a fixed is because all BHs accrete almost all of their mass in short,
fraction of the halo virial radius. As long as the sphere en- (near) Eddington-limited bursts of accretion and thus that
closes the central object, our results are insensitive to this our treatment of accretion in low-density environments is
choice. Halo mass, m , is calculated as the total mass lessimportant.Theaccretionmodelinhigh-densityenviron-
halo
enclosed within a sphere, centred on the most bound par- mentsisnecessarilyverycrude,butwenotethatourresults
ticle in the halo, that has a mean density of 200 times the are insensitive to the details of the accretion model as long
mean density of the Universe and the virial radius r is asαissufficientlylargethattheBHsbecomemoremassive
halo
theradiusofthissphere.Becauseitisnotexpectedthatthe thanobservedintheabsenceoffeedback andiftworeason-
samephysicsholdsforboththecentralgalaxyofahaloand ableconditionsaremetthatarenecessaryforself-regulation
its satellites, which are expected to rapidly have their gas to be possible. Firstly, the BH accretion rate must increase
supplystrippedwhentheybecomesatellites, ouranalysisis with increasing density, and secondly it must increase with
restrictedtoBHsidentifiedasresidinginthecentralgalaxy BH mass (see BS09).
in a DM halo, defined as thegalaxy closest to thecentre of EnergyfeedbackisimplementedbyallowingBHstoin-
theDM potential well of each halo. ject a fixed fraction of the rest mass energy of the gas they
accreteintothesurroundingmedium.Theenergydeposition
rate is given by
2.1 The black hole model
ǫ ǫ
SeedBHsofmassmseed =10−3mg ≈105M⊙ –wheremg is E˙ =ǫfǫrm˙accrc2 = 1−f rǫrm˙BHc2, (1)
thesimulationgasparticlemass–areplacedintoeveryDM
where m˙ is the rate at which the BH is accreting gas,
halo that contains more than 100 DM particles (which cor- accr
and m˙ is therate of BH mass growth.
responds to a DM halo mass of 4.1×1010M⊙/h) and does WBeHset ǫ to be 0.1, the mean value for ra-
not already contain a BH particle. Haloes are identified by r
diatively efficient accretion onto a Schwarzschild BH
regularlyrunningafriends-of-friendsgroupfinderduringthe
(Shakura& Sunyaev1973)anduseǫ =0.15asourfiducial
simulation.Afterforming,BHsgrowbytwoprocesses:accre- f
value. It was shown in BS09 that, for ǫ = 0.15, this simu-
tionofambientgasandmergers.Gasaccretionoccursatthe f
minimumoftheEddingtonrate,m˙ =4πGm m /ǫ σ c lation reproducestheobserved redshift zero mBH−m∗ and
andm˙ =α4πG2m2 ρ/(c2+v2)E3d/d2,wheremBHistpherprTo- mBH−σ∗relations.Energyisreturnedtothesurroundingsof
accr BH s p the BH ‘thermally’, by increasing the temperature of N
ton mass, σ is the Thomson cross-section, ǫ is the radia- heat
T r oftheBH’sneighbouringSPHparticlesbyatleast∆T .A
tive efficiency of the BH, c is the speed of light, c and ρ min
s BHperformsnoheatinguntilithasbuiltupenoughofanen-
are thesound speed and gas density of thelocal medium,v
ergyreservoirtoheatbythisamount.Imposingaminimum
is the velocity of the BH relative to the ambient medium,
temperatureincreaseensuresthattheradiativecoolingtime
and αis a dimensionless efficiency parameter. Theparame-
is sufficiently long for the feedback to be effective. We set
terαaccountsforthefact that oursimulations possess nei-
N =1and∆T =108Kbuttheresultsareinsensitive
ther the necessary resolution nor the physics to accurately heat min
to theexact valuesof theseparameters (see BS09).
model accretion onto a BH on small scales. Note that for
α = 1 this accretion rate reduces to the so called Bondi-
Hoyle(Bondi & Hoyle1944) rate.
2.2 The cosmological simulation
As long as we resolve the scales and physics relevant
to Bondi-Hoyle accretion, we should set α = 1. If a simu- The simulation employed in the current work uses a cu-
lation resolves the Jeans scales in theaccreting gas, then it bic box of size 50 comoving Mpc/h and assumes periodic
will also resolve the scales relevant for Bondi-Hoyle accre- boundaryconditions.Thesimulationcontains2563 particles
tionontoanyBHlargerthanthesimulationmassresolution ofbothgasandcollisionlesscoldDMandisevolveddownto
(BS09). We therefore generally set α equal to unity. How- redshift zero. The DM and initial baryonic particle masses
ever,thisargumentbreaksdowninthepresenceofamulti- are 4.1×108M⊙/h and 8.7×107M⊙/h, respectively. Co-
phase interstellar medium, because our simulations do not movinggravitational softenings are set to1/25 ofthemean
resolve the properties of the cold, molecular phase, and as interparticleseparationdowntoz=2.91,belowwhichthey
such the accretion rate may be orders of magnitude higher switch to a fixed proper scale of 2kpc/h. The simulation
than the Bondi-Hoyle rate predicted by our simulations for employed in this study was previously also analysed as the
star-forming gas. We therefore use a power-law scaling of fiducialsimulation in BS09 and BS10
the accretion efficiency such that α = (n /n∗)β in star- Comparison of the simulation employed in this study
H H
forming gas, where n∗ = 0.1cm−3 is the critical density to an otherwise identical one with eight (two) times lower
H
fortheformation ofacold,star-forming gasphase.Thepa- mass (spatial) resolution informs us in what mass and red-
rameter β is a free parameter in our simulations. We set shift range therelations between m and galaxy and halo
BH
β = 2, but note that the results shown here are insensitive properties are numerically converged.The relation between
to changes in this parameter when β &2. We note that we mBH and both m∗ and mhalo is numerically converged up
do not resolve the Bondi radius of BHs less massive than to redshift two for all haloes with m > 10m . Mea-
BH seed
theparticle mass in oursimulations, and thatfor BHswith surements of stellar velocity dispersion are, however, only
m ∼m theBondiradiusisunresolvedunlessthedensity convergedforz<1andm >102m .Wewillonlygive
BH g BH seed
is low or the temperature high. Our choice of α = 1 there- resultsforredshiftsandBHmassesforwhichtheresultsare
fore provides an underestimate of thetrueaccretion ratein converged with respect to numerical resolution.
these regimes. However, even setting α= 100 for all densi- Wenotethatthesesimulationsdonotresolvethescales
(cid:13)c 2010RAS,MNRAS000,1–8
4 C. M. Booth & J. Schaye
Table 1. Evolution of the normalisation of the power-law rela-
tionsbetweenBHmassandthepropertiesofitshost.Thesample
includesallgalaxiesthatcontainaBHwithmBH>102mseed≈
107M⊙, corresponding to 162 (132) haloes at z = 0 (z = 1).
The median BH, stellar, and halo mass are 107.6, 1010.6, and
1012.8M⊙, respectively. The central column gives the median
change in log10mBH at redshift one relative to z = 0 for fixed
valuesofthequantitylistedintheleftcolumn.Therightcolumn
showstheslope,αs,ofthebestfitpower-lawdescribingtherate
ofevolutionofthescalingrelation(Eq.2)overtheredshiftrange
0−1.Fromtoptobottom,weconsiderevolutionofmBHforfixed
stellarmass,halomass,centralstellarvelocitydispersion,galaxy
bindingenergy (m∗σ∗2), and halobindingenergy (Eq. 3). Errors
arecalculated from103 bootstrapresamplingsofthedata.
Variable ∆log10 mmBBHH((zz==01)) αs
Figure 1. The median mBH/m∗ ratio as a function of redshift m∗ 0.20±0.05 0.52±0.05
for galaxies of different stellar masses (left panel) and for BHs mhalo 0.23±0.03 0.65±0.06
above a given mass (right panel) predicted by the cosmological σ∗ −0.09±0.04 −0.32±0.05
simulation. The points with error bars show the measurements U∗ 0.02±0.03 0.05±0.06
of Merlonietal. (2010, red triangles)for asample of galaxies in Uhalo 0.01±0.03 0.03±0.05
the mass range 10.5 < log(m∗/M⊙) < 11.5 and Decarlietal.
(2010,bluediamonds) forasampleof galaxies withtypical stel-
larmasses∼1011M⊙.Weshowanumberofdifferentmasscuts 1 of the amplitude of the relations between m and the
BH
forthe simulationresultstodemonstrate that theresultsarein-
masses,velocitydispersionsandbindingenergiesofboththe
sensitivetotheparticularmasscutchosen.Boththesimulations
host galaxies and the host DM haloes. The central column
and observations in this plot show the total stellar mass of the
gives log m (z = 1)−log m (z = 0), calculated by
galaxies.ThesimulationpredictsthatthemedianmBH/m∗ ratio 10 BH 10 BH
fitting power-law relations under the assumption that the
increaseswithredshift.Thepredictionsareinexcellentagreement
withtheobservations,regardlessoftheparticularmasscutmade. slopesofthescalingrelationsdonotevolve,whichisagood
approximationintheredshiftrangestudiedhere1.Theright-
mostcolumnofTable1givestheslope,α ,ofthepower-law
s
on which the BH is the gravitationally dominant compo- evolution in theamplitude of each scaling relation
nent in thegalaxy and so cannot be used to study BH self-
m
regulation on the smallest scales. However, the simulations BH ∝(1+z)αs, (2)
Xn0
dohavesufficientresolutionforbaryonstobegravitationally
whereX isoneofthevariableslistedintheleftcolumn,and
dominant in the centres of haloes. We cannot conclusively
n is the slope of the m −X relation at z = 0. We find
rule out that if we increased our mass resolution signifi- 0 BH
cantlyandusedmoresophisticatedsub-gridmodelsthatthe that αs = 0.52±0.05 for the mBH−m∗ relation, in good
agreement with DiMatteo et al. (2008), who found α =
BHwouldself-regulate ondifferentscales. However,sugges- s
0.5.
tively,inBooth & Schaye(2010)weverifiedthatinsimula-
tionswithaspatialresolutionof0.5kpc/hthat,atz =2,the The evolution of the mBH−σ∗ relation is smaller but
significant, with α = −0.32 ± 0.05. This is in apparent
BH masses scale in thesame way as in the lower-resolution s
disagreement with various observational studies that either
simulations.
infer a positive (Salviander et al. 2006; Treu et al. 2007;
Woo et al.2008;Gu et al.2009)ornegligible(Shields et al.
2003; Gaskell 2009) evolution in the normalisation of the
3 SIMULATION RESULTS mBH−σ∗ relation. The predicted evolution does, however,
3.1 The evolving relations between black holes agree with other simulation studies (Robertson et al. 2006)
and galaxies and semi-analytic models (Malbon et al. 2007). Taken to-
gether, the simulation predicts that the mBH − m∗ and
Fig. 1 compares the predicted evolution of the median
mBH/m∗ ratiofordifferentminimumstellar(leftpanel)and
BH(rightpanel)masseswithobservationsofAGNingalax- 1 At z = 0 (z = 1) the slope of the relation between mBH and
ies with m∗ ∼ 1011M⊙. The mBH/m∗ ratio increases with Uhalois1.01±0.14(0.96±0.17)andtheslopeoftherelationbe-
redshift,incloseagreementwithobservations(Merloni et al. tweenmBHandU∗is0.93±0.07(0.96±0.09),consistentwiththe
2010;Decarli et al.2010).Atredshiftzerothisagreementis z=0observationalresultsofFeoli&Mele(2007)andFeolietal.
unsurprising because the efficiency of AGN feedback in the (2010).Atallredshiftstheseslopesareconsistentwithunity.The
simulation was tunedtoreproducethenormalisation of the slopes of the relations between mBH and m∗, σ∗ and mhalo are
1.16±0.06(1.2±0.2),4.6±0.8(4.4±0.8)and1.5±0.2(1.5±0.3).
z=0BHscalingrelations.Theagreementathigherredshift
Thereisthusnoevidenceforevolutioninanyoftheslopes,which
represents, however, a non-trivial prediction of a model in
agreeswiththeresultsofRobertsonetal.(2006)forthemBH−σ∗
whichBHsself-regulatetheiraccretionthroughthecoupling
relationandwithDiMatteoetal.(2008)forthemBH−m∗ rela-
of a small fraction of the radiative energy to the ambient tion.Theslopesofthez=0relationsareconsistent withobser-
medium. vations(H¨aring&Rix2004;Tremaineetal.2002;Bandaraetal.
In Table 1 we show the predicted evolution up to z = 2009).
(cid:13)c 2010RAS,MNRAS000,1–8
Evolution of the black hole scaling relations 5
mBH −σ∗ relations evolve such that the relation between
mBH andstellarbindingenergy(∝m∗σ∗2)isindependentof
redshift (α =0.05±0.06).
s
It is tempting to conclude from the finding that the
ratio mBH/(m∗σ∗2) does not evolve that theBH mass is de-
termined by the binding energy of the galaxy. However, we
demonstratedexplicitlyinBS10thattheBHmassisinstead
controlled by the binding energy of the DM halo. This im-
pliesthatthebindingenergyofthegalaxytracksthebinding
energyofthehalo,whichwewillconfirmandexplainbelow.
3.2 The evolving relations between black holes
and dark matter haloes
Wenowturnourattentiontotherelationsbetweenthemass
of theBH and theDM haloin which it resides. InBS10 we
argued that a BH grows until it has injected an amount of
energy into its surroundings that scales with the binding Figure2.PredictedevolutionofthenormalisationofthemBH−
energy of its host DMhalo. Wetherefore donot expect the mhalorelation.Thered,dottedcurveshowsthemedianevolution
predictedbythecosmologicalsimulationwhenallBHsforwhich
mBH−Uhalorelationtoevolve.Indeed,inthesimulationthe bothBHandhalopropertiesarewellresolved(mBH>102mseed)
amplitude of this relation is independent of redshift, with areincluded.Thegreyregionrepresentstheallowedrangeinevo-
αs = 0.03±0.05, as would be expected for a fundamental lution predicted by the analytic model of BS10, which assumes
link between mBH and the binding energy of the host DM thattheBHmassiscontrolledbythebindingenergyoftheDM
halo. halo.ThebindingenergyofanNFWhalodependsonmass,red-
We do, however, expect the relation between m and shift,concentration (whichitselfdepends onbothmassandred-
BH
mhalo to evolve. At higher redshift, haloes of a given mass shift) and on the radius, rej, at which it is evaluated. The grey
aremorecompact thantheirredshiftzerocounterpartsand region corresponds to rej/rhalo =0.1−1.0 (bottom to top) and
are thus more strongly gravitationally bound. This means the solid black line to rej = 0.22rhalo, the value for which we
that, at a fixed halo mass, more energy is required to eject which we predict mBH ∝ m1h.a5l5o at z = 0, in accord with both
observations (Bandara et al. 2009) and simulation (BS10). For
gasfromhaloesathighredshiftandinordertoself-regulate,
comparison, the black dashed line shows the evolution that our
BHsmustgrowtobemoremassive.Fig.2showsthenormal- analytic model would have predicted if we had ignored the evo-
isation ofthemBH−mhalo relation asafunctionofredshift lutionofthec(mhalo)relation.Atallredshiftsthenormalisation
and confirms that the simulation predicts the amplitude of ofthesimulatedmBH−mhalo relation(red,dottedcurve)agrees
this relation to increase with redshift (red, dashed curve), withthatpredictedbytheanalyticmodel basedontheassump-
with α =0.65±0.06. tion that the fundamental relation is between BH mass and the
s
bindingenergyoftheDMhalo.
4 EXPLAINING THE EVOLUTION thefunction
As we already noted, the idea that the binding energy of c
f(c,x)= ×
thedarkhalocontrolsthemassoftheBHexplainsourfind- ln(1+c)−c/(1+c) 2
ing that the m −U relation is independent of red-
BH halo
shift. We will now show that the analytic model of BS10 (cid:0)1− 1 − 2ln((cid:1)1+cx) . (4)
also reproduces the evolution of the mBH−mhalo relation (1+cx)2 1+cx !
andthatitcanexplaintheobservedevolutioninthescaling
Simulationshaveshownthatcisafunctionofbothredshift
relations with thestellar properties if theobserved galaxies
and halo mass, and scales approximately as2 (Duffy et al.
evolvepredominantly through dry mergers, as predicted by
2008)
thesimulation.
c∝m−0.1(1+z)−0.5. (5)
halo
4.1 The mBH−mhalo relation Combining Eqs. 3-5, BS10 found the slope of the mBH −
m relation to be weakly dependent on r . At z = 0 it
If the energy injected by a BH is proportional to the halo halo ej
variesfromn =1.50forx=0.1ton =1.61forx=1.0.In
gravitationalbindingenergy,U ,then,foraDMhalowith 0 0
halo ordertoexactlymatchtheslopeof1.55thatisbothobserved
an NFW (Navarroet al. 1997) density profile(BS10)
(Bandara et al. 2009 find1.55±0.31) and predicted by the
m2
m ∝U ∝ halo ∝f(c,x)(1+z)m5/3 , (3)
BH halo r halo
halo
wheremhalo isthehalomass,c(mhalo,z)isthehaloconcen- 2 Notethattheslopeofthepower-lawdependenceofconcentra-
tration, x is defined to be x ≡ rej/rhalo, rej is the physical tiononredshiftdependsonthehalodefinitionused(Duffyetal.
scale onwhich BHself-regulation takesplace,andf(c,x)is 2008).
(cid:13)c 2010RAS,MNRAS000,1–8
6 C. M. Booth & J. Schaye
simulations (BS10 find 1.55±0.05 for the same simulation
as is analyzed here3), we would need to use x=0.22.
By using an NFW density profile and Eq. (5), we have
implicitly assumed that the dark matter profile is well de-
scribed by the results obtained from simulations that in-
clude only dark matter. Duffy et al. (2010) have recently
shownthat,onthescales ofinteresthere,theback-reaction
of the baryons onto the dark matter is in fact very small
if feedback from AGN is included,as required to reproduce
theobservedstellarandgaspropertiesofgroupsofgalaxies
(McCarthy et al. 2010; Puchwein et al. 2008; Fabjan et al.
2010;Duffy et al. 2010).
If,asarguedinBS10,theBHmassiscontrolled bythe
DM halo binding energy, then we expect the m −m
BH halo
relation to evolve because the halo binding energy depends
not only on halo mass, but also on the virial radius and
concentration, both of which vary with redshift for a fixed
halo mass.
If the c−m relation did not evolve, then we would
halo
expectm (m )∝(1+z)(Eq.3).However,becausehalo
BH halo
concentration decreases with redshift (Eq. 5) we expect the
actual evolution of the m −m relation to be weaker,
BH halo
i.e.α <1.TheresultingrelationbetweenBHmassandDM
s Figure 3. Galaxy properties as a function of redshift for m∗ ≈
halobindingenergypredictsthat,atagivenmhalo,mBH in- 1011M⊙ (ateachredshiftweselectedthe20galaxieswithstellar
creases with redshift and that by z = 2 BHs are between massesclosesttothisvalue).Theyellowshadedregionshowsthe
1.5 (for rej/rhalo = 0.1) and 2.6 (for rej/rhalo = 1.0) times area that contains the 25th to 75th percentiles of the data and
more massive than at redshift zero. For our fiducial radius theblack,solidcurveshowsthemedian.Thetoppanelshowsthe
of self-regulation of x=0.22, BHs are 2.1 times more mas- ratioofstellartohalobindingenergyandthebottompanelshows
sive, in excellent agreement with the simulation prediction thespecificstarformationrate(SSFR≡m˙∗/m∗).Thedashedline
of α =0.65±0.06 (Table 1). inthebottompanelshowstheinverseoftheageoftheUniverse.
s
GalaxieswithSSFRsbelowthislinemaybeconsideredpassive.
TheevolutionpredictedbyEqs.3-5isshowninFig.2.
The grey shaded region outlines the analytic prediction for
theevolutioninBHmassovertherangerej/rhalo =0.1−1.0 and galaxy stellar properties evolve. BS10 showed that the
and thesolid black line shows theprediction for rej/rhalo = BH mass is determined by the binding energy of the DM
0.22(thevaluethatreproducestheslopeoftheredshiftzero halo,whichexplainswhythem −U relation doesnot
BH halo
mBH−mhalorelation).Thered,dottedcurveshowsthesim- evolve. We find that the mBH −U∗ relation also does not
ulation prediction for the evolution of the mBH−mhalo re- evolve,implyingthat,overtherangeofredshiftsandmasses
lation, including all BHs with mBH >102mseed. At all red- investigated here, U∗ ∝Uhalo. The top panel of Fig. 3 con-
shifts thenormalisation of thesimulated mBH−mhalo rela- firmsthat thisis indeed thecase in our simulation.
tioniscompatiblewiththatpredictedbytheanalyticmodel. Forthebindingenergyofthegalaxytotrackthatofthe
For comparison, the dashed, black line shows the predicted halo, we require the two to grow through the same mech-
evolutionofthemBH−mhalo relation ifc(mhalo)wereinde- anism. This condition is met if the galaxies grow primarily
pendentofredshift. Theanalyticmodelcan onlyreproduce throughdrymergers.Intheabsenceofsignificantin-situstar
the simulation result if the evolution of the concentration- formation, both the stellar component, which is predomi-
mass relation is taken intoaccount. nantlyspheroidalformassivegalaxies,andtheDMhaloare
TheevolutionofthemBH−mhalorelationthusprovides collisionless systemsandarethereforeexpectedtoevolvein
additional evidence for theidea that themasses of BHs are a similar manner.
determined by the binding energies of the haloes in which The bottom panel of Fig. 3 shows that at z ≪
they reside. 1 the specific star formation rates (SSFR≡ m˙s/m∗) of
galaxies with m∗ ∼ 1011M⊙ are significantly lower than
the inverse of the Hubble time, 1/t , implying that
H
4.2 The relations between m and galaxy stellar
BH the galaxies are indeed not growing significantly via in-
properties
situ star-formation, in agreement with various observa-
Considering now only the stellar masses for which the evo- tions (e.g. Schawinski et al. 2006; van derWel et al. 2009;
lution has been measured observationally (m∗ ∼ 1011M⊙) Bezanson et al. 2009; van Dokkum et al. 2010). The analy-
and the redshift range for which all of the stellar and BH sis in Fig. 3 was carried out for galaxies at a fixed stellar
propertiesofthegalaxiesareconvergednumerically(z<1), mass,butthesameresultsholdifwetraceindividualgalax-
weaskifwecanexplainhowtherelationsbetweenBHmass iesthroughtime.Whilethestellar masses ofthemost mas-
sive z = 0 galaxies grew on average by a factor 2.65 since
z=1, thefraction of stars in theredshift zero objects with
3 Wequotedaslopeof1.5±0.2.Ourerrorbarisgreaterbecause birth redshifts below z=1 is, on average, only 15%.
BS10usedmBH >10mseedwhereaswerequiremBH >100mseed. Merloni et al. (2010) studied the evolution of the BH
(cid:13)c 2010RAS,MNRAS000,1–8
Evolution of the black hole scaling relations 7
scalingrelationintheredshiftrange1<z<2.2,andfound mBH/m∗ ∝(1+z)αs,withαs ≈0.5,andmBH/σ∗4 ∝(1+z)αs
that at z = 1 a large fraction of galaxies would be identi- with α ≈ −0.3, in apparent conflict with recent observa-
s
fied as star-forming. This is consistent with our results as tions. The ratio between the BH mass and the binding en-
theSSFRsofourgalaxysampleareincreasingwithincreas- ergyofthedarkhaloisindependentofredshift,inagreement
ing redshift so that a significant fraction of them would be withBS10whoarguedthattheBHmassiscontrolledbythe
identified as star-forming at z ≈ 1. The fraction would be halo bindingenergy.The simple analytic model of BS10, in
evenhigherifwehadrequiredthegalaxiestocontainactive whichtheBHmassisassumedtoscaleinproportion tothe
AGN, as is the case for the objects selected in the observa- binding energy of the dark halo, not only reproduces the
tions, because AGN activity is typically accompanied by a simulated redshift zero m −m relation, but also its
BH halo
temporary increase in thestar formation rate. evolution. For a fixed halo mass BHs are more massive at
One might naively think that the mBH −m∗ relation higher redshift because the haloes are more compact and
should not evolve at all if the galaxies grew predominantly thus more tightly bound. Assuming an NFW halo density
through dry mergers. However, the progenitors of galaxies profileandtheevolutionofthehaloconcentration-massrela-
withm∗ ∼1011M⊙atz=0typicallyformedtheirstarsand tion predicted bysimulations, themodel can quantitatively
grew their BHs later than the progenitors of galaxies that account for thepredicted evolution.
alreadyhavethesamestellarmassatz=1.Becausethepro- The simulation predicts that the ratio between the
genitorsofhigherredshiftsgalaxiesformedearlier,theyhave BH mass and the binding energy of the stellar component
higherbindingenergiesandthusgreaterBHmassesrelative of the galaxy is also independent of redshift (at least for
to their halo masses. Thus, even if m∗ ∼ 1011M⊙ galaxies m∗ ∼1011M⊙ andz<1),eventhoughBS10demonstrated
aregrowingpredominantlybydrymergersatbothredshifts, explicitly that the correlations between BH mass and stel-
theymayhavedifferentBHmasses. Observe,however,that lar properties are not fundamental. This result is, however,
the evolution in the mBH−m∗ relation that we predict for consistent with a picture in which massive galaxies grow
thesemassivegalaxiesisonlymild(about0.2dex;seeFig.1) primarily through dry mergers at low redshift, which we
and that in situ star formation thus become important for showedtobethecaseinthesimulation.Combinedwiththe
z&1 (see Fig. 3). observed evolution in the m∗ −σ∗ relation, this idea can
If the ratio of the stellar to halo binding energies re- quantitatively account for the evolution in the mBH −m∗
mains constant, we can write relation.
One interesting implication of this scenario is that the
mBH ∝Uhalo∝U∗ ∝m∗σ∗2. (6) evolutionoftherelationsbetweenBHsandthepropertiesof
To explain the evolution of mBH/m∗ that is observed for their host galaxies may differ for galaxies that do not grow
m∗ ∼ 1011M⊙, and which the simulation reproduces, we predominantly through dry mergers, as would be expected
need to know how the m∗ −σ∗ relation evolves for such forlowermassesandathigherredshifts.Wewillinvestigate
galaxies. thisfurtherintheafuturework,employinghigherresolution
Measurements of the evolution of the m∗−σ∗ relation simulations.
have so far only been undertaken for a small number of
objects. Cappellari et al. (2009) presented stacked observa-
tions of seven galaxies with m∗ ∼ 1011M⊙ in the redshift
range 1.6<z <2.0 and found that these galaxies typically ACKNOWLEDGEMENTS
have the same σ∗ as the very highest velocity dispersion We would like to thank Marcel Haas for a careful read-
early-type galaxies of the same mass in the local Universe.
ing of the manuscript and the anonymous referee for com-
This is in agreement with observations showing that galax-
ments that improved its clarity. The simulations presented
ies of a given stellar mass are more compact at higher red-
here were run on the Cosmology Machine at the Institute
shifts (see e.g. Williams et al. 2010). Indeed, for early-type
for Computational Cosmology in Durham as part of the
galaxies with stellar masses ∼ 1011M⊙ Cenarro & Trujillo Virgo Consortium research programme, on Stella, the LO-
(2009) find that typical velocity dispersions decrease from
FARBlueGene/LsysteminGroningen,andonHuygens,the
≈ 240 km/s at z = 1.6 to ≈ 180 km/s at z = 0, which
Dutchnationalsupercomputer.Thisworkwassupportedby
implies σ∗ ∝ (1+z)0.3. Combined with Eq. (6) this yields an NWO Vidi grant.
mBH/m∗ ∝(1+z)0.6, in good agreement with theαs ≈0.5
that is observed for galaxies of this mass.
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