Growth of Early Supermassive Black Holes and the High-Redshift Eddington Ratio Distribution

0.21 MB·English

by  Colin DeGraf

Draftversion January27,2012
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
GROWTH OF EARLY SUPERMASSIVE BLACK HOLES AND THE HIGH-REDSHIFT EDDINGTON RATIO
DISTRIBUTION
C. DeGraf, 1 T. Di Matteo, 1 N. Khandai, 1 R. Croft 1
Draft version January 27, 2012
ABSTRACT
Using a new large-scale ( 0.75Gpc)3 hydrodynamic cosmological simulation we investigate the
2 ∼
growth rate of supermassive black holes in the early universe (z > 4.75). Remarkably we find a clear
1
0 peakinthetypicalEddingtonratio(λ)atblackholemassesof4 ∼8 107M⊙ (typicallyfoundinhalos
− ×
2 of 7 1011 1 1012M⊙), independentofredshiftandindicativethatmostofBHgrowthoccursin
∼ × − ×
the cold-flow dominated regime. Black hole growth is by and large regulated by the evolution of gas
n
density. ThetypicalEddingtonratioatagivenmassscalessimplyascosmologicaldensity(1+z)3and
a
the peak is causedby the competition between increasedgas density available in more massive hosts,
J
andadecreaseduetostrongAGNfeedbackthatdeprivestheblackholeofsufficientgastofuelfurther
5
rapid growth in the high mass end. In addition to evolution in the mean Eddington ratio, we show
2
thatthedistributionofλamongbothmass-selectedandluminosity-selectedsamplesisapproximately

log-normal. We combine these findings into a single log-normalfitting formula for the distribution of
]
O Eddington ratios as a function of (M ,z). This formula can be used in analytic and semi-analytic
BH
models for evolving black hole populations, predicting black hole masses of observedquasars, and, in
C
conjunction with the observed distribution of Eddington ratios, can be used to constrain the black
.
h hole mass function.
p Subject headings: quasars: general—galaxies: active —blackhole physics— methods: numerical—
- galaxies: evolution
o
r
t
s 1. INTRODUCTION 2006;Jiang et al.2009). Inthispaperwetakeadvantage
a ofa new, verylargesimulationto investigatethe growth
[ It has been well established that supermassive black
historiesofearlyuniverseblackholesacrossawiderange
holes are present in the center of most galaxies
1 (Kormendy & Richstone 1995), and that they are corre- ofmasses,probingboththemeanandthedistributionof
v latedwiththe propertiesoftheirhosts(Magorrian et al. growthratesforblackholesacrossawiderangeofmasses
3 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; andluminosities,andprovidefits forthesedistributions.
8 Tremaine et al. 2002; Graham & Driver 2007). These
3 correlations provide strong evidence that the growth of 2. METHOD
5
a black hole and the evolutionof its hostgalaxydirectly In this paper we use a new cosmological hydrody-
1. influence one another, such that black hole growth is a namic simulation of a 533 h−1 Mpc box specifically
0 importantaspectofunderstandinggalacticevolutionand intended for high-redshift investigations. The simula-
2 vice versa. tionusesthemassivelyparallelcosmolocialTreePM-SPH
1 In general, the link between black hole and galactic code P-GADGET (an updated version of GADGET-
: evolution is attributed to some form of quasar feedback 2, see Springel 2005) incorporating a multi-phase ISM
v
i (Burkert & Silk2001;Sazonov et al.2004;Springel et al. model with star formation (Springel & Hernquist 2003)
X 2005b; Churazov et al. 2005; Di Matteo et al. 2005; and black hole accretion and feedback (Springel et al.
r Bower et al. 2006; Ciotti & Ostriker 2007; Sijacki et al. 2005a; Di Matteo et al. 2005), has a gravitational soft-
a 2007; Hopkins et al. 2007b) which can result in the self- ening length of 5h−1 kpc and mass resolution of 2.8
regulation of the growth of the black hole (see, e.g. 108M⊙ for dark matter and 5.7 107M⊙ for gas. ×
Di Matteo et al. 2005). In this model we would expect Within the simulation,black h×oles aremodeled as col-
black holes to grow rapidly during their early lifetime lisionless sink particles which form in newly emerging
(i.e. while at low mass), until some point at which the and resolved dark matter halos. These halos are found
blackholefeedbackbeginstosignificantlyaffectitsenvi- by calling a friends of friends group finder at regular in-
ronment,resultinginanoticeabledeclineingrowthrate. tervals (in time intervals spaced by ∆loga = log1.25).
This effect has been observed in individual black hole Any group above a threshold mass of 5 1010h−1M⊙
histories, but such investigations (see, e.g. Sijacki et al. not already containing a black hole is pro×vided one by
2009; Di Matteo et al. 2011) have tended to focus on converting its densest particle to a sink particle with a
the largest mass black holes, primarily to explain how seed mass of MBH,seed = 5 105h−1M⊙. This seeding
blackholescouldgrowrapidlyenoughtoproducetheex- prescription is chosen to rea×sonably match the expected
tremelylargemasses( 109M⊙byz 6)foundinobser- formationofsupermassiveblackholesbygasdirectlycol-
∼ ∼
vations by the Sloan Digital Sky Survey (e.g. Fan et al. lapsing to BHs with M M (e.g. Bromm & Loeb
BH seed
∼
2003; Begelman et al. 2006) or by PopIII stars collaps-
1McWilliamsCenterforCosmology,CarnegieMellonUniver- ing to 102M⊙ BHs at z 30 (Bromm & Larson
sity,5000ForbesAvenue, Pittsburgh,PA15213, USA 2004; Y∼oshida et al. 2006) foll∼owed by sufficient expo-
2 DeGraf et al.
nential growth to reach M by the time the host halo and large scale high-resolution imaging (Feng et al.
seed
reaches 1010M⊙. Following insertion, BHs grow in 2011). For further details on the simulation methods
∼
mass by accretion of surrounding gas and by merging and convergence studies done for similar simulations,
with other black holes. Gas is accreted according to see Di Matteo et al. (2008).
M˙ =α4πG2MB2HρBH,whereρ isthelocalgasdensity Becausethe simulationsavesthe complete setofblack
BH (c2s+v2)3/2 BH hole properties (mass, accretion rate, position, local gas
(determined fromthe gas particles within the black hole density, sound speed, velocity, and BH velocity relative
kernel), cs is the local sound speed, v is the velocity of tolocalgas)foreachBHateverytimestep,theblackhole
the BH relative to the surrounding gas, and α is intro-
output for such a large simulation is prohibatively diffi-
ducedtocorrectforthereductionofthegasdensityclose
cult to analyze using previous techniques. For this rea-
to the BH due to our effective sub-resolution model for son, Lopez et al. (2011) developed a relational database
the ISM. To allow for the initial rapid BH growth nec-
management system specifically for this simulation. A
essary to produce sufficiently massive BHs at early time similar strategy has also been followed in the analysis of
( 109M⊙byz 6)weallowformildlysuper-Eddington the Millenium simulation (Lemson & Virgo Consortium
∼ ∼
accretion,butlimitittoamaximumof3 M˙Edd topre- 2006). In addition to providing a substantially more
×
vent artificially high values. efficient query system for extracting information, this
The BH is assumedto radiate with a bolometric lumi- database is significantly more flexible than traditional
nosity proportional to the accretion rate, L = ηM˙BHc2 approaches. For a complete summary of the database
(Shakura & Sunyaev 1973), where the radiative effi- format and its efficiency, please see Lopez et al. (2011).
ciency η is fixed to 0.1 throughout the simulation and
our analysis. To model the expected coupling between 3. RESULTS
the liberated radiation and the surrounding gas, 5 per 3.1. Typical Black Hole Growth Rates
cent of the luminosity is isotropically deposited to the
To quantify the growth rate of black holes, we use the
localblackholekernelasthermalenergy. The5percent
value for the coupling factor is based on galaxy merger mean Eddington ratio (λ = MM˙˙eBdHd) which we calculate
simulations such that the normalization of the M σ for each black hole over a finite time interval. Because
BH
−
relation is reproduced (Di Matteo et al. 2005). we have the complete BH growth history, we are able to
Thesecondmodeofblackholegrowthisthroughmerg- compute this quantity based solely on the gas accretion,
erswhichoccurwhendarkmatterhalosmergeintoasin- andneglectanymass gainedthroughblack hole mergers
glehalo,suchthattheirblackholesfalltowardthecenter (though we find the mass gained by mergers to be small
ofthenewhalo,eventuallymergingwithoneanother. In enoughtohaveanegligibleeffectonourresults). InFig-
cosmologicalvolumes,itisnotpossibletodirectlymodel ure 1 we plot λ /(1+z)3 as a function of M for
BH,initial
thephysicsoftheinfallingBHsatthesmallestscales,soa severalredshifhtrianges. We plot λ /(1+z)3 ratherthan
sub-resolutionmodelisused. Sincethemergerstypically λ for two reasons: First, to shohwithat the dependence
occur at the center of a galaxy (i.e. a gas-rich environ- hofiλ on M is independent of redshift (at least for
BH
ment),weassumethefinalcoalescencewillberapid(e.g. z h4i.75), and second to show that λ (1+z)3.
Mayer et al. 2007), so we merge the BHs once they are ≥ h i∝
Regardless of redshift considered, we find similar be-
withinthe spatialresolutionofthe simulation. However,
havior for Eddington ratio with respect to mass: more
topreventmergingofBHswhicharerapidlypassingone
massiveblackholesgrowfasterthanlowmassblackholes
atinvoethtoero,nmeearngeortshearreisptroeovehnitgehd(ifcotmhepBarHabs’levetlooctihteylroeclaa-l uptoapeakgrowthrateatMBH ∼4−8×107M⊙,while
the blackholesabovethischaracteristicmassgrowmore
sound speed).
slowly. Thus black holes grow fastest (relative to their
The model used for black hole creation, accretion
current mass) while at intermediate masses, and grow
and feedback has been investigated and discussed
slower at higher mass.
in Sijacki et al. (2007); Di Matteo et al. (2008);
We find this peak in the Eddington ratio to be caused
Colberg & di Matteo (2008); Sijacki et al. (2009);
by the change in the local gas density available for fuel-
DeGraf et al. (2010); Degraf et al. (2011), finding it
ing BH growth. We plot the evolution in local gas den-
does a good job reproducing the M σ relation,
the total black hole mass density B(DHi−Matteo et al. sity (ρBH, the density of gas contributing to M˙BH) with
2008), the QLF (DeGraf et al. 2010), and the expected mass in Figure 1, showing a clear peak at 5 107M⊙.
∼ ×
We note that neither the sound speed nor the BH veloc-
black hole clustering behavior (Degraf et al. 2011).
This simple model thus appears to model the growth, ity (the other factors in the calculation of M˙BH) exhibit
activity, and evolution of supermassive black holes in a peak with respect to MBH, confirming that the peak
a cosmological context surprisingly well (though the Eddington ratio is caused by the evolution in the local
detailed treatment of the accretion physics is infeasible gas density. To show how the gas density evolves, in
for cosmological scale simulations). We also note that Figure 2 we show the gas density profiles around BHs
Booth & Schaye(2009)andJohanssonet al.(2008)have below the Eddington ratio peak ( 107M⊙ - blue), at
∼
adopted a very similar model, and have independently the peak ( 5 107M⊙ - green), and above the peak
∼ ×
investigated the parameter space of the reference model ( 4 108M⊙ – red), each averagedacross 100 BHs. In
∼ ×
ofDi Matteo et al.(2008),aswellasvaryingsomeofthe generalwefindthegasdensityprofiletogrowwithM
BH
underlying prescriptions. In addition, this simulation until MBH 5 107M⊙ (as expected for BHs found in
∼ ×
haspreviouslybeenusedtoinvestigatethegrowthofthe moremassivehalos). Above 5 107M⊙thegasdensity
first very massive black holes (Di Matteo et al. 2011), away from the BH continues∼to ×grow,but the innermost
statistical properties of quasars (DeGraf et al. 2011), densityissuppressed,withthesuppressiongrowingwith
Growth of Early Supermassive BHs 3
Fig. 3.—RedshiftevolutionoftheEddingtonratioforblackholes
withMBH>107M⊙ (shadedregionshows1-σstandarddeviation
in log(λ)) and i-band magnitude mi < 21 (green line) compared
with data from Shen&Kelly (2011) (black asterisks). We also
show the evolution inthe gas density aroundBHs forcomparison
(bluedashedline).
virialtemperatureofthehalo. Dekel & Birnboim(2006)
Fig.1.— Colored lines: The mean Eddington ratio (hλi) as a suggestthat inthese halosAGNfeedbackbecomes more
functionofMBHforseveralredshiftranges,scaledby (1+1z)3,with ssiugsnciefipctaibnlte,stioncheeathtiendgilauntdesphuoschki-nhgeabtyedthgeascewnitlrlableAmGoNre.
Poisson error bars [Note that the datapoints’ x-positions for each
z-bin have been shifted to the right (3% increase for each z-bin) Thiswouldthusproduceasuppressioninthegasdensity
such that the error bars are distinguishable]. We also show the profile, consistent with the picture described above and
typicalhosthalomasscorrespondingtothegivenBHmassonthe the downturn in Figure 1.
top axis. Filled circles: Average gas density at the BHs position
In addition to the evolution in λ with M , Figure 1
(ρBH)forz=4.75−5. also shows that λ evolves with redshift asBH (1+z)3,
∼
which is also caused by the evolution in the local gas
density. In Figure 3 we show the evolution in log(λ)
h i
withredshiftamongMBH >107M⊙ BHs(shadedregion,
showing 1-σ standard deviation). We plot the average
gas density at the BH (blue dashed line), showing the
evolutioninλisprimarilycausedbytheevolutioninρ
BH
(recall M˙ ρ ). We also compare to observational
BH BH
∝
measurements of Shen & Kelly (2011) (black asterisks),
showingthat this generalredshift evolutionis consistent
with current observations, and the normalization is ap-
proximatelyconsistentif we use a similarmagnitude cut
(i-band magnitude m <21 - green line).
i
3.2. Eddington Ratio Distributions
In addition to investigating the mean Eddington ra-
Fig. 2.— Gas density profiles averaged among 100 black holes tio, we also study the distribution of λ among com-
withmass ∼107M⊙ (blue), ∼5×107M⊙ (green), ∼4×108M⊙ parable BHs. Previous work on the λ-distribution
(red). Dottedlineshowsthegravitational softeninglength.
has often found roughly log-normal distributions using
M in both magnitude and distance. This suppres- both observational (Kollmeier et al. 2006; Netzer et al.
BH
sion of the localgas density is caused by the feedback of 2007; Netzer & Trakhtenbrot 2007; Willott et al. 2010;
the black hole, with the stronger feedback of high-mass Trakhtenbrotet al. 2011) and phenomenological ap-
BHs producing the strongesteffect (see Di Matteo et al. proaches(Shankar et al.2011)[thoughAird et al.(2011)
(2011)fordetailedinvestigationoffeedbackamongmas- findλtofollowapowerlawwhenselectedforhoststellar
sive BHs). mass, rather than BH mass]. However, these observa-
We alsoshowthe typicalmassofhaloshostingagiven tional studies necessarily incorporate several uncertain-
M on the top axis, noting that the Eddington ratio ties, such as sample selection and scatter in black hole
BH
peaksatahosthalomassof 7 1011 1 1012M⊙. This mass estimators, which we can bypass, using our sim-
∼ × − ×
massverycloselymatchesthecriticalshockheatingscale ulation to probe our black holes’ Eddington ratios di-
of 6 1011M⊙ (Dekel & Birnboim 2006; Dekel et al. rectly. InFigure4weshowthedistributionofEddington
∼ ×
2009, and consistent with our simulation), above which ratios among black holes selected by M (black his-
BH
infalling gas is shock heatednear the virialradius to the tograms). We find that the distribution produced by
4 DeGraf et al.
our simulation is indeed log-normal, in keeping with ob-
servational findings (Kollmeier et al. 2006; Netzer et al.
2007; Netzer & Trakhtenbrot 2007; Willott et al. 2010;
Trakhtenbrot et al.2011). Inparticular,wenotethatthe
distribution remains log-normal regardless of the mass
considered,withFigure4showingthisholdsamongblack
holes thatarebelow, at, andabovethe peak observedin
Figure 1.
Because we find λ to follow a log-normal distribution
and the mean of that distribution obeys a well-defined
curve with M (Figure 1), we are able to provide a
BH
general fitting formula for P(λM ,z), the probability
BH
|
distribution of black hole Eddington ratios as a function
of redshift and black hole mass:
1 −(ln(λ)−µm)2
P(λMBH,z)= e 2σm2 (1)
| λσ √2π
m
where µ and σ are the mean and standard deviation
m m
of ln(λ), respectively, and are fit by σ 0.39 and
m
∼
µm =(1+z)3Ae−(cid:16)log10(cid:16)MMBµH(cid:17)(cid:17)2/2σ02, (2)
with A .00094, Mµ = 5 107M⊙, and σ0 0.85. In
∼ × ∼
Figure4weplotthe distributionpredictedbyEquations
1&2 (red curve) compared the the actual distribution,
showing that this simple formula is capable of reproduc-
ingthedistributionofλforBHsinoursimulationacross
a wide range of masses and redshifts, without requiring
knowledge of individual black hole environments.
Inadditiontothedistributionforamass-selectedsam-
ple, in Figure 5 we show the Eddington ratio distribu-
tion from our simulation (red histogram) compared to
the observed distribution from Kollmeier et al. (2006)
(black histogram) for two luminosity selected samples.
We again note that the distribution is described by a
roughlylog-normaldistribution,andthatoursimulation
is approximately consistent with observational results.
Furthermore, by combining P(λM ,z) with the
BH
|
black hole mass function (Φ ) we can obtain the Ed-
BH
dington ratio probability distribution for a luminosity-
selected sample: Fig. 4.— Eddington ratio distribution for black holes at three
different mass scales (black) and the predicted distribution from
Φ (M )P(λM ,z) Equations1&2(redcurves).
BH BH BH
P(λLBH,z)= ∞ | (3)
| R Φ (M )P(λM ,z)dλ of the black hole mass function at high redshift, even
0 BH BH | BH
without measurements of the black hole masses.
where M = σTLBH . In Figure 5 we plot this pre-
BH 4πGmpcλ 4. CONCLUSIONS
dicted probability distribution (using our simulation’s
mass function) in red, showing P(λL ,z) is well pre- With a new large-scale simulation, we show that the
BH
dicted in this manner. We note th|at this approach is growth of black holes tends to follow a typical growth
significant as it provides a potentially powerful tool for pattern. In general, we find that black holes grow more
constrainingthe black hole mass function using observa- rapidlyathigherredshiftthancomparableblackholesat
tions of the Eddington ratio distribution. We show this lower redshift, characterized by λ (1+z)3. This scal-
∝
inFigure5byplottingP(λL ,z)basedonthreediffer- ing is causedby the redshift evolutionin the gas density
BH
ent local mass functions: th|e Shankar et al. (2009) mass about the black holes, and is comparable to current ob-
function (dashed green); the Shankar et al. (2009) mass servational data from Shen & Kelly (2011).
function derived from the Hopkins et al. (2007b) lumi- ThetypicalEddingtonratioalsoscaleswithMBH such
nosity function (dashed blue), and the mass function of that λ peaks at MBH 4 8 107M⊙ (typically found
∼ − ×
Hopkins et al.(2007a)(dashedpink). BecauseP(λLBH) inhalosof 7 1011 1 1012M⊙). Thispeakiscaused
| ∼ × − ×
is sensitive to the slope of Φ , the distribution of λ at byevolutioninthedensityofthegasathalocentersthat
BH
high L (where the mass function is steepest) varies is available to fuel black hole growth. In general, more
BH
substantially with the mass function used, suggesting massiveblackholesarefoundinmoremassivehaloswith
thatwithimprovedstatisticsfromupcomingsurveys,we correspondinglyhighergasdensities,henceλgrowswith
could use the observed P(λLBH) to constrain the slope MBH forlowmasses. However,aboveMBH 5 107M⊙
| ∼ ×
Growth of Early Supermassive BHs 5
blackholefeedbackhasasufficientlystrongeffectonthe
local environmen to suppress the density of the nearby
gas. Thus although these more massive black holes are
foundinmoremassivehaloswithcorrespondinglyhigher
gas densities in general, the feedback has significantly
lessened the density of the innermost gas where accre-
tion occurs. This suppression of the local gas density is
strongerformoremassiveBHs,andcausesλtodecrease
for MBH > 5 107M⊙.
Althoug∼h t×he local environment is important for the
accretion rate of individual black holes, we show that
the distribution of Eddington ratios follows a roughly
log-normaldistribution regardlessof the black hole pop-
ulationconsidered,consistentwithcurrentobservational
findings. We usethis,togetherwiththeevolutionin λ ,
h i
to provide a simple fitting formula for the distribution
of Eddington ratio with (M ,z). This general forumla
BH
canbe used for predicting the growth/evolutionof black
holepopulations intheoreticalandsemi-analyticmodels
(such as the evolution of the black hole mass function),
forpredictingthemassofobservedhigh-redshiftquasars,
and,inconjunctionwithupcomingobservationsoftheλ-
distribution, to constrain the slope of the high-redshift
black hole mass function.
Fig.5.— Distribution of Eddington ratios for BHs in our sim-
ulation (red histogram) compared with observational data from
ACKNOWLEDGMENTS
Kollmeieretal. (2006) (black histogram) for two luminositybins.
Wealsoshowthepredicteddistributionbasedonourfittingfunc- This work was supported by the National Science
tion(Equations3)usingoursimulation’smassfunction(solidred), Foundation, NSF Petapps,OCI-0749212and NSF AST-
the Shankar etal. (2009) base mass function (dashed green), the
1009781. Thesimulationsusedinthispaperwerecarried
Shankaretal.(2009)massfunctionderivedfromtheHopkinsetal.
(2007b)luminosityfunction(dashed blue), andthe massfunction out on Kraken at the National Institute for Computa-
ofHopkinsetal.(2007a)(dashedpink). tional Sciences (http://www.nics.tennessee.edu/).
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