A simple model to link the properties of quasars to the properties of dark matter halos out to high redshift

0.46 MB·English

by  Darren J. Croton

Mon.Not.R.Astron.Soc.000,000–000 (2005) Printed27January2009 (MNLATEXstylefilev2.2)
A simple model to link the properties of quasars to the
properties of dark matter halos out to high redshift
9
Darren J. Croton1,2
0
0
2 1Department of Astronomy, Universityof California, Berkeley,CA, 94720, USA
2Centre for Astrophysics & Supercomputing, Swinburne Universityof Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia
n
a Accepted —.Received—;inoriginalform—
J

6
2 ABSTRACT
We present a simple model of how quasars occupy dark matter halos from z = 0 to
O] z =5usingthe observedmBH−σ relationandquasarluminosity functions.Thispro-
videsawayforobserverstostatisticallyinferhosthalomassesforquasarobservations
C
using luminosity and redshift alone. Our model is deliberately simple and sidesteps
. any need to explicitly describe the physics. In spite of its simplicity, the model re-
h
p produces many key observationsand has predictive power:1) model quasarshave the
- correctluminosityfunction(by construction)andspatialclustering(by consequence);
o
2) we predict high redshift quasars of a given luminosity live in less massive dark
r
t matterhalosthanthe sameluminosityquasarsatlowredshifts;3)wepredictafactor
as of ∼5 more 108.5M⊙ black holes at z ∼2 than is currently observed; 4) we predict a
[ factor of ∼20 evolution in the amplitude of the mBH−Mhalo relation between z =5
and the present day; 5) we expect luminosity dependent quasar lifetimes of between

1 t ∼ 107−8yr, but which may become as short as 105−6yr for quasars brighter than
Q
v L∗; 6) while little luminosity dependent clustering evolution is expected at z ∼< 1, in-
4
creasingly strong evolution is predicted for L>L∗ quasars at higher redshifts. These
0
last two results arise from the narrowing distribution of halo masses that quasars
1
4 occupy as the Universe ages. We also deconstruct both “downsizing” and “upsizing”
. trends predicted by the model at different redshifts and space densities. Importantly,
1
thisworkillustrateshowcurrentobservationscannotdistinguishbetweenmorecompli-
0
catedphysicallymotivatedquasarmodelsandoursimplephenomenologicalapproach.
9
It highlights the opportunities such methodologies provide.
0
:
v Key words: quasars: general, galaxies: active, cosmology: dark matter, methods:
i statistical
X
r
a
1 INTRODUCTION and the mBH −mbulge relation (Ferrarese & Merritt 2000;
Tremaine et al. 2002; Marconi & Hunt 2003; H¨aring & Rix
Quasars represent a unique population of objects in the 2004). Although initially surprising (why should the sub-
Universe that encapsulate many otherwise diverse areas parsec physics of black hole growth correlate with the kilo-
of physics. These include extreme environments of gravity parsec properties of the galactic bulge?), it is now be-
(black holes), sub-to-kiloparsec-scale dynamics (black hole lieved that these relationships simply reflect the physics
two and three body interactions and host galaxy merg- of a common formation mechanism (Silk & Rees 1998;
ersorseculartriggers ofquasaractivity),sub-to-kiloparsec- Hopkinset al. 2006a). For example, galaxy major mergers
scale hydrodynamics (gas infall, accretion disks and quasar maysimultaneously drivegrowthinthebulgeandforcegas
winds), and quasars as cosmological probes of the evolving into the central regions of the galaxy to fuel the black hole
large-scale cosmic web. They are among themost luminous (and hence a quasar). Secular processes may be operating
objectsintheUniverse.Theenergyliberatedduringasingle tosimilareffect,suchasbarinstabilities(Sellwood & Moore
quasareventcanoutshinetheentirestellarlightofthehost 1999).
galaxy. After fading, their presence can still be measured
There is a need to understand the phenomenology of
through the local quiescent black hole population.
blackholesandgalaxiesingreaterdetail.Activeblackholes
Although much work has been done to describe the are thought to have great impact on the evolution of their
physics of black holes and their evolution, the majority of hosts. Duringa quasar event,rapid hole growth occurs and
what we know remains primarily phenomenological. Two outflows drive winds that may liberate gas from the galaxy
fundamentalcorrelationsareobserved:themBH−σrelation (DiMatteo et al.2005;Hopkinset al.2006a;Thacker et al.
2 D. J. Croton
2006).Suchgas is considered fuel for futurestar formation, ogy. Analytic methods, such as those recently described in
andwithoutitnewstarsnolongerformandthehostgalaxy Neistein & Dekel(2008)andZhang et al.(2008),arenotas
subsequently reddens and fades, a kind of galactic extinc- usefulhereastheyproducehalomergertreeslackingspatial
tion. However, supermassive black holes do not always die andvelocityinformation.Wewilllaterneedtheseproperties
withtheirgalaxy,butareoftenlaterfoundinalowluminos- in our analysis.
itystate.Heatingfrom lowluminosityactivegalactic nuclei Instead, we turn to a numerical N-body simula-
(AGN) provide a long term energy source that suspends tion of dark matter evolution, the Millennium Simula-
the cooling of gas from the surrounding hot halo (the so tion (Springel et al. 2005c). This simulation, run using
called “radiomode”solutiontothe“coolingflow”problem: a WMAP1+2DFGRS cosmology, follows the evolution of
Croton et al.2006;Bower et al.2006).Thisheating/cooling 10 billion dark matter particles in a box of side-length
balanceis knowntomaintain theaged appearanceof many 500h−1Mpc from z=127 to z=0. Within the simulation,
massivelocalgalaxies. UnderstandingAGNandblackholes bothhalosandsubhalos(i.e.theboundsub-structurewithin
has now become essential to understanding galaxies, and a given halo) are accurately resolved down to virial masses
hencemodellingtheco-evolutionofbothhasbecomeasub- of less than 1011M⊙, more than sufficient for our purposes
ject of great interest. here. Note that in what follows we do not discriminate be-
In this paper we build a general model of how black tween halos and subhalos when populating the simulation
holes and quasars occupy dark matter halos and how this with quasars1. Thus, it is possible in our model for given
occupationevolveswithtime.Ourmethodissimilarinspirit halotohostmorethanonequasaratanygiventime.Small-
to that by Marinoni & Hudson (2002) and Vale& Ostriker scale clustering measures of quasars indicate that this may
(2004) but for quasars rather than galaxies. We work from indeed be the case (e.g. Hennawi et al. 2006; Myers et al.
aminimalsetofassumptionsandusetwokeyobservational 2008; Padmanabhan et al. 2008), although it is not critical
constraints:themBH−σ relationandthequasarluminosity for our current analysis.
function.Undersuchconstraintsthemodelnaturallyrepro-
duces many quasar and black hole observations out to red-
shiftsasdistantasz ∼5.Anumberofpredictionsaregiven. 2.1 Linking quasar luminosity with halo mass
This model is deliberately simple; it sidesteps any attempt
We start with the dark matter halo virial mass, Mvir, of
to explicitly describe the quasar triggering mechanism, the
each Millennium Simulation halo. Ourgoal is to first relate
detailsofblackholeaccretion,orthehydrodynamicsofsub-
itspropertiestotheexpectedvelocitydispersionofanoccu-
sequentquasarwindsandoutflows.Althoughunderstanding
pyinggalaxy, σ,andthen tothequasarluminosity through
such detail is certainly desirable, we show that the current themBH−σ correlation.Readerswhoareonlyinterestedin
observations do not discriminate between the more com- the final relations should skip ahead to the equation sum-
plicated physically motivated modelling of quasars and our
mary in Section 2.1.1 (see also equation 13 in Section 4.2).
simple phenomenological model.
Dark matter halos are identified at each redshift as re-
The outline of this paper is as follows. In Section 2 we
gionsofthesimulationwhosemeandensityinsideaspherical
describetheconstructionofthemodelandtheobservational
aperture exceeds 200 times the critical density of the Uni-
data used to constrain it. In Section 3 we explore the vari-
verse. The virial mass of a halo and its virial velocity, Vvir,
ous consequences of this model, comparing to observations
are then related by
whereavailableandmakingpredictionswherenot.Section4
provides some discussion, placing the model into a broader Vvir =[10GH(z)Mvir]1/3 , (1)
context of black hole and galaxy co-evolution. Finally, Sec- where Mvir has units of h−1M⊙, G = 4.3 ×
tion5summarisesourmainresults.Unlessotherwisestated, 10−9(km/s)2MpcM−1 is Newtons gravitational constant,
we assume a standard WMAP first year ΛCDM cosmology ⊙
and
with ΩΛ = 0.75, Ωm =0.25, σ8 =0.9 and Hubble constant
H0 = 100hkm/sMpc−1 (Spergel et al. 2003; Seljak et al. H(z)≡H0E(z)=H0[Ωm(1+z)3+ΩΛ]1/2 (2)
2005). We choose the value of the Hubble parameter, h, to
isthevalueoftheHubbleconstantatredshiftz(Hogg1999).
be either h=0.7 or h=1.0 dependingon the context; this
will be clearly marked. Here,H0 =100hkm/sMpc−1 isthelocalvalueoftheHub-
bleconstant(withhthedimensionlessHubblevalue),while
Ωm and ΩΛ are the universal mass and dark energy densi-
ties. Note, that both Mvir and H(z) above must share the
2 ACCURATELY POPULATING DARK same h valuefor dimensional consistency.
MATTER HALOS WITH QUASARS Halo virial velocity is typically related to the observed
galaxy circular velocity,v ,by
c
We build our quasar model in two parts. First, we map
quasar luminosity onto dark matter halos using the halo vc =γVvir , (3)
virial properties. Second, we determine which of these ha-
with γ a parameterof orderunity(see Section 2.3).Thisin
los actually host a quasar as a function of luminosity at
turncan berelated tothevelocity dispersion of thegalaxy,
any given redshift. Both parts are undertaken using obser-
σ,through theobserved correlation (Baes et al. 2003)
vationalconstraintsonly,notablythemBH−σ relation and
quasar luminosity function.
Tobegin werequireknowledgeofthedarkmatterhalo 1 Throughout this paper our use of the term “halo” includes
population and its evolution. There are several ways this bothhalos(i.e.quasarswithincentralgalaxies)andsubhalos(i.e.
can be achieved at a given redshift for a given cosmol- quasarswithinsatellitegalaxies).
Linking quasars to dark matter halos 3
σ v
log =(−0.22±0.03)+(1.04±0.12)log c , (4) convert between bolometric quasar luminosity and bJ-band
(cid:0)u0(cid:1) (cid:0)u0(cid:1) absolute magnitude, Croom et al. (2005) provide
whereu0 =200km/s.Equations1-4connectthevirialmass M =−2.66 log(L )+79.42 , (9)
ofadarkmatterhaloatredshiftz tothevelocitydispersion bJ Q
of the occupyinggalaxy. where L here is in watts and assumes h = 0.7. Second,
Q
A well defined correlation between stellar velocity dis- toconvertbetweenB-band,bJ-band(usedbythe2dFQSO
persionandblackholemassisobservedinthelocalUniverse, RedshiftSurvey,hereafter2QZ),andi-band(theSloanDig-
themBH−σ relation (Tremaine et al. 2002): italSkySurveyQuasarSurvey,hereafterSDSS)(k-corrected
log mBH =(8.13±0.09)+(4.02±0.44)log σ , (5) to z = 2) quasar luminosities we use the conversions given
(cid:0)h−701M⊙(cid:1) (cid:0)σ0(cid:1) in Croom et al. (2005) and Richardset al. (2006)
where h70 is the Hubble parameter with H0 = MB =MbJ +0.06 ,
70km/sMpc−1, and σ0 =200km/s. For practical purposes, M [z =2]=M −0.71 . (10)
whenusingequation5inourmodelweincludetheobserved i bJ
0.3dexdispersioninmBH (forsimplicityweapplythesame
dispersion at all values of σ). We assume no evolution with 2.2 Deciding which halos host quasars
redshiftineitheramplitudeorslopeoftherelation.Wedis-
cuss this furtherin Section 3.2. AlthoughSection2.1providesamappingbetweenhalomass
With a valueof mBH for each dark matterhalo wecan and quasar luminosity, not all halos will actually host a
now determine the luminosity of a quasar that may occupy quasar at any given time. To decide which do we use the
thehalo.Wetakethis(bolometric)luminosityassomefrac- quasar luminosity function.
tion, η, of theEddington luminosity: Specifically,weconstrainthemodeltohavethecorrect
luminosity function at a given epoch using the technique
L = ηL
Q edd of abundance matching (e.g. Conroy et al. 2006 and refer-
= η 3.3×104mBH(h−701L⊙) ences therein). First, we calculate the observed cumulative
= η 1.3×1038mBH(h−701erg/s) . (6) luminosityfunction,frombrighttofaintquasars,whichpro-
vides a smoother representation of the data than the lumi-
Equation 6 completes our sought after connection between nosity function alone. Then, starting at the bright-end, we
virial mass and quasar luminosity, which we will now sum- move faint-ward down the cumulative luminosity function
marise. in magnitude bins of ∆M = 0.01 and randomly dilute the
numberdensityofmodelquasarsinthesamemagnitudebin
toensuretheymatchthesamecumulativeabundanceasthe
2.1.1 A summary of the important equations and
observations.Hence,byconstruction, themodelreproduces
relationships
theobserved quasar luminosity function.
Equations 1-6 provide a mapping between halo virial mass The observed quasar luminosity function is only mea-
and quasar bolometric luminosity. These equations can be sured at discrete redshifts, and we would like to be able to
reduced to thefollowing relation buildourmodelatanyarbitrary redshift.Hence,insteadof
L /η using the data itself, we abundance match to a functional
log(cid:16)1012hQ−701L⊙(cid:17) = (−1.99±0.33) fiwteowfiltlhleatdeartcaomacproasrsemwiutlhtiaprlee trhedossheioftfs.CTrohoemtwetoadl.at(a20s0e4t)s
+ (1.39±0.22) log γ3H(z) Mvir , (7) (2QZ), covering 0.4 < z < 2.1, and Richardset al. (2006)
h (cid:16)1013h−1M⊙(cid:17)i (SDSS),covering 0.5<z<4.8.
whereH(z)isdefinedbyequation2.Notethat,forinternal Croom et al.(2004)modeltheirdatausinganevolving
consistency, H(z) must be calculated using the same value doublepower-law with theform
of h as Mvir, butfor LQ we haveassumed thestandard ob- Φ(L∗ )
servervalueofh=0.7.Quasarluminosityhereisdependent Φ(L ,z)= bJ , (11)
bJ (L /L∗ )−α+(L /L∗ )−β
on both parameters γ, describing the relationship between bJ bJ bJ bJ
virialandcircularvelocities(equation3),andη,theEdding- wherethecharacteristicluminosity,L∗ ,isafunctionofred-
bJ
tonluminosityfractionthatquasarsareassumedtoshineat shift
(equation 6). Observational errors from equations 4 and 5
have been propagated throughout; their impact on the re- L∗bJ(z)=L∗bJ(z=0)10k1z+k2z2 . (12)
sults are discussed in Section 4.3.
Theauthorsusedthisfunctionalformtofitthe2QZquasar
Equations 1-5 can also be reduced to map directly be- luminosity function out to z∼2.
tween halo mass and black hole mass
Interestingly however, the Croom et al. double power-
log MBH = (−2.66±0.33) law is a good fit to the quasar luminosity function well be-
(cid:16)108h−1M⊙(cid:17) yonditsoriginal fittingrange. This can beseen in figure19
+ (1.39±0.22) log γ3H(z) Mvir . (8) ofRichardset al.(2006),wheretheCroom et al.resultcon-
h (cid:16)1013h−1M⊙(cid:17)i tinues to provide a good match to the SDSS quasar lumi-
nosity function out to redshift three2. Hence, to maintain
ThisequationcontainsaredshiftdependencethroughH(z),
something we will investigate later in Section 3.6.
Finally, we provide a few useful equations from thelit- 2 Note that Richardsetal. fit their SDSS data using a simpler
erature for converting to different optical filters. First, to singlepower-law,whichdoesnotrealisticallyrepresentthequasar
4 D. J. Croton
Table1.Parametersforthedoublepower-lawfittothejoint2QZandSDSSquasarluminosityfunctionsatz<3andz>3(Section2.2
andequations 11and12).AsinCroometal.(2004),h=0.7wasassumedinthefitwhererelevant.
redshiftrange α(brightslope) β (faintslope) Mb∗J(z=0) k1 k2 Φ∗(Mpc−3mag−1)
z<3.0 −3.31 −1.09 −21.61 1.39 −0.29 1.67×10−6
z>3.0 −3.31+0.5(z−3) −1.09 −21.61 1.22 −0.23 1.67×10−6
the most realistic shape of the quasar luminosity function
across thewidest possible redshift rangewe adopt thefit of
Croom et al. (2004) to z = 3. Above this redshift we mod-
ifytheirparametersslightly,addingevolutiontobright-end
power-law slope and softening the decline of L∗ with red-
bJ
shift. The fitting values we adopt are given in Table 1. The
fitsare shown below in Section 3.3.
2.3 Model assumptions and parameters
In order to keep our model as simple as possible while re-
maining accurate and (most importantly) understandable
we list below our main simplifying assumptions:
• Both the local mBH−σ relation (equation 5) and the
local correlation between σ and v (equation 4) exist at all
c
redshifts and donot evolvewith time.
• Therelationshipbetweenhalovirialvelocityandgalaxy Figure 1.Therelationshipbetweenquasarluminosityanddark
halo virial mass at various redshifts from z =0 to z = 5 (equa-
circular velocity can be described by equation 3 with γ =
tion7).MagnitudesareprovidedinboththebJ-bandusedbythe
1.0(seeAppendixA5ofPorciani et al.2004foradiscussion
2QZsurveyandi-bandusedbytheSDSS(k-correctedtoz=2–
on thispoint).
seeequation 10).Notethat quasarsofafixedhalomassbecome
• Quasars can be modelled as a simple “light bulb”, brighterwithincreasinglook-backtime.
where, at any given time, they are either on or off.
• “On” quasars shine at half the Eddington luminosity
(i.e.η = 0.5inequation6).Thisisthemeanobservedvalue
measured byMcLure & Dunlop(2004) duringtheheight of
quasar activity at z=2 (see also Marconi et al. 2004). for select redshifts out to z = 5. On the left axis we plot
bJ-bandquasarabsolutemagnitudeforcomparisonwiththe
The impact of the uncertainty of these assumptions on our 2QZ survey results, and on the right axis we show i-band
results is further discussed in Section 4.3. magnitudes for comparison with the SDSS quasar results
Althoughtheabovechoicesarereasonabletheyarenot (see equation 10).
necessarily correct in detail. For example, quasar luminosi- Figure 1 reveals that model quasars hosted by halos of
tiesarenoteither“on”or“off”,butfollowalightcurvewith agivenmassgetbrighterthefurtherbackintimeyoulook.
a peak luminosity that is likely dependent on the specifics Forexample,inhalosof1012.5M⊙,quasarsbrighteninlumi-
of thequasartrigger, theproperties of thehost galaxy, and nosity by about two magnitudes between z = 0 and z = 2,
redshift(e.g.Hopkinset al.2006a).Morecomplexmodelling and by another one-and-a-half magnitudes between z = 2
is required to capture these processes. Our goal here is not and z =5. One can see this by eye directly in the observa-
to model the specifics of the quasar population in detail, tionaldata,however,withoutdeferringtotheoryormodels.
but rather we aim to find the simplest model possible that The results of Croom et al. (2004) show similar amounts
can match a set of key observations, in spite of the missing
of brightening in the 2QZ quasar luminosity function be-
detail. We will discuss this furtherin Section 4. tween z = 0.4 and z = 2.1, while for the same catalogue,
Croom et al. (2005) use clustering to infer that the masses
ofL∗ quasardark halohostsacross thesame redshift inter-
3 CONSTRAINTS, CONSEQUENCES AND val remain approximately constant at 1012−13M⊙. In this
PREDICTIONS TO HIGH REDSHIFT sense, our model is simply mimicking the data from which
it was constrained.
3.1 The quasar luminosity-halo mass relation From a theoretical point-of-view, the evolution in am-
plitude seen in figure 1 originates from a redshift depen-
Therelationshipbetweenquasarluminosityanddarkmatter
dence in equation 1, where, at fixed halo mass, the virial
virialmassisgivenbyequation7.Thisisplottedinfigure1
velocityofahaloincreaseswithincreasingredshift(seealso
Wyithe& Loeb 2003). This behaviour simply tells us that
faint-end.Theirdatadoes notprobebelowL∗ attheseredshifts halos of equivalent mass live in higher σ-peaks at higher
(seehowever Fontanot etal.2007). redshifts.
Linking quasars to dark matter halos 5
Figure2.ThemBH−σrelationbetweenquasarblackholemass
andgalaxyvelocitydispersion.Weassumeinthemodelthatthis
relationship is redshift independent (see Section 2.1). A fit to
theobservedlocalrelationmeasuredbyTremaineetal.(2002)is Figure 3.2QZquasarluminosityfunctions (Croometal.2004)
shownbythesolidthickline. covering z = 0.40 to z = 2.10 (symbols with error-bars). The
(difficult to see) dotted lines show the fit to the data described
byequation11withdoublepower-lawparametersgivenintable1.
Solidlinesshowthemodelluminosityfunctionsdeterminedfrom
abundance matchingtotheobservations (Section2.2).
3.2 The mBH−σ relation 3.3 The quasar luminosity function to z∼5
The mBH −σ relation is the primary observation used in As discussed in Section 2.2, we use abundance mapping to
ourmodeltomapquasarluminosityontomass.Toproduce selectthosehaloswithquasarsthatwillproducealuminos-
a realistic quasar mock catalogue we include the observed ity function identical to that observed at each redshift. To
0.3dex of scatter reported in Tremaine et al. (2002) when maximise its versatility we match the model to a fit of the
performing this mapping. Our model relation is plotted in datathatvariessmoothlywithredshift(Section2.2andta-
figure 2, where the points are a randomly selected sample ble1),ratherthantothedataitself(whichlimitsustowork
of quasars from the model, and the thick solid line shows at theobserved redshifts only).
the best fit to the observed correlation. It is important to Model luminosity functions are shown for z < 2
note that, being a model quasar population, we are 100% in figure 3, comparing with the 2QZ survey results of
completedowntolowdispersion,unlikethemeasureddata. Croom et al. (2004), and for z > 2 in figure 4, compar-
Suchmeasures,especially atlowdispersion,stillremainob- ingwith theSDSSresultsof Richardset al. (2006).Ineach
servationally formidable. panel of both figures, the symbols show the observed data,
When constructingthemodel it was interestingtofind thedottedlinethedoublepower-lawfittothedata,andthe
that including the above dispersion does not appear im- solid line is themodel.
portant for its success in any way. All results presented in It is no coincidence that the model line and double
this paper are essentially unchanged if we had rather sim- power-law fits are essentially indistinguishable for all but
ply made a direct mapping between mBH and σ, ignoring thelowest space densitiesat thehighest redshifts(at which
the scatter. Evidence is emerging to suggest that a tight we are limited by the finite size of the Millennium Simu-
correlation exists between L and M (e.g. White et al. lation). Note that our extension to the Croom et al. (2004)
Q halo
2007).Understandingthisresult will becritical when using doublepower-lawrepresentationofthe2QZluminosityfunc-
quasar clustering measures to observationally constrain the tion(table1)providesagoodfittothedatauptothehigh-
host masses of high redshift quasars. est redshifts probed by theSDSS quasarsurvey in figure4.
Wehavemadeacritical assumptionwhenconstructing
ourmodelthatthemBH−σdoesnotevolvewithredshift,ei- 3.4 Quasar lifetimes
therinamplitudeorslope.Anevolvingrelationwouldplace
quasars of a given magnitude in either more massive halos The quasar lifetime, t , is defined here as N = t /t ,
Q Q Q H
(foradecreasing amplitudewith increasing redshift) orless where N is the fraction of halos at a given redshift that
Q
massive halos (for an increasing amplitude with increasing host a quasar, and t is the Hubble time at that redshift.
H
redshift).Thischangewillbereflectedintheclusteringprop- N isafunctionofbothmassandredshiftandisdetermined
Q
ertiesof thequasarsthemselves. Asweshow inSection 3.7, inourmodelthroughabundancematchingmodelquasarlu-
our non-evolving mBH−σ relation assumption produces a minosities to the observed quasar luminosity function (Sec-
quasar population whose 2-point function matches the ob- tion2.2).Itis,inthiscontext,anormalisedquasarselection
servations extremely well out to at least z ∼ 4. The obser- function.
vationalpictureastowhetherthemBH−σ relation evolves In figure 5 we present quasar lifetimes as a function of
remains unclear (see Croton 2006 for a discussion), so for limiting faint quasar magnitude for five redshifts, ranging
thecurrent work we retain the non-evolvingassumption. from z = 0.5 to z = 4. Lifetimes in general range from be-
6 D. J. Croton
Figure 5. Predictions for quasar lifetimes, tQ, as a function of
limiting faint quasar magnitude, at five different redshifts from
z = 0.5 to z = 4.0. At z∼<2 and LQ > L∗ the quasar lifetime
shortens considerably. At higher redshifts, tQ remains approxi-
matelyconstant ormayevenincreaseforthebrightest.
keydifferencebetweenourmodelandmanypreviousworks,
and may explain its ability to simultaneously match such
a wide range of observations. For example, Wyithe& Loeb
(2003)findanover-productionofbrightquasarsatlowred-
shift. In our model, such quasars have very short lifetimes,
Figure 4. Similar to figure 3, but this time for quasars in the andhencearenotcommonlyseeninsurveys.Thismaysim-
SDSSsurveycoveringz=2.01toz=4.75(Richardsetal.2006). plyarise duetothedwindlingsupplyofcoldgasinmassive
Again,dottedlinesindicatethedoublepower-lawfittothedata
systems at late times (Fabian 1994).
(equation 11 and table 1), while the solid lines show the model
quasarluminosityfunction(Section2.2).
3.5 The active and passive black hole mass
tween 107 to 108 years. At lower redshifts we see a distinct functions
turn-over in tQ, where brighter quasars have much shorter Observations suggest that a black hole gains the majority
lifetimes(asshortas105−6 years)thanfainterquasars(lev- ofitsmasswhileintheactivehighaccretion(quasar)phase
ellingoutat∼107.5 years).Thisturn-overoccursatapprox- (Heckman et al. 2004). The cumulative effect of such mass
imately L∗ inthequasarluminosityfunction (equation12). growthovercosmictimeismeasurableinthelocal(passive)
Theturn-overislesspronouncedforhigherredshiftquasars, black hole population. Our model makes a prediction for
flattening somewhat and even increasing to high luminosi- theactive black hole mass function. It is important to note
ties at z=4. that thisprediction arises viathedual constraint of linking
Duetothesimplicityandtransparencyofthemodelwe quasar luminosity to halo virial mass through the mBH−σ
knowexactlywhymodelquasarlifetimesbehaveinthisway: relation (equation 7), and from matching the abundance of
quasarsoccupyanarrowerrangeofhalomassesatlatetimes blackholestothequasarluminosityfunction(Section2.2)at
relativetoearlytimes(skipforwardtofigure10toseethis). various redshifts. We did not tunethe model in this regard
Becauseofthis,lowredshiftbrightquasarsarerareamongst to force a particular outcome.
theabundant∼1013M⊙ masshalostheyoccupy(hencethe In figure 6 we show the observed cumulative black
turnoveratbrightluminosities),whereashighredshiftbright hole mass function for both z = 0 passive and z = 2 ac-
quasarsare frequentamong theirrare ∼1013M⊙ mass halo tive black holes. This figure is adopted from figure 6 of
hosts (hence tQ remains flat). Fainter quasars (those with McLure & Dunlop(2004).Theupperthinsolid anddashed
∼ LQ < L∗) tend to always commonly populate mostly lines show the local observed results inferred from the
abundant halos, again resulting in a relatively constant tQ. mBH−bulge luminosity relation and mBH −σ relation, re-
Wewill return to this point in Section 4.2. spectively.ThethreedatapointsshowthecumulativeSDSS
Physically speaking, it is important to realise that the quasar mass function at z = 2 for three different limiting
trends seen in figure 5 do not result from the explicit mod- black hole masses. The upward pointing arrows on each in-
elling of a changing Eddington accretion fraction, as has dicatethatthespacedensitymeasuredineachbinisincom-
often been explored (e.g. Hopkinset al. 2005b). Quasars in plete,andhenceprovideonlyalowerlimittothetruemass
ourmodelareassumed toalwaysaccrete at afixedfraction density.
of the Eddington rate across the quasar lifetime3. This is a The three thick lines in figure 6 show the model pre-
diction for our complete sample of quasar black holes at
3 NotethatwhentQ becomes longerthanthedoublingtimefor
theblackhole(∼>108yr)blackholemassandquasarluminosity ourwork,thisismostrelevantforextremeluminosityquasarsat
change non-trivially across the quasar lifetime. This is an inter- the highest redshifts probed. Ignoring this effect for the sake of
nal inconsistency that all light bulb models must navigate. For simplicitydoesnotchangeourconclusionsinanyqualitativeway.
Linking quasars to dark matter halos 7
Figure7.EvolutioninthemBH−Mhalorelationandhalomass-
to-bolometric luminosity ratio out to z = 5 (equation 8). There
aretwo model predictions here: 1) that, at a give redshift,more
massive halos should show an increasingly larger mBH−Mhalo
ratio(drivenbyacceleratedblackholegrowth),and2)theoverall
amplitude of this ratio should evolve with look-back time (see
Figure 6. The cumulative black hole mass function. The lower Section3.6).
solid, dashed and dotted lines show the model result at three
different redshifts, while the filled circles indicate lower lim-
its measured from the SDSS z = 2 quasar mass function of 3.6 Evolution in the mBH−Mhalo relation and
McLure&Dunlop (2004). Note that the model predicts a fac- mass-to-light ratios
tor of ∼ 5 more 108.5M⊙ black holes (quasars) than currently
be seen in the data at z = 2. The upper solid and dashed lines Section 2.1 and equation 8 provide an analytic prediction
showthelocal(passive)cumulativeblackholemassfunction,cal- for how black hole and dark matter halo mass are related.
culated intwo different ways. The passivefunction is builtfrom This relationship includes a redshift dependence through
thecontinual productionofquasarsacrosscosmictime. E(z)=[Ωm(1+z)3+ΩΛ]1/2,implyingthatthemBH−Mvir
ratioshouldevolvewithtime,withmoremassiveblackholes
occupying dark matter halos of a fixed mass at higher red-
shifts relative to lower redshift.
redshifts 0.5, 2.0, and 4.0, as indicated in the legend. For
In figure 7 we use equation 8 to plot the hole-to-halo
black holes more massive than logmBH ∼> 9.0 at z = 2 our
mass relation at five different epochs, from z = 0 to z = 5.
model is close to the lower limit found in the SDSS data.
At any given redshift the ratio increases with increasing
If true, the model indicates (perhaps unsurprisingly) that
halomass,implyingthatblackholesbecomeproportionally
much of the massive end of the black hole mass function
largerthemoremassivethehalois.Theevolutionintheam-
forms solely from accretion during this time of peak activ-
ity. At lower black hole masses, logmBH ∼ 8.5, our quasar pwliitthudteheoframtiBoHc/hMavnigrinwgithbyreadsfhacifttorcaonfa∼lso5bbeetcwleeaernlyzse=en0,
modelpredictsanexcessoffainterquasarsyettobeseenin
the z ∼ 2 data. At higher redshift, z =4, the model shows andz=2,increasingtoafactorof20byz=5.Previousau-
thorshaveattemptedtoquantifythechangeinblackholeto
thecontinuingbuild-upofmassiveblackholesbutwithless
hostgalaxy/halopropertieswithtime(e.g.Robertson et al.
activity at lower masses. By late times, z = 0.5, massive
2006; McLure et al. 2006; Croton 2006). Evolution of this
black hole mass growth has largely stopped while the low
typeisaclearpredictionofourmodel.Ourresultsaresimi-
mass holes continueto grow.
lartothosefoundbyWyithe& Padmanabhan(2006a)using
This last behaviour is a black hole manifestation of
different techniques(see also Wyithe& Loeb 2003).
the popular “downsizing” paradigm (Heckman et al. 2004;
Wecanequivalentlyrecastfigure7asachangingmass-
Merloni & Heinz 2008). It is interesting that such shift in
to-light ratio using Equation 7. The right axis in figure 7
black hole mass growth with time arises naturally from the
model constraints alone. If we skip ahead to figure10 (Sec- shows this result. Halos with masses greater than 1014M⊙
host quasars with mass-to-light ratios less than unity re-
tion4.2)wecanseethereasonwhy.Here,dashedlinesshow
gardless of the redshift of interest. This is also true of all
thechangingspacedensityofhalos/quasarswithtime,while
horizontal dotted lines show black hole mass. At high red- lower mass (Mvir ∼< 1013M⊙) quasar/halo systems at red-
shifts z∼> 2.
shift,verylowspacedensitycontoursaremostlyflat(oronly
slowly rising) and correspond to a fixed black hole mass of
∼ 109−10M⊙. At lower redshifts all density contours turn 3.7 Quasar clustering to z∼4
over sharply, and black hole mass decreases by up to a few
orders-of-magnitudeatfixedspacedensity.Hence,downsiz- Theclustering of a given population of quasarswill depend
ingis predictedfor all objects at z∼< 2. Atz∼> 2themodel bothonthemassesofquasarhostsandtheluminosityrange
predicts that low space density objects (i.e. the most mas- whichdefinesthequasarsample.Bothkeyobservationsused
sive) should show no downsizing trends (relative to z =2), to constrain our quasar model, the mBH −σ relation and
while higher space density objects (i.e. closer to L∗) should quasarluminosityfunction,arerelevanttoshapethemodel
be“upsizing”, especially aboveredshifts z∼4. 2-point function.
8 D. J. Croton
Figure8.Modelcomparisonstothe2QZ(Porcianietal.2004)andSDSS(Shenetal.2007)projectedredshift-spacequasarcorrelation
functions,fromz∼1toz∼4.Atallredshiftsourmodelisagoodmatchtothedata(howeverwithsomesmallunder-predictionofthe
clusteringamplitudeatz>3).Wealsofindahintofsteepening intheclusteringonsmallscalescales(rp∼<1h−1Mpc).
In figure 8 we present the observed and model pro-
jected correlation functions at five discrete redshifts from
z = 1 to z = 4. The left three panels are taken from
Porciani et al.(2004)usingthe2QZdata,whereastheright
twopanelsarethosefromShen et al.(2007)withtheSDSS
data. The marked redshift in each panel indicates the me-
dian redshift of the measured quasars. For panels left-to-
right, the absolute magnitude range defining each quasar
sample is: the 2QZ survey M ∈[−25.32,−21.72], M ∈
bJ bJ
[−25.97,−22.80],andM ∈[−26.44,−23.37],andtheSDSS
bJ
survey4 M <−26.1 and M <−26.7 (all magnitudes have
i i
unitsof 5logh70).
The default model produces a very good fit to the ob-
Figure 9. Model predictions for the luminosity dependence of
servedquasarclusteringatallredshiftsconsidered.Atz>3
quasarbias,forquasarsamplesdefinedbyafaintmagnitudelimit.
thereisasmallunder-predictionoftheclusteringamplitude,
Results at five redshifts are shown, from z = 0.5 to z = 4.0.
however the abundance of model quasars here is extremely Quasarsatz∼<1showlittleornoluminositydependentbias.At
lowandthecorrelationfunctionsomewhatnoisy.Thisover-
higher redshifts, a strong luminosity dependence in the bias is
all success gives us confidence that the use of the mBH−σ predictedforbrighterquasarsrelativetofaint.
relationandabundancemappingtechniquetoconstructour
model is actually placing quasars in the correct halos, at
least in a statistical sense. At z <2 where the clustering is 3.8 Luminosity dependent quasar bias at high
wellmeasured,anupturnisseenatsmallscales(r∼< 1Mpc). redshift
HintsofsuchexcesspairshavebeenfoundintheSDSSLRG-
For most redshift bins our statistics are far superior to
QSOcrosscorrelationofPadmanabhan et al.(2008),aswell
that which can be measured observationally. To test the
as work by Hennawi et al. (2006) and Myers et al. (2008).
model further we look for signatures of evolution in the
We leave a more detailed analysis of this result to future
clustering amplitude as a function of luminosity, captured
work.
here through the quasar bias. Such evolution, or lack
there of, is a more detailed probe of how quasars oc-
cupy halos, and is currently the focus of much observa-
tionalscrutiny(Porciani & Norberg2006;Myerset al.2007;
daAˆngela et al. 2008).
In figure 9 we show predictions for the luminosity de-
pendentbias(Mo & White1996)measuredfromourmodel,
4 Note that Shenetal. (2007) do not state the absolute magni-
calculatedusingtheJenkins et al.(2001)massfunctionand
tuderangethatdefinesthequasarsintheirtwohighredshiftbins.
afittothemodel quasarhalooccupation fraction. Thecal-
Hence, for simplicity we select model quasars brighter than the
faint absolute magnitude corresponding to i = 20.2 (the SDSS culation is performed analytically to remove the noise of
quasarapparentmagnitudelimit)atthemedianredshiftofeach small number statistics from rare objects at the high mass
sample(M.Strauss,priv.comm.). end. Samples are defined with a faint limiting magnitude
Linking quasars to dark matter halos 9
(i.e.applyingequation7toahalomasscut),andwepresent Fromanalyticargumentsalone,Silk & Rees(1998)pos-
results for fiveredshift binsranging from z=0.5 to z=4. tulate a critical black hole mass, determined by the sur-
At z∼< 1 weak (or no) luminosity dependent bias is rounding halo properties, above which star formation has
present in the model quasar population. At z > 1, how- been suppressed due to an expanding quasar wind that
ever, bright quasars show an increased bias with respect to sweepsthegalaxycleanofitsstarforminggas.Blackholesin
faintquasars.Thisrangesfromb∼3tob∼7betweenlumi- thispictureeitherform early inthecollapsing proto-galaxy
nosityextremesatz=2,andb∼5tob∼15atz>2.Such at above the critical mass, or form close the the critical
increases are only marginally observable with current data, mass and are maintained at this mass by the hierarchical
but will certainly be testable in future surveys as quasar growth of the system. This model provides an elegant ex-
numbersincrease and theluminosity baseline widens. planationformanyobservedblackhole–galaxy/halocorre-
Both weak luminosity dependence at low redshift and lationsbutdoesnotpredictotherquasarpropertiessuchas
significantluminositydependenceathighredshiftconstitute theirluminosities or evolvingspace density.Regardless, the
afirmprediction fortheclustering ofquasarsin ourmodel. Silk & Rees (1998) model has become somewhat of a seed
Thesepredictionsariseasaconsequenceofthechangingoc- from which a number of more detailed models have devel-
cupation statistics of quasars in halos with time. At high oped.
redshiftquasarsarespreadoveramuchwiderrangeofhalo One popular extension of these ideas is discussed in a
masses relative to low redshift for a comparable (to L∗) lu- series of papers by Wyithe& Loeb (Wyithe& Loeb 2002,
minosity range (see figure 10). This produces the stronger 2003). In their model, quasars and their subsequent rapid
clusteringgradient seenin figure9.Saidanotherway,lumi- black hole growth are triggered from major mergers, as
nosity dependent clustering evaporates at late times due to discussed above. Under the assumption that the local gas
the narrowing of the range of halo masses that host quasars traps much of the quasar energy without radiating it away,
as the Universe ages. We will discuss this result further in and that the subsequent quasar luminosity is some fixed
Section 4.2. fraction of the Eddington luminosity, they derive a series
of equations that allow them to predict quasar luminosity
andblackhole/hostcorrelationsatvariousredshifts.Witha
smallnumberoffree(butphysicallymotivated)parameters
4 DISCUSSION their model is tuned to provide a good fit to the high red-
shift quasar luminosity function, although it over-predicts
4.1 An (incomplete) overview of other popular
theabundanceof bright low redshift quasars. The modelof
quasar models
Wyithe& Loeb is somewhat similar to ours but differs in
Quasars have been modelled in a number of ways one critical way. The abundance of quasars in their model
in the past several years (Ciotti & Ostriker 1997; is determined from halos who undergo rapid growth (i.e.
Silk & Rees 1998; Fabian 1999; Kauffmann & Haehnelt major mergers). Our modelling makes no such assumption,
2000; Haehnelt & Kauffmann 2000; Cattaneo 2001; but rather forces the correct quasar number by abundance
Ciotti & Ostriker 2001; Wyithe & Loeb 2002, matching to the actual quasar luminosity function. We dis-
2003; Granato et al. 2004; Kawata & Gibson 2005; cuss theadvantages of this below.
Begelman & Nath 2005; Springel et al. 2005a,b; Tofurtherexplorethispictureofblackholeandgalaxy
Di Matteo et al. 2005; Hopkinset al. 2005a; growth a numberof authors have turned to performing hy-
Cattaneo et al. 2005a,b; Hopkinset al. 2006a; drodynamic simulations of the complex merger and accre-
Thacker et al. 2006; Lidzet al. 2006; Fontanot et al. tion processes themselves. It should be remembered, how-
2006; Wyithe& Padmanabhan 2006b; Malbon et al. 2007; ever, that the physics of such processes cannot actually
Sijacki et al.2007;Hopkinset al.2008a,b;Merloni & Heinz be resolved with current computing power, as the scales
2008; DiMatteo et al. 2008). The goal of such works has involved lie orders-of-magnitude below what is required.
usually been to understand the cosmological evolution of Never-the-less, a combination of the phenomenological and
quasars and their triggering mechanism and black hole gas hydrodynamicmethodologies allow for an increased levelof
accretion rates. This is typically achieved by matching the detail and accuracy in the models that is unreachable by
model output to observables like the quasar luminosity analytic methods alone.
function, the evolving space density of bright quasars, and OneexampleofthisistheworkbyHopkinset al..Ina
the mBH − σ relation. More recently, quasars (and more seriesofpapers(Hopkinset al.2005a,b,2006a,b,c,2007a,b,
generally AGN) have been linked to the quenching of star 2008a,b), these authors explore the co-evolution of black
formation in massive elliptical galaxies, and quasar models holes and galaxies triggered by mergers using high resolu-
haveadapted to reflect these new found appreciations. tion hydrodynamic simulations which include many realis-
In fact, most modern AGN models are built upon tic physical processes. The most important aspect of their
the current belief that quasars are triggered by major work is the inclusion of merger driven star formation and
merging events of gas rich galaxies (DiMatteo et al. 2005; the redistribution of disk gas which drives both galactic
Hopkinset al. 2006a). This is a reasonable assumption to bulge growth and, simultaneously, growth in the black hole
make. Something significant must be happening to the gas throughaccretion.Onceconvolvedwithcosmological statis-
in the galaxy to cause it to lose so much angular momen- tics5, e.g. the evolution of merger rates with time, their
tum, a necessary condition to drive gas into the central re- model produces results that match a large number of ob-
gions where the black hole resides. Without such angular
momentumloss it is hardtoimagine howtherequirednear
Eddington accretion rates can be achieved. 5 Hopkinsetal.simulateindividualmergerevents,albeitalarge
10 D. J. Croton
Figure 10. Halo virial mass predictions (solid lines) for quasars with measured luminosity and redshift, as defined by equation 13.
Magnitudes are provided in both the bJ-band used by the 2QZ survey and i-band used by the SDSS (k-corrected to z = 2 – see
equation10).Dashedlinesindicatetheevolutionofthehosthalospacedensity,asinferredfromtheobservedquasarluminosityfunction.
These contours show that the distribution of mass for quasar host halos narrows as the Universe ages. Horizontal dotted lines show
the corresponding black hole mass, given by equation 6. The downturn of density contours relative to black hole mass at low redshifts
indicatedownsizingintheblackholepopulation.Bothupsizingandno-sizingareseenathigherredshifts,dependingonthespacedensity
considered.
servables: luminosity and mass functions, quasar lifetimes, + (0.72±0.11) log LQ/η , (13)
Eddington ratios, and host galaxy properties. (cid:16)1012h−701L⊙(cid:17)
In a critical deviation from past models (including our
where H(z) is defined by equation 2. Equation 13 can be
workhere),thesimulationsofHopkinset al.explicitlytrack
used to take a quasar with observed bolometric luminosity
thevaryingblackholeaccretionrateduringthemerger.This
L at redshift z and predict its halo virial mass, assuming
produces a range of predictions for how the quasar/AGN Q
valuesforparametersγ (therelationship betweenvirialand
light curve should appear as a function of time, from first
circularvelocities)andη(theEddingtonluminosityfraction
interaction to final merger remnant. Their simulations sug-
of thequasar).
gest a possible common origin for the many observed AGN
Equation 13 is plotted in figure 10. From this figure
types, a so called “unified” merger driven model of AGN
alone thequasarhost virial mass (eitherindividualoraver-
andgalaxies (Hopkinset al.2006a),thatdependsprimarily
aged over a group) for a given quasar luminosity and red-
on the time during themerger that the system is observed.
shiftmaybesimplyreadoff(solidlines).Weemphasise,fig-
The beauty of their model is that it presents a well defined
ure 10 allows observers to determine statistically measured
and testable picture of black hole and galaxy growth. The
halo masses to complement their observations without the
drawback to their work is that its detail and complexity
need for large quasar surveys or clustering measures. We
sometimes cloud its interpretation (see Section 4.4 below).
discuss theuncertainty on such masses below.
Also over-plotted in figure 10 are the corresponding
mean space densities of quasar hosts (dashed lines), calcu-
4.2 A tool for observers latedfromthequasarluminosityfunction6(Section2.2),and
the black hole mass at fixed luminosity (horizontal dotted
The quasar model presented in this paper provides a rela-
lines), as given by equation 6.
tionship between quasar luminosity, redshift, dark matter
The spacing of density contours relative to halo mass
halo mass, and black hole mass. The essence of this model
highlights a main result of this work, which is that the dis-
is captured by equation 7 (Section 2.1.1). For convenience,
tributionofthemassesofdarkmatterhaloshostingquasars
we invert this equation to obtain
narrows with decreasing redshift. This leads to behaviour
log Mvir = (1.43±0.32)− log γ3H(z) such as luminosity dependent clustering at high redshift
(cid:16)1013h−1M⊙(cid:17) (cid:16) (cid:17)
6 Note that the artificial bump at z = 3 is due to a change in
number of them, and not the evolution of structure in a cosmo- theassumedfittingparametersofthequasarluminosityfunction
logicalcontext. (table1)andhasnobearingontheresults.