The EAGLE simulations of galaxy formation: calibration of subgrid physics and model variations

2.9 MB·English

by  Robert A. Crain

Mon.Not.R.Astron.Soc.000,1–26(2011) Printed6April2015 (MNLATEXstylefilev2.2)
The EAGLE simulations of galaxy formation: calibration of subgrid
physics and model variations
Robert A. Crain1,2(cid:63), Joop Schaye1, Richard G. Bower3, Michelle Furlong3,
5
Matthieu Schaller3, Tom Theuns3, Claudio Dalla Vecchia4,5, Carlos S. Frenk3,
1
0 Ian G. McCarthy2, John C. Helly3, Adrian Jenkins3, Yetli M. Rosas-Guevara6,
2
Simon D. M. White7 and James W. Trayford3
r
p
A 1LeidenObservatory,LeidenUniversity,POBox9513,2300RALeiden,theNetherlands
2AstrophysicsResearchInstitute,LiverpoolJohnMooresUniversity,146BrownlowHill,Liverpool,L35RF
3 3InstituteforComputationalCosmology,DepartmentofPhysics,UniversityofDurham,SouthRoad,Durham,DH13LE
4InstitutodeAstrof´ısicadeCanarias,C/V´ıaLa´cteas/n,38205LaLaguna,Tenerife,Spain
] 5DepartamentodeAstrof´ısica,UniversidaddeLaLaguna,Av.delAstrof´ısicoFrancisoSa´nchezs/n,38206LaLaguna,Tenerife,Spain
A 6CRAL,ObservatoiredeLyon,Universite´Lyon1,9AvenueCh.Andre´,F-69561SaintGenisLavalCedex,France
7Max-Planck-Institutfu¨rAstrophysik,Karl-Schwarzschild-Str.1,D-85748Garching,Germany
G
.
h
p 6April2015
-
o
r
t ABSTRACT
s
We present results from thirteen cosmological simulations that explore the parameter space
a
[ ofthe“EvolutionandAssemblyofGaLaxiesandtheirEnvironments”(EAGLE)simulation
project. Four of the simulations follow the evolution of a periodic cube L = 50cMpc on a
2
side, and each employs a different subgrid model of the energetic feedback associated with
v
starformation.Therelevantparameterswereadjustedsothatthesimulationseachreproduce
1
the observed galaxy stellar mass function at z = 0.1. Three of the simulations fail to form
1
discgalaxiesasextendedasobserved,andweshowanalyticallythatthisisaconsequenceof
3
1 numerical radiative losses that reduce the efficiency of stellar feedback in high-density gas.
0 Such losses are greatly reduced in the fourth simulation - the EAGLE reference model - by
. injectingmoreenergyinhigherdensitygas.Thismodelproducesgalaxieswiththeobserved
1
size distribution, and also reproduces many galaxy scaling relations. In the remaining nine
0
simulations, a single parameter or process of the reference model was varied at a time. We
5
1 findthatthepropertiesofgalaxieswithstellarmass(cid:46) M(cid:63) (the“knee”ofthegalaxystellar
: massfunction)arelargelygovernedbyfeedbackassociatedwithstarformation,whilethoseof
v
moremassivegalaxiesarealsocontrolledbyfeedbackfromaccretionontotheircentralblack
i
X holes.Bothprocessesmustbeefficientinordertoreproducetheobservedgalaxypopulation.
Ingeneral,simulationsthathavebeencalibratedtoreproducethelow-redshiftgalaxystellar
r
a massfunctionwillstillnotformrealisticgalaxies,buttheadditionalrequirementthatgalaxy
sizesbeacceptableleadstoagreementwithalargerangeofobservables.
Keywords: cosmology:theory–galaxies:formation–galaxies:evolution–galaxies:haloes
1 INTRODUCTION theregimewherethoseoutcomesaredeterminedprimarilybygrav-
itationalforces,havebeencorroboratedbyadiverserangeofob-
Theformation,assemblyandevolutionofcosmicstructuresisor-
servationaltests.Theseinclude,butarenotlimitedto,cosmicshear
chestratedbygravitationalcollapse.Thenon-linearityofthispro-
inducedbylarge-scalestructure(e.g.Fuetal.2008),theabundance
cess precludes a fully-predictive analytic theory of structure for-
ofbrightestclustergalaxies(BCGs,e.g.Rozoetal.2010),testsof
mation,requiringthattheconfrontationoftheoreticalexpectations
thecosmicexpansionrate(e.g.Blakeetal.2011a)andthedistance-
with observational measurements must generally proceed via nu-
redshiftrelation(e.g.Blakeetal.2011b),redshift-spacedistortions
merical simulations. The predictions of cosmological simulations
of the 2-point correlation function (e.g. Beutler et al. 2012) and
basedontheprevailingΛ-colddarkmatter(ΛCDM)paradigm,in
theluminosity-distancerelationoftypeIasupernovae(e.g.Suzuki
etal.2012).
(cid:63) E-mail:[email protected] The formation and evolution of galaxies is governed ulti-
(cid:13)c 2011RAS
2 R.A.Crainetal.
mately, however, by the interaction of the diverse physical pro- thetypicalsizeofanISMresolutionelementinsimulationsoflarge
cesses that, in addition to gravity, influence baryonic matter. The cosmologicalvolumes.
inclusionoftheseprocessesinsimulationsisrecognisedasamajor Sincetheselossescannotbemodelleddirectlybycosmolog-
challenge,owingbothtothecomplexityofthephysicalprocesses, icalsimulations,theirimpactonresolvedscalesmustbeincorpo-
and the difficulty of developing numerical algorithms able to ac- ratedintophenomenological“subgrid”treatmentsthatapproximate
curatelymodeltheireffectsinacomputationallyefficientmanner. the action of unresolved processes, and couple them to resolved
Thischallengehas,byandlarge,impededcosmologicalhydrody- scales1. The implementation and parametrisation of subgrid rou-
namicalsimulationsfromyieldinggalaxypopulationswhoseprop- tinesisthereforethegreatestsourceofuncertaintyincosmological
erties are consistent with observational measurements. Although simulations,andadjustmentofthesecharacteristicscanresultinthe
imperfectmodelscanproveinstructive,greaterconfidenceisgen- dramaticalterationofsimulationoutcomes(Okamotoetal.2005;
erallyascribedtothosethatmoreaccuratelyresembletheobserved Schayeetal.2010;Haasetal.2013a,b;Scannapiecoetal.2012;
Universe.Moreover,thereproductionofkeyobservablesisoftena Vogelsbergeretal.2013;LeBrunetal.2014;Torreyetal.2014).
prerequisitefortestingparticularaspectsofgalaxyformationthe- Untilsmall-scalelossescanbeaccuratelycomputedandappropri-
ory.Forexample,whenwishingtostudytheevolutionofangular atelyincorporatedintosubgridroutines,itwillremainimpossible
momentumindiscgalaxies,amodelthatreproducestheirobserved toformulateatrulypredictivecosmologicalsimulationthatcan,for
sizeandrotationvelocityisclearlydesirable. example,yieldabinitioestimatesofthestellarmassofgalaxiesor
The reproduction of these particular diagnostics has in fact themassoftheircentralBH.
become a cause ce´le`bre for the simulation community, owing to Inacompanionpaper,Schayeetal.(2015,hereafterS15)ar-
thelong-standingneedtoaddressthecloselyrelated“overcooling” guethattheappropriatemethodologyisthereforetocalibratethe
(Cole 1991; White & Frenk 1991; Blanchard et al. 1992; Balogh parametersofsubgridroutines,inorderthatsimulationsreproduce
etal.2001)and“angularmomentum”(Katz&Gunn1991;Navarro well characterised observables. The calibrated observables them-
& White 1994) problems. In the absence of feedback, gas effi- selves cannot then be advanced as predictions of the model, but
ciently radiates the heat it acquires from thermalising its gravita- thosenotconsideredduringthecalibrationcanreasonablybecon-
tional potential. This excess dissipation has two principal conse- sideredasconsequencesoftheimplementedastrophysics.Anob-
quences: i) the fraction of gas that is converted into stars by the viousadvantageofthisapproachisthat,byensuringthatkeyprop-
present epoch is much higher than observed, and ii) the forma- ertiesofthegalaxypopulationarereproduced,simulationscanbe
tion and coalescence of dense clumps spuriously drains angular usedtoaddressthewidestrangeofproblems.Arelatedadvantage
momentum from the baryons. Simulated galaxies therefore form isthat,sinceanyalterationtotheresolutionofacalculationwillin
toomanystars(anddosotooearly),theyaremorecompactthan generalnecessitatearecalibrationofthemodel,theadoptedsub-
observed, and they exhibit insufficient rotational support. The in- grid routines need not sacrifice physical detail in order to realise
clusionofprescriptionsforenergeticfeedbackprocessesinmod- numericalconvergence.
elshasbeenshowntoalleviatetheseproblems(Abadietal.2003; Adoptingthisphilosophy,S15introducedthe“Evolutionand
Sommer-Larsenetal.2003;Springel&Hernquist2003),andhas AssemblyofGaLaxiesandtheirEnvironments”(EAGLE)project,
enabledseveralgroupstoconductsimulationsoftheΛCDMcos- asuiteofcosmologicalhydrodynamicalsimulationsoftheΛCDM
mogonythatformgalaxieswithsizesandrotationcurvesthatare, cosmogonyconductedbytheVirgoConsortium2.Feedbackfrom
for particular galaxy masses, consistent with observational mea- starformationandAGNisimplementedthermally,suchthatout-
surements(e.g.Governatoetal.2004;Okamotoetal.2005;Sales flowsdevelopasaresultofpressuregradientsandwithouttheneed
etal.2010;Guedesetal.2011;McCarthyetal.2012;Brooketal. toimposewinds‘byhand’,forexamplebyspecifyingtheirveloc-
2012;Munshietal.2013;Aumeretal.2013;Marinaccietal.2014). ity and mass loading with respect to the star formation rate. The
In spite of this success, the detailed behaviour of the multi- parametersofthesubgridroutinesgoverningfeedbackassociated
phase interstellar medium (ISM) when subject to energetic feed- withstarformationandthegrowthofBHsarecalibratedtorepro-
back remains ill-understood, and the community has yet to con- ducetheobservedz = 0.1galaxystellarmassfunction(GSMF)
vergeonuniquesolutionstotheovercoolingandangularmomen- and the relation between the mass of galaxies and their central
tumproblems(Scannapiecoetal.2012).Theprincipaluncertainty BH, respectively, whilst also seeking to yield galaxies with sizes
isarguablyoneofaccounting.Firstly,itisnotknownwhatarethe (i.e. effective radius) similar to those observed. S15 focussed on
energy,momentumandmassfluxesincidentupontheISMandstar- theEAGLEreferencemodel(“Ref”),andacomplementarymodel
formingcomplexestherein(butseeLopezetal.2011;Rosenetal. designedtomeetthecalibrationcriteriaathigherresolution(“Re-
2014), due to mechanisms such as radiation pressure and winds cal”)3. Besides demonstrating that cosmological simulations can
fromO-classstars,asymptoticgiantbranch(AGB)starsandactive becalibratedtoreproducethesediagnosticssuccessfullywithun-
galacticnuclei(AGN);thephotoionisationandphotoelectricheat- precedentedaccuracy,thestudyshowedthatthesimulationsrepro-
ingofHIIregionsbyradiationassociatedwithstars(includingX- duce a diverse and representative set of low-redshift observables
raybinaries)andtheaccretiondiscsofblackholes(BHs);andther- thatwerenotconsideredduringthecalibrationprocess.Inasepa-
monuclear and core collapse supernovae (SNe). A second, often ratepaper,Furlongetal.(2014)showthattheEAGLEsimulations
overlookedissueisthatitisunknownwhatfractionoftheincident
energyisdissipatedbyradiativeprocessesandthermalconduction
(e.g.Orlandoetal.2005),andwhatfractionoftheincidentmomen- 1 Werefertolossesonthesescalesas“subgridlosses”.Lossesinduced
byprocessesactingonscalesthatareresolvedbycosmologicalsimulations
tumislostduetocancellation.Estimatingtheseinitial“losses”isa
canalsobesignificant,anddependentuponthesubgridimplementation;we
long-standingprobleminthestudyoftheISM,notleastbecauseof
termthese“macroscopiclosses”.
theextremeresolutionanddynamicrangedemandsoftheproblem: 2 Seealsohttp://eagle.strw.leidenuniv.nlandhttp://icc.dur.ac.uk/Eagle
thelossesaretypicallyestablishedonscalessignificantlysmaller 3 S15alsointroducedathirdmodelthatbetterreproducestheobserved
thanaparsec(e.g.Mellemaetal.2002;Fragileetal.2004;Yirak propertiesofintragroupgasatintermediateresolutionbyadoptingahigher
etal.2010).Thisisatleastthreeordersofmagnitudesmallerthan AGNheatingtemperature(“AGNdT9”).
(cid:13)c 2011RAS,MNRAS000,1–26
EAGLE:calibrationandmodelvariations 3
alsobroadlyreproducetheobservedGSMFasearlyasz =7,and conditionsisprovidedinAppendixBofS15,andthetoolsneces-
accuratelytrackitsevolutiontothepresentday. sarytogeneratethemindependentlyareavailableonline4.
This paper introduces many more simulations from the EA- The simulations were evolved by a modified version of the
GLEproject.Thekeyaimsofthisstudyaretoillustratethereasons N-body TreePM smoothed particle hydrodynamics (SPH) code
for the parametrisation adopted by the EAGLE reference model, GADGET3, last described by Springel (2005). The modifications
andthesensitivityofitsoutcomestothevariationofthekeysub- comprise updates to the hydrodynamics algorithm and the time-
grid parameters. The simulations explored here are naturally di- stepping criteria, and the addition of subgrid routines governing
videdintotwocategories.Thefirstcomprisesfoursimulationscal- the phenomenological implementation of processes occurring on
ibrated to yield the z = 0.1 GSMF and central BH masses as a scales below the resolution limit of the simulations. The updates
function of galaxy stellar mass. The models, one of which is the tothehydrodynamicsalgorithm,whichwecollectivelyrefertoas
EAGLE reference model, differ in terms of the adopted subgrid “Anarchy” (Dalla Vecchia in prep.), comprise an implementation
efficiencyoffeedbackassociatedwithstarformation,andthefash- of the pressure-entropy formulation of SPH derived by Hopkins
ionbywhichthisefficiencydepends(ifatall)upontheproperties (2013),theartificialviscosityswitchproposedbyCullen&Dehnen
ofthelocalenvironment.Thesuccessfulreproductionofthecali- (2010),anartificialconductionswitchsimilartothatproposedby
brationdiagnosticsbyeachofthemodelshighlightsthattheseob- Price(2008),theC2smoothingkernelofWendland(1995),andthe
servablesalonedonotidentifyaunique“solution”,andindicates time-steplimiterofDurier&DallaVecchia(2012).
that complementary constraints are necessary to break modelling Thesubgridroutinesrepresentanevolutionofthoseusedfor
degeneraciesand,potentially,motivatetheinclusionofadditional the GIMIC (Crain et al. 2009), OWLS (Schaye et al. 2010) and
complexity. cosmo-OWLS(LeBrunetal.2014)projects,andincludeelement-
The second category comprises simulations each featuring a by-element radiative cooling and photoionisation heating for 11
variation of a single subgrid parameter value with respect to the species,starformation,stellarmassloss,energyfeedbackfromstar
referencemodel.Thesecalculationsenableanexaminationofthe formation,gasaccretionontoandmergersofBHs,andAGNfeed-
roleoftheseparametersinafashionsimilartotheOWLSproject back.ThekeyupdateswithrespecttotheroutinesusedbyOWLS
(Schaye et al. 2010), and highlight the sensitivity of outcomes to aretheinclusionofametallicitydependenceinthestarformation
thevariationoftheseparameters.Incommonwithcomplementary law, the implementation of energy feedback associated with star
studies,thesesimulationsindicatethatthepropertiesofthesimu- formationviastochasticthermalheating,andtheinclusionofavis-
latedgalaxypopulationaremostsensitivetothesubgridparameters coustransportlimitontheBHaccretionrate.
governing the efficiency of energy feedback (Schaye et al. 2010; S15 introduced the resolution nomenclature of the EAGLE
Scannapieco et al. 2012; Haas et al. 2013a,b; Vogelsberger et al. project.“Intermediate-resolution”simulationshaveparticlemasses
2013). correspondingtoanL = 100comovingMpc(hereafter cMpc)
Thispaperisstructuredasfollows.Thesimulationinitialcon- volume realised with 2 × 15043 particles (an equal number of
ditions, and the algorithms used to evolve them, are described in baryonic and dark matter particles), such that the initial gas par-
§2.Theparametrisationofthefourmodelsthatarecalibratedtore- ticle mass is mg = 1.81×106M(cid:12), and the mass of dark mat-
producethez = 0.1GSMFisdescribedin§3.Resultsfromthese terparticlesismdm = 9.70×106M(cid:12).ThePlummer-equivalent
simulations, which serve as a motivation for the development of gravitational softening length is fixed in comoving units to 1/25
thereferencemodel,arepresentedin§4,whereresultsfromsimu- of the mean interparticle separation (2.66 comovingkpc, here-
lationsfeaturingsingle-parametervariationsofthereferencemodel after ckpc)untilz = 2.8,andinproperunits(0.70properkpc,
arealsoshown.Finally,theresultsaresummarisedanddiscussed hereafter pkpc) at later times. The intermediate-resolution simu-
in§5. lations marginally resolve the Jeans scales at the star formation
threshold (n (cid:39) 10−1cm−3) in the warm (T (cid:39) 104K) ISM.
H
“High-resolution”simulationsadoptparticlemassesandsoftening
lengths that are smaller by factors of eight and two, respectively.
The SPH kernel size, specifically its support radius, is limited to
aminimumofone-tenthofthegravitationalsofteningscale.This
study focusses on intermediate-resolution simulations using vol-
2 SIMULATIONSANDSUBGRIDPHYSICS
umesofsideL=25,50and100cMpc,whichthereforecomprise
This section comprises an overview of the simulation setup and 2×3763,2×7523and2×15043particles,respectively.
subgridphysicsimplementation.Itincludessimilarinformationto Galaxiesandtheirhosthaloesareidentifiedbyamulti-stage
Sections3&4ofS15,soreadersfamiliarwiththesimulationsmay process, beginning with the application of the friends-of-friends
wish to skip this section. A relatively comprehensive description (FoF)algorithm(Davisetal.1985)tothedarkmatterparticledistri-
ofthesubgridimplementationsofstarformationandfeedbackis bution,withalinkinglengthofb=0.2timesthemeaninterparticle
retainedhere,becausethesedetailsareanecessaryfoundationfor separation.Gas,starandBHparticlesareassociatedwiththeFoF
latersections. group,ifany,oftheirnearestneighbourdarkmatterparticle.The
The cosmological parameters assumed by the EAGLE sim- SUBFIND algorithm (Springel et al. 2001; Dolag et al. 2009) is
ulations are those recently inferred by the Planck Collaboration thenusedtoidentifyself-boundsubstructures,orsubhaloes,within
(2014a,b),thekeyparametersbeingΩ = 0.307,Ω = 0.693, the full particle distribution (gas, stars, BHs and dark matter) of
m Λ
Ω = 0.04825,h = 0.6777andσ = 0.8288.Initialconditions FoFhaloes.Thesubhalocomprisingtheparticlewiththeminimum
b 8
adoptingtheseparametersweregeneratedusingatransferfunction gravitationalpotential,whichisalmostexclusivelythemostmas-
createdwiththeCAMBsoftware(Lewisetal.2000),the2nd-order sivesubhalo,isdefinedasthecentralsubhalo,theremainderbeing
LagrangianperturbationtheorymethodofJenkins(2010),andthe
Gaussian white noise field Panphasia (Jenkins 2013; Jenkins &
Booth2013).Acompletedescriptionofthegenerationoftheinitial 4 Seehttp://eagle.strw.leidenuniv.nl.
(cid:13)c 2011RAS,MNRAS000,1–26
4 R.A.Crainetal.
satellite subhaloes. The coordinate of the particle with the mini- 1998),
mum potential also defines the position of the halo, about which (cid:18) Σ (cid:19)n
is computed the spherical overdensity (SO; Lacey & Cole 1994) Σ˙(cid:63) =A 1 M gpc−2 , (1)
mass,M ,fortheadopteddensitycontrastof200timesthecrit- (cid:12)
200
ical density, ρc. Satellite subhaloes separated from their central whereΣ(cid:63) andΣg arethesurfacedensitydensityofstarsandgas,
galaxy by less than the minimum of 3pkpc and the stellar half- respectively,canbeexpressedas:
mass radius of the central galaxy are merged into the latter; this
stepeliminatesasmallnumberoflow-masssubhaloesdominated m˙∗ =mgA(cid:0)1 M(cid:12)pc−2(cid:1)−n(cid:16)GγfgP(cid:17)(n−1)/2, (2)
bysingle,high-densitygasparticlesorBHs.Finally,whenquoting
wherem isthegasparticlemass,γ =5/3istheratioofspecific
the properties of galaxies (e.g. stellar mass, star formation rate), g
heats(andshouldnotbeconfusedwithγ ),Gisthegravitational
only those subhalo particles within a spherical aperture of radius eos
constant,f isthemassfractioningas(assumedtobeunity),and
30pkpcareconsidered.S15(theirFigure6)demonstratedthatthis g
P is the total pressure. This pressure law implementation is ad-
practiceyieldsstellarmassescomparabletothoserecoveredwithin
vantageousfortworeasons.Firstly,thefreeparametersofthestar
aprojectedcircularaperturewiththePetrosianradiusatz=0.1.
formationlaw(A,n)arespecifiedexplicitlybyobservations:the
valuesA = 1.515×10−4 M yr−1kpc−2 andn = 1.4(n = 2
(cid:12)
forn > 103cm−3)areadopted,wherethevalueofAhasbeen
2.1 Radiativeprocesses H
adjustedfromthatreportedbyKennicutt(1998)toconvertfromthe
Radiativecoolingandheatingratesarecomputedonanelement- Salpeterinitialstellarmassfunction(IMF)totheChabrier(2003)
by-elementbasisbyinterpolatingtables,generatedwithCLOUDY form adopted by the simulations. Secondly, this implementation
(version 07.02, Ferland et al. 1998), that specify cooling rates as guaranteesthattheobservedKennicutt-Schmidtrelationisrepro-
afunctionofdensity,temperatureandredshift,undertheassump- ducedforanyequationofstate(i.e.anycombinationofT and
eos
tionthatthegasisopticallythin,isinionisationequilibrium,and γ ) applied to star-forming gas. This is in contrast to volumet-
eos
is exposed to the cosmic microwave background and a spatially- ric star formation laws, which must be recalibrated whenever the
uniform, temporally-evolving Haardt & Madau (2001) UV/X-ray equationofstateisaltered.
background (for further details, see Wiersma et al. 2009a). The Starformationoccursonlyincold,densegas,requiringthat
UV/X-raybackgroundisimposedinstantaneouslyatz =11.5.To adensitythresholdforstarformation,n(cid:63),beimposed.Sincethe
H
account for enhanced photoheating rates (relative to the optically transition from a warm, neutral phase to a cold, molecular one
thin rates assumed here) during the epochs of reionisation, 2eV occurs at lower densities and pressures in metal-rich (and hence
perprotonmassisinjected,rapidlyheatinggasto∼ 104K.This dust-rich) gas, we adopt the metallicity-dependent star formation
isdoneinstantaneouslyatz = 11.5(consistentwithPlanckcon- thresholdproposedbySchaye(2004),whichwasimplementedin
straints)forHIreionisation,butforHeIItheenergyinjectionisdis- theOWLSsimulation“SFTHRESHZ”:
tributedinredshiftwithaGaussianfunctioncentredaboutz=3.5 (cid:34) (cid:18) Z (cid:19)−0.64 (cid:35)
withawidthofσ(z)=0.5.Thisensuresthatthethermalevolution n(cid:63)(Z)=min 0.1cm−3 ,10cm−3 (3)
oftheintergalacticmediummimicsthatinferredbySchayeetal. H 0.002
(2000).
where Z is the gas metallicity. Hydrogen number density, n , is
H
relatedtotheoverallgasdensity,ρ,vian ≡ Xρ/m ,whereX
H H
isthehydrogenmassfractionandm isthemassofahydrogen
H
2.2 TheISMandstarformation
atom.Toexaminetheeffects,ifany,ofadoptingthismetallicity-
Simulations of large cosmological volumes lack, in general, the dependent threshold, the EAGLE suite includes a simulation that
resolution and physics to model the cold (T (cid:28) 104K) inter- adopts a constant threshold of n(cid:63)H = 0.1cm−3, which was the
stellar gas phase from which molecular clouds and stars form. A fiducialapproachofOWLS.Inbothcases,topreventstarformation
globaltemperaturefloor,T (ρ)isthereforeimposed,correspond- inlow-overdensitygasathighredshift,star-forminggasisrequired
eos
ing to a polytropic equation of state, P ∝ ργeos, normalised tohaveanoverdensityδ>57.7.
eos
to T = 8000K at n = 0.1cm−3. A fiducial polytrope of
eos H
γ = 4/3isadopted,sincethisensuresthattheJeansmass,and
eos 2.3 Stellarevolutionandmassloss
the ratio of the Jeans length to the SPH support radius, are inde-
pendentofdensity,thusinhibitingspuriousfragmentation(Schaye The implementation of stellar evolution and mass loss is based
& Dalla Vecchia 2008). The effect of varying γ is explored in upon that described by Wiersma et al. (2009b). Star particles are
eos
§4.2, where simulations conducted using isothermal (γ = 1) treatedassimplestellarpopulations(SSPs)withanIMFoftheform
eos
andadiabatic(γ =5/3)equationsofstateareexamined. proposedbyChabrier(2003),spanningtherange0.1−100 M .
eos (cid:12)
Asecondtemperaturefloorof8000Kisimposedforgaswith Ateachtimestepandforeachstellarparticle,thosestellarmasses
n > 10−5cm−3, which prevents metal-rich gas from cooling reaching the end of the main sequence phase are identified using
H
toverylowtemperatures,sincethephysicalprocessesrequiredto metallicity-dependentlifetimes,andthefractionoftheinitialparti-
modeldense,low-temperaturegasarenotincludedhere.Thisfloor clemassreachingthisevolutionarystageisused,togetherwiththe
doesnotapplytolow-density(i.e.intergalactic)gas,sincesuchgas particleinitialelementalabundancesandnucleosyntheticyieldta-
coolsadiabaticallyandismodelledaccuratelybythehydrodynam- bles,tocomputethemassofeachelementthatislostthroughwinds
icsscheme. fromAGBstars,windsfrommassivestars,andtypeIISNe.Eleven
Star formation is implemented stochastically, based on the elementsaretracked,andthemassandenergylostthroughtypeIa
pressurelawschemeofSchaye&DallaVecchia(2008).Underthe SNe is also computed, assuming the rate of type Ia SNe per unit
(reasonable) assumption that star-forming gas is self-gravitating, stellar mass is specified by an empirically-motivated exponential
the observed Kennicutt-Schmidt star formation law (Kennicutt delayfunction.
(cid:13)c 2011RAS,MNRAS000,1–26
EAGLE:calibrationandmodelvariations 5
2.4 Energyfeedbackfromstarformation forheatedresolutionelements,DallaVecchia&Schaye(2012)de-
rived an estimate for the maximum gas density, n , at which
StarsinjectenergyandmomentumintotheISMviastellarwinds, H,tc
theirstochasticheatingschemecanbeefficient(theirequation18),
radiation,andSNe.Theseprocessesareparticularlyimportantfor
massive,short-livedstarsand,ifstarformationissufficientlyvig- (cid:18) T (cid:19)3/2(cid:18) m (cid:19)−1/2
n ∼10cm−3 g , (4)
orous, the associated feedback can drive large-scale galactic out- H,tc 107.5 K 106 M
(cid:12)
flows (e.g. Veilleux et al. 2005). At present, simulations of large
cosmological volumes lack the resolution necessary to model the where T > ∆TSF is the post-heating temperature. For simplic-
self-consistentdevelopmentofoutflowsfromfeedbackinjectedon ity,thecoolingisassumedtobeBremsstrahlung-dominated,sothe
thescalesofindividualstarclusters,andmustappealtoasubgrid value of nH,tc will be over-estimated in the temperature regime
treatment. where(metal-)linecoolingdominates(T (cid:28)107K).S15notedthat
Inthesimplestimplementationofenergyfeedbackbythermal somestarsdoformintheEAGLEsimulationsfromgaswithden-
heating,theenergyproducedateachtimestepbyastarparticleis sitygreaterthanthecriticalvalue,inwhichcasethespurious(nu-
distributedtoanumberofitsneighbouringhydrodynamicresolu- merical)radiativelossesaresignificant.Inthecaseofsuchlosses
tionelements,supplementingtheirinternalenergy.DallaVecchia beingsignificant,theenergybudgetusedtoreproducetheGSMF
&Schaye(2012,seealsoDallaVecchia&Schaye2008,Creasey inthesimulationwilllikelybeanoverestimateofthatrequiredin
etal.2011,Kelleretal.2014andCreaseyetal.2015)arguethatthe Nature.
feedbackenergy(canonically∼ 1051 ergper100M forastan- Equation 4 indicates that numerical losses associated with
(cid:12)
dardIMFifconsideringSNeasthesoleenergysource)isinitially stars forming from high-density gas can be mitigated by appeal-
distributedovertoomuchmass:themassofatleastoneresolution ingtoahigherheatingtemperature,∆TSF.However,thisisnotan
element,O(104 −107)M ,ratherthanthatoftheactualejecta, idealsolution.Forafixedquantityoffeedbackenergyperstellar
(cid:12)
O(100−101)M(cid:12).Theresultingtemperatureincrementisthenfar massformed(i.e.constantfth),theprobabilitythatastarparticle
smallerthaninreality,andbyextensiontheradiativecoolingtime triggersaheatingeventisinverselyproportionalto∆TSF.Based
oftheheatedgasismuchtooshort.Pressuregradientsestablished ontheenergybudgetdescribedabove,theexpectationvalueofthe
bytheheatingaretooshallowand,perhapsmoreimportantly,are numberofresolutionelements(inthiscase,SPHparticles)heated
typically erased on a (radiative) timescale shorter than the sound by a star particle is specified by Dalla Vecchia & Schaye (2012,
crossingtimeofaresolutionelement. equation8)as:
The EAGLE simulations adopt the stochastic thermal feed- (cid:18) ∆T (cid:19)−1
backschemeofDallaVecchia&Schaye(2012),inwhichthetem- (cid:104)Nheat(cid:105)≈1.3fth 107.5SFK . (5)
peratureincrement,∆T ,ofheatedresolutionelementsisspeci-
SF
fied.Besidesenablingonetomitigatetheproblemdescribedabove, In the regime of ∆T (cid:29) 107.5K, the probability of heating a
SF
astochasticimplementationoffeedbackisadvantageousbecauseit singleresolutionelementbecomessmall,leadingtopoorsampling
enables the quantity of energy injected per feedback event to be ofthefeedbackcycle.Wethereforeonlyconsidermodelsadopting
specified, even if the mean quantity of energy injected per unit ∆T =107.5K.
SF
stellar mass formed is fixed. Having specified ∆TSF, the proba- If ∆TSF is sufficiently high to ensure that numerical losses
bilitythataresolutionelementneighbouringayoungstarparticle aresmall,thephysicalefficiencyoffeedbackcanbecontrolledby
is heated,is determined bythe fraction of theenergy budget that adjusting f . This mechanism is a means of modelling the sub-
th
isavailableforfeedback5.Forconsistencywiththenomenclature gridradiativelossesthatarenotaddressedbyoursimpletreatment
introducedbyDallaVecchia&Schaye(2012),werefertothisfrac- of the ISM. Because these losses should depend on the physical
tionasfth. conditionsintheISM,thereisphysicalmotivationtospecifyfth
We adopt the convention that fth = 1 equates to an ex- asafunctionofthelocalpropertiesofthegas.Primarily,itisthe
pectation value of the injected energy of 1.736×1049 erg M−1 freedomtoadjustf thatenablesthesimulationstobecalibrated.
(cid:12) th
(8.73×1015 ergg−1)ofstellarmassformed.Thiscorrespondsto
the energy available from type II SNe resulting from a Chabrier
IMF,subjecttotwoassumptions.Firstly,that6−100M(cid:12)starsare 2.5 BlackholesandAGNfeedback
the progenitors of type II SNe (6−8M stars explode as elec-
(cid:12)
FeedbackassociatedwiththegrowthofBHsisanessentialingredi-
troncapturesupernovaeinmodelswithconvectiveovershoot;e.g.
Chiosietal.1992),andsecondlythateachSNliberates1051erg. entoftheEAGLEsimulations.Besidesregulatingthegrowthofthe
BHs,AGNfeedbackquenchesstarformationinmassivegalaxies
Weinjectenergyonceforeachstarparticle,whenitreachesanage
andshapesthegasprofilesoftheirhosthaloes.Theimplementa-
of30Myr.
tionadoptedhereconsistsoftwoelements,namelyi)aprescription
Forthermalfeedbacktobeeffective,thepressuregradientes-
forseedinggalaxieswithBHsandforfollowingtheirgrowthvia
tablishedbytheheatingmustbeabletoperformworkonthegas
mergersandgasaccretion,andii)aprescriptionforcouplingthe
(viasoundwaves,orshocksforsupersonicflows)onatimescale
radiatedenergy,liberatedbyBHgrowth,totheISM.Themethod
thatisshorterthanthatrequiredtoerasethegradientviaradiative
fortheformerisbasedonthatintroducedbySpringeletal.(2005)
cooling.Bycomparingthesoundcrossingandcoolingtimescales
andmodifiedbyBooth&Schaye(2009)andRosas-Guevaraetal.
(2013),whilethemethodforthelatterissimilartothatdescribed
byBooth&Schaye(2009).
5 Durier&DallaVecchia(2012)showthatimplementationsofkineticand
FollowingSpringeletal.(2005),seedBHsareplacedatthe
thermalfeedbackconvergetosimilarsolutionsinthelimitthatthecool-
centreofeveryhalomoremassivethan1010 M /hthatdoesnot
ingtimeislong.Weadoptathermalimplementationhereforconsistency (cid:12)
withourAGNfeedbackimplementation,andbecausenumericaltestsindi- alreadycontainaBH.Candidatehaloesareidentifiedbyrunninga
catethatitislesssusceptibletoproblemsstemmingfrompoornumerical FoFalgorithmwithlinkinglengthb = 0.2onthedarkmatterdis-
samplingofoutflows. tribution.Whenaseedisrequired,thehighestdensitygasparticle
(cid:13)c 2011RAS,MNRAS000,1–26
6 R.A.Crainetal.
is converted into a collisionless BH particle, inheriting the parti- that couples to the ISM. In common with the efficiency of feed-
cle mass and acquiring a subgrid BH mass m = 105 M /h back associated with star formation, f , the value of (cid:15) must be
BH (cid:12) th f
which,fortheintermediateresolutionsimulationsconsideredhere, chosenbycalibratingtoobservations.Becauseofself-regulation,
issmallerthantheinitialgasparticlemassbyafactorof12.3.Cal- the value of (cid:15) only affects the masses of BHs (Booth & Schaye
f
culationsofBHpropertiesarethereforefunctionsofm ,whilst 2009),whichvaryinverselywith(cid:15) ,andithaslittleeffectonthe
BH f
gravitational interactions are computed using the particle mass. stellar mass of galaxies (provided its value is non-zero). The pa-
WhenthesubgridBHmassexceedstheparticlemass,BHparticles rameter(cid:15) canbecalibratedbyensuringthenormalisationofthe
f
stochasticallyaccreteneighbouringgasparticlessuchthatparticle observedrelationbetweenBHmassandstellarmassisreproduced
andsubgridBHmassesgrowinconcert.BHsarepreventedfrom at z = 0. Although implemented as a single heating mode, Mc-
“wandering”outoftheirparenthaloesbyforcingthosewithmass Carthyetal.(2011)demonstratethatthisschememimicsquiescent
< 100m tomigratetowardsthepositionoftheminimumofthe ‘radio’-likeandvigorous‘quasar’-likeAGNmodeswhentheBH
g
gravitationalpotentialintheirhalo. accretionrateisasmall((cid:28) 1)orlarge(∼ 1)fractionoftheEd-
BHsaremergedifseparatedbyadistancethatissmallerthan dingtonrate,respectively.
both the BH kernel size, h , and three gravitational softening S15demonstratedthattheefficiencyadoptedbyOWLS,(cid:15) =
BH f
lengths,andiftheirrelativevelocityissmallerthanthecircularve- 0.15,remainsasuitablechoiceatthehigherresolutionofEAGLE.
(cid:112)
locityatthedistanceh ,v < Gm /h ,whereh and Therefore, a fraction (cid:15) (cid:15) = 0.015 of the accreted rest mass en-
BH rel BH BH BH f r
m are,respectively,theSPHkernelsizeandthesubgridmass ergyiscoupledtothelocalISMasfeedback.EachBHmaintains
BH
of the most massive BH in the pair. The relative velocity thresh- a “reservoir” of feedback energy, E . After each time step ∆t,
BH
oldpreventsBHsfrommergingduringtheinitialstagesofgalaxy anenergy(cid:15) (cid:15) m˙ c2∆tisaddedtothereservoir.OnceaBHhas
f r accr
mergers. storedsufficientenergytoheatatleastonefluidelementofmass
m ,itbecomeseligibletoheat,stochastically,itsSPHneighbours
g
byincreasingtheirtemperatureby∆T .Foreachneighbourthe
AGN
2.5.1 Gasaccretionontoblackholes heatingprobabilityis
Thegasaccretionrate,m˙ ,isspecifiedbytheminimumofthe E
accr P = BH , (10)
Eddingtonrate, ∆(cid:15) N (cid:104)m (cid:105)
AGN ngb g
m˙ = 4πGmBHmp, (6) where∆(cid:15)AGN isthechangeininternalenergyperunitmasscor-
Edd (cid:15) σ c respondingtothetemperatureincrement,∆T (theparameter
r T AGN
∆T isconvertedinto∆(cid:15) assumingafullyionisedgasof
and AGN AGN
primordialcomposition),N isthenumberofgasneighboursof
m˙accr =min(cid:0)m˙Bondi[(cs/Vφ)3/Cvisc],m˙Bondi(cid:1), (7) the BH and (cid:104)mg(cid:105) is their mngeban mass. The value of EBH is then
reducedbytheexpectationvalueoftheinjectedenergy.Thetime
wherem˙ istheBondi-Hoyle(1944)rateapplicabletospheri-
Bondi stepoftheBHsislimitedtoaimforprobabilitiesP <0.3.
callysymmetricaccretion,
Larger values of ∆T yield more energetic feedback
AGN
4πG2m2 ρ events,generallyresultinginreducedradiativelosses(asperequa-
m˙ = BH . (8)
Bondi (c2+v2)3/2 tion4).However,largervaluesalsomakethefeedbackmoreinter-
s
mittent.Ingeneral,theambientdensityofgaslocaltothecentral
Herem istheprotonmass,σ theThomsoncrosssection,cthe
p T BHofgalaxiesisgreaterthanthatofstar-forminggasdistributed
speedoflight,(cid:15) theradiativeefficiencyoftheaccretiondisc,andv
r throughouttheirdiscs,soahigherheatingtemperatureisrequired
therelativevelocityoftheBHandthegas.Finally,Vφisthecircu- tominimisenumericallosses6.TheEAGLEreferencemodelpre-
lationspeedofthegasaroundtheBHcomputedusingequation16 sentedbyS15adopts∆T =108.5 K.Inthatstudyanalterna-
AGN
ofRosas-Guevaraetal.(2013)andC isafreeparameterrelated
visc tiveintermediate-resolutionmodelwasalsopresented(model“AG-
totheviscosityofanotionalsubgridaccretiondisc.Thegrowthof NdT9”)which,byappealingto∆T = 109 K,wasfoundto
AGN
theBHisspecifiedby
more accurately reproduce the observed gas fractions and X-ray
m˙ =(1−(cid:15) )m˙ . (9) luminositiesofgalaxygroups.Thishighertemperatureincrement
BH r accr
wasalsofoundtobenecessaryinhigh-resolutionsimulations,since
We assume a radiative efficiency of (cid:15)r = 0.1. The factor theyresolvehigherambientdensitiesclosetoBHsandhenceex-
(cs/Vφ)3/Cvisc bywhichtheBondirateismultipliedinequation hibithighercoolingrates.
7 is equivalent to the ratio of the Bondi and viscous time scales Thevaluesoftherelevantsubgridparametersadoptedbyall
(seeRosas-Guevaraetal.2013).ThecriticalratioofVφ/cs above simulationsfeaturedinthisstudyarelistedinTable1.
whichangularmomentumisassumedtosuppresstheaccretionrate
scalesasC−1/3.LargervaluesofC thereforecorrespondtoa
visc visc
lowersubgridkineticviscosity,andsoacttodelaythegrowthof
3 CALIBRATEDSIMULATIONS
BHsbygasaccretionand,byextension,theonsetofquenchingby
AGNfeedback. AsdiscussedbyS15(seetheir§2),ifsubgridmodelsforenergy
feedbackofferanincompletedescriptionoftheprocessestheyare
2.5.2 AGNfeedback
6 BHsareinprincipleabletoinjectfeedbackenergyatalltimes,unlike
A single mode of AGN feedback is adopted, whereby energy is
starparticleswhichinjectpromptfeedbackonlyonce.TheAGNfeedback
injectedthermallyandstochastically,inamanneranalogoustoen- cyclecanthereforeremainwellsampledforhigherheatingtemperatures
ergyfeedbackfromstarformation(see§2.4).Theenergyinjection thanisthecaseforstarformationfeedback,aslongastheintervalbetween
rateis(cid:15)f(cid:15)rm˙accrc2,where(cid:15)f isthefractionoftheradiatedenergy heatingeventsisshorterthanaSalpetertime(Booth&Schaye2009).
(cid:13)c 2011RAS,MNRAS000,1–26
EAGLE:calibrationandmodelvariations 7
Identifier Sidelength N γeos n(cid:63)H fth-scaling fth,max fth,min nH,0 nn Cvisc/2π ∆TAGN
[cMpc] [cm−3] [cm−3] log [K]
10
Calibratedmodels
FBconst 50 752 4/3 Eq.3 − 1.0 1.0 − − 103 8.5
FBσ 50 752 4/3 Eq.3 σ2 3.0 0.3 − − 102 8.5
DM
FBZ 50 752 4/3 Eq.3 Z 3.0 0.3 − − 102 8.5
Ref(FBZρ) 50 752 4/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 100 8.5
Refvariations
eos1 25 376 1 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 100 8.5
eos5/3 25 376 5/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 100 8.5
FixedSfThresh 25 376 4/3 0.1 Z,ρ 3.0 0.3 0.67 2/ln10 100 8.5
WeakFB 25 376 4/3 Eq.3 Z,ρ 1.5 0.15 0.67 2/ln10 100 8.5
StrongFB 25 376 4/3 Eq.3 Z,ρ 6.0 0.6 0.67 2/ln10 100 8.5
ViscLo 50 752 4/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 102 8.5
ViscHi 50 752 4/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 10−2 8.5
AGNdT8 50 752 4/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 100 8.0
AGNdT9 50 752 4/3 Eq.3 Z,ρ 3.0 0.3 0.67 2/ln10 100 9.0
Table1.Parametersthatarevariedinthesimulations.Columnsare:thesidelengthofthevolume(L)andtheparticlenumberperspecies(i.e.gas,DM)per
dimension(N),thepowerlawslopeofthepolytropicequationofstate(γeos),thestarformationdensitythreshold(n(cid:63)H),thescalingvariableoftheefficiency
ofstarformationfeedback(fth),theasymptoticmaximumandminimumvaluesoffth,theRefmodel’sdensitytermdenominator(nH,0)andexponent
(nn)fromequation14,thesubgridaccretiondiscviscosityparameter(Cvisc)fromequation7,andthetemperatureincrementofstochasticAGNheating
(∆TAGN).Theuppersectioncomprisesthefourmodelsthathavebeencalibratedtoreproducethez=0.1GSMF,andthelowersectioncomprisesmodels
featuringasingle-parametervariationsofRef(variedparameterhighlightedinbold).AllmodelsalsoadoptnZ =2/ln10withtheexceptionsofFBσ,for
whichtheparameternZisreplacedbynTwiththesamenumericalvalue(seeequation12),andFBconst,forwhichtheparameterisinapplicable.
designed to model, are subject to numerical losses, or if the out- GSMF,apracticecommonlyadoptedbythesemi-analyticgalaxy
comesoftheprescriptionsaresensitivetoresolution,thenthetrue formationmodellingcommunity.Low-redshiftgalaxysurveysen-
efficiencies of feedback processes cannot be predicted from first abletheGSMFtobecharacterisedinthelocalUniverseacrossfive
principles.Itwasthereforearguedthatcosmologicalhydrodynami- decadesinmassscale,withaprecisionthatis,assumingauniver-
calsimulationsarepresentlyunabletoyieldabinitiopredictionsof salIMF,limitedprimarilybysystematicuncertaintiesinthestellar
thestellarmassofgalaxies,northemassoftheircentralBH.Sub- evolutionmodelsusedtoinferthemasses(Conroyetal.2009;Pforr
gridparametersshouldthereforebecalibratedsuchthatsimulations etal.2012,butseeTayloretal.2011),peculiarvelocitycorrections
reproducedesireddiagnosticquantities,stellarandBHmassesbe- for faint galaxies (e.g. Baldry et al. 2012), and the method used
inggermaneexamples. tosubtracttheskybackgroundfrombrightgalaxies(e.g.Bernardi
etal.2013;Kravtsovetal.2014).Anadditionalmotivationforap-
Theoptimalapproachtocalibratingsubgridmodelsisnotun-
pealing to the GSMF as the calibration diagnostic is that its re-
ambiguous,sincetherecanbemultiplemeasurableoutcomesthat
productionbythesimulationsisaprerequisiteforexaminationof
aresensitivetotheadjustmentofsubgridparameters,someorall
manyobservablescalingrelations.
ofwhichmightreasonablybeconsideredvalidconstraints.Forex-
ample,inthecaseoffeedbackefficiencies,onemightcalibratethe
Whilstcalibratingthesimulations,attentionwasalsopaidto
modeltoreproducethevelocityand/ormassloadingofoutflowing
thesizesofgalaxies.Aswiththecalibrationdiagnostic,thischoice
gas.However,thesequantitiesremainill-characterisedobservation-
issomewhatarbitrary,butitisreadilymotivatedsincetheformation
ally,andaresensitivetothephysicalscaleonwhichtheyaremea-
ofunrealisticallycompactorextendedgalaxieswouldlimittheutil-
sured(whichisgenerallynotevenwellknown).Reproducingthe
ityofthesimulationsandlikelyindicatephysicaland/ornumerical
propertiesofoutflowsonaparticularspatialscaleoffersnoguar-
inaccuracies.Theformationofdiscgalaxieswithrealisticsizesin
anteethattheyarereproducedonotherscales,since,forexample,
cosmologicalsimulationshasproventobeanon-trivialchallenge,
theinteractionofoutflowswiththecircumgalacticmediummaybe
leadingtotheidentificationandrectificationofmanyshortcomings
inadequately modelled. The choice of calibration diagnostic(s) is
ofnumericaltechniques(e.g.Sommer-Larsenetal.1999;Ritchie&
thereforesomewhatarbitrary,butsomechoicescanbemoreread-
Thomas2001;Marri&White2003;Okamotoetal.2003;Agertz
ily motivated. Clearly, it is necessary that any diagnostic be well
et al. 2007; Springel 2010; Hopkins 2013, 2014). In terms of the
characterised observationally on the scales resolved by the simu-
physicsofgalaxyassembly,Navarro&White(1994)identifiedthe
lation.Perhapsthemostelegantexampleisthestarformationlaw
transferofangularmomentumfromcold,denseclumpsofbaryons
which,onthe∼102pcscaleswefollowintheISM,canbeaccu-
tothenon-dissipativedarkmatterresidingattheoutskirtsofgalaxy
ratelyrepresentedbytheKennicutt-Schmidtrelation;asdescribed
haloesasthemajorconcern.Inacomprehensivesimulationcom-
in§2.2,thefreeparametersofthesubgridstarformationlaware
parisonproject,Scannapiecoetal.(2012)highlightedthattheangu-
unambiguouslyprescribedbyobservations.Inaddition,itisdesir-
larmomentumproblem(andalsotheovercoolingproblem)remains
able to confront calibrated models with complementary observa-
withoutaconsistentsolution,andthereforethatgalaxysizescannot
tional constraints, to minimise the risk of overlooking modelling
yetbeuniquelypredicted,evenwhentheassemblyhistoryoftheir
degeneracies.
parent halo is fully specified. It cannot be assumed that the sizes
Wechosetocalibratethefeedbacksimulationstothez=0.1 ofgalaxydiscswillbeaccuratelyreproducedbysimulations,even
(cid:13)c 2011RAS,MNRAS000,1–26
8 R.A.Crainetal.
iftheysuccessfullyreproduceasuitablecalibrationdiagnostic.For havebeenusedtoinvestigatetheestablishmentoftheGSMF(Op-
thisreason,werequireamodeltoreproduceboththeGSMFand penheimer et al. 2010; Puchwein & Springel 2013; Vogelsberger
theobservedsize-massrelationofdiscgalaxies7atlow-redshift,in etal.2013)andtheformationofdiscgalaxiessimilartotheMilky
orderforthecalibrationprocesstobedeemedsuccessful. Way(Marinaccietal.2014),toexaminetheobservedpropertiesof
The subgrid model governing feedback associated with star someintergalacticabsorptionsystems(Oppenheimer&Dave´2006,
formationinEAGLEisprimarilydependentupontheIMF,f and 2009;Vogelsbergeretal.2014b)andtoreproducetheLocalGroup
th
∆T .AuniversalIMFisadoptedthroughout,andtheadoptionof satellitepopulation(Okamotoetal.2010).
SF
afixed∆T =107.5Kwasmotivatedin§2.4.Therefore,energy Thisimplementationiswell-motivatedsince,bytemporarily
SF
feedback associated with star formation is calibrated exclusively decouplingwindsandspecifyingtheirpropertiesasafunctionof
byvaryingf ,thefractionofthetotalavailableenergyfromtype darkmatterproperties,itaffordssimulationsthebestopportunity
th
IISNethatcouplestotheISM.ThemaineffectontheGSMFof to achieve numerical convergence. However, it precludes the full
increasing(decreasing)f istolower(raise)itsnormalisationin exploitationofthehydrodynamicscalculation.Thephysicalprop-
th
termsofthecomovingnumberdensityofgalaxieswithstellarmass erties of outflows are almost certainly dependent upon the local
belowthe“knee”oftheSchechter(1976)function(demonstrated (baryonic) conditions of the ISM, and these properties are avail-
in§4.2.3). able to use as inputs to subgrid feedback models. Since the phi-
ThesubgridmodelgoverningAGNfeedbackinEAGLEispri- losophyadoptedfortheEAGLEprojectistocalibratethefeedback
marilydependentupon(cid:15) ,C and∆T .Theretentionofthe scheme,theconvergencedemandsplacedupontheadoptedsubgrid
f visc AGN
AGNfeedbackefficiencyadoptedbyOWLS((cid:15) =0.15)wasmo- modelsarerelaxed,presentingtheappealingopportunitytocouple
f
tivatedin§2.5.2.Incontrastto∆T ,whosevalueisfixedbythe thevalueofsubgridparameters(e.g.f )tothebaryonicproperties
SF th
needtosuppressnumericalradiativelossesandtoadequatelysam- ofthelocalISM9.
plethefeedbackprocess,thefreedomtoadjust∆T ,whichde-
AGN
terminestheenergeticsandintermittencyofAGNfeedbackevents,
canbemotivated.Wethereforeexploresimulationswithdifferent
3.1 Calibratingthestarformationfeedbackefficiency
AGNheatingtemperaturesbut,intermsofthecalibrationofgalaxy
properties,∆TAGNonly(weakly)affectsthestellarmassofgalax- The role of fth in shaping the z = 0.1 GSMF is investigated in
ieswithM (cid:38)1011M (itdoes,however,impactmarkedlyupon thissection.Examinationofthepreviousgenerationofsimulations
(cid:63) (cid:12)
thepropertiesoftheintragroupandintraclustermedia,e.g.LeBrun uponwhichtheEAGLEprojectisbasedindicatesthatAGNfeed-
etal.2014;Schayeetal.2015).Forthepurposesofreproducingthe back is a necessary ingredient for regulating the growth of mas-
present-dayGSMF,AGNfeedbackisprimarilycalibratedbyvary- sive galaxies (Crain et al. 2009; Schaye et al. 2010; Haas et al.
ing C , which broadly governs the mass scale at which AGN 2013a)andestablishingthegas-phasepropertiesofgalaxygroups
visc
feedbackbecomesefficient.Sincethismassscaleisweaklydepen- (McCarthyetal.2010,2011;LeBrunetal.2014).Fourcalibrated
dent upon C , models can adopt values of this parameter that simulations are explored, each featuring energy feedback associ-
visc
differbyordersofmagnitude. atedwithbothstarformationandthegrowthofBHs.Allfoursim-
The calibration of phenomenological components of galaxy ulations adopt ∆TAGN = 108.5K, and each features a constant
formationmodelsisapracticethatwasestablishedduringthede- value of Cvisc that is allowed to differ between simulations such
velopmentofthefirstgenerationofsemi-analyticgalaxyformation that,whencombinedwiththefunctionspecifyingfth,theresult-
models8 (Kauffmann et al. 1993; Cole et al. 1994; Somerville & ingz = 0.1GSMFfeaturesabreakclosetotheobservedscaleof
Primack1999;Coleetal.2000;Hattonetal.2003).Semi-analytic M(cid:63) ∼1010.5M(cid:12).
models are built upon the framework of dark matter halo merger Inthreeofthemodels,asymptoticefficienciesoffth,max =3
trees,sotheparameterstheyadoptforgoverningfeedbackmustbe andfth,min = 0.3areused.Valuesgreaterthanunitycanbemo-
coupledtosimplifiedmodelsforthestructureofgalaxiesandtheir tivatedphysically,sincetherearesourcesofenergyfeedbackother
interstellarandcircumgalacticmedia.Incontrast,hydrodynamical thantypeIISNe,andindeedsuchsourcesareofteninvokedinsim-
simulations enable feedback properties to be specified, if so de- ulationsofgalaxyformation,forexamplestellarwindsandradia-
sired,basedonthephysicalconditionslocaltonewlyformedstar tionpressure(Stinsonetal.2013;Hopkinsetal.2014)orcosmic
particles. rays (Jubelgas et al. 2008; Booth et al. 2013). However the pri-
Thusfarthecapabilitytospecifyfeedbackparametersinthis marymotivationforappealingtofth > 1hereistooffsetnumer-
fashion has only been exploited by a limited number of groups, icallossesthatresultfromthefiniteresolutionofthesimulations.
each using a very similar feedback implementation: kinetically- Therearetwomeansbywhichfiniteresolutionintroducesartificial
drivenwindsthatarelaunchedoutsideoftheISM(bydecoupling losses.Thefirst,asdiscussedin§2.4(equation4),beingthatthere
thewindparticlesfromhydrodynamicforces),andwhoseproper- existsamaximumdensityabovewhichstochasticthermalfeedback
tiesareimposedbyspecifyingtheirinitialvelocityandmassload- isinefficient,becausethepressuregradientestablishedbyfeedback
ingfactor(η = M˙ /M˙ )asafunctionofthepropertiesofthe iserasedbyradiativecoolingbeforeitisabletoexertmechanical
wind (cid:63)
dark matter environment, for example the gravitational potential workonthegas.Thesecondstemsfromtheinabilityoflargecos-
or velocity dispersion. Simulations adopting this implementation mologicalsimulationstomodeltheformationoftheearliestgen-
eration of stars. As discussed by Haas et al. (2013a), this means
thatthefirstgenerationofgalaxiesthatforminsimulationsdoso
7 Thesizeevolutionofbothlate-andearly-typegalaxieswillbeexplored
indetailinaforthcomingpaper(Furlongetal.inprep).
8 Modern semi-analytic models often adopt distribution sampling tech- 9 S15 introduced the nomenclature “weak convergence” to describe the
niques to efficiently calibrate their many free parameters (Kampakoglou consistencyofsimulationoutcomesinthecasethatsubgridparametersare
etal.2008;Henriquesetal.2009;Boweretal.2010;Luetal.2012;Mutch recalibratedwhentheresolutionischanged,asopposedto“strongconver-
etal.2013;Henriquesetal.2013,2014). gence”inthecaseofholdingtheparametersfixed.
(cid:13)c 2011RAS,MNRAS000,1–26
EAGLE:calibrationandmodelvariations 9
Okamoto et al. 2010; Oppenheimer et al. 2010; Puchwein &
Springel2013;Vogelsbergeretal.2013;Khandaietal.2014;Vo-
gelsbergeretal.2014a).However,becausethesestudiesalladopt
kinetically-drivenoutflowsimposedoutsideoftheISM,andspec-
ifythepropertiesoftheoutflowsbyimposingamassloadingand
initial wind velocity, our implementation is significantly differ-
ent. Rather than specifying the properties of the outflows, sim-
ilar behaviour on macroscopic scales is sought as an outcome
of the stochastic heating implementation, by calibrating the effi-
ciency, f , as a function of σ2 . The latter is the square of the
th DM
3-dimensional velocity dispersion of dark matter particles within
thesmoothingkernelofastarparticleattheinstantitisborn,and
is a proxy for the characteristic virial scale of the star particle’s
environment,
µm σ2 (cid:16) σ (cid:17)2
T = p DM (cid:39)(4×105K)µ DM . (11)
DM 3k 100kms−1
Forsimplicity,weassumeatalltimesthemeanmolecularweight
of a fully ionized gas with primordial composition, µ = 0.591.
The fit to the z = 0.1 GSMF for this model is achieved with a
Figure1.ThefractionoftheenergybudgetduetotypeIISNefeedback slightlyhighersubgridviscosityforBHaccretionthanisthecase
thatisusedforthermalheating,fth,intheFBconst,FBσ,FBZandRef forFBconst,C =2π×102.
visc
models.TheFBconstmodel,representedbythedashedgreyline,adopts
Since the properties of star formation-driven outflows are
fth=1,independentoflocalconditions.fthdeclinessmoothlyasafunc- linkedtothestateoflocaldarkmatteronlyviagravitationalforces,
tionofTDMintheFBσmodel(equation12),andasafunctionofZinthe
nophysicalmotivationforf (T )issought.Theadoptedfunc-
FBZmodel(equation13).Theupperaxisisalignedandscaledsuchthat th DM
bothFBσ(upperaxis)andFBZ(loweraxis)aredescribedbythedarkblue tionalformsimplymaximisesthefeedbackefficiency(byminimis-
curve(nophysicalcorrespondencebetweenTDMandZisimpliedbythis ingputativesubgridradiativelosses)inlow-massgalaxies,whilst
alignment).TheRefmodeladdsadensitydependencetoFBZ(equation reducing the feedback efficiency in more massive counterparts,
14),suchthatforstarsformingfromgaswithnH<nH,0thefthfunction where the conversion of gas into stars is known to be most ef-
isshiftedtolowervalues(e.g.cyancurvefornH=nH,0/3)andviceversa ficient (e.g. Eke et al. 2005; Behroozi et al. 2013; Moster et al.
(e.g.redcurvefornH=3nH,0). 2013).Athighermassesstill,AGNfeedbackisassumedtodomi-
natetheregulationofstarformation,soalowstarformationfeed-
backefficiencyforhigh-dispersionenvironmentsisreasonable.The
withinhaloesthathavenotbeensubjecttofeedback,andhenceex-
adoptedfunctionalformoff isalogistic(sigmoid)functionof
th
hibitunrealisticallyhighgasfractionsandstarformationefficien-
log T ,
cies.Appealingtof >1forthesegalaxiespartiallycompensates 10 DM
th
forthisunavoidableshortcoming. f =f + fth,max−fth,min, (12)
th th,min 1+(cid:16)TDM(cid:17)nT
105K
FBconst shown in Figure1 (dark blue curve, corresponding to the upper
x-axis). The function asymptotes to f and f in the
ThesimplestmodelinjectsintotheISMafixedquantityofenergy th,max th,min
limits T (cid:28) 105K and T (cid:29) 105K, respectively, and
perunitstellarmassformed,independentoflocalconditions.This DM DM
varies smoothly between these limits about T = 105K (or
valuecorrespondstothetotalenergyliberatedbytypeIISNe,i.e. DM
σ (cid:39)65 km s−1).Theparametern >0controlshowrapidly
f =1.TheadoptedsubgridviscosityparameterforBHaccretion DM T
isthC =2π×103.Althoughtheinjectedenergyisindependent fth varies as the dark matter “temperature” scale deviates from
visc 105K.Theratherunnaturalvaluen = 2/ln10 (cid:39) 0.87follows
of local conditions, scale-dependent macroscopic radiative losses T
fromanearlyimplementationofthefunctionalformadoptedinthe
cannonethelessdevelopself-consistently,forexampleduetodif-
feedbackroutine;anexponentofunitywouldyieldsimilarresults.
ferencesinthemetallicity(andhencecoolingrate)ofoutflowing
gas, the ram pressure at the disc-CGM interface, or the depth of
thepotentialwell.Thismodelthereforeprovidesabaselineagainst
FBZ
whichitispossibletoassessthedegreetowhichtheoverallphys-
icallossesneedtobeestablishedbycalibratinglossesonsubgrid Adjusting the subgrid radiative losses with the metallicity of the
scales. Because fth = 1 represents the uncalibrated case, there ISMassignsaphysicalbasistothefunctionalformoffth.Phys-
is no reason to expect that FBconst will reproduce the observed icallossesassociatedwithstarformationfeedback10 arelikelyto
z = 0.1GSMF.However,wewillseelaterthatitdoesdoso,but be more significant when the metallicity is sufficient for cooling
failstoreproducetheobservedsizesofdiscgalaxies. frommetallinestodominateoverthecontributionfromHandHe.
Fortemperatures105K < T < 107K,characteristicofoutflow-
ing gas in the simulations, the transition is expected to occur at
FBσ
This model adopts the popular convention of prescribing feed- 10 Ametallicitydependencefor(cid:15)f(§2.5.2)isnotexplored,becausemetals
backparametersaccordingtolocalconditionsinferredfromneigh- arenotexpectedtodominatetheradiativelossesatthehighertemperatures
bouringdarkmatterparticles(Oppenheimer&Dave´ 2006,2008; associatedwithAGNfeedback.
(cid:13)c 2011RAS,MNRAS000,1–26
10 R.A.Crainetal.
Z ∼ 0.1Z (Wiersma et al. 2009b). This qualitative behaviour remainclosetounity(theyare1.06and0.70,respectively,forRef-
(cid:12)
iscapturedbythesamefunctionalformasequation12,replacing L0100N1504).ThesubgridviscosityparameterforBHaccretionis
(T ,n ,105K)with(Z,n ,0.1Z )toobtain, lowerinRefthanintheothercalibratedmodels,C =2π×100.
DM T Z (cid:12) visc
f −f
f =f + th,max th,min, (13)
th th,min 1+(cid:16) Z (cid:17)nZ
0.1Z(cid:12) 4 RESULTS
where Z = 0.0127isthesolarmetallicity(AllendePrietoetal.
(cid:12) Thissectionbeginswithanexaminationofthecalibratedsimula-
2001) and n = n = 2/ln10. This function corresponds to
Z T tions.In§4.2,simulationsfeaturingsingleparametervariationsto
the dark blue curve and the lower x-axis in Figure1, with f
th the reference model are briefly introduced, and the impact of the
asymptotingtof andf inthelimitsZ (cid:28) 0.1Z and
th,max th,min (cid:12) changesontheresultinggalaxypopulationareexplored.
Z (cid:29)0.1Z ,respectively.Sincegalaxiestendtofollowatightre-
(cid:12)
lationbetweentheirmassandmetallicity,thefeedbackefficiency
is, as in FBσ, greatest for low-mass galaxies. Moreover, because 4.1 Examinationofthecalibratedsimulations
metallicity characteristically decreases with redshift at fixed stel-
We begin by examining the models calibrated to reproduce the
lar mass, the feedback efficiency is weighted towards early cos-
z =0.1GSMF.BesidestheGSMF,theevolutionofthecomoving
micepochs.Thishelpstopartiallydecouplethegrowthofgalaxies
stellarmassdensityofeachsimulationisexamined,asarethesizes
fromthegrowthoftheirparenthalo,whichisthoughttobeanec-
andspecificstarformationrates(SSFRs)ofgalaxiesatz=0.1.As
essaryconditionforreproducingtheobservednumberdensityevo-
discussedbyS15,theGSMFandgalaxysizeswereusedforthecal-
lutionoflow-massgalaxiesinaCDMcosmogony(e.g.Weinmann
ibrationprocess,soarenotpresentedaspredictions.Althoughthe
etal.2012;Henriquesetal.2013;Mitchelletal.2014;Whiteetal.
modelparameterswerenotadjustedinordertoimprovethecorre-
2014).ThesubgridviscosityparameterforAGNisthesameasper
spondencebetweenthesimulatedandobservedSSFRs,theformer
FBσ,C =2π×102.
visc were inspected throughout the calibration process, and hence the
predictionsoftheSSFRsarenot“blind”.Theaimofthisexercise
Ref(equivalently,FBZρ) istoexaminehowdifferentimplementationsofphysicalprocesses
impactuponthepropertiesofgalaxiesandtheirenvironments.As
AsignificantfractionofthestarparticlesintheFBσandFBZmod-
partofthisprocedure,theconditionsoftheISMfromwhichstars
elsformatdensitiesgreaterthann ,thecriticaldensityabove
H,tc arebornarealsoexplored.
whichfeedbackenergyisquicklyradiatedaway(equation4).The
feedbackassociatedwiththesehigh-densitystarformationevents
isthereforenumericallyinefficient.Theconsequencesofthisover- 4.1.1 Thegalaxystellarmassfunction
coolingareexploredlaterin§4.1.Theoverestimatedlossescanbe
Thez = 0.1GSMFsofthefourcalibratedsimulations,FBconst
compensated by introducing a density dependence in the expres-
(redcurve),FBσ(green),FBZ(cyan)andRef(darkblue),runus-
sionforf :
th ingL050N0752initialconditions,areshownintheleft-handpanel
f =f + fth,max−fth,min , (14) ofFigure2.Theright-handpanelalsoshowstheGSMFoftheref-
th th,min 1+(cid:16) Z (cid:17)nZ(cid:16)nH,birth(cid:17)−nn erencemodelatintermediateresolutioninaL = 100cMpcvol-
0.1Z(cid:12) nH,0 ume(Ref-L100N1504,redcurve),whichwasintroducedbyS15,
where n is the density of a gas particle at the instant it is andinasmallerL=25cMpcrealisation(Ref-L025N376,green),
H,birth
convertedintoastarparticle.Thefeedbackefficiencythereforein- whichisusedin§4.2asabaselineagainstwhichtocomparesev-
creaseswithdensityatfixedmetallicity,whilstrespectingtheorig- eralsingleparametervariationsofthereferencemodel.Maintain-
inal asymptotic values. The choice of n = 0.67cm−3 was ingtheconventionestablishedbyS15,curvesaredrawnwithdot-
H,0
guided by a suite of small test simulations, which also indicated ted lines where galaxies are less massive than 100 (initial mass,
thattheadoptionofn =n issufficienttoreproducethez=0.1 m )baryonicparticles,asresolutiontests(presentedinS15)indi-
n Z g
GSMF. The effect of the additional density term is illustrated in catethatsamplingerrorsduetofiniteresolutionaresignificantin
Figure1,whereitcanbeseenthatforn = n thefunctional thisregime.Atthehigh-massend,curvesaredrawnwithdashed
H H,0
formadoptedbyRefisidenticaltothatofFBZ,butthef curve lines where the GSMF is sampled by fewer than 10 galaxies per
th
is shifted to lower (higher) values for stars forming from lower (∆log M = 0.2) bin. Data points represent the GSMF in-
10 ∗
(higher)densitygas.Onlytheshifttohigherf forhigherdensity ferredfromtheSloanDigitalSkySurvey(Li&White2009,SDSS,
th
gas(atfixedmetallicity)isrequiredtooffsetnumericallosses,but filled circles) and the Galaxy and Mass Assembly (Baldry et al.
forsimplicitywechoosetoincludetheshifttolowerefficiencyat 2012, GAMA, open circles) survey. In both cases, the data have
lowdensitythatisimpliedbytheadoptedfunction.Suchadensity beenadjustedforconsistencywiththevalueoftheHubbleparam-
dependence may also have a physical basis: because the star for- eter adopted by the simulations. GAMA measurements on scales
mation law has asupra-linear dependence on surface density, the M < 108M are drawn as lower limits, since the data are ex-
(cid:63) (cid:12)
feedbackenergyinjectionrateperunitvolumeincreaseswithden- pectedtobeincompleteatthetypicalsurfacebrightnessassociated
sity.Atfixeddensity,ahigherenergyinjectionratecorrespondsto withthesegalaxies(Baldryetal.2012).
highertemperaturesandlongercoolingtimes.Therefore,feedback Intherangeofstellarmasseswherethemassresolutionand
associatedwithclusteredstarformationisexpectedtoleadtolower volume of the simulation enable a robust measurement of the
radiativelosses(e.g.Heiles1990;Creaseyetal.2013;Krauseetal. GSMF(108 <M (cid:46)1011M ),thenumberdensityofgalaxiesat
(cid:63) (cid:12)
2013;Nath&Shchekinov2013;Royetal.2013;Kelleretal.2014), fixedstellarmassproducedbyeachofthefourcalibratedmodels
andvice-versa.Thedensity-dependentfeedbackefficiencyadopted isconsistentwiththeobservationaldatato< 0.31dex.Thispre-
here(equation14)ensuresthat,whenintegratingoverallstarfor- cisioniscomparabletothesystematicuncertaintyassociatedwith
mationeventsinthesimulation,themeanandmedianvaluesoff spectrophotometrictechniquesforinferringgalaxystellarmasses,
th
(cid:13)c 2011RAS,MNRAS000,1–26