Galaxy evolution in the infra-red: comparison of a hierarchical galaxy formation model with SPITZER data

1 MB·English

by  C. G. Lacey

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed1February2008 (MNLATEXstylefilev2.2)
Galaxy evolution in the infra-red: comparison of a hierarchical
galaxy formation model with SPITZER data
⋆ 1 1 1 2 3 3
C. G. Lacey , C. M. Baugh, C.S. Frenk, L. Silva, G.L. Granato, and A. Bressan,
8 1InstituteforComputationalCosmology,DepartmentofPhysics,UniversityofDurham,SouthRoad,Durham,DH13LE,UK
0 2INAF,OsservatorioAstronomicodiTrieste,ViaTiepolo11,I-34131Trieste,Italy
0 3INAF,OsservatorioAstronomicodiPadova,Vicolodell’Osservatorio2,I-35122Padova,Italy.
2

n
a 1February2008
J

8
1 ABSTRACT
Wepresentpredictionsfortheevolutionofthegalaxyluminosityfunction,numbercountsand

] redshiftdistributionsintheIRbasedontheΛCDMcosmologicalmodel.Weusethecombined
h
GALFORMsemi-analyticalgalaxyformationmodelandGRASILspectrophotometriccodeto
p
- computegalaxySEDs includingthe reprocessingof radiationby dust. The model,which is
o the same as thatin Baughetal. (2005), assumestwo differentIMFs: a normalsolar neigh-
r
bourhoodIMFforquiescentstar formationindisks, anda verytop-heavyIMFin starbursts
t
s triggeredbygalaxymergers.Wehaveshownpreviouslythatthetop-heavyIMFseemstobe
a
necessary to explain the number counts of faint sub-mm galaxies. We compare the model
[
with observationaldata from the Spitzer Space Telescope, with the model parametersfixed

2 at values chosen before Spitzer data became available. We find that the model matches the
v observedevolutionintheIRremarkablywelloverthewholerangeofwavelengthsprobedby
2
Spitzer.Inparticular,theSpitzerdatashowthatthereisstrongevolutioninthemid-IRgalaxy
6 luminosityfunctionoverthe redshiftrangez ∼ 0−2,andthisisreproducedbyourmodel
5
withoutrequiringany adjustmentof parameters. On the other hand,a modelwith a normal
1
IMF in starbursts predicts far too little evolution in the mid-IR luminosity function, and is
.
4 thereforeexcluded.
0
7 Key words: galaxies: evolution – galaxies: formation – galaxies: high-redshift– infrared:
0 galaxies–ISM:dust,extinction
:
v
i
X
r 1 INTRODUCTION a complete picture of galaxy evolution. In particular, it is essen-
a
tialtounderstandIRemissionfromdustinordertounderstandthe
Inrecent years, theevolution of galaxies atmid- andfar-infrared
cosmichistoryofstarformation,sincemostoftheradiationfrom
wavelengthshasbeenopenedupfordirectobservationalstudyby
young stars must have been absorbed by dust over the history of
infraredtelescopesinspace.Alreadyinthe1980s,theIRASsatel-
the universe, in order to account for the far-IR background (e.g.
lite surveyed the local universe in the IR, showing that much of
Hauseretal. 1998).
present-day starformation isopticallyobscured, revealing apop-
ulation of luminous and ultra-luminous infrared galaxies (LIRGs
with total IR luminosities LIR ∼ 1011 −1012L⊙ and ULIRGs Followingtheseearlydiscoveries,theISOsatelliteenabledthe
withLIR & 1012L⊙),andprovidingthefirsthintsofstrongevo- first deep surveys of galaxies in the mid- and far-IR. The deep-
lutioninthe number density of ULIRGsat recent cosmic epochs estof thesesurveyswereinthemid-IRat 15µm, andprobed the
(e.g. Wrightetal. 1984; Soiferetal. 1987a; Sanders&Mirabel evolution of LIRGs and ULIRGs out to z ∼ 1, showing strong
1996).ThenextmajoradvancecamewiththediscoverybyCOBE evolutioninthesepopulations, and directlyresolving most of the
of the cosmic far-IR background which has an energy density cosmicinfraredbackgroundatthatwavelength(Elbazetal. 1999,
comparabletothatintheoptical/near-IRbackground(Pugetetal. 2002; Gruppionietal. 2002). Deep ISO surveys in the far-IR at
1996;Hauseretal. 1998).Thisimpliesthat,overthehistoryofthe 170µm (Doleetal. 2001; Patrisetal. 2003) probed lower red-
universe, asmuchenergyhasbeenemittedbydust ingalaxiesas shifts,z∼0.5.Aroundthesametime,sub-mmobservationsusing
reaches us directly in starlight, after dust extinction is taken into theSCUBAinstrumentontheJCMTrevealedahugepopulationof
account.Thisdiscoverymadeapparenttheneedtounderstandthe high-zULIRGs(Smail,Ivison&Blain 1997;Hughesetal. 1998)
IRasmuchastheopticalemissionfromgalaxiesinordertohave whichweresubsequentlyfoundtohavearedshiftdistributionpeak-
ingatz ∼2(Chapmanetal. 2005),confirmingthedramaticevo-
lutioninnumberdensityforthispopulationseenatshorterwave-
⋆ E-mail:[email protected](CGL) lengthsandlowerredshifts.Thesub-mmgalaxieshavebeenstud-
2 Lacey et al.
ied in more detail in subsequent SCUBA surveys (e.g. SHADES, withthemaximumpossibletheoreticalself-consistency,andallof
Mortieretal. 2005). themodelparametersrelatedirectlytophysicalprocesses.Forex-
Now observations using the Spitzer satellite (Werneretal. ample, the typical dust temperature and the shape of the SED of
2004), with its hugely increased sensitivity and mapping speed dustemissiondependonthestellarluminosityandthedustmass,
arerevolutionizingourknowledgeofgalaxyevolutionatIRwave- andevolutioninallofthesequantitiesiscomputedself-consistently
lengths from 3.6 to 160 µm. Spitzer surveys have allowed direct inthistypeofmodel.Followingthismodellingapproachthusal-
determinations of the evolution of the galaxy luminosity func- lows more rigorous testing of the predictions of physical models
tion out to z ∼ 1 in the rest-frame near-IR and to z ∼ 2 in forgalaxyformationagainstobservationaldataatIRwavelengths,
the mid-IR (LeFloc’hetal. 2005; Perez-Gonzalezetal. 2005; as well as shrinking the parameter space of the predictions. Ex-
Babbedgeetal. 2006;Franceschinietal. 2006).Individualgalax- amplesofsuchmodelsareGranatoetal. (2000)andBaughetal.
ieshavebeendetectedbySpitzerouttoz ∼6(Eylesetal. 2005). (2005).(Analternativemodellingapproachalsobasedontheoret-
In the near future, the Herschel satellite (Pilbratt 2003) should ical IR SEDs but with a simplified treatment of the assembly of
makeitpossible tomeasure thefar-IRluminosityfunctionout to galaxiesandhaloshasbeenpresentedbyGranatoetal. (2004)and
z∼2,andthusdirectlymeasurethetotalIRluminositiesofgalax- Silvaetal. (2005).)
iesovermostofthehistoryoftheuniverse. Inthispaper,wefollowthethirdapproach,withphysicalmod-
Accompanyingtheseobservationaladvances,varioustypesof elling both of galaxy formation and of the galaxy SEDs, includ-
theoreticalmodelshavebeendevelopedtointerpretorexplainthe ing the effects of dust. This paper is the third in a series, where
observationaldataongalaxyevolutionintheIR.Wecandistinguish wecombinetheGALFORMsemi-analyticalmodelofgalaxyforma-
threemainclassesofmodel: tion(Coleetal. 2000)withtheGRASILmodelforstellaranddust
(a) Purely phenomenological models: In these models, emissionfromgalaxies(Silvaetal. 1998).TheGALFORMmodel
the galaxy luminosity function and its evolution are described computestheevolutionofgalaxiesintheframeworkoftheΛCDM
by a purely empirical expression, and this is combined with modelforstructureformation,basedonphysicaltreatmentsofthe
observationally-based templates for the IR spectral energy dis- assembly of dark matter halos by merging, gas cooling in halos,
tribution (SED). The free parameters in the expression for the starformationandsupernovafeedback,galaxymergers,andchem-
luminosity function are then chosen to obtain the best match icalenrichment.TheGRASILmodelcomputestheSEDofamodel
to some set of observational data, such as number counts galaxyfromthefar-UVtotheradio,basedontheoreticalmodelsof
and redshift distributions in different IR bands. These pa- stellarevolutionandstellaratmospheres,radiativetransferthrough
rameters are purely descriptive and provide little insight into atwo-phasedustmediumtocalculateboththedustextinctionand
the physical processes which control galaxy evolution. Exam- dustheating,andadistributionofdusttemperaturesineachgalaxy
ples of these models are Pearson&Rowan-Robinson (1996); calculated from a detailed dust grain model. In the first paper in
Xuetal. (1998); Blainetal. (1999); Franceschinietal. (2001); theseries(Granatoetal. 2000),wemodelledtheIRpropertiesof
Chary&Elbaz (2001); Rowan-Robinson (2001); Lagacheetal. galaxiesinthelocaluniverse.Whilethismodelwasverysuccess-
(2003);Gruppionietal. (2005). fulinexplainingobservationsofthelocaluniverse,wefoundsub-
(b) Hierarchical galaxy formation models with phenomeno- sequentlythatitfailedwhenconfrontedwithobservationsofstar-
logicalSEDs:Inthesemodels,theevolutionoftheluminosityfunc- forminggalaxiesathighredshifts,predictingfartoofewsub-mm
tions of the stellar and total dust emission are calculated from a galaxies (SMGs) at z ∼ 2 and Lyman-break galaxies (LBGs) at
detailed model of galaxy formation based on the cold dark mat- z ∼ 3. Therefore, in the second paper (Baughetal. 2005), we
ter(CDM)modelofstructureformation,includingphysicalmod- proposed anew versionofthemodel whichassumes atop-heavy
elling of processes such as gas cooling and galaxy mergers. The IMFinstarbursts(withslope x = 0,compared toSalpeter slope
stellar luminosity of a model galaxy is computed from its star x = 1.35), but anormal solar neighbourhood IMF for quiescent
formation history, and the stellar luminosity absorbed by dust, star formation. In this new model, the star formation parameters
whichequalsthetotalIRluminosityemittedbydust,iscalculated werealsochangedtoforcemorestarformationtohappeninbursts.
from this based on some treatment of dust extinction. However, Thisrevisedmodel agreed wellwithboththenumber countsand
the SED shapes required to calculate the distribution of the dust redshift distributions of SMGs detected at 850µm, and with the
emission over wavelength from thetotal IR dust emissionare ei- rest-frame far-UV luminosity function of LBGs at z ∼ 3, while
therobservationally-basedtemplates(e.g.Guiderdonietal. 1998; stillmaintainingconsistencywithgalaxypropertiesinthelocaluni-
Devriendt&Guiderdoni 2000) or are purely phenomenological, versesuchastheoptical,near-IRandfar-IRluminosityfunctions,
e.g. a modified Planck function with an empirically chosen dust andgasfractions,metallicities,morphologiesandsizes.
temperature(e.g.Kavianietal. 2003).Inthisapproach,theshape This same model of Baughetal. (2005) was found by
of theIRSEDassumed foramodel galaxy maybeincompatible LeDelliouetal. (2005a,2006)toprovideagoodmatchtotheob-
withitsotherpredictedproperties,suchasitsdustmassandradius. served evolution of the population of Lyα-emitting galaxies over
(c) Hierarchical galaxy formation models with theoretical theredshiftrangez ∼3−6.Supportforthecontroversialassump-
SEDs: These models are similar to those of type (b), in that the tionofatop-heavyIMFinburstscamefromthestudiesofchem-
evolution of the galaxy population is calculated from a detailed icalenrichmentinthismodelbyNagashimaetal. (2005a,b),who
physical model of galaxy formation based on CDM, but instead foundthatthemetallicitiesof boththeintergalacticgasingalaxy
ofusingphenomenological SEDsforthedust emission,thecom- clustersandthestarsinellipticalgalaxieswerepredictedtobesig-
pleteSEDofeachmodelgalaxy, fromthefar-UVtotheradio,is nificantlylowerthanobservedvaluesifanormalIMFwasassumed
calculatedbycombining atheoretical stellarpopulation synthesis forallstarformation,butagreedmuchbetterifatop-heavyIMFin
modelforthestellaremissionwithatheoreticalradiativetransfer burstswasassumed, asinBaughetal.. Inthisthird paper inthe
and dust heating model to predict both the extinction of starlight series, weextend the Baughetal. (2005) model tomake predic-
by dust and the IR/sub-mm SED of the dust emission. The ad- tionsforgalaxyevolutionintheIR,andcomparethesepredictions
vantages of thistype of model are that it iscompletely ab initio, withobservationaldatafromSpitzer.Weemphasizethatallofthe
GalaxyevolutionintheIR 3
modelparametersforthepredictionspresentedinthispaperwere shock-heatingandradiativecoolingofgasinsidedarkhalos,lead-
fixedbyBaughetal. prior tothepublication of anyresultsfrom ingtotheformationofgalaxydisks;(iii)quiescentstarformation
Spitzer, and we have not tried to obtain a better fit to any of the ingalaxydisks;(iv)feedbackbothfromsupernovaexplosionsand
Spitzerdatabyadjustingtheseparameters1. fromphoto-ionizationoftheIGM;(v)chemicalenrichmentofthe
Our goals in this paper are to test our model of galaxy evo- stars and gas; (vi) galaxy mergers driven by dynamical friction
lution witha top-heavy IMF in starburstsagainst observations of withincommondarkmatterhalos,leadingtotheformationofstel-
dust-obscured star-forming galaxies over the redshift range z ∼ larspheroids,andalsotriggeringburstsofstarformation.Theend
0−2,andalsototestourpredictionsfortheevolutionofthestellar productofthecalculationsisapredictionofthenumbersandprop-
populationsofgalaxiesagainstobservationaldataintherest-frame ertiesofgalaxiesthatresidewithindarkmatterhaloesofdifferent
near-andmid-IR.Theplanofthepaperisasfollows:InSection2, masses.Themodelpredictsthestellarandcoldgasmassesofthe
we give an overview of the GALFORM and GRASIL models, fo- galaxies,alongwiththeirstarformationandmergerhistories,their
cusing on how the predictions wepresent later on arecalculated. sizesandmetallicities.
InSection3,wecomparethegalaxynumber countspredictedby The prescriptions and parameters for the different processes
ourmodelwithobservationaldatainall7Spitzerbands,from3.6 which we use in this paper are identical to those adopted by
to 160 µm. In Section 4, we investigate galaxy evolution in the Baughetal. (2005),butdifferinseveral importantrespectsfrom
IR in more detail, by comparing model predictions directly with Coleetal. (2000). All of these parameters werechosen by com-
galaxyluminosityfunctionsconstructedfromSpitzerdata.InSec- parison withpre-Spitzer observational data. Thebackground cos-
tion 5, we present the predictions of our model for the evolution mologyisaspatiallyflatCDMuniversewithacosmologicalcon-
ofthegalaxystellarmassfunctionandstarformationratedistribu- stant, with “concordance” parameters Ωm = 0.3, ΩΛ = 0.7,
tion,andinvestigatetheinsightourmodeloffersonhowwellstel- Ωb = 0.04, and h ≡ H0/(100kms−1Mpc−1) = 0.7. Theam-
larmassesandstarformationratescanbeestimatedfromSpitzer plitudeoftheinitialspectrumofdensityfluctuationsissetbythe
data.WepresentourconclusionsinSection6.IntheAppendix,we r.m.s.linearfluctuationinasphereofradius8h−1Mpc,σ8 =0.93.
present model predictions for galaxy redshift distributions in the Forcompleteness,wenowsummarizetheprescriptionsandparam-
differentSpitzerbands,toassistininterpretingdatafromdifferent etersused,butgivedetailsmainlywheretheydifferfromthosein
surveys. Coleetal. (2000),or where theyareparticularlyrelevant topre-
dictingIRemissionfromdust.
2 MODEL
2.1.1 Haloassemblyhistories
In this paper use the GALFORM semi-analytical model to predict
the physical properties of the galaxy population at different red- AsinColeetal. (2000),wedescribetheassemblyhistoriesofdark
shifts,andcombineitwiththeGRASILspectrophotometricmodel matterhalosthroughhalomergertreeswhicharecalculatedusing
to predict the detailed SEDs of model galaxies. Both GALFORM aMonteCarlomethodbasedontheextendedPress-Schechterap-
and GRASIL have been described in detail in various previous proach (e.g. Lacey&Cole 1993). The process of galaxy forma-
papers, but since the descriptions of the different model compo- tionisthencalculatedseparatelyforeachhalomergertree,follow-
nents,aswellasofourparticularchoiceofparameters,arespread ingthebaryonicphysicsinalloftheseparatebranchesofthetree.
amongdifferentpapers,wegiveanoverviewofbothofthesehere. AshasbeenshownbyHellyetal. (2003),thestatisticalproperties
GALFORMisdescribedin§2.1,andGRASILin§2.2.Particularly ofgalaxiescalculatedinsemi-analyticalmodelsusingtheseMonte
importantfeaturesofourmodelarethetriggeringofstarburstsby Carlomergertreesareverysimilartothosecomputedusingmerger
mergers (discussed in §2.1.4) and the assumption of a top-heavy treesextracteddirectlyfromN-bodysimulations.
IMF in starbursts (discussed in §2.1.7). We further discuss the
choice of model parameters in §2.3. Readers who are already fa-
miliarwiththeBaughetal. (2005)modelcanskipstraighttothe 2.1.2 Gascoolinginhalos
results,startingin§3.
The cooling of gas in halos is calculated using the same simple
sphericalmodelasinColeetal. (2000).Thediffusegasinhalos
2.1 GALFORMgalaxyformationmodel (consisting of all of the gas which has not previously condensed
intogalaxies)isassumedtobeshock-heatedtothehalovirialtem-
We compute the formation and evolution of galaxies within the peraturewhenthehaloisassembled,andthentocoolradiativelyby
framework of the ΛCDM model of structure formation using the atomicprocesses.Thecoolingtimedependsonradiusthroughthe
semi-analytical galaxy formation model GALFORM. The general gasdensityprofile,whichisassumedtohaveacoreradiuswhich
methodologyandapproximationsbehindtheGALFORMmodelare growsasgasisremovedfromthediffusephasebycondensinginto
setoutindetailinColeetal. (2000)(seealsothereviewbyBaugh galaxies.Thegasatsomeradiusr inthehalothencoolsandcol-
(2006)).Insummary, theGALFORMmodel followsthemainpro- lapsestothehalocentreonatimescalewhichisthelargerofthe
cesseswhichshapetheformationandevolutionofgalaxies.These coolingtimet andthefree-falltimet atthatradius.Thus,for
cool ff
include:(i)thecollapseandmergingofdarkmatterhalos;(ii)the t (r)>t (r),wehavehotaccretion,andfort (r)<t (r),
cool ff cool ff
wehavecoldaccretion2.Inourmodel,gasonlyaccretesontothe
centralgalaxyinahalo,notontoanysatellitegalaxieswhichshare
1 A closely related model of galaxy formation obtained by applying
GALFORMprinciplestotheMillenniumsimulationofSpringeletal. (2005)
has recently been published by Boweretal. (2006). This model differs
fromthecurrentoneprimarilyinthatitincludesfeedbackfromAGNactiv- 2 NotethatcontrarytoclaimsbyBirnboim&Dekel (2003),theprocess
ity,butdoesnothaveatop-heavyIMFinbursts.Weplantoinvestigatethe of“coldaccretion”,ifnotthename,hasalwaysbeenpartofsemi-analytical
IRpredictionsofthisalternativemodelinasubsequentpaper. models(seeCrotonetal. (2006)foradetaileddiscussion)
4 Lacey et al.
thathalo.Wedenoteallofthediffusegasinhalosas“hot”,andall Weadopt fdyn = 50andτ∗,burst,min = 0.2Gyr(theseparame-
ofthegaswhichhascondensedintogalaxiesas“cold”. terswerechosen by Baughetal. (2005) toallow asimultaneous
match to the sub-mm number counts and to the local 60µm lu-
minosityfunction). Thestarformationrateinaburstthusdecays
2.1.3 Starformationtimescaleindisks exponentiallywithtimeafterthegalaxymerger.Itisassumedtobe
truncatedafter3e-foldingtimes(wherethee-foldingtimetakesac-
Theglobalrateofstarformationψingalaxydisksisassumedtobe
countofstellarrecyclingandfeedback-seeGranatoetal. (2000)
relatedtothecoldgasmass,Mgas,byψ = Mgas/τ∗,disk,where for details), withthe remaining gas being ejected intothe galaxy
thestarformationtimescaleistakentobe
haloatthattime.
τ∗,disk=τ∗0`Vc/200kms−1´α∗, (1)
where Vc is the circular velocity of the galaxy disk (at its half- 2.1.5 Feedbackfromphoto-ionization
massradius)andτ∗0 isaconstant.Weadoptvaluesτ∗0 = 8Gyr
andα∗ = −3,chosentoreproducetheobservedrelationbetween Aftertheintergalacticmedium(IGM)hasbeenreionizedatredshift
gasmassandB-bandluminosityforpresent-daydiskgalaxies.As zreion,theformationoflow-massgalaxiesisinhibited,bothbythe
discussed in Baughetal. (2005), this assumption means that the effectoftheIGMpressureinhibitingcollapseofgasintohalos,and
diskstarformationtimescaleisindependentofredshift(atagiven bythereductionofgascoolinginhalosduetothephoto-ionizing
Vc),resultingindisksathighredshiftthataremuchmoregas-rich background.Wemodelthisinasimpleway,byassumingthatfor
thanatlowredshift,andhavemoregasavailableforstarformation z < zreion,coolingofgasiscompletelysuppressedinhaloswith
inburststriggeredbygalaxymergersathighredshift. circularvelocitiesVc <Vcrit.WeadoptVcrit =60kms−1,based
onthedetailedmodellingbyBensonetal. (2002).Weassumein
this paper that reionization occurs at zreion = 6, for consistency
2.1.4 Galaxymergersandtriggeringofstarbursts with Baughetal. (2005), but increasing this to zreion ∼ 10 in
linewiththeWMAP3-yearestimateofthepolarizationofthemi-
Inthemodel,allgalaxiesoriginateascentralgalaxiesinsomehalo,
crowavebackground(Spergeletal. 2006)hasnosignificanteffect
butcanthenbecomesatellitegalaxiesiftheirhosthalomergesinto
onthemodelresultspresentedinthispaper.
anotherhalo.Mergerscanthenoccurbetweensatelliteandcentral
galaxieswithinthesamehalo,afterdynamicalfrictionhascaused
thesatellitegalaxytosinktothecentreofthehalo.Galaxymergers 2.1.6 Feedbackfromsupernovae
canproducechangesingalaxymorphologyandtriggerbursts.We
classifygalaxymergersaccordingtotheratioofmasses(including Photo-ionization feedback only affects very low mass galaxies.
stars and gas) M2/M1 6 1 of the secondary to primary galaxy Moreimportantformostgalaxiesisfeedbackfromsupernovaex-
plosions.Weassumethatenergyinputfromsupernovaecausesgas
involved. We define mergers to be major or minor according to
whetherM2/M1 > fellip orM2/M1 < fellip (Kauffmannetal. tobeejectedfromgalaxiesatarate
1993).Inmajormergers,anystellardisksineithertheprimaryor M˙ej =β(Vc)ψ=[βreh(Vc)+βsw(Vc)] ψ (3)
secondaryareassumedtobedisrupted,andthestarsrearrangedinto
aspheroid.Inminormergers,thestellardiskintheprimarygalaxy Thesupernova feedbackisassumedtooperateforbothquiescent
isassumedtoremainintact,whileallofthestarsinthesecondary starformationindisksandforstarburststriggeredbygalaxymerg-
areassumedtobeaddedtothespheroidoftheprimary.Weadopta ers,withtheonlydifferencebeingthatwetakeVctobethecircular
thresholdf =0.3formajormergers,consistentwiththeresults velocityatthehalfmassradiusofthediskintheformercase,and
ellip
ofnumericalsimulations(e.g.Barnes 1998),whichreproducesthe atthehalf-massradiusofthespheroidinthelattercase.Forsim-
observed present-day fraction of spheroidal galaxies. We assume plicity,wekeepthesamefeedbackparametersforstarburstsasfor
thatmajormergersalwaystriggerastarburstifanygasispresent. quiescentstarformation.
Wealsoassumethatminormergerscantriggerbursts,iftheysat- The supernova feedback has two components: the reheating
isfybothM2/M1 > fburst andthegasfractioninthediskofthe term βrehψ describes gas which is reheated and ejected into the
primary galaxy exceeds fgas,crit. Following Baughetal. (2005), galaxyhalo, fromwhereitisallowedtocool againafterthehalo
weadoptfburst = 0.05andfgas,crit = 0.75.Theparametersfor masshasdoubledthroughhierachicalmassbuild-up.Forthis,we
bursts in minor mergers were motivated by trying to explain the usetheparametrizationofColeetal. (2000):
ninugmebqenr.(o1f)sfuobr-tmhemstgaarlfaoxrimesa.tiAonntiimmpeosrctaalneticnodnissekqsu,ecnocmeboinfeadsswumith- βreh =(Vc/Vhot)−αhot, (4)
thetriggeringofstarburstsinminormergers,isthattheglobalstar whereweadoptparametervaluesV =300kms−1andα =
hot hot
formationrateathighredshiftsisdominatedbybursts,whilethatat 2.Thereheatingtermhasthelargesteffectonlow-massgalaxies,
lowredshiftsitisdominatedbyquiescentdisks(seeBaughetal. forwhichejectionofgasfromgalaxiesflattensthefaint-endslope
foradetaileddiscussionofthesepoints). ofthegalaxyluminosityfunction.
Ineitherkindofstarburst,weassumethattheburstconsumes Thesecondtermβswψineqn.(3)isthesuperwindterm,which
allofthecoldgasinthetwogalaxiesinvolvedinthemerger,and describesejectionofgasoutofthehaloratherthanjustthegalaxy.
thatthestarsproducedareaddedtothespheroidofthemergerrem- Onceejected,thisgasisassumednevertore-accreteontoanyhalo.
nant.Duringtheburst,weassumethatstarformationproceedsac- Wemodelthesuperwindejectionefficiencyas
wcoerdaisnsgumtoetherelationψ=Mgas/τ∗,burst.Forthebursttimescale, βsw =fswminˆ1,(Vc/Vsw)−2˜ (5)
τ∗,burst =max[fdynτdyn,sph;τ∗,burst,min], (2) abnadseVdsown=Be2n0so0nkemtas−l.1(,2a0s0i3n).BWauegahdoepttapl.ar(a2m00e5te)r.vTahlueessufpsewrw=in2d
whereτ isthedynamicaltimeinthenewly-formedspheroid. termmainlyaffectshighermassgalaxies,wheretheejectionofgas
dyn,sph
GalaxyevolutionintheIR 5
fromhaloscauses anincreaseinthecooling timeof gasinhalos neighbourhoodIMF,suchasthatofKroupa (2001).)Burstsofstar
by reducing the gas densities. This brings the predicted break at formation triggeredby galaxy mergers areassumed toformstars
thebright endof thelocal galaxyluminosityfunction intoagree- with a top-heavy IMF with slope x = 0. As discussed in detail
mentwithobservations,asdiscussedinBensonetal. (2003).The inBaughetal. (2005),thetop-heavyIMFinburstswasfoundto
variousparametersfor supernova feedback arethuschoseninor- berequiredinordertoreproducetheobservednumbercountsand
dertomatchtheobservedpresent-dayopticalandnear-IRgalaxy redshift distributions of the faint sub-mm galaxies. Furthermore,
luminosityfunctions, aswellasthegalaxymetallicity-luminosity as shown by Nagashimaetal. (2005a,b), the predicted chemical
relation. abundancesoftheX-rayemittinggasingalaxyclustersandofthe
We note that the galaxy formation model in this paper, un- starsinellipticalgalaxiesalsoagreebetterwithobservationaldata
like some other recent semi-analytical models, does not include inamodelwiththetop-heavyIMFinbursts,ratherthanauniversal
AGNfeedback.Instead,theroleofAGNfeedbackinreducingthe solarneighbourhoodIMF.
amount of gas cooling to form massive galaxies is taken by su- A variety of other observational evidence has accumulated
perwindsdrivenbysupernovaexplosions.Thefirstsemi-analytical which suggests that the IMF in some environments may be top-
modeltoincludeAGNfeedbackwasthatofGranatoetal. (2004), heavy compared to the solar neighbourhood IMF. Riekeetal.
who introduced a detailed model of feedback from QSO winds (1993) argued for a top-heavy IMF in the nearby starburst M82,
duringtheformationphaseofsupermassiveblackholes(SMBHs), based on modelling its integrated properties, while Parraetal.
withtheaimofexplainingtheco-evolutionofthespheroidalcom- (2007) found possible evidence for atop-heavy IMFintheultra-
ponents of galaxies and their SMBHs. The predictions of the luminous starburst Arp220 from the relative numbers of super-
Granatoetal. modelfornumbercountsandredshiftdistributions novae of different types observed at radio wavelengths. Evidence
in the IR have been computed by Silvaetal. (2005) using the has been found for a top-heavy IMF in some star clusters in in-
GRASIL spectrophotometric model, and compared to ISO and tensely star-forming regions, both in M82 (e.g. McCradyetal.
Spitzer data. However, the Granatoetal. (2004) model has the 2003),andinourownGalaxy(e.g.Figeretal. 1999;Stolteetal.
limitations that it does not include the merging of galaxies or of 2005; Harayamaetal. 2007). Observations of both the old and
darkhalos,anddoesnottreattheformationandevolutionofgalac- young stellar populations in the central 1 pc of our Galaxy
ticdisks.Morerecently,severalsemi-analyticalmodelshavebeen alsofavouratop-heavyIMF(Paumardetal. 2006;Manessetal.
published which propose that heating of halo gas by relativistic 2007).Fardaletal. (2006)foundthatreconcilingmeasurementsof
jetsfromanAGN inanopticallyinconspicuous or “radio”mode theopticalandIRextragalacticbackgroundwithmeasurementsof
canbalanceradiativecoolingofgasinhigh-masshalos,thussup- thecosmicstarformationhistoryalsoseemedtorequireanaverage
pressing hot accretion of gas onto galaxies (Boweretal. 2006; IMF that was somewhat top-heavy. Finally, vanDokkum (2007)
Crotonetal. 2006; Cattaneoetal. 2006; Monacoetal. 2007). foundthatreconcilingthecolourandluminosityevolutionofearly-
However,theseAGNfeedbackmodelsdifferindetail,andallare type galaxies in clusters also favoured a top-heavy IMF. Larson
fairlyschematic. Noneof thesemodels hasbeenshown torepro- (1998)summarizedotherevidenceforatop-heavyIMFduringthe
ducetheobservednumbercountsandredshiftsofthefaintsub-mm earlier phases of galaxy evolution, and argued that this could be
galaxies. anaturalconsequenceofthetemperature-dependenceoftheJeans
Theeffectsofoursuperwindfeedbackarequalitativelyquite massforgravitationalinstabilityingasclouds.Larson (2005)ex-
similartothoseoftheradio-modeAGNfeedback.Bothsuperwind tendedthistoarguethatatop-heavyIMFmightalsobeexpected
andAGNfeedbackmodelscontainfreeparameters,whicharead- instarburstregions,wherethereisstrongheatingofthedustbythe
justed in order to make the model fit the bright end of the ob- youngstars.
servedpresent-daygalaxyluminosityfunctionatopticalandnear- In our model, the fraction of star formation occuring in the
IRwavelengths.However,sincethephysicalmechanismsaredif- burstmodeincreaseswithredshift(seeBaughetal. (2005)),sothe
ferent, they make different predictions for how the galaxy lumi- averageIMFwithwhichstarsarebeingformedshiftsfrombeing
nosityfunctionshouldevolvewithredshift.Currentmodelsforthe closetoasolarneighbourhoodIMFatthepresent-daytobeingvery
radio-mode AGN feedback arevery schematic, but theyhave the top-heavy at high redshift. In this model, 30% of star formation
advantageoverthesuperwindmodelthattheenergeticconstraints occuredintheburstmodewhenintegratedoverthepasthistoryof
are greatly relaxed, since accretion onto black holes can convert theuniverse, but only 7%of thecurrent stellar mass wasformed
massintoenergywithamuchhigherefficiencythancansupernova inbursts,becauseofthemuchlargerfractionofmassrecycledby
explosions.WewillinvestigatethepredictionsofmodelswithAGN dyingstarsforthetop-heavyIMF.Wenotethatourpredictionsfor
feedbackfortheIRandsub-mmevolutionofgalaxiesinafuture the IR and sub-mm luminosities of starbursts are not sensitive to
paper. thepreciseformofthetop-heavyIMF,butsimplyrequirealarger
fractionofm∼5−20M⊙starsrelativetoasolarneighbourhood
IMF.
2.1.7 TheStellarInitialMassFunctionandChemicalEvolution
Inthispaper,wecalculatechemicalevolutionusingtheinstan-
StarsinourmodelareassumedtoformwithdifferentInitialMass taneousrecyclingapproximation,whichdependsonthetotalfrac-
Functions(IMFs),depending onwhether theyformindisksorin tionofmassrecycledfromdying stars(R),andthetotalyieldof
bursts.BothIMFsaretakentobepiecewisepowerlaws,withslopes heavyelements(p).BothoftheseparametersdependontheIMF.
xdefinedbydN/dlnm∝m−x,withN thenumberofstarsand We use the results of stellar evolution computations to calculate
mthestellarmass(sotheSalpeterslopeisx=1.35),andcovering valuesofRandpconsistentwitheachIMF(seeNagashimaetal.
astellarmassrange0.15 < m < 120M⊙.Quiescentstarforma- (2005a)fordetailsofthestellarevolutiondataused).Thus,weuse
tioningalaxydisksisassumedtohaveasolarneighbourhoodIMF, R=0.41andp=0.023forthequiescentIMF,andR=0.91and
forwhichweusetheKennicutt (1983)paramerization,withslope p = 0.15 for the burst IMF. Our chemical evolution model then
x=0.4form<M⊙andx=1.5form>M⊙.(TheKennicutt predicts the masses and total metallicities of the gas and stars in
(1983)IMFissimilartootherpopularparametrizationsofthesolar eachgalaxyasafunctionoftime.
6 Lacey et al.
2.1.8 Galaxysizesanddustmasses withacontinuousdistributionofgrainsizes(varyingbetween8A˚
and0.25µm),andalsoPolycyclicAromaticHydrocarbon (PAH)
Forcalculatingtheextinctionandemissionbydust,itisessentialto
moleculeswithadistributionofsizes.Theequilibriumtemperature
haveanaccuratecalculationofthedustopticaldepthsinthemodel
in the local interstellar radiation field is calculated for each type
galaxies, which inturn depends on themass of dust and the size
andsizeofgrain,ateachpointinthegalaxy,andthisinformation
ofthegalaxy. Thedust massiscalculated fromthegasmassand
isthenusedtocalculatetheemissionfromeachgrain.Inthecaseof
metallicitypredictedbythechemicalenrichmentmodel,assuming
verysmallgrainsandPAHmolecules,temperaturefluctuationsare
thatthedust-to-gasratioisproportionaltometallicity,normalized
important,andtheprobabilitydistributionofthetemperatureiscal-
tomatchthelocalISMvalueatsolarmetallicity.Thesizesofgalax-
culated.ThedetailedspectrumofthePAHemissionisobtainedus-
iesarecomputedexactlyasinColeetal. (2000):gaswhichcools
ingthePAHcross-sectionsfromLi&Draine (2001),asdescribed
inahaloisassumedtoconserveitsangular momentum asitcol-
inVegaetal. (2005).Thegrainsizedistributionischosentomatch
lapses,formingarotationally-supportedgalaxydisk;theradiusof
themeandustextinctioncurveandemissivityinthelocalISM,and
this disk is then calculated from its angular momentum, includ-
isnotvaried,exceptthatthePAHabundanceinmolecularclouds
ingthegravityofthedisk,spheroid(ifany)anddarkhalo.Galaxy
isassumedtobe10−3 ofthatinthediffusemedium(Vegaetal.
spheroidsarebuiltupbothfrompre-existingstarsingalaxymerg-
2005).
ers,andfromthestarsformedinburststriggeredbythesemergers;
(vi)RadioemissionfromionizedgasinHIIregionsandfromsyn-
the radii of spheroids formed in mergers are computed using an
chrotronradiationfromrelativisticelectronsacceleratedinsuper-
energyconservationargument.Incalculatingthesizesofdisksand
novaremnantshocksarecalculatedasdescribedinBressanetal.
spheroids,weincludetheadiabaticcontractionofthedarkhalodue
(2002).
tothegravityofthebaryoniccomponents.Thismodelwastested
The output from GRASIL is then the complete SED of a
fordisksbyColeetal. (2000)andforspheroidsbyAlmeidaetal.
galaxy from the far-UV to the radio (wavelengths 100A˚ . λ .
(2007)(seealsoCoendaetal.inpreparation,andGonzalezetal.
1m).TheSEDofthedustemissioniscomputedasasumoverthe
in preparation). During a burst, we assume that the gas and stars
differenttypesofgrains,havingdifferenttemperaturesdepending
involved in the burst have a distributionwith the same half-mass
ontheirsizeandtheirpositioninthegalaxy.ThedustSEDisthus
radiusasthespheroid(i.e.η = 1inthenotationofGranatoetal.
intrinsicallymulti-temperature.GRASILhasbeenshowntogivean
(2000),whousedavalueη=0.1).
excellentmatchtothemeasuredSEDsofbothquiescent(e.g.M51)
andstarburst(e.g.M82)galaxies(Silvaetal. 1998;Bressanetal.
2002).
2.2 GRASILmodelforstellaranddustemission
The assumption of axisymmetry in GRASIL is a limitation
Foreachgalaxyinourmodel,wecomputethespectralenergydis- whenconsideringstarburststriggeredbygalaxymergers.However,
tributionusingthespectrophotometricmodelGRASIL(Silvaetal. observationsoflocalULIRGsimplythatmostofthestarformation
1998;Granatoetal. 2000).GRASILcomputestheemissionfrom happensinasingleburstcomponentafterthegalaxymergerissub-
thestellarpopulation,theabsorptionandemissionofradiationby stantiallycomplete,sotheassumptionofaxisymmetryfortheburst
dust, and also radio emission (thermal and synchrotron) powered componentmaynotbesobad.
bymassivestars(Bressanetal. 2002).
2.2.1 SEDmodel 2.2.2 GRASILparameters
ThemainfeaturesoftheGRASILmodelareasfollows: Themain parameters in the GRASILdust model are the fraction
(i)Thestarsareassumedtohaveanaxisymmetricdistributionina fmcofthecoldgaswhichisinmolecularclouds,thetimescaletesc
diskandabulge.Giventhedistributionofstarsinageandmetal- fornewly-formedstarstoescapefromtheirparentmolecularcloud,
licity(obtained fromthe star formation and chemical enrichment andthecloudmassesMc andradiirc inthecombinationMc/rc2,
history), the SED of the stellar population is calculated using a whichdeterminesthedustopticaldepthoftheclouds.Weassume
population synthesis model based onthePadova stellar evolution fmc = 0.25,Mc = 106M⊙ andrc = 16pcasinGranatoetal.
tracksandKuruczmodelatmospheres(Bressanetal. 1998).This (2000),andalsoadoptthesamegeometricalparametersasinthat
isdoneseparatelyforthediskandbulge. paper. We make the following two changes in GRASIL parame-
(ii)Thecoldgasanddustinagalaxyareassumedtobeina2-phase tersrelativetoGranatoetal.,asdiscussedinBaughetal. (2005):
medium,consistingofdensegasingiantmolecularcloudsembed- (a) We assume tesc = 1Myr in both disks and bursts (instead
ded in a lower-density diffuse component. In a quiescent galaxy, of the Granatoetal. values tesc = 2 and 10Myr respectively).
thedustandgasareassumedtobeconfinedtothedisk,whilefor Thisvaluewaschoseninordertoobtainabettermatchofthepre-
agalaxy undergoing aburst, thedust and gasare confined tothe dictedrest-framefar-UVluminosityfunctionofgalaxiesatz ∼ 3
spheroidalburstcomponent. tothatmeasuredforLyman-breakgalaxies.(b)Thedustemissivity
(iii)Starsareassumedtobeborninsidemolecularclouds,andthen law inburstsat long wavelengths ismodified from ǫν ∝ ν−2 to
toleakoutintothediffusemediumonatimescaletesc.Asaresult, ǫν ∝ ν−1.5 forλ > 100µm. Thiswasdoneinorder toimprove
theyoungestandmostmassivestarsareconcentratedinthedustiest slightlythefitofthemodeltotheobservedsub-mmnumbercounts.
regions,sotheyexperiencelargerdustextinctionsthanolder,typ- In applying GRASIL to model the SEDs of a sample of nearby
icallylower-massstars,anddust inthecloudsisalsomuchmore galaxies, Silvaetal. (1998) found that a similar modification (to
stronglyheatedthandustinthediffusemedium. ǫν ∝ν−1.6)seemedtoberequiredinthecaseofArp220(theonly
(iv)Theextinctionofthestarlightbydustiscomputedusingara- ultra-luminousstarburstintheirsample),inordertoreproducethe
diativetransfercode;thisisusedalsotocomputetheintensityof observedsub-mmdataforthatgalaxy.Thismodificationinfacthas
thestellarradiationfieldheatingthedustateachpointinagalaxy. littleeffectontheIRpredictionspresentedinthepresentpaper,but
(v)Thedustismodelledasamixtureofgraphiteandsilicategrains weretainitforconsistencywithBaughetal. (2005).
GalaxyevolutionintheIR 7
2.2.3 InterfacewithGALFORM functionsintheB-andK-bandsandat60µm,therelationsbetween
gasmassandluminosityandmetallicityandluminosity,thesize-
For calculating the statistical properties of the galaxy population
luminosityrelationforgalaxydisks,andthefractionofspheroidal
from the combined GALFORM+GRASIL model, we follow the
galaxies.Inaddition,themodelwasrequiredtoreproducetheob-
same strategy as described in Granatoetal. (2000). We first run
served rest-frame far-UV(1500A˚) luminosity function at z = 3,
theGALFORMcodetogeneratealargecatalogueofmodelgalaxies
andtheobservedsub-mmnumbercountsandredshiftdistribution
atanyredshift,andthenruntheGRASILcode onsubsamplesof
at850µm(Baughetal. 2005).Thesub-mmnumbercountsarethe
these.Forthequiescentgalaxies,weselectasubsamplewhichhas
mainfactordrivingtheneedtoincludeatop-heavyIMFinbursts.
equalnumbersofgalaxiesinequallogarithmicbinsofstellarmass,
Theparametersforourstandardmodelareexactlythesameas
whilefortheburstinggalaxies,weselectasubsample withequal
inBaughetal. (2005),whichwerechosenbeforeSpitzerdatabe-
numbersofgalaxiesinequallogarithmicbinsofburstmass.Forthe
cameavailable.Sincetheseparameterswerenotadjustedtomatch
burstsample,wecomputeSEDsatseveraldifferentrepresentative
anydataobtainedwithSpitzer,thepredictionsofourmodelinthe
stages in the burst evolution, while for the quiescent sample, we
Spitzer bands are genuine predictions. We could obviously have
only compute SEDsat a single epoch. Using thissampling strat-
fine-tunedourparametersinordertomatchbettertheobservational
egy, weobtain agood coverage of allthe different masses, types
data we considered in this paper, but this would have conflicted
andevolutionarystagesofgalaxies,whileminimizingthecompu-
withourmaingoal,whichistopresentpredictionsforawidesetof
tationalcostofrunningtheGRASILcode.Thestatisticalproperties
observablepropertiesbasedonasinglephysicalmodelinaseries
ofthegalaxypopulationarethenobtainedbyassigningthemodel
ofpapers.
galaxiesappropriateweightsdependingontheirpredictednumber
Sinceourassumptionofatop-heavyIMFinburstsisacon-
densityinarepresentativecosmologicalvolume.
troversialone,wewillalsoshowsomepredictionsfromavariant
TheoutputsfromtheGALFORMgalaxyformationmodel re-
model, which is identical to the standard model, except that we
quiredbyGRASILtocalculatethegalaxySEDsare:thecombined
assume the same solar neighbourhood (Kennicutt) IMF in bursts
starformationhistoryandmetallicitydistributionforthediskand
andindisks.Comparingthepredictionsforthestandardandvari-
bulge,theradiiofbothcomponents,andthetotalmassofdust.The
ant models then shows directly the effects of changing the IMF
dust massiscalculated fromthemassandmetallicityof thecold
inbursts.Wenotethatthevariantmodelmatchesthepresent-day
gasinthegalaxy,assumingthatthedust-to-gasratioisproportional
opticalandnear-IRluminosityfunctionsalmostaswellasthestan-
tothemetallicity.Sincethegasmassandmetallicitybothevolve,
dardmodel,thoughitisapoorerfittothelocal60µmluminosity
sodoesthedustmass,andthisevolutionisfullytakenintoaccount
functionforthebrightestgalaxies(seeFig.9).Thevariantmodel
inGRASIL.Forsimplicity,weassumethatthesizedistributionof
underpredictsthe850µmcountsbyafactorof10–30.
thedustgrainsandPAHmoleculesdoesnotevolve,apartfromthe
normalization.
OncewehavecalculatedtheSEDsforthemodelgalaxies,we
computeluminositiesindifferentobservedbands(e.g.theoptical 3 NUMBERCOUNTS
B-bandortheSpitzer24µmband)byconvolvingtheSEDwiththe
Webeginourcomparisonofthepredictionsofourgalaxyforma-
filter+detectorresponsefunctionforthatband.Forcomputingthe
tion model against Spitzer data with the galaxy number counts.
predictedfluxesfromgalaxiesinafixedobserver-frameband,we
Fig.1showsnumbercountsinthefourIRACbands(3.6,4.5,5.8
redshifttheSEDbeforedoingtheconvolution.
and8.0µm),andFig.2doesthesameforthethreeMIPSbands
The GRASIL code is quite CPU-intensive, requiring several
(24, 70 and 160 µm). Each panel is split in two: the upper sub-
minutesofCPUtimepergalaxy.Consequently, wearelimitedto
runningsamplesofafewthousandgalaxiesateachredshift.Asa panel plots the counts per logarithmic flux interval, dN/dlnSν,
result, quantities such as luminosity functions and redshift distri- whilethelowersub-panel insteadplotsSνdN/dlnSν.Thelatter
is designed to take out much of the trend with flux, in order to
butions still show some small amount of noise, rather than being
showmoreclearlythedifferencesbetweenthemodelandtheon-
completelysmoothcurves,ascanbeseeninmanyofthefiguresin
servationaldata.Ineachcaseweplotthreecurvesforourstandard
thispaper.
model: the solid blue line shows the total number counts includ-
ingbothextinctionandemissionbydust,thesolidredlineshows
thecontributiontothisfromgalaxiescurrentlyformingstarsina
2.3 ChoiceofparametersintheGALFORM+GRASILmodel
burst,andthesolidgreenlineshowsthecontributionfromallother
Thecombined GALFORM+GRASILmodel hasasignificant num- galaxies(star-formingornot),whichwedenoteas“quiescent”.In
berofparameters,butthisisinevitablegiventheverywiderange Fig.1,wealsoplotadashedbluelinewhichshowsthepredicted
ofphysicalprocesseswhichareincluded.Theparametersarecon- totalcountsifweignoreabsorptionandemissionfrominterstellar
strainedbyrequiringthemodelpredictionstoreproducealimited dust(emissionfromdustintheenvelopesofAGBstarsisstillin-
set of observational data - once this is done, there is rather little cludedinthestellarcontribution,however).IntheMIPSbands,the
freedominthechoiceofparameters.Wehavedescribedabovehow predictedcounts arenegligibleintheabsenceof interstellardust,
the main parameters are fixed, and more details can be found in sowedonotplottheminFig.2.Inthelowersub-panels,wealso
Coleetal. (2000)andBaughetal. (2005).Forbothofthesepa- showbyadashedmagentalinethepredictionfromavariantmodel
pers,largegridsofGALFORMmodelswererunwithdifferentpa- whichassumesanormal(Kennicutt)IMFforallstarformation,but
rameters,inordertodecidewhichsetofparametersgavethebest isotherwiseidenticaltoourstandardmodel(whichhasatop-heavy
overallfittothesetofcalibratingobservationaldata.Thesepapers IMFinbursts).ThisvariantmodelfitsthelocalB-andK-bandand
alsoshowtheeffectsofvaryingsomeofthemainmodelparameters 60µmluminosityfunctionsabout aswellasourstandardmodel,
aroundtheirbest-fitvalues.Theparametersinthestandardmodel butdramaticallyunderpredictsthe850µmnumbercounts.Theob-
forwhichwepresentresultsinthispaperwerechosentoreproduce servednumbercountsareshownbyblacksymbolswitherrorbars.
the following properties for present-day galaxies: the luminosity Overall, the agreement between the predictions of our stan-
8 Lacey et al.
Figure1.GalaxydifferentialnumbercountsinthefourIRACbands.Thecurvesshowmodelpredictions,whilethesymbolswitherrorbarsshowobservational
datafromFazioetal. (2004)(withdifferentsymbolsfordatafromdifferentsurveyfields).Eachpanelissplitintwo:theuppersub-panelplotsthecounts
asdN/dlnSν vsSν,whilethelowersub-panelplotsSνdN/dlnSν (inunitsmJydeg−2)onthesamehorizontalscale.Theuppersub-panelsshowfour
differentcurvesforourstandardmodel-solidblue:totalcountsincludingdustextinctionandemission;dashedblue:totalcountsexcludinginterstellardust;
solidred:ongoingbursts(includingdust);solidgreen:quiescentgalaxies(includingdust).Thelowersub-panelscomparethetotalcountsincludingdustfor
thestandardmodel(solidblueline)withthoseforavariantmodelwithanormalIMFforallstars(dashedmagentaline).Theverticaldashedlineshowsthe
estimatedconfusionlimitforthemodel.(a)3.6µm.(b)4.5µm.(c)5.8µm.(d)8.0µm.
dardmodelandtheobservedcountsisremarkablygood,whenone dustemission(duetostrongPAHfeaturesatλ ∼ 6−9µm) be-
takesaccountofthefactthatnoparametersofthemodelweread- coming very important at bright fluxes (which correspond tolow
justedtoimprovethefittoanydatafromSpitzer.Considerfirstthe average redshifts - see Fig. A1(b) in the Appendix). The 8.0 µm
resultsfortheIRACbands,showninFig.1.Here,theagreement counts thusarepredicted tobe dominated by dust emission from
ofthemodelwithobservationsseemsbestat3.6and8.0µm,and quiescentlystar-forminggalaxies,exceptatthefaintestfluxes.The
somewhatpoorerat5.8µm.Themodelpredictssomewhattoofew countsat5.8µmshowbehaviourwhichisintermediate,withmild
objectsatfainterfluxesinalloftheIRACbands.Comparingthered emissioneffectsatbrightfluxesandmildextinctionatfaintfluxes.
andgreencurves,weseethatquiescentgalaxiesratherthanbursts Comparingthesolidblueanddashedmagentalines,weseethatthe
dominatethecountsatallobservedfluxesinalloftheIRACbands, predicted number counts in the IRAC bands are almost the same
butespeciallyattheshorterwavelengths,consistentwiththeexpec- whether or not we assume a top-heavy IMF in bursts, consistent
tationthatat3.6and4.5µm,weareseeingmostlylightfromold withthecountsbeingdominatedbyquiescentgalaxies.
stellarpopulations.Comparingthesolidanddashedbluelines,we
ConsidernexttheresultsfortheMIPSbands,showninFig.2.
seethattheeffectsofdustaresmallat3.6and4.5µm,withasmall
We again see remarkably good agreement of the standard model
amount of extinctionatfaint fluxes(andthushigher averagered-
with the observational data. The agreement is especially good at
shifts),butnegligibleextinctionforbrighterfluxes(andthuslower
faint fluxes (corresponding to higher redshifts). In particular, the
redshifts).Ontheotherhand,dusthaslargeeffectsat8.0µm,with
model matches well the observed 24 µm counts at the “bump”
GalaxyevolutionintheIR 9
aroundfluxesSν ∼ 0.1−1mJy.AccuratemodellingofthePAH
emission features is obviously crucial for modelling the 24 µm
number counts, since the PAH features dominate the flux in the
24 µm band as they are redshifted into the band at z & 0.5.
On the other hand, the standard model overpredicts the number
countsatbrightfluxes(correspondingtolowredshifts)inallthree
MIPSbands.Theevolutionatthesewavelengthspredictedbyour
ΛCDM-basedmodelthusseemstobenotquiteasstrongasindi-
catedbyobservations.
In the MIPS bands, emission from galaxies is completely
dominatedbydust,whichiswhynodashedbluelinesareshownin
Fig.2.Comparingtheredandgreencurves,weseethatquiescent
(butstar-forming)galaxiestendtodominatethenumbercountsin
thesebandsatbrighterfluxes,andburstsatfainterfluxes.Thisre-
flects the increasing dominance of bursts in the mid- and far-IR
luminosity function at higher redshifts. Comparing thesolid blue
anddashedmagentacurves,weseethatourstandardmodelwitha
top-heavyIMFinburstsprovidesasignificantlybetteroverallfitto
theobserved 24µmcountsthanthevariantmodel withanormal
IMF inbursts(although at the brightest fluxes, the variant model
fitsbetter).Thefaintnumbercountsat70µmalsofavourthetop-
heavy IMF model, while the number counts at 160 µm cover a
smallerfluxrange,anddonotusefullydistinguishbetweenthetwo
variantsofourmodelwithdifferentburstIMFs.
We can use our model to predict the flux levels at which
sources should become confused in the different Spitzer bands.
We estimate the confusion limit using the source density cri-
terion (e.g. Vaisanenetal. 2001; Doleetal. 2003): if the tele-
scope has an FWHM beamwidth of θFWHM, we define the ef-
fective beam solid angle as ω = (π/(4ln2))θ2 =
beam FWHM
1.13θ2 , and then define the confusion limited flux S
FWHM conf
to be such that N(> S ) = 1/(N ω ), where
conf beam beam
N(> S) is the number per solid angle of sources brighter than
flux S. We choose N = 20 for the number of beams
beam
per source, which gives similar results to more detailed analy-
ses(e.g.Vaisanenetal. 2001;Doleetal. 2004b).Weusevalues
of the beamsize θFWHM = (1.66,1.72,1.88,1.98) arcsec for
the four IRAC bands (Fazioetal. 2004b) and (5.6,16.7,35.2)
arcsec for the three MIPS bands (Doleetal. 2003). Our stan-
dard model then predicts confusion-limited fluxes of S =
conf
(0.62,0.62,0.69,0.70)µJy in the (3.6,4.5,5.8,8.0)µm IRAC
bands,andS = (0.072,2.6,43)mJyinthe(24,70,160)µm
conf
MIPS bands. These confusion estimates for the MIPS bands are
similar tothose of Doleetal. (2004b),which were based on ex-
trapolatingfromtheobservedcounts.Thesevaluesfortheconfu-
sionlimitsareindicatedinFigs.1and2byverticaldashedlines.
Ourgalaxyevolutionmodeldoesnotcomputethecontribution
ofAGNtotheIRluminositiesofgalaxies.Ontheotherhand,the
observednumbercountstowhichwecompareincludebothnormal
galaxies, in which the IR emission is powered by stellar popula-
tions, and AGN, in which there is also IR emission from a dust
torus,whichisexpectedtobemostprominentinthemid-IR.How-
ever,multi-wavelengthstudiesusingoptical,IRandX-raydatain-
Figure2.GalaxydifferentialnumbercountsinthethreeMIPSbands.The
dicate that even at 24 µm, the fraction of sources dominated at
curvesshowmodelpredictionswhilethesymbolswitherrorbarsshowob-
thatwavelengthbyAGNisonly10-20%(e.g.Franceschinietal.
servational data.Themeaningofthedifferentmodellinesisthesameas
2005),andthecontributionofAGN-dominatedsourcesintheother
inFig.1.(a)24µm,withobservationaldatafromPapovichetal. (2004).
Spitzerbandsislikelytobesmaller.Thereforeweshouldnotmake (b)70µm,withobservational datafromDoleetal. (2004a)(filledsym-
anyseriouserrorbycomparingourmodelpredictionsdirectlywith bols),Frayeretal. (2006a)(crosses),andFrayeretal. (2006b)(opensym-
thetotalnumbercounts,aswehavedonehere. bols).(c)160µm(bottompanel),withobservationaldatafromDoleetal.
(2004a)(filledsymbols)andFrayeretal. (2006a)(crosses).
10 Lacey et al.
Figure3.Predictedevolutionofthegalaxyluminosityfunctioninourstandardmodel(includingdust)atrest-framewavelengthsof(a)3.6and(b)8.0µmfor
redshiftsz=0,0.5,1,1.5,2and3,asshowninthekey.
4 EVOLUTIONOFTHEGALAXYLUMINOSITY Babbedgeetal. (2006) and Franceschinietal. (2006)4. The
FUNCTION model predictions are given for redshifts z = 0, 0.5 and 1. For
theobservationaldata,themeanredshiftsforthedifferentredshift
Whilegalaxynumbercountsprovideinterestingconstraintsonthe-
binsuseddonotexactlycoincidewiththemodelredshifts,sowe
oretical models, it is more physically revealing to compare with
plotthemwiththemodeloutputclosestinredshift5.Theobserva-
galaxyluminosityfunctions,sincetheseisolatebehaviouratpartic-
tionalestimatesofthe3.6µmLFrelyonthemeasuredredshifts.
ularredshifts,luminositiesandrest-framewavelengths.Inthefol-
InthecaseofBabbedgeetal. (2006),thesearemostlyphotomet-
lowingsubsections,wecompareourmodelpredictionswithrecent
ric,usingopticalandNIR(including3.6and4.5µm)fluxes,while
estimates of luminosity function (LF) evolution based on Spitzer
fortheFranceschinietal. sample,about 50%oftheredshiftsare
data.
spectroscopicandtheremainderphotometric.Inbothsamples,the
measured3.6µmfluxeswerek-correctedtoestimatetherest-frame
3.6µmluminosities.
4.1 Evolutionofthegalaxyluminosityfunctionat3-8µm Weseefromcomparingthebluecurvewiththeobservational
datainFig.4thatthe3.6µmLFpredictedbyourstandardmodel
We consider first the evolution of the luminosity function in the
isinverygoodagreementwiththeobservations.Inparticular,the
wavelength range covered by the IRAC bands, i.e. 3.6-8.0 µm.
observationaldatashowverylittleevolutioninthe3.6µmLFover
Fig. 3 shows what our standard model with a top-heavy IMF in
the redshift range z = 0 − 1. The largest difference seen is at
bursts predicts for LF evolution at rest-frame wavelengths of 3.6
z = 1, where the Babbedgeetal. data show a tail of objects to
and8.0µmforredshiftsz = 0−33.Weseethatatarest-frame
veryhighluminosities,whichisnotseeninthemodelpredictions.
wavelengthof3.6µm,themodelLFhardlyevolvesatalloverthe
However,thistailisnotseenintheFranceschinietal. dataatthe
wholeredshiftrangez = 0−3.Thislackofevolutionappearsto
sameredshift, and isalsonot present intheobservational dataat
besomewhatfortuitous.Galaxyluminositiesatarest-framewave-
thelowerredshifts.Morespectroscopicredshiftsareneededforthe
lengthof3.6µmaredominatedbytheemissionfrommoderately
Babbedgeetal. sampletoclarifywhetherthishigh-luminositytail
oldstars,butthestellarmassfunctioninthemodel evolvesquite
is real. Comparing the red, green and blue lines for the standard
stronglyovertherangez = 0−3(asweshowin§5).Theweak
model shows thatthemodel luminosityfunction isdominated by
evolutioninthe3.6µmLFresultsfromacancellationbetweena
quiescentgalaxiesatlowluminosity,butthecontributionofbursts
decliningluminosity-to-stellar-massratiowithincreasingtimeand
becomescomparabletothatofquiescentgalaxiesathighluminosi-
increasing stellar masses (see Figs. 13(a) and (e)). On the other
ties. We have not shown model LFs excluding dust extinction in
hand,atarest-framewavelengthof8.0µm,themodelLFbecomes
thisfigure,sincetheyarealmostidenticaltothepredictionsinclud-
significantly brighter ingoing from z = 0 to z = 3. Galaxy lu-
ingdust.ThedashedmagentalinesshowthepredictedLFsforthe
minositiesatarest-framewavelengthof8.0µmaredominatedby
emissionfromdustheatedbyyoungstars,sothisevolutionreflects
theincreaseinstarformationactivitywithincreasingredshift(see 4 Babbedgeetal. (2006)alsocomparedtheirmeasuredLFsat3.6,8.0and
Fig.13(b)in§5). 24µmwithpredictionsfromapreliminaryversionofthemodeldescribed
In Fig. 4, we compare the model predictions for evolu- inthispaper
tion of the LF at 3.6 µm with observational estimates from 5 Specifically, for z = 0, we compare with the z = 0.1 data from
Babbedgeetal., for z = 0.5 we compare with the z = 0.5 data
fromBabbedgeetal. andz = 0.3datafromFranceschinietal.,andfor
z = 1,wecompare withthez = 0.75(opensymbols)andz = 1.25
3 Inthisfigure,andinFigs.4,5,8,and10,theluminositiesLν arecalcu- (filled symbols) data from Babbedgeetal. and z = 1.15 data from
latedthroughthecorrespondingSpitzerpassbands. Franceschinietal.