The galaxy stellar mass-star formation rate relation: Evidence for an evolving stellar initial mass function?

0.48 MB·English

by  Romeel Davé

Mon.Not.R.Astron.Soc.000,000–000(2007) Printed2February2008 (MNLATEXstylefilev2.2)
The galaxy stellar mass–star formation rate relation: Evidence for
an evolving stellar initial mass function?
Romeel Dave´
8 AstronomyDepartment,UniversityofArizona,Tucson,AZ85721
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2February2008

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ABSTRACT

1 Theevolutionofthegalaxystellarmass–starformationraterelationship(M∗−SFR)provides
1 keyconstraintsonthestellarmassassemblyhistoriesofgalaxies.Forstar-forminggalaxies,
M∗−SFR is observed to be fairly tight with a slope close to unity from z ∼ 0 → 2, and
]
h itevolvesdownwardsroughlyindependentlyofM∗.Simulationsofgalaxyformationrepro-
p ducethesetrends,broadlyindependentofmodelingdetails,owingtothegenericdominance
- ofsmoothandsteadycoldaccretioninthesesystems.Incontrast,theobservedamplitudeof
ro theM∗−SFRrelationevolvesmarkedlydifferentlythaninmodels,indicatingeitherthatstel-
t larmassassemblyispoorlyunderstoodorthatobservationshavebeenmisinterpreted.Stated
as intermsofastarformationactivityparameterαsf ≡(M∗/SFR)/(tHubble−1Gyr),models
[ predict a constant αsf ∼ 1 out to redshifts z ∼ 4+, while the observed M∗−SFR relation
indicatesthatαsf increasesby∼ ×3fromz ∼ 2untiltoday.Thelowαsf (i.e.rapidstarfor-
2
mation)athigh-z notonlyconflictswithmodels,butisalsodifficulttoreconcilewith other
1v observationsofhigh-z galaxies,suchasthesmallscatterinM∗−SFR,theslowevolutionof
star forminggalaxiesat z ∼ 2−4, andthe modestpassive fractionsin mass-selected sam-
8
3 ples.SystematicbiasescouldsignificantlyaffectmeasurementsofM∗ andSFR,butdetailed
0 considerationssuggestthatnoneareobviouscandidatestoreconcilethediscrepancy.Aspec-
. ulativesolutionisconsideredinwhichthestellarinitialmassfunction(IMF)evolvestowards
0
morehigh-massstarformationatearlierepochs.FollowingLarson(1998),amodelisinves-
1
7 tigatedinwhichthecharacteristicmassMˆ wheretheIMFturnsoverincreaseswithredshift.
0 PopulationsynthesismodelsareusedtoshowthattheobservedandpredictedM∗−SFRevo-
v: lutionmaybebroughtintogeneralagreementifMˆ =0.5(1+z)2M⊙outtoz ∼2.SuchIMF
evolutionmatchesrecentobservationsofcosmicstellarmassgrowth,andtheresultingz =0
i
X cumulativeIMFissimilartothe“paunchy”IMFfavoredbyFardaletal. (2007)toreconcile
r theobservedcosmicstarformationhistorywithpresent-dayfossillightmeasures.
a
Keywords: galaxies:evolution,galaxies:formation,galaxies:high-redshift,cosmology:the-
ory,stars:massfunction
1 INTRODUCTION fallingfromtheintergalacticmedium(IGM)doesnotpassthrough
an accretion shock on its way to forming stars. Simulations and
How galaxies build up their stellar mass is a central question in analytic models show that cold mode dominates global accretion
galaxy formation. Within the broadly successful cold dark mat- at z & 2 (Keresetal. 2005), and dominates accretion in all ha-
ter scenario, gas is believed to accrete gravitationally into grow- loswithmasses.1012M⊙(Birnboim&Dekel 2003;Keresetal.
ingdarkmatterhalos,andthenradiativeprocessesenablethegas 2005).
todecouplefromnon-baryonicmatterandeventuallysettleintoa Thecentral featuresof cold mode accretion arethat itis(1)
star-formingdisk(e.g.Mo,Mao,&White1998).However, many rapid (2) smooth, and (3) steady. It is rapid because it is limited
complicationsarisewhentestingthisscenarioagainstobservations, bythefree-falltimeandnotthecoolingtime;itissmoothbecause
asthestellarassemblyofgalaxiesinvolvesahostofotherprocesses most oftheaccretion occursinsmalllumps andnot through ma-
including starformation, kineticandthermal feedback fromvari- jormergers(Muralietal. 2001;Keresetal. 2005;Guo&White
ous sources, and merger-induced activity, all of which arepoorly 2007);anditissteadybecauseitisgovernedbythegravitational
understoodincomparisontohaloassembly. potentialoftheslowlygrowinghalo.Aconsequenceofcoldmode
Akeyinsightintogasaccretionprocessesistherecentrecog- accretionisthatthestarformationrateisafairlysteadyfunction
nition of the importance of “cold mode” accretion (Katzetal. of time (e.g. Finlatoretal. 2006; Finlator,Dave´&Oppenheimer
2003;Birnboim&Dekel 2003;Keresetal. 2005),wheregasin- 2007), which results in a tight relationship between the stellar
2 R. Dave´
mass M∗ and the star formation rate SFR. Hence simulations of ofIGMandgalaxyobservations(e.g.Oppenheimer&Dave´ 2006;
star-forming galaxies generically predict a tight relationship with Dave´,Finlator&Oppenheimer 2006; Finlator&Dave´ 2007).
M∗ ∝SFR that evolves slowly with redshift. In detail, a slope ThenewsimulationsincludestellarrecyclingassumingaChabrier
slightlybelowunityoccursowingtothegrowthofhothalosaround IMF, by returning gas (and metals) into the ambient reservoir, in
higher-massgalaxiesthatretardsaccretion. additiontotheusualheating,cooling,andstarformation,andfeed-
The stellarmass–star formation rate(M∗−SFR)relationfor back recipes (Springel&Hernquist 2003; Oppenheimer&Dave´
starforminggalaxieshasnowbeenobservedouttoz ∼ 2,thanks 2006). Outflow parameters are computed based on properties of
toimprovingmultiwavelengthsurveys.Asexpectedfrommodels, galaxiesidentifiedwithanon-the-flygroupfinder.Theseandother
therelationshipisseentobefairlytightfromz∼0−2,withaslope improvements that go towards making simulations more realistic
justbelowunityandascatterof. 0.3dex(Noeskeetal. 2007a; down to z = 0 are described in Oppenheimer&Dave´ (2007).
Elbazetal. 2007; Daddietal. 2007). It also evolves downwards Tworunstoz = 0areconsideredhere,eachwith2563 darkmat-
in amplitude to lower redshift in lock step fashion, suggesting a ter and an equal number of gas particle, in cubic volumes of 32
quiescent global quenching mechanism such asaloweringof the and 64h−1Mpc (comoving) on a side. The assumed cosmology
ambient cosmic density, again as expected in models. The range is ΛCDM with Ωm = 0.3, ΩΛ = 0.7, H0 = 69 km/s/Mpc,
ofdatausedtoquantifythisevolutionisimpressive:Noeskeetal. Ωb = 0.048, and σ8 = 0.83. The resolved galaxy population,
(2007a)usedtheAEGISmulti-wavelengthsurveyintheExtended definedhereasthosewithstellarmassesgreater than64gaspar-
Groth Strip to quantify the M∗-SFR relation from z ∼ 1 → 0; ticle masses, is limited by stellar masses of 2.5 × 109M⊙ and
Elbazetal. (2007) used GOODS at z ∼ 0.8 − 1.2 and SDSS 2×1010M⊙ inthetwovolumes(thisissomewhat moreconser-
spectraatz ∼ 0;andDaddietal. (2007)usedstar-formingBzK- vativethaninFinlatoretal. 2006,inordertoensureaccuratestar
selectedgalaxieswithSpitzer,X-ray,andradiofollow-uptostudy formationhistories).
therelationatz ∼1.4−2.5.Ineachcase,acarefulaccountingwas To explore the impact of feedback parameters, simulations
doneofbothdirectUVandre-radiatedinfraredphotonstomeasure with two other outflow models are considered, namely one with
thetotalgalaxySFR,payingparticularattentiontocontamination nooutflows,andonewiththeSpringel&Hernquist (2003)“con-
fromactivegalacticnuclei(AGN)emission(discussedin§4).Mul- stant wind” outflow model, taken from the suite of outflow
tiwavelengthdatacoveringtherest-opticalwereemployedtoaccu- runs described in Oppenheimer&Dave´ (2006). These runs have
ratelyestimateM∗.ThisbroadagreementinM∗−SFRslope,scat- 32h−1Mpc box sizes with 2 × 2563 particles, and they were
ter,andqualitativeevolutionbetweenmodelpredictionsandthese evolved only down to z = 2. Their cosmology isslightly differ-
datalendssupporttotheideathatcoldmodeaccretiondominates ent,havingσ8 =0.9fromfirst-yearWMAPparameters,butthisis
inthesegalaxies. notalargeeffectattheepochsconsideredhere.
Yet all is not well for theory. Closer inspection reveals that Inhydro simulations, starformation ratesandstellar masses
theamplitudeoftheM∗−SFRrelationevolveswithtimeinaway are obtained by summing over particles in galaxies identified us-
thatisinconsistentwithmodelexpectations.Thisdisagreementis ing the group finder SKID (Spline Kernel Interpolative DEN-
fairlygeneric,asshownin§2,arisinginbothhydrodynamicsimu- MAX;http://www-hpcc.astro.washington.edu/tools/skid.html);see
lationsandsemi-analyticmodels,andisfairlyinsensitivetofeed- Keresetal. (2005)fordetails.Gasparticlesthathavehighenough
backimplementation.Thesenseofthedisagreementisthatgoing densitytoformstarshaveinstantaneousstarformationratescom-
to higher redshifts, the observed SFRs are higher, and/or stellar putedinGadget-2; itisthisSFRthatissummedtoobtainthein-
masseslower,thanpredictedincurrentmodels. stantaneousSFRofagalaxy.Ofcourse,observedgalaxySFR’sand
The purpose of this paper is to understand the implications M∗’saremeasuredquitedifferentlythaninsimulations,apointthat
oftheobservedM∗−SFRrelationforourtheoreticalviewofhow willfigureprominentlyin§4.Notethatthesehydrosimulationsdo
galaxiesaccumulatestellarmass.In§3itisarguedthatM∗−SFR notincludeanyfeedbackfromactivegalacticnuclei(AGN),orany
amplitudeevolutionimpliesatypicalgalaxystarformationhistory process that explicitly truncates star formation in massive galax-
thatisdifficulttoreconcilewithnotonlymodelexpectations,but ies,hencetheydonotreproducetheevolutionofpassivegalaxies.
alsootherobservationsofhigh-z galaxies.Possiblesystematicef- Herethefocusisonstar-forminggalaxiesonly,withtheassump-
fectsthatmaybiastheestimationofSFRandM∗ areconsidered tionthatgalaxiesarequicklysquelchedtobecomepassiveandso
in §4, and it is argued that none of them are obvious candidates do not appear in star-forming selected samples (e.g. Salimetal.
toexplainthediscrepancy.Finallyaspeculativeavenueforrecon- 2007).Sinceallgalaxiesinoursimulationswouldbeobservation-
ciliationisconsidered,namelythatthestellarinitialmassfunction allyclassified’asstar-forming,nocutsaremadetothesimulated
(IMF)hasacharacteristicmassthatevolveswithredshift(§5).This galaxypopulation.
modelisconstrainedbasedontheM∗-SFRrelationin§6.Theim- To complement the hydro simulations, results for M∗−SFR
plications for such an evolving IMF are discussed in §7. Results aretakenfromtheMilleniumsimulationplussemi-analyticmodel
aresummarized in§8.A ΛCDMcosmology withΩ = 0.25 and (“Millenium SAM”) lightcones of Kitzbichler&White (2007),
H0 = 70km/s/Mpcisassumedthroughoutforcomputingcosmic as fit by Daddietal. (2007) and Elbazetal. (2007). Besides
timescales. being a completely different modeling approach, the Millenium
SAMalsoincludesAGNfeedbackthatbroadlyreproducestheob-
served passive galaxy population and its evolution. Additionally,
a different semi-analytic model based on the Millenium simula-
2 M∗−SFR:MODELSVS.OBSERVATIONS
tion by DeLucia&Blaizot (2007) was examined, to see if vari-
To begin, the observed M∗−SFR relation is compared with ations in semi-analytic prescriptions can result in significant dif-
model predictions, in order to understand the origin of the ferences in M∗−SFR. It was found that the DeLucia&Blaizot
relation. The primary simulations used are cosmological hy- (2007) model produced a somewhat shallower slope than the
drodynamic runs with Gadget-2 (Springel 2005) incorporating Kitzbichler&White (2007)model, whiche.g. atz ∼ 2resulted
momentum-drivenoutflows,whichuniquelymatchawidevariety in a smaller offset from observations at M∗ ∼ 1010M⊙, but a
SFR-M andan evolvingIMF 3
∗
SAMproducesexcellentagreementatz =0,havingbroadlybeen
tunedtodoso.Byz =1,thepredictedhydrosimulationslopesare
similartothatobservedbyElbazetal. (2007)thoughsteeperthan
seenbyNoeskeetal. (2007a),butmoreimportantlytheamplitude
is low by ∼ ×2−3. At z = 2, this trend continues, with a ∼
×4− 5 amplitude offset. The Millenium SAM has a slope that
remainsnearthez =0value,henceitissomewhatshallowerthan
observedatz ∼ 2,buttheamplitudeevolvesslowlymuchlikein
thehydrosimulations.TherapidevolutioninobservedM∗−SFR
amplitudehasbeennotedbyseveralauthors(Papovichetal. 2006;
Noeskeetal. 2007a),alongwiththefactthatitismorerapidthan
intheMilleniumSAM(Elbazetal. 2007;Daddietal. 2007).The
newaspectdemonstratedhereisthatthedisagreementpersistsfor
currenthydrodynamicsimulations,andaswillbeshowninthenext
section this is true regardless of the main free parameter in such
models,namelyfeedbackimplementation.
Theslopeandscatterareadirectreflectionofthesimilarityin
theshapeofgalaxystarformationhistories(SFH)amongvarious
models and at various masses. For a galaxy observed at time t0,
neglectingmasslossfromstellarevolution,
t0 t0 SFR(t)
M∗(t0)=Z0 SFR(t)dt=SFR(t0)Z0 SFR(t0)dt. (1)
Theintegral depends purely on theform of theSFHup tot0.So
long as the SFH form does not vary significantly with M∗, the
Figure1.M∗−SFRrelation forsimulations versusobservations atz = slope of M∗−SFR will be near unity, and so long as the current
0,1,2. Red points show a 32h−1Mpc hydro simulation box, magenta SFRissimilartotherecentpast-averagedone,thescatterwillbe
show64h−1Mpcbox.Arunningmean(withstandarddeviations shown small.AspointedoutbyNoeskeetal. (2007b),theobservedtight
as vertical bars) is shown separately for each simulation; agreement in scatterof. 0.3deximpliesthatthebulkofstar-forminggalaxies
overlap regionisexcellent. Greenlines showresults fromtheMillenium cannotbeundergoinglargebursts;thiswillbediscussedfurtherin
SAM.Bluelinesshowfitstoobservations atz = 2(Daddietal. 2007), §3.Thisisconsistent withafairlysteady modeof galaxyassem-
andz = 0,1(Elbazetal. 2007),withdashedlinesindicating ±0.3dex blyasgenericallypredictedbycoldmodeaccretion.Notehowever
scatter(Noeskeetal. 2007a).Themodels’slopesandscatterareingood
thatthereverseconclusiondoesnotnecessarilyhold,inthesense
agreement,buttheamplitudedeviatesfromobservationsathigh-z.Lower
thataslopenearunityisnotuniquelyindicativeofasteadymodeof
right: Star formation histories in 100 Myr bins for four star forming
galaxies selected from the 32h−1Mpcrun closest to a final logM∗ = galaxyformation;otherscenarios(suchasthestagedgalaxyforma-
11,10.5,10,9.5(toptobottomcurves).Notethelackoflargeburstsand tionmodelofNoeskeetal. 2007b)canbeconstructedtoreproduce
thesimilarityofthestarformationhistoryshapes,whichyieldsthepredic- aslopenearunity,asdiscussedin§3.
tionofatightM∗ ∝SFR(roughly)relation.Themagentalineshowsthe Example galaxy SFHs from the 32h−1Mpc momentum-
formoftheSFHassumedinPEGASEmodelingdiscussedin§6. drivenoutflowhydrosimulationareshowninthelowerrightpanel
ofFigure1,forarangeofz = 0stellarmassesfrom1011M⊙ −
109.5M⊙,in100Myrbins.Theyareremarkablyself-similar,and
larger one at M∗ ∼ 1011M⊙. Since the results are qualitatively generallyquitesmooth.Theincreasedvarianceinsmallergalaxies
similar,onlytheKitzbichler&White (2007)modelsarediscussed owestodiscretenesseffectsinspawningstarparticles(theseSFHs
further.Whiletherearemanymoresemi-analyticmodelsinthelit- arecomputedfromthehistoryofgalaxystarparticlesratherthan
erature,andperhapsevensomethatareinbetteragreementwiththe the instantaneous SFR at each time); the true scatter in instanta-
M∗−SFRrelation,itisbeyondthescopeofthisworktoconsider neousSFRincreasesonlymildlyatsmallmasses,from∼0.2dex
themall. atM∗ &1011 M⊙to∼0.3dexatM∗ &109.5 M⊙,asshownin
Figure 1 (first three panels) shows the M∗−SFR relation Figure1.
at z = 0,1,2 in simulations, compared to parameterizations The early simulated SFHs are characterized by a rapid rise
from observations (blue lines) by Elbazetal. (2007) (z ≈ in star formation to z ∼ 6, corresponding to rapid early halo
0,1) and Daddietal. (2007) (z ≈ 2). The observed scatter growth (Lietal. 2007). Next, there is a period of roughly con-
of 0.3 dex (Noeskeetal. 2007a; Elbazetal. 2007; Daddietal. stant star formation from z ∼ 6 → 2; this will be the key
2007)isindicatedbythedashedbluelines.Itisworthnotingthat epoch for the present work. Finally, at z . 2 the star forma-
other z ∼ 2 observations suggest a larger scatter (Shapleyetal. tion rates decline modestly, with more massive systems showing
2005; Papovichetal. 2006), so the observational picture on greater decline. This pattern reflects the changing a balance be-
this is not entirely settled. Finally, the green lines show the tweenthegrowthofpotentialwellsbeingabletoattractmoremat-
Kitzbichler&White (2007) Millenium SAMresults, which gen- ter, and cosmic expansion which lowers the ambient IGM den-
erallytrackthehydrosimulationresults,particularlyinamplitude sityandinfallrates.ItisqualitativelysimilartotheSFHinferred
whichwillbethekeyaspect. for the Milky Way. In the models, the fact that the SFHdepends
At z = 0, the relations are generally in agreement, though weaklyonM∗ givesrisetoM∗ ∝SFR,andthefactthatmerger-
theobserved slopeofM∗ ∝SFR0.77 issomewhat shallower than drivenstarburstsarenotasignificantgrowthpathforgalaxystel-
that predicted by the hydro simulations (≈ 0.9). The Millenium larmass(Keresetal. 2005;Guo&White 2007)givesrisetothe
4 R. Dave´
smallscatter.Indetailaslopeslightlybelowunityoccursbecause
more massive galaxies tend to have their accretion curtailed by
hot halos (e.g. Keresetal. 2005), resulting in “natural downsiz-
ing”(Neistein,vandenBosch&Dekel 2006).
Note that the SFRs of individual galaxies in simulations do
notdropbyalargeamountoveraHubbletime.Evenforthemost
massivestarforminggalaxiestoday,theSFRinsimulationsdrops
only be a factor of few since z = 2, and smaller galaxies show
essentially no drop. Meanwhile, the amplitude of M∗-SFRdrops
bynearlyanorderofmagnitudeinthemodels,andevenmoreinthe
data.Sotheamplitudeevolutionowespredominantlytothegrowth
ofM∗ ofindividualgalaxies.Inotherwords,modelsindicatethat
theM∗−SFRrelationdoesn’tevolvedownwardsinSFRsomuch
asupwardsinM∗.
Overall,whiletheslopeandscatteroftheM∗−SFRrelation
areinbroadagreement,thereisastarkandgenericdiscrepancyin
theamplitudeevolutionbetweenthemodelsanddata.Theagree-
ment in slope and scatter is an encouraging sign that our under-
standingofgasaccretionprocessesisreasonablysound.Examining
thediscrepancyinamplitudeevolutionisthefocusoftheremainder
ofthispaper.
3 EVOLUTIONOFTHESTARFORMATIONACTIVITY
PARAMETER
TheamplitudeevolutionofM∗−SFRcanberecastintermsofthe
timescaleforstarformationactivity.Modelsgenerallypredictthat
galaxieshaveM∗/SFR∼tH,wheretHistheHubbletime,perhaps
reducedby1Gyrbecausefewstarswereformingatz&6,asseen
e.g.inthelowerrightpanelofFigure1.
Aconvenientparameterizationisprovidedbythestarforma- Figure 2. Top panel: Star formation activity parameter αsf ≡
tionactivityparameter1definedhereasthedimensionlessquantity (M∗/SFR)/(tH −1Gyr)atz=0−4,evaluatedatM∗ =1010.7M⊙.
αsf ≡(M∗/SFR)/(tH−1Gyr).Physically,thismayberegarded Redcirclesshowresultsfromhydrosimulationswithamomentum-driven
asthefractionoftheHubbletime(minusaGyr)thatagalaxyneeds outflowmodel,cyancircles(z >2)showhydrosimulationswithnoout-
tohaveformedstarsatitscurrentrateinordertoproduceitscurrent flows,andmagenta circles (z > 2show “constant wind”outflow simu-
stellarmass.Notethatαsf isingeneralafunctionofM∗. lations (see Oppenheimer&Dave´ 2006, for details of outflow models).
Greentriangles showtheMillenium SAMresults,andbluecrossesshow
Figure2showsαsf(z) forsimulationsand observations, top
panelshowingvaluesatM∗ =1010.7M⊙,bottomshowingM∗ = tzhe≈o0b.s4e5rv),atainodnsDbayddEilebtaazl.et(2al0.07(2,0z0≈7,2z).≈Ina0l,l1c)a,sNeso,ersekgearedtleasl.so(2f0m0o7da-,
1010M⊙. The values for models have been obtained by fitting elingtechnique oroutflowimplementation, αsf ispredicted tobearound
power laws to simulated galaxies’ M∗−SFR. The momentum- unityorlargeratallepochsandatallmasses.Theobservedevolutionof
driven outflow hydro simulation results are shown as the red cir- αsf outtoz ∼ 2ismarkedlydiscrepantfrommodelpredictions.Bottom
cles,whilecyanandmagentacirclesshowresultsfornowindsand panel:Sameasabove forM∗ = 1010M⊙,showingthat thetrends are
the wind model of Springel&Hernquist (2003) at z = 2−4, broadlyinsensitivetostellarmass.
respectively. The outflow model has only a mild effect on αsf,
because the form of the SFHs are broadly similar regardless of
outflows, even though thetotal amount of starsformedissignifi- stellarmass.Thissuggeststhatαsf ∼1isagenericconsequenceof
cantlydifferent(Oppenheimer&Dave´ 2006).Themostdramatic smoothandsteadycoldaccretiondominatingthegrowthofthese
outlier among models is the high-mass bin of the no-wind simu- galaxies.
lation, where the rapid gas consumption in large galaxies results In contrast, the observed M∗−SFR relation yields a rapidly
inlower current star formation rates relativeto thepast-averaged rising αsf = 0.19 → 0.31 → 0.75 from z = 2 → 1 → 0 at
one(i.e.naturaldownsizing);thisactuallymakesαsflargerandthe 1010.7M⊙. Values at 1010M⊙ are slightly lower but show simi-
discrepancy withobservationsworse. MilleniumSAMresultsare larrapidevolution.Thedramaticdifferencebetweenobservedand
quite similar to the hydro runs, again showing little evolution in predictedamplitudeevolutioneitherindicatesasignificantmisun-
αsf,althoughtheshallowersloperesultsinasomewhatsmallerαsf derstanding in our basic picture of galaxy assembly, or else that
forthelowermassbin.Overall,itisremarkablethatregardlessof observationshavebeenmisinterpretedinsomeway.Giventhecur-
methodology or feedback processes, all models predict αsf to be rentlyuncertainstateofgalaxyformationtheory,onemightbein-
aroundunitywithlittleevolutionfromz = 4 → 0,regardlessof clinedtowardstheformer,andcallintoquestiontheentirepicture
of cold mode-dominated accretion. However, it turns out that the
rapidly declining αsf also results in uncomfortable conflicts with
1 This is similar but not identical to the “maturity parameter” of otherobservationsofhigh-redshiftgalaxies.Toseethis,letusdis-
Scovilleetal. (2007). cardtheideaofcoldmodeaccretion,oranypreconceivedtheoreti-
SFR-M andan evolvingIMF 5
∗
calnotion,andask:Whatpossiblestarformationhistoriesforthese dominantgrowthmodeforgalaxies.Moreover,thisactuallyexac-
galaxiescouldyieldαsf risingwithtimeasobserved? erbatestheM∗−SFRproblem,becauseitprovidesamodetoform
Onescenarioistopostulatethatobservedhigh-zgalaxiesare manymorestarsbeyondwhatisformedquiescently,whichwould
progressivelymoredominatedbyburstsofstarformation.Indeed, make the expected stellar masses even higher than just from the
somemodelshavesuggestedthatburstsmaycompriseasignificant quiescentmode.
portion of the high-z galaxy population (e.g. Kolattetal. 1999). Hencetheunseengalaxiesmustbepassive, orformingstars
However,asNoeskeetal. (2007a)pointsout,thisseemsunlikely at some undetectably low rate. In the currently popular downsiz-
given the tightness of the M∗-SFR relation. They argue that the ingscenario(Cowieetal. 1996),thesesystemswouldhaveformed
observedscatterof0.3dexatz ∼ 1impliesthatmoststarform- theirstarsearlierinarapidmode,andhavenowattenuatedtheirstar
ing galaxieshave current star formation rates. ×2−3of their formationthroughsomefeedbackprocesssoasnottobeselected
quiescent value, and that the mean of the relation represents the instar-formingsamples.Perhapsbyz ∼ 2,downsizingisalready
quiescent “main sequence” of galaxyassembly. Fewgalaxies be- wellunderway,andtheobservedgalaxiesonlyrepresentthesmall
tweenthismainsequenceandthepassivepopulationareseen,ar- fractionofsystemsthathavejustrecentlybegunformingstarsand
guingforrapidmovementbetweenthetwopopulations.Atz ∼2, willsoonbequenched.
theconstraintsareevenmoresevere,asDaddietal. (2007)mea- While qualitatively sensible, the “unseen passive galaxies”
sure a 1σ scatter of 0.29 dex (0.16 dex interquartile), while the model is once again quantitatively inconsistent with other obser-
observedαsf ≈0.2impliestheyareburstingat∼×5theirquies- vations of high-z galaxies. Perez-Gonzalezetal. (2007) used a
centrate.SincetheseobservationsareabletoprobeSFRsdownto Spitzer/IRAC-selected sample to determine that the star-forming
∼ 10 M⊙/yr,moderatestellarmass(M∗ & 1010.5M⊙)galaxies BzK (sBzK) selection technique, i.e. the one employed by
with αsf ∼ 1 would have been seen if present, but are not. The Daddietal. (2007) to measure M∗-SFR at z ∼ 2, encompasses
onlywaytheburstscenariowouldworkisifallobservedgalaxies 88%oftheirgalaxies(2691/3044)from1.6<z <2.5.Thisisre-
somehowconspiretobeburstingatapproximatelythesamelevel, markablyefficient,significantlymoresothane.g.theLymanbreak
which seems highly contrived. Hence bursts do not seem to be a or Distant Red Galaxy selection techniques in obtaining a stellar
viablewaytoproducelowαsf. mass-limitedsample.Theirstellarmasscompletenessisestimated
Another possible scenario is that models simply need tobe- tobe∼75%aboveM∗ &1010.3M⊙forpassivelyevolvinggalax-
gin forming stars at a later epoch. This would effectively lower ies. Puttingthese together, thismeans that conservatively at least
αsf asdefined here, whilestillmaintaining afairlyconstant SFR two-thirds of all galaxies above this mass threshold are selected
as required by the observed scatter. Theoretically, such a sce- as sBzK galaxies, and hence less than one-third are unseen pas-
nario is easier said than done, as the epoch where galaxy forma- sivegalaxies.Incontrast,theobservedM∗−SFRrelationatz ∼2
tion begins is not a free parameter even in semi-analytic mod- implies αsf ≈ 0.2, meaning that ∼ 80% of galaxies should be
els, but rather is governed by halo formation times that are well- unseen(passive),orelsethedutycycleofsuchobjectsshouldbe
constrained by current cosmological parameters. Such a scenario ∼ 20%.Thisassumesgalaxiesbegansignificantactivity∼ 1Gyr
would require overwhelmingly strong feedback at early epochs, aftertheBigBang,andthatαsf =0.2atz >2;thelatterisagain
far stronger than even the strong feedback present in e.g. hydro conservative,asanextrapolationwouldindicatethatαsf shouldbe
simulations that match observed z ∼ 4 − 7 galaxy star forma- evenloweratz >2.HenceitdoesnotappearthatsBzKselection
tion rates (Dave´,Finlator&Oppenheimer 2006; Bouwensetal. missesnearlyenoughsystemstomaketheunseenpassivegalaxies
2007),inorderforcopiousdensegastobepresent inearlyhalos scenarioviabletoexplainαsf(z≈2).
yetnot formstars.Still,inthespiritofdiscardinganytheoretical Anotherwaytoaccomodatethelowscatter,unityslope,and
preconceptions,“delayedgalaxyformation”appearstobeaviable rapid evolution in M∗−SFR is to propose that galaxies evolve
solution. alongtheM∗−SFRrelation,asproposed byDaddietal. (2007).
However,quantitativeconsiderationofdelayedgalaxyforma- Atz ∼ 2,αsf = 0.2impliesthatgalaxiesmustbegrowingvery
tionleadstosignificantproblemswhenconfrontedbyotherobser- quickly. Rapid growth means that for any given galaxy, the am-
vations.Atz =2,αsf ≈0.2impliesthat(constant)starformation plitudeof M∗−SFRisessentiallyunchanging. Now since SFR∝
beganroughly0.5Gyrprior–whichcorrespondstoz ≈ 2.3!Of M∗ = SFRdt,anunchangingamplitudeimpliesasolutionof
courseitcannotbetruethatallgalaxiesbeganformingstarsatz ∼ R
2.3,asmanystar-forminggalaxiesareseenatsignificantlyhigher SFR=Cexp(t/τ) =⇒ M∗ =Cτexp(t/τ)=τ SFR. (2)
redshifts(e.g.Bouwensetal. 2007).Indeed,boththecosmicstar Inotherwords,inthisscenario,galaxieshaveexponentiallygrow-
formationrate(e.g.Hopkins&Beacom 2006;Fardaletal. 2007), ingstarformationratesandstellarmasses!Thisgrowthwouldpre-
and the rest-UV luminosity function (Reddyetal. 2007) are ob- sumably continue until star formation istruncated by some feed-
served to be fairly constant between z ∼ 4 → 2. So it appears back mechanism, probably when M∗ & 1011M⊙. While at first
galaxies were forming stars quite vigorously at z ≫ 2.3. Even this “exponentially growing galaxies” idea may seem incredible,
ifgalaxiessmoothlyrampeduptheirstarformationtovaluesob- ofallthescenariositactuallycomesclosesttoreconcilingallthe
servedatz∼2,theywouldstillneedtostartatz≈2.7. high-zgalaxydata.
Ifmostgalaxiescannothavesuchalowαsf,perhapstheob- Onewaytotestthisscenarioistoconsidertheevolutionofthe
servedsystemsrepresentonlyasmallsubsetofgalaxiesatz ∼2, numberdensitiesnofgalaxies.Thee-foldingtimeofstellarmass
andtherestaresimplyunseen.Sincecurrentsurveysidentifyboth growthisgivenbyτ = M∗/SFR.Assuming(conservatively)that
UV-luminousandred,dustystarforminggalaxies,itmustbethat αsf = 0.2 remains constant at z > 2, then τ ∝ tH −1 Gyr2.
theunseengalaxiesarepassive,orelsesoextremelyextinctedthat
they only show up in e.g. sub-millimeter surveys. For the latter
case, Reddyetal. (2005) found that the cosmic SFR has a rela- 2 Technically,thefactthatτchangeswithtimemeansthattheassumption
tivelyminorcontributionfromsuchseverelyextinctedsystemsthat ofaconstantM∗-SFRamplitudeinequation2isinvalid;however,asimple
escapeBzK/LBGselectionatz∼2,sotheyarenotlikelytobethe evolutionarymodelaccountingforthisyieldsvirtuallyidenticalresults.
6 R. Dave´
Asimplecalculationshowsthattherearee.g.fivee-foldingtimes corrected UV or 24µ estimate, respectively; however, only one-
(or2.2decadesofgrowth)fromz = 3.5 → 2.Foragivenstellar thirdoftheirsampleareanomalousinthatsense.Meanwhile,stel-
massM∗,n(M∗)atz =2shouldthenbesimilarton(M∗/102.2) larmassesareestimatedfromSEDfitting:Usingmulti-bandpho-
at z = 3.5 (neglecting mergers). Fontanaetal. (2006) measured tometryorspectra,onefitsmodelstellarpopulationtemplates(in
thenumberdensityofgalaxieswithM∗ = 1011M⊙ atz ∼ 2to theircase,Bruzual&Charlot 2003),varyingparametersuntilthe
be7×10−4Mpc−3,andthosewithM∗ = 108.8M⊙ atz = 3.5 best fit is obtained. In detail, Daddietal. (2007) used a calibra-
have n = 3× 10−3Mpc−3 (based on an extrapolation of their tionofBzKcolortostellarmassbasedonfittingK20surveyspec-
STYfits),andstellarmassfunctionsfromElsneretal. (2007)yield tra(Fontanaetal. 2004).ThekeyassumptionsinSEDfittingare
similar values. Hence at face value current data suggests that the populationsynthesismodels,dustextinctionmodels,andtheform
stellarmassfunctiondoesnotevolvequiteasrapidlyasexpectedin ofthestarformationhistory,allofwhichintroducesystematicun-
thisscenario.Stilltherearesubstantialuncertaintiesintheobserved certainties.
stellar mass function at such low masses and high redshifts, so a Currentuncertaintiesinobtainingaccuratestellarmassesfrom
factor of a few discrepancy is not compelling enough to rule out restnear-IRdatarevolvearoundthepoorlyknowncontributionof
theexponentiallygrowinggalaxiesscenario. thermally-pulsatingasymptoticgiantbranch(TP-AGB)starstothe
AnotherscenariothathasbeenshowntoreproduceM∗-SFR K-band light (Marastonetal. 2006). This is found to lower the
evolution is the staged galaxy formation model of Noeskeetal. stellar mass estimates for high-z galaxies by a factor of two or
(2007b).Here,adelayisintroducedintheonsetofgalaxyforma- more (Marastonetal. 2006; Bruzual 2007). However, the sense
tionthatisinverselyrelatedtothegalaxymass.Hencelarge-mass would be to exacerbate the problem by lowering M∗ and hence
objectsformearlyandquickly,whilesmallonesstartforminglater furtherreducingαsf,comparedwiththeBruzual&Charlot (2003)
overlongertimescales.Thisissimilartotheexponentially-growing models utilized in Fontanaetal. (2004). Hence while uncertain-
galaxies scenario in the sense that the latter implicitlyassumes a ties in population synthesis models are not negligible, they seem
sortofdelay,becausegalaxiesmustaccumulatelotsofgasintheir unlikelytomakea&×3differenceintheproperdirection.
halowithout formingstarsinorder tohaveasufficientgasreser- It is possible to hide more stellar mass in objects if
voir to then consume exponentially. Hence it may be possible to one postulates a very old underlying stellar population,
constrain staged galaxy formation with similar measurements of in other words by modifying the assumed SFH form.
thestellarmassfunctiontohigher-z;thisisbeinginvestigatedby Papovich,Dickinson&Ferguson (2001) found that for UV-
K.Noeske(priv.comm.).Ifeitherexponentially-growingorstaged selected galaxies, a mass increase of ∼ ×3 − 8 was possible
galaxy formation is correct, this would indicate that star forma- without violating rest-optical constraints. However, K-selected
tion at early epochs does not scale with gas density as seen lo- sampleslikethatofDaddietal. (2007)aregenerallyredderwith
cally(Kennicutt 1998),orelsethatcoldgasissomehowprevented morecontributionfromoldstellarlight,sotheamountthatonecan
fromaccumulatinginhalosasexpectedfromgrowthofstructure. hideismorelimited(C.Papovich,priv.comm.).Moreproblematic
Insummary,thesmallαsf atz ∼2impliedbytheM∗−SFR is the fact that doing so requires assuming that the bulk of the
amplitude is difficult to reconcile with the observed scatter in galaxy’s stars formed very early in the Universe, and hence the
M∗−SFR, the observed evolution of star-forming galaxies to current SFR is much lower than at that early epoch. It turns out
higher redshifts, and the observed fraction of passive galaxies. thatthisisnot aself-consistent solutionfor reducing αsf.Thisis
Taken together, the latter data all point to the idea that observed becauseifsuchaSFHhistoryweretrue,onewouldexpectthatthe
star-forminggalaxiesrepresentthemajorityofgalaxiesatinterme- currentM∗wouldbesignificantlymorethanobtainedbyassuming
diatemasses(∼ 1010 −1011M⊙)at z ∼ 2, andhavebeen qui- a constant SFH over a Hubble time. This would imply αsf ≫ 1
escentlyformingstarsovermuchofaHubbletime,i.e.αsf ∼ 1. for those galaxies. Hence even hiding 5× as much stellar mass
Whiledesignermodelsofstellarassemblycannotbeexcluded,they ascurrentlyinferredwouldbeinsufficient,becauseitwouldonly
wouldbehighlyunexpected,andwouldrequiresubstantialrevision make αsf ∼ 1, whereas such a SFH implies αsf ≫ 1. In order
toourunderstandingofhowgasaccumulatesandformsstarsinhi- for there to be a large old stellar population while still having a
erarchicalstructureformation.Thegeneraldifficultyinreconciling highcurrentSFR,itmustbethatsuchgalaxiesformedmanystars
thedatatogetherwiththemodels’preferenceforαsf ∼1suggests early, then sat around not forming stars until a recent flare-up;
anuncomfortablesituationthatbearsinvestigationintoalternative however, thisscenariosuffersthesamedifficultiesasthedelayed
solutions. galaxyformation model. Henceone cannot self-consistentlyalter
theSFHstohidealargeamountofveryoldstarsinthesegalaxies.
Instead, it may be possible to appeal to some systematic re-
ductionintheinferredstarformationrate.WhymightthetrueSFR
4 SYSTEMATICEFFECTSINM∗ANDSFR
belowerthanestimated?Onepossibilityisthattheextinctioncor-
DETERMINATIONS
rectionsbeingusedforUV-estimatedSFRsaretooexcessive.Now
DeterminingM∗andSFRfrombroad-bandobservationsisfraught recall that for the majority of galaxies, Daddietal. (2007) esti-
with a notoriously large number of systematic uncertainties that matedtheSFRfromtheuncorrectedUVplus24µflux,soforthose
could alter the values of αsf inferred from observations. Here cases the extinction correction is not relevant; it is only relevant
it is considered whether such systematics could be responsible forthe. 25%thatare“mid-IRexcess”systems.Forthose,there
for the low value of αsf at high-z, particularly focusing on the may be room to alter the extinction law such that the extinction
Daddietal. (2007) analysis at z ∼ 2 since that is where the inferredfromtheUVspectralslopeisoverestimated.Daddietal.
discrepancies from model expectations and the range of possi- (2007),likemoststudiesofactivelystarforminggalaxies,usethe
ble systematics aregreatest. Starformation rates there aregener- Calzettietal. (2000)law.UsinginsteadaMilkyWayorLMClaw
ally estimated by summing rest-UV flux (uncorrected for extinc- would indeed yieldlessextinctionfor agiven UV spectral slope,
tion)with24µflux.Forobjectsthatareanomalouslyhighorlow but the typical reduction islessthan a factor of two. Soit seems
in 24µ to dust-corrected UV SFRestimations, they use the dust- unlikelythatextinctionaloneiscausingthelowαsf eveninthemi-
SFR-M andan evolvingIMF 7
∗
nority mid-IRexcess objects, unless theextinction law weredra- (2003)orKroupa (2001)IMFwouldresultinshiftingbothM∗and
maticallydifferentthananythingseeninthelocaluniverse. SFRdownby≈0.15dex(Elbazetal. 2007).Hencetofirstorderit
Another possibilityisthatthereisanadditional contribution makesnodifferenceaslongastheM∗−SFRslopeisaroundunity
totherest-UVfluxbesidesstarformation.Theobviouscandidate (E. Daddi, priv. comm.). Fundamentally, this is because at high-
areAGN,whicharecommoninz∼2starforminggalaxies.While z both stellar mass and star formation rate indicators are driven
Papovichetal. (2006)foundthatonly∼25%ofM∗ >1011M⊙ by light from stars & 0.5M⊙, and favored present-day Galactic
galaxiesatz∼2haveX-raydetectableAGN,Daddietal. (2007b) IMFsgenerally agree on the shape above this mass. Hence more
identifiedobscured AGN through their mid-IR excess, and found substantialIMFvariationswouldberequired.
thattheyappearin∼50%ofgalaxiesatM∗ >4×1010M⊙,and Insummary,therearenoobviouspathstosystematicallyalter
dominatetheoverallAGNpopulationatz ∼ 2.Importantly,they M∗andSFRdeterminationtomakeupuptherequired∼×3−5
alsofound(inagreementwithpreviousstudies,e.g.Papovichetal. differenceinαsf atz ∼2.Itistruethatmanyofthesesystematics
2006)thatsmallermassgalaxieshavealowerAGNfraction,asex- cannotberuledoutentirely,anditmaybepossiblethatacombina-
pectedtheoretically(Hopkinsetal. 2007).RecallthatDaddietal. tionofsucheffectscouldexplainpartorevenallofthedifference.
(2007)usedUVfluxestoestimateSFRinsuchmid-IRexcessob- Itisalsopossiblethat locallycalibratedrelations toestimateM∗
jects,andbycomparisonwithX-raydatatheyestimatedthatsuch andSFRmaybesubstantiallydifferentathigh-zforunknownand
an inferred SFR is unlikely to be wrong by a significant factor. unanticipatedreasons.Thereaderislefttoassesstheplausibilityof
Moreover, since AGN are increasingly rare in smaller mass sys- suchscenarios.Theviewtakenhereisthatthedifficultyinobtain-
tems,itwouldbeanoddcoincidenceiftheymimickedtherelation ingastraightforwardsolutionmotivatestheconsiderationofmore
M∗ ∝SFR0.9 so tightly. Like the burst model, this represents a exoticpossibilities.
fine-tuningproblem,whereAGNhavetocontributeaveryspecific
amounttotheinferredSFRateachmassscalefornoobviousphys-
icalreason.Thiscannotberuledout,butseemsaprioriunlikely.
5 ANEVOLVINGIMF?
ItisalsopossiblethatAGNcouldbepresentingalaxiesthat
arenotmid-IRexcesssystems.ThestrongdichotomyintheX-ray OnepossiblewaytoalterM∗-SFRevolutionistoinvokeanIMF
propertiessuggestthereisnotacontinuumofAGN,andthatAGNs thatisinsomedirectorindirectwayredshift-dependent. Thekey
are limited to mid-IR excess systems (e.g. Daddietal. 2007b). point is that most direct measures of the star formation rate ac-
However,thereexistAGNtemplateswherethecontibutiontorest- tuallytracehigh-massstar formation, whilestellarmassisdomi-
UVandrest-8µfluxarebroadlycomparabletothatinstar-forming nated by lower-mass stars. Hence by modifying the ratio of high
galaxies(Daleetal. 2006).Thesecouldbemistakenfor“normal” tolow massstarsformed, i.e.theshape oftheIMF,itispossible
star-forming systems, whereas in fact the dominant flux is from to alter the M∗−SFR relation. The possibility of a varying IMF
AGN.Still,onceagain,tomimicthetightM∗−SFRrelationwould has been broached many times in various contexts (e.g. Larson
require AGN to be contaminating the SFR estimate at a similar 1998;Fergusonetal. 2002;Laceyetal. 2007;Fardaletal. 2007),
levelatallstellarmasses.Moreover,iftheyaredominatingtheUV thoughtypicallyasalastresortscenario.Itishighlyspeculativeas
flux then they should appear as central point sources, but Hub- there is no clear-cut evidence at present that supports a time- or
ble/ACS imaging of sBzK galaxies generally do not show such space-varyingIMF,butitisworthinvestigatinginlightofgrowing
bright central sources (M. Dickinson, priv. comm.). Hence while observationalandtheoreticalargumentsinitsfavor.
AGNarecertainlypresent,itseemsunlikelythattheyareamajor As Kroupa (2001) points out, a universal IMF is not to be
contaminantforUV-derivedSFRs. expected theoretically, though no variations have been unequivo-
Aperhapsevenmoretroublesomeissueistheuseofrest-8µ cally detected in local studies of star-forming regions. The aver-
(i.e.24µatz ∼ 2)asastarformationrateindicator.Thisisquite ageGalacticIMFappearstobewell-representedbyseveralpower
uncertain,becausethedominantfluxatthosewavelengthsisfrom laws,steepestat>1M⊙,thensignificantlyshallowerat<0.5M⊙,
polycyclicaromatichydrocarbon (PAH)emissionfeatures,whose withaturnoverat<0.1M⊙ (seereviewbyKroupa 2007).Forms
nature is not well understood. As Smithetal. (2007) points out, that follow thisincludethe currently-favored Kroupa (2001)and
thereismorethanafactoroftwoscatterinstarformationrateindi- Chabrier (2003) IMFs. But deviations are seen, at least at face
catorsthatassumeafixed(local)PAHtemplate.Moreover,owing value. The nearby starburst galaxy M82 appears to have an IMF
toPAHfeaturesmovinginandoutofbands,standardphotometric deficientinlow-massstars(Riekeetal. 1980,1993).A similarly
redshiftuncertaintiesof∆z/(1+z) ∼ 0.1canresultinuptoan bottom-lightIMFisinferredinthehighlyactiveArchesclusternear
orderofmagnitudeerrorininferredSFRs.Ontheotherhand,the theGalacticcenter,suggestingthatvigorousstarformationmakes
tightscatterinthez ∼ 2M∗−SFRrelationwouldseemtolimit theIMFtop-heavy(e.g.Figer 2005).Also,theyoungLMCcluster
theamountofscatterowingtoPAHfeaturevariations,orelsebean R136showsaflatteningintheIMFbelow∼2M⊙(Siriannietal.
indicationthatUVlightdominatesoverPAHemission(thoughthe 2000). There are hints that presently active star clusters form
latterisnotwhatisseenlocally;Smithetal. 2007).Calzettietal. more low-mass stars compared to the disk-averaged 5 Gyr-old
(2007)notesthat8µemissionhasasubstantialmetallicitydepen- IMF (Kroupa 2007), suggesting that the IMF was weighted to-
dence,butitisinthesensethatlowermetallicitieshaveloweremis- wardsheavierstarsatlowermetallicitiesand/orearliertimes.Even
sion,sotheirSFRwouldbeunderestimatedbyusingsolarmetal- iftheIMFshapeisuniversal,largerstarformingregionsmaypro-
licity calibrations. So while it is conceivable that some unknown ducemorehigh-massstarssimplybecausetheyhaveenoughmate-
reasoncausesPAHemissionperunitstarformationtobesystem- rialtoaggregateintolargerstars(Weidner&Kroupa 2006).
aticallyhigherinhigh-z galaxiescomparedtolocalones,thereby HencetherearehintsofIMFvariations,albeitcontroversial,
causingtheSFRtobeoverestimated,buttherearenoobviousindi- and interestingly they consistently go in the direction of having
cationsthatthiswouldmimicatightM∗-SFRrelation. a higher ratio of massive to low-mass stars in conditions sim-
Finally,theobservedSFRandM∗’sinDaddietal. (2007)are ilar to those in high-z galaxies. Compared to present-day star-
inferred assuming a Salpeter IMF. Assuming instead a Chabrier formingsystems,galaxiesathigh-z havefarhigherstarformation
8 R. Dave´
surface densities than the Galactic disk (Erbetal. 2006c), have αsf(z > 0)isinferredtobelowerpurelybecauseoftheassump-
highergascontentandhencepresumablymoremassivestarform- tionofanon-evolvingIMF,whereasinrealityαsf(z)(atanygiven
ingclouds(Erbetal. 2006b),andhavesomewhatlowermetallic- mass)isconstantasexpectedinmodels.
ities (Savaglioetal. 2005; Erbetal. 2006a). Larson (1998) and An“evolvingKroupa”IMFisdefinedinwhich
Fardaletal. (2007)listotherpiecesofcircumstantialevidencethat
favorearlytop-heavyIMFs.Sowhiletherearenodirectobserva- MˆIMF =0.5(1+z)γM⊙. (3)
tionalevidencesorstrongtheoreticalargumentsforit,itisnotin- AboveMˆIMF,theIMFhasaformdN/dlogM ∝ M−1.3,while
conceivablethattheIMFinhigh-zgalaxiesmaybemoretop-heavy belowitscalesasM−0.3.NotethatthecurrentGalacticIMFsalso
orbottom-light3thanthepresent-dayGalacticIMF. haveaturn-downatmassesbelow0.1M⊙;thepresentanalysispro-
SuchanIMFwouldhavethreemaineffectsontheM∗−SFR videsnoconstraintsinthisregime,asthestellarmassformedthere
relation: It would increase the output of UV flux per unit stellar isa small fraction of the total. The exact form of the IMF is not
mass formed; it would cause more stellar mass loss from stellar critical here; Kroupa (2001) ischosen for its convenient explicit
evolution;andthemassrecyclingwouldprovidealargergasreser- parameterizationintermsofMˆIMF,but theanalysisbelow could
voirwithwhichtoformstars.Allthreeeffectsgotowardsrecon- equallywellhavebeendonewiththesimilarChabrierIMF.
ciling theoretical expectations with observations of the M∗-SFR ThegoaloftheevolvingIMFistoincreasetheratioofthestar
relation.TheSFRinferredbyassumingastandardIMFwouldbe formation rate of high-mass stars (SFRhiM) to total stellar mass
anoverestimate,sincemanyfewerlow-massstarsareformedper formed, such that the inferred αsf(z) is non-evolving. Note that
high-massstar.Increasedrecyclinglosseswouldlowerthetheoret- differentprobessuchasUVluminosity,Hαemission,andmid-IR
icalexpectationsfortheamountofstellarmassremaininginthese emissiontraceslightlydifferentregimesofhigh-massstarforma-
galaxies,bringingthemmoreintolinewithobservedstellarmasses. tion, but so long as the high-mass IMF slope remains fixed, the
Recycledgaswouldincreasetheavailablereservoirfornewstars, relativeratiosof such emissionshould beconstant. Sofor exam-
effectivelydelayingstarformationingalaxiesandpushingmodels ple,inorder toproduce no evolutioninαsf fromz ∼ 0 → 2at
towardsthe“delayedgalaxyformation”scenario.Henceevenmod- 1010M⊙,thefactorsbywhichf ≡SFRhiM/M∗mustberaisedare
estevolutionintheIMFcouldinprinciplereconciletheobserved 1.9atz = 1and3.3atz = 2,relativetoastandardKroupaIMF
andtheoretically-expectedαsf evolution. (cf.Figure2).Henceγmustbechosentomatchthesefactors.
What value of γ would produce such evolution? To deter-
minethis,thePEGASE.2(Fioc&Rocca-Volmerange 1997)pop-
ulationsynthesis model isemployed, taking advantage of itsfea-
6 ASIMPLEIMFEVOLUTIONMODEL
ture allowing user-definable IMFs. PEGASE outputs a variety of
HowistheIMFexpectedtoevolvewithtime?Thisisdifficultto quantities as a function of time from the onset of star formation,
predictsincenoabinitiotheoryfortheIMFexiststoday.Butasim- given an input star formation history. The assumed SFH is taken
pleconjecturebyLarson (1998,2005)mayhavesomeempirical tobeexponentially-declining witha10Gyrdecaytimeandstart-
value.Henotesthatthereisamassscaleof∼0.5M⊙belowwhich ing 1 Gyr after the Big Bang, in order to approximate a typical
theGalacticIMFbecomesflatter(Kroupa,Tout,&Gilmore 1993; modelSFH.Anexampleof suchaSFHisshown asthemagenta
Kroupa 2001;Chabrier 2003),andatwhichpointthemasscontri- lineinthelowerrightpanelFigure1,normalizedtoastarformation
butionperlogarithmicbin(MdN/dlogM)ismaximized.Larson rateof2M⊙yr−1today;asisevident,itisquitesimilartotypical
uses simple thermal physics arguments to show that this charac- SFHsfromthesimulations,atleastformoderatestellarmasses.A
teristic IMF mass scale (MˆIMF) is related to the minimum tem- constantSFHwasalsotried,andtheresultswerenotsignificantly
perature Tmin of molecular clouds, measured today to be around different.BesidestheinputSFH,allotherparametersaretakenas
8K.Inthatcase,ifthetemperatureintheinterstellarmedia(ISM) thePEGASEdefaultvalues.Nodustextinctionisassumed.
ofhigh-z galaxiesishigher,thischaracteristicmasswillshiftup- ThePEGASEoutputquantitiesspecificallyusedherearethe
wards,resultinginanIMFthatismorebottom-light. stellar mass M∗ formed, and the un-extincted 1650A˚ luminosity
TogenerateacrudeIMFevolutionmodel,letusassumethat (L1650).ThelatterisusedasaproxyforSFRhiM.Usingthese,it
theMˆIMF ∝(1+z)γ,whereγissomeunknownparameterwhose ispossibletocomputef foranygivenassumedIMFatanygiven
valueissetbythevariouseffectsabove.Larson (1985)predicted time.NotethatPEGASEaccountsforbothstellarmassaccumula-
analytically that MˆIMF ∝ Tm3.i3n5, while recent numerical experi- tionandmasslossowingtostellarevolution,butdoesnotaccount
mentsbyJappsenetal. (2005)foundalesssteepscalingofTm1.i7n. fortheimpactofrecycledgas.ByvaryingMˆIMF,itispossibleto
So for example an evolution that scaled purely with CMB tem- estimatefforanyassumedγ,andtherebydeterminewhatvalueof
peraturewouldyieldγ = 1.7. However,other factorsmaycause γwouldyieldnoαsf evolution.
anincreasedISMtemperature,e.g.,elevateddiskSFRratescaus- In practice, determining f ≡SFRhiM/M∗ at any epoch re-
ingmoresupernovaheatinput;lowercoolingratesowingtolower quiresintegratingthemassgrowthuptothatepochforanevolving
metallicities;and/or photoionization fromastronger metagalactic IMF.Todoso,firstPEGASEmodelsarecomputedforawiderange
UVflux(e.g.Scottetal. 2002). ofMˆIMFvalues.Thenthefollowingprocedureisused:
The approach adopted hereisto treat γ asa freeparameter, (1)Avalueofγisselected;
tobeconstrainedusingtheobservedevolutionofαsf.Theassump- (2)thefollowingintegraliscomputed:
tionisthatαsf(z=0)reflectsthepresent-dayGalacticIMF,while t0 ∆M
M∗(t0)=Z ∆t (MˆIMF,t)dt, (4)
0
3 Thedistinctionbetweentop-heavyandbottom-lightIMFsismostlyse- where∆M/∆tisthestellarmassgrowthrateattimettakenfrom
mantic.Heretheconventionisusedthattop-heavymeansahigh-massslope PEGASEmodelsinterpolatedtotheappropriateMˆIMFasgivenin
thatislesssteepthanthelocalSalpetervalue,whilebottom-lightmeansthat equation3;
thehigh-masssloperemainssimilarbutlow-massstarsaresuppressed. (3) L1650 is taken at time t from PEGASE, and fev(t0) =
SFR-M andan evolvingIMF 9
∗
notingthatstellarlifetimesscaleasM−3,andthattheSunhasa
lifetime of 10 Gyr; this yields Mt3urnoff = 10/(tH −1), (tH in
Gyr),whichis1.24M⊙ and1.58M⊙ atz =1,2,respectively.In-
serting values of x = 1.9,3.3 at z = 1,2 yields MˆIMF = 2.4
and5.2M⊙.Thesevaluesarequitesimilartothoseobtainedfrom
PEGASEmodeling,showingthattheresultsarenotcriticallyde-
pendentondetailsofPEGASE.
Thesameprocedurecanbeappliedtothescenarioproposed
byLarson (1998),where theCMB temperature issolelyrespon-
sible for setting a floor to the ISM temperature. In that case,
Tmin = MAX[TCMB,8K], and MˆIMF ∝ Tmβin. Larson (1985)
predictedβ = 3.35, whileJappsenetal. (2005)found β = 1.7.
Theresultsofthisscenarioforthesetwovaluesofβareshownas
thelongandshortdot-dashedlines,respectively,inFigure3.Asis
evident,thisIMFevolutionisnotnearlysufficienttoreconcilethe
observedandpredictedαsf evolutionouttoz∼2,mainlybecause
TCMB onlyexceeds8Katz > 1.93, andhenceMˆIMF doesnot
Figure3.Evolution oftheratioofhigh-mass starformationrate tostel-
changefromz∼0−2.
larmass,fortheevolvingKroupaIMF(equation3)relativetoastandard
Note that since CMB heating of the ISM is subdominant at
KroupaIMF,normalizedtounityatz=0.Thecurvesarecomputedfrom
allredshifts,theassumedformofIMFevolution(eq.3,scalingas
PEGASE.2modelsfordifferentvaluesofγ,asdescribedin§6.Thedata
pointsarefromfitstotheobservedM∗−SFRrelationatz=0.45,1,2(de- TCγMB)isnotphysicallywellmotivated.Whiletheevolutionoutto
scribedandshowninFigure2)atstellarmassesof1010.5M⊙ (squares), z∼2iswell-fitbysuchaform,itcouldbethattheformchangesat
1010M⊙ (triangles), and 109.5M⊙ (circles). The required evolution is higherz,orthatitshouldbeparameterizedassomeotherfunction
reasonably well fit by γ = 2, i.e. MˆIMF = 0.5(1+ z)2M⊙. Long ofz.Forinstance,ifitisthevigorousnessofstarformationactivity
andshortdashed-dotmodelsshowresultsfortheLarson (1998)scenario thatdeterminesMˆIMF,perhapsIMFevolutionactuallyreversesat
wherethe minimumtemperature ofmolecular clouds is setbythe CMB veryhighredshifts.Forlackofbetterconstraints,theformforIMF
temperature, andMˆIMF ∝ Tmβin,whereβ = 3.35(Larson 1985)and evolutioninequation3willbeassumedatallz,butthisshouldbe
β=1.7(Jappsenetal. 2005),respectively.BecausetheCMBtemperature
takenasanillustrativeexampleratherthanawell-motivatedmodel.
doesnotexceedthethelocalminimumtemperatureinmolecularcloudsof
8Kuntilz&2,thisscenariodoesnotyieldsubstantialevolutionatz.2. This analysis also makes predictions for αsf at z > 2 that
wouldbeinferredassumingastandard(non-evolving)IMF.Forin-
stance,Figure3impliesthatat1010M⊙,αsf(z = 3) = 0.13and
L1650(t0)/M∗(t0) is computed (the subscript “ev” stands for αsf(z = 4) = 0.12. Notably,theevolutionslowssignificantlyat
“evolving”); z & 3,andactuallyreversesatz & 4,thoughthesimpleSFHas-
(4) Similarly, fst(t0) is computed for a standard Kroupa IMF, sumedinthePEGASEmodelingmaybeinsufficientlyrealisticto
whereMˆIMF =0.5M⊙; yieldvalidresultsatz&4.AnysuchextrapolationofthisIMFevo-
(5) The ratio fev(t0)/fst(t0) is computed, and compared to the lutionshouldbemadewithcaution,astheM∗-SFRrelationonly
factorsneededtoproducenoαsf evolution. constrains itout toz ∼ 2. However, inthenext section thiswill
Theresultsof thisprocedure areshown in Figure3,namely becomparedtoobservationsthatsuggestthatsuchIMFevolution
fev/fst asafunctionofredshiftfor severalvaluesofγ.Theval- maybereasonableouttoz ∼3+.
ues by which the M∗−SFR relationneeds to be adjusted to pro- Figure4illustratestheevolvingIMF,intermsofstellarmass
ducenoαsf evolutionatz = 0.45,1,2areindicatedbythesym- formed per unit logarithmic mass bin. A Salpeter IMF is shown
bols:SquaresareforM∗ = 109.5M⊙,trianglesfor1010M⊙,and forreference,andevolvingKroupaIMFsareshownatz = 0,1,2
circlesfor 1010.5M⊙. Theseare computed frompublished fitsto withcharacteristicmasses MˆIMF = 0.5M⊙,2M⊙, and 4.5M⊙,
M∗−SFR (Noeskeetal. 2007a; Elbazetal. 2007; Daddietal. respectively.Asexpected,theevolutionresultsinmorehighmass
2007). Because the M∗−SFR relation becomes shallower with starscomparedtolowmassonesathigherredshifts.Whilesucha
time,therequiredadjustmentfactorsaregenerallysmalleratlower dramaticincreaseinMˆIMF mayseemsurprising,recallthatstud-
masses. iesoflocalhighlyactivestar-formingregionssuggestacharacter-
Figure 3 shows that γ = 2 provides a reasonable fit to the istic mass scale of a few M⊙ (Riekeetal. 1993; Siriannietal.
requiredIMFevolutionfortypicalstar-forminggalaxiesatM∗ = 2000; Figer 2005). Note that an IMF with a steeper high-mass
1010M⊙. The implied values of MˆIMF are 2M⊙ and 4.5M⊙ at slope(e.g.Miller&Scalo 1979;Kroupa,Tout,&Gilmore 1993)
z = 1,2,respectively.Notethatthereisnoapriorireasonwhya wouldresultinevenstrongerIMFevolutionbeingrequiredthanin
singlevalueofγ shouldfitatallredshiftsconsidered,sothegood theKroupa (2001)case.
fitsupportstheformofMˆIMFevolutionassumedinequation3. In summary, an evolving Kroupa IMF whose characteristic
As a check, an independent method for determining MˆIMF massincreaseswithredshiftasMˆIMF = 0.5(1+z)2M⊙ isable
evolution was explored that is less comprehensive, but conceptu- to alter the observed αsf evolution from z ∼ 2 → 0 into one
allymorestraightforward.Here,thehigh-massstarformationrate with approximately no evolution, as predicted by models or in-
is assumed to be unchanged, while observations of stellar mass ferred fromcomplementary observations. The precise IMF shape
are assumed to reflect stars near the main sequence turnoff mass is not well constrained; it is certainly possible that an IMF with
(Mturnoff),sinceredopticallightisdominatedbygiantstars.For different behavior, for instance an evolving high-mass slope (e.g.
anevolvingKroupaIMF,toobtainareductionofafactorofxin Baldry&Glazebrook 2003), could produce the same αsf evolu-
the amount of stars produced at Mturnoff, it is easy to show that tion.Theonlyrequirementsarethatitproducesthecorrectratioof
logMˆIMF = logMturnoff +logx.Mturnoff maybeestimatedby high-massstarsrelativetolow-massstarsasafunctionofredshift,
10 R. Dave´
Figure4.Initialmassfunctions,multipliedbyMtoshowthemassformed
perunitlogarithmicmassbin,andarbitrarilynormalizedat1M⊙.Salpeter Figure 5. Cosmic star formation history. Data points are from
(dashed) and Kroupa (2001) (green solid) IMFs are shown; Chabrier is Hopkins (2004); best-fit relation to an updated compilation from
similartoKroupa.EvolvingKroupaIMFsareshownatz = 1(magenta Hopkins&Beacom (2006)isshownasthegreenline. Aschematic rep-
got-dashed)andz =2(reddashed),withcharacteristicmassesMˆIMF = resentation of the cosmic star formation history assuming an evolving
2M⊙ and 4.5M⊙, respectively, as required to remove evolution in αsf IMF with γ = 2 is shown as the blue line, which is simply the
(see§6).Thecumulativez = 0IMF(cyandotted)showsthesummation Hopkins&Beacom (2006)fitdivided bytheratioofhigh-massstarfor-
ofanevolvingKroupaIMFwithMˆIMF = 0.5(1+z)2overthehistory mationtostellarmassgrowthforanevolvingIMFversusaconstantone,
oftheUniverse,assuminganevolvingIMFcosmicstarformationhistory namelythesolid(γ = 2)lineinFigure3.Magentaandredpointsshow
asshowninFigure5.Alsoshownisthe“paunchy”IMF(blue)favoredby theresultofdifferentiating thecosmicstellarmassdensitymeasurements
Fardaletal. (2007)forreconciling thecosmicstarformationhistory,the as a function redshift by Perez-Gonzalezetal. (2007) and Elsneretal.
present-dayK-bandluminositydensity,andextragalacticbackgroundlight (2007), respectively. Cyan dashed line shows the fit to cosmic star for-
constraints.Thecumulativez =0evolvingIMFissimilartothepaunchy mation rate inferred from acompilation ofcosmic stellar mass densities
IMF,showing that it would gowell towards reconciling fossil light data byWilkins,Trentham&Hopkins (2007).Foreaseofcomparison,alldata
withtheintegratedcosmicSFH(see§7). hereareshownhereassumingaSalpeterIMF.TheevolvingIMFcaseisin
goodagreementwithdirectmeasuresofstellarmassgrowth.
andthat(unlikee.g.themerger-inducedtop-heavyIMFsuggested
byLaceyetal. 2007)itappliestothebulkofstarforminggalaxies versusrecentmodels(Finlator&Dave´ 2007;Brooksetal. 2007;
atanyepoch. Kobayashi,Springel&White 2007)wouldperhapsnotstand,and
in particular many more metals would have to be driven out of
galaxiesofagivenM∗viaoutflows(i.e.the“missingmetals”prob-
lemwouldgetworse;Dave´&Oppenheimer 2007).TypeIasuper-
7 IMPLICATIONSOFANEVOLVINGIMF
novaratesathigh-z wouldbelowered;theapparentturn-downin
Whataretheobservational implicationsof suchanevolving IMF TypeIaratesatz&1arguedbyHopkins&Beacom (2006)asbe-
at early epochs? Surprisingly few, as it turns out, so long as ingindicativeofalongTypeIadelaytimeof3Gyr,mayinsteadbe
the massive end of the IMF remains unaffected. This is be- accomodated viaamorecanonical short delaytimecoupled with
cause, as mentioned before, virtually all measures of high-z star anevolvingIMF.Theseareexamplesofthetypesofobservations
formation, be they UV, IR or radio, trace light predominantly thatcouldpotentiallyconstrainIMFevolution.
from high-mass stars. Feedback energetics and metallicities like- Thestarformationratesinferredforhigh-redshiftgalaxiesby
wise reflect predominantly massive star output. Hence so long assumingpresent-dayIMFswouldhavetobereviseddownwards
as an IMF preserves the same high mass star formation rate, for an evolving IMF. A quantitative estimate of the reduction in
it is expected to broadly preserve the successes of understand- starformationratesfromthestandardtoevolvingIMFcaseispro-
ing the relationship between the rest-UV properties of galax- videdbytheratiofev/fst asafunctionofredshiftfromFigure3.
ies (e.g. Dave´,Finlator&Oppenheimer 2006; Bouwensetal. Specifically, the “true” cosmic star formation rate (assuming the
2007), cosmic metal pollution (e.g. Oppenheimer&Dave´ 2006; evolving IMFiscorrect) istheone currently inferredassuming a
Dave´&Oppenheimer 2007), feedback strength, and Type II su- non-evolvingIMF,dividedbythecurveforaparticularvalueofγ,
pernovarates. sayγ =2.
ThemaindifferencecausedbysuchIMFevolutionisinstellar Figure 5 shows the cosmic star formation history with
massaccumulationrates.High-zmeasuresofstellarmassarethere- data points compiled by Hopkins (2004), and the fit by
fore critical for testing this type of scenario. Droryetal. (2005) Hopkins&Beacom (2006).Italsoschematicallyillustratestheef-
determinedthestellarmassfunctionevolutionfromaK-bandsur- fectofanevolvingIMFbydividingtheHopkins&Beacom (2006)
vey out toz ∼ 5, but at even moderately high redshiftsthe rest- fitbythefev/fst curveforγ = 2.NotethatHopkins&Beacom
frame light is actually quite blue, so their stellar mass estimates (2006)usedaSalpeterIMFtoinferstarformationrates,sothese
implicitlyinvolvesasignificantIMFassumption.Thehigh-zstellar correction factors (derived for a Kroupa IMF) are not exact, but
mass–metallicityrelation(Erbetal. 2006a)hasthepotentialtotest should be reasonably close. Figure 5 shows that the cosmic SFR
theIMF;iftheevolvingIMFiscorrect,thegoodagreementfound peakshiftstolaterepochsintheevolvingIMFcase.Thetotalstel-