Can galaxy outflows and re-accretion produce the downsizing in specific star formation rate of late-type galaxies?

0.29 MB·English

by  C. Firmani

Mon.Not.R.Astron.Soc.000,1–??(2002) Printed18January2010 (MNLATEXstylefilev2.2)
Can galaxy outflows and re–accretion produce the downsizing in
specific star formation rate of late–type galaxies?
0
C. Firmani1,2⋆, V. Avila-Reese2†, A. Rodr´ıguez-Puebla2
1
0 1OsservatorioAstronomicodiBrera,viaE.Bianchi46,I-23807Merate,Italy
2 2InstitutodeAstronom´ıa,UniversidadNacionalAuto´nomadeMe´xico,A.P.70-264,04510,Me´xico,D.F.

n
a
J 18January2010

8
1
ABSTRACT

]
O
Anincreasingamountofrecentobservationalevidenceshowsthatlessmassivegalaxies
C are, the higher on average is their specific star formation rates (SSFR = SFR/M , M is
∗ ∗
. the stellar mass). Such a trend, called the ’SSFR downsizing’ (SSFR–DS) phenomenon,is
h
seen forlocaland high–redshift(backto z ∼ 1−2) galaxysamples. We use observational
p
datarelatedonlytodiscgalaxiesandexplorehowdoestheaverageSSFRchangewithz for
-
o differentmasses. For allthe masses in the range∼ 109.5−1010.5 M , the SSFR increases
⊙
r with(1+z)toapowerthatseemsnottodependonM ,andatallredshiftssmallergalaxies
st have ever higher SSFRs; galaxies less massive than M∗∗ ∼ 1010 M⊙ are formingstars at a
a greaterratethaninthepastassumingconstantSFRsoveraHubbletimetobuildstellarmass.
[
We show that these features strongly disagree with the Λ Cold Dark Matter (ΛCDM) halo

2 hierarchicalmassaccretionrates.Further,bymeansofself–consistentmodelsofdiscgalaxy
v evolutioninsidegrowingΛCDMhalos,theeffectsthatdiscfeedback–drivenoutflowsandgas
8 re–accretionhaveonthegalaxySSFRhistoriesareexplored.Theparametersoftheoutflow
8 andre–accretionschemesaretunedtoreproducethepresent–dayMh–M∗relation(Mhisthe
1 halomass)inferredfromtheobservationally–basedM functionofdiscgalaxies.Inthecase
∗
5 ofoutflowsonly,theSSFRofindividualmodelgalaxiesincreaseswithzroughlyas(1+z)2.2
.
9 for all the masses (somewhat shallower than observations) with a normalization factor that
0 dependsonmassasM0.1,i.emoremassivegalaxieshaveslightlylargerSSFRs, contraryto
∗
9 theobservedstrongSSFR–DStrend.Forthere–accretioncases,thedependenceonzremains
0
approximatelythe same aswithoutgasre–infall,butthe correlationon massincreaseseven
:
v for most of the reasonable values of the model parameters. The comparison of models and
i observationsintheSSFR–M∗planeatz ∼0(wherethedataaremorereliable),clearlyshows
X
the divergenttrend in SSFR as the masses are lower (upsizing vs downsizing). We explain
r whyourmodelsshowthereportedtrends,andconcludethattheSSFR–DSphenomenonfor
a
low–massgalaxiesposesasharpchallengeforΛCDM–baseddiscgalaxyevolutionmodels.
Keywords: cosmology:theory—galaxies:evolution—galaxies:haloes—galaxies:high-
redshift—galaxies:spiral—galaxies:starformation
1 INTRODUCTION overashortertime-span,thanlessmassive/latertypegalaxies(e.g.,
Faberetal.1992;Wortheyetal.1992;Carollo,Danziger&Buson
The inference of the assembling history of stellar populations in
1993; Bell & de Jong 2000). With the advent of large and com-
galaxiesasafunctionoftheirmorphological typeandluminosity
pletesurveysoflocalgalaxiesandthesignificantimprovementof
(mass)isamajortopicofpresent–dayextragalacticastronomy,as
the population synthesis models, the average ages, and even the
well as a key probe for cosmologically–based models of galaxy
wholestarformationrate(SFR)historiesofallkindsofgalaxies,
formationandevolution.
havebeeninferred(e.g.,Kauffmanetal.2003;Heavensetal.2004;
Based on chemical and spectro-photometric studies of local
Jimenezetal.2005;Gallazzietal.2005,2008;CidFernandezetal.
galaxies, it has long been known that more massive/earlier type
2005; Panter et al. 2007). The general conclusion of these works
galaxiesholdonaverageolderstellarpopulationsandwereformed
confirmsand extendstheprevious results:moremassive galaxies
assembledtheirstarsatearlierepochsandonshortertimescales,
haltingtheirstarformation(SF)atlaterepochs.Thisphenomenon
⋆ E–mail:fi[email protected]
† E–mail:[email protected]
2 Firmaniet al.
hasbeennowdubbed as’archaeological downsizing’ (Thomaset analytical models (SAMs) several authors have found systemati-
al.2005;Fontanotetal.2009). callythattheSFinmodeledlow–massgalaxieshappenstooearly
In thelast decade, the observational capabilitiesallowed the and isover–quenched at later times, showing these galaxies have
extensionofgalaxypopulationstudiestohighredshifts.Withthese toolowSSFRsatlowredshiftsascomparedwithobservationalin-
’look-back studies’, thecurrent properties ofgalaxiesat theirob- ferences(seeforrecentresultsandmorereferencesSomervilleet
servedtimes,aswellastheirpasthistories,wereinferred.Inapi- al. 2008 and Fontanot et al. 2009). The problem is sharpened by
oneeringwork,Cowieetal.(1996)havefoundthatthemaximum thefact that in the SAMs, thebaryon mass fraction of low–mass
rest–frame K band luminosity of actively star–forming galaxies galaxieshas tobedecreased systematically inorder to reproduce
−
has declined with time in the redshift range z = 1.7 0.2, i.e. thelow–luminosityendoftheluminosityfunction(e.g.,Bensonet
−
theSFefficiencyshiftstolowermassgalaxieswithtime.Thisphe- al.2003).Thisisaccomplishedbyintroducingappropriatelytuned
nomenonhasbeenconfirmedbysubsequentworksbasedonwider schemes of very strong stellar ejective feedback as a function of
surveys with multi-wavelenght information (Juneau et al. 2005; mass,whichworsenstheSSFR–DSproblem( 4).Latere–infallof
§
Feulneretal2005;Belletal.2007;Noeskeetal.2007a,b;Zheng theejectedgas(Bertone,DeLucia&Thomas2007;Oppenheimer
etal.2007;Cowie&Berger2008;Mobasheretal.2009;Chenet &Dave´2008)couldofferapartialcuretothisproblem.
al.2009;Damenetal.2009;seeFontanotetal.2009formoreref- In this work we aim to ’isolate’ the SSFR–DS problem for
erences).TheseworksallowedtoinferthespecificSFrates(SSFR onlylow–massdiscgalaxiesandexploreinatransparentwaythe
=SFR/M ,M isthegalaxystellarmass)forrelativelycomplete effects of ejective stellar feedback and gas re–accretion on the
∗ ∗
samplesofgalaxiesatdifferentredshiftsuntilz 1 2.Thegen- SSFRhistoriesofgalaxiesbymeansofevolutionaryhydrodynami-
∼ −
eralresultisthatthemeasuredSSFRstendtobehigherassmaller calmodelsofdiscgalaxiesformedinsidegrowingCDMhalos.We
isM ,aphenomenon called‘SSFRdownsizing’, thistrendbeing willshowthatasolutiontotheSSFR–DSproblemisnotsimplein
∗
observed at all thestudied redshifts.Recent look–back studiesof thecontextofthehierarchicalΛCDMscenario.
galaxy luminosity functions have also confirmed the local infer- In Section 2, the observational data to be used in this paper
ences of archaeological downsizing mentioned above: the high– arepresentedanddiscussed.Preliminarytheoreticalpredictionsre-
massendofthegalaxymassfunctionseemstobemostlyinplace garding the SSFR histories of galaxies as a function of mass are
since z 2, while the abundances of galaxies of smaller stellar presented in Section 3. A brief review of our galaxy disc evolu-
∼
massesgrowgraduallywithtime(Daddietal.2004,2007;Bundyet tionarymodelsandtheparametricschemesintroducedformodel-
al.2004,2006;Droryetal.2005;Conseliceetal.2007;Marchesini ingejectivefeedbackandgasre–infallaregiveninSection4.The
et al. 2009; Pe´rez-Gonza´lez et al. 2008 and see more references model results and their comparison toobservations are presented
therein). inSection5.Finally,asummaryoftheresultsandadiscussionare
Theemergingobservationalpictureofdownsizinginthestel- presentedinSection6.
larmassgalaxyassemblyhasbeenconfrontedwiththehierarchical Thecosmologicalmodelusedthroughoutthispaperisthecon-
clusteringscenarioofgalaxyformationandevolution,basedonthe cordanceonewithh = 0.7,Ω = 0.7,Ω = 0.3,Ω = 0.042,
Λ m b
ΛColdDarkMatter(ΛCDM)cosmological model.Atthispoint, andσ =0.8.
8
asshowninNeisteinetal.(2006;seealsoFontanotetal.2009),it
isimportant to realizethat the downsizing effect towhich obser-
vationsrefergenerically,hasactuallymanymanifestations,related
2 THEOBSERVATIONS
to different phenomena, involving different types of galaxies and
differentepochsintheirhistories.Hereinweemphasizeatleastthe Ouraimistotargetthepotential problemofSSFR–DS,andthen
twodistinctdownsizingphenomenamentionedabove: compareobservationswithdetailedtheoreticalpredictions.Wefo-
cusourstudyhereonlyonnormal(notdwarf)low–massdiscgalax-
(i) the ’archaeological downsizing’ related to the early (z>2)
∼ ies(109.5M <1010.5 M ).Ononehand,duetocompletenessis-
assemblyofmoststarsinmassive/earlytypegalaxies,and ∗∼ ⊙
sues, dwarf galaxiesarenot taken intoaccount inthez 0and
(ii) the’SSFRdownsizing’ (hereafter SSFR–DS)relatedtothe ∼
high–redshiftobservationalsamplestobeusedhere.Ontheother
later growth of the relative stellar mass as less massive are the
hand,sincemostofdwarfgalaxiesaresatellitesinbiggersystems,
galaxies.
thephysicsofthemisaffectedbyseveralenvironmentalprocesses
Thearchaeologicaldownsizinghasapartialexplanationinthe (rampressure,tidalstripping,interactions,etc.),whicharenotcon-
frame of the hierarchical clustering process of dark–matter halos sideredinourmodels.
(Mouri&Taniguchi2005;Neisteinetal.2006;Guo&White2008; InordertodeterminetheSSFR–M relationatdifferentred-
∗
Keresˇ etal.2009),andastrophysicalprocesseslikeActiveGalac- shifts,multi–wavelengthphotometric andspectroscopic extensive
ticNuclei(AGN)–feedbackhavebeenintroducedalsotogetbetter samplesarenecessary.ThestellarmassesandSFRsofgalaxiesare
agreementwiththeobservations(e.g.,Boweretal.2006;Crotonet inferredcommonlybyusingfittingtechniquestostellarpopulation
al.2006;Cattaneoetal.2006;Monaco,Fontanot&Taffoni2007; synthesis(SPS)models;theSFRsofgalaxiescanbeinferredalso
Hopkins et al. 2008; Somerville et al. 2008, and see more refer- byusingemissionlinesthattraceSFwhenavailable.Theuncertain-
encestherein). tiesassociatedtotheseinferencesarelargeduetoselectioneffects,
RegardingtheSSFR–DS,whichisthefocusof thiswork, if sampleincompleteness,andthemethodsusedtoinferM andthe
∗
confirmed, it will turn as a pervasive problem for current mod- SFR(seeforadiscussion 6.1).Forourwork,werequiresamples
§§
els. This problem refers mainly to the high SSFRs of low–mass wherelate–typegalaxiesareseparatedfromearly–typeones.This
(M 109 3 1010M )star–formingdiscgalaxies.Forthese isthecaseoftheobservationalworksbySalimetal.(2007;z 0)
∗ ⊙
∼ − × ∼
galaxiestheAGNfeedbackisnotexpectedtobeimportantbecause andbyBelletal.(2007;highredshifts).
eithertheydonothaveAGNs(e.g.,Kauffmannetal.2003;Salimet Belletal.(2007;seealsoZhengetal.2007andDamenetal.
al.2007)ortheirAGNsaretooweakastoproducesignificantfeed- 2009) analyze the COMBO-17 photometric redshift survey com-
back(e.g.,Boweretal.2006;Crottonetal.2006).Byusingsemi– bined with Spitzer 24µm data (0.2 < z<1.0). Stellar masses
∼
Galaxyoutflowsandre–accretion 3
AGN,whichformawelldefinedlinearsequenceintheSSFRvs.
M plot,fittedlinearlybyLogSSFR=–0.35(logM –10)–9.83.
∗ ∗
InFig.1thesamefourmassesrelatedtothedatafromBelletal.
(2007)areusedfortheSalimetal.(2007)resultsatz 0.
∼
FromFig.1weseethattheSSFRofwhatroughlycanbecon-
sidered late–type galaxies declines fromz 1 toz 0 almost
∼ ∼
withthesameslopeforallthemasses.However,thenormalization
strongly depends onM , being on average theSSFRsat all red-
∗
shiftshigheraslowerisM .ThisshowstheSSFR-DSbehavior.A
∗
crudeapproximationtotherangeofthedatadisplayedinFig.1is:
SSFR=0.13M−0.53(1+z)2.6Gyr−1, (1)
∗,10
where M is M in units of 1010M . Recently, for all (late–
∗,10 ∗ ⊙
andearly–type)galaxiesandbacktoz 2.8,Damenetal.(2009)
≈
havereportedatrendofSSFRonredshiftalsonotstronglydepend-
ingonM ,withatrendroughlyproportionalto(1+z)5(seealso
∗
Feulneretal.2005; Zhengetal.2007; Martinetal.2007; Pe´rez-
Gonza´lezetal.2008).Therearesomepiecesofevidencethatthe
SSFRsofmassivegalaxiesstarttoincreasesignificantlyandover-
comestheSSFRsoflessmassivegalaxiesonlyatredshiftslarger
than2(Pe´rez-Gonza´lezetal.2008)
Noticethateachofthepairsofcurvesinferredfromobserva-
tionsfor z>0.3 inFig.1 refer toagiven M that isthesameat
∼ ∗
Figure1.EvolutionoftheSSFRoflate–type galaxies fordifferent fixed eachredshift,i.e.thesecurvesdonotrefertotheevolutionofin-
values of M∗. The data at z ∼ 0 (squared dots) are from Salim et al. dividual galaxies (evolutionary tracks), but to galaxy populations
(2007). The solid curves were inferred from Bell et al. (2007) for their
at different epochs. We have carried out an exercise for recover-
two fields, CDFS and A901 (corresponding upper and lower curves, re-
ingapproximategalaxyevolutionarytracksfromtheobservational
spectively;seetext).Fromtoptobottom,thedatacorrespondtomassbins
dataindiagramofFig.1.WeassumethatagalaxyofmassM at
centeredatLog(M∗/M⊙)=9.5(cyan),10.0(blue),10.5(green),and11.0 ∗
redshiftzincreasesitsmassduringthetimeinterval(dt/dz)dzby
(red), respectively. The thin dot–long–dashed green line is our estimate
fromthedatafortheindividual evolution ofanaveragegalaxythatends formingstarsatarateM˙∗ =(1 R)SSFR(M∗,z)M∗(z),where
−
atz = 0withM∗ = 1010.5M⊙ (seetext). Thesolidblacklineshows SSFRisgivenbyeq.(1),andR=0.4takesintoaccountforgas
thecurve1/[tH(z)–1Gyr](1-R)correspondingtoconstantSFRalongthe returnduetostellarmassloss.Thisway,weareabletocalculate
time.Thedashedcurvesare’dark–matterhalo’SSFRhistoriescalculatedas M˙ /M ateachz foran“average”galaxyevolutionarytrackthat
∗ ∗
M˙h/[Mh(1−R)](seetext);fromtoptobottomthecorrespondingmasses ends at z 0 with a given M∗. The dot-dashed line shows the
areMh=1012.6M⊙(red),Mh=1012.1M⊙(green),Mh=1011.7M⊙ average ev∼olutionary track for a present–day galaxy stellar mass
(blue),andMh=1011.4M⊙(cyan).Thecorrespondingstellarmassesare of1010.5M .Theevolutionarytracksintherangeofthedataare
⊙
roughly as thoseofthedata. Thecosmological specific infall rates drive
roughlyparallelandatz 0intercepttheSSFRaxisatthelevelof
SSFRsthatareslightlyhigherformassivegalaxiesandstronglylowerfor ∼
thecorrespondingmasses.AsseeninFig.1,theevolutionarytrack
low–massivegalaxiesthanobservationalinferences.
inferredthiswayisonlyslightlysteeperthantheobservedconstant
masscurves.
were estimated using 17 pass-bands in conjunction with a non-
evolvingtemplatelibraryderivedfromthePe´gaseSPSmodel,and
SFRs were determined from the combined UV and IR fluxes of
3 PRELIMINARYTHEORETICALPREDICTIONS
galaxies. In Fig. 1, we reproduce the averaged data presented in
their Fig.3(SSFRvs. M ),where four redshift bins centered on Fromatheoretical point ofview,inorder toestimatethelevelof
∗
z 0.9,0.7,0.5,and0.3aregiven.Wereproducethedataasso- SF activity of galaxies as a function of their masses and epochs,
∼
ciated only with the blue cloud (late–type) galaxies and for four let consider first the case of a constant SFR history. Then, M
∗
masses [Log(M /M ) = 9.5, 10.0, 10.5, and 11.0 from top to at the cosmic time t (z) should be given by M SFR (1
∗ ⊙ H ∗
∼ ×
bottom, respectively]. Their data are separated according to two – R) (t (z) – 1 Gyr); 1 Gyr is subtracted in order to take into
H
×
COMBO-17fieldsobservedbySpitzer:theextendedChandraDeep account the (average) delay in the formation of halos that host
FieldSouth(CDFS),andthefieldaroundtheAbell901/902(A901) galaxies. Ifthe measured SSFRat z = 0issmaller (larger) than
galaxycluster(upperandlowercurvesinFig.1,respectively).Note 1/[t (z) – 1 Gyr] (1–R), then the average SFR of the given
H
∼
that the SSFRs from CDFS are systematically higher than those galaxy has been higher (lower) in the past than in the current
around the denser field of a cluster. It is well known that the SF epoch.InFig.1thecurve1/[t (z)–1Gyr](1-R)forthecosmol-
H
activitytendstobequenched ingalaxiesinhigh–density regions. ogyusedhereandforR=0.4isplotted(thinsolidline).Thecon-
Thus,forcomparisonwithourtheoreticalpredictions,thedatacor- stantSFRcaseisjustindicativeofthesituation:low–massgalaxies
respondingtoCDFSarelikelymoreappropriate. (M <1010M )showcurrentSFRshigherthantheirpastaverage
∗∼ ⊙
Thedata inFig.1corresponding toz 0weretaken from SFR.
∼
Salimetal.(2007),whoobtainedM anddust–correctedSFRsfor Now,withinthecontext oftheΛCDMcosmogony, onemay
∗
50,000galaxiesbyfittingtheSDSSandGALEX photometry anticipateapotentialdisagreementwithobservationsrelatedtothe
≈
toalibraryofdust–attenuatedSPSmodels.Theywereabletosep- SSFR–DSphenomenon.ByusinganextendedPress–Schechterap-
aratefrom their sample the“pure” star–forming galaxies withno proach (Avila-Reese et al. 1998; Firmani & Avila-Reese 2000 –
4 Firmaniet al.
hereafter FA2000; see 4), we calculate for a given present–day forthecosmologyadoptedhere.Thefinalgalaxybaryonfraction,
§
M its averaged mass aggregation history (MAH), M (z), from f M /M ,isdeterminedbythefurtherdiscgasejecting
h h b,gal gal h
≡
tens of thousands of Monte-Carlo extractions. By assuming that andinfallingprocesses.
the SSFR is driven by the halo specific mass aggregation rate, Theaccretingmassshellsatthetimeoftheirvirialization,t ,
v
(M˙ /M )(z),andcorrectingbythegasreturnfactor(1–R),we are assumed to rotate rigidly with a specific angular momentum
h h
havecalculatedthe’darkmatter’–drivenSSFRhistoriesfordiffer- calculatedasj (t )=∆J /∆M ,where∆representsadiffer-
sh v h h
enthalomasses.InFig.1thedashedlinesshowfromtoptobottom encebetweentwotimesteps,andJ =λ GM5/2/ E 1/2,M ,
the SSFRhistories for halo masses from Mh = 1012.6M⊙ (red) and Eh, are the halo total angularhmomehntum,hmass|, ahn|d energhy
to Mh = 1011.4M⊙ (cyan), which correspond roughly to stellar respectively; λh is the halo spin parameter, assumed to be con-
massesfromM∗ =1011M⊙toM∗ =109.5M⊙,respectively(see stant in time. As a result of the assembling of these mass shells,
4.1andFig.2).Agoodinterpolatingformulatoourresultssince a present–day halo ends with an angular momentum distribution
§§
z 1is: close to the (universal) distribution measured in N-body simula-
∼
1 M˙ 10−1.5 tions(Bullocketal.2001).Theradialmassdistributionofthelayer
SSFR≈ (1 R)Mh = (1 R)M∗1,/182(1+z)2.25Gyr−1(2) iscalculatedbyequatingitsspecificangularmomentumtothatof
h
− − itsfinal circular orbit incentrifugal equilibrium (detailed angular
whereM∗,12isM∗inunitsof1012M⊙. momentumconservation). Thesuperposition oftheselayersform
From this very preliminary calculation, we conclude that agaseousdisc,whichtendstobesteeper inthecentreandflatter
while the dark matter–driven SSFR histories of massive galaxies attheperipherythantheexponentiallaw.Thegravitationalinterac-
donotshowasignificantdiscrepancywithobservations,forlow– tionofdiscandinnerhaloduringtheirassemblyiscalculatedusing
massgalaxies(M∗<∼1010.5M⊙)thediscrepancybecomessignifi- theadiabaticinvarianceformalism.
cantanditincreasesassmalleristhemass.ThepredictedSSFRs Afurtherstepisthecalculationofthestellarsurfacedensity
oflow–massgalaxiesarelowerthantheobservedonesbeyondthe profile. The previously mentioned processes and the fact that SF
uncertaintylevel. Afirstguesstosolvesuch aconflict withinthe islessefficientattheperipherythaninthecentreproducesstellar
contextofthehierarchicalcosmogonyistoproposethatthemain discswithanearlyexponentialsurfacedensitydistribution,andthe
stellarformationepochoflowmassgalaxiesisdelayedbysomeas- sizemainlydeterminedbyλ .ThediscSFatagivenradius(as-
h
trophysicalprocessestolowerredshifts.Cangasoutflowsandlater sumingazimuthalsymmetry)istriggeredbytheToomregasgrav-
re–accretion work inthis direction? In what follows, weturn out itational instability criterion and self–regulated by a balance be-
toself–consistentmodelsofdiskgalaxyformationandevolutionto tweentheenergyinputduetoSNeandtheturbulentenergydissipa-
explorethisquestion. tionintheISM.Bothingredientsdeterminethegasdischeightand
theSFrate.ThisphysicalprescriptionnaturallyyieldsaSchmidt-
Kennicutt-like law. TheSFefficiency depends on the gassurface
4 THEMODEL density determined mainly by λh, and on the gas infall history,
whichinitsprimaryphaseisproportionaltothehaloMAH.
Theformationandevolutionofdiscgalaxieswithingrowingdark
matter halos can be followed in a transparent and self-consistent
way by using simplified hydrodynamic models of discs in cen- 4.1 Feedback–drivenoutflows
trifugalandverticalhydrostaticequilibrium,andwithaSFmecha-
nismtriggeredbyadiscinstabilitycriterionandself–regulatedby Inourpreviousworks,wedidnottakeintoaccountdiscmassout-
anenergybalanceprocess(thesemi-numericalapproach;FA2000; flows(galacticsuper-winds)duetoSNandradiationpressurefeed-
Avila-Reese &Firmani2000; seefor similar approaches van den back.Therefore,thedrivingfactor fordiscgrowthwassolelythe
Bosch2000;Naab&Ostriker2006;Stringer&Benson2007;Dut- gasinfall rateassumedproportional tothehalomassaggregation
ton&vandenBosch2009–hereafterDvdB09). rate,wheretheproportionalitycoefficientisthegalaxybaryonmass
Themainphysicalingredientsofthemodelsusedhereareas fraction,fb,gal.Thisfractionwasfixedtoaconstantvaluesmaller
follows.AspecialextendedPress–Schechterapproachbasedonthe thantheuniversalbaryonfraction,fb,Univ,byafactor 4.With
∼
conditionalprobability(Lacey&Cole1993)isusedtogeneratethe thissimpleassumption,nearlyflatrotationcurvesatthepresentand
halo mass accretion histories (MAHs) fromthe primordial Gaus- athigherredshifts,andgoodagreementwithobserveddiscscaling
siandensityfluctuationfield.Ageneralizedsecondaryinfallmodel relationswereobtained (e.g., FA00; Avila-Reese et al.2008; Fir-
withellipticalorbitsisappliedtocalculatethetime–by–timeviri- mani&Avila-Reese2009).
alizationoftheaccretingmassshells(Avila-Reeseetal.1998).The Here we introduce an algorithm for outflows from the disc
orbitellipticityparameterisfixedinsuchawaythatthestructureof similartothoseusedintheSAMsbutappliedlocallytotheevolv-
theΛCDMhaloesagreeswithresultsfromcosmological N–body ing disc–halo system as a function of the radius (DvdB09). The
simulations(Avila-Reeseetal.1999;FA2000).A(baryon)fraction outflowisassumedtomoveatthelocalescapevelocityofthedisc–
of the massof each accreting shell isassumed tocool down ina halosystem,Vesc(r).Therefore,theefficiencyofmassejectionde-
dynamicaltimeandformadisclayer1.Wecall’primary’cosmo- pendsonmass.Forhaloslessmassivethan 1012 M⊙,theout-
∼
logicalgasaccretionthatonerelatedtotheseaccretingmassshells flowswerefoundtobeabletoreducetheinitialuniversal baryon
withanuniversal baryonfractionf Ω /Ω = 0.163 fractionbyfactors 2andmoreasthemassissmaller (c.f.van
b,Univ ≡ B DM denBosch2002;B∼ensonetal.2003;DvdB09).Themassejected
perunitofareaandtimefromdiscradiusrisgivenby
1eveFrosrhhoarltoermthaassnesthleowHeurbtbhlaenti∼me5,1tH01.1TMhe⊙refthoerer,atdhiiastiavsesucmoopltiinogntiwmoerkiss Σ˙ (r)=Σ˙ (r)ǫ 1000kms−1 n, (3)
wellforhalosofthismassandsmaller.Forlargermasses,thecoolingtime eject SFR (cid:18) Vesc(r) (cid:19)
becomeslargerthantH.Then,themassbaryonfractionavailabletoform
thegalaxydecreasessystematicallywithmass. where Σ˙SFR is the SFR surface density, and ǫ is a free param-
Galaxyoutflowsandre–accretion 5
isη = 0.009M −1,one obtainsnamely thevalue of 1000
SN ⊙
∼
kms−1 reported ineq. (3) and theparameter ǫcan be interpreted
as the fraction of SN kinetic energy transferred into the outflow.
For the latter case (n = 1), by assuming that each SN produces
amomentum p = 3104 M kms−1 andusing thesamevalue
SN ⊙
of η as above, the normalization velocity in eq. (3) should be
SN
300kms−1;bykeeping1000kms−1ineq.(3),theoutflowmo-
∼
mentumisalready3.3timesthemomentumproducedbyoneSN;
thisshouldbestillmultipliedbyourǫ.However,weshouldsaythat
the’momentum–driven’caseisthoughtfortheeffectofmassive–
starsratherthanfortheSNcontributions(e.g.Murray,Quataert&
Thompson2005;Oppenheimer&Dave´2008;Grimesetal.2009);
thetotalmomentum(energy)injectedbymassivestarsduringtheir
lifeismuchhigherthantheoneinjectedbySNe.
BecausetheoutflowrateisproportionaltotheSFR(eq.3),the
outflowsaremoreefficientinremovinggasatearlierepochs,when
the SFR is higher. The early accreted gas has lower angular mo-
mentumthanthelateraccretedgas.Thus,astheresultofdiscgas
ejectionbyoutflows,thespinparameterλofthefinaldiscbecomes
largerthanthespinincaseofnooutflows,which,byassumption,
isequaltothehalospinparameter(seealsoDvdB09).
Thestudyofthegalaxy–sizeoutflowphysicsisacomplicated
hydrodynamicandradiationtransferproblem,anditrequiresavery
large dynamical range, from parsec to megaparsec scales (e.g.,
Dalla Vecchia & Schaye 2008; Oppenheimer & Dave´ 2008 and
more references therein). The model described above is certainly
anoversimplification,whichoverestimatesthefeedbackefficiency
(see for a discussion DvdB09). In most of the previous works it
was assumed that the ejected gas is lost forever by the galaxy–
halosystemandtheproportionalitycoefficientandexponentineq.
(3) were varied in order to reproduce the low–luminosity side of
the observed luminosity function as well as galaxy scaling laws
like the Tully–Fisher relation (c.f. Benson et al. 2003; DvdB09).
Feedback–drivenoutflowsseemtobealsonecessarytoexplainthe
steepmass–metallicityrelationofgalaxiesatredshiftsz 1 2,
∼ −
andtheobservedmetallicitiesoftheIGMathighredshifts(Aguirre
Figure2.Panel(a):Semi–empirical andmodeledstellarvshalomasses. et al. 2001; Oppenheimer & Dave´ 2006; Finlator & Dave´ 2008;
Theformerareinferredbydifferentauthorsbymatchingagivenobserved DvdB09).
luminosity (M∗)function withthe theoretical halo mass function forall The full modeling of feedback–driven outflows isout of the
galaxiesandhalos(non-continuouscurves),foronlylate–typegalaxiesand
scopeofthispaper.Instead,weaimtoexploreinasemi–empirical
halosthatlikelyhostlate–typegalaxies (continuousredcurvefortheav-
way the effects of different kinds and levels of feedback on the
erage,anddottedcurvesencompassingthedottedhorizontallinesforthe
SFRhistoryofdiscgalaxiesasafunctionofmass.Theexploration
1σuncertainty),andfromdirectweakgravitationallensingstudies(cyan
solidsquareswithverticalerrorbars).Thecorrespondingliteraturesources issemi–empiricalinthesensethatwetakecaretoagreewiththe
areindicatedinsidethefigure.Forthemodelpredictions(thickdot–dashed present–day Mh–M∗ relation inferred from matching the galaxy
curves),thediscoutflowcasesforn = 1,2,and5(respectivelyinclock- stellar(luminosity)andhalomasscumulativefunctions.InFig.2
wisesense)areshown.Themodelsareforcedtoagreewiththecontinuous weshow thisrelationasinferredbydifferentauthors(Shankar et
redlineat1010 M⊙ bytuningthefreeparameterǫineq.(3);seeforthe al.2006;Baldry,Glazebrook&Driver2008;Conroy&Wechsler
valuesTable1.Then=2’energy–driven’outflowcasebestreproducesthe 2009;Mosteretal.2009;Rodr´ıguez-Pueblaetal.2009,inprep.).
observationalinferences.Panel(b):Sameaspanel(a)butformodelsthat
Differentobservedgalaxyluminosityfunctions,differentmethods
includegasre–accretion.Onlycaseswithn=1andMpcg=const.(µ=0, topassfromluminositytoM ,anddifferenthalomassfunctions
seeeq.5)areplotted.Forlargervaluesofn,there–accretionproducestoo ∗
(analytical fits or halos directly from N–body numerical simula-
shallowslopesintheMh–M∗relation.Thethreecasesplottedwiththick
tions), as well asdifferent corrections to account for halo groups
solid,dashed,anddot–dashedlines,correspondtovaluesforthe(Log(m),
α)parametersof(10.0,1.0),(10.5,1.0),and(10.0,0.5),respectively(see andsub-halos, wereusedbyeachoneof theseauthors. Forcom-
Table1). pleteness,wereproduceinFig.2alsothemoredirect(butyetvery
uncertain) inferences basedonweaklensing studiesforlate–type
galaxies(squareswithlargeverticalerrorbars;Mandelbaumetal.
eter. In the literature, values for n from 1 to 5 were commonly 2006).
usedinordertoreproducethelow–massluminosityfunction.The TheM –M relationhasbeeninferredcommonlyusingthe
h ∗
values n = 2 and 1 are expected for the ’energy–driven’ and whole galaxy population and all the halos. In the case of Baldry
’momentum–driven’ outflows, respectively. For the former case, etal.(2008;longdashedblueline),theusedgalaxysamplerefers
assuming that each SN produces an energy E = 1051 erg, to galaxies in the field, where disc galaxies dominate. Since our
SN
and that the the number of SNe per solar mass of stars formed study refers to disc galaxies, the M –M relation should be in-
h ∗
6 Firmaniet al.
ferredfromastellarmassfunctionforonlydiscgalaxiesandfrom timeand,inanycase,uncertainduetothestochasticnatureofthe
a halo mass function related to halos that will host disc galax- problem.Instead,weadopthereasimpleapproachthatcanbeeas-
ies.ThisinferencehasbeenperformedinRodr´ıguez-Pueblaetal. ilyimplementedinournumericalcode.Letusconsiderthatduring
(2009),whousedtheM functionforSDSSlate–type(blue)cen- thetimeintervalelapsedbetweentheejectionofagaselementand
∗
tral galaxies as reported in Yang, Mo & van den Bosch (2009), itsre–infallontothedisc,acertainamountM ofprimarycos-
pcg
andthehaloM functioncorrectedforhalosthatdidnotsuffera mological gas (see above for a definition) accretes onto the disc.
h
majormerger(massratiolargerthan0.2)sincez = 0.85;Gover- Theinformationrelatedtothesurroundinggasthatbreakstheout-
natoetal.(2009)haveshownthatafteramajormergertookplace flow(mostlyrelatedtotheinfallingcosmologicalgas),aswellasof
beforez 0.8, thegalaxycanregenerateasignificant discuntil thedynamicsinthegravitationalfield,areactuallysummarizedin
∼
the present epoch. Rodr´ıguez-Puebla et al. checked that the frac- theparameterM (seeforasimilarstatementDubois&Teyssier
pcg
tions of late–type galaxies and ’quiet’ halos with respect to their 2008). In other words, the physics concerning to the outflow/re–
corresponding distributionfunctionsweresimilar,around55% in accretion process can be parametrized through M . With this
pcg
both cases. The solid red line in Fig. 2 reproduces the results by ideainmind, wefollow thenext simpleapproach for calculating
Rodr´ıguez-Puebla et al.including the1σ uncertainty (dotted red gasre–accretioninourcode.Foragivendiscgaselement,assoon
lineswhichencompasstheareaofdottedhorizontallines);thisun- asisejected,theacumulatedamountoftheaccretedprimarycos-
certaintyismainlyduetotheuncertaintiesinthepopulation syn- mologicalgasisnumericallycalculatedtimebytime.Whensuch
thesismodelsusedtoestimateM . amountreachesthevalueofM ,thenthegivenmasselementis
∗ pcg
WeusetheM –M relationbyPuebla-Rodr´ıguezasindica- reintegratedtothedisc.Thisprocedure isappliedtoeachejected
h ∗
tive, intending that the feedback–driven mass ejection shall help gaselement at anytime.Adopting such anapproach, thephysics
reproduce roughly such a (semi–empirically derived) relation. It reducestofixM inaparametricfashion.
pcg
shouldbestressedthatourconclusionsbelowregardingSSFRsare Actuallyweintroducealognormaldistributionaroundourpa-
unchanged if our models are calibrated to reproduce any of the rameterM inordertotakeintoaccountthestochasticalnature
pcg
otherM –M relationsshowedinFig.2.Wefocusourstudyonly oftheproblem.Suchdistributionisdefinedby
h ∗
on galaxies with M <1010.5M , those for which the SSFR-DS
phenomenonbecome∗s∼strong(se⊙e 2). dP = α e−α2 ln Mp′cg −hln Mp′cg i 2dln M′ , (4)
Inpanel(a)ofFig.2,thethic§kdot–dashedlinesshowourre- √π (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (cid:0) pcg(cid:1)
sultsforthecasesn=1,2,and5inclockwisesense,respectively. where α regulates the width of the distribution around the loga-
In each model the outflow efficiency ǫ isfixed in order to obtain rithmicmeanvalue ln(M′ ) ;αandM =e(hln(Mp′cg)i) are
thebest agreement withtheM –M relationatM = 1010M . h pcg i pcg
h ∗ ∗ ⊙ introducedasfreeparameters.
Thehighernis,theshallowertheM –M relationatlowmasses;
h ∗ Bymeansofcosmological hydrodynamical simulations,Op-
forexample,theslopesintherange9.5< LogM <10.5are0.79,
∼ ∗ ∼ penheimer & Dave´ (2008) have found that the re–accretion time
0.62, and0.36 for n = 1,2, and 5,respectively (thecorrespond- scaleswithgalaxymassroughlyasM−1/2 andtheyinterpretthis
ing slope for the Rodr´ıguez-Puebla et al. inference is 0.65). We gal
tobethecasewhenenvironmentaleffectsaredominatingtheretar-
haveexperimentedalsowithanoutflowmodelwithconstantmass
dationofoutflows.SuchabehaviorimpliesinourcasethatM
loading (i.e., n = 0); as expected, the models follow a correla- pcg
shouldberoughlyconstant ormoderatelydecreasingwithmass2.
tion steeper than the case n = 1. If outflows are not introduced
This way, for massive galaxies, M is attained soon and gas
(f =0.04=const. is used), then the correlation is only slightly pcg
b,gal re–accretion becomes consequently efficient (galactic fountains),
shallowerthanthecaseM M inFig.2(slope 0.94);thisis
becauseinourmodelsthehlo∝wert∗hediscmassis,th≈elessefficient whileforlowermassgalaxies,attainingMpcg takeslongertimes;
for the smallest galaxies, it may takes eventually periods longer
istheprocessofgastransformationintostars.
than the current Hubble time in which case the gas does not re-
AsseenfromFig.2,areasonableagreementbelow1010.5M
⊙ turn.Giventhelargeuncertaintiesinthephysicsoftheprocess,we
with the observational inferences is obtained for n = 2 (’SN
introduceaparametricexpressionforM :
energy–driven’ outflows) with a high SN energy transference ef- pcg
ficiency,ǫ = 0.62. Thevaluesofǫfordifferentcases,aswellas m M µ
M = h , (5)
theobtainedvaluesofM∗ atz = 0fordifferenthalomasses,are pcg M⊙ (cid:18)21011M⊙(cid:19)
presentedinTable1.
wheremandµareparameterstobeprobed.
4.2 Re–accretionoftheejectedgas
The assumption that gas islost forever is strong and likely unre- 2 ThepropertiesofMpcgcanbeestimatedalsobysimplemomentumor
energy conservation arguments. Letus consider agiven outflow ofmass
alistic. Theoutflow propagates along theintra–halo and/or intra–
filament media and even if gas escapes from the current gravita- Mo movinginagiven solidangle attheejected velocity Vo inthehalo
tionalpotential,itwillbesloweddownandstoppedbyfurtherin- potential;then,MoVoa = (Mo+Mp)Va,whereMpisthemasspiled
bytheoutflow,V istheshockvelocity,anda= 1or2formomentumor
teractionwiththeintergalacticmedium,andthenre–accretedlater energyconservation,respectively.AroughandgeneralestimateforV could
by thehalo–galaxy system(see e.g.,Bertone et al.2007; Oppen- bethetypicalhalocircularvelocity,Vc,whichapproximatelyscaleswith
heimer&Dave´2008). thehalomassasVc ∝ Mh1/3.Therefore,themasspiledbytheoutflow
takingThinetmooatcicoonuonftatnheejheacltoe–dgdailsacxgyagsrealveimtaetinotncaalnpboetecnatlicaullaanteddtbhye bisacMkptot≈he(dMisco.VIno)oath/eMrwha/o3rd.sS,uMchomisarsesinwteigllradteradgtothtehemdaisscswelheemne,nwtitMhino
viscosityofthesurroundinggas.Oncetheejectedgaselementstops thegivensolidangle,theamountofmassMpofprimarygasisaccreted.
withrespecttothegasaround,boththeejectedgaselementandthe Thus,MpcgisjustMpscaledtotheentiresolidangle4π.Fromthiscrude
circungalacticgaswillinfalltogetherontothediscgalaxy.Thisis approximation,Mpcg∝Mh−1/3orMh−2/3,dependenciesthatarewithin
arathercomplexhydrodynamicprocess,demandingincomputing therangeexploredbelowbyus.
Galaxyoutflowsandre–accretion 7
LogMh
n ǫ Log(m) α µ 11.2 11.4 11.7 12.1 11.2 11.4 11.7 12.1
LogM∗ LogSSFR
1 1.83 9.38 9.63 10.00 10.48 -1.39 -1.37 -1.34 -1.32
2 0.62 9.22 9.54 10.00 10.60 -1.36 -1.34 -1.31 -1.28
5 0.024 8.65 9.18 10.00 10.90 -1.30 -1.26 -1.19 -1.20
1 7.23 10.0 1.0 0 9.05 9.43 10.00 10.77 -1.25 -1.17 -1.13 -1.12
1 3.18 10.5 1.0 0 9.20 9.51 10.00 10.70 -1.33 -1.26 -1.19 -1.15
1 5.95 10.0 0.5 0 9.15 9.49 10.00 10.69 -1.28 -1.22 -1.18 -1.17
1 9.61 10.0 1.0 -1 8.73 9.15 10.00 11.08 -1.28 -1.05 -0.92 -1.07
1 6.46 10.0 1.0 1 9.38 9.63 10.00 10.48 -1.44 -1.37 -1.34 -1.35
2 2.32 10.0 1.0 1 9.21 9.53 10.00 10.62 -1.40 -1.33 -1.31 -1.30
Table1.PredictedM∗andSSFRatz =0asafunctionofhalomassMhfordifferentmodelswithonlyoutflows(lineswithcolumns3–5empty)andwith
bothoutflowsandgasre–accretion.AllthemassesareinunitsofM⊙,andtheSSFRisinunitsofGyr−1.
Regardingtheangularmomentumofthere–accretedgas,the We first study models with galactic outflows only, and then
situationisevenmoreuncertain.Weassumethatthespecificangu- includethepossibilityofgasre–accretion.Inallthecasesthepa-
larmomentumofthisgasisafractionf oftheonecorresponding rametersofthefeedback–drivenoutflowsandgasre–accretionare
j
tothecurrentlyhierarchicallyinfallingcosmicgas.Theresultsto chosen appropriately to reproduce the M –M relation inferred
h ∗
bepresentedheredependpoorlyontheassumedfractionwithinthe semi–empirically(Fig.2).Ouraimistoexplorewhetherthemodel
reasonablerangeoff 0.3 0.7 SSFRhistoriesfordifferentmassesagreeornotingeneralwiththe
j
≈ −
In panel (b) of Fig. 2, the thick black curves show our re- observationaltrendofSSFR-DSdiscussedin 2(Fig.1).
§
sults for the outflow model with n = 1 (see eq. 3) and constant
values of the re–accretion M parameter, i.e. with µ = 0 (see
pcg
eq. 5): m = 1010 (solid line) and m = 1010.5 (dashed line), in 5 RESULTS
both cases using α = 1 in the probability distribution of M
pcg
(eq.4);andm = 1010 butα = 0.5(dot–dashedline).Thecorre- TheobservationaldataintheSSFRvs.(1+z)diagram(Fig.1)re-
spondingslopesoftheMh–M∗relationintherange9.5<∼ LogM∗ fertocurvesofconstantM∗ correspondingtogalaxypopulations
<10.5are0.58,0.57,and0.52,closetotheslopeoftheobserva- atdifferentredshiftsratherthantocurvesequivalenttotheevolu-
∼
tions. For outflow models with n larger than 1, after taking into tionofindividualgalaxies.However,asdiscussedin 2,atleastfor
§
account re–accretion, the Mh–M∗ relations become significantly z<∼1,thelatercurvesseemtobeonlyslightlysteeperthanthefor-
shallower than the observational inferences. InTable 1 wereport meronesintheSSFRvs.(1+z)diagram.Havinginmindthissmall
ourresultsatz = 0(M andSSFRfordifferenthalomasses)for difference(notethatatz 0,thecomparisonofmodelsandob-
∗ ∼
there–accretionparametersmentionedaboveaswellasforsome servationsiscompletelyfair),wecomparebelowtheobservational
caseswithµ = 0,andforn = 2.Asexpected,byincludingre– datawithresultsforindividualgalaxyevolutionmodels,whichend
accretionandt6ryingtoreproducetheM –M relationimpliesvery atz = 0withstellarmassessimilartothosecorrespondingtothe
h ∗
highoutflowefficiencies.Weseealsothattheresultsdependactu- data:Log(M∗/M⊙)=9.5,10.0,and10.5.Themassivecase,Log
allyweakly on thedispersion parameter α (e.g., compare rows 4 (M∗/M⊙)=11.0,ashasbeenexplainedin 4,isoutofthescopeof
§
and6fromTable1). theproblemstudiedherein.Recallthatourmodelsareconstrained
heretoagreewiththeM –M relationinferredsemi–empirically
h ∗
(Fig.2).
4.3 Strategy
5.1 Galacticoutflowsonly
Agivengalaxymodelisdefinedbythehalomass,M ,itsMAH,
h
anditsλh.Theinitialbaryonfractionisassumedequaltotheuni- We first experiment with the usual ejecting SN feedback model,
versalone,fb,gal=fb,Univ,butthefeedback–drivenoutflowandgas where the gas in the outflows (galactic super-winds) is assumed
re–accretionparametersdefinetheactualvalueoffb,gal.Sincewe to be lost forever from the halo ( 4.1). In Fig. 2 we showed
areinterestedhereingenericevolutionarytrendsrelatedtotheSFR thatthesemi–empiricallyderivedM§§–M relationisbetterrepro-
h ∗
andM∗historiesasafunctionofmass,westudyonlythe“central” duced by the n = 2 outflow model (’energy–driven’ outflows).
modelsofdifferentmassescharacterizedby: TheSSFRevolutionfor thiscaseandforthepresent–day masses
Log(M /M )=9.5and10.5areplottedinFig.3withthedotted
theaveragedMAHcorrespondingtothegivenhalomassM , ∗ ⊙
h
• cyan(lowercurve)andgreen(uppercurve)lines,respectively.The
a value of λ = 0.03, which is somewhat smaller than the
h
• SSFRtrackscorresponding tomassesinbetweenthismassrange
meanofallrelaxedhaloesmeasuredinnumericalsimulations(e.g.,
liewithinthesetwocurves.
Bettetal.2007).
Forallthemasses,theSSFRincreaseswithzwithatrendin
Note that these physical ingredients have in fact probability den- roughagreementwithobservations,butthereisasmallbutsystem-
sitydistributions(nottakenintoaccounthere)that,ofcourse,will aticaltrendofhigherSSFRsasthemassislarger,oppositetothe
produce anintrinsic scatter inthe values of the galaxy properties strong observed SSFR-DS phenomenon. A good fit to the model
studied. resultsfromz =0toz 3is:
∼
8 Firmaniet al.
5.2 Galacticoutflows+re–accretion
In 4.2wedescribedourparametricmodeltoaccountforthere–
§§
accretionoftheejectedgasbyfeedback–driven galacticoutflows.
Themodelswithre–accretionagreewiththeM –M relationonly
h ∗
foroutflowswithn= 1(seepanelbofFig.2).InFig.3,thecase
withn = 1,µ = 0,m = 1010 (seeeq.5)andα = 1(seeeq.4)
isplottedfortwomodelsthatendatz = 0withLog(M /M )=
∗ ⊙
9.5 (upper dashed cyan line) and 10.5 (lower dashed green line).
Wehaveexperimentedalsowithmanyotherre–accretionparame-
ters(seeTable1).Asexpected,theSSFRsarehigherforthemod-
elswithre–accretion.Thisisbecausetherateoflaterre–accretion
oftheejectedgassumsuptothehierarchical haloaccretionrate,
raisingtheSFR.FromFig.3weseethattheSSFRincreaseswith
(1+z) toa power not strongly dependent on massand actually,
similartothecaseofnore–accretion(dottedcurves;seeeq.[6]).
It is obvious that if re–accretion isincluded, then the mass load-
ingintheoutflowmodelshouldbeincreasedinordertoreproduce
thesameM –M relationthanwithoutre–accretion.Forexample,
h ∗
comparethemodelsgiveninrows1and4inTable1(n = 1out-
flowcase),whereǫincreasedfrom1.83to7.23;notethattheSSFR
increasedforallthemasses,butmoreforthemoremassivemodels.
The physics of the model is rather complex. For M =
pcg
Figure3.SameasinFig.1butincludingevolutionarytracksofindividual const., the larger themass, theshorter the timeperiod of gas re–
galaxies.Thedottedcurvesareformodelswithoutflowsandforthefavorite incorporationintothediscbecauseMpcgisequaledquicklybythe
casen = 2;theuppergreenandlowercyanlinesareforM∗ = 1010.5 accretedprimarycosmologicalmassinmoremassivegalaxies.Fur-
M⊙andM∗=109.5M⊙,respectively.Thedashedcurvesareformodels thermore,agivenmasselementcanbeejectedandre–accretedsev-
withoutflows(n=1)andgasre–accretion(µ=0,m=1010,α=1.0); eraltimesdependingontheoutflowefficiency.Giventhatmassive
theuppergreenandlowercyanlinesareforM∗=1010.5M⊙andM∗= galaxies recover their ejected gassooner than low mass galaxies,
109.5M⊙,respectively. thenthesamestellarmass(M –M relation) isreacheddecreas-
h ∗
ing the power n of eq. (3), i.e. decreasing the low mass galaxies
outflowwithrespecttomassivegalaxies.
The behavior of the SSFR with mass is again against the
SSFR=0.005M 0.1(1+z)2.2Gyr−1. (6) SSFR–DS.Forthe value of Mpcg= 1010 M⊙ used here, models
∗,10 withM 1010.5 M raisetheirSSFRtovalues slightlylower
∗ ⊙
∼
Suchabehaviorisconsequenceof(i)mainlytheCDMhaloMAHs on average than observational inferences, while the lowest mass
whichdrivethegasinfallprocessand,consequently,theSFhisto- models,onlyveryslightlyraisetheirSSFRatlateepochs,faraway
ries:thelessmassivethehalos,thelowerthespecificmassaggre- fromthevaluesrequiredtoreproducethehighSSFRsinferredfor
gationratesatlateepochs(seeFig.1);and(ii),inalessextent,of low–massdiscgalaxies.
thefactthatthediscoutflowgasejectionismoreefficientforlower Further,wehaveexperimentedwithcaseswhereµineq.(5)
masshalos.TheSSFRhistoriesoflow–massmodelshavesignif- isvaried.Forµ < 0andforafixedvalueofm,theepochwhen
icantlyover–quenched present–day SFRs,i.e.their stellarmasses theSSFRbecomeshigherduetogasre–accretionisdelayedwith
hadtobeassembledwithSFRsinthepasthigherthanthecurrent respecttothecaseµ=0.Thisincreaseisearlyandlargeformore
one. In general, all the models since z 2 lie below the curve massivegalaxies,butwithtimetheSFR–ejection–re–accretionpro-
∼
1/[t (z) – 1 Gyr] (1-R) of approximately constant SFR history cessisstabilized.Forsmallgalaxies,theinfluenceofre–accretion
H
(thinblacksolidlineinFig.1;see 3). over theSSFRbeginsatlateepochs. Overall,takingasanexam-
The most reliableobservation§s arefor local galaxy samples. ple the case µ = 1, galaxies that end with M∗>∼1010.5 M⊙
−
InFig.4weplotthefitandits1σdispersiontotheSSFR–M cor- haveSSFRhistoriesclosetothoseinferredfromobservations.For
∗
relationgivenforlocalSDSSgalaxiesbySalimetal.(2007).They smallergalaxies,whiletheirSSFRsarenowhigherthaninprevi-
reportthefitforonlystarforminggalaxies(solidblueline)andfor ouscases,theyremainbeingfarfromtheobservationalconstraints.
boththissampleandtheoneofgalaxieswithAGN(short–dashed Modelskeepshowingatrendindisagreement withtheSSFR-DS
red line). In both cases the galaxy SSFRs significantly decrease trend.Regardingthecaseswithµ> 0,theSSFRsoflessmassive
asM increases.Thethreelong–dashedmagentacurvesshownin galaxiesresultevenlowerthanintheµ=0case.
∗
thisplotcorrespond tomodelswithonlyoutflow andwithvalues In Fig.4 the loci of thedifferent re–accretion models inthe
ofn=5,2,and1fromtoptobottom,respectively(seealsoTable z = 0SSFR–M diagramareplotted(seealsoTable1).Theµ =
∗
1).Thicklineisusedforourpreferredn=2case(seepanel(a)of 0,m=1010fiducialcaseisshownwiththethickdot–dashedblack
Fig.2).Thedifferencebetweenmodels(weakupsizing)andobser- line.Theuppermostdot–dashedcurvecorrespondstoourµ= 1
−
vations(strongdownsizing)inthetrendoftheSSFR–M relation case.Thedot–dashedcurveimmediatelylowertothefiducialcase
∗
issignificant.Inparticular,forM <1010.5M ,thelowerthemass, isforthesamevaluesofthiscasebutforα=0.5.Thenextlower
∗∼ ⊙
thebiggerthedifferencesbetweenthemodelSSFRsandthosein- curveisasthefiducialcasebutform = 1010.5.Thenfollowthe
ferred from observations. For M > 1010.5M , the differences curvescorrespondingtoourµ=1andn=2cases(seeTable1).
∗ ⊙
tendtodisappear. Asseen,forareasonablerangeofvaluesforthere–accretionmodel
Galaxyoutflowsandre–accretion 9
thatdidnotsufferamajormergersincez = 0.85.Wehavefound
thattheslopeoftheM –M relationatmassesM <31010M is
h ∗ ∗∼ ∗
better attainedfor theoutflow model withn = 2(energy–driven
SNfeedback,eq.3)andforahighSNenergyefficiency,ǫ=0.62
(Fig.2,panela).
For the n = 2 outflow case, the SSFR of all masses raises
withredshift proportional to(1+z)2.2 upto z 3withalittle
dependence on(present–day) mass,asM 0.1 (eq∼.6).TheSSFRs
∗
at low redshifts for models with M <1010.5M are well below
∗∼ ⊙
the constant SFR curve 1/[t (z) – 1 Gyr](1-R) (see 3), which
H
§
meansquiescentlateSFactivity,andmiserablyfailinreproducing
theobservationalinferences(Fig.3).Inparticular,themodelsshow
a trend contrary to the observed SSFR-DS phenomenon. Such a
conflictisclearlyseenintheSSFRvsM diagramforlocalgalax-
∗
ies(z 0),wheretheobservationaldataaremorereliable:while
∼
theSSFRofobservedgalaxiessignificantlydecreaseswithM ,the
∗
modelsshowaslightincreasewithM .Thesituationissimilarfor
∗
modeloutflowswithn=1andn=5(Fig.4).
Westressthat theconflict isnot relatedtotheobserved low
SSFRsofmassivediscgalaxies(>1010.5M )withrespecttomod-
∼ ⊙
els,butinsteadisrelatedto(i)theobservedtrendofSSFRincreas-
ingasthemassdecreases(SSFR–DS),and(ii)tothetoohighval-
ues of the SSFRs of galaxies with masses M < 1010M , both
∗ ⊙
Figure 4. Local (z ∼ 0)SSFR–M∗ relation. The fitto the SDDS data itemsapplyingsincez 1.
∼
(averageand1σscatter)bySalimetal.(2007)isplottedwithsolidblue Inthemodels,ononehand,theSFRhistoryislargelydriven
linesforonlystarforminggalaxies,andwithshortdashedredlinesforboth bythehierarchicalhalomassaggregationrate3,whichonaverage
thestarformingandAGNgalaxies.Thelong–dashedmagentacurvesare increaseswithredshiftatnearlythesamerateforallmasses,butat
formodelswithonlyoutflowsandn=5,2and1(respectivelyfromtopto
agivenz,massivehaloshaveratesslightlyhigherthanthoseofless
bottom).Ourpreferredcase,n=2,ishighlightedwithbold.Thedot–long–
massivehalos(seeFig.1andeq.2).Ontheotherhand,theejective
dashedblackcurvesareformodelswithoutflowsandgasre–accretion.Our
feedbackschemeusedhereproducesproportionallymoregasloss
preferredcase,n = 2,µ = 0,m = 1010,α = 1.0ishighlightedwith
(hence,lesslaterSF)forthelowermassdiscsthanforthemassive
bold.Fortheother curves, seetext andTable1.Inallthecases, models
showaweak’upsizing’behaviorcontrarytothestrong’downsizing’ofthe ones. We conclude that the low SSFRs of low–mass disc galaxy
observations. modelswithoutflows,andthe(slight)increaseofsuchSSFRswith
M ,opposed totheobservedstrongSSFR–DSphenomenon –the
∗
lowerthemass,thehighertheSSFR,isanaturalconsequence of
parameters,theSSFRasafunctionofM∗remainsinconflictwith theΛCDMhaloassemblingand,inaminorlevel,ofthefeedback–
observations,andtheSSFRsoflow–massmodelsaretoolow. drivenoutflowschemescommonlyusedtoreproducethelocalM –
h
M relation.
∗
6 SUMMARYANDDISCUSSION
6.2 Thecontributionofgasre–accretion
Bymeansofself–consistentmodelsofdiscgalaxyevolutioninside
We further explored the possibility that the gas ejected from the
growingΛCDMhalos,theeffectsoffeedback–drivendiscejective
outflows(galacticsuper-winds)andgasre–accretionontheevolu- disc–halosystemcanbere–accretedlaterontothedisc.Ourscheme
tionoftheSSFRforlow–massgalaxies(M <1010.5M atz=0) forsuchaprocessisverygeneralandencompassesalargerangeof
∗∼ ⊙
possibilitiesfortheunknownandcomplexoutflowhydrodynamic
havebeen explored. Wehave studied only the’central’ (average)
modelscorrespondingtodifferenthalomassessinceourmaingoal and radiation transfer processes. Instead of introducing as a key
was to probe general trends regarding the evolution of the SSFR parameter the time elapsed between when a given mass element
isejectedanditisfurtherre–accreted, weuseasaparameter the
anditsdependenceonmass.
primarycosmologicalgasmassthatwillbeaccretedduringthispe-
riod,M .Thisway,wetakeintoaccountinsomewaytheamount
pcg
6.1 Theeffectofgasoutflows of circumgalactic massthat interactswiththeoutflow, amount of
massthatinour models isrelatedtotheMAHof thegivenhalo.
The rate of mass ejection was assumed proportional to the local
WeuseaparametrizationforM givenbyeq.(5),andalognor-
pcg
SFRandinverselyproportionaltothelocalescapevelocityV to
esc maldistribution(eq.4)forsuchaparameterisconsideredinorder
agivenpowern(seeeq.3,forn = 2andn = 1theoutflowsare
tointroducea(natural)dispersionaroundthere–accretiontimesof
expectedtobeenergy–andmomentum–driven,respectively).The
theejectedmassshell.
parametersoftheoutflowmodelswereexploredinthelightofthe
M –M relationatz =0(galaxyefficiency)inferredfromasemi–
h ∗
empiricalapproachthatmakesuseoftheobservedluminosity(M∗) 3 Inourmodels,theSFRhistorydependsalsoonthelocaldiscgassur-
function(Fig.2). facedensityandonsomediscdynamicalandhydrodynamicalproperties.
Our best outflow models werechosen tobe thosethat agree Nevertheless,thedominantfactorintheSFRhistoriesofourmodeleddisc
withtheMh–M∗relationinferredfordisc(blue)galaxiesandhalos galaxiesisthecosmologicalgasinfallrate.
10 Firmaniet al.
Ourresultsshowthatthegasre–accretionmayraisetheSSFR byepisodicprocesses(externallikemergersorinternallikestatisti-
of disc galaxies but it does it in the incorrect direction regarding calfluctuationsinmassivestarformation);ifamongthelow–mass
mass:whileforlowmassgalaxiestheincreaseinSSFRissmall, galaxies,thosewithlow–SFRsaremissedduetodetectionlimits,
for the massive galaxies the increase is significant at all epochs thentheSSFRsoflow–massgalaxieswillbebiasedonaverageto-
(Fig.3).Wehaveexperimentedwithseveralcases:M constant ward higher values of SSFR,a bias that increases for samples at
pcg
forallmasses,andM decreasingorincreasingwithmass.The higher redshifts. Nevertheless, in most of the observational stud-
pcg
decreaseofSSFRasM decreases(upsizing)isseeninallthecases iesreportingtheSSFR–DSphenomenon theauthorsdiscussthat,
∗
(forz 0,seeFig.4). whilethisispossibleatsomelevel,hardlywoulditbethedominant
∼
We conclude that for models of disc galaxy formation and factorintheSSFR–DSphenomenonobservedsincez 1(seefor
∼
evolutioninthecontextoftheΛCDMcosmogony,theproblemof exampleNoeskeetal.2007a).
the SSFR–DS is not solved by an interplay of outflows and re– Ontheotherhand,theanalysisofverylocalsurveyscertainly
accretionofgas. helpstodisentanglewhetherepisodicstarburstingeventsdominate
ornot theSFhistoryoflow–massdiscgalaxies.Fromastudyof
theSFactivityofgalaxieswithinthe11MpcLocal Volume,Lee
6.3 IstheSSFR–DSphenomenonwellestablished?
etal.(2007;seealsoBothwell,Kennicutt&Lee2009)havefound
Wehaveshowninaclearandtransparentwaythedifficultygalaxy thatintermediate–luminosity discgalaxies( 19<∼MB<∼ 15or
− −
evolutionmodelsinthecontextoftheΛCDMcosmologyhavefor 50<∼Vm/kms−1<∼120)showrelativelylowscatterintheirSFac-
explainingtheSSFR–DSphenomenon, relatedmainlytosub–L∗, tivity,implyingfactors2–3fluctuationsintheirSFRs;aboveVm ≈
star–forming(late–type)galaxies.Beforediscussingsomecaveats 120 km/s the sequence turns off toward lower levels of SSFRs
of the models and possible cures to the problem, it should be andlargerbulge–to–discratios.Theseresultsarefornearbygalax-
stressedthatatthelevelofobservationalinferencestherearestill ies, where selection effects are minimal, and imply that the SS-
largeuncertainties. FRsofdiscgalaxieswithM∗>∼5108M⊙ followarelativelytight
TheuseofSPSmodelstofitobservationaldataandinferM sequence, without strong fluctuations. For galaxies smaller than
∗
andSFRshouldbetakenstillwithcautionduemainlytotheuncer- Vm 50km/s(dwarfs)thesituationseemsdifferent.Theresults
∼
taintiesinstellarevolution(forexampleinthethermally–pulsating byLeeetal.(2007)showthatasignificantfractionofsuchgalaxies
asymptoticgiant branch, TP-AGB,andhorizontal branchphases) areundergoingstrongepisodicSFfluctuationsduetothelargescat-
andtoourpoorknowledgeoftheinitialmassfunction,IMF,aswell terintheirSSFRs.Anotherobservationalstudyofnearbygalaxies
as due to degeneracies like the one between age and metallicity byJamesetal.(2008),concludedalsothatthereislittleevidencein
(seeforrecentextensivediscussionsMarastonetal.2006;Bruzual theirsampleofpredominantlyisolatedfieldgalaxiesofsignificant
2007; Tonini et al.2009; Conroy, Gunn &White2009a; Conroy, SFthroughbriefbutintensestar-burstphases.Therefore,itseems
White & Gunn 2009b). For example, Conroy et al. (2009a) esti- thatthetightsequencefoundfornormalstar–forming(disc)galax-
matethatstellarevolutionuncertaintiescanintroduceupto 0.3 iesintheSSFR–M∗ planeinlargesurveysasSDSS(Brinchman
dexstatisticalerrorinM atz 0.Regardingsystematicale∼rrors, etal.2004;Salimetal.2007;Schiminovichetal.2007)isintrin-
∗
theyaremoredifficulttoevalua∼te.Marastonetal.(2006)claimthat sicandduetoahighdegreeoftemporalself–regulatedSFwithin
thestellarmassesof galaxieswithdominating stellarpopulations individual galaxies. Thissequence (calledthe’mainsequence’ in
of 1Gyr agecould be onaverage 60% lower if theirassump- Noeske et al. 2007a) seemsto persist back toredshifts z 1 as
tion∼sfor convective overshooting during theTP–AGBphases are discussedabove. ∼
used.Itisnoteasytoestimatethedirectioninwhichthesestatis-
ticalandsystematicaluncertaintiescouldinfluencetheSSFR–M
∗ 6.4 Outlook
dependencies at different redshifts; at least, it is not obvious that
theSSFR–DSphenomenoncouldbeeliminated. If the SSFR–DSphenomenon is definitively confirmed by obser-
It is healthy to stress that the approach we follow for com- vations, then, as we have argued, it poses a serious difficulty for
paring models and observations is the recommended one in the current discgalaxy evolutionmodels inthecontext of thehierar-
senseofminimizinguncertainties.Conroyetal.(2009b)showthat chical ΛCDMscenario. Thisdifficulty, atone oranother level,is
itisbettertofirstinferthephysicalpropertiesofobservedgalaxies presentalsoinothergalaxyevolutionapproaches,e.g.,theSAMs
(e.g.,stellarmassandSFR)byusingtheSPStechnique,andthen (Somervilleetal.2008;Fontanotetal.2009;LoFaroetal.2009).
comparethemtogalaxyevolutionmodels,ratherthanapplyingthe Thesemodels show, morefromastatisticalpoint of view thanat
SPStechniquetomodelsinordertocomparetheirpredictionswith the galaxy individual evolution level, that the population of rela-
thedirectobservables. tivelysmallgalaxies(M 109 1010.5 M )ismostlyassem-
∗ ⊙
≈ −
OthersignificantsourcesofuncertaintyintheinferredSSFR– bled at higher redshifts becoming older, redder, and with(much)
M relationsatdifferent redshiftsaretheselection effectsdueto lowerSSFRsatlaterepochsthantheobservedgalaxiesinthesame
∗
theincompleteness of thesample, dustabsorption, environmental massrange.
effects, and/or limit detections of the tracers of SF due to flux– Ourmodelsclearlyshowthatlessmassivegalaxiesassemble
limitsorlowemission–linesignal–to–noiseratios;inaddition,ob- their stars early because their dark halos assemble early (earlier
scuredAGNemissioncouldbecontaminatingtheinfraredfluxand onaveragethanmoremassiveones),andthatSN–drivenoutflows,
some of theoptical linesused toestimateSFR(e.g.,Daddi et al. whicharemoreefficientaslessmassiveisthegalaxy,contributeto
2007;Chenetal.2009). quenchlaterSF.Gasre–accretionhelpstoincreasemoderatelythe
The main concern regarding the SSFR–DS phenomenon SSFR,butnotenoughastoagreewithobservationsforlow–mass
amongtheseissuesisthatselectionandenvironmentaleffectscould galaxies.TheSF–feedbackefficienciesusedheretoproducegalac-
biastheobservedtrendthattheSSFRincreasesasM issmaller. ticoutflowsabletorecover therequiredM –M relationaretoo
∗ h ∗
Forexample,thehighSSFRsoflow–massgalaxiescouldbedueto largeaccordingtohydrodynamical simulations(c.f.Oppenheimer
transientstarburstsiftheSFregimeofthesegalaxiesisdominated &Dave´2008;Dubois&Teyssier2008).Besides,itshouldbetaken