Log-normal star formation histories in simulated and observed galaxies

1.9 MB·

by  Benedikt Diemer

TobesubmittedtotheAstrophysicalJournal
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
LOG-NORMALSTARFORMATIONHISTORIESINSIMULATEDANDOBSERVEDGALAXIES
BenediktDiemer1,MartinSparre2,3,LouisE.Abramson4,andPaulTorrey5,6
1InstituteforTheoryandComputation,Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cambridge,MA02138,USA;
[email protected]
2HeidelbergerInstitutfu¨rTheoretischeStudien,Schloss-Wolfsbrunnenweg35,69118Heidelberg,Germany
3DarkCosmologyCentre,NielsBohrInstitute,UniversityofCopenhagen,JulianeMariesVej30,2100Copenhagen,Denmark
4DepartmentofPhysics&Astronomy,UCLA,430PortolaPlaza,LosAngeles,CA90095-1547,USA
5MITKavliInstituteforAstrophysicsandSpaceResearch,77MassachusettsAve.37-241,CambridgeMA02139,USA
6CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
7 TobesubmittedtotheAstrophysicalJournal
1
0 ABSTRACT
2
Gladders et al. have recently suggested that the star formation histories (SFHs) of individual galaxies are

n characterized by a log-normal function in time, implying a slow decline rather than rapid quenching. We
a test their conjecture on theoretical SFHs from the cosmological simulation Illustris and on observationally
J inferred SFHs. While the log-normal form necessarily ignores short-lived features such as starbursts, it fits

9 the overall shape of the majority of SFHs very well. In particular, 85% of the cumulative SFHs are fitted to
within a maximum error of 5% of the total stellar mass formed, and 99% to within 10%. The log-normal

] performssystematicallybetterthanthecommonlyuseddelayed-τmodel,andissupersededonlybyfunctions
A
withmorethanthreefreeparameters. Poorfitsaremostlyfoundingalaxiesthatwererapidlyquenchedafter
G becoming satellites. We explore the log-normal parameter space of normalization, peak time, and full width
at half maximum, and find that the simulated and observed samples occupy similar regions, though Illustris
.
h predictswider,later-formingSFHsonaverage. Theensembleoflog-normalfitscorrectlyreproducescomplex
p metricssuchastheevolutionofIllustrisgalaxiesacrossthestarformationmainsequence,butoverpredictstheir
- quenchingtimescales. SFHsinIllustrisareadiversepopulationnotdeterminedbyanyonephysicalproperty
o
r of galaxies, but follow a tight relation where width ∝ (peaktime)3/2. We show that such a relation can be
t explained qualitatively (though not quantitatively) by a close connection between the growth of dark matter
s
a halosandtheirgalaxies.
[
Keywords:cosmology: theory-methods: numerical-galaxies: starformation

1
v
8 1. INTRODUCTION kins & Beacom 2006; Madau & Dickinson 2014). At any
0 Oneofthemostfundamentalaspectsofgalaxyformationis redshiftalongthisglobaltrajectory,star-forminggalaxiesex-
3 the star formation history (SFH), both in individual galaxies hibit a correlation between their stellar mass and SFR, lead-
2 andintheuniverseoverall. Unfortunately, thetime-resolved ing to the concepts of a “star formation main sequence” and
0 SFHsofindividualgalaxiesaredifficulttomeasureobserva- aquiescentpopulation(Brinchmannetal.2004;Noeskeetal.
. 2007; Elbaz et al. 2007; Karim et al. 2011; Whitaker et al.
1 tionally in all but the most local galaxies where we have ac-
2012; Speagle et al. 2014). Massive galaxies tend to form
0 cesstoresolvedstellarpopulations(Weiszetal.2011,2014;
theirstarsearlier,atrendknownasdownsizing(Cowieetal.
7 Skillmanetal.2014;Williamsetal.2011;Lewisetal.2015).
1996;Heavensetal.2004;Treuetal.2005;Bundyetal.2006;
1 In more remote systems, we need to rely on stellar archae-
Neistein et al. 2006; Kriek et al. 2007; Conroy & Wechsler
: ology, i.e. measurements of stellar ages based on photomet-
v 2009). Finally, the SFR has been connected to a number of
ric or spectroscopic observations and stellar population syn-
i physicalpropertiesofgalaxiessuchasmorphology(Postman
X thesis models (Tinsley 1968; Gallagher et al. 1984; Sandage
1986;Kauffmannetal.2003;Thomasetal.2005;Kennicutt & Geller 1984; Wuyts et al. 2011) and environment, which
r turnsouttobecloselyrelatedtowhetheragalaxyisasatel-
a &Evans2012;McDermidetal.2015;Lejaetal.2016). The
liteorcentral(Oemler1974;Dressler1980;Pengetal.2010,
numberofindependenttimebinsinsuchSFHmeasurements
2012).
isusuallysmall,thoughbettertimeresolutioncanbeachieved
One might hope that these global observations would
withmoreflexibleparameterizationsoftheSFH(Tojeiroetal.
strongly constrain the SFHs of individual galaxies, but this
2007, 2009; Pacifici et al. 2013, 2016a,b). Instead of focus-
connection is not easily established. For example, average
ing on individual galaxies, one can try to measure the star
formation rate (SFR) of galaxy populations at different red- SFHs can be inferred by integrating the main-sequence SFR
over time (e.g., Leitner 2012), but this approach leads to in-
shifts and connect them to their progenitors statistically, but
consistencies (Leja et al. 2015). Instead, the most success-
such inferences are complicated by scatter and merging in a
ΛCDM universe (Behroozi et al. 2013a; Torrey et al. 2015, ful theoretical models link the growth of stellar mass to the
growthofthedarkmatterhalosthatgalaxiesinhabit, forex-
2016;Wellons&Torrey2016).
ample via subhalo abundance matching (e.g., Kravtsov et al.
In a global sense, however, many fundamental aspects of
2004;Conroyetal.2006;Behroozietal.2013b),halooccupa-
star formation in the universe are now well established. The
tiondistributions(e.g.,Peacock&Smith2000;Seljak2000;
star formation rate density (SFRD) peaks around z ≈ 2 and
Hearin et al. 2016), semi-analytic models (Kauffmann et al.
declinesthereafter(Lillyetal.1996;Madauetal.1998;Hop-
2 Diemeretal.
1993; Somerville et al. 2001; Guo et al. 2011), or other as-
sumptions (Bouche´ et al. 2010; Dave´ et al. 2012; Lilly et al. Earlypeak, smallwidth Latepeak, smallwidth
2013;Tacchellaetal.2013;Mitraetal.2017). Oneimportant
conclusion from these models is that there has to be signifi-
cantscatterbetweenhaloandgalaxymasses(andthusgrowth
histories)inordertoexplainobservations(Moreetal.2009; e
t
Behroozi et al. 2013b; Reddick et al. 2013; Gu et al. 2016). a
r
Asaresult,evenmodelsthatagreeonglobalconstraintscan n
lead to orthogonal interpretations of the evolution of indi- o
i
t
vidual galaxies. A good example for such disagreement are a
m
the“rapidquenching1”frameworkwheregalaxiesfollowthe
r Earlypeak, largewidth Latepeak, largewidth
mainsequenceuntiltheysharplyfallbelowthemainsequence o
f
(Pengetal.2012;Wetzeletal.2013;Tinkeretal.2016;Tac- r
a
chellaetal.2016b)andthe“stochastic”frameworkwherecor- t
S
relatedscatterandthecentrallimittheoremleadtothemain
sequence(Kelson2014;Kelsonetal.2016).
Recently, another rather different picture has been pro-
posed. Inspired by the fact that the global SFRD is well fit
byalog-normalintime,Gladdersetal.(2013,hereafterG13)
suggestedthatthisformmightalsodescribetheSFHsofindi- Time since the Big Bang
vidualgalaxies(G13;Dressleretal.2013,2016;Oemleretal.
2013a;Abramsonetal.2015,2016). Thelog-normalSFRis Figure1. Illustrationofthelog-normalSFH.Theshapeisdeterminedby
givenbytheexpression thepeaktimetpeakandwidthσSFR(definedasfullwidthathalfmaximum).
Thekeyfeatureofthelog-normalisthat,inlineartime,itrisesquicklyand
(cid:32) (cid:33)
A (lnt−T )2 declinesslowly,withthetwotimescalesinextricablylinked.
SFR(t)= √ exp − 0 (1)
2πτ2×t 2τ2
talassumptionofG13,namelywhetherthelog-normalfunc-
whereA,T ,andτarefreeparameters(throughoutthepaper,
0 tional form is a good fit for SFHs in the Illustris simulation
t refers to the time since the Big Bang, not lookback time).
and for the inferred SFHs of Pacifici et al. (2016b, hereafter
G13 emphasize that the key assumption need not be the ex-
P16). We find that log-normals fit the majority of Illustris
actfunctionalformofthelog-normal,butratheritssteeprise
galaxies very well, particularly in the mass range studied by
andslowdeclineinlineartime,suggestingaphysicalpicture
G13. We investigate the log-normal parameter space of nor-
different from main sequence star formation interrupted by
malization, peak time, and width as a common language for
suddenquenching. Whilethereissomeevidencethatthema-
simulatedandobservedSFHs. Ourgoalisnottotestthefit-
jorityofgalaxiesceasetheirstarformationgradually(Noeske
tingprocedureofG13indetail,ortocomparethegalaxypop-
et al. 2007; Schawinski et al. 2014; Peng et al. 2015), the
ulation in Illustris to that of G13. Instead, we study which
log-normal SFR is no more than an assumption. The appeal
physicalpropertiesofIllustrisgalaxiestranslateintoparticu-
of this assumption is that it allows for wide-ranging predic-
larvaluesofthelog-normalparameters. Wealsocomparethe
tions if the log-normal parameters for a sample of galaxies
log-normaltootherfittingfunctionsanddiscusstheimplica-
can be inferred. This procedure was implemented by G13
tionsofthelog-normalframeworkintermsofquenchingand
whofoundsurprisinglygoodagreementofthepredictedstel-
theglobalstarformationpropertiesoftheuniverse.
larmassfunctionsandtheirevolution,thestarformationmain
Thepaperisorganizedasfollows.InSection2,wedescribe
sequence, downsizing, and many more complicated metrics
thesimulatedandobserveddataouranalysisisbasedon. In
(G13; Abramson et al. 2016). These successes cannot triv-
Section 3, we investigate the quality of the log-normal fits
iallybereproducedwithsymmetricformsoftheSFHsuchas
and the resulting parameter space. In Section 4, we discuss
aGaussianinlineartime(G13).
theimplicationsofourfindingsfortheglobalstarformation
Given the scarcity of reliable, time-resolved SFH obser-
properties of the universe and the quenching of star forma-
vations, cosmological simulations of galaxy formation can
tion. WesummarizeourresultsinSection5. InAppendices
help to differentiate between the different physical pictures.
AandBwegivemathematicalexpressionsofvariousfitting
These simulations have recently reached a reasonable agree-
functions and discuss additional details regarding fits and fit
ment with a number of observables, providing some level of
results.
confidence in their predictions for individual galaxies (Vo-
Throughout the paper, we extensively use the language of
gelsbergeretal.2014b;Torreyetal.2014;Schayeetal.2015;
log-normalfunctions(Equation1). Thisfunctionalformim-
Dave´ et al. 2016). More specifically, Sparre et al. (2015b)
posestheconstraintthattheriseanddeclineofstarformation
showed that the galaxy population of the Illustris simula-
aresymmetricinlogarithmictime. Hence,theparametersT
tionbroadlymatchestheobservedmainsequence,andinves- 0
andτareinunitsoflogarithmictime,andcanbeinterpreted
tigated individual SFHs using principal component analysis
asthetimewhenhalfthestarshaveformedandtheduration
(seealsotherelatedanalysesofSimhaetal.2014andCohn
ofthegalaxy’sstarformation. However,thelog-normaldoes
&vandeVoort2015). not peak at t = exp(T ), and we thus substitute the more in-
In this paper, we systematically investigate the fundamen- 0
tuitive parameter space of the SFR’s peak time, t , and its
peak
full width at half maximum, σ . Figure 1 illustrates how
1 Theterm“quenching”issomewhatambiguous. Inthispaper, weuse SFR
these parameters determine the shape of the log-normal. Fi-
it to mean the cessation of star formation, without any presumption as to
whetherthedecreasehappensquicklyorslowly,andwhetherithappensdue nally,thenormalization Acorrespondstothetotalintegrated
toadiminishinggassupplyorotherphysicalprocesses. star formation, a quantity we re-cast as the total stellar mass
StarFormationHistories 3
formed, tohavespecificstarformationrates
Mfinal = A×109× fret (2) 0.2
sSFR(z )≤ (3)
where fret = 0.6 is the retention factor due to stellar evolu- obs 109×tobs
tion(seeSection2.3). WehavetakenintoaccountthatSFRs
wherethesSFRisinunitsofyr−1 andt isthecosmictime
are measured in M /yr whereas times are in Gyr. Detailed obs
(cid:12) at observation in Gyr. For the comparisons with the Illustris
expressions for a number of useful properties of log-normal and G13 samples, we set z = 0. We consider the median
SFHsaregiveninAppendixA.1. obs
SFHs in 6 bins in redshift and 6 bins in mass given by P16
(seetheirFigure5).
2. SIMULATION,DATA,ANDMETHODS
In this section we introduce the observed and simulated 2.3. TheIllustrisSimulation
datasets our analysis is based on, as well as our method for TheIllustrissimulation(Vogelsbergeretal.2014b)follows
extractingandfittingSFHsfromsimulations. a comoving cosmological volume of 106.5 cubic Mpc. The
cosmological parameters were set according to the WMAP9
2.1. TheGladdersetal. GalaxySample cosmologyofHinshawetal.(2013),andthesamecosmology
was adapted for all calculations in this paper. The simula-
Ourprimaryobserveddatasetconsistsofthestellarmasses,
tion was run using the moving-mesh code Arepo (Springel
SFRs,andbest-fitlog-normalparametersofG13. Theunder-
2010) which includes physical models for gas cooling, star
lyinggalaxyobservationsweretakenfromanumberofgalaxy
formation, metal enrichment, black hole growth, as well as
surveys. Inparticular,thestellarmassesandSFRsforgalax-
ieswithM >4×1010M weredrawnfromtheSloanDigital feedback from stellar winds, supernovae, and AGN (Vogels-
∗ (cid:12)
berger et al. 2013, 2014a,b; Genel et al. 2014; Torrey et al.
SkySurvey(seeOemleretal.2013b,andreferencestherein),
2014; Sijacki et al. 2015). For the purposes of this paper,
whereasthePG2MCsurvey(Calvietal.2011)wasusedfor
galaxieswith M < 4×1010 M . Duetothesmallervolume thestarformationprescriptionisofparticularinterest. InIl-
∗ (cid:12)
lustris,starsareformedaccordingtoasub-gridmodelwhich
of the latter survey, the samples were re-normalized to cre-
stochasticallyplaces“star”particlesingascellsthatexceeda
ateequalweights. Theresultingsamplecontains2094galax-
thresholdnumberdensityof0.13cm−3(Springel&Hernquist
ies with a mean redshift of 0.0678, and is complete above
M =1010 M . WereferthereadertoG13andthereferences 2003;Vogelsbergeretal.2013). Eachstarparticlerepresents
∗ (cid:12)
apopulationofstarsbornwithaChabrierinitialmassfunction
thereinfordetailsonhowthestellarmassesandstarformation
(Chabrier2003). Thetimescaleofstarformationdependson
rateswerecomputed.
the inverse root of the density, leading to a relation between
The fundamental assertion of G13 is that the log-normal,
surface density and star formation that roughly follows the
afunctionalformthatdescribestheglobalSFRDoftheuni-
observedKennicutt-Schmidtrelation(e.g.,Kennicutt1998).
verse, might also be a good description of the SFHs of indi-
Astheyage,simulationstarparticlesreturnmasstothein-
vidualgalaxies. However,foreachobservedgalaxy,onlytwo
terstellar medium via stellar winds and supernovae (Vogels-
variables are known (M and SFR) whereas the log-normal
∗
berger et al. 2013). Star particles therefore have masses that
hasthreefreeparameters(Equation1). Thus,G13usedaddi-
evolvewithtimetovalueslowerthantheirinitialbirthmass.
tional global constraints, namely the SFRD (accounting for
the contribution from galaxies with M < 1010 M ) and Foranygivengalaxy,thisdecreasecausestheintegratedSFR
∗ (cid:12)
tobehigherthanthesumofstellarparticlemasses. Through-
SFR distributions back to z ≈ 1. Some of these observa-
outthepaper,wedistinguishthesetwostellarmassdefinitions
tions were drawn from the IMACS Cluster Building Survey
carefully. The mass decrease in a stellar population can be
(ICBS,Oemleretal.2013b;Dressleretal.2013;Oemleretal.
expressedasaretentionfactorwhichwefindtovarybetween
2013a). Inacombinedfitovertheglobalconstraintsandthe
0.54 for the oldest and 0.62 for the youngest stellar popula-
stellarmassesandSFRsofeachgalaxy, theyfoundthebest-
tions in Illustris, with a mean of 0.57. We note that this is
fit T and τ for the 2094 galaxies (see their Figure 9). The
0 veryclosetothevalueof0.6assumedbyG13,meaningthat
normalization of the log-normals was set such that the inte-
wecandirectlycomparethecumulativeSFRsoftheobserved
grated SFR matches the observed M , meaning that one has
∗
andsimulatedgalaxysampleswithoutincurringalargeerror.
to assume a retention factor, f , i.e. the ratio between the
ret Dark matter halos, and the galaxies that inhabit them, are
stellar mass initially formed to the stellar mass that survives
identified using the Subfind algorithm (Davis et al. 1985;
untilthegalaxyisobserved. G13assumedaretentionfactor
Springel et al. 2001; Dolag et al. 2009). The various prop-
of0.6whichweadoptthroughoutthispaper.
erties of galaxies are computed based on the bound matter
within twice the the stellar half-mass radius of a subhalo as
2.2. ThePacificietal. InferredSFHs identified by Subfind (Vogelsberger et al. 2014b). We use a
P16 derived SFHs by fitting the multi-band photometry numberofgalaxypropertieslistedintheSubfindcatalogs,in-
of 845 quiescent galaxies with spectral energy distributions cluding various mass and size definitions, metallicity, black
(SEDs) computed from a large library of theoretical SFHs. hole mass, and properties of the parent group if applicable.
TheSEDlibraryconsistsof500,000simulatedgalaxySEDs, Furthermore, we use the stellar assembly data provided by
created by applying the semi-analytic model of De Lucia & Rodriguez-Gomezetal.(2016b),forexamplethefractionof
Blaizot(2007)tothemergertreesfromtheMillenniumSim- agalaxy’sstellarmassthatwasformedinsitu,exsitu,andin
ulation(Springeletal.2005,seealsoPacificietal.2012),and varioustypesofmergers.
predictingtheemissionusingthestellarpopulationsynthesis We consider all galaxies with M ≥ 109 M , a sample of
∗ (cid:12)
model of Bruzual & Charlot (2003). This library is flexible 29203 galaxies (19375 of which are centrals at z = 0, and
enough not to introduce a particular shape of the SFH a pri- 9828 satellites). For the comparisons with the observational
ori. P16 restricted themselves to quiescent galaxies, defined galaxy sample described in Section 2.1, we use the sub-set
4 Diemeretal.
ofhigh-massgalaxieswith M ≥ 1010 M ,asampleof6947 correlated which a standard χ2 statistic would not account
∗ (cid:12)
galaxies(4643centralsand2304satellites). Inbothsamples, for. Instead, we compute the Kolmogorov-Smirnov statistic
thesatellitefractionisalmostexactly1/3. Giventhattheini- D, equivalent to the maximum difference of the cumulative
tialmassofstarparticlesinIllustrisisabout1.6×106M ,all functionsatanytime. Wenormalizethisnumberbythetotal
(cid:12)
galaxiesareresolvedby600ormoreparticles,andthehigh- stellarmasscreatedinagalaxysuchthat
masssampleby6000ormoreparticles. Sparreetal.(2015b)
max(|cSFR (t)−cSFR (t)|)
testedtheresolutiondependenceofstarformationinIllustris D≡ sim fit (5)
cSFR(t )
usingalower-resolutionrun(Vogelsbergeretal.2014a),and 0
foundgoodconvergenceinkeypropertiessuchasthestarfor- wheret istheageoftheuniversetoday.WeuseDtoquantify
0
mationmainsequence,theSFHs,andtheirvariance. thequalityofourfitsinSection3.1.
Finally, wenoteaninconvenientfeatureofthelog-normal
2.4. StarFormationHistoriesandLog-normalFitting
functional form: when only data from times well before the
We extract the SFHs of Illustris galaxies using the same peak is available, the peak time and width are not well con-
procedureasSparreetal.(2015b). Inparticular,weconsider strained, resulting in a strong degeneracy between T and τ.
0
all stellar particles in the corresponding friends-of-friends ThisissuemanifestsitselfinasmallfractionofIllustrisgalax-
subgroupandbinthembytheirbirthtimes,weightedbytheir ies that experience rising star formation rates z = 0 and are
initialmass,in100equaltimebinsbetweenthefirstandlast sometimesassignedunphysicallyhighvaluesofT andτ.We
0
snapshotsofthesimulation(att=54Myrandt=13.75Gyr, shall later return to the question whether such galaxies exist
respectively). Definedinthisway, theSFHincludesthestar intherealuniverse,butinafittingsense,theycauseinconve-
formation in all progenitors that have merged with a galaxy nientartifactssuchasSFHsthatpeakhundredsofGyrinthe
during its lifetime. This definition matches observations of future.
stellar mass and SFR at z = 0 which do not distinguish be- Thus, we impose a prior on t such that the fit is unaf-
peak
tweenstarsformedinsitu(inthegalaxyinquestion)orexsitu fected at t < t (the time at z = 0). Later peak times
peak 0
(inasmallergalaxythatmerged). However,itdoesnotmatch are penalized by multiplying χ2 by the square of the loga-
the observational definitions at higher redshift if a galaxy is rithmic difference of t and t . This term effectively cuts
peak 0
toaccreteasignificantamountofstarsatlatertimes, though offthedistributionat50Gyr,correspondingtoascalefactor
the effect of such mergers is significant only for high-mass a =10intheIllustriscosmology. Asimilar,lessinfor-
peak,max
galaxies(Conroy&Wechsler2009;Rodriguez-Gomezetal.
mative prior penalizes extreme widths of σ > 102.5. The
2016b). In some figures, we use the star formation rate at SFR
vastmajority,over90%ofthegalaxiesinoursample,areun-
z = 0 which was computed from the SFR in all gas cells in affectedbythesepriors. Galaxiesinthehigh-t tailexperi-
a simulated galaxy. We have verified that this instantaneous encedrasticallydifferentbest-fitparameters,npaemakelychanges
SFRmatchestheSFHaveragedover20Myrverywell.
of 50% or greater for 4% of the galaxies. However, due to
WenowinvestigatewhetherthetabulatedSFHsarewellfit
the strong degeneracy of T and τ, the actual fitted SFHs of
bylog-normalfunctions. Wenotethattheterm“fit”takeson 0
thesegalaxiesarealmostidenticaltothosefitwithoutaprior.
a slightly unusual meaning in this context because the log- Thus,thefitqualityDisbarelyaffected:only3%oftheSFHs
normalformisnotintendedtodescribethedetailsofaSFH.
experienceachangeinfitqualityof20%orgreater. Theprior
First, theSFHsincosmologicalsimulationssufferfromshot
is even less important for the high-mass sample. For the in-
noise due to the limited number of stellar particles formed
ferencesandcomparisonsshowninthispaper,thetailsinthe
in a given time bin. This noise grows as the bin size de-
T andτdistributionsarenotimportant.
creases, making it difficult to define a meaningful goodness- 0
of-fit statistic such as χ2. Second, the spatial resolution (or 2.5. HaloMassAccretionHistories
force softening length) in Illustris is such that giant molecu-
BesidestheSFH,wealsoconsiderthemassaccretionhis-
larcloudsarenotresolved,whichiswhythesub-gridmodel
tory(MAH)ofagalaxy’sdarkmatterhalowhichweextract
described in Section 2.3 is employed. The short-term struc-
fromtheSublinkmergertreesprovidedbyRodriguez-Gomez
ture of a SFH is thus significantly influenced by the charac-
etal.(2015). Foreachhaloatz = 0,wefollowitsmostmas-
teristicsofthesub-gridISMmodelandresolution, asshown
sive (or main branch) progenitor back in time and record its
byhigh-resolutionzoom-insimulations(e.g.,Haywardetal.
mass. In order to reduce the complex shape of the MAHs
2011; Torrey et al. 2012; Sparre et al. 2015a). We note that
to characteristics such as a halo’s formation time, we fit the
othercosmologicalhydrodynamicalsimulationsusesimilarly
MAHswiththeexponentialformofWechsleretal.(2002),
motivatedsub-gridstarformationmodels(Schayeetal.2015;
Dave´etal.2016),meaningtheirSFHsmaybesubjecttocom- M(z)= M e−αz (6)
0
parableuncertaintiesonshorttimescales.
Forthesereasons, wedisregardtheshort-termbehaviorof where M0 is the halo mass at z = 0. This function was de-
thesimulatedSFHsandinsteadfocusontheoverall,cumula- signedtofittheMAHsofisolatedhalosandisnotequipped
tiveevolutionofagalaxy’sstellarmass.Wefitthecumulative tocapturetheeffectsofsubhaloaccretionontoalargerhost.
SFR, Thus, we restrict the fits to epochs when the halos were not
(cid:90) t subhaloswhichimprovesthecorrelationbetweenhaloforma-
cSFR(t)≡ SFR(t(cid:48))dt(cid:48) (4) tion redshift and the SFH of their respective galaxies (Sec-
0 tion 3.3). Following Wechsler et al. (2002), we define a
with the integral of the log-normal SFR which turns out to formation redshift where the halo has formed half its mass,
begivenbyacomplementaryerrorfunction(AppendixA.1). z =ln(2)/α. Wehavealsoexperimentedwiththemore
wechsler
We use the Levenberg-Marquardt algorithm to minimize the flexible, three-parameter function of Tasitsiomi et al. (2004)
squareresidualsbetweenthisfunctionalformandthecumula- and McBride et al. (2009), but find that the resulting forma-
tiveSFRin100timebins. Thevaluesinthesebinsarehighly tionredshiftcorrelateslessstronglywiththeSFHs.
StarFormationHistories 5
2000 7
a)Massive, early-forming b)Slowlyquenchedsatellite 3.5 c)Star-formingtoday d)RisingSFRtoday
6
3.0
1.5
1500 5
yr) yr) yr) 2.5 yr)
/M(cid:12)1000 /M(cid:12) 1.0 /M(cid:12) 2.0 /M(cid:12) 4
R( R( R( 1.5 R( 3
F F F F
S 500 S 0.5 S 1.0 S 2
0.5 1
0 0.0 0.0 0
1012 109 1010 1010
× × × ×
2.0
6 3.5
M)(cid:12) 5 M)(cid:12) 1.5 M)(cid:12) 1.5 M)(cid:12) 3.0
R( 4 R( R( R( 2.5
F F F F
S S S 1.0 S 2.0
ve 3 ve 1.0 ve ve
ati ati ati ati 1.5
Cumul 2 Mtpefiankal==21.92.6Cumul 0.5 Mtpefiankal==29.8.1 Cumul 0.5 Mtpefiankal==81.30.7Cumul 1.0 Mtpefiankal==5172.5.4
1 σSFH=3.4 σSFH=3.1 σSFH=31.4 0.5 σSFH=259.9
D=0.019 D=0.073 D=0.020 D=0.009
0 0.0 0.0 0.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
t(Gyr) t(Gyr) t(Gyr) t(Gyr)
2.0
20
e)Earlystarburst f)Latestarburst 30 g)Majormerger h)Rapidlyquenchedsatellite
500
400 15 25 1.5
/yr) /yr) /yr) 20 /yr)
(M(cid:12)300 (M(cid:12) 10 (M(cid:12) 15 (M(cid:12) 1.0
R R R R
F 200 F F F
S S S 10 S
5 0.5
100 5
0 0 0 0.0
1011 109 1011 109
× × × 2.0 ×
4 1.0
M)(cid:12) 1.5 M)(cid:12) M)(cid:12) M)(cid:12) 1.5
( ( ( 0.8 (
R R 3 R R
F F F F
veS 1.0 veS veS 0.6 veS 1.0
Cumulati 0.5 MtσpSefiFanHkal===211..011.0Cumulati 12 MtσpSefiFanHkal===809..13.4 Cumulati 00..24 MtσpSefiFanHkal===611.121.5.0Cumulati 0.5 MtσpSefiFanHkal===439..42.0
D=0.168 D=0.019 D=0.039 D=0.152
0.0 0 0.0 0.0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
t(Gyr) t(Gyr) t(Gyr) t(Gyr)
Figure2. Representativelog-normalfitstoSFHsfromIllustris.ThedarkbluelinesshowtheSFRandcumulativeSFHofasimulatedgalaxy(topandbottomof
eachsetofpanels),andthelightbluelinethebest-fitlog-normal.Thegrayshadedareasindicatetimeswhenthegalaxywasasatellite.Thebest-fitparameters
arelistedinthebottomrightcornersoftheSFHpanels,whereMfinalisinlog10unitsandtpeakandσSFRareinGyr.Fromtoplefttorightbottom:a)amassive,
early-forminggalaxythatstoppedformingstarsafewGyrago;b)asatellitethatexperiencedaburstofstarformationafterinfallandslowlyquenchedthereafter;
c)averybroadSFH;d)aSFHthatisstillrisingtoday,resultinginapeaktimeinthefarfuture;e)theworstfittedSFHinthehigh-masssample,asystemthat
formsalmostallitsstarsintwoearlystarbursts;f)a“latebloomer”thatformedinoneintensestarburst;g)agalaxythatexperiencedastarburstduetoamajor
merger;h)asatellitethatexperiencedastarburstuponbeingaccreted,andthenquenchedabruptly. Exceptforthecaseofrapidquenching,alargevarietyof
cumulativeSFHsiswellfitbythelog-normalform,eveniftheSFRisnoisyorbursty.
Finally, we attempt to draw a more direct connection to description, and if the MAH is well described by a pure ex-
the SFH fits by fitting the MAHs with a log-normal in time. ponential, thethirdfreeparameterisunconstrained. Forthis
Wefindthatthelog-normalisnotparticularlyadaptedtothe reason, aboutaquarterofthefitsfailtoconverge, evenwith
shape of MAHs. The exponential rise at early times is gen- thepriorsdescribedinSection2.4. Wereturntotherelation
erallywellfit, buttheadditionalfreeparametercomparedto betweenSFHandMAHlog-normalfitsinSection4.5.
theWechsleretal.(2002)functionrarelyimprovesthefitsig-
3. RESULTS
nificantly. IftheMAHflattensorevendecreases,thepower-
law term in the Tasitsiomi et al. (2004) formula is a better In this section, we consider the quality of log-normal fits
to Illustris and observationally inferred SFHs, and compare
6 Diemeretal.
All 100 Gladders13 0.8
4000 Centrals Illustris
Satellites
0.6
3000 /dx 10−1
N 0.4
d
N 10−2
2000 0.2
10−3 0.0
1000 10.0 10.5 11.0 11.5 12.0 12.5 4 3 2 1 0 1 2
− − − −
log M (M ) log SFR(M /yr)
10 10
∗ (cid:12) (cid:12)
0 Figure4. ComparisonofthegalaxypopulationsintheG13(orange)and
0.00 0.02 0.04 0.06 0.08 0.10 0.12 Illustris(blue)samples(withM∗ >1010M(cid:12)). Whilethestellarmassfunc-
tions(leftpanel)aresimilar,Illustriscontainsfewerquiescentandmorestar-
D forminggalaxiesthantheG13sample(rightpanel).
Figure3. Distributionoffitquality,definedasthemaximumdeviationfrom
thecumulativeSFH,D. Theplotcutsoutasmallfractionofoutlierswith SFHs of P16 (see Appendix B.1 and Figure 15). We find fit
D>0.13).ThevastmajorityofSFHsisfitwellbythelog-normalform.The qualitiesverysimilartothoseoftheIllustrisSFHs,withDbe-
tailatD>0.08isalmostentirelyduetolow-masssatellitesthatexperienced
tween1.4and5%,andamedianof2.8%(comparedto3.1%
rapidquenching.
fortheIllustrissample).
the best-fit parameters from the observational and simulated Insummary,thelog-normalfunctionalformprovidesanex-
galaxysamples. cellentdescriptionofthemajorityofthesimulatedandobser-
vationallyinferredcumulativeSFHsweconsidered. Thekey
3.1. TheQualityofLog-normalFits predictionofthelog-normalisasteepriseandslowdeclinein
lineartime,andthatthetwotimescalesarecoupled.Thisfun-
Figure 2 shows an array of example log-normal fits to a
damental characteristic is shared with the delayed-τ model,
rangeoftypesofSFH,namelythoseofearlyandlate-forming
meaningthatmanyofthefollowingresultscouldprobablybe
galaxies, centrals and satellites, slow and fast decliners, and
obtainedbasedondelayed-τfitsaswell. Weconcludethata
galaxies that underwent a major merger. Despite the ex-
tremelydifferentshapesofthecorrespondingSFHs, thelog- steepriseandslowdeclinearekeyfeaturesoftheSFHscon-
sideredhere,exceptforgalaxieswhosestarformationissud-
normal captures the main features in the cumulative distri-
denlytruncatedbyexternalprocessessuchassatellitequench-
bution while glossing over the noise and spikes in the SFR.
ing. Models where the rise and decline timescales are inde-
Figure2highlightsalargerangeoffitqualities,includingthe
worstfittoanySFHinthehigh-masssample(M >1010M ). pendent (such as the double power law) can achieve a better
∗ (cid:12)
In the lower-mass sample (M < 1010 M ), one particularly fitinsomecases,thoughatthecostofpredictivepower.
∗ (cid:12)
difficultcaseforthelog-normalfunctionisasharpcut-offin
3.2. TheLog-normalParameterSpace
star formation, for example the rapid quenching of satellites
after infall into their host (see Figure 2 for an example, or We now consider the distribution of best-fit parameters in
Mistanietal.2016forananalysisofthiseffectinIllustris). theG13,P16,andIllustrissamples. Beforeattemptingsucha
Figure 3 shows the distribution of D, the maximum frac- comparison,weneedtocheckthatthegalaxypopulationsin
tionalresidualbetweenthefitandthecumulativeSFR.While IllustrisandintheG13sampleareatleastroughlycompara-
Drangesfromalmostzeroto34%,thevastmajorityofgalax- ble. Figure 4 shows the abundance of galaxies as a function
ies,namely85%(92%ofcentralsand73%ofsatellites),are ofstellarmassandSFRinIllustris(blue)andtheG13sample
fit to 5% or better at all times. Only 1% of the fits are off (orange). The distributions of M are relatively similar (left
∗
by 10% or more at any time (0.4% of centrals and 2.7% of panel), but the SFRs (right panel) differ significantly. The
satellites).Thestatisticsareevenbetterinthehigh-masssam- G13samplecontainsamuchlargerfractionofentirelyquies-
ple,whereonly0.3%ofgalaxies(0.1%ofcentralsand0.8% cent,non-starforminggalaxies,whereasIllustriscontainsan
of satellites) exhibit a D of 10% or greater. Figure 3 also excessofgalaxieswithSFRsbetween1and10M /yr. This
(cid:12)
demonstratesthatthetailtowardpoorfitqualitiesconsistsal- disagreementisnotsurprising,astheG13sampleismoreor
mostentirelyofsatellites. Visualinspectionoftheseobjects lessrepresentativeofthequiescentandstar-formingfraction
confirmsthatmostofthemarelow-masssatellitesthatexperi- inthez = 0universe(Abramsonetal.2015),whereasanex-
encedrapidquenchingafterbeingaccretedintoalargerhalo. cess of blue, star-forming galaxies in Illustris at z = 0 has
Thegenerallyexcellentqualityofthelog-normalfitsisby been noted before (Vogelsberger et al. 2014a). Furthermore,
no means guaranteed, as we are fitting 100 time bins with a Sparre et al. (2015b) detected a paucity of starburst galax-
function of only three free parameters. For comparison, we ies in Illustris compared to observations. Conversely, we re-
alsoconsiderthedelayed-τmodel(e.g.,Gavazzietal.2002) mind the reader that the log-normal parameters of G13 are
andthedoublepowerlaw(e.g.,Behroozietal.2013b)inAp- inferredfromasemi-globalfitratherthanmeasuredfromre-
pendixA.Weshowthat,onaverage,thelog-normalfunction solvedSFHs. Withthesecaveatsinmind,weemphasizethat
performs slightly better than the delayed-τ model which has ourgoalisnottotestthegalaxypopulationofIllustrisorthe
the same number of free parameters. In contrast, the dou- fittingprocedureofG13,butrathertocomparetheregionsof
ble power law has four free parameters (normalization, peak log-normalparameterspaceoccupiedbythetwosamples,and
time,risingandfallingslope),leadingtoasignificantlymore to investigate the physical properties of galaxies that occupy
accuratefit,particularlyforrapidlyquenchedsatellites. those regions. For those purposes, the populations shown in
Wealsoapplyourfittingproceduretotheinferredmedian Figure4aresufficientlysimilar.
StarFormationHistories 7
Illustris
Gladders13
x x x
d d d
/ / /
N N N
d d d
13
Illustris
(peaktimeinGyr)peak 101 σ(widthinGyr)SFH 110012 Mlogfinal10 1112 011...505 10log(+1)N
t
100 10
100 0.0
13
Gladders13
(peaktimeinGyr) 101 (widthinGyr)FH 110012 z=4 z=2 z=1 z=0 Mlogfinal10 1112 011...505 10log(+1)N
peak σS
t
100 10
100 0.0
10 11 12 13 100 101 100 101 102
log10Mfinal tpeak(peaktimeinGyr) σSFH(widthinGyr)
Figure5. Log-normalparametersoftheIllustris(blue,M∗>1010M(cid:12))andG13(orange)samples.Thetoprowshowsthedistributionsofthethreefitparameters,
namelythetotalstellarmassformed,peaktime,andfullwidthathalfmaximum.Thehistogramsarenormalizedtothesamesamplesize.Thebottomtworows
showthedensityofgalaxiesasafunctionofpairsoftheseparameters. TheIllustrisandG13samplesoccupysimilarareasinthisparameterspace,butIllustris
SFHsexhibitlargerwidthsandamorepronouncedcorrelationbetweentpeakandσSFR.ThetailoftheIllustrispopulationtowardhightpeakandσSFR(i.e.,SFHs
whichwillpeakinthefuture)isinfluencedbyourpriorandshouldnotbetakentooseriously(Section2.4).SeeAppendixB.3foracomparisonintheoriginal
T0-τparameterspaceofG13.
Wecomparethebest-fitparametersoftheIllustrisandG13 tributetoM .Theleftandrightcolumnsshowtheexpected
final
samplesinFigure5. Thetoprowshowshistogramsforeach correlations,butalsoasignatureofthetailsdiscussedabove.
parameter. While the distributions of final stellar masses Because the extremely late-forming SFHs in Illustris predict
roughly agree, we notice that Illustris galaxies peak slightly significant future star formation, they lead to very high final
later(highert )andhavesignificantlylargerwidths(higher masses. Thesegalaxiesmanifestthemselvesastailsinthetop
peak
σ ). These differences are partially due to the excess of righthandcornersoftheM -t (leftcolumn)andM -τ
SFR final peak final
blue, star-forming galaxies in Illustris compared to the G13 (rightcolumn)planes. Ignoringthesetails,wefindthatlarger
sample (Figure 4). Moreover, we notice tails of a few per- galaxiesformearlier(havelowert )anddeclinerelatively
peak
cent of the population toward very high values in all three fast(havelowerσ ),inagreementwiththedownsizingpic-
SFR
parameters in the Illustris sample (Mfinal ∼> 1012, tpeak ∼> t0, ture. Atlowmasses,thescatterinpeaktimeincreases. Inter-
σSFR ∼> 30). The corresponding fits are poorly constrained estingly, these trends are less significant in the G13 sample,
andinfluencedbyourpriorwhichdiscouragest >t (Sec- especiallytherelationbetweenmassandpeaktime.
peak 0
tion2.4). Wenowconsidertherelationbetweenpeaktimeandwidth
ThebottomrowsofFigure5showthecorrelationsbetween (center column of Figure 5). Once again ignoring the Illus-
each pair of parameters. We first investigate the trends with tris population with a peak time beyond z = 0, the G13 and
stellar mass in the Illustris sample. M is a reflection of Illustrissamplesoccupyasimilarregionofparameterspace.
final
the stellar mass of galaxies, modulo their current age: for Inbothsamples,earlier-forminggalaxiesformfaster,butthe
early-forming galaxies, M ≈ M (z = 0), whereas late- correlation is more significant in Illustris. Furthermore, at
final ∗
forminggalaxieshaveyettoformsomeofthestarsthatcon- fixed t Illustris galaxies have broader widths, i.e. form
peak
8 Diemeretal.
z
obs
0.0 0.5 1.0 1.5
2.0
102
Illustris
)
r
y
G 1.5 lo
) n g
nGyr 101 widthi 10(+N
widthi σ(SFH 101 1.0 1)
(
H 0.5
F
S
σ
100 100 log10(σSFH)=1.5log10(tpeak)−0.08 0.0
100 101
Gladders13 tpeak(peaktimeinGyr)
Figure7. CorrelationbetweenpeaktimeandwidthfortheentireIllustris
r) sample. Thedashedlineshowsthebest-fitpowerlaw,thedottedlinesthe
y
G 68%scatterof0.18dex(about50%).
n 101
i galaxiesbelowthecorrespondingline. WenotethattheP16
h
t pointsthemselveslieslightlyabovethelinesateachredshift,
d
wi indicating that the log-normal fits overestimate the sSFR at
(
H theredshiftofobservation(seeAppendixB.1andFigure15).
SF Nevertheless, the agreement between the observational sam-
σ
plesisreassuring,andlendscredibilitytotheinferredparam-
etersofG13.
100
100 101 3.3. WhichGalaxyPropertiesDeterminetheSFH?
Havingestablishedthatoursimulatedandobservedgalaxy
t (peaktimeinGyr)
peak
samples occupy similar areas of the log-normal parameter
Figure6. Comparison of the log-normal parameters of the Pacifici et al. space,wenowaskwhatgalaxyformationphysicscausespar-
(2016b,roundpoints)samplewiththoseofIllustris(top)andG13(bottom). ticular shapes of the SFH: are they pre-determined by initial
TheP16sampleisrestrictedtoquiescentgalaxieswhich,inlog-normalspace,
conditionssuchasthedensityenvironment,influencedbythe
liebelowtheredshift-dependentlines(samecolorscaleastheroundpoints).
Ateachredshift,theSFRoftheP16fitsslightlyexceedsthequiescentcut, history of a galaxy’s halo, or do they arise from the compli-
indicatingthatthelog-normalfitsoverestimatetheactuallyobservedSFR. catedphysicsofgalaxyformation? Forthisinvestigation,we
considertheentireIllustrissamplewithM >109 M .
∗ (cid:12)
starsforalongertimethaninferredbyG13.
We note another potentially revealing difference between 3.3.1. ThePeakTime-WidthRelation
theG13andIllustrissamples: theabsenceofyounggalaxies Figure7reproducesthet -σ relationfortheentireIl-
peak SFR
or“latebloomers”(G13,Oemleretal.2013a;Dressleretal. lustrissample. ThecorrelationswithM areverysimilarto
final
2016) in Illustris. Those are galaxies that form the majority those in the high-mass sample shown in Figure 5. Figure 7
of their stars relatively quickly after z ≈ 1. The correspond- demonstratesthatthet -σ relationiswellapproximated
peak SFR
ing parameter space of t between z = 1 and z = 0 and byapowerlaw,
peak
lσuSsFtrRis∼<(5thoGuygrhisavfieswiblysupcohpgualalatexdieisneGx1is3t,ibnutthdeesleorwteedr-imnaIsls- σSFR =0.83tp3e/a2k, (7)
sample,Section3.3). Thisdifferenceisanothermanifestation with a standarddeviation of 0.18 dex (about 50%). The line
of the relatively wide SFHs in Illustris: if galaxies peak as fitandscatterfortheM >1010M samplearevirtuallyiden-
∗ (cid:12)
late as z = 1, they are most likely still active today (at least tical. According to this relation, the SFHs in Illustris follow
accordingtothelog-normalfits). a scaling relation between their peak time and width, where
As an independent, observational check, we compare the early-forming galaxies also form quickly. We notice a few
G13andIllustrissamplestothelog-normalparametersofthe outliers,forexamplerelativelylate,shortstarburstSFHs(see
P16 median SFHs in Figure 6. The color scale of the P16 Figure 2 for an example), but overall the relation is surpris-
pointsindicatestheirobservationredshift. TheG13andP16 inglytight. P16foundasimilarrelationbetweenSFHwidth
samplesmatchverywellandoccupythesamet -σ re- and the age of the universe, but they referred to the time of
peak SFR
lation. ThelowerscatterintheP16sampleisexpectedsince observationratherthanpeaktime(theirFigure6).
thoseparameterswerederivedfrommedianSFHsratherthan GivenhowwellthepeaktimeofaSFHpredictsitswidth,
individualgalaxies. However,P16consideredonlyquiescent thequestionwhichgalaxypropertiesdeterminetheSFHcan
galaxies as defined in Equation 3. The corresponding limits be divided into two separate questions: which properties
at each redshift are shown with lines in Figure 6, and points change peak time and width along their degeneracy, and
ofagivencolorshouldbecomparedtotheIllustrisandG13 whichproperties(ifany)influencewidthatfixedpeaktime?
StarFormationHistories 9
All
)
ts Centrals
uni Satellites
y
r
a
r
t
bi
r
a
(
x
d
/
N
d
9 10 11 12 13 100 101 100 101 102 0.5 0.0 0.5
−
log10Mfinal tpeak(peaktimeinGyr) σSFH(widthinGyr) log10∆σSFH(deviation)
Figure8. Dependence of the log-normal parameters on whether a galaxy is a central (orange) or a satellite (purple) at z = 0. The entire galaxy sample
(M∗ >109M(cid:12))isshowningray. Thoughcentralsare,onaverage,moremassivethansatellites,satellitesformearlierandfaster,contrarytothegeneraltrend
thatmoremassivegalaxiesformearlier.ThetailstowardveryhightpeakandσSFRarepredominantlycausedbyextremelylate-formingcentrals.Therightpanel
showsthedeviationfromthecorrelationbetweentpeakandσSFR(Equation7),i.e.whetheraSFHisrelativelywideornarrowatfixedpeaktime.
3.3.2. Centralsvs.Satellites imum).Thecorrespondingrelationswithstellarmassarevery
similar. For large, early-forming galaxies, M is roughly
Partoftheanswertothesequestionsmightbefoundinthe final
differences between central and satellite galaxies. Figure 8 equal to M∗ today which manifests itself in a well-defined
stellarmass-halomassrelationatthehigh-massend. Asex-
shows the distributions of their parameters compared to the
pected,galaxiesinmoremassivegalaxiesformearlier.Atlow
overall sample. The distributions do not strongly depend on
haloorstellarmasses,however,thecorrelationdisappears(in
mass, except at very high masses where most galaxies are
agreementwith Dressleretal. 2016). Atfirstsight, this cor-
early-forming.
relationseemstobeinconflictwiththeexpectationthatlarge
Figure8showsthateventhoughsatellitesare,onaverage,
dark matter halos form late in hierarchical structure forma-
slightlylessmassivethancentrals(firstpanel),theyformear-
tion. However, while the halos of massive clusters are still
lier (second panel), in contrast with the downsizing expec-
growingtoday,theircentralgalaxiesaregrowingbymergers
tation that larger galaxies form earlier. Moreover, satellites
rather than star formation, meaning that their stellar popula-
form significantly faster than centrals (third panel), both be-
tions whose ages we investigate here were largely formed in
cause they are an intrinsically older population and because
otherhalos(Rodriguez-Gomezetal.2016b). Thus,largeha-
their star formation may be quenched after infall into their
los form late whereas large central galaxies form early (e.g.,
hosts(e.g.,vandenBoschetal.2008;Hearin&Watson2013;
Neisteinetal.2006).
Wetzeletal.2013).
However, the differences between the central and satellite Thenextvariableweconsideristheenvironmentatz = 3,
i.e. whetheragalaxywasborninanoverdenseorunderdense
populationsdonotappeartoberesponsibleforthescatterin
regionoftheuniverse. Wequantifydensitybythedistanceto
thet -σ relation. TherightpanelofFigure8showsthe
peak SFR the 5th nearest neighbor, but using a smoothed galaxy over-
deviationfromthisrelationwhichismoreorlesssymmetric
density gives similar results. The trend that galaxies born
for both centrals and satellites, though centrals have a slight
in overdense environments form earlier holds for all stellar
tendencytolieabovetherelation(i.e.,exhibitrelativelywide
masses,andisamanifestationoftheearliercollapsetimesof
SFHs at fixed t ) and satellites below. The scatter around
peak halos in overdense regions, known as “assembly bias” (Gao
therelationiscomparableforthetwosub-samples, 0.16dex
et al. 2005; Wechsler et al. 2006; Croton et al. 2007; Dalal
forcentralsand0.2dexforsatellites.
etal.2008;Zentneretal.2014;Miyatakeetal.2016;Tinker
3.3.3. TheGalaxy-HaloConnection etal.2016,seealsotheargumentsinDressler1980,Abram-
sonetal.2016, andKelsonetal.2016). Theenvironmentat
As discussed in Section 1, we expect the MAH of a halo
earlier and later redshifts shows similar relations with t ,
tohavesignificantinfluenceontheevolutionaryhistoryofits peak
but the correlation weakens at late (z < 1) and very early
galaxy. Thus, we compare the log-normal parameters to a
(z>4)redshifts.
number of halo properties in Figure 9. The first impression
Weexpectthattheageofagalaxyshouldbestronglyinflu-
from the gray histograms is that all correlations are subject
encedbytheageofitshalo. Wehavemeasuredhaloageina
tolargescatter,particularlyatlowmass. Thisscatterimplies
number of ways: as the redshift where half (or a quarter) of
thatgalaxySFHsarediverseandnotdeterminedbyanyone
today’smasshasbeenaccreted,byfittingwiththemassaccre-
parameter (see Dressler et al. 2016 for an observational in-
tion history models of Wechsler et al. (2002) and Tasitsiomi
vestigation). Nevertheless, some trends are well-defined in
etal.(2004),andasthestarvationredshiftz suggestedby
anaveragesense. Manyhaloandgalaxypropertiescorrelate starve
Hearin&Watson(2013)whichisdefinedasthemaximumof
with halo and stellar mass, introducing a trivial correlation
z ,theredshiftwherethehalofirstgrewto1012M ,and
withpeaktimeandwidth. Thus, wesplittheoverallsample wechsler (cid:12)
theredshiftwhenitwasaccretedifitisasatellite. Thecorre-
into three stellar mass bins. We have also verified that the
lationswithz andz areshownintherightcolumns
central and satellite samples are similarly distributed for the wechsler starve
ofFigure9. Forallstellarmasses,earlierhaloformationdoes
correlationsshown.
correspond to lower t as expected, though the correlation
TheleftcolumnofFigure9showsthecorrelationswiththe peak
exhibitslargescatter. Thisscatterisnotsurprisinggiventhat
maximummassahalohasobtainedoveritshistory(forsub-
the stellar mass-halo mass relation is subject to a scatter of
halos,themasstodaycanbesignificantlylowerthanthismax-
10 Diemeretal.
Maximum halo mass Environment (z = 3) Halo age Halo age
x x x x
d d d d
/ / / /
N N N N
d d d d
12 12 12 12
nal 11 nal 11 nal 11 nal 11
fi fi fi fi
M M M M
g10 g10 g10 g10
o o o o
l 10 l 10 l 10 l 10
9<M <10
10<M∗ <11
11<M∗<
9 ∗ ∞ 9 9 9
yr) 101 yr) 101 yr) 101 yr) 101
G G G G
n n n n
i i i i
e e e e
m m m m
ti ti ti ti
k k k k
a a a a
e e e e
p p p p
( ( ( (
tpeak tpeak tpeak SFR tpeak
100 100 100 ∝MAR 100
0.5 0.5 0.5 0.5
n) n) n) n)
o o o o
ati ati ati ati
vi vi vi vi
e e e e
d d d d
( 0.0 ( 0.0 ( 0.0 ( 0.0
H H H H
F F F F
S S S S
σ σ σ σ
∆ ∆ ∆ ∆
g10 g10 g10 g10
o o o o
l l l l
0.5 0.5 0.5 0.5
− − − −
11 12 13 14 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
log10Mhalo,max log10d5thnearest(kpc/h) zwechsler zstarve
Figure9. Thegalaxy-haloconnectioninthelog-normalparameterspace. Eachcolumnreferstoahaloorgalaxyproperty,namelyhalomass,environment
(quantifiedbythedistancetothe5thnearestneighboratz=3),thehaloageaccordingtoWechsleretal.(2002),andthe“starvationredshift”whereahalois
expectedtoquenchintheagematchingformalismofHearin&Watson(2013).Ineachcolumn,thetoppanelshowsthedistributionoftherespectivepropertyin
oursimulatedgalaxysample(gray). ThebottomthreerowsshowthedistributionofMfinal,tpeak,andthedeviationfromthetpeak-σSFRrelationasafunctionof
thehaloproperty(grayshading).Thelinesandshadedareasshowthemedianandstatisticaluncertaintyforeachofthreebinsinstellarmass.Theblackdashed
anddottedlinesinthebottompanelshighlightthemeanand68%scatterofthetpeak-σSFRrelation.Inthethirdcolumn,thegraydashedlinesshowtherelation
expectediftheSFRwasproportionaltothehalomassaccretionrate(MAR).SeeSections3.3and4.5foradetailedinterpretationofthisfigure.
about 0.2 dex (e.g., More et al. 2009; Behroozi et al. 2013b; massgalaxies(purplelinesinFigure9). Allotherdefinitions
Reddicketal.2013;Guetal.2016). of formation redshift are less correlated with t than z
peak starve
Interestingly, z exhibitsamorepronounced(andmore andz .
starve wechsler
orlessmass-independent)relationwitht .Byconstruction, Sofar,wehavediscussedhowt dependsonstellarmass,
peak peak
z issupposedtocorrespondtotheredshiftwhenahalo’s environment,andhaloage. Butwhatinfluencestheduration
starve
galaxy is expected to quench (Hearin & Watson 2013). The of a galaxy’s star formation at fixed peak time? From the
strong correlation with t means that the Illustris galaxy bottom panels in Figure 9, we conclude that mass and en-
peak
population supports the age matching picture in which the vironment have little effect on whether a SFH lies above or
SFR (or color) of a galaxy is directly connected to its halo’s belowthepeaktime-widthrelation. Itappearsthatverymas-
accretion history. However, this connection is non-trivial: sivegalaxies(M >1011 M )generallylieslightlybelowthe
∗ (cid:12)
massive, early-forming galaxies tend to live in halos with a relation, but this trend is not very strong. However, z
wechsler
highz (becausethehaloreached1012M early),whereas does correlate significantly, and for all stellar masses, with
starve (cid:12)
z forthesamehalomayindicatealateformationtime, the deviation from the relation. In particular, at fixed peak
wechsler
leadingtothelargedisagreementbetweenthetrendsforhigh time, galaxies in early-forming halos tend to have relatively