Table Of ContentTheAstrophysicalJournal,826:214(21pp),2016August1 doi:10.3847/0004-637X/826/2/214
©2016.TheAmericanAstronomicalSociety.Allrightsreserved.
THE ANGULAR MOMENTUM DISTRIBUTION AND BARYON CONTENT OF
STAR-FORMING GALAXIES AT z∼1–3*
A.Burkert1,2,N.M.FörsterSchreiber2,R.Genzel2,3,4,P.Lang2,L.J.Tacconi2,E.Wisnioski2,S.Wuyts2,5,K.Bandara2,
A. Beifiori1,2, R. Bender1,2, G. Brammer6, J. Chan2, R. Davies2, A. Dekel7, M. Fabricius1,2, M. Fossati1,2, S. Kulkarni2,
D. Lutz2, J. T. Mendel1,2, I. Momcheva8, E. J. Nelson8, T. Naab9, A. Renzini10, R. Saglia1,2, R. M. Sharples11,
A. Sternberg12, D. Wilman1,2, and E. Wuyts2
1Universitäts-SternwarteLudwig-Maximilians-Universität(USM),Scheinerstr.1,München,D-81679,Germany;[email protected]
2Max-Planck-InstitutfürextraterrestrischePhysik(MPE),Giessenbachstr.1,D-85748Garching,Germany;[email protected],[email protected]
3DepartmentofPhysics,LeConteHall,UniversityofCalifornia,Berkeley,CA94720,USA
4DepartmentofAstronomy,NewCampbellHall,UniversityofCalifornia,Berkeley,CA94720,USA
5DepartmentofPhysics,UniversityofBath,ClavertonDown,Bath,BA27AY,UK
6SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA
7RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel
8DepartmentofAstronomy,YaleUniversity,P.O.Box208101,NewHaven,CT06520-810,USA
9Max-PlanckInstituteforAstrophysics,KarlSchwarzschildstrasse1,D-85748Garching,Germany
10INAF-OsservatorioAstronomicodiPadova,Vicolodell’Osservatorio5,Padova,I-35122,Italy
11DepartmentofPhysics,DurhamUniversity,ScienceLaboratories,SouthRoadDurhamDH13LE,UK
12SchoolofPhysicsandAstronomy,TelAvivUniversity,TelAviv69978,Israel
Received2015October11;revised2016May13;accepted2016May15;published2016August1
ABSTRACT
We analyze the angular momenta of massive star-forming galaxies (SFGs) at the peak of the cosmic star formation
epoch(z∼0.8–2.6).Oursampleof∼360log(M*/Me)∼9.3–11.8SFGsismainlybasedontheKMOS3DandSINS/
zC-SINF surveys of Hα kinematics, and collectively provides a representative subset of the massive star-forming
population.Theinferredhaloscaleangularmomentumdistributionisbroadlyconsistentwiththattheoreticallypredicted
for their dark matter halos, in terms of mean spin parameter álñ∼0.037 and its dispersion (σlogλ ∼ 0.2). Spin
parameterscorrelatewiththediskradialscaleandwiththeirstellarsurfacedensity,butdonotdependsignificantlyon
halomass, stellar mass, orredshift. Ourdata thussupportthelong-standingassumption thaton average,even at high
redshifts,thespecificangularmomentumofdiskgalaxiesreflectsthatoftheirdarkmatterhalos(j =j ).Thelackof
d DM
correlation between λ×(j /j ) and the nuclear stellar density Σ*(1 kpc) favors a scenario where disk-internal
d DM
angularmomentumredistributionleadsto“compaction”insidemassivehigh-redshiftdisks.Foroursample,theinferred
averagestellartodarkmattermassratiois∼2%,consistentwithabundancematchingresults.Includingthemolecular
gas,thetotalbaryonicdisktodarkmattermassratiois∼5%forhalosnear1012Me,whichcorrespondsto31%ofthe
cosmologically available baryons, implying that high-redshift disks are strongly baryon dominated.
Key words: cosmology: observations – galaxies: evolution – galaxies: high-redshift – infrared: galaxies
1. INTRODUCTION Here G is the gravitational constant, H(z) and H are the
0
In the cold dark matter (CDM) paradigm, baryonic disk Hubble constants at z and z=0, and ΩΛ,0 and Ωm,0 are the
galaxies form at the centers of dark matter halos (e.g., Fall & energy densities of Λ and total matter at z=0, relative to the
Efstathiou 1980; Fall 1983; see Mo et al. 2010 for a review). closuredensity.Tidaltorquetheory(Hoyle1951;Peebles1969;
Defining the halo radius as the region within which the White 1984) suggests that within the virial radius, the
virialized dark matter particles have on average 200 times the centrifugal support of baryons and dark matter (labeled
mean mass density of the universe, the halo’s virial velocity, “DM” fromhere on)issmall andgiven bythespin parameter,
v ,itsmassM ,anditsvirialradiusR aregiveninthe
virial DM virial
spherical collapse model (for a flat ΛCDM universe) by the w J M J ´ E1 2
following well known relations (Peebles 1969; Gunn & Gott l = virial = e DM DM ~ DM DM , (2)
1972; Bertschinger 1985; Mo et al. 1998) wvirial,cs Rvirial ´ vvirial GMD5M2
R = vvirial and where ω=vrot/R is the angular speed (vrot is the rotational/
virial 10H(z) tangential velocity) at R, and “virial” and “cs” stand for “within
MDM= vv2irialRvirial = vv3irial , where tThheevciroinalstraandtiuòsi”sa∼nd2“,ceJnatnridfujgaarlestuhpeptoortta”l(aωnrdots,cpse=cifi(GcM(j/=R3J)1//M2)).
G 10GH(z)
angular momenta, and E∼GM2/R is the absolute value of the
H(z)=H (W + W ´ (1 + z)3)1 2. (1)
0 L,0 m,0 total gravitational energy. Building on earlier work by Peebles
(1969) and Barnes & Efstathiou (1987), simulations have shown
* thattidaltorquesgenerateauniversal,near-lognormaldistribution
BasedonobservationsobtainedattheVeryLargeTelescopeoftheEuropean
SouthernObservatory,Paranal,Chile(ESOProgrammeIDs075.A-0466,076.A- function of halo spin parameters, withálñ=0.035–0.05 and a
0527,079.A-0341,080.A-0330,080.A-0339,080.A-0635,081.B-0568,081.A- dispersionof±0.2inthelog(Bullocketal.2001a;Hetznecker&
0672,082.A-0396,183.A-0781,087.A-0081,088.A-0202,088.A-0209,091.A-
0126,092.A-0091,093.A-0079,094.A-0217,095.A-0047,096.A-0025). Burkert 2006; Bett et al. 2007; Maccio et al. 2007).
1
TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
If the baryons are dynamically cold, or they can cool after becomes even dominant for dwarf spheroidal galaxies and
shock heating at R , they fall inwards and form a lowermassdisks(Martinssonetal.2013a,2013b).Intheouter
virial
centrifugallysupporteddiskof(exponential)radialscalelength regions (on scales of 10–30 kpc) z=0 disks are dark matter
R , given by (e.g., Mo et al. 1998; see also Fall 1983, their dominated, as demonstrated by their flat rotation curves (e.g.,
d
Equation (4)) Sofue & Rubin 2001; Courteau & Dutton 2015).
At high-redshift little is known empirically so far about the
1 ⎛ f ⎞ 1 ⎛ j ⎞ baryonicangularmomentumdistribution(seeFörsterSchreiber
Rd = 2 ⎝⎜mJd⎠⎟ ´ l ´ Rvirial = 2 ⎝⎜j d ⎠⎟ ´ l ´ Rvirial. et al. 2006 for a first attempt). Look-back studies have shown
d DM that most “normal,” massive star-forming galaxies (SFGs)
(3) (selected from rest-frame UV/optical imaging surveys) from
z∼0 to z∼3 are located on or near a star formation “main-
Here md=Md/MDM is the ratio of the baryonic disk mass to sequence” in the stellar mass (M*) versus star formation rate
that of the dark matter halo and fJd is the fraction of the total (SFR)plane.Itsslopeisapproximatelyindependentofredshift
dark halo angular momentum in the disk, Jd=fJd JDM. and slightly sub-linear (SFR∼ M0.7-1), but its amplitude
*
In the literature it has generally been assumed that the strongly increases with redshift to z∼2.5 (such that the
specificangularmomentumofthebaryonsandthedarkmatter specific SFR, sSFR = SFR/M* ∼ (1 + z)2.5–3; Daddi
is thesame, suchthat j =j (e.g.,Dutton &van denBosch
d DM et al. 2007; Noeske et al. 2007; Schiminovich et al. 2007;
2012). Indeed, models adopting j =j have been very
d DM Rodighiero et al. 2010, 2011; Whitaker et al. 2012, 2014;
successful in explaining the scaling relations of low-redshift Speagle et al. 2014). The location of galaxies in the stellar
disk galaxies (e.g., Fall 1983; Mo et al. 1998; Dutton & van mass–sSFRplanecorrelateswiththeirinternalstructure.Outto
den Bosch 2012; Romanowsky & Fall 2012; Fall & atleastz∼2.5,typicalSFGsonthe“main-sequence”arewell
Romanowsky 2013). The situation is, however, different for approximated by exponential light and mass profiles with
passive spheroids. Fall & Romanowsky (2013) found jd/ Sérsic index n ∼1 while passive galaxies below the main-
j ∼0.8 for late-type, star-forming disks, but only ∼0.1 for S
DM sequence, outlier starbursts well above the main-sequence, as
early-type passive spheroids, with Sa and S0 galaxies in well as the most massive (log(M*/Me)>11) main-sequence
between these two extremes. Early numerical simulations of SFGstendtoexhibitcuspierprofileswithSérsicindicesn >2
cosmological disk galaxy formation suffered fromcatastrophic (e.g., Wuyts et al. 2011b, 2012; Bell et al. 2012; SLang
angular momentum loss, leading to disk galaxies with scale
et al. 2014; Bruce et al. 2014a, 2014b; Nelson et al. 2016;
lengths that were an order of magnitude smaller than observed
Whitaker et al. 2015).
(Navarro & Benz 1991; Navarro & White 1994; Navarro &
The ionized gas kinematics of these SFGs are broadly
Steinmetz 1997). More recent simulations using improved
consistent with these structural properties (e.g., Genzel et al.
numerical schemes and including stellar feedback, however,
2006, 2008; Förster Schreiber et al. 2009; Épinat et al. 2009,
confirmed the assumption j =j (e.g., Übler et al. 2014;
d DM 2012; Law et al. 2009; Wright et al. 2009; Jones et al. 2010;
Danovich et al. 2015; Teklu et al. 2015). We note, however,
Wisnioski et al. 2015; see Glazebrook 2013 for a more
that this result is not at all trivial. The infalling baryons can
complete review). The majority (>70%) of massive (log(M*/
both lose and gain angular momentum between the virial and
disk scale. In addition, the baryon fraction of galaxies, Me)>10) main-sequence SFGs at z∼2.5 are rotationally
supported disks (e.g., Newman et al. 2013; Wisnioski et al.
including disk galaxies, is much smaller than the cosmic
2015), albeit with large velocity dispersions and often clumpy
baryon fraction, indicating that substantial amounts of gas
and irregular rest-frame UV/optical morphologies (Cowie
either never entered the galactic plane or were blown out
afterwards. In this case, the specific angular momentum of the et al. 1995, 1997; van den Bergh et al. 1996; Elmegreen
et al. 2004, 2009; Elmegreen 2009; Förster Schreiber
gas that is retained in the disks could be very different
comparedtothespecificangularmomentumofthegasentering et al. 2011b; Wuyts et al. 2012).
In this paper we want to take the next step and explore the
the virial radius.
angular momentum distribution and baryon to dark matter
Most studies so far concentrated on galaxies in the low-
fractions in z∼0.8–2.6 SFGs, at the peak of cosmic star
redshift universe. Recent high-resolution simulations of high-
redshiftdiskgalaxyformationbyDanovichetal.(2015)found formationactivity,bytakingadvantageoftherecentgrowthin
that the gas entering the virial radius in cold streams has ∼3 sample sizes and coverage of the M*–SFR plane with Hα
λ .Subsequentangularmomentumredistributionandlossby kinematicsintegralfieldunit(IFU)datasets.Thisprogresshas
toDrqMues and feedback-driven outflows however leads to disk started in the last few years, for instance with the SINS/zC-
spins that are similar to the halo spins. Clearly, given this SINF (e.g., Förster Schreiber et al. 2006, 2009; Mancini
complexity, it is of great interest to empirically study the etal.2011,N.M.FörsterSchreiberetal.2016,inpreparation),
baryonic angular momentum distributions of galaxies as a MASSIV (e.g., Épinat et al. 2009, 2012; Contini et al. 2012)
function of cosmic epoch. and HiZELS (e.g., Swinbank et al. 2012) surveys with
Anotherimportantphysicalparameterofgalaxyformationis SINFONI on the Very Large Telescope (VLT), as well as
the relative fraction m of baryonic to dark matter mass in the with surveys with OSIRIS on the Keck telescope (e.g., Law
d
half-light regions R1/2 of z∼0 galaxies, which depends on etal.2009;Wrightetal.2009;WiggleZ,Wisnioskietal.2011,
type and mass. Massive early-type spheroidal systems and 2012).MostimportantlywehaverecentlystartedtheKMOS3D
massivedisks,includingtheMilkyWay,arebaryondominated survey (e.g., Wisnioski et al. 2015) with the multiplexed near-
within ∼1.2 R1/2 (called “maximal disks” if MDM/Mbaryon < infrared IFU spectrometer KMOS on the VLT (Sharples et al.
0.3 within that radius; Barnabè et al. 2012; Bovy & Rix 2013; 2008, 2012), which will deliver IFU data for at least
Cappellari et al. 2013; Dutton et al. 2013; Courteau & Dutton ∼600z∼0.6–2.7 SFGs (see also, e.g., Sobral et al. 2013;
2015). In contrast, the dark matter fraction is significant and Stott et al. 2014, 2016; Mendel et al. 2015; Harrison et al.
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
2016;Magdisetal.2016forotherexamplesofKMOSsurveys cuts. Our Hα kinematic galaxy sample thus is unbiased in
of distant galaxies). The combined data of these surveys termsofsizeandplacementrelativetothemain-sequenceline.
currently provide a sample of over 1000 galaxies, with a good AppendixA.1givesmoredetailsonthesourceselection,and
coverage of massive (log(M*/Me)>10) SFGs in the Appendix A.2 summarizes how the key global, structural, and
z∼0.8–2.6 redshift range. kinematic properties of the galaxies were derived (including
Throughout the paper, we adopt a flat ΛCDM cosmology stellar and gas masses, SFRs, half-light radii R1/2, rotation
with Ω =0.27, Ω =0.046, and H =70 km s−1 velocities v , and velocity dispersions σ ). The IFU observa-
m,0 b,0 0 rot 0
Mpc−1(Komatsu et al. 2011), and a Chabrier (2003) initial tions, and the Hα kinematic maps, profiles, and basic
stellar mass function (IMF). measurements, have been presented and discussed in detail in
themainreferencepapersdescribingthesurveysconsideredfor
our study; we refer the interested reader to these papers for
2. OBSERVATIONS AND ANALYSIS extensive examples of the data (Wright et al. 2007, 2011; van
Starkenburg et al. 2008; Épinat et al. 2009, 2012; Förster
2.1. Galaxy Sample
Schreiber et al. 2009; Law et al. 2009; Mancini et al. 2011;
We base this study on IFU observations of the Hα Contini et al. 2012; Swinbank et al. 2012; Genzel et al. 2013,
kinematics and distribution in a large initial sample of 433 2014; Wisnioski et al. 2015; N. M. Förster Schreiber et al.
z=0.76–2.6massive,star-forming diskgalaxies.Thedatafor 2016, in preparation; Wuyts et al. 2016).
thesegalaxiescomefromdifferentIFUsurveys,eitherongoing
or in the literature, with the 2 year sample of the KMOS3D
survey (Wisnioski et al. 2015) constituting the strong majority 2.2. Estimating the Halo Masses and Halo λ Parameters
(∼3/4:316ofthe433galaxies).Sincethesubjectofthisstudy
isananalysisofangularmomenta,wefirsteliminatedfromthe Withthediskparametersinhand(R1/2,vrot=vrot(R∼R1/2),
initial sample all major mergers (23 galaxies), all dispersion- σ0=σ (R∼2 R1/2), M*, Mgas=Mmolgas,13 Mbaryon(R1/2) =
dominatedgalaxies(v /σ <1.5,31galaxies;seebelow),and (M*+Mgas)(R1/2)), the next step is to estimate the halo
20 galaxies without wroetll-d0efined kinematics and/or with very masses,spinparameters,andspecificdiskangularmomentafor
large beam smearing corrections (see below). This leaves us the individual galaxies. Following Mo et al. (1998) we used
four independent methods to reach this goal:
with a disk sample of 359 SFGs, which we will denote
henceforthasthe“full”sample.Wealsocreatedasecond,still
1. ourprimaryapproachistodeterminethedarkmatterhalo
more restricted “best” sample of the 233 highest quality, well-
mass and angular momentum parameter, as well as the
resolved rotating disks, by retaining only vrot/σ0
2 SFGs. baryonic to dark matter mass ratio, from fitting an
We also eliminated minor mergers and SFGs with obviously exponential disk embedded in a Navarro–Frenk–White
perturbedmorphologies/kinematics,aswellasgalaxieswithan halo(NFW;Navarroetal.1997),extractingtheoptimum
offset between the morphological major axis (continuum or parameters from a Monte-Carlo (MC) search of the
Hα) and the kinematic major axis of greater than 40°. Finally
parameter space;
we culled all insufficiently resolved disks with a half-light/
2. we also estimate the halo angular momentum parameter
mass radius R1/2<2 kpc and a ratio of R1/2 to the HWHM for a dark matter dominated, isothermal halo;
beam size R1/2,beam less than unity, and SFGs with still 3. we determine the dark matter angular momentum
significant beam smearing corrections. SFGs in this “best” parameter from fitting an exponential disk embedded in
sample exclusively come from SINS/zC-SINF and KMOS3D
anNFWhalo,thistimeadoptingaconstantbaryonicdisk
and were all analyzed in a consistent manner.
mass to dark matter halo mass ratio; and
Figure 1 shows that our final IFU sample yields a good
4. as a variant, we determine the dark matter mass and
representation of the mass-selected main-sequence SFG angularmomentumparameterfromfittinganexponential
population at log(M*/Me)>10.1 and sSFR/sSFR(ms, z)> disk embedded in an NFW halo, by inverting the stellar
0.1 in the range 0.8z2.6, as drawn from the 3D-HST
masstodarkmattermassratioasafunctionofhalomass
survey catalogs (Brammer et al. 2012; Skelton et al. 2014;
from the abundance matching work of Moster et al.
Momcheva et al. 2016) in the CANDELS extragalactic survey (2013) and Behroozi et al. (2013b).
fields (Grogin et al. 2011; Koekemoer et al. 2011)—hereafter
the “reference” galaxy sample or population. Our kinematic We discuss in Appendix B the details of our methods,
sample by design is heavily incomplete for lower mass SFGs includingcorrectionoftheobservedrotationvelocitiesanddisk
(log(M*/Me)<10) and does not cover the passive popula- sizes for asymmetric drift, and deviations from pure exponen-
tion. The inhomogeneous redshift coverage in the upper left tial surface density distributions.
panel of Figure 1, with a significant lack of SFGs at The four methods yield independent estimates of the mass,
1.2<z<2,reflectstheintervalswhereHαisshiftedbetween radius, and angular momentum parameter of the dark matter
the J- and H-, and H- and K-band atmospheric transmission halofromthemass,size,andkinematicsofthecentralbaryonic
windows, and the emphasis on z∼0.76–1.1 and z∼2–2.6 disk.Allassumeimplicitlythatthespecificangularmomentum
slices in the first two years of the KMOS3D survey (Wisnioski ofthebaryonsonthescaleofthedarkhalo isthesameasthat
et al. 2015). ofthedarkmattercomponent.Moreimportantlyandprecisely,
Perhaps most importantly for the results in this paper, the
bottom right panel of Figure 1 demonstrates that the 13 As explained in Appendix A.2, our M estimates are derived from the
gas
distribution of galaxy half-light radii is indistinguishable from scaling relations between Mgas, M*, SFR, and z for main-sequence SFGs
presented by Genzel et al. (2015), assuming that at z∼1–3 the molecular
that of the underlying reference galaxy population, when
componentdominatesandtheatomicfractioncanbeneglected.Assuch,the
selectedaccordingtothesameredshift,stellarmass,andsSFR M massestimatesmaybelowerlimits.
gas
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure1.Redshift(topleft),stellarmass(bottomleft),sSFR(topright,relativetothemain-sequencerelationatagivenz),anddiskhalf-lightradius(bottomright)
distributions of 0.7 < z < 2.7 galaxies from the 3D-HST source catalogs (red squares and dashed lines; Brammer et al. 2012; Skelton et al. 2014; Momcheva
etal.2016)intheCANDELSextragalacticsurveyfields(Groginetal.2011;Koekemoeretal.2011),andthe“full”HαIFUsampleassembledforthispaper(filled
bluecirclesandsolidlines).Theleftpanelsrefertoall359SFGsinoursample.Theupperrightpanelcomparesallgalaxieswithlog(M*/Me)>10.1(abovewhich
oursampleisagoodrepresentationofthe3D-HSTsample),leaving276SFGs.Thebottomrightpanelcomparesthesizedistributionsofthe3D-HSTandHαsamples
inthatpartofthestellarmass–SFRplanewhereoursampleisrepresentativeofthe3D-HSTreferencesample(log(M*/Me)>10.1,sSFR/sSFR(ms,z)>0.1,261
SFGs).Theblackdashedlineinthebottomleftpanelshowsthesame3D-HSTdistributionasplottedinredbutscaledsoastomatchthatofourHαIFUsample
(shown in blue) above log(M*/Me)=10.5; similarly, the black dashed line in the top right panel shows the 3D-HST distribution now restricted to log(M*/
Me)>10.1andscaledtomatchourIFUsampleabovesSFR/sSFR(ms,z)=0.1.
all methods deliver an estimate of the product we then use Equations (1) and (3) to express the specific
angular momentum of the stellar/baryonic disk, j , as
d
l ´ (j j )=l (R ~ R ) ´ (j j )(R ~ R )
d DM DM virial baryon DM virial
⎛ ⎞ ⎛ ⎞
´(j (R ~ R ) j (R ~ R )). j j
baryon 1 2,disk baryon virial ⎜ d ⎟=1177 ´ ⎜ d ⎟ ´ f-2 3
⎝kms-1kpc⎠ ⎝j ⎠ *
DM
Hence, our results depend on the angular momentum distribu- ⎛H(z)⎞-1 3 ⎛ M ⎞2 3
tiononthehaloscale(ofbothbaryonsanddarkmatter),aswell ´⎜ ⎟ ´ l ´ ⎜ * ⎟ .
⎝ H ⎠ 0.035 ⎝1011M ⎠
as on any redistribution of angular momentum between 0
different baryonic components (inner and outer disk, outflow, (4)
bulge, etc.). The former is a measure of the total angular
momentumstateofthehalo(“nature”),whilethelatterdepends Here f*=M*/(0.17 × MDM) is the fraction of the
on intra-halo baryonic processes (“nurture”). cosmologically available baryons that are tied up in the stellar
disk. The assumption of constant λ for all SFGs cannot be
correct for each SFG. However, it is plausibly correct on
3. RESULTS AND DISCUSSION
averagesincethehalospinparametersareexpectedtofollowa
3.1. The Specific Angular Momenta of z=1–2 Star-forming lognormal distribution about the mean (see Section 1), such
Disk Galaxies that the average trend of the data as a function of stellar mass
can then be compared to theory and other observations. The
3.1.1. Correlation of AngularMomentumwith Stellar Mass assumption of constant λ in Equation (4) should just lead to a
In this section we begin with an empirical investigation of scatter in the data, but no trends with parameters such as M*,
the observed baryonic angular momentum distributions of our z, etc.
disks. Following Romanowsky & Fall (2012) we assume for Equation (4) ties an easily observable quantity (j ) to the
d
simplicity that all disks in our sample have the same dark product of the another easily observable quantity, M2 3, and
*
matter angular momentum parameter, λ =λ/0.035, and theratioj /j .Foradiskofconstantrotationvelocityv and
0.035 d DM rot
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure2.Leftpanel:specificangularmomentuminthestellardisk(adoptingEquation(5))asafunctionofstellarmass,afterremovaloftheredshiftdependencein
Equation(4)(multiplyingtheobservedj*withH(z)1/3)forthedisksofthe“full”(opencircles)andthe“best”(filledcircles)disksamples.Thetypicaluncertaintyis
shownasacrossinthelowerright.Thedashedredlineisthebestlinearfitofslope2/3(Equation(4))tothe“full”sample(zeropoint−3.33).Thethickdashedblack
andsolidbluelinesmarktheaveragetrendlineofthe“full”and“best”samplesinbinsof0.25dexinlog(M*).Rightpanel:samesymbolsastheleftpanel,butnow
emphasizingasfilledredcirclesSFGsintheupper25%percentileofcentralstellarsurfacedensityΣ*(R1kpc)>109.7Mekpc−2(equivalenttonS>3),andas
blackfilledcirclesallgalaxiesfromourdisksamplewithv /σ <2,thatis,SFGswithlowrotationalsupportanddispersion-dominatedSFGs.
rot 0
of effective radius R1/2 the specific angular momentum is traced by the ionized gas distribution at/near R1/2, do not
strongly depend on central stellar surface density, or Sérsic
jd = kd ´ vrot ´ R1 2, (5) index.Thedensest,cuspiestgalaxiesaremorecommonamong
themoremassive(log(M*/Me)>10.6)SFGs,consistentwith
with the constant kd=1.19 for a thin exponential disk (Sérsic massive bulges being present (e.g., Lang et al. 2014) but the
index n = 1). The disks of our sample are thick, dispersion specific angular momentum of these systems appears uncorre-
S
truncated, their rotation velocities are not constant, and their latedwiththeircentralstellarproperties. Thissuggeststhatthe
Sérsicindices n 1 (see Appendices A and B), inwhichcase formation/presence of central bulges at z∼0.8–2.6 does not
S
k becomesafunctionofn andv /σ .Thiscaseisdiscussedin mainly depend on the main galaxy disk having a low angular
d S rot 0
Appendix B.7 and we used the fitting function given in momentum.
Equation (19). Since the correction function in Equation (19) The black filled circles in the right panel of Figure 2 denote
all SFGs in our initial disk sample but with v /σ <2 (see
uses Sérsic indices derived from the optical continuum stellar rot 0
Section 2.1), that is, SFGs with relatively low rotational
light as well as v /σ values inferred from the ionized gas
rot 0 supportandincludingdispersion-dominatedSFGs.TheseSFGs
kinematics, and Equation (5) also mixes information obtained appear to form the low tail of the specific angular momentum
from stars (R1/2) and gas (vrot), in what follows we assume distribution, and have predominantly low stellar masses and
jd=j*=jgas.The3D-HSTworkofNelsonetal.(2016)shows small effective radii (<3 kpc), as pointed out earlier by
thatthisassumptionistofirstordercorrectwhencomparingthe Newman et al. (2013). These low observed specific angular
disksizesofgasandstars,butthattheionizedgasdiskstendto momentaarenottheresultofsystematicallylowerinclinations
be somewhat larger than those of the stars, most notably at the (face-ondisks)ofthedispersion-dominatedSFGsascompared
highest stellar masses, where stellar bulges become prominent to the entire “full” sample, nor are they correlated with larger
((R1/2,Hα/R1/2,*)=1.1×(M*/1010Me)0.05). beam smearing corrections, both of which might suggest a
In the left panel of Figure 2 we plot the specific angular systematicunderestimateoftheintrinsicrotationvelocities.We
thus conclude that the low specific angular momentum of
momenta of our SFGs as a function of stellar mass, adopting
Equations (4) and (5) and λ = 0.035 for all galaxies. The data dispersion-dominated systems most likely is an intrinsic
followthetheoreticallyexpectedM2 3dependence.Thereisno property.
*
significant difference between the “full” and “best” samples,
otherthanthatthe“best”samplebydesign(Section2.1)lacksa
3.1.2. Comparison toObservations at z∼0andto
number of low mass, small SFGs with low rotational support,
Recent Simulations
includingdispersion-dominatedgalaxies.Thedispersionofthe
data around the trend line is ±0.031 dex for the “best” and Aftereliminationoftheredshiftandstellarmassdependencies
±0.035 dex for the “full” sample. of the specific angular momentum, by multiplying j with
d
In right panel of Figure 2 we emphasize as filled red circles H(z)1/3×M-2 3 (see Equation (4)), we compare in the left
*
SFGs in the upper 25% percentile of central stellar surface panelofFigure3thehigh-zSFGsofthe“full”sampletothelate-
density Σ*(R 1kpc)>109.7Me kpc−2 (equivalent to type disks (filled red squares), early-type disks(brown triangles),
n > 3). The observed specific angular momenta of SFGs, as and E/S0 galaxies (black crossed squares) in the local universe
S
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure3.Leftpanel:distributionoftheobservedspecificangularmomentaofthez=0.8–2.6SFGsinthe“full”sample(grayopencircles,andthickgraysolidtrend
lineofthebinnedaverages,bothasinFigure2),aftermultiplyingthedatainFigure2withM-2 3inordertoremovethestellarmassdependence,aswellasthe
*
redshiftdependence(seeEquation(4)).Filledredsquaresdenotethez=0late-typedisks(Sbcd),brownfilledtrianglestheearly-typedisks(Sa),andblackcrossed
squaresthespheroidalgalaxies(S0,E)inthecompilationofFall&Romanowsky(2013).Green,blue,andmagentalinesarethepredictionsofangularmomentaof
SFGsfromtheIllustrishydrodynamicalsimulation(Geneletal.2015;S.Genel2016,privatecommunication).Rightpanel:stellarmassdependenceoflog(j/j )in
d DM
thehigh-zandlow-zdataofFigure2(samesymbolsasintheleftpanel),withtheassumptionofλ=0.035,afterremovingthef*(M*)dependencewiththefitting
functionofDuttonetal.(2010;similartoMosteretal.2013).
from the compilation by Fall & Romanowsky (2013). The baryontodarkmatterspecificangularmomentainourgalaxies.
observed angular momentum distributions of high-z and z∼0 For this purpose, we need to eliminate the dependency on the
disk galaxies (which include Sa galaxies) are in excellent function f*(M*). To do so, and again following Romanowsky
agreement. The tail of low angular momentum SFGs at high-z, & Fall (2012), we use the fitting function of Dutton et al.
including the dispersion-dominated SFGs, stretches down to 0.8 (2010), who empirically derived the stellar baryon fractions of
dexbelowthetrendline,inthesameregionoccupiedbythez∼0 local universe galaxies from a combined analysis of stellar
spheroidal galaxies in the Fall & Romanowsky (2013) compila- kinematics, weak lensing, and abundance matching. The
tion.Thisagreementoflow-andhigh-zgalaxiesintermsoftheir Dutton et al. (2010) fitting function (for late-type SFGs at
angularmomentumdistributionsisbynomeanstrivial,giventhat z=0) is given by
high-zSFGsare4–7timesmoregas-richintermsoftheircentral
(molecular)gasreservoirs(Daddietal.2010;Tacconietal.2010, ⎛ M ⎞0.5
2013), experience more frequent perturbations from (dissipative) f*(M*)D10=0.29 ´ ⎝⎜5 ´ 10*10M ⎠⎟
minor and major mergers (Fakhouri & Ma 2008; Genel
et al. 2009; Rodriguez-Gomez et al. 2015), and exhibit much ⎡ ⎛ M ⎞⎤-0.5
more commonly powerful galactic outflows that might alter the ´⎢1 + ⎜ * ⎟⎥ . (6)
⎣ ⎝5 ´ 1010M ⎠⎦
angular momentum distribution of the disk (Übler et al. 2014).
The observed average specific angular momenta as a
Dutton et al. (2010)also gave a similar fitting function for
function of stellar mass in the left panel of Figure 3 are also
in impressive agreement with the recent generation of early-type galaxies. A similar correction function (in terms of
hydrodynamical simulations (Illustris, Genel et al. 2015; mass dependence and zero point) is obtained from the Moster
EAGLE, Zavala et al. 2016). We show in the left panel of et al. (2013) or Behroozi et al. (2013a, 2013b) abundance
Figure3thepredictedspecificangularmomentaintheIllustris matching. We decided to take the Dutton et al. (2010) fitting
simulation as green (z = 2), blue (z = 1), and magenta (z= 0) functioninEquation(6),sinceitextendstolog(M*/Me)>11
lines (Genel et al. 2015; S. Genel 2016, private communica- where we have many SFGs in our sample.
tion),whichagreequitewellwithboththelow-andhigh-zdata TherightpanelofFigure3showstheresultsofapplyingthis
(butfallonaverage∼0.1–0.2dexbelowtheaveragetrend-lines correction, which should now give a quantitative estimate of
of the data). theratioofdisktodarkmatterspecificangularmomentumasa
function of stellar mass, all again under the simplifying
assumption of a constant dark matter angular momentum
3.1.3. Eliminating f* parameter.Withtheseassumptionswefind〈log(j /j )〉−0.2
d DM
Following themotivationofSection1,ournextgoalisnow dex, independent of stellar mass between log(M*/Me)=9.5
tousethedataandEquation(4)togaininsightsontheratioof and 11.6, and including the tail of lower angular momentum
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure4.Distributionofinferredλ×(j /j )parameters.Left:distributionforall312z=0.8–2.6rotationdominatedSFGsinthe“full”samplewithconverged
d DM
NFWMCmodeling,undertheassumptionofnoadiabaticcontractionofthedarkmatterhalo(histogram,with1σPoissonianuncertainties,AppendixB.5).Thecyan
squares,redtriangles,andgreencirclesdenotethesameNFWMCmodeling,butthistimewiththe“best”sample,withthe“full”samplebutincludingadiabatichalo
contraction,aswellaswithoutadiabaticcontractionbutincludingthecorrectionsforvaryingSérsicindices,respectively(AppendixB.7).Right:forcomparisonwith
thehistogramintheleftpanelweshowthreeother,simplerestimatesoftheλdistribution.Cyansquaresdenotethedistributionofλvaluesthatisobtainedforthe
“best”disksampleifinsteadoftheNFWMCmodelingthesimpleassumptionMDM=(M*+Mmolgas)d/mdismadeandλisestimatedfromEquation(17)with
m =0.05.Ifinsteadthefullsampleisused,thecentroidissimilar,butthescatterislarger.WhenusingtheMosteretal.(2013)fittingfunctionstoinferthehalomass
d
from thestellar mass (and then obtaining λfrom Equation(17)again),the resulting λdistribution isgiven bygreen circles. Finally, theresults ofthe simplest,
isothermalmodelaremarkedbyredtriangles(Equation(9)).
galaxies discussed before. Leaving out these extreme outliers The intrinsic dispersion of these distributions varies between
byselectingthe“best”sampleyieldsanaverageof−0.09.For 0.16 and 0.19 dex in logλ.
comparison, the average of all Fall & Romanowsky (2013) Red triangles denote the same NFW MC modeling
star-forming spiral galaxies (including Sa types) is −0.17, methodology for pure exponential disks but this time with
while the late-type systems (Sbcd) have an average of −0.03. adiabatic contraction, as described by Mo et al. (1998). The
We conclude that SFGs between z=0 and 2.6 plausibly have resulting distribution again is lognormal with a similar
onaverageretainedbetween60%and90%oftheirdarkmatter dispersion, but with a greater mean,ál ´ (j j )ñ = 0.044
d DM
specific angular momentum in their main baryonic disk. to 0.071, depending on whether we use an unweighted or
weightedestimator(fourthrowofTable1).Greencirclesagain
denotethesameNFWMCfitting,withoutadiabaticcontraction
3.2. Angular Momentum Parameter (as for the black shaded diagram), but now implementing
individual corrections to λ values for surface density distribu-
Based on the analysis in Section 2.2 and in the Appendices
tionsdeviatingfromthepureexponentialdistributionsassumed
A and B, we now have several estimates of λ parameters, or so far, and obtained from free n fits to the rest-frame optical
more precisely of λ×(jd/jDM), for rotation dominated SFGs continuumdistributionsforeachSSFG(seeAppendixB.7).The
ofthe“full”and“best”samples.Uncertaintiesoftheindividual resulting mean, given in the fifth row of Table 1, is
measurements range from ±0.06 to ±0.33 dex in logarithmic ál ´ (j j )ñ = 0.051. The intrinsic dispersions of these
units, with a median of ±0.2 dex. distribudtionDsMrange between 0.16 and 0.22 dex in logλ.
Figure 4(leftpanel)showsthedistributionfunctionofthese IntherightpanelofFigure4wecompareour“primary”NFW
λparametersforour“primary”NFWMCmodeling(method1 MC modeling (again depicted as histogram) with the other,
in Section 2.2, described in detail in Appendices B.4 and B.5) simpler methods discussed in Section 2.2 and Appendix B. All
and assuming pure (baryonic) exponential disks. Thisdistribu- result in smaller mean values ofál ´ (j j )ñ. Red triangles
tion (plotted as a histogram) is well fitted by a lognormal denote the results for the “best” disk sadmpDlMe if instead of the
function of intrinsic dispersion ∼0.17 dex in logλ, after NFW MC modeling the simplest, isothermal model
subtracting the measurement uncertainties in quadrature from (Equation (9)) is adopted. This yields ál ´ (j j )ñ ~ 0.02
the measured dispersion of the distribution. Taking the “full” withintrinsicdispersionσ(logλ)∼0.22(sixthrodwDoMfTable 1).
and “best” samples with converged NFW models yields error- Greencirclesdenotethedistributionofλvaluesthatisobtained
weighted averages of ál ´ (j j )ñ = 0.039 and 0.041, from Equation (17), and when M =(M* + M ) /m
d DM DM molgasd d
respectively (the first two rows of Table 1). An unweighted valuesareestimatedbyinvertingtheMosteretal.(2013)fitting
Gaussian fit to the modeling results for the “full” sample in functions to infer the halo mass from the stellar mass. The
Figure4yieldsál ´ (j j )ñ = 0.032(thirdrowofTable1). resulting λ distribution has ál ´ (j j )ñ = 0.016 and σ
d DM d DM
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Table1
SummaryofDeterminationsofál´(j j )ñ
d DM
Sub-sample N ál´(j j )ña d (ál´(j j )ñ)b σ(logλ)c
d DM 2 d DM
NFWMC,“full,”errorweighted 312 0.039 0.003 0.17
noadiab.contraction
NFWMC,“best,”errorweighted 220 0.041 0.004 0.19
noadiab.contraction
NFWMC,“full,”equalweight 312 0.032 0.003 0.16
noadiab.contraction
NFWMC,“full,”errorweighted 304 0.071 0.004 0.16
withadiab.contraction
NFWMC,“full,”errorweighted 256 0.051 0.0044 0.22
noadiab.contraction
withSérsiccorrections
Isothermal,“best,”equalweight 233 0.020 0.002 0.22
“Moster,”“best,”equalweight 233 0.016 0.002 0.3
m =0.05,“best,”equalweight 233 0.023 0.002 0.24
d
finaloverallaverage 0.037 0.004(stat) 0.2
0.018(syst)
Notes.
aWeightedmeanoflogλdistribution.
bTwicetheuncertaintyofthemeanofweightedlogλdistribution.
cDispersionofweightedlogλdistribution,aftersubtractioninquadratureofthemedianmeasurementerror.
(logλ)=0.3 (row 7 in Table 1). Finally, if in Equation (17) a and 6. In each panel we show the “best” (filled blue circles) and
constant value of m =0.05 is adopted, motivated by the “full” (filled and open blue circles) samples, and give the equal-
d
average ám ñ value obtained from our primary NFW MC weight trend line of binned averages (thick continuous gray
d
method, the resulting distribution is shown by the cyan squares curve), as well as the best error-weighted linear fit in log–log
and hasál ´ (j j )ñ = 0.023 and σ(logλ)∼0.24. If in all space (i.e., a power law) to the “full” sample (dashed red line).
d DM
thesethreecasesthe“full”insteadofthe“best”sampleisused, The fit parameters (zero point and slope) for these weighted fits
the centroid is similar, but the scatter is larger. are listed, as are the correlation coefficients (R). In Figure 5 the
In our view the “primary” NFW MC fitting constitutes the panels at the top and bottom show the strongest and weakest
besttechniqueforestimatinghalomassesandthecorresponding correlations, respectively. In Figure 6 the plots are sorted by
angular momentum parameters because it makes the fewest increasing correlation strength from left to right.
assumptions and takes into account all the relevant measured Figure 5 shows that the inferred λ×(j /j ) distribution,
d DM
properties and their observational uncertainties. However, there centroid,anddispersiondonotdependmuchonredshiftinthe
is a significant degeneracy between λ and M /M in this interval covered by our data, on the stellar or halo masses, on
baryon DM
technique (see Figure 11 in Appendix B: logλ ∼ −0.8 + thecentralconcentrationsofthegalaxiesastracedbytheSérsic
0.58 × log(M /M )). Looking at Table 1 the agreement index of the rest-frame light distribution, nor on the stellar
baryon DM
of the results of the different methods is quite encouraging, surface density in the central 1 kpc.
keepinginmindthesubstantiallydifferentassumptionsinvolved Figure 6 shows the three strongest correlations (with slopes
in each of the entries. In terms of the mean, the results of the differingfromzeroatthe10–15σlevel)betweenλ×(j /j )
d DM
different methods scatter both to larger, as well as to smaller, and the disk scale length, the stellar surface density within the
values than our “primary” NFW MC modeling. In terms of effectiveradius,andtherotationvelocityattheeffectiveradius.
scatter, the “simpler” methods yield somewhat larger scatter, as The strongest correlation is between λ×(j /j ) and disk
d DM
might have been expected. The comparison in Table 1 gives a radius. This is interesting as the specific angular momentum is
good indication of the systematic uncertainties, which are essentially a product of rotational velocity and scale radius.
significantlylarger thanthe formalfituncertainty ineachofthe Equation (3), however, shows that for galaxies with a given
entries in Table 1. Dispersion-dominated objects, with low virial radius or virial mass, one would expect a linear
inferredspecificangularmomenta(seeSection3.1andFigure2), correlation of λ×(j /j ) with disk radius. We also find
d DM
were excluded in the ál ´ (j j )ñ distributions discussed significant correlations with the stellar surface density within
d DM
here, which may therefore be biased in their mean and scatter. the half-light radius and with the rotation velocity at R1/2
However,inourdatasetstheseobjectsrepresentonlyabout10% (middleandleftpanelsofFigure6).Sincestellarmassandspin
of the size of our “full” sample such that the bias is small parameter are not significantly correlated, the correlation
compared to other uncertainties from the assumptions or between stellar surface density and λ (dlogΣ*/dλ ∼ −3.4) is
methodology described above. probablylargelyinducedbythestrongcorrelationbetweendisk
We adopt as the final mean inferred parameter radius and λ (dR /dλ ∼ 1.33) since Σ*∼M*/R2. The anti-
d d
ál ´ (j j )ñ = 0.037, with a small statistical uncertainty correlation between λ×(j /j ) and v , although it seems
d DM d DM rot
of ±0.004 and a dominant systematic uncertainty of ±0.018. counter-intuitive, follows from Equations (1) and (3) as
We now explore the correlation of the inferred angular described in Appendix B.1.
momentawithredshift,andwithvariousdiskandhaloproperties. In the λ–R and λ–Σ correlations in Figure 6 we have
d
ThemostimportantparametercorrelationsareshowninFigures5 removed the mean redshift dependence of the disk sizes. The
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TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure5.Dependenceoftheinferredλ×(j /j )parametersfromtheNFWMCmodelingforthe“best”(filledbluecircles)and“full”(filledplusopenbluecircles)
d DM
samplesasafunctionof(fromtoplefttobottomright)stellarsurfacedensityinthecentral1kpc,halomass,stellarmass,redshift,andSérsicindexoftherest-frameR-
bandcontinuumlight.Ineachpanellargecrossesdenotethetypicaluncertainties,thickgraycurvesdenotethetrendlineofbinnedequal-weightaveragesofthe“full”
sampledata,andthereddottedlineisthebestlinearerror-weightedfitinlog–logspace(i.e.,apowerlaw)tothe“full”sampledata.Zeropointsandslopes,andtheir
1σuncertainties,alongwiththecorrelationcoefficients(R),arelistedaswell.Thepanelsineachrowareorderedinascendingcorrelationstrengthfromlefttoright
(withthetwostrongercorrelationsinthetoprow).Alltrendsexploredinthisfigureshowlittle,ifany,correlation,andhaveaslopeconsistentwithzeroatthe2σlevel
orlessexceptforthetrendwithM ,forwhichtheslopediffersfromzeroatthe4.8σlevel).
DM
fittingfunctionforthedependenceofpopulationaverageddisk A similar relation of slightly larger scatter and somewhat
scalelengthonredshiftforSFGsobtainedbyvanderWeletal. poorer correlation is obtained without the redshift correction A
(2014b) from CANDELS near-IR HST imagery14 gives (z).Inthatcasethezeropointandslopeare−1.71(±0.03)and
+0.68 (±0.05).
A(z) = (1 + z)-0.75. (7) FortheSFGswithhigheststellarsurfacedensities,log(Σ*(R
Dividingtheobserveddiskhalf-lightradiibyEquation(7)then R1/2) [Me kpc−2])
9, the spin parameters are about half
yieldsthecorrelationinFigure6,whichhasascatterof±0.17 (〈λ× (jd/jDM)〉∼0.018) that of the median value of all SFGs.
dex.Onecanturnthisfindingaroundandusethiscorrelationas Including the Sérsic index corrections discussed in
Appendix B.7 makes no significant difference for the average
aoneparameterestimatoroftheangularmomentumparameters
λ×(j /j ) value or for the (lack of) mass dependence. The
of SFGs without kinematic data, correlatdionDsMof λ×(j /j ) with R and Σ* become slightly
d DM d
⎡ ⎛ ⎞⎤ flatter,butwithintheuncertaintiesofthefitsshowninFigure6.
j
log⎢l ´ ⎜ d ⎟⎥=-1.96(0.04) Insummaryofthissectiontherearethreemainconclusions.
⎣⎢ ⎝j ⎠⎦⎥ First, we find a near-universal, lognormal distribution of
DM
⎛ ⎞ λ×(j /j ), whose centroid (0.037) and dispersion (0.2 in
R d DM
+0.75(0.046) ´ log⎜ d ⎟. (8) logλ) are very similar to that inferred for the dark matter
⎝A(z)⎠
component as determined from CDM simulations. The stellar
surface density within the half-light radius and rotation
velocity exhibit a significant negative correlation with
λ×(j /j ). More compact and denser SFGs have lower
(1M4*E)quraetliaotinon(7sh)iipsboafseSdFGonstfhreomredtshheiftsaemvoelu3tiDon-HoSfTth/eCAlinNeDarElLoSg(Rre1f/e2r)e–nlocge valuesdofDMλ×(jd/jDM), either because they had initially
populationdiscussedinSection2.1;Figure1showsthatthesizedistributionof smaller dark matter λ values, or because a fraction of the
ourIFUsampleandthisreferenceSFGpopulationisessentiallyidenticalover baryons suffered significant angular momentum loss between
the same redshift and mass range, when excluding galaxies well below the
the halo and circum-nuclear scale. This result is in very good
main sequence, such that the same redshift evolution should apply for our
galaxies. agreement with previous work in disk-dominated galaxies in
9
TheAstrophysicalJournal,826:214(21pp),2016August1 Burkertetal.
Figure6.Dependenceoftheinferredλ×(j /j )parametersfromtheNFWMCmodelingforthe“best”(filledbluecircles)and“full”(filledplusopenbluecircles)
samplesasafunctionof(fromlefttoright)rdotaDtiMonvelocityatR∼R1/2,stellarsurfacedensitywithinR1/2andRd/(1+z)−0.75.Thenomenclatureisthesameasin
Figure5.ThethreeplotsinthisfigureshowthestrongestparametercorrelationsamongthetrendsexploredinSection3.2(withslopesdifferingfromzeroatthe
10–15σlevel).
Figure7.StellartototalhalomassratiosasafunctionofM forourdisksofthe“full”(openandfilledbluecircles)and“best”(filledbluecircles)samplesfromour
DM
NFWMCmodeling.Thebluecrossintheupperrightdenotesthetypical1σuncertainties.Thecontinuouscyanlinedenotestheaveragetrendofthe“full”samplein
bins of0.25 dexinlog(M ).Thered filledsquare marksthemedian valueofour measurements. Thered arrowdenotesthe averagecorrection fromstellarto
DM
baryonicdiskmasses,includingthemoleculargascontribution(áM (M +M )ñ~0.58).Therightverticalaxisdenotestheratioofthestellardiskmasstothe
gas gas *
totalbaryonmass,adoptingacosmicbaryontodarkhalofractionof0.17.ThethickgrayandblackcurvesgivethefittingfunctionobtainedbyMosteretal.(2013)
fromrankorderedabundancematchingofstellarmassfunctionsanddarkmatterhalosimulationsforz=0.8andz=2.6,respectively.Theblackdashedcurves
denotethe±1σuncertaintyofthecombinedz=0.8andz=2.6Mosteretal.(2013)models.Magentaandgreenthickdashedlinesmarkthez=1andz=2
abundancematchingresultsofBehroozietal.(2010,2013b).
the local universe (Fall 1983; Dutton & van den Bosch 2012; halo. We thus can compare our results of the ratio of
Romanowsky & Fall 2012; Fall & Romanowsky 2013; M*/M asa function ofM ,obtained with kinematic data,
DM DM
Courteau & Dutton 2015). Third, the lack of correlation with the totally independent results from abundance matching
betweenλ×(jd/jDM)andΣ*(1kpc)(ornS)suggeststhatthe (Conroy & Wechsler 2009; Behroozi et al. 2010, 2013b;
central bulges are decoupled from the kinematic properties of Mosteretal.2010,2013).Theresultingdependenceoftheratio
the outer disk to which our data are sensitive. M*/M as a function of halo mass is shown in Figure 7,
DM
again for our “full” and “best” samples with converged NFW
3.3. Baryon to Dark Matter Mass Ratios
MCmodels.Forcomparisonweshowtheabundancematching
Our NFW MC modeling gives the ratio of the baryonic results at z∼1 and 2 (Behroozi et al. 2013b; Moster
and stellar masses in the disk to the mass of the dark matter et al. 2013).
10
Description:The inferred halo scale angular momentum distribution is broadly consistent with that theoretically predicted for their dark matter halos, in terms of
