DDraft version May 11, 2017
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2: AN EPOCH OF DISK ASSEMBLY
Raymond C. Simons , Susan A. Kassin , Benjamin J. Weiner , Sandra M. Faber , Jonathan R. Trump , TimothyM. Heckman , David C. Koo , Camilla Pacifici , Joel R. Primack , Gregory F. Snyder , Alexander de laVega Johns Hopkins University, Baltimore, 3400 North Charles St., MD, 21218, USA Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD, 21218, USA Steward Observatory, 933 N. Cherry St., University of Arizona, Tucson, AZ 85721, USA UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA Department of Physics, University of Connecticut, 2152 Hillside Rd Unit 3046, Storrs, CT 06269, USA NASA Postdoctoral Program Fellow Physics Department, University of California, Santa Cruz, CA 95064, USA Giacconi Fellow
Draft version May 11, 2017
ABSTRACTWe explore the evolution of the internal gas kinematics of star-forming galaxies from the peak ofcosmic star-formation at z ∼ V rot , which quantifyordered motions, and gas velocity dispersion σ g , which quantify disordered motions, are adopted fromthe DEEP2 and SIGMA surveys. This sample covers a continuous baseline in redshift from z = 2 . z = 0 .
1, spanning 10 Gyrs. At low redshift, nearly all sufficiently massive star-forming galaxies arerotationally supported ( V rot > σ g ). By z = 2, the percentage of galaxies with rotational support hasdeclined to 50% at low stellar mass (10 − M (cid:12) ) and 70% at high stellar mass (10 − M (cid:12) ).For V rot > σ g , the percentage drops below 35% for all masses. From z = 2 to now, galaxies exhibitremarkably smooth kinematic evolution on average. All galaxies tend towards rotational support withtime, and it is reached earlier in higher mass systems. This is mostly due to an average decline in σ g by a factor of 3 since a redshift of 2, which is independent of mass. Over the same time period, V rot increases by a factor of 1.5 for low mass systems, but does not evolve for high mass systems. Thesetrends in V rot and σ g with time are at a fixed stellar mass and should not be interpreted as evolutionarytracks for galaxy populations. When galaxy populations are linked in time with abundance matching,not only does σ g decline with time as before, but V rot strongly increases with time for all galaxymasses. This enhances the evolution in V rot /σ g . These results indicate that z = 2 is a period of diskassembly, during which the strong rotational support present in today’s massive disk galaxies is onlyjust beginning to emerge. Subject headings: galaxies: evolution - galaxies: formation -galaxies: fundamental parameters - galax-ies: kinematics and dynamics INTRODUCTION
The peak of cosmic star-formation at z ∼ σ g ) in addition to ordered rotation ( V rot ) (e.g.,Genzel et al. 2006; F¨orster Schreiber et al. 2006; Law etal. 2007; Wright et al. 2007; Genzel et al. 2008; Shapiro etal. 2008; Law et al. 2009; F¨orster Schreiber et al. 2009;Swinbank et al. 2011; Law et al. 2012; Newman et al.2013; Glazebrook 2013; Wisnioski et al. 2015; Price etal. 2016; Simons et al. 2016; Mason et al. 2017; Straat-man et al. 2017). Even still, the majority of massivestar-forming galaxies, log M ∗ /M (cid:12) >
10, at this redshiftare rotation-dominated ( V rot > σ g ), with at least 70%showing disk-like kinematic signatures (Wisnioski et al.2015). ∗ [email protected] However, these galaxies are unlike local disks. Locally,massive star-forming galaxies have much stronger rota-tional support ( V rot /σ g ∼
10) and low values of σ g ( ∼ − ). At a redshift of 2, the quantity V rot /σ g rarelyexceeds a few, indicating that these galaxies mature sig-nificantly in the 10 Gyrs between z = 2 and now.The evolution of V rot and σ g from z = 1 . V rot /σ g byincreasing in V rot and declining in σ g . This evolutionis a function of mass, with massive galaxies developingstrong rotational support first (i.e., “kinematic downsiz-ing”; Kassin et al. 2012).Although significant progress has been made in un-derstanding the kinematic state of star-forming galaxiesat even higher redshifts, it is unclear how these lowerredshift trends from Kassin et al. (2012) relate to thelarge amounts of disordered motions found in galaxiesat z = 2. One might not expect a smooth extension to z = 2, where the processes governing galaxy assemblyare violent and inhospitable to disk formation.To examine this, or any kinematic evolution, it is im- a r X i v : . [ a s t r o - ph . GA ] M a y Simons et al.portant to have a large and homogenous sample. In ad-dition, the sample must have minimal biases, cover thesame galaxy mass range at all redshifts, and ideally havekinematics measured using the same technique. To ad-dress these factors, we combine the kinematics samplesfrom the DEEP2 survey (Kassin et al. 2012), which spans0 . < z < .
2, and the SIGMA survey (Simons et al.2016), which spans 1 . < z < .
5. These samples havesimilar selections. They are both morphologically unbi-ased (i.e., they do not select only disk-like systems) andthey trace the star-formation main sequence. Both sam-ples contain enough galaxies to overcome the large intrin-sic scatter in galaxy kinematics. Furthermore, measure-ments were made using the same fitting routine in bothsurveys. The combined sample covers a large redshiftrange, 0 . < z < .
5, and spans a significant portionof the age of universe, from 2.6 to 12.4 Gyr. In this pa-per, we quantify the evolution of the following properties: V rot , σ g and their contributions to the total dynamicalsupport. Moreover, we examine how these properties de-pend on stellar mass.The format of this paper is as follows. First, in § § §
5, we linkgalaxy populations in time and examine their kinematicevolution. Finally, we summarize our conclusions in § m ,Ω Λ ) = (0.704, 0.272, 0.728). DATA
Measurements of the internal kinematics of star-forming galaxies are adopted from Kassin et al. (2012)(from the DEEP2 survey) and Simons et al. (2016) (fromthe SIGMA survey). Both samples are morphologicallyunbiased (i.e., not only disk-like systems are selected)and kinematics are measured from rest-optical emissionslines which trace the hot ionized T ∼ K gas in galax-ies. The galaxies used in this paper are shown in a stel-lar mass vs. star formation rate ( M ∗ − SF R ) diagramin Figure 1. We compare with the star-formation mainsequence at their respective redshifts (from Speagle et al.2014) and show that our sample is representative of typ-ical star-forming galaxies over 0 . < z < .
5. We notethat the galaxies over the redshift range 0 . < z < . σ scatter of therelation at all redshifts.Below we describe the DEEP2 and SIGMA samples( § § § DEEP2 sample
The intermediate redshift sample (0 . < z < . (cid:48)(cid:48) wide and the spectral resolution is R ∼ σ instr = 26 km s − ). The nominal on-sourceexposure times were 1 hour. The spectra were observedin natural optical seeing conditions, which varied be-tween 0.5 (cid:48)(cid:48) and 1.2 (cid:48)(cid:48) . Galaxies are selected to have brightenough emission lines to measure kinematics ( (cid:38) − erg s − cm − ). Additionally, galaxies are required tohave V - and I -band imaging from the Hubble Space Tele-scope/Advanced Camera for Surveys (HST/ACS) instru-ment. Inclinations were measured from the Hubble imag-ing using the SExtractor program (Bertin & Arnouts1996) in Lotz et al. (2008). Galaxies are selected to haveinclinations between 30 ◦ < i < ◦ , avoiding dust ef-fects in edge-on systems and highly uncertain inclinationcorrections in face-on systems. Furthermore, galaxies arerequired to have slits aligned within 40 ◦ of the photo-metric major axes (Weiner et al. 2006a; Covington etal. 2010). Following Simons et al. (2015), we apply aconservative selection on the rest-optical HST half-lightdiameter ( D > . SIGMA sample
The high redshift (1 . < z < .
5) sample used in thispaper is from the SIGMA kinematics survey (Simons etal. 2016; hereafter S16). The sample is briefly discussedhere and we refer to S16 for further details. The galax-ies in SIGMA are located in the UDS, GOODS-S andGOODS-N fields. Spectra were taken with the MOS-FIRE spectrograph (McLean et al. 2010, 2012) on theKeck-I telescope as a part of the TKRS-2 survey (Wirthet al. 2015) and were also published in Trump et al.(2013) and Barro et al. (2014). The slits were 0.7 (cid:48)(cid:48) wideand the spectral resolution is R ∼ σ inst = 35 kms − ). The on-source exposure times ranged between 1.5- 2 hours and the near-infrared seeing varied between 0.45- 0.85 (cid:48)(cid:48) . HST/Wide Field Camera 3 (WFC3) imaging isavailable for all of the galaxies in SIGMA through theCANDELS survey (Grogin et al. 2011; Koekemoer et al.2011). Axis ratios were measured from the HST/WFC3 H -band image using the GALFIT software (Peng et al.2010) by van der Wel et al. (2012) and are used to de-rive inclinations. As in the K12 sample, all galaxies arerequired to have at least one slit aligned with the photo-metric position angle and intrinsic emission sizes whichare large enough to resolve kinematics ( D > . Kinematic Measurements
Kinematics (rotation velocity V rot , and gas velocitydispersion σ g ) were measured from strong nebular emis-sion lines (H α and [O iii ] λ ROTCURVE pro-gram (Weiner et al. 2006a) in Kassin et al. (2007) andS16 for the DEEP2 and SIGMA surveys, respectively.The measurement technique is briefly discussed here andthe reader is referred to Weiner et al. (2006a) for details. ∼
2: An Epoch of Disk Assembly 3
Fig. 1.—
Galaxies in the DEEP2 (black circles) and SIGMA (grey diamonds) kinematics samples lie along the star-formation mainsequence at their respective redshifts. The panels show redshift bins in equal intervals of lookback time. The solid and dashed lines are themain sequence fit and rms scatter for each redshift bin, respectively, from Speagle et al. 2014 and renormalized to a Chabrier IMF.
Seeing blurs rotation and artificially elevates the veloc-ity dispersion in the center of the galaxy (due to the risein the rotation curve) and needs be taken into accountin the kinematic modeling.
ROTCURVE models the veloc-ity profile of the emission line, taking into account see-ing, with two components: an arctangent rotation curveand a constant dispersion term. The rotation velocityuncorrected for inclination ( V rot × sin i ) is measured atthe flat portion of the rotation curve. Recent evidencehas suggested that the rotation curves of high redshiftstar-forming galaxies fall, instead of flatten, beyond 1.5effective radii (Lang et al. 2017; Genzel et al. 2017). Ourhigh redshift data do not extend far enough to distin-guish between these two cases. However, there is only asmall difference in the maximum velocity derived froma fit to either model. Adopting a value of V rot whichis measured at a smaller radius, e.g., at 1.5 R e , wouldlower the rotation velocities by ∼ . V rot asmeasured at the flat part of the modeled rotation curvein all of our galaxies.Typical uncertainties on measurements of V rot × sin i are approximately 10 km s − for DEEP2 and 30 km s − for SIGMA. Uncertainties on σ g are approximately 15km s − for DEEP2 and 25 km s − for SIGMA. Inclina-tion corrections are applied to the ROTCURVE values usingthe rest ∼ V -band axis ratio measured from the Hubbleimages (ACS V - and I -bands in K12 for 0 . < z < . . < z < .
2, respectively, and WFC3 H -band inS16 for 1 . < z < . σ g integrates small scalevelocity gradients below the seeing limit (Covington etal. 2010; Kassin et al. 2014). It is mostly due to non-ordered motions among HII regions, but also includessmaller contributions from thermal broadening ( ∼
10 kms − for H gas at 10 K) and the internal turbulence inHII regions ( ∼
20 km s − in local HII regions, Shields1990). As in K12, we refer to σ g as tracing “disorderedmotions”.The quantity V rot measures the ordered motions of agalaxy while the quantity σ g measures its disordered mo-tions. We follow Weiner et al. (2006a) and Kassin etal. (2007) by combining both forms of dynamical sup-port into the quantity S . = (cid:113) . V rot + σ g . This termserves as a kinematic tracer of the total mass in anisothermal potential (Weiner et al. 2006a). Stellar Mass and SFR Measurements
Stellar masses ( M ∗ ) were measured for the galaxies inthe K12 sample using absolute B-band magnitudes andrest B − V colors (Bell & de Jong 2001; Bell et al. 2005)with empirical corrections from spectral energy distri-bution (SED) fitting (Bundy et al. 2006), as in Lin etal. (2007). A Chabrier (2003) initial mass function wasadopted and the uncertainties in stellar mass are approx- Simons et al.
Fig. 2.—
Star-forming galaxies evolve in V rot and σ g with time. The small faint background points are measurements for individualgalaxies. The large points and associated error bars show medians of the individual points in bins of lookback time and their standarderror. Solid lines are the best-fit relations to the median points. Low mass and high mass galaxies are shown in blue and red, respectively.Top left: The average rotation velocity, V rot , increases with time since z = 2 . σ g , which traces disordered motions, decreases precipitously from z = 2 . V rot and σ g are normalized by S . , which traces the total dynamical supportof galaxies. This allows us to examine the fraction of total dynamical support that V rot and σ g provide to galaxies. At z ∼
2, low massgalaxies have a significant fraction of their total support in disordered motions. With time, all galaxies on average increase in rotationalsupport (i.e., increase in V rot /S . ) and decrease in dispersion support (i.e., decrease in σ g /S . ). This happens earlier for higher massgalaxies. imately 0.2 dex. Star-formation rates (SFR) were de-rived in Noeske et al. (2007). For galaxies with 24 µ mdetection, the obscured SFR was measured from the IRluminosity using the SED templates of Chary & Elbaz(2001) and is added to the unobscured component of theSFR, derived from the DEEP2 emission line luminosities(Weiner et al. 2007). Otherwise, the SFR was measuredfrom the extinction corrected emission line luminositiesfollowing the calibration in Kennicutt (1998).Stellar masses for the galaxies in SIGMA were mea-sured from SED fitting to the UV-optical-NIR data avail-able in the CANDELS fields, as described in Barro etal. (2014). The fit was performed with FAST (Kriek etal. 2009) assuming a Chabrier (2003) initial mass func-tion, Bruzual & Charlot (2003) stellar population syn-thesis models and a Calzetti extinction law (Calzetti etal. 2000). The uncertainties in stellar mass are approxi-mately 0.3 dex (Mobasher et al. 2015). For galaxies withdetections in the mid-to-far IR, the SFR was calculatedusing both the obscured (IR) and unobscured (UV) com- ponents (Kennicutt 1998). Otherwise, SFRs are derivedfrom the extinction corrected UV using the dust param-eters of the best fit SED model. KINEMATIC EVOLUTION AT FIXED STELLAR MASS
We use measurements of V rot , σ g , and S . from theK12 and S16 samples to explore the kinematic evolu-tion of star-forming galaxies from z ∼ . z ∼ . t L ) are adopted,each spanning 1 Gyr. Our conclusions are insensitiveto the size of these intervals. We first examine evolu-tion at fixed stellar mass for two stellar mass ranges:9 < log M ∗ /M (cid:12) <
10 (hereafter referred to as “lowmass”) and 10 < log M ∗ /M (cid:12) <
11 (hereafter referredto as “high mass”). The z ∼ . t L ≈
10 Gyr) bin inS16 does not include galaxies below log M ∗ /M (cid:12) = 9 . . < log M ∗ /M (cid:12) <
10. Thehigh stellar mass bin at z ∼ . ∼
2: An Epoch of Disk Assembly 5
Fig. 3.—
The evolution of S . , a tracer of the galaxy potentialwell depth, is shown. This quantity mildly declines since z = 2 . S . has not changedsignificantly in high mass galaxies (red). The points and lines arethe same as in Figure 2. the subsections for details.We find a large amount of scatter in the individualmeasurements of V rot , σ g and S . at a given mass butsmooth average trends with mass and time, highlight-ing the need for a large sample. On average, the rota-tion velocities of low mass galaxies rise by a factor of1.5 from z ∼ . V rot over the same pe-riod. The gas velocity dispersion smoothly declines from z ∼ . V rot and σ g to the total dynamical support, as quantified by S . .While all galaxies tend towards rotational support withtime, it is obtained earlier for galaxies with higher stellarmasses. This phenomenon, dubbed “kinematic downsiz-ing”, was first shown by K12 to z = 1 . z = 2 .
5. This result follows from the facts that σ g does not depend significantly on mass and low massgalaxies sit in shallower potential wells than high massgalaxies.Low mass galaxies are delayed in their kinematic de-velopment, taking an additional 4 - 5 Gyr to reach thesame level of rotational support as high mass galaxies.From z = 2 to z = 1, low mass galaxies are stronglysupported by dispersion ( σ g /S . > . z = 0 . σ g /S . < . Increase in V rot With Time for Low Mass Galaxies
The evolution of the rotation velocity, V rot , is shown inthe top left panel of Figure 2. The galaxies in our samplespan a large range in V rot at fixed epoch. This scatter is mostly intrinsic but is due in part to our relatively widebins in stellar mass (1 dex) and to a lesser extent frommeasurements uncertainties.At all times, the average rotation velocity is higher forgalaxies with higher stellar mass. We find a mild increaseof the average V rot with time in galaxies with low stellarmass, rising from 50 km s − at z = 2 to 70 km s − at z = 0 .
2. High mass galaxies show little evolution in V rot over the same time period. We estimate the standarderror on the median V rot in each bin through bootstrapresampling and perform an uncertainty weighted leastsquares fit to the median points with a line:log (cid:18) V rot km s − (cid:19) = a (cid:18) t L Gyr (cid:19) + b (1)The best-fit values of the slope and intercept are a = − . ± .
01 (i.e., slow rise) and b = 1 . ± .
07 for ourlow stellar mass bin, and a = − . ± .
014 (i.e., noevolution) and b = 2 . ± .
09 for our high stellar massbin. The intercept value of each mass bin, i.e., V rot at z = 0, is consistent with the local stellar mass Tully-Fisher relation (90 and 175 km s − at log M ∗ /M (cid:12) = 9 . .
5, respectively; Reyes et al. 2011).
Smooth Decay of σ g With Time for All Galaxies
The evolution of the gas velocity dispersion, σ g , isshown in the top right panel of Figure 2. As with V rot ,galaxies in our sample span a wide range in σ g at allepochs, largely due to measurement uncertainties.The galaxies at the highest redshifts have values of σ g which are roughly 3 times larger than the galaxiesat the lowest redshifts (see also Wisnioski et al. 2015).The quantity σ g decreases from 70 km s − to 20 km s − over 0 . < z < .
0. As first shown in K12, σ g roughlydoubles over 0 . < z < .
2, which spans approximately8 Gyrs. Although the time from z = 1 to z = 2 is muchshorter, only 3 Gyrs, we find significant evolution in σ g during this period, with the median increasing from 40km s − to 70 km s − . We perform a fit to the median t L − σ g relation for our two mass bins with:log (cid:16) σ g km s − (cid:17) = a (cid:18) t L Gyr (cid:19) + b (2)The best fit values are a = 0 . ± .
010 and b =1 . ± .
04 for the low mass bin and a = 0 . ± . b = 1 . ± .
07 for the high mass bin. There isno significant difference between the evolution of σ g inthe low mass and high mass bins. This result impliesa mass-independent half-life timescale for σ g , the timeover which it declines by a factor of 2, of approximately6 Gyrs. The intercept values, i.e., σ g at z = 0, areconsistent with measurements of the ionized gas velocitydispersion in local star-forming disk galaxies ( ∼ − − , e.g., Epinat et al. 2008). Evolution of Dynamical Support, S . , With Time In Figure 3, we examine the evolution of S . , whichtraces the total dynamical support of galaxies. We fit aline to the median S . vs. lookback time:log (cid:18) S . km s − (cid:19) = a (cid:18) t L Gyr (cid:19) + b (3) Simons et al. Fig. 4.—
The fraction of star-forming galaxies with disk-like kinematics declines with increasing redshift and decreasing mass. In theleft panel, the fractions of galaxies with V rot /σ g > z = 2, only ∼ −
70% of all galaxies meet the very lenient criteria of V rot /σ g >
1. In the right panel, the fractions of galaxies with V rot /σ g > z = 2, less than 40% of all galaxies meet this criterion. Measurements from kinematics surveys in the literatureare shown as open squares and are color-coded by their mass ranges with the same stellar mass bins used for our data. Samples which spanmultiple bins in mass are shown with two or more colors. We find good agreement among all surveys once stellar mass is accounted for. The best-fit slope and intercept values are a = 0 . ± .
008 and b = 1 . ± .
06 for the low mass bin and a = 0 . ± .
01 and b = 2 . ± .
09 for the high mass bin.The quantity S . has marginally declined in low massgalaxies since z = 2 .
5, i.e., they now sit in shallowerpotential wells, but we measure no significant evolutionin high mass galaxies.While the fit to the low mass galaxies indicates asmooth evolution, we caution that the trend is largelydriven by the high redshift measurements. The medianpoints below z = 1 . z = 1 . Fraction of Total Dynamical Support Given By V rot and σ g Over Time
We normalize V rot and σ g by S . and measure the rel-ative contributions of ordered and disordered motions,respectively, to the total dynamical support. A thindisk which is supported by rotation will tend toward V rot /S . = √
2, and a system which is supported bydispersion will tend toward σ g /S . = 1.The evolution of V rot /S . is shown in the bottom leftpanel of Figure 2. K12 showed that V rot /S . rises from z = 1 . z = 2 . V rot /S . , i.e., they are more rotationally-supported,than low mass galaxies. This phenomenon, dubbed“kinematic downsizing” (K12, S16), is present out to z = 2 .
5. High mass galaxies reach a strong level of rota-tional support ( V rot /S . > .
3) by a lookback time of5 − z ∼ .
7. Low mass galaxies reach the samedegree of rotational support a few Gyrs later, by z ∼ .
2. We fit a line to the median V rot /S . using only the risingpart of this relation, which covers 0 . < z < . . < z < . V rot S . = a (cid:18) t L Gyr (cid:19) + b (4)The best fit slope and intercept are a = − . ± . b = 1 . ± .
04 for low mass galaxies and a = − . ± .
020 and b = 1 . ± .
08 for high mass galax-ies. The values for the best-fit slopes are similar andsuggest that low mass and high mass galaxies developrotational support at similar rates. We refit the highmass galaxies with a slope fixed to the best-fit slope ofthe low mass galaxies. At fixed slope, the high mass andlow mass trends are offset by 4.3 Gyr. In other words, theassembly of rotational support in the high mass galax-ies is, on average, followed 4 − σ g /S . (Figure 2, bottom right). On average, σ g /S . declines with time for all galaxies and is always larger forlow mass galaxies. Given that σ g itself does not dependon stellar mass (top right panel), its relative contributionto the total dynamical support will always be larger ingalaxies with lower stellar mass. Beyond z = 1.0, or alookback time of ∼ σ g /S . (cid:38) . z = 0, both low mass andhigh mass galaxies have low levels of dispersion sup-port ( σ g /S . ≤ .
4) and high levels of rotational sup-port ( V rot /S . ≥ . σ g /S . over the declining part of the relation, covering0 . < z < . . < z < . ∼
2: An Epoch of Disk Assembly 7 σ g S . = a (cid:18) t L Gyr (cid:19) + b (5)The best fit slope and intercept values are a = 0 . ± .
01 and b = 0 . ± .
04 for low mass galaxies and a = 0 . ± .
02 and b = − . ± .
11 for high massgalaxies. As before, the slopes are statistically identi-cal and suggest that the dispersion support declines atsimilar rates in both mass bins. We refit to the highmass galaxies with a slope fixed to low mass fit and mea-sure an offset between the mass bins of 4.6 Gyrs. Theseresults mirror our earlier conclusions and once again sug-gest that kinematic assembly in low mass galaxies is de-layed from that in high mass galaxies by ∼ − THE FRACTION OF GALAXIES WITH ROTATIONALSUPPORT
We now investigate the fraction of star-forming galax-ies with V rot /σ g > V rot /σ g > M ∗ /M (cid:12) >
10) star-forming disk galaxies in the local universe tend to havevalues of V rot /σ g > α (Epinat et al. 2008) and V rot /σ g >
10 when measured through a cold neutral gastracer such as HI (Walter et al. 2008; de Blok et al. 2008).We divide our sample into 3 overlapping bins in fixedstellar mass: 9 < log M ∗ /M (cid:12) <
10 (“low mass”),9 . < log M ∗ /M (cid:12) < . < log M ∗ /M (cid:12) <
11 (“high mass”).Figure 4 (left) shows the evolution of the fraction ofgalaxies with V rot /σ g >
1. As first reported by K12 to z = 1 .
2, this fraction increases with time for all galax-ies and is always higher at higher stellar masses. TheSIGMA sample extends these trends to z = 2. The frac-tion of galaxies in SIGMA with V rot /σ g > ± ± ± z = 2. That is, abouthalf of the star-forming galaxies in the low and interme-diate mass bins have dominant contributions from σ g .The standard error on the reported fractions are cal-culated via bootstrap resampling. Measurement uncer-tainties will push individual galaxies above or below thethreshold in V rot /σ g and we account for this by perturb-ing the values of V rot and σ g by their associated errorson each draw.In Figure 4 (right), the fraction of galaxies which have V rot /σ g > z = 2 that meet this benchmark for thelow, intermediate and high mass bins are only 15( ± ± ± F , with: F (cid:18) V rot σ g > x (cid:19) = a (cid:18) t L Gyr (cid:19) + b (6)The best-fit slope and intercept for F ( V rot /σ g >
1) are a = − . ± .
006 and b = 1 . ± .
04 for low massgalaxies, a = − . ± .
009 and b = 1 . ± .
066 forintermediate mass galaxies and a = − . ± .
009 and b = 1 . ± .
071 for high mass galaxies. Similarly, thebest-fit values for F ( V rot /σ g >
3) are a = − . ± . Fig. 5.—
Galaxies in our sample are linked in time using anabundance matching model from Moster et al. 2013. Three galaxypopulations with z = 0 halo masses of 11 < M h /M (cid:12) < . . < M h /M (cid:12) < . . < M h /M (cid:12) < . and b = 0 . ± .
07 for low mass galaxies, a = − . ± .
007 and b = 1 . ± .
05 for intermediate mass galaxiesand a = − . ± .
01 and b = 1 . ± .
09 for high massgalaxies.By z = 0 .
2, more than 70% of star-forming galaxieswith log M ∗ /M (cid:12) > V rot /σ g > V rot /σ g >
1. Even at z = 2, most of the sample shows some degree of rota-tion. More than half of galaxies at this redshift have V rot /σ g >
1. However, these galaxies are unlike localrotating galaxies, whose V rot /σ g often exceeds 5. Theredshift at which 35% of galaxies have V rot /σ g > z = 1, 1.5 and 2.6 for the low, intermediate and highmass bins, respectively. Kinematic analogs of local well-ordered disk galaxies are rare at z = 2. At this redshift,only 7 (16%) and 2 (4%) of the galaxies in our samplereach V rot /σ g > >
10, respectively.
Comparison With the Literature
In Figure 4 (left), we compare our measurements withthose in the literature. Any comparison between surveysis complicated due in large part to differences in sampleselection and measurement techniques. We refrain fromperforming detailed corrections between the samples andonly report here a straightforward comparison. We in-clude measurements from the following surveys: KROSSat z ∼ z ∼ z ∼ z ∼ . z ∼ z ∼ . . Fig. 6.— The same as Figure 2, but for galaxy populations linked in time via abundance matching. All galaxy populations on averagespin-up with time and decrease in disordered motions, i.e., increase in V rot and decline in σ g . High mass galaxies reach strong levels ofrotational support, i.e., V rot /S . > . σ g /S . < . 4, by z = 1 . M ∗ ,z =0 /M (cid:12) ∼ . M ∗ ,z =0 /M (cid:12) ∼ . 3) and high (log M ∗ ,z =0 /M (cid:12) ∼ . 1) mass abundance matched populations,respectively. The small points are individual galaxies and large points are the medians in bins of lookback time. The lines are the best-fitrelations to the median points of each mass bin. . 0. A catalog of kinematic measurements for 586 typ-ical star-forming galaxies was released in Harrison etal. (2017). We use this catalog to calculate the frac-tion of resolved galaxies in KROSS which have rota-tion velocities that exceed their intrinsic velocity dis-persions. We adopt mass bins which overlap our sam-ple and these are shown as separate points in Figure 4.The first-year sample from the KMOS-3D survey con-tains 191 star-forming galaxies in two redshift intervals:0 . < z < . . < z < . 7. These galaxiescover a stellar mass range of 9 . (cid:46) log M ∗ /M (cid:12) (cid:46) . z ∼ (cid:46) log M ∗ /M (cid:12) (cid:46) . z ∼ z ∼ z ∼ 2, respectively. Weinclude the sample of 48 MASSIV galaxies from Ver-gani et al. (2012), covering 1 < z < . . (cid:46) log M ∗ /M (cid:12) (cid:46) 11. We adoptthe V /σ measurements in their Table 2 and calculatea rotation-dominated fraction of 65%. We include thehigh redshift sample from the AMAZE/LSD survey of33 galaxies over 2 . < z < . . (cid:46) log M ∗ /M (cid:12) (cid:46) . 0. Gnerucci et al. (2011) report that 33% of the AMAZE galaxies can beclassified as rotationally-supported. Finally, we in-clude results from the KMOS Deep survey (KDS) at z ∼ . < log M ∗ /M (cid:12) < 11. Turner et al. 2017 reportthat 13/32 of the isolated field galaxies in this sampleare rotation-dominated. We remove the 4 galaxies withlog M ∗ /M (cid:12) < 9, 3 of which are dispersion-dominated,and compute a rotation-dominated fraction of 43% overour mass range. TRACING GALAXY POPULATIONS WITH ABUNDANCEMATCHING In previous sections of this paper, galaxies trends arereported for a fixed stellar mass. However, as a naturalconsequence of ongoing star formation and mass accre-tion, the galaxies in our sample will grow in stellar masswith time and migrate between mass bins. As such, thetrends discussed so far should not be interpreted as tracksfor individual galaxy populations. In order to trace theevolution of a galaxy population from z = 2 . z = 0,we adopt a model linking high redshift galaxies with theirmore massive low redshift descendants.Galaxy populations are tracked in time using the multi- ∼ 2: An Epoch of Disk Assembly 9 Fig. 7.— The same as Figure 3, but for galaxy populations linkedin time via abundance matching. The quantity S . increases withtime for all galaxy populations, indicating that their potential wellsgrow with time. The lines, points and color scheme are the sameas in Figure 6. epoch abundance matching (MEAM) model (Moster etal. 2013), following Papovich et al. (2015). The con-clusions of this section are similar if we adopt otherabundance matching models from the literature (e.g.,Rodriguez-Puebla et al. 2017). The abundance match-ing technique assumes that there is a monotonic relationbetween the stellar mass of a galaxy and its halo mass,with the most massive galaxies residing in the most mas-sive halos. Observed stellar mass functions are then rankassigned to simulated halo mass functions at all redshiftsand galaxy populations are tracked in time.Moster et al. (2013) adopt halo and subhalo evolutionfrom the Millenium simulation and stellar mass functionsup to z ∼ z = 0 halo massesof 11 < M h /M (cid:12) < . 5, 11 . < M h /M (cid:12) < . . < M h /M (cid:12) < . 5. These correspond to typical z = 0 stellar masses of log M ∗ /M (cid:12) ∼ M ∗ /M (cid:12) < z ∼ 1. Similarly, theintermediate mass bin only extends to z ∼ . 5. We usethese tracks to link galaxies in our sample with their ap-propriate descendants. Increasing V rot , Decreasing σ g , and Increasing S . With Time Using the abundance matching technique to linkgalaxy populations in time, we now revisit the evolutionof V rot , σ g and S . in Figures 6 and 7. The most striking difference is in the evolution of V rot .At fixed stellar mass we measured mild to no evolutionin V rot . However, in Figure 6 (top left) we show that V rot strongly evolves for the evolving galaxy population: star-forming galaxies spin-up with time on average . Weuse Eq. 1 and fit to the median evolution of V rot in eachmass bin. The best-fit slope and intercept values are a = − . ± . 012 and b = 0 . ± . 054 for low massgalaxies, a = − . ± . 010 and b = 1 . ± . 04 forintermediate mass galaxies, and a = − . ± . 090 and b = 1 . ± . 070 for high mass galaxies. The best-fitslopes are similar and indicate that star-forming galaxiesin this mass range have a typical V rot doubling timescaleof ∼ σ g at fixedstellar mass ( § σ g dra-matically declines with time (Figure 6, top right). In-termediate mass and high mass galaxies trace each otherback to z = 1 . 5, rising from 30 km s − at z = 0 . − at z = 1 . 5. The high mass galaxies, for whichwe have measurements out to high redshift, reach typicalvalues of σ g = 60 km s − at z = 2.The evolution of S . for the abundance matched pop-ulations is shown in Figure 7. At fixed stellar mass, wefound a mild decline in S . with time. At fixed galaxypopulation, we find that S . increases with time forall galaxies. This result suggests that, perhaps unsur-prisingly, galaxy potential wells steepen as they acquiremass. The quantity σ g declines with time on averageand so the additional dynamical support comes from theincrease in V rot .As before, we use Eq. 3 and fit to the median S . evo-lution for each mass bin. The best-fit slope and interceptvalues are a = − . ± . 019 and b = 1 . ± . 08 forthe fixed population at low mass, a = − . ± . b = 2 . ± . 05 at intermediate mass, and a = − . ± . 04 and b = 2 . ± . 28 at high mass.We normalize V rot and σ g by S . (Figure 6, bot-tom panels) and find the same general behavior as be-fore: while all galaxy populations increase in rotationalsupport with time (rising in V rot /S . and declining in σ g /S . ), massive galaxies are the most rotationally sup-ported at all times on average. The high mass populationreaches a strong level of rotational support ( V rot /S . > . 3) as far back as z = 1 . 5, while the intermediate massand low mass populations reach the same degree of rota-tional support at z = 0 . z = 0 . 2, respectively. Fraction of galaxies with V rot /σ g > and 3 withtime In Figure 8, we use the linked galaxy populations torevisit how the fraction of galaxies with V rot /σ g > σ g with time and its weak dependance onmass. Additionally, unlike the case at fixed stellar mass,there is a contribution from the steep rise in V rot withtime in all galaxy populations and its strong dependanceon mass.0 Simons et al. Fig. 8.— The same as Figure 4, but for galaxy populations linked in time via abundance matching. For all three galaxy populations(low, intermediate and high mass), the fractions of galaxies with V rot /σ g > V rot /σ g > z ∼ . z ∼ . We use Eq. 6 to fit to the evolution in each mass bin.The best-fit slope and intercept for F ( V rot /σ g > 1) are a = − . ± . 012 and b = 0 . ± . 05 at low mass, a = − . ± . 007 and b = 1 . ± . 04 at intermediatemass and a = − . ± . 009 and b = 1 . ± . 07 athigh mass. For F ( V rot /σ g > 3) the best-fit values are a = − . ± . 013 and b = 0 . ± . 06 at low mass, a = − . ± . 01 and b = 0 . ± . 06 at intermediatemass and a = − . ± . 01 and b = 1 . ± . 09 at highmass.The majority of galaxies in these populations havesome degree of rotational support (Figure 8, left). Morethan half of the galaxies in the low mass bin have V rot /σ g > z = 1 and the intermediate and highmass bins reach similar values at z = 2.However, it is much less common for galaxies to reach V rot /σ g > V rot /σ g > z ∼ . z ∼ . M ∗ ,z =0 /M (cid:12) > . 0; i.e., Milky-Way and M31-mass galaxies) were likely weakly rotationally-supported, V rot /σ g < 3, at the peak of cosmic star-formation. DISCUSSION AND CONCLUSIONS To explore the evolution of internal galaxy kinematicsover a continuous period of time from z = 2 . . < z < . . < z < . 5; Simons et al. 2016).The full sample contains 507 star-forming galaxies andcovers a mass range of 9 < log M ∗ /M (cid:12) < 11. Our sam-ple is homogenous and representative of galaxies on thestar-formation main sequence. There is no selection on morphology. Kinematics were measured using the samesoftware in both surveys. This data set is the largestcontiguous and morphologically unbiased sample of kine-matics in the literature and it extends the work of Kassinet al. (2012) from z = 1 . z = 2.Kinematics are measured from rest-optical emissionlines (H α , [O iii ] λ K gas. Two kinematic quantities are mea-sured: the rotation velocity V rot , which traces orderedmotions, and the gas velocity dispersion σ g , which tracesdisordered motions. Both of these quantities provide dy-namical support, and when combined into the quantity S . = (cid:113) . V rot + σ g , trace the depth of galaxy poten-tial wells (Weiner et al. 2006a; Kassin et al. 2007; Cov-ington et al. 2010).Average trends in σ g and V rot with time are presentedfor galaxies at fixed stellar mass. We find a smooth anddramatic decline in σ g since z = 2.5. The time for σ g to decline by a factor of two, or the half-life time, is ∼ V rot increases ingalaxies with low stellar mass (9 < log M ∗ /M (cid:12) < < log M ∗ /M (cid:12) < V rot and σ g to S . . At a fixed stellar mass, all galaxieson average rise in V rot /S . with time, i.e. they becomeincreasingly supported by rotation. In the same vein, allgalaxies on average decline in σ g /S . with time, i.e., theybecome decreasingly supported by dispersion. While theslopes of these trends ( V rot /S . and σ g /S . with time)are similar for all masses, high mass galaxies have higher V rot /S . and lower σ g /S . at all times. This indicatesa time delay in the development of rotational supportbetween low mass and high mass galaxies, with low mass ∼ 2: An Epoch of Disk Assembly 11trailing high mass by a few Gyrs (i.e., “kinematic down-sizing”; Kassin et al. 2012; Simons et al. 2016). Thisresult arises from the independence of σ g on mass: agiven value of σ g assumes a more dominant role in theshallower potential wells that host low mass galaxies.These results are complimentary to studies which trackmorphological regularity, tracing the distribution of starsin galaxies, as a function of mass and redshift. Morpho-logically regular stellar disks are increasingly common atlow redshift ( z ≤ . 5) and high mass (e.g., Mortlock etal. 2013; van der Wel et al. 2014; Huertas-Company etal. 2016). For example, Huertas-Company et al. (2016)find that as much as 80% of the stellar mass density at z = 2 resides in galaxies with disturbed/irregular mor-phologies.Galaxies grow in stellar mass with time and so thesetrends, which are at fixed stellar mass, should not be in-terpreted as tracks for individual galaxy populations. Tolink galaxy populations in time, we adopt an abundancematching model. Doing so, we find that V rot increases, σ g decreases, and the potential well depth S . increaseswith time for all galaxies. The simultaneous rise in S . and V rot indicates that galaxies spin-up as they assembletheir mass. On average, a star-forming galaxy of Milky-Way mass which is today strongly supported by rotation( σ g /S . < . 3) was likely strongly supported by disper-sion at z = 2 ( σ g /S . = 0 . z = 0 . 2, the vast majority (90%) of star-forming galaxies are rotation-dominated ( V rot > σ g ). By z = 2,the percentage of galaxies with V rot > σ g has declinedto 50% for low mass systems (10 − M (cid:12) ) and 70%for high mass systems (10 − M (cid:12) ). These measure-ments are consistent those in the literature once stellarmass is taken into account. We consider a stronger cri-terion for rotational support, V rot > σ g , and find thatthe fractions of galaxies meeting this threshold drop be-low 35% for all masses. For perspective, most sufficientlymassive galaxies in the local universe (log M ∗ /M (cid:12) > V rot /σ g > V rot /σ g > z = 2 and the kinematic characteristics oflocal disks, i.e., low σ g and high V rot /σ g , were only justbeginning to emerge. This epoch is one of disk assem-bly, as star-forming galaxies are rapidly assembling stel-lar mass and beginning to develop the first well-ordereddisks. ACKNOWLEDGEMENTS The data presented herein were obtained at the W.M.Keck Observatory, which is operated as a scientific part-nership among the California Institute of Technology, theUniversity of California and the National Aeronauticsand Space Administration. We wish to extend thanksto those of Hawaiian ancestry on whose sacred moun-tain we are privileged guests. We thank the referee forproviding a useful report which has improved this pa-per. RCS and SAK would like to acknowledge an RSACgrant from STScI. GFS appreciates support from a Giac-coni Fellowship at the Space Telescope Science Institute,which is operated by the Association of Universities forResearch in Astronomy, Inc., under NASA contract NAS5-26555. SF and DK acknowledge support from NSFgrant AST-0808133 to UCSC. This work has made useof the Rainbow Cosmological Surveys Database, whichis operated by the Universidad Complutense de Madrid(UCM), partnered with the University of California Ob-servatories at Santa Cruz (UCO/Lick,UCSC). This re-search made use of Astropy, a community-developed corePython package for Astronomy (Astropy Collaborationet al. 2013). REFERENCESAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al.2013, A&A, 558, A33 6Aumer, M., White, S. D. M., Naab, T., & Scannapieco, C. 2013,MNRAS, 434, 3142 6Barro, G., Trump, J. R., Koo, D. C., et al. 2014, ApJ, 795, 1452.2, 2.4Bell, E. F., & de Jong, R. S. 2001, ApJ, 550, 212 2.4Bell, E. 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