z ∼ 2: An Epoch of Disk Assembly

The intermediate redshift sample ($0.1\,\lt \,z\,\lt \,1.2$) used in this paper is from Kassin et al. (2007) and Kassin et al. (2012, hereafter K12). The sample is briefly discussed here and the reader is referred to K12 for further details. The galaxies in the K12 sample are located in the Extended Groth Strip. Spectra, from which the internal kinematics are measured, are from the DEEP2 survey (Newman et al. 2013a) and were taken with the DEIMOS instrument (Faber et al. 2003) on the Keck II telescope. The slits used were 1'' wide and the spectral resolution was $R\,\sim \,5000$ (${\sigma }_{\mathrm{instr}}=26$ km s⁻¹). The nominal on-source exposure times were 1 hr. The spectra were observed in natural optical seeing conditions, which varied between 05 and 12. Galaxies are selected to have bright enough emission lines to measure kinematics (≳10⁻¹⁷ erg s⁻¹ cm⁻²). In addition, galaxies are required to have V- and I-band imaging from the Hubble Space Telescope/Advanced Camera for Surveys (HST/ACS) instrument. Inclinations were measured from the HST imaging using the SExtractor program (Bertin & Arnouts 1996) in Lotz et al. (2008). Galaxies are selected to have inclinations between $30^\circ \,\lt \,i\,\lt \,70^\circ$, avoiding dust effects in edge-on systems and highly uncertain inclination corrections in face-on systems. Furthermore, galaxies are required to have slits aligned within 40° of the photometric major axes (Weiner et al. 2006; Covington et al. 2010). Following Simons et al. (2015), we apply a conservative selection on the rest-optical HST half-light diameter (${D}_{50}\,\gt \,0.8$ times the FWHM of the seeing). This size selection removes 50 galaxies. The sample used in this paper contains 462 galaxies.

The high-redshift ($1.3\,\lt \,z\,\lt \,2.5$) sample used in this paper is from the SIGMA kinematics survey (Simons et al. 2016; hereafter S16). The sample is briefly discussed here and we refer to S16 for further details. The galaxies in SIGMA are located in the UDS, GOODS-S, and GOODS-N fields. Spectra were taken with the MOSFIRE spectrograph (McLean et al. 2010, 2012) on the Keck I telescope as a part of the TKRS-2 survey (Wirth et al. 2015) and were also published in Trump et al. (2013) and Barro et al. (2014). The slits were 07 wide and the spectral resolution was $R\,\sim \,3630$ (${\sigma }_{\mathrm{inst}}=35$ km s⁻¹). The on-source exposure times ranged between 1.5 and 2 hr and the near-infrared (NIR) seeing varied between 045 and 085. HST/Wide Field Camera 3 (WFC3) imaging is available for all of the galaxies in SIGMA through the CANDELS 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 were used to derive inclinations. As in the K12 sample, all galaxies are required to have at least one slit aligned with the photometric position angle and intrinsic emission sizes that are large enough to resolve kinematics (${D}_{50}\,\gt \,0.8$ times the FWHM of the seeing; Simons et al. 2016). We remove four galaxies from the original sample for which inclination corrections were not applied. The sample used in this paper contains 45 galaxies, and a catalog of their physical properties and kinematic measurements is available in S16.

Kinematics (rotation velocity V_rot and gas velocity dispersion ${\sigma }_{g}$) were measured from strong nebular emission lines (Hα and [O iii] λ5007) with the ROTCURVE program (Weiner et al. 2006) in Kassin et al. (2007) and S16 for the DEEP2 and SIGMA surveys, respectively. The measurement technique is briefly discussed here and the reader is referred to Weiner et al. (2006) for details.

Seeing blurs rotation and artificially elevates the velocity dispersion in the center of the galaxy (due to the rise in the rotation curve) and needs to be taken into account in the kinematic modeling. ROTCURVE models the velocity profile of the emission line, taking into account seeing, with two components: an arctangent (arctan) rotation curve and a constant dispersion term. The rotation velocity uncorrected for inclination (${V}_{\mathrm{rot}}\times \sin i$) is measured at the flat portion of the rotation curve. Recent evidence has suggested that the rotation curves of high-redshift star-forming galaxies fall, instead of flatten, beyond 1.5 effective radii (Genzel et al. 2017; Lang et al. 2017). Our high-redshift data do not extend far enough to distinguish between these two cases. However, there is only a small difference in the maximum velocity derived from a fit to either model. While the flat part of the rotation curve is measured as the asymptote of the arctan model, the change in velocity beyond a few effective radii is negligible (<10%). If we adopt a value of V_rot that is measured at 1.5 R_e, the rotation velocities are lowered by ∼0.1 dex. For consistency across redshift, we adopt V_rot as measured at the flat part of the modeled rotation curve in all of our galaxies.

Typical uncertainties on measurements of ${V}_{\mathrm{rot}}\times \sin i$ are approximately 10 km s⁻¹ for DEEP2 and 30 km s⁻¹ for SIGMA. Uncertainties on ${\sigma }_{g}$ are approximately 15 km s⁻¹ for DEEP2 and 25 km s⁻¹ for SIGMA. Inclination corrections are applied to the ROTCURVE values using the rest $\sim V$-band axis ratio measured from the HST images (the ACS V- and I-bands in K12 for $0.1\,\lt \,z\,\lt \,0.6$ and $0.6\,\lt \,z\,\lt \,1.2$, respectively, and the WFC3 H-band in S16 for $1.3\,\lt \,z\,\lt \,2.5$).

The gas velocity dispersion ${\sigma }_{g}$ integrates small-scale velocity gradients below the seeing limit (Covington et al. 2010; Kassin et al. 2014). It is mostly due to nonordered motions among H ii regions but also includes smaller contributions from thermal broadening (∼10 km s⁻¹ for H gas at 10⁴ K) and the internal turbulence in H ii regions (∼20 km s⁻¹ in local H ii regions; Shields 1990). As in K12, we refer to ${\sigma }_{g}$ as tracing "disordered motions."

The quantity V_rot measures the ordered motions of a galaxy while the quantity ${\sigma }_{g}$ measures its disordered motions. We follow Weiner et al. (2006) and Kassin et al. (2007) by combining both forms of dynamical support into the quantity ${S}_{0.5}=\sqrt{0.5\,{V}_{\mathrm{rot}}^{2}+{\sigma }_{g}^{2}}$. This term serves as a kinematic tracer of the total mass in an isothermal potential (Weiner et al. 2006).

Stellar masses (${M}_{* }$) were measured for the galaxies in the K12 sample using absolute B-band magnitudes and rest B−V colors (Bell & de Jong 2001; Bell et al. 2005) with empirical corrections from spectral energy distribution (SED) fitting (Bundy et al. 2006), as in Lin et al. (2007). A Chabrier (2003) IMF was adopted, and the uncertainties in stellar mass are approximately 0.2 dex. SFRs were derived in Noeske et al. (2007). For galaxies with a 24 μm detection, the obscured SFR was measured from the IR luminosity using the SED templates of Chary & Elbaz (2001) and was added to the unobscured component of the SFR, derived from the DEEP2 emission line luminosities (Weiner et al. 2007). Otherwise, the SFR was measured from the extinction-corrected emission line luminosities following the calibration in Kennicutt (1998).

Stellar masses for the galaxies in SIGMA were measured from SED fitting to the UV-optical-NIR data available in the CANDELS fields, as described in Barro et al. (2014). The fit was performed with FAST (Kriek et al. 2009) assuming a Chabrier (2003) IMF, Bruzual & Charlot (2003) stellar population synthesis models, and a Calzetti extinction law (Calzetti et al. 2000). The uncertainties in stellar mass are approximately 0.3 dex (Mobasher et al. 2015). For galaxies with detections in the mid- to far-IR, the SFR was calculated using both the obscured (IR) and unobscured (UV) components (Kennicutt 1998). Otherwise, SFRs were derived from the extinction-corrected UV using the dust parameters of the best-fit SED model.

The evolution of the rotation velocity, V_rot, is shown in the top left panel of Figure 2. The galaxies in our sample span a large range in V_rot at a fixed epoch. This scatter is mostly intrinsic but is due in part to our relatively wide bins in stellar mass (1 dex) and to a lesser extent from measurement uncertainties.

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At all times, the average V_rot is higher for galaxies with a higher stellar mass. We find a mild increase of the average V_rot with time in galaxies with low stellar mass, rising from 50 km s⁻¹ at $z=2$ to 70 km s ⁻¹ at $z=0.2$. High-mass galaxies show little evolution in V_rot over the same time period. We estimate the standard error on the median V_rot in each bin through bootstrap resampling and perform an uncertainty weighted least-squares fit to the median points with a line:

$$\begin{eqnarray}&&\mathrm{log}\,\left(\displaystyle \frac{{V}_{\mathrm{rot}}}{\mathrm{km}\,{{\rm{s}}}^{-1}}\right)=a\left(\displaystyle \frac{{t}_{L}}{\mathrm{Gyr}}\right)+b.\end{eqnarray} \tag{ 1 }$$

The best-fit values of the slope and intercept are $a=-0.016\pm 0.01$ (i.e., slow rise) and $b=1.91\pm 0.07$ for our low stellar mass bin, and $a=-0.004\pm 0.014$ (i.e., no evolution) and $b=2.18\pm 0.09$ for our high stellar mass bin. The intercept value of each mass bin, i.e., a V_rot at $z=0$, is consistent with the local stellar mass Tully–Fisher relation (90 and 175 km s⁻¹ at $\mathrm{log}\,{M}_{* }/{M}_{\odot }=9.5$ and 10.5, respectively; Reyes et al. 2011).

The evolution of the gas velocity dispersion, ${\sigma }_{g}$, is shown in the top right panel of Figure 2. As with V_rot, galaxies in our sample span a wide range in ${\sigma }_{g}$ at all epochs, largely due to measurement uncertainties.

The galaxies at the highest redshifts have values of ${\sigma }_{g}$ that are roughly three times larger than the galaxies at the lowest redshifts (see also Wisnioski et al. 2015). The quantity ${\sigma }_{g}$ decreases from 70 to 20 km s⁻¹ over $0.2\,\lt \,z\,\lt \,2.0$. As first shown in K12, ${\sigma }_{g}$ roughly doubles over $0.1\,\lt z\,\lt 1.2$, which spans approximately 8 Gyr. Although the time from $z=1$ to $z=2$ is much shorter, only 3 Gyr, we find a significant evolution in ${\sigma }_{g}$ during this period (see also e.g., Mieda et al. 2016). The median value of σ_g increases from 40 to 70 km s⁻¹. over this redshift range. We perform a fit to the median ${t}_{L}\mbox{--}{\sigma }_{g}$ relation for our two mass bins with:

$$\begin{eqnarray}&&\mathrm{log}\,\left(\displaystyle \frac{{\sigma }_{g}}{\mathrm{km}\,{{\rm{s}}}^{-1}}\right)=a\left(\displaystyle \frac{{t}_{L}}{\mathrm{Gyr}}\right)+b\end{eqnarray} \tag{ 2 }$$

The best-fit values are $a=0.055\pm \,0.010$ and $b=1.19\,\pm \,0.04$ for the low-mass bin and $a=0.043\pm 0.010$ and $b=1.31\pm 0.07$ for the high-mass bin. There is no significant difference between the evolution of ${\sigma }_{g}$ in the low-mass and high-mass bins. This result implies a mass-independent half-life timescale for σ_g, the time over which it declines by a factor of 2, of approximately 6 Gyr. The intercept values, i.e., ${\sigma }_{g}$ at $z=0$, are consistent with measurements of the ionized gas velocity dispersion in local star-forming disk galaxies ($\sim 15\mbox{--}25$ km s⁻¹, e.g., Epinat et al. 2008).

In Figure 3 we examine the evolution of ${S}_{0.5}$, which traces the total dynamical support of galaxies. We fit a line to the median ${S}_{0.5}$ versus lookback time:

$$\begin{eqnarray}&&\mathrm{log}\,\left(\displaystyle \frac{{S}_{0.5}}{\mathrm{km}\,{{\rm{s}}}^{-1}}\right)=a\left(\displaystyle \frac{{t}_{L}}{\mathrm{Gyr}}\right)+b.\end{eqnarray} \tag{ 3 }$$

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The best-fit slope and intercept values are $a=0.017\,\pm 0.008$ and $b=1.70\pm 0.06$ for the low-mass bin and $a=0.01\pm 0.01$ and $b=2.00\pm 0.09$ for the high-mass bin. The quantity ${S}_{0.5}$ has marginally declined in low-mass galaxies since $z=2.5$—i.e., they now sit in shallower potential wells. However, we measure no significant evolution in high-mass galaxies.

While the fit to the low-mass galaxies indicates a smooth evolution, we caution that the trend is largely driven by the high redshift measurements. The median points below $z=1.5$ are consistent with a flat slope. The points above $z=1.5$ are all above this flat line, consistent with a zero-point shift in the modified Tully–Fisher relation at high redshift (Cresci et al. 2009; Price et al. 2016; Straatman et al. 2017; Übler et al. 2017).

We normalize V_rot and ${\sigma }_{g}$ by ${S}_{0.5}$ and measure the relative contributions of ordered and disordered motions, respectively, to the total dynamical support. A thin disk that is supported by rotation will tend toward ${V}_{\mathrm{rot}}/{S}_{0.5}=\sqrt{2}$, and a system that is supported by dispersion will tend toward ${\sigma }_{g}/{S}_{0.5}=1$.

The evolution of ${V}_{\mathrm{rot}}/{S}_{0.5}$ is shown in the bottom left panel of Figure 2. K12 showed that ${V}_{\mathrm{rot}}/{S}_{0.5}$ rises from $z=1.2$ to now and is a strong function of mass. We find a smooth extension of these results to $z=2.5$.

At all redshifts, high-mass galaxies have higher values of ${V}_{\mathrm{rot}}/{S}_{0.5}$—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 rotational support (${V}_{\mathrm{rot}}/{S}_{0.5}\,\gt \,1.3$) by a lookback time of $5\mbox{--}6\,\mathrm{Gyr}$, or $z\,\sim \,0.7$. Low-mass galaxies reach the same degree of rotational support a few Gyr later, by $z\,\sim \,0.2$. We fit a line to the median ${V}_{\mathrm{rot}}/{S}_{0.5}$ using only the rising part of this relation, which covers $0.5\,\lt \,z\,\lt \,2.5$ in the high-mass bin and $0.1\,\lt \,z\,\lt \,2.5$ in the low-mass bin:

$$\begin{eqnarray}&&\displaystyle \frac{{V}_{\mathrm{rot}}}{{S}_{0.5}}=a\left(\displaystyle \frac{{t}_{L}}{\mathrm{Gyr}}\right)+b.\end{eqnarray} \tag{ 4 }$$

The best-fit slope and intercept are $a=-0.074\pm 0.010$ and $b=1.53\pm 0.04$ for the low-mass galaxies and $a=-0.042\pm 0.020$ and $b=1.59\pm 0.08$ for the high-mass galaxies. The values for the best-fit slopes are similar and suggest that the low-mass and high-mass galaxies develop rotational support at similar rates. We refit the high-mass galaxies with a slope fixed to the best-fit slope of the low-mass galaxies. At a fixed slope, the high-mass and low-mass trends are offset by 4.3 Gyr. In other words, the assembly of rotational support in the high-mass galaxies is, on average, followed 4–5 Gyr later by a similar kinematic assembly in the low-mass galaxies.

As expected, we find inverted trends in the evolution of ${\sigma }_{g}/{S}_{0.5}$ (Figure 2, bottom right). On average, ${\sigma }_{g}/{S}_{0.5}$ declines with time for all galaxies and is always larger for low-mass galaxies. Given that ${\sigma }_{g}$ itself does not depend on stellar mass (top right panel), its relative contribution to the total dynamical support will always be larger in galaxies with a lower stellar mass. Beyond $z=1.0$, or a lookback time of $\sim 9\,\mathrm{Gyr}$, low-mass galaxies have large contributions from dispersion (${\sigma }_{g}/{S}_{0.5}\,\gtrsim \,0.8$). They progressively decline in dispersion support with time up to the present day. At $z=0$, both low-mass and high-mass galaxies have low levels of dispersion support (${\sigma }_{g}/{S}_{0.5}\leqslant 0.4$) and high levels of rotational support (${V}_{\mathrm{rot}}/{S}_{0.5}\geqslant 1.3$). We fit a line to the median ${\sigma }_{g}/{S}_{0.5}$ over the declining part of the relation, covering $0.5\,\lt \,z\,\lt \,2.5$ for high-mass galaxies and $0.1\,\lt \,z\,\lt \,2.5$ for low-mass galaxies:

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{g}}{{S}_{0.5}}=a\left(\displaystyle \frac{{t}_{L}}{\mathrm{Gyr}}\right)+b.\end{eqnarray} \tag{ 5 }$$

The best-fit slope and intercept values are $a=0.063\pm 0.01$ and $b=0.20\pm 0.04$ for low-mass galaxies and $a=0.056\,\pm 0.02$ and $b=-0.026\pm 0.11$ for high-mass galaxies. As before, the slopes are statistically identical and suggest that dispersion support declines at similar rates in both mass bins. We refit to the high-mass galaxies with a slope fixed to the low-mass fit and measure an offset between the mass bins of 4.6 Gyr. These results mirror our earlier conclusions and once again suggest that kinematic assembly in low-mass galaxies is delayed from that in high-mass galaxies by $\sim 4\mbox{--}5$ Gyr.

In Figure 4 (left) we compare our measurements with those in the literature. Any comparison between surveys is complicated due in large part to differences in sample selection and measurement techniques. We refrain from performing detailed corrections between the samples and only report here a straightforward comparison. We include measurements from the following surveys: IMAGES at z ∼ 0.6 (Flores et al. 2006; Yang et al. 2008; Neichel et al. 2008; Puech et al. 2007, 2008), KROSS at $z\,\sim 1$ (Stott et al. 2016), KMOS-3D at $z\,\sim 1$ and $z\,\sim 2$ (Wisnioski et al. 2015), MASSIV at $z\,\sim 1.3$ (Contini et al. 2012), AMAZE/LSD at $z\,\sim 3$ (Gnerucci et al. 2011), and the KMOS Deep survey (KDS) survey at $z\,\sim \,3.5$ (Turner et al. 2017). The adopted measurements are briefly described below. To account for differences in the stellar mass of galaxies in each sample, we color-code each by its approximate mass coverage. Our sample and the literature measurements are in good agreement once we account for stellar mass.

The IMAGES sample contains 68 galaxies at 0.4 < z < 0.8 over a stellar mass range 10.0 < log M_*/M_⊙ < 11.0. Measurements of V_rot and σ_g are taken from Puech et al. 2007, 2008; M. Puech, private communication. The KROSS sample covers the redshift range of $0.6\,\lt \,z\,\lt \,1.0$ and a stellar mass range of $9.0\,\lesssim \,\mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lesssim \,11.0$. A catalog of kinematic measurements for 586 typical star-forming galaxies was released in Harrison et al. (2017). We use this catalog to calculate the fraction of resolved galaxies in KROSS that have rotation velocities that exceed their intrinsic velocity dispersions. We adopt mass bins that overlap our sample and these are shown as separate points in Figure 4. The first-year sample from the KMOS-3D survey contains 191 star-forming galaxies in two redshift intervals: $0.7\lt z\lt 1.1$ and $1.9\lt z\lt 2.7$. These galaxies cover a stellar mass range of $9.6\lesssim \mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lesssim 11.1$ at $z\,\sim 1$ and $10\,\lesssim \,\mathrm{log}\,{M}_{* }/{M}_{\odot }\lesssim \,11.2$ at $z\,\sim 2$. Wisnioski et al. (2015) report the fraction of galaxies in their sample with rotationally supported kinematics as 93% and 74% at $z\,\sim 1$ and $z\,\sim 2$, respectively. We include the sample of 48 MASSIV galaxies from Vergani et al. (2012), covering $1\,\lt z\,\lt \,1.6$ and a stellar mass range of $9.5\,\lesssim \,\mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lesssim \,11$. We adopt the $V/\sigma$ measurements in their Table 2 and calculate a rotation-dominated fraction of 65%. We include the high redshift sample from the AMAZE/LSD survey of 33 galaxies over $2.6\,\lt z\lt \,4.9$ and a stellar mass range of $9.0\,\lesssim \mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lesssim \,11.0$. Gnerucci et al. (2011) report that 33% of the AMAZE galaxies can be classified as rotation-dominated. Finally, we include results from the KDS at $z\,\sim \,3.5$ (Turner et al. 2017), which covers a mass range of $8\,\lt \,\mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lt \,11$. Turner et al. (2017) report that 13/32 of the isolated field galaxies in this sample are rotation-dominated. We remove the four galaxies with $\mathrm{log}\,{M}_{* }/{M}_{\odot }\,\lt 9$, three of which are dispersion-dominated, and compute a rotation-dominated fraction of 43% over our mass range.

Using the abundance matching technique to link galaxy populations in time, we now revisit the evolution of ${V}_{\mathrm{rot}},{\sigma }_{g}$, and ${S}_{0.5}$ in Figures 6 and 7.

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The most striking difference is in the evolution of V_rot. At a fixed stellar mass we measured mild to no evolution in 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. We use Equation (1) and fit to the median evolution of V_rot in each mass bin. The best-fit slope and the intercept values are $a=-0.053\pm 0.03$ and $b=1.93\pm 0.12$ for low-mass galaxies, $a=-0.034\,\pm 0.02$ and $b=2.21\pm 0.09$ for intermediate-mass galaxies, and $a=-0.056\pm 0.03$ and $b=2.54\pm 0.24$ for high-mass galaxies. The best-fit slopes are similar and indicate that star-forming galaxies in this mass range have a typical V_rot doubling timescale of ∼6 Gyr, similar to the half-life timescale of ${\sigma }_{g}$ at a fixed stellar mass (Section 3.2).

At a fixed galaxy population we again find that ${\sigma }_{g}$ dramatically declines with time (Figure 6, top right). The intermediate- and high-mass galaxies trace each other back to $z=1.5$, rising from 30 km s⁻¹ at $z=0.2$ to 45 km s⁻¹ at $z=1.5$. The high-mass galaxies, for which we have measurements out to high redshift, reach typical values of ${\sigma }_{g}=60$ km s⁻¹ at $z=2$.

The evolution of S_0.5 for the abundance matched populations is shown in Figure 7. At fixed stellar mass we measured a mild decline in ${S}_{0.5}$ with time. At fixed galaxy population we find that ${S}_{0.5}$ increases with time for all galaxies. This result suggests that, perhaps unsurprisingly, galaxy potential wells steepen as they acquire mass. The quantity ${\sigma }_{g}$ declines with time on average, so the additional dynamical support comes from the increase in V_rot.

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As before, we use Equation (3) and fit to the median ${S}_{0.5}$ evolution for each mass bin. The best-fit slope and intercept values are $a=-0.015\pm 0.019$ and $b=1.73\pm 0.08$ for the fixed population at low mass, $a=-0.011\pm 0.001$ and $b=2.0\pm 0.05$ at intermediate mass, and $a=-0.026\,\pm 0.04$ and $b=2.24\pm 0.28$ at high mass.

We normalize V_rot and ${\sigma }_{g}$ by ${S}_{0.5}$ (Figure 6, bottom panels) and find the same general behavior as before: while all galaxy populations increase in rotational support with time (rising in ${V}_{\mathrm{rot}}/{S}_{0.5}$ and declining in ${\sigma }_{g}/{S}_{0.5}$), massive galaxies are the most rotationally supported at all times on average. The high-mass population reaches a strong level of rotational support (${V}_{\mathrm{rot}}/{S}_{0.5}\,\gt \,1.3$) as far back as $z=1.5$, while the intermediate- and low-mass populations reach the same degree of rotational support at $z=0.5$ and $z=0.2$, respectively.

In Figure 8, we use the linked galaxy populations to revisit how the fraction of galaxies with ${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt \,1$ and 3 evolve with time and galaxy mass.

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As before, at a fixed stellar mass, we find trends that are smooth in both mass and redshift. These results are consistent with kinematic downsizing: more galaxies tend to become rotation-dominated with time and at all times galaxies are more likely to be rotation-dominated at higher mass. As before, this result is due in part to the large decline in ${\sigma }_{g}$ with time and its weak dependence on mass. In addition, unlike the case at a fixed stellar mass, there is a contribution from the steep rise in V_rot with time in all galaxy populations and its strong dependence on mass.

We use Equation (6) to fit to the evolution in each mass bin. The best-fit slope and intercept for F(${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt 1$) are $a=-0.047\pm 0.012$ and $b=0.90\pm 0.05$ at low mass, $a=-0.053\pm 0.007$ and $b=1.11\pm 0.04$ at intermediate mass, and $a=-0.06\pm 0.009$ and $b=1.27\pm 0.07$ at high mass. For F(${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt 3$) the best-fit values are $a=-0.07\pm 0.013$ and $b=0.59\pm 0.06$ at low mass, $a=-0.052\pm 0.01$ and $b=0.78\pm 0.06$ at intermediate mass, and $a=-0.10\pm 0.01$ and $b=1.32\pm 0.09$ at high mass.

The majority of galaxies in these populations have some degree of rotational support (Figure 8, left). More than half of the galaxies in the low-mass bin have ${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt \,1$ at $z=1$, and the intermediate- and high-mass bins reach similar values at $z=2$.

However, it is much less common for galaxies to reach ${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt \,3$ at high redshifts (Figure 8, right). Less than 30% of galaxies in the low-mass population have ${V}_{\mathrm{rot}}/{\sigma }_{g}\,\gt \,3$ beyond $z\,\sim \,0.4$ (Figure 8, right), indicating that it would be exceedingly rare for today's LMC-mass galaxies to have strong rotational support at redshifts greater than 1. The intermediate- and high-mass populations drop below 30% at around $z\,\sim \,1.5$. The most massive star-forming disks in the local universe ($\mathrm{log}\,{M}_{* ,z=0}/{M}_{\odot }\,\gt \,10.0;$ i.e., the Milky Way and M31-mass galaxies) were likely weakly rotationally supported, ${V}_{\mathrm{rot}}/{\sigma }_{g}\,\lt \,3$, at the peak of cosmic star formation.