The Evolution and Origin of Ionized Gas Velocity Dispersion from z ∼ 2.6 to z ∼ 0.6 with KMOS^3D∗

We derive two-dimensional projected Hα velocity and velocity dispersion fields for all ${\mathrm{KMOS}}^{3{\rm{D}}}$ galaxies using linefit (Davies et al. 2009, 2011; Förster Schreiber et al. 2009), a code that takes into account the instrument line-spread function and fits a Gaussian model for each spaxel of the reduced data cube after continuum subtraction. From these maps, we exclude spaxels with S/N ≤ 2, uncertainties on the velocity or velocity dispersion of ≥100 km s⁻¹, as well as off-source fits to noise features. We determine the maximum and minimum of the velocity map through a weighted average of either the 5% of spaxels of both the highest and lowest velocity values for galaxies with ≥50 suitable spaxels, or otherwise of the five spaxels with highest and lowest velocities. The kinematic major axis is defined as the line going through the maximum and minimum of the velocity map. The kinematic center is defined as the midpoint on the kinematics major axis connecting the maximum and minimum of the velocity map. This method follows the procedures outlined by Wisnioski et al. (2015), and in the ${\mathrm{KMOS}}^{3{\rm{D}}}$ data release and final survey paper by E. Wisnioski et al. (2019, in preparation).

Along the kinematic major axis, we then extract spectra in circular apertures of diameter 2 × FWHM of the model-independent point-spread function (PSF) associated with each individual galaxy. Here, the flux from all spaxels within an aperture is integrated to create a single spectrum. For the dynamical modeling of our galaxies (see Section 3.2), we repeat this same procedure for each iteration of the model fitting to properly account for any effects related to this integration process. We consider a galaxy to be spatially resolved if we can measure its kinematics over a total of at least 3 × PSF_FWHM along the kinematic major axis. We fit the Hα velocity and velocity dispersion from the resulting spectra, providing us with the one-dimensional rotation curve v_rot(r) · sin(i) and dispersion profile σ(r), uncorrected for beam-smearing. Uncertainties for each data point are derived using Monte Carlo analysis and have typical values of 6 km s⁻¹ and 10 km s⁻¹ for the velocity and dispersion values, respectively.

With this methodology, we have successfully extracted kinematic profiles for all 535 ${\mathrm{KMOS}}^{3{\rm{D}}}$ Hα-detected galaxies with secure redshifts.

As a next step, we consider all galaxies with kinematic profile extractions that are resolved—a total of 456 SFGs. We further exclude targets for which we detect multiple systems within the IFU, and we eliminate merging or potentially interacting systems with larger separations based on projected distances, redshift separations, and mass ratios, as informed through the 3D-HST catalog (J. T. Mendel et al. 2019, in preparation). Galaxies that are strongly contaminated by sky features, have prominent broad-line regions, or have very strong outflows affecting the recovery of the galaxies' velocity and dispersion are also excluded. This results in 356 galaxies.

We exploit the dynamical fitting code dysmal (Cresci et al. 2009; Davies et al. 2011; Wuyts et al. 2016; Übler et al. 2018) to model our galaxies. dysmal is a forward-modeling code that allows for a flexible number of components (disk, bulge, halo, etc.) and free parameters. It accounts consistently for finite scale heights and flattened spheroidal potentials (Noordermeer 2008), and it includes the effects of pressure support on the rotation velocity. It also accounts for the instrument line-spread function and for beam-smearing effects by convolving with the two-dimensional PSF of each galaxy.

For our modeling, we assume a velocity dispersion that is isotropic and constant throughout the disk, motivated by deep adaptive optics imaging spectroscopy on kiloparsec scales of 35 z = 1–2.6 SFGs in the SINS/zC-SINF sample (Genzel et al. 2006, 2008, 2011, 2017; Cresci et al. 2009; Förster Schreiber et al. 2018; see also Section 5.2). We note that for nearby galaxies, radially declining velocity dispersions have been observed for atomic and molecular gas (van der Kruit & Freeman 1984; Dickey et al. 1990; Boulanger & Viallefond 1992; Kamphuis & Sancisi 1993; Meurer et al. 1996; Petric & Rupen 2007; Tamburro et al. 2009; Wilson et al. 2011; Caldú-Primo et al. 2013; Mogotsi et al. 2016; Sun et al. 2018; Koch et al. 2019), where the velocity dispersion usually reaches a constant level only in the disk outskirts. The observed radial changes in velocity dispersion are, however, rarely larger than 10–20 km s⁻¹, and such variations on small scales are likely washed out through the coarser spatial resolution of typical high-z observations (but see Section 5.2 for a high-resolution example).

We create a three-dimensional mass model of each galaxy consisting of an exponential disk with the effective radius R_e adopted from the H-band measurements, with the ratio of scale height to scale length q₀ = 0.25, and with a central bulge (${R}_{e,\mathrm{bulge}}=1\,\mathrm{kpc}$, Sérsic index ${n}_{{\rm{S}},\mathrm{bulge}}=4$; e.g., Lang et al. 2014; Tacchella et al. 2015b). The value of q₀ = 0.25 is motivated by the falloff in the q = b/a distribution of SFGs at the mass and redshift of our sample (van der Wel et al. 2014a). For galaxies without an H-band-based measurement of the bulge mass (see Section 2; ca. 30%), we use average values of B/T =[0.25; 0.35; 0.45; 0.5] for total stellar masses of log(M_⋆/M_⊙) =[<10.8; 10.8–11; 11–11.4; >11.4], following Lang et al. (2014). We fix the physical size of the bulge because individual measurements of ${R}_{e,\mathrm{bulge}}$ are very uncertain, in contrast to measurements of B/T (see Tacchella et al. 2015b). In a population-averaged sense, however, ${R}_{e,\mathrm{bulge}}=1\,\mathrm{kpc}$ is a robust choice (see Lang et al. 2014). We calculate the galaxy inclination i as $\cos (i)={[({q}^{2}-{q}_{0}^{2})/(1-{q}_{0}^{2})]}^{1/2}$. The mass model is then rotated to match the galaxy's observed orientation in space, convolved with the line-spread function and the PSF of the observation to take into account beam-smearing, and subsequently pixelated to resemble the spatial sampling of the observation. We approximate the PSF as a two-dimensional Moffat function that has been fitted to the standard star observations associated with each KMOS detector and pointing. For our modeling, we assume that light traces mass.

Using dysmal, we simultaneously fit the one-dimensional velocity and velocity dispersion profiles of our galaxies in observed space. The best-fitting intrinsic rotation velocity, v_rot, is constrained both through the mass model and the intrinsic velocity dispersion via pressure support. We apply Markov Chain Monte Carlo (MCMC) sampling to determine the model likelihood based on comparison to the observed one-dimensional kinematic profiles and assuming Gaussian measurement noise. To ensure convergence of the MCMC chains, we model each galaxy with 400 walkers, a burn-in phase of 50–100 steps, followed by a running phase of another 50–100 steps (>10 times the maximum autocorrelation time of the individual parameters). For each free parameter, we adopt the median of all model realizations as our best-fit value, with asymmetric uncertainties corresponding to the 1σ confidence ranges of the one-dimensional marginalized posterior distributions.

In order to recover the intrinsic velocity dispersion as best as possible, we consider a total of three setups with varying free parameters and treatment of the kinematic profiles:

• 1.  
  In our first setup, we feed the kinematic profiles obtained as described in Section 3.1, with free parameters being the total dynamical mass in the range log(M_tot/M_⊙) = [9; 13], and the intrinsic velocity dispersion in the range σ₀ = [5; 300] km s⁻¹. M_tot is the total mass distributed in the three-dimensional disk plus bulge structure necessary to reproduce the observed kinematics. Other parameters are fixed, specifically i, R_e, and B/T.
• 2.  
  Due to extinction, skyline contamination, and noise limitations, some galaxies display asymmetric kinematic profiles. Therefore, we employ a symmetrization technique in a second setup, where the one-dimensional profiles are folded (for the dispersion profile) or rotated (for the rotation curve) around the kinematic center, interpolated onto a common grid, and averaged by calculating the mean at each radial grid point to obtain symmetric profiles, with uncertainties added in quadrature. Again, the free parameters are M_tot and σ₀, allowed to vary within the same ranges as for setup 1.
• 3.  
  As noted in Section 2, the R_e and B/T of our galaxies are derived from H-band imaging. It is known that the mass distribution derived from the H-band light might differ from the corresponding Hα flux profiles (e.g., Wuyts et al. 2012; Tacchella et al. 2015a; Nelson et al. 2016; D. J. Wilman et al. 2019, in preparation). In particular, the dispersion profiles can be sensitive to the central mass concentration. In the third setup, we therefore proceed as in setup 2, but additionally leave the disk effective radius R_e and the bulge-to-total fraction B/T as free parameters. For R_e, we use an effectively flat prior centered on the fiducial value, and truncated at ±2.5 kpc, with hard bounds of R_e = [0.1; 20] kpc. For B/T, we use an effectively flat prior with hard bounds of B/T = [0; 1].

Comparing results from the three setups, we generally find good agreement for both the derived intrinsic dispersions and the dynamical masses, as listed in Table 1. For setup 3, the model-derived (mass/Hα) effective radii are systematically higher compared to the H-band measurements by ∼0.6 kpc. For the range of R_e ≈ 2−10 kpc and log(M_*/M_⊙) ≈ 9.2–11.5 in our kinematic sample, this is in agreement with the results by Nelson et al. (2016) and D. J. Wilman et al. (2019, in preparation), who find R_e,Hα/R_e,H ≈ 1.1–1.2 from high-resolution HST observations and from our full ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample, respectively. The average agreement between the H-band-derived B/T and the model-derived B/T is better; however, the model-derived value is likely more realistic for cases with only a grid-based B/T.

We tested a fourth setup for a subset of our sample, including not only the bulge and disk components but in addition an NFW halo (Navarro et al. 1996), with a prior on the expected dark matter halo mass (Moster et al. 2018) and the concentration parameter fixed to the theoretically expected value (Dutton & Macciò 2014). The resulting best-fit velocity dispersions are robust in that they agree within the uncertainties with the results from the other three setups with a standard deviation of 5.9 km s⁻¹, and there are no systematic effects. However, the limited field-of-view of KMOS (compared to e.g., SINFONI) together with our typical integration times of 5–9 hr per target constrain our ability to map the faint outskirts of galaxies where the kinematics are most sensitive to additional dynamical components with a different mass distribution. Therefore, we do not include fits from this fourth setup in our final sample.

We inspect the fits from all three model setups to create our best-fit sample. By default, we choose the fit to setup 1, but if it is bad or poorly constrained, we consider setups 2 and 3 in this order. Galaxies with poor fits in all setups are excluded. With this strategy, we stay as closely as possible to the original data, but at the same time do not need to disregard galaxies with one-sided extinction or skyline contamination that otherwise show good data quality, and we can choose fits from setup 3 with a more appropriate mass distribution, if necessary.

Finally, we impose a v_rot/σ₀ ≥ 1 cut to focus on rotation-dominated systems. Here, v_rot is the model intrinsic rotation velocity at 1.38 R_e, which is the location of the peak of the rotation curve for a Noordermeer disk with n_S = 1. Our final sample consists of 175 galaxies, with 80, 47, and 48 galaxies in the redshift slices z = 0.6–1.1, z = 1.2–1.7, and z = 1.9–2.6. Of those galaxies, 56% are from setup 1, 31% from setup 2, and 13% from setup 3. We show examples of galaxies and their fits from different setups in Appendix A. The averaged uncertainties on our derived σ₀ values cover the range δσ₀ =2–29 km s⁻¹, with 68th percentiles of δσ₀ = 5–15 km s⁻¹, and mean values in the three redshift slices z ∼ 0.9, 1.5, and 2.3 of δσ₀ = 8, 10, and 13 km s⁻¹. Asymmetric uncertainties can be as high as δσ₀ = 37 km s⁻¹.

In Figure 1, we compare physical properties of our final sample (blue shading) to the underlying representative population of SFGs from the 3D-HST survey (gray shading) and to the full ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample (pink lines). Compared to our full ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample, we have not selected preferentially in redshift. In terms of stellar mass, both our full ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample and our kinematic sample include fewer lower mass systems compared to the 3D-HST galaxies, such that our sample is not mass-complete. This is mainly a consequence of the K_AB ≲ 23 cut. With respect to the main sequence of SFGs, however, our kinematic sample follows the distribution of both the full ${\mathrm{KMOS}}^{3{\rm{D}}}$ and the 3D-HST sample. The fraction of systems with effective radii below the population average is smaller for our kinematic sample compared to the 3D-HST and ${\mathrm{KMOS}}^{3{\rm{D}}}$ samples. This is due to our conservative definition of resolved kinematics, where we request measurements over at least 3 × PSF_FWHM, with the primary effect of reducing the number of galaxies with R_e < 2 kpc. Generally, for very small systems, it is more challenging to recover the intrinsic velocity dispersion, because the kinematics are often unresolved (but see Wisnioski et al. 2018 for a detailed study of the kinematics of compact galaxies in the ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey). Axis ratios of our galaxies are homogeneously distributed, following the ${\mathrm{KMOS}}^{3{\rm{D}}}$ and 3D-HST parent samples (see also Section 3.6).

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In Figure 2, we show SFR (top) and size (bottom) both as a function of stellar mass for the 3D-HST parent sample (gray density histogram), the full ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample (purple diamonds), and our final kinematic sample at redshifts z ∼ 0.9 (blue circles), z ∼ 1.5 (green pentagons), and z ∼ 2.3 (red hexagons). The figure illustrates the homogeneous coverage of the ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey of typical main-sequence galaxies over more than two orders of magnitude in stellar mass. Similarly, the galaxies from our final sample are distributed along the main sequence and have typical sizes for their redshifts, with a tendency toward higher-than-average sizes particularly at z ∼ 2.3. This bias at the highest redshifts is introduced through our conservative definition of resolved galaxies and ensures robust σ₀ measurements even at these high redshifts.

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Our final sample contains 28 galaxies for which the best-fit ${\sigma }_{0}$ value within the 1σ uncertainties is lower than 10 km s⁻¹. In using the Hα line, we are supposedly tracing emission originating from ionized H ii regions. Due to thermal broadening (σ_th ≈ 10 km s⁻¹) as well as the expansion of H ii regions (v_ex ≳ 10 km s⁻¹), we expect some minimum velocity dispersion for the average galaxy of σ₀ ≈ 10–15 km s⁻¹ (Shields 1990).

This minimum value is lower than the typical spectral resolution of KMOS: the effective FWHM spectral resolution at the Hα line measured from the reduced data of galaxies in our ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey is ΔR = λ/Δλ ∼ 3515, 3975, and 3860 in the YJ, H, and K bands, respectively (E. Wisnioski et al. 2019, in preparation). For our kinematic sample, the corresponding mean spectral resolutions are σ_instrumental ∼ 37, 32, and 34 km s⁻¹. However, as discussed in more detail in E. Wisnioski et al. (2019, in preparation), within the bands there are variations of the spectral resolution of up to ΔR = 1000 for individual IFUs. It is therefore crucial to measure the associated spectral resolution at Hα for each individual galaxy from sky or arc lines in order to reliably recover the velocity dispersion, as it is done for KMOS^3D.

Our line-fitting procedure can recover intrinsic velocity dispersions that are a fraction of the instrumental resolution. However, these measurements get increasingly uncertain for decreasing intrinsic velocity dispersions. For galaxies for which the best-fit ${\sigma }_{0}$ value within the 1σ uncertainties is lower than 10 km s⁻¹, we adopt as a conservative upper limit the upper 2σ boundary of the marginalized posterior distribution derived from the MCMC chain. The resulting upper limits lie between 18 and 53 km s⁻¹.

Before we investigate in detail the redshift evolution of ${\sigma }_{0}$ and its potential drivers, we want to exclude any residual effects of beam-smearing. Therefore, we consider ${\sigma }_{0}$ as a function of the effective radius, R_e, and of the ratio of the outermost measured data point to the effective radius, R_max/R_e.

We do not find significant correlations with R_e or R_max/R_e, as listed in Table 2 (for R_e see also Figure 16). We would expect correlations with these parameters if unresolved rotation enters our measure of the velocity dispersion. As mentioned in Sections 3.1 and 3.3, we only consider galaxies for our final sample for which we can extract kinematics over a distance of at least 3 × PSF_FWHM, with a mean value of 4 × PSF_FWHM. However, the extracted kinematics can still be affected by beam-smearing even in the outer parts of the galaxies. The fact that we do not find correlations with size implies that our forward-modeling procedure properly accounts for beam-smearing even for the smaller systems we include.

Similarly, we test for correlations of ${\sigma }_{0}$ with instrumental resolution and again we do not find a significant correlation, indicating that both our kinematic fitting code and forward-modeling procedure properly account for the instrumental line-spread function (see Table 2).

For local galaxies, there exists a correlation between galaxy inclination and line-of-sight velocity dispersion. This is due to the transition from measuring predominantly vertical velocity dispersion in face-on systems to measuring predominantly radial velocity dispersion in edge-on systems, with a typical ratio of σ_z/σ_r ∼ 0.6 (van der Kruit & Freeman 2011; Glazebrook 2013). For instance, Leroy et al. (2008) find for the THINGS sample that the H i line-of-sight velocity dispersion increases for galaxies with i > 60° (b/a < 0.5), as does the variation of velocity dispersion in individual galaxies. Intriguingly, evidence for higher velocity dispersions in more edge-on systems has been found in the higher resolution z ∼ 1–2 data from the SINS survey (Genzel et al. 2011). For our ${\mathrm{KMOS}}^{3{\rm{D}}}$ kinematic sample, and in agreement with the earlier results by Wisnioski et al. (2015), we do not find a correlation between σ₀ and b/a, as shown in Figure 3 and Table 2, possibly due to the coarser spatial resolution of our data.

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In Figure 4, we show the intrinsic velocity dispersion of our ${\mathrm{KMOS}}^{3{\rm{D}}}$ galaxies in the kinematic sample as a function of redshift, where upper limits are indicated by arrows (Section 3.4). Our data reflect the known trend of increasing average velocity dispersions with increasing redshift.

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To quantify the evolution, we fit a linear relation in σ₀−z space to our best-fit data.¹⁸ We use the Bayesian approach to linear regression by Kelly (2007), which allows for the inclusion of censored data (i.e., upper limits). The routine requires symmetric uncertainties, which we calculate as the mean of the asymmetric uncertainties on σ₀ from our MCMC modeling.¹⁹ Figure 4 shows the derived fit as a solid line, with average values of σ₀ ∼ 31.1, 38.3, and 46.7 km s⁻¹ at z ∼ 0.9, 1.5, and 2.3. The corresponding fit is described by the equation

$$\begin{eqnarray}&&{\sigma }_{0}\,[\mathrm{km}\ {{\rm{s}}}^{-1}]=(21.1\pm 3.0)+(11.3\pm 2.0)\cdot z.\end{eqnarray} \tag{ 1 }$$

We also perform a "robust" fit where the upper limit cases are not included, but entirely left out. We find a slightly shallower evolution indicated by the dashed line. If, instead, for these galaxies we do not assign upper limits but take the formal median of the posterior distribution at face value, we find a slightly steeper evolution indicated by the dashed–dotted line. In Table 3, we list our fit parameters and uncertainties.

The σ₀ evolution we derive between z ∼ 0.9 and z ∼ 2.3 is slightly shallower than what has been reported for the first year of data from the ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey by Wisnioski et al. (2015). In particular, the authors cite σ₀ ∼ 24.9 km s⁻¹ at z ∼ 0.9 and σ₀ ∼ 47.5 km s⁻¹ at z ∼ 2.3, i.e., a difference of 6–7 km s⁻¹ for the lowest redshift bin. We partly attribute this difference to our treatment of upper limits. Indeed, if we take the formal best-fit values of the upper limit cases at face value, we find through linear regression a value of σ₀ ∼ 28.4 km s⁻¹ at z ∼ 0.9 (see Table 3), reducing the difference to ∼4 km s⁻¹. This difference is smaller than the uncertainty on the average σ₀ value we derive through our fitting based on the standard deviation of the posterior distribution of the zero point and slope, which is δσ₀ = 4.8 km s⁻¹ for the z ∼ 0.9 bin.

Figure 4 shows substantial scatter in ${\sigma }_{0}$ at fixed redshift with values from σ₀ ≈ 20 km s⁻¹ to σ₀ ≈ 100 km s⁻¹. The question is whether this scatter is physical or purely driven by observational uncertainties.

As listed in Table 3, our robust best-fit relation has an intrinsic scatter around the regression with a standard deviation of 10.4 km s⁻¹, suggesting that part of the scatter is indeed due to real variations of the intrinsic dispersion values and not just due to measurement uncertainties. To quantify the intrinsic variance in each redshift slice, we first calculate the observed variance around the robust best-fit relation, i.e., the variance of the redshift-normalized dispersion values excluding upper limits. The redshift-normalized values are defined as

$$\begin{eqnarray}&&{\sigma }_{0,\mathrm{norm}}={\sigma }_{0}-(a+b\cdot z),\end{eqnarray} \tag{ 2 }$$

with coefficients a and b as listed in Table 3. Then, we perform a Monte Carlo analysis of the scatter due to uncertainties: for each measurement i, we draw 1000 times from a normal distribution ${ \mathcal N }(0,\delta {\sigma }_{0,i})$, where δσ_0,i is the symmetric uncertainty of σ_0,i derived from our dysmal MCMC modeling and calculate the corresponding sample variance per redshift slice.

We calculate this intrinsic variance as

$$\begin{eqnarray}&&{\mathrm{VAR}}_{\mathrm{int}}(z)={\mathrm{VAR}}_{\mathrm{obs}}(z)-{\mathrm{VAR}}_{\delta {\sigma }_{0}}(z)\end{eqnarray} \tag{ 3 }$$

and list the corresponding values in Table 4. From this analysis, we conclude that at least ∼40%–50% of the observed variance, i.e., ∼60%–70% of the observed scatter, is due to real variations of the intrinsic dispersion values, mostly independent of redshift. We also show a histogram of the redshift-normalized dispersion values in Figure 5, σ_0,norm, in black, together with a histogram of the Monte Carlo draws from the uncertainty distribution in red. Again, this clearly shows that, even though the uncertainties are substantial, there is residual scatter in our ${\sigma }_{0}$ distribution beyond what can be accounted for by uncertainties. Further, if we focus on the absolute values listed in Table 4, the intrinsic variance increases above z ∼ 1.5, such that at z ∼ 2.3, it has doubled compared to z ∼ 0.9 and z ∼ 1.5. This suggests that the population of galaxies in our highest redshift bin is more diverse in ISM conditions compared to the lower redshift samples.

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Table 4.  Variances of ${\sigma }_{0}$ around the Robust Best-fit Relation: Observed Variance VAR_obs, Variance Due to Measurement Uncertainties, ${{\rm{V}}{\rm{A}}{\rm{R}}}_{\delta {\sigma }_{0}}$, and Intrinsic Variance, VAR_int

Measure	z ∼ 0.9	z ∼ 1.5	z ∼ 2.3	0.6 < z < 2.6
VARobs (km2 s−2)	171	208	357	237
${{\rm{V}}{\rm{A}}{\rm{R}}}_{\delta {\sigma }_{0}}$ (km2 s−2)	87	130	194	133
VARint (km2 s−2)	85	78	163	104
VARint/VARobs	0.50	0.38	0.46	0.44

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However, no significant residual trend with σ_0,norm and the physical properties related to SFR, mass, size, or rotation velocity remains, as we show in detail in Figure 15 in Appendix C. That means that we cannot identify a physical source for the scatter in ${\sigma }_{0}$ at fixed redshift. This might be due to the limited dynamical range of our data, or it could imply that the intrinsic scatter is driven through the interplay of more than one parameter. Alternatively, the scatter could be due to real variations of the velocity dispersion on short timescales, for instance caused by a dynamic driver such as minor mergers or variations in gas accretion from the cosmic web. This has recently been proposed by Hung et al. (2019) based on results from the FIRE simulations, where variations of the intrinsic dispersion are connected to variations of the gas inflow rate on timescales ≲100 Myr.

The results presented above and in the remainder of the paper are based on our kinematic sample as defined in Section 3.3, i.e., 175 resolved and rotation-dominated disk galaxies that are well fit by our dynamical model, without strong contamination from OH lines or outflows, and without close neighbors. If we instead consider all modeled galaxies from setup 1, which is about twice as many compared to the kinematic sample (see Section 3.2), we find a similar median evolution of σ₀ ≈ 31, 40, and 49 km s⁻¹ at z ∼ 0.9, 1.5, and 2.3; however, the mean values in the three redshift slices are systematically higher with σ₀ ≈ 34, 45, and 58 km s⁻¹. While at all redshifts the scatter is substantially increased due to galaxies with higher observational uncertainties or poor fits (VAR_obs ≈ 730, 850, and 1560 km² s⁻²), the systematic increase of the mean values is mostly due to the inclusion of dispersion-dominated systems (see, e.g., Newman et al. 2013 for a discussion of such galaxies).

To put the evolution of velocity dispersion from z = 2.6 to z = 0.6 based on our ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample into a broader context, we collect measurements reported in the literature from z ∼ 4 to z = 0, covering 12 Gyr of cosmic history (Table 5).

Table 5.  Literature Data of the 0 < z < 4 Velocity Dispersion Measurements Shown in Figure 6

Sample/References	z	Primary Target Line	N Included	Instrument/Method	BS Correction	Comment on Selection
KDS	3.8–3.1	[O iii]	14	KMOS/IFU	(1)	their "RD"
MOSDEF	3.8–1.4	Hα, Hβ, [O iii]	108	MOSFIRE/slit	(2)	their "resolved/aligned"
AMAZE-LSD	3.7–3.1	[O iii]	11	SINFONI/IFU	(1)	their "rotating'
Livermore et al. (2015)	3.7–1.3	Hα, Hβ	8	SINFONI+NIFS+OSIRIS/IFU+lensing	(3)	their 'Disk" with vrot/σ0 > 1
${\mathrm{KMOS}}^{3{\rm{D}}}$	2.6–0.6	Hα	175	KMOS/IFU	(2)	see Section 3.3
SINS/zC-SINF	2.5–1.4	Hα	25	SINFONI/IFU+AO	(2)	see Section 3.3
SIGMA	2.5–1.3	Hα, [O iii]	49	MOSFIRE/slit	(4)
Genzel et al. (2017)	2.4–0.9	Hα	6	KMOS+SINFONI/IFU+AO	(2)	see Section 3.3
MASSIV	1.6–0.9	Hα, [O iii]	53	SINFONI/IFU	(1)
DEEP2	1.2–0.1	Hα, Hβ, [O ii], [O iii]	544	DEIMOS/slit	(4)
KROSS	1.0–0.8	Hα	171	KMOS/IFU	(4)	their "sigma0_flag = O" with v2.2/σ0 > 1
DYNAMO	∼0.1	Hα	25	SPIRAL+WiFeS/IFU	(1)	their "RD"
GHASP	local	Hα	153	scanning Fabry–Perot	(1)
Übler et al. (2018)	1.4	Hα, CO(3–2)	1	LUCI/slit+NOEMA/interferometry	(2)	see Section 3.3
Swinbank et al. (2011)	2.4	CO(6–5), CO(1–0)	1	IRAM+EVLA/interferometry	(3)
PHIBSS	1.5–0.7	CO(3–2)	7	IRAM+NOEMA/interferometry	(4)	see Section 3.3
HERACLES	local	CO(2–1)	13	IRAM/single dish	(5)
EDGE-CALIFA	local	CO(1–0)	17	CARMA/interferometry	(1)
THINGS	local	H i	35	VLA/interferometry	(5)
Dib et al. (2006)	local	H i	13	literature compilation

Note. Beam-smearing (BS) correction methods: (1) correction map derived from the model velocity field including two-dimensional PSF, (2) simultaneous forward-modeling of both the velocity and dispersion including two-dimensional PSF; see Section 3.2, (3) subtraction in quadrature of the velocity gradient across each spaxel, (4) model-based look-up grid, and (5) exclusion of regions most strongly affected based on a model grid.

References. KDS, Turner et al. (2017); MOSDEF, Kriek et al. (2015); Price et al. (2019); AMAZE-LSD, Gnerucci et al. (2011); ${\mathrm{KMOS}}^{3{\rm{D}}}$, Wisnioski et al. (2015; E. Wisnioski et al. 2019, in preparation); SINS/zC-SINF, Förster Schreiber et al. (2006, 2009, 2018); SIGMA, Simons et al. (2016, 2017); MASSIV, Contini et al. (2012), Epinat et al. (2012); DEEP2, Davis et al. (2003), Kassin et al. (2007, 2012); KROSS, Stott et al. (2016), Johnson et al. (2018); DYNAMO, Green et al. (2014); GHASP, Epinat et al. (2008, 2010); PHIBSS, Tacconi et al. (2013), Freundlich et al. (2019); HERACLES, Leroy et al. (2009), Tamburro et al. (2009), Caldú-Primo et al. (2013), Mogotsi et al. (2016); EDGE-CALIFA, Bolatto et al. (2017), Levy et al. (2018); THINGS, Walter et al. (2008), Tamburro et al. (2009), Ianjamasimanana et al. (2012), Caldú-Primo et al. (2013), Mogotsi et al. (2016).

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In the top panel of Figure 6, we show again our ${\mathrm{KMOS}}^{3{\rm{D}}}$ kinematic sample as clouds of gray circles, including upper limits as arrows, in the σ₀−z space. The median values at z ∼ 0.9, 1.5, and 2.3, shown as large circles in blue, green, and red, are based on the best fit plotted in Figure 4 and its uncertainties (see Table 3). We include other individual intrinsic dispersion measurements or averages from ionized gas as colored symbols, and atomic and molecular data as black symbols, which are listed in Table 5. Error bars show the mean uncertainty of the individual systems in those samples. In our comparison, we do not apply any corrections or normalizations in mass (see Wisnioski et al. 2015), which are expected to be small for main-sequence galaxies (Simons et al. 2017).

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In Table 5, we also list the different techniques used to correct for beam-smearing effects. As explained in Section 3.2 and in the references listed there, we account for beam-smearing effects through a full forward-modeling of both the velocity and velocity dispersion fields with a unique PSF model for each galaxy. Techniques based on only the velocity information, or on grid-based models or look-up tables, might perform less well in their beam-smearing corrections generally resulting in overestimated intrinsic velocity dispersions. For slit surveys, systematic offsets toward higher values might be expected, due to the sometimes uncertain galaxy position angle and the resulting difficulties in disentangling rotational and turbulent motions (see Price et al. 2016, 2019 for a discussion and solution approach). Similarly, the methods chosen to calculate or model the intrinsic velocity dispersion might further introduce systematic differences. We note that recent work by Varidel et al. (2019) on a sample of 20 local SFGs suggests that complex structure in the gas distribution may further impact the derived dispersion values.

Figure 6 shows generally good agreement of the various σ₀ measurements reported in the literature. Comparing slit versus IFU techniques, the slit measurements shown here, i.e., data from DEEP2, SIGMA, and MOSDEF, tend toward higher values compared to the averages derived from our ${\mathrm{KMOS}}^{3{\rm{D}}}$ and SINS/zC-SINF surveys, likely for the reasons discussed above, but agree within their uncertainty with the IFU data where available. Interestingly, the deep measurements obtained for individual targets by Genzel et al. (2017), and particularly for the lensed systems by Livermore et al. (2015) and Jones et al. (2010) at 1.5 < z < 3 also tend toward higher σ₀ values, but have moderate values at z > 3, in agreement with the averages obtained from seeing-limited IFU and slit spectroscopy by Gnerucci et al. (2011), Turner et al. (2017), and Price et al. (2019). Generally, the statistical power of these time-intensive and challenging individual measurements is still very limited. Systematic differences in σ₀ may arise through selection effects: for instance, the nearby galaxies from the DYNAMO sample are selected to be z ∼ 2 analogs and have many physical properties, including dispersions, similar to high-z SFGs (see Green et al. 2014; White et al. 2017; Fisher et al. 2019).

In contrast, the molecular and atomic data indicated by black points suggest somewhat lower values on average, particularly at z ≈ 0. Levy et al. (2018) study 17 nearby, rotation-dominated SFGs in CO and ionized gas. They found consistently higher rotation velocities ($\langle$v_CO − v_Hα$\rangle$ ≈ 14 km s⁻¹) and lower velocity dispersions ($\langle$σ_CO − σ_Hγ$\rangle$ ≈ −17 km s⁻¹) for the molecular gas compared to the ionized gas (see also Cortese et al. 2017 for a comparison at z ∼ 0.2). At high redshift, there exist only few multiphase measurements of the intrinsic gas velocity dispersion. Detailed observations reveal comparable values for ionized and molecular gas (Genzel et al. 2013; Übler et al. 2018); however, the uncertainties are larger such that differences like those found locally could be washed out.

We quantify the difference between the atomic+molecular and the ionized gas velocity dispersions over cosmic time in the bottom panel of Figure 6. Fitting a robust linear relation²⁰ to the average local and individual high-z measurements of atomic+molecular gas, we find a zero point of a = 10.9 ± 0.6 km s⁻¹ and a slope of b = 11.0 ± 2.0 km s⁻¹ (gray dashed line). For the ionized velocity dispersion, we choose in addition to our own averages from the ${\mathrm{KMOS}}^{3{\rm{D}}}$ and SINS/zC-SINF surveys the other large KMOS surveys, KROSS and KDS, and the local average from the GHASP survey. This choice maximizes the redshift range and avoids systematic effects at z > 0 through different instrumentation. We find a higher zero-point offset of a = 23.3 ± 4.9 and a somewhat shallower slope of b = 9.8 ± 3.5, while the extrapolation of our best fit to the ${\mathrm{KMOS}}^{3{\rm{D}}}$ data only gives a = 21.1 ± 3.0 and b = 11.3 ± 2.0 (Table 3). Fixing the slope to that of the atomic+molecular fit, the zero point shifts in between these measurements, with a = 22.8. In Table 6, we list our fit parameters and uncertainties.

Table 6.  Results and Standard Deviations from the Robust Least-squares Linear Regression Fits of the Form σ₀/km s = a + b · z to the Data Sets Shown in the Bottom Panel of Figure 6

Sample	a (km s−1)	b (km s−1)
Ionized Gas (Best Averages)	23.3 ± 4.9	9.8 ± 3.5
... Fixing Slope to Atomic+Molecular	22.8	11.0 (fixed)
${\mathrm{KMOS}}^{3{\rm{D}}}$ Incl. Upper Limits (Table 3)	21.1 ± 3.0	11.3 ± 2.0
Atomic+Molecular Gas	10.9 ± 0.6	11.0 ± 2.0

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This suggests that the redshift evolution of the intrinsic velocity dispersion in all gas phases is quite comparable, but their normalization differs. The typical thermal broadening of the atomic/molecular and the ionized gas due to their characteristic temperatures is ∼5 km s⁻¹ and ∼10 km s⁻¹, respectively, meaning the measured velocity dispersions are superthermal even in the local universe. Part of the difference between atomic+molecular and ionized gas velocity dispersions can be explained through the expansion of H ii regions from which the ionized emission originates, with typical values of 10–25 km s⁻¹ (Shields 1990), accounting for another ∼5–15 km s⁻¹ when added in quadrature. In combination, these effects can explain the difference in the local normalizations of the gas-phase velocity dispersions, as well as their average offset of ∼10–15 km s⁻¹ at fixed redshift.

Clearly, more studies of high-z molecular kinematics are warranted to corroborate our result, which potentially has important implications for work on ionized gas kinematics.

Figure 6 shows a smooth evolution of velocity dispersion with redshift over the past ∼12 Gyr, likely connected to decreasing accretion rates and gas fractions with cosmic time (see Sections 5.3 and 5.4). This evolution also suggests that the typical thickness of the young, star-forming gas disk is lower for lower redshift SFGs, as has also been found in state-of-the-art cosmological simulations (Pillepich et al. 2019).

This is potentially interesting in the context of Galactic archeology: early research of the vertical structure of our Milky Way found evidence for two main, distinct exponential disks with scale heights of ∼300 and ∼1450 pc (Gilmore & Reid 1983). This was confirmed through later work on the Milky Way as well as nearby edge-on galaxies (e.g., Dalcanton & Bernstein 2002; Yoachim & Dalcanton 2006; Jurić et al. 2008). The thick disk components have been found to be generally older (>6 Gyr) than those of the thin disks, raising the question of distinct formation periods. Naturally, observations of the typically thick high-z disks also prompted the question of the connection between these early thick disks and modern disk structure (e.g., Elmegreen & Elmegreen 2006).

To explicitly address the question of distinct formation periods of thin versus thick disks, we make the simple assumption of a step function describing the ${\sigma }_{0}$ of the ionized gas below and above z = 1. Unsurprisingly, the resulting fit with σ₀ = 28 km s⁻¹ at z < 1 and σ₀ = 42 km s⁻¹ at z > 1 is not a good description of the compiled data, with a goodness of fit that is a factor of ∼20 worse compared to the linear fit shown in the bottom panel of Figure 6.

Our results suggest that in the absence of recent major mergers, it should depend primarily on the star formation history (connected to gas accretion) if present-day galaxies have distinct disks of different age and scale height, or if there is instead one component with a vertical age gradient (see also Leaman et al. 2017). This interpretation is in agreement with the recent work by Bovy et al. (2012, 2016) and Rix & Bovy (2013), who argue based on elemental abundances that the Milky Way has a continuous range of different scale heights, with no sign of a thin–thick disk bimodality. Simulations by, e.g., Burkert et al. (1992), Aumer et al. (2016), Aumer & Binney (2017), and Grand et al. (2016) support this picture.

However, in this context, it is important to remember that based on the stellar and gas masses of our galaxies and results from comoving number density studies (e.g., Brammer et al. 2011), only the lower mass, lower redshift systems in our sample may evolve into present-day disk galaxies, while the galaxies that already have high baryonic masses at high redshift will most likely evolve into present-day's early-type galaxies. With our data, we, therefore, do not necessarily track the change in star-forming scale height over time for progenitor–descendant populations, but rather the change in average star-forming scale height of main-sequence galaxies at different epochs.

The redshift dependence of ${\sigma }_{0}$ suggests that one or more physical galaxy properties that are themselves redshift-dependent drive velocity dispersion. Consistent with previous findings in the literature (e.g., Johnson et al. 2018), we find several properties positively correlating with σ₀, particularly the SFR, SFR surface density Σ_SFR, gas and stellar mass, and their surface densities. We list direct and residual (after correcting for redshift dependence) Spearman rank correlations in Table 7 and show plots for several quantities in Figure 16 in Appendix C. In Table 7, we also list the SFR_Hα and Σ_SFR,Hα derived from the Hα fluxes (see E. Wisnioski et al. 2019, in preparation), tracing the more recent star formation history, but find no appreciable difference in correlations compared to our fiducial SFR properties (see Section 2).

We emphasize that due to the limited dynamical range covered by the individual redshift slices, we do not find significant correlations of ${\sigma }_{0}$ within one redshift slice with any of the above properties, such that we cannot readily connect the scatter in ${\sigma }_{0}$ at fixed redshift to a physical driving source. Similarly, if we remove the redshift dependence of ${\sigma }_{0}$ by normalizing with our best-fit relation, we do not find any significant correlations of the normalized ${\sigma }_{0}$ with physical properties (see Section 4.2 and Figure 15).

Over the full redshift range covered by our ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey, the SFR and gas mass correlate most strongly and significantly with intrinsic velocity dispersion. In order to identify which of these two physical quantities is most directly tied to the elevated velocity dispersions at high redshift, we discuss in the following sections the physical mechanisms through which quantities such as SFR and gas mass may affect velocity dispersion, namely stellar feedback and gravitational instabilities, and we comment on the tentative connection to active galactic nucleus (AGN) feedback for individual galaxies.

Turbulence driving can be provided through thermal and momentum feedback from massive stars. Correlations between intrinsic velocity dispersion and SFR properties have previously been reported in the literature (e.g., Dib et al. 2006; Lehnert et al. 2009, 2013; Green et al. 2010, 2014; Moiseev et al. 2015; Johnson et al. 2018), and often invoked the argument for stellar feedback-driven turbulence.

From a theoretical point of view, feedback-driven turbulence is mainly generated through momentum injection from supernovae into the ISM (contributions to the momentum injection from, e.g., expanding H ii regions or stellar winds are minor; see Mac Low & Klessen 2004; Ostriker & Shetty 2011). Feedback-driven turbulence should therefore primarily depend on the decay rate of turbulence, the momentum injected per supernova, and the supernova rate, where the latter is the quantity connecting turbulence to the SFR and Σ_SFR. Ostriker & Shetty (2011) and Shetty & Ostriker (2012) derived a weak dependence of σ₀ on star formation rate surface density. Even considering the case where feedback-driven turbulence vertically stabilizes the disk, the resulting velocity dispersions are low (Equation (22) by Ostriker & Shetty 2011):

$$\begin{eqnarray}&&{\sigma }_{z}=5.5\,\mathrm{km}\ {{\rm{s}}}^{-1}\cdot \displaystyle \frac{{f}_{p}}{{(1+\chi )}^{1/2}}\left(\displaystyle \frac{{\epsilon }_{\mathrm{ff}}({\rho }_{0})}{0.005}\right)\left(\displaystyle \frac{{p}_{* }/{m}_{* }}{3000\,\mathrm{km}\ {{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 4 }$$

Here, f_p is a factor characterizing the evolution of turbulence, with f_p = 1 for strong dissipation and f_p = 2 for weak dissipation. χ is a measure of the importance of the gas disk's self-gravitational weight and is below 0.5 for marginally stable disks, such that the first factor is in the range ∼0.8–2. The mean star formation efficiency _ff(ρ₀) is assumed to be approximately constant with a fiducial value of _ff(ρ₀) = 0.005. p_*/m_* = 3000 km s⁻¹ is the fiducial value of momentum injection per supernova (but see, e.g., Fisher et al. 2019 for arguments for a z-dependent p_*/m_*). As a result, the gas velocity dispersion is expected to vary only mildly due to supernova explosions.

Similar results are obtained by other theoretical models investigating stellar feedback as the sole driver of the turbulence in the ISM, for instance the models discussed by Dib et al. (2006), Joung et al. (2009), and Kim et al. (2013). In fact, the resulting velocity dispersions in the ISM do not even seem to depend much on the supernova rate. Rather, very high supernova rates might create superbubble structures that, instead of stirring the ambient medium, will eventually blow out of the galactic disk, thus transferring energy and metals into the circumgalactic medium (Mac Low et al. 1989; Joung et al. 2009; Baumgartner & Breitschwerdt 2013; Kim & Ostriker 2018). This is an important result because at higher redshift, the supernova rate is also higher. However, other work indicates that it is not only the rate but also the location of supernovae that is crucial for the efficiency of stellar feedback turbulence driving: considering peak driving, where supernovae go off in the densest ISM regions (e.g., their birth clouds), Gatto et al. (2015) found local Hα velocity dispersions of up to 60 km s⁻¹ for gas mass surface densities of Σ_gas ∼ 100 M_⊙ pc⁻². This is similar to high-z conditions and therefore suggests that stellar feedback can more easily maintain elevated velocity dispersions at higher redshift. Also, some idealized simulations of isolated galaxies are able to produce a velocity dispersion of ∼50 km s⁻¹ from strong stellar feedback (Hopkins et al. 2011).

If stellar feedback is an important factor in powering turbulence, then it would not only be the (observed) global scaling of velocity dispersion with SFR or Σ_SFR that will be expected, but in particular locally elevated velocity dispersion in regions of high star formation rate density (see Gatto et al. 2015). We exploit the high-resolution data from the SINS-zC/SINF AO survey (Förster Schreiber et al. 2018) to study local correlations between Σ_SFR and σ₀. In Figure 7 we show the local intrinsic velocity dispersion per spaxel of galaxy Q2346-BX482 as a function of radius, color-coded by Σ_SFR (adopted from Figure A1 by Genzel et al. 2011). The local intrinsic velocity dispersion is derived from the observed dispersion map, after correcting all instrumental and beam-smearing effects through modeling. In the vicinity of the giant star-forming clump ∼6.5 kpc southeast from the center (inset), no elevated velocity dispersion can be registered.

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In Figure 8, we show the local intrinsic velocity dispersion per spaxel as a function of local Σ_SFR for 10 SINS/zC-SINF galaxies. The velocity dispersions of these galaxies with a mean redshift of z ∼ 2.2 have somewhat higher values compared to our ${\mathrm{KMOS}}^{3{\rm{D}}}$ sample, consistent with their higher average SFR and Σ_SFR. Only two of these galaxies show an intrinsic scaling of σ₀ with Σ_SFR. The best-fit power-law relation derived from this subgalactic, high-quality data shows a very weak dependence of local ${\sigma }_{0}$ on Σ_SFR,²¹ confirming the earlier findings by Genzel et al. (2011; see also Patrício et al. 2018; Tadaki et al. 2018; but see Swinbank et al. 2012a). Similar results are found for both ionized gas (Varidel et al. 2016; Zhou et al. 2017) and molecular gas (Caldú-Primo & Schruba 2016) in local galaxies. For atomic gas, several studies of local galaxies find correlations with SFR or Σ_SFR that are too weak to explain the turbulent velocities in the galaxy outskirts (e.g., Tamburro et al. 2009; Ianjamasimanana et al. 2015; Utomo et al. 2019).

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In summary, while global σ₀ correlates with SFR properties, we do not find a direct connection between high, local star formation activity and elevated σ₀, as suggested by some simulations. Generally, however, simulations and models agree that stellar feedback is able to maintain galaxy-wide turbulence on scales of 10–20 km s⁻¹.

Turning to gravity-driven turbulence, an empirical model to describe the redshift evolution of velocity dispersion is that of marginally stable disks, where (non-interacting) galaxies are subject to gas replenishment from the halo or the cosmic web, and to gas loss through either outflows or star formation (Noguchi 1999; Silk 2001; Immeli et al. 2004a, 2004b; Förster Schreiber et al. 2006; Elmegreen et al. 2007; Genzel et al. 2008; Dekel et al. 2009a; Bouché et al. 2010; Krumholz & Burkert 2010; Cacciato et al. 2012; Davé et al. 2012; Lilly et al. 2013; Saintonge et al. 2013; Dekel & Burkert 2014; Krumholz & Burkhart 2016; Rathaus & Sternberg 2016; Leaman et al. 2017). In this framework, the (in)stability of the disk directly corresponds to the level of turbulence in the ISM, where turbulence is fed through external (accretion) and internal (radial flows, clump formation) events via the release of gravitational energy, creating a self-regulation cycle to maintain marginal stability (Dekel et al. 2009a; Genel et al. 2012a; but see Elmegreen & Burkert 2010).

For a snapshot in time that represents the observation of a high-z galaxy, this equilibrium situation is captured through the Toomre-Q parameter (Toomre 1964), where generally Q < Q_crit ≈ 1 indicates gravitational instability. Considering the one-component approximation for a gas disk, we can write (Binney & Tremaine 2008; Escala & Larson 2008; Dekel et al. 2009b)

$$\begin{eqnarray}&&{Q}_{\mathrm{gas}}=\displaystyle \frac{{\sigma }_{0}\kappa }{\pi G{{\rm{\Sigma }}}_{\mathrm{gas}}}=\displaystyle \frac{{\sigma }_{0}}{\pi G{{\rm{\Sigma }}}_{\mathrm{gas}}}\displaystyle \frac{{{av}}_{c}(r)}{r}.\end{eqnarray} \tag{ 5 }$$

Here, κ is the epicyclic frequency, a is a constant taking values of 1 and $\sqrt{2}$ for Keplerian and constant rotation velocity, respectively, and v_c is the circular velocity tracing the dynamical mass.

The framework of Toomre (in)stability generally refers to the linear regime, where perturbations are assumed to be axisymmetric. The galaxies studied here, however, are in the nonlinear limit where the ISM is turbulent and many stars have formed (Mandelker et al. 2014). Inoue et al. (2016) investigated the stability of simulated high-z disks, finding that large parts of the disks are in the nonlinear regime with Q > 1–3. This result, however, depends on the gas fraction, which is generally too low in the simulations. Indeed, for those simulated galaxies with the highest gas fractions (f_gas ∼ 0.4; still lower than for typical z ∼ 2 SFGs), Inoue et al. (2016) found values more compatible with observational findings. Meng et al. (2019) argued in recent work that the Toomre-Q linear stability analysis is still applicable to simulated high-z galaxies, with values of Q ∼ 0.5–1 in gas-rich regions (see also Behrendt et al. 2015 for simulations of isolated gas-rich disks).

Generally, for a multicomponent system, an effective Q parameter has to be computed, ${Q}_{\mathrm{eff}}^{-1}={\sum }_{i}{Q}_{i}^{-1}$, where i refers to, e.g., stars or different gas phases (e.g., Wang & Silk 1994; Escala & Larson 2008; Genzel et al. 2011; Romeo & Falstad 2013; Obreschkow et al. 2015, and references therein). Simulations of galaxy formation support a picture where Q_eff ∼ 1 for high-z galaxies, and Q_eff ∼ 2−3 for low-z galaxies where the increasing impact of a stellar disk increases Q_crit (Hohl 1971; Athanassoula & Sellwood 1986; Bottema 2003; Immeli et al. 2004a; Kim & Ostriker 2007; Agertz et al. 2009a, 2009b; Aumer et al. 2010; Ceverino et al. 2010; Hopkins et al. 2011; Genel et al. 2012b; Danovich et al. 2015). For gas-rich, thick or clumpy disks Q_crit decreases instead, such that for z ≳ 1 galaxies, values of Q_crit ≈ 0.7 are expected (e.g., Goldreich & Lynden-Bell 1965; Kim & Ostriker 2007; Wang et al. 2010; Romeo & Agertz 2014; Behrendt et al. 2015).

It has been shown that the gas-rich, star-forming disks observed at high redshift are at most marginally stable to gravitational fragmentation (Genzel et al. 2011; see also Swinbank et al. 2017; Johnson et al. 2018; Tadaki et al. 2018; and Fisher et al. 2017 for local high-z analogs), and Wisnioski et al. (2015) have shown that the redshift evolution predicted by Equation (5) for Q ∼ 1 gas disks is in remarkable agreement with observations (see also, e.g., Green et al. 2014; Turner et al. 2017; White et al. 2017; Johnson et al. 2018). In addition, Genzel et al. (2011) have shown that on subkiloparsec spatially resolved scales, values of Q ∼ 0.2 can be reached in regions of star-forming clumps, possibly demonstrating gravitational fragmentation at work.

We calculate Q_gas for our galaxies following Equation (5) by evaluating the circular velocity at v_c(r = 1.38R_e). As mentioned in Section 3.3, this radius corresponds to the theoretical peak of a Noordermeer disk with n_S = 1, such that the local gradient of the rotation curve is flat, leading to $a=\sqrt{2}$. The circular velocity v_c is computed from the model rotation velocity corrected for pressure support from the turbulent motions (Burkert et al. 2010, 2016; Wuyts et al. 2016). In Figure 9, we show f_gas = M_gas/M_bar, with M_bar = M_* + M_gas, as a function of Q_gas, color-coded by redshift as in Figure 4. Despite the large scatter, an anti-correlation between f_gas and Q_gas is evident, such that galaxies with higher gas fractions have lower Q (Spearman rank correlation coefficient ρ_S = −0.30 with significance σ_ρ = 3.6). This is in agreement with the theoretical prediction that SFGs that are more gas rich have lower Q values. The average Q_gas for our galaxies in the redshift bins z ∼ 0.9; 1.5; 2.3 is Q_gas = 1.2; 0.7; 0.5 (arrows in Figure 9). Our results on the average offset of ionized versus atomic+molecular gas from Section 4.5 suggest that the cold gas tracing the bulk of the gas mass might have a velocity dispersion lower by 10–15 km ⁻¹. This would lower the Q_gas values by a factor of ∼1.2–2. While our calculation of the Toomre-Q parameter is simplified through the omission of the stellar component, this suggests that thick high-z disks with high gas fractions of ≳50% can be marginally stable even down to Q_gas < 0.7.

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While gravitational instabilities are likely important drivers of the elevated velocity dispersions at z > 1, the contribution from stellar feedback-driven turbulence of the order of 10–20 km s⁻¹ could become comparable or even dominant for lower z, low-σ₀ galaxies. Therefore, one must consider both processes to get a complete picture.

The combination of stellar feedback and gravitational processes for turbulence driving has recently been investigated through the analytic model for structure and evolution of gas in galactic disks by Krumholz et al. (2018), who combine prescriptions for star formation, stellar feedback, and gravitational instabilities into a unified "transport+feedback" model to explain the range of observed dispersions from z = 3 to the present day. In their model, gas is in vertical hydrostatic equilibrium and energy equilibrium. This model assumes (isolated) rotating galactic disks built of gas and stars within a quasi-spherical dark matter halo over a wide redshift range. Disks are stable or marginally stable to gravitational collapse, regulated by mass transport through the disk. The gas is in vertical hydrostatic equilibrium, and in energy equilibrium such that losses through the decay of turbulence are balanced by energy input into the system via stellar feedback and the release of gravitational energy via mass transport through the disk.

Consistent with the discussion above, Krumholz et al. (2018) showed in their model that stellar feedback may maintain velocity dispersions of ∼10 km s⁻¹, creating a dispersion floor, while gravitational instabilities, for instance created through radial mass transport through the disk, are necessary to constantly drive velocity dispersions beyond σ₀ ∼ 20 km s⁻¹ for moderate star formation rates (see also Figure 4 by Krumholz et al. 2018). They make a prediction for galactic gas velocity dispersion and its correlation with SFR. In particular, they show that (see their Equation (60))

$$\begin{eqnarray}&&\mathrm{SFR}=\displaystyle \frac{0.42}{\pi G}\displaystyle \frac{1}{Q}\cdot {f}_{\mathrm{gas}}{v}_{\mathrm{circ}}^{2}{\sigma }_{0},\end{eqnarray} \tag{ 6 }$$

where we have substituted appropriate constants for high-z galaxies following Krumholz et al. (2018). Specifically, we adopt a rotation curve slope of β = 0, an offset between resolved and unresolved star formation law normalizations of ϕ_a = 3, a fraction of ISM in the star-forming phase f_sf = 1, a ratio of total pressure to turbulent pressure at the midplane of ϕ_mp = 1.4, a star formation efficiency per freefall time of _ff = 0.0015, an orbital period of t_orb,out = 200 Myr, and a maximum star formation timescale of t_sf,max = 2 Gyr.

We make two adjustments to our data to properly compare to the model: here, and for all of Section 5.4, we subtracted 15 km s⁻¹ in quadrature from our intrinsic dispersion values, denoted by σ_0,15, to ensure consistency with the theoretical model (see Krumholz & Burkhart 2016 and Krumholz et al. 2018, Appendix B). This 15 km s⁻¹ represents the average combination of thermal motions and expansion of H ii regions that enter our ionized gas velocity dispersion measurement (see also Sections 3.4 and 4.5). We also modify our gas mass fractions: the corresponding parameter used by Krumholz et al. (2018) describes an effective gas fraction at the midplane. This has typically higher values than our gas fraction f_gas because of the larger stellar scale heights compared to the gas scale heights. For the comparison here, we adopt a scaling factor of 1.5 for our gas mass fractions, motivated by measurements in the solar neighborhood (McKee et al. 2015; Krumholz et al. 2018; M. Krumholz 2019, private communication).

To compare the model prediction from Equation (6) to our data, we group correlated quantities and separate the star formation properties SFR and f_gas from the kinematic tracers v_circ and σ₀. We show the result for our kinematic sample in Figure 10, specifically the SFR divided by gas fraction as a function of circular velocity squared times intrinsic velocity dispersion. Figure 10 reveals a clear trend between the displayed quantities, with a Spearman rank correlation of ρ_S = 0.57 with significance σ_ρ = 6.8. We also show model predictions from Krumholz et al. (2018) as quoted in Equation (6) for three values of Q. There is a tendency for higher z galaxies to have a predicted Q ≲ 1, consistent with our results presented in Figure 9. Generally, however, our galaxies scatter around Q = 1 at all redshifts. This suggests that SFGs self-regulate at all times such that the population of SFGs evolves roughly along lines of constant Q. This result is largely independent from the specific choices of parameters such as ϕ_a or f, which will only affect the average Q value. Note that the above correlation between SFR and velocity dispersion is predicted for both the combined "transport+feedback" model and a model without feedback, but not for models lacking the "transport" component accounting for gravitational instabilities (see also Krumholz & Burkhart 2016).

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In the following, we now investigate separately changes of circular velocity and gas fraction in the σ₀–SFR parameter space. In Figure 11, we show for our kinematic sample the intrinsic velocity dispersion as a function of SFR, color-coded by circular velocity. As expected from the main sequence and Tully Fisher relation (Tully & Fisher 1977), which is in place for our data set at all redshifts (Übler et al. 2017), our data display a gradient such that circular velocity on average increases with increasing SFR. We plot the high-z model by Krumholz et al. (2018) as lines, but we modify it such that we vary the galaxy circular velocity from v_circ = 50 km s⁻¹ to v_circ = 450 km s⁻¹ in order to appropriately cover the range of observed velocities in our kinematic sample. In the model framework, stellar feedback creates and sustains a dispersion floor, represented through the horizontal regime of the model lines. The predicted rapid increase of velocity dispersion with SFR, the exact location here dependent on circular velocity, requires the release of gravitational energy through radial transport through the disk (see Krumholz et al. 2018 for details). The agreement between the theoretical model and our data is remarkably good: ∼60% of our data are matched by the model for this simple variation of only the circular velocity, with all other parameters being fixed to the fiducial "transport+feedback high-z" parameters.

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For the gas fraction, we can make only an approximate comparison. As mentioned in Section 2, gas masses for our galaxies are calculated by applying the scaling relation by Tacconi et al. (2018), since direct gas mass measurements are not available for most of our galaxies. With this, we get the total gas mass over the total baryonic mass per galaxy. Again, we use a scaling factor of 1.5 for our gas mass fractions. In Figure 12, we show the same parameter space as in Figure 11 but now color-coded by gas fraction. While galaxies with SFR ≲ 10 M_⊙ yr⁻¹ have on average lower gas fractions, no strong trend is apparent at higher SFRs. We show again lines based on the "transport+feedback high-z" model by Krumholz et al. (2018), but now we vary the gas fraction (and with it f_g,P) from f_g,Q = 0.2 to f_g,Q = 1.0 in order to explore the range of scaled gas fractions of galaxies in our kinematic sample. With solid lines, we show models with v_ϕ = 400 km s⁻¹, and dashed lines show v_ϕ = 200 km s⁻¹. It becomes clear that in the model framework, galaxies at fixed SFR and σ₀ can have higher f_g,Q and lower v_ϕ, or lower f_g,Q and higher v_ϕ, but rotation velocity has to be varied to cover the full range of SFRs in our observations.

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Horizontal variations of the model predictions can be reached by changing the fraction of gas assumed to be in the star-forming ISM, and through changes in the outer rotation curve slope. For instance, assuming only 20% of the gas to be in the star-forming phase pushes the horizontal floor of the model below 10 km s⁻¹ and lowers the predicted SFR by almost an order of magnitude. Assuming a dropping rotation curve, on the other hand, lifts the horizontal floor and increases the predicted SFR. Assuming an outer rotation curve slope of β = −0.5 increases the horizontal saturation of the model to ∼32 km s⁻¹, while increasing the star formation rate only marginally. Lang et al. (2017) have shown that the typical outer rotation curve slope of galaxies in our sample is negative. This is more pronounced at higher redshift, possibly offering an additional reason for the elevated velocity dispersions at z ≳ 2 in this model framework.

Considering these analytic model prescriptions and the typical uncertainty of the intrinsic dispersion measurements of δσ₀ ∼ 10 km s⁻¹ in our kinematic sample, we conclude that galaxies with σ₀ ≳ 35 km s⁻¹ are dominated by gravitational instability-driven turbulence. This encompasses more than 60% of galaxies in our sample, underlining the importance of gravity-driven turbulence in SFGs at z ∼ 1–3.

As a final remark, we briefly want to comment on AGN feedback as a potential additional source for elevated velocity dispersions in the SFGs in our kinematic sample. While we excluded galaxies, or regions of galaxies, that are so strongly affected by the AGN and associated outflows that the disk kinematics cannot be recovered, we do not entirely exclude AGNs. This ensures that we can explore the full mass range covered by the ${\mathrm{KMOS}}^{3{\rm{D}}}$ survey, including the high-mass end where at log(M_*/M_⊙) > 11, above the Schechter mass, the fraction of AGNs increases rapidly (Förster Schreiber et al. 2014, 2018; Genzel et al. 2014).

While we do not find significant correlations between z-normalized σ₀ and mass properties (Table 7), we do note a cloud of galaxies from all redshifts with dispersions above average for the highest stellar (log(M_*/M_⊙) > 11) and baryonic masses (log(M_bar/M_⊙) ≳ 11.3) as shown in Figure 13. About half of the log(M_bar/M_⊙) ≳ 11.3 above-average dispersion galaxies are known to host an AGN (stars in Figure 13). We speculate that the energy deposited by strong AGN feedback in the form of nuclear outflows could induce turbulence in the disk via the reaccretion of material at larger radii.

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It is important to keep in mind that outflow components with velocities similar to the galaxy rotation velocity can broaden the line width but may not be distinguishable from the star-forming regions, due to S/N limitations. Comparing to the deep AO data from the SINS/zC-SINF survey that we show in Figure 8, one of the three identified log(M_*/M_⊙) ≳ 11 AGN (Q2343-BX610), shows above-average velocity dispersions (after excluding the regions clearly affected by the nuclear outflow), while the other two (D3a-6004, D3a-15504) have average dispersions.