2.2.1 Rest-frame UV-colour-selected galaxies

A sample of 23 rest-frame UV-colour-selected (BX) galaxies in the redshift range $2.0<z<2.5$ that overlap with our parent $z\sim2$ photometric catalogue were selected from the SINS survey sample of Förster Schreiber et al. (Reference Förster Schreiber2009, hereafter FS09) and the AO-assisted IFS survey of Law et al. (Reference Law2009, hereafter LA09). Twenty-one of these galaxies had net Ly $\alpha$ EWs in the spectroscopic catalogue of Reddy et al. (Reference Reddy2008).

The BX galaxies targeted by FS09 and LA09 have stellar masses in the range $9.0 < \mathrm{log}({M}_{\star }/{M}_{\odot }) < 10.7$ and are drawn from the near-IR spectroscopic sample of Erb et al. (2006a,b,c). Although they have a number of galaxies in common, the LA09 galaxies tend to have stellar masses in the less-massive to typical-mass range (mean $\mathrm{log}({M}_{\star }/{M}_{\odot }) \approx 10.1$ ) compared to the FS09 sources that favour the higher mass end of the BX sample (mean $\mathrm{log}({M}_{\star }/{M}_{\odot }) \approx 10.42$ ).

The SINS survey used the SINFONI instrument at the ESO VLT in natural-seeing and AO-assisted modes to extract spatially resolved maps of the velocity-integrated flux, relative velocity, and velocity dispersion of the H $\alpha$ emission line. To facilitate a general analysis of all the SINS H $\alpha$ galaxies, FS09 defined a working criterion involving the observed velocity gradient ( ${v}_{\mathrm{obs}}$ ) and the integrated line width ( ${\sigma}_{\mathrm{int}}$ ) by which galaxies with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $> 0.4$ were classified as ‘rotation-dominated’, and those with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $< 0.4$ were classified as ‘dispersion-dominated’. Using either quantitative kinemetric analysis Shapiro et al. Reference Shapiro2008) or qualitative assessment of the asymmetry in the velocity field and dispersion map, galaxies with kinematics consistent with rotation were further classified as either ‘discs’ or ‘mergers’. Updated kinematic classifications for eight of the SINS objects were derived from the deep AO-assisted data collected as part of the SINS/zC-SINF survey (Förster Schreiber et al. Reference Förster Schreiber2018, hereafter FS18). Galaxies identified by FS18 as possibly hosting an AGN (e.g., Q2343-BX610) were rejected from our sample.

The LA09 AO study utilised the OSIRIS near-infrared integral field spectrograph (Larkin et al. Reference Larkin2006) at the W. M. Keck Observatory. LA09 quantified the rotational dynamic support using the ratio of shear velocity to intrinsic velocity dispersion ( $v_{\mathrm{shear}}/{\sigma}_{\mathrm{mean}}$ ) and used detailed morphological analysis in combination with the 2D kinematic maps to characterise the kinematic properties of each galaxy. For the purposes of kinematic classification, we equate $v_{\mathrm{shear}}/{\sigma}_{\mathrm{mean}}$ with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ in the nomenclature of FS09, and, except for sources identified by LA09 as merging systems, assign the LA09 galaxies as either rotation or dispersion-dominated.

The grand design spiral galaxy Q2343-BX442 reported by Law et al. (Reference Law2012a, hereafter LA12) was also included in our $z\sim2$ kinematic sample. Net Ly $\alpha$ EW and kinematic parameters for Q2343-BX442 were supplied by Law (private communication). With ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ = 0.83, and clear disc-like morphology, we classify Q2343-BX442 as ‘rotation-dominated’. Details of the $z\sim2$ UV-selected BX galaxies that comprise our kinematic sample are summarised in Table 1 along with references to the source studies in each case.

2.2.2 KMOS $^{\mathrm{3D}}$ galaxies

We supplement our $z\sim2$ kinematic sample with a redshift-selected subset ( $2.0<z<2.5$ ) of 13 galaxies from the COSMOS field pointings of KMOS $^{\mathrm{3D}}$ (K $^{\mathrm{3D}}$ )—an integral field survey of over 600 mass-selected galaxies at 0.7 $<z<$ 2.7 using the KMOS instrument at the ESO VLT (Wisnioski et al. Reference Wisnioski2019). The K $^{\mathrm{3D}}$ survey combined galaxy dynamics derived from H $\alpha$ , near-IR continuum, velocity, and velocity dispersion maps with structural parameters and multi-band imaging to establish a set of criteria by which robust kinematic classifications could be determined (Wisnioski et al. Reference Wisnioski2015). The K $^{\mathrm{3D}}$ sources that we employ are part of the ‘high S/N disc sample’ of rotation-dominated galaxies reported by Wisnioski et al. (Reference Wisnioski2015, hereafter WI15) that focused on massive galaxies with $\mathrm{log}({M}_{\star }/{M}_{\odot }) \gtrsim 10$ (see Table 1 for details). These galaxies all meet the less exacting FS09 criterion of ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ > 0.4 for classification as rotation-dominated in our study. The K $^{\mathrm{3D}}$ galaxies were cross-matched with the D2 field of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS Hudelot et al. Reference Hudelot2012) to obtain $u^* g^\prime r^\prime i^\prime z^\prime$ multi-band photometry. The $u^* g^\prime r^\prime i^\prime z^\prime$ photometric data were transformed into $U_nG\mathcal{R}$ magnitudes to facilitate direct comparison with the $z\sim2$ sources of FS09, LA09, FS18, and LA12.

The transformation was achieved by performing spectrophotometry on rest-frame UV composite spectra derived from a sample of $z\sim3$ $U_nG\mathcal{R}$ LBGs divided into quartiles on the basis of net Ly $\alpha$ EW (A. Shapley, private communication). The composite spectra are representative of the average LBG spectrum in each quartile and thus accurately trace the colour–colour evolution and colour–magnitude distribution of each quartile and its component galaxies (Cooke et al. Reference Cooke, Omori and Ryan-Weber2013). For each K $^{\mathrm{3D}}$ galaxy, the composite spectra were first redshifted to the observation frame, and flux density in the Lyman- $\alpha$ forest corrected for redshift-dependent absorption through the IGM. The spectra were then convolved with the bandpasses of the $u^* g^\prime r^\prime$ filters, the resulting integrated flux densities normalised to the observed $g^\prime $ -band magnitude and simulated $u^* g^\prime r^\prime$ -band magnitudes calculated. From these, the composite spectrum that best fit the observed $u^* g^\prime r^\prime i^\prime z^\prime$ photometry for each galaxy was determined. A reddening correction (Calzetti et al. Reference Calzetti2000) was applied to the best fit spectrum in each case as required to optimise the fit. The best fit normalised and reddened spectrum was then convolved with the bandpassess of the $U_nG\mathcal{R}$ filters, and $U_nG\mathcal{R}$ -band magnitudes estimated for each galaxy. In all cases, the quartile 1 (strongest net Ly $\alpha$ EW absorption) or the quartile 2 (next strongest net Ly $\alpha$ EW absorption with some emission) composite spectra provided the best fit to the observed photometry, suggesting that the K $^{\mathrm{3D}}$ galaxies are best classified as aLBG or G $_a$ spectral types (see Section 2.4). Using the transformed $U_nG\mathcal{R}$ photometry, a standard colour–colour test (Steidel et al. Reference Steidel2004; Adelberger et al. Reference Adelberger2004) was applied to confirm that the K $^{\mathrm{3D}}$ sample satisfied (within the photometric uncertainties) the criteria to be selected as $z\sim2$ LBGs.

With the goal of measuring net Ly $\alpha$ EWs, seven of the ‘high S/N disc sample’ K $^{\mathrm{3D}}$ galaxies were included as secondary science targets on our multi-object slitmasks using the LRIS instrument (Oke et al. Reference Oke1995; Steidel et al. Reference Steidel2004) at Keck on 2016 December 26, 27 and 2020 January 20–22. These data were reduced in the conventional manner using IRAF and in-house code. Net Ly $\alpha$ EWs were measured following the procedure of Kornei et al. (Reference Kornei2010). We successfully measured net Ly $\alpha$ EW for three WI15 galaxies from the 2020 January data. However, weather conditions and primary science constraints during the 2016 December run resulted in low S/N spectra for the remaining four WI15 galaxies that were too poor to enable the reliable measurement of net Ly $\alpha$ EW. The 2D and 1D spectra of these four galaxies show no evidence of a Ly $\alpha$ emission component of $\gtrsim$ 10 Å. The q1 and q2 quartiles of Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003) have average net Ly $\alpha$ EWs $-$ 14.9 Å and $-$ 1.1 Å, respectively, and show Ly $\alpha$ emission components of $<$ 10 Å. Consequently, we can reasonably postulate that the net Ly $\alpha$ EWs for the four galaxies are similar to the q1 and q2 quartile galaxies. On this basis, we report net Ly $\alpha$ EW $<0.0$ for these four sources and provisionally assign them spectral types aLBG/G $_a$ in our system

2.3.1 AMAZE and LSD galaxies

Using the SINFONI integral field spectrograph (Eisenhauer et al. Reference Eisenhauer, Iye and Moorwood2003) on the ESO Very Large Telescope (VLT) in natural-seeing and natural guide star adaptive optics (AO) observation modes, respectively, the related AMAZE (Maiolino et al. Reference Maiolino2008) and LSD (Mannucci et al. Reference Mannucci2009) surveys conducted near-IR IFU spectroscopic observations on LBGs at redshifts $z\gtrsim3$ with $\mathcal{R}$ $\simeq$ 24.5 (L^* and brighter) corresponding to a mass range of $\mathrm{log}({M}_{\star }/{M}_{\odot })\approx 10-11$ .

Gnerucci et al. (Reference Gnerucci2011, hereafter GN11) derived nebular emission-line kinematics for a subset of amaze and lsd galaxies by fitting the profile and shift of the [Oiii] $\lambda\lambda$ 4959, 5007 doublet. Due to limited signal-to-noise, GN11 used a plane-fitting method to assign kinematic classifications according to the following criteria: galaxies for which the velocity map shows a non-zero gradient after plane-fitting were classified as ‘rotating’; galaxies for which the velocity map could not be fitted with a plane were classified as ‘not-rotating’; and galaxies with velocity maps well-fitted by a plane but with inclination consistent with zero were labelled as ‘not classifiable’. GN11 further employed a rotating disc modelling approach to estimate maximum rotation velocities and intrinsic velocity dispersions for galaxies classified as ‘rotating’ in their sample.

We extract 18 GN11 galaxies that overlap with our parent photometric catalogue. Details of the AMAZE and LSD galaxies used in this work are summarised in Table 2.

2.3.2 KDS galaxies

As part of the KMOS Deep Survey (KDS), Turner et al. (Reference Turner2017, hereafter TU17) investigated the kinematics of typical isolated field SFGs at $z \simeq$ 3.5 in the mass range $9.0 < \mathrm{log}({M}_{\star }/{M}_{\odot }) < 10.5$ using the KMOS instrument at the ESO VLT (Sharples et al. Reference Sharples2013). With natural-seeing measurements of the [Oiii] $\lambda$ 5007 emission line, TU17 extracted 2D kinematic maps and used beam-smearing corrections derived from dynamical modelling to determine values of intrinsic rotation velocity (V $_\mathrm{C}$ ) and intrinsic velocity dispersion ( ${\sigma}_\mathrm{int}$ ) for the spatially resolved target galaxies in their sample. Dictated by the signal-to-noise ratio of their data, TU17 used a simple empirical diagnostic based on the ratio V $_\mathrm{C}$ / $\sigma_\mathrm{int}$ to kinematically classify their sample. Galaxies were classified as ‘rotation-dominated’ if V $_\mathrm{C}$ / $\sigma_\mathrm{int} > 1$ , and as ‘dispersion-dominated’ if V $_\mathrm{C}$ / $\sigma_\mathrm{int} < 1$ .

The SSA22-P2 pointing of the KDS survey targeted a field environment to the south of the main SSA22 spatial overdensity (Steidel et al. Reference Steidel2000) and yielded five morphologically isolated (non-merger) field galaxies that were in common with our parent photometric catalogue. Details of the five KDS galaxies used in this work are given in Table 2.

2.3.3 DSF2237a-C2

With a redshift of $z \simeq 3.3$ and kinematics derived from measurements of the [Oiii] $\lambda$ 5007 emission line, we include the LA09 galaxy, DSF2237a-C2, in our $z\sim3$ kinematic sample. Drawn from the $z\sim3$ LBG catalogue of Steidel et al. (Reference Steidel2003), DSF2237a-C2 has $v_{\mathrm{shear}}/{\sigma}_{\mathrm{mean}} = 0.6 \pm 0.2$ , and is the only isolated LA09 galaxy where the observed velocity gradient ‘is consistent with rotation and unambiguously aligned with the morphological major axis’ (LA09). For the purposes of kinematic classification, we treat DSF2237a-C2 similarly to the $z\sim2$ LA09 galaxies (see Section 2.2.1) and assign a ‘rotation-dominated’ classification to this galaxy.

3.2.1 $z\sim2$ kinematics on the CMD

Fig. 2 shows the 36 galaxies in the $z\sim2$ kinematic sample overlaid on the parent population of 557 $z\sim2$ LBGs dispersed on a $(U_n-\mathcal{R})$ / $\mathcal{R}$ CMD adapted from Paper I. We note that most of the $z\sim2$ kinematic sample is distributed towards the redder half of the CMD; 31 of 36 galaxies lie above the primary cut, and only five galaxies lie below it. To a large degree, this tendency reflects the non-random selection bias in the source IFU studies, and the observational difficulty associated with obtaining high-quality IFU-based data for faint and/or compact sources. For example, the early observations of FS09 targeted galaxies with ‘higher mass, spatially resolved velocity gradients, large velocity dispersions, or spatially extended emission’, and WI15 reports specifically on a ‘high S/N disc sample’ drawn from the larger K $^{\mathrm{3D}}$ survey. Due to the known relationship between Ly $\alpha$ and $z\sim2$ LBG morphology (Law et al. Reference Law2012b), and that between rest-frame UV colour, Ly $\alpha$ absorption/emission strength, and rotational dynamic support that we report herein, both of these selection biases inevitably result in samples that are redder than the average of the SFG population accessible by ground-based IFU spectrographs and our LBG parent sample. That being said, the aims of this paper do not require that the kinematic galaxies sample the parent LBG population in an unbiased way. It is sufficient that the kinematic sample spans a wide enough range in apparent magnitude and colour to identify a relationship with the broadband colour and magnitudes of LBGs, and that this relationship can be used to statistically segregate the galaxies by their Ly $\alpha$ properties and general kinematic type.

Figure 2. Similar to the left panel of Fig. 1 but for the $z\sim2$ kinematic sample plotted on a $(U_n-\mathcal{R})$ vs $\mathcal{R}$ CMD adapted from Paper I. Galaxies classified as ‘rotation-dominated’ (rd, circles) follow the form of the aLBG population distribution, and galaxies classified as ‘dispersion-dominated’ (dd, diamonds) follow the eLBG population to the extent that can be estimated given the scarcity of Ly $\alpha$ -emitting galaxies in the sample, and at $z\sim2$ in general. Galaxy mergers (m, stars) are typically found central in colour and towards the bright end of the CMD. The mean positions of the rd, dd, and m sub-samples are marked by a red circle, blue diamond, and orange star, respectively.

This ‘red bias’ of the $z\sim2$ kinematic sample notwithstanding, all but one of the 22 rotation-dominated galaxies lie above the primary cut, and all are above the dotted-dashed line, in the white and red regions that enclose $\gtrsim$ 95% of the aLBG population. The ‘high-confidence’ aLBG (red) region is free from contamination by non-merger dispersion-dominated galaxies, and only one such source (Q1217-BX95, see Section 3.2.2) is found above the primary cut. Although few in number, all dispersion-dominated galaxies lie below the dashed line (in the white and blue regions) where $\gtrsim$ 97% of the eLBGs are located. All but one galaxy (Q1623-BX502, see Section 3.2.2) below the primary cut are dispersion-dominated systems (including mergers). Thus, as illustrated by the mean positions of the ‘rd’ and ‘dd’ sub-samples in Fig. 2 (red circle and blue diamond, respectively), the rotation- and dispersion-dominated subsets of the kinematic sample are coincident with, and may follow, the colour distributions of aLBGs and eLBGs on the CMD.

To test these propositions, we use the non-parametric two-sided Kolmogorov–Smirnov (KS) test to evaluate whether the distributions of the rotation and dispersion-dominated sub-samples in $(U_n-\mathcal{R})$ colour are statistically consistent with the null hypotheses that they are drawn from the $(U_n-\mathcal{R})$ colour distributions of the parent aLBG and eLBG populations. For consistency in comparison, we limit the magnitude range of the underlying aLBGs and eLBGs to be the same as that spanned by the kinematic sample ( $23.31 < \rm{M_{AB}} < 24.85$ ), and the kinematic sample to galaxies that derive from the UV-colour-selected $z\sim2$ LBG sample used to establish the photometric segregation and selection criteria described in Paper I, and shown on the CMDs in Figs. 2 & 3. Histogram plots in $(U_n-\mathcal{R})$ colour of the sub-samples used in the KS test are shown in the top panel of Fig. 3. The bifurcation in $(U_n-\mathcal{R})$ colour is more readily visualised in the lower panel of Fig. 3 which shows colour histograms of the difference in normalised fraction of the same aLBG & eLBG, and rd & dd sub-samples (i.e., $\mathrm{nFrac}_{aLBG} - \mathrm{nFrac}_{eLBG}$ and $\mathrm{nFrac}_{rd} - \mathrm{nFrac}_{dd}$ ).

Figure 3. Top: Histograms in $(U_n-\mathcal{R})$ colour of the aLBG, eLBG, ‘rotation-dominated’ (rd) and ‘dispersion-dominated’ (dd) sub-samples used in the KS tests. Bottom: Histograms of the difference in normalised fraction of the same aLBG/eLBG and rd/dd sub-samples showing clear bifurcation in $(U_n-\mathcal{R})$ colour.

The KS tests yield a probability (p) of 0.969 that the dispersion-dominated galaxies in our kinematic sample and the underlying eLBG population derive from the same distribution of $(U_n-\mathcal{R})$ colours. Moreover, we can reject with greater than 95% confidence ( $p = 0.046$ ) the null hypothesis that the dispersion-dominated sub-sample and aLBGs are drawn from the same colour distribution. The analogous hypothesis that the rotation-dominated sub-sample and eLBGs share a common distribution of $(U_n-\mathcal{R})$ colours can be similarly rejected with high confidence ( $p = 0.0002$ ), and there is a probability of $p = 0.181$ that they derive from the same distribution of aLBG colours.

The lower confidence of the KS result for the rotation-dominated sub-sample with respect to aLBGs can be plausibly explained in terms of the non-random selection bias in the kinematic sample described above. Nevertheless, a KS test between the rotation and dispersion-dominated sub-samples indicates that we can reject with $\gtrsim$ 95% confidence ( $p = 0.0425$ )—and $\sim$ 98% ( $p = 0.0203$ ) if we include the K $^{\mathrm{3D}}$ discs—the null hypothesis that they are drawn from the same distribution in $(U_n-\mathcal{R})$ colour.

Sources classified as mergers will often contain two galaxies and will have brighter magnitudes compared to the non-merging fraction of the kinematic sample. The mean $\mathcal{R}$ -band magnitude of the ten merging systems is m( $\mathcal{R}$ ) $_{merger}$ = 23.51 (orange star in Fig. 2), which is $\sim$ 0.6 mag brighter than the mean magnitude of the non-merging fraction (m( $\mathcal{R}$ ) $_{non-merger}$ = 24.10). Moreover, galaxies in the kinematic sample segregate in colour depending on their classification. Mergers would statistically contain two galaxies roughly randomly selected from the aLBG, eLBG, Ga, and Ge populations (modulo any environmental effects). The mean $(U_n-\mathcal{R})$ colour (and standard deviation) of the mergers is 1.03 (0.38) compared to 1.43 (0.37) and 0.70 (0.37) for the rotation and dispersion-dominated sub-samples, respectively.

The statistical evidence connects the kinematic properties of $z\sim2$ LBGs to Ly $\alpha$ spectral type via their mutual association with rest-frame UV colour on the CMD. We next look to investigate this trend directly using the 28 galaxies in the kinematic sample for which spectroscopically determined net Ly $\alpha$ EWs are available. The top panel of Fig. 4 shows this sub-sample with their assigned Ly $\alpha$ spectral types overlaid on the parent $z\sim2$ LBGs. Seven out of eight rotation-dominated galaxies are aLBG or G $_a$ spectral types, and all seven K $^{\mathrm{3D}}$ rotators are net Ly $\alpha$ -absorbers (see Section 2.2.2). Three out of four dispersion-dominated galaxies are net Ly $\alpha$ -emitters (eLBG or G $_e$ spectral types. The lower panel of Fig. 4 shows just those members of the kinematic sample that meet the criteria for classification as aLBGs or eLBGs overlaid on the parent sample dispersed in colour-magnitude space with symbols colour-coded on a red-blue scale according to their measured net Ly $\alpha$ EW as described in Paper I. Seven of the 10 aLBGs are rotation-dominated and three are mergers. All rotation-dominated aLBGs lie above the primary cut, with six of the seven above the dashed blue line in the ‘high-confidence’ aLBG region. The three eLBGs are located near or below the $z\sim2$ eLBG distribution mean, with two galaxies (including the dispersion-dominated Q2343-BX418) below the dotted-dashed red line in the ‘high-confidence’ eLBG region.

Figure 4. Top: Similar to the right panel of Fig. 1 but for the $z\sim2$ kinematic sample plotted on a $(U_n-\mathcal{R})$ vs $\mathcal{R}$ CMD. Members of the interacting close-pair identified by FS09 (Q2346-BX404 & Q2346-BX405) are indicated by an asterisk ( $^*$ ). Other galaxies highlighted in Section 3.2.2 are labelled as follows: (a) Q1217-BX95, (b) Q2343-BX418, (c) Q2343-BX660, and (d) Q1623-BX502. Bottom: Similar to the top panel, but with members of the kinematic sample that meet the criteria for classification as aLBGs and eLBGs overlaid on the parent $z\sim2$ LBGs (squares) colour-coded on a red-blue gradient according to their net Ly $\alpha$ EW. Black crosses indicate the mean colour and magnitude positions for the $z\sim2$ aLBG and eLBG distributions bisected by the solid green line (primary cut). Dashed blue and dotted-dashed red lines indicate a 1- $\sigma$ dispersion in colour from the primary cut for the aLBG and eLBG distributions, respectively, and define one choice of photometric selection criteria for the isolation of pure Ly $\alpha$ -absorbing and Ly $\alpha$ -emitting sub-samples as determined in Paper I.

Overall, not only do the rotation and dispersion-dominated sub-samples lie on the CMD in a way that is statistically associated with the distribution in colour of the Ly $\alpha$ -absorbing and Ly $\alpha$ -emitting populations, they have spectroscopically determined Ly $\alpha$ spectral types that are consistent with this trend. Moreover, reviewing both panels of Fig. 4, the broadband photometric selection criteria select nearly pure samples of rotation- and dispersion-dominated systems and potentially 100% pure selection for galaxies meeting our net Ly $\alpha$ EW criteria.

3.2.2 eLBGs and dispersion-dominated galaxies

The absence of galaxies with large, disc-like rotating structures in the Ly $\alpha$ -emitting half of the CMD, and the fact that the ‘high-confidence’ aLBG region is almost completely devoid of net Ly $\alpha$ -emitting and/or dispersion-dominated sources, suggest that strong Ly $\alpha$ emission is linked to dispersion-dominated kinematics. The strength of such a claim is, however, challenged by the relative scarcity of faint, blue Ly $\alpha$ -emitting galaxies in our sample. In this regard, a closer examination of the eLBG and dispersion-dominated galaxies in the kinematic sample is instructive.

The four dispersion-dominated galaxies in the $z\sim2$ kinematic sample (diamond symbols in Fig. 2 and the top panel of Fig. 4) have a mean net Ly $\alpha$ EW of $+$ 14.4 Å. Of these, three are net Ly $\alpha$ -emitters (eLBG or G $_e$ spectral types). The exception is Q2346-BX405 which has a net Ly $\alpha$ EW of $-8.61$ Å (G $_a$ spectral type). While FS09 ascribe a ‘dispersion-dominated’ classification to Q2346-BX405, they note that it also has some kinematic features that are consistent with disc rotation. Moreover, they point out that Q2346-BX404 and Q2346-BX405 are an interacting pair—a feature that could result in the partial disruption of rotation kinematics and/or a subsequent classification as a dispersion-dominated system. In any event, Q2346-BX405 and its partner Q2346-BX404 (marked by asterisks in the top panel of Fig. 4) lie near the centre of the CMD at positions we find to be typical of mergers, and would be excluded from selection as aLBGs or eLBGs in our photometric method.

The other dispersion-dominated galaxy of interest within our framework is Q1217-BX95 (galaxy ‘a’ in the top panel of Fig. 4). While its location on the CMD is within the eLBG colour–magnitude distribution, it has a colour ( $(U_n-\mathcal{R})$ = 1.2) that is more typical of a rotation-dominated net Ly $\alpha$ -absorber. Q1217-BX95 is relatively massive ( $\mathrm{log}({M}_{\star }/{M}_{\odot })=10.08$ ), but is compact ( $R_e$ = 0.6 kpc), has low gas-phase metallicity ( $12 + \mathrm{log}({O}/{H}) = 8.15 \pm 0.02$ ), and displays a $5\sigma$ detection of the auroral [Oiii] $\lambda$ 4363 emission line.Footnote ¹ On the basis of this ensemble of properties, and a net Ly $\alpha$ EW of $+$ 10.2 Å, we would predict Q1217-BX95 to have dispersion-dominated kinematics, consistent with the classification for this galaxy reported by LA09. We note that photometric errors deriving from poor seeing in the Q1217 field images ( $1.56^{\prime\prime}$ in $U_n$ ) may be responsible for the $(U_n-\mathcal{R})$ colour of this galaxy (C. Steidel, private communication).

The three spectroscopic eLBGs in our kinematic sample (Q2343-BX418, Q2343-BX660, and Q1623-BX502) are the three sources furthest below the primary cut on the CMD, and lie close to the eLBG mean (see bottom panel of Fig. 4). They are, however, kinematically diverse. The dispersion-dominated Q2343-BX418 (galaxy ‘b’ in the top panel of Fig. 4) has low stellar mass ( $\mathrm{log}({M}_{\star }/{M}_{\odot })= 9.0$ ), has low metallicity ( $12 + \mathrm{log}({O}/{H}) = 7.9 \pm0.2$ ), is very blue ( $(U_n-\mathcal{R})$ = 0.32), is very compact ( $R_e$ = 0.8 kpc), and has strong Ly $\alpha$ emission (net Ly $\alpha$ EW = $+$ 53.5 Å). Although having kinematic structure consistent with merging compact galaxies (LA09), Q2343-BX660 (galaxy ‘c’ in the top panel of Fig. 4) is otherwise similar to Q2343-BX418; it has relatively low stellar mass ( $\mathrm{log}({M}_{\star }/{M}_{\odot })= 9.9$ ), low metallicity ( $12 + \mathrm{log}({O}/{H}) = 7.99 \pm0.05$ ), compact morphology ( $R_e$ = 1.6 kpc), very blue colour ( $(U_n-\mathcal{R})$ = 0.36), and is a strong Ly $\alpha$ emitter (net Ly $\alpha$ EW = $+$ 20.4 Å).

With a net Ly $\alpha$ EW of $+$ 28.5 Å, and a ‘rotation-dominated’ kinematic classification derived from the deep AO-assisted observations of FS18, Q1623-BX502 (galaxy ‘d’ in the top panel of Fig. 4) appears to be inconsistent with our proposition that large net Ly $\alpha$ EW is a useful predictor of dispersion-dominated kinematics. Although its magnitude and redder colour $(U_n-\mathcal{R})$ = 0.72) than the other eLBGs place it in the ‘white strip’ on the CMD that contains both aLBGs and eLBGs, its relatively low stellar mass ( $\mathrm{log}({M}_{\star }/{M}_{\odot })= 9.4$ ), compact morphology ( $\rm{R_e} = 1.1 \pm 0.7$ kpc), low gas phase metallicity ( $12 + \mathrm{log}({O}/{H}) = 8.09$ ), are typical of what we would predict for a dispersion-dominated galaxy with such strong Ly $\alpha$ emission. Earlier reports based on seeing-limited and AO observations classify Q1623-BX502 as ‘dispersion-dominated’ (FS09, LA09 and Newman et al. Reference Newman2013) and the ratios ${\rm{\Delta}}{v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{tot}}$ ( $0.50 \pm 0.16$ ) and ${V}_{\mathrm{rot}}/{\sigma}_{0}$ ( ${2.0}_{-0.8}^{+1.5}$ ) for Q1623-BX502 reported by FS18 are equivalent (within the stated uncertainties) to the threshold values (0.4 and $\sqrt{3.36} \approx 1.83$ , respectively) used by FS09 and FS18 to classify galaxies as either rotation or dispersion-dominated (see Section 3.3.2). Thus, Q1623-BX502 is a borderline case in terms of its kinematics, and, by its position on the CMD, would not be selected by our kinematic broadband photometric criteria (see Paper I).

To summarise, although some of the Ly $\alpha$ -emitting sources in the $z\sim2$ kinematic sample are not classified as ‘dispersion-dominated’ in the source IFU studies, they are characteristically compact, blue in colour, have relatively low stellar mass, low gas-phase metallicity, and highly disordered kinematics in which the rotation signature (if any) is weak and more typical of low angular momentum ‘orbital-type’, rather than the strong ‘disc-like’ rotation characteristic of the Ly $\alpha$ absorbers. We note that those cases that are borderline or complex do not meet our broadband photometric criteria and would not be selected.

3.3.1 Kinematic classifications vs net Ly $\alpha$ EW

With even a naive application of the kinematic classifications derived from the literature, there is a clear bifurcation in the average Ly $\alpha$ spectral properties of the non-merger rotation and dispersion-dominated populations in the $z\sim2$ and $z\sim3$ kinematic samples. The mean net Ly $\alpha$ EW for rotation-dominated galaxies in the $z\sim2$ sample is $-$ 7.0 Å, and $+$ 14.4 Å for the dispersion-dominated sub-sample. (cf. $+$ 1.1 and $+$ 19.6 Å for the $z\sim3$ rotating/rotation-dominated and not-rotating/dispersion-dominated sub-samples, respectively). A two-sided KS test on a combined ( $z\sim2$ + $z\sim3$ ) sample comprising eighteen rotation- and 14 dispersion-dominated galaxies rejects with $\sim$ 99% confidence ( $p = 0.012$ ) the null hypothesis that the rotation-and dispersion-dominated galaxies were drawn from the same distribution of net Ly $\alpha$ EW values.

3.3.2 Quantitative kinematics versus Ly $\alpha$

We now look to investigate the relationship between net Ly $\alpha$ EW and galaxy kinematics using quantitative parameters derived from the IFU kinematic maps of the respective source studies in addition to the classifications that we have used up to this point. In doing so, we note that methods to measure rotation velocity and global velocity dispersion differ between surveys, Although we have taken care to compare like with like as much as possible, homogenising these methods is beyond the scope of this paper, and we acknowledge that this is a potential source of systematic errors.

For this purpose, we invoke the ratio ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ used by FS09 (and with modified notation by FS18) where ${v}_{\mathrm{obs}}/2$ is half the maximum observed H $\alpha$ velocity gradient across the source, and ${\sigma }_{\mathrm{int}}$ ( ${\sigma }_{\mathrm{tot}}$ in FS18) is the integrated velocity dispersion derived from the linewidth of the H $\alpha$ emission in the spatially collapsed object spectrum. We note that ${v}_{\mathrm{obs}}/2$ and ${\sigma}_{\mathrm{int}}$ are equivalent to the shear velocity ( ${v}_{\mathrm{shear}}$ ) and net velocity dispersion ( ${\sigma}_{\mathrm{net}}$ ), respectively, in the nomenclature of LA09. We similarly equate for the purpose of this analysis, the observed rotation velocity (V $_{\mathrm{obs}}$ ) and velocity dispersion ( $\sigma_{\mathrm{obs}}$ ) of TU17 with ${v}_{\mathrm{obs}}/2$ and ${\sigma}_{\mathrm{int}}$ , respectively.

As a quantity derived from the observed velocity field, ${v}_{\mathrm{obs}}/2$ is related to the actual rotation velocity by a factor of $1/\sin(i)$ , where i is the galaxy inclination angle, and ${\sigma }_{\mathrm{int}}$ incorporates contributions to the integrated linewidth from both large-scale velocity gradients and intrinsic (random) gas motions. Nevertheless, ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ has proven to be a useful (if approximate) probe of the nature of the dynamical support (FS09) and provides here a ready means to compare kinematic results from different studies. Values of ${v}_{\mathrm{obs}}/2$ , ${\sigma}_{\mathrm{int}}$ and the ratio ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ for a subset of 21 $z\sim2$ and five $z\sim3$ galaxies in our kinematic samples derived from their respective source studies are listed in Tables 3 & 4.

Table 3. Kinematic and Ly $\alpha$ properties of UV colour-selected $z\sim2$ LBGs.

^aHalf the observed difference between the maximum and minimum relative H $\alpha$ velocities across the source, uncorrected for inclination. Equivalent to the shear velocity ( ${v}_{\mathrm{shear}}$ ) in the LA09 and LA12 nomenclature.

^bIntegrated velocity dispersion derived from the width of the H $\alpha$ emission line in the spatially collapsed object spectrum. Equivalent to ${\sigma }_{\mathrm{tot}}$ in FS18 and ${\sigma }_{\mathrm{net}}$ in LA09 and LA12.

^cWe use ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ as an empirical measure of dynamical support and adopt the criterion of FS09 in which galaxies with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $> 0.4$ are rotation-dominated, and those with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $< 0.4$ are dispersion-dominated.

^dIntrinsic rotation velocity corrected for beam smearing and inclination angle according to ${v}_{\mathrm{rot}} \cdot \sin(i) = C_{\mathrm{PSF,v}} \cdot v_{\mathrm{obs}}/2$ (FS18). For LA09 galaxies, we use $C_{\mathrm{PSF,v}} = 1.3$ , the average correction factor of the FS18 sample, and $\sin(i) = \pi/4$ , the mean inclination angle correction for a distribution of randomly inclined discs (LA09).

^eIntrinsic local velocity dispersion determined from correction of the observed local velocity dispersion ( $\sigma_{\mathrm{0obs}}$ ) for beam smearing according to $\sigma_{\mathrm{0}} = C_{\mathrm{PSF,}\sigma} \cdot \sigma_{\mathrm{0obs}}$ (FS18). For the purposes of this comparison $\sigma_{\mathrm{mean}}$ in LA09 approximates to $\sigma_{\mathrm{0obs}}$ and we apply to the LA09 sample a correction of $C_{\mathrm{PSF},\sigma} = 0.85$ , the average correction factor for the FS18 sample.

^fRatio of intrinsic rotation velocity and velocity dispersion corrected for beam smearing and inclination angle. Within the rotating-disc framework of FS18, ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}} \approx \sqrt{3.36}$ corresponds to the point above which rotation starts to dominate over velocity dispersion in the dynamical support of turbulent discs.

^gKinematic classification assigned in the source studies (see Section 22): rd = rotation-dominated, dd = dispersion-dominated, m = merger.

^hObservation mode employed in the source studies: NS = natural seeing, AO = adaptive optics assisted.

ⁱSource references for kinematic data: FS09 = Förster Schreiber et al. (Reference Förster Schreiber2009), LA09 = Law et al. (Reference Law2009), LA12 = Law et al. (Reference Law2012a), FS18 = Förster Schreiber et al. (Reference Förster Schreiber2018).

Table 4. Kinematic and Ly $\alpha$ Properties of UV Colour-Selected $z\sim3$ LBGs.

^aObserved rotation velocity measured directly from the 2D kinematic maps uncorrected for inclination (V $_{\mathrm{obs}}$ in TU17).

^bObserved velocity dispersion corrected for instrumental resolution ( $\sigma_{\mathrm{obs}}$ in TU17().

^cIntrinsic rotation velocity (V $_{\mathrm{max}}$ in GN11 and V $_{\mathrm{C}}$ in TU17).

^dIntrinsic velocity dispersion ( $\sigma_{\mathrm{int}}$ in both GN11 and TU17). Listed $\sigma_{\mathrm{0}}$ values and uncertainties for GN11 galaxies are estimated from Fig. 20 in GN11.

^eRatio of intrinsic rotation velocity and velocity dispersion corrected for beam smearing and inclination angle. TU17 classifies galaxies as rotation- or dispersion-dominated if ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ is greater than or less than 1, respectively.

^fKinematic classification assigned in the source studies (see Section 2.3): rd = ‘rotation-dominated’, rot = ‘rotating’ and dd = ‘dispersion-dominated’.

^gObservation mode employed in the source studies: NS = natural seeing, AO = adaptive optics assisted.

^hSource references for kinematic data: TU17 = Turner et al. (Reference Turner2017) and GN11 = Gnerucci et al. (Reference Gnerucci2011).

Fig. 5 shows ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ (with supplied uncertainties) for these galaxies plotted versus net Ly $\alpha$ EW. Galaxies that are common between the different source studies and observation modes, and for which we have multiple estimates of ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ , lie on vertical dotted lines to identify them for clarity. LA09 conclude that observational bias is an unsatisfactory explanation for differences in the global kinematic properties of the Keck/OSIRIS (LA09) and VLT/SINFONI (FS09) samples, and similar kinematics were found for the galaxies that were successfully observed by both surveys. There is also excellent agreement between the kinematic and morphological properties of Q1623-BX502 determined by both seeing-limited (FS09) and AO-assisted observations (FS09, LA09, FS18). Moreover, all the galaxies plotted in Fig. 5 are drawn from the related $z\sim2$ and $z\sim3$ UV-colour-selected catalogues of Steidel et al. (Reference Steidel2003, Reference Steidel2004) and are, therefore, comparable in terms of their intrinsic properties (see Section 2.1). We can be confident, therefore, that the dispersion versus net Ly $\alpha$ EW shown in Fig. 5 derives from a genuine range in the physical properties of the galaxies (see Section 3.3.3), and that the kinematics probed by our sample, from the multiple surveys, reliably reflect the true continuum of kinematic character ranging from genuinely dispersion-dominated to rotationally supported systems.

The horizontal dashed line in Fig. 5 at ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ = 0.4 marks the threshold for the classification of galaxies as rotation or dispersion-dominated as established by FS09 for both seeing-limited and AO-assisted observations at typical spatial resolution. Where this criterion is applied to mergers, ${v}_{\mathrm{obs}}/2$ can be interpreted as the projected ‘orbital velocity’ of the system.

The upper right-hand corner of Fig. 5 is unpopulated; there are no net Ly $\alpha$ -emitters with strong rotational signatures. All galaxies with strong rotational support are Ly $\alpha$ absorbers and occupy the upper left corner of the plot. The lower left-hand corner ( ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $\lesssim 0.4$ and net Ly $\alpha$ EW $\lesssim$ 0 Å) is dominated by merging systems. The only salient exceptions to these trends are: Q2346-BX405 (net Ly $\alpha$ EW $= -8.6$ ), which is classified as dispersion-dominated ( ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ $= 0.21 \pm 0.06$ ), but shows kinematic features consistent with disc rotation and is part of an interacting pair (FS09); and Q1623-BX502 (net Ly $\alpha$ EW = $+$ 28.5 Å) and SSA22a-D3 (net Ly $\alpha$ EW = $+$ 11.47 Å), both of which have low angular momentum ( $v_{\mathrm{obs}}/2 = \sim$ 40 and 53 km s $^{-1}$ , respectively) and are borderline for classification as rotation- or dispersion-dominated.

Parameterisation of the kinematics of our sample via ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ allows us to statistically test the strength of the association between galaxy kinematics and net Ly $\alpha$ EW independent of the inherited kinematic classifications. For galaxies with multiple estimates of ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ , we calculate average values and derive a sample of 17 discrete non-merger galaxies. A Spearman rank order correlation test on this sub-sample yields a coefficient ( $\rho$ ) of $\sim$ 0.577 and a probability (p) of 0.0154 ( $\sim2.2\sigma$ significance) that this moderate to strong degree of association could arise from an uncorrelated sample.

While ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ is a useful empirical measure of rotational dynamic support, a more robust and physically intuitive parameter for this purpose is the ratio ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ invoked by FS18, where $v_{\mathrm{rot}}$ is the intrinsic rotation velocity corrected for beam smearing and inclination angle according to:

(1) \begin{equation}\mbox{$v_{\mathrm{rot}}$} \cdot \mbox{$\sin(i)$} = \mbox{$C_{\mathrm{PSF,}v}$} \cdot \mbox{$v_{\mathrm{obs}}/2$}\end{equation}

and $\sigma_{\mathrm{0}}$ is the intrinsic local velocity dispersion determined from correction of the observed local velocity dispersion ( $\sigma_{\mathrm{0obs}}$ ) for beam smearing according to:

(2) \begin{equation}\mbox{$\sigma_{\mathrm{0}}$} = \mbox{$C_{\mathrm{PSF,}\sigma}$} \cdot \mbox{$\sigma_{\mathrm{0obs}}$,}\end{equation}

where $\sin(i)$ is the correction for inclination of a galaxy from the plane of the sky, and $C_{\mathrm{PSF,}v}$ and $C_{\mathrm{PFS,}\sigma}$ are the beam-smearing correction factors for rotation velocity and velocity dispersion, respectively, in the rotating disc framework of FS18.

For the SINS/zC-SINF galaxies of FS09 and FS18, and Q2343-BX442 from LA12, we use the $v_{\mathrm{rot}}$ , $\sigma_{\mathrm{0}}$ and ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ values as published. Applying Equation (1) to the LA09 galaxies, we use the average correction factor of the FS18 sample (i.e., $C_{\mathrm{PSF,v}} = 1.3$ ) in a manner similar to Tacconi et al. (Reference Tacconi2013), and $\sin(i) = \pi/4$ , the mean inclination angle correction for a distribution of randomly inclined discs (LA09) to estimate $v_{\mathrm{rot}}$ . For the purposes of this comparison, $\sigma_{\mathrm{mean}}$ in LA09 approximates to $\sigma_{\mathrm{0obs}}$ in FS18. Following Equation (2), we apply to the LA09 sample a correction of $C_{\mathrm{PSF},\sigma} = 0.85$ , the average correction factor for the FS18 sample. Uncertainties are propagated from the LA09 estimates for $v_{\mathrm{shear}}$ and $\sigma_{\mathrm{mean}}$ and added in quadrature for the ratio ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ .

For the $z\sim3$ galaxies, we similarly extract from the source references values for intrinsic rotation velocity (V $_{\mathrm{max}}$ in GN11 and V $_{\mathrm{C}}$ in TU17) and intrinsic velocity dispersion ( $\sigma_{\mathrm{int}}$ in both GN11 and TU17), and equate these to ${v}_{\mathrm{rot}}$ and ${\sigma}_{\mathrm{0}}$ respectively for the purposes of this analysis. Values of $v_{\mathrm{rot}}$ , $\sigma_{\mathrm{0}}$ and ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ used to construct Fig. 6 are listed with supplied or derived uncertainties, and associated net Ly $\alpha$ EWs, in Tables 3 & 4.

Figure 5. Ly $\alpha$ versus the ratio of observed rotation velocity to integrated velocity dispersion ( ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ ) for UV-colour-selected $z\sim2-3$ LBGs from FS09 (orange), LA09 (green), LA12 (purple), FS18 (dark orange), and TU17 (blue) plotted as a function of net Ly $\alpha$ EW and kinematic classification. Uncertainties are as given in the source publications or, for Q2343-BX442 (LA12) as supplied by D. Law (private communication). Galaxies with ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ estimates from multiple sources are aligned in net Ly $\alpha$ EW and indicated with vertical dotted lines. Rotation-dominated galaxies are shown as circles, dispersion-dominated systems as diamonds, and mergers as stars. The horizontal dashed line indicates the threshold value ( ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ = 0.4) used by FS09 to classify galaxies as either rotation- or dispersion-dominated.

Fig. 6 shows the ratio of intrinsic rotation velocity to intrinsic velocity dispersion ( ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ ) thus derived for subsets of our kinematic samples, plotted as a function of net Ly $\alpha$ EW and rest-frame UV colour, that is, $(U_n-\mathcal{R})$ for $z\sim2$ (left) and $(G-\mathcal{R})$ for $z\sim3$ (right). Fig. 7 shows the ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ versus net Ly $\alpha$ EW relationship for the $z\sim2$ and $z\sim3$ samples with galaxies colour-coded according to their source survey. A Spearman rank correlation test for the ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ versus net Ly $\alpha$ EW relationship for a combined non-merging subset of 18 $z\sim2$ and $z\sim3$ galaxies gives a $\rho$ -value of -0.69, and a p-value of 0.0014 that allows us to reject the null hypothesis that there is no association between ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ and net Ly $\alpha$ EW with $\gtrsim$ 99.8% ( $\sim3\sigma$ ) confidence.

The $z\sim2$ subset of strongly rotating ‘disc-like’ sources ( ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ $\gtrsim3$ ) have the most negative values of net Ly $\alpha$ EW, and exclusively populate the top left corner of the left panel in Fig. 6, significantly above the horizontal dashed line at ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ $= \sqrt{3.36}$ ( $\sim$ $1.83$ ) that corresponds to the point above which rotation starts to dominate over velocity dispersion in the dynamical support of turbulent discs (FS18). Similarly, non-merger galaxies with the lowest values of ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ have high net Ly $\alpha$ EWs and lie towards the bottom right. Mergers are confined to the bottom left corner of the plot. In general, there is an increase in the dispersion of the sample in the ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ dimension relative to the ${v}_{\mathrm{obs}}/2{\sigma }_{\mathrm{int}}$ case for the $z\sim2$ sample (cf. Fig. 5) indicating that as the rigour of the kinematic analysis is increased, the relationship between net Ly $\alpha$ EW and the degree of rotational dynamic support is strengthened.

Differences in the details of the colour-colour selection criteria of the parent photometric catalogues (see Paper I) preclude direct comparison of the colour dispersion of the $z\sim2$ and $z\sim3$ kinematic samples. However, in both redshift ranges (see Fig. 6), there is a clear trend from red to blue with decreasing rotational dynamic support (as measured by ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ ) that provides quantitative verification, at a galaxy-by-galaxy level, of the statistical relationship between kinematic type, rest-frame UV colour, and net Ly $\alpha$ EW, as described in Sections 3.1 & 3.2.

3.3.3 Other galaxy properties

In Fig. 6, we present net Ly $\alpha$ EW as it relates to broadband colour and kinematics. Fig. 8 plots net Ly $\alpha$ EW and kinematics in relation to several other properties previously reported for the galaxies in our sample: specifically, stellar mass ( $M_{\star }$ ), star-formation rate from SED fitting (SFR $_{\mathrm{SED}}$ ), galactic size ( $R_e$ ), age, gas fraction ( $\mu$ ), and dynamical mass ( $M_{dyn}$ ). In each case, the plots show the full sample (left panels), as well as the sample with the mergers removed (right panels), in order to better understand the behaviour associated with discrete and isolated galaxies and those involved in interactions. Symbol sizes are scaled as indicated in the caption and respective plot legends. The range of colour gradients omits outliers to help highlight any systematic trends. Table 5 lists the kinematic, spectroscopic, and physical properties used to construct the plots.

Figure 6. Ratio of intrinsic rotation velocity to intrinsic velocity dispersion ( ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ ) for $z\sim2$ (left) and $z\sim3$ (right) LBGs plotted as a function of net Ly $\alpha$ EW, rest-frame UV colour, and kinematic classification. Rotation-dominated (rd) galaxies are shown as circles, dispersion-dominated (dd) systems as diamonds, and mergers (m) as stars. Symbols are colour-coded on a blue/green/red gradient according to their rest-frame UV colours from $(U_n-\mathcal{R})$ = 0.3 to 1.5, and from $(G-\mathcal{R})$ = 0.21 to 1.15 for the $z\sim2$ and $z\sim3$ samples, respectively. The horizontal dashed lines mark the thresholds used by FS18 ( $z\sim2$ ) and TU17 ( $z\sim3$ ) to classify galaxies as either rotation- or dispersion-dominated. Vertical dotted lines indicate galaxies with kinematic data from multiple surveys. Dynamical support due to rotation (as measured by ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ ) correlates strongly with net Ly $\alpha$ EW and rest-frame UV colour as predicted from the CMD results described in Sections 3.1 & 3.2.

Figure 7. Same as Figure 6 but with the $z\sim2$ and $z\sim3$ samples plotted together and colour-coded according to their respective source surveys. Also shown are the 14 $z\sim0.03$ LARS galaxies (grey/white symbols) plotted using kinematic and Ly $\alpha$ EW data from Herenz et al. (Reference Herenz2016). LARS galaxies classified as ‘rotating discs’ (R) are plotted as circles, ‘perturbed rotators’ (P) as hexagons, and galaxies with ‘complex kinematics (C) as triangles. The two LARS galaxies listed by Herenz et al. (Reference Herenz2016) with a Ly $\alpha$ EW of zero (LARS04 & LARS06) are net absorbers at all apertures within the LARS field of view. Grey arrows indicate the direction that these galaxies would move on the plot if this net absorbing character was reflected in the quoted Ly $\alpha$ EWs. (see Section 4.3).

In all cases, the quoted stellar properties ( $M_{\star }$ , SFR $_{\mathrm{SED}}$ and stellar population age) were derived from evolutionary synthesis modeling of optical to near-IR broadband spectral energy distributions (SEDs) supplemented with mid-IR photometry when available. LA09, FS09 and FS18 used Bruzual & Charlot (Reference Bruzual and Charlot2003) models with a Chabrier (Reference Chabrier2003) initial mass function (IMF), the Calzetti et al. (Reference Calzetti2000) reddening law, and solar metallicity. Best fits were achieved with either constant or exponentially declining SFRs (see source studies for details). The modelling procedures are described in detail by Shapley et al. (Reference Shapley2005) (LA09) and in Appendix A of FS09 (FS09, FS18). For Q2343-BX442, LA12 used modelling procedures described by Erb et al. (2006c), Shapley et al. (Reference Shapley2005) and Reddy et al. (Reference Reddy, Erb, Pettini, Steidel and Shapley2010) to fit a SED constructed from ground-based $U_nG\mathcal{R}JK_s$ , HST/WFC3 F160W and Spitzer IRAC photometry, with a Chabrier IMF, Calzetti reddening, and a constant star formation history. LA12 used updated Charlot & Bruzual population synthesis models that resulted in slightly lower estimates for $M_{\star }$ , SFR $_{\mathrm{SED}}$ and stellar population age than estimates in the literature for similar samples (cf. Erb et al. 2006c).

For the SINS/zC-SINFONI galaxies of FS09 and FS18, we use the galactic half-light radius (R $_{e}$ ) values estimated from HST rest-frame optical (H-band) imaging where available from Tacchella et al. (Reference Tacchella2015). Otherwise, sizes shown are H $\alpha$ radii derived from IFU maps. In any event, the difference in sizes between H $\alpha$ and H-band emission is small, such that the choice made here is inconsequential (FS18). For LA09 galaxies, R $_e$ is the radius of nebula emission. For Q2343-BX442, we use the total luminous radius of $\sim$ 8 kpc estimated by LA12 from HST/WFC3 infra-red imaging of the stellar continuum.

To facilitate direct comparison across our sample, we use cold gas fractions ( $\mu$ = $M_{gas}$ /( $M_{gas}$ + $M_{\star }$ )) from Erb et al. (2006c) who estimated the mass of the gas associated with star formation using the local empirical correlation between star formation rate per unit area and gas surface density (Kennicutt Reference Kennicutt1998). Dynamical masses ( $M_{dyn}$ ) for SINS/zC-SINFONI galaxies are derived from modelling in the ‘rotating disc’ framework of FS18, or by one of the related methods described in detail by FS09 for galaxies unsuitable for modelling as discs. LA12 used a similar kinematic disc-fitting algorithm to model the velocity field of Q2343-BX442 and derive estimates for $M_{dyn}$ . For LA09 galaxies, $M_{dyn}$ is the single-component dynamical mass within the radius probed by nebular emission derived from the integrated velocity dispersion as per Erb et al. (2006c).

The top panels show that rotation-dominated galaxies are larger and most are older than their dispersion-dominated counterparts. There is no clear trend among the Ly $\alpha$ -absorbing rotators, but the Ly $\alpha$ -emitting dispersion-dominated galaxies are all similarly young. The centre panels show that the more strongly rotating Ly $\alpha$ -absorbers typically have lower gas fractions and higher stellar masses than most of the dispersion-dominated Ly $\alpha$ -emitters, but there is no apparent trend in either of these properties within the rotation-dominated sub-sample. Although the dispersion-dominated galaxies have consistently high gas fractions and usually lower stellar masses, there is significant scatter in the stellar mass/net Ly $\alpha$ EW trend in our small sample. For example, galaxy Q1217-BX95 (large grey diamond in the centre panels) has relatively strong Ly $\alpha$ emission (net Ly $\alpha$ EW = $+$ 10.2 Å), dispersion-dominated kinematics ( ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}=0.34)$ , but a relatively large stellar mass ( $\sim3\times10^{10}$ M $_{\odot}$ ) in the context of our sample. Finally, the bottom panels show larger dynamical masses and star formation rates for rotation-dominated sources compared to dispersion-dominated galaxies.

Figure 8. Physical properties of a sub-set of UV-colour-selected z $\sim2$ LBGs from FS09, LA09, LA12 and FS18 plotted as a function of the ratio of intrinsic rotation velocity to intrinsic velocity dispersion ( ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ ) and net Ly $\alpha$ EW. Kinematic classifications (symbol shapes), uncertainties, and broken lines are as described in Fig. 6. Illustrative symbol sizes and colours encode the relative magnitude of galaxy physical properties (listed in Table 5 and described in the text). In each case, the plots show the full sample (left panels), as well as the sample with the mergers removed (right panels). For what follows, the range of values rendered in each case for the non-merger sub-sample (filled symbols) are given in parentheses. Top: Galactic radius ( $0.6-8.0$ kpc) denoted by symbol size, and age from SED fitting ( $0.10-2.75$ Gyr) on a light to dark orange scale. Centre: Stellar mass ( $0.1-6.0 \times 10^{10}$ M $_{\odot}$ ) denoted by symbol size, and gas fraction ( $0.22-0.93$ ) on a light to dark purple scale. Grey symbols denote galaxies for which no value of $\mu$ is available. Bottom: Dynamical mass ( $0.3-29 \times 10^{10}$ M $_{\odot}$ ) denoted by symbol size, and star formation rate from SED fitting ( $11-52$ M $_{\odot}$ /yr) on a light to dark green scale.

Table 5. Physical and Kinematic Properties of UV Colour-Selected $z\sim2$ LBGs.

^aRatio of intrinsic rotation velocity to intrinsic velocity dispersion corrected for beam smearing and inclination angle (see Section 3.3.2).

^bGalactic half-light radius.

^cStellar properties from SED modelling (mass ( $m_{\star }$ ), age, and star-formation rate (SFR $_{\mathrm{SED}}$ )) collated from the respective source studies.

^dCold gas fraction ( $\mu$ = $M_{gas}$ /( $M_{gas}$ + $M_{\star}$ )) from Erb et al. (2006b).

^eDynamical mass.

^fKinematic classification assigned in the source studies: rd = rotation-dominated, dd = dispersion-dominated, m = merger.

^gSource references for kinematic and other data: FS09 = Förster Schreiber et al. (Reference Förster Schreiber2009), LA09 = Law et al. (Reference Law2009), LA12 = Law et al. (Reference Law2012a), FS18 = Förster Schreiber et al. (Reference Förster Schreiber2018).

The small size of the kinematic sample that meets the necessary selection criteria for this work (see Section 2.1) prevents the robust investigation of the relationship between net Ly $\alpha$ EW and kinematics at fixed values of other galactic properties, but even with this sample, we can see the emergence of trends that are consistent with relationships reported separately between kinematics, net Ly $\alpha$ EW and the respective physical properties. These illustrative plots suggest the potential value of a holistic multi-dimensional approach to the study of galaxy evolution in large samples and on large scales, and provide motivation to collect larger samples with consistent data that would inform the strength of these (and other) relations and how they are connected causally or otherwise.

4.4.1 Kinematics and large-scale structure at $z\sim2-3$

A key finding of this work, Paper I, and C09 is that Ly $\alpha$ spectral types in populations of $z\sim2-3$ LBGs segregate consistently with a range of intrinsic galactic properties. For example, we show in Sections 3.2.2 & 3.3.3 that eLBGs are characteristically, compact, blue, low-mass, low-metallicity, presumably young, dispersion-dominated systems, and that aLBGs are typically, red, spatially diffuse, high-mass rotation-dominated galaxies, usually with disc-like morphology.

Cooke et al. (Reference Cooke, Omori and Ryan-Weber2013) use the same photometric selection method to isolate large samples ( $\sim$ 10 $^5$ ) of $z\sim3$ aLBGs and eLBGs, and investigated their respective dark matter halo mass and spatial distribution on large scales using two-point correlation function analysis. They find that aLBGs preferentially reside in group and cluster environments, and eLBGs are typically found on the outskirts of groups and clusters and in the field. Moreover, cross-correlation function results showed that the two spectral types avoid each other on single halo to cluster halo scales. Similarly, and consistent with simulations that predict a decrease in Ly $\alpha$ EW with increasing overdensity in $z\sim2$ protoclusters (Muldrew et al. Reference Muldrew, Hatch and Cooke2015), spectroscopic follow-up of protoclusters identified at $z\sim3-4$ show that protocluster members have lower average Ly $\alpha$ EW compared to equivalent coeval field galaxies (Toshikawa et al. Reference Toshikawa2016; Lemaux et al. Reference Lemaux2018). There is also a growing body of work indicating that galaxies with large net Ly $\alpha$ EW are over-represented in under-dense regions, while galaxies with lower net Ly $\alpha$ EW are mainly located in over-dense environments (e.g., Ouchi et al. Reference Ouchi2010; Díaz et al. 2014; Bielby et al. Reference Bielby2016; Guaita et al. Reference Guaita2017, Reference Guaita2020; Shi et al. Reference Shi2019).

Combining these results with our findings, it would appear that we are seeing at $z\sim2-3$ , a spatial segregation of galaxies with rotation- and dispersion-dominated kinematics on large ( $\sim$ 100 Mpc) scales. That is, larger more massive rotating disc galaxies (aLBGs in our system) are located preferentially in group environments, and dispersion-dominated eLBGs tend to be found on group outskirts and in the field.

4.4.2 Bimodality at high redshift and the Morphology–Density Relation

The demonstrated segregation of Ly $\alpha$ spectral types with a wide range of galactic properties—including the large-scale clustering behaviour of aLBGs and eLBGs described above—is suggestive of a non-homogeneous galaxy population at high redshift. Indeed, it is reminiscent of the bimodal distribution of blue, star-forming spirals and large, massive red and quiescent ellipsoids of the modern-day universe (see e.g., Blanton & Moustakas Reference Blanton and Moustakas2009).

Independent evidence of galaxy bimodality at high redshift is observed in damped Ly $\alpha$ systems (DLAs). Wolfe et al. (Reference Wolfe, Prochaska, Jorgenson and Rafelski2008) found a bimodality in the DLA population based on [CII] 158 $\mu$ m cooling rates that is independent of HI column density. DLAs with high cooling rates have significantly higher velocity line profiles, metallicity, dust to gas ratios, ISM line widths, and star formation rates compared to those with low cooling rates. Intriguingly, the properties of the high and low cooling-rate DLAs map well with the properties of aLBG and eLBG Ly $\alpha$ spectral types, respectively, as discussed in this work and by Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003), Du et al. (Reference Du2018), and Pahl et al. (Reference Pahl2020).

These results caution against assuming that high-redshift galaxy populations are homogeneous, and emphasise the need to consider heterogeneity, and likely bimodality, in the distribution of galactic properties. Accordingly, from a holistic consideration of the many relationships between Ly $\alpha$ and galaxy properties discussed above, we envisage a model in which massive rotating disc systems (aLBGs) that are preferentially found in groups and clusters merge to form present-day elliptical galaxies (e.g., Toomre & Toomre Reference Toomre and Toomre1972). Compact eLBGs may be either super-star clusters in faint, low-mass galaxies, or the precursors of bulges in present-day spiral galaxies, which are known to have dispersion-supported kinematics and are expected to form at high redshift. Such a model suggests that we are observing at $z\sim2-3$ signs of a nascent morphology-density relation (Dressler Reference Dressler1980; Postman et al. Reference Postman2005) traced by the aLBG and eLBG populations.