# Lyman-$\alpha$ at Cosmic Noon II: The relationship between kinematics and
Lyman-$\alpha$ in $z\sim$ 2–3 Lyman Break Galaxies

Garry Foran  Centre for Astrophysics and Supercomputing, Swinburne University
of Technology, PO Box 218, H29, Hawthorn, VIC 3122, Australia
<EMAIL_ADDRESS>Jeff Cooke  Centre for Astrophysics and Supercomputing,
Swinburne University of Technology, PO Box 218, H29, Hawthorn, VIC 3122,
Australia  Emily Wisnioski  Research School of Astronomy and Astrophysics,
Australian National University, ACT 2611, Australia  Naveen Reddy  Department
of Physics & Astronomy, University of California, Riverside, 900 University
Avenue, Riverside, CA 92521, USA  Charles Steidel  Cahill Center for Astronomy
and Astrophysics, California Institute of Technology, MS 249-17, Pasadena, CA,
92115, USA

(dd Mmm YYYY; dd Mmm YYYY; dd Mmm YYYY)

###### Abstract

We report for the first time a relationship between galaxy kinematics and net
Lyman-$\alpha$ equivalent width (net Ly$\alpha$ EW) in star forming galaxies
during the epoch of peak cosmic star formation. Building on the previously
reported broadband imaging segregation of Ly$\alpha$-emitting and
Ly$\alpha$-absorbing Lyman-break galaxies (LBGs) at $z\sim 2$ (Paper I in this
series) and previously at $z\sim 3$, we use the Ly$\alpha$ spectral type
classification method to study the relationship between net Ly$\alpha$ EW and
nebular emission-line kinematics in samples of $z\sim 2$ and $z\sim 3$ LBGs
drawn from the literature, for which matching rest-frame UV photometry,
consistently measured net Ly$\alpha$ EWs, and kinematic classifications from
integral field unit spectroscopy are available. we show that $z\sim 2$ and
$z\sim 3$ LBGs segregate in colour-magnitude space according to their
kinematic properties and Lyman-$\alpha$ spectral type, and conclude that LBGs
with Ly$\alpha$ dominant in absorption (aLBGs) are almost exclusively
rotation-dominated (presumably disc-like) systems, and LBGs with Ly$\alpha$
dominant in emission (eLBGs) characteristically have dispersion-dominated
kinematics. We quantify the relationship between the strength of rotational
dynamic support (as measured using
${v}_{\mathrm{obs}}/2{\sigma}_{\mathrm{int}}$ and
${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$) and net Ly$\alpha$ EW for subsets
of our kinematic sample where these data are available, and demonstrate the
consistency of our result with other properties that scale with net Ly$\alpha$
EW and kinematics. Based on these findings, we suggest a method by which large
samples of rotation- and dispersion-dominated galaxies might be selected using
broadband imaging in as few as three filters and/or net Ly$\alpha$ EW alone.
If confirmed with larger samples, application of this method will enable an
understanding of galaxy kinematic behaviour over large scales in datasets from
current and future large-area and all-sky photometric surveys that will select
hundreds of millions of LBGs in redshift ranges from $z\sim 2-6$ across many
hundreds to thousands of Mpc. Finally, we speculate that the combination of
our result linking net Ly$\alpha$ EW and nebular emission-line kinematics with
the known large-scale clustering behaviour of Ly$\alpha$-absorbing and
Ly$\alpha$-emitting LBGs is evocative of an emergent bimodality of early
galaxies that is consistent with a nascent morphology-density relation at
$z\sim 2-3$.

###### keywords:

galaxies: evolution; galaxies: fundamental parameters; galaxies: high-
redshift; galaxies: kinematics and dynamics; galaxies: photometry

Australian Research Council Centre of Excellence for All-sky Astrophysics in 3
Dimensions (ASTRO-3D) Australian Research Council Centre of Excellence for
All-sky Astrophysics in 3 Dimensions (ASTRO-3D) Australian Research Council
Centre of Excellence for All-sky Astrophysics in 3 Dimensions (ASTRO-3D)

##  1 INTRODUCTION

Kinematics are a characteristic feature of the intrinsic galaxy population
dichotomy that exists in the modern-day Universe (see for example Blanton &
Moustakas, 2009), and as such, are a key constraint for simulations that aim
to understand the mechanisms by which galaxies evolve over cosmic time (e.g.,
El-Badry et al., 2018; Hung et al., 2019; Pillepich et al., 2019; Meng et al.,
2019; Vogelsberger et al., 2020; Dubois et al., 2020). This intimate
relationship between galaxy kinematics and their environment is manifest in
the well-studied Morphology–Density Relation (MDR; e.g., Dressler, 1980;
Postman et al., 2005; Houghton, 2015; Paulino-Afonso et al., 2019; Sazonova et
al., 2020), and has been investigated for a range of kinematic types and
environments out to intermediate redshifts up to $z\sim 1.5$ (e.g., Pérez-
Martínez et al., 2017; Pelliccia et al., 2019; Böhm et al., 2020; Cole et al.,
2020; Tiley et al., 2020).

Developments in the capability and efficiency of optical and near-IR integral
field unit (IFU) and slit spectrographs over the past fifteen years have
enabled numerous observational campaigns to probe in exquisite detail the
nebular emission-line kinematics of star-forming galaxies (SFGs) at redshifts
that span the peak of the cosmic star formation rate density ($z\approx 1-4$;
Madau & Dickinson, 2014). The synthesis of these results has produced a
picture of the high-redshift SFG population in which the majority ($\gtrsim
75$%) of massive ($\mathrm{log}({M}_{\star}/{M}_{\odot})\gtrsim 10$) SFGs
appear to have assembled primitive discs with characteristically large ionised
gas velocity dispersions and rotation-dominated kinematics (e.g., Wisnioski et
al., 2015, 2019; Simons et al., 2016, 2017; Förster Schreiber et al., 2018).
Lensed and other deep surveys targeting lower mass
($\mathrm{log}({M}_{\star}/{M}_{\odot})\lesssim 9.5$) SFGs report a
significantly lower ($<50\%$) rotation-dominated fraction, and a higher
proportion of galaxies that are kinematically disordered (dispersion-
dominated) or have kinematic structure and/or morphologies that identified
them as mergers (e.g., Gnerucci et al., 2011; Livermore et al., 2015;
Leethochawalit et al., 2016; Mason et al., 2017; Turner et al., 2017; Girard
et al., 2018). Additionally, the nebular emission-line kinematics of SFGs in
the redshift range $z\sim 2-3$, have been shown to correlate with a range of
galactic physical properties including: stellar and dynamical mass; size and
morphology; specific and total star formation rates; gas outflows and gas
fraction; and nebular conditions including the degree and spatial distribution
of metals (see for example Glazebrook (2013) and references therein, and more
recently, Wisnioski et al. (2015, 2018, 2019); Leethochawalit et al. (2016);
Simons et al. (2016, 2017); Mason et al. (2017); Molina et al. (2017); Turner
et al. (2017); Girard et al. (2018); Förster Schreiber et al. (2018, 2019) and
Förster Schreiber & Wuyts (2020)).

Despite these advances, the ability of currently available samples to inform
the relationship between kinematics and large-scale structure at redshifts
$z\gtrsim 2$ is challenged by a range of factors including: small sample
sizes; observational biases that are a function of survey depth and redshift
range; specific sample selection criteria; the spatial and spectral resolution
of the observations; and the diversity of kinematic analysis methods and
classification criteria (e.g., Glazebrook, 2013; Leethochawalit et al., 2016;
di teodoro et al., 2016; Rodrigues et al., 2017; Simons et al., 2019; Hung et
al., 2019). To better inform simulations that model the relationship between
galactic kinematics and different formation and evolution pathways, and to
facilitate the proper statistical study of relationships between the global
kinematic properties of galaxy populations and large scale structure out to
high redshifts, there is a need for much larger samples spanning a wider range
of intrinsic and environmental galaxy properties, mapped over large scales,
and to redshifts above $z\sim 4$, at which redshifts IFU-based kinematic
measurements are currently not possible.

Contemporaneously with progress in understanding the kinematics of SFGs at
$z\gtrsim 2$, rest-frame UV spectroscopic studies of star-forming UV-colour-
selected Lyman break galaxies (LBGs) in the same redshift range have reported
the sensitivity of Lyman-$\alpha$ (Ly$\alpha$) visibility to a wide range of
galactic properties. Due to the resonant character of the Ly$\alpha$
transition (see M. Dijkstra in Verhamme et al. (2019) for a comprehensive
description), Ly$\alpha$ transmission and spectral morphology are modulated by
neutral gas properties such as optical depth, covering fraction, dust content,
and kinematics (see the review of Hayes (2015) and more recently e.g., Trainor
et al. (2015, 2019); Gronke & Dijkstra (2016); Reddy et al. (2016); Steidel et
al. (2018); Du et al. (2018); Remolina-Gutiérrez & Forero-Romero (2019);
Byrohl & Gronke (2020) and Pahl et al. (2020)). In addition to these HI gas
properties which directly control the absorption and scattering of Ly$\alpha$
during radiative transfer, it has been demonstrated for $z\sim 2-4$ LBGs that
larger Lyman-$\alpha$ transmission, or net Lyman-$\alpha$ equivalent width
(net Ly$\alpha$ EW), is associated with galaxies with bluer UV colours, lower
metallicities, lower stellar masses, lower total UV luminosities, lower star
formation rates, harder ionising field strengths, and more compact
morphologies (e.g., Shapley et al., 2003; Erb et al., 2006a; Law et al., 2007;
Steidel et al., 2010; Pentericci et al., 2010; Law et al., 2012b; Erb et al.,
2016; Du et al., 2018; Marchi et al., 2019).

Deep observational surveys of the spatial redistribution of Ly$\alpha$ into
the circumgalactic medium (CGM) have established the apparent ubiquity of so-
called ‘Lyman-$\alpha$ halos’ around early SFGs (e.g., Steidel et al., 2011;
Hayes et al., 2013; Momose et al., 2014; Wisotzki et al., 2016, 2018), and
there is a growing body of observational and computational work suggesting
that Ly$\alpha$ visibility in the early universe reflects, and is modulated
by, the galactic environment on small and large scales (Ouchi et al., 2010,
2018, 2020; Cooke et al., 2010, 2013; Jose et al., 2013; Díaz et al., 2014;
Muldrew et al., 2015; Toshikawa et al., 2016; Bielby et al., 2016; Guaita et
al., 2017, 2020; Lemaux et al., 2018; Shi et al., 2019). Given this
established sensitivity of Ly$\alpha$ to a wide range of intrinsic and
environmental galactic properties, and the trends that have been demonstrated
linking many of the same properties to galaxy kinematics, it is reasonable to
ask the question: “Is there a relationship between Ly$\alpha$ and galaxy
kinematics and how might this be used to inform our understanding of galaxy
formation and evolution?” – especially on large scales and at high redshifts
where Ly$\alpha$ is frequently the only spectroscopic indicator available.

Radiative transfer simulations have investigated the influence of solid body
rotation on Ly$\alpha$ observables (Garavito-Camargo et al., 2014; Remolina-
Gutiérrez & Forero-Romero, 2019) and predict the sensitivity of Ly$\alpha$
spectral line morphology to the bulk rotation of neutral gas and the viewing
angle relative to the rotation axis. They also predict, however, that there
should be no observable difference in the integrated Ly$\alpha$ line flux, the
Ly$\alpha$ escape fraction, or the average number of scatterings for each
Ly$\alpha$ photon caused by changes in the radiative transfer mechanism under
the influence of rotation or dispersion-dominated kinematics alone. The only
direct observational study of a relationship between Ly$\alpha$ and galaxy
kinematics reported to date is the low-$z$ work of Herenz et al. (2016) who
derived values for shear velocity and intrinsic velocity dispersion from the
H$\alpha$ kinematic maps of galaxies in the Lyman-$\alpha$ Reference Sample
(LARS; Östlin et al., 2014; Hayes et al., 2014). The LARS collaboration
surmise a causal connection between turbulence in actively star forming
systems and interstellar medium conditions that favour an escape of Ly$\alpha$
radiation, and further speculate that dispersion-dominated kinematics are a
necessary requirement for a galaxy to have a significant amount of escaping
Ly$\alpha$.

In the first paper in this series (Foran et al., 2023, hereafter referred to
as Paper I), we report the photometric segregation of $z\sim 2$ LBGs versus
net Ly$\alpha$ EW in colour-magnitude space, and derive criteria for the
selection of pure samples of LBGs with Ly$\alpha$ dominant in absorption and
Ly$\alpha$ dominant in emission using broadband imaging alone. Together with
the analogous $z\sim 3$ result of Cooke (2009, hereafter C09), we have
suggested the utility of this method to study a wide range of properties known
to be associated with Ly$\alpha$ in large samples and over large scales in
data from current and future large-area photometric campaigns. In particular,
we foresee application of this approach to datasets from the all-sky LSST
survey by the Vera C. Rubin Observatory that will select hundreds of millions
of LBGs in redshift ranges from $z\sim 2-6$ across many hundreds to thousands
of Mpc.

In this paper we report a direct relationship between nebular emission-line
kinematics and net Ly$\alpha$ EW in samples of $z\sim 2$ and $z\sim 3$ LBGs
drawn from the literature, and extend the results of Paper I and C09 to
propose a method by which the generalised kinematics of large samples of LBGs
might be predicted using broadband imaging in as few as three filters, and
studied on large scales in data from large-area and all-sky photometric
surveys. Finally, we combine our result with known relationships between
Ly$\alpha$ and galactic environment, and speculate on how these findings might
be used to inform our understanding of galaxy formation and evolution in the
early Universe.

This paper is structured as follows: In Section 2, we describe the
photometric, spectroscopic, and kinematic data used in the subsequent
sections. A relationship between net Ly$\alpha$ EW and the nebular emission-
line kinematics of LBGs at $z\sim 3$ and $z\sim 2$ is presented in Section 3.
In Section 4, we discuss these results, their potential utility, and their
implications for galaxy evolution science. The important conclusions of the
paper are summarised in Section 5. We assume a $\Lambda$CDM cosmology with
$\Omega_{M}$= 0.3, $\Omega_{\Lambda}$= 0.7 and H0= 70 km s-1 Mpc-1. All
magnitudes are quoted in the AB system of Oke & Gunn (1983).

##  2 DATA

###  2.1 Overview

We assemble from the literature complementary kinematic samples of $z\sim 2$
and $z\sim 3$ LBGs with consistent multi-band rest-frame UV broadband
photometry, uniformly measured net Ly$\alpha$ EWs, and kinematic
classifications quantitatively and comparably determined from IFU-based
spectroscopy. The $z\sim 2$ kinematic sample consists of 23 UV-colour-selected
(BX) LBGs and 13 $K_{s}$-band (mass) selected SFGs in the range $2.0<z\lesssim
2.5$. Twenty-two of these are classified as ‘rotation-dominated’ (including
all the $K_{s}$-band selected SFGs), four are ‘dispersion-dominated’, and ten
are classified as ‘mergers’ in the source studies. (see Section 2.2 and Table
1 for details). The $z\sim 3$ kinematic sample consists of 24 LBGs in the
range $2.6<z\lesssim 3.4$, of which ten are classified as ‘rotating’ or
‘rotation-dominated’, eleven are ‘not-rotating’ or ‘dispersion-dominated’, and
three galaxies are labelled as ‘not classified’ in the source study (see
Section 2.3 and Table 2 for details). The LBGs in the kinematic samples all
had broadband optical photometry available from the catalogs of Steidel et al.
(2004, $z\sim 2$) and Steidel et al. (2003, $z\sim 3$). Broadband photometry
for the $K_{s}$-band selected SFGs was transformed to match the
$U_{n}G\cal{R}$ photometric system of these catalogs (see Section 2.2.2).

The parent photometric catalogs derive from an observational campaign that
targeted 14 uncorrelated fields with a total survey area of 1900 arcmin2,
resulting in a sample that is minimally affected by systematic biases due to
cosmic variance or clustering. The survey used the rest-frame UV colour
selection criteria of Steidel et al. (2003, $z\sim 3$) and Steidel et al.
(2004, $z\sim 2$); Adelberger et al. (2004, $z\sim 2$). These criteria were
designed to recover galaxies with intrinsic properties – particularly rest-
frame UV luminosity and reddening by dust – that were similar across both
redshift ranges. The faint end magnitude cuts of $\cal{R}$ $\leq 25.5$ (and
$\cal{R}$ $\leq 26.0$ for one $z\sim 3$ field) in the parent LBG samples were
chosen so as to facilitate spectroscopic redshift determinations using the
rich complement of strong interstellar and stellar lines in the rest-frame UV
continuum between Ly$\alpha$ and $\sim$1700 Å (Steidel et al., 2003, 2004).
Moreover, the faintest galaxies in the kinematic samples are brighter than
$\cal{R}$ $=25$, thus mitigating any potential bias at the faint end due to
over-reliance on Ly$\alpha$ in emission for redshift determination.

The $z\sim 2$ and $z\sim 3$ parent samples have $\cal{R}$-band apparent
magnitudes in the range $22.0<\cal{R}$$<25.5$ and $22.7<\cal{R}$$<26.0$,
corresponding to rest-frame UV luminosities (absolute magnitudes) of
$-22.6<\mathrm{M}_{UV}<-19.1$ and $-22.6<\mathrm{M}_{UV}<-19.5$, respectively.
The bulk of galaxies in the parent samples have stellar masses in the range
$9\lesssim\mathrm{log}({M}_{\star}/{M}_{\odot})\lesssim 11$ (Shapley et al.,
2003, 2005; Erb et al., 2006c; Reddy et al., 2006; Reddy & Steidel, 2009) and
star formation rates (inferred from rest-frame UV luminosities uncorrected for
extinction) in the range $3\lesssim\mathrm{M}_{\odot}$ yr${}^{-1}\lesssim 60$
(median 9.9 $\mathrm{M}_{\odot}$ yr-1) and $5.5\lesssim\mathrm{M}_{\odot}$
yr${}^{-1}\lesssim 66$, (median 10.3 $\mathrm{M}_{\odot}$ yr-1), respectively
(Steidel et al., 2004). Accordingly, our parent and kinematic samples are
typical of LBGs/SFGs at these redshifts (Álvarez-Márquez et al., 2016, and
references therein), and the $z\sim 2$ LBGs lie (though with a range of
properties (see Reddy et al., 2006)) on the main sequence of stellar mass and
star formation rate for $z\sim 2$ SFGs (Daddi et al., 2007).

Throughout this work we use consistently determined net Ly$\alpha$ EW as a
measure of Ly$\alpha$ visibility. Net Ly$\alpha$ EW incorporates information
about Ly$\alpha$ in emission, Ly$\alpha$ in absorption, (even for strong to
weak emitters) and their combined effects in the observed Ly$\alpha$ spectral
feature. This is critically important for our LBG samples – especially at
$z\sim 2$ where Ly$\alpha$ in absorption dominates the population (see Paper
I). Net Ly$\alpha$ EWs for galaxies in the parent LBG catalogs were measured
uniformly at $z\sim 2$ (see Reddy et al. (2008)) and $z\sim 3$ (Shapley et
al., 2003, priv. comm.) using the method described by Kornei et al. (2010).
The rest-frame UV colour criteria used to select the $z\sim 2$ and $z\sim 3$
LBGs result in a net Ly$\alpha$ EW distribution for the $\cal{R}$ $<$ 25.5
samples that is representative of the intrinsic distribution for the parent
population of galaxies (Reddy et al., 2008).

###  2.2 $z\sim 2$ Kinematic sample

####  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 $z\sim 2$ photometric catalog
were selected from the SINS survey sample of Förster Schreiber et al. (2009,
hereafter FS09) and the AO-assisted IFS survey of Law et al. (2009, hereafter
LA09). Twenty-one of these galaxies had net Ly$\alpha$ EWs in the
spectroscopic catalog of Reddy et al. (2008).

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., 2008) 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., 2018, 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., 2006) 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. (2012a,
hereafter LA12) was also included in our $z\sim 2$ kinematic sample. Net
Ly$\alpha$ EW and kinematic parameters for Q2343-BX442 were supplied by D. Law
(priv. comm.). 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 $z\sim 2$ UV-selected BX galaxies that comprise our kinematic
sample are summarised in Table 1 along with references to the source studies
in each case.

Table 1: $z\sim 2$ SFGs used to establish the relationship between galaxy
kinematics and net Ly$\alpha$ EW in colour-magnitude space

ID |  RA |  DEC |  $z$ |  $\cal{R}$-banda Mag. |  $(U_{n}-\cal{R})$a Colour |  Ly$\alpha$b type |  net Ly$\alpha$ EW (Å) |  Kin. c class. |  Ref.d
---|---|---|---|---|---|---|---|---|---
Q1623-BX376 | 16:25:45.59 | 26:46:49.26 | 2.4085 | 23.31 | 0.99 | aLBG | -12.03 | rd | FS09
Q1623-BX447 | 16:25:50.37 | 26:47:14.28 | 2.1481 | 24.48 | 1.31 | aLBG | -10.05 | rd | FS09
Q1623-BX455 | 16:25:51.66 | 26:46:54.88 | 2.4074 | 24.80 | 1.22 | Ga | -6.54 | rd | FS09,NE13,FS18
Q1623-BX502 | 16:25:54.38 | 26:44:09.25 | 2.1558 | 24.35 | 0.72 | eLBG | 28.49 | dd/rde | FS09,LA09,FS18
Q1623-BX528 | 16:25:56.44 | 26:50:15.44 | 2.2682 | 23.56 | 0.96 | aLBG | -11.53 | m | FS09
Q1623-BX599 | 16:26:02.54 | 26:45:31.90 | 2.3304 | 23.44 | 1.02 | Ga | -0.66 | m | FS09,NE13,FS18
Q2343-BX389 | 23:46:28.90 | 12:47:33.55 | 2.1716 | 24.85 | 1.54 | Ga | -8.30 | rd | FS09,NE13,FS18
Q2343-BX513 | 23:46:11.13 | 12:48:32.14 | 2.1092 | 23.93 | 0.61 | Ge | 18.43 | m | FS09,LA09,NE13,FS18
Q2346-BX404f | 23:48:21.40 | 00:24:43.07 | 2.0282 | 23.39 | 0.59 | Ge | 2.39 | dd | FS09
Q2346-BX405f | 23:48:21.22 | 00:24:45.46 | 2.0300 | 23.36 | 0.68 | Ga | -8.61 | dd | FS09
Q2346-BX416 | 23:48:18.21 | 00:24:55.30 | 2.2404 | 23.49 | 1.15 | aLBG | -17.22 | rd | FS09
Q2346-BX482 | 23:48:12.97 | 00:25:46.34 | 2.2569 | 23.32 | 1.12 | aLBG | -14.98 | rd | FS09,NE13,FS18
Q0449-BX93 | 04:52:15.417 | -16:40:56.88 | 2.0067 | 22.99 | 0.63 | aLBG | -13.31 | m | LA09
Q1217-BX95 | 12:19:28.281 | 49:41:25.90 | 2.4244 | 24.11 | 1.21 | Ge | 10.17 | dd | LA09
HDF-BX1564 | 12:37:23.47 | 62:17:20.02 | 2.2225 | 23.28 | 1.28 | aLBG | -11.12 | m | LA09
Q1623-BX453 | 16:25:50.84 | 26:49:31.40 | 2.1816 | 23.38 | 1.47 | Ga | -0.16 | m | LA09
Q1700-BX490 | 17:01:14.83 | 64:09:51.69 | 2.3960 | 22.88 | 1.28 | Ge | 5.90 | m | LA09
Q2343-BX418 | 23:46:18.57 | 12:47:47.38 | 2.3052 | 23.99 | 0.32 | eLBG | 53.49 | dd | LA09
Q2343-BX660 | 23:46:29.43 | 12:49:45.54 | 2.1735 | 24.36 | 0.36 | eLBG | 20.38 | m | LA09
Q1623-BX543 | 16:25:57.70 | 26:50:08.59 | 2.5211 | 23.11 | 1.40 | Ga | -7.80 | m | FS09,LA09,NE13,FS18
Q1623-BX663 | 16:26:04.58 | 26:48:00.20 | 2.4333 | 24.14 | 1.26 | – | – | m | FS09,NE13
SSA22a-MD41 | 22:17:39.97 | 00:17:11.04 | 2.1713 | 23.31 | 1.50 | – | – | rd | FS09
Q2343-BX442 | 23:46:19.36 | 12:47:59.69 | 2.1760 | 24.48 | 1.54 | aLBG | -15.0 | rd | LA12
COS4_04519 | 10:00:28.65 | 02:13:27.0 | 2.2290 | 23.80∗ | 1.46∗ | Ga | -9.1 | rd | WI15
COS4_05094 | 10:00:33.69 | 02:13:48.5 | 2.1720 | 24.72∗ | 1.47∗ | – | – | rd | WI15
COS4_05389g | 10:00:17.59 | 02:13:58.7 | 2.3060 | 23.91∗ | 1.26∗ | – | – | rd | WI15
COS4_08775 | 10:00:16.54 | 02:16:09.3 | 2.1630 | 23.42∗ | 0.95∗ | aLBG | -13.2 | rd | WI15
COS4_10347 | 10:00:20.03 | 02:17:08.8 | 2.0630 | 23.85∗ | 1.44∗ | – | – | rd | WI15
COS4_13174 | 10:00:26.96 | 02:18:50.0 | 2.0980 | 26.20∗ | 1.72∗ | – | – | rd | WI15
COS4_13701 | 10:00:27.05 | 02:19:09.9 | 2.1670 | 23.62∗ | 1.50∗ | aLBG/Ga | ¡0.0 | rd | WI15
COS4_15813 | 10:00:34.98 | 02:20:28.8 | 2.3590 | 24.14∗ | 1.87∗ | aLBG | -15.1 | rd | WI15
COS4_15820 | 10:00:15.56 | 02:20:29.7 | 2.0910 | 23.63∗ | 1.16∗ | – | – | rd | WI15
COS4_16342 | 10:00:29.41 | 02:20:49.6 | 2.4690 | 24.27∗ | 2.09∗ | aLBG/Ga | ¡0.0 | rd | WI15
COS4_18209 | 10:00:20.16 | 02:21:53.3 | 2.0870 | 23.74∗ | 2.14∗ | aLBG/Ga | ¡0.0 | rd | WI15
COS4_19680 | 10:00:18.06 | 02:22:45.8 | 2.1670 | 24.54∗ | 1.37∗ | aLBG/Ga | ¡0.0 | rd | WI15
COS4_24763 | 10:00:13.61 | 02:26:04.8 | 2.4650 | 24.08∗ | 1.97∗ | – | – | rd | WI15

  * a

Magnitudes and colours marked with an asterisk (∗) are derived from
$u^{*}g^{\prime}r^{\prime}i^{\prime}z^{\prime}$ photometry using the method
outlined in Section 2.2.2.

  * b

See Section 2.4 for definitions of Ly$\alpha$ spectral types.

  * c

Kinematic classification: rd = ‘rotation-dominated’, dd = ‘dispersion-
dominated’ and m = ‘merger’ according to the criteria described in Section
2.2.

  * d

Source references for kinematic classifications: FS09 = Förster Schreiber et
al. (2009), LA09 = Law et al. (2009), LA12 = Law et al. (2012a), NE13 = Newman
et al. (2013), WI15 = Wisnioski et al. (2015), FS18 = Förster Schreiber et al.
(2018)

  * e

See Section 3.2.2 for notes regarding the kinematic classification of
Q1623-BX502.

  * f

Q2346-BX404 and Q2346-BX405 are an interacting pair with angular separation
corresponding to a projected distance of 30.3 kpc at the redshift of the
sources (FS09).

  * g

Photometry for COS4_05389 is derived from that of the two CFHTLS sources of
which it is comprised (CFHTLS IDs 259119 and 259367).

####  2.2.2 KMOS3D Galaxies

We supplement our $z\sim 2$ kinematic sample with a redshift-selected subset
($2.0<z<2.5$) of 13 galaxies from the COSMOS field pointings of KMOS3D (K3D) –
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., 2019). The K3D
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., 2015). The K3D sources
that we employ are part of the ‘high S/N disc sample’ of rotation-dominated
galaxies reported by Wisnioski et al. (2015, 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 K3Dgalaxies were cross-matched with the
D2 field of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS Hudelot
et al., 2012) 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 $U_{n}G\cal{R}$ magnitudes to
facilitate direct comparison with the $z\sim 2$ 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 $z\sim 3$ $U_{n}G\cal{R}$ LBGs
divided into quartiles on the basis of net Ly$\alpha$ EW (A. Shapley, priv.
comm.). 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., 2013). For each K3D 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., 2000) 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 $U_{n}G\cal{R}$ filters, and $U_{n}G\cal{R}$-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 K3D galaxies are best classified as aLBG or Ga
spectral types (see Section 2.4). Using the transformed $U_{n}G\cal{R}$
photometry, a standard colour–colour test (Steidel et al., 2004; Adelberger et
al., 2004) was applied to confirm that the K3D sample satisfied (within the
photometric uncertainties) the criteria to be selected as $z\sim 2$ LBGs.

With the goal of measuring net Ly$\alpha$ EWs, seven of the ‘high S/N disc
sample’ K3D galaxies were included as secondary science targets on our multi-
object slitmasks using the LRIS instrument (Oke et al., 1995; Steidel et al.,
2004) at Keck on 26, 27 December 2016 and 20–22 January 2020. 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. (2010).
We successfully measured net Ly$\alpha$ EW for three WI15 galaxies from the
January 2020 data. However, weather conditions and primary science constraints
during the December 2016 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$10 Å. The q1 and q2
quartiles of Shapley et al. (2003) have average net Ly$\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/Ga in our system

###  2.3 $z\sim 3$ Kinematic sample

####  2.3.1 AMAZE and LSD Galaxies

Using the SINFONI integral field spectrograph (Eisenhauer et al., 2003) 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., 2008) and LSD (Mannucci et al., 2009) surveys conducted
near-IR IFU spectroscopic observations on LBGs at redshifts $z\gtrsim 3$ with
$\cal{R}$ $\simeq$ 24.5 (L* and brighter) corresponding to a mass range of
$\mathrm{log}({M}_{\star}/{M}_{\odot})\approx 10-11$.

Gnerucci et al. (2011, hereafter gn11) derived nebular emission-line
kinematics for a subset of amaze and lsd galaxies by fitting the profile and
shift of the [O III] $\lambda\lambda$ 5007, 4959 å 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’. they 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 catalog.
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. (2017, 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., 2013). With natural-seeing
measurements of the [O III] $\lambda$ 5007 emission line, TU17 extracted 2D
kinematic maps and used beam-smearing corrections derived from dynamical
modelling to determine values of intrinsic rotation velocity (VC) 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 VC/$\sigma_{\mathrm{int}}$ to kinematically classify their sample.
Galaxies were classified as ‘rotation-dominated’ if
VC/$\sigma_{\mathrm{int}}>1$, and as ‘dispersion-dominated’ if
VC/$\sigma_{\mathrm{int}}<1$.

The SSA22-P2 pointing of the KDS survey targeted a field environment to the
south of the main SSA22 spatial overdensity (Steidel et al., 2000) and yielded
five morphologically isolated (non-merger) field galaxies that were in common
with our parent photometric catalog. 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 [O III] $\lambda$ 5007 emission line, we include the LA09 galaxy,
DSF2237a-C2, in our $z\sim 3$ kinematic sample. Drawn from the $z\sim 3$ LBG
catalog of Steidel et al. (2003), 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 $z\sim 2$ LA09 galaxies (see Section 2.2.1), and assign a ‘rotation-
dominated’ classification to this galaxy.

Table 2: $z\sim 3$ LBGs used to establish the relationship between galaxy
kinematics and Ly$\alpha$ EW in colour-magnitude space

ID |  RA |  DEC |  $z$ |  $\cal{R}$ Mag. |  $U_{n}-\cal{R}$ Colour |  Ly$\alpha$ type a |  net Ly$\alpha$ EW (Å) |  Kin. b class. |  Obs.c mode |  Ref.d
---|---|---|---|---|---|---|---|---|---|---
SSA22a-M38 | 22:17:17.69 | +00:19:0.70 | 3.2880 | 24.11 | 1.15 | aLBG | -16.52 | rot | NS | GN11
SSA22a-C16 | 22:17:31.95 | +00:13:16.10 | 3.0650 | 23.64 | 0.98 | aLBG | -21.24 | rot | NS | GN11
SSA22a-D17 | 22:17:18.87 | +00:18:16.80 | 3.0890 | 24.27 | 0.45 | Ge | 18.14 | rot | NS | GN11
CDFA-C9 | 00:53:13.71 | +12:32:11.10 | 3.2095 | 23.99 | 0.89 | Ga | -6.48 | rot | NS | GN11
3C324-C3 | 15:49:47.10 | +21:27:5.40 | 3.2890 | 24.14 | 0.85 | Ge | 0.97 | rot | NS | GN11
DSF2237b-D29 | 22:39:32.70 | +11:55:51.70 | 3.3670 | 23.70 | 0.96 | Ge | 1.79 | not-rot | NS | GN11
DSF2237b-C21 | 22:39:29.00 | +11:50:58.10 | 3.4030 | 23.50 | 1.10 | Ga | -0.12 | not-rot | NS | GN11
SSA22a-aug96M16 | 22:17:30.85 | +00:13:10.70 | 3.2920 | 23.83 | 0.70 | Ge | 19.40 | not-rot | NS | GN11
SSA22a-C36 | 22:17:46.09 | +00:16:43.00 | 3.0630 | 24.06 | 0.78 | aLBG | -13.67 | NC | NS | GN11
SSA22a-D3 | 22:17:32.41 | 00:11:32.9 | 3.0860 | 23.37 | 0.97 | Ge | 11.47 | rd | NS | TU17
SSA22b-C20 | 22:17:48.81 | 00:10:14.0 | 3.2020 | 24.60 | 0.68 | Ga | -4.18 | rd | NS | TU17
SSA22b-D5 | 22:17:35.75 | 00:06:10.5 | 3.1880 | 24.42 | 0.66 | Ge | 1.04 | rd | NS | TU17
SSA22b-D9 | 22:17:22.27 | 00:08:04.2 | 3.0880 | 25.14 | 0.21 | eLBG | 37.83 | dd | NS | TU17
SSA22b-MD25 | 22:17:41.65 | 00:06:20.5 | 3.3090 | 24.53 | 0.56 | eLBG | 32.81 | dd | NS | TU17
SSA22b-C5 | 22:17:47.06 | +00:04:25.70 | 3.1140 | 25.16 | 0.71 | eLBG | 85.25 | not-rot | AO | GN11
SSA22a-C6 | 22:17:40.93 | +00:11:26.00 | 3.0950 | 23.44 | 0.79 | Ge | 6.19 | not-rote | AO | GN11
SSA22a-M4 | 22:17:40.91 | +00:11:27.90 | 3.0930 | 24.83 | 0.76 | Ga | -7.70 | NCe | AO | GN11
SSA22a-C30 | 22:17:19.28 | +00:15:44.70 | 3.1010 | 24.22 | 0.82 | Ge | 3.60 | not-rot | AO | GN11
Q0302-C131 | 03:04:35.04 | -00:11:18.30 | 3.2340 | 24.48 | 0.54 | eLBG | 27.60 | rot | AO | GN11
Q0302-C171 | 03:04:44.31 | -00:08:23.20 | 3.3350 | 24.63 | 0.60 | Ga | -0.91 | NC | AO | GN11
DSF2237b-D28 | 22:39:20.25 | +11:55:11.30 | 2.9320 | 24.46 | 0.32 | Ge | 1.03 | not-rot | AO | GN11
Q0302-M80 | 03:04:45.70 | -00:13:40.60 | 3.4160 | 24.12 | 1.01 | Ge | 0.81 | not-rot | AO | GN11
DSF2237b-MD19 | 22:39:21.08 | +11:48:27.70 | 2.6150 | 24.48 | 0.44 | Ge | 13.94 | not-rot | AO | GN11
DSF2237a-C2 | 22:40:8.29 | +11:49:4.80 | 3.3172 | 23.55 | 1.13 | Ge | 0.03 | rd | AO | LA09

  * a

See Section 2.4 for definitions of Ly$\alpha$ spectral types.

  * b

Kinematic classification: rot = ‘rotating’, not-rot = ‘not-rotating’, rd =
‘rotation-dominated’, dd = ‘dispersion-dominated’ and NC = ‘not classifiable’
according to the criteria of the respective source studies (see Section 2.3).

  * c

NS = natural seeing, AO = adaptive optics assisted

  * d

Source references for kinematic classifications: GN11 = Gnerucci et al.
(2011), LA09 = Law et al. (2009), TU17 = Turner et al. (2017)

  * e

GN11 identifies SSA22a-C6 and SSA22a-M4 as a close-pair. They have a projected
separation of 12 kpc. and SSA22a-M4 is redshifted by $\sim$90 km s-1 relative
to SSA22a-C6.

###  2.4 Ly$\alpha$ spectral type classifications

We employ herein the same Ly$\alpha$ spectral type classification scheme as
that used by C09 at $z\sim 3$ and in Paper I at $z\sim 2$ to demonstrate the
photometric segregation of LBGs with respect to net Ly$\alpha$ EW. For this
purpose we define galaxies with Ly$\alpha$ dominant in absorption (net
Ly$\alpha$ EW $<$ $-$10.0Å) as ‘aLBGs’, and galaxies with Ly$\alpha$ dominant
in emission (net Ly$\alpha$ EW $>$ 20.0 Å) as ‘eLBGs’. We further divide the
remaining $z\sim 2$ LBGs into Ga and Ge spectral types with net Ly$\alpha$ EWs
in the range $-$10.0 $<$ net Ly$\alpha$ EW $<$ 0.0 Å and 0.0 $<$ net
Ly$\alpha$ EW $<$ 20.0 Å respectively.

##  3 ANALYSIS AND RESULTS

###  3.1 $z\sim$3 LBGs

Figure 1: Rest-frame UV colour–magnitude ($(G-\cal{R})$ vs $\cal{R}$) diagrams
adapted from Cooke (2009). In both plots, the parent sample of 775
spectroscopic $z\sim 3$ LBGs is shown in light grey: squares indicate aLBGs,
triangles eLBGs, and plus signs galaxies with intermediate values of net
Ly$\alpha$ EW. Left: Points labelled q1–q4 mark the colour and magnitude
distribution means of the parent LBG sample divided into numerical quartiles
on the basis of net Ly$\alpha$ EW. The primary cut (green line) bisects the
aLBG and eLBG distributions. The dashed (red) and dotted-dashed (blue) lines
indicate an offset of 1.5$\sigma$ in colour dispersion from the primary cut
for the aLBG and eLBG distributions respectively, and define the shaded red
and blue regions within which pure samples of Ly$\alpha$-absorbing and
Ly$\alpha$-emitting LBGs can be selected with high confidence (see text). The
$z\sim 3$ kinematic sample is overlaid as white symbols. Galaxies classified
as ‘rotation-dominated’ or ‘rotating’ (rd/rot) are shown as circles, and
galaxies classified as ‘dispersion-dominated’ or ‘not-rotating’ (dd/not rot)
as diamonds. The mean colour–magnitude values of the rotation-
dominated/rotating and dispersion-dominated/not-rotating sub-samples are
indicated by the red circle and blue diamond respectively. Black crosses (X)
are galaxies designated as ‘not classifiable’ by GN11, and galaxies marked
with an asterisk (∗) are members of an interacting close-pair. Right: Similar
to the left panel, but with Ly$\alpha$ spectral types identified. Filled red
symbols denote aLBGs, filled blue symbols are eLBGs, and hollow red and blue
symbols are Ga and Ge spectral types respectively (see Section 2.4).

C09 discovered that $z\sim 3$ LBGs segregate in colour–magnitude space
according to their net Ly$\alpha$ EW, and determined photometric criteria to
select pure sub-samples with Ly$\alpha$ dominant in absorption (aLBGs) and
Ly$\alpha$ dominant in emission (eLBGs) based on broadband imaging. Figure 1
shows our $z\sim 3$ kinematic sample overlaid on rest-frame UV
colour–magnitude diagrams (CMDs) of $z\sim 3$ LBGs segregated according to
their Ly$\alpha$ spectral type adapted from C09. In the left panel, points
labelled q1-q4 show the monotonic trend of the parent sample divided into
numerical quartiles on the basis of net Ly$\alpha$ EW. The more positive net
Ly$\alpha$ EW quartiles (weaker absorption and stronger emission) trend
consistently toward fainter $\cal{R}$-band magnitudes and bluer $(G-\cal{R})$
colours. The primary cut (solid green line) passes through the midpoint
between the mean colour and magnitude values of the aLBG and eLBG
distributions, and has slope that maximises the segregation of aLBGs and
eLBGs. The dashed (red) and dotted-dashed (blue) lines indicate an offset of
1.5$\sigma$ in colour dispersion from the primary cut for the aLBG and eLBG
distributions respectively, and define one choice of photometric criteria for
the selection of pure Ly$\alpha$-absorbing and Ly$\alpha$-emitting sub-samples
that lie in the shaded red and blue regions respectively.

The aLBG and eLBG distributions segregate on the CMD such that $\gtrsim 90$
percent of aLBGs (grey squares) are located above the red dotted-dashed line
(in the white and red regions), and $\lesssim 10$ percent of eLBGs (grey
triangles) are located in the red region above the blue dashed line. The
reverse is true for eLBGs; $\gtrsim 90$ percent of eLBGs lie below the dashed
line (in the white and blue regions), and $\lesssim 10$ percent of aLBGs are
found in the blue region below the dotted-dashed line. Thus, the aLBG and eLBG
distributions partially overlap in the central part of the CMD, but relatively
pure subsets of aLBGs and eLBGs can be selected from the ‘high-confidence’ red
and blue regions respectively (see C09 and Paper I for quantitative details of
the Ly$\alpha$ spectral type photometric selection method).

Galaxies in the $z\sim 3$ kinematic sample classified as ‘rotating’ or
‘rotation-dominated’ lie at or above the primary cut coincident with the
majority of aLBGs, and dominate in the ‘high-confidence’ aLBG (red) region.
Galaxies classified as ‘not-rotating’ or ‘dispersion-dominated’ are scattered
on the CMD, but all galaxies below the primary cut, and in the ‘high-
confidence’ eLBG (blue) region are classified as ‘not-rotating’ or
‘dispersion-dominated’. In addition, the not-rotating/dispersion-dominated
sub-sample is on average fainter and bluer than their rotating/rotation-
dominated counterparts, as indicated by the red circle and blue diamond in the
left0 panel of Figure 1, and we note that the ‘not-rotating’ sub-sample may
include late-stage mergers (GN11). Thus, the rotating/rotation-dominated and
not-rotating/dispersion-dominated subsets of the kinematic sample follow the
aLBG and eLBG distributions within their known dispersion characteristics.

This trend is reinforced when we include the spectroscopically-determined net
Ly$\alpha$ EW data and assign a Ly$\alpha$ spectral type to each galaxy in our
kinematic sample according to the definitions given in Section 2.4. The right
panel of Figure 1 shows the $z\sim 3$ kinematic sample colour-coded according
to their Ly$\alpha$ spectral type overlaid on the parent LBGs. The three aLBGs
lie well above the primary cut; of these, two are confirmed rotators, and the
third is ‘not classifiable’ and could be a face-on disc (GN11). The four
eLBGs, of which three are ‘not-rotating’ or ‘dispersion-dominated’, lie on or
below the primary cut. The association between kinematics and Ly$\alpha$
spectral type is less clear for intermediate Ga and Ge LBGs – particularly for
those that lie toward the centre of the CMD – and may be obscured by the
‘rotating’/‘not-rotating’ classification scheme of the source study that
precludes an interpretation in terms of late-stage merging systems (GN11).

Of the ten not-rotating/dispersion-dominated galaxies in the $z\sim 3$
kinematic sample, three are eLBGs, six have Ge spectral type, and one is
borderline Ga with a net Ly$\alpha$ EW of $-$0.1 Å. The mean net Ly$\alpha$ EW
for these 10 galaxies is $+$19.6 Å. The ten rotating/rotation-dominated
sources consist of two aLBGs, two Ga LBGs, three borderline Ga/Ge spectral
types with net Ly$\alpha$ EW $\approx 0$ Å, two Ge LBGs, and one eLBG. Two
‘rotating’ galaxies (SSA22a-D17 and Q0302-C131) have significant net
Ly$\alpha$ emission ($+$18.14 and $+$27.60 Å respectively). The average net
Ly$\alpha$ EW for the rotating/rotation-dominated sub-sample is $+$1.1 Å.

These results suggest an empirical relationship between rest-frame UV colour,
Ly$\alpha$ spectral type, and the kinematic properties of $z\sim 3$ LBGs. In
the following section we extend our study to $z\sim 2$ where a larger number
of LBGs with kinematic classifications, appropriate imaging, and Ly$\alpha$
spectroscopic data are available.

###  3.2 $z\sim$2 LBGs

####  3.2.1 $z\sim 2$ Kinematics on the CMD

Figure 2 shows the 36 galaxies in the $z\sim 2$ kinematic sample overlaid on
the parent population of 557 $z\sim 2$ LBGs dispersed on a
$(U_{n}-\cal{R})$/$\cal{R}$ CMD adapted from Paper I. We note that most of the
$z\sim 2$ kinematic sample is distributed toward 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 K3D survey. Due to the known relationship
between Ly$\alpha$ and $z\sim 2$ LBG morphology (Law et al., 2012b), 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 theSFG
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.

This ‘red bias’ of the $z\sim 2$ 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
$\gtrsim 95$ percent 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 $\gtrsim 97$
percent 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 Figure 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.

Figure 2: Similar to the left panel of Fig. 1 but for the $z\sim 2$ kinematic
sample plotted on a $(U_{n}-\cal{R})$ vs. $\cal{R}$ 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 $z\sim 2$ in general. Galaxy mergers (m, stars) are typically found
central in colour and toward 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.

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 $(U_{n}-\cal{R})$ colour are
statistically consistent with the null hypotheses that they are drawn from the
$(U_{n}-\cal{R})$ 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 $z\sim 2$ LBG sample used to establish the
photometric segregation and selection criteria described in Paper I, and shown
on the CMDs in Figures 2 & 4. Histogram plots in $(U_{n}-\cal{R})$ colour of
the sub-samples used in the KS test are shown in the top panel of Figure 3.
The bifurcation in $(U_{n}-\cal{R})$ colour is more readily visualised in the
lower panel of Figure 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
$\mathrm{nFrac}_{rd}-\mathrm{nFrac}_{dd}$).

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 $(U_{n}-\cal{R})$ 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 $(U_{n}-\cal{R})$ 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 K3D discs – the null hypothesis that they are drawn
from the same distribution in $(U_{n}-\cal{R})$ colour.

Figure 3: Top: Histograms in $(U_{n}-\cal{R})$ 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
$(U_{n}-\cal{R})$ colour.

Galaxies classified as mergers tend to be found at brighter magnitudes
compared to the non-merging fraction of the kinematic sample. The mean
$\cal{R}$-band magnitude of the ten merging systems is m($\cal{R}$)merger =
23.51 (orange star in Figure 2), which is $\sim 0.6$ mag brighter than the
mean magnitude of the non-merging fraction (m($\cal{R}$)non-merger = 24.10).
Moreover, galaxies in the kinematic sample segregate in colour depending on
their classification; the mean $(U_{n}-\cal{R})$ 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 $z\sim 2$ 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 Figure 4 shows this sub-sample
with their assigned Ly$\alpha$ spectral types overlaid on the parent $z\sim 2$
LBGs. Seven out of eight rotation-dominated galaxies are aLBG or Ga spectral
types, and all seven K3D 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 Ge 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 $z\sim 2$ eLBG distribution mean, with two galaxies (including the
dispersion-dominated Q2343-BX418) below the dotted-dashed red line in the
‘high-confidence’ eLBG region.

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 Figure 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.

Figure 4: Top: Similar to the lower panel of Fig. 1 but for the $z\sim 2$
kinematic sample plotted on a $(U_{n}-\cal{R})$ vs $\cal{R}$ 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 $z\sim 2$ 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 $z\sim 2$ 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.

####  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 $z\sim 2$ kinematic sample
(diamond symbols in Figure 2 and the top panel of Figure 4) have a mean net
Ly$\alpha$ EW of $+$14.4 Å. Of these, three are net Ly$\alpha$-emitters (eLBG
or Ge spectral types). The exception is Q2346-BX405 which has a net Ly$\alpha$
EW of $-8.61$ Å (Ga 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 Figure 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 Figure 4). While its location on
the CMD is within the eLBG colour–magnitude distribution, it has a colour
($(U_{n}-\cal{R})$ = 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 [O III] $\lambda$4363 Å
emission line111Gas-phase metallicities supplied by C. Steidel (priv. comm.)
determined using the O3N2 index and the calibration described by Steidel et
al. (2016) and Strom et al. (2017). Values are quoted using the scale of
Asplund et al. (2009) in which solar metallicity has the value
$12+\mathrm{log}({O}/{H})=8.69$. 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 $(U_{n}-\cal{R})$ colour of this galaxy (C. Steidel, priv.
comm.).

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 Figure 4).
They are, however, kinematically diverse. The dispersion-dominated Q2343-BX418
(galaxy ‘b’ in the top panel of Figure 4) has low stellar mass
($\mathrm{log}({M}_{\star}/{M}_{\odot})=9.0$), low metallicity
($12+\mathrm{log}({O}/{H})=7.9\pm 0.2$), is very blue ($(U_{n}-\cal{R})$ =
0.32), very compact ($R_{e}$ = 0.8 kpc), and has strong Ly$\alpha$ emission
(net Ly$\alpha$ EW = $+$53.5 Å). Although having kinematic structure
consistent with merging compact galaxies (LA09), Q2343-BX660 (galaxy ‘c’ in
the top panel of Figure 4) is otherwise similar to Q2343-BX418; it has
relatively low stellar mass ($\mathrm{log}({M}_{\star}/{M}_{\odot})=9.9$), low
metallicity ($12+\mathrm{log}({O}/{H})=7.99\pm 0.05$), compact morphology
($R_{e}$ = 1.6 kpc), very blue colour ($(U_{n}-\cal{R})$ = 0.36), and is a
strong Ly$\alpha$ emitter (net Ly$\alpha$ EW = $+$20.4 Å).

With a net Ly$\alpha$ EW of $+$28.5 Å, and a ‘rotation-dominated’ kinematic
classification derived from the deep AO-assisted observations of FS18,
Q1623-BX502 (galaxy ‘d’ in the top panel of Figure 4) appears to be
inconsistent with our proposition that large net Ly$\alpha$ EW is a useful
predictor of dispersion-dominated kinematics. Although its magnitude and
redder colour $(U_{n}-\cal{R})$ = 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., 2013) 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 $z\sim
2$ 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 Kinematics vs net Ly$\alpha$ EW – Direct Comparisons

####  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 $z\sim 2$ and $z\sim 3$ kinematic samples. The mean net
Ly$\alpha$ EW for rotation-dominated galaxies in the $z\sim 2$ sample is
$-$7.0 Å, and $+$14.4 Å for the dispersion-dominated sub-sample. (cf. $+$1.1
and $+$19.6 Å for the $z\sim 3$ rotating/rotation-dominated and not-
rotating/dispersion-dominated sub-samples respectively). A two-sided KS test
on a combined ($z\sim 2$ \+ $z\sim 3$) 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 (Vobs) 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 $z\sim 2$ and
five $z\sim 3$ galaxies in our kinematic samples derived from their respective
source studies are listed in Tables 3 & 4.

Figure 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 Figure 5 are drawn from the related $z\sim 2$ and $z\sim
3$ UV-colour-selected catalogs of Steidel et al. (2003, 2004) 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 Figure 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.

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 $\sim 2-3$ 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 (priv.
comm.). 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.

The horizontal dashed line in Figure 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 Figure 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$0.577 and a probability (p) of 0.0154 ($\sim
2.2\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:

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

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:

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

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. (2013), 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 $z\sim 3$ galaxies, we similarly extract from the source references
values for intrinsic rotation velocity (Vmax in GN11 and VC 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 Figure 6 are listed with supplied or derived uncertainties, and
associated net Ly$\alpha$ EWs, in Tables 3 & 4.

Figure 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, i.e., $(U_{n}-\cal{R})$ for $z\sim 2$ (left) and
$(G-\cal{R})$ for $z\sim 3$ (right). Figure 7 shows the
${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$ versus net Ly$\alpha$ EW
relationship for the $z\sim 2$ and $z\sim 3$ 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 $z\sim 2$ and $z\sim 3$
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% ($\sim 3\sigma$) confidence.

Figure 6: Ratio of intrinsic rotation velocity to intrinsic velocity
dispersion (${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$) for $z\sim 2$ (left)
and $z\sim 3$ (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 $(U_{n}-\cal{R})$ = 0.3
to 1.5, and from $(G-\cal{R})$ = 0.21 to 1.15 for the $z\sim 2$ and $z\sim 3$
samples respectively. The horizontal dashed lines mark the thresholds used by
FS18 ($z\sim 2$) and TU17 ($z\sim 3$) 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 $z\sim 2$ and $z\sim 3$ samples plotted together and colour-coded
according to their respective source surveys. Also shown are the 14 $z\sim
0.03$ LARS galaxies (grey/white symbols) plotted using kinematic and
Ly$\alpha$ EW data from Herenz et al. (2016). 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. (2016) 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).

The $z\sim 2$ subset of strongly rotating ‘disc-like’ sources
(${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$$\gtrsim 3$) have the most negative
values of net Ly$\alpha$ EW, and exclusively populate the top left corner of
the left panel in Figure 6, significantly above the horizontal dashed line at
${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$$=\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 the highest net Ly$\alpha$ EWs
and lie toward the bottom right. Mergers are isolated in 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 $z\sim 2$
sample (cf. Figure 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 $z\sim 2$ and $z\sim 3$ kinematic samples. However,
in both redshift ranges (see Figure 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.

Table 3: Kinematic and Ly$\alpha$ Properties of UV Colour-Selected $z\sim 2$
LBGs

ID |  ${v}_{\mathrm{obs}}/2$a (km s-1) |  ${\sigma}_{\mathrm{int}}$b (km s-1) |  ${v}_{\mathrm{obs}}/2{\sigma}_{\mathrm{int}}$c |  ${v}_{\mathrm{rot}}$d (km s-1) |  ${\sigma}_{\mathrm{0}}$e (km s-1) |  ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$f |  net Ly$\alpha$ EW (Å) |  Kin. g class. |  Obs.h mode |  Ref. i
---|---|---|---|---|---|---|---|---|---|---
HDF-BX1564 | $12^{+12}_{-12}$ | $103^{+14}_{-14}$ | $0.12^{+0.12}_{-0.12}$ | $20^{+20}_{-20}$ | $51^{+11}_{-11}$ | $0.39^{+1.02}_{-1.02}$ | -11.12 | m | AO | LA09
Q0449-BX93 | $13^{+8}_{-8}$ | $71^{+5}_{-5}$ | $0.18^{+0.11}_{-0.11}$ | $22^{+13}_{-13}$ | $48^{+17}_{-17}$ | $0.45^{+0.71}_{-0.71}$ | -13.31 | m | AO | LA09
Q1217-BX95 | $12^{+13}_{-13}$ | $61^{+5}_{-5}$ | $0.20^{+0.21}_{-0.21}$ | $20^{+22}_{-22}$ | $59^{+20}_{-20}$ | $0.34^{+1.13}_{-1.13}$ | 10.17 | dd | AO | LA09
Q1623-BX376 | $60^{+18}_{-18}$ | $99^{+11}_{-9}$ | $0.60^{+0.19}_{-0.19}$ | — | — | — | -12.03 | rd | NS | FS09
Q1623-BX447 | $100^{+30}_{-30}$ | $144^{+17}_{-17}$ | $0.69^{+0.22}_{-0.22}$ | — | — | — | -10.05 | rd | NS | FS09
Q1623-BX453 | $29^{+10}_{-10}$ | $94^{+5}_{-5}$ | $0.31^{+0.11}_{-0.11}$ | $48^{+17}_{-17}$ | $66^{+20}_{-20}$ | $0.72^{+0.45}_{-0.45}$ | -0.16 | m | AO | LA09
Q1623-BX455 | $55^{+17}_{-17}$ | $130^{+28}_{-28}$ | $0.42^{+0.16}_{-0.16}$ | — | — | — | -6.54 | rd | NS | FS09
—"— | $125^{+15}_{-15}$ | $145^{+15}_{-16}$ | $0.86^{+0.14}_{-0.14}$ | $175^{+54}_{-17}$ | $56^{+15}_{-24}$ | $3.10^{+2.40}_{-0.50}$ | -6.54 | rd | AO | FS18
Q1623-BX502 | $45^{+14}_{-14}$ | $72^{+7}_{-5}$ | $0.62^{+0.19}_{-0.20}$ | — | — | $0.80^{+0.20}_{-0.20}$ | 28.49 | dd | NS | FS09
—"— | $35^{+11}_{-11}$ | $68^{+3}_{-3}$ | $0.51^{+0.16}_{-0.16}$ | $58^{+18}_{-18}$ | $57^{+17}_{-17}$ | $1.02^{+0.43}_{-0.43}$ | 28.49 | dd | AO | LA09
—"— | $33^{+10}_{-10}$ | $66^{+5}_{-5}$ | $0.50^{+0.16}_{-0.16}$ | $77^{+50}_{-32}$ | $40^{+16}_{-14}$ | $2.00^{+1.50}_{-0.80}$ | 28.49 | rd | AO | FS18
Q1623-BX528 | $67^{+20}_{-20}$ | $141^{+8}_{-8}$ | $0.48^{+0.14}_{-0.15}$ | — | — | — | -11.53 | m | NS | FS09
Q1623-BX543 | $55^{+17}_{-17}$ | $149^{+22}_{-23}$ | $0.37^{+0.12}_{-0.12}$ | — | — | — | -7.80 | m | NS | FS09
—"— | $39^{+4}_{-4}$ | $153^{+7}_{-7}$ | $0.25^{+0.03}_{-0.03}$ | $65^{+7}_{-7}$ | $118^{+27}_{-27}$ | $0.55^{+0.25}_{-0.25}$ | -7.80 | m | AO | LA09
—"— | $93^{+15}_{-15}$ | $163^{+8}_{-11}$ | $0.57^{+0.10}_{-0.10}$ | $128^{+29}_{-21}$ | $70^{+15}_{-25}$ | $1.80^{+0.90}_{-0.30}$ | -7.80 | m | AO | FS18
Q1623-BX599 | $49^{+15}_{-15}$ | $153^{+9}_{-9}$ | $0.32^{+0.10}_{-0.10}$ | — | — | — | -0.66 | m | NS | FS09
—"— | $80^{+20}_{-20}$ | $180^{+8}_{-10}$ | $0.44^{+0.11}_{-0.11}$ | $139^{+62}_{-36}$ | $71^{+18}_{-27}$ | $2.00^{+1.40}_{-0.40}$ | -0.66 | m | AO | FS18
Q1700-BX490 | $50^{+10}_{-10}$ | $122^{+6}_{-6}$ | $0.41^{+0.08}_{-0.08}$ | $83^{+17}_{-17}$ | $76^{+21}_{-21}$ | $1.09^{+0.34}_{-0.34}$ | 5.90 | m | AO | LA09
Q2343-BX389 | $205^{+62}_{-62}$ | $245^{+70}_{-50}$ | $0.84^{+0.30}_{-0.35}$ | — | — | $3.10^{+0.40}_{-0.40}$ | -8.30 | rd | NS | FS09
—"— | $260^{+23}_{-23}$ | $258^{+23}_{-27}$ | $1.01^{+0.13}_{-0.14}$ | $299^{+40}_{-21}$ | $56^{+13}_{-15}$ | $5.30^{+1.70}_{-0.70}$ | -8.30 | rd | AO | FS18
Q2343-BX418 | $23^{+12}_{-12}$ | $70^{+5}_{-5}$ | $0.33^{+0.17}_{-0.17}$ | $38^{+20}_{-20}$ | $52^{+14}_{-14}$ | $0.73^{+0.59}_{-0.59}$ | 53.49 | dd | AO | LA09
Q2343-BX442 | $150^{+25}_{-25}$ | $180^{+20}_{-20}$ | $0.83^{+0.17}_{-0.17}$ | $234^{+49}_{-29}$ | $66^{+6}_{-6}$ | $3.55^{+0.23}_{-0.15}$ | -15.00 | rd | AO | LA12
Q2343-BX513 | $27^{+8}_{-8}$ | $101^{+24}_{-19}$ | $0.26^{+0.09}_{-0.10}$ | — | — | — | 18.43 | m | NS | FS09
—"— | $65^{+16}_{-16}$ | $102^{+6}_{-6}$ | $0.64^{+0.16}_{-0.16}$ | $108^{+26}_{-26}$ | $85^{+21}_{-21}$ | $1.27^{+0.35}_{-0.35}$ | 18.43 | m | AO | LA09
—"— | $60^{+15}_{-15}$ | $139^{+11}_{-11}$ | $0.43^{+0.11}_{-0.11}$ | $102^{+64}_{-26}$ | $55^{+24}_{-28}$ | $1.80^{+2.10}_{-0.50}$ | 18.43 | m | AO | FS18
Q2343-BX660 | $40^{+9}_{-9}$ | $73^{+5}_{-5}$ | $0.55^{+0.13}_{-0.13}$ | $66^{+15}_{-15}$ | $60^{+14}_{-14}$ | $1.10^{+0.32}_{-0.32}$ | 20.38 | m | AO | LA09
Q2346-BX404 | $20^{+6}_{-6}$ | $97^{+4}_{-3}$ | $0.21^{+0.06}_{-0.06}$ | — | — | — | 2.39 | dd | NS | FS09
Q2346-BX405 | $32^{+10}_{-10}$ | $83^{+5}_{-3}$ | $0.39^{+0.12}_{-0.12}$ | — | — | — | -8.61 | dd | NS | FS09
Q2346-BX416 | $70^{+21}_{-21}$ | $138^{+9}_{-9}$ | $0.51^{+0.16}_{-0.16}$ | — | — | — | -17.22 | rd | NS | FS09
Q2346-BX482 | $233^{+70}_{-70}$ | $132^{+8}_{-9}$ | $1.76^{+0.54}_{-0.54}$ | $239^{+40}_{-40}$ | $52^{+13}_{-21}$ | $4.40^{+1.00}_{-1.00}$ | -14.98 | rd | NS | FS09
—"— | $225^{+20}_{-20}$ | $123^{+6}_{-7}$ | $1.83^{+0.19}_{-0.19}$ | $287^{+63}_{-30}$ | $58^{+14}_{-15}$ | $4.90^{+1.60}_{-0.70}$ | -14.98 | rd | AO | FS18

  * a

Half 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.

  * b

Integrated 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.

  * c

We 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.

  * d

Intrinsic 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).

  * e

Intrinsic 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.

  * f

Ratio 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.

  * g

Kinematic classification assigned in the source studies (see Section 2.2): rd
= rotation-dominated, dd = dispersion-dominated, m = merger

  * h

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

  * i

Source references for kinematic data: FS09 = Förster Schreiber et al. (2009),
LA09 = Law et al. (2009), LA12 = Law et al. (2012a), FS18 = Förster Schreiber
et al. (2018)

Table 4: Kinematic and Ly$\alpha$ Properties of UV Colour-Selected $z\sim 3$
LBGs

ID |  ${v}_{\mathrm{obs}}/2$a (km s-1) |  ${\sigma}_{\mathrm{int}}$b (km s-1) |  ${v}_{\mathrm{obs}}/2{\sigma}_{\mathrm{int}}$ |  ${v}_{\mathrm{rot}}$c (km s-1) |  ${\sigma}_{\mathrm{0}}$d (km s-1) |  ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$e |  net Ly$\alpha$ EW (Å) |  Kin. f class. |  Obs.g mode |  Ref. h
---|---|---|---|---|---|---|---|---|---|---
SSA22a-D3 | $53\pm 17$ | $115\pm 13$ | $0.46\pm 0.16$ | $109^{+52}_{-32}$ | $93^{+20}_{-36}$ | $1.16^{+0.72}_{-0.43}$ | 11.47 | rd | NS | TU17
SSA22b-C20 | $66\pm 18$ | $97\pm 17$ | $0.68\pm 0.22$ | $139^{+33}_{-30}$ | $48^{+34}_{-51}$ | $2.87^{+3.12}_{-2.14}$ | -4.18 | rd | NS | TU17
SSA22b-D5 | $27\pm 10$ | $38\pm 20$ | $0.71\pm 0.46$ | $57^{+26}_{-23}$ | $23^{+23}_{-18}$ | $2.45^{+3.44}_{-2.41}$ | 1.04 | rd | NS | TU17
SSA22b-D9 | $32\pm 14$ | $83\pm 8$ | $0.39\pm 0.17$ | $46^{+22}_{-15}$ | $73^{+10}_{-17}$ | $0.63^{+0.33}_{-0.23}$ | 37.83 | dd | NS | TU17
SSA22b-MD25 | $44\pm 15$ | $87\pm 19$ | $0.51\pm 0.20$ | $57^{+34}_{-24}$ | $75^{+21}_{-30}$ | $0.76^{+0.56}_{-0.39}$ | 32.81 | dd | NS | TU17
SSA22a-M38 | — | — | — | $346^{+51}_{-45}$ | $50^{+15}_{-15}$ | $6.92^{+2.31}_{-2.26}$ | -16.52 | rot | NS | GN11
SSA22a-C16 | — | — | — | $129^{+95}_{-24}$ | 90 | $1.43^{+1.06}_{-0.27}$ | -21.24 | rot | NS | GN11
SSA22a-D17 | — | — | — | $260^{+75}_{-65}$ | $130^{+16}_{-10}$ | $2.00^{+0.63}_{-0.52}$ | 18.14 | rot | NS | GN11
CDFA-C9 | — | — | — | $129^{+206}_{-31}$ | $117^{+3}_{-4}$ | $1.10^{+1.76}_{-0.27}$ | -6.48 | rot | NS | GN11
3C324-C3 | — | — | — | $86^{+63}_{-20}$ | $95^{+3}_{-5}$ | $0.91^{+0.66}_{-0.22}$ | 0.97 | rot | NS | GN11
Q0302-C131 | — | — | — | $>117$ | $43^{+5}_{-6}$ | $2.72^{+0.32}_{-0.38}$ | 27.60 | rot | AO | GN11

  * a

Observed rotation velocity measured directly from the 2D kinematic maps
uncorrected for inclination (Vobs in TU17).

  * b

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

  * c

Intrinsic rotation velocity (Vmax in GN11 and VC in TU17).

  * d

Intrinsic 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.

  * e

Ratio 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.

  * f

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

  * g

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

  * h

Source references for kinematic data: TU17 = Turner et al. (2017) and GN11 =
Gnerucci et al. (2011)

Figure 8: Physical properties of a sub-set of UV-colour-selected z $\sim 2$
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 Figure 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⊙) 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⊙) denoted by
symbol size, and star formation rate from SED fitting($11-52$ M⊙/yr) on a
light to dark green scale.

####  3.3.3 Other galaxy properties

In Figure 6, we present net Ly$\alpha$ EW as it relates to broadband colour
and kinematics. Figure 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
(SFRSED), 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.

In all cases, the quoted stellar properties ($M_{\star}$, SFRSED 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
(2003) models with a Chabrier (2003) initial mass function (IMF), the Calzetti
et al. (2000) 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.
(2005) (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.
(2005) and Reddy et al. (2010) to fit a SED constructed from ground-based
$U_{n}G\mathcal{R}JK_{s}$, 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 $M_{\star}$, SFRSED 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 (Re) values estimated from HST rest-frame optical (H-band)
imaging where available from Tacchella et al. (2015). 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, Re 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, 1998). 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}}$0̄.34), but a relatively large
stellar mass ($\sim 3\times 10^{10}$ M⊙) 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.

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.

Table 5: Physical and Kinematic Properties of UV Colour-Selected $z\sim 2$
LBGs

ID |  ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$a |  net Ly$\alpha$ EW (Å) |  $(U_{n}-\cal{R})$ Colour |  $R_{e}$b (kpc) |  Agec (Myr) |  $M_{\star}$c (${10}^{10}$ M⊙) |  $\mu$d |  $M_{dyn}$e ($\times{10}^{10}$ M⊙) |  SFRSEDc (M⊙ yr-1) |  Kin. f class. |  Ref. g
---|---|---|---|---|---|---|---|---|---|---|---
HDF-BX1564 | $0.39^{+1.02}_{-1.02}$ | -11.12 | 1.28 | $1.3^{+0.2}_{-0.2}$ | $571^{+285}_{-285}$ | $2.90^{+1.16}_{-1.16}$ | 0.55 | 0.5 | $53^{+31}_{-31}$ | m | LA09
Q0449-BX93 | $0.45^{+0.71}_{-0.71}$ | -13.31 | 0.63 | $1.3^{+0.1}_{-0.1}$ | $203^{+101}_{-101}$ | $0.90^{+0.36}_{-0.36}$ | — | 0.5 | $47^{+28}_{-28}$ | m | LA09
Q1217-BX95 | $0.34^{+1.13}_{-1.13}$ | 10.17 | 1.21 | $0.6^{+0.1}_{-0.1}$ | $571^{+285}_{-285}$ | $3.00^{+1.20}_{-1.20}$ | — | 0.3 | $52^{+31}_{-31}$ | dd | LA09
Q1623-BX453 | $0.72^{+0.45}_{-0.45}$ | -0.16 | 1.47 | $1.4^{+0.1}_{-0.1}$ | $404^{+202}_{-202}$ | $3.10^{+1.24}_{-1.24}$ | 0.52 | 1.0 | $77^{+46}_{-46}$ | m | LA09
Q1623-BX455 | $3.1^{+2.4}_{-0.5}$ | -6.54 | 1.22 | $2.1^{+0.8}_{-0.4}$ | $1015^{+885}_{-611}$ | $1.03^{+0.52}_{-0.39}$ | 0.87 | $3.9^{+2.1}_{-0.7}$ | $15^{+10}_{-1}$ | rd | FS18
Q1623-BX502 | $0.80^{+0.20}_{-0.20}$ | 28.49 | 0.72 | $1.7^{+0.5}_{-0.5}$ | $227^{+177}_{-137}$ | $0.23^{+0.14}_{-0.10}$ | 0.92 | $0.85^{+0.36}_{-0.36}$ | $14^{+9}_{-1}$ | dd | FS09
—"— | $1.02^{+0.43}_{-0.43}$ | 28.49 | 0.72 | $1.4^{+0.2}_{-0.2}$ | $203^{+101}_{-101}$ | $0.30^{+0.12}_{-0.12}$ | 0.92 | 0.7 | $15^{+9}_{-9}$ | dd | LA09
—"— | $2.0^{+1.5}_{-0.8}$ | 28.49 | 0.72 | $1.1^{+0.7}_{-0.7}$ | $227^{+177}_{-137}$ | $0.23^{+0.14}_{-0.10}$ | 0.92 | $0.58^{+0.77}_{-0.21}$ | $14^{+9}_{-1}$ | rd | FS18
Q1623-BX543 | $0.55^{+0.25}_{-0.25}$ | -7.80 | 1.40 | $1.1^{+0.1}_{-0.1}$ | $10^{+5}_{-5}$ | $0.40^{+0.16}_{-0.16}$ | 0.92 | 2.5 | $431^{+258}_{-258}$ | m | LA09
—"— | $1.8^{+0.9}_{-0.3}$ | -7.80 | 1.40 | $3.3^{+1.2}_{-0.8}$ | $81^{+21}_{-9}$ | $0.94^{+0.20}_{-0.09}$ | 0.92 | $5.0^{+2.4}_{-2.0}$ | $150^{+2}_{-5}$ | m | FS18
Q1623-BX599 | $2.0^{+1.4}_{-0.4}$ | -0.66 | 1.02 | $2.4^{+0.6}_{-0.6}$ | $2750^{+18}_{-1141}$ | $5.66^{+0.07}_{-0.24}$ | 0.37 | $4.1^{+3.3}_{-1.7}$ | $34^{+19}_{-1}$ | m | FS18
Q1700-BX490 | $1.09^{+0.34}_{-0.34}$ | 5.90 | 1.28 | $1.6^{+0.1}_{-0.1}$ | $10^{+5}_{-5}$ | $0.40^{+0.16}_{-0.16}$ | 0.93 | 1.5 | $441^{+264}_{-264}$ | m | LA09
Q2343-BX389 | $3.1^{+0.4}_{-0.4}$ | -8.30 | 1.54 | $4.2^{+1.1}_{-1.1}$ | $2750^{+224}_{-250}$ | $4.40^{+0.19}_{-0.31}$ | 0.22 | $14^{+1}_{-1}$ | $26^{+1}_{-2}$ | rd | FS09
—"— | $5.3^{+1.7}_{-0.7}$ | -8.30 | 1.54 | $6.2^{+1.2}_{-1.2}$ | $2750^{+224}_{-250}$ | $4.40^{+0.19}_{-0.31}$ | 0.22 | $29^{+9}_{-6}$ | $26^{+1}_{-2}$ | rd | FS18
Q2343-BX418 | $0.73^{+0.59}_{-0.59}$ | 53.49 | 0.32 | $0.8^{+0.1}_{-0.1}$ | $102^{+51}_{-51}$ | $0.10^{+0.04}_{-0.04}$ | 0.93 | 0.3 | $11^{+6}_{-6}$ | dd | LA09
Q2343-BX442 | $3.55^{+0.23}_{-0.15}$ | -15.00 | 1.54 | $\sim 8$ | $1100^{+1000}_{-500}$ | $6^{+2}_{-1}$ | 0.33 | 11 | $52^{+37}_{-21}$ | rd | LA12
Q2343-BX513 | $1.27^{+0.35}_{-0.35}$ | 18.43 | 0.61 | $1.4^{+0.1}_{-0.1}$ | $3000^{+1500}_{-1500}$ | $7.80^{+3.12}_{-3.12}$ | 0.19 | 1.6 | $26^{+15}_{-15}$ | m | LA09
—"— | $1.8^{+2.1}_{-0.5}$ | 18.43 | 0.61 | $2.6^{+1.5}_{-0.8}$ | $806^{+209}_{-166}$ | $2.70^{+0.56}_{-0.05}$ | 0.19 | $2.5^{+3.2}_{-1.2}$ | $10^{+1}_{-4}$ | m | FS18
Q2343-BX660 | $1.10^{+0.32}_{-0.32}$ | 20.38 | 0.36 | $1.6^{+0.1}_{-0.1}$ | $1609^{+804}_{-804}$ | $0.80^{+0.32}_{-0.32}$ | 0.69 | 0.9 | $5^{+3}_{-3}$ | m | LA09
Q2346-BX482 | $4.4^{+1.0}_{-1.0}$ | -14.98 | 1.12 | $4.2^{+0.9}_{-0.9}$ | $286^{+35}_{-83}$ | $1.69^{+0.30}_{-0.25}$ | 0.31 | $13^{+2}_{-1}$ | $50^{+32}_{-1}$ | rd | FS09
—"— | $4.9^{+1.6}_{-0.7}$ | -14.98 | 1.12 | $6.0^{+0.8}_{-0.8}$ | $286^{+35}_{-83}$ | $1.69^{+0.30}_{-0.25}$ | 0.31 | $26^{+12}_{-5}$ | $50^{+32}_{-1}$ | rd | FS18

  * a

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

  * b

Galactic half-light radius

  * c

Stellar properties from SED modelling (mass ($m_{\star}$), age, and star-
formation rate (SFRSED)) collated from the respective source studies

  * d

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

  * e

Dynamical mass

  * f

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

  * g

Source references for kinematic and other data: FS09 = Förster Schreiber et
al. (2009), LA09 = Law et al. (2009), LA12 = Law et al. (2012a), FS18 =
Förster Schreiber et al. (2018)

##  4 DISCUSSION

###  4.1 Context

The results presented here are part of a broader project that aims to examine
the intrinsic and environmental properties of high redshift galaxies in a
holistic manner to lend insight into their evolution. C09 and Paper I describe
a method by which large samples of $z\sim 2-3$ LBGs with known Ly$\alpha$
spectral type can be selected from optical broadband imaging datasets, and by
so doing suggest a means by which a wide range of other spectral, physical,
and environmental galaxy properties known to correlate with Ly$\alpha$ might
be studied in large samples and on large scales using optical broadband
imaging and/or Ly$\alpha$ data alone. The power of this approach was
demonstrated by Cooke et al. (2013) who studied the large-scale clustering
properties and halo mass distributions of photometrically selected $z\sim 3$
aLBG and eLBG populations via correlation function analysis (see Section
4.4.1).

Ly$\alpha$-sensitive properties accessible by this approach include: size and
morphology (e.g., Law et al., 2007, 2012b; Pentericci et al., 2010; Marchi et
al., 2019); gas fraction (e.g., Erb et al., 2006c; Law et al., 2012b);
parameters derived from SED fitting such as stellar mass, SFR, UV continuum
slope/reddening and age (e.g., Shapley et al., 2003; Erb et al., 2006a; Law et
al., 2007; Kornei et al., 2010; Steidel et al., 2010; Law et al., 2012b; Hathi
et al., 2016; Reddy et al., 2016; Du et al., 2018; Marchi et al., 2019); gas
covering fraction and outflow kinematics (e.g., Shapley et al., 2003; Steidel
et al., 2010; Law et al., 2012b; Reddy et al., 2016; Guaita et al., 2017;
Steidel et al., 2018; Du et al., 2018; Trainor et al., 2019; Marchi et al.,
2019); low and high ionisation absorption line and nebula emission line
strengths (e.g., Shapley et al., 2003; Steidel et al., 2010; Law et al.,
2012b; Stark et al., 2014; Trainor et al., 2016; Erb et al., 2016; Guaita et
al., 2017; Du et al., 2018); and, dark matter halo mass, and large-scale
spatial distribution (e.g., Cooke et al., 2013; Díaz et al., 2014; Guaita et
al., 2017, 2020; Shi et al., 2019).

In this work we add the relationships between Ly$\alpha$, broadband colour,
and nebular emission-line kinematics to this list.

###  4.2 The importance of kinematics

Galaxy kinematics are a characteristic feature of the Morphology–Density
Relation (MDR) in the local Universe (Bamford et al., 2009, and references
therein), and are a key constraint for simulations that aim to understand the
mechanisms by which galaxies evolve over cosmic time (Somerville & Davé,
2015). Despite its critical role in reconciling observations with physical and
computational models, acquiring kinematic information for thousands to
millions of galaxies over large scales from high to low redshift is not
possible for the foreseeable future – in particular for galaxies at $z\gtrsim
4$. Thus, there is strong motivation to explore connections between kinematics
and other properties that may serve as proxies for predicting kinematic type
where such detail is not directly obtainable.

At $z\gtrsim 4$, the bulk of the galaxy population is only accessible via deep
imaging and, for galaxies reachable with deep spectroscopy, often Ly$\alpha$
is the only accessible feature at optical/near-IR wavelengths. This is also
the case for the fainter and lower-mass $z\sim 2-3$ galaxies. In this work we
have reported a fundamental relationship between galaxy kinematics and net
Ly$\alpha$ EW that has potential to serve as a predictor of kinematic type in
such cases. In addition, we have proposed a broadband imaging method whereby
samples of known kinematic type can be selected via the photometric
segregation of Ly$\alpha$ spectral types on the CMD, and from which the most
promising galaxies can be efficiently selected for expensive follow-up
observations. We do not propose that the coarse kinematic behaviour derived
from this method is a substitute for the detailed and more accurate kinematic
information available from IFU-based studies. Rather, we suggest this method
as a complementary approach that can elucidate the typical kinematic character
of large samples for which individual IFU measurements are not feasible, and
at redshifts $z\gtrsim 4$ where such measurements are currently not possible.

Together, these results provide a means to explore all the above
Ly$\alpha$-sensitive properties, and their relation to generalised kinematic
type in very large samples of galaxies on large scales and potentially out to
high redshift in datasets from current and future large-area photometric
campaigns. For example, we envisage application of our method to datasets from
the all-sky LSST that will select hundreds of millions of LBGs in redshift
ranges from $z\sim 2-6$ across many hundreds to thousands of Mpc.

###  4.3 Comparison with low-redshift analogues

The fourteen low redshift ($0.03<z<0.2$) star-forming galaxies comprising the
Lyman-$\alpha$ Reference Sample (LARS) have continuum size, stellar mass, and
rest-frame absolute magnitudes typical of Lyman break analogues in the local
Universe selected to have properties similar to star-forming galaxies at
$2<z<3$ (Östlin et al., 2014; Hayes et al., 2014). Accordingly, they provide a
useful low-z reference sample for comparison with our $z\sim 2-3$ results.

Herenz et al. (2016, hereafter HE16) derive values for shear velocity (vshear)
and intrinsic velocity dispersion ($\sigma_{0}$) from the H$\alpha$ kinematic
maps of the LARS galaxies in a manner similar to the methods used by FS09,
LA09 and FS18 at $z\sim 2$. They classify each galaxy as either a ‘rotating
disc’, ‘perturbed rotator’, or as having ‘complex kinematics’ based on the
qualitative appearance of their velocity fields and the classification scheme
introduced by Flores et al. (2006). HE16 show that the LARS galaxies are
characterised by high intrinsic velocity dispersions in the range 40$-$100 km
s-1 (54 km s-1 median), low shear velocities (30$-$180 km s-1, 65 km s-1
median), and v${}_{\rm{shear}}/\sigma_{0}$ ratios ranging from 0.5 to 3.2. In
this respect the LARS galaxies are kinematically similar to turbulent star-
forming galaxies observed at high redshift (HE16), including the $z\sim 2-3$
LBGs studied herein (cf. the kinematic parameters in Tables 3 & 4).

The LARS team used a synthetic narrow-band imaging method to measure total
flux in the region of Ly$\alpha$, and the subtraction from this of a modelled
stellar continuum spectrum to derive the Ly$\alpha$ flux (or flux decrement).
Pixel-wise SED fitting of the same spectral models as were used for continuum
subtraction was then used to estimate the continuum flux density at 1216 Å by
which the flux differential was divided in order to calculate a Ly$\alpha$ EW.
The bandpass used to measure Ly$\alpha$ sampled rest-frame wavelengths between
about 1205 and 1230 Å (depending on the exact redshift of the target galaxy).
Accordingly, the measured flux included both Ly$\alpha$ in emission and at
least part of the Ly$\alpha$ absorption signal blueward of 1216 Å. For
comparison with our $z\sim 2-3$ samples, we use the integrated values of
Ly$\alpha$ EW given by He16 and Hayes et al. (2014) that were calculated using
fluxes and flux densities integrated over a circular aperture with twice the
isophotal Petrosian radius determined for each galaxy from the image that
transmits Ly$\alpha$ and the Far-UV continuum (see Hayes et al. (2013) and
Hayes et al. (2014)).

LARS galaxies with higher shearing velocities (v${}_{\rm{shear}}\gtrsim 50$ km
s-1) have preferentially lower Ly$\alpha$ EWs and lower Ly$\alpha$ escape
fractions than their lower angular momentum counterparts. Moreover, Ly$\alpha$
EW and Ly$\alpha$ escape fraction correlate with v${}_{\rm{shear}}/\sigma_{0}$
in the sense that the LARS galaxies with ‘complex kinematics’ have higher
Ly$\alpha$ EWs and higher Ly$\alpha$ escape fractions than systems with a
kinematic signature indicative of a ‘perturbed rotator’ or a ‘rotating disc’.

While these observations of HE16 are in good qualitative agreement with our
findings, the confidence with which we can directly compare the LARS
Ly$\alpha$ EW values with the net Ly$\alpha$ EWs of the $z\sim 2-3$ LBGs is
moderated by a number of factors:

  1. (i)

Due to the relatively broad bandpass of the synthetic filters used for the
LARS imaging, a large amount of stellar continuum light is transmitted along
with the Ly$\alpha$ signal. The reliability of the measured Ly$\alpha$ flux
is, therefore, critically dependent on the quality of the stellar continuum
modelling, and in cases where stellar light dominates over Ly$\alpha$, small
errors in the modelling can result in large errors in the derived properties
(M. Hayes, priv. comm.).

  2. (ii)

Hayes et al. (2014) found that due to the spatial redistribution of Ly$\alpha$
under the influence of resonant scattering processes, the computation of
integrated Ly$\alpha$ quantities for their galaxies is a strong function of
the radius over which values were summed. Figure 4 in Hayes et al. (2014)
shows the aperture-dependent behaviour of Ly$\alpha$ EW (and other properties)
for the LARS galaxies.

  3. (iii)

The LARS team chose to adopt a convention of reporting Ly$\alpha$ EW in
emission only such that even if an integrated Ly$\alpha$ EW with net
absorption was measured, it has been reported only in terms of its emission
component, i.e., with a Ly$\alpha$ EW of zero (M. Hayes, priv. comm.).

Figure 7 shows a v${}_{\rm{shear}}/\sigma_{0}$ versus Ly$\alpha$ EW plot of
the HE16 galaxies overlaid on a ${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$
versus net Ly$\alpha$ EW plot of our combined $z\sim 2$ and $z\sim 3$
kinematic samples. The curve-of-growth results of Hayes et al. (2014) indicate
that Two LARS galaxies (LARS04 & LARS06) are net absorbers at all apertures
within their field of view, but these are reported as having Ly$\alpha$ EW = 0
under the convention adopted by the LARS team. While the magnitude of any
offset is difficult to quantify, grey arrows in Figure 7 indicate the
direction in which these two galaxies would move on the plot if this net
absorbing character was reflected in the quoted Ly$\alpha$ EWs. Taking into
account this caveat, and the other sources of systematic uncertainty
notwithstanding, the relationship between galaxy kinematics and Ly$\alpha$
observables in the LARS sample is almost indistinguishable from our result at
$z\sim 2-3$.

HE16 surmise that there is a causal relationship between turbulence in
actively star-forming galaxies and ISM conditions that facilitate the escape
of Ly$\alpha$ photons, and further speculate that dispersion-dominated
kinematics are a necessary requirement for a galaxy to have a significant
amount of escaping Ly$\alpha$ radiation. HE16 also note that, like the $z\sim
2$ samples of FS09 and LA09, galaxies in their sample with lower stellar mass
($M_{\star}$) typically have lower v${}_{\rm{shear}}/\sigma_{0}$ ratios and
that the strongly Ly$\alpha$-emitting LARS-LAEs are preferentially found among
the systems with $M_{\star}$ $\lesssim 10^{10}\,{M}_{\odot}$.

These conclusions for the low-z LARS galaxies are consistent with our results
that show a link between higher net Ly$\alpha$ EW and dispersion-dominated
kinematics in $z\sim 2-3$ LBGs. Moreover, they support our general proposition
that Ly$\alpha$ emission may be a useful diagnostic of galaxy kinematics and
other properties, particularly at high redshifts where the number density of
LAEs is higher (e.g., Wold et al., 2014), and low-mass star-forming
dispersion-dominated galaxies are more prevalent (e.g., Turner et al., 2017;
Mason et al., 2017; Girard et al., 2018).

###  4.4 Implications for galaxy evolution science

####  4.4.1 Kinematics and Large-Scale Structure at $z\sim 2-3$

A key finding of this work, Paper I, and C09 is that Ly$\alpha$ spectral types
in populations of $z\sim 2-3$ 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. (2013) use the same photometric selection method to isolate large
samples ($\sim$105) of $z\sim 3$ 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 $z\sim 2$ protoclusters (Muldrew et al., 2015),
spectroscopic follow-up of protoclusters identified at $z\sim 3-4$ show that
protocluster members have lower average Ly$\alpha$ EW compared to equivalent
coeval field galaxies (Toshikawa et al., 2016; Lemaux et al., 2018). 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., 2010; Díaz et al., 2014; Bielby et al., 2016; Guaita et al., 2017, 2020;
Shi et al., 2019).

Combining these results with our findings, it would appear that we are seeing
at $z\sim 2-3$, 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 for example Blanton &
Moustakas (2009)).

Independent evidence of galaxy bimodality at high redshift is observed in
damped Ly$\alpha$ systems (DLAs). Wolfe et al. (2008) 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. (2003), Du et al. (2018), and
Pahl et al. (2020).

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, 1972). 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 $z\sim 2-3$ signs of a nascent
morphology-density relation (Dressler, 1980; Postman et al., 2005) traced by
the aLBG and eLBG populations.

##  5 Summary and Conclusions

In this paper we report a direct relationship between nebular emission-line
kinematics and net Ly$\alpha$ EW in samples of $z\sim 2$ and $z\sim 3$ LBGs
drawn from the literature for which matching rest-frame UV broadband
photometry, consistently measured net Ly$\alpha$ EWs, and kinematic
classifications from IFU-based spectroscopy are available. We conclude that
LBGs with Ly$\alpha$ dominant in absorption (aLBGs) are almost exclusively
rotation-dominated (presumably disc-like) systems, and LBGs with Ly$\alpha$
dominant in emission (eLBGs) characteristically have dispersion-dominated
kinematics.

The key results of this paper are summarised below:

  1. (i)

In Sections 3.1 & 3.2 we show that rotation- and dispersion-dominated $z\sim
2-3$ LBGs segregate consistently with rest-frame UV colour, and that their
distributions on a rest-frame UV colour-magnitude diagram (CMD) are coincident
with the aLBG and eLBG distributions respectively of the parent LBG samples
from which they were drawn.

  2. (ii)

This congruent behaviour is reinforced by the assignment to the kinematic
samples of Ly$\alpha$ spectral types based on spectroscopically-determined net
Ly$\alpha$ EWs (see Section 2.4), and is statistically supported by the
results of two-sided Kolmogorov–Smirnov (KS) tests.

  3. (iii)

Galaxies located in the strongly Ly$\alpha$-absorbing part of the CMD (aLBGs)
are characteristically massive, red, spatially diffuse disc-like systems.
Galaxies with photometric properties akin to those with Ly$\alpha$ dominant in
emission (eLBGs) are most likely to be compact, blue, and dispersion-
dominated, with low (if any) rotational dynamic support, free from large or
luminous rotating disc structures. Moreover, we find that sources in our
kinematic samples that have been positively identified as merging systems
reside toward the bright side, and centrally in colour, on the CMD.

  4. (iv)

In Section 3.3.1 we report the segregation in average net Ly$\alpha$ EW for
subsets of rotation- and dispersion-dominated galaxies in our $z\sim 2$ and
$z\sim 3$ kinematic samples, and show, for a combined $z\sim 2-3$ sample of 32
galaxies, a clear bifurcation (KS-test confidence $\sim$99%) in the average
Ly$\alpha$ spectral properties of rotation- and dispersion-dominated LBGs.

  5. (v)

In Section 3.3.2 we quantify the relationship between the strength of
rotational dynamic support (as measured using
${v}_{\mathrm{obs}}/2{\sigma}_{\mathrm{int}}$ and
${v}_{\mathrm{rot}}/{\sigma}_{\mathrm{0}}$) and net Ly$\alpha$ EW for subsets
of our kinematic sample where these data are available. Both results show a
statistically significant non-linear negative correlation between rotational
dynamic support and net Ly$\alpha$ EW.

  6. (vi)

In Section 3.3.3 we confirm that the relationship between net Ly$\alpha$ EW
and kinematics that we report is consistent with relationships reported
separately between kinematics, net Ly$\alpha$ EW and other galactic properties
such as stellar mass, star-formation rate, gas fraction, age, and size.

  7. (vii)

We demonstrate in Section 4.3 the consistency of our result with the low-$z$
kinematics versus Ly$\alpha$ study of Herenz et al. (2016): a result that
suggests the utility of Ly$\alpha$ emission as a diagnostic of galaxy
kinematics and other properties over a wide range of redshifts.

In Paper I in this series (Foran et al., 2023) we report the photometric
segregation of $z\sim 2$ LBGs versus net Ly$\alpha$ EW in rest-frame UV
colour–magnitude space, and derive criteria for the selection of pure samples
of LBGs with Ly$\alpha$ dominant in absorption and Ly$\alpha$ dominant in
emission on the basis of optical broadband imaging alone. Together with the
analogous $z\sim 3$ result of Cooke (2009), we have suggested the utility of
this method to study a wide range of properties known to be associated with
Ly$\alpha$ (see Section 3.3.3), in large samples and over large scales in
datasets from current and future large-area and all-sky photometric surveys
such as the Vera Rubin Observatory Legacy Survey of Space and Time (LSST:
Ivezić et al., 2019).

Here we add nebular emission-line kinematics to the list of properties that
might be studied by such an approach. We propose a method by which the
generalised kinematic type of large samples of LBGs might be determined on the
basis of net Ly$\alpha$ spectral types determined from broadband imaging, and
their relation to other properties studied on large scales and at redshifts
beyond the range accessible by current IFU spectrographs.

The small size of the kinematic sample that meets the necessary selection
criteria for this work (see Section 2.1) precludes a robust investigation of
the relationship between net Ly$\alpha$ EW and kinematics at fixed values of
other galactic properties. Nevertheless, these results (i) suggest the
potential value of a holistic interpretation of galaxy evolution in terms of
these many correlated properties, including kinematics and their relation to
galaxy environment, and (ii) provide motivation for a dedicated high-
resolution AO observational campaign that specifically targets a larger,
uniformly-selected and analysed sample that would inform the strength of all
these relations, and enable application of multi-variate regression techniques
to determine how they are related causally or otherwise (cf. the low-$z$ LARS
study of Runnholm et al. (2020)).

Finally, in Section 4.4 we speculate that the combination of our result
linking net Ly$\alpha$ EW and nebular emission-line kinematics with the known
large-scale clustering behaviour of Ly$\alpha$-absorbing and
Ly$\alpha$-emitting LBGs (Cooke et al., 2013), is evocative of an emergent
bimodality of early galaxies that is consistent with a nascent morphology-
density relation being observed at $z\sim 2-3$ (Section 4.4).

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