#  The KBSS-KCWI Survey: The connection between extended Ly$\alpha$ halos and
galaxy azimuthal angle at $z\sim 2-3$ ††thanks: Based on data obtained at the
W. M. Keck Observatory, which is operated as a scientific partnership among
the California Institute of Technology, the University of California and the
National Aeronautics and Space Administration. The Observatory was made
possible by the generous financial support of the W. M. Keck Foundation.

Yuguang Chen (陈昱光)1, Charles C. Steidel1, Dawn K. Erb2, David R. Law3, Ryan F.
Trainor4, Naveen A. Reddy5, Alice E. Shapley6, Anthony J. Pahl6, Allison L.
Strom7, Noah R. Lamb8, Zhihui Li (李智慧)1 and Gwen C. Rudie9
1Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, 1200 E California Blvd, MC249-17, Pasadena, CA 91125, USA
2Center for Gravitation, Cosmology, and Astrophysics, Department of Physics,
University of Wisconsin-Milwaukee, 3135 N. Maryland Avenue,
Milwaukee, WI 53211, USA
3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
4Department of Physics and Astronomy, Franklin & Marshall College, 637 College
Ave., Lancaster, PA 17603, USA
5Department of Physics and Astronomy, University of California, Riverside, 900
University Avenue, Riverside, CA 92521, USA
6Department of Physics and Astronomy, University of California, Los Angeles,
430 Portola Plaza, Los Angeles, CA 90095, USA
7Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA
8Department of Physics, Drexel University, 32 S. 32nd Street, Philadelphia, PA
19104, USA
9The Observatories of the Carnegie Institution for Science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
Email<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the first statistical analysis of kinematically-resolved,
spatially-extended $\rm Ly\alpha$ emission around $z=2-3$ galaxies in the Keck
Baryonic Structure Survey (KBSS) using the Keck Cosmic Web Imager (KCWI). Our
sample of 59 star-forming galaxies ($z_{\mathrm{med}}=2.29$) comprises the
subset with typical KCWI integration times of $\sim 5$ hours and with existing
imaging data from the Hubble Space Telescope and/or adaptive optics-assisted
integral field spectroscopy. The high resolution images were used to evaluate
the azimuthal dependence of the diffuse $\rm Ly\alpha$ emission with respect
to the stellar continuum within projected galactocentric distances of
$\lesssim 30$ proper kpc. We introduce cylindrically-projected 2D spectra
(CP2D) that map the averaged $\rm Ly\alpha$ spectral profile over a specified
range of azimuthal angle, as a function of impact parameter around galaxies.
The averaged CP2D spectrum of all galaxies shows clear signatures of $\rm
Ly\alpha$ resonant scattering by outflowing gas. We stacked the CP2D spectra
of individual galaxies over ranges of azimuthal angle with respect to their
major axes. The extended $\rm Ly\alpha$ emission along the galaxy principal
axes are statistically indistinguishable, with residual asymmetry of $\leq 2$%
($\sim 2\sigma$) of the integrated $\rm Ly\alpha$ emission. The symmetry
implies that the $\rm Ly\alpha$ scattering medium is dominated by outflows in
all directions within 30 kpc. Meanwhile, we find that the blueshifted
component of $\rm Ly\alpha$ emission is marginally stronger along galaxy minor
axes for galaxies with relatively weak $\rm Ly\alpha$ emission. We speculate
that this weak directional dependence of $\rm Ly\alpha$ emission becomes
discernible only when the $\rm Ly\alpha$ escape fraction is low. These
discoveries highlight the need for similar analyses in simulations with $\rm
Ly\alpha$ radiative transfer modeling.

###### keywords:

galaxies: evolution — galaxies: ISM — galaxies: high-redshift

††pagerange:  The KBSS-KCWI Survey: The connection between extended Ly$\alpha$
halos and galaxy azimuthal angle at $z\sim 2-3$ ††thanks: Based on data
obtained at the W. M. Keck Observatory, which is operated as a scientific
partnership among the California Institute of Technology, the University of
California and the National Aeronautics and Space Administration. The
Observatory was made possible by the generous financial support of the W. M.
Keck Foundation.– The KBSS-KCWI Survey: The connection between extended
Ly$\alpha$ halos and galaxy azimuthal angle at $z\sim 2-3$ ††thanks: Based on
data obtained at the W. M. Keck Observatory, which is operated as a scientific
partnership among the California Institute of Technology, the University of
California and the National Aeronautics and Space Administration. The
Observatory was made possible by the generous financial support of the W. M.
Keck Foundation.

##  1 Introduction

Extended “halos” of diffuse Lyman-alpha (Ly$\alpha$) emission, extending to
many times larger radii than starlight, are nearly ubiquitous around rapidly-
star-forming galaxies at redshift $z>2$ (e.g. Steidel et al., 2011; Wisotzki
et al., 2016; Leclercq et al., 2017; Wisotzki et al., 2018). While there is
not yet consensus on the dominant physical mechanism giving rise to diffuse
$\rm Ly\alpha$ halos, there is general agreement on the list of potential
sources, all of which depend on substantial cool hydrogen gas in the
circumgalactic medium (CGM; e.g., Ouchi et al. 2020)): (1) resonant scattering
of $\rm Ly\alpha$ produced by recombination of gas photoionised by massive
stars or active galactic nuclei (AGN), i.e., a central source, where $\rm
Ly\alpha$ photons are subsequently scattered until they find optically-thin
channels to escape (2) in situ photoionization of H i by the metagalactic
ultra-violet (UV) ionising radiation field combined with local sources and
followed by recombination (sometimes called “fluorescence”) (3) accreting gas
losing energy via collisional excitation of $\rm Ly\alpha$ (sometimes referred
to as “gravitational cooling” (4) emission from unresolved satellite galaxies
in the halos of larger central galaxies. In principle, the observed surface
brightness, spatial distribution, and kinematics of $\rm Ly\alpha$ emission
can discriminate between the various mechanisms and, perhaps more importantly,
can provide direct information on the degree to which gas in the CGM is
accreting, outflowing, or quiescent. $\rm Ly\alpha$ emission from the CGM, if
interpreted correctly, can provide a detailed map of the cool component of the
dominant baryon reservoir associated with forming galaxies, as well as
constraints on large-scale gas flows that are an essential part of the current
galaxy formation paradigm.

Because a $\rm Ly\alpha$ photon is produced by nearly every photoionisation of
hydrogen, the intrinsic $\rm Ly\alpha$ luminosity of a rapidly star-forming
galaxy can be very high, and thus easily detected (Partridge & Peebles, 1967).
However, due to its very high transition probability, $\rm Ly\alpha$ is
resonantly scattered until the last scattering event gives it an emitted
frequency and direction such that the optical depth remains low along a
trajectory that allows it to escape from the host galaxy. When the $\rm
Ly\alpha$ optical depth is high in all directions, the vastly increased
effective path length – due to large numbers of scattering events during the
time the photon is radiatively trapped – increases the probability that the
photon is destroyed by dust or emitted via two-photon mechanism. But $\rm
Ly\alpha$ photons that are not absorbed by dust grains or converted to two-
photon radiation must eventually escape, with the final scattering resulting
in the photon having a frequency and direction such that the photon can freely
stream without further interaction with a hydrogen atom. The radiative
transfer of $\rm Ly\alpha$ thus depends in a complex way on the distribution,
clumpiness, and kinematics of H i within the host galaxy, as well as on where
and how the photon was produced initially. But the added complexity is
counter-balanced by the availability of a great deal of information about the
scattering medium itself that is otherwise difficult or impossible to observe
directly: i.e., the neutral hydrogen distribution and kinematics in the CGM.

Since the commissioning of sensitive integral-field spectrometers on large
ground-based telescopes – the Multi Unit Spectroscopic Explorer (MUSE; Bacon
et al. 2010) on the Very Large Telescopes (VLT) of the European Southern
Observatory (ESO) and the Keck Cosmic Web Imager (KCWI; Morrissey et al. 2018)
at the Keck Observatory – it has become possible to routinely detect diffuse
$\rm Ly\alpha$ emission halos around individual galaxies at high redshift
(e.g., Wisotzki et al. 2016; Leclercq et al. 2017), and to simultaneously
measure the spatially-resolved $\rm Ly\alpha$ kinematics (e.g., Erb et al.
2018; Claeyssens et al. 2019; Leclercq et al. 2020). Such observations can
then be interpreted in terms of simple expectations based on $\rm Ly\alpha$
radiative transfer; for example, a generic expectation is that most $\rm
Ly\alpha$ emission involving scattered photons (i.e., those that must pass
through an H i gas distribution before escaping their host) will exhibit a
“double-peaked” spectral profile, where the relative strength of the blue-
shifted and red-shifted peaks may be modulated by the net velocity field of
the emitting gas. In the idealised case of a spherical shell of outflowing
(infalling) gas, one predicts that an external observer will measure a
dominant red (blue) peak (Verhamme et al., 2006). The fact that most ($\sim
90$%) of star-forming galaxy “down the barrel” (DTB) spectra with $\rm
Ly\alpha$ in emission in the central portions exhibit dominant red peaks
(e.g., Pettini et al. 2001; Steidel et al. 2010; Kulas et al. 2012; Trainor et
al. 2015; Verhamme et al. 2018; Matthee et al. 2021) has led to the conclusion
that outflowing gas dominates $\rm Ly\alpha$ radiative transfer, at least at
small galactocentric distances.

Essentially every simulation of galaxy formation (Faucher-Giguère et al. 2010)
predicts that gaseous accretion is also important – particularly at high
redshifts ($z\lower 2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}2$) – and this
has focused attention on systems in which a double $\rm Ly\alpha$ profile with
a blue-dominant peak is observed, often cited as evidence for on-going
accretion of cool gas (e.g., Vanzella et al. 2017; Martin et al. 2014, 2016;
Ao et al. 2020). Quantitative predictions of $\rm Ly\alpha$ emission from
accreting baryons depend sensitively on the thermal state and the small-scale
structure of the gas (e.g., Kollmeier et al. 2010; Faucher-Giguère et al.
2010; Goerdt et al. 2010), leading to large uncertainties in the predictions.
The role played by “local” sources of ionising photons over and above that of
the metagalactic ionising radiation field is likely to be substantial for
regions near QSOs (Cantalupo et al. 2014; Borisova et al. 2016; Cai et al.
2019; O’Sullivan et al. 2020) but much more uncertain for star-forming
galaxies, where the escape of scattered $\rm Ly\alpha$ photons is much more
likely than that of ionising photons.

Models of $\rm Ly\alpha$ radiative transfer have attempted to understand the
dominant physics responsible for producing $\rm Ly\alpha$ halos around
galaxies. Using photon-tracing algorithms with Monte-Carlo simulations,
Verhamme et al. (2006); Dijkstra (2014); Gronke & Dijkstra (2016); Gronke et
al. (2016) have explored the effects of resonant scattering on the emergent
central $\rm Ly\alpha$ line profile using various idealised H i geometries and
velocity fields; in most cases simple models can be made to fit the observed
1-D profiles (Gronke, 2017; Song et al., 2020). There have also been attempts
to model or predict spatially-resolved $\rm Ly\alpha$ which almost certainly
depends on a galaxy’s immediate environment (Zheng et al. 2011; Kakiichi &
Dijkstra 2018), including both outflows and accretion flows, as well as the
radiative transfer of $\rm Ly\alpha$ photons from the site of initial
production to escape (e.g. Faucher-Giguère et al., 2010; Lake et al., 2015;
Smith et al., 2019; Byrohl et al., 2020). However, the conclusions reached as
to the dominant process responsible for the extended $\rm Ly\alpha$ emission
have not converged, indicating that more realistic, high resolution,
cosmological zoom-in simulations may be required to capture all of the
physical processes. Despite the variety of $\rm Ly\alpha$ radiative transfer
models to date, as far as we are aware, no specific effort has been made to
statistically compare full 2-D model predictions (simultaneous spatial and
kinematic) to observed $\rm Ly\alpha$ halos.

Some insight into the relationship between galaxy properties and the
kinematics and spatial distribution of cool gas in the CGM has been provided
by studies at lower redshifts, where galaxy morphology is more easily
measured. Statistical studies using absorption line probes have clearly shown
that the strength of low-ionization metal lines such as Mg ii, Fe ii depends
on where the line of sight passes through the galaxy CGM relative to the
projected major axis of the galaxy – the “azimuthal angle” – (e.g. Bordoloi et
al., 2011; Bouché et al., 2012; Kacprzak et al., 2012; Nielsen et al., 2015;
Lan & Mo, 2018; Martin et al., 2019), and the inclination of the galaxy disk
relative to the line of sight (e.g., Steidel et al. 2002; Kacprzak et al.
2011). More recently, clear trends have also been observed for high ions (O
vi) (Kacprzak et al., 2015). In general, these trends support a picture of
star-forming galaxies in which high-velocity, collimated outflows
perpendicular to the disk are responsible for the strongest absorption lines
in both low and high ions, with low ions also being strong near the disk
plane. Theoretically at least, accretion flows might also be quasi-collimated
in the form of cold streams of gas that would tend to deposit cool gas near
the disk plane (see, e.g., Tumlinson et al. 2017). It is less clear how such a
geometry for gas flows in the CGM would manifest as emission in a resonantly-
scattered line like $\rm Ly\alpha$. One might expect that $\rm Ly\alpha$
photons would escape most readily along the minor axis, since the large
velocity gradients and lower H i optical depths of outflowing material both
favour $\rm Ly\alpha$ escape from the host galaxy. This picture is consistent
with Verhamme et al. (2012), who showed that $\rm Ly\alpha$ escape is enhanced
when the simulated galaxies are viewed face-on.

Observations of low-redshift, spatially resolved $\rm Ly\alpha$ emission have
so far been limited to small samples – e.g., in the local universe ($z<0.2$),
using the Hubble Space Telescope (HST), the “$\rm Ly\alpha$ reference sample”
(LARS; Östlin et al. 2014) has obtained images probing $\rm Ly\alpha$ emission
around galaxies in great spatial detail, reaffirming the complex nature of
$\rm Ly\alpha$ radiative transfer and its relation to the host galaxies. In
most cases LARS found evidence that extended $\rm Ly\alpha$ emission is most
easily explained by photons produced by active star formation that then
diffuse into the CGM before a last scattering event allows escape in the
observer’s direction. Although there are small-scale enhancements associated
with outflows, even for galaxies observed edge-on the $\rm Ly\alpha$ emission
is perhaps smoother than expected (Duval et al., 2016).

At $z>2$, $\rm Ly\alpha$ emission is more readily observed but detailed
analyses are challenged by the relatively small galaxy sizes (both physical
and angular) and the need for high spatial resolution to determine the
morphology of the stellar light. In this paper, we present a statistical
sample of $z>2$ galaxies drawn from a survey using KCWI of selected regions
within the Keck Baryonic Structure Survey (KBSS; Rudie et al. 2012; Steidel et
al. 2014; Strom et al. 2017). Since the commissioning of KCWI in late 2017, we
have obtained deep IFU data ($\sim 5$ hour integrations) for $>100$ KBSS
galaxies with $z=2-3.5$, so that the $\rm Ly\alpha$ line is covered within the
KCWI wavelength range; some initial results from the survey have been
presented by Erb et al. (2018); Law et al. (2018). The 59 galaxies included in
our current analysis are those that, in addition to the KCWI data, have also
been observed at high spatial resolution by either HST or adaptive-optics-
assisted near-IR spectroscopy using Keck/OSIRIS. The overarching goal of the
study is to evaluate the spatial and spectral distribution of $\rm Ly\alpha$
emission within $\simeq 5$ arcseconds as compared to the principle axes
defined by the galaxy morphology on smaller angular scales by each galaxy’s
UV/optical continuum emission. In particular, we seek to use the observed
kinematics and spatial distribution of $\rm Ly\alpha$ emission to evaluate
whether the cool gas in the CGM of forming galaxies shows evidence for
directional dependence – e.g,, inflows or outflows along preferred directions
– with respect to the central galaxy.

This paper is organised as follows. In §2, we describe the KBSS-KCWI sample;
§3 introduces the high-resolution imaging and IFU dataset; §4 covers the
details on the measurement of the galaxy principle axes providing the
definition of the galactic azimuthal angle; §5 presents results on the
connection between Ly$\alpha$ halos and galactic azimuthal angle. §6 looks
into the connection between the $\rm Ly\alpha$ azimuthal asymmetry and the
overall $\rm Ly\alpha$ emission properties. §7 checks higher order azimuthal
asymmetry of $\rm Ly\alpha$ emission by dividing the sample into finer
azimuthal bins. §8 discusses the implications of the results, with a summary
in §9. Throughout the paper, we assume a $\Lambda$CDM cosmology with
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $h=0.7$. Distances are given in
proper units, i.e., physical kpc (pkpc).

##  2 The KBSS-KCWI Galaxy Sample

In late 2017, we began using the recently-commissioned Keck Cosmic Web Imager
(KCWI; Morrissey et al. 2018) on the Keck II 10m telescope to target selected
regions within the survey fields of the Keck Baryonic Structure Survey (KBSS;
Rudie et al. 2012; Steidel et al. 2014; Strom et al. 2017). The main goal of
the KCWI observations has been to detect diffuse emission from the CGM (within
impact parameter, $D_{\rm tran}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}{}100$ pkpc) of a substantial
sample of rapidly star-forming galaxies and optically-faint AGN host galaxies,
reaching surface brightness sensitivity of $\sim 5\times 10^{-20}$ ergs s-1
cm-2 arcsec-2 ($1\sigma$) for unresolved emission lines. Such limiting surface
brightness would allow detection of the extended $\rm Ly\alpha$ halos of
individual galaxies at redshifts $z\sim 2-3$ (e.g., Steidel et al. 2011), and
would be capable of detecting extended diffuse UV metallic cooling line
emission as predicted by simulations of galaxies with comparable mass and
redshift (e.g., Sravan et al. 2016).

KCWI offers three selectable image slicer scales, each of which can be used
with three different regimes of spectral resolving power,
$R\equiv\lambda/\Delta\lambda$. All of the observations used in the present
study were obtained using the “medium” slicer scale and low resolution grating
(BL), providing integral field spectra over a contiguous field of view (FoV)
of
20${}^{\prime\prime}\mskip-7.6mu.\,$3$\times$16${}^{\prime\prime}\mskip-7.6mu.\,$5
covering the common wavelength range 3530-5530 Å with resolving power $\langle
R\rangle=1800$.

Given the relatively small solid angle of the KCWI FoV, and the total
integration time desired for each pointing of $\sim 5$ hours, it was necessary
to choose the KCWI pointings carefully. In general, we chose KCWI pointings to
maximize the number of previously-identified KBSS galaxies with $2\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3.4$ within the field of view, so
that the KCWI spectra would include the $\rm Ly\alpha$ line as well as many
other rest-frame far-UV transitions. Most of the targeted galaxies within each
pointing were observed as part of KBSS in both the optical (Keck/LRIS; Oke et
al. 1995; Steidel et al. 2004) and the near-IR (Keck/MOSFIRE; McLean et al.
2012; Steidel et al. 2014). The pointings were chosen so that the total sample
of KBSS catalog galaxies observed would span the full range of galaxy
properties represented in the KBSS survey in terms of stellar mass (M∗), star
formation rate (SFR), $\rm Ly\alpha$ emission strength, and rest-optical
nebular properties. Most of the KCWI pointings were also directed within the
regions of the KBSS survey fields that have been observed at high spatial
resolution by HST.

As of this writing, the KBSS-KCWI survey comprises 39 pointings of KCWI,
including observations of 101 KBSS galaxies, of which 91 have $2\leq z\leq
3.4$ placing $\rm Ly\alpha$ within the KCWI wavelength range. In this work, we
focus only on galaxies without obvious spectroscopic or photometric evidence
for the presence of an AGN, and have therefore excluded 14 objects with
spectroscopic evidence for the presence of AGN, to be discussed elsewhere. The
remaining objects show no sign of significant AGN activity – e.g., they lack
emission lines of high ionisation species in the rest-UV KCWI and existing
LRIS spectra, their nebular line ratios in the rest-frame optical are
consistent with stellar excitation based on spectra taken with Keck/MOSFIRE
(see e.g. Steidel et al. 2014), they lack power-law SEDs in the NIR-MIR, etc.
Because measurements of galaxy morphology are important to the analysis, we
considered only the subset of the star-forming (non-AGN) galaxies that have
also been observed at high spatial resolution using HST imaging or Keck/OSIRIS
IFU spectroscopcy behind adaptive optics (§3.2).

In addition to the known KBSS targets, many of the KCWI pointings include
continuum-detected serendipitous galaxies whose KCWI spectra constitute the
first identification of their redshifts. A total of 50 galaxies with $z>2$
(most of which are fainter in the optical continuum than the KBSS limit of
R$=25.5$) have been identified. We have included 10 such objects in our
analysis sample, based on their having HST observations with sufficient S/N
for determination of morphology.

A minimum total integration of 2.5 hours (5 hours is more typical) at the
position of a galaxy in the KCWI data cube was also imposed, in order to
ensure the relative uniformity of the data set. Ultimately, after inspection
of the high resolution images (§3.2), 6 galaxies were removed because of
source ambiguity or obvious contamination from nearby unrelated objects in the
images, and two were excluded because they were not sufficiently resolved by
HST to measure their position angle reliably (see §4).

Figure 1:  Redshift distribution of the galaxy sample. The blue histogram
represents the full sample of 59 galaxies. The orange histogram shows the
distribution for the 38 galaxies with nebular redshift measurements from
MOSFIRE near-IR spectra, while the rest are calibrated based on Chen et al.
(2020b) using rest-UV absorption lines or $\rm Ly\alpha$ emission from
Keck/LRIS and/or KCWI spectra. The mean (median) redshift of the full sample
is 2.42 (2.29).

The final sample to be considered in this work contains 59 galaxies, listed in
Table 1. The redshifts given in Table 1 are based on MOSFIRE nebular spectra
for 38 of the 59 galaxies, which have a precision of $\sim\pm 20$ km
s${}^{-1}\,$ and should accurately reflect the galaxy systemic redshift. In
the remaining cases, features in the rest-frame UV spectra including $\rm
Ly\alpha$ emission and strong interstellar absorption features (e.g., Si ii,
Si iv, C ii, C iv, O i) were used to estimate the systemic redshift of the
galaxy, using the calibration described by Chen et al. (2020b). Briefly, the
method uses the statistics – based on several hundred KBSS galaxies with both
nebular and UV observations – of the velocity offsets between nebular
redshifts and redshifts defined by UV spectral features for samples divided by
their UV spectral morphology, i.e., $\rm Ly\alpha$ emission only, $\rm
Ly\alpha$ emission and interstellar absorption, or interstellar absorption
only. The mean offsets for the appropriate sub-sample were applied to the UV-
based redshifts in cases where nebular redshifts are not available; systemic
redshifts obtained using such calibrations have an uncertainty of $\simeq 100$
km s${}^{-1}\,$ when the rest-UV spectra are of high quality (see, e.g.,
Steidel et al. 2018.) Figure 1 shows the redshift distribution of the KCWI
sample, which has $z_{\mathrm{med}}=2.29\pm 0.40$ (median and standard
deviation), for which the conversion between angular and physical scales is
8.21 pkpc/" with our assumed cosmology.

Reliable SED fits are available for 56 of the 59 galaxies using the BPASSv2.2
stellar population synthesis model (Stanway & Eldridge, 2018) and SMC
extinction curve. This choice of SED model has been shown to predict
internally consistent stellar mass ($M_{\ast}$) and star-formation rate (SFR)
for high-redshift galaxies having properties similar to those in our sample
(see, e.g., Steidel et al. 2016; Strom et al. 2017; Theios et al. 2019), i.e.,
$8.5\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}{\rm
log}~{}(M_{\ast}/M_{\odot})\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}11$, and ${\rm 1\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}SFR/(M_{\odot}~{}yr^{-1})\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}100}$. The distributions of $M_{*}$
and SFR (Figure 2) are similar to those of the full KBSS galaxy sample, albeit
with a slight over-representation of ${\rm log}~{}(M_{\ast}/M_{\odot})\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}9$ galaxies 111For direct
comparison of our sample with the so-called “star-formation main-sequence”
(SFMS), we note that SED fits that assume solar metallicity SPS models from
Bruzual & Charlot (2003) and attenuation curve from Calzetti et al. (2000)
result in a distribution of SFR vs. $M_{\ast}$ entirely consistent with the
published SFMS at $z\sim 2$ (e.g., Whitaker et al. 2014)..

Figure 2:  Distribution of SFR and $M_{*}$ of 56/59 galaxies in this sample;
the remaining 3 galaxies have insufficient photometric measurements for
reliable SED fitting. The normalised distributions of the parent KBSS sample
are shown in the orange 1-D histograms. The SFR and $M_{*}$ of galaxies used
in this work are similar to those of the parent KBSS sample; the values are
all based on the BPASS-v2.2-binary spectral synthesis models (Stanway &
Eldridge, 2018), assuming stellar metallicity $Z=0.002$, SMC extinction as
described by Theios et al. (2019), and a Chabrier (2003) stellar initial mass
function. Figure 3:  Distribution of $W_{\lambda}(\rm Ly\alpha)$ for the
sample (blue histogram). The orange skeletal histogram shows the normalised
$W_{\lambda}(\rm Ly\alpha)$ distribution from Reddy & Steidel (2009), which is
a subset of the current KBSS sample large enough to be representative. The
sample discussed in this work is slightly biased toward Ly$\alpha$-emitting
galaxies compared to the parent sample of $z\sim 2-3$ KBSS galaxies.

In order to facilitate comparison of the $\rm Ly\alpha$ emission line strength
of sample galaxies with those in the literature, Table 1 includes the rest-
frame $\rm Ly\alpha$ equivalent width ($W_{\lambda}(\rm Ly\alpha)$) based on
extraction of 1-D spectra from the KCWI data cubes over a spatial aperture
defined by the extent of the UV continuum light of each galaxy222In general,
the $\rm Ly\alpha$ emission evaluated within a central aperture represents
only a fraction of the total that would be measured in a large aperture that
accounts for the diffuse $\rm Ly\alpha$ halos with spatial extent well beyond
that of the FUV continuum (see, e.g., Steidel et al. 2011; Wisotzki et al.
2016); however, the central $W_{\lambda}(\rm Ly\alpha)$ is a closer
approximation to values measured in most spectroscopic galaxy surveys.. The
values of $W_{\lambda}(\rm Ly\alpha)$ in Table 1 were measured using the
method described in Kornei et al. (2010) (see also Reddy & Steidel 2009),
where positive values indicate net emission and negative values net
absorption. Aside from a slight over-representation of galaxies with the
strongest $\rm Ly\alpha$ emission ($W_{\lambda}(\rm Ly\alpha)$$\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}40$ Å), the sample in Table 1 is
otherwise typical of UV-continuum-selected galaxies in KBSS, by construction.
The $W_{\lambda}(\rm Ly\alpha)$ distribution for the sample in Table 1 is
shown in Figure 3. In addition to $W_{\lambda}(\rm Ly\alpha)$, we also
measured the total $\rm Ly\alpha$ flux ($F_{\mathrm{Ly}\alpha}$) and the ratio
between the blue- and red-shifted components of emission for the entire $\rm
Ly\alpha$ halo
($F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$),
which are discussed further in §6.

Table 1: Summary of the galaxy sample and the observations.  Objecta | RA | DEC |  $t_{\mathrm{exp}}$ (KCWI) | Redshiftb |  $W_{\lambda}(\rm Ly\alpha)$c |  $F_{\mathrm{Ly}\alpha}(\mathrm{tot})$d | $\frac{F_{\mathrm{Ly}\alpha}(\mathrm{blue})}{F_{\mathrm{Ly}\alpha}(\mathrm{red})}$e | Imagingf |  $\rm PA_{0}$g |  $\rm PA_{0}$h
---|---|---|---|---|---|---|---|---|---|---
Identifier | (J2000.0) | (J2000.0) | (hr) |  | (Å) | $10^{-17}~{}\mathrm{erg~{}s}^{-1}\mathrm{cm}^{-2}$ | Data | (deg) | Method
Q0100-BX210 | 01:03:12.02 | +13:16:18.5 | 5.4 | 2.2769 | $-3\pm 2$ | $6.3\pm 0.5$ | $0.08\pm 0.05$ | F160W | 37 | (i), (ii)
Q0100-BX212 | 01:03:12.51 | +13:16:23.2 | 5.2 | 2.1063 | $-4\pm 3$ | $2.3\pm 0.7$ | $0.5\pm 0.29$ | F160W | -57 | (i), (ii)
Q0100-C7 | 01:03:08.24 | +13:16:30.1 | 5.1 | 3.0408 | $12\pm 1$ | $9.0\pm 0.4$ | $0.15\pm 0.03$ | F160W | 16 | (i), (ii)
Q0100-D11 | 01:03:08.25 | +13:16:37.6 | 5.1 | 2.5865 | $6\pm 1$ | $1.7\pm 0.4$ | $0.11\pm 0.14$ | F160W | -12 | (i), (ii)
Q0142-BX165 | 01:45:16.87 | -09:46:03.5 | 5.0 | 2.3576 | $51\pm 3$ | $37.5\pm 0.5$ | $0.28\pm 0.01$ | F160W | 14 | (ii)
Q0142-BX188 | 01:45:17.79 | -09:45:05.6 | 5.3 | 2.0602 | $-6\pm 1$ | $4.6\pm 0.7$ | $0.47\pm 0.15$ | F160W | 66 | (i), (ii)
Q0142-BX195-CS10 | 01:45:17.11 | -09:45:06.0 | 4.5 | 2.7382 (UV) | $-1\pm 3$ | $2.2\pm 0.5$ | $0.47\pm 0.24$ | F160W | 29 | (ii)
Q0142-NB5859 | 01:45:17.54 | -09:45:01.2 | 5.8 | 2.7399 (UV) | $20\pm 2$ | $3.7\pm 0.4$ | $0.65\pm 0.14$ | F160W | 42 | (i), (ii)
Q0207-BX144 | 02:09:49.21 | -00:05:31.7 | 5.0 | 2.1682 | $25\pm 3$ | $44.9\pm 0.6$ | $0.29\pm 0.01$ | F140W | -74 | (i), (ii)
Q0207-MD60 | 02:09:53.69 | -00:04:39.8 | 4.3 | 2.5904 | $-27\pm 2$ | $1.4\pm 0.5$ | $-0.16\pm 0.21$ | F140W | -64 | (ii)
Q0449-BX88 | 04:52:14.94 | -16:40:49.3 | 6.3 | 2.0086 | $-7\pm 3$ | $3.3\pm 0.8$ | $0.16\pm 0.16$ | F160W | 10 | (ii)
Q0449-BX88-CS8 | 04:52:14.79 | -16:40:58.6 | 5.2 | 2.0957 (UV) | $4\pm 1$ | $3.9\pm 0.8$ | $0.34\pm 0.17$ | F160W | -39 | (i), (ii)
Q0449-BX89 | 04:52:14.80 | -16:40:51.1 | 6.3 | 2.2570 (UV) | $22\pm 2$ | $7.6\pm 0.4$ | $0.05\pm 0.03$ | F160W | -32 | (i), (ii)
Q0449-BX93 | 04:52:15.41 | -16:40:56.8 | 6.3 | 2.0070 | $-7\pm 2$ | $7.4\pm 0.8$ | $0.6\pm 0.13$ | F160W, OSIRIS | -18 | (i), (ii), (iii)
Q0449-BX110 | 04:52:17.20 | -16:39:40.6 | 5.0 | 2.3355 | $35\pm 2$ | $20.4\pm 0.6$ | $0.47\pm 0.03$ | F160W | -48 | (i), (ii)
Q0821-MD36 | 08:21:11.41 | +31:08:29.4 | 2.7 | 2.583 | $75\pm 10$ | $23.1\pm 0.6$ | $0.16\pm 0.02$ | F140W | 37 | (i), (ii)
Q0821-MD40 | 08:21:06.96 | +31:07:22.8 | 4.3 | 3.3248 | $27\pm 3$ | $11.0\pm 0.4$ | $0.34\pm 0.03$ | F140W | -7 | (ii)
Q1009-BX215 | 10:11:58.71 | +29:41:55.9 | 4.0 | 2.5059 | $1\pm 1$ | $3.7\pm 0.8$ | $0.34\pm 0.15$ | F160W | -20 | (i), (ii)
Q1009-BX218 | 10:11:58.96 | +29:42:07.5 | 5.3 | 2.1091 | $-6\pm 3$ | $4.0\pm 0.6$ | $0.18\pm 0.11$ | F160W | -43 | (i), (ii)
Q1009-BX222 | 10:11:59.09 | +29:42:00.5 | 5.3 | 2.2031 | $-4\pm 1$ | $4.7\pm 0.5$ | $0.28\pm 0.08$ | F160W | -83 | (i), (ii)
Q1009-BX222-CS9 | 10:11:58.92 | +29:42:02.6 | 5.3 | 2.6527 (UV) | $-1\pm 4$ | $1.0\pm 0.4$ | — | F160W | 76 | (i), (ii)
Q1009-D15 | 10:11:58.73 | +29:42:10.5 | 5.3 | 3.1028 (UV) | $-18\pm 6$ | $4.2\pm 0.4$ | $0.28\pm 0.08$ | F160W | 28 | (i), (ii)
Q1549-BX102 | 15:51:55.98 | +19:12:44.2 | 5.0 | 2.1934 | $50\pm 3$ | $19.5\pm 0.5$ | $0.51\pm 0.03$ | F606W | -87 | (i), (ii)
Q1549-M17 | 15:51:56.06 | +19:12:52.7 | 3.3 | 3.2212 (UV) | $27\pm 5$ | $4.3\pm 0.5$ | $0.0\pm 0.05$ | F606W | 72 | (ii)
Q1623-BX432 | 16:25:48.74 | +26:46:47.1 | 3.6 | 2.1825 | $17\pm 1$ | $10.8\pm 0.7$ | $0.46\pm 0.06$ | F160W | 16 | (ii)
Q1623-BX436 | 16:25:49.10 | +26:46:53.4 | 3.6 | 2.0515 (UV) | $-12\pm 2$ | $2.6\pm 1.0$ | — | F160W | 12 | (ii)
Q1623-BX453 | 16:25:50.85 | +26:49:31.2 | 4.8 | 2.1821 | $10\pm 2$ | $2.8\pm 0.5$ | $0.33\pm 0.14$ | F160W, OSIRIS | 31 | (iii)
Q1623-BX453-CS3 | 16:25:50.35 | +26:49:37.1 | 4.7 | 2.0244 (UV) | $17\pm 2$ | $5.6\pm 0.9$ | $0.3\pm 0.11$ | F160W | -66 | (i), (ii)
Q1623-C52 | 16:25:51.20 | +26:49:26.3 | 4.8 | 2.9700 (UV) | $4\pm 1$ | $8.5\pm 0.4$ | $0.22\pm 0.03$ | F160W | -13 | (ii)
Q1700-BX490 | 17:01:14.83 | +64:09:51.7 | 4.3 | 2.3958 | $-3\pm 4$ | $12.8\pm 0.5$ | $0.39\pm 0.03$ | F814W, OSIRIS | 86 | (i), (ii), (iii)
Q1700-BX561 | 17:01:04.18 | +64:10:43.8 | 5.0 | 2.4328 | $-3\pm 3$ | $8.5\pm 0.6$ | $0.51\pm 0.08$ | F814W | 9 | (ii)
Q1700-BX575 | 17:01:03.34 | +64:10:50.9 | 5.0 | 2.4334 | $0\pm 2$ | $5.2\pm 0.6$ | $0.14\pm 0.08$ | F814W | -34 | (ii)
Q1700-BX581 | 17:01:02.73 | +64:10:51.3 | 4.7 | 2.4022 | $9\pm 4$ | $10.6\pm 0.7$ | $0.28\pm 0.05$ | F814W | 27 | (i), (ii)
Q1700-BX710 | 17:01:22.13 | +64:12:19.3 | 5.0 | 2.2946 | $-10\pm 3$ | $8.1\pm 0.6$ | $0.51\pm 0.09$ | F814W | -19 | (i), (ii)
Q1700-BX729 | 17:01:27.77 | +64:12:29.5 | 4.7 | 2.3993 | $14\pm 2$ | $12.9\pm 0.5$ | $0.17\pm 0.03$ | F814W | -3 | (i), (ii)
Q1700-BX729-CS4 | 17:01:28.95 | +64:12:32.4 | 3.0 | 2.2921 (UV) | $14\pm 3$ | $5.8\pm 1.1$ | $0.21\pm 0.12$ | F814W | 40 | (ii)
Q1700-BX729-CS9 | 17:01:27.49 | +64:12:25.1 | 4.7 | 2.4014 (UV) | $35\pm 2$ | $1.1\pm 0.5$ | $-0.19\pm 0.21$ | F814W | -59 | (i), (ii)
Q1700-MD103 | 17:01:00.21 | +64:11:55.6 | 5.0 | 2.3151 | $-24\pm 1$ | $-0.5\pm 0.5$ | — | F814W | 55 | (ii)
Q1700-MD104 | 17:01:00.67 | +64:11:58.3 | 5.0 | 2.7465 (UV) | $6\pm 2$ | $8.2\pm 0.4$ | $0.11\pm 0.03$ | F814W | 7 | (i), (ii)
Q1700-MD115 | 17:01:26.68 | +64:12:31.7 | 4.7 | 2.9081 (UV) | $33\pm 7$ | $7.8\pm 0.5$ | $0.05\pm 0.03$ | F814W | 3 | (i), (ii)
Q2206-MD10 | 22:08:52.21 | -19:44:13.9 | 5.0 | 3.3269 | $5\pm 2$ | $4.0\pm 0.5$ | $0.07\pm 0.08$ | F160W | 41 | (i), (ii)

Table 1: —-continued.

Objecta | RA | DEC |  $t_{\mathrm{exp}}$ (KCWI) | Redshiftb |  $W_{\lambda}(\rm Ly\alpha)$c |  $F_{\mathrm{Ly}\alpha}(\mathrm{tot})$d | $\frac{F_{\mathrm{Ly}\alpha}(\mathrm{blue})}{F_{\mathrm{Ly}\alpha}(\mathrm{red})}$e | Imagingf |  $\rm PA_{0}$g |  $\rm PA_{0}$h
---|---|---|---|---|---|---|---|---|---|---
Identifier | (J2000.0) | (J2000.0) | (hr) |  | (Å) | $10^{-17}~{}\mathrm{erg~{}s}^{-1}\mathrm{cm}^{-2}$ | Data | (deg) | Method
DSF2237b-MD38 | 22:39:35:64 | +11:50:27.5 | 3.7 | 3.3258 | $2\pm 2$ | $3.0\pm 0.5$ | $0.44\pm 0.16$ | F606W | 52 | (i), (ii)
Q2343-BX379 | 23:46:28.96 | +12:47:26.0 | 4.0 | 2.0427 (UV) | $-7\pm 2$ | $6.4\pm 1.3$ | $0.64\pm 0.23$ | F140W | -60 | (i), (ii)
Q2343-BX389 | 23:46:28.90 | +12:47:33.5 | 4.0 | 2.1712 | $-16\pm 1$ | $1.6\pm 0.6$ | $0.65\pm 0.49$ | F140W | -50 | (ii)
Q2343-BX391 | 23:46:28.07 | +12:47:31.8 | 4.0 | 2.1738 | $-16\pm 1$ | $3.3\pm 0.7$ | — | F140W | 21 | (i), (ii)
Q2343-BX417 | 23:46:26.27 | +12:47:46.7 | 3.0 | 2.2231 (UV) | $-9\pm 6$ | $4.8\pm 0.7$ | $0.49\pm 0.15$ | F160W | -38 | (ii)
Q2343-BX418 | 23:46:18.57 | +12:47:47.4 | 4.9 | 2.3054 | $46\pm 3$ | $40.5\pm 0.6$ | $0.46\pm 0.01$ | F140W, OSIRIS | 2 | (i), (ii), (iii)
Q2343-BX418-CS8 | 23:46:18.73 | +12:47:51.6 | 5.1 | 2.7234 (UV) | $66\pm 15$ | $6.7\pm 0.4$ | $0.15\pm 0.04$ | F140W | -67 | (i), (ii)
Q2343-BX429 | 23:46:25.26 | +12:47:51.2 | 5.2 | 2.1751 | $-8\pm 5$ | $3.0\pm 0.5$ | $0.31\pm 0.15$ | F160W | 28 | (i), (ii)
Q2343-BX442 | 23:46:19.36 | +12:47:59.7 | 4.6 | 2.1754 | $-18\pm 4$ | $1.6\pm 0.7$ | $0.38\pm 0.52$ | OSIRIS | -12 | （iv)
Q2343-BX513 | 23:46:11.13 | +12:48:32.1 | 4.8 | 2.1082 | $10\pm 1$ | $18.6\pm 0.7$ | $0.71\pm 0.05$ | F140W, OSIRIS | -9 | (i), (ii), (iii)
Q2343-BX513-CS7 | 23:46:10.55 | +12:48:30.9 | 4.8 | 2.0144 (UV) | $-22\pm 2$ | $4.2\pm 1.0$ | $0.53\pm 0.26$ | F140W | 71 | (i), (ii)
Q2343-BX587 | 23:46:29.17 | +12:49:03.4 | 5.2 | 2.2427 | $-4\pm 3$ | $7.2\pm 0.5$ | $0.22\pm 0.05$ | F160W | 44 | (i), (ii)
Q2343-BX587-CS3 | 23:46:28.24 | +12:49:07.2 | 4.8 | 2.5727 (UV) | $2\pm 2$ | $1.3\pm 0.4$ | $0.17\pm 0.26$ | F140W | -53 | (i), (ii)
Q2343-BX587-CS4 | 23:46:28.62 | +12:49:04.8 | 5.2 | 2.8902 (UV) | $-13\pm 1$ | $2.8\pm 0.4$ | $0.08\pm 0.08$ | F140W | -59 | (i), (ii)
Q2343-BX610 | 23:46:09.43 | +12:49:19.2 | 5.3 | 2.2096 | $8\pm 2$ | $13.5\pm 0.7$ | $0.26\pm 0.04$ | F140W | 22 | (ii)
Q2343-BX660 | 23:46:29.43 | +12:49:45.6 | 5.0 | 2.1742 | $20\pm 3$ | $25.7\pm 0.6$ | $0.26\pm 0.02$ | F140W, OSIRIS | 37 | (i), (ii), (iii)
Q2343-BX660-CS7 | 23:46:29.82 | +12:49:38.7 | 4.2 | 2.0788 (UV) | $41\pm 1$ | $2.3\pm 0.9$ | — | F140W | 89 | (ii)
Q2343-MD80 | 23:46:10.80 | +12:48:33.2 | 4.8 | 2.0127 | $-26\pm 3$ | $2.8\pm 0.8$ | $0.22\pm 0.24$ | F140W | 9 | (i), (ii)

  * a

The “CS” objects are continuum serendipitous objects discussed in §2. Their
naming follows nearby KBSS galaxies that are previously known, and may not be
physically associated to the CS objects.

  * b

If marked as “UV”, the systemic redshift was estimated from features in the
rest-UV spectra, calibrated as described by Chen et al. (2020b). The typical
uncertainties on UV-estimated systemic redshifts are $\delta v\equiv c\delta
z_{\textrm{s}ys}/(1+z_{\rm sys})\simeq 100$ km s${}^{-1}\,$. Otherwise,
$z_{\textrm{s}ys}$ was measured from nebular emission lines in rest-optical
(MOSFIRE) spectra, with $\delta v\simeq 20$ km s${}^{-1}\,$.

  * c

Rest-frame $\rm Ly\alpha$ equivalent width. Details discussed in §2.

  * d

Total $\rm Ly\alpha$ flux. See §6 for more details.

  * d

Flux ratio between the blueshifted and redshifted components of $\rm Ly\alpha$
emission. Details in §6.

  * f

F140W and F160W images obtained using HST-WFC3-IR, F606W and F814W images from
HST-ACS.

  * g

Typical uncertainty: $\pm 10^{\circ}$.

  * h

Methods used to measure $\rm PA_{0}$: (i) using GALFIT on HST images; (ii)
using the pixel intensity second moment on HST images; (iii) pixel intensity
second moment on OSIRIS H$\alpha$ map; (iv) kinematics of H$\alpha$ emission.
Details on the methods are shown in §4.2.

##  3 Observations and Reductions

###  3.1 KCWI

The KCWI data discussed in the present work were obtained between 2017
September and 2020 November, in all cases using the medium-scale slicer made
up of 24 slices of width 0${}^{\prime\prime}\mskip-7.6mu.\,$69 and length
20${}^{\prime\prime}\mskip-7.6mu.\,$3 on the sky. The instrumental setup uses
the BL volume phase holographic (VPH) grating with an angle of incidence that
optimises the diffraction efficiency near 4200 Å, with the camera articulation
angle set to record the spectra of each slice with a central wavelength of
$\sim 4500$ Å. A band-limiting filter was used to suppress wavelengths outside
of the range 3500-5600 Å to reduce scattered light; the useful common
wavelength range recorded for all 24 slices in this mode is 3530-5530 Å, with
a spectral resolving power ranging from $R\simeq 1400$ at 3530 Å to $R\simeq
2200$ at 5530 Å. At the mean redshift of the sample ($\langle z\rangle=2.42$),
$\rm Ly\alpha$ falls at an observed wavelength of $\sim 4160$ Å, where $R\sim
1650$.

The E2V 4k$\times$4k detector was binned 2$\times$2, which provides spatial
sampling along slices of 0${}^{\prime\prime}\mskip-7.6mu.\,$29 pix-1. Because
each slice samples 0${}^{\prime\prime}\mskip-7.6mu.\,$68 in the dispersion
direction, the effective spatial resolution element is rectangular on the sky,
with an aspect ratio of $\sim 2.3:1$. We adopted the following approach to the
observations, designed to ensure that the slicer geometry with respect to the
sky is unique on each 1200s exposure so that the effective spatial resolution
on the final stacked data cube is close to isotropic. Typically, a total
integration of $\sim 5$ hours is obtained as a sequence of 15 exposures of
1200s, each obtained with the sky position angle (PA) of the instrument
rotated by 10-90 degrees with respect to adjacent exposures. Each rotation of
the instrument field of view is accompanied by a small offset of the telescope
pointing before the guide star is reacquired. In this way, a given sky
position is sampled in 15 different ways by the slicer.

####  3.1.1 KCWI Data Reduction

Each 1200s exposure with KCWI was initially reduced using the data reduction
pipeline (DRP) maintained by the instrument team and available via the Keck
Observatory website333https://github.com/Keck-DataReductionPipelines/KcwiDRP.
The DRP assembles the 2D spectra of all slices into a 3D data cube (with
spaxels of
0${}^{\prime\prime}\mskip-7.6mu.\,$29$\times$0${}^{\prime\prime}\mskip-7.6mu.\,$69,
the native scale) using a suite of procedures that can be customised to suit
particular applications. The procedures include cosmic-ray removal, overscan
subtraction and scattered light subtraction, wavelength calibration, flat-
fielding, sky-subtraction, differential atmospheric refraction (DAR)
correction, and flux-calibration. Wavelength calibration was achieved using
ThAr arc spectra obtained using the internal calibration system during the
afternoon prior to each observing night. Flat-fielding was accomplished using
spectra of the twilight sky at the beginning or end of each night, after
dividing by a b-spline model of the solar spectrum calculated using the
information from all slices. For each frame, the sky background was subtracted
using the sky-modeling feature in the DRP, after which the sky-subtracted
image (still in the 2-D format) is examined in order to mask pixels, in all 24
slices, that contain significant light from sources in the field. The frame
was then used to make a new 2-D sky model using only unmasked pixels, and the
sky-subtracted image is reassembled into a wavelength-calibrated (rebinned to
1 Å per wavelength bin) data cube, at which time a variance cube, an exposure
cube, and a mask cube are also produced.

Next, we removed any remaining low frequency residuals from imperfect sky
background subtraction by forming a median-filtered cube after masking obvious
continuum and extended emission line sources using a running 3D boxcar filter.
The typical dimensions of the filter are 100 Å (100 pixels) in the wavelength
direction, 16 pixels (4${}^{\prime\prime}\mskip-7.6mu.\,$6) along slices, and
1 pixel (0${}^{\prime\prime}\mskip-7.6mu.\,$69) perpendicular to the slices,
with the last ensuring slice-to-slice independence. Minor adjustments to the
filter dimensions were made as needed. If the running boxcar encounters a
large region with too few unmasked pixels to allow a reliable median
determination, then the pixel values in the filtered cube were interpolated
from the nearest adjacent regions for which the median was well-determined.
Finally, the median-filtered cube for each observed frame was subtracted from
the data. We found that this method proved effective for removing scattered
light along slices caused by bright objects within the KCWI field of view.

Because neither Keck II nor KCWI has an atmospheric dispersion corrector, each
cube is corrected for differential atmospheric refraction (DAR) (i.e.,
apparent position of an object as a function of wavelength) using the
elevation and parallactic angle at the midpoint of the exposure and a model of
the atmosphere above Maunakea. Finally, each cube was flux-calibrated using
observations obtained with the same instrument configuration of one or more
spectrophotometric standard stars selected from a list recommended by the KCWI
documentation.

Prior to stacking reduced data cubes of individual exposures covering the same
sky region, they must be aligned spatially and rotated to account for
differences in the PA of the instrument with respect to the sky on different
exposures. To accomplish this, we averaged the DAR-corrected cubes along the
wavelength axis to create pseudo-white-light images, rotated each to the
nominal sky orientation (N up and E left) based on the World Coordinate System
(WCS) recorded in the header, and cross-correlated the results to determine
the relative offsets in RA and Dec, which we found to have a precision of
$\simeq 0.03$-arcsec (root mean square, RMS). The offsets were applied to the
WCS in the header of each cube, and all cubes for a given pointing were then
resampled to a common spatial grid with spatial sampling of
$0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}3\times
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}3$ using a 2D drizzle algorithm in
the Montage package444http://montage.ipac.caltech.edu/, with drizzle factor of
0.7. If the wavelength grid is different among cubes, the spectrum in each
spaxel was resampled to a common grid using cubic spline. Finally, we stacked
the resampled cubes by averaging, weighted by exposure time. A white light
image of the final stacked data cube was used to determine small corrections
to the fiducial RA and Dec to align with other multiwavelength data of the
same region.

###  3.2 High Spatial Resolution Imaging

For galaxies in the mass range of our sample at $z\sim 2.5$, the typical half-
light diameter is $\sim 4$ pkpc, which corresponds to $\simeq
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}5$ (Law et al., 2012c). To obtain
reliable measurements of the orientation of the galaxy’s projected major and
minor axes (see §4), high resolution images are crucial. We gathered existing
data from two sources: space-based optical or near-IR images from the HST/ACS
or HST/WFC3; alternatively, H$\alpha$ maps obtained using the Keck/OSIRIS
integral field spectrometer (Larkin et al. 2006) behind the WMKO laser guide
star adaptive optics (LGSAO) system, which typically provides spatial
resolution of
$0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}11-0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}15$
(Law et al. 2007, 2009, 2012a).

Table 2: Summary of HST Observations

Field | Inst. | Filter | Prog. ID | PI |  $t_{\mathrm{exp}}$ (s)a
---|---|---|---|---|---
DSF2237b | ACS | F606W | 15287 | A. Shapleyb | 6300
Q0100 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q0142 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q0207 | WFC3 | F140W | 12471 | D. Erb | $~{}~{}800$
Q0449 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q0821 | WFC3 | F140W | 12471 | D. Erb | $~{}~{}800$
Q1009 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q1549 | ACS | F606W | 12959 | A. Shapleyd | $12000$
Q1623 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q1700 | ACS | F814W | 10581 | A. Shapleye | $12500$
Q2206 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$
Q2343 | WFC3 | F140W | 14620 | R. Trainor | $~{}5200$
Q2343 | WFC3 | F160W | 11694 | D. Lawc | $~{}8100$

  * a

Defined as the median exposure time across the whole FoV.

  * b

Pahl et al. (2021).

  * c

Law et al. (2012b, c).

  * d

Mostardi et al. (2015).

  * e

Peter et al. (2007).

The HST/WFC3-IR images in this work were obtained using either the F140W or
F160W filters, from programs listed in Table 2, with spatial resolution of
$\sim 0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}16$ and$\sim
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}18$, respectively. The HST/ACS
images were taken in either the F606W or F814W filters, with spatial
resolution of $\sim 0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}09$ (full width
at half maximum; FWHM). In all cases, overlapping exposures were aligned and
combined using DrizzlePac555https://www.stsci.edu/scientific-
community/software/drizzlepac. For the Q2343 field, where galaxies have been
observed with comparably deep observations in two filters, we selected the
image with the smaller estimated uncertainty in the measured position angle
(see §4.2). The Keck/OSIRIS $\rm H\alpha$ maps of six galaxies included in the
sample are presented by Law et al. (2007, 2009, 2012a, 2018); details of the
observations and data reduction can be found in those references.

##  4 Galaxy Azimuthal Angle

###  4.1 Motivation

Figure 4:  A schematic diagram of how the galaxy azimuthal angle ($\phi$) is
defined in this work and how it might be related to the origin and kinematics
of gas in the CGM under common assumptions of a bi-conical outflow with
accretion along the disk plane. Suppose that we are viewing a galaxy projected
on the sky in this diagram, naively, one would expect to see inflow aligning
with the projected galaxy major axis, and outflow aligning with the minor
axis. Impact parameter, $D_{\mathrm{tran}}$ is defined as the projected
distance from the center of the galaxy. The galaxy azimuthal angle, $\phi$ is
defined as the projected angle on the sky with respect to the center of the
galaxy, and starts from the projected galaxy major axis.

The physical state of the CGM is controlled by the competition between
accretion of new material onto the galaxy, and outflows driven by processes
originating at small galactocentric radii. In what one might call the
“classic” picture (see Veilleux et al. 2005; Tumlinson et al. 2017 for review
articles), based on observations of nearby starburst galaxies, inflowing and
outflowing gas occurs preferentially along the major and minor axes,
respectively, as in the schematic diagram in Figure 4. It is well-established
around star-forming galaxies at $z\lesssim 1$ that outflows driven by energy
or momentum from stellar feedback (radiation pressure from massive stars,
supernovae) originating near the galaxy center tend to escape along the
direction that minimizes the thickness of the ambient interstellar medium
(ISM) that would tend to slow or prevent outflows from escaping the galactic
disk. Meanwhile, accretion of cool gas from the intergalactic medium (IGM) is
believed to occur in quasi-collimated filamentary flows that carry significant
angular momentum and thus would tend to approach the inner galaxy in a
direction parallel to the disk plane (Nelson et al., 2019; Péroux et al.,
2020). If feedback processes are sufficiently vigorous, the accretion is also
most likely along directions that do not encounter strong outflows. When
projected onto the plane of the sky, these considerations would tend to
intersect outflowing material along the polar direction (the projected minor
axis) and accreting material when the azimuthal angle (see Figure 4) is closer
to the PA of the projected major axis.

At redshifts $z\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1$, there is
strong support for this general geometric picture from the statistics of the
incidence and strength of rest-UV absorption lines observed in the spectra of
background sources, as a function of the azimuthal angle of the vector
connecting the sightline and the center of the galaxy. As might be expected
from a picture similar to Fig. 4, sightlines with $\phi$ close to the minor
axis intersect gas with a large range of velocities, which kinematically
broadens the observed absorption complexes comprised of many saturated
components (e.g., Bordoloi et al. 2011; Bouché et al. 2012; Kacprzak et al.
2012; Schroetter et al. 2019). For azimuthal angles $\phi$ close to the major
axis, the absorption features are strong due to the high covering fraction and
column density of low-ionization gas near the disk plane, and broadened by the
kinematics of differential rotation. The same picture has been supported by
cosmological simulations where the gas metallicity, radial velocity, and the
amount of outflowing mass all show significant differences along the galaxy
major and minor axes (Nelson et al., 2019; Péroux et al., 2020).

It remains unclear, however, whether this geometric picture should be applied
to galaxies at $z\sim 2$, where a large fraction of galaxies, particularly
those with ${\rm log}(M_{\ast}/M_{\odot})\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10$, appear to be “dispersion
dominated”, where the rotational component of dynamical support ($V_{\rm
rot}$) is significantly smaller than that of apparently random motion
($\sigma$) (Law et al. 2009; Förster Schreiber et al. 2009), and where the
central dynamical mass of the galaxy may be dominated by cold gas rather than
stars – and therefore any disk would be highly unstable. For such galaxies,
there is not always a clear connection between the morphology of starlight and
the principal kinematic axes (e.g., Erb et al. 2004; Law et al. 2012c).
Meanwhile, a WFC3 survey conducted by Law et al. (2012c) for similar galaxies
strongly supports 3D triaxial morphology, instead of inclined disk, to be the
most suitable model to describe these galaxies. The fact that the vast
majority of galaxies with “down the barrel” spectra at $z>2$ have
systematically blueshifted interstellar absorption lines and systematically
redshifted $\rm Ly\alpha$ emission suggests that outflows cannot be confined
to a small range of azimuth (Shapley et al. 2003; Steidel et al. 2010; Jones
et al. 2012 etc.).

The spatial and spectral distribution of the $\rm Ly\alpha$ emission
surrounding galaxies may provide crucial insight into the degree of
axisymmetry and the dominant magnitude and direction of gas flows in the CGM.
In any case, IFU observations of $\rm Ly\alpha$, where both the geometry and
kinematics of the extended emission can be mapped, complement information
available from absorption line studies.

In the analysis below, we define the galaxy azimuthal angle ($\phi$) as the
absolute angular difference between the vector connecting the galaxy centroid
and a sky position at projected angular distance of $\theta_{\rm tran}$ (or
projected physical distance $D_{\rm tran}$) and $\rm PA_{0}$, the PA of the
galaxy major axis measured from the galaxy stellar continuum light, as
illustrated schematically in Figure 4,

$\displaystyle\phi=|\mathrm{PA}-\rm PA_{0}|.$ (1)

The zero point for measurements of position angle is arbitrary, but for
definiteness we measure PA as degrees east (E) of north (N), so that angles
increase in the counter-clockwise direction when N is up and E to the left.
The use of equation 1 to define $\phi$ implies that $0\leq\phi/{\rm deg}\leq
90$, and that $\phi=0$ (90) deg corresponds to the projected major (minor)
axis.

Figure 5:  HST images of the 35 galaxies whose $\rm PA_{0}$ were determined
from Methods i and ii. For each image, shown in the four corners, from top-
left in clockwise direction, are the KBSS identifier, the FWHM of PSF,
redshift, and the instrument and filter that the image was taken with. The
dashed red line, dash-dotted yellow line, and the solid white line indicate
the direction of $\rm PA_{0}$ measured from GALFIT, pixel moment, and the
average between the two. Figure 6:  Same as Figure 5, except that the galaxies
do not have a clear central SB peak. Therefore, their $\rm PA_{0}$ were
determined only from the pixel moment (white solid line).  Figure 7:  Similar
to Figure 5, showing the $\rm PA_{0}$ of galaxies with both Keck/OSIRIS
H$\alpha$ maps from Law et al. (2009) and HST continuum images. For each
panel, the left image shows the OSIRIS H$\alpha$ map, in which the cyan dashed
line is $\rm PA_{0}$ measured from this map using second pixel moment. The
right image shows the HST image, where the red dashed line and the yellow
dash-dotted line show the $\rm PA_{0}$ measured from GALFIT fitting and second
moment from this image when present. The white solid lines are the final $\rm
PA_{0}$ determined for each galaxy by averaging the OSIRIS and HST
measurements777Since the HST WFC3-IR/F160W image of Q1623-BX453 is unresolved,
its $\rm PA_{0}$ is only derived from the OSIRIS H$\alpha$ map..  Figure 8:
Left: HST WFC3-F160W image of Q2343-BX442. Right: H$\alpha$ velocity map of
Q2343-BX442 by Law et al. (2012a). The $\rm PA_{0}$ (white solid line) is
defined to be perpendicular to its rotational axis.

###  4.2 Methods

In order to obtain reliable measurements of $\phi$, it is important to measure
$\rm PA_{0}$ accurately and consistently. We used up to four different methods
to determine $\rm PA_{0}$ for each galaxy. The choice of method depends on the
information available; these are briefly summarised as follows:

  * •

(i) Sérsic profile fitting of HST images: In this method, we fit a 2D Sérsic
profile (Sérsic, 1963) to the host galaxy by using GALFIT (Peng et al., 2002,
2010), and determined $\rm PA_{0}$ from the best-fit model parameters. The
point-spread function (PSF) was measured by selecting stellar sources over the
full HST pointing using the star classifier in SExtractor, which calculates a
“stellarity index”for each object based on a neural network. We then examined
sources with the highest 3% stellarity indices by eye, and normalised and
stacked them to form an empirical PSF. Our fiducial model consists of a 2D
elliptical Sérsic profile convolved with the PSF. We also included the first-
order Fourier mode to handle the asymmetric morphology in most cases. However,
over- or under-fitting can cause failure of convergence or unreasonably large
fitting errors and residuals. In most cases, the cause of the failure and the
required adjustment are obvious in the original galaxy image and the model
residual. For example, if the residual reveals an additional source, we would
add an additional Sérsic component or a simple scaled PSF to the model,
depending on the size of the additional source. Meanwhile, if the primary
source shows a triangular morphology, we would add the third-order Fourier
mode888The second-order Fourier mode is degenerate with the ellipticity.
associated with the Sérsic profile. However, in certain cases, obtaining a
successful fit requires experimenting with the model by adding or removing
certain degrees of freedom. The rule of thumb is that we add or remove degrees
of freedom one at a time, and adopt the adjustment if it makes the fit
converge, or significantly diminishes the reduced $\chi^{2}$ and the fitted
error of $\rm PA_{0}$. In the end, 23 galaxies were fit with the fiducial
model. Nine galaxies (Q0100-C7, Q0821-MD36, Q1009-BX222, Q1009-D15,
Q1623-BX453-CS3, Q1700-BX729-CS9, Q2343-BX418, Q2343-BX418-CS8, Q2343-BX660)
were fit without the Fourier modes. Five galaxies (Q1009-BX218, Q1700-BX490,
Q1700-BX710, Q1700-BX729, Q2343-BX513-CS7) were fit with additional sources.
Three galaxies (Q2343-BX391, Q2343-BX587-CS3, Q2343-BX587-CS4) were fit with
third-order Fourier modes. Galaxies with unsuccessful fits require using
alternative methods, detailed below. Successful Sérsic fits were obtained for
40 of 59 galaxies, shown in Figures 5 and 7; galaxies with successful
applications of this method tend to be isolated and to have a dominant central
high-surface-brightness component.

  * •

(ii) Second moment of pixel intensity on HST images: This method calculates
the flux-weighted pixel PA arithmetically derived from the second moment of
the pixel intensity,

$\displaystyle\tan(2{\rm PA_{0}})=\frac{2\langle xy\rangle}{\langle
x^{2}\rangle-\langle y^{2}\rangle},$ (2)

where $x$ and $y$ are the pixel positions relative to the center of the galaxy
in the x- and y-directions, and “$\langle\textrm{...}\rangle$” indicates the
arithmetic mean of the pixel coordinates weighted by pixel flux. Galaxy
centers were defined as the point where $\langle x\rangle=0$ and $\langle
y\rangle=0$. Especially for the galaxies that are morphologically complex, we
found that the $\rm PA_{0}$ measurements using this method are sensitive to
the surface brightness threshold used to define the outer isophotes, and to
the spatial resolution of the HST image. Therefore, all ACS images (which have
higher spatial resolution than those taken with WFC3-IR) were convolved with a
2D Gaussian kernel to match the PSF to those of the WFC3-IR/F160W images. We
tested various surface brightness thresholds, and found that a threshold of
50% of the peak SB after convolution provides the most consistent measurements
between the $\rm PA_{0}$ measured using (i) and (ii) has an RMS of 10.8
degrees.

  * •

(iii) Second moment of pixel intensity of OSIRIS H$\alpha$ maps: This method
uses the same algorithm as in (ii), applied to Keck/OSIRIS H$\alpha$ maps
rather than HST continuum images. The SB threshold follows Law et al. (2009).
There are 6 galaxies whose major axes were determined using this method,
including one (Q1623-BX453) whose F160W image is unresolved. For the remaining
5 galaxies, the RMS between $\rm PA_{0}$ measured from the HST images and this
method is $\sim 15$ degrees. All 6 galaxies are shown in Figure 7.

  * •

(iv) H$\alpha$ Kinematics: As the only galaxy in this sample that demonstrates
not only rotational kinematics from H$\alpha$ emission, but also clear disk
morphology (Law et al., 2012a), the galaxy major axis of Q2343-BX442 is
defined to be perpendicular to its rotational axis (Figure 8). Because of its
complex morphology, we did not apply method (i) to this galaxy. Method (ii) is
likely dominated by the inner spiral structure and leads to a $\rm PA_{0}$
that is $\sim 90$ deg apart from that determined from kinematics, while method
(iii) is consistent with this method within 15 deg.

Table 1 lists the adopted measurement of $\rm PA_{0}$ for each galaxy in the
sample, as well as the methods used to determine it. In cases where multiple
methods were applied, we used the average $\rm PA_{0}$ of the OSIRIS and HST
measurements, where the latter value is an average of the results obtained
using methods (i) and (ii). This way, more weight is given to method (iii)
results when available, since it utilises entirely independent data from a
different instrument. The robustness of $\rm PA_{0}$ measurements is discussed
in the following two subsections.

###  4.3 Relationship between the Kinematic and Morphological Major Axes

An important issue for the interpretation of morphological measurement of $\rm
PA_{0}$ is the extent to which it is likely to be a proxy for the kinematic
major axis. Two of the 17 galaxies in Figure 6 (Q2343-BX389 and Q2343-BX610)
were studied as part of the SINS IFU survey by Förster Schreiber et al. (2018,
hereafter FS18), and were subsequently re-observed with the benefit of
adaptive optics by FS18; the latter found that both have clear rotational
signatures and that the inferred difference between their morphological major
axis $\rm PA_{0}$ and the kinematic major axis $\rm PA_{\rm kin}$ are
$\Delta\rm PA\equiv|\rm PA_{0}-\rm PA_{\rm kin}|=2\pm 5$ deg and $27\pm 10$
deg for Q2343-BX389 and Q2343-BX610, respectively. From their full sample of
38 galaxies observed at AO resolution, FS18 found $\langle\Delta{\rm
PA}\rangle=23$ deg (mean), with median $\Delta{\rm PA}_{\rm med}=13$ deg.

Among the 6 galaxies which also have H$\alpha$ velocity maps from Law et al.
(2009) (used for method (iii) above), two (Q0449-BX93 and Q1623-BX453) show no
clear rotational signature, so that $\rm PA_{{\rm kin}}$ is indeterminate.
Small amounts of velocity shear were observed for Q2343-BX513 and Q2343-BX660,
both having $\rm PA_{\rm kin}$ consistent with $\rm PA_{0}$999Q2343-BX513 was
also observed by FS18, who found $\rm PA_{\rm kin}=-35$ deg and $\Delta{\rm
PA}=40\pm 12$ deg; our adopted $\rm PA_{0}=-9$ deg differs by 26 deg from
their $\rm PA_{\rm kin}$ value. However, for Q2343-BX418, the implied $\rm
PA_{\rm kin}$ is nearly perpendicular to its morphology-based $\rm PA_{0}$.
For Q1700-BX490, the kinematic structure is complicated by the presence of two
distinct components: the brighter, western component appears to have $\rm
PA_{\rm kin}\simeq 20$ deg, which would be consistent with $\rm PA_{0}$
measured from its $\rm H\alpha$ intensity map; however, if the eastern
component, which has a slightly blue-shifted velocity of $\sim 100\rm
km~{}s^{-1}\,$, is included as part of the same galaxy, we found $\rm
PA_{0}=86$ deg.

As summarised in Table 1 and Figures 5-7, most of the galaxy sample has $\rm
PA_{0}$ measured using 2 or more of the methods described above, and generally
the morphologically-determined major axis $\rm PA_{0}$ agree with one another
to within $\sim 10$ deg. We caution that no high-spatial-resolution kinematic
information is available for most of the sample; based on the subset of 9 that
do have such measurements, approximately two-thirds show reasonable agreement
between $\rm PA_{0}$ and $\rm PA_{\rm kin}$.

###  4.4 Distribution and robustness of $\rm PA_{0}$ measurements

Figure 9:  Comparison of $\rm PA_{0}$ values measured using different methods.
Blue points compare methods (i) and (ii), while orange points compare methods
(i) and (iii). The red shaded regions indicates where the absolute differences
between the two measurements are greater than 90 degrees, arithmetically
forbidden because of the rotational symmetry. The overall RMS $=11.4$ deg.

To estimate the systematic uncertainty of the $\rm PA_{0}$ measurements, we
compare values measured using different methods for the same objects in Figure
9. Between methods (i) and (ii), which are both based on HST images, the
measured $\rm PA_{0}$ for 40 galaxies are well-centred along the 1-to-1 ratio,
with RMS $\simeq 10.8$ deg. Values of $\rm PA0$ measured using method (iii)
are based on spectral line maps from an IFU rather than continuum light in a
direct image, so that the 5 $\rm PA_{0}$ values measured using methods (i) and
(iii) exhibit larger scatter, with RMS$\simeq 15.6$ deg relative to a 1:1
ratio. Therefore, we conclude that the uncertainty in final $\rm PA_{0}$
measurements is $\lesssim 15$ deg.

Figure 10:  The kernel density estimate (KDE; blue shaded region) of $\rm
PA_{0}$ for the galaxy sample, normalised so that a uniform distribution would
have a constant $\mathrm{KDE}=1$. The KDE was constructed using Gaussian
kernels of fixed $\sigma=10^{\circ}$, corresponding to an opening angle
represented by the black block at the top-right. The orange solid lines
indicate the values of $\rm PA_{0}$ for the individual galaxies. There is an
apparent excess in the KDE of galaxies with $\rm PA_{0}\simeq 10-40^{\circ}$,
which we attribute to sample variance.

Figure 10 shows the normalised Kernel Density Estimator (KDE) and the
individual measurements of $\rm PA_{0}$. There is an apparent excess in the
occurrence rate of values between $\rm PA_{0}\sim 10-40^{\circ}$. To evaluate
its possible significance, we conducted 1000 Monte-Carlo realisations of a
sample of 58 galaxies with randomly assigned $\rm PA_{0}$; we find that there
is a $\simeq 5$% probability that a similar excess is caused by chance, thus
is not statistically significant, and is consistent with that expected from
sample variance.

Furthermore, because the HST and OSIRIS images used to measure $\rm PA_{0}$
were rotated to the nominal North up and East left orientation prior to
measurement, we tested whether significant bias might result from the choice
of pixel grid by re-sampling all images with pixel grids oriented in five
different directions. We found that the values measured for $\rm PA_{0}$ were
consistent to within 10 deg (RMS).

(a)  (b)

Figure 11:  (a) Histogram of the relative contribution of measurements made at
different slicer azimuthal angles ($\phi_{\rm slicer}$) in units of total
exposure time. The distribution of $\phi_{\mathrm{Slicer}}$ is relatively
uniform, with a small excess near $\phi_{\mathrm{Slicer}}\sim 10^{\circ}$. (b)
Stacks of the galaxy continuum images for which the major and minor axes of
each galaxy were aligned with the X and Y axes prior to averaging. Each panel
shows (i) the pseudo-narrow-band image (rest frame $1230\pm 6$ Å) of the KCWI
galaxy continuum, (ii) the stacked HST continuum image, after aligning the
principal axes in the same way, (iii) the HST image convolved with a Gaussian
kernel of $\mathrm{FWHM}=1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}02$ to
match the KCWI continuum, (iv) the residual between the KCWI continuum and the
HST image convolved with the KCWI PSF, (v) a 2D circular Gaussian profile with
$\mathrm{FWHM}=1.21~{}\mathrm{arcsec}$ as the best symmetric Gaussian profile
from a direct fit of the KCWI continuum image, and (vi) the residual between
the KCWI continuum and the model in (v) isotropic Gaussian profile. The colour
map of (i), (ii), (iii), and (v) is in log scale, with linear red contours in
the dcrement of 0.17. The colour map of (iv) and (vi) is in linear scale. The
residual map in panel (vi) shows a clear dipole residual in the Y (minor axis)
direction that is not present in (iv). The RMS values in panels (iv) and (vi)
were calculated within $|\Delta x|<1~{}\mathrm{arcsec}$ and $|\Delta
y|<1~{}\mathrm{arcsec}$ to reflect the dipole residual. The “boxiness” of the
KCWI stack is likely due to the undersampling of KCWI in the spatial
direction. Taken together, (b) demonstrates that the KCWI PSF is axisymmetric
(with ${\rm FWHM}=1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}02$) , and that
the KCWI continuum image is capable of distinguishing the galaxy major and
minor axes.

The possible systematic bias introduced by an uneven $\rm PA_{0}$ distribution
is mitigated by our observational strategy of rotating the KCWI instrument PA
between individual exposures. We define $\rm PA_{\mathrm{Slicer}}$ as the
position angle along the slices for each 1200 s exposure, and

$\displaystyle\phi_{\mathrm{slicer}}=|\rm PA_{\mathrm{Slicer}}-\rm PA_{0}|.$
(3)

Figure 11a shows the distribution of $\phi_{\mathrm{slicer}}$ in units of
total exposure time. There is a slight tendency for the observations to align
with the galaxy major axis (i.e., $\phi_{\rm slicer}\sim 0$) that results from
our usual practice of beginning a sequence of exposures of a given pointing
with one at $\rm PA_{\rm slicer}=0$ deg and the aforementioned excess of
galaxies with $\rm PA_{0}\sim 20^{\circ}$.

In order to characterise the PSF of our final KCWI data cubes, and to show
that it is effectively axisymmetric, Figure 11b shows various composite images
of the galaxy sample. Prior to forming the stack, each individual data cube
was rotated to align the galaxy major (minor) axis with the X (Y) coordinate
of the stacked image. The KCWI stacked galaxy continuum image [panel (i)] was
made by integrating each aligned data cube along the wavelength axis over the
range $1224\leq\lambda_{0}/{\rm\AA}\leq 1236$ in the galaxy rest frame (i.e.,
$2000<\Delta v/\rm km~{}s^{-1}\,\leq 5000$). This integration window was
chosen to be representative of the UV continuum near $\rm Ly\alpha$ without
including the $\rm Ly\alpha$ line itself, and to be unaffected by $\rm
Ly\alpha$ absorption from the IGM. The center of the galaxy is determined with
high confidence (see §5.1) by fitting a 2D Gaussian function to the individual
galaxy continuum image. A similar approach – aligning the principal axes in
the high-resolution HST images prior to stacking – was used to produce the HST
stacked continuum image shown in panel (ii), except that the centers in the
HST images were measured using the first moment of pixel intensity. Both
stacks were conducted in units of observed surface brightness. The FWHM of the
stacked HST image along the major and minor axes are
$0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}55$ and
$0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}35$ respectively. The stacked KCWI
continuum image is well reproduced by the convolution of the $\rm
PA_{0}$-aligned stacked HST image [panel (ii) of Fig. 11b] with an
axisymmetric 2-D Gaussian profile with
$\mathrm{FWHM}=1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}02$ [see panels (iii)
and (iv) of Fig. 11b].

Comparing the convolution of HST and KCWI [panel (iii)] with the best direct
fit of a symmetric Gaussian profile to the KCWI continuum image ($\rm
FWHM=1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}21$) [panel (v)] show that even
at the $\simeq 1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}02$ resolution of
KCWI one can clearly distinguish the major axis elongation. The residual map
assuming a symmetric Gaussian profile [panel (vi)] shows a clear dipole
residual compared to panel (iv). Thus, Figure 11 shows that (1) the PSF of the
KCWI cubes is axisymmetric and thus has not introduced a bias to the azimuthal
light distribution measurements and (2) the spatial resolution is sufficient
to recognize non-axisymmetry even on sub-arcsec angular scales of the
continuum light.

##  5 Analyses

###  5.1 $\rm Ly\alpha$ Spatial Profile

Figure 12:  Stacked images of the galaxy continuum (Left) and the continuum-
subtracted $\rm Ly\alpha$ emission (Right) with the X- and Y-axes aligned with
the galaxy major and minor axes, respectively. The color coding is on a log
scale, while the contours are linear. The intensity scales have been
normalized to have the same peak surface brightness intensity at the center.
The $\rm Ly\alpha$ emission is more extended than the continuum emission.
Figure 13:  Top panel: The average $\rm Ly\alpha$ surface brightness profile
of the continuum-subtracted composite $\rm Ly\alpha$ image shown the righthand
panel of Figure 12. Red points represent the median surface brightness
evaluated over all azimuthal angles ($0^{\circ}<\phi\leq 90^{\circ}$) as a
function of projected distance from the galaxy center. Orange and purple
curves show the profiles evaluated over $0^{\circ}<\phi\leq 45^{\circ}$ (
major axis) and $45^{\circ}<\phi<90^{\circ}$ (minor axis) azimuthal angles.
Dashed cyan curve shows the best-fit profile of the two-component exponential
model. Dotted cyan curves show the two component separately. The grey profile
shows the normalised continuum for comparison. Bottom: The residual surface
brightness profile formed by subtracting the all-azimuth average from the
major and minor axis profiles. The residuals are consistent with zero aside
from a marginally-significant difference at $\theta_{\rm
tran}<1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}0$, where the the $\rm
Ly\alpha$ emission is slightly stronger along the major axis. Unless otherwise
noted, the conversion between $\theta_{\mathrm{tran}}$ and $D_{\mathrm{tran}}$
for this and later figures assumes a redshift of 2.3, the median redshift of
the sample.

To study the dependence of the $\rm Ly\alpha$ emission profile on galaxy
azimuthal angle, we first analyse the $\rm Ly\alpha$ surface brightness (SB)
profile as a function of the impact parameter (or transverse distance,
$D_{\mathrm{tran}}$). Figure 12 compares the stacked continuum and continuum-
subtracted narrow-band $\rm Ly\alpha$ emission, composed in the same way as in
Figure 11. The integration window of the $\rm Ly\alpha$ image is $-700<\Delta
v/(\mathrm{kms}^{-1})\leq 1000$ (1213 Å – 1220 Å) to include most of the $\rm
Ly\alpha$ emission (as shown later in Figure 14). Continuum subtraction
throughout this work was done spaxel-by-spaxel in the data cube of each target
galaxy by subtracting the average flux density in two windows flanking the
position of $\rm Ly\alpha$, with $2000<|c\Delta v_{\rm
sys}/\mathrm{km~{}s}^{-1}|<5000$, where $\Delta v_{\rm sys}$ is the velocity
separation relative to rest-frame $\rm Ly\alpha$ (i.e., two windows each of
width $\simeq 12$ Å in the rest frame, [1195-1207] and [1224-1236] Å. Similar
to §4.4, the center of the galaxy in the KCWI data was determined by fitting a
2D Gaussian profile to the KCWI continuum image. The fitting box was chosen by
eye to include all of the signal from the galaxy, but excluding nearby
contamination. Despite the arbitrary box size, we found that the derived
centroid is very robust: varying the box size by 4 pixels
($1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}2$), the fit result does not
change by more than $0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}01$. The
typical fitting error propagated from the reduced $\chi^{2}$ is also $\sim
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}01$ (median), i.e., much smaller
than the seeing disk.

A 2D Gaussian fit to the profiles finds that the FWHM values are
$1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}279\pm
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}003(\textrm{major axis})\times
1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}182\pm
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}003(\textrm{minor axis})$ (or a
$\sim 8\%$ difference) for the continuum emission and
$1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}689\pm
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}005\times
1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}705\pm
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}005$ ($<1\%$ difference between
major and minor axes) for the $\rm Ly\alpha$ emission. Therefore, the $\rm
Ly\alpha$ emission in the stacked image is both more symmetric and more
spatially extended than the continuum emission.

Figure 13 shows the median $\rm Ly\alpha$ SB as a function of
$D_{\mathrm{tran}}$ (red). Each point represents the median SB of a bin of
pixels with $\Delta D_{\mathrm{tran}}=0.1~{}\mathrm{pkpc}$. The $\rm Ly\alpha$
surface brightness profile falls off much more slowly than the continuum
(grey). Following Wisotzki et al. (2016), we fit the $\rm Ly\alpha$ SB profile
with a two-component model – a compact “core” component and an extended “halo”
component. Both components are exponential profiles convolved with the KCWI
PSF with the amplitudes, exponential radii, and a uniform background term as
free parameters. Further details of the model fitting will be described in a
future work (R. Trainor & N. Lamb, in prep.). The best-fit exponential radii
$r_{\mathrm{exp}}=3.71^{+0.06}_{-0.04}$ pkpc and $15.6^{+0.5}_{-0.4}$ pkpc.
The $r_{\mathrm{exp}}$ of the halo component is close to Steidel et al. (2011)
(for KBSS galaxies observed with narrow-band $\rm Ly\alpha$ imaging), which
found the median-stacked $\rm Ly\alpha$ profile has $r_{\mathrm{exp}}=17.5$
pkpc, but slightly more extended than Wisotzki et al. (2018) for SF galaxies
at $z>3$ (see also Matsuda et al. 2012; Momose et al. 2014; Leclercq et al.
2017). Dividing the SB profiles into two subsamples with
$0^{\circ}\leq\phi<45^{\circ}$ (purple) and $45^{\circ}\leq\phi\leq
90^{\circ}$ (orange) that represent the galaxy major and major axes
respectively, one can see that the resulting profiles are consistent with one
another to within 1$\sigma$, or within $\lesssim 2\times
10^{-19}\mathrm{~{}erg~{}s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2}$. The
possible exception is at the smallest projected distances
($D_{\mathrm{tran}}<1~{}\mathrm{arcsec}$, or $\lesssim 8$ pkpc), where the
$\rm Ly\alpha$ emission is marginally enhanced; if real, the difference in
profiles (the asymmetry) represents $<2\%$ of the total $\rm Ly\alpha$ flux.
Thus, the composite $\rm Ly\alpha$ intensity is remarkably symmetric,
suggesting an overall lack of a strong statistical connection between the
morphology of the starlight and that of the extended $\rm Ly\alpha$ emission
surrounding individual star-forming galaxies at $z\sim 2-3$.

###  5.2 Cylindrical Projection of 2D Spectra

The similarity of the $\rm Ly\alpha$ surface brightness profile along galaxy
major and minor axes suggests that extended $\rm Ly\alpha$ emission depends
little on the galaxy orientation. However, the KCWI data cubes allow for
potentially finer discrimination through examination of both the surface
brightness and kinematics of $\rm Ly\alpha$ emission as a function of
projected galactocentric distance. To facilitate such comparison, we introduce
“cylindrical projections” of 2D $\rm Ly\alpha$ emission. The basic idea behind
cylindrical projection, illustrated in Figure 14, is to provide an intuitive
visualisation of spatial and spectral information simultaneously.

Figure 14:  Left: A schematic diagram explaining cylindrically projected 2D
(CP2D) spectra. Spaxels with similar $D_{\mathrm{tran}}$ are averaged to
create the emission map in $D_{\mathrm{tran}}$-$\Delta v$ space. Right: The
composite CP2D spectra of the continuum-subtracted $\rm Ly\alpha$ emission
line map averaged over all 59 galaxies, at all azimuthal angles ($\phi$). The
colour-coding of the $\rm Ly\alpha$ surface intensity is on a log scale to
show the full extent of the emission, whereas the contours are spaced linearly
and marked as white lines in the colourbar. The stack was formed by shifting
the wavelengths of each galaxy data cube to the rest frame, leaving the
surface brightness in observed units. The black ellipse at the top right shows
the effective resolution of the stacked maps, with principal axes
corresponding to the spectral resolution FWHM and the spatial resolution FWHM
(see §4.4). Pixels with $\theta_{\mathrm{tran}}<0.1~{}\mathrm{arcsec}$ have
been omitted to suppress artifacts owing to the singularity in the cylindrical
projection.

Compared to the standard 2D spectrum one obtains from slit spectroscopy, the
cylindrical 2D spectrum replaces the 1D spatial axis (i.e., distance along a
slit) with projected distance, by averaging spaxels in bins of
$D_{\mathrm{tran}}$ or, equivalently, $\theta_{\rm tran}$. When projected as
in the righthand panel of Figure 14, it can also be viewed as the $\rm
Ly\alpha$ spectrum at every projected radial distance (averaged, in this case,
over all azimuthal angles) or as the average radial profile at each slice of
wavelength or velocity.

Figure 14 shows the stacked cylindrical 2D spectrum formed by averaging the
continuum subtracted data cubes at wavelengths near rest-frame $\rm Ly\alpha$
for all 59 galaxies in Table 1. This composite cylindrical 2D spectrum,
analogous to a “down-the-barrel” $\rm Ly\alpha$ spectrum in 1D, but evaluated
as a function of galactocentric distance, shows that the $\rm Ly\alpha$
emission line is comprised of distinct redshifted and blueshifted components
extending to $\pm 1000$ km s${}^{-1}\,$ with respect to $v_{\rm sys}=0$, with
a minimum close to $v_{\rm sys}=0$. The vast majority of individual galaxies,
and therefore also the average in the stacked profile, has $F_{\rm
Ly\alpha}(\mathrm{blue})/F_{\rm Ly\alpha}(\mathrm{red})\simeq 0.3$, and is
thus “red peak dominated”. The two-component spectral morphology extends to at
least $\theta_{\rm tran}\simeq 3~{}\mathrm{arcsec}$ or $D_{\rm tran}\simeq
25~{}\mathrm{pkpc}$. This overall spectral morphology is most readily
explained as $\rm Ly\alpha$ photons being resonantly scattered by outflowing
material, whereby redshifted photons are scattered from the receding
(opposite) side are more likely to escape in the observer’s direction than
blue-shifted photons (e.g. Pettini et al., 2001; Steidel et al., 2010;
Dijkstra, 2014).

As $D_{\mathrm{tran}}$ increases and the $\rm Ly\alpha$ SB decreases
exponentially, the two $\rm Ly\alpha$ peaks become less distinct and merge
into a symmetric “halo” centered on $\Delta v=0$. The vast majority of the
$\rm Ly\alpha$ emission is within $-700<\Delta v/\mathrm{km~{}s}^{-1}<1000$.
However, we caution that the apparent blue edge at $\Delta
v\sim-700~{}\mathrm{km~{}s}^{-1}$ of $\rm Ly\alpha$ emission in this composite
2D spectrum is likely caused by continuum over-subtraction resulting from the
relatively simple technique that we used in this work. The continuum
subtraction assumed a linear interpolation of the continuum spectrum
underneath the $\rm Ly\alpha$ emission (see §5.1), which tends to over-
estimate the continuum flux blueward of the systemic redshift of the galaxy
due to intrinsic $\rm Ly\alpha$ absorption in the stellar spectrum and
residual effects of the often-strong $\rm Ly\alpha$ absorption damping wings
on which $\rm Ly\alpha$ emission is superposed. To improve on this would
require a more sophisticated continuum-subtraction method in the inner $\simeq
1\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}0$ of the galaxy profile; however,
since most of the remainder of this work will involve comparison of 2D
cylindrical projections with one another, the imperfections in continuum
subtraction at small $\theta_{\rm tran}$ are unlikely to affect the results.

###  5.3 Dependence on azimuthal angle of cylindrically projected 2D (CP2D)
spectra

To investigate how $\rm Ly\alpha$ emission depends on the galaxy azimuthal
angle, we split each CP2D spectrum of individual galaxies averaged over two
independent bins of azimuthal angle ($\phi$) with respect to the galaxy’s
major axis: $0^{\circ}\leq\phi<45^{\circ}$ (“major axis”) and
$45^{\circ}\leq\phi\leq 90^{\circ}$ (“minor axis”), as in §5.1 and Figure 13.
The CP2D stacks covering these azimuth ranges were combined separately to form
CP2D composites that we refer to as “Major Axis” and “Minor Axis”. To reveal
subtle differences in surface brightness and/or velocity along these two
directions, we subtracted one from the other – Figure 15 shows the result.

Figure 15:  Left: The stacked CP2D spectra along the galaxy major
($0^{\circ}\leq\phi<45^{\circ}$; top) and minor ($45^{\circ}\leq\phi\leq
90^{\circ}$; bottom) axes. Both the colour-coding and the contours are on
linear scales. Right: The residual CP2D maps: the top panel shows the
difference between the Major axis and Minor axis maps, in the same units of
surface intensity as in the lefthand panels, where blue colours indicate
regions with excess $\rm Ly\alpha$ surface intensity along the Major axis;
orange colours indicate regions where $\rm Ly\alpha$ is brighter in the Minor
axis map. The bottom panel shows the same residual map in units of the local
noise level. The most prominent feature is excess $\rm Ly\alpha$ emission
along the galaxy major axis relative to that along the minor axis), at $\Delta
v\sim+300~{}\mathrm{kms}^{-1}$, extending to $\theta_{\rm tran}\sim
2\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}0$ or $D_{\rm tran}\sim 15$ pkpc.

The difference between the CP2D spectra along the major and minor axes (top
right of Figure 15) shows excess emission along the galaxy major axis at
$\Delta v\simeq+300~{}\mathrm{kms}^{-1}$ – consistent with the velocity of the
peak of the redshifted component in the full composite CP2D spectrum – within
$\theta_{\rm tran}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2$${}^{\prime\prime}~{}$
($D_{\mathrm{tran}}\lesssim 15~{}\mathrm{pkpc}$). The $\rm Ly\alpha$ flux of
this asymmetric component of $\rm Ly\alpha$ amounts to $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2$% of the total, and has a peak
intensity $\simeq 5$% that of the peak of the redshifted $\rm Ly\alpha$
component shown in Fig. 13.

The significance map in the bottom-right panel of Figure 15 is based on the
standard deviation of 100 independent mock CP2D stacks, each made by assigning
random $\rm PA_{0}$ to the galaxies in our sample before combining. The mock
stacks were then used to produce a 2D map of the RMS residuals evaluated in
the same way as the observed data. Considering the effective resolution, the
overall significance (compared to the standard deviation) of the most
prominent feature in the top-right panel of Figure 15 is $\simeq 2-2.5\sigma$
per resolution element. Thus, while the residual (excess) feature may be
marginally significant statistically, the level of asymmetry relative to the
total $\rm Ly\alpha$ flux is in fact very small.

###  5.4 The robustness of residual $\rm Ly\alpha$ aysmmetry

Despite the marginally significant detection of the excess $\rm Ly\alpha$
emission along galaxy major axes, its robustness is subject to scrutiny. In
particular, we would like to determine whether the apparent detection is
typical of the population or is caused by a few outlier objects having very
asymmetric $\rm Ly\alpha$ as a function of azimuthal angle.

Figure 16:  Distribution of the difference in $\rm Ly\alpha$ flux integrated
over velocity and angular distance in the bins of azimuthal angle
corresponding to “major” and “minor” axes. Positive (negative) values indicate
that $\rm Ly\alpha$ emission is stronger along the major (minor) axis. The
integration is conducted within $0<\Delta v/(\mathrm{km~{}s}^{-1})<500$ and
$\theta_{\mathrm{tran}}\leq 2~{}\mathrm{arcsec}$ (top) and $0<\Delta
v/(\mathrm{km~{}s}^{-1})<1000$ and $\theta_{\mathrm{tran}}\leq
3~{}\mathrm{arcsec}$ (bottom). There are two outliers in the first integration
(top panel), while one remains in the second (bottom panel).

Figure 16 shows histograms of the difference in integrated $\rm Ly\alpha$ flux
between the major and minor axis bins of azimuthal angle. We calculated the
differences integrated over two different ranges of $\Delta v$ and
$\theta_{\rm tran}$; (1) the range where the excess shown in Figure 15 is most
prominent, $\theta_{\rm tran}\leq 2$${}^{\prime\prime}~{}$ and $0\leq(\Delta
v)/\rm km~{}s^{-1}\,\leq 500$, shown in the top panel, and (2) the range which
encapsulates most of the redshifted component of $\rm Ly\alpha$ $\theta_{\rm
tran}\leq 3$${}^{\prime\prime}~{}$ and $0\leq(\Delta v)/\rm km~{}s^{-1}\,\leq
1000$, shown in the bottom panel. Two galaxies – Q0142-BX165 and Q2343-BX418 –
are clearly outliers in (1), while only Q0142-BX165 stands out in (2). The
distribution of $\Delta F_{\rm Ly\alpha}$ for the other 56 galaxies in the
sample is relatively symmetric around $\Delta F_{\mathrm{Ly}\alpha}=0$.

Figure 17:  Same as Figure 15, but without Q0142-BX165, which is the strongest
outlier in terms of excess $\rm Ly\alpha$ emission along the galaxy major
axis.  Figure 18:  Same as Figure 15, but without Q0142-BX165 and Q2343-BX418,
the two most significant outliers in the top panel of Figure 16. Significant
excess emission that is larger than a resolution element for the redshifted
peak no longer exists.

Figures 17 and 18 show the stacked $\rm Ly\alpha$ profiles as in Figure 15,
but with the strongest outliers removed from the stack. After removing both
outliers (Figure 18), the excess $\rm Ly\alpha$ emission along the galaxy
major axis at $\Delta v\simeq 300~{}\mathrm{km~{}s}^{-1}$ becomes consistent
with noise. Although Q2343-BX418 is not an extreme outlier in terms of the
overall $\rm Ly\alpha$ asymmetry of the integrated redshifted component of
$\rm Ly\alpha$ emission (bottom panel of Fig. 16), when only Q0142-BX165 is
removed from the stack (Fig. 17) the composite 2D spectra still show obvious
excess $\rm Ly\alpha$ emission along the galaxy major axis, albeit with
slightly reduced significance. Meanwhile, when both outliers are removed from
the stack (Figure 18), one can see an emerging excess $\rm Ly\alpha$ emission
along the minor axis, for the blue peak, with $-700\lesssim\Delta
v/(\mathrm{km~{}s}^{-1})\lesssim-200$ and $\theta_{\rm tran}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}5$
with integrated significance $\sim 2\sigma$. The flux of the excess
blueshifted emission comprises $\sim 10$% of the total blueshifted $\rm
Ly\alpha$ flux, with a peak amplitude $\sim 1$% of the peak $\rm Ly\alpha$
intensity (i.e., the redshifted peak).

We conducted an analysis on the blueshifted emission similar to that done for
the redshifted asymmetry, with results summarised in Figure 19. There is no
obvious outlier in the difference in integrated $\rm Ly\alpha$ flux between
the major and minor axis azimuth bins except for Q0142-BX165, for which the
excess again favors the major axis (i.e., it is in the direction opposite to
the apparent blueshifted asymmetry identified in Figure 18). We also
consecutively removed from the $\rm Ly\alpha$ stack galaxies with extreme
excess emission along the minor axis, and found no sudden and significant
changes in the composite spectra. Evidently, the blueshifted excess along the
minor axis, while of about the same significance as the redshifted excess in
the major axis direction, is a general property of the full sample rather than
a result of a small number of outliers.

Figure 19:  Same as Figure 16, but with a different velocity range of
$-700<\Delta v/(\mathrm{km~{}s}^{-1})\leq-200$ and $\theta_{\mathrm{tran}}\leq
2.5~{}\mathrm{arcsec}$ that focuses on the blueshifted component of $\rm
Ly\alpha$ emission. No individual galaxy is an extreme outlier in terms of
excess blueshifted $\rm Ly\alpha$ along the minor axis.

In summary, we found excess emission along the galaxy major axis for the
redshifted component of $\rm Ly\alpha$ near $\Delta v\simeq
300~{}\mathrm{km~{}s}^{-1}$. However, this particular excess emission appears
to be caused by galaxy outliers with extreme emission along the major axis.
After removing them from the composite CP2D spectra we found excess emission
along the galaxy minor axis for the blue peak within $-700\lesssim\Delta
v/(\mathrm{km~{}s}^{-1})\lesssim-200$ that is not apparently affected by the
extreme scenarios. Both detections are not particularly significant at the
$\sim 2\sigma$ level.

###  5.5 A Closer Look at the Extreme Cases

Figure 20:  Top: Same as the top-right panel of Figure 15, but for a single
galaxy, Q0142-BX165, which has the strongest excess $\rm Ly\alpha$ emission
along the galaxy major axis. Note that the colour scale is 10 times that of
Figure 15. Bottom: The HST F160w image of Q0142-BX165, overlaid with contours
from the KCWI continuum image (left) and the narrow-band $\rm Ly\alpha$ image
(right).

Figure 20 shows the residual between the cylindrically projected 2D spectra
extracted along the major and minor axes of Q0142-BX165, as well as continuum
images from both HST and KCWI. Q0142-BX165 has two comparably bright
components separated by $\simeq 3.5~{}\mathrm{pkpc}$ ($\simeq
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}4$). Careful inspection of Keck/LRIS
and Keck/MOSFIRE spectra of this system revealed no sign of an object at a
different redshift. There is no significant offset between the KCWI continuum
and $\rm Ly\alpha$ centroids ($\leq 0.5~{}{\rm pix}\sim
0\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}15$ separation), indicating that the
apparent directional asymmetry in the CP2D spectra is not caused by a spatial
shift between the continuum and $\rm Ly\alpha$ emission. Instead, the narrow-
band $\rm Ly\alpha$ map shows that the $\rm Ly\alpha$ emission is elongated
approximately along the N-S direction. However, after aligning the KCWI and
HST astrometry with reference to a nearby compact galaxy, we found that both
the KCWI stellar continuum near $\rm Ly\alpha$ and the narrow-band $\rm
Ly\alpha$ emission are centered near the SW component (see Figure 20). It is
possible that the SW component alone is responsible for the $\rm Ly\alpha$
emission, in which case its $\rm PA_{0}$ would be $-59^{\circ}$, $\sim
70^{\circ}$ off from what was determined in §4. However, adopting $\rm
PA_{0}=-59^{\circ}$ would cause BX165 to become an outlier with excess $\rm
Ly\alpha$ emission along the minor axis. Meanwhile, the elongation of the KCWI
continuum aligns with the direction of the separation of the two components,
and is roughly consistent with the direction of the $\rm Ly\alpha$ elongation
as well. This seems to suggest that the $\rm Ly\alpha$ elongation simply
reflects the asymmetry of the continuum source, albeit on a larger angular
scale; however, as shown in Figure 6, many galaxies in the sample possess
similar morphologies, but Q0142-BX165 is the only one that shows extraordinary
asymmetry in $\rm Ly\alpha$ emission. In any case, Q0142-BX165 has a uniquely
asymmetric $\rm Ly\alpha$ halo, possibly due to source confusion.
Consequently, we exclude it from most of the analysis that follows.

Figure 21:  Same as Figure 20 but for Q2343-BX418, the object with the second
strongest major axis $\rm Ly\alpha$ asymmetry.

The KCWI data cube for Q2343-BX418 has been analysed previously by Erb et al.
(2018); here, we consider it in the context of the analysis of Q0142-BX165
above (see Figure 21). The difference in peak SB between the major and minor
axis CP2D spectra is nearly equal to that of Q0142-BX165 ($2.4\times
10^{-18}~{}\mathrm{erg~{}s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2}\textrm{\AA}^{-1}$
for both). However, the spatial extent of the excess emission is significantly
smaller for Q2343-BX418. The HST/WFC3, KCWI continuum, KCWI $\rm Ly\alpha$
images, and OSIRIS-H$\alpha$ images all show Q2343-BX418 comprise a single
component whose centroids in the various images are consistent with one
another. Despite its extreme SB asymmetry in $\rm Ly\alpha$, Q2343-BX418
exhibits no other obviously peculiar property compared to the rest of the
sample.

##  6  Integrated Line Flux and Azimuthal Asymmetry

Figure 22:  Relationship between the flux measurements of anisotropic (excess)
$\rm Ly\alpha$ emission ($\Delta
F_{\mathrm{Ly}\alpha}=F_{\mathrm{major}}-F_{\mathrm{minor}}$) of the
blueshifted component (left) and redshifted component (right) of $\rm
Ly\alpha$ emission and properties of the integrated $\rm Ly\alpha$ halo [Top:
central $\rm Ly\alpha$ equivalent width, $W_{\lambda}(\rm Ly\alpha)$; Middle:
total $\rm Ly\alpha$ flux, $F_{\mathrm{Ly}\alpha}(\mathrm{tot})$; Bottom: the
ratio between the total blueshifted and redshifted components,
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$].
Galaxies without reliable $F_{\mathrm{red}}$ are omitted in the bottom panel
since their
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$ are
dominated by noise. The pink lines and shaded regions show the results and
their 1$\sigma$ uncertainties of a linear regression accounting for the errors
in both x- and y-directions. The vertical dashed line in each panel marks the
median value of the (x-axis) property for the full sample. The yellow diamond
in each panel marks the location of Q2343-BX418, the outlier that caused the
excess emission of the redshifted peak along the galaxy major axis as
discussed in §5.5.

As shown in the previous sections, the degree of $\rm Ly\alpha$ halo azimuthal
asymmetry varies from case to case in our $z\simeq 2.3$ sample, but the
correlation with the morphology of the central galaxy is sufficiently weak
that, on average, $\rm Ly\alpha$ halos are remarkably symmetric and appear to
be nearly independent – both kinematically and spatially – of the apparent
orientation of the galaxy at the center.

Thus far we have treated the blueshifted and redshifted components of $\rm
Ly\alpha$ emission separately. However, the overall $\rm Ly\alpha$ profile is
expected to provide clues to the geometry and velocity field of circumgalactic
H i. In this section, we compare the dependency between excess $\rm Ly\alpha$
emission and $W_{\lambda}(\rm Ly\alpha)$, total $\rm Ly\alpha$ flux
$F_{\mathrm{Ly}\alpha}(\mathrm{tot})$, and the ratio of the total flux of
blueshifted and redshifted components of $\rm Ly\alpha$ emission
[$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$].
The integration windows used to compute the values in Table 1 are
$F_{\mathrm{Ly}\alpha}(\mathrm{tot})$: $\theta_{\mathrm{tran}}\leq
3~{}\mathrm{arcsec}$ and $-700<\Delta v/(\mathrm{km~{}s}^{-1})\leq 1000$;
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})$: $\theta_{\mathrm{tran}}\leq
2.5~{}\mathrm{arcsec}$ and $-700<\Delta v/(\mathrm{km~{}s}^{-1})\leq 0$;
$F_{\mathrm{Ly}\alpha}(\mathrm{red})$: $\theta_{\mathrm{tran}}\leq
3~{}\mathrm{arcsec}$ and $0<\Delta v/(\mathrm{km~{}s}^{-1})\leq 1000$.
Different integration windows were used for the two components in order to
optimise the S/N of the integral; they were chosen based on the detected
extent of each component in Figure 17, in both $\theta_{\rm tran}$ and $\Delta
v$.

Figure 22 examines whether or not there is a connection between major
axis/minor axis asymmetry in $\rm Ly\alpha$ flux and the overall $\rm
Ly\alpha$ halo properties mentioned above.For each pair of variables in Figure
22, we indicate the value of the Pearson coefficient ($r$) and the
corresponding probability $p$ that the observed data set could be drawn from
an uncorrelated parent sample. We also performed a linear regression using
Orthogonal Distance Regression (ODR) in SciPy, which accounts for the
estimated uncertainty in both x- and y-variables. As can be seen from the
figure, most of the Pearson tests show no significant correlation and linear
regression yields slopes consistent with zero. However, the Pearson test for
the relation between
$F_{\mathrm{major}}(\mathrm{blue})-F_{\mathrm{minor}}(\mathrm{blue})$ and
$F_{\mathrm{Ly}\alpha}(\mathrm{tot})$ (middle left panel of Figure 22) yields
$p=0.02$, suggesting a marginally significant trend in which the asymmetry of
the blueshifted component of $\rm Ly\alpha$ favors the minor axis when $F_{\rm
Ly\alpha}(\rm tot)$ is weak, and the major axis when $F_{\rm Ly\alpha}({\rm
tot})$ is strong.

The second relationship that stands out is
$F_{\mathrm{major}}(\mathrm{red})-F_{\mathrm{minor}}(\mathrm{red})$ and
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$
(bottom right panel of Figure 22), where a non-parametric test for correlation
is not significant. The linear regression results in a marginally-significant
positive slope, indicating that as the blueshifted component of $\rm Ly\alpha$
approaches the strength of the redshifted component, there is a tendency for
excess emission along the major axis; for galaxies with
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.3$ (i.e., smaller than the median
value for the sample), the tendency is for excess $\rm Ly\alpha$ emission
along the minor axis.

Figure 23:  The difference between the CP2D spectra of $\rm Ly\alpha$ emission
for the major and minor axes. The maps show the residual for CP2D stacks for
two sub-samples representing those below (left) and above (right) the sample
median, From top to bottom, the $\rm Ly\alpha$ halo properties are the central
$\rm Ly\alpha$ equivalent width ($W_{\lambda}(\rm Ly\alpha)$), the integrated
$\rm Ly\alpha$ flux ($F_{\mathrm{Ly}\alpha}({\rm tot})$), and the flux ratio
between the blueshifted and redshifted components
($F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$).
This figure confirms that the blueshifted excess $\rm Ly\alpha$ emission
favours weak $\rm Ly\alpha$ emitting galaxies.

To further explore the reliability of the correlations, we split the sample in
two halves according to the overall halo properties, and compared the
subtracted CP2D spectra between the major and minor axes. Figure 23 shows the
result. As discussed in §5.5, the excess $\rm Ly\alpha$ emission along the
galaxy major axis for the red peak at $\Delta v\simeq
300~{}\mathrm{km~{}s}^{-1}$ within $\theta_{\mathrm{tran}}\lesssim
1~{}\mathrm{arcsec}$ can be attributed to a single outlier (Q2343-BX418),
which happens to fall above the median in all 3 quantities considered in
Figure 23 (i.e., on the righthand panels of the figure). The subtracted CP2D
spectra also indicate that essentially the entire excess $\rm Ly\alpha$
emission for the blue peak along the minor axis – as identified earlier (§5.5)
– is contributed by galaxies below the median $W_{\lambda}(\rm Ly\alpha)$ and
$F_{\rm Ly\alpha}({\rm tot})$ (top and middle lefthand panels of Figure 23).
In particular, the integrated significance within $-700<\Delta v/(\rm
km~{}s^{-1}\,)\leq-200$ and $\theta_{\mathrm{tran}}\leq
2\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}5$ exceeds $2.5\sigma$ for the
$W_{\lambda}(\rm Ly\alpha)<\mathrm{Median}$ bin. Comparison of the top two
panels also illustrate the same trend of
$F_{\mathrm{major}}(\mathrm{blue})-F_{\mathrm{minor}}(\mathrm{blue})$ vs.
$W_{\lambda}(\rm Ly\alpha)$ and $F_{\mathrm{Ly}\alpha}$ correlations suggested
by Figure 22.

For the subsamples divided based on the value of
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$,
the differences between the major axis and minor axis range of azimuthal angle
are less significant: there is a marginally significant excess of $\rm
Ly\alpha$ emission, more noticeable in the redshifted component, where it
appears to extend over the range $\theta_{\rm tran}\simeq 1-3$ arcsec, and the
bin with higher
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$
appears to have a major axis excess over approximately the same range of
angular distances. If Q2343-BX418 is removed from the stack of larger
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$
galaxies, the residual remains, showing that it is not attributable to a
single outlier. However, none of the residuals in the bottom panels of Figure
23 reaches a threshold of $2\sigma$ per resolution element.

In summary, a small statistical azimuthal asymmetry of $\rm Ly\alpha$ halos
persists when the galaxy sample is divided into two according to central
$W_{\lambda}(\rm Ly\alpha)$, total $F_{\rm Ly\alpha}$, and the flux ratio of
blueshifted and redshifted emission. Perhaps most intriguing is that galaxies
with small or negative central $W_{\lambda}(\rm Ly\alpha)$ have a tendency to
exhibit excess $\rm Ly\alpha$ emission along galaxy minor axes extending over
a fairly large range of both $\theta_{\rm tran}$ and velocity ($-700<(\Delta
v/\rm km~{}s^{-1}\,)<-200$). The apparent excess along the major axis of
redshifted $\rm Ly\alpha$ for the subsample with stronger $\rm Ly\alpha$
emission and larger
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$, on
the other hand, is confined to a smaller range of (redshifted) velocities,
again roughly coincident with the typical location of the“red peak”.

##  7 Finer Division of Galaxy Azimuthal Angles

At lower redshifts ($z<1$), the covering fraction and column density of gas in
various ionisation stages are commonly found to be related to the orientation
of the gaseous disk relative to the line of sight. Many authors have used used
background QSO or galaxy sightlines to detect strong Mg ii absorbers
associated with galaxies at redshifts to allow the foreground galaxy
orientations to be measured (e.g., Steidel et al. 2002; Bordoloi et al. 2011;
Bouché et al. 2012; Schroetter et al. 2019; Lundgren et al. 2021). A common
conclusion is that the sightlines giving rise to strong absorption tend to be
those located at azimuthal angles corresponding to both the galaxy major and
minor axes, but fewer (strong) absorbers are found at intermediate angles,
$30^{\circ}<\phi<60^{\circ}$. The kinematics of the absorbing gas also appear
to be related to $\phi$ (e.g., Ho et al., 2017; Martin et al., 2019), with the
broadest (and therefore strongest) systems found along the minor axis,
presumably due to fast bi-conical outflows oriented perpendicular to the disk,
followed by major axis sightlines sampling accreting or galactic fountain gas
sharing the halo’s angular momentum and thus exhibiting disk-like rotation.

Figure 24:  Similar to the right panel of Figure 15 but residual maps are
between the major and minor (left), major and intermediate (middle), and
intermediate and minor (right) axes with each bin size of only
$\Delta\phi=30^{\circ}$. The strong residual beyond
$\theta_{\mathrm{tran}}>4~{}\mathrm{arcsec}$ is caused by a contaminating
source near a single object. No sign of a bimodal distribution of $\rm
Ly\alpha$ emission is present. The sample in this figure is the same as in
Figure 17. (Q0142-BX165: discarded; Q2343-BX418: included).

If the $\rm Ly\alpha$ emission around $z=2$–3 galaxies arises in CGM gas with
properties similar to that of the low-ionization metallic absorbers at lower
redshift, one might expect to see similar evidence for asymmetries along the
two principal axes relative to intermediate azimuthal angles. We tested this
possibility by expanding our analysis in §5.2 by dividing $\phi$ into three
azimuthal bins – major axis ( $0^{\circ}\leq\phi<30^{\circ}$), minor axis
($60^{\circ}\leq\phi\leq 90^{\circ}$), and intermediate
($30^{\circ}\leq\phi\leq 60^{\circ}$). CP2D difference spectra among these 3
bins are shown in Figure 24.

Figure 24 shows no obvious sign of a bimodal distribution of $\rm Ly\alpha$
emission with respect to $\phi$, in which case the middle and right panels
would show residuals of opposite sign. Instead, the residual maps of “Major
$-$Intermediate” and “Intermediate $-$ Minor” suggest a gradual transition of
the small asymmetries between major and minor axes in the larger bins of
$\phi$ shown previously.

In any case, the main conclusion to draw from Figure 24 is once again that
$\rm Ly\alpha$ emission halos are remarkably similar in all directions with
respect to the projected principal axes of $z\sim 2-3$ galaxies.

##  8 Discussion

###  8.1 Comparison to previous work

The morphology of $\rm Ly\alpha$ emission from the CGM and its relation to the
host galaxies has been analysed in various works between $z=0$ and $z\lesssim
4$. However, analyses quantifying the $\rm Ly\alpha$ emission with respect to
the host-galaxy orientation is limited. In this section, we attempt to compare
the existing research on $\rm Ly\alpha$ halo morphology with our findings.

At very low redshifts ($z=0.02-0.2$), Guaita et al. (2015) studied the $\rm
Ly\alpha$ halos of 14 $\rm Ly\alpha$-emitting galaxies in the Lyman Alpha
Reference Sample (LARS; Östlin et al. 2014) and their connection to the host-
galaxy morphologies. They found that the $\rm Ly\alpha$ halos for the subset
of the sample that would be considered LAEs are largely axisymmetric, and
there is no single galaxy morphological property that can be easily connected
to the overall shape of the $\rm Ly\alpha$ emitting regions. Indeed, the $\rm
Ly\alpha$ images of the individual galaxies in Hayes et al. (2014) appear to
be independent of galaxy orientation beyond $D_{\rm tran}\sim 2$ pkpc.
Meanwhile, the stacked $\rm Ly\alpha$ image of the LARS galaxies is elongated
in the same direction as the far-UV continuum, i.e., along the major axis.
However, this stack could be significantly affected by sample variance from
galaxies that are particularly bright in rest-UV and $\rm Ly\alpha$, as we saw
for the KBSS sample before removing outliers. Interestingly, Duval et al.
(2016) studied a special galaxy in the LARS sample which is almost perfectly
edge-on – they found two small $\rm Ly\alpha$ emitting components (each of
extent $<1$ pkpc) near the disk consistent with $\rm Ly\alpha$ emission
escaping from the ISM through “holes” in the galactic disk, akin to that
expected in a classical Galactic fountain (Bregman, 1980). While this is
likely to be driven by stellar feedback allowing $\rm Ly\alpha$ to escape in
the direction perpendicular to the galaxy disk, the observed $\rm Ly\alpha$
features are far closer to the galaxy than could be measured in our $z\sim
2-3$ sample. Moreover, most of the galaxies in our sample do not have
organised thin disks, and so likely have much lower dust column densities
obscuring active star forming regions.

At $z>3$, Bacon et al. (2017) (and subsequent papers from the same group)
conducted a systematic survey of $\rm Ly\alpha$ emitting galaxies in the
Hubble Ultra Deep Field using VLT/MUSE. For example, Leclercq et al. (2017)
found significant variation of $\rm Ly\alpha$ halo morphology among 145
galaxies, and identified correlations between the halo size and the size and
brightness of the galaxy UV continuum. However, its connection with the galaxy
morphological orientation remain to be investigated.

Meanwhile, Chen et al. (2020a) examined a case of a strongly-lensed pair of
galaxies with extended $\rm Ly\alpha$ emission at $z>3$. The continuum image
in the reconstructed source plane of their system A has at least three
subcomponents extending over $>1$ arcsec, which may be similar to the
subsample of galaxies shown in our Figure 6, in terms of morphological
complexity. In the context of the analysis we describe in the present work,
this arrangement of $\rm Ly\alpha$ with respect to continuum emission would be
classified as excess minor axis $\rm Ly\alpha$ emission. Unfortunately, the
$\rm Ly\alpha$ halo of a second $z>3$ system is truncated in the source plane
reconstruction.

In summary, we compared our result with galaxies and their $\rm Ly\alpha$
emission morphology at $z\sim 0$ and $z>3$ in previous work, finding that
although our results are qualitatively consistent with earlier results, the
comparison is hampered by limited sample sizes, as well as by differences in
redshift and intrinsic galaxy properties represented in each sample.

###  8.2 Theoretical predictions

Many existing studies of the distribution and kinematics of CGM gas have
focused on simulated galaxies within cosmological hydrodynamic simulations.
Péroux et al. (2020) analysed how inflowing and outflowing gas might
distribute differently as a function of the galactic azimuthal angle within
the EAGLE and IllustrisTNG simulations, finding significant angular dependence
of the flow rate and direction, as well as the CGM metallicity, with outflows
of higher metallicity gas favoring the galaxy minor axis, and accretion of
more metal-poor gas tending to occur along the major axis, at $z<1$. Although
Péroux et al. (2020) focused on $z\sim 0.5$ for their study, they made clear
that the predicted trends would weaken significantly with increasing redshift.

At much higher redshifts ($z=5-7$) around galaxies in the FIRE suite of
simulations, Smith et al. (2019) found that $\rm Ly\alpha$ escape is highly
correlated with the direction of the H i outflow. Naively, one might expect
that more $\rm Ly\alpha$ would be found along the galaxy minor axis, which is
the direction along which gaseous outflows would encounter the least
resistance to propagation to large galactocentric radii. However, the galaxies
experiencing the most active star formation at these redshifts tend to be
altered on short timescales ($\sim 10^{7}$ yrs) by episodic accretion, star
formation, and feedback events. Thus, the direction of outflows may change on
similar timescales, while the CGM will evolve on a longer timescale, possibly
erasing any clear signatures of alignment of outflows and $\rm Ly\alpha$
emission. For the same reason, rapidly star-forming galaxies at $z\sim 2-3$,
most of which have not yet established stable stellar disks, are likely to be
surrounded by gas that is similarly turbulent and disordered.

Meanwhile, analytic or semi-analytic models of $\rm Ly\alpha$ resonant
scattering for idealised outflow geometries have focused primarily on the
integrated $\rm Ly\alpha$ emission profile from the entire galaxy or $\rm
Ly\alpha$ halo. Although many models account for the impact of the geometry
and kinematics of gaseous outflows or accretion on the integrated $\rm
Ly\alpha$ emission profiles, there have been fewer efforts to predict the two-
dimensional spatial and spectral profiles for detailed comparison to IFU
observations.

For example, Carr et al. (2018) constructed a model to predict the $\rm
Ly\alpha$ spectral morphology assuming biconical H i outflows with resonant
scattering. The model predicts the integrated spectral profile without spatial
information. In the context of the biconical outflow model, an integrated
profile resembling our observation is predicted when the minor axis is
perpendicular to the line of sight, with a large outflow having a small
opening angle. However, this particular configuration would likely give rise
to a highly asymmetric spatial distribution of $\rm Ly\alpha$ emission, which
we have shown is unlikely to be consistent with our observations.

Our results highlight the need for 3-D models of the cool gas in the CGM
around rapidly-star-forming galaxies at high redshifts, prior to the
development of stable disk configuration, for which the assumption of
axisymmetry of outflows, at least on average, may be closer to reality. In any
case, predicting the spatial and spectral properties of $\rm Ly\alpha$
emission will require realistic treatment of the kinematics, small-scale
structure, and radiative transfer of $\rm Ly\alpha$ photons from the sites of
production to their escape last scattering from the CGM toward an observer.

Gronke et al. (2016) has devised a model that assumes a two-phase CGM,
composed of optically-thick clumps embedded in a highly-ionised diffuse
“inter-clump” medium. This method has been used to fit the $\rm Ly\alpha$
profiles at multiple locations within a spatially-resolved “$\rm Ly\alpha$
blob” at $z\simeq 3.1$ (LAB) (Li et al., 2020). More recently, Li et al., in
prep, have shown that the clumpy outflow models can be applied successfully to
fit multiple regions within a spatially resolved $\rm Ly\alpha$ halo
simultaneously, i.e., using a central source producing $\rm Ly\alpha$ which
then propagates through a clumpy medium with an axisymmetric outflow (see also
Steidel et al. 2011, who showed that $\rm Ly\alpha$ emission halos similar in
extent to those presented in this paper are predicted naturally given the
observed velocity fields of outflows viewed “down the barrel” to the galaxy
center and the same radial dependence of clump covering fraction inferred from
absorption line studies of background objects).

###  8.3 Implications for H i kinematics and $\rm Ly\alpha$ Radiative Transfer

A spectral profile with a dominant redshifted component of $\rm Ly\alpha$
emission line and a weaker blueshifted component – with peaks shifted by
similar $|\Delta v|$ relative to the systemic redshift – is a typical
signature of an expanding geometry, a central $\rm Ly\alpha$ source function,
and resonant scattering. It has also been shown that, for an ensemble of
galaxies also drawn from the same KBSS redshift survey, the mean $\rm
Ly\alpha$ absorption signature measured in the spectra of background objects
within $D_{\rm tran}\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}50$ pkpc
($\theta_{\rm tran}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}6\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}$)
are outflow-dominated (Chen et al., 2020b). However, although the observed
$\rm Ly\alpha$ halos and their dependence on galaxy properties may be
naturally explained by central $\rm Ly\alpha$ sources scattering through the
CGM, many authors have emphasized that collisionally-excited $\rm Ly\alpha$
emission from accreting gas (i.e., graviational cooling – see e.g., Faucher-
Giguère et al. 2010; Goerdt et al. 2010; Lake et al. 2015) and in situ
photoionization by the UV background and/or local sources of ionizing photons
(e.g., Leclercq et al. 2020) may also contribute significantly to extended
$\rm Ly\alpha$ halos.

Due to the complex nature of $\rm Ly\alpha$ radiative transfer, our results
cannot resolve this issue definitively. However, the fact that the stacked
CP2D $\rm Ly\alpha$ spectra show asymmetry of $<2$% between the major and
minor axis for the ensemble, combined with (1) the empirical correlation
between the central $\rm Ly\alpha$ line strength ($W_{\lambda}(\rm Ly\alpha)$)
and the $\rm Ly\alpha$ flux integrated within the entire halo, and (2) the
consistently red-peak-dominated kinematics of both the central and integrated
$\rm Ly\alpha$ line, all favor scattering of $\rm Ly\alpha$ photons produced
near the galaxy center through an outflowing, clumpy medium – at least within
$D_{\mathrm{tran}}\lesssim 30$ pkpc. The remarkable statistical symmetry of
the full 2D profiles, both spatially and spectrally, suggests that most of the
galaxies in our sample lack persistent disk-like configurations, an inference
supported also by the ubiquity of blue-shifted absorption profiles in DTB
spectra of similar galaxies and their lack of dependence on HST morphology
(Law et al. 2012a, c). As a consequence, outflows do not behave in the manner
expected for central starbursts in disk galaxies; in other words, $z\sim 2-3$
galaxies on average appear to be more axisymmetric than their lower-redshift
counterparts. This may have important implications for the cycling of gas and
metals into and out of forming galaxies.

On the other hand, we have detected marginal ($2\sigma$) excess $\rm Ly\alpha$
emission along the galaxy major axis for the red peak, and along the galaxy
minor axis for the blue peak. While most of the excess emission along the
major axis can be easily explained by the relatively small sample size and the
presence of one or two extreme cases, the excess blueshifted emission along
the galaxy minor axis cannot be. While it is possible that the observed
asymmetries indicate the prevalence of outflows along the major axis and
inflows along the minor axis (i.e., the opposite of the behavior of galaxies
at $z<1$), we regard such an interpretation as unlikely. An important clue may
be that most of the blueshifted, minor-axis excess is contributed by galaxies
with weaker than the median $\rm Ly\alpha$ emission strength – many in that
subset have central $W_{\lambda}(\rm Ly\alpha)<0$, meaning that $\rm Ly\alpha$
photons must scatter from higher-velocity material or in directions with a
more porous distribution of optically thick gas to have a high probability of
escaping. Since we can only observe the photons that escape, galaxies with
lower overall $\rm Ly\alpha$ escape fractions will mean that those that do
escape must take more extreme paths on average. As shown in §6, the galaxies
with relatively weak emission also tend to have lower values of
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$,
which makes smaller absolute differences in emission strength vs. azimuthal
angle more noticeable.

Finally, we would like to emphasise the fact that the lack of a strong
statistical correlation between the morphology of extended $\rm Ly\alpha$
emission and the galaxy orientation does not imply that the individual $\rm
Ly\alpha$ halos are symmetric. In fact, as shown in Figures 16, 19, and 22,
individual $\rm Ly\alpha$ halos are often asymmetric, particularly for objects
with weak central $\rm Ly\alpha$ emission. Rather, our results indicate that
morphological variations of $\rm Ly\alpha$ halos are uncorrelated with the
apparent orientation of the host galaxy starlight.

##  9 Summary

In this paper, we have presented the first statistical results of an IFU
survey of star-forming galaxies at $\langle z\rangle=2.43$ drawn from the Keck
Baryonic Structure Survey and observed with the Keck Cosmic Web Imager on the
Keck 2 telescope. The 59 galaxies, with stellar mass and SFR typical of the
full KBSS galaxy sample, comprise the subset of the KBSS-KCWI survey with both
deep KCWI observations (typical exposure times of $\sim 5$ hours) and existing
high-spatial-resolution images from Hubble Space Telescope and/or Keck/OSIRIS.
The high resolution images were used to determine the direction of the
projected major axis of the stellar continuum light of each galaxy; the KCWI
IFU data cubes were used to detect spatially- and spectrally-resolved $\rm
Ly\alpha$ emission from the CGM around each galaxy to a limiting surface
brightness of $\lesssim 1\times 10^{-19}$ ergs s-1 cm-2 arcsec-2 in the
composite data (Figure 13), enabling detection of diffuse $\rm Ly\alpha$
emission halos to projected distances of $\theta_{\rm tran}\simeq
4$${}^{\prime\prime}~{}$ ($D_{\rm tran}\simeq 30$ pkpc).

Our major findings are summarised below:

  1. 1.

We introduced “cylindrically projected 2D spectra” (CP2D) in order to
visualise and quantify $\rm Ly\alpha$ spectra as a function of projected
galactocentric distance $D_{\rm tran}$. The CP2D spectra are averages of
spaxels over a specified range of azimuthal angle ($\phi$) at a common
galactocentric distance, enabling statistical analyses of $\rm Ly\alpha$
spectral profiles and their spatial variation simultaneously. The CP2D spectra
clearly show distinct redshifted and blueshifted components of $\rm Ly\alpha$
emission that remain distinct out to projected distances of at least 25 pkpc,
with rest-frame velocity extending from $-700\leq(\Delta v_{\rm sys}/\rm
km~{}s^{-1}\,)\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1000$ with
respect to the galaxy systemic redshift, with blue and red peaks at
$\simeq-300$ km s${}^{-1}\,$ and $\simeq+300$ km s${}^{-1}\,$, respectively.
(§5.2)

  2. 2.

We stacked the CP2D spectra of individual galaxies after aligning their
continuum major axes, in bins of azimuthal angle $\phi$ measured with respect
to the major axis. By creating difference images of the CP2D projections in
independent ranges of $\phi$, we showed that residual differences between
“major axis” and “minor axis” – which would reflect asymmetries in either the
spatial or spectral dimension along different ranges of $\phi$ – are very
small, with amplitude: $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2\times
10^{-20}~{}\mathrm{erg~{}s}^{-1}\mathrm{cm}^{-2}\mathrm{arcsec}^{-2}\textrm{\AA}^{-1}$,
corresponding to asymmetries in $\rm Ly\alpha$ flux amounting to $\leq 2\%$ of
the total, between galaxy major and minor axis directions. (§5.3)

  3. 3.

We found little evidence of statitically significant assymetry of the $\rm
Ly\alpha$ emission, except for an excess ($\simeq 2\sigma$) of $\rm Ly\alpha$
emission along galaxy major axes for the redshifted component of $\rm
Ly\alpha$ emission, with a peak near $\sim+300~{}\mathrm{km~{}s}^{-1}$.
However, closer inspection revealed that most of the signal was caused by two
galaxies with unusually asymmetric $\rm Ly\alpha$ halos. After discarding
these outliers, another excess emission feature, with integrated significance
$\simeq 2\sigma$, manifests as excess emission in the blueshifted component of
$\rm Ly\alpha$ along galaxy minor axes. This feature extends over a large
range of velocity, and appears to be contributed primarily by galaxies with
weaker than the sample median $\rm Ly\alpha$ emission, and central $\rm
Ly\alpha$ equivalent width $W_{\lambda}(\rm Ly\alpha)<0$. The same weak-$\rm
Ly\alpha$ subsample includes many of highest $M_{\ast}$ galaxies in the
sample, as well as many of the galaxies with the smallest flux ratio between
blueshifted and redshifted components
($F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$).
We speculate that this asymmetry may indicate that significant azimuthal
variation of $\rm Ly\alpha$ emission morphology exists only for galaxies with
the smallest $\rm Ly\alpha$ escape fractions within the sample, i.e., weaker
$\rm Ly\alpha$ emitting galaxies possess more developed rotational structure.
Evidently, one sees only the photons managing to find rare low-$N_{\rm HI}$
holes or that scatter from the highest velocity material, both of which are
more likely along the minor axis ( i.e., similar to expectations based on the
standard picture of biconical starburst-driven outflows from disk-like
systems). (§5.5, §6)

  4. 4.

Taken together, the results show that, statistically, the $\rm Ly\alpha$ halo
around galaxies in this sample (and, by extension, the population of
relatively massive star-forming galaxies at $z\sim 2-3$) have remarkably
little correlation – either kinematically or spatially – with the
morphological distribution of stellar continuum light of the host galaxy. The
observations suggest that most of the galaxies do not conform to expectations
in which outflows are bi-conical and oriented along the minor axis of disk-
like configurations, with accretion occurring preferentially along the major
axis, suggested by observations of CGM gas in star-forming galaxies with
$z<1$. Instead, the lack of systematic variation in the kinematics and spatial
extent with azimuthal angle of $\rm Ly\alpha$ emission, together with the fact
that
$F_{\mathrm{Ly}\alpha}(\mathrm{blue})/F_{\mathrm{Ly}\alpha}(\mathrm{red})$ is