# The CGM-GRB Study II: Outflow-Galaxy Connection at $z\sim 2-6$

Pradip Gatkine NASA Hubble Fellow  Division of Physics, Mathematics and
Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Dept.
of Astronomy, University of Maryland, College Park, MD 20742, USA  Sylvain
Veilleux Dept. of Astronomy, University of Maryland, College Park, MD 20742,
USA  Joint Space Science Institute, University of Maryland, College Park, MD
20742, USA  Daniel Perley Astrophysics Research Institute, Liverpool John
Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool
L3 5RF,UK  Joseph Durbak  Dept. of Physics, University of Maryland, College
Park, MD 20742, USA  Simone Dichiara Dept. of Astronomy, University of
Maryland, College Park, MD 20742, USA  Astrophysics Science Division, NASA
Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  S.
Bradley Cenko Joint Space Science Institute, University of Maryland, College
Park, MD 20742, USA  Astrophysics Science Division, NASA Goddard Space Flight
Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Eleonora Troja Dept. of
Astronomy, University of Maryland, College Park, MD 20742, USA  Astrophysics
Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA

(Accepted 2021 November 30)

###### Abstract

We use a sample of 27 GRBs at redshift $z=2-6$ to probe the outflows in their
respective host galaxies ($\mathrm{log(M_{*}/M_{\odot})}~{}\sim~{}9-11$) and
search for possible relations between the outflow properties and those of the
host galaxies such as $\mathrm{M_{*}}$, SFR, and specific SFR. First, we
consider three outflow properties $-$ outflow column density
($\mathrm{N_{out}}$), maximum outflow velocity ($\mathrm{V_{max}}$), and
normalized maximum velocity ($\mathrm{V_{norm}}$ =
$\mathrm{V_{max}/V_{circ,halo}}$, where $\mathrm{V_{circ,halo}}$ is the halo
circular velocity). We observe clear trends of $\mathrm{N_{out}}$ and
$\mathrm{V_{max}}$ with increasing SFR in high-ion-traced outflows, with a
stronger ($>~{}3\sigma$) $\mathrm{V_{max}}-$SFR correlation. We find that the
estimated mass outflow rate and momentum flux of the high-ion outflows scale
with SFR and can be supported by the momentum imparted by star formation
(supernovae and stellar winds). The kinematic correlations of high-ion-traced
outflows with SFR are similar to those observed for star-forming galaxies at
low redshifts.

The correlations with SFR are weaker in low-ions. This, along with the lower
detection fraction in low-ions, indicates that the outflow is primarily high-
ion dominated. We also observe a strong ($>~{}3\sigma$) trend of normalized
velocity ($\mathrm{V_{norm}}$) decreasing with halo mass and increasing with
sSFR, suggesting that outflows from low-mass halos and high-sSFR galaxies are
most likely to escape and enrich the outer CGM and IGM with metals. By
comparing the CGM-GRB stacks with those of starbursts at $z\sim 2$ and $z\sim
0.1$, we find that over a broad redshift range, the outflow strength strongly
depends on the main-sequence offset at the respective redshifts rather than
simply the SFR.

galaxies: evolution, high-redshift, star formation

††journal: ApJ††facilities: Very Large Telescopes, Lowell Discovery Telescope,
Spitzer Space Telescope, Hubble Space Telescope, Neil Gehrels Swift
Observatory††software:  galfit (Peng et al., 2002), astropy (Robitaille et
al., 2013), ESOreflex (Freudling et al., 2013), lifelines (Davidson-Pilon,
2019)

##  1 Introduction

Galactic inflows and outflows shape the evolution of galaxies as well as
enrich the circumgalactic medium (CGM) and intergalactic medium (IGM). The gas
inflows fuel star formation while stellar winds, supernova (SN) explosions,
and active galactic nuclei inject energy and metal-enriched matter (as well as
entrained cold gas) at large distances into the interstellar medium (ISM) and
CGM (Veilleux et al., 2005; Benson, 2010; Booth & Schaye, 2013; Tumlinson et
al., 2017; Rupke, 2018; Veilleux et al., 2020). The recycling flows from the
CGM bring back the metal-enriched gas to refuel the star formation
(Christensen et al., 2016). At the same time, removal of cold gas from the ISM
can quench the star formation activity. Thus, galactic outflows regulate
stellar buildup and are an important piece of the galactic feedback puzzle.
AGN-driven outflows are thought to be the dominant feedback process in massive
galaxies (Veilleux et al., 2005; Fabian, 2012; Heckman & Best, 2014; King &
Pounds, 2015; Nelson et al., 2019) whereas SN-driven outflows are thought to
be more important in low-mass, star-forming galaxies (Sharma & Nath, 2012).

Supernova-driven outflows at high-redshift are important for the early
enrichment of the CGM and IGM (Tumlinson et al., 2017; Veilleux et al., 2020).
The low-mass star forming galaxies are of particular interest in this context
since their outflows are most likely to escape their shallower potential
wells. The relationship between the outflows and their host galaxies in the
early universe holds the key to tune the models of galactic feedback and
understand the history of galaxy growth and cosmic metal enrichment.

Observations at high redshift ($z>2$) using various techniques have shown the
presence of ubiquitous outflows in star-forming galaxies. The prominent
techniques include down-the-barrel absorption-line studies (Frye et al., 2002;
Shapley et al., 2003; Sugahara et al., 2017; Du et al., 2018; Rudie et al.,
2019), outflows at larger radii using background quasar or galaxy sightlines
(Steidel et al., 2010; Lehner et al., 2014; Turner et al., 2014; Rudie et al.,
2019), quasar-quasar pairings (Hennawi et al., 2006; Prochaska et al., 2014),
observing lensed galaxy spectra (Rigby et al., 2018), spatially-resolved
spectroscopy in optical or radio (Harrison et al., 2012; Swinbank et al.,
2015; Nielsen et al., 2020; Pizzati et al., 2020), and GRB afterglow
sightlines (Fox et al., 2008; Gatkine et al., 2019). Galactic as well as
cosmological zoom-in simulations provide the framework to understand the
outflow mechanisms (for instance, Hirschmann et al. 2013; Shen et al. 2013;
Muratov et al. 2015; Nelson et al. 2019; Mitchell et al. 2020). The high-$z$
outflow-galaxy relation and its evolution with redshift has recently been
studied in Sugahara et al. (2017, 2019).

However, the outflow-galaxy relation in low-mass galaxies in the early
universe remains poorly understood due to observational challenges. Two key
challenges are: determining the redshift of the galaxy (in case of background
QSO/galaxy sightlines) and obtaining high quality absorption spectra of these
faint galaxies (for down-the-barrel technique). Apart from this, reliably
removing the continuum spectrum of the background object can be a challenge.

Use of GRB sightlines to probe the outflows and CGM of its host galaxy offers
a promising solution to these problems. In Gatkine et al. (2019), we described
this method in detail. The main idea here is to use the bright GRB afterglow
to probe the kinematics/outflows in the CGM of its host galaxy. GRB hosts at
$z>2$ are typically low-mass galaxies (log($\mathrm{M_{*}/M_{\odot}}$) $<$
10.5), which makes them ideally suited for exploring the low-mass outflows
which are difficult to probe using other techniques. The key advantages
include: 1) clear identification of the host-galaxy redshift, 2) high signal-
to-noise ratio (SNR) and high-resolution spectra due to the bright GRB
afterglow, and 3) the featureless continuum of the GRB afterglow eliminates
the problem of continuum subtraction.

In this paper, we use the CGM-GRB sample compiled in Gatkine et al. (2019) to
explore the correlations between outflow and galaxy properties. The CGM-GRB
sample consists of 27 GRBs at $z\sim 2-6$ with high SNR (median SNR $\sim$ 10)
and high-resolution ($\delta v<50$ km s-1) spectra. Multi-component Voigt-
profiles were fit to the absorption spectra of various high- and low-ion
species (including C IV, Si IV, Si II, Fe II, and O VI). The CGM kinematics of
this sample were studied in Gatkine et al. (2019). In this paper, we report
the observations of their host galaxies in the optical and near-IR to estimate
their star formation rate (SFR) and stellar mass ($\mathrm{M_{*}}$). These
observations and their analyses are described in Section 2. We then discuss
the techniques used for visualizing and inferring correlations in Section 3.
The key correlations between outflow properties and galaxy properties such as
$\mathrm{M_{*}}$, SFR, specific star formation rate (sSFR =
SFR/$\mathrm{M_{*}}$), and halo mass are detailed in Section 4. Finally, the
implications of our results are discussed in Section 5.

Throughout this paper, we use the following model of cosmology: $H_{0}=$ 70
$\mathrm{km~{}s^{-1}Mpc^{-1}}$, $\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$

##  2 Observations and Methods

As described earlier, we measure the galaxy properties in the CGM-GRB sample.
The sample is selected strictly on the criterion of availability of a high-
resolution ($\delta v$ $<$ 50 km s-1) and high-SNR (SNR $>$ 5) afterglow
spectrum. No cuts are made to the sample based on galaxy properties. The
redshift distribution of the sample is shown in the first panel of Fig. 2.

###  2.1 Optical Photometry

We performed optical photometry of previously unpublished or unobserved GRB
hosts in the CGM-GRB sample. We observed GRB hosts using the 4.3-meter Lowell
Discovery Telescope (LDT). We also obtained deep archival imaging of two GRB
hosts using the FORS instrument on the Very Large Telescope (VLT) and one each
using HST WFC3 (program ID 15644), the Kilo-Degree Survey (KiDS) (Kuijken et
al., 2019), and PanSTARSS survey (Flewelling et al., 2016). We consider a GRB
host as detected if the offset of the potential host and the GRB location is
within 1″. The probability of a chance alignment of a galaxy brighter than the
typical depth in our observations ($i$-band $\sim$ 24.7 AB mag) within 1″is
approximately 0.01 (see Fig. 6 in Beckwith et al. 2006). At $z\sim$ 3, 1″
roughly corresponds to 7.5 kpc. From previous HST observations of other GRB
host samples at (Bloom et al., 2002; Fruchter et al., 2006; Lyman et al.,
2017), more than $90\%$ of the GRBs occur within this offset from their host
galaxies. All the GRBs are localized with a $<0.5\arcsec$ precision. The
resulting magnitudes are further corrected for Milky Way Galactic extinction
using the dust maps of Schlafly & Finkbeiner (2011) and the extinction law
with $R_{V}$ = 3.1 from Cardelli et al. (1989). The photometry results are
presented in Table 1.

The LDT imaging was performed using the Large Monolithic Imager (LMI, Massey
et al. (2013)). The LMI data was detrended with a custom python-based pipeline
(Toy et al., 2016). Individual fields were astrometrically aligned and co-
added using SCAMP and SWARP respectively. The aperture photometry of the co-
added images was performed using Sextractor with an aperture radius of
$\sim$1.5″, which is typical of the average seeing in our observations. The
magnitudes were calibrated against the SDSS (Alam et al., 2015) and GAIA
catalogs (Evans et al., 2018). Conversion of GAIA magnitudes to Sloan
magnitudes was performed using the conversion tables provided in GAIA data
release 2 (Brown et al., 2018).

The FORS data was flat-fielded using the ESO pipeline ESOreflex (Freudling et
al., 2013) and was further aligned, co-added, and calibrated as described
above. PanSTARSS and KiDS surveys provide reduced, stacked, and zero-point
calibrated images, which were used to determine the science magnitudes/upper
limits. The HST photometry was performed using archived drizzled and
calibrated images and the AB magnitude was derived using the provided zero
point. A 1″ aperture was used for HST images given the diffraction-limited
imaging.

Table 1: Summary of new observations

GRB | $z^{c}$ | Tel./Instr. | Filter | AB Mag
---|---|---|---|---
000926Aa | 2.0377 |  Spitzer/IRAC | 3.6 $\mu m$ | 25.2 $\pm$ 0.15
021004a | 2.3281 |  Spitzer/IRAC | 3.6 $\mu m$ | 24.22 $\pm$ 0.18
071031b | 2.6912 |  Spitzer/IRAC | 3.6 $\mu m$ |  $>$ 25.3
080310b | 2.4274 |  Spitzer/IRAC | 3.6 $\mu m$ | 23.74 $\pm$ 0.24
090926A | 2.106 |  Spitzer/IRAC VLT/FOR2 |  3.6 $\mu m$ RSpecial |  22.96 $\pm$ 0.05 23.9 $\pm$ 0.1
111008Ab | 4.989 |  Spitzer/IRAC HST/WFC3 |  3.6 $\mu m$ F110W |  24.73 $\pm$ 0.3 25.5 $\pm$ 0.07
120327A | 2.813 | LDT/LMI | SL-r | 24.9 $\pm$ 0.2
130606A | 5.911 |  Spitzer/IRAC | 3.6 $\mu m$ | 24.91 $\pm$ 0.25
130610A | 2.091 |  Spitzer/IRAC LDT/LMI |  3.6 $\mu m$ SL-r |  23.46 $\pm$ 0.05 23.7 $\pm$ 0.1
141028A | 2.333 |  Spitzer/IRAC LDT/LMI |  3.6 $\mu m$ SL-r |  $>$ 25.1 $>$ 25.8
141109A | 2.993 |  Spitzer/IRAC LDT/LMI |  3.6 $\mu m$ SL-i |  23.4 $\pm$ 0.1 24.1
151021A | 2.329 |  Spitzer/IRAC KiDS Survey |  3.6 $\mu m$ SL-r |  $>$ 25.7 24.4 $\pm$ 0.2
151027B | 4.0633 |  Spitzer/IRAC LDT/LMI |  3.6 $\mu m$ SL-r SL-i |  $>$ 22.66 $>$ 24.3 24.8 $\pm$ 0.4
160203A | 3.518 |  Spitzer/IRAC PanSTARRS |  3.6 $\mu m$ PS1-i |  21.74 $\pm$ 0.02 $>$ 22.7
161023A | 2.709 |  Spitzer/IRAC VLT/FORS2 |  3.6 $\mu m$ RSpecial |  $>$ 25.9 $>$ 25.7
170202A | 3.645 | LDT/LMI |  SL-r SL-i |  $>$ 25.4 $>$ 23.4

aafootnotetext: Spitzer Prog ID 40599, PI : R. Chary.

Figure 1:  Contamination-subtracted images of GRB fields from Spitzer-IRAC in
3.6 $\mu m$ band. Each thumbnail is 8” $\times$ 8” in size. The central red
circle is the 1.8″aperture used to define the source flux and the outer
annulus is used to define the background flux. The circle is centered on the
best-known position of the GRB or of the detected host galaxy. References for
GRB positions: 000926 (Fynbo et al., 2001), 021004 (Henden & Levine, 2002),
071031 (Krühler et al., 2009), 080310 (Littlejohns et al., 2012), 111008A
(Bolmer et al., 2018), 130606A (Castro-Tirado et al., 2013), 141109A (Xu et
al., 2014), 151021A (McCauley & Melandri, 2015), 151027B (Greiner et al.,
2018), 161023A (de Ugarte Postigo et al., 2018).

###  2.2 Spitzer IRAC Photometry

We obtained deep archival imaging of GRB hosts using Spitzer Infrared Array
Camera (IRAC) channel 1 ($3.6\mu m$). Out of a total of 27 GRB hosts, we
present new Spitzer IRAC photometry of 14 hosts in this paper and 11 were
previously published as a part of the SHOALS survey (Perley et al., 2016). The
remaining two GRBs remained unobserved by the end of Spitzer mission. The
newly presented data have been collected as a part of various previous
programs which are summarized in Table 1.

By analyzing the new data the same way as Perley et al. (2016), we ensure
procedural consistency with the previously published data. The reduction and
photometry method is described in detail in Perley et al. (2016). Here, we
briefly summarize the key points. We acquired the Level-2 PBCD (Post-Basic
Calibrated Data) from the Spitzer Legacy Archive. We use the default
astrometry provided with the Level-2 products (with an accuracy of 0.3″). Due
to the large PSF of Spitzer IRAC ($\sim$ 1.8″ at 3.6 $\mu m$), source
confusion and flux contamination from neighboring sources is an important
issue. We compare each IRAC image with deep ground-based optical images (as
described in Section 2.1) to identify the primary source and any neighboring
contaminants. We used the galfit tool (Peng et al., 2002) over several
iterations to model the sources (using the PSF and PRF files provided in
Spitzer
documentation111https://irsa.ipac.caltech.edu/data/Spitzer/docs/irac/calibrationfiles/psfprf/)
and subtract the neighboring sources which may contaminate the host or sky
background regions. The subtracted image is then used for performing aperture
photometry.

We implemented the IRAC handbook recommendations for aperture photometry using
a custom IDL wrapper around the aper procedure in the Astronomy User’s
Library222https://idlastro.gsfc.nasa.gov/ (see Perley et al. (2016) for
details). For aperture photometry, we place a 1.8″ aperture on the host galaxy
location (guided by deep optical imaging) and a sky annulus with an inner
radius of 3.6″ and outer radius of 6″. The source aperture and sky annulus are
marked in red and yellow respectively in Fig. 1. In the case of optical
detection and IR non-detection, we specify a 2-$\sigma$ limit. However, in the
case of optical as well as IR non-detection, we evaluate a 3-$\sigma$ upper
limit to account for the uncertainty (typically $<$ 1″) in the GRB host
location.

###  2.3 Stellar Mass

We use the Spitzer IRAC 3.6$\mu m$ photometry to infer the stellar mass of
galaxies in our sample. At $z$ $\sim$ $2-6$, Spitzer IRAC measures the rest-
frame optical light (beyond the Balmer break) from long-lived stars in the
host galaxies. Here we follow the methodology used in Perley et al. (2016) to
derive the stellar masses. SED fitting is a more accurate method to estimate
$\mathrm{M_{*}}$ (by breaking the degeneracy between age and extinction).
However, this requires extensive, ultra-deep optical observations of faint GRB
hosts in multiple filters, which is resource intensive. Instead, we use
Spitzer single-band (3.6 $\mu m$) photometry, which can still provide a
reasonable estimate of stellar mass, particularly for galaxies at $z>2$.

We calculate the absolute magnitude at $\lambda_{rest}=3.6\mu m/(1+z)$ as
$\mathrm{M_{AB}}$ = $m_{\mathrm{AB,3.6\mu m}}-$ DM + 2.5log(1+$z$), where DM
is distance modulus. In Perley et al. (2016), a grid of model galaxy SEDs is
constructed for an array of redshifts ($z\sim 0-10$) and each decade in
$\mathrm{M_{*}}$ ($10^{8}$ to $10^{11}\mathrm{M_{\odot}}$) by summing Bruzual
& Charlot (2003) galaxy SED templates (using Chabrier (2003) initial mass
function). The models also incorporate a modest dust attenuation to validate
the single-band stellar mass conversion function against the more accurate
SED-fit (optical $+$ Spitzer multiband) stellar masses in the MODS (Kajisawa
et al., 2009) and UltraVISTA samples (Caputi et al., 2015). We then evaluate
the stellar mass by interpolating on the $\mathrm{M_{*}}$, redshift, and AB
magnitude grid (see Perley et al. (2016) for more details). While the single-
band method suffers from uncertainties associated with various model
assumptions such as the IMF, dust extinction ($\mathrm{A_{V}}$), and star
formation history, this method is consistent with the masses obtained from SED
fitting at the $\sim$ 0.3 dex level. Further, by using the same method
throughout our sample, we ensure that the correlations derived here are on an
equal footing. The $\mathrm{M_{*}}$ of our GRB hosts are summarized in Table
2.

Figure 2:  Properties of the CGM-GRB sample. Panel 1: The redshift
distribution of the sample. Panel 2: The SFR $vs$ $\mathrm{M_{*}}$ of the GRB
host galaxies in our sample. The lines show the main sequence curves (yellow:
$z=1$, blue: $z=2$, red: $z=4$) as described in Speagle et al. (2014). Panel
3: The cumulative distriution of the stellar mass in the CGM-GRB sample. The
spread shows 95% confidence interval around the value by incorporating any
upper limits. The dotted vertical lines show the value of characteristic mass,
$\mathrm{M^{*}}$ in the mass function (written as a Schechter function) at the
respective redshifts. The horizontal line shows the median (i.e. CDF = 0.5).
Panel 4: Same as panel 3, for SFR.

###  2.4 Dust Correction

The ultraviolet (UV) dust extinction of the host galaxies needs to be
estimated to convert the absolute magnitudes into intrinsic rest-frame UV
luminosities. Following Greiner et al. (2015), we perform the dust correction
using empirical correlations of the spectral index of the UV continuum $\beta$
(where $f_{\lambda}=\lambda^{\beta}$), rest-frame absolute UV magnitude at
$\lambda_{rest}$ = 1600 $\mathrm{\AA}$ ($\mathrm{M_{UV}}$), and the dust
extinction at rest-frame $1600\mathrm{\AA}$ ($A_{1600}$). Here we assume that
GRB hosts at high redshift follow a power law SED
($f_{\lambda}=\lambda^{\beta}$) in the UV (redward of Ly$\alpha$) and the same
correlations as the extensive high-$z$ ($2.5-6$) star forming galaxy sample of
$>4000$ galaxies from HST HUDF and CANDELS surveys studied in Bouwens et al.
(2009, 2014). They derive the following empirical relation for star forming
galaxies at $\left\langle z\right\rangle$ = 3.8:

$\beta=-1.85-0.11(\mathrm{M_{UV}}+19.5)$ (1)

The uncertainties on the numerical coefficients here are small
($-1.85\pm{0.06}$ and $-0.11\pm{0.01}$). Then, we iteratively solve for
$\mathrm{M_{UV}}$ and $\beta$. The typical $\beta$ for the high-$z$ star
forming sample in Bouwens et al. (2009, 2014) is $\beta$ $\sim$ $-2$. In
equation 1, this corresponds to $\mathrm{M_{UV}=-18.1}$. Hence, we use $\beta$
$=$ $-2$ for our weaker upper limits (where $\mathrm{M_{UV,lim}>-18.1}$),
where the value of $\beta$ is more uncertain. For stronger upper limits (i.e.
$\mathrm{M_{UV,lim}<-18.1}$, we use the $\beta$ corresponding to the limit.
Finally, the $A_{1600}$ is evaluated using the following relation from Meurer
et al. (1999):

$A_{1600}=4.43~{}\mathrm{mag}+1.99\beta$ (2)

This dust-correction method is described in detail in Greiner et al. (2015).

###  2.5 Star formation rate

We use single-band photometry in the rest-frame UV to calculate UV-based SFR.
To compute the SFR from dust-corrected UV luminosity ($L_{UV,corr}$), we
follow the relations described in Savaglio et al. (2009) where they
simultaneously compare the emission-line and dust-corrected UV luminosities of
GRB hosts to derive the conversion factor between dust-corrected UV luminosity
and SFR. We use the $A_{1600}$ and $\mathrm{M_{UV}}$ values calculated in
Section 2.4 to compute $L_{UV,corr}$. The SFR is then calculated as follows:

$\mathrm{SFR_{1500}}=1.62~{}\mathrm{M_{\odot}yr^{-1}}\times\frac{L_{\mathrm{1500,corr}}}{\mathrm{10^{40}~{}erg~{}s^{-1}~{}\AA^{-1}}}$
(3)

As a validation step, we compare the $\mathrm{A_{1600}}$ evaluated using the
$\beta$ method with that using the afterglow-derived $\mathrm{A_{V}}$
(assuming an SMC extinction law). The resulting SFRs derived using the two
methods are consistent with each other within a factor of two except for GRBs
130408A and 080810 where the afterglow $\mathrm{A_{V}}$ is larger, leading to
a higher SFR estimate (for the afterglow $\mathrm{A_{V}}$ method) by a factor
of 3. Nonetheless, it is important to note that the afterglow-derived
extinction corresponds to a single sightline, while the extinction derived
using the $\beta$ method is an average value for the host. The star formation
rates of our GRB hosts are summarized in Table 2.

For GRBs 071031, 080804, and 120815, photometric observations are either
unavailable or too shallow. In the cases of GRBs 080804 and 120815, we have
used H$\alpha$ emission-line-based SFRs from Krühler et al. (2011) since they
are more robust compared to UV-luminosity. For GRB 071031, we use the
Ly$\alpha$-based SFR from Milvang-Jensen et al. (2012). While less robust,
this measurement is consistent with the upper limit of 3
$\mathrm{M_{\odot}yr^{-1}}$ from an archival HST WFC3 (F160W filter)
observation.

Note however that our sample naturally has low line-of-sight dust extinction
compared to the general GRB host population since we only select the
afterglows that are bright enough for high-resolution rest-frame UV
spectroscopy. While there may be a systematic bias in the dust correction, we
have used the same SFR-tracer and analysis procedure for the entire sample
(except GRBs 071031, 080804, and 120815), thus minimizing any relative bias.
Our sample may contain a small number of heavily dust enshrouded galaxies, for
which we may underestimate the SFR. However, we have minimized this
possibility by ruling out heavy dust obscuration in 4 massive GRB hosts in our
sample (where the probability of heavy dust obscuration is high) by using deep
VLA observations (Gatkine et al., 2020) and hence, the typical dust
corrections described here can be used for estimating their star formation
rates. These GRBs are marked with asterisk in Table 2.

Table 2: Summary of GRB host properties in the CGM-GRB sample

GRB | $z$ | log(NHI)a | AVbbSpitzer Prog ID 80054, PI : E. Berger. All other Spitzer observations are taken from Spitzer Prog IDs 11116, 13104, 90062 PI: D. Perley. | M3.6/(1+z) | log($\mathrm{M_{*}/M_{\odot}}$) | MUV |  SFR ($\mathrm{M_{\odot}yr^{-1}}$) | References
---|---|---|---|---|---|---|---|---
000926 | 2.0385 | 21.3 $\pm$ 0.25 | 0.15 | $-19.6$ | 9.3$\pm$0.3 | $-19.5$ | 4.0${}^{+1.3}_{-1.0}$ |  Castro et al. (2003) Chen et al. (2009)
021004∗ | 2.3281 | 19.0 $\pm$ 0.2 | 0.2 | $-20.9$ | 9.5$\pm$0.1 | $-21.4$ | 11.8${}^{+3.7}_{-2.8}$ |  Fiore et al. (2005) Fynbo et al. (2005)
050730 | 3.9672 | 2.1 $\pm$ 0.1 | 0.12 |  $>$ $-20.5$ |  $<$ 9.46ccRedshifts taken from Gatkine et al. (2019) | $-18.1$ | 0.8${}^{+0.2}_{-0.1}$ |  D’Elia et al. (2007) Toy et al. (2016)
050820A | 2.6137 | 21.1 $\pm$ 0.1 | 0.08ddFrom Perley et al. (2016) | $-20.42$ | 9.4$\pm$0.15ddFrom Perley et al. (2016) | $-19.1$ | 2.4${}^{+1.3}_{-0.9}$ |  Prochaska et al. (2007) Chen et al. (2009)
050922C | 2.1996 | 21.55 $\pm$ 0.1 | 0.10 | $-19.6$ |  $<$ 9.0ddFrom Perley et al. (2016) |  $>$ $-18.3$ |  $<$ 1.0 |  Prochaska et al. (2008) Covino et al. (2013)
060607A | 3.0738 | 16.95 $\pm$ 0.03 | 0.08 |  $>$ $-20.52$ |  $<$ 9.4ddFrom Perley et al. (2016) |  $>$ $-17.5$ |  $<$ 0.4 |  Prochaska et al. (2008) Schady et al. (2012)
071031 | 2.6912 | 22.15 $\pm$ 0.05 | 0.14 |  $>$ $-20.1$ |  $<$ 9.2 | $-$ | 1.4${}^{+0.3}_{-0.3}$ee071031: SFR using Ly$\alpha$ Milvang-Jensen et al. (2012), 080804: SFR using H$\alpha$ Krühler et al. (2015), 120815: SFR using H$\alpha$ Krühler et al. (2015).
∗ GRBs with deep VLA observations from Gatkine et al. (2020) |  Fox et al. (2008) Li et al. (2018)
080310∗ | 2.4274 | 18.7 $\pm$ 0.1 | 0.10 | $-21.3$ | 9.8$\pm$0.1ddFrom Perley et al. (2016) | $-19.0$ | 2.4${}^{+1.4}_{-0.9}$ |  Fox et al. (2008) Perley et al. (2009)
080804 | 2.205 | 21.3 $\pm$ 0.1 | 0.17 | $-20.2$ | 9.3$\pm$0.15ddFrom Perley et al. (2016) | $-$ | 15.1${}^{+20}_{-7}$ee071031: SFR using Ly$\alpha$ Milvang-Jensen et al. (2012), 080804: SFR using H$\alpha$ Krühler et al. (2015), 120815: SFR using H$\alpha$ Krühler et al. (2015).
∗ GRBs with deep VLA observations from Gatkine et al. (2020) |  Fynbo et al. (2009) Toy et al. (2016)
080810∗ | 3.351 | 17.5 $\pm$ 0.15 | 0.40 | $-22.15$ | 10.24$\pm$0.1ddFrom Perley et al. (2016) | $-22.9$ | 173${}^{+45}_{-36}$ |  Page et al. (2009) Wiseman et al. (2017)
090926A | 2.106 | 21.73 $\pm$ 0.07 |  $<$ 0.04 | $-21.9$ | 9.8$\pm$0.1 | $-20.5$ | 11.6${}^{+3.7}_{-2.8}$ |  D’Elia et al. (2010) Zafar et al. (2018)
100219A | 4.665 | 21.13 $\pm$ 0.12 | 0.13 |  $>$ $-20.4$ |  $<$ 9.4ddFrom Perley et al. (2016) | $-20.0$ | 6.7${}^{+5.5}_{-3.2}$ |  Thöne et al. (2012) Toy et al. (2016)
111008A | 4.989 | 22.3 $\pm$ 0.06 | 0.12 | $-20.9$ | 9.5$\pm$0.2 | $-20.5$ | 12.3${}^{+3.5}_{-2.7}$ |  Sparre et al. (2014) Zafar et al. (2018)
120327A | 2.813 | 22.01 $\pm$ 0.09 |  $<$ 0.03 | $-23.2$ | 10.8$\pm$0.1 | $-21.2$ | 28.1${}^{+12.5}_{-8.7}$ |  D’elia et al. (2014) Heintz et al. (2019)
120815A | 2.358 | 21.95 $\pm$ 0.1 | 0.19 $\pm$ 0.04 |  $>$ $-21.2$ |  $<$ 9.7ddFrom Perley et al. (2016) | $-$ | 2.3${}^{+2}_{-1}$ee071031: SFR using Ly$\alpha$ Milvang-Jensen et al. (2012), 080804: SFR using H$\alpha$ Krühler et al. (2015), 120815: SFR using H$\alpha$ Krühler et al. (2015).
∗ GRBs with deep VLA observations from Gatkine et al. (2020) |  Krühler et al. (2015) Zafar et al. (2018)
120909A | 3.929 | 21.20 $\pm$ 0.10 | 0.16 $\pm$ 0.04 |  $>$ $-20.2$ |  $<$ 9.5ddFrom Perley et al. (2016) | $-20.8$ | 17.9${}^{+6.2}_{-4.6}$ |  Cucchiara et al. (2015) Heintz et al. (2019)
121024A∗ | 2.298 | 21.50 $\pm$ 0.10 | 0.56ddFrom Perley et al. (2016) | $-21.8$ | 10.15$\pm$0.15 | $-21.7$ | 37${}^{+20}_{-15}$ |  Friis et al. (2015) Toy et al. (2016)
130408A | 3.757 | 21.70 $\pm$ 0.10 | 0.2 | $-$ | $-$ |  $>$ $-21.1$ |  $<$ 13.4 |  Zafar et al. (2018)
130606A | 5.911 | 19.93 $\pm$ 0.2 |  $<$ 0.07 | $-21.8$ | 10.0$\pm$0.2 | $-19.9$ | 6.3${}^{+2.4}_{-1.7}$ |  Hartoog et al. (2015) Zafar et al. (2018)
130610A | 2.091 | $-$ | 0.01 | $-21.3$ | 9.7$\pm$0.05 | $-20.6$ | 13${}^{+4.1}_{-3.1}$ |  Smette et al. (2013) Littlejohns et al. (2015)
141028A | 2.333 | 20.60 $\pm$ 0.15 | 0.13 |  $>$ $-20.0$ |  $<$ 9.2 |  $>$ $-19.2$ |  $<$ 2.3 |  Wiseman et al. (2017)
141109A | 2.993 | 22.10 $\pm$ 0.10 | 0.11 | $-22.1$ | 10.1$\pm$0.1 | $-20.9$ | 19.7${}^{+14}_{-8}$ |  Heintz et al. (2018) Heintz et al. (2019)
151021A | 2.329 | 22.3 $\pm$ 0.2 | 0.2 |  $>$ $-19.4$ |  $<$ 9.0 | $-20.3$ | 9.6${}^{+4.3}_{-3}$ |  Heintz et al. (2018)
151027B | 4.0633 | 20.5 $\pm$ 0.2 |  $<$ 0.12 | $-23.45$ |  $<$ 10.8 | $-21.9$ | 58${}^{+40}_{-24}$ |  Heintz et al. (2018) Zafar et al. (2018)
160203A | 3.518 | 21.75 $\pm$ 0.10 |  $<$ 0.1 | $-24.2$ | 11.2$\pm$0.05 |  $>$ $-22.9$ |  $<$ 71 |  Heintz et al. (2018)
161023A | 2.709 | 20.96 $\pm$ 0.05 | 0.09 |  $>$ $-19.5$ |  $<$ 9.1 |  $>$ $-19.6$ |  $<$ 3.4 |  Heintz et al. (2018) de Ugarte Postigo et al. (2018)
170202A | 3.645 | 21.55 $\pm$ 0.10 |  $<$0.12 | $-$ | $-$ |  $>$ $-21.0$ |  $<$ 11.5 |  Selsing et al. (2019) Zafar et al. (2018)

Column Descriptions: M3.6/(1+z): AB magnitude in rest-frame optical/NIR from
Spitzer data; MUV: Absolute magnitude at $\lambda_{rest}$ = 1600Å; SFR: in
units of $\mathrm{M_{\odot}/yr}$ aafootnotetext: Neutral hydrogen column
densities (in cm-2) measured from the damped Ly-$\alpha$ absorption, unless
noted otherwise.  bbfootnotetext: Extragalactic dust extinction in magnitude,
derived assuming SMC extinction law (Gordon et al., 2003) ccfootnotetext:
Derived Using host galaxy SED

##  3 Sample properties and analysis

###  3.1 Comparison with star formation main sequence

Figure 2 shows the distribution of $\mathrm{M_{*}}$, SFR, and $z$ of the CGM-
GRB sample. We compare the relative position of our sample with respect to the
star forming main sequence at $z=$ 2 and 4. The star forming main sequence and
its scatter is computed using Equation (28) in Speagle et al. (2014). The key
characteristics of our sample in terms of galaxy properties are summarized
below.

1\. We divide the sample in two groups $-$ $z1$: 2-2.7 and $z2$: 2.7-5.9 $-$
which have equal number of objects and roughly equal cosmological timescale (1
and 1.4 Gyrs). We highlight that there is no significant difference in the two
groups in terms of SFR distribution. On the other hand, the host-galaxy
stellar mass distribution of the high-$z$ group is biased towards higher
masses as shown in Fig. 2 (panel 3). However, note that this is not an
intrinsic bias in the sample selection since our sample is selected based only
on the afterglow properties. Regardless, from Fig. 2, we conclude that our
sample primarily traces the low-mass end of the galaxy mass function at the
respective redshifts (by comparing against the characteristic stellar mass in
the Schechter function).

2\. While there is a significant spread, the majority of the GRB hosts in our
sample are within 0.5 dex (i.e. 3$\times$) of the star formation main sequence
at their respective redshifts (within observational uncertainties). It should
also be noted that majority of the GRB hosts here are below the main sequence.
Thus, our sample traces a moderately sub-main sequence galaxy population at
$z\sim 2-6$.

###  3.2 Blue-wing column density and outflows

To quantify outflows, we use the multi-component Voigt-profile fits to the
high-resolution GRB afterglow absorption spectra (in the rest-frame UV) and
the resulting column densities from Gatkine et al. (2019). We then integrate
the apparent column density (derived from the fit) bluewards of $-100$ km s-1.
We define this quantity as blue wing column density ($\mathrm{N_{out}}$),
which is a measure of the galactic outflow. This velocity threshold is
carefully chosen to minimize any contamination from the line-of-sight
absorption in the interstellar medium. A detailed justification for this limit
is provided in Gatkine et al. (2019) through kinematic and geometric modeling
of the ISM $+$ CGM of a representative galaxy in this sample (see sections
3.1, 5, 7.4, and Appendix B in Gatkine et al. 2019). This is similar to down-
the-barrel observations of outflows, albeit with random sightlines and using
high-resolution and high signal-to-noise spectra.

We compare the blue wing column density as described above with host galaxy
properties ($\mathrm{M_{*}}$ and SFR). In particular, we focus on four
species. These include two high ionization potential species (high-ion) $-$ C
IV and Si IV and two low ionization potential species (low-ion) $-$ Si II and
Fe II. Primarily, we used C IV 1550, Si IV 1402, Si II 1526, and Fe II 1608
absorption lines to trace the outflow-galaxy relations (summarized in Figs.
3$-$10). These species are selected for three reasons. 1) Their absorption
lines fall within the passbands over a large redshift range at $z>2$. 2) These
lines are not too weak (leading to underestimates) or not too strong
(saturated). In most cases, we do not have saturation in the blue wings. 3)
They allow us to compare the differences between the relations of high-ion and
low-ion species with host galaxy properties.

In particular, for low-ion lines, other alternatives have been used in the
literature including Si II 1260, O I 1302, and C II 1334. However, we did not
use them as the primary focus of the correlation investigation to avoid
potential blending issues. Notably, Si II 1260 has the most severe blending
issue due to S II 1259 that is essentially at a velocity offset of -200 km/s.
This can be seen in Fig. 11 showing the stacks of the respective lines in the
CGM-GRB sample. Therefore, it is difficult to reliably integrate Si II 1260
for measuring the outflows. For C II 1334 and O I 1302, the blending issue
(due to C II∗ 1335 and Si II 1304, respectively) is less severe for the
outflows. Hence we used Si II 1526 and Fe II 1608 which are free from such
blending issues for the primary investigation of the correlations. We further
conducted a secondary investigation with O I 1302 and C II 1334 as a
consistency check. Those results are summarized in Appendix C.

###  3.3 Inferring correlations and hypothesis testing

To investigate the presence of correlations between outflow and galaxy
properties, we primarily focus on the parameter space of logarithms of
$\mathrm{M_{*}}$, SFR, outflow column density, and maximum outflow velocity.
First, we perform Kendall-$\tau$ test by using a null hypothesis that there is
no intrinsic correlation between the two parameters. The $1-p$-value from
Kendall-$\tau$ test gives us the confidence level at which the null hypothesis
is rejected (i.e. a smaller p-value implies higher probability of the
existence of a correlation). Second, we perform a linear regression to infer
the best-fit line for each investigated correlation using Schmitt’s binned
regression (Schmitt, 1985). Note that we include all the upper (and lower)
limits in both of these analyses using the astronomy survival analysis code
called ASURV (Feigelson & Nelson, 1985; Isobe et al., 1986; Isobe & Feigelson,
1990). The resulting best-fit and Kendall-$\tau$ p-value are shown in the
correlation figures.

Due to multiple upper limits in the stellar mass and/or star formation rates
in the sample, simply using linear regression does not provide complete
information on the underlying correlations and/or their spread. Therefore, we
also divide the sample in two equal parts (around the median) based on the
galaxy property under consideration ($\mathrm{M_{*}}$ or SFR) and investigate
whether the sample distribution of the outflow property (eg:
$\mathrm{N_{out}}$) in the two bins is consistent with being drawn from the
same population. Therefore, for this hypothesis testing, our null hypothesis
is that there is no correlation between the galaxy properties and outflow
properties. If the null hypothesis is true, the two samples of outflow
properties (eg: column density) split based on galaxy property (eg:
$\mathrm{M_{*}}$ or SFR) are consistent with being drawn from the same
population, which would imply an absence of correlation between the given
outflow property and the galaxy property.

We plot the cumulative densities of both the samples, which further visually
shows the distinction or similarity between the two sample distributions. To
accommodate the upper limits in our observations, we calculate the cumulative
distribution function (CDF) and its spread is evaluated using a survival
analysis method called Kaplan-Meier estimator (Isobe et al. (1986); Feigelson
& Nelson (1985)). A python package called lifelines (Davidson-Pilon, 2019) is
used to calculate the CDF and its 1-$\sigma$ spread using the Kaplan-Meier
method. For two-sample hypothesis testing, we use the log-rank test in
lifelines. The resulting $p$-values describe the probability of the two
samples being drawn from the same distribution (and hence, no correlation).
With a stronger intrinsic correlation, the p-value is expected to be smaller.
In figures 3$-$10, the split in samples is shown with a vertical dotted line.
The median of the sample on either side and its 68-percentile spread are shown
in large square points for comparison (on X-axis, it is 68-percentile spread
in the sample, on Y-axis, it is 68-percentile spread in the inferred median).

##  4 Outflow correlations

In this section, we describe various outflow-galaxy correlations. A wide
variety of correlations were investigated. In this section, we only describe
the ones where at least one of the species show a $\gtrsim 2\sigma$
correlation. The rest of the correlations are described in Appendix B for the
sake of completeness and a visual comparison. All the investigated
correlations are summarized in Table 3.

###  4.1 Outflow column density $vs$ galaxy properties

In Figures 3 and 14, we plot the blue wing column densities of C IV, Si IV, Fe
II, and Si II with SFR and $\mathrm{M_{*}}$, respectively. The key findings
are summarized in the following subsections.

####  4.1.1 Blue-wing detection fraction

We define the detection fraction as the number of objects with detected blue-
wing absorption divided by the total number of objects in each sample. The
detection fractions in the left and right samples in Fig. 3
($\mathrm{N_{out}}$ $vs$ SFR) are: C IV (1 and 0.92), Si IV (0.92 and 1), Fe
II (0.62 and 0.71), and Si II (0.66 and 0.77). The relative difference in the
left and right samples is insignificant (i.e. contributed by an excess of one
non-detection in one of the samples). We find the same result Fig. 14
($\mathrm{N_{out}}$ $vs$ $\mathrm{M_{*}}$). Thus, we conclude that the
detection fractions in the outflow do not strongly depend on the galaxy’s
stellar mass or star formation rate. However, we note that the blue-wing
detection fraction is significantly higher in high-ion species (C IV, Si IV)
compared to the low-ion species, hinting at a prevalence of outflows primarily
traced by the warm phase ($\mathrm{10^{4.5}-10^{5.5}}$ K, Tumlinson et al.
(2017); Gatkine et al. (2019)) assuming collisional ionization equilibrium.

####  4.1.2 $N_{out}$ $vs$ SFR

From visual inspection of $\mathrm{N_{out}}$ $vs$ SFR panels in Fig. 3, we
note that there is a greater prevalence of high $\mathrm{N_{out}}$ in the
high-SFR sample compared to the low-SFR sample. The CDF plots provide a
quantitative measure of any such relation. First, we focus on the high-ion
species (C IV and Si IV). The low-SFR and high-SFR samples are most distinct
(i.e. small $p$-value) in high-ion species. We can reject the hypothesis of
absence of correlation (between $\mathrm{N_{out}}$ $vs$ SFR) for C IV and Si
IV with 87% and 98% confidence ($1-p$) respectively. Thus, a higher star
formation rate is likely to be correlated with a higher column density of C IV
and Si IV in the outflows (albeit with a relatively lower confidence). In
addition, we note that the spread of $\mathrm{N_{out}}$ (as evident from the
68-percentile errorbars on the median points) is considerably higher in the
high-SFR sample compared to the low-SFR sample (by $\sim$ 0.3-0.5 dex). This
effect is discussed in more detail in Section 5.4.

Figure 3:  Column density in the outflows in GRB hosts traced by high-ion (C
IV, Si IV) and low-ion (Fe II, Si II) species $vs$ their SFR. The
Kendall-$\tau$ $p$-value indicates the strength of correlation ($1-p$ is the
confidence level of the correlation). The vertical dotted line splits the
sample into two equal groups around the median SFR. The CDF of each group is
shown on the right to compare the distributions of low-SFR and high-SFR parts
of the sample. The log-rank-test $p$-value shown in the CDF plot measures the
extent to which the distributions are similar and hence consistent with no
correlation. The median and 68-percentile spread of the median column density
is shown using the blue and orange squares. The best-fit line (including
limits in the data) is also shown here. Apart from a weak correlation, there
is a significant increase in the spread of column density at high SFR
(particularly for the high-ion lines).

Unlike the case of high-ion species, the $p$-value is high, indicating a weak
(for Si II) or no correlation (for Fe II) of low-ion $\mathrm{N_{out}}$ with
SFR. The fact that we see a stronger high-ion outflow with star formation but
only a weak change in the low-ion species indicates that a higher SFR leads to
a stronger high-ion traced outflow but does not significantly affect the low-
ion traced outflow phase. In other words, a higher SFR selectively enriches
the outflow with high-ions.

###  4.2 Outflow kinematics and galaxy properties

We study the relationship between outflow kinematics and galaxy properties
using the maximum velocity of the outflow (i.e. maximum velocity in the blue
wing), $\mathrm{V_{max}}$. We define $\mathrm{V_{max}}$ as the velocity of the
most blue-shifted absorption component $+$ the half-power width of that
component. The maximal velocity is a key determinant of the outflow energy and
mass outflow rate and hence, the enrichment of the CGM (and IGM). Therefore,
in this paper, we use $\mathrm{V_{max}}$ as a proxy for outflow kinematics to
investigate the effect of galaxy properties.

####  4.2.1 Outflow $\mathrm{V_{max}}$ $vs$ SFR

From Fig. 4, a strong correlation is observed between $\mathrm{V_{max}}$ and
SFR for high-ion species (2-$\sigma$ for C IV and 3-$\sigma$ for Si IV). The
best-fit relations are given by $\mathrm{V_{max}}$ $\propto$
$\mathrm{SFR^{0.12}}$ and $\mathrm{SFR^{0.29}}$ for C IV and Si IV,
respectively. This correlation is much tighter than the SFR $-$ column density
relation. The smaller variation would mean the velocity gain due to higher SFR
is mostly independent of the sightline being probed. By combining this with
previous results from Section 4.1.2, it can be said that star formation
uniformly drives up the high-ion outflow velocity, but also imparts a large
variance in the overall amount of outflowing material (column density) that is
being driven. We discuss this aspect in more detail in Section 5.4.

Figure 4:  Same as Figure 3, for the maximum outflow velocity,
$\mathrm{V_{max}}$ $vs$ $\mathrm{SFR}$. The horizontal dashed line in the
panels shows the 100 km s-1 level, which we treat as the threshold for
outflow.

On the other hand, for low-ion species, the correlation is weaker (39% and 93%
confidence for Fe II and Si II respectively), primarily due to a larger spread
in the $\mathrm{V_{max}}$ compared to high-ion species (which can be easily
seen by comparing their CDF plots). This shows a larger variance in the
kinematics of low-ion traced outflows relative to the high-ion traced
outflows.

In addition, Fig. 4 also shows that the high-redshift and the low-redshift
populations (groups $z1$ and $z2$) follow the same trend for
$\mathrm{V_{max}-SFR}$ correlation. We do not observe any significant
evolution in the relation of outflow kinematics and SFR. This further
corroborates a previous result from Gatkine et al. (2019) which shows no
evolution in the CGM kinematics in the same two redshift bins.

####  4.2.2 Outflow kinematics $vs$ Halo mass

The halo mass is an important factor in determining whether the outflow will
eventually escape and enrich the intergalactic medium or it will virialize and
enrich the CGM. Therefore, it is important to understand how the outflow
velocity compares with the characteristic velocity of the halo. To study how
the outflow kinematics relate to the halo mass, we define a normalized
velocity, $\mathrm{V_{norm}}$ = $\mathrm{V_{max}}$/$\mathrm{V_{circ,halo}}$,
where $\mathrm{V_{circ,halo}}$ is the halo circular velocity. The
$\mathrm{V_{circ,halo}}$ is calculated using the following equations from Mo &
White (2002):

$V_{circ,halo}=\left(\frac{GM_{halo}}{r_{halo}}\right)^{1/2}$ (4)

$r_{halo}=\left(\frac{GM_{halo}}{100\Omega_{m}H_{0}^{2}}\right)^{1/3}(1+z)^{-1}$
(5)

Here, $\mathrm{M_{halo}}$ is calculated using the redshift-dependent stellar-
to-halo mass ratio from Wechsler & Tinker (2018).

The $\mathrm{V_{norm}}-\mathrm{M_{halo}}$ relation is summarized in Fig. 5. We
observe a clear inverse correlation in both high-ion and low-ion outflows. The
inverse correlation is slighlty stronger in high-ions (confidence: 99.7% in C
IV, 99.9% in Si IV) compared to low-ions (confidence: 86.4% in Fe II, 99.2% in
Si II). We note that most of the low-redshift points appear in the
low-$\mathrm{M_{halo}}$ group while the high-redshift points appear in the
high-$\mathrm{M_{halo}}$ group. This is because the stellar-to-halo mass ratio
is larger at higher redshifts. Also, $\mathrm{V_{circ}}$ scales as
$\mathrm{M_{halo}^{1/3}}$, thus lowering the value of $\mathrm{V_{norm}}$ for
the high-redshift objects.

The key takeaway from the $\mathrm{V_{norm}}-\mathrm{M_{halo}}$ relation is
that the outflows in low-mass halos have a greater probability of reaching
and/or escaping the outer CGM and enriching the intergalactic medium. Assuming
that $\mathrm{V_{max}}$ reflects the gas motion at the largest radii of the
outflows, as interpreted in Martin & Bouché (2009) (with or without
acceleration at larger radii), we can infer that outflows with
$\mathrm{V_{max}}$ $>$ $2\times\mathrm{V_{circ}}$ (i.e.
$\mathrm{log(V_{norm})}>0.3$) are most likely to escape the CGM and enrich the
intergalactic medium at high redshifts.

###  4.3 Outflow $\mathrm{V_{norm}}$ $vs$ specific SFR

Following the strong $\mathrm{V_{norm}-M_{halo}}$ relation observed in Section
4.2.2, we set out to explore whether $\mathrm{V_{norm}}$ (which is a gauge of
whether the outflow can escape) is impacted by the sSFR. These results are
summarized in Fig. 6. We clearly observe a strong correlation between
$\mathrm{V_{norm}}$ and sSFR for both high- and low-ions (except Fe II).

Our results are analogous to those found in Heckman & Borthakur (2016) for
extreme starbursts at $z\sim 0-0.7$ using Si II line. They find that
$\mathrm{V_{norm}}$ $\propto$ $\mathrm{sSFR^{0.25}}$ with a $>$ 3-$\sigma$
confidence. Our scaling relations are consistent with this slope
($\mathrm{V_{norm}}$ $\propto$ sSFR0.26, sSFR0.24, sSFR0.26, and sSFR0.39 for
C IV, Si IV, and Si II, with 3-$\sigma$, 3.7-$\sigma$, and 2.6-$\sigma$
confidence, respectively).

We note that $\mathrm{V_{norm}}$ is larger in Heckman & Borthakur (2016). This
is because they study extreme starburst galaxies with typical SFRs that are
higher by at least an order of magnitude compared to our sample. This leads to
a $2-3$x boost in outflow velocities. Regardless, the correlation is fairly
robust in the sSFR range : $\mathrm{log(sSFR)}\sim-10$ to $-7.5$, similar to
our range of interest. Given the strong agreement between the slopes obtained
from the low-redshift results (Heckman & Borthakur, 2016) and our high-
redshift results, we argue that the $\mathrm{V_{norm}-sSFR}$ proportionality
may be a redshift-independent fundamental property of star-forming galaxies.
This should be investigated further using low- and intermediate-redshift
analogs of high-$z$ star-forming galaxies.

This is the first such evidence of $\mathrm{V_{norm}-sSFR}$ correlation at
$z>2$. The combined $\mathrm{V_{norm}-sSFR}$ and
$\mathrm{V_{norm}}$$-$$\mathrm{M_{halo}}$ relations (Section 4.2.2) imply that
the outflows from the low-mass halos and high-sSFR galaxies have the highest
probability of escaping the halo and transferring matter to the intergalactic
medium, and thus enriching it with metals.

Figure 5:  Same as Figure 3, for the normalized velocity,
$\mathrm{V_{max}/V_{circ,halo}}$ $vs$ $\mathrm{M_{halo}}$. Figure 6:  Same as
Figure 15, for the the scaling relations of normalized maximum velocity
($\mathrm{V_{max}/V_{circ,halo}}$) with specific SFR (= SFR /
$\mathrm{M_{*}}$).

###  4.4 Outflow rate and mass loading correlations

Assuming a spherical outflow geometry, the time-averaged mass outflow rate
($\dot{M}_{out}$) across a cross-section area $A$ (over a dynamical timescale
$\sim$ $\mathrm{R/V_{mean}}$) can be written as:

$\dot{M}_{out}=m\frac{A}{R}\int{N_{a}(v_{out})v_{out}dv}$ (6)

where $m$ is the mean atomic mass per H atom, $R$ is the radius at which the
cross-section is evaluated, and $N_{a}(v)$ is the apparent column density (per
unit velocity) at a velocity $v$.

Here, we use the observable parameter $\int{N_{a}(v_{out})v_{out}dv}$
(abbreviated as N$\cdot$V) as a proxy for the mass outflow rate since the
radial profile of the outflow is unknown. We use the best-fit Voigt profiles
for the spectra in our sample as derived in Gatkine et al. (2019) and evaluate
$N_{a}(v_{out})$ using the apparent optical depth method ( Savage & Sembach
1991, also see section 4.2 in Gatkine et al. 2019). The integration is
performed leftwards of $v$ = -100 $\mathrm{km~{}s^{-1}}$ to consider only the
outflows. We note that using the N$\cdot$V of various high- and low-ion
species as the proxy for $\mathrm{\dot{M}_{out}}$ has certain limitations. The
ionization fraction of the species, the metallicity of the outflow, and the
outflow geometry can vary within the redshift range and as a function of
galaxy properties. Despite these variations, the N$\cdot$V provides a useful
insight into the comparative outflow dynamics of the high-ion and low-ion
traced outflows. Keeping these strengths and limitations in mind, we
investigate the following correlations to trace the outflow dynamics.

####  4.4.1 Outflow rate vs SFR

In Fig. 7, we plot N$\cdot$V (a proxy for the mass outflow rate) against SFR.
We observe an increasing trend in the high-ions (C IV and Si IV) with an
approximate log-log slope of $\sim 0.5$, suggesting that the mass outflow rate
of high-ion traced outflows is driven by star formation. However, we do not
observe such a trend in the low-ions. This is expected given the large scatter
(or absence) of $\mathrm{N_{out}}$ and $\mathrm{V_{max}}$ correlations with
SFR in the low-ion species.

Figure 7:  Same as Figure 3, for the relation of a proxy of outflow rate
(N$\cdot$V = $\int N_{a}vdv$) vs $\mathrm{SFR}$. The units for Y-axis are:
$\mathrm{cm^{-2}km~{}s^{-1}}$. The regression fit and slope are shown where
the Kendall-$\tau$ p-value is less than 0.15 (i.e. 1.5-$\sigma$ or higher
level for correlation).

The high-ion correlation has a slope (0.4$-$0.7, see Fig. 7) slightly
shallower than the slope obtained by Heckman et al. (2015)
($\mathrm{log(\dot{M}_{out})}-$log(SFR) slope of $\sim$ 1) for local
starbursts using both high- and low-ion lines, albeit with a larger scatter
(see Fig. 5 in their paper). Chisholm et al. (2015) also find a similar
correlation (slope $\sim$ 0.5) in a sample of local star-forming galaxies
using low-ion lines. These similarities indicate a common star-formation-
driven outflow mechanism from $z$ $\sim$ 0 to $z$ $\sim$ 6\. We note here,
however, that the marked difference that we see between the high- and low-ion
correlations imply a systematic difference in the outflow mechanisms of the
phases traced by them. The blue wings in the high-ion absorption appear to be
dominated by outflows driven by star formation (as supported by the
correlations) while the blue wings in the low-ions are more complex.

To gauge how the mass outflow rate compares with the star formation of the
sample, we take C IV species as an example and estimate the total outflow rate
as follows:

$\dot{M}_{out,H}=\dot{M}_{out,CIV}\frac{1}{f_{CIV}Z_{out}}\left(\frac{N_{H}}{N_{C}}\right)_{solar}$
(7)

Here, $f_{CIV}$ is the ionization fraction of C IV, $Z_{out}$ is an estimate
of the outflow metallicity, and $N_{C}/N_{H}$ is the carbon abundance
(Lodders, 2003). Using equations 6 and 7,

$\displaystyle\dot{M}_{out}\approx
1~{}\mathrm{M_{\odot}~{}yr^{-1}}\left(\frac{(N.V)_{CIV}}{10^{16.5}cm^{-2}km/s}\right)\left(\frac{5~{}\mathrm{kpc}}{R_{out}}\right)$
(8)
$\displaystyle\left(\frac{0.3}{f_{CIV}}\right)\times\left(\frac{0.25}{Z_{outflow}}\right)\left(\frac{2.8\times
10^{-4}}{(N_{C}/N_{H})_{\odot}}\right)$

Here, a representative N$\cdot$V value is taken for C IV species at SFR $\sim$
10 $\mathrm{M_{\odot}yr^{-1}}$ (see Fig. 7). There is a scatter of about an
order of magnitude in either direction in the observed N$\cdot$V values for C
IV. The values of $Z_{outflow}$ and $f_{CIV}$ are assumed to be the same as
those in Gatkine et al. (2019) and the radius of the outflowing shell is
assumed to be 5 kpc. Note that this mass outflow rate is a lower limit given
the conservative assumptions of C IV ionization fraction (see Sec 6.1.2 in
Gatkine et al. 2019 for a detailed explanation) and outflow radius. The
measured outflow properties can arise from a radius as large as 50 kpc (i.e.
the typical virial radius of these galaxies). Thus, the $\dot{M}_{out}$
estimated in Eqn 8 can be larger by almost an order of magnitude. Regardless,
given the scatter in the observed N$\cdot$V, many of the observed systems have
an estimated mass outflow rate comparable or even greater than the SFR.

Given that N$\cdot$V $\propto$ $\mathrm{SFR^{0.4}}$ for C IV (or,
$\mathrm{SFR^{0.7}}$ for Si IV), it is evident that the lower-SFR galaxies in
our sample experience a higher mass outflow rate as a fraction of their SFR
and thus undergoing an efficient removal of the gas compared to high-SFR
galaxies.

####  4.4.2 Mass loading factor

Similar to section 4.4.1, we use N$\cdot$V/SFR as the proxy for mass loading
factor ($\eta$ = $\mathrm{\dot{M}_{out}/SFR}$) to measure how efficiently the
galaxy removes gas (in comparison to star formation rate). We study its
relation with SFR and $\mathrm{M_{halo}}$ in Figs. 8 and 9 respectively to
understand the potential drivers of the mass-loading factor.

For the $\eta~{}vs$ SFR plot (Fig. 8), we do not find a clear correlation.
However, by splitting the sample into two equal high-SFR and low-SFR bins and
comparing the medians, we observe a declining trend with SFR. This suggests a
higher outflow efficiency in the low-SFR galaxies, particularly for the
outflow traced by low-ion lines. We note that this result has to be seen in
conjunction with the N$\cdot$V $vs$ SFR plot (Fig. 7). The weaker decline in
the high-ions is driven by the intrinsic correlation between N$\cdot$V and
SFR. On the other hand, the stronger decline seen in the low-ions is due to a
lack of such an intrinsic N$\cdot$V$-$SFR correlation, leaving the 1/SFR as
the dominant term. The observed slow decline in the high-ions is in agreement
with the simulated slope for stellar feedback ($\eta$ $\sim$ SFR0.35 for winds
leaving the ISM) in the star-forming galaxies at $z~{}>~{}2$ in the EAGLE
simulations (Mitchell et al. 2020, see Fig. 3 therein).

In the $\eta~{}vs~{}\mathrm{M_{halo}}$ plot (Fig. 9), we observe a large
scatter in this relation and thus, we cannot find any statistically
significant correlation for either low- or high-ion species. The large scatter
in this relation makes a comparison with cosmological simulations difficult.
There is a possible hint for a declining outflow efficiency at high halo
masses, particularly for the low-ions (as seen from the binned sample),
however, a significantly larger sample is needed in the future to constrain
any underlying correlation.

Figure 8:  Same as Figure 3, for the relation of a proxy of mass loading
factor (N$\cdot$V/SFR = $\int N_{a}vdv$/SFR) vs $\mathrm{SFR}$. The units for
Y-axis are: $\mathrm{cm^{-2}km~{}s^{-1}M_{\odot}^{-1}yr}$.  Figure 9:  Same as
Figure 3, for the relation of a proxy of the mass loading factor
(N$\cdot$V/SFR = $\int N_{a}vdv$/SFR) vs $\mathrm{M_{halo}}$. The units for
Y-axis are: $\mathrm{cm^{-2}km~{}s^{-1}M_{\odot}^{-1}yr}$.

####  4.4.3 Momentum flux vs SFR

The stellar processes (supernovae and stellar winds) can drive momentum flux
as a combination of both radiation pressure (Murray et al., 2005) and ram
pressure of a hot outflow collectively created from the ejecta of massive
stars (Chevalier & Clegg, 1985; Veilleux et al., 2005; Heckman & Borthakur,
2016). Similar to Eqn. 6, the time-averaged momentum flux across a cross-
section area $A$ (over a dynamical timescale $\sim$ $\mathrm{R/V_{mean}}$) can
be written as:

$\dot{p}_{out}=m\frac{A}{R}\int{N_{a}(v_{out})v^{2}_{out}dv}$ (9)

We use the observable parameter $\int{N_{a}(v_{out})v^{2}_{out}dv}$
(abbreviated as N$\cdot$V2) as a proxy for the momentum flux. In Fig. 10, we
plot N$\cdot$V2 against SFR. We observe a correlation for high-ions:
$\dot{p}_{out}$ $\propto$ $\mathrm{SFR^{0.3}}$ for C IV and
$\mathrm{SFR^{0.9}}$ for Si IV, with a stronger correlation for Si IV.
However, we find no correlation for the low-ions. This is expected, given the
strong correlation between $\mathrm{V_{max}}$ and SFR for high-ions observed
in Sec. 4.2.1 and a lack of such a correlation for low-ions.

Next, we compare the momentum flux with star-formation-driven momentum flux to
examine whether it is sufficient to drive the observed momentum flux in the
high-ion traced outflows. For a standard Kroupa/Chabrier initial mass function
and a constant star-formation rate, the estimated stellar-driven momentum flux
is given by the sum of radiation pressure and ram pressure components (Heckman
et al., 2015) as:

$\dot{p}_{*}=4.8\times
10^{33}~{}\mathrm{dynes}\times\mathrm{SFR}~{}\mathrm{(\mathrm{in~{}M_{\odot}yr^{-1}})}$
(10)

To compare this with the observed momentum flux, we use C IV ion as a probe
and rewrite Eqn. 9 as follows:

$\displaystyle\dot{p}_{out}=5\times
10^{32}~{}\mathrm{dynes}\left(\frac{N.V^{2}}{10^{18.5}cm^{-2}km^{2}/s^{2}}\right)\left(\frac{0.3}{f_{CIV}}\right)$
(11)
$\displaystyle\times\left(\frac{0.25}{Z_{outflow}}\right)\left(\frac{2.8\times
10^{-4}}{(N_{C}/N_{H})_{\odot}}\right)$

Here, a representative N$\cdot$V2 value is taken for C IV species at SFR
$\sim$ 10 $\mathrm{M_{\odot}yr^{-1}}$ (see Fig. 10). There is a scatter of
about an order of magnitude in either direction in the observed N$\cdot$V2
values for C IV. The values of $Z_{outflow}$ and $f_{CIV}$ are assumed to be
the same as those in Gatkine et al. (2019) and the radius of the outflowing
shell is assumed to be 5 kpc. We issue the same caution as in Sec. 4.4.1 about
this estimate being a lower limit given the conservative assumptions of C IV
ionization fraction and outflow radius.

Despite the scatter in the observed N$\cdot$V2, it is clear that the estimated
momentum flux (from Eqn. 11) is smaller than that driven by star formation
(from Eqn. 10). Thus, it is evident that the observed high-ion traced outflows
are primarily driven by the momentum directly injected by supernovae and
stellar winds. This is different from the results on the low-redshift
starburst and star-forming galaxies (Heckman et al. 2015 and Chisholm et al.
2017, respectively), where the momentum flux in the outflows is comparable or
even greater than that imparted by star formation for a significant fraction
of the sample. However, the slope of the N$\cdot$V${}^{2}-$SFR (or
$\dot{p}_{out}-$SFR) correlation for high-ions (particularly Si IV) is
consistent with the local starburst result from Heckman et al. (2015). In the
future, we will perform detailed ionization modeling to further examine these
correlations.

Figure 10:  Same as Figure 3, for the relation of a proxy of momentum flux
(N$\cdot$V2 = $\int N_{a}v^{2}dv$) vs $\mathrm{SFR}$. The units for Y-axis
are: $\mathrm{cm^{-2}(km~{}s^{-1})^{2}}$. The regression fit and slope are
shown where the Kendall-$\tau$ p-value is less than 0.15 (i.e. 1.5-$\sigma$ or
higher level for correlation).

##  5 Discussion

###  5.1 Star-formation-driven outflow

The strong correlation of the blue-wing column density and maximum outflow
velocity with SFR, as seen in Figs. 3 and 4 provides an important evidence for
a star-formation-driven outflow. If we consider Si IV as the outflow tracer,
the best-fit lines suggest $\mathrm{N_{SiIV}}$ $\propto$ SFR0.25 and
$\mathrm{V_{max,SiIV}}$ $\propto$ SFR0.29. Our $\mathrm{V_{max}}$ slope is in
close agreement with the slope derived in Sugahara et al. (2017) for $z\sim 2$
star-forming galaxies (= 0.25).

Similarly, the (weak) trends of high-ion column density and $\mathrm{V_{max}}$
with $\mathrm{M_{*}}$ are consistent with previous observations of C IV-traced
outflows from Du et al. (2018) at $z\sim 1-1.35$. However, our sample extends
to 0.4 dex lower mass, where we start to see a systematic decline in
$\mathrm{N_{out}}$ as well as $\mathrm{V_{max}}$ which is not seen in these
previous observations. We argue that this decline is driven by the lower star
formation rates. Similarly, the large spread (in $\mathrm{N_{out}}$ and
$\mathrm{V_{max}}$ ) observed at log($\mathrm{M_{*}/M_{\odot}}$) $>$ 9.4 is
due to the large spread in SFR at this mass range in our sample, as shown in
Fig. 2. Hence, we conclude that the apparent trends between $\mathrm{V_{max}}$
or $\mathrm{N_{out}}$ and $\mathrm{M_{*}}$ are almost entirely modulated by
the SFR.

For low-ion outflows, the best-fit relation for Si II is
$\mathrm{V_{max,SiII}}$ $\propto$ SFR0.14. Various surveys from low to high
redshifts have reported correlation between some form of $\mathrm{V_{max}}$
for low-ion species and SFR. Our results agree with the slopes observed in
Weiner et al. (2009); Bradshaw et al. (2013); Bordoloi et al. (2014); Chisholm
et al. (2015) with redshifts ranging from $z\sim 0-1.6$. Our correlations also
qualitatively agree with $z\sim 0.5$ sample of Rubin et al. (2014) (only for
their galaxies without systemic absorption). From the $\mathrm{V_{max}-SFR}$
slope derived here, we further corroborate the suggestion from Sugahara et al.
(2019) that the $\mathrm{V_{max}-SFR}$ could be a more fundamental relation
over a wide redshift range ($z\sim 0-6$) for star-forming main sequence.
However, we caution that there is a significant spread in the relation
depending on the species used for deriving them (for instance, we get a slope
of 0.29 for Si IV and 0.14 for C IV).

Both Erb et al. (2012) and Rubin et al. (2012) find a strong correlation of
$\mathrm{V_{max}}$ with stellar mass and weak correlation with SFR for
galaxies of similar mass and SFR range to our sample in the redshift ranges
$z\sim 1-2$ and $z\sim 0.3-0.7$ for low-ion species. Similarly, Bordoloi et
al. (2014) and Chisholm et al. (2015) find a high-significance correlation
with $\mathrm{M_{*}}$ using Mg II and Si II species respectively. Rubin et al.
(2012) argue that this could be because star formation history and/or galaxy
dynamics have a more direct physical link to maximum wind velocities than
current star formation activity. However, we find that the trends flip $-$ a
stronger correlation with SFR compared to stellar mass (for instance, consider
Si II in Figs. 4 and 16). This indicates that for low-mass galaxies at $z>2$,
the current star formation has a greater impact on the observed low-ion
outflows than its star formation history. This is interesting from the
perspective of causal connection. The timescale required for a 250 km s-1
outflow to travel 50 kpc (roughly the virial radius of typical galaxies at
$z\sim 3$) is about 200 Myr, while the UV-based SFR that we measure is from
the past 100 Myr. This could indicate a long-lasting star formation activity.
This is in line with our typical depletion timescale of 500 Myr (using
($\mathrm{M_{gas}}\sim\mathrm{M_{*}}$)/SFR $\sim$ 1/sSFR).

We compare our scaling relations with the recently published results from the
TNG50 simulations (Nelson et al., 2019). We find that the slopes of our
$\mathrm{V_{max}}-\mathrm{SFR}$ relation (for both high and low ions, slope
$\sim$ 0.12$-$0.28) are consistent with the slope of $0.15-0.2$ in Nelson et
al. (2019) (see Fig. 15 therein). However, our results do not fully agree with
the predicted $\mathrm{V_{max}}-\mathrm{M_{*}}$ relation. While their slope of
0.2 at $z=2$ is consistent with our high-ion results (slope = 0.19 for Si IV),
Nelson et al. (2019) show a slope that steepens with redshift (eg: slope = 0.3
at $z=4$). We do not observe such steepening in our
$\mathrm{V_{max}}-\mathrm{M_{*}}$ plots (Fig. 16). In fact, the slope appears
to be shallower in the high-redshift group (group $z2\sim 2.7-6$). This
discrepancy may be explained by a combination of two factors: a) the
increasing contribution of hotter phases in the outflow with increasing
$\mathrm{M_{*}}$ (see Fig. 10 in Nelson et al. (2019)) and b) the outflow
scaling relations shown in Nelson et al. (2019) include all the phases while
our observations might only cover the warm phases for high ions (assuming
collisional ionization equilibrium).

Given the strong $\mathrm{N_{out}-SFR}$ and $\mathrm{V_{max}-SFR}$
correlations, particularly for high-ion outflows, it is expected that the mass
outflow rate would be strongly correlated with SFR. In Sec. 4.4.1 and Fig. 7,
we use N$\cdot$V (= $\int{N_{a}(v_{out})v_{out}dv}$) as a measure of the mass
outflow rate and obtain the following relations,

$\displaystyle\mathrm{\dot{M}_{out}}\propto\mathrm{SFR^{0.4}}~{}\mathrm{for~{}C~{}IV}$
(12)
$\displaystyle\mathrm{\dot{M}_{out}}\propto\mathrm{SFR^{0.7}}~{}\mathrm{for~{}Si~{}IV}$

The slope obtained here is in good agreement with the slope observed in the
FIRE simulations ($\sim$ 0.6) at $4>z>2$ with the rate evaluated at
0.25$\mathrm{R_{vir}}$ (see Fig. B2 in Muratov et al. (2015)). This slope is
also consistent with EAGLE simulations at similar redshift range ($z\sim
2.4-4.7$) for gas particles ejected out of the ISM through galactic winds (see
Fig. 3 in Mitchell et al. (2020)). Both of these slopes have been obtained in
the log(SFR) range of -0.5 to 1.5, similar to our SFR range.

###  5.2 Stacking comparison with previous studies

We compare the stacked profiles of the high- and low-ion species in the CGM-
GRB sample against the publicly available stacks of high- and low-redshift
down-the-barrel outflow studies from Rigby et al. (2018). This includes a
$z\sim 2$ sample of 14 gravitationally lensed starburst galaxies (MEGaSaURA
sample) and a sample of 41 starburst / star-forming (majority starburst) at
$z\sim 0.1$ (the COS-sample , Chisholm et al. 2016).

We observe that the outflows in both MEGaSaURA and COS-sample stacks $-$ which
consist of starbursts (SFR $\sim$ 10$\times$ main sequence) at their
respective redshifts $-$ are considerably broader and stronger compared to the
CGM-GRB sample, where majority of the galaxies at $\sim$0.5 dex below the main
sequence at their respective redshifts (see Fig. 2). Further, we note that the
galaxies in the $z\sim 0.1$ COS-sample have a similar $\mathrm{M_{*}}$ and SFR
to our CGM-GRB host galaxies at $z>2$ (see Chisholm et al. 2016), yet the COS-
sample outflows are significantly stronger and broader than the CGM-GRB
outflows for both high- and low-ions. This shows that, over a broad redshift
range, the strength of the outflow is correlated with the main-sequence offset
at the respective redshifts rather than the simple star-formation rate. This
should be a critical consideration while comparing outflow samples at
different redshifts.

On the other hand, the MEGaSaURA sample galaxies have a similar redshift range
($z\sim$ 1.68$-$3.6) to our CGM-GRB sample, but are significantly above the
main sequence (and hence, starburst). Similar to the COS- sample, their
outflows are significantly broader in velocity and stronger in absorption
depth for both high- and low-ions. This comparison further corroborates that
the SFR-driven outflows are strongly correlated with the main-sequence offset
at the respective redshifts.

We should, however, caution that a direct stacking comparison of the line
profiles with previous down-the-barrel samples in the literature is an unfair
comparison due to several reasons. The most important is the selection effect.
By definition, down-the-barrel samples at $z>2$ are selected to be galaxies
bright enough in rest-frame UV for absorption spectroscopy ($g_{AB}$ $<$ 21,
for MEGaSaURA). On the other hand, the GRB hosts in our sample are faint, with
apparent magnitude $>$ 23 in most cases. In fact, the low-mass, low-SFR
galaxies at high-z are extremely difficult to probe at high resolution using
down-the-barrel technique. However, GRB sightlines offer a unique opportunity
to probe this population, which is not feasible at scale by any other method.

Lastly, due to a combination of selection effect and galaxy properties, the
down-the-barrel observations at high-z preferentially look directly down the
outflows. The GRB sightlines, on the other hand, are random. These reasons
also contribute to the marked difference seen in the stacks of CGM-GRB sample
and down-the-barrel high-z stacks (eg: MEGaSaURA) and should be studied in
detail in the future.

Figure 11:  The median absorption line profile for the CGM-GRB sample (in
green) of the individual spectra (in blue) in the CGM-GRB sample. The red line
in the bottom panel shows the rms noise (shifted vertically by $-1.5$). The
overlaid traces are stacked median profiles of the MEGaSaURA $z\sim 2$ sample
of 14 lensed starburst galaxies (in orange) and $z\sim$ 0.1 COS-sample of 41
starburst / star-forming galaxies (in magenta), both sourced from Rigby et al.
(2018).

###  5.3 Evidence for High-ion traced outflows

We observe three key differences between high-ion and low-ion outflows: 1) The
detection fraction of low-ion outflows is lower than high-ion outflows ($\sim$
65% $vs$ 95%), irrespective of the SFR. 2) The correlations of high-ion
$\mathrm{N_{out}}$ and $\mathrm{V_{max}}$ with SFR are stronger compared to
low-ions (see Figs. 3 and 4). Also, the spread in $\mathrm{V_{max}}$ is higher
in low-ions. 3) Low-ion outflows show a much steeper decline in
$\mathrm{V_{norm}}$ with higher halo masses compared to high-ion outflows (see
Fig. 5). These difference point towards a systematic difference in the high-
ion and low-ion outflows. In addition, we can infer that the outflows in our
sample are primarily traced by high-ions.

It is well known that galactic outflows are multiphase in nature and the
aforementioned differences can help understand the phase structure of the
outflows. Cosmological simulations find a complex shift in the temperature
distribution of the outflow with the dominant phase shifting towards higher
temperatures as the stellar mass increases at $z\sim 2$ (see Fig. 10 in Nelson
et al. (2019)). This trend is also seen in FIRE simulations with the
contribution of $\mathrm{T}>10^{5.3}$ K phase increasing and that of
$10^{4}<\mathrm{T}<10^{4.7}$ K phase decreasing with an increasing halo mass
at $z\sim 2$ (in the range $\mathrm{M_{halo}}$ $\sim$ $10^{11}-10^{12}$
$\mathrm{M_{\odot}}$ (see Fig. A1 in Hafen et al. (2019)). These effects can
potentially explain the sharp decline in low-ion $\mathrm{V_{norm}}$ and their
weaker and shallower correlations in terms of $\mathrm{N_{out}}$. Photo-
ionization modeling of the observations would help understand whether this
explanation is correct. Regardless, we infer that the correlations seen in the
star-forming galaxies using random, narrow sightlines (as offered by GRBs)
imply high-ion dominated outflows in the star-forming galaxies at $z>2$.

###  5.4  Outflow Geometry

The sharp increase in the spread of outflow column density at SFR $\gtrsim$ 10
$\mathrm{M_{\odot}yr^{-1}}$, particularly for high-ion lines is an indication
of systematic variance in the outflow properties with SFR.

There are various possible causes that could lead to an elevated spread in the
apparent outflow column density. Some of the scenarios include a variance in
the metallicity of the outflowing clouds due to inefficient mixing of metals
(Schaye et al., 2007), a variance in the entrainment efficiency of the ISM, or
the onset of wind-stimulated condensation in the CGM (see Heckman et al.
(2017)) at high star formation rates. A more careful treatment of the physical
processes in the outflows is warranted to explain this phenomenon.

Another possible explanation for such a variance is outflow geometry. Given
that GRBs sample a very narrow beam in a random direction offset from the
galactic center, a spherical outflow is less likely to lead to the observed
spread. In a non-spherical outflow, with certain regions of high gas
entrainment efficiency compared to others (eg: a biconical outflow), there
will be preferred sightlines that exhibit high column density and others will
exhibit a relatively lower column density. This is illustrated in Fig. 12. In
addition, projection effects become more important in a non-spherical outflow
as opposed to a spherical outflow. Therefore, we speculate that the increased
spread could indicate development of a non-spherical outflow in the high-SFR
systems, while the low-SFR galaxies having more uniform/spherical outflows.

Such development of outflow collimation naturally (hydrodynamically) emerges
in the recent TNG simulation results along the minor axis of the galaxy
despite isotropic injection of the stellar feedback (Nelson et al., 2019).
Thus, they come to a similar conclusion in their simulations: the mass outflow
rate of winds is not directionally isotropic, even for
$\mathrm{M_{*}}=10^{10}\mathrm{M_{\odot}}$ at $z=1$. While they suggest that
the effect is more pronounced as we go below $z\sim 2$ due to the rise of
ordered rotation (and resulting emergence of galactic discs), it will be
interesting to probe how this effect evolves in the redshift-SFR space.

###  5.5 Evolution with redshift

While we see that $\mathrm{V_{max}}$ for high-ion lines is correlated with SFR
and $\mathrm{M_{*}}$, we do not observe any systematic evolution in
$\mathrm{V_{max}}$ in our two redshift groups ($z1:2-2.7$ and $z2:2.7-6$).
This is consistent with a weak or no evolution ($\mathrm{V_{max}}$ $\sim$
$(1+z)^{0.5}$) suggested in Sugahara et al. (2017, 2019). However, we do not
observe as high $\mathrm{V_{max}}$ values as seen in these studies. It is
possible that this discrepancy is due to the small sample size (7 galaxies) in
Sugahara et al. (2019) or due to difference in the observational technique
(down-the-barrel $vs$ GRB sightlines).

Figure 12:  A schematic showing a stronger biconical outflow along the minor
axis and weaker spherical outflow elsewhere. The narrow GRB sightlines are
shown as arrows. Some of the narrow and randomly pointed GRB sightlines will
trace the strong biconical outflow, giving a larger outflow column density,
while others will trace the weaker outflow, giving a smaller outflow column
density. This will produce a scatter in $\mathrm{N_{out}-SFR}$ relation.

##  6 Summary

We explored the outflow-galaxy correlations in low-mass ($\mathrm{M_{*}}$
$\sim$ $10^{9}-10^{11}$ $\mathrm{M_{\odot}}$) star-forming galaxies at $z\sim
2-6$ using GRB sightline spectroscopy. This technique offers a narrow, off-
centered, and randomly oriented sightline as opposed to a broad, down-the-
barrel beam to probe the outflows in absorption. We summarize our results as
follows:

1\. SFR-driven outflows: We find strong correlations between outflow column
density ($\mathrm{N_{out}}$), outflow maximum velocity ($\mathrm{V_{max}}$)
and SFR. This correlation is stronger for high-ion outflows compared to low-
ion outflows.

2\. Correlation with $\mathrm{M_{*}}$: We find a large spread, and therefore,
weaker correlations of $\mathrm{N_{out}}$ and $\mathrm{V_{max}}$ in high-ion
outflows with stellar mass. We find that this spread is almost entirely driven
by the spread in SFR in a given stellar mass bin. On the other hand, we find
that low-ion outflows are not correlated with stellar mass.

3\. We observe a higher detection fraction in high-ions compared to low-ions
(irrespective of the SFR or $\mathrm{M_{*}}$) as well as typically higher
$\mathrm{V_{max}}$ and $\mathrm{N_{out}}$ in high-ions, indicating that the
outflow is dominated by the phase traced by high-ionization lines.

4\. We investigated how the normalized velocity ($\mathrm{V_{norm}}$ =
$\mathrm{V_{max}/V_{circ,halo}}$) depends on the halo mass
($\mathrm{M_{halo}}$). We find a decline in $\mathrm{V_{norm}}$ with
increasing halo mass and thereby, infer that the outflows from the low-mass
halos are more likely to escape and enrich the outer CGM and/or IGM than those
in the halos of higher masses. We also observe a steeper decline in
$\mathrm{V_{norm}}$ for low-ions potentially hinting towards a systematic
shift in the temperature distribution of the outflow as the halo mass
increases.

5\. sSFR and metal enrichment: While neither $\mathrm{N_{out}}$ nor
$\mathrm{V_{max}}$ are correlated with sSFR, we observe a strong correlation
between $\mathrm{V_{norm}}$ (= $\mathrm{V_{max}/V_{circ,halo}}$) and sSFR at
the 3-$\sigma$ and 3.7-$\sigma$ levels for C IV and Si IV, respectively, with
$\mathrm{V_{norm}}$ $\propto$ sSFR0.25. This power law is consistent with the
low-redshift results from Heckman & Borthakur (2016). From this result, we
infer that the outflows in galaxies with higher sSFR have higher velocities
relative to the characteristic velocity of their halos. Thus, the outflows
from high sSFR galaxies are more likely to escape and enrich the outer CGM and
IGM compared to low sSFR galaxies.

6 Outflow dynamics: Using N$\cdot$V and N$\cdot$V2 as proxies for
$\dot{M}_{out}$ and $\dot{p}_{out}$ respectively, we find that they are both
correlated to the SFR for the high-ion traced outflows. We do not find such a
correlation for low-ions. While the high-SFR galaxies have a higher outflow
rate, the low-SFR galaxies are more efficient in driving outflows normalized
to their SFRs. In addition, we find that the estimated momentum flux of the
outflow can easily be supported by the momentum injected by stellar process
(i.e. supernovae and stellar winds). This, in concert with the observed
correlations, implies that the blue wings in the high-ion absorption are
dominated by star-formation-driven outflows.

7\. Redshift evolution: We do not observe any systematic difference in the
$\mathrm{V_{max}-SFR}$ and $\mathrm{N_{out}-SFR}$ correlations (for high-ions)
in our two redshift groups ($z1$: 2$-$2.7 and $z2:$ $2.7-6$) of similar size.
This suggests that the correlations of the outflow with SFR do not
significantly evolve with redshift and are more fundamental in nature.

8\. Stacking comparison: By comparing the stacks of our CGM-GRB sample with
the down-the-barrel studies of starbursts at $z\sim 2$ (MEGaSaURA) and at
$z\sim 0.1$ (COS-sample) in Section 5.2, we find that over a broad redshift
range, the strength of the outflow is correlated with the main-sequence offset
at the respective redshifts rather than the simple star-formation rate.

9\. Structure of the outflow: We observe a larger spread in the high-ion
correlation with SFR beyond an SFR of $\sim$10 $\mathrm{M_{\odot}yr^{-1}}$. We
speculate that this spread could arise due to an emergence of non-spherical
outflows (eg: biconical) at high-SFRs, leading to some sightlines cutting
across a larger section of the outflow while others probing a weaker outflow.
This characteristic can be uniquely probed using the GRB sightline technique
due to the random orientation (i.e. not down the barrel) and narrow beam of
the sightline (as shown in Fig. 12. However, a more detailed investigation of
the physical processes in the outflows is needed to explain the observed
spread in the outflow column densities at high SFR.

These results highlight the unique potential of GRB afterglow spectroscopy to
explore the nature and importance of stellar feedback at high redshifts.

The authors are grateful to Drs. A. Cucchiara and V. Toy for their useful
comments in the early stages of this paper. P.G. was supported by NASA Earth
and Space Science Fellowship (ASTRO18F-0085), NASA Hubble Fellowship (HST-
HF2-51478.001-A), and David & Ellen Lee Fellowship at Caltech for this
research. S.V. acknowledges partial support from the National Science
Foundation under grant 1711377 and National Aeronautics and Space
Administration under grants 16-APRA 16-0064 and ADAP NNX16AF24G.

These results made use of the Lowell Discovery Telescope (LDT) at Lowell
Observatory. Lowell is a private, non-profit institution dedicated to
astrophysical research and public appreciation of astronomy and operates the
LDT in partnership with Boston University, the University of Maryland, the
University of Toledo, Northern Arizona University and Yale University. The
Large Monolithic Imager was built by Lowell Observatory using funds provided
by the National Science Foundation (AST-1005313).

Based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and
179.A-2004, and on data products produced by the KiDS consortium. The KiDS
production team acknowledges support from: Deutsche Forschungsgemeinschaft,
ERC, NOVA and NWO-M grants; Target; the University of Padova, and the
University Federico II (Naples).

##  Appendix A Covering fraction

A covering fraction ($C_{f}$) of 1 was assumed in deriving the Voigt-profile
fits and column densities used in this paper Gatkine et al. (2019). Here we
present a more quantitative rationale for that assumption. A typical feature
of fractional covering fraction seen in partial covering fraction cases is
line saturation with residual positive flux. We do not see that in any of the
GRB sightlines in this sample, suggesting a high covering fraction. All the
line saturations go to zero flux (see Figures 17-43 in Gatkine et al. 2019).

Secondly, in Gatkine et al. (2019), all the doublets were fit using a joint
Bayesian fitting for the doublets with an assumption of $C_{f}$ = 1 resulting
in reasonable fits for both the components of the doublets, which suggest that
a $C_{f}$ of 1 is a consistent assumption.

Lastly, the UV emitting region for a GRB during the first 1-2 days after the
GRB explosion (typically the timescale when the spectra are obtained) is
extremely tiny ($\sim$ $1-2$ light days $\sim$ 100$-$200 AU). This is much
more compact than the typical cloud sizes expected for the outflows which can
be estimated as $\mathrm{\frac{total~{}column~{}density}{density}}$ $\sim$
$10^{18}/10^{-2}$ cm $\sim$ 1020 cm $>>$ 100-200 AU. Here, we take typical HI
column density as the total column density and 10-2 cm-3 as a typical density
for the outflow absorbers. Similarly, Liang & Remming (2020) suggest a cloud
diameter of 1-2 pc. Hence, the GRB sightline can be treated as a pencil beam
and therefore, a covering fraction of 1 is a reasonable assumption.

Furthermore, we computed the $C_{f}$ for the C IV and Si IV doublets, and
approximated it for the Si II 1526 $-$ Si II 1190 pair using the formalism
described in Hamann et al. (1997). An example is shown in Fig. 13. Note that
the ratio of (oscillator strength $\times$ $\lambda$) needs to be $\sim$2 for
the formalism in Hamann et al. (1997) and hence, the $C_{f}$ for Si II 1526 is
only an approximation. The covering fraction, shown in green is always
$\gtrsim$ 0.75 for the line components. Therefore, the assumption of $C_{f}$ =
1 is a reasonable assumption for our CGM-GRB sample probed by the GRB
sightline.

Figure 13:  Covering fractions calculated for C IV, Si IV doublets and
approximated for Si II 1526 in GRB 080810. The blue, red, and green lines show
the normalized flux, Voigt-profle fit, and covering fraction, respectively.
The error in the flux is shown in the bottom panel offset by $-1.5$.

##  Appendix B Additional outflow-galaxy correlations

In this appendix section, we discuss the weaker ($<$ 2$\sigma$) outflow-galaxy
correlations to provide the full scope of correlations investigated for the
CGM-GRB sample. Deeper observations of the host galaxies and a larger sample
will be required to evaluate these correlations in the future.

###  B.1 ${N_{out}}$ $vs$ ${M_{*}}$

From visual inspection of the $\mathrm{N_{out}}$ $vs$ $\mathrm{M_{*}}$ panels
in Fig. 14, there is a minor rise in $\mathrm{N_{out}}$ of high-ions with
stellar mass (better seen in C IV and Si IV), albeit with weak statistical
significance (i.e. a high value of Kendall-$\tau$ p-value). More
quantitatively, the correlation exists with a confidence level of 80% and 74%
in C IV and Si IV respectively. Si IV plot also shows a significantly larger
spread in column density at a higher stellar mass. On the other hand, the low-
ions do not show any difference between the low-mass and high-mass samples (as
evident from the CDF plots and high $p$-values). Thus, the overall column
density of the low-ion outflow remains independent of the stellar mass and
shows a weak correlation for the column density of high-ion outflow. This
finding implies that the prevalence of low-ion traced outflows is largely
independent of the stellar mass and for high-ion outflows (particularly Si
IV), it is only moderately boosted at high $\mathrm{M_{*}}$. Given the large
spread ($\sim$ 2 dex) in the column densities at any mass (for both high- and
low-ions), it is clear that the relation between outflow column density and
stellar mass, if any, is complex with multiple contributing factors such as
halo mass, SFR, and ionization state.

Figure 14:  Same as Figure 3, for the outflow column density $vs$
$\mathrm{M_{*}}$

###  B.2 ${N_{out}}$ $vs$ sSFR

We investigated the correlations, if any, between outflow column density
($\mathrm{N_{out}}$) and specific SFR (sSFR = SFR/$\mathrm{M_{*}}$) in Fig.
15. We do not see any statistically significant correlation. Previous studies
at lower redshifts have shown only a weak or no correlation between outflow
column density and sSFR for either low-ion or high-ion outflows. For instance,
Du et al. (2016) see only a weak correlation for C IV-traced outflows at
$z\sim 1.25$ while Bradshaw et al. (2013) see no correlation for Mg II-traced
outflows at $z\sim 0.7-1.63$. Thus, the lack of $\mathrm{N_{out}-sSFR}$
relations for both high and low ions at $z\sim 2-6$ in our data, are
consistent with previous results at lower redshifts.

Figure 15:  Same as Figure 3, for outflow column density $vs$ specific SFR.
The objects where both SFR and $\mathrm{M_{*}}$ are not detected are shown in
a lighter shade.

###  B.3 Outflow $\mathrm{V_{max}}$ $vs$ $\mathrm{M_{*}}$

This relation is summarized in Fig. 16. We observe a weak correlation in C IV
(84% confidence or $\sim$ 1.5$\sigma$) and a slightly stronger correlation in
Si IV (98% confidence or $\sim$ 2.5$\sigma$). We note a larger spread in the
low-$\mathrm{M_{*}}$ group, as evident from the errorbars around the
low-$\mathrm{M_{*}}$ in Fig. 16 (top panels). This spread can be directly
explained by the larger spread of SFR in the low-$\mathrm{M_{*}}$ group
compared to the high-$\mathrm{M_{*}}$ group in Fig. 2 (see panel 2). It is the
variance in SFR that is directly causing the spread in the
$\mathrm{V_{max}}-\mathrm{M_{*}}$ plot for high ions. By combining this with
the result from Section B.1, we can deduce that the stellar mass by itself
does not significantly affect the kinematics or column density of the high-ion
outflows, but is almost entirely driven by the SFR instead.

For low-ion species, we do not see any correlation between $\mathrm{V_{max}}$
and $\mathrm{M_{*}}$. We observe a large spread in both high-mass and low-mass
groups. Overall, our high-$z$ results for the low-ion traced outflows show a
different picture compared to the results in low-$z$ studies. For instance,
Rubin et al. (2014) find a 3.5-$\sigma$ correlation between $\mathrm{V_{max}}$
and $\mathrm{M_{*}}$ and no correlation with current SFR. In contrast, for our
sample at $z>2$, we find a weak correlation ($\sim$ 2$\sigma$ for Si II) with
SFR and no correlation with the stellar mass.

Figure 16:  Same as Figure 3, for the maximum outflow velocity,
$\mathrm{V_{max}}$ $vs$ $\mathrm{M_{*}}$. The horizontal dashed line in the
panels shows the 100 km s-1 level, which we treat as the threshold for
outflow.

###  B.4 Outflow $\mathrm{V_{max}}$ $vs$ sSFR

We summarize the results of $\mathrm{V_{max}}$ $vs$ sSFR correlation in Fig.
17. We do not find any statistically significant correlation of sSFR with
$\mathrm{V_{max}}$. A weak, 1-$\sigma$ correlation appears to be present for
$\mathrm{V_{max}}$ $vs$ sSFR for low-ion outflows, albeit with large scatter.

Correlations between outflow kinematics and sSFR have been seen in past
observations. For instance, Heckman et al. (2015); Heckman & Borthakur (2016)
report a strong ($>$ 2 $\sigma$) correlation between outflow velocity of warm
ionized gas and sSFR over 2.5 orders of magnitude in sSFR for starburst
galaxies at $z<0.2$. Bradshaw et al. (2013) suggest a $\mathrm{V_{max}-sSFR}$
correlation at $z\sim 0.7-1.63$, but shallower compared to $z\sim 0$
correlation. Sugahara et al. (2017) also report $\mathrm{V_{max}-sSFR}$
correlations at $z\sim 0-1$, albeit with a significant variation with redshift
and over only one order of magnitude in sSFR. For comparison, we explore this
correlation over almost 2 orders of magnitude in sSFR and do not find any
statistically significant correlation.

This lack of correlation maybe due to a number of reasons: 1) an intrinsic
weakening of the correlation at high redshifts, 2) presence of a large scatter
in the correlation, thus requiring observations over a larger range in sSFR to
see the correlation, and 3) the upper and lower limits in sSFR may mask the
underlying correlation. In all the three cases, deeper observations of the
host galaxy are needed to better constrain the sSFR and thereby, its relation
with the outflow properties.

Figure 17:  Same as Figure 15, for the the scaling relations of the maximum
outflow velocity ($\mathrm{V_{max}}$) with specific SFR (= SFR /
$\mathrm{M_{*}}$). The horizontal dashed line in the panels shows the 100 km
s-1 level, which we treat as the threshold for outflow.

##  Appendix C Consideration of other low-ion lines: C II 1334 and O I 1302

As previously explained in section 3.2, we did not use C II 1334 and O I 1302
for the primary investigation of the outflow-galaxy correlations due to mild
concerns about possible blending due to neighboring lines. However, we
performed the same analysis for these two lines as a consistency check for the
low-ion correlations observed in the main paper. Here we mainly focus on
$\mathrm{N_{out}}$ and $\mathrm{V_{max}}$ $vs$ $\mathrm{M_{*}}$ and SFR. These
plots are show in Figs. 18 and 19. We observe that the $\mathrm{N_{out,CII}}$
$-$ SFR correlation has a smaller p-value ($\sim$ $2\sigma$) compared to that
observed for other low-ion lines. Other than this deviation, we do not see any
significant departure in the results compared to those obtained for Si II 1526
and Fe II 1608.

Figure 18:  Same as Figure 3, for the correlations of the C II 1334-traced
outflow with galaxy properties. Figure 19:  Same as Figure 3, for the
correlations of the O I 1302-traced outflow with galaxy properties.

##  Appendix D Relation of O VI absorption with galaxy properties

O VI traces the warm-hot medium ($\mathrm{10^{5}-10^{6}}$ K, assuming
collisional equilibrium). It is likely that O VI absorption traces a different
phase than the gas traced by C IV and Si IV. Here we look at the same
relations studied above with O VI. Due to the location of the O VI doublet in
the far UV ($\lambda_{rest}$ = 1031.9 and 1037.6 Å), fewer afterglow spectra
in our sample cover the O VI band. Therefore, we caution that the relations
observed for O VI are based on a smaller sample than other lines. Regardless,
O VI correlations can provide useful insights into the warm-hot phase of the
CGM.

From Fig. 20, it is clear that both the outflow column density and
$\mathrm{V_{max}}$ correlate with the stellar mass. The outflow column density
is also correlated to the SFR, albeit less so than with the stellar mass. The
O VI correlations appear to follow the correlations of high-ion species (C IV,
Si IV) described earlier. With a caution of limited sample size, the data
indicates that star formation activity also drives outflows in the OVI-traced
phase at $z\sim 2-6$.

Figure 20:  Same as Figure 3, for the correlations of the O VI-traced outflow
with galaxy properties. Table 3: Summary of correlations between CGM and
galaxy properties investigated for this sample

Outflow Property | Galaxy Property | Ion | Slope | Intercept | K$-\tau$ p-value | Median-split p-value
---|---|---|---|---|---|---
Outflow Column Density (Nout) | M∗ | C IV | 0.23 | 12.08 | 0.20 | 0.44
Outflow Column Density (Nout) | M∗ | Si IV | 0.20 | 11.57 | 0.26 | 0.26
Outflow Column Density (Nout) | M∗ | Fe II | 0.22 | 11.91 | 0.61 | 0.88
Outflow Column Density (Nout) | M∗ | Si II | 0.18 | 12.24 | 0.80 | 0.92
Outflow Column Density (Nout) | SFR | C IV | 0.22 | 13.99 | 0.14 | 0.13
Outflow Column Density (Nout) | SFR | Si IV | 0.25 | 13.25 | 0.08 | 0.02
Outflow Column Density (Nout) | SFR | Fe II | 0.12 | 13.86 | 0.53 | 0.98
Outflow Column Density (Nout) | SFR | Si II | 0.27 | 13.65 | 0.20 | 0.53
Maximum outflow velocity (Vmax) | M∗ | C IV | 0.07 | 1.64 | 0.16 | 0.4
Maximum outflow velocity (Vmax) | M∗ | Si IV | 0.18 | 0.49 | 0.02 | 0.07
Maximum outflow velocity (Vmax) | M∗ | Fe II | 0.0095 | 2.03 | 0.87 | 0.9
Maximum outflow velocity (Vmax) | M∗ | Si II | -0.03 | 2.54 | 0.87 | 0.75
Maximum outflow velocity (Vmax) | SFR | C IV | 0.12 | 2.19 | 0.05 | 0.33
Maximum outflow velocity (Vmax) | SFR | Si IV | 0.28 | 2.07 | 0.0007 | 0.0001
Maximum outflow velocity (Vmax) | SFR | Fe II | 0.06 | 2.08 | 0.61 | 0.66
Maximum outflow velocity (Vmax) | SFR | Si II | 0.14 | 2.10 | 0.07 | 0.82
Normalized Velocity (Vmax/Vcirc,halo) | Mhalo | C IV | -0.32 | 3.92 | 0.003 | 0.24
Normalized Velocity (Vmax/Vcirc,halo) | Mhalo | Si IV | -0.45 | 5.34 | 0.0012 | 0.98
Normalized Velocity (Vmax/Vcirc,halo) | Mhalo | Fe II | -0.73 | 8.34 | 0.14 | 0.12
Normalized Velocity (Vmax/Vcirc,halo) | Mhalo | Si II | -0.51 | 5.85 | 0.0085 | 0.04
Outflow Column Density (Nout) | sSFR | C IV | 0.12 | 15.34 | 0.66 | -
Outflow Column Density (Nout) | sSFR | Si IV | 0.098 | 14.31 | 0.90 | -
Outflow Column Density (Nout) | sSFR | Fe II | -0.17 | 12.59 | 0.41 | -
Outflow Column Density (Nout) | sSFR | Si II | 0.06 | 14.30 | 0.97 | -
Maximum outflow velocity (V_max) | sSFR | C IV | 0.03 | 2.67 | 0.97 | -
Maximum outflow velocity (V_max) | sSFR | Si IV | 0.13 | 3.48 | 0.86 | -
Maximum outflow velocity (V_max) | sSFR | Fe II | 0.32 | 5.08 | 0.33 | -
Maximum outflow velocity (V_max) | sSFR | Si II | 0.21 | 4.03 | 0.34 | -
Normalized Velocity (Vmax/Vcirc,halo) | sSFR | C IV | 0.26 | 2.42 | 0.003 | -
Normalized Velocity (Vmax/Vcirc,halo) | sSFR | Si IV | 0.24 | 2.19 | 0.0002 | -
Normalized Velocity (Vmax/Vcirc,halo) | sSFR | Fe II | 0.18 | 1.38 | 0.51 | -
Normalized Velocity (Vmax/Vcirc,halo) | sSFR | Si II | 0.39 | 3.32 | 0.010 | -
Outflow rate (N·V = $\int N_{a}vdv$) | SFR | C IV | 0.41 | 15.81 | 0.12 | 0.12
Outflow rate (N·V = $\int N_{a}vdv$) | SFR | Si IV | 0.74 | 14.80 | 0.032 | 0.01
Outflow rate (N·V = $\int N_{a}vdv$) | SFR | Fe II | 0.90 | 14.40 | 0.48 | 0.91
Outflow rate (N·V = $\int N_{a}vdv$) | SFR | Si II | 0.31 | 14.96 | 0.46 | 0.77
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | SFR | C IV | 0.39 | 14.39 | 0.94 | 0.42
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | SFR | Si IV | 0.59 | 13.66 | 0.79 | 0.82
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | SFR | Fe II | 1.47 | 12.31 | 0.77 | 0.04
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | SFR | Si II | 0.94 | 12.94 | 0.78 | 0.05
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | Mhalo | C IV | -0.45 | 20.33 | 0.80 | 0.75
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | Mhalo | Si IV | 0.05 | 13.61 | 0.87 | 0.33
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | Mhalo | Fe II | 1.47 | 12.31 | 0.77 | 0.39
Mass loading factor (N·V/SFR = $\int N_{a}vdv$/SFR) | Mhalo | Si II | 0.94 | 12.94 | 0.78 | 0.07
Momentum flux (N·V2= $\int N_{a}v^{2}dv$) | SFR | C IV | 0.32 | 18.11 | 0.074 | 0.13
Momentum flux (N·V2 = $\int N_{a}v^{2}dv$) | SFR | Si IV | 0.91 | 16.90 | 0.017 | 0.001
Momentum flux (N·V2 = $\int N_{a}v^{2}dv$) | SFR | Fe II | 0.97 | 16.54 | 0.46 | 0.96
Momentum flux (N·V2 = $\int N_{a}v^{2}dv$) | SFR | Si II | 0.25 | 17.39 | 0.48 | 0.81
Outflow Column Density (Nout) | M∗ | O VI | 0.37 | 10.74 | 0.099 | 0.48
Outflow Column Density (Nout) | SFR | O VI | 0.06 | 14.02 | 0.70 | 0.27
Maximum outflow velocity (Vmax) | M∗ | O VI | 0.06 | 1.77 | 0.35 | 0.69
Maximum outflow velocity (Vmax) | SFR | O VI | -0.06 | 2.30 | 0.94 | 0.21

aafootnotetext:

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