# Evolution of Civ Absorbers. II. Where does Civ live?

Farhanul Hasan Department of Astronomy, New Mexico State University, Las
Cruces, NM 88003, USA  Christopher W. Churchill Department of Astronomy, New
Mexico State University, Las Cruces, NM 88003, USA  Bryson Stemock Department
of Astronomy, New Mexico State University, Las Cruces, NM 88003, USA  Nikole
M. Nielsen Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria 3122, Australia  ARC Centre of Excellence for
All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Glenn G. Kacprzak
Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria 3122, Australia  ARC Centre of Excellence for
All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Mark Croom
Department of Astronomy, New Mexico State University, Las Cruces, NM 88003,
USA  Michael T. Murphy Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, Hawthorn, Victoria 3122, Australia

(Received Aug 10 2021; Revised Oct 13 2021; Accepted Oct 16 2021)

###### Abstract

We use the observed cumulative statistics of Civ absorbers and dark matter
halos to infer the distribution of Civ-absorbing gas relative to galaxies at
redshifts $0\\!\leq\\!z\\!\leq\\!5$. We compare the cosmic incidence $dN/dX$
of Civ absorber populations and galaxy halos, finding that massive
$L\\!\geq\\!{L_{\star}}$ halos alone cannot account for all the observed
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers. However, the $dN/dX$ of lower
mass halos exceeds that of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers. We
also estimate the characteristic gas radius of absorbing structures required
for the observed Civ $dN/dX$, assuming each absorber is associated with a
single galaxy halo. The ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ gas radii are $\sim$30–70%
($\sim$20–40%) of the virial radius of $L_{\star}$ ($0.1L_{\star}$) galaxies,
and the ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ gas radius is $\sim$100–150%
($\sim$60–100%) of the virial radius of $L_{\star}$ ($0.1L_{\star}$) galaxies.
For stronger absorbers, the gas radius relative to virial radius rises across
Cosmic Noon and falls afterwards, while for weaker absorbers, the relative gas
radius declines across Cosmic Noon and then dramatically rises at $z\\!<\\!1$.
A strong luminosity-dependence of gas radius implies highly extended Civ
envelopes around massive galaxies before Cosmic Noon, while a luminosity-
independent gas radius implies highly extended envelopes around dwarf galaxies
after Cosmic Noon. From available absorber-galaxy and Civ evolution data, we
favor a scenario in which low-mass galaxies enrich the volume around massive
galaxies at early epochs and propose that the outer halo gas
($>\\!0.5{R_{v}}$) was produced primarily in ancient satellite dwarf galaxy
outflows, while the inner halo gas ($<\\!0.5{R_{v}}$) originated from the
central galaxy and persists as recycled accreting gas.

galaxies: abundances – halos – formation – evolution – intergalactic medium –
quasars: absorption lines

††journal: Astrophysical Journal

##  1 Introduction

According to the standard $\Lambda$CDM paradigm of cosmology, dark matter (DM)
comprises $\sim$25% of the total energy density and $\sim$85% of the matter
density of the universe (Planck Collaboration et al., 2020). In overdense
regions, DM gravitationally accumulates into halos and groupings and clusters
of halos (e.g., Navarro et al., 1996, 2010; Springel et al., 2005; Guo et al.,
2010; Klypin et al., 2011; Wechsler & Tinker, 2018). As baryonic gas accretes
into the DM halos, the gas cools and condenses, transforming into the luminous
components of the universe, including stars, galaxies, and the groups and
clusters in which they amass (e.g., Vale & Ostriker, 2004; Moster et al.,
2013; Somerville & Davé, 2015; Naab & Ostriker, 2017; Vogelsberger et al.,
2020).

Galaxy formation and evolution is largely dictated by how baryons
gravitationally and hydrodynamically respond as they are cycled through DM
halos. This “baryon cycle” not only produces the luminous components of the
universe, but also generates extended gaseous structures. As much as 50% of
the baryonic mass bound to a galaxy can be contained in a highly extended
surrounding region commonly referred to as the circumgalactic medium (CGM;
e.g., Steidel et al., 2010; Tumlinson et al., 2011, 2017; Peeples et al.,
2014). Additionally, large scale gaseous filaments connecting galaxies form a
baryonic intergalactic medium (IGM; e.g., Davé et al., 2001; McQuinn, 2016;
Danforth et al., 2016; Burchett et al., 2020).

Through the star formation and stellar feedback stages in the baryon cycle,
the otherwise pristine gaseous universe is enriched by metals originally fused
in stellar cores. Starting at early cosmic epochs, galactic scale winds can
eject metals, distributing them into the CGM and the IGM (e.g., Oppenheimer &
Davé, 2006, 2008; Oppenheimer et al., 2012; Ford et al., 2014; Muratov et al.,
2015, 2017; Christensen et al., 2016; Anglés-Alcázar et al., 2017; Hafen et
al., 2019, 2020; Nelson et al., 2019). As such, these metal enriched gaseous
structures serve as diverse tracers of the baryon cycle across much of cosmic
time, and, for that reason, have been widely studied in order to chart their
cosmic incidence, redshift clustering, relationships to galaxy properties,
kinematics, chemical and ionization conditions, and the evolution in these
conditions (e.g., Sargent et al., 1988; Steidel, 1990; Churchill et al., 1999;
Nestor et al., 2005; Simcoe et al., 2006; Schaye et al., 2007; Cooksey et al.,
2013; Nielsen et al., 2013b; Bordoloi et al., 2014; Shull et al., 2014;
Kacprzak et al., 2015; Burchett et al., 2015; Lehner et al., 2016; Codoreanu
et al., 2018; Rudie et al., 2019; Dutta et al., 2020; Hasan et al., 2020;
Péroux & Howk, 2020).

Studies which target quasar absorption lines arising in the CGM of their host
galaxies have revealed metal-bearing gas out to large galactocentric distances
for both low-ionization Mgii $\lambda\lambda 2796,2803$ selected absorbers
(e.g., Bergeron & Boissé, 1991; Steidel et al., 1994; Kacprzak et al., 2008;
Chen et al., 2010; Nielsen et al., 2013a; Schroetter et al., 2019; Huang et
al., 2021) and for higher-ionization Ovi $\lambda\lambda 1031,1037$ selected
absorbers (e.g., Tumlinson et al., 2011; Stocke et al., 2013; Mathes et al.,
2014; Kacprzak et al., 2015; Nielsen et al., 2017; Ng et al., 2019). Such
studies targeting the intermediate ionization gas commonly observed via the
Civ $\lambda\lambda 1548,1550$ doublet, though not as numerous, have shown
that this ion can also exist at large galactocentric distances (e.g., Chen et
al., 2001; Borthakur et al., 2013; Bordoloi et al., 2014; Liang & Chen, 2014;
Burchett et al., 2016; Rudie et al., 2019; Schroetter et al., 2021).

However, to the degree that these ions and their spatial distribution around
galaxies have been characterized, it remains unclear (1) how far out within
the DM halo (or beyond the DM halo) metals are distributed, (2) what the
origins of these metals are, i.e., what percent comes from outside the central
galaxy and what percent from inside, and (3) what accretion modes, stellar
feedback mechanisms, and recycling processes are distributing metals to such
large galactocentric distances. It is likely that the answers to these
questions will change in their details as a function of both redshift and DM
halo mass.

Global insights into the statistical relationship between metal-line absorbing
structures at different redshifts, DM halos, and their central luminous
galaxies, can be inferred by comparing their cosmic distributions. Over the
last decade, tremendous progress has been made in characterizing the DM halo
mass function (e.g., Murray et al., 2013; Behroozi et al., 2013; Bocquet et
al., 2016) and galaxy luminosity function (e.g., Weisz et al., 2014; Parsa et
al., 2016; Bouwens et al., 2021) over the redshift ranges corresponding to
those for which metal-line absorbers have been characterized.

In Hasan et al. (2020, hereafter Paper I), we used Keck/HIRES (Vogt et al.,
1994) and VLT/UVES (Dekker et al., 2000) spectra, including from the KODIAQ
(O’Meara et al., 2015) and UVES SQUAD (Murphy et al., 2019) projects, to
expand and extend the survey work of Cooksey et al. (2013) and constrained the
global statistics of Civ absorbers to a limiting equivalent width detection
sensitivity of ${W_{r,{\rm lim}}}\\!=\\!0.05$ Å over the redshift range
$1.0\\!\leq\\!z\\!\leq\\!4.75$. Using the works of Cooksey et al. (2010) and
Burchett et al. (2016), we extended our investigation to cover
$0\\!\leq\\!z\\!\leq\\!1$. We found that the cosmic incidence of Civ absorbers
increases with time across $0\\!\leq\\!z\\!\leq\\!5$ (corresponding to the
last $\sim$12.5 Gyr, or $\sim$90% of cosmic time), with the rise being more
rapid as the $W_{r}$ of the subsample is increased. The evolution of Civ
absorbers with ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ occurred most prominently prior to and
throughout the so-called “Cosmic Noon” epoch ($1.5\\!\leq\\!z\\!\leq\\!3$),
when cosmic star formation, black hole activity, galactic scale winds, and gas
accretion rates reached their peak (e.g., van de Voort et al., 2011; Madau &
Dickinson, 2014; Feldmann et al., 2016; Rupke, 2018). On the contrary, Civ
absorbers with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ evolved very little until
after Cosmic Noon was completed, at which time their cosmic incidence,
$dN/dX$, rose by a factor of $\sim$4.5.

In this paper, we use the observed evolution of Civ absorber populations in
concert with the evolution of galaxy populations to better constrain the
statistical extent of metal-enriched gas as a function of galaxy properties
and to elucidate the connection between enriched gas, galaxies, and DM halos
as manifested by the baryon cycle. In the most general sense, properties of
gaseous structures denser than the cosmic mean should be causally linked to
galaxies, just as galaxy growth is causally connected to halo growth (e.g.,
Kravtsov et al., 2004; More et al., 2009; Moster et al., 2018; Behroozi et
al., 2019; Whitler et al., 2020). We exploit this connection to understand how
the spatial distribution of metals relative to galaxies changes with galaxy
luminosity, halo mass, and redshift.

This paper is organized as follows. In Section 2, we connect the populations
of galaxies to that of DM halos and investigate the connection between
absorbers and halos. We leverage our measurements to estimate the statistical
extent of Civ absorbing gas around galaxies of different luminosities in
Section 3. We discuss our results in the context of theory and observations
and their implications for galaxy evolution and the baryon cycle in Section 4.
Finally, the conclusions are presented in Section 5. As in Paper I, we adopt
the WMAP5 cosmology, with $H_{0}=71.9$ $\,{\rm km\,s}^{-1}~{}{\rm Mpc}^{-1}$,
$\Omega_{\hbox{\tiny M}}=0.258$, and $\Omega_{\Lambda}=0.742$ (Komatsu et al.,
2009).

##  2 Gas, Galaxies, and Halos

In this section, we investigate the cosmic incidence of DM halos and compare
them to the cosmic incidence of Civ populations that we measured in Paper I.
For a given population of objects, the observed cosmic incidence, or “co-
moving path density,” is proportional to the product of the cosmic number
density, $n(z)$, and the physical cross-section, $\sigma(z)$,

$\frac{dN}{dX}(z)=\frac{c}{H_{0}}n(z)\sigma(z)\,.$ (1)

In general terms, $dN/dX$ is the number of structures intercepted by a line of
sight per unit of “absorption distance,” $X(z)$ (Bahcall & Peebles, 1969). For
absorbing systems, this quantity is interpreted as the number of intervening
absorption systems observed per unit co-moving line-of-sight path length.
Similarly, we can interpret $dN/dX$ for a population of DM halos as the number
of virial halos, defined by $R_{v}(M_{h},z)$, the virial radius of a DM halo
of mass $M_{h}$ at redshift $z$, intercepted per unit co-moving path of a
given line of sight. A comparison of the evolution of cosmic incidence of gas
absorbers and DM halos provides insight into the spatial distribution of gas
structures in the universe.

To examine the cosmic incidence of Civ absorbers relative to DM halos, we
define the “relative gas incidence,”

$\displaystyle{f_{\hbox{\tiny
X}}}(z)=\frac{\displaystyle\frac{dN}{dX}(z)\Big{|}_{\hbox{\small{\hbox{{\rm
C}\kern 0.90005pt{\sc iv}}}}(W_{r}\geq W_{r,{\rm
lim}})}}{\displaystyle\frac{dN}{dX}(z)\Big{|}_{\hbox{\small$R_{v}$}(M_{h}\geq{M_{h,\mathrm{min}}}(z))}}\,,$
(2)

as the ratio of co-moving path density of Civ absorbers with
$W_{r}\\!\geq\\!W_{r,{\rm lim}}$ to that of spherical DM halos with
${M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}(z)}$.

Under the assumption that Civ absorbers are co-spatial with DM halos, the
relative gas incidence quantifies the relative cross-sectional size of Civ
absorbers with $W_{r}\\!\geq\\!W_{r,{\rm lim}}$ with respect to the halo mass-
weighted cross-section of DM halos with
$M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}(z)$. A value of ${f_{\hbox{\tiny
X}}}(z)\\!<\\!1$ implies that the sizes of “Civ halos” with
$W_{r}\\!\geq\\!W_{r,{\rm lim}}$ are smaller than the halo mass-weighted
virial radii of DM halos, thus suggesting that such Civ absorbers, on average,
spatially reside within the virial radii of DM halos. A value of
${f_{\hbox{\tiny X}}}(z)\\!>\\!1$ implies that the sizes of “Civ halos” with
$W_{r}\\!\geq\\!W_{r,{\rm lim}}$ are larger than the halo mass-weighted virial
radii of DM halos, thus suggesting that such Civ absorbers, on average, reside
outside the virial radii of DM halos. This assumption of equal abundance of
Civ absorbers and DM halos is quite restrictive and is likely to be incorrect,
if for instance, there are significant numbers of satellite galaxies
surrounding massive centrals in a halo (e.g., Zheng et al., 2005; Mandelbaum
et al., 2006). We discuss this assumption in section 4 and elsewhere below.

If instead, we assume that absorbing halos and DM halos have the same physical
extent, $f_{\hbox{\tiny X}}$ measures the relative abundance of Civ absorbers
relative to DM halos. In this scenario, ${f_{\hbox{\tiny X}}}(z)\\!>\\!1$
implies that absorbers with $W_{r}\\!\geq\\!W_{r,{\rm lim}}$ are more abundant
than DM halos with ${M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}(z)}$ and
${f_{\hbox{\tiny X}}}(z)\\!<\\!1$ implies the converse. Without either of the
above assumptions, the relative gas incidence quantifies the product of
abundance and cross-section of absorbers relative to that of DM halos.

The numerator of Eq. 2 has been measured directly from the data, survey path
length, and completeness functions for ${W_{r,{\rm lim}}}\\!=\\!0.05,0.3$, and
$0.6$ Å as described in Paper I. There are no assumptions required for the
calculation, for example, about the gas physical conditions, cosmic number
densities, or absorber cross-sections, as it is simply an exercise in counting
absorbers. The computation of the denominator, on the other hand, relies on
our cosmic census of DM halos and their evolution. We computed the denominator
of Eq. 2 from Eq. 1 using

$n(z)\sigma(z)=\int_{M_{h,\mathrm{min}}}^{\infty}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\phi(M_{h},z)\,\pi
R^{2}_{v}(M_{h},z)\,dM_{h}\,,$ (3)

where $\phi(M_{h},z)$ is the DM halo mass function (number density of DM halos
with mass $M_{h}$ per unit halo mass) at redshift $z$, and
${\sigma(M_{h},z)=\pi R^{2}_{v}(M_{h},z)}$ is the assumed spherical cross-
sectional area.

Figure 1: Comparison of cosmic incidence of Civ absorbers and DM halos. The
measured $dN/dX$ of different populations of Civ absorbers are shown by light
blue (${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$), purple
(${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$), and orange
(${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$) data points; the uncertainties are $\pm
1\sigma$. The computed $dN/dX$ of halos (Eq. 3) are represented by solid
curves; halos hosting galaxies with ${L\geq{0.01L_{\star}}}$ are shown in
pink, ${L\geq{0.1L_{\star}}}$ is dark blue, and ${L\geq{L_{\star}}}$ is grey.
The shading on these curves reflects the uncertainty in the halo mass obtained
from abundance matching to luminosity (see Appendix A).

Historically, the sizes of absorbing structures tracing various ionization
phases of gas have been linked to the directly-observed luminosity of galaxies
around which they are found (e.g., Bergeron & Boissé, 1991; Lanzetta, 1993;
Steidel, 1995; Chen et al., 2001, 2010; Kacprzak et al., 2008; Ribaudo et al.,
2011; Nielsen et al., 2013a). In order to map Eq. 3 to a minimum central
galaxy luminosity, $L_{\mathrm{min}}$, we apply the technique of abundance
matching (e.g., Kravtsov et al., 2004; Trujillo-Gomez et al., 2011; Behroozi
et al., 2013, 2019), thus mapping the cumulative number density of halos with
$M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}$ at redshift $z$ to that of galaxies with
$L\\!\geq\\!{L_{\mathrm{min}}}$. We adopted the 1500 Å UV luminosity functions
of Parsa et al. (2016) and the DM halo mass functions of Behroozi et al.
(2013) across $0\\!\leq\\!z\\!\leq\\!5$. Details on how we perform abundance
matching and compute Eq. 3 are presented in Appendix A.

Figure 2:  The Civ absorption to DM virial halo ratios (Eq. 2) as functions of
redshift for halos with (a) $L\\!\geq\\!{0.01L_{\star}}$, (b)
$L\\!\geq\\!{0.1L_{\star}}$, and (c) $L\\!\geq\\!{L_{\star}}$. The blue,
purple and orange data represent ${f_{\hbox{\tiny X}}}$ for Civ gas with
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$, ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$, and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$, respectively.

In Figure 1, we compare the measured $dN/dX$ for Civ absorbers with the
thresholds ${W_{r,{\rm lim}}}=0.05$, 0.3, and 0.6 Å (Paper I) to the $dN/dX$
of DM halos with ${M_{h,\mathrm{min}}}(z)$ corresponding to fixed
${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$, $0.1L_{\star}$, and $L_{\star}$.
The shaded bands represent the uncertainty in luminosity obtained from
abundance matching. We added a data point for the $dN/dX$ of
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ absorbers at $\langle
z\rangle\\!\simeq\\!0.5$, from Cooksey et al. (2010), which was not presented
in Paper I.

Our choice of ${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$ follows from the COS-
Dwarfs survey, which found Civ around galaxies with UV $1500$ Å luminosities
$L\\!\sim\\!0.005$–0.14$L_{\star}$ at $z\\!\leq\\!0.1$ (Bordoloi et al.,
2014). We choose ${L_{\mathrm{min}}}\\!=\\!{0.1L_{\star}}$ as one of the
luminosity cutoffs based on the findings of Burchett et al. (2016) that at
$z\\!\simeq\\!0$, the Civ covering fraction within 1 $R_{v}$ declines below
10% for galaxies with stellar mass below $\log M_{\star}\\!\sim\\!9.5$
$\mathrm{M}_{\odot}$. This translates to $M_{h}\\!\approx\\!3\times 10^{11}$
$\mathrm{M}_{\odot}$ according to the stellar-halo mass relation of Girelli et
al. (2020), and is roughly the halo mass for a $0.1L_{\star}$ galaxy at
$z\\!\simeq\\!0$ (see Figure 7). Finally, we choose the
${L_{\mathrm{min}}}\\!=\\!{L_{\star}}$ to examine the most massive galaxies at
each redshift.

The results in Figure 1 indicate that the cumulative co-moving path density of
halos hosting $L\\!\geq\\!{0.01L_{\star}}$ and ${L\\!\geq\\!{0.1L_{\star}}}$
galaxies exceeds that of Civ absorbers at all redshifts for thresholds of
${W_{r,{\rm lim}}}\\!=\\!0.05$ Å and above. At $z\\!>\\!2$, the
$L\\!\geq\\!{L_{\star}}$ halo $dN/dX$ is smaller than the
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ $dN/dX$, though they roughly trace each
other for $z\\!<\\!2$. Our exercise would suggest that the product of the
number density and cross-section, $n(z)\sigma(z)$, of the weakest Civ
absorbers exceeds that of the most massive halos at early epochs.

Applying Eq. 2, we then compute the gas incidence ${f_{\hbox{\tiny X}}}(z)$ of
Civ absorbers relative to the virial radius as a function of redshift for our
six combinations of ${W_{r,{\rm lim}}}\\!=\\!0.05$, $0.3$ and 0.6 Å and
${L_{\mathrm{min}}}\\!=\\!0.01{L_{\star}}$, $0.1{L_{\star}}$, and $L_{\star}$
and present them in Figure 2. Each panel shows a different $L_{\mathrm{min}}$
halo population.

For Civ absorbers with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ we find
${f_{\hbox{\tiny X}}}\\!\simeq\\!0.1$–$0.5$ for halos hosting galaxies with
$L\geq{0.01L_{\star}}$ and $0.1L_{\star}$. For $L\\!\geq\\!{L_{\star}}$, we
find ${f_{\hbox{\tiny X}}}\\!\\!\simeq\\!0.8$–$3$. The value of
${f_{\hbox{\tiny X}}}\\!>\\!1$ for $L\\!\geq\\!{L_{\star}}$ halos at
$z\\!=\\!0$ and $z\\!\geq\\!2$ means that all the observed
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers cannot be accounted for by
massive halos with $L\\!\geq\\!{L_{\star}}$ galaxies at those redshifts. If
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers are spatially coincident with
$L\\!\geq\\!{L_{\star}}$ halos, ${f_{\hbox{\tiny X}}}\\!>\\!1$ implies that
the absorbing structures extend beyond the virial radii of the halos, likely
in the IGM between halos. If ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbing
structures have the same physical extent as $L\\!\geq\\!{L_{\star}}$ virial
halos, then ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ is more abundant than
$L\\!\geq\\!{L_{\star}}$ halos, again implying that some Civ absorbing systems
must live in the IGM. We can also conclude that $L\\!\geq\\!{0.01L_{\star}}$
and $L\\!\geq\\!{0.1L_{\star}}$ halos are more abundant and/or larger in
physical size than ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbing
structures.

At early epochs, from $z\\!=\\!4$ to $z\\!=\\!2$, the evolution of
$f_{\hbox{\tiny X}}$ for Civ absorbers with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
declines by a factor of three for the most massive halos. This evolution is
less pronounced for halos hosting $L\\!\geq\\!{0.1L_{\star}}$ galaxies, and
virtually absent for $L\\!\geq\\!{0.01L_{\star}}$ halos. This behavior would
suggest that, at early epochs, the product $n(z)\sigma(z)$ of the most massive
halos grows at a rate that exceeds that of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
Civ absorbing structures, but that this difference in evolution diminishes as
lower mass halos are included in the halo population.

What is interesting for ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers is
that their relative gas incidence has a minimum at $z\\!\simeq\\!1.5$ followed
by an increase toward the present epoch, $z\\!=\\!0$; this evolution is
observed for all halo masses. There are two noteworthy inferences that can be
drawn from this behavior. First, this may suggest that the spatial extent of
weak Civ absorbers in the outer regions of galactic halos is decreasing during
the epoch of Cosmic Noon, when the global star formation is at its peak (Madau
& Dickinson, 2014). Second, following Cosmic Noon, the typical physical extent
of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbing structures likely grows
faster than the virial radius of halos.

For stronger Civ absorbers, ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$, we find ${f_{\hbox{\tiny X}}}\\!<\\!1$ for
all halo masses. This may suggest that higher column density (and/or
kinematically complex) Civ absorbing structures reside within the virial radii
of galaxy halos of all masses. It may also suggest that all halo populations
are significantly more abundant than stronger absorbers.

For ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
absorbers, the early epoch evolution of relative gas incidence becomes most
pronounced as lower mass halos are included in the halo population. For
$L\\!\geq\\!{0.01L_{\star}}$ galaxies, ${f_{\hbox{\tiny X}}}$ for
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ absorbers increases by a factor of a few
from $z\\!=\\!4$ to $z\\!=\\!2$ and then flattens out. Thus, when the lowest
mass halos are included, the ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ absorbing
structures are growing in size and/or number density at a rate that exceeds
that of galaxy halos prior to Cosmic Noon, but then after Cosmic Noon, the
halo and absorber sizes remain in lock step. Similar behavior is found for
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ Civ absorbers, though the early epoch
evolution is less pronounced. When we limit our consideration to
$L\\!\geq\\!{0.1L_{\star}}$ halos, we find that the early epoch evolution is
less pronounced.

The takeaways from Figure 2 are that (a) $L\\!\geq\\!{L_{\star}}$ galaxy halos
cannot account for all the observed ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ
absorbers, but $L\\!\geq\\!{0.01L_{\star}}$ and $L\\!\geq\\!{0.1L_{\star}}$
halos can, (b) if Civ absorbers are co-spatial with DM halos, then all
absorbers should live inside the virial radius of $L\\!\geq\\!{0.01L_{\star}}$
and $L\\!\geq\\!{0.1L_{\star}}$ halos, but ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
absorbers may live beyond the virial radius of $L\\!\geq\\!{L_{\star}}$ halos,
(c) as lower mass halos (lower luminosity galaxies) are accounted for, the
physical extent of Civ absorbing gas resides progressively in the inner
regions of a halo and/or absorbers become much less common than halos, and (d)
for ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers, there is a minimum in the
incidence of the Civ absorbing structures relative to virial halos at
$z\\!\simeq\\!1.5$, shortly after Cosmic Noon, followed by a remarkable
relative increase at $z\\!\leq\\!1$.

As informative as these concluding remarks may be, some of them require key
assumptions about the association of Civ absorbers and galaxy halos. The
assumption that each individual Civ absorbing structure is coincident with a
single DM halo hosting a single galaxy may not be an adequate or accurate
description of absorber-galaxy associations, as we discuss later. Regardless,
the analysis above does provide insights into the relative global evolution of
the product of cosmic number density and cross-sectional area of cumulative
populations of Civ absorbers and galaxy halos.

##  3 The extent of Civ around galaxies

In this section we aim to characterize the redshift evolution of the “gas
radius” as a function of galaxy luminosity, which we refer to as
$R_{g}({L/L_{\star}},z)$. We first describe how we estimate the gas radius,
then we examine its evolution for a fixed galaxy luminosity. Finally, we
examine the redshift evolution as a function of galaxy luminosity. Our focus
is to obtain some insight on how $R_{g}({L/L_{\star}},z)$ compares and evolves
relative to the virial radii of halos hosting galaxies of various
luminosities.

###  3.1 Estimating the Gas Radius

Figure 3:  Evolution of characteristic Civ gas radius $R_{\star}$ (Eq. B2),
and virial radius $R_{v}$ (Eq. A2), for $L_{\star}$ galaxies, assuming
galaxies with ${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$ are associated with
the absorbers. Top panels: $R_{g}\\!=\\!{R_{\star}}$ as a function of redshift
are shown for (a) ${W_{r,{\rm lim}}}\\!=\\!0.05$ Å absorbers (blue data points
and curves); (b) ${W_{r,{\rm lim}}}\\!=\\!0.3$ Å absorbers (purple data points
and curves); and (c) ${W_{r,{\rm lim}}}\\!=\\!0.6$ Å (orange data points and
curves). The black curves and grey shading represent $R_{v}$($L_{\star}$) and
$\pm 1\sigma$ uncertainties as a function of redshift. Bottom panels:
$f_{R}\\!=\\!{R_{\star}}/{R_{v}}({L_{\star}})$ shown as a function of redshift
for each population. In each panel, the data points and error bars represent
the value obtained when $f_{c}=1$ is assumed, while the solid, dash-dotted,
and dotted curves show the values obtained if we assume $f_{c}\\!=\\!0.75$,
$0.50$, and $0.25$, respectively. The horizontal dotted black line shows
$f_{R}\\!=\\!1$, above which absorbers live outside the DM halo. We adopt
$\beta\\!=\\!0.5$ based on Chen et al. (2001).

Under the assumption that each Civ absorber is associated with a single galaxy
halo, the cosmic number density of gas structures, $n(z)$ in Eq. 1, is given
by the number density of galaxies above a minimum luminosity,
$L_{\mathrm{min}}$. The cross-section of absorbing gas is $\sigma(z)=\pi
R_{g}^{2}({L/L_{\star}},z)$, where the characteristic gas absorption radius
for a given $W_{r,{\rm lim}}$ is assumed to scale with galaxy luminosity as

$R_{g}({L/L_{\star}},z)=R_{\star}(z)~{}({L/L_{\star}})^{\beta}\,,$ (4)

where ${R_{\star}}(z)$ is the characteristic absorption radius for an
$L_{\star}$ galaxy at redshift $z$, and $\beta$ is the Holmberg scaling
parameter (e.g., Holmberg, 1975), which is commonly employed to parameterize
how the gas radius scales with galaxy luminosity (e.g., Bergeron & Boissé,
1991; Steidel, 1995; Chen et al., 2001; Kacprzak et al., 2008; Nielsen et al.,
2013a).

To compute ${R_{\star}}(z)$, we use the method employed in Paper I using the
observed Civ incidence, which we summarize in Appendix B. Importantly,
${R_{\star}}(z)$ is not a function of $dN/dX$ and $z$ only, but also of the
assumed minimum galaxy luminosity ${L_{\mathrm{min}}}/{L_{\star}}$, the
Holmberg power-law luminosity scaling $\beta$, the multi-parameter luminosity
function, $\phi(\phi_{\star},{L/L_{\star}},\alpha,z)$, and the absorbing gas
covering fraction $f_{c}(z)$.

With regard to the covering fraction, sparse data are available for Civ
absorbers. At $z\\!\sim\\!0$, Bordoloi et al. (2014) measured the covering
fraction within $\sim$0.5$R_{v}$ of galaxies with
$L\\!\sim$0.005–0.14$L_{\star}$ to be $\langle f_{c}\rangle\\!\simeq\\!0.5$,
while Burchett et al. (2016) found $\langle f_{c}\rangle\\!\simeq\\!0.33$
within $R_{v}$ of $L\\!\geq\\!{0.01L_{\star}}$ galaxies. From stacked spectra
of $L\\!\geq\\!{0.01L_{\star}}$ galaxies, Liang & Chen (2014) estimated a mean
within $R_{v}$ of $\langle f_{c}\rangle\\!\simeq\\!0.45$ at $z\\!<\\!0.18$.
These studies all had a detection threshold of ${W_{r,{\rm
lim}}}\\!\simeq\\!0.05$ Å. Schroetter et al. (2021) constrained the covering
fraction of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers at $z\\!\sim$1–1.5
around $L\\!\geq\\!{0.01L_{\star}}$ galaxies. Based on their model for Civ
covering fraction, we estimate $\langle f_{c}\rangle\\!\simeq\\!0.25$ within
$R_{v}$. Finally, for $z\\!\sim\\!2\\!-\\!3$, Rudie et al. (2019) estimated
the covering fraction of Civ around eight $L\\!\geq\\!{0.1L_{\star}}$
galaxies. Adopting their column density threshold of $\log
N/\mathrm{cm}^{-2}\\!=\\!13$ as the rough equivalent of ${W_{r,{\rm
lim}}}\\!=\\!0.05$ Å, we estimate the average of their measurements to be
$\langle f_{c}\rangle\\!\simeq\\!0.75$ within 100 kpc. As for the uncertainty
in the covering fraction, ${R_{\star}}(z)$ scales as $f_{c}^{-1/2}(z)$. Our
fiducial calculations employ $f_{c}(z)\\!=\\!1$, but we also explore the
values $f_{c}(z)\\!=\\!0.25$, 0.50, 0.75, and assume no redshift evolution.

We assume ${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$ based on both Bordoloi et
al. (2014) and Burchett et al. (2016) finding absorption around
$L\\!\geq\\!{0.01L_{\star}}$ galaxies. For these plots, we assume
$\beta\\!=\\!0.5$ following the findings of Chen et al. (2001). Note that
$\beta$ could be a function of $z$, ${L/L_{\star}}$, and/or $W_{r,{\rm lim}}$.
As there are no observational constraints on such dependencies with $\beta$,
we assume it is a single valued constant. For Civ absorbers with ${W_{r,{\rm
lim}}}\\!\simeq\\!0.1$ Å, Chen et al. (2001) reports $\beta\\!=\\!0.5$ for
$B$-band luminosity at $z\\!\leq\\!1$. We use $\beta\\!=\\!0.5$ for our
fiducial calculations, however, we explore a range of values between
$0\\!\leq\\!\beta\\!\leq\\!0.5$. Once we compute ${R_{\star}}(z)$, we can
obtain $R_{g}({L/L_{\star}},z)$ from Eq. 4 as a function of galaxy luminosity
for the assumed value of $\beta$.

###  3.2 Evolution of the Gas Radius

####  3.2.1 Redshift Evolution for $\beta=0.5$

In Figure 3 (upper panels), we present the redshift evolution of the gas
radius around $L_{\star}$ galaxies for ${W_{r,{\rm lim}}}\\!=\\!0.05$, 0.3,
and 0.6 Å Civ absorbers, respectively. In this special case the gas radius
given by Eq. 4 is $R_{\star}(z)$. We also show the virial radius for
$L_{\star}$ galaxies, $R_{v}({L_{\star}})$, as the black curve with gray
shading. The virial radius $R_{v}({L_{\star}})$ is calculated from Eq. A2 with
$M_{h}(z)$ corresponding to ${L_{\star}}$ using abundance matching at redshift
$z$ (see Appendix A). Recall that the halo mass of an $L_{\star}$ galaxy
evolves as shown in Figure 7(b). The characteristic gas radius
${R_{\star}}(z)$ for a covering fraction of $f_{c}(z)\\!=\\!1$ is given by the
data points as computed directly from $dN/dX(z)$ using Eq. B2 with
$\beta\\!=\\!0.5$. We show the sensitivity to the assumed covering fractions
by also presenting $f_{c}(z)\\!=\\!0.75$ (dash-dotted curves),
$f_{c}(z)\\!=\\!0.50$ (solid curves), and $f_{c}(z)\\!=\\!0.25$ (dotted
curves).

In Figure 3 (lower panels), we plot the ratio of the absorbing gas radius of
absorbers to the virial radius of the host galaxy’s DM halo, which we will
refer to as the “relative gas radius,”

$f_{R}({L/L_{\star}},z)=\frac{R_{g}({L/L_{\star}},z)}{R_{v}({L/L_{\star}},z)}\,,$
(5)

which quantifies the extent of Civ-absorbing gas relative to the virial radius
as a function of redshift.

We find that, for $L_{\star}$ galaxies, absorbers with
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ reside well within the virial radius at all
redshifts, independent of the assumed covering fraction. The relative gas
radius appears to slightly increase from high redshift and peak across Cosmic
Noon, following which it then decreases toward the current epoch. For
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ absorbers, the relative gas radius is well
within the virial radius at all redshifts for $f_{c}\\!>\\!0.25$ and is
suggestive of a mild decrease with redshift from $z\\!=\\!3$ toward the
current epoch. For ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers, the gas
extends out to beyond the virial radius (especially at $z\\!\simeq\\!0$ and
$z\\!\geq\\!3$) for $f_{c}\\!\leq\\!0.75$. Interestingly, for these weakest
absorbers, the relative gas radius declines from $z\\!=\\!4$, minimizes across
Cosmic Noon, and then increases toward the current epoch.

Using Eq. 4 to estimate $R_{g}({L/L_{\star}},z)$, we can examine
$f_{R}({L/L_{\star}},z)$ for arbitrary $L/L_{\star}$. As an example, in Figure
4 (upper panels), we present the redshift evolution of the gas radius for
$0.1L_{\star}$ galaxies. The virial radius $R_{v}({0.1L_{\star}})$ is shown as
the black curve with gray shading. For $\beta\\!=\\!0.5$, we have
$R_{g}({0.1L_{\star}},z)\\!\approx\\!0.32R_{g}({L_{\star}},z)\\!=\\!0.32{R_{\star}}(z)$.
In Figure 4 (lower panels), we show $f_{R}({0.1L_{\star}},z)$ for ${W_{r,{\rm
lim}}}\\!=\\!0.05$, 0.3, and 0.6 Å absorbers, respectively. Note that the
redshift evolutionary behavior of the relative gas radius follows similar
trends as for $L_{\star}$ galaxies; this is because the gas radius depends on
the $dN/dX$ of absorbers and the growth of the virial radius is highly similar
for $0.1L_{\star}$ galaxies. However, $f_{R}({0.1L_{\star}},z)$ is
systematically smaller than $f_{R}({L_{\star}},z)$, suggesting that the extent
of Civ absorbing gas relative to the virial radius is smaller in lower
luminosity galaxy halos. In fact, for ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
absorbers, the gas radius may not quite extend out to the virial radius,
instead only extending beyond $R_{v}$ for $f_{c}\\!\geq\\!0.25$.

Figure 4:  Same as Figure 3, but for $0.1L_{\star}$ galaxies.

Our analysis would suggest that for $L_{\star}$ galaxies,
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ Civ absorption typically resides in the
inner 50–80% of the virial halo, and ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
absorption in the inner 30–50% of the virial halos for covering fractions of
0.5 or greater. For $0.1L_{\star}$ galaxies, these radii are slightly smaller,
30–50% and 10–30% for ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ absorption, respectively. However, for
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorption, the gas radius may extend, on
average, beyond the virial radius for $L_{\star}$ galaxies and between 60% and
100% $R_{v}$ for $0.1L_{\star}$ galaxies.

####  3.2.2 Luminosity Dependent Evolution

We now examine ${f_{R}}({L/L_{\star}},z)$ as a function of luminosity and
redshift for ${W_{r,{\rm lim}}}\\!=\\!0.05$, 0.3, and 0.6 Å Civ absorbers for
three different values of $\beta$. Our fiducial value of $\beta\\!=\\!0.5$ is
motivated by the findings of Chen et al. (2001) for ${W_{r,{\rm
lim}}}\\!=\\!0.1$ Å Civ absorbers at $z\leq 1$. This value represents a strong
luminosity dependence for the gas radius $R_{g}({L/L_{\star}},z)$ and will
yield smaller gas radii around lower luminosity galaxies. It is not clear if
such a strong dependence holds for all galaxy luminosities, redshifts, or
${W_{r,{\rm lim}}}$. In the absence of constraining data, we adopt a simple
bracketing of the luminosity scaling by exploring $\beta\\!=\\!0$, 0.25, and
0.5. For $\beta\\!=\\!0$, there is no luminosity scaling and we have
$R_{g}({L/L_{\star}},z)\\!=\\!R_{\star}(z)\\!\propto\\!{(dN/dX(z))/f_{c}(z)}^{1/2}$
for a given ${W_{r,{\rm lim}}}$. In this case, the Civ absorber gas radius is
not physically coupled to the stellar mass and/or star formation rate (SFR).
The value $\beta\\!=\\!0.25$ represents an intermediate luminosity
sensitivity.

In Figure 5, we present the $\beta$ dependence of ${f_{R}}({L/L_{\star}},z)$
as a function of $L/L_{\star}$ and redshift for the three $W_{r,{\rm lim}}$
populations of Civ absorbers. The solid curves represent
$f_{R}({L/L_{\star}},z)$ for $f_{c}(z)\\!=\\!0.5$, while dashed curves are for
$f_{c}(z)\\!=\\!1$. We show ${f_{R}}({L/L_{\star}},z)$ for $\beta=0$
(magenta), $\beta=0.25$ (light blue), and $\beta=0.5$ (dark blue). As
mentioned above, the Chen et al. (2001) value of $\beta\\!=\\!0.5$ indicates a
strong dependence of gas radius on galaxy luminosity, while $\beta\\!=\\!0.25$
indicates a weaker scaling, and $\beta\\!=\\!0$ means that the gas radius has
no luminosity dependence. The horizontal dotted line in all panels of Figure 5
represents ${f_{R}}({L/L_{\star}},z)\\!=\\!1$, where gas radius equals virial
radius; ${f_{R}}\\!>\\!1$ means Civ is outside $R_{v}$ on average,
${f_{R}}\\!<\\!1$ means Civ is inside $R_{v}$ on average.

In Figure 5, we see that the luminosity and redshift dependence of
$f_{R}({L/L_{\star}},z)$ for a given $W_{r,{\rm lim}}$ is strongly determined
by the choice of $\beta$. For $\beta\\!=\\!0.5$, the relative gas radius
increases with luminosity, such that brighter (more massive) galaxies have
larger Civ envelopes relative to their virial radii than do fainter (less
massive) galaxies. This luminosity dependence is stronger as redshift
increases. For $\beta\\!=\\!0$, the trend seen for $\beta=0.5$ is reversed;
${f_{R}}({L/L_{\star}},z)$ increases toward lower galaxy luminosity and the
sensitivity to luminosity increases as redshift decreases. For $\beta=0.25$,
the luminosity dependence of $f_{R}({L/L_{\star}},z)$ is weak and changes
little with redshift.

Figure 5:  $f_{R}\\!=\\!R_{g}/R_{v}$ (Eq. 5) as a function of ${L/L_{\star}}$
for absorbers with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ (top panels),
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ (middle panels), and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ (bottom panels). Each column is a different
redshift, from $z\\!=\\!0$ to $z\\!=\\!5$. The magenta, light blue, and dark
blue curves represent $f_{R}$ for $\beta\\!=\\!0$, $0.25$, and $0.5$,
respectively. For solid curves, we assume $f_{c}(z)\\!=\\!0.5$, and dash-
dotted curves assume $f_{c}(z)\\!=\\!1$.
${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$ is adopted in every case. The
horizontal dotted black line shows $f_{R}\\!=\\!1$ in each panel.

For $f_{c}(z)\\!=\\!0.5$, our analysis suggests some fraction of the weakest
absorbers (with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$) likely populate regions
beyond the virial radius of galaxy halos. If $\beta\\!=\\!0.5$, this
predominantly occurs around higher luminosity galaxies, depending on redshift;
e.g. $f_{R}({L/L_{\star}},z)\\!\geq\\!1$ for $L\\!\geq\\!0.06{L_{\star}}$ at
$z\\!=\\!0$ and $L\\!\geq\\!0.2{L_{\star}}$ at $z\\!=\\!2$. If
$\beta\\!=\\!0$, super-virial gas halos are predominantly around lower
luminosity galaxies; e.g. $f_{R}({L/L_{\star}},z)\\!\geq\\!1$ for
$L\\!\leq\\!0.4{L_{\star}}$ at $z\\!=\\!0$ and $L\\!\leq\\!0.06{L_{\star}}$ at
$z\\!=\\!2$.

On the other hand, for $f_{c}(z)\\!\\!=\\!\\!0.5$, stronger absorbers
typically live within the inner 50% of the virial halos regardless of the
choice of $\beta$. However, if $\beta\\!=\\!0.5$, our analysis would predict
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ absorbers populate halos out to the outer
edge of the virial radius for $L\\!\geq\\!{L_{\star}}$ at $z\\!\geq\\!3$. For
a given galaxy luminosity at redshift $z$, the relative gas radius decreases
with increasing $W_{r,{\rm lim}}$ (primarily because ${dN/dX}(z)$ decreases
with increasing ${W_{r,{\rm lim}}}$). The luminosity and redshift dependence
of $f_{R}({L/L_{\star}},z)$ for ${W_{r,{\rm lim}}}\\!=\\!0.3$ Å and
${W_{r,{\rm lim}}}\\!=\\!0.6$ Å absorbers are similar to that of ${W_{r,{\rm
lim}}}\\!=\\!0.05$ Å absorbers.

The above analysis has a high degree of potential for statistically
constraining the spatial distribution of Civ absorbers relative to the viral
radius of galaxies as a function of $W_{r,{\rm lim}}$, galaxy luminosity, and
redshift. However, the resulting behavior of $f_{R}({L/L_{\star}},z)$ depends
strongly on the luminosity scaling of the gas radius and covering fraction.
Neither is observationally constrained as a complete function of luminosity or
redshift. Further, the covering fraction depends on $W_{r,{\rm lim}}$, and
this is not observationally established as a function of redshift. To date, we
can draw only from the pioneering observational programs that examine these
relationships Chen et al. (2001); Steidel et al. (2010); Bordoloi et al.
(2014); Liang & Chen (2014); Burchett et al. (2016); Rudie et al. (2019);
Schroetter et al. (2021). In this sense, until additional observations are
conducted, our analysis should be viewed as an averaged approximation,
assuming that such dependencies, if present (see Chen, 2012), average out.

Our estimates provide a statistical relative gas radius as a function of
galaxy luminosity and redshift assuming a non-evolving luminosity scaling,
$\beta$, and covering fraction, $f_{c}$. For ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
absorbers, it is fairly uncontroversial that, statistically, the gas radius
generally resides within the inner half of the virial halo for a wide range of
luminosities over the redshift range $z\\!=\\!5$ to $z\\!=\\!0$. For
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ absorbers the regions occupied by the Civ
bearing gas are slightly more extended such that the $\beta$ dependence
predicts the outer half of the virial radius is occupied in the low luminosity
regime following Cosmic Noon ($\beta\\!=\\!0$) or the high luminosity regime
prior to Cosmic Noon ($\beta\\!=\\!0.5$).

In the case of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers, we see that
the luminosity scaling parameter strongly affects whether highly extended,
i.e., ${f_{R}}({L/L_{\star}},z)\\!\geq\\!1$, absorbing structures reside
around low luminosity galaxies after Cosmic Noon or around high luminosity
galaxies prior to Cosmic Noon.

###  3.3 Comparison with Observations

Figure 6: $f_{R}({L/L_{\star}},z)$ for absorbers with
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ as a function of galaxy luminosity, at (a)
$z\\!\simeq\\!0$, (b) $z\\!\simeq\\!1$, and (c) $z\\!\simeq\\!2.5$. As in
Figure 5, the magenta, light blue, and dark blue curves in each panel show the
$f_{R}$ estimated from eq. 5 with $\beta\\!=\\!0$, $0.25$, and $0.5$,
respectively. The solid and dash-dotted curves represent covering fractions of
$50\%$ and $100\%$, respectively. (a) The circles are observed $f_{R}$ values
for COS-Dwarfs galaxies (Bordoloi et al., 2014), while the triangles are from
the sample of Burchett et al. (2016). All the detections are color-coded by
Civ absorber $W_{r}$, while the non-detections are shown by open circles and
triangles. (b) The squares are observed $f_{R}$ values from Schroetter et al.
(2021), color-coded by absorber $W_{r}$. (c) The diamonds are observed $f_{R}$
values from Rudie et al. (2019), color-coded by absorber column density, $\log
N~{}(\mathrm{cm}^{-2})$. The yellow shaded areas represent the effective
search space of (a) the COS-Dwarfs ($\\!D\leq\\!150$ kpc and
${L/L_{\star}}\\!\leq\\!0.15$), (b) MEGAFLOW ($D\\!\leq\\!250$ kpc), and (c)
KBSS ($D\\!\leq\\!100$ kpc and ${L/L_{\star}}\\!\geq\\!0.1$) surveys,
respectively. All these studies had a detection threshold of ${W_{r,{\rm
lim}}}\\!\simeq\\!0.05$ Å.

Hereafter, we refer to ${f_{R}}({L/L_{\star}},z)$ as $f_{R}$, such that the
luminosity and redshift are implied in the ratio ${R_{g}}/{R_{v}}$. In Figure
6, we compare the available observational data to our estimates of $f_{R}$ for
three different $\beta$ values ($\beta\\!=\\!0$ is magenta, $\beta\\!=\\!0.25$
is light blue, and $\beta\\!=\\!0.5$ is dark blue) and two covering fractions
($f_{c}(z)\\!=\\!1$ are dash-dotted and $f_{c}(z)\\!=\\!0.5$ are solid).

In Figure 6(a), we plot the absorber-galaxy pairs of Bordoloi et al. (2014,
circles) and Burchett et al. (2016, triangles) at $z\\!\simeq\\!0$. Since both
studies adopted $R_{200}$ for the virial radius, we applied a scale factor of
$\sim$1.2 to enable direct comparison with $R_{v}$ (see Appendix C). The COS-
Dwarfs survey (Bordoloi et al., 2014) selected for
$L\\!\leq\\!0.14{L_{\star}}$ galaxies probed within impact parameters of
$D\\!\leq\\!150$ kpc (their search space is shaded light yellow). The blind
survey of Burchett et al. (2016) searched for galaxies associated with
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers with no a priori selection
for galaxy luminosity or impact parameter. Note that the measured $W_{r}$ is
color coded over the range shown on the color bar.

Given the number of non-detections (open data points) below the $f_{R}$
curves, it is clear that $f_{c}\\!=\\!1$ does not apply. For
$f_{c}\\!=\\!0.5$, taking a census of the data, we find that for
$\beta\\!=\\!0$ the observed covering fraction within ${f_{R}}$, i.e., under
the magenta curve, is $f_{c}\\!=\\!N_{det}/N_{tot}\\!=\\!0.46$. Similarly for
$\beta\\!=\\!0.25$, the light blue curve, we count $f_{c}\\!=\\!0.48$, and for
$\beta\\!=\\!0.5$, the dark blue curve, we count $f_{c}\\!=\\!0.56$. However,
dwarf galaxies with $L\\!\leq\\!{0.1L_{\star}}$ are found to typically contain
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ in the inner half of the virial halo
(${f_{R}}\\!\leq\\!0.5$) and only one such galaxy has ${f_{R}}\\!>\\!1$ as is
predicted by a $\beta\\!=\\!0$ scenario. Thus, the data at $z\\!\simeq\\!0$
are consistent with $\beta\\!=\\!0.25$ and $\beta\\!=\\!0.5$ for
$f_{c}\\!=\\!0.5$, but not with $\beta\\!=\\!0$ at low luminosities. These
covering fractions are in reasonable agreement with those found by Bordoloi et
al. (2014), Liang & Chen (2014), Burchett et al. (2016), and Manuwal et al.
(2021) of $f_{c}\\!=\\!0.5$, $0.45$, $0.33$, and $0.43$ respectively, within
the virial radius. Chen et al. (2001) found $\beta\\!=\\!0.5$ for a sample of
galaxies with $\langle z\rangle\\!=\\!0.4$, but this was for $B$-band
luminosity (we adopted the UV-band) and for a Civ absorption sensitivity of
$W_{r}\\!\geq\\!0.2$ Å. Nielsen et al. (2013a) found the Mgii $\beta$ to vary
significantly with luminosity band and $W_{r,{\rm lim}}$. At $z\\!\simeq\\!0$,
the most leverage that could be applied to constrain our models would occur by
increasing the number of detections at $f_{R}\\!\sim\\!2$ for both
$L\\!<\\!0.1{L_{\star}}$ and $L\\!\simeq\\!{L_{\star}}$.

In Figure 6(b), we compare to the observations of Schroetter et al. (2021)
covering $1.0\\!\leq\\!z\\!\leq\\!1.5$. These data correspond to the Civ
absorbers in their study for which the authors identified a single galaxy
“counterpart.” The methods by which we computed the virial radii and UV band
luminosities of these galaxies are presented in Appendix C. The Schroetter et
al. (2021) search space was 250 pkpc (proper kpc) projected from the
background quasar at the redshift of the absorption and their minimum
luminosities extend below 0.01$L_{\star}$ (their search space is shaded light
yellow). Further, the absorption was known a priori, so we are plotting
detections only. However, from their survey data, they presented an impact
parameter dependent model of the covering fraction, which we estimate to have
an impact parameter weighted average of $f_{c}\\!=\\!0.25$ within $R_{v}$.

Of the nine galaxies appearing on Figure 6(b), two are inconsistent with
$\beta\\!=\\!0$ for $f_{c}\\!=\\!0.5$. However, these two galaxies are
consistent with $\beta\\!=\\!0.25$ and 0.5, especially if the covering
fraction is lower than $f_{c}\\!=\\!0.5$. At $1.0\\!\leq\\!z\\!\leq\\!1.5$,
the most leverage that could be applied to constrain our models would occur by
increasing the number of detections to $f_{R}\\!>\\!1$ for
$L\\!<\\!{0.1L_{\star}}$. However, continued examination of
$L\simeq{L_{\star}}$ galaxies out to $f_{R}\\!>\\!1.2$ would also apply
leverage. As our formalism associates a single absorber to an individual
galaxy, we present only the single galaxy counterparts to Civ absorbers from
Schroetter et al. (2021), which constituted just $\approx$17% of their
detected Civ sample. In contrast, $\approx$30% of their absorbers had multiple
galaxy counterparts within 250 kpc, indicating the role that clustering or
grouping of galaxies may play in Civ absorption. We revisit this observational
fact and its implications in detail below.

Finally, in Figure 6(c), we compare to the seven galaxies observed by Rudie et
al. (2019), covering $2.1\\!\leq\\!z\\!\leq\\!2.7$. The impact parameter range
for this targeted sample of galaxies was 35–100 pkpc projected from the
background quasar at the redshift of the absorption (their search space is
shaded light yellow). The methods by which we computed the virial radii and UV
band luminosities of these galaxies are presented in Appendix C. The measured
absorption strengths are not $W_{r}$, but are the total Civ column densities
(sum of Voigt profile components), color coded over the range shown on the
color bar. For reference, a single-component Civ absorber with $\log N/{\rm
cm}^{-2}\\!=\\!14.1$ would have $W_{r}\\!=\\!0.13$, $0.2$, and $0.3$ Å for
$b\\!=\\!8$, $15$, and $30$ km s-1, respectively.

Weighting by column density, we estimate that the Rudie et al. (2019) sample
has a mean covering fraction of $f_{c}\\!\simeq\\!0.75$ for
$\approx$$L_{\star}$ galaxies at $2.1\\!\leq\\!z\\!\leq\\!2.7$ (see their
Figure 5). Though the galaxies appear nearly consistent with $\beta\\!=\\!0$
and $f_{c}\\!=\\!0.5$, this is an artifact of the galaxies being selected
within a luminosity independent, fixed impact parameter range (100 pkpc).
Unfortunately, the data available at the Cosmic Noon era shown on Figure 6(c)
fill too limited of a parameter space to discriminate between the free-
parameters $\beta$ and $f_{c}$ in our models. At these redshifts, the derived
UV luminosities of the galaxies are sensitive to $k$-corrections, which can
range from 0.2 for an Im galaxy, 1.2 for an Scd galaxy, and 2.2 for an Sbc
galaxy (e.g., Nielsen et al., 2013b). As described in Appendix C, we adopted
the average of Im and Scd galaxies, appropriate for star forming galaxies at
these redshifts.

To improve the observational constraints on $\beta$ and $f_{c}$, much larger
samples of direct detections to small equivalent width detection thresholds
($W_{r}\\!\simeq\\!0.05$ Å) will be necessary. In summary, at $z\\!\sim\\!0$,
observations are needed to $f_{R}\\!\sim\\!2$ for $L\\!<\\!{0.1L_{\star}}$ and
$L\\!\simeq\\!{L_{\star}}$. At $1.0\\!\leq\\!z\\!\leq\\!1.5$, observations to
$f_{R}\\!>\\!1$ for $L\\!<\\!{0.1L_{\star}}$ would be useful. At
$2.1\\!\leq\\!z\\!\leq\\!2.7$, the data remain highly incomplete; at Cosmic
Noon it will remain a challenge to constrain the $\beta$ luminosity dependence
unless $\beta\\!\geq\\!0.5$ and deep surveys down to
${L/L_{\star}}\\!\leq\\!0.1$ out to ${f_{R}}\\!=\\!1.2$ can be conducted.
These data are critical as they would provide direct information on where Civ
lives around galaxies of different luminosities as a function of redshift.
Such data would provide essential observational constraints on cosmological
simulations that theoretically model the baryon cycle of galaxies.

###  3.4 Uncertainties, Assumptions, and Caveats

Our primary assumption in calculating the Civ gas radius is that each absorber
is found around a single galaxy, so that the abundance of absorbers is the
same as the abundance of galaxies above a minimum luminosity and of DM halos
above a minimum mass. However, several pieces of observational evidence, which
we discuss in Section 4.1, would indicate that this may not be an accurate
picture of galaxy-absorber association.

The Holmberg luminosity scaling, $\beta$, used to estimate gas radius (Eq. B2)
is quite uncertain. To date, only Chen et al. (2001) has constrained
$\beta=0.5$ for $B$-band luminosity at $z\\!<\\!1$ for Civ with $W_{r}\geq
0.2$ Å. Due to the lack of sufficient constraints, we have set $\beta$ to be a
fixed quantity for all $W_{r,{\rm lim}}$ at all $L_{\star}$ and all $z$.
However, for Mgii, Nielsen et al. (2013a) established that $\beta$ varies with
$z$, $W_{r,{\rm lim}}$, galaxy color, and luminosity band. Furthermore,
Churchill et al. (2013) reported different halo mass $\beta$ scalings as Mgii
$W_{r,{\rm lim}}$ is varied. Lastly, Civ absorption is known to be more
extended for AGN-host galaxies than other galaxy types (e.g. Prochaska et al.,
2014), which would imply that these galaxies may have a different Holmberg
scaling.

The covering fraction $f_{c}$ is an additional uncertainty. In Eq. B2, we have
assumed a “mean” covering fraction for gas halos at a given redshift. In fact,
the Civ covering fraction is observed to have a profile that declines with
increasing distance from the central galaxy (e.g., Bordoloi et al., 2014;
Burchett et al., 2016; Schroetter et al., 2021). Furthermore, at fixed impact
parameter, $f_{c}$ is observed to decrease with increasing $W_{r,{\rm lim}}$
(Bordoloi et al., 2014; Rudie et al., 2019). The few existing measurements are
also limited by small sample sizes, especially at $z\\!>\\!2$ (e.g., Rudie et
al., 2019). The covering fraction is also found to be dependent on star
formation activity and stellar/halo mass (Bordoloi et al., 2014; Burchett et
al., 2016), implying that $f_{c}$ is also potentially a function of luminosity
or halo mass. In addition, the covering fraction of gas around quasar-host
galaxies is known to be higher than around other types of galaxies (e.g.
Prochaska et al., 2014; Landoni et al., 2016). For example, Prochaska et al.
(2014) found ${W_{r}}\\!>\\!0.2$ Å Civ to persist beyond $\sim$500 kpc from
$M_{h}\\!\gtrsim\\!10^{12.5}$ $\mathrm{M}_{\odot}$ quasar-host galaxies at
$z\\!\sim\\!2$. However, given that such galaxies are very massive
($L\\!>\\!{L_{\star}}$), they represent a much rarer population of galaxies
than the overall population of galaxies at $L\\!\sim\\!0.01$–2$L_{\star}$ that
we study here. Finally, Burchett et al. (2016) also reported a dependence of
covering fraction on galaxy environment. Given the lack of robust constrains
on $f_{c}(z)$, we assumed a constant $f_{c}$ for all $L/L_{\star}$ and for all
$W_{r,{\rm lim}}$, without regard to impact parameter.

The minimum galaxy luminosity, $L_{\mathrm{min}}$, as applied in both Eq. B2
and our DM halo abundance matching is also not well-constrained. The low-$z$
studies of Bordoloi et al. (2014) and Burchett et al. (2016) suggest that
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ is found around galaxies as faint as
$\sim\\!{0.01L_{\star}}$, but this limit is found to be about
$\sim\\!0.4{L_{\star}}$ at $z\\!\sim\\!2.5$ (Rudie et al., 2019, see Figure
6). It remains unclear whether $L_{\mathrm{min}}$ decreases at higher redshift
or if an apparent decrease is simply due to the difficulty in identifying
fainter galaxies with Civ absorbers at higher redshifts. We assumed
${L_{\mathrm{min}}}\\!=\\!{0.01L_{\star}}$ for all redshifts, based on Civ
detections around these lower luminosity galaxies at low redshifts. Assuming
${L_{\mathrm{min}}}\\!=\\!{0.1L_{\star}}$ instead yields $\approx\\!25$–$40$%
smaller $f_{R}$ values at $z\\!=\\!2$–5, but the redshift-evolution is
qualitatively the same. Hence, the magnitude of gas radius values is sensitive
to $L_{\mathrm{min}}$, but the evolution is not.

Central to our model is the assumption that a single luminous galaxy resides
in each DM halo. To equate luminous galaxies to DM halos, we adopted abundance
matching, which provides a one-to-one mapping between galaxy luminosity and DM
halo mass for a given adopted luminosity function and halo mass function (see
Appendix A). While we do incorporate appropriate scatter to account for the
vast array of baryonic processes involved in galaxy formation in halos, there
are several additional sources of systematic and statistical uncertainty in
abundance matching that merit detailed discussion (see Behroozi et al., 2010;
Wechsler & Tinker, 2018).

The model also does not take into account any halo mass dependence of
$W_{r,{\rm lim}}$, such that all populations of absorbers, regardless of
absorption strength, are assumed to have the same abundance as halos above a
minimum mass. This neglects any possible underlying relationship between halo
(or galaxy) mass and absorption strength, for example if strong Civ absorbers
were found only around high-mass galaxies.

The modelled behavior of the of gas radius relative to the virial radius
depends on the adopted definition of the virial radius. While many authors
employ a fixed virial overdensity of $\Delta_{v}\\!=\\!200$, we use the
spherical collapse approximation of Bryan & Norman (1998), where $\Delta_{v}$
ranges from $\approx$370 (at $z\\!=\\!0$) to $\approx$178 (at $z\\!>\\!4$) for
our cosmology (Komatsu et al., 2009). Using the most recently published set of
cosmological parameters (Planck Collaboration et al., 2020) would change the
resulting $R_{v}$ by up to $\sim$4%. Some authors have used alternative
definitions for the extent of influence of a DM halo. More et al. (2015)
advocated the use of a “splashback radius,” which corresponds to the apocenter
of the orbits of the most recently accreted matter in a halo, as a more
physically motivated halo boundary than the virial radius. Shull (2014)
instead considered the redshift at which half the mass in a halo had
collapsed/virialized, which yields a $\approx$40–50% smaller virial radius.
Finally, Diemer et al. (2013) demonstrated that at $z\\!<\\!1$, a large
fraction of the growth of halo mass and virial radius is due to pseudo-
evolution, i.e. the evolution of a reference density (such as the critical
density), rather than the actual physical accretion of matter.

Finally, we note that our model is based on many average properties of
galaxies and DM halos as a function of redshift. We have taken no account of
galaxy environment nor the rich diversity of individual galaxy formation and
evolution histories (including the star formation history which would dictate
the strengths and rates of outflows; see below) that certainly have a
correspondence with how Civ absorbing gas is distributed on a case by case
basis.

##  4 Discussion

We investigated the statistical connection between galaxy halos and their
enriched gas envelopes across $\sim$12.5 Gyr of cosmic time, based on observed
statistics of Civ absorption selected systems and galaxies. While we assume a
simplified model in which each individual Civ absorber is associated with a
single galaxy DM halo, we recognize that this scenario represents only one of
many possible scenarios for how metal-enriched absorbers observed in quasar
spectra are related to galaxies and DM halos. Our model does serve, however,
to provide a working framework on which to examine this relationship as more
data become available in the future. In this section, we examine our model in
the context of the baryon cycle governing galaxy evolution and then discuss
several alternative scenarios and how they compare.

###  4.1 Our Model in Context

The assumption that each absorber is found around a different galaxy, hence of
equal abundance of these populations, is a stringent criterion of our model.
Civ absorbers found in quasar spectra are sometimes clustered close to each
other in redshift space, indicating they could arise in the CGM of the same
galaxy (e.g., Péroux et al., 2004; Cooksey et al., 2013; Burchett et al.,
2015; Hasan et al., 2020; Schroetter et al., 2021). Significant clustering of
absorbers in physical space could result in degeneracies in the absorber
abundance relative to galaxy abundance. However, the clustering of systems
does depend on the velocity window across which a single system is defined,
and this varies among different studies. For context, we defined a single
absorber to be within a velocity window of $\Delta v\\!=\\!500$ km s-1 (Paper
I). In our current sample, $\sim$1% and $\sim$5% of the absorbers are found
within 1000 and 3000 km s-1, respectively, of another absorber along the same
line-of-sight. While this could imply possible physical connection between
those absorbers, such as associated via a large-scale structure such as a
cluster or group, the large velocities are consistent with expectations that a
negligible number of our absorbers overlap in a single DM halo.

Transverse absorber studies have found a non-negligible fraction of metal
absorbers beyond the virial radii of galaxies (e.g., Prochaska et al., 2011;
Turner et al., 2014; Mathes et al., 2014; Liang & Chen, 2014; Huang et al.,
2021) and/or gravitationally unbound from DM halos due to their apparent
velocities (e.g., Tumlinson et al., 2011; Bordoloi et al., 2014; Rudie et al.,
2019). On the other hand, Schroetter et al. (2021) found that a majority of
their Civ absorbers and $\sim$73% of the absorbers with only Civ (but no Mgii)
detected did not have an associated galaxy counterpart within 250 kpc down to
a sensitivity of ${L/L_{\star}}\\!\simeq\\!0.01$ at the redshift of the
absorbers ($1\\!\leq\\!z\\!\leq\\!1.5$). If a substantial fraction of Civ
absorbers live in the IGM, then the assumption of each absorber being
associated with a galaxy DM halo would break down. However, this phenomenon
would conform with our inference that some weak absorbers reside beyond the
virial radius of DM halos. Alternatively, the Schroetter et al. (2021) result
could be explained by absorption from the CGM of lower mass or quiescent
galaxies below their luminosity detection threshold.

The model also does not account for systematic relationships between Civ
absorber strength and stellar activity, such as specific star formation rate.
For example, Borthakur et al. (2013) found a higher incidence rate of Civ
absorption around star-bursting galaxies than around their control sample.
Incorporating the sample of Borthakur et al. (2013), Bordoloi et al. (2014)
showed that higher $W_{r}$ Civ absorbers are preferentially found around
galaxies with higher specific star formation rates.

However, it is likely that the biggest challenge to our adopted model arises
from the lack of accounting for overlap between virial halos. We treat DM
halos as individual, isolated objects with their separate abundances and
cross-sections. If there is significant overlap between different halos, the
abundances would be different from what we calculated for isolated low-mass
and high-mass halos. Several studies of the spatial distribution of
intergalactic metals using semi-analytical outflow models have concluded that
strong clustering of enriching sources typically leads to significant overlap
between enriched regions (e.g., Scannapieco, 2005; Scannapieco et al., 2006;
Samui et al., 2008). Furthermore, observations indicate that Civ absorption is
affected by the large-scale environment of galaxies (e.g., Adelberger et al.,
2003, 2005; Burchett et al., 2016). We further discuss this issue below.

###  4.2 Baryon Cycle and the CGM

From a phenomenological standpoint, as star formation increases, more metals
are produced, and more metal-enriched gas is therefore driven out of galactic
centers via outflows. Therefore, an a priori expectation for metal-line
absorbers is that their cosmic incidence should increase in tandem with global
star formation activity of the universe. Indeed, strong Mgii absorber
incidence is found to increase from $z\\!\sim\\!5$ to a peak around
$z\\!\sim\\!2$ and fall afterwards (e.g., Matejek & Simcoe, 2012), in
correspondence with the global cosmic star formation rate density (e.g., Madau
& Dickinson, 2014) and the increased incidence of outflows (e.g., Rupke, 2018)
at Cosmic Noon. Similarly, the redshift-evolution of
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ
absorbers is qualitatively consistent with the evolution of cosmic star
formation activity at $z\\!>\\!2$, implying a possible causal connection
between strong Civ and global star formation rate at early epochs, as was
argued by Chen et al. (2016).

However, as seen in Figure 1, the cosmic incidence of
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers does not evolve in this
fashion. Instead, it rises rapidly in the last 8 Gyr following Cosmic Noon,
having been effectively constant across ${1.5\\!\leq\\!z\\!\leq\\!2.5}$. In
the context of our models, the relative extent ($f_{R}=R_{g}/R_{v}$) of this
population of absorbers around galaxies may even be at its global minimum just
following the epoch of Cosmic Noon (see Figures 3 and 4). In contrast, the
incidences of both stronger Civ with ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ monotonically, yet slowly, increase across
Cosmic Noon.

The most rapid evolution is for ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ at
$z\\!<\\!1$. This late-time rapid evolution is difficult to explain
physically, as neither the metal density relative to the cosmic mean nor the
expected amount of metals based on star formation has increased rapidly in the
last 8 Gyr (e.g., Péroux & Howk, 2020). Furthermore, the ionizing background
at the Heii ionization edge (which is similar to the ionization energy of Civ)
has softened by $z\\!\sim\\!1$, meaning Civ photoionization conditions have
not been particularly favorable in these last 8 Gyr (e.g., Haardt & Madau,
2012). Borthakur et al. (2013) suggested a model for $z\\!\leq\\!0.2$ in which
Civ would arise in pre-existing clumpy clouds or filaments in the CGM that are
being shock-ionized and accelerated by the ram pressure of winds from nearby
starburst galaxies. They argue that the ensemble of such clouds must have a
covering fraction of $\sim$80%. It remains to be worked out whether such a
scenario can be made consistent with the rapid evolution in very weak Civ at
$z\\!<\\!1$.

However, the rapid rise in $dN/dX$ (and subsequently rapid rise in $f_{R}$) of
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ at $z\\!<\\!1$ could be an effect of
increasing collisional ionization of Civ instead of photoionization.
Cosmological hydrodynamical simulations have predicted the increasing
prevalence of collisional ionization of metals with decreasing redshift, as
halos that grow to higher masses
($M_{h}\\!\gtrsim\\!10^{12}~{}{\mathrm{M}_{\odot}}$) have higher virial
temperatures ($T_{v}\\!\gtrsim\\!10^{5}$ K) able to collisionally ionize
metals to higher ionization states (e.g., Oppenheimer & Davé, 2006;
Oppenheimer et al., 2016, 2018). In the EAGLE simulations, at
$z\\!\leq\\!0.25$, the Ovi ionization fraction peaks at $\sim$1–2$R_{v}$ from
$L\\!\sim\\!{L_{\star}}$ halos owing to the optimal temperature for
collisional ionization to Ovi (Oppenheimer et al., 2016). The lower virial
temperature of sub-$L_{\star}$ halos results in a lower Ovi fraction, while
higher virial temperatures of group-sized ($L\\!\gtrsim\\!2{L_{\star}}$) halos
are able to further ionize Ovi to higher states such as Ovii and also result
in a lower Ovi fraction. Oppenheimer & Davé (2006) found large fractions of
collisionally ionized Civ absorbers in their simulations due to favorable
virial temperatures of $T\\!\simeq\\!10^{5}$ K, a trend accentuated with
decreasing redshift (especially at $z\\!<\\!2.5$).

Collisional ionization induced by hot gas at or outside the virial radius of
massive galaxies could explain how ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ
incidence increases in the last $\sim$8 Gyr and how the relative gas radius
grows significantly from $z\\!=\\!1$ to $z\\!=\\!0$. Additionally, the fact
that the ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ $f_{R}$ peaks at
$L\\!\approx\\!{L_{\star}}$ and drops for higher luminosities at $z\\!=\\!0$
(see top left panel of Figure 5, $\beta\\!=\\!0.5$ curve) might be explained
by the Oppenheimer et al. (2016) conclusion that more massive halos would
collisionally ionize metals to higher states (although the collisional
ionization temperature of Civ is $\sim\\!0.5$ dex lower than that of Ovi). On
the other hand, lower ionization metals are primarily photoionized and subject
to very little collisional ionization, even at lower redshifts (e.g., Ford et
al., 2013). Empirically, Rosenwasser et al. (2018) showed that photoionization
cannot produce detectable Ovi absorption at $z\\!\leq\\!0.2$ due to the
decreasing strength of the UVB at low redshifts, instead requiring collisional
ionization to generate such high-ionization absorbers at late epochs. For Civ,
there is a similar expectation of collisional ionization dominating at low $z$
(however see Manuwal et al. (2021)), while lower ionization metals such as
Mgii and Feii remain mostly photoionized. The increasing prevalence of
collisional ionization of higher ionization metals but not lower ionization
metals, in tandem with the decreasing UVB at later times, could potentially
explain why the Civ $dN/dX$ rises rapidly at $z\\!<\\!1$ but Mgii $dN/dX$
falls at those redshifts (Matejek & Simcoe, 2012; Hasan et al., 2020).

The evolution of the statistical gas radii provides a more detailed look into
how Civ absorbing structures grow relative to galaxies (see Figures 3 and 4).
For $L_{\star}$ and $0.1{L_{\star}}$ galaxies, we find that the relative gas
radius for ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ absorbers slightly grows from
$z\\!\sim\\!3$ to a peak at $z\\!\sim$1.5–2, and then declines afterwards.
This suggests a causal physical connection between
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ around galaxies and star formation
processes, similar to, yet more mild, than what is found for strong Mgii
absorbers (Matejek & Simcoe, 2012; Chen et al., 2017).

The intermediate ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ Civ population exhibits
relatively little evolution. The relative gas radii evolves similarly if
somewhat less dramatically as that of the ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
Civ absorbers. Both evolve differently from the
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ population. In contrast, the relative gas
radii of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers appear to evolve
independent of the star formation peak across the Cosmic Noon epoch.

These differences in evolution may be suggestive of different origin scenarios
for Civ absorbers of different absorption strength. In particular, star
formation-driven outflows could be a major pathway to populating the inner
regions of DM halos with ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ absorbers. And
in fact, there is some observational evidence connecting strong Civ absorbers
to galactic winds (Fox et al., 2007; Steidel et al., 2010; Nielsen et al.,
2020).

Since the gas radius of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers
relative to the virial radius appears to evolve independently of (if not
opposite to) the global cosmic star formation history, the weakest absorbers
would seem less likely to be tracing intrinsic feedback processes like
outflows. Though this statement is counter to the interpretations of Borthakur
et al. (2013) at $z\\!\simeq\\!0$, a possible alternative scenario is that the
gas halos of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers may grow via
extrinsic processes, such as changing environmental conditions in the IGM in
the vicinity of DM halos that favor the C+3 ionization state and small column
densities (for example, local overdensity evolution of the outer halo/IGM
interface or enrichment by star-forming sub-halos). Alternatively, the
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ gas radius may trace distant-past star
formation activity that is eventually manifested (post Cosmic Noon) as an
evolving increase in weak Civ absorption in the outer extremes of the virial
halo.

Such scenarios seem reasonable given the remarkably similar evolution of the
gas radius of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers and the virial
radius until $z\\!\sim\\!1$ for $L_{\star}$ galaxies (see Figure 3(a)). Global
statistics of absorbers, galaxies, and DM halos would suggest that, depending
on the covering fraction, the gas radius for this population of absorbers
resides near or beyond the virial radius of $L_{\star}$ galaxies for the
majority of cosmic time. For $0.1L_{\star}$ galaxies, we find similar results
with the exception that the gas radius is not as extended relative to the
virial radius (see Figure 4(a)). However, the rapid growth of the radius of
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ absorbers relative to the virial radii of
both $0.1L_{\star}$ and $L_{\star}$ galaxies over the last 8 Gyr suggests a
shift in the dominant mechanism for the creation and sustenance of a weak Civ
absorber population following Cosmic Noon.

Using a similar formalism to ours, Schaye et al. (2007) calculated the
characteristic gas radius of Civ clouds with super-solar metallicity at
$z\\!\sim\\!2.3$. Assuming ${L_{\mathrm{min}}}\\!=\\!{0.1L_{\star}}$, they
estimated ${R_{\star}}\\!\approx\\!80$ kpc, which is remarkably close to the
value of $R_{\star}$ we find for ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ absorbers
(see Figure 3(c)). This correspondence of absorbing halo sizes indicates that
the strongest Civ absorbers are metal-rich and therefore likely inhabit denser
regions of the CGM. On the contrary, $R_{\star}$ for
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ is about a factor of three larger at this
redshift, suggesting weak Civ absorbers live in less enriched and therefore
more tenuous regions.

In a recent comparative analysis of a diverse suite of cosmological
simulations, Fielding et al. (2020) found that different physical processes
shape the inner and outer regions of the CGM of $\sim$$L_{\star}$ galaxies at
low redshifts. Intrinsic galactic feedback plays the dominant role in shaping
the inner CGM, while extrinsic processes such as cosmological accretion and
interaction with satellites dominate the makeup of the outer CGM. Li et al.
(2021) also established the distinction between an outflow-dominated inner CGM
and accretion-dominated outer CGM in dwarf galaxies in the FIRE simulations.
Our result that the characteristic extent of strong Civ is statistically
confined to the inner CGM and very weak Civ extends to the outer halo and
beyond, is consistent with these theoretical works that suggest stronger
absorbers are primarily associated with intrinsic feedback while weaker
absorbers mostly trace extrinsic IGM accretion/satellite interaction
processes.

In a comprehensive study of absorbing gas dynamics, origins, and fates at
$z\\!=\\!0.25$, Ford et al. (2014) quantified the fraction of each absorbing
species originating as outflow, newly accreted, recycled, or ambient gas from
the IGM, as a function of both halo mass and projected galactocentric distance
(impact parameter). The Civ absorbing gas in the inner halos is dominated by
“recycled accretion” (accreting gas that was ejected in a wind at least once
in the past), whereas the outer halos are dominated by ancient outflows (non-
accreting gas ejected into the CGM by a galactic wind that occurred more than
$1$ Gyr in the past). At $100$ kpc, in the CGM of both $10^{11}$
$\mathrm{M}_{\odot}$ and $10^{12}$ $\mathrm{M}_{\odot}$ halos, more than 50%
of the weak Civ absorption originated from ancient outflows.

Interestingly, Ford et al. (2014) also found that “young outflows” (non-
accreting gas younger than 1 Gyr) significantly contribute to strong Mgii
absorption (especially in the inner halo where strong Civ absorbing gas is
predominantly recycled accretion). These simulations naturally explain why the
strong Mgii absorber cosmic incidence closely traces the global cosmic star
formation (Matejek & Simcoe, 2012) and why the strong Civ absorber incidence
does not (Cooksey et al. 2013; Paper I, also see Figure 1 of this work). That
the gas radius of strong Civ is confined to the inner halo and that, relative
to the virial radius, it shows only a mild growth during the Cosmic Noon
epoch, would be consistent with Civ tracing mostly recycled accretion, as
opposed to young outflows. Oppenheimer et al. (2020) found that the Civ
covering fraction shows a delay of $\sim$0.5–2.5 Gyr to respond to gas
ejection into the CGM, in contrast to the much quicker response from covering
fraction of low-ionization phases.

A general picture appears in which higher column density Civ bearing gas is
predominantly recycling back onto the galaxy after $\sim$1 to a few billion
years of transiting through the inner halo, while lower-column density Civ
bearing gas resides in the outer halo primarily as non-accreting ancient
outflow material. One aspect of this picture is that star-forming (satellite)
sub-halos are not taken into account in the work of Ford et al. (2014). There
is clearly the possibility that Civ bearing gas can be seeded into the outer
regions of more massive central halos from nearby lower-mass satellites.

Taking a broader view, simulations find that CGM and IGM gas is substantially
mixed so that a given background quasar line of sight may intercept absorbing
gas with multiple origins and fates (e.g., Ho et al., 2020; Li et al., 2021).
As such, our attempts to infer that absorbers are associated with a single
origin or fate are probably naive due to the true complexity of the baryon
cycle. There may be very little causal correlation between the origin and
future evolution of absorbing gas traced by a single ion (e.g., Hafen et al.,
2019, 2020). Yet, it is of continued interest to attempt to understand the
origin and fate of the bulk of the selected metal-line absorbing gas in DM
halos that host galaxies.

###  4.3 Baryon Cycle and Galaxy Evolution

Models that track the chemical enrichment history of the universe generally
agree that low-mass galaxies are the principal polluters of the IGM,
especially at early times. Booth et al. (2012) determined that observational
constraints on the median Civ optical depth can be satisfied only if the IGM
was primarily enriched as early as $z\\!=\\!3$ by low-mass halos with
$M_{h}\\!<\\!10^{10}$ $\mathrm{M}_{\odot}$ ($L\\!<\\!{0.01L_{\star}}$
according to Figure 7) ejecting metals out to ${R_{g}}\\!\gtrsim\\!100$ kpc.
Using cosmological hydrodynamic simulations, Wiersma et al. (2010) reported
that at least half of the mass in metals in the IGM at $z\\!=\\!2$ was ejected
by halos with $M_{h}\\!<\\!10^{11}$ $\mathrm{M}_{\odot}$
($0.01\\!\leq\\!{L/L_{\star}}\\!\leq\\!0.1$), with even lower masses
potentially dominating enrichment at higher redshifts. Scannapieco (2005) used
a simple outflow model to show that the smallest, most numerous, and earliest
sources were the most efficient at enriching the IGM, as they are able to
enrich a larger comoving volume (corresponding to smaller physical volume) at
early times. Similarly, Porciani & Madau (2005) found that the observed high
cross-correlation of Lyman Break Galaxies (LBG) and Civ absorbers (Adelberger
et al., 2003) can be explained by IGM enrichment from
$M_{h}\\!=\\!10^{8}$–$10^{10}$ $\mathrm{M}_{\odot}$ dwarf galaxies at
$6\\!\leq\\!z\\!\leq\\!12$. From a detailed treatment including star
formation, Samui et al. (2008) corroborated these general results and also
suggested that $M_{h}\\!>\\!10^{10}$ $\mathrm{M}_{\odot}$ halos enrich a
significant volume of the IGM only after $z\\!=\\!3$.

As gleaned from Figure 5, our models suggest that for $\beta\\!=\\!0.5$, high-
mass halos have more extended weak Civ envelopes (${f_{R}}\\!>\\!1)$ than low-
mass halos, a trend that is accentuated as redshift increases. This is
seemingly at odds with the expectation of efficient outflows and extended IGM
enrichment at higher redshifts from low-mass halos as opposed to high-mass
halos. In the case of gas radius being independent of galaxy luminosity
($\beta\\!=\\!0$), our models suggest low-mass halos have more extended weak
Civ envelopes (${f_{R}}\\!>\\!1)$ than high-mass halos. However, this trend is
accentuated as redshift decreases, which is again at odds with early IGM
enrichment from low-mass halos.

One possibility for the mismatch between observations, theory, and our models
is that $\beta$ could evolve with redshift. As Chen et al. (2001) has reported
$\beta\\!=\\!0.5$ for $z\\!\leq\\!1$, we could adopt that value at low
redshift and this yields higher mass galaxies having larger gas radii at low
redshift in our models. We can infer then, that if the luminosity scaling
evolved such that $\beta\\!=\\!0$ at $z\\!\simeq\\!5$, our model predictions
of ${f_{R}}({L/L_{\star}},z)$ could, in principle, be aligned with current
observations and theoretical expectations. However, there is no physical basis
by which to assume $\beta$ evolves in this contrived fashion.

As stated in Section 4.1, our formalism treats each halo as isolated, and
therefore does not allow for overlap between halos. Scannapieco (2005) showed
how the clustering of intergalactic metals implies a volume overlap of
individual enriched “bubbles” from galaxies. They posited that volume overlap
may explain the large observed sizes of enriched regions, which would
otherwise require extremely large energies to be ejected by a single source.
It is well known that the clustering of galaxies relative to dark matter (the
bias) increases with halo mass; high-mass halos are substantially more
clustered than low-mass halos (e.g., Wechsler & Tinker, 2018). Thus, the
volume of gas within or surrounding a massive halo may have been enriched by a
galaxy other than the central galaxy in that halo. Significant overlap of gas
envelopes from low-mass halos clustered around a high-mass halo would result
in the smaller gas envelopes being “absorbed” into the gas envelope of the
higher mass halo. Our formalism would then break down for high-mass halos,
especially for higher redshifts where the physical volume of the universe was
smaller and halo volume overlap was probably more severe.

There is a growing body of evidence suggesting that some phases of the CGM are
more extended for galaxies in groups. Galaxies residing in groups have a
higher covering fraction of Mgii at $z\\!\sim\\!1$ (Dutta et al., 2020) and of
Civ at $z\\!\sim\\!3$ (Galbiati et al., in prep) than isolated galaxies,
implying higher volume filling factors in the CGM of group galaxies.
Additionally, Schroetter et al. (2021) found that the majority of the Civ+Mgii
systems they observed at ${z\\!=\\!1\\!-\\!1.5}$ may be associated with
multiple galaxies within 250 kpc of the quasar line of sight, suggesting
absorbers might also exist in group environments. Similarly, Hamanowicz et al.
(2020) found most of their detected Mgii systems to be associated with pairs
or groups of galaxies. Nielsen et al. (2018) found that, compared to isolated
galaxies, Mgii absorbers in group environments at
$0.1\\!\leq\\!z\\!\leq\\!0.9$ have both a higher median $W_{r}$ and a higher
covering fraction. They suggested that Mgii-bearing gas forms a kinematically
complex intragroup medium in group environments. These measurements of Mgii
covering fractions led Lan (2020) to conclude that the evolution of cool gas
may reflect the star formation activity of an entire halo consisting of the
central galaxy and its satellites, as opposed to just the central. Such ideas
hark back to the original suggestions of York et al. (1986).

Chen et al. (2010), Bordoloi et al. (2011), and Nielsen et al. (2018) all
reported that the $W_{r}$ of Mgii absorbers in group environments does not
decline with impact parameter out to about 200 kpc, unlike the very strong
anti-correlation between $W_{r}$ and impact parameter observed for isolated
galaxies. This fact is highly suggestive of gas halo overlap or an intragroup
medium in overdense environments. Lopez et al. (2008) found an excess of
strong Mgii absorbers and a paucity of weak Mgii absorbers (${W_{r}}\leq
0.3~{}$Å) in $0.3\\!\leq z\leq\\!0.9$ cluster environments, which they suggest
is due to a population of absorbers associated with overdense cluster
environments in which weak systems are destroyed (though they could be blended
together).

In both the FIRE-1 (Anglés-Alcázar et al., 2017) and FIRE-2 (Hafen et al.,
2020) simulations, winds from satellite galaxies constitute an increasing
fraction to the central galaxy CGM mass and this effect is more prominent with
increasing halo mass of the central galaxy. In fact, Hafen et al. (2020)
reported that in $\sim\\!10^{12}$ $\mathrm{M}_{\odot}$ halo progenitors, the
central halo can actually accrete more gas from satellite winds than from
recycled winds originating in the central galaxy at both $z\\!=\\!0.25$ and
$z\\!=\\!2$.

However, for Ovi absorbers, Pointon et al. (2017) found that the average
$W_{r}$ in $z\\!\leq\\!0.25$ low-mass galaxy groups is smaller and the
absorption velocity profiles narrower than for isolated galaxies. They
attributed this to higher virial temperatures in group environments (with more
massive halos) that are able to ionize oxygen to higher-ionization states,
thus diminishing Ovi absorption properties in groups. In general, Ovi
absorption does not behave similarly to Mgii absorption in isolated galaxies,
kinematically (Nielsen et al., 2017; Kacprzak et al., 2019) or with halo mass
(Ng et al., 2019).

Crucially, the association of metals within the virial radius of a galaxy
living a high-mass DM halo does not require that the metals originated in that
galaxy. As Oppenheimer et al. (2012) demonstrated, at high redshifts, metals
ejected in winds in regions of lower overdensities will gravitate into denser
regions of the IGM and progressively into higher mass DM halos with cosmic
time. This “outside-in” enrichment of the CGM of higher mass halos results in
higher ionization metals like Civ tracing lower density, older gas. This gas
would be unique to the “ancient outflow” gas seen in the simulation of Ford et
al. (2014). It would be interesting to examine the simulations to determine
whether this gas that has fallen in from the IGM is consistent with
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers.

The upshot is, highly extended envelopes of enriched gas surrounding galaxies
should not reflexively be interpreted as evidence for powerful winds
originating from the central galaxies. Material traveling at a constant speed
of $500$ km s-1 for $300$ Myr would only be able to traverse up to $225$
comoving kpc at $z\\!=\\!2$, compared to $450$ comoving kpc at $z\\!=\\!5$.
Whereas these winds escape more easily from the potential wells of early low-
mass halos, in higher mass halos the winds slow down due to hydrodynamic
interactions with the denser CGM (e.g., Oppenheimer & Davé, 2006). This is
further theoretical corroboration that winds originating in higher mass halos
may not reach the most extended regions of their own CGM. Of course, AGN
outflows are believed to be an important mode of feedback in some massive
galaxies (e.g., Nelson et al., 2019), but the low space density of AGN,
coupled with relatively brief duty cycles suggests that this mode of feedback
is not nearly as prevalent as less energetic SN-driven winds (AGN do not
dominate the measured $dN/dX$ of Civ absorbers).

So instead of outflows emanating from $L\\!\gtrsim\\!{L_{\star}}$ galaxies at
various epochs, lower velocity outflows at $z\\!>\\!5$ transferred from the
low-mass progenitors in the cosmic vicinity of such galaxies could explain the
extended gaseous envelopes surrounding higher mass halos. In fact, Díaz et al.
(2014) found evidence for this scenario; they demonstrated that their sample
of Civ absorbers at $z\\!\sim\\!5.7$ likely trace low-to-intermediate density
environments of faint, low-mass galaxies, rather than massive
($L\\!\geq\\!{L_{\star}}$) galaxies at that redshift. This scenario would
imply that the luminosity scaling parameter of $\beta\\!=\\!0.5$ in our models
(which predicts highly extended gas radii for
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers at all redshifts, see Figure
5), does not, in reality, describe a causal link between the central galaxy
luminosity and the gas radius (relative to the virial radius). It would imply
that, if the gas radius appears to behave in this fashion, the causality is
due to enrichment from local lower mass halos clustered around the higher
luminosity galaxy.

In view of these considerations, we are faced with an alternate explanation of
our $\beta\\!=\\!0.5$ prediction that high-mass halos have more extended Civ
envelopes than low-mass halos and that the most massive halos may have hosted
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ gas out to as much as twice their virial
radius at $z\\!\geq\\!4$. If there is significant overlap between high-mass
halos and their surrounding low-mass halos, it would render our model too
simplistic and the definition of an isolated spherical “gas envelope” as
meaningless. Halo overlap implies the importance of lower mass satellites
and/or environmental effects on the CGM of massive galaxies and the nearby
IGM.

###  4.4 So, How Does Civ Evolve?

The preceding discussion leads to the emergence of a picture where low-mass
dwarf galaxies at early epochs disperse carbon-enriched gas to the overdense
regions surrounding massive central galaxies. This results in large, extended
Civ gas envelopes around massive, luminous galaxies, likely extending beyond
their virial halos. The metals found at large distances (near the edge of, or
outside, the virial radius) from their centers thus likely form in their
satellites. This clearly poses a challenge for observational studies which
attempt to derive constraints on galaxy outflows based on physical proximity
of absorbers to a luminous galaxy, since absorbers detected at impact
parameters may not have even originated inside the galaxy in question.

In this picture, gas in the outskirts of enriched gas halos are optically thin
and mostly produced in ancient winds expelled from lower mass galaxies. Deeper
in the halos, the gas becomes optically thick and/or kinematically complex and
its origins lie in more recent outflows ejected by the more massive central
galaxies. This picture is consistent with a steep scaling of gas radius with
luminosity, even though the physical origin of gas at the edge of the halos
may not be from the central galaxies themselves. Obtaining conclusive evidence
of this scenario requires significantly increasing observations of gas radius
across parameter space, supplementing the currently limited data described in
Section 3.3. One focus of this observational effort could be to target high-
luminosity galaxies at high redshifts ($z\\!>\\!3$) so that the
$\beta\\!=\\!0.5$ scenario can be tested at early epochs. Furthermore,
simulations can be utilized to comprehensively characterize $\beta$ as a
function of halo mass, redshift, $W_{r,{\rm lim}}$, and galaxy properties such
as color, SFR, and environment. To the best of our knowledge, no such
theoretical characterization of $\beta$ exists. An evolving or luminosity-
dependent $\beta$ would muddy the waters of the physical picture we have
outlined.

The evolution of Civ gas halos tells a story of metal enrichment and changing
ionization conditions across the vast majority of cosmic time. Optically thick
Civ envelopes approximately grow with the rise in metal enrichment deeper in
the halo, caused by the rise in star formation activity across Cosmic Noon,
and grow more gradually as metal enrichment slows down. In contrast, optically
thin Civ envelopes decline relative to virial halos across Cosmic Noon. Large
extended envelopes are in place at high redshifts, but some of this gas
transitions into virial halos, which grow steadily with cosmic time. The
extragalactic UV background (UVB), which becomes harder with time from
$z\\!\sim\\!5$ to a peak around $z\\!\sim\\!1$ (Haardt & Madau, 2012), causes
a change in carbon ionization fractions, and favors higher ionization states
like Cv and Cvi as Cosmic Noon progresses (Simcoe, 2011, see their Figures 11
and 12).

It is therefore reasonable that in the outskirts of gas halos, optically thin
carbon is ionized to higher ionization states than Civ between $z\\!\sim\\!5$
and $z\\!\sim\\!1$, thus reducing the relative sizes of the weak Civ halos.
Optically thick and/or kinematically complex Civ in denser regions closer to
galactic centers would be shielded from these ionization effects so that the
strong absorbing halo sizes do not respond to these universal changes in
background ionization conditions. The relative growth of
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ halos at $z\\!<\\!1$ is puzzling,
nevertheless, as the gradually softening UVB favors lower ionization states of
carbon than Civ at these late times. We speculate that this growth may be due
to increased collisional ionization of lower ionization carbon to Civ in the
outskirts of hot halos of massive galaxies at lower redshifts. Unfortunately,
there is no strong empirical evidence for this idea currently.

##  5 Conclusion

We investigated the statistical extent of metal-enriched gas relative to DM
halos and their central galaxies by leveraging the observed statistics of Civ
absorption lines, DM halos, and luminous galaxies across $\sim$12.5 Gyr of
cosmic time. We draw from the results of our Paper I in which we developed a
census of Civ absorber populations with minimum limiting equivalent widths of
${W_{r,{\rm lim}}}\\!=\\!0.05$, $0.3$, and $0.6$ Å over the redshift range
$0\\!\leq\\!z\\!\leq\\!5$. Here, we applied abundance matching using the
observed UV luminosity functions (Parsa et al., 2016) and DM halo mass
functions (Murray et al., 2013) to obtain a census of DM halos hosting
$L\\!\geq\\!{0.01L_{\star}}$ galaxies over the redshift range
$0\\!\leq\\!z\\!\leq\\!5$.

###  5.1 General Findings from Cumulative Statistics

We first compared the cosmic incidence, i.e., the product of number density
and cross-sectional area, of Civ absorber populations with that of galaxy
populations (see Figures 1 and 2).

  * •

$L\\!\geq\\!{L_{\star}}$ galaxy halos alone cannot account for all the
observed ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ absorbers, but they can
account for all ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ absorbers.

  * •

Assuming Civ absorbers are co-spatial with DM halos, all absorbers live inside
the virial radius of ${L\\!\geq\\!{0.01L_{\star}}}$ and
${L\\!\geq\\!{0.1L_{\star}}}$ halos. Under this assumption, the weakest
population of absorbers with ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ on average
reside beyond the virial radius of $L\\!\geq\\!{L_{\star}}$ halos, and inside
the inner $\approx$30% of $L\\!\geq\\!{0.01L_{\star}}$ halos and inner
$\approx$50% of $L\\!\geq\\!{0.1L_{\star}}$ halos. Strong Civ with
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
typically resides in the inner $\approx$10% ($\approx$50%) of the virial halo
of $L\\!\geq\\!{0.01L_{\star}}$ and $L\\!\geq\\!{0.1L_{\star}}$
($L\\!\geq\\!{L_{\star}}$) galaxies.

  * •

Accounting for lower mass halos (lower luminosity galaxies) results in Civ
absorbing gas residing further inside the inner regions of a halo and/or
becoming less abundant than halos.

  * •

The incidence of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ gas halos relative to
that of virial halos decreases during the Cosmic Noon epoch to a minimum at
$z\\!\sim\\!1$, regardless of minimum luminosity of the cumulative galaxy
population. Following $z\\!\sim\\!1$ (over the last $\sim$8 Gyr), this
relative incidence increases. On the contrary, the relative incidence of
${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ Civ
halos evolve very mildly, with some indications that they grow with cosmic
time for $L\\!\geq\\!{0.01L_{\star}}$ galaxy populations.

Next, assuming that each population of $W_{r,{\rm lim}}$ absorbers has the
same cosmic number density as $L\\!\geq\\!{0.01L_{\star}}$ galaxies, we
estimated the Civ absorbing gas radius, $R_{g}$, and the ratio of this gas
radius relative to halo virial radius, $f_{R}$, for three Civ absorber
populations, ${W_{r,{\rm lim}}}\\!=\\!0.05$, $0.3$, and $0.6$ Å, over the
redshift range $0\\!\leq\\!z\\!\leq\\!5$ (see Figures 3 and 4).

  * •

In $L_{\star}$ galaxy halos, the ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ absorbing gas halo sizes are
${f_{R}}\\!\sim\\!30$–$70$% of the virial radius. The
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ gas halos are larger than virial halos
(${f_{R}}\\!>\\!1$) at $z\\!=\\!0$ and $z\\!>\\!2.5$, implying that some of
the weakest Civ absorbers may live in the IGM surrounding massive galaxies.
For $0.1L_{\star}$ halos, the relative gas radii are smaller, with
${f_{R}}\\!\sim\\!20$–$40$% for ${W_{r}\\!\geq\\!0.3}~{}{\rm\AA}$ and
${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ gas and ${f_{R}}\\!\sim\\!60$–$100$% for
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ gas.

  * •

We find that $f_{R}$ for ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$ absorbers increases
up to the end of Cosmic Noon ($z\\!\sim\\!1.5$) and then declines afterwards.
In stark contrast, the relative radii of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
gas halos decrease from $z\\!\sim\\!5$ to $z\\!\sim\\!1$ and increase again at
$z\\!<\\!1$. The evolution of gas radii are very similar between
$0.1L_{\star}$ and $L_{\star}$ halos.

To fully constrain relative gas radius as a function of luminosity and
redshift, we require better constraints on the gas covering fraction
$f_{c}(z)$ and luminosity-scaling of gas radius $\beta$. Because we have
adopted halo abundance matching, larger galaxy luminosity implies larger halo
mass, though the exact proportionality evolves with redshift (see Appendix A).

  * •

If we assume a redshift-independent value of the only available observational
constraint of $\beta\\!=\\!0.5$ (Chen et al., 2001), then we find that the
relative gas radius for all absorber populations generally increases with
galaxy luminosity (halo mass). In this scenario, the
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ gas radius exceeds the virial radius of
massive $L\\!\geq\\!{L_{\star}}$ halos at virtually all redshifts for
$f_{c}\\!=\\!0.5$ (see Figure 5).

  * •

If instead $\beta\\!=\\!0$, where the physical gas radius is not proportional
to the central galaxy luminosity, the cumulative statistics yield a gas radius
that decreases with increasing galaxy luminosity (halo mass). In this case,
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ extends past virial radius for the
least massive halos with $L\\!\leq\\!0.05{L_{\star}}$ for $f_{c}\\!=\\!0.5$ at
all redshifts. In both these scenarios, the ${W_{r}\\!\geq\\!0.6}~{}{\rm\AA}$
gas halos are always contained in the inner half of the virial halo.

In summary, a strong scaling of gas radius with galaxy luminosity requires
highly extended, super-virial gas structures around high-luminosity galaxies
before Cosmic Noon and in the present day, while a luminosity-independent gas
radius requires highly extended structures around high-luminosity galaxies
after Cosmic Noon.

###  5.2 State of the Observations

We examined which of the two scenarios outlined above ($\beta\\!\simeq\\!0$ or
$\beta\\!\simeq\\!0.5$) better agrees with the observed spatial location of
gas as a function of galaxy luminosity, redshift, and $f_{c}$ (see Figure 6).
We compared our model predictions of the ratio $f_{R}$ with observed data for
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ at $z\\!\simeq\\!0$ (Bordoloi et al.,
2014; Burchett et al., 2016), $z\\!\simeq\\!1$ (Schroetter et al., 2021), and
$z\\!\simeq\\!2.5$ (Rudie et al., 2019). Unfortunately, the small sample sizes
and limitations in survey search space make it impossible to conclusively
constrain $\beta$ and $f_{c}$ with the currently available data.

  * •

Observational leverage would be most useful in constraining the model at high
redshifts; no constraints on $f_{R}$ exist at $z\\!>\\!3$, and at the Cosmic
Noon epoch of $2\\!\leq\\!z\\!\leq\\!3$, we require observations at
${f_{R}}\\!>\\!0.5$ for $L\\!\geq\\!{L_{\star}}$ galaxies and at
${f_{R}}\\!>\\!1$ for $L\\!\leq\\!{0.1L_{\star}}$ galaxies (both corresponding
to impact parameters $D\\!\gtrsim\\!125$ pkpc). The latter observations are
certainly more challenging to acquire, owing to the difficulty of locating
faint dwarf galaxies at high redshift.

  * •

At $1\\!\leq\\!z\\!\leq\\!1.5$, observations at ${f_{R}}\\!>\\!1$ for
$L\\!\leq\\!{0.1L_{\star}}$ galaxies ($D\\!\gtrsim\\!150$ pkpc) would be most
useful, while at $z\\!\simeq\\!0$, observations at ${f_{R}}\\!>\\!1$ for
$L\\!\sim\\!{L_{\star}}$ galaxies ($D\\!\gtrsim\\!400$ pkpc) and for
$L\\!\leq\\!{0.1L_{\star}}$ galaxies ($D\\!\gtrsim\\!180$ pkpc) would be
required. Theoretical simulations could also be of great utility in
understanding the spatial distribution of ${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$
Civ in these regions.

###  5.3 Improving Our Understanding of Civ

From our analysis, we conclude that optically thin Civ lives in the outskirts
of virial halos (near or beyond $R_{v}$) and in the IGM, while optically thick
and/or kinematically complex Civ lives in the inner half of virial halos.

  * •

In the inner halos, Civ gas is likely to be primarily produced via stellar
feedback processes generating outflows which recycle back into the central
galaxies. We suggest that the weak redshift evolution of optically thick
and/or kinematically complex Civ gas halos is mostly a consequence of the
longer timescales than are seen for low-ionization gas, such as that traced by
Siii and Mgii.

  * •

We propose that Civ in the outer halos of galaxies is primarily produced in
ancient outflows, likely originating in lower mass satellite galaxies at
earlier epochs. The strong evolution in the relative gas radius ($f_{R}$) of
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ is very interesting and presents
theoretical challenges. This $f_{R}$ decreases from high redshift through
Cosmic Noon and then grows rapidly from $z\\!=\\!1$ to $z\\!=\\!0$. The
cumulative statistics appear to favor a scenario in which
${W_{r}\\!\geq\\!0.05}~{}{\rm\AA}$ Civ gas halos are more extended for more
massive galaxies; this may in part be due to a population of lower mass
satellites at earlier epochs that enrich their surroundings.

We stress that the model we presented is but one of several possible scenarios
of the connection between galaxies and their metal-enriched halo gas. To
definitively discriminate between competing scenarios, we require a
significant increase in observational data as well as improved
characterization in theoretical simulations. Interestingly, our model and its
implications for the baryon cycle could be also explored for the more
thoroughly studied Mgii absorber-galaxy population, which has excellent
measurements for $dN/dX$ to stringent $W_{r}$ sensitivity limits (e.g.,
Matejek & Simcoe, 2012; Zhu & Ménard, 2013; Mathes et al., 2017; Chen et al.,
2017).

Larger samples of observed galaxy-absorber pairs across a wide range of galaxy
luminosities, impact parameters, and redshifts, will be crucial for
constraining the nature of Civ absorbers and their connection to galaxy
evolution. To the degree that our model applies, these observations would
constrain $f_{c}(z)$ and $\beta$.

In the last few years, Multi-Object and Integral Field Spectroscopy-based
programs like MEGAFLOW (e.g., Schroetter et al., 2021), KBSS (e.g., Rudie et
al., 2019), CGM at Cosmic Noon with KCWI (e.g., Nielsen et al., 2020), MUDF
(e.g., Fossati et al., 2019), and MAGG (e.g., Dutta et al., 2020) have lead
the way in mapping the CGM of galaxies across the vast majority of cosmic time
and providing detailed information on the spatial distribution of gas halos
relative to galaxies. In addition, measurements of galactic outflows at high
redshifts, which will be enabled by the soon-to-be-launched James Webb Space
Telescope, will be a direct empirical test of the scenario of early enrichment
by low-mass halos we proposed. Finally, state-of-the-art cosmological
hydrodynamical simulations such as EAGLE (e.g., Schaye et al., 2015), FIRE-2
(e.g., Hafen et al., 2019), and IllustrisTNG (e.g., Nelson et al., 2019) can
be important tools in constraining the nature of highly uncertain model
parameters such as $\beta$ across a broad range of physical properties and
testing their sensitivity to adopted feedback prescriptions including outflow
energetics, mass loading factors, and sources of feedback. These predictions
can then be tested against observations as they become available.

## Acknowledgements

We would like thank Cameron Hummels and Peter Behroozi for valuable input and
discussions that contributed to this work. We would also like to thank
Benjamin Oppenheimer, Joop Schaye, Freeke van de Voort, and Dylan Nelson for
insightful comments. We are very grateful to Gwen Rudie, Ilane Schroetter,
Nicolas Bouché, and Joe Burchett for generously contributing unpublished
quantities from their galaxy-absorber survey work for use in this paper. We
also thank the anonymous referee for valuable comments which helped improve
this manuscript. N.M.N., G.G.K., and M.T.M. acknowledge the support of the
Australian Research Council through Discovery Project grant DP170103470.
M.T.M. also thanks the Australian Research Council for Discovery Project grant
DP130100568 for their partial support of this work. Parts of this research
were supported by the Australian Research Council Centre of Excellence for All
Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013.

##  Appendix A Mapping Galaxies Onto Halos

Figure 7: Relationship between DM halo mass and galaxy luminosity, obtained
from abundance matching. (a) The solid curves and shaded regions show the
median UV magnitude-halo mass relations and $\pm 1\sigma$ standard deviations,
respectively, at a given redshift. On each curve, the circle, triangle, and
square represents the magnitude and halo mass for a $0.01L_{\star}$,
$0.1L_{\star}$, and $L_{\star}$ galaxy, respectively. (b) Median halo mass
corresponding to $0.01L_{\star}$ (blue), $0.1L_{\star}$ (purple), and
$L_{\star}$ (red) galaxies, as functions of redshift. The shaded regions
represent the $\pm 1\sigma$ standard deviations on the median.

The DM halo mass function (HMF), $\phi(M_{h},z)$, which describes the number
density of DM halos per unit halo mass at a given redshift, can be integrated
from a minimum halo mass $M_{h,\mathrm{min}}$ to infinity, to obtain
$n({M_{h,\mathrm{min}}},z)$, the cumulative number density of halos with
$M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}$. We used the code
hmf111https://hmf.readthedocs.io/en/latest/index.html (Murray et al., 2013) to
calculate the HMF at redshifts $0\\!\leq\\!z\\!\leq\\!5$, adopting the fitting
functions of Behroozi et al. (2013) for the halo multiplicity function. The
choice of multiplicity function has a small effect ($\sim$1%) on the
abundances calculated.

Since galaxies populate halos, we expect there to be a mapping between halo
and galaxy properties. While a number of observables have been used to probe
the galaxy-halo connection, here we use galaxy luminosity $L$ to connect
galaxies to their host halos. As stated in section 2, this is primarily
because historically, absorbing gas structures have been linked to the
directly-observed luminosity of galaxies around which they are found (e.g.,
Chen et al., 2001, 2010; Ribaudo et al., 2011; Nielsen et al., 2013a).

To that end, we apply the technique of abundance matching (e.g., Kravtsov et
al., 2004; Trujillo-Gomez et al., 2011; Behroozi et al., 2013, 2019; Klypin et
al., 2015), where we match the cumulative number density of halos with
${M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}}$ to that of galaxies with
$L\\!\geq\\!{L_{\mathrm{min}}}$,

$n(M_{h}\\!\geq\\!M_{h,\mathrm{min}},z)=n(L\\!\geq\\!L_{\mathrm{min}},z)\,.$
(A1)

This method allows us to compute the minimum halo mass corresponding to the
minimum luminosity, thereby providing a link between galaxies and their host
halos across cosmic time. Our operating assumption is that every DM halo more
massive than $M_{h,\mathrm{min}}$ hosts one galaxy more luminous than
$L_{\mathrm{min}}$. Similar to DM halos, the cumulative number density of
galaxies with luminosities $L_{\mathrm{min}}$ and above,
$n({L_{\mathrm{min}}},z)$, is obtained by integrating the luminosity function
(LF) $\phi(L,z)$ from $L_{\mathrm{min}}$ to infinity. We adopted the best-fit
parameters of the 1500 Å UV Schechter LF provided by Parsa et al. (2016) for
$z\\!=\\!0$–$8$. By abundance matching observed LFs to the HMFs, we generate
halo mass-luminosity relations (HMLR) at redshifts $z\\!=\\!0$–$5$. Baryonic
processes, such as the halo merger history, angular momentum relaxation,
cooling, and stellar feedback can produce scatter in the HMLR. This scatter is
often incorporated with a log-normal conditional luminosity function with a
constant standard deviation $\sigma_{M}$ (e.g., More et al., 2009; Ren et al.,
2019; Whitler et al., 2020).

To obtain the HMLR accounting for scatter in the abundance matching, we used
the package AbundanceMatching222https://github.com/yymao/abundancematching,
which utilizes an iterative deconvolution technique described by Behroozi et
al. (2010). Following Whitler et al. (2020), we applied a standard deviation
of $\sigma_{M}=0.5$ in the HMLR (this value is approximately invariant with
redshift). We ran $100,000$ random abundance matching realizations and
extracted the median and standard deviation of UV magnitude,
$M_{\mathrm{UV}}$, as functions of $M_{h}$, over redshifts $z\\!=\\!0$–$5$.

The results of abundance matching are presented in Figure 7. In Figure 7(a),
we present $M_{\mathrm{UV}}$ as a function of $M_{h}$ and $z$. The solid
curves are the median HMLR and the shaded regions are the $\pm 1\sigma$
standard deviation for a given redshift. On each curve, we indicate the halo
mass and UV magnitude corresponding to $0.01L_{\star}$, $0.1L_{\star}$, and
$L_{\star}$ galaxies at that redshift. In Figure 7(b), we show the redshift
evolution of the median $M_{h}$ and its standard deviation for
$0.01L_{\star}$, $0.1L_{\star}$, and $L_{\star}$ galaxies.

Due to the evolution in the HMF and LF, the HMLR also evolves. For instance,
an $L_{\star}$ galaxy is only $\sim$0.5 magnitudes fainter at $z\\!=\\!2$ than
at $\\!z=\\!5$, while a $\\!z=\\!0$ $L_{\star}$ galaxy is roughly $2.5$ mags
fainter than its $z\\!=\\!1$ counterpart. The brightest galaxies show the
steepest rise in halo mass. The halo mass for an $L_{\star}$ galaxy rises by
$\sim$13 times from $z\\!\sim\\!5$ to $z\\!\sim\\!0$, while the halo mass for
$0.01L_{\star}$ and $0.1L_{\star}$ galaxies increase by $\sim\\!7$–$8$ times.
We also find that the HMLRs plateau at large halo masses
($M_{h}\\!\gtrsim\\!10^{13}$ $\mathrm{M}_{\odot}$), and that the uncertainties
in the HMLR increase with luminosity and redshift, as was noted previously
(e.g., Behroozi et al., 2010; Whitler et al., 2020).

From abundance matching, we know the $M_{h,\mathrm{min}}$ corresponding to a
given $L_{\mathrm{min}}$ at any redshift. We can then compute the cosmic
incidence, $dN/dX$, of DM halos with $M_{h}\\!\geq\\!{M_{h,\mathrm{min}}}$ at
redshift $z$, from Eq. 1 and Eq. 3 for a fixed $L_{\mathrm{min}}$. The virial
radius $R_{v}(M_{h},z)$ in Eq. 3 is computed from the analytical approximation
of Besla et al. (2007),

$R_{v}(M_{h},z)=\frac{206}{h^{2/3}}\left[\left(\frac{M_{h}}{10^{12}}\right)\frac{97}{\Omega_{\hbox{\tiny
M}}(z)~{}\Delta_{v}(z)}\right]^{1/3}\,,$ (A2)

where $R_{v}(M_{h},z)$ is in kiloparsecs, $M_{h}$ is in solar masses,
$\Omega_{\hbox{\tiny M}}(z)$ is the matter density of the universe at redshift
$z$ and $\Delta_{v}(z)$ is the virial overdensity approximated using Eq. 6 of
Bryan & Norman (1998).

##  Appendix B The Characteristic Gas Radius

We estimate ${R_{\star}}(z)$, the characteristic gas radius of an
${L_{\star}}$ galaxy, by constraining the product $n(z)\sigma(z)$ for Civ
absorbers defined by $W_{r}\geq{W_{r,{\rm lim}}}$. Given that these absorbers
have measured $dN/dX(z)$, we apply the relationship

$\begin{array}[]{rcl}\displaystyle\frac{dN}{dX}(z)&\\!\\!=\\!\\!\\!&\displaystyle\frac{c}{H_{0}}n(z)\sigma(z)\,,\\\\[15.0pt]
\displaystyle
n(z)\sigma(z)&\\!\\!=\\!\\!\\!&\displaystyle\bigintsss_{{L_{\mathrm{min}}}/{L_{\star}}}^{\infty}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\phi(\phi_{\star},t,\alpha,z)\,f_{c}(z)\pi
R^{2}_{g}(t,z)\,dt\,,\end{array}$ (B1)

where $t={L/L_{\star}}$, and where $\phi(\phi_{\star},t,\alpha,z)$ is the 1500
Å UV luminosity function (Parsa et al., 2016), $L_{\mathrm{min}}$ is the
minimum galaxy luminosity, $f_{c}(z)$ is the covering fraction, and we assume
gas absorbing cross-section is $\pi R^{2}_{g}({L/L_{\star}},z)$, where
$R_{g}({L/L_{\star}},z)={R_{\star}}(z)({L/L_{\star}})^{\beta}$ (Eq. 4). Here,
we assume that the number density $n(z)$ of an absorbing population is equal
to the number density of the population of galaxies with
$L\\!\geq\\!{L_{\mathrm{min}}}$. Integrating and solving for ${R_{\star}}(z)$,
we obtain

$R_{\star}(z)=\left[\frac{H_{0}}{\pi
cf_{c}(z)\phi_{\star}(z)}\frac{dN/dX(z)}{\Gamma\left[x(z),{L_{\mathrm{min}}}/{L_{\star}}\right]}\right]^{1/2}\,,$
(B2)

where $\Gamma[x(z),{L_{\mathrm{min}}}/{L_{\star}}]$ is the upper incomplete
Gamma function (Abramowitz & Stegun, 1972),
$x(z)\\!=\\!2\beta\\!+\\!\alpha(z)\\!+\\!1$, and $\alpha(z)$ and
$\phi_{\star}(z)$ are the faint-end slope and normalization, respectively, of
the luminosity function at a given $z$.

##  Appendix C Observed gas radius data

For $z\\!\simeq\\!0$, we obtained the published impact parameter $D$, $R_{v}$,
and $r$-band luminosity $L/L_{\star}$ for COS-Dwarfs galaxies directly from
Bordoloi et al. (2014), and $D$, $R_{v}$, and halo mass $M_{h}$ for the
Burchett et al. (2016) sample from J.N. Burchett . We applied a scale factor
of $\sim\\!1.2$ to both these $R_{v}$ to enable direct comparison with our
chosen form of $R_{v}$, since we do not use their overdensity definition of
$\Delta_{v}=200$ (see Appendix A). We also applied a scale factor of $\sim$1.2
to the halo masses of Burchett et al. (2016) for the same reason. We converted
these $M_{h}$ to UV $L/L_{\star}$ using the HMLR from abundance matching.
Finally, we converted the Bordoloi et al. (2014) $r$-band $L/L_{\star}$ to
$1500$ Å UV $L/L_{\star}$ by abundance matching the Parsa et al. (2016) LF at
$z\\!=\\!0$ to the $z\\!=\\!0$ SDSS $r$-band LF from Blanton et al. (2003).

For $z\\!\sim\\!1$, we obtained $R_{g}$ and stellar masses $M_{\star}$ of the
galaxies in the Schroetter et al. (2021) sample from I. Schroetter and N.
Bouché . The $M_{\star}$ were obtained from Spectral Energy Distribution (SED)
fitting as described in Schroetter et al. (2019). These masses were converted
to $M_{h}$ using the stellar mass-halo mass relation (SHMR) parameterized by
Girelli et al. (2020). Specifically, we used the best-fit parameters of the
$z\\!=\\!1$ SHMR with a relative scatter of 0.2 dex as presented in their
Table 2. Their SHMR is broadly consistent with other published works (e.g.,
Moster et al., 2013; Behroozi et al., 2013; Bocquet et al., 2016). Using these
$M_{h}$ values, we then calculated $R_{v}$ from eq. A2, and used abundance
matching to obtain $L/L_{\star}$. Two of the 11 galaxies had
${L/L_{\star}}\\!<\\!0.01$ and were therefore below the minimum luminosity
cutoff of our models.

For $z\\!\sim\\!2.5$, we obtained the published $R_{g}$ from Rudie et al.
(2019), and unpublished apparent $\mathcal{R}$-band galaxy AB magnitudes from
G. C. Rudie . We converted the apparent magnitude to an absolute $R$-band
magnitude with the average of $k$-corrections that assumes Sbc and Im galaxy
SEDs (see, e.g. Nielsen et al., 2013b). These are reasonable assumptions for
galaxy morphological type as the KBSS sample was composed of star-forming
galaxies at $z\\!\sim\\!2.5$ (e.g., Steidel et al., 2014; Strom et al., 2017).
We do not make extinction corrections as these are on the order of the
uncertainty in the $k$-correction (Strom et al., 2017). Once again, we
converted the $R$-band luminosity to the 1500 Å UV band and obtained halo mass
from abundance matching.

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