# CGM2 $+$ CASBaH: The Mass Dependence of H I Ly$\alpha$-Galaxy Clustering and
the Extent of the CGM

Matthew C. Wilde University of Washington, Department of Astronomy, Seattle,
WA 98195, USA  Kirill Tchernyshyov University of Washington, Department of
Astronomy, Seattle, WA 98195, USA  Jessica K. Werk University of Washington,
Department of Astronomy, Seattle, WA 98195, USA  Todd M. Tripp Department of
Astronomy, University of Massachusetts, 710 North Pleasant Street, Amherst, MA
01003-9305, USA  Joseph N. Burchett University of California, Santa Cruz; 1156
High St., Santa Cruz, CA 95064, USA  Department of Astronomy , New Mexico
State University, PO Box 30001, MSC 4500, Las Cruces, NM 88001  J. Xavier
Prochaska University of California, Santa Cruz; 1156 High St., Santa Cruz, CA
95064, USA  Kavli Institute for the Physics and Mathematics of the Universe
(Kavli IPMU) The University of Tokyo; 5-1-5 Kashiwanoha, Kashiwa, 277-8583,
Japan  Division of Science, National Astronomical Observatory of Japan,2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan  Nicolas Tejos Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile
Nicolas Lehner Department of Physics and Astronomy, University of Notre Dame,
Notre Dame, IN 46556  Rongmon Bordoloi North Carolina State University,
Department of Physics, Raleigh, NC 27695-8202  John M. O’Meara W. M. Keck
Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA  Jason Tumlinson
Space Telescope Science Institute, Baltimore, MD, USA  J. Christopher Howk
Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN
46556, USA

###### Abstract

We combine datasets from the CGM2 and CASBaH surveys to model a transition
point, $R_{\rm cross}$, between circumgalactic and intergalactic media (CGM
and IGM, respectively). In total, our data consist of 7244 galaxies at z $<$
0.5 with precisely measured spectroscopic redshifts, all having impact
parameters of 0.01 $-$ 20 comoving Mpc from 28 QSO sightlines with high-
resolution UV spectra that cover H I Ly$\alpha$. Our best-fitting model is an
exclusionary two-component model that combines a 3D absorber-galaxy cross
correlation function with a simple Gaussian profile at inner radii to
represent the CGM. By design, this model gives rise to a determination of
$R_{\rm cross}$ as a function of galaxy stellar mass, which can be interpreted
as the boundary between the CGM and IGM. For galaxies with $10^{8}\leq
M_{\star}/M_{\odot}\leq 10^{10.5}$, we find that $R_{\rm
cross}(M_{\star})\approx 2\pm 0.6R_{\rm vir}$. Additionally, we find excellent
agreement between $R_{\rm cross}(M_{\star})$ and the theoretically-determined
splashback radius for galaxies in this mass range. Overall, our results favor
models of galaxy evolution at z $<$ 0.5 that distribute $T\approx 10^{4}$K gas
to distances beyond the virial radius.

##  1 Introduction

The formation and evolution of galaxies involves a complex interplay between
gravitational collapse of gas from the intergalactic medium (IGM), galaxy
mergers, and feedback due to stellar evolution and active galactic nuclei
(AGN) that drive gaseous outflows and change the ionization state of the
galaxies’ gaseous halos. Together, these processes drive the “cosmic baryon
cycle” which takes place largely in the region of a galaxy referred to as the
circumgalactic medium (CGM). Indeed, understanding the CGM is critical for
developing a complete theory of galaxy evolution, as highlighted by the recent
decadal survey (National Acadamy of Sciences, 2021). In particular, the extent
of the gaseous CGM relative to the extent of the dark matter halo is a subject
of great interest for models that aim to reproduce the properties of gaseous
halos.

The existence of the CGM, first predicted by Bahcall & Spitzer (1969), was
initially revealed by detection of Mg II and H I absorption at large projected
distances ($R_{\perp}>20$ kpc) from $L*$ galaxies (Bergeron, 1986; Morris et
al., 1993; Bergeron & Boissé, 1991; Lanzetta et al., 1995; Chen et al., 2005),
and subsequently traced via higher-energy metal-line transitions such as Si
III, C IV and O VI that are observed to correlate with galaxies and their
global properties (e.g. Tripp & Savage, 2000; Tripp et al., 2008; Prochaska et
al., 2011; Tumlinson et al., 2011; Werk et al., 2013). Within 0.5 $R_{\rm
vir}$ of L $\sim$ L* galaxies, the metal line incidence is found to be $60-90$
% for a range of ionized metal species (Werk et al., 2013). Conversely, Berg
et al. (2022) find an 80% chance of finding a massive galaxy nearby to any
high-metallicity absorber. The CGM of $M_{\star}>$ 108 $M_{\odot}$ galaxies is
now well-established to be metal-enriched (Liang & Chen, 2014; Bordoloi et
al., 2014; Prochaska et al., 2017; Berg et al., 2022), and to extend to at
least 1 $R_{\rm vir}$, and very likely beyond it (Wakker & Savage, 2009;
Burchett et al., 2015; Finn et al., 2016; Wilde et al., 2021; Borthakur,
2022).

Generally, hydrodynamical simulations of galaxy evolution, which exhibit
complex interactions between gravitational collapse from the cosmological
large scale structure and subsequent feedback from supernovae and AGN-driven
winds that heat and enrich the CGM and IGM (EAGLE, Schaye et al. 2015;
IllustrisTNG, Pillepich et al. 2018; SIMBA, Davé et al. 2019; and CAMELS,
Villaescusa-Navarro et al. 2022), are consistent with the range of
observations of the CGM in absorption. Yet these models still rely on
simplistic implementations of the “sub-grid” physics in order to model entire
galaxies (e.g. Ford et al., 2013; Hummels et al., 2013), and physical
properties of the CGM are dependent on the simulation resolution (Hummels et
al., 2019; Peeples et al., 2019). More sensitive observations of the CGM,
including the ability to detect the diffuse gas in emission, are needed both
to break degeneracies in these models, e.g., between heating and cooling
mechanisms, and to develop a flexible parametric model of the CGM (Singh et
al., 2021).

The two-point correlation function between H I absorption along QSO sightlines
and galaxies has proven to be an essential tool to understand the connection
of galaxies to the IGM (e.g. Morris et al., 1993; Chen et al., 2005; Ryan-
Weber, 2006; Prochaska et al., 2011; Tejos et al., 2014; Prochaska et al.,
2019). The primary advantages of leveraging the clustering of these two
entities over one-to-one association analyses is that it provides results for
large scales (1-10 Mpc) as well as the relatively smaller scales where the
baryonic processes associated with the CGM play out, and the correlation
function statistically characterizes absorber-galaxy relationships when
multiple galaxies are close to the sightline and a one-to-one assignment is
ambiguous. Since H I traces both enriched material from galaxies as well as
primordial accretion from the IGM, observations of the CGM, IGM, and galaxies
in the same volume are fundamental to both testing the predictions of galaxy
evolution models and providing a means to differentiate between them (e.g.
Fumagalli et al., 2011; Oppenheimer et al., 2012; Stinson et al., 2012; Ford
et al., 2013; Hummels et al., 2013; Butsky et al., 2020; Singh et al., 2021).

Understanding the physical profile and size of the CGM sheds light on the non-
linear processes of galaxy formation: on what spatial scale(s) do
virialization, accretion, and feedback transform these galactic atmospheres?
Astronomers have long used some version of the virial radius as an estimator
for the size of galaxy halos, but this estimate is somewhat arbitrary and is
based on the distibution of unobservable dark matter. By observing the radial
gas profile around galaxies out to large scales, we can effectively map the
gaseous halo, which in turn constrains the physics of galaxy-scale feedback
processes. Observationally determining the galactic atmosphere’s extent has
additional implications for constraining galaxy evolution and assembly models.
For example, the galaxy baryon and metal budgets require a scale to integrate
the total mass (e.g. Peeples et al., 2014; Werk et al., 2014). Furthermore,
the gaseous halo likely plays an important role in the quenching of dwarf
satellite galaxies as they become stripped by ram-pressure in a low-density
CGM (Putman et al., 2021), and it is useful to constrain where this occurs,
i.e., the extent of the CGM, and how this depends on central galaxy mass.

The presence of H I absorption beyond the virial radius is now widely accepted
for a range of galaxy stellar masses (e.g. Prochaska et al., 2011; Tejos et
al., 2012, 2014; Wilde et al., 2021; Bouma et al., 2021; Borthakur, 2022). In
Wilde et al. (2021) (Paper I) we found an empirical relation between galaxy
stellar mass and the extent of the CGM as indicated by H I covering fractions.
For galaxies with stellar masses $10^{8}<M_{\star}/M_{\odot}<10^{11.5}$, we
found that the CGM extends to two times the virial radius. In this paper, we
focus on the functional forms of the mass dependence of the H I-traced CGM
using a power-law model similar to the 2-halo correlation function. We also
investigate other two-component models that differentiate the CGM from the
IGM. We combine the CGM2 Survey, which focuses on sightlines at low galaxy
impact parameters ($<1$ Mpc), with the _COS Absorption Survey of Baryon
Harbors_ (CASBaH) that probes larger spatial scales ($<20$ Mpc). In doing so,
we greatly increase the absorber-galaxy sample from 543 spectroscopically-
confirmed absorber-galaxy pairs to 7244 pairs spanning $0.003<z<0.48$. Our
goal is to provide the most reliable constraints to date on the spatial extent
of the CGM as traced by H I absorption.

The paper is structured as follows: In Section 2, we briefly review each of
the galaxy-absorber surveys and discuss their combined properties. In Section
3, we introduce two models of the H I-galaxy correlation functions and cover
our main results in Section 4. We compare our results with simulations and
previous results and discuss their implications for galaxy evolution models in
Section 5. Finally, we summarize our results in Section 6.

##  2 Data - Combining CGM2 and CASBaH

Both surveys feature far-ultraviolet spectroscopy of QSOs with _HST_ , using
both the _Cosmic Origins Spectrograph_ (COS, Green et al., 2012) and the
_Space Telescope Imaging Spectrograph_ (STIS, Woodgate et al., 1998), and
deep, ground-based optical spectroscopy of foreground galaxies in the QSO
fields. CASBaH is well suited to the study of the interface between the CGM
and the IGM, at scales $\gtrsim 1$ Mpc. CGM2 provides a relatively more
complete mapping of the inner CGM at scales $\lesssim 1$ Mpc. By combining
CGM2 and CASBaH data, we leverage the strengths of each survey, as described
below. Figure (1) shows the distributions of galaxy stellar masses and impact
parameters versus redshift from both surveys out to z $=0.5$. Together, the
surveys allow us to probe the CGM as it transitions into the IGM for a large
sample of galaxies.

$\begin{array}[]{c}\includegraphics[scale={0.6}]{combined_data_z_rho_log.pdf}\\\
\includegraphics[scale={0.6}]{combined_data_mass_z.pdf}\\\ \end{array}$

Figure 1: Top: Distribution of the combined CGM2 (blue dots) and CASBaH
(purple dots) data sets in both logarithmic impact parameter, and redshift.
The data are roughly uniform in redshift space but we can see the relative
contributions of the data sets in impact parameter space; CGM2 is highly
concentrated at lower impact parameters while CASBaH explores much greater
impact parameters. Bottom: Galaxy stellar mass distribution as a function of
redshift for the two data sets.

###  2.1 CGM2

The CGM2 survey, first presented in Wilde et al. (2021), includes precise
spectroscopic redshifts and bulk galaxy properties (e.g. stellar masses, M∗,
and star formation rates, SFR) from a combination of Gemini GMOS spectra and
deep, broadband photometry for $\sim$1000 galaxies in the foreground of 22
QSOs, each with S/N $\approx$10 HST/COS G130M$+$G160M spectra. By matching
galaxy and absorber redshifts in $\pm$500 km s-1 windows, the CGM2 survey is
ultimately a large collection of measurements pertaining to the CGM of z $<$ 1
galaxies over a wide range of stellar masses, 108 $\lesssim
M_{\star}/M_{\odot}\lesssim 10^{11.5}$. The data acquisition and analysis are
explained in detail in Wilde et al. (2021). Here we present a brief overview
of the survey data relevant to the present analysis.

The CGM2 galaxy spectra were obtained using Gemini-GMOS spectrographs on the
twin Gemini North and South telescopes (Hook et al., 2004; Gimeno et al.,
2016). Galaxy redshifts were inferred from the template fitting code,
Redrock111https://github.com/desihub/redrock (v0.14) and manually inspected
with VETRR222https://github.com/mattcwilde/vetrr. The typical statistical
uncertainly of our redshifts is $\sigma_{z}\sim 50$-$100$ km s-1 ($z\simeq
0.00016$-$0.00030$). Photometry of the CGM2 galaxy catalog was obtained from
the Gemini-GMOS pre-imaging in $g$ and $i$ bands as well as all available
bands from DESI Legacy Imaging Surveys Data Release 8 (DR8) (Dey et al.,
2019), WISE (Cutri et al., 2013), Pan-STARRS Data Release 2 (Chambers et al.,
2016), and SDSS DR14 (Abolfathi et al., 2018).

The 22 QSOs included in the CGM2 survey have _HST_ /COS spectra selected from
the COS-Halos (GO11598, GO13033; Tumlinson et al., 2013) and COS-Dwarfs
(GO12248; Bordoloi et al., 2014) surveys. In general, the CGM2 QSO targets
have $z_{\rm QSO}>$ 0.6 and available HST imaging, which permits detailed
analysis of absorption-hosting galaxies with $z<0.5$. All COS spectra include
both the G130M and G160M gratings, and have a S/N $\simeq 8-12$ per resolution
element (FWHM $\simeq$ 16-18 km s-1) or better over $1150$-$1800$ Å. The COS
data and their reduction are presented in detail in Tumlinson et al. (2013)
and Bordoloi et al. (2014) and follows the same method used by Tripp et al.
(2011), Meiring et al. (2011), Tumlinson et al. (2011) and Thom et al. (2012).

###  2.2 CASBaH

The CASBaH program was designed to take advantage of the multitude of
resonance transitions at rest-frame wavelengths $<$ 912 Å to probe the
physical conditions, metallicity, and physics of the multiphase CGM. A wide
variety of elements and ionization stages have resonance lines only at
$\lambda<$ 912 Å (see, e.g., Verner et al., 1994), so observations of this
wavelength range provide new diagnostics and precise constraints using banks
of adjacent ions such as N i through N v, O i through O vi, and Ne ii through
Ne viii (see Tripp et al., 2011, for examples of lines detected by CASBaH).
The Ne viii 770.4, 780.3 Å doublet has received particular attention as a
probe of warm-hot gas at $\approx 10^{5}-10^{6}$ K (e.g., Savage et al., 2005;
Burchett et al., 2019; Wijers et al., 2020). In many contexts such as the
Milky Way interstellar medium, these lines are inaccessible because they are
blocked by the H i Lyman limit. CASBaH overcomes this limitation by observing
QSO absorbers with sufficient redshift to bring the lines into the observable
band of HST.

The motivation and design of the CASBaH program is summarized in section 1 of
Haislmaier et al. (2021), and the CASBaH galaxy redshift survey is presented
in Prochaska et al. (2019). Briefly, CASBaH obtained both HST/COS and HST/STIS
spectra of nine QSOs at 0.92 $<z_{\rm QSO}<$ 1.48, with two primary selection
criteria. First, since some of the most important target lines (e.g., Ne viii)
are weak, the QSOs were required to be UV-bright so that good signal-to-noise
and sensitivity to weak lines would be attained. Second, the targets were
required to have $z_{\rm QSO}>0.9$ to provide a total redshift path that is
sufficient to accumulate a statistically useful sample of absorbers of
interest. No considerations were given to known foreground galaxies or
absorbers, so the targets were not selected in a way that would favor
particular types of foreground absorbers or galaxies, except that sightlines
with known black Lyman limits at $\lambda_{\rm ob}>1150$ Å were excluded to
avoid using HST time on sightlines that would not contribute useful
pathlengths to the samples (see Burchett et al., 2019). The CASBaH UV spectra
were reduced in the same way as the CGM2 data.

The CASBaH galaxy-redshift survey (Prochaska et al., 2019) measured thousands
of redshifts in the fields of seven of the CASBaH QSOs using the Keck DEIMOS
and MMT Hectospec spectrographs, with typical redshift uncertainties of
$\approx$ 30 km s-1. The survey used a wedding-cake strategy with the
Hectospec covering galaxies in the $\approx 1^{\circ}$ fields centered on the
QSOs and the DEIMOS survey providing a deeper survey with a smaller field of
view (81.5 arcmin2) (see Prochaska et al., 2019). Using the CASBaH galaxy
database, supplemented with data from public surveys such SDSS, we selected a
sample of 6701 galaxies with spectroscopic redshifts $z<0.481$ and comoving
impact parameters less than 13 cMpc, appropriate for the H I analysis
presented here.

###  2.3 Synergy of CGM2 \+ CASBaH

The CASBaH and CGM2 surveys have complementary designs. On the one hand, CGM2
is built on COS-Halos and thus favors at least one $L*$ galaxy close to the
sightline. CGM2 also covers a smaller FOV. On the other hand, CASBaH is a
blind survey that covers a larger FOV. Consequently, CASBaH provides more
information about galaxies and large-scale structures at larger impact
parameters, but as a blind survey, it is cross-section weighted in favor of
galaxies at larger impact parameters. Also, since CASBaH avoided sightlines
with black Lyman limits in the HST band (i.e., at $\lambda_{\rm{ob}}\geq 1150$
Å), it will not include galaxies at $z_{\rm gal}>0.26$ that harbor absorbers
with $N$(H i) $\gtrsim 10^{17}$ cm-2. Thus, CGM2 probes the inner CGM
including higher $N$(H i) absorbers, while CASBaH complements CGM2 by adding
very large samples of galaxies and structures at larger distances.

###  2.4 Galaxy Properties

To estimate the galaxy properties for both surveys, we used CIGALE (Noll et
al., 2009; Boquien et al., 2019) to fit the spectral energy distribution (SED)
and retrieve stellar mass and star formation rates (SFR). We used the Bruzual
& Charlot (2003) stellar population models, assuming a Chabrier (2003) initial
mass function (IMF). We chose a grid of metallicities ranging from
$0.001$-$2.5Z_{\odot}$. A delayed star formation history (SFH) model was
employed with an exponential burst. The e-folding time of the main stellar
population models ranged from 0.1-8 Gyr. We varied the age of the oldest stars
in the galaxy from 2-12 Gyr. We included an optional late burst with an
e-folding time of 50 Myr and an age of 20 Myr. The burst mass fraction varied
from 0.0 or 0.1 to turn this feature on or off. Nebular emission and
reprocessed dust models (Dale et al., 2014) were also included with the
default values. The dust models have slopes ranging from $1-2.5$ and the
nebular models include no active galactic nuclei.

We employed the Calzetti et al. (1994) dust attenuation law, but we also
included a “bump” in the UV (see discussion in Prochaska et al., 2019) at
217.5 nm with a FWHM of 35.6 nm. The bump amplitude is set at 1.3 and the
power law slope is -0.13 (Lo Faro et al., 2017). We varied the color excess of
the stellar continuum from the young population, E(B-V), from 0.12-1.98.
Finally, we used a reduction factor of 0.44 to the color excess for the old
population compared to the young stars.

CIGALE then provides us with Bayesian estimates for the stellar mass and SFR
for each galaxy in the combined catalog. In order to calculate the virial
radius we used the abundance matching method of Moster et al. (2013) with the
modifications used in Burchett et al. (2016). We adopt the convention of using
$R_{\rm vir}$ = $R_{200m}$, the radius within which the average mass density
is 200 times the mean matter density of the universe, as the virial radius
($R_{\rm vir}$) of a galaxy halo.

###  2.5 Combining the CGM2 and CASBaH Surveys

In order to combine the surveys, we modified both catalogs to ensure the same
matching criteria between galaxies and absorbers. In the original CGM2 survey,
we measured the $2\sigma$ upper limit on absorption within $\delta v=\pm 30$
km s-1 of the galaxies redshift using the normalized error of the quasar flux
when no absorption system was found within our $|\delta v|<500$ km s-1 window.
In order to match the CASBaH survey, we adjusted this to a $3\sigma$ upper
limit. This did not change our results in a meaningful way. The original
CASBaH survey used a velocity window of $|\delta v|<400$ km s-1 to match the
galaxies to absorption systems. We adjusted the window for this work to
$|\delta v|<500$ km s-1 to match the CGM2 survey. As in Paper I, we restrict
our H I measurements to those less than $z<0.481$ since at this redshift, the
Lyman-$\alpha$ line redshifts out of the G160 grating band, and thus we are
only sensitive to higher order transitions at higher redshifts.

Having made these two small changes to each survey, both could be combined to
give us a total survey that includes 7244 galaxies spanning $\sim 0.01-8$
comoving Mpc in impact parameter around 28 QSO sightlines. The distributions
of impact parameter, redshift, and stellar mass are shown in Figure 1. In this
paper, we will focus on galaxies with $8<\log M_{\star}/M_{\odot}<10.5$, a
stellar mass range with good coverage in both surveys, which trims our galaxy
sample to 6136 galaxies from CASBaH and 453 galaxies from CGM2 for a total
sample of 6589 absorber-galaxy pairs. The number of absorber-galaxy pairs is
summarized in Table 1.

Table 1: Number of Absorber-Galay Pairs

Survey | $10^{7-11.3}M_{*}/M_{\odot}$ | $10^{8-10.5}M_{*}/M_{\odot}$ | $10^{8-9}M_{*}/M_{\odot}$ | $10^{9-10}M_{*}/M_{\odot}$ | $10^{10-10.5}M_{*}/M_{\odot}$
---|---|---|---|---|---
CGM2 | 543 | 453 | 103 | 271 | 79
CASBaH | 6701 | 6136 | 1265 | 3545 | 1326
Total | 7244 | 6589 | 1368 | 3816 | 1405

Note. — Summary of absorber-galaxy pairs used in this manuscript. (1) The
number of absorber-galaxy pairs in each survey and total of the combined
surveys; (2) the number of absorber-galaxy pairs in the entire mass range; (3)
the mass range used to perfom the model fitting; (4, 5, 6) the number of
absorber-galaxy pairs within each mass bin used for model verification.

##  3 Modeling Absorber-Galaxy Clustering

We model the CGM using an absorber-galaxy cross-correlation analysis. This
technique is based on modeling the covering fraction, $f_{c}$, as a binomial
probability distribution of detections. To ensure high completeness in the
absorber sample, based on the S/N of the data, we require a total column
density $N_{\rm HI}$ $\geq 10^{14}$ cm-2 to consider the sightline to have a
“detection”. Likewise, a non-detection is the case where we do not detect gas
above this threshold. The models used here are based on the models employed in
Paper I, which was inspired by the model developed by Hennawi & Prochaska
(2007) and Prochaska et al. (2019). A more detailed explanation can be found
in those three papers. In Paper I, we found a mass dependence of the extent of
the CGM based on dividing the data into three mass bins. In this work, we wish
to quantify the mass dependence of the clustering as well as determine the
redshift dependence given our data.

###  3.1 Single Power-Law Model

Figure 2: Corner plots showing the posterior parameter probabilities for the
parameters in the single power-law clustering model. We find a non-zero,
positive mass dependence term in the two-halo absorber-galaxy clustering,
$\beta^{2h}$.

The single power-law model consists of two terms: the base rate of detection
due to the random incidence of absorbers greater than this threshold and an
excess above this base rate due to the clustering of galaxy-absorber pairs.

Much like Prochaska et al. (2019), we define the 3D absorber-galaxy cross-
correlation function, $\xi_{ag}(r)$ as

$\xi_{ag}(r)=\left(\frac{r}{r_{0}}\right)^{-\gamma}.$ (1)

To model the galaxy mass dependence of the clustering, we add a new mass
dependence to the clustering scale, $r_{0}$,

$r_{0,m}(m)=r_{0}\left(\frac{M_{\star}}{M_{0}}\right)^{\beta}.$ (2)

As before, we examine the projected 2-D correlation function, which is
obtained by integrating the 3-D correlation function over the line of sight

$\chi_{\perp}(r)=\frac{1}{\Delta
r_{\parallel}}\int_{r_{\parallel}}\xi_{ag}(\sqrt{r_{\parallel}^{2}+r_{\perp}^{2}}~{})dr_{\parallel},$
(3)

where $r_{\parallel}$ is the line-of-sight distance, $r_{\perp}$ is the
transverse distance, and $\Delta r_{\parallel}$ is the size of the redshift
window.

For simplicity of notation, $r$ is equivalent to $r_{\perp}$ in the following
analysis.

In the following definitions, we label the single power law clustering terms
“2-halo,” as the galaxy clustering method we adopt here describes the
clustering of separate dark matter halos. This approach distinguishes the
“two-halo” only method from the two-component model we develop later in this
manuscript.

In order to model $f_{c}$, we assume that the number of detected absorbers
above the column-density threshold has a Poisson distribution. We consider two
cases: (1) one or more absorbers detected, and (2) the case where no absorbers
are detected. In this framework the probability of seeing no absorbers is

$P^{\rm miss}=\frac{\lambda^{0}\exp(-\lambda)}{0!}$ (4)

where we denote the rate of incidence (see below) as $\lambda$. The
probability of finding one or more absorbers is just the complement of
Equation 4,

$f_{c}=1-P^{\rm miss}.$ (5)

We model the rate of absorber incidence as the projected correlation function,
the 2-halo term, as the excess over the probability of intersecting an
absorber with $N_{\rm HI}$ $>10^{14}$ cm-2 in the redshift window,

$\lambda=(1+\chi_{\perp}^{2h})~{}\langle d\mathcal{N}/dz\rangle\delta z,$ (6)

where $\langle dN/dz\rangle$ is the base rate of detection due to the random
incidence of absorbers greater than this threshold and $deltaz$ is the line-
of-sight redshift window.

In addition to parameterizing the mass dependence as in Equation (2), we also
parameterize the redshift dependence of $\langle dN/dz\rangle$ as follows:

$\frac{d\mathcal{N}(\rm N_{\rm HI}\geq\rm N_{\rm
HI}^{14},z)}{dz}=C_{0}(1+z)^{\alpha},$ (7)

where $N_{\rm HI}^{14}$ denotes absorbers with column densities of $10^{14}$
cm-2, $C_{0}$ is the random rate of incidence at $z=0$, and $\delta z$ is the
redshift window. We adopt a redshift window to be $\pm 500$ km s-1 in velocity
units.

Thus, we have a rate of incidence of the form

$\begin{multlined}\lambda=(1+[\chi_{\perp}^{2h}(r,m|r^{2h}_{0},\gamma^{2h},\beta^{2h})])~{}\langle
d\mathcal{N}(z|C_{0},\alpha)/dz\rangle~{}\delta
z.\end{multlined}\lambda=(1+[\chi_{\perp}^{2h}(r,m|r^{2h}_{0},\gamma^{2h},\beta^{2h})])~{}\langle
d\mathcal{N}(z|C_{0},\alpha)/dz\rangle~{}\delta z.$ (8)

Finally, we construct the likelihood function,

$\mathcal{L}=\prod_{i}P^{\rm hit}(r_{i},z_{i},m_{i}|\theta)\prod_{j}P^{\rm
miss}(r_{j},z_{j},m_{j}|\theta),$ (9)

where $\theta=[r^{2h}_{0},\gamma^{2h},\beta^{2h},C_{0},\alpha]$.

In constructing our Bayesian model, we must choose priors. For the single
power law parameters, we chose the priors based on the results of cross-
correlation analysis by Tejos et al. (2014) except for our new mass dependent
term, $\beta^{2h}$, which was motivated by physical arguments:

  * •

$r^{2h}_{0}\sim\mathcal{N}(\mu=3.2,\sigma=0.3),~{}r^{2h}_{0}>0$

  * •

$\gamma^{2h}\sim\mathcal{N}(\mu=1.7,\sigma=0.1)$, $\gamma^{2h}>0$

  * •

$\beta^{2h}>0$,

where $\mathcal{N}$ is the normal distribution with mean $\mu$ and variance
$\sigma^{2}$.

The priors for the redshift dependence were chosen based on the findings in
Kim et al. (2021):

  * •

$C_{0}\sim\rm{Lognormal}(\mu=1.25,\sigma=0.11)$ , $C_{0}>0$

  * •

$\alpha\sim\mathcal{N}(\mu=0.97,\sigma=0.87)$ , $-3<\alpha<3$

We note that we chose to use the more recent results of Kim et al. (2021) in
modeling the redshift evolution instead of that from Danforth et al. (2016),
as were used in Paper I.

As in Paper I, we apply the Bayesian Markov Chain Monte Carlo (MCMC) sampler
emcee (Foreman-Mackey et al., 2013) to generate samples from the posterior
probability distribution function to estimate the parameters of interest and
their distributions, using Equation (9) and the priors described above.

In Figure 2, we show the posterior distributions of our single power-law model
with $M_{0}=10^{9.5}M_{\odot}$. These were fit only to data with $8<\log
M_{\star}/M_{\odot}<10.5$, as above this range there is a change in the virial
radius due to the $M_{\star}-M_{\rm halo}$ relation from abundance matching
(Moster et al., 2013). Below this mass range we find a very flat covering
fraction profile, which does not show a clustering signal.

###  3.2 Two-component Models

The single power-law model used in galaxy-galaxy clustering and adapted above
to model the galaxy-absorber clustering makes no assumption of a CGM or
overlapping (in projection) gaseous halos. However, the existence of the CGM
is now well-established (Tumlinson et al., 2017). In particular, the trends of
ionized metal species with impact parameter around L* and sub-L* galaxies from
$z=0-3.5$ distinctly show that metal-enriched gaseous atmospheres are a
fundamental component of galaxies (e.g. Werk et al., 2013; Lehner et al.,
2014; Bordoloi et al., 2014; Borthakur et al., 2015; Rudie et al., 2019). In
the following section, we therefore assume the existence of the CGM and use a
simple Gaussian profile to model the excess clustering signal due to the
presence of the CGM. In addition, we investigated several other functional
forms of the CGM component, which we describe in §3.2.2. We find that the
particular functional form of this component has little impact on the results.

####  3.2.1 The Gaussian CGM Two-Component Model

We now add a third term to the detection rate: a Gaussian 1-halo component.
The detection rate now consists of a baseline random incidence rate, an
enhancement due to large-scale absorber-galaxy clustering, and an additional
enhancement due to the CGM. We employ an exclusion model where the
contribution from the 2-halo term terminates at the distance it reaches the
1-halo component. This scheme, shown in Figure 3, also allows us to determine
a natural estimate of the extent of the CGM: the crossing point of the 1-and
2-halo components. More explicitly, within some radius, the galaxy has a CGM
that we define as the gas of that galaxy and any other satellite galaxies
within its halo. Our formalism then defines the $R_{\rm cross}$ where this CGM
component exceeds the 2-halo.

Figure 3: A schematic depiction of our two-component exclusion model and the
determination of $R_{\rm cross}$. The 2-halo component cuts off interior to
$R_{\rm cross}$.

The model is similar to that single power-law we introduced before with a few
key differences. We introduce a Gaussian one-halo term defined as:

$G(r)^{1h}=Ae^{-(r/\sigma)^{2}}.$ (10)

Where the two models intersect, $R_{\rm cross}$, we can solve for $\sigma$ as

$\sigma=\sqrt{\frac{1}{2}\frac{R_{\rm cross}^{2}}{\ln(A)+\gamma\ln(R_{\rm
cross}/r_{0})}}.$ (11)

It should be noted that $R_{\rm cross}$ here is the 3-D distance and not the
projected distance. In order to characterize the mass dependence of $R_{\rm
cross}$ we define

$R_{\rm cross}=R_{cross,0}\left(\frac{M_{\star}}{M_{0}}\right)^{\beta^{1h}},$
(12)

where $R_{cross,0}$ is the 1-halo term extent for a galaxy at the fixed pivot
mass $M_{0}$. The galaxy mass dependence of $\sigma$ includes contributions
from the mass dependencies of $R_{cross}$ and $r_{0}$.

This parameterization allows us to compare the mass dependence of the 1-halo
term, $\beta^{1h}$ with that of the 2-halo term, $\beta^{2h}$.

In order to solve for the projected clustering signal, $\xi$, we first make
some definitions to ease the notation. We use $s=r_{\parallel}$ in the
remainder of the analysis. The integration is performed over different
portions of the line of sight distance, s, corresponding to the 1 and 2-halo
components. We define the line of sight crossing point $s_{\rm cross}$ as

$s_{\rm cross}=\sqrt{\max(R_{\rm cross}^{2}-r_{\perp}^{2},0)},$ (13)

and we can then integrate Equation 10 to $s_{\rm eval}=\min(s_{\rm
cross},s_{\rm max})$, where $s_{\rm max}$ is the maximum interval we wish to
integrate over, which in our case is $[-500,500]$ km s-1. Thus we have

$\chi(r_{\perp})\propto 2\int_{0}^{s_{\rm
eval}}G(r_{\perp},s)^{1h}ds+2\int_{s_{\rm eval}}^{s_{\rm
max}}\xi(r_{\perp},s)^{2h}ds$ (14)

where the factor of 2 comes from the fact that both components are symmetric.
Here we integrate the one-halo component over the more nearby regime out to
$s_{\rm eval}$ and only integrate the 2-halo term beyond $s_{\rm eval}$ out to
the maximum line of sight distance, thus excluding the regimes in which the
models do not apply. For the two-component model, we choose fairly weak priors
on unknown parameters based on physical arguments while following the same
priors as described above for the parameters in the single power-law model:

  * •

$\beta^{1h}>-3$

  * •

$A>0$

  * •

$R_{\rm cross}>0$

Figure 4: Posterior probabilities for the parameters in the two-component
clustering model. We again recover a non-zero, positive mass dependence term
in the two-halo absorber-galaxy clustering, $\beta^{2h}$ but find an even
stronger one-halo CGM clustering mass dependence $\beta^{1h}\simeq 0.14\pm
0.07$.

We can then follow the same MCMC fitting procedure described above to
determine the posteriors for the parameters in this model as well as the
crossing radius, $R_{\rm cross}$. These are shown in Figure 4. As before, we
only fit data with $8<\log M_{\star}/M_{\odot}<10.5$ and use
$M_{0}=10^{9.5}M_{\odot}$.

Figure 5: Comparison of our two models to the empirical covering fraction as a
function of impact parameter in comoving kpc in mass bins of
$10^{8-9}M_{\odot}$, $10^{9-10}M_{\odot}$ and $10^{10-10.5}M_{\odot}$. The
data are shown in black with 1$\sigma$ error bars. The single power-law model
is shown in pink while the two-component model is shown in purple. The
vertical dotted line denotes $R_{\rm cross}$ in each mass bin. Both models
recreate the covering fraction of the data in all mass bins except for the
lowest mass bin where the clustering signal disappears. The two-component
model provides a better match to the data for galaxies of
$M_{\star}>10^{9}M_{\odot}$ at the lowest impact parameters where the single
power law model underestimates the covering fraction.

####  3.2.2 Other Two-Component Models

While the single power-law clustering model does an adequate job reproducing
the data on large spatial scales, its contribution is insufficient at
$R_{\perp}$ $\lesssim$ 200 kpc as can be seen in Figure 5 (pink curve).
Furthermore, the primary goal of our study is to find the boundary between the
CGM and IGM, and thus including a CGM component is essential for this purpose.
We explored several candidate functional forms for this CGM component.

We first investigated a two-component model where each component is
represented by a power law, inspired by the 1-halo and 2-halo terms that are
used to model the clustering of galaxies. The 3D and projected forms of the
two absorber-galaxy correlation functions are given by Equations 1 and 3,
respectively, and the two-component correlation function is the sum of these
parts. We also considered a model where the two-component correlation function
is, in 3D, the maximum of the two power laws. This is similar to our chosen
model, but with an inner power law rather than an inner Gaussian profile.

To rise above the outer power law component at small radii, the inner power
law has to be steeper. In practice, the two power law indices turned out to be
similar, yielding essentially the same result as a single power law fit. This
outcome is not unexpected: the enhancement in the incidence rate or surface
density of gas near galaxies often does not resemble a steepening power law at
small radii (Zhu et al., 2014; Lan, 2020).

In those studies, the enhancement is better described by a function that
declines gradually (compared to a power law) at small radii and quickly at
large radii. The top-hat function, which has amplitude $A$ inside a boundary
and amplitude 0 outside the boundary, is an extreme example of this class. Our
adopted Gaussian profile allows a smoother transition between the CGM-like and
outer components of the model. However, we note that a fit to the data
combining a inner 3D top-hat with an outer power law yields an $R_{\rm
cross}$($M_{*})$ that is effectively indistinguishable from the one that
emerges from the Gaussian component model.

###  3.3 Model Comparison

In addition to comparing the two models to each other, Figure 5 compares the
models to the empirical covering fraction as a function of impact parameter
and mass. The data are shown in black with 1$\sigma$ error bars. The single
power-law model is shown in pink while the two component model is shown in
purple. Both models recreate the covering fractions in all mass bins at all
values of $R_{\perp}$ except for one data point in the $\log
M_{*}/M_{\odot}=9-10$ bin at $R_{\perp}$ $\approx$ 200 kpc. Moreover, the two
models make different predictions at low $R_{\perp}$ except for in the lowest
mass bin ($\log M_{*}/M_{\odot}<9$) where there is no discernible excess above
the clustering signal. This does not preclude the presence of a CGM around
these galaxies, but rather suggests that we require more data at lower
$R_{\perp}$ for galaxies with $\log M_{*}/M_{\odot}<9$ to be able to constrain
$R_{\rm cross}$ at these masses.

Figure 6: Comparison of the two-halo 3D cross correlation posteriors between
the two-component model ($r_{0}=3.99^{+0.28}_{-0.24}$ cMpc, $\gamma=1.62\pm
0.07$) and the single power-law model ($r_{0}=3.58^{+0.28}_{-0.24}$ cMpc,
$\gamma=1.55\pm 0.05$). The two models are consistent with each other within
the 1$\sigma$ limits and have a power-law slope consistent with the absorber-
galaxy 3D cross correlation found in the literature (e.g. Tejos et al., 2014)
of $\gamma=1.7\pm 0.1$.

The two-halo only model under-predicts the observed signal for galaxies at
intermediate masses ($\log M_{*}/M_{\odot}=9-10$). The two component model
does better for galaxies of $M_{\star}=10^{9-10}M_{\odot}$ at the lowest
impact parameters where the single power law model underestimates the covering
fraction, although not significantly so. For $R_{\rm cross}<300$ kpc, one
detects 52 H I systems where 46 systems are predicted. Assuming Poisson
statistics, the two-halo only model is consistent with the data at 1$\sigma$
level. Analogous to the one-halo term of galaxy-galaxy clustering, the data
themselves do not require an enhanced covering fraction of H I absorption that
we identify as the CGM.

We find the 1-halo component has a stronger clustering mass dependence,
$\beta^{1h}\simeq 0.14\pm 0.07$, than the two-halo term, $\beta^{2h}\simeq
0.08\pm 0.03$. We also find the 2-halo clustering terms in each model to be
internally consistent with each other as seen in Figure 6.

##  4 Results

###  4.1 Clustering Mass Dependence

As seen in Figure 2, we find the clustering parameters to be $r_{0}=3.6\pm
0.3$ cMpc, $\gamma=1.6\pm 0.5$. $r_{0}$ and $\gamma$ are consistent with those
found in Tejos et al. (2014) who find $r_{0}=3.7\pm 0.1$ cMpc and
$\gamma=1.7\pm 0.3$. We also find a mass dependence of the absorber-galaxy
clustering of $\beta^{2h}=0.07^{+0.3}_{-0.2}$.

We find the the two component model better fits the data as can be seen in
Figure 5. Specifically, the two component model better matches the covering
fraction for galaxies of $M_{\star}>10^{9-10}M_{\odot}$ at the lower impact
parameters where the single power law model underestimates the covering
fraction. In addition, we find the two-component model reproduces the mass
dependence of the 2-halo clustering term, $\beta^{2h}\simeq 0.07$ while also
producing a stronger mass dependence of the 1-halo clustering term,
$\beta^{1h}\simeq 0.14$.

###  4.2 Physically-Motivated Extent of the CGM

As mentioned above, using the two-component model produces an estimate of
$R_{\rm cross}$, a natural metric for the extent of the CGM. This 3-D distance
demarcates where the contribution to the clustering begins to be dominated by
the CGM above the expected two-halo clustering due to isolated galaxy halos
traced by H I. $R_{\rm cross}$ can be viewed as the maximum radius to which an
enhancement from the CGM could extend without over-predicting the data at
large radii.

In Figure 7, we see $R_{\rm cross}$ (blue) compared with the spread in virial
radii of the galaxy sample (grey filled region). The filled blue region
represents the 1$\sigma$ limits of the distribution in $R_{\rm cross}$ while
the blue line denotes the median of this distribution. We find $R_{\rm cross}$
is $\sim 2\pm 0.6R_{\rm vir}$ for galaxies in the range
$8<\log(M_{\star}/M{\odot})<10.5$. The black crosses correspond to the values
published in Paper I defined as the extent where there is 50%chance to see H I
absorption above $10^{14}$ cm-2. The vertical dotted lines denote the mass
range of $8<\log(M_{\star}/M{\odot})<10.5$ that was used in our MCMC analysis.
Above this range, we see a change in the relation of the virial radius with
stellar mass, and below this mass range, we find little to no correlation
between absorbers and galaxies. 5).

We also calculated the splashback radius, $R_{\rm sp}$, using the method from
Diemer (2018) and encoded in the
COLOSSUS333https://bdiemer.bitbucket.io/colossus/ package. This radius denotes
the location at which particles reach the apocenter of their first orbit. We
find excellent agreement of $R_{\rm cross}$ with the results in Paper I and
$R_{\rm cross}$ neatly matches the splashback radius for galaxies in this mass
range. We discuss these results in more detail below.

Figure 7: A comparison of $R_{\rm cross}$ with the virial radius ($R_{\rm
vir}$, grey filled region) as well as the splashback radius ($R_{\rm splash}$,
pink shaded region) of the galaxy sample. The filled regions in $R_{\rm vir}$
and $R_{\rm splash}$ denote the redshift range for the galaxies in our sample
($0.1\lesssim z\lesssim 0.48$). The filled blue region represents the
1$\sigma$ limits of the distribution in $R_{\rm cross}$ while the blue line
denotes the median of this distribution. The black crosses correspond to the
values published in Paper I. The vertical dotted lines denote the mass range
of $8<\log(M_{\star}/M{\odot})<10.5$ to which we limited the fitting in our
MCMC analysis in Figure 5.

##  5 Discussion

Both of the models we investigate do an adequate job of recreating the cross
correlation signal at all impact parameters and masses
$10^{8}<M_{\star}<10^{10.5}M_{\odot}$ as seen in Figure 5. It is not entirely
clear that the single power law model has any physically-consistent meaning,
however. Effectively, it would seem to signify that every time one measures H
I absorption at the same redshift as a particular galaxy ($|\Delta v|<$ 500 km
s-1), the absorption is always due to _another galaxy’s CGM_. Note, we would
conclude this for all galaxies, i.e. each has no CGM and only neighbors with a
CGM. This is clearly impossible. The two-halo-only model for the CGM
effectively breaks down when the galaxies lie within the halo under
consideration, i.e. when they “mix.” We cannot and do not try to distinguish
between the two. However, our formalism does allow one to identify the outer
extent of this “mixing.”

The two-component model asserts that galaxies with $M_{\star}>10^{8}M_{\odot}$
have a CGM, an assumption that is motivated by previous survey results (e.g.
Werk et al., 2013). Additionally, this model is able to better recreate the
data – from the combined datasets of CGM2 $+$ CASBaH, which together represent
the largest sample of galaxies with confirmed spectroscopic redshifts in the
foregrounds of UV-bright QSOs with high-resolution absorption spectroscopy –
both at smaller impact parameters and at $M_{\star}>$ 109 $M_{\odot}$.

The much larger number of galaxies at larger impact parameters drives the fit
of the models to the data. There is, however, a $>1\sigma$ inconsistency
between the two-halo only model and the data at $R_{\perp}\sim 200$ and for
both models at $R_{\perp}\sim 600$ in the log$M_{\star}=9-10M_{\odot}$ mass
range. The latter inconsistency may be due to cosmic variance or the
assumption that the absorber-galaxy measurements are independent and are not
correlated, which would increase the scale of the error bars at $R_{\perp}\sim
600$.

###  5.1 Comparing the mass dependence of the single and two-component models

Our galaxy sample includes a large number of galaxies at low ($<500$ kpc)
impact parameters which allows us to better model the regime in which the two-
halo galaxy clustering becomes dominated by the signal of galaxies that
inhabit the same dark matter halo, the one-halo term. By separating these two
terms in the manner presented here, we can disentangle the large scale
clustering as well as the contribution of the CGM to the 3D correlation of
absorbers and galaxies.

Our analysis finds nearly identical terms for the mass dependence of the
clustering at large scales, $\beta^{2h}$ as well as the contribution of
absorbers at random, $C_{0}$ and $\alpha$. We do find a stronger mass
dependence in the one-halo term, $\beta^{1h}$ than at larger scales. This can
be seen in Figure 5 where the correlation steepens in higher mass bins.

###  5.2 Absorber-Galaxy Bias

Our covering fraction analyses provide an estimate of the galaxy-absorber
correlation function, $\xi_{ag}$ (eq. 1). Here, we test if the mass dependence
of $\xi_{ag}$ outside the CGM is consistent with absorption systems and
galaxies simply being two independent tracers of the same underlying dark
matter distribution. Assuming both tracers have linear bias, $\xi_{ag}$ should
be equal to $b_{a}b_{g}\xi_{\rm DM}$, where $b_{a}$ and $b_{g}$ are the
absorber and galaxy bias, respectively, and $\xi_{\rm DM}$ is the dark matter
3D correlation function. Following Tinker et al. (2010) (hereafter, T10), we
assume the dark matter correlation function can be described by a power-law
function of radius with index $\gamma=1.62$. We fix the power-law index in the
$\xi_{ag}$ determined by fitting a single power-law to the data to this same
value, with which it is consistent. With the above assumptions,
$\xi_{ag}=(r/r_{0}(M))^{-\gamma}=b_{a}b_{g}\xi_{\rm DM}(r)$. The radial
dependence cancels, leaving the proportionality $r_{0}(M)^{\gamma}\propto
b_{a}b_{g}$.

We show a scaled $r_{0}(M)^{\gamma}$ in Figure 8 along with the galaxy bias as
a function of stellar mass from T10 and implemented in the COLOSSUS package
(Diemer, 2018). If $b_{a}$ is constant and the assumptions stated above hold,
$r_{0}(M)^{\gamma}$ should have the same mass dependence as galaxy bias. While
there is a visually apparent difference between the galaxy bias and the best-
fit $r_{0}(M)^{\gamma}$, this difference is not significant at a $2\sigma$
level and so is merely suggestive. If the difference is real, it could be a
consequence of the H I mass per dark matter mass being a function of
overdensity. Up to the overdensities at which $M_{star}=10^{10.5}$ $M_{\odot}$
galaxies tend to be found, this function would be increasing: H I would be
less common in low density regions than in higher density filaments. This
behavior would be consistent with theoretical expectations (e.g., Hui & Gnedin
1997; Schaye 2001; Davé et al. 2010) and observations (e.g., Rudie et al.
2012; Burchett et al. 2020).

Figure 8: A comparison of the slopes of the relative bias as a function of
mass derived from our analysis (orange) compared to Tinker et al. (2010) (T10,
black). The dashed lines correspond to the ranges spanned by the $1\sigma$
limits in in $\beta^{2h}$. The relative bias,
$r_{0}(M)\propto(M_{\star}/M_{0})^{\gamma\beta}$, are normalized to the value
of T10 at log $M_{\star}/M_{\odot}=9.5$. We find a steeper mass dependence
than T10, but the significance of the difference is less than $2\sigma$.

###  5.3 Comparison to Previous Work

One of the key aspects of this analysis is determining the mass dependence of
the extent of the $N_{\rm HI}$ $>10^{14}$ cm-2 for which our model provides a
direct metric, $R_{\rm cross}(M_{\star})$. We compare our resulting $R_{\rm
cross}(M_{\star})$ to the method and results from Paper I in Figure 7. The
result of Paper I, $R^{14}_{\rm CGM}$, which are based only on the CGM2 survey
are shown as black crosses in the mass bins they span in that paper. We also
compare the method used in that paper to determine $R^{14}_{\rm CGM}$, the
radius at which the probability of detecting $N_{\rm HI}$ $>10^{14}$ cm-2 is
$>50\%$, calculated with the two-component model using the combined CGM2 $+$
CASBaH surveys and find it to be consistent within 1$\sigma$ with our newer
model for $R_{\rm cross}(M_{\star})$. We find that our mass dependent estimate
of the extent of the CGM, $R_{\rm cross}(M_{\star})$ corroborates the findings
of Paper I that the $N_{\rm HI}$ $>10^{14}$ cm-2 extends to approximately
twice the virial radius ($\sim 2\pm 0.6R_{\rm vir}$).

One of the main strengths of the CGM${}^{2}+$ CASBaH sample is the large
number of galaxies at small projected separations ($<$1 Mpc). This allows us
to investigate the smaller scale regime in more detail within the context of
similar studies such as Tejos et al. (2014) (hereafter, T14) who uses a single
power law model to measure the two-point correlation between H I and galaxies
above $N_{\rm HI}$ $>10^{14}$ cm-2. In this work they break up their
measurements into SF vs non-SF samples while we do not. Our sample however is
dominated by the more common SF galaxies and we will compare our results to
their SF sample. Comparing our cross-correlation results with T14, we find
good agreement between the results in T14, $r_{0}^{\rm T14}=3.8\pm 0.2$ Mpc,
$\gamma=1.7\pm 0.1$ and the results from both models presented here,
$r_{0}=3.99^{+0.28}_{-0.24}$ Mpc, $\gamma=1.62\pm 0.07$) and the single power-
law model ($r_{0}=3.58^{+0.28}_{-0.24}$ Mpc, $\gamma=1.55\pm 0.05$. We find a
mass dependence of this cross-correlation, however as parameterized by
$\beta^{2h}$.

Our results are slightly in tension with Momose et al. (2021) who find
galaxies in the $10^{9-10}M_{\odot}$ range dominate their H I-galaxy cross
correlation signal. We find the largest mass bin sample to have the most
elevated covering fractions at low impact parameter.

###  5.4 Physical Extent of Galaxy Halos

Astronomers often use the viral radius as a means to describe the
characteristic size of galaxy halos and it is convenient to compare this to
the extent of the gaseous galactic atmosphere as we have done here and in
Paper I. The virial radius is typically defined in terms of the spherical
overdensity mass definition which is based on the radius which encloses an
overdensity of 200 times the critical or mean density, i.e., $R200_{c}$ and
$R200_{m}$. Because the mean and critical densities are decreasing over cosmic
time, this can lead to a pseudo-evolution as pointed out in Diemer et al.
(2013). In addition, subhalos show evidence of being stripped outside the
virial radius of clusters (Behroozi et al., 2014).

An alternative physically motivated halo scale is the splashback radius,
$R_{\rm sp}$ (Diemer & Kravtsov, 2014; Adhikari et al., 2014; More et al.,
2015). This radius effectively distinguishes infalling material from matter
orbiting in the halo. We compare our results to the splashback radius in
Figure 7 and find that our estimate of the extent of the H I CGM, $R_{\rm
cross}$, neatly aligns with $R_{\rm sp}$ over the mass range
$10^{8}<M_{\star}/M_{\odot}<10^{10.5}$. This result implies that $R_{\rm sp}$
is a better approximation of the CGM extent than the more commonly used viral
radius.

O’Neil et al. (2021) compared $R_{\rm sp}$ as estimated from dark matter and
gas profiles in the IllustrisTNG simulations and found that the gas $R_{\rm
sp}$ is consistently smaller than the dark matter $R_{\rm sp}$. However, they
were looking at much more massive halos $M_{\rm halo}>10^{13}$ in which shocks
dominate the gas distribution. Nonetheless, the fact that $R_{\rm
cross}\approx R_{\rm sp}$ at the mass ranges considered here ($M_{\rm
halo}~{}10^{10-12}M_{\odot}$) is intriguing. The halo mass accretion rate
generally sets whether $R_{\rm sp}$ exceeds $R_{\rm vir}$; a rapid accretion
rate will impact the growth of the gravitational potential well, leading to
$R_{\rm sp}<R_{\rm vir}$. If the location of $R_{\rm cross}$ reflects the
extent of orbiting gas in a halo, then our observational results imply a halo
mass accretion rate that is slow enough to keep the apocenters of orbiting
structures at large radii.

Another way of defining the extent of the CGM is to use the boundary of the
pressure-supported CGM. For galaxies with halo masses $\gtrsim
10^{11.5}M_{\odot}$ ($M_{\star}\approx 10^{9.8}M_{\odot}$), this pressure
support comes from fact that the gas that has fallen into the gravitational
potential well is virially shocked and cannot cool within a Hubble time
(Binney, 1977; Rees & Ostriker, 1977; Silk, 1977). For the galaxies in our
survey, which are predominately below this halo mass, however, the gas would
rapidly cool and thus this pressure support might come from galactic winds.
Fielding et al. (2017) and Lochhaas et al. (2018) show that supernovae winds
with reasonable mass loading efficiencies could shock the gas to distances
past the virial radius and account for the survival of cool gas at these large
radii. Using a more comprehensive model of the multiphase CGM, Fielding &
Bryan (2022) show that SF in the galactic disk can slow cooling and accretion
as part of a global preventive self-regulation mechanism. In addition, the
winds can transport cold clouds to large radii, consistent with these
constraints from our combined survey data.

##  6 Summary

Herein, we have examined the associations of galaxies with Ly$\alpha$
absorption $z<0.48$ to explore the spatial profile of this gas and the mass
dependence of the profile. Specifically, we have combined the CGM2 and CASBaH
H I measurement and constructed a catalog of 7244 absorber-galaxy pairs around
28 QSO sightlines (6589 absorber-galaxy pairs when we restrict our galaxy
sample to galaxies with $8<\log M_{\star}/M_{\odot}<10.5$). The CGM2 survey
has better sampling of galaxies at low impact parameter while CASBaH samples
galaxies out to 20 cMpc. This allows us to characterize the H I profile via
the covering fraction as a tracer of the gas.

  1. 1.

By modeling the covering fraction as a power-law with a mass dependent length
scale, we find good agreement with previous studies, such as T14, of our
clustering amplitude and power law slope parameters.

  2. 2.

In Section 3.1, we find the clustering scale has a mass dependence with a
power-law slope of $\beta^{2h}=0.08\pm 0.03$.

  3. 3.

We compare the slope of our absorber-galaxy bias to the galaxy-dark matter
bias of Tinker et al. (2010). The absorber-galaxy bias is a steeper function
of galaxy mass than the galaxy-dark matter bias. However, this difference is
only significant at a sub-$2\sigma$ level.

  4. 4.

We model the data with an exclusionary two-component model where we adopt an
inner-CGM Gaussian profile to describe the data at smaller impact parameters
and the customary two-halo single power-law model at larger impact parameters.
This model faithfully reproduces the data for galaxies
$M_{\star}>10^{8}M_{\odot}$.

  5. 5.

The two component model allows us to calculate the crossover radius, $R_{\rm
cross}(M_{\star})$, where the models are equal. $R_{\rm cross}(M_{\star})$
represents a soft upper estimate of the furthest impact parameter needed to
optimally fit the inner CGM component. We then use $R_{\rm cross}$ as an
estimate of the extent of the CGM and find $R_{\rm cross}(M_{\star})\approx
2\pm 0.6R_{\rm vir}$ for galaxies $10^{8}\leq M_{\star}/M_{\odot}\leq
10^{10.5}$. Additionally, we find excellent agreement between $R_{\rm
cross}(M_{\star})$ and the splashback radius, $R_{\rm sp}$ for galaxies in
this mass range.

##  7 Acknowledgments

MCW, KT, and JKW acknowledge support for this work from NSF-AST 1812521, NSF-
CAREER 2044303, the Research Corporation for Science Advancement, grant ID
number 26842. Support for the CASBaH HST programs HST-GO-11741 and HST-
GO-13846 was provided through grants from the Space Telescope Science
Institute under NASA contract NAS5-26555.

Support for the CASBaH HST programs HST-GO-11741 and HST-GO-13846 was provided
through grants from the Space Telescope Science Institute under NASA contract
NAS5-26555.

The CGM2 Survey would not have been possible without the substantial
contributions from a dedicated group of UW undergraduate Student Quasar
Absorption Diagnosticians, the Werk SQuAD, with over 50 individual
undergraduate research assistants since 2016. The SQuAD confirmed all auto-
fitted galaxy spectroscopic redshifts by eye, identified absorption systems
along every quasar line of sight, and measured various spectroscopic
properties (e.g. SFRs) of the nearly 1000 galaxies included in the survey. We
are deeply grateful to work with such motivated and enthusiastic students.

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