# Kinematics of MgII Absorbers from the Redshift-space Distortion Around
Massive Quiescent Galaxies

Ying Zu1,2
1Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China
2Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao
Tong University, Shanghai 200240, China E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The kinematics of MgII absorbers is the key to understanding the origin of
cool, metal-enriched gas clouds in the circumgalactic medium of massive
quiescent galaxies. Exploiting the fact that the cloud line-of-sight velocity
distribution is the only unknown for predicting the redshift–space distortion
(RSD) of MgII absorbers from their 3D real–space distribution around galaxies,
we develop a novel method to infer the cool cloud kinematics from the
redshift–space galaxy–cloud cross–correlation $\xi^{s}$. We measure $\xi^{s}$
for ${\sim}10^{4}$ MgII absorbers around ${\sim}8{\times}10^{5}$ CMASS
galaxies at $0.4{<}z{<}0.8$. We discover that $\xi^{s}$ does not exhibit a
strong Fingers-of-God effect, but is heavily truncated at velocity
${\sim}300\,\mathrm{km}\,s^{-1}$. We reconstruct both the redshift and
real–space cloud number density distributions inside haloes, $\xi^{s}_{1h}$
and $\xi_{1h}$, respectively. Thus, for any model of cloud kinematics, we can
predict $\xi^{s}_{1h}$ from the reconstructed $\xi_{1h}$, and
self–consistently compare to the observed $\xi^{s}_{1h}$. We consider four
types of cloud kinematics, including an isothermal model with a single
velocity dispersion, a satellite infall model in which cool clouds reside in
the subhaloes, a cloud accretion model in which clouds follow the cosmic gas
accretion, and a tired wind model in which clouds originate from the galactic
wind–driven bubbles. All the four models provide statistically good fits to
the RSD data, but only the tired wind model can reproduce the observed
truncation by propagating ancient wind bubbles at
${\sim}250\,\mathrm{km}\,s^{-1}$ on scales ${\sim}400\,h^{-1}\mathrm{kpc}$.
Our method provides an exciting path to decoding the dynamical origin of metal
absorbers from the RSD measurements with upcoming spectroscopic surveys.

###### keywords:

galaxies: evolution — galaxies: formation — intergalactic medium — quasars:
absorption lines

††pubyear: 2021††pagerange: Kinematics of MgII Absorbers from the Redshift-
space Distortion Around Massive Quiescent Galaxies–Kinematics of MgII
Absorbers from the Redshift-space Distortion Around Massive Quiescent Galaxies

##  1 Introduction

The circumgalactic space in dark matter haloes is permeated by multi-phase gas
with temperatures ranging from $10^{2}$ to $10^{8}$ K and densities between
$10^{-6}$ to $1$ cm-3 (Tumlinson et al., 2017). One of the most intriguing
puzzles of the circumgalactic medium (CGM) is the apparent abundance of cool
(${\sim}10^{4}$ K), metal-enriched gas clouds (e.g., detected via the MgII
absorption doublets in rest-frame UV along quasar sightlines; Bergeron &
Boissé, 1991; Steidel & Sargent, 1992; Steidel et al., 1994; Churchill et al.,
1999), embedded in the hot ($T{>}10^{6}\,$K) gaseous corona (Spitzer, 1956;
Anderson et al., 2013) surrounding massive quiescent galaxies (Thom et al.,
2012; Prochaska et al., 2013; Werk et al., 2014; Keeney et al., 2017).
Particularly, it is unclear whether the cool clouds are governed by random,
inflowing, or outflowing motion (Lan & Mo, 2019). In this paper, we explore
the kinematics of MgII absorbers around a large sample of luminous red
galaxies (LRGs) observed by the Baryon Oscillation Spectroscopic Survey (BOSS;
Dawson et al., 2013) at $z{\sim}0.55$, in hopes of paving the path for a
self–consistent redshift–space distortion (RSD) modelling of metal absorbers
with upcoming spectroscopic surveys like the DESI (Levi et al., 2019) and PFS
(Takada et al., 2014).

While the hot corona can be naturally explained by the shock-heating of
inflowing gas to the virial temperature of massive dark matter haloes
(Birnboim & Dekel, 2003; Kereš et al., 2005; Dekel et al., 2009), the physical
origin of the cool, metal-enriched CGM gas surrounding massive quiescent
galaxies remains largely elusive (Chen, 2017, and references therein). This is
mainly due to the lack of galactic winds driven by recent star formation in
those systems, and winds are the dominant means of transporting enriched
material into the CGM (Oppenheimer & Davé, 2006; Peeples & Shankar, 2011; Davé
et al., 2011; Hummels et al., 2013; Ford et al., 2014; Ma et al., 2016;
Fielding et al., 2017; Muratov et al., 2017; Fielding et al., 2020). Around
star-forming galaxies, cool gas clouds are commonly seen as blue-shifted
absorption and emission in galactic winds (see Heckman & Thompson, 2017;
Veilleux et al., 2020, and references therein). They could be dragged into the
CGM via entrainment in a fast-moving hot wind (but see Zhang et al., 2017),
accelerated by radiation or cosmic ray pressure (Murray et al., 2005; Kim &
Ostriker, 2018), or formed out of radiative cooling of the hot wind material
(Wang, 1995; Thompson et al., 2016). Without ongoing star-forming or quasar
(Cai et al., 2019) activities, it is unclear whether galactic winds remain a
viable channel for producing cool CGM gas in massive quiescent galaxies.

One possibility is that the galactic winds launched during the star formation
episodes prior to quenching may survive more than a few Gyrs. Recently,
Lochhaas et al. (2018) developed an 1D semi-analytic model of CGM as long-
lived bubbles blown by galactic winds, which then shock and sweep up ambient
halo gas as they propagate outward from the host galaxy (also see Samui et
al., 2008; Sarkar et al., 2015). They showed that the shocked wind can
radiatively cool to $10^{4}$ K and the cooled massive shells can travel to
several hundred $\mathrm{kpc}$s with $v{\sim}100{-}300\mathrm{km}\,s^{-1}$
within a span of $1{-}10$ Gyrs. Therefore, the bubble effectively “hangs” at a
large distance from the galaxy for a long time, explaining the presence of
wind material in the CGM surrounding quiescent galaxies. Since Lochhaas et al.
(2018) mainly focused on the Milky Way-sized haloes with
$M_{h}{\simeq}10^{12}M_{\odot}$ probed by COS-HALO (Tumlinson et al., 2013),
it is unclear if the results also applies to the LRG-size haloes with
$M_{h}{\gtrsim}10^{13}M_{\odot}$.

Apart from galactic winds, cool clouds could also appear in the CGM of massive
quiescent galaxies via cosmological gas accretion (Kereš et al., 2009;
Fumagalli et al., 2011; van de Voort et al., 2012), gas stripping from
satellite infall (Gauthier et al., 2010; Gauthier, 2013; Lee et al., 2021),
and condensation due to thermal instability in the hot medium (Mo & Miralda-
Escude, 1996; Maller & Bullock, 2004; Sharma et al., 2012; McCourt et al.,
2012; Voit & Donahue, 2015; Voit, 2018; Nelson et al., 2020). All three
scenarios above involve the infall of cool clouds through the hot halo gas
while experiencing effects of hydrodynamic drag and cloud evaporation
(Armillotta et al., 2017), modulo differences in the initial cloud conditions.
For example, Afruni et al. (2019) developed a semi-analytic model for cloud
accretion from the intergalactic medium, finding that the accretion of clouds
with cloud mass $M_{\mathrm{cl}}{\sim}10^{5}M_{\odot}$ could reproduce the
velocity distribution and column densities of the cool clouds observed by COS-
LRG (Chen et al., 2018; Zahedy et al., 2019). Meanwhile, those clouds will
evaporate before reaching the host galaxy, therefore unlikely rejuvenating any
star formation. However, there is no appreciable difference between the
observed metallicity distributions of the star–forming and quiescent galaxies
(Lehner et al., 2013; Berg et al., 2019). Therefore, without the enrichment
from stellar outflows, it is unclear how the newly accreted cool gas becomes
significantly metal-enriched while in pressure equilibrium with the hot gas.

The key to distinguishing between the wind vs. no–wind origins lies in the
kinematics of the cool clouds. The most common probe of the cloud kinematics
is the distribution of redshifts of MgII absorbers relative to their host
galaxies $\Delta z$. In particular, $\Delta z$ is usually interpreted in the
literature as the relative line-of-sight (LOS) velocity of MgII absorbers
(normalized by the speed of light $c$), so that $c\Delta
z\simeq{v}_{\mathrm{los}}(1+z)$. This approximation, however, ignores the fact
that $\Delta z$ also depends on the comoving distance between the two systems
along the LOS, $y$ , so that $c\Delta z={v}_{\mathrm{los}}(1+z)+H_{z}y$111$y$
and ${v}_{\mathrm{los}}$ are both vectors, which could be negative or positive
depending on whether it is pointed toward or away from the observer,
respectively., where $H_{z}$ is the Hubble parameter at $z$. The approximation
is more than adequate for comparing the LOS velocity dispersion of the MgII
absorbers to that of the dark matter (see e.g., Huang et al., 2016), but for a
more stringent test of cloud kinematic models we need to explicitly model the
redshift-space galaxy-absorber cross–correlation functions (Lanzetta et al.,
1998; Chen et al., 2005; Ryan-Weber, 2006; Wilman et al., 2007; Chen &
Mulchaey, 2009; Tejos et al., 2012; Borthakur et al., 2016). In this paper, we
will first reconstruct the 3D spatial distribution of cool clouds from the
observed projected LRG-MgII cross-correlation function, and then infer the
velocity distribution from the observed RSD of MgII absorbers around LRGs. A
similar method was first developed by Zu & Weinberg (2013) for inferring the
galaxy infall kinematics around massive galaxy clusters.

This paper is organized as follows. We describe the BOSS LRG and the MgII
absorber catalogues in §2. The RSD measurements are presented in §3. We
present our reconstruction of the 3D distribution of MgII absorbers within the
LRG haloes and extract the small-scale RSD signal of MgII absorbers in §4. We
compare the theoretical predictions of RSD from different kinematic models in
§5 before we summarise our results and look to the future in §6. Throughout
this paper, we assume the Planck cosmology (Planck Collaboration et al.,
2020). All the length and mass units in this paper are scaled as if the Hubble
constant is $100\,\mathrm{km}\,s^{-1}\mathrm{Mpc}^{-1}$. In particular, all
the separations are co-moving distances in units of $h^{-1}\mathrm{Mpc}$, and
the halo mass is in the unit of $h^{-1}M_{\odot}$. We use $\lg x{=}\log_{10}x$
for the base-$10$ logarithm.

##  2 Data

Figure 1:  The redshift distributions of CMASS LRGs (red) and MgII absorbers
(blue) between $z{=}0.4$ and $2.5$. The gray dashed vertical line indicates
the maximum redshift ($z{=}0.8$) of our cross-correlation analysis.

###  2.1 Luminous Red Galaxies from BOSS CMASS

We employ the BOSS CMASS galaxy sample derived from the Data Release 12 (DR12;
Alam et al., 2015) of Sloan Digital Sky Survey (SDSS; York et al., 2000). As
part of the SDSS-III (Eisenstein et al., 2011), BOSS (Dawson et al., 2013)
measured the spectra of 1.5 million galaxies over a sky area of ${\sim}$10,000
deg2 using the BOSS spectrographs (Smee et al., 2013; Bolton et al., 2012)
onboard the 2.5-m Sloan Foundation Telescope at the Apache Point Observatory
(Gunn et al., 1998, 2006). The BOSS LRGs were selected from the Data Release 8
(DR8; Aihara et al., 2011) of SDSS five-band imaging (Fukugita et al., 1996)
using two separate sets of colour and magnitude cuts for the LOWZ ($z{<}0.4$)
and CMASS ($z{>}0.4$) samples, respectively (Reid et al., 2016). For maximal
redshift overlap with the MgII absorbers, we select $823193$ CMASS galaxies
with $z\in[0.4,0.8]$ as our LRG sample for cross-correlating with the MgII
absorbers. We generate a random galaxy sample based on the standard CMASS
random galaxy catalogue provided in the public SDSS DR12 directory, by
applying the redshift distribution of the observed galaxy sample to the random
catalogue (red curve in Figure 1). To minimize the shot noise from the random,
we use 50 times more galaxies in the random than in the observed data set.

###  2.2 MgII Absorbers

We select a sample of $9088$ MgII absorbers within the CMASS footprint with
$z\in[0.4,0.8]$ from the JHU-SDSS MgII absorber
catalogue222https://www.guangtunbenzhu.com/jhu-sdss-metal-absorber-catalog
derived by Zhu & Ménard (2013), who applied an automatic absorption-line
detection algorithm to ${\sim}140000$ quasar spectra from the SDSS DR5
(Schneider et al., 2010) and DR12 (Pâris et al., 2017). Using a novel
nonnegative matrix factorization technique to estimate the quasar continua,
Zhu & Ménard (2013) detected $61974$ unique MgII absorbers between
$0.4<z<5.5$. The completeness and purity of the absorber catalogue are further
validated with the visually inspected Pittsburgh MgII catalogue (Quider et
al., 2011) derived from SDSS DR4. For cross-correlating with the CMASS
galaxies, we limit our sample to absorbers within the CMASS footprint and with
$z\in[0.4,0.8]$, yielding $9088$ MgII absorbers in the final sample. We do not
select absorbers by their equivalent width (EW), but will apply the values of
equivalent width as individual weights when computing the correlation
functions.

The MgII absorbers were detected along the sightlines of SDSS quasars, which
generally were given a higher target priority by the SDSS tiling algorithm
(Blanton et al., 2003) than galaxies. Therefore, in order to build a truly
random catalogue of MgII absorbers, we need to mimic the spectroscopic survey
strategy of SDSS, so that the conditional fibre assignment probability of
quasars at any angular separation away from a foreground galaxy is exactly
preserved in the random absorber catalogue. Since this conditional probability
is analytically intractable, we make use of the parent quasar catalogue that
Zhu & Ménard (2013) searched for MgII absorbers, and randomly assign redshifts
of the detected MgII absorbers to the angular coordinates of the observed
quasars. To remove any potential systematics associated with the difference
between DR7 and DR12 quasars, we assign the redshifts of DR7 absorbers to the
coordinates of DR7 quasars, and vice versa for DR12. This random generation
scheme also eliminates any systematic bias due to the intrinsic clustering of
quasar sightlines (Gauthier et al., 2010). Finally, we shuffle the redshifts
of the detected MgII absorbers ten times to build a random absorber catalogue
ten times the size of the data catalogue.

Figure 1 shows the redshift distributions of CMASS LRGs (red) and MgII
absorbers (blue) between $z{=}0.4$ and $2.5$. The dashed vertical line
indicates the maximum redshift ($z{=}0.8$) of our analysis, beyond which the
number density of CMASS LRGs declines rapidly. We note that the number density
of MgII absorbers stays relative flat within the vast redshift range between
$z{=}0.8$ and $1.8$, leaving huge statistical potential to future surveys that
will target a high number density of LRGs and emission lines galaxies within
the same redshift range.

##  3 Galaxy-cloud Cross-correlation Function in Redshift-space
$\mathbf{\xi_{\mathrm{\tt{LRG-MgII}}}^{s}}$

Figure 2: Schematic diagram of the redshift-space distortion of a MgII
absorber (dark-shaded blue circle) in the halo of an LRG (red ellipse) at
redshift $z$. The observer is at the bottom of the diagram, making the line-
of-sight direction vertically upward. The LOS velocity ${v}_{\mathrm{los}}$
shifts the cloud from its real-space LOS distance $y$ by
${v}_{\mathrm{los}}(1+z)/H_{z}$ to $r_{\pi}$ (light-shaded blue circle), where
$H_{z}$ is the Hubble parameter at $z$.

Before presenting the redshift-space cross-correlation function between CMASS
LRGs and MgII absorbers $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$, we introduce the
notations relevant to describing cloud kinematics in the galacto-centric
coordinates in Figure 2. The red ellipse indicates the position of the LRG at
redshift $z$, with the LOS direction pointing vertically upward. The dark-
shaded blue circle indicates the real-space position of a cool cloud, with 3D
distance $r$ away from the LRG, which can be further decomposed into a
projected separation $r_{p}$ and an LOS distance $y$. However, unlike $r_{p}$,
$y$ is not directly accessible through observations, because the relative
velocity of the cloud $v$ has an LOS component ${v}_{\mathrm{los}}$ that
displaces the observed position of the cloud (light-shaded blue circle) to
$r_{\pi}$ along the LOS by ${v}_{\mathrm{los}}(1+z)/H_{z}$. Obviously, to
recover the kinematics of the cool clouds (${v}_{\mathrm{los}}$), we need to
infer both the real and redshift–space distributions of the cool clouds ($y$
and $r_{\pi}$).

###  3.1 Correlation Function Measurements

Figure 3: Left: Redshift-space cross-correlation function between LRGs and
MgII absorbers; Right: Redshift-space auto-correlation function of LRGs. The
logarithmic values of the correlation functions are indicated by the colour
bars on top. The $0.25$ dex shift between the two colour bars accounts for the
difference in the large-scale bias between MgII absorbers and LRGs. The small-
scale Fingers-of-God effect is pronounced in the right panel, but weak in the
left panel.

We measure $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ using the Landy & Szalay (1993)
estimator,

$\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}(r_{p},r_{\pi})=\frac{D_{g}D_{c}-D_{g}R_{c}-R_{g}D_{c}+R_{g}R_{c}}{R_{g}R_{c}},$
(1)

where $D_{g}D_{c}$ and $R_{g}R_{c}$ are the number counts of data galaxy–data
cloud pairs and random galaxy–random cloud pairs at 2D separation
$(r_{p},r_{\pi})$, respectively. By the same token, $D_{g}R_{c}$ and
$R_{g}D_{c}$ indicate the corresponding number counts of data galaxy-random
cloud pairs and random galaxy-data cloud pairs. For computing pair counts
associated with the data clouds, we weigh each pair by the EW of the MgII
absorber (the sum of the doublet EWs). We also repeat our analysis in this
paper using measurements without the EW weights, and discover that the impact
of weights on our conclusions is negligible.

To compare the RSD of MgII absorbers with that of galaxies, we also measure
the redshift–space galaxy auto–correlation functions via

$\xi_{\mathrm{\tt{LRG-
LRG}}}^{s}(r_{p},r_{\pi})=\frac{D_{g}D_{g}-2D_{g}R_{g}+R_{g}R_{g}}{R_{g}R_{g}}.$
(2)

The difference between $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ and
$\xi_{\mathrm{\tt{LRG-LRG}}}^{s}$ on large scales should be a constant shift
in amplitude by a factor of $b_{g}/b_{c}$, where $b_{g}$ and $b_{c}$ are the
large-scale clustering biases of the LRGs and MgII absorbers, respectively. On
small scales, however, any difference between the two would be primarily
induced by the deviation of cloud kinematics from that of the luminous red
satellites within the haloes.

For reconstructing the 3D number density profile of the clouds, we measure the
projected galaxy–cloud cross–correlations $w_{p}$ from the same data and
random galaxy samples using an integration length of
$r_{\pi}^{\mathrm{max}}=40\,h^{-1}\mathrm{Mpc}$, so that

$w_{p}(r_{p})=\int_{-r_{\pi}^{\mathrm{max}}}^{+r_{\pi}^{\mathrm{max}}}\\!\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}(r_{p},r_{\pi})\,\mathrm{d}r_{\pi}.$ (3)

Finally, we estimate the measurement uncertainties on $\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}$ and $w_{p}$ using the standard Jackknife re–sampling method by
dividing the CMASS sky coverage into $200$ approximately equal-area regions
(Norberg et al., 2009).

###  3.2 Comparison between $\mathbf{\xi_{\mathrm{\tt{LRG-MgII}}}^{s}}$ and
$\mathbf{\xi_{\mathrm{\tt{LRG-LRG}}}^{s}}$

Figure 3 compares the RSD signature of the MgII absorbers (left) to that of
the galaxies (right) around LRGs, with the logarithmic values of the
correlation functions indicated by the respective color bars on top. We shift
the two color bars by $0.25$ index to account for the bias difference between
the two tracers, so that the Kaiser effect (Kaiser, 1987) on scales beyond
$r_{p}{\sim}5\,h^{-1}\mathrm{Mpc}$ looks similar between the two panels,
despite the much larger uncertainties in the measurement of
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$.

The most striking difference between the two 2D correlation functions is the
absence (presence) of a strong Fingers-of-God (FOG; Jackson, 1972) effect in
the RSD of MgII absorbers (galaxies) around LRGs. This is probably not
surprising, given that we know the velocity dispersion of MgII absorbers is
smaller than that of dark matter within the LRG haloes (Zhu et al., 2014;
Huang et al., 2016; Anand, Nelson, & Kauffmann, 2021; Huang et al., 2021).
However, it is unclear whether the lack of FOG is caused simply by a reduced
velocity dispersion, or some other kinematic features that may allow us to
distinguish between wind vs. no–wind origins of the cool clouds.

Figure 4: Comparison between the $\xi(r_{p},r_{\pi})/w_{p}(r_{p})$ profiles of
LRG–MgII cross–correlation (blue circles) and LRG auto–correlation (red
squares), within the LRG haloes (left main panel;
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$) and on large scales beyond the 1–halo
regime (right panels; $r_{p}{=}[0.75,1.25,1.75,3.50]\,h^{-1}\mathrm{Mpc}$),
respectively. For each panel, the bottom and upper x–axes indicate the LOS
separation using the relative velocity $\Delta V$ and the LOS distance
$r_{\pi}$, respectively. The two sets of profiles closely track each other at
large $r_{p}$ (right panels), but differ significantly at $r_{p}{=}0.26$ (left
panel), where a sharp truncation occurs at $\Delta
V{\sim}300\,\mathrm{km}\,s^{-1}$ (i.e.,
$r_{\pi}{\sim}0.34\,h^{-1}\mathrm{Mpc}$) in the LRG–MgII cross–correlation but
not in the LRG auto–correlation.

To look more closely into the small–scale differences in RSD, we calculate the
ratio between $\xi^{s}(r_{p},r_{\pi})$ and $w_{p}(r_{p})$ at fixed $r_{p}$,
and compare the two sets of ratios as functions of $r_{\pi}$ in Figure 4. In
essence, the $\xi^{s}(r_{p},r_{\pi})/w_{p}(r_{p})$ ratio profiles measure the
redshift–space LOS probability distribution function of clouds or galaxies
around LRGs at any $r_{p}$. In the left panel of Figure 4, we compare the
ratio profiles of the LRG–MgII cross–correlation (blue circles) and LRG
auto–correlation (red squares) at $r_{p}{\simeq}0.26\,h^{-1}\mathrm{Mpc}$
(i.e., $0.01\,h^{-1}\mathrm{Mpc}<r_{p}<0.5\,h^{-1}\mathrm{Mpc}$). We limit our
lowest $r_{p}$ bin below $r_{p}{=}0.5\,h^{-1}\mathrm{Mpc}$ because we want to
focus exclusively on the so–called “1-halo” term of the correlation signals —
the typical halo radius of the LRG host haloes is estimated to be
${\sim}0.5{-}0.6\,h^{-1}\mathrm{Mpc}$ from either fitting to the stacked MgII
absorption profiles (Zhu et al., 2014) or the weak lensing of massive
quiescent galaxies (Mandelbaum et al., 2016; Zu & Mandelbaum, 2016). We also
compare the two sets of ratio profiles computed at larger projected distances,
$r_{p}{=}[0.75,1.25,1.75,3.50]\,h^{-1}\mathrm{Mpc}$, in the four right panels.
In each panel, we label the bottom x–axis using the widely–used relative
velocity $\Delta V$, while showing the redshift–space LOS distance $r_{\pi}$
in the upper x–axis.

The four right panels of Figure 4 indicate that the LOS distribution of MgII
absorbers is in good agreement with that of galaxies at
$r_{p}{>}0.6\,h^{-1}\mathrm{Mpc}$, i.e., in the “2–halo” regime. This is
unsurprising, because the cool clouds should follow the distribution of
galaxies on large scales regardless of wind or no–wind origins. Within the LRG
haloes (left panel of Figure 4), however, the LOS distribution of MgII
absorbers is drastically different from that of galaxies, exhibiting a sharp
truncation at $\Delta V{\sim}300\,\mathrm{km}\,s^{-1}$ (i.e.,
$r_{\pi}{\sim}3.4\,h^{-1}\mathrm{Mpc}$). This sharp truncation explains the
missing of FOG effect in the RSD of MgII absorbers, and can not be reproduced
simply by reducing the velocity dispersion of the cool clouds (as will be
demonstrated later in § 5). We note that the luminous red satellites do not
exhibit the typical satellite kinematics within the LRG haloes — they are
subject to stronger dynamic friction and likely had an earlier infall,
therefore travelling more slowly than the avearge satellites. We thus expect
the cross–correlation between the LRGs and typical galaxies would produce an
even stronger FOG effect than the LRG auto–correlation. We also note that
Kauffmann et al. (2017) also measured the redshift–space cross–correlation
function between the CMASS LRGs and MgII absorbers. However, they used a
different method of generating random catalogues, a different estimator than
Equation 1, and a different binning, so it is difficult to make direct
comparison between Figure 4 and their results. To extract the kinematic
information encoded in the $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ profile at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$, in the next step we will reconstruct the
real–space LOS distribution of MgII absorbers around the LRGs from the
measurements of $w_{p}(r_{p})$.

##  4 Spatial Distribution of Cool Clouds Around LRGs

Figure 5: Left panel: Projected EW profiles of MgII absorbers around LRGs.
Gray circles with errorbars are the measurements of the total absorption from
stacked quasar spectra by Zhu et al. (2014), while red squares with errorbars
indicate our measurements from the projected cross–correlation function
between LRGs and the resolved MgII absorbers. Solid curves are the
best–fitting predictions for the total (blue) and resolved (red) absorption
signals, with dashed and dotted curves indicating the 1–halo and 2–halo
components, respectively. The ratio profiles between the data measurements and
best-fitting predictions are shown in the bottom panel for the total (gray
circles) and resolved (red squares) absorption signals, respectively. Right
panel: Similar to the left panel, but for the 3D isotropic cross–correlation
functions between LRGs and MgII absorbers. The bottom panel shows the
predicted radius dependence of absorber detection fraction in the Zhu & Ménard
(2013) catalogue.

###  4.1 Predict Projected EW Profile $\mathbf{\left\langle
W\right\rangle(r_{p})}$

Zhu et al. (2014) measured the projected MgII EW profile around the same
sample of LRG galaxies, from the flux decrements induced in the stacked quasar
spectra by MgII absorbers at the redshifts of the foreground LRGs. The (Zhu et
al., 2014) profile is subsequently confirmed by other stacked MgII absorption
measurements (also see Pérez-Ràfols et al., 2015; Lan & Mo, 2018), and is
roughly consistent with the stacked H$\alpha$ emission measurements,
suggesting the metal absorption and H$\alpha$ emission likely originate from
the same CGM gas (Zhang et al., 2016; Zhang, Zaritsky, & Behroozi, 2018). The
stacked signal of Zhu et al. (2014) is analogous to our measurement of the
projected cross–correlation between LRGs and MgII absorbers $w_{p}(r_{p})$,
but with one key difference. That is, by stacking the quasar spectra, Zhu et
al. (2014) included the contribution from all the MgII absorbers, while our
$w_{p}$ measurement only includes the spectrally resolved absorbers. Comparing
to our method, the stacking technique is more useful for characterizing the
total gas mass density profile, but at the expense of the individual redshifts
of gas clouds, therefore less useful for inferring cloud kinematics. To
reconstruct the spatial distribution of the resolved MgII absorbers, we will
utilize the stacked absorption signal to infer the total gas density profile,
which can be separated into contributions from the non-resolved vs. resolved
MgII absorbers by fitting to the $w_{p}$ measurement assuming a
radius–dependent detection fraction of the resolved absorbers.

To facilitate our joint analysis with the stacked MgII EW profile measured by
Zhu et al. (2014), we convert our $w_{p}(r_{p})$ measurement into a projected
MgII EW profile $\left\langle W\right\rangle$,

$\left\langle
W\right\rangle(r_{p})=w_{p}(r_{p})\frac{\mathrm{d}W}{\mathrm{d}r_{\pi}},$ (4)

where $\mathrm{d}W/\mathrm{d}r_{\pi}$ is the average EW of MgII absorption per
$h^{-1}\mathrm{Mpc}$ along the LOS. Ideally, this quantity should be computed
by summing the EWs of all the MgII absorbers within the redshift range probed.
However, the Zhu & Ménard (2013) MgII absorber catalogue suffers severe
incompleteness at the low EW end, and an extra redshift–dependent
incompleteness due to strong sky lines and sensitivity variation of the SDSS
spectrographs. Therefore, we empirically infer the value of
$\mathrm{d}W/\mathrm{d}r_{\pi}{=}2.6\times 10^{-4}\,h\mbox{\AA}/\mathrm{Mpc}$,
from the ratio between the stacked MgII EW profile measured by Zhu et al.
(2014) and our measurement of $w_{p}$ on scales beyond
$3\,h^{-1}\mathrm{Mpc}$.

The left panel of Figure 5 compares the $\left\langle W\right\rangle$ profiles
inferred from $w_{p}$ via Equation 4 (red squares) and from the stacking
technique of Zhu et al. (2014) (gray circles). The two measurements are
roughly consistent on scales above $0.5\,h^{-1}\mathrm{Mpc}$ — this is by
design because we have adjusted the value of $\mathrm{d}W/\mathrm{d}r_{\pi}$
to match the two. However, the two profiles start to diverge below
$0.5\,h^{-1}\mathrm{Mpc}$. In particular, the $w_{p}$–inferred $\left\langle
W\right\rangle$ of resolved absorbers exhibits a flat trend with decreasing
$r_{p}$, while the stacked $\left\langle W\right\rangle$ of total absorption
is consistent with a projected NFW (Navarro-Frenk-White; Navarro et al., 1997)
profile (Zhu et al., 2014). The discrepancy between the two $\left\langle
W\right\rangle$ profiles on small scales implies that there is a radius-
dependent detection inefficiency of the Zhu & Menard catalogue, so that MgII
absorbers that are closer to the LRGs are less likely to be detected. At fixed
EW, the close proximity of quasar sightlines to LRGs shoud not affect the
detection probability of MgII absorbers, because the scattered light from the
LRGs into the quasar fibres is minimal. We speculate that the radius-dependent
inefficiency is induced by the fact that the average EW of the MgII absorbers
decreases from the outer CGM to the center of the LRGs. Since the detection
probability decreases rapidly at $\mathrm{EW}<2\mbox{\AA}$ (see figure 7 of
Zhu & Ménard (2013)), the stacked MgII absorption profile is much steeper than
the number density profile of the resolved MgII absorbers. The overall shape
of our $w_{p}$–inferred $\left\langle W\right\rangle$ profile is consistent
with the excess absorber number density profile measured by Anand, Nelson, &
Kauffmann (2021), who utilized a larger MgII absorber catalogue derived from
the SDSS DR16.

For the purpose of our analysis, it is more convenient to work with
correlation functions than density profiles. Therefore, instead of
reconstructing the radial number density profile of MgII absorbers $n(r)$, we
focus on the “1–halo” component of the 3D isotropic cross–correlation function
between the LRGs and absorbers $\xi_{1h}(r)$. The two are related via

$\xi_{1h}(r)=\frac{n(r)-\overline{n}}{\overline{n}},$ (5)

where $\overline{n}$ is the mean number density of cool clouds.

Following Zhu et al. (2014), we assume that the total cool gas distribution
follows the dark matter distribution within haloes, which we model as

$\xi_{1h}^{\mathrm{gas}}(r)\equiv\xi_{1h}^{\mathrm{dm}}(r)=\frac{\rho_{\mathrm{NFW}}(r\mid
M_{h},c)}{\rho_{m}}f_{\mathrm{trans}}(r)-1,$ (6)

where $\rho_{\mathrm{NFW}}(r\mid M_{h},c)$ is the NFW density profile of a
dark matter halo with halo mass $M_{h}$ and concentration $c$, $\rho_{m}$ is
the mean density of the Universe, and $f_{\mathrm{trans}}$ describes the
so–called “spashback” transition near the halo boundary (Diemer & Kravtsov,
2014),

$f_{\mathrm{trans}}(r)=\left[1+\left(\frac{r}{1.495\,r_{200m}}\right)^{4}\right]^{-2}.$
(7)

We emphasize that although the result from Zhu et al. (2014) shows that an NFW
profile provides an adequate description of the average cool gas density
profile, the spatial distribution of the cool gas clouds in individual haloes
could deviate significantly from the dark matter, depending on the dynamical
state of the cool gas. For example, Wild et al. (2008) discovered that quasars
could destroy MgII absorbers up to comoving distances of ${~{}}800$ $kpc$
along their lines of sight, so that the spatial distribution of cool clouds
could depend strongly on the duty cycle of the supermassive blackholes in the
galactic centre. However, it is plausible that the gas density profiles
averaged over many haloes of various dynamical states may converge to an
NFW–like profile. This possibility can be tested in the future against the
increasingly fine-grained hydrodynamic simulations of the CGM (Peeples et al.,
2019; Hafen et al., 2019; Suresh et al., 2019; Hummels et al., 2019; Fielding
et al., 2020).

For describing the resolved MgII absorbers in our sample, we further include a
radius–dependent inefficiency

$\xi_{1h}(r)=\frac{\rho_{\mathrm{NFW}}(r\mid
M_{h},c)}{\rho_{m}}f_{\mathrm{trans}}(r)f_{\mathrm{det}}(r)-1,$ (8)

where $f_{\mathrm{det}}$ is the detection fraction

$f_{\mathrm{det}}(r)=\left(\frac{r}{\kappa\,r_{200m}}\right)^{\alpha},$ (9)

and we set $f_{\mathrm{det}}$ to be unity when $r{>}\kappa\,r_{200m}$.

We model the “2–halo” component of the 3D isotropic LRG–MgII cross–correlation
as

$\xi_{2h}(r)=b_{g}b_{c}\xi_{mm}(r)f_{\mathrm{ex}}(r),$ (10)

where $\xi_{mm}$ is the dark matter auto–correlation function. Since we do not
need to distinguish between $b_{g}$ and $b_{c}$, we merge them to form a
single parameter $b{\equiv}b_{g}b_{c}$. The last term on the right-hand side,
$f_{\mathrm{ex}}(r)$, describes the so–called “halo exclusion” effect, which
arises from the fact that the centre of one halo cannot sit within the virial
radius of another. In particular, $f_{\mathrm{ex}}(r)$ is unity at
$r{>}r^{\mathrm{max}}_{200m}$, as there is no halo exclusion when the distance
is larger than twice the radius of the largest haloes
$r^{\mathrm{max}}_{200m}$. By the same token, $f_{\mathrm{ex}}(r)$ declines to
zero at some characteristic distance that corresponds to twice the radius of
the smallest haloes.

For analytically solving for $f_{\mathrm{ex}}$ at any given distance, we need
to explicitly integrate over all pairs of haloes that are too small to run
into each other when separated by that distance(e.g., see equation B10 of
Tinker et al., 2005). However, such analytic calculation can only be performed
if the halo occupation distributions (HODs) of both the LRGs and MgII
absorbers are assumed a priori. Since a comprehensive HOD modelling of the
MgII absorbers is beyond the scope of this paper, for our first–cut analysis
we choose to use a simple error function to mimic the behavior of halo
exclusion

$f_{\mathrm{ex}}(r)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{r-\beta\,r_{200m}}{\sigma_{\mathrm{ex}}}\right)\right],$
(11)

where $\beta\,r_{200m}$ is the characteristic scale above which
$f_{\mathrm{ex}}$ asymptotes to unity, and $\sigma_{\mathrm{ex}}$ describes
the width of the transition of $f_{\mathrm{ex}}$ from unity to zero, which
depends on the widths of the halo mass distributions of the LRGs and MgII
absorbers. We have verified that for the HOD of a typical stellar–mass
thresholded sample in (Zu & Mandelbaum, 2015), the analytic halo exclusion
profile can be reasonably well described by Equation 11.

We derive the overall $\xi(r)$ signals by directly summing the two components,
so that

$\xi^{\mathrm{gas}}(r)=\xi_{1h}^{\mathrm{gas}}(r)+\xi_{2h}(r),$ (12)

for the LRG–gas cross–correlation, and

$\xi(r)=\xi_{1h}(r)+\xi_{2h}(r),$ (13)

for the LRG–MgII absorbers cross–correlation, respectively. Note that we adopt
the same “2–halo” component for the total and resolved absorption, assuming
that the EW–dependence of cloud bias $b_{c}$ is weak (Tinker & Chen, 2008;
Gauthier et al., 2009). Finally, we integrate the $\xi$ profiles along the LOS
to obtain $w_{p}(r_{p})$ via Equation 3, and then multiply by
$\mathrm{d}W/\mathrm{d}r_{\pi}$ to predict the $\left\langle
W\right\rangle(r_{p})$ profiles.

###  4.2 Reconstruct 3D Galaxy–cloud Cross–correlation

In total, we have seven free parameters, including
$\\{M_{h},c,b,\kappa,\alpha,\beta,\sigma_{\mathrm{ex}}\\}$. We perform simple
$\chi^{2}$–minimization fitting to the observed total and resolved
$\left\langle W\right\rangle$ profiles, using the diagonal errors provided by
Zhu et al. (2014) for the stacked signal and the full covariance matrix
measured from Jackknife re-sampling for the resolved profile, respectively. We
enforce the 2–halo term to be zero at $r{<}0.2h^{-1}\mathrm{Mpc}$ during the
fit, in order to eliminate the false solution that artificially boosts the
2–halo term on very small scales to fit the data. We find the best–fitting
values to be
$\\{\lg\,M_{h}{=}13.19,c{=}5.1,b{=}1.15,\kappa{=}0.63,\alpha{=}1.2,\beta{=}1.15,\sigma_{\mathrm{ex}}{=}0.14\\}$,
and the best–fitting predictions of the total (blue solid) and resolved (red
solid) signals are shown in the left main panel of Figure 5. The bias value is
slightly higher than predicted for haloes with $\lg\,M_{h}{=}13.19$, probably
because a good fraction of the LRGs are satellite galaxies inside massive
haloes (Zu, 2020). Both predictions are in excellent agreement with the
measurements, as demonstrated by the ratios between data and predictions in
the left bottom panel of Figure 5.

In many observations, MgII absorbers detected at small impact parameters
($r_{p}{<}0.2\,h^{-1}\mathrm{Mpc}$) to a foreground galaxy are usually
considered part of the CGM of that galaxy, but some of them should be
interlopers from other galaxies, i.e., part of the “2–halo” component. To
obtain an estimate of the interloper fraction, we show the best-fitting 1–halo
components of the total (blue dashed) and resolved (red dashed) absorption in
the left panel of Figure 5. For the stacked measurement, the strong absorption
signal at $r_{p}{<}0.2\,h^{-1}\mathrm{Mpc}$ is almost entirely contributed by
the gas clouds within the LRG haloes (i.e., the blue dashed curve is much
higher than the blue dotted curve). For the resolved absorbers, however,
${\sim}1/3{-}1/5$ of the strong MgII absorbers with impact parameters
(${<}0.2\,h^{-1}\mathrm{Mpc}$) are interlopers from the CGM of other galaxies
(compare the red dashed curve with the blue dotted curve). This is roughly
consistent with the expectation from hydrodynamic simulations, where Ho et al.
(2020) found that a ${\pm}500\,\mathrm{km}\,s^{-1}$ cut by
${v}_{\mathrm{los}}$ could include many gas clouds physically outside of the
virial radius. Fortunately, our cross–correlation method is able to
statistically remove the impact of those interloper on our analysis.

In the right panel of Figure 5, we show the best–fitting 3D isotropic
cross–correlation functions for the total signal (blue solid), which is
further decomposed into 1–halo (blue dashed; Equation 6) and 2–halo (blue
dotted; Equation 10) components; The 1–halo component of the best–fitting
$\xi$ profile of the resolved clouds is indicated by the red dashed curve
(Equation 8). In the bottom sub-panel, we show the best–fitting profile of
$f_{\mathrm{det}}$, indicating that the relatively flat trend of $\left\langle
W\right\rangle(r_{p})$ below $r_{p}{\sim}0.3\,h^{-1}\mathrm{Mpc}$ is due to
the cancellation of the steep increase in total gas density by the rapid
decline of detection sensitivity toward small radii.

###  4.3 Subtract Two–halo Term from $\mathbf{\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}}$

Figure 6: Comparison between the redshift–space LRG–MgII cross–correlation
functions at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ (blue) vs.
$r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$ (red). Circles (squares) with errorbars are
the observed $\xi^{s}$ at $r_{p}{=}0.26$ ($r_{p}{=}0.75$). Dashed and dotted
curves indicate the 1–halo and 2–halo components of $\xi^{s}(r_{\pi}|r_{p})$,
respectively, predicted from $\xi(r{=}\sqrt{r_{p}^{2}{+}r_{\pi}^{2}})$, i.e.,
assuming zero peculiar velocities. We expect the 2–halo component of the
observed $\xi^{s}$ signal at $r_{p}{=}0.26$ can be well approximated by the
$\xi^{s}$ signal observed at $r_{p}{=}0.75$.

The kinematics of cool clouds within the LRG haloes is encoded in the
redshift–space cross–correlation function between the LRGs and MgII absorbers
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ at small $r_{p}$. From the modelling of
$w_{p}$ we infer that the typical radius of the LRG haloes is
${\sim}0.6\,h^{-1}\mathrm{Mpc}$ (for haloes with $\lg\,M_{h}{=}13.19$).
Therefore, we will focus on the average RSD signal measured between
$r_{p}{=}0.01\,h^{-1}\mathrm{Mpc}$ and $0.5\,h^{-1}\mathrm{Mpc}$, i.e.,
$\xi^{s}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$.

Ideally, one would model the full RSD signal on both small and large
$r_{\pi}$, i.e., including both the 1–halo and 2–halo components of
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$, as
was previously done in, e.g., Zu & Weinberg (2013). However, since we are only
concerned with the CGM kinematics inside the haloes, we will isolate the
1–halo term of the $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ signal by subtracting
the 2–halo term from the full RSD signal observed at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. The isolated 1–halo RSD can then be
modelled as the convolution between $\xi_{1h}$, the quantity we have
reconstructed in § 4.2, and the LOS velocity distribution of the cool clouds,
the quantity we want to infer in this paper.

Figure 7: Top panel: Blue circles and red squares are the redshift–space
LRG–MgII cross–correlation functions measured at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ and $0.75\,h^{-1}\mathrm{Mpc}$,
respectively. Red solid curve is our fiducial prediction of $\xi^{s}_{2h}$
from the best–fitting powered exponential function (Equation 14). Red dotted
and dashed curves are the upper and lower bounds of $\xi^{s}_{2h}$,
respectively, inferred from adjusting the small–scale slope parameter of
Equation 15. Bottom panel: The 1–halo components of
$\xi^{s}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$, obtained by subtracting
the predicted $\xi^{s}_{2h}$ curves shown in the top panel from the observed
signal at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. Black circles with errorbars
indicate our fiducial estimate of the
$\xi^{s}_{1h}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$ profile.

Accurate subtraction of the 2–halo term requires theoretically modelling the
large-scale infall of cool clouds outside the halo radius, which is highly
challenging and therefore beyond the scope of this paper. For a first–cut
analysis, we derived the 2–halo term
$\xi^{s}_{2h}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$ empirically, by
making use of the RSD signal observed at $r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$.
The rationale is as follows. Firstly, without the peculiar velocities, the
cross–correlation function along the LOS is simply
$\xi^{s}_{v{=}0}(y|r_{p})=\xi(\sqrt{y^{2}{+}r_{p}^{2}})$. Since $\xi(r)$ is a
slow–varying function of $r$ on large scales, the underlying cloud
distribution along the LOS should be similar between $r_{p}{=}0.26$ and $0.75$
for $y{\gtrsim}2\,h^{-1}\mathrm{Mpc}$. Secondly, the LOS velocity distribution
is the projection of the isotropic 2D velocity distribution along the LOS. The
two average “sightlines” at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ and
$0.75\,h^{-1}\mathrm{Mpc}$ are almost parallel at the mean redshift of the LRG
sample ($z{\sim}0.55$), differing only by 1 arc min. Therefore, the LOS
velocity distributions of clouds beyond the halo radius should be almost the
same between the two $r_{p}$ bins. Combining the two factors together, we
expect that the 2–halo components of the redshift–space cross–correlation
signals are similar between the two adjacent $r_{p}$ bins at
$0.26\,h^{-1}\mathrm{Mpc}$ and $0.75\,h^{-1}\mathrm{Mpc}$.

Figure 6 illustrates the rationale behind our expectation that the two
adjacent $r_{p}$ bins should share similar 2–halo terms of the RSD signal
around LRGs. The dashed and dotted curves indicate the 1–halo and 2–halo
contributions to $\xi^{s}_{v{=}0}(r_{\pi}|r_{p})$, predicted by the
best–fitting $\xi(r)$ in the absence of peculiar velocities, at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ (blue) and $0.75\,h^{-1}\mathrm{Mpc}$
(red), respectively. Circles with errorbars are the measured redshift–space
correlation functions $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ for the two $r_{p}$
bins (also shown in Figure 4). Unsurprisingly, the 1–halo term (dashed blue
curve) at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ is ${\sim}20$ times higher than
that at $r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$ (dashed red curve), driving the
strong tophat–like enhancement of the observed redshift–space correlation on
scales below $r_{\pi}{\simeq}4\,h^{-1}\mathrm{Mpc}$ (i.e., $\Delta
V{\simeq}300\,\mathrm{km}\,s^{-1}$) for the $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$
bin. Furthermore, the 2–halo terms of the two $r_{p}$ bins are indeed
comparable when peculiar velocities are turned off (dotted curves). More
important, the observed redshift–space correlations of the two $r_{p}$ bins
are highly consistent with each other at $r_{\pi}{>}4\,h^{-1}\mathrm{Mpc}$.
This consistency is very encouraging — the underlying 2–halo component of the
observed $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ signal at $r_{p}{=}0.26$ (blue
circles) should be well approximated by the total observed
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ signal at $r_{p}{=}0.75$ (red squares).

Therefore, in order to estimate the underlying 1–halo term at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$, we fit an analytic curve to the total
observed $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ signal at
$r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$ and subtract it from the total observed
signal at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. Zu & Weinberg (2013) discovered
that the $\xi^{s}(r_{\pi}|r_{p})$ profiles can be well described by a powered
exponential function

$\xi^{s}(r_{\pi}|r_{p})=\xi^{s}_{0}\exp\left\\{-\left|\frac{r_{\pi}}{r_{\pi,c}}\right|^{\beta}\right\\},$
(14)

where $r_{\pi,c}$ is the characteristic $r_{\pi}$ at which $\xi^{s}$ declines
to $1/e$ of its maximum value $\xi^{s}_{0}$ at $r_{\pi}{=}0$, and $\beta$ is
the shape parameter that controls the rate of the decline. We apply Equation
14 to the RSD signal at $r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$, and the results
are shown in Figure 7.

In the top panel of Figure 7, blue circles (red squares) with errorbars are
the observed total $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ signals at
$r_{p}{=}0.26\,(0.75)\,h^{-1}\mathrm{Mpc}$. Red solid curve is the prediction
from Equation 14 using the best–fitting parameters of $r_{\pi,c}{=}6.71$,
$\beta{=}1.36$, and $\xi^{s}_{0}{=}41.67$. We omit the data point from the
lowest $r_{\pi}$ bin during the fit, where we expect a relatively larger
discrepancy between the 2–halo term at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ and
the observed signal at $r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$ than other $r_{\pi}$
bins (see Figure 6). We also show two other model curves (dotted and dashed)
in which we use a different function form

$\xi^{s}(r_{\pi}|r_{p})=\xi^{s}_{0}\left|\frac{r_{\pi}}{r_{\pi,c}}\right|^{\alpha}\exp\left\\{-\left|\frac{r_{\pi}}{r_{\pi,c}}\right|^{\beta}\right\\},$
(15)

where the extra parameter $\alpha$ modifies the slope at small $r_{\pi}$. We
adjust the two model fits so that the dashed and dotted curves roughly
delineate the lower and upper bounds of the observed data points at
$r_{\pi}{<}5\,h^{-1}\mathrm{Mpc}$, respectively.

We show the derived 1–halo components of the $\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}$ at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ in the bottom panel of
Figure 7, by subtracting each of the three model curves of $\xi_{2h}^{s}$
shown in the top panel from the observed $\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$
data points. Circles with errorbars represent our fiducial estimates for
$\xi^{s}_{1h}(r_{\pi}|r_{p}{=}0.26)$, obtained from the subtraction of the
best fit using Equation 14, while the dashed and dotted curves from the
subtraction of the respective curves predicted by Equation 15. We emphasize
that, despite the large discrepancies in $\alpha$, all three estimates of
$\xi^{s}_{1h}$ exhibit a sharp truncation at
$r_{\pi}{\sim}3{-}4\,h^{-1}\mathrm{Mpc}$, echoing the missing–FOG effect seen
in Figure 3. This also suggests that the sharp truncation is physical, and
most likely driven by some kinematic feature in the velocity distribution of
cool clouds. In the next section, we will describe the fiducial measurements
of $\xi^{s}_{1h}(r_{\pi}|r_{p}{=}0.26)$ (black open circles with errorbars in
the bottom panel of Figure 7) using four different cloud kinematic models. For
our first–cut analysis, we directly apply the Jackknife uncertainties of
$\xi^{s}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$ for the error matrix of
the fiducial measurements, i.e., assuming zero uncertainty from the
subtraction of the 2–halo term. Despite that the errorbars are underestimated,
they are roughly consistent with the allowed range sandwiched between the
lower and upper bounds, indicated by the gray dotted and dashed curves,
respectively.

##  5 Comparison of Cloud Kinematic Models

Armed with the 1–halo components of the 3D real–space ($\xi_{1h}(r)$) and
redshift–space ($\xi^{s}_{1h}(r_{\pi}|r_{p}{=}0.26\,h^{-1}\mathrm{Mpc})$;
hereafter shortened as $\xi^{s}_{1h}(r_{\pi})$) cross–correlation functions
between the LRGs and MgII absorbers, we now investigate different cloud
kinematic models by examining their predicted RSD signals and comparing to the
observations. In particular, we compute $\xi^{s}_{1h}$ by convolving
$\xi_{1h}(r)$ with the LOS velocity distribution predicted by each kinematic
model

$\displaystyle\xi^{s}_{1h}(r_{\pi}\mid r_{p})$ $\displaystyle=$
$\displaystyle\frac{H_{z}}{1+z}\int_{-\infty}^{+\infty}\left[\xi_{1h}\big{(}\sqrt{r_{p}^{2}+y^{2}}\big{)}\right]\times\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(16) $\displaystyle
p\big{(}{v}_{\mathrm{los}}{=}\frac{H_{z}}{1+z}\left(r_{\pi}-y\right)\mid
r_{p},y,\bm{\theta}\big{)}\,\mathrm{d}y,$

where $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$,
$p({v}_{\mathrm{los}}|r_{p},y,\bm{\theta})$ is the LOS velocity distribution
of cool clouds at fixed $y$, and $\bm{\theta}$ is the parameter set of each
kinematic model.

Following the discussion in § 1, we consider four different kinematic models:

  * •

Isothermal model described by a single LOS velocity dispersion
${\sigma}_{\mathrm{los}}$.

  * •

Satellite infall model in which the MgII absorbers reside in the CGM of
satellite galaxies and follow the motion of satellite infall.

  * •

Cloud accretion model in which clouds follow the cosmological gas accretion as
prescribed by Afruni et al. (2019).

  * •

Tired wind model in which ancient wind bubble “hangs” in the outer halo,
inspired by the semi–analytic model of Lochhaas et al. (2018).

Among the four models, the first three are no–wind models that assume random
or inflowing motion of the clouds, while the last one assumes outflowing
clouds. Although our ultimate goal is to distinguish the four kinematic
models, the current measurement uncertainties of $\xi^{s}_{1h}$ are quite
large (${\sim}30{-}40\%$ on scales below $3\,h^{-1}\mathrm{Mpc}$), rendering
any $\chi^{2}$–based model comparison highly inconclusive. Nonetheless, we
will compute the Akaike Information Criterion (AIC) value for each
best–fitting model as AIC$=2k{+}\chi^{2}$, where $k$ is the number of model
parameters, and compare the AIC of the four models.

More important, we will assess the capability of each model to reproduce the
sharp truncation of the observed $\xi^{s}_{1h}$ profile at
$r_{\pi}{\sim}3.4\,h^{-1}\mathrm{Mpc}$, i.e., the missing–FOG effect in the
RSD of MgII absorbers around LRGs. Despite the large statistical uncertainties
in individual $r_{\pi}$ bins, the truncation feature itself is robust — the
observed $\xi^{s}_{1h}$ profile stays roughly constant below
$r_{\pi}{\sim}3\,h^{-1}\mathrm{Mpc}$, but drops rapidly from $16.9{\pm}5.8$ at
$r_{\pi}{=}3.15\,h^{-1}\mathrm{Mpc}$ to $1.2{\pm}2.6$ at
$r_{\pi}{=}3.85\,h^{-1}\mathrm{Mpc}$. We will describe each model in turn
below.

###  5.1 Isothermal

Figure 8: Comparison of the observed 1–halo components of the LRG–MgII
absorber redshift–space cross–correlation function $\xi_{1h}^{s}$ (circles
with errorbars) with the best–fitting prediction by the isothermal model
(solid curve). The $\chi^{2}$ value is indicated in the top left corner.
Dashed line shows the predicted signal when peculiar velocity is off.

The isothermal model was implicitly assumed in many CGM studies, where a
velocity dispersion was usually computed from the $\Delta V$ distribution and
compared to the virial or escape velocity of the haloes. Similarly, we model
$p({v}_{\mathrm{los}})$ as a Gaussian distribution with zero mean and
dispersion ${\sigma}_{\mathrm{los}}$, and predict the $\xi^{s}_{1h}$ profile
by convolving $p({v}_{\mathrm{los}})$ with our reconstructed $\xi_{1h}(r)$
profile via Equation 16. We derive the best–fitting value of
${\sigma}_{\mathrm{los}}$ via $\chi^{2}$ minimization. Figure 8 compares the
best–fitting prediction from the isothermal model (solid black curve) to the
observed $\xi^{s}_{1h}$ profile (circles with errorbars). The best–fitting
value of ${\sigma}_{\mathrm{los}}$ is $187\,\mathrm{km}\,s^{-1}$, lower than
the virial LOS dispersion (${\simeq}247\,\mathrm{km}\,s^{-1}$ for
$\lg\,M_{h}{=}13.19$) and consistent with the results from previous studies
(e.g., Zhu et al., 2014; Huang et al., 2016; Anand, Nelson, & Kauffmann, 2021;
Huang et al., 2021). The solid curve has a $\chi^{2}$ value of $8.21$, hence a
statistically good fit to the eight data points. However, the model predicts a
gradual decline of the $\xi_{1h}^{s}$ profile, without the sharp truncation
seen in the data profile. This is unsurprising — the isothermal model can only
stretch the relative redshift distribution of MgII absorbers around LRGs, but
cannot produce a sharp feature in the distribution.

###  5.2 Satellite Infall

Figure 9: Similar to Figure 8, but for the satellite infall model. Inset
panels (a) and (b) indicate the average radial and LOS velocity profiles
predicted by the best–fitting model, respectively.

Many of the LRGs are the central galaxies of group–size haloes with halo mass
above $10^{13}\,h^{-1}M_{\odot}$, which frequently accrete lower–mass systems
that likely host star-forming galaxies that could drive cool clouds into the
CGM via strong outflows. In this scenario, the observed MgII absorbers around
LRGs are actually associated with the CGM of infalling satellite galaxies.
Hydrodynamic simulations predict that it usually takes a couple Gyrs before
the satellite galaxies become quenched after being accreted onto the larger
halo (Simha et al., 2009; Wetzel et al., 2013). Therefore, we could model the
cloud kinematics based on the velocity distribution of recently accreted
satellites inside haloes.

After examining the galaxy infall kinematics in the Millennium simulation, Zu
& Weinberg (2013) found that the average radial velocity of the infalling
galaxies reaches maximum at some characteristic radius
(${\sim}1\,h^{-1}\mathrm{Mpc}$) and then declines rapidly towards zero at the
halo center. Therefore, we describe the radial velocity profile of cool clouds
as

$\left\langle
v_{r}(r)\right\rangle=\left(\frac{r}{1.2}\right)^{\beta}v_{\mathrm{max}},$
(17)

where $v_{\mathrm{max}}$ is the maximum radial velocity at
$1.2\,h^{-1}\mathrm{Mpc}$, about twice the $r_{200m}$ of LRG haloes. We do not
assume any rotational component in the cloud kinematics, which is more
important for disky galaxies (Zabl et al., 2019; DeFelippis et al., 2021).
Adopting an average projected distance of $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$,
we then convert the radial velocity profile $\left\langle
v_{r}(r)\right\rangle$ to an average LOS velocity profile
$\left\langle{v}_{\mathrm{los}}(y)\right\rangle$ and assume a constant LOS
velocity dispersion ${\sigma}_{\mathrm{los}}$, yielding a simple satellite
infall model with three parameters
$\\{v_{\mathrm{max}},\,\beta,\,{\sigma}_{\mathrm{los}}\\}$. By minimizing the
$\chi^{2}$, we obtain the best–fitting parameters as
$v_{\mathrm{max}}{=}{-}479.0\,\mathrm{km}\,s^{-1}$, $\beta{=}0.42$, and
${\sigma}_{\mathrm{los}}{=}71.1\,\mathrm{km}\,s^{-1}$, respectively.

Figure 9 shows the best–fitting prediction from our satellite infall model
(solid black curve), with the two inset panels indicating the best–fitting
average radial (top) and LOS (bottom) velocity profiles, respectively.
Compared to the isothermal model ($k{=}1$), the minimum $\chi^{2}$ decreases
from $8.21$ to $6.47$ with two more parameters ($k{=}3$). On the one hand,
based on the classical Akaike information criterion (AIC$=2k+\chi^{2}$), the
satellite infall model has a slightly higher AIC value ($12.47$) than the
isothermal model ($10.21$), hence a slightly less preferred model; But on the
other hand, the satellite infall model produces a more flattened profile on
small scales and a steeper decline, hence a better match to the shape of the
$\xi^{s}_{1h}$ profile than the isothermal model. However, our satellite
infall model still cannot reproduce the observed sharp truncation at
$r_{\pi}{\sim}3.4\,h^{-1}\mathrm{Mpc}$.

###  5.3 Cloud Accretion

Figure 10: Similar to Figure 8, but for the cloud accretion model. Inset
panels (a), (b), and (c) show the cloud mass evaporation, average radial
velocity profile, and average LOS velocity profile, respectively.

Apart from infalling with the satellite galaxies, the observed cool clouds
could also be part of the cosmological inflow that sustains the growth of LRG
haloes. We adopt the same semi–analytic model of cloud accretion developed
Afruni et al. (2019) to predict the radial and LOS velocity profiles of gas
clouds; But instead of assuming a theoretical cool gas accretion rate as in
Afruni et al. (2019), we convolve the LOS velocity distribution with our
reconstructed $\xi_{1h}(r)$ profile to predict the $\xi^{s}_{1h}$ profile via
Equation 16. In this way, our comparison with the observation is
self–consistent, because both $\xi_{1h}(r)$ and $\xi^{s}_{1h}$ are derived
from observations and the cloud kinematics is the only unknown in the problem.
We will briefly describe the relevant components of the Afruni et al. (2019)
model below and refer the reader to Afruni et al. (2019) for more technical
details.

The Afruni et al. (2019) model describes the radial equation of motion of the
cool clouds as

$\frac{\mathrm{d}v_{r}}{\mathrm{d}r}=\frac{1}{v_{r}(r)}\frac{GM(<r)}{r^{2}}-\frac{\pi\,r^{2}_{cl}\,\rho_{\mathrm{cor}}(r)\,v_{r}(r)}{m_{cl}(r)},$
(18)

where $M(<r)$ is the halo mass enclose within $r$, $\rho_{\mathrm{cor}}$ is
the density of the hot corona gas, and $r_{cl}$ ($m_{cl}$) is the radius
(mass) of a typical gas cloud. The second term on the r.h.s. of Equation 18
describes the deceleration of the cool clouds due to ram pressure drag, and
$\rho_{\mathrm{cor}}$ is derived from the NFW potential by assuming
hydrostatic equilibrium of the hot gas with the dark matter halo. The mass and
radius of the gas cloud are related via

$r_{cl}=\left(\frac{3m_{cl}}{4\pi\rho_{cl}}\right)^{1/3},$ (19)

where $\rho_{cl}$ is the density of the cool clouds, calculated from assuming
pressure equilibrium between the cool and hot phases of the CGM. Additionally,
the mass loss due to hydrostatic instabilities (Armillotta et al., 2017) is
modelled as

$\frac{\mathrm{d}\,m_{cl}}{\mathrm{d}\,r}=-\alpha\frac{m_{cl}(r)}{v_{r}(r)},$
(20)

where $\alpha$ is the cloud evaporation rate. Following Afruni et al. (2019),
we assume all the clouds start with zero velocity at
$r_{200m}{=}0.6\,h^{-1}\mathrm{Mpc}$ and initial cloud mass $m_{0,cl}$, and
then glide through an LRG halo with $\lg\,M_{h}{=}13.19$. By combining
Equations 18, 19, and 20 together, we can solve for the radial velocity
profile $\left\langle v_{r}(r)\right\rangle$ of cool clouds for any given
values of $\alpha$ and $m_{0,cl}$, hence the LOS velocity profile
$\left\langle{v}_{\mathrm{los}}(y)\right\rangle$ at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. Finally, assuming a constant LOS velocity
dispersion ${\sigma}_{\mathrm{los}}$, we can predict the $\xi^{s}_{1h}$
profile from $\xi_{1h}(r)$ using Equation 16. By minimizing the $\chi^{2}$, we
find the best–fitting parameters of the cloud accretion model to be
$\alpha{=}2.10\,\mathrm{G}yr^{-1}$, $\lg\,m_{0,cl}{=}6.47$, and
${\sigma}_{\mathrm{los}}{=}98.5\,\mathrm{km}\,s^{-1}$, respectively. Compared
to the constraints from Afruni et al. (2019), our mass loss rate is similar to
theirs ($\alpha{=}1.8\,\mathrm{G}yr^{-1}$), but our cloud mass is larger by a
factor of 30 ($\lg\,m_{0,cl}{=}4.85$). The cloud mass discrepancy is
unsurprising because we rely only on the kinematics to infer $m_{cl}$ while
Afruni et al. (2019) used both the velocity distribution and the hydrogen
column densities.

Figure 10 shows the best–fitting prediction from the cloud accretion model
(solid black curve), with the three inset panels indicating the best–fitting
average mass evaporation (top), radial velocity (middle), LOS velocity
(bottom) profiles, respectively. Despite significant differences in the
predicted radial velocity profiles, the cloud accretion model is almost
indistinguishable from the satellite infall model, with nearly the same values
of minimum $\chi^{2}$ ($6.47$ vs. $6.50$) and the same number of parameters
(hence similar AIC values $12.47$ vs. $12.50$). Meanwhile, similar to the
satellite infall model, the cloud accretion model produces a flattened
small–scale profile followed by a decline that is less sharp than seen in the
data profile.

###  5.4 Tired Wind

Figure 11: Similar to Figure 8, but for the tired wind model. Inset panels
(a), (b), and (c) indicate the LOS velocity dispersion, average radial
velocity, and average LOS velocity profiles, respectively.

The three no–wind models we have explored so far could all provide
statistically good fits to the RSD data, but are unable to reproduce the sharp
truncation in the observed $\xi^{s}_{1h}$ profile. In particular, the two
inflow–based models could produce a similarly flattened profile on small
scales, but the decline at $r_{\pi}{\sim}3{-}4\,h^{-1}\mathrm{Mpc}$ is not as
steep as in the data profile. An outflow–based model may be able to obtain a
better match by propagating a wind bubble to the outer CGM, as predicted by
the model of Lochhaas et al. (2018).

To mimic the kinematic features of ancient wind bubbles driven by the early
star formation of LRGs, we adopt a simple “tired wind” prescription consisting
of a dispersion–dominated component at small radii and a radially outflowing
component at large radii. In particular, the radial velocity profile is

$\displaystyle\left\langle v_{r}(r)\right\rangle$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0&\quad\mbox{if $r<r_{\mathrm{min}}$
},\\\ \left(1-\left(r/1.2\right)^{\gamma}\right)v_{0}&\quad\mbox{if
$r\geqslant r_{\mathrm{min}}$ },\end{array}\right.$

where $v_{0}$ is the launch velocity of the wind bubbles from the halo center
and $r_{\mathrm{min}}$ is the minimum radius of the radiatively cooled wind
bubble, driven by the last star formation episode of the LRGs before
quenching. On scales below $r_{\mathrm{min}}$, we assume random motion with
LOS velocity dispersion ${\sigma}_{\mathrm{los}}$ for the cool clouds, which
could be formed via shell fragmentation due to various instabilities; On
scales above $r_{\mathrm{min}}$, we assume a constant LOS velocity dispersion
of $30\,\mathrm{km}\,s^{-1}$ for the outflow component. This choice of
dispersion is small enough so that each wind bubble remains a coherent
thin–shell without fragmentation, and large enough to avoid overfitting the
data. Therefore, we have in total four parameters: $r_{\mathrm{min}}$,
${\sigma}_{\mathrm{los}}$, $v_{0}$, and $\gamma$, and their best–fitting
values from $\chi^{2}$ minimization are $0.35\,h^{-1}\mathrm{Mpc}$,
$161.0\,\mathrm{km}\,s^{-1}$, $317.0\,\mathrm{km}\,s^{-1}$, and $2.18$,
respectively.

Figure 11 shows the best–fitting prediction from the tired wind model (solid
black curve), with the three inset panels indicating the best–fitting LOS
velocity dispersion (top), radial velocity (middle), and LOS velocity (bottom)
profiles, respectively. The tired wind model provides an excellent description
to the data with $\chi^{2}{=}3.1$ and a reasonable AIC value of $11.10$.
Clearly, the significant improvement in $\chi^{2}$ from the other three models
is mainly driven by the better match to the observed truncation of
$\xi^{s}_{1h}$, thanks to the survived wind bubble moving at
$250\,\mathrm{km}\,s^{-1}$ at $r{\sim}400\,h^{-1}\mathrm{kpc}$. This tired
wind scenario is also roughly consistent with the theoretical expectation from
the Lochhaas et al. (2018) model. While the Lochhaas et al. (2018) model
tracks the radial motion of a single wind–bubble as a function of time for an
individual galaxy, our radial velocity profile effectively describes the
synthetic snapshot of a continuous series of wind–bubbles launched at
different epochs by different galaxies.

Thus far, all the four best–fitting models we explored provide statistically
good fits to the RSD of MgII absorbers around LRGs, yielding similar AIC
values that render no particular model more preferable than the other three.
However, if we focus on reproducing the truncation feature in the observed
$\xi^{s}_{1h}$, the tired wind model clearly stands out, by interpreting the
truncation as the kinematic signature of ancient wind bubbles.

##  6 Summary and Look to the Future

In this paper, we have measured the LRG–MgII absorber cross–correlation
function in the redshift space and projected along the LOS,
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}(r_{p},r_{\pi})$ and $w_{p}(r_{p})$,
respectively. Compared to the LRG auto–correlation, $\xi_{\mathrm{\tt{LRG-
MgII}}}^{s}(r_{p},r_{\pi})$ exhibits a similar Kaiser effect on large scales,
but a much weaker FOG effect on small scales. In particular, the
$\xi^{s}(r_{\pi})$ profile measured at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$ is
heavily truncated at $r_{\pi}{\sim}3.4\,h^{-1}\mathrm{Mpc}$, likely due to a
nontrivial feature in the motion of cool clouds around the LRGs.

Combining $w_{p}(r_{p})$ and the stacked MgII absorption profile measured by
Zhu et al. (2014), we successfully reconstructed the 3D isotropic
cross–correlation function $\xi(r)$ between the MgII absorbers and LRGs. We
found that the detection completeness of the resolved MgII absorbers can be
described by a simple power–law function of distance away from the LRGs
(${\sim}r^{1.2}$), yielding a flat covering fraction profile despite the
stacked MgII absorption follows a projected NFW profile. Furthermore, we
extracted the 1–halo component of the 3D correlation function $\xi_{1h}(r)$,
which provides us the real–space LOS distribution of cool clouds within the
LRG haloes, i.e., $\xi_{1h}(y{=}(r^{2}-r_{p}^{2})^{1/2})$ at
$r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$.

To obtain the redshift–space LOS distribution of cool clouds within the LRG
haloes, we have estimated the 1–halo component of the $\xi^{s}(r_{\pi})$
profile of MgII absorbers at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. In
particular, we subtracted the 2–halo component empirically derived from the
$\xi^{s}(r_{\pi})$ profile at $r_{p}{=}0.75\,h^{-1}\mathrm{Mpc}$, i.e.,
externally tangent to the LRG haloes. The sharp truncation feature seen in
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$ persists in the estimated 1–halo term
$\xi^{s}_{1h}(r_{\pi})$ at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$, which can be
compared with predictions from convolving different kinematic models of the
cool clouds with $\xi_{1h}(y)$.

We consider four kinematic models (three without wind and one with wind) of
the MgII absorbers, including an isothermal model in which cool clouds move
randomly with a single LOS velocity dispersion, a satellite infall model in
which cool clouds are associated with the infalling satellites, a cloud
accretion model in which cool clouds join the cosmological inflow of dark
matter (Afruni et al., 2019), and a tired wind model in which cool clouds
originate from the fragmentation and propagation of ancient wind bubbles
(Lochhaas et al., 2018). For each model, we derived the best–fitting
parameters by minimizing the $\chi^{2}$ between the observed and predicted
$\xi^{s}_{1h}(r_{\pi})$ at $r_{p}{=}0.26\,h^{-1}\mathrm{Mpc}$. Based on the
$\chi^{2}$ values, all the four models provide statistically good fits to the
data. They also yield similar values of AIC, suggesting that the current
measurement uncertainties are too large to statistically distinguish between
the no–wind vs. wind models.

Interestingly, we found that only the tired wind model is capable of
reproducing the observed sharp truncation in the redshift–space LOS
distribution of MgII absorbers within the LRG haloes, while the other three
fail to do so. The tired wind model explains the truncation with an ancient
wind bubble expanding at ${\sim}250\,\mathrm{km}\,s^{-1}$ on scales around
$400\,h^{-1}\mathrm{kpc}$. This physical picture is roughly consistent with
the theoretical expectation from the Lochhaas et al. (2018) model, in which
they discovered that galactic wind–driven bubble could cool radiatively and
then “hang” in the outer halo for several Gyrs.

Although our finding supports the tired wind scenario, it is far from being
conclusive. Due to the large uncertainties in the measurement of
$\xi_{\mathrm{\tt{LRG-MgII}}}^{s}$, our results are also consistent with both
the dispersion and inflow–dominated cloud kinematics. However, we have
convincingly demonstrated the efficacy of our method for inferring cool cloud
kinematics from the RSD of MgII absorbers. In particular, the cloud kinematics
is the missing link between the projected and redshift–space cross–correlation
functions between the MgII absorbers and galaxies, and our method is able to
exploit the spherical symmetry in cross–correlation analysis to infer such
missing link in a self–consistent manner. With ongoing and future
spectroscopic surveys like the DESI and PFS, we will obtain a greater number
of quasar sightlines, hence greater number of MgII absorbers (Zhao et al.,
2019; Anand, Nelson, & Kauffmann, 2021), and apply the method to massive
quiescent galaxies as well as a large number of emission line galaxies.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## Acknowledgements

We thank Cassi Lochhaas, Ting-Wen Lan, and David Weinberg for helpful
discussions, and the anonymous referee for comments that have greatly improved
the manuscript. YZ acknowledges the support by the National Key Basic Research
and Development Program of China (No. 2018YFA0404504), National Science
Foundation of China (11873038, 11621303, 11890692), the science research
grants from the China Manned Space Project (No. CMS-CSST-2021-A01, CMS-
CSST-2021-B01), the National One-Thousand Youth Talent Program of China, and
the SJTU start-up fund (No. WF220407220). YZ thanks the stimulating
discussions with Cathy Huang during his quarantine at the Zhangjiang Hi-Tech
Park during the pandemic.

## References

  * Afruni et al. (2019) Afruni A., Fraternali F., Pezzulli G., 2019, A&A, 625, A11
  * Aihara et al. (2011) Aihara H., et al., 2011, ApJS, 193, 29
  * Alam et al. (2015) Alam S., et al., 2015, ApJS, 219, 12
  * Anand, Nelson, & Kauffmann (2021) Anand A., Nelson D., Kauffmann G., 2021, MNRAS, 504, 65. doi:10.1093/mnras/stab871
  * Anderson et al. (2013) Anderson M. E., Bregman J. N., Dai X., 2013, ApJ, 762, 106
  * Armillotta et al. (2017) Armillotta L., Fraternali F., Werk J. K., Prochaska J. X., Marinacci F., 2017, MNRAS, 470, 114
  * Berg et al. (2019) Berg M. A., et al., 2019, ApJ, 883, 5
  * Bergeron & Boissé (1991) Bergeron J., Boissé P., 1991, A&A, 243, 344
  * Birnboim & Dekel (2003) Birnboim Y., Dekel A., 2003, MNRAS, 345, 349
  * Blanton et al. (2003) Blanton M. R., Lin H., Lupton R. H., Maley F. M., Young N., Zehavi I., Loveday J., 2003, AJ, 125, 2276
  * Bolton et al. (2012) Bolton A. S., et al., 2012, AJ, 144, 144
  * Borthakur et al. (2016) Borthakur S., et al., 2016, ApJ, 833, 259
  * Cai et al. (2019) Cai Z., Cantalupo S., Prochaska J. X., Arrigoni Battaia F., Burchett J., Li Q., Chisholm J., et al., 2019, ApJS, 245, 23. doi:10.3847/1538-4365/ab4796
  * Chen (2017) Chen H.-W., 2017, The Circumgalactic Medium in Massive Halos. p. 167, doi:10.1007/978-3-319-52512-9_8
  * Chen & Mulchaey (2009) Chen H.-W., Mulchaey J. S., 2009, ApJ, 701, 1219
  * Chen et al. (2005) Chen H.-W., Prochaska J. X., Weiner B. J., Mulchaey J. S., Williger G. M., 2005, ApJ, 629, L25
  * Chen et al. (2018) Chen H.-W., Zahedy F. S., Johnson S. D., Pierce R. M., Huang Y.-H., Weiner B. J., Gauthier J.-R., 2018, MNRAS, 479, 2547
  * Churchill et al. (1999) Churchill C. W., Rigby J. R., Charlton J. C., Vogt S. S., 1999, ApJS, 120, 51
  * Davé et al. (2011) Davé R., Finlator K., Oppenheimer B. D., 2011, MNRAS, 416, 1354
  * Dawson et al. (2013) Dawson K. S., et al., 2013, AJ, 145, 10
  * DeFelippis et al. (2021) DeFelippis D., Bouché N. F., Genel S., Bryan G. L., Nelson D., Marinacci F., Hernquist L., 2021, arXiv e-prints, p. arXiv:2102.08383
  * Dekel et al. (2009) Dekel A., et al., 2009, Nature, 457, 451
  * Diemer & Kravtsov (2014) Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1
  * Eisenstein et al. (2011) Eisenstein D. J., et al., 2011, AJ, 142, 72
  * Fielding et al. (2017) Fielding D., Quataert E., McCourt M., Thompson T. A., 2017, MNRAS, 466, 3810
  * Fielding et al. (2020) Fielding D. B., et al., 2020, ApJ, 903, 32
  * Ford et al. (2014) Ford A. B., Davé R., Oppenheimer B. D., Katz N., Kollmeier J. A., Thompson R., Weinberg D. H., 2014, MNRAS, 444, 1260
  * Fukugita et al. (1996) Fukugita M., Ichikawa T., Gunn J. E., Doi M., Shimasaku K., Schneider D. P., 1996, AJ, 111, 1748
  * Fumagalli et al. (2011) Fumagalli M., Prochaska J. X., Kasen D., Dekel A., Ceverino D., Primack J. R., 2011, MNRAS, 418, 1796
  * Gauthier (2013) Gauthier J.-R., 2013, MNRAS, 432, 1444
  * Gauthier et al. (2009) Gauthier J.-R., Chen H.-W., Tinker J. L., 2009, ApJ, 702, 50
  * Gauthier et al. (2010) Gauthier J.-R., Chen H.-W., Tinker J. L., 2010, ApJ, 716, 1263
  * Gunn et al. (1998) Gunn J. E., et al., 1998, AJ, 116, 3040
  * Gunn et al. (2006) Gunn J. E., et al., 2006, AJ, 131, 2332
  * Heckman & Thompson (2017) Heckman T. M., Thompson T. A., 2017, arXiv e-prints, p. arXiv:1701.09062
  * Ho et al. (2020) Ho S. H., Martin C. L., Schaye J., 2020, ApJ, 904, 76
  * Huang et al. (2016) Huang Y.-H., Chen H.-W., Johnson S. D., Weiner B. J., 2016, MNRAS, 455, 1713
  * Huang et al. (2021) Huang Y.-H., Chen H.-W., Shectman S. A., Johnson S. D., Zahedy F. S., Helsby J. E., Gauthier J.-R., Thompson I. B., 2021, MNRAS, 502, 4743
  * Hummels et al. (2013) Hummels C. B., Bryan G. L., Smith B. D., Turk M. J., 2013, MNRAS, 430, 1548
  * Jackson (1972) Jackson J. C., 1972, MNRAS, 156, 1P
  * Kaiser (1987) Kaiser N., 1987, MNRAS, 227, 1
  * Kauffmann et al. (2017) Kauffmann G., Nelson D., Ménard B., Zhu G., 2017, MNRAS, 468, 3737
  * Keeney et al. (2017) Keeney B. A., et al., 2017, ApJS, 230, 6
  * Kereš et al. (2005) Kereš D., Katz N., Weinberg D. H., Davé R., 2005, MNRAS, 363, 2
  * Kereš et al. (2009) Kereš D., Katz N., Fardal M., Davé R., Weinberg D. H., 2009, MNRAS, 395, 160
  * Kim & Ostriker (2018) Kim C.-G., Ostriker E. C., 2018, ApJ, 853, 173
  * Lan & Mo (2018) Lan T.-W., Mo H., 2018, ApJ, 866, 36
  * Lan & Mo (2019) Lan T.-W., Mo H., 2019, MNRAS, 486, 608
  * Landy & Szalay (1993) Landy S. D., Szalay A. S., 1993, ApJ, 412, 64
  * Lanzetta et al. (1998) Lanzetta K. M., Webb J. K., Barcons X., 1998, in D’Odorico S., Fontana A., Giallongo E., eds, Astronomical Society of the Pacific Conference Series Vol. 146, The Young Universe: Galaxy Formation and Evolution at Intermediate and High Redshift. p. 175
  * Lee et al. (2021) Lee J. C., Hwang H. S., Song H., 2021, MNRAS, 503, 4309
  * Lehner et al. (2013) Lehner N., et al., 2013, ApJ, 770, 138
  * Levi et al. (2019) Levi M., et al., 2019, in Bulletin of the American Astronomical Society. p. 57 (arXiv:1907.10688)
  * Lochhaas et al. (2018) Lochhaas C., Thompson T. A., Quataert E., Weinberg D. H., 2018, MNRAS, 481, 1873
  * Ma et al. (2016) Ma X., Hopkins P. F., Faucher-Giguère C.-A., Zolman N., Muratov A. L., Kereš D., Quataert E., 2016, MNRAS, 456, 2140
  * Maller & Bullock (2004) Maller A. H., Bullock J. S., 2004, MNRAS, 355, 694
  * Mandelbaum et al. (2016) Mandelbaum R., Wang W., Zu Y., White S., Henriques B., More S., 2016, MNRAS, 457, 3200
  * McCourt et al. (2012) McCourt M., Sharma P., Quataert E., Parrish I. J., 2012, MNRAS, 419, 3319
  * Mo & Miralda-Escude (1996) Mo H. J., Miralda-Escude J., 1996, ApJ, 469, 589
  * Muratov et al. (2017) Muratov A. L., et al., 2017, MNRAS, 468, 4170
  * Murray et al. (2005) Murray N., Quataert E., Thompson T. A., 2005, ApJ, 618, 569
  * Navarro et al. (1997) Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493
  * Nelson et al. (2020) Nelson D., et al., 2020, MNRAS, 498, 2391
  * Norberg et al. (2009) Norberg P., Baugh C. M., Gaztañaga E., Croton D. J., 2009, MNRAS, 396, 19
  * Oppenheimer & Davé (2006) Oppenheimer B. D., Davé R., 2006, MNRAS, 373, 1265
  * Pâris et al. (2017) Pâris I., et al., 2017, A&A, 597, A79
  * Peeples & Shankar (2011) Peeples M. S., Shankar F., 2011, MNRAS, 417, 2962
  * Peeples et al. (2019) Peeples M. S., Corlies L., Tumlinson J., O’Shea B. W., Lehner N., O’Meara J. M., Howk J. C., et al., 2019, ApJ, 873, 129. doi:10.3847/1538-4357/ab0654
  * Hafen et al. (2019) Hafen Z., Faucher-Giguère C.-A., Anglés-Alcázar D., Stern J., Kereš D., Hummels C., Esmerian C., et al., 2019, MNRAS, 488, 1248. doi:10.1093/mnras/stz1773
  * Hummels et al. (2019) Hummels C. B., Smith B. D., Hopkins P. F., O’Shea B. W., Silvia D. W., Werk J. K., Lehner N., et al., 2019, ApJ, 882, 156. doi:10.3847/1538-4357/ab378f
  * Suresh et al. (2019) Suresh J., Nelson D., Genel S., Rubin K. H. R., Hernquist L., 2019, MNRAS, 483, 4040. doi:10.1093/mnras/sty3402
  * Fielding et al. (2020) Fielding D. B., Tonnesen S., DeFelippis D., Li M., Su K.-Y., Bryan G. L., Kim C.-G., et al., 2020, ApJ, 903, 32. doi:10.3847/1538-4357/abbc6d
  * Pérez-Ràfols et al. (2015) Pérez-Ràfols I., Miralda-Escudé J., Lundgren B., Ge J., Petitjean P., Schneider D. P., York D. G., Weaver B. A., 2015, MNRAS, 447, 2784
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Prochaska et al. (2013) Prochaska J. X., Hennawi J. F., Simcoe R. A., 2013, ApJ, 762, L19
  * Quider et al. (2011) Quider A. M., Nestor D. B., Turnshek D. A., Rao S. M., Monier E. M., Weyant A. N., Busche J. R., 2011, AJ, 141, 137
  * Reid et al. (2016) Reid B., et al., 2016, MNRAS, 455, 1553
  * Ryan-Weber (2006) Ryan-Weber E. V., 2006, MNRAS, 367, 1251
  * Samui et al. (2008) Samui S., Subramanian K., Srianand R., 2008, MNRAS, 385, 783
  * Sarkar et al. (2015) Sarkar K. C., Nath B. B., Sharma P., Shchekinov Y., 2015, MNRAS, 448, 328
  * Schneider et al. (2010) Schneider D. P., et al., 2010, AJ, 139, 2360
  * Sharma et al. (2012) Sharma P., McCourt M., Quataert E., Parrish I. J., 2012, MNRAS, 420, 3174
  * Simha et al. (2009) Simha V., Weinberg D. H., Davé R., Gnedin O. Y., Katz N., Kereš D., 2009, MNRAS, 399, 650
  * Smee et al. (2013) Smee S. A., et al., 2013, AJ, 146, 32
  * Spitzer (1956) Spitzer Lyman J., 1956, ApJ, 124, 20
  * Steidel & Sargent (1992) Steidel C. C., Sargent W. L. W., 1992, ApJS, 80, 1
  * Steidel et al. (1994) Steidel C. C., Dickinson M., Persson S. E., 1994, ApJ, 437, L75
  * Takada et al. (2014) Takada M., et al., 2014, PASJ, 66, R1
  * Tejos et al. (2012) Tejos N., Morris S. L., Crighton N. H. M., Theuns T., Altay G., Finn C. W., 2012, MNRAS, 425, 245
  * Thom et al. (2012) Thom C., et al., 2012, ApJ, 758, L41
  * Thompson et al. (2016) Thompson T. A., Quataert E., Zhang D., Weinberg D. H., 2016, MNRAS, 455, 1830
  * Tinker & Chen (2008) Tinker J. L., Chen H.-W., 2008, ApJ, 679, 1218. doi:10.1086/587432
  * Tinker et al. (2005) Tinker J. L., Weinberg D. H., Zheng Z., Zehavi I., 2005, ApJ, 631, 41. doi:10.1086/432084
  * Tumlinson et al. (2013) Tumlinson J., et al., 2013, ApJ, 777, 59
  * Tumlinson et al. (2017) Tumlinson J., Peeples M. S., Werk J. K., 2017, ARA&A, 55, 389
  * Veilleux et al. (2020) Veilleux S., Maiolino R., Bolatto A. D., Aalto S., 2020, A&ARv, 28, 2
  * Voit (2018) Voit G. M., 2018, ApJ, 868, 102
  * Voit & Donahue (2015) Voit G. M., Donahue M., 2015, ApJ, 799, L1
  * Wang (1995) Wang B., 1995, ApJ, 444, 590
  * Werk et al. (2014) Werk J. K., et al., 2014, ApJ, 792, 8
  * Wetzel et al. (2013) Wetzel A. R., Tinker J. L., Conroy C., van den Bosch F. C., 2013, MNRAS, 432, 336
  * Wilman et al. (2007) Wilman R. J., Morris S. L., Jannuzi B. T., Davé R., Shone A. M., 2007, MNRAS, 375, 735
  * Wild et al. (2008) Wild V., Kauffmann G., White S., York D., Lehnert M., Heckman T., Hall P. B., et al., 2008, MNRAS, 388, 227. doi:10.1111/j.1365-2966.2008.13375.x
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Zabl et al. (2019) Zabl J., et al., 2019, MNRAS, 485, 1961
  * Zahedy et al. (2019) Zahedy F. S., Chen H.-W., Johnson S. D., Pierce R. M., Rauch M., Huang Y.-H., Weiner B. J., Gauthier J.-R., 2019, MNRAS, 484, 2257
  * Zhang et al. (2017) Zhang D., Thompson T. A., Quataert E., Murray N., 2017, MNRAS, 468, 4801
  * Zhang et al. (2016) Zhang H., Zaritsky D., Zhu G., Ménard B., Hogg D. W., 2016, ApJ, 833, 276. doi:10.3847/1538-4357/833/2/276
  * Zhang, Zaritsky, & Behroozi (2018) Zhang H., Zaritsky D., Behroozi P., 2018, ApJ, 861, 34. doi:10.3847/1538-4357/aac6b7
  * Zhao et al. (2019) Zhao Y., Ge J., Yuan X., Zhao T., Wang C., Li X., 2019, MNRAS, 487, 801
  * Zhu & Ménard (2013) Zhu G., Ménard B., 2013, ApJ, 770, 130
  * Zhu et al. (2014) Zhu G., et al., 2014, MNRAS, 439, 3139
  * Zu (2020) Zu Y., 2020, arXiv e-prints, p. arXiv:2010.01143
  * Zu & Mandelbaum (2015) Zu Y., Mandelbaum R., 2015, MNRAS, 454, 1161. doi:10.1093/mnras/stv2062
  * Zu & Mandelbaum (2016) Zu Y., Mandelbaum R., 2016, MNRAS, 457, 4360
  * Zu & Weinberg (2013) Zu Y., Weinberg D. H., 2013, MNRAS, 431, 3319
  * van de Voort et al. (2012) van de Voort F., Schaye J., Altay G., Theuns T., 2012, MNRAS, 421, 2809