# Halo mass-observable proxy scaling relations and their dependencies on
galaxy and group properties

Ziwen Zhang CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China; Email<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China  Huiyuan Wang CAS Key Laboratory for Research in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui 230026, China; Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Wentao Luo CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China; Email<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China  Houjun Mo Department of Astronomy, University of
Massachusetts, Amherst, MA 01003-9305, USA  Jun Zhang Department of Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai Key Laboratory
for Particle Physics and Cosmology, Shanghai 200240, China  Xiaohu Yang
Department of Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240,
China  Hao Li  CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei,
Anhui 230026, China; Email<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China  Qinxun Li CAS Key Laboratory for Research in
Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei, Anhui 230026, China; Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China

###### Abstract

Based on the DECaLS shear catalog, we study the scaling relations between halo
mass($M_{\rm h}$) and various proxies for SDSS central galaxies, including
stellar mass($M_{*}$), stellar velocity dispersion($\sigma_{*}$), abundance
matching halo mass($M_{\rm AM}$) and satellite velocity
dispersion($\sigma_{\rm s}$), and their dependencies on galaxy and group
properties. In general, these proxies all have strong positive correlations
with $M_{\rm h}$, consistent with previous studies. We find that the $M_{\rm
h}$-$M_{*}$ and $M_{\rm h}$-$\sigma_{*}$ relations depend strongly on group
richness($N_{\rm sat}$), while the $M_{\rm h}$-$M_{\rm AM}$ and $M_{\rm
h}$-$\sigma_{\rm s}$ relations are independent of it. Moreover, the dependence
on star formation rate(SFR) is rather weak in the $M_{\rm h}$-$\sigma_{*}$ and
$M_{\rm h}$-$\sigma_{\rm s}$ relations, but very prominent in the other two.
$\sigma_{\rm s}$ is thus the best proxy among them, and its scaling relation
is in good agreement with hydro-dynamical simulations. However, estimating
$\sigma_{\rm s}$ accurately for individual groups/clusters is challenging
because of interlopers and the requirement for sufficient satellites. We
construct new proxies by combining $M_{*}$, $\sigma_{*}$, and $M_{\rm AM}$,
and find the proxy with 30% contribution from $M_{\rm AM}$ and 70% from
$\sigma_{*}$ can minimize the dependence on $N_{\rm sat}$ and SFR. We obtain
the $M_{\rm h}$-supermassive black hole(SMBH) mass relation via the SMBH
scaling relation and find indications for rapid and linear growth phases for
SMBH. We also find that correlations among $M_{\rm h}$, $M_{*}$ and
$\sigma_{*}$ change with $M_{*}$, indicating that different processes drive
the growth of galaxies and SMBH at different stages.

gravitational lensing - galaxies: halos

##  1 Introduction

In the standard $\Lambda$CDM paradigm, dark matter halos, approximately in
dynamical equilibrium, form hierarchically through gravitational instability.
Galaxies form and evolve in the potential wells of dark matter halos (e.g.
White & Rees, 1978; Fall & Efstathiou, 1980; Mo et al., 2010). Galaxies
residing in one halo are regarded as a galaxy group or a cluster. In recent
years, many studies have concentrated on establishing galaxy - halo
connections, manifested as scaling relations between halo mass and galaxy and
group properties (proxies). These scaling relations can be used to estimate
the halo mass of galaxy groups and clusters (Yang et al., 2005; Zahid et al.,
2016; Seo et al., 2020), constrain cosmology (Bocquet et al., 2015; Caldwell
et al., 2016), and study galaxy formation models (see Mo et al., 2010;
Wechsler & Tinker, 2018; Chen et al., 2021a, b, for references).

In these scaling relations, halo mass ($M_{\rm h}$) is not a direct
observable. $M_{\rm h}$ estimation usually requires high-quality data and/or
sophisticated techniques, and sometimes model assumptions. Numerous studies
have developed various methods to constrain the halo mass, including the
virial mass estimator (e.g. Carlberg et al., 2001; Rines et al., 2013;
Abdullah et al., 2020), X-ray emission (Ruel et al., 2014), Sunyaev–Zel’dovich
(SZ) effects (Ruel et al., 2014; Rines et al., 2016), caustics technique (e.g.
Rines & Diaferio, 2006; Rines et al., 2013), abundance matching (Yang et al.,
2007) and weak lensing (e.g. Mandelbaum et al., 2006, 2016; Hoekstra, 2007;
Leauthaud et al., 2012; Velander et al., 2014; Gonzalez et al., 2015; Han et
al., 2015; Hudson et al., 2015; Viola et al., 2015; Luo et al., 2018; Gonzalez
et al., 2021; Rana et al., 2022; Zhang et al., 2021, 2022b). Various halo mass
proxies have been investigated, such as stellar mass, group richness, stellar
velocity dispersion, group total stellar mass or luminosity, and satellite
velocity dispersion.

Among these studies, the most widely discussed halo mass proxy is the stellar
mass ($M_{*}$) of central galaxies. And the corresponding scaling relation is
referred to as stellar mass-halo mass relation (hereafter SHMR) (e.g. Yang et
al., 2003; Moster et al., 2010; Leauthaud et al., 2012; Li et al., 2013;
Mandelbaum et al., 2016; Kravtsov et al., 2018; Wechsler & Tinker, 2018;
Behroozi et al., 2019; Zhang et al., 2021, 2022b). The SHMR can be used to
constrain galaxy formation processes. For example, $M_{*}/M_{\rm
h}/(\Omega_{\rm b}/\Omega_{\rm m})$ is usually regarded as the baryon-to-star
conversion efficiency. The mean efficiency peaks around the Milky-way like
galaxies, with a value of about 20%, and declines quickly toward both lower
and higher $M_{*}$ ends. At lower $M_{*}$, the efficiency is believed to be
suppressed by supernova feedback and stellar winds (e.g. Dekel & Silk, 1986;
Kauffmann & Charlot, 1998; Cole et al., 2000; Hopkins et al., 2012), while the
feedback from active galactic nuclei (AGN) is suggested to be responsible for
suppressing the efficiency at higher $M_{*}$ (e.g. Silk & Rees, 1998; Croton
et al., 2006; Fabian, 2012; Heckman & Best, 2014; Cui et al., 2021). The SHMR
has large scatter, which depends on galaxy properties, such as color, star
formation rate and morphology (e.g. More et al., 2011; Rodríguez-Puebla et
al., 2015; Mandelbaum et al., 2016; Behroozi et al., 2019; Lange et al.,
2019a; Romeo et al., 2020; Zhang et al., 2021; Bilicki et al., 2021; Posti &
Fall, 2021; Xu & Jing, 2022; Wang et al., 2023a). Research based on
simulations suggests that the dependence is related to both halo assembly and
AGN feedback (Cui et al., 2021). More recently, several studies found a number
of massive star-forming/spiral galaxies that exhibit very high star formation
efficiency, about 50%-60% (e.g. Posti et al., 2019; Zhang et al., 2022b).
Their results indicate that these galaxies can convert most of their halo gas
into stars, implying that AGN feedback is inefficient in them. The scatter of
SHMR depends also strongly on satellite properties (Lu et al., 2016; Zhou &
Han, 2022). All these results suggest that the SHMR and its scatter contain
valuable information about galaxy formation and halo assembly.

Stellar velocity dispersion ($\sigma_{*}$) directly reflects the gravitational
potential of the galaxy, and is well correlated with the stellar mass. Given
the SHMR, one may expect a strong correlation between stellar velocity
dispersion and halo mass. The $\sigma_{*}$ measurement is straightforward, and
its systematic uncertainty may be lower than the stellar mass, making the
stellar velocity dispersion an ideal alternative to stellar mass as a halo-
mass proxy. Previous studies on this connection find that the stellar velocity
dispersion is indeed a robust proxy for halo mass (e.g. Zahid et al., 2016,
2018; Seo et al., 2020; Sohn et al., 2020). The $M_{\rm h}$-$\sigma_{*}$
relation also has other important applications. For instance, Shankar et al.
(2020) found that the $M_{\rm h}$-$\sigma_{*}$ relation and the large-scale
clustering of AGNs can be used to constrain the shape and amplitude of the
scaling relation between supermassive black hole (SMBH) mass and galaxy
properties, as well as to constrain the radiative efficiency of AGNs. The halo
masses in these observational studies are estimated via abundance matching and
satellite kinematics. It is required to have a weak lensing based
investigation as an independent check. Moreover, it is also interesting to
investigate if the SHMR and $M_{\rm h}$-$\sigma_{*}$ relation are independent
of each other. Studying the bivariate correlation may be particularly
important for understanding the connection between AGN feedback, galaxy growth
and halo environments.

Previous studies have also adopted the total stellar mass or total luminosity
of group members as a halo mass proxy, and found good correlations with halo
mass (e.g. Yang et al., 2005; Conroy et al., 2007; Han et al., 2015). Han et
al. (2015) performed a maximum-likelihood weak-lensing analysis and found that
the group luminosity has the tightest relation with weak lensing halo mass
among all single proxies they investigated. The group luminosity and stellar
mass have been used to estimate the halo mass based on abundance matching
(e.g. Yang et al., 2005, 2007). The inferred halo mass is referred to as the
abundance matching halo mass, denoted by $M_{\rm AM}$, in the following. Luo
et al. (2018) and Gonzalez et al. (2021) studied the relation between $M_{\rm
AM}$ and halo mass based on weak lensing. Gonzalez et al. (2021) further
investigated the dependence of the $M_{\rm h}$-$M_{\rm AM}$ relation on galaxy
morphology and found that the relation for early-type centrals is closer to
the one-to-one relation than the relation based on all types of centrals.

Satellite kinematics, which results from the equilibrium of a gravitational
system, can also provide a measurement of halo mass. Many investigations have
been carried out to study the relation between satellite velocity dispersion
($\sigma_{\rm s}$) and halo mass using both simulations (e.g. Evrard et al.,
2008; Li et al., 2012; Munari et al., 2013; Saro et al., 2013) and
observational data (e.g. Rines & Diaferio, 2006; Yang et al., 2007; Hoekstra,
2007; Rines et al., 2013; Ruel et al., 2014; Gonzalez et al., 2015; Han et
al., 2015; Viola et al., 2015; Rines et al., 2016; Abdullah et al., 2020;
Gonzalez et al., 2021; Rana et al., 2022). Evrard et al. (2008) used dark
matter particles to calculate $\sigma_{\rm s}$ and found that the relation has
small scatter. Munari et al. (2013) and Saro et al. (2013) found a weak bias
(less than 10$\%$) when using simulated subhalos or satellite galaxies to
derive $\sigma_{\rm s}$. They all found that halo mass is roughly proportional
to $\sigma_{\rm s}^{3}$, consistent with theoretical expectations.

Observationally, various methods have been applied to establish the $M_{\rm
h}$-$\sigma_{\rm s}$ relation. For example, some studies estimated the halo
mass based on the caustics technique (e.g. Rines & Diaferio, 2006; Rines et
al., 2013), SZ effect (e.g. Ruel et al., 2014; Rines et al., 2016) and X-ray
observation (Ruel et al., 2014). These studies usually focused on individual
galaxy clusters and the obtained relations are broadly consistent with those
from simulations. However, the results based on weak lensing data appear
complex. Some studies obtained similar slopes but different amplitudes in
comparison to relations obtained from simulations (e.g. Gonzalez et al., 2021;
Zhang et al., 2022b), and some obtained even different slopes (e.g. Han et
al., 2015; Gonzalez et al., 2015; Viola et al., 2015; Rana et al., 2022).
Hoekstra (2007) obtained the relation for massive clusters and found it in
agreement with simulations but with larger scatter. In addition, very few
studies extended the $M_{\rm h}$ \- $\sigma_{\rm s}$ relation to $M_{\rm
h}<10^{13}\rm M_{\odot}$ (e.g. Han et al., 2015; Zhang et al., 2022b).

The main goal of this paper is to study various halo mass proxies (stellar
mass $M_{*}$, stellar velocity dispersion $\sigma_{*}$, abundance matching
halo mass $M_{\rm AM}$, satellite velocity dispersion $\sigma_{\rm s}$, and
combinations of them) and their relations with the halo mass obtained from
weak lensing. We also investigate their dependencies on other galaxy/group
properties to shed light on underlying physical processes, and evaluate the
performance of these halo mass proxies. The outline of this paper is as
follows. In Section 2, we describe sample selections and the weak lensing
shear catalog to be used. Section 3 presents measurements of weak lensing and
satellite kinematics. We also introduce in this section an proxy based on
combinations of parameters and the way to evaluate the performance of halo
mass proxies. In Section 4, we show the scaling relations we obtain and their
dependencies on other galaxy and group properties. Finally, we summarize our
results in Section 5. Throughout this paper, we assume a flat $\Lambda$CDM
cosmology with $\rm\Omega_{m}$ = 0.307, $\rm\Omega_{b}$ = 0.048,
$\rm\Omega_{\Lambda}$ = 0.693 and h = 0.678. Here, h = $\rm H_{0}/100$ km $\rm
s^{-1}$ $\rm Mpc^{-1}$ and $\rm H_{0}$ is the Hubble constant (Planck
Collaboration et al., 2016).

##  2 Observational data

###  2.1 Central galaxy samples

The galaxies used in this paper are drawn from the New York University Value
Added Galaxy Catalog (NYU-VAGC111http://sdss.physics.nyu.edu/vagc/ Blanton et
al., 2005) of the Sloan Digital Sky Survey Data Release 7 (SDSS DR7)
(Abazajian et al., 2009). We select galaxies with $r$-band Petrosian magnitude
$r\leq 17.72$, with redshift in the range of 0.01 $\leq$ $z$ $\leq$ 0.2 and
with redshift completeness $C_{z}>$ 0.7. We use the group catalog222
https://gax.sjtu.edu.cn/data/Group.html of Yang et al. (2005, 2007) to
classify central and satellite galaxies. Central galaxies are defined as the
most massive galaxies in galaxy groups. As shown in the next section, we
obtain the halo mass by using the Dark Energy Camera Legacy Survey (DECaLS,
Dey et al., 2019). About 32$\%$ of SDSS DR7 galaxies do not overlap with the
DECaLS. This leaves a sample of 323,448 central galaxies.

$M_{*}$ of the central galaxies are obtained by cross-matching with the MPA-
JHU DR7 catalog 333https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/ (Kauffmann et
al., 2003; Brinchmann et al., 2004). They are calculated by fitting the SDSS
$ugriz$ photometry to models of galaxy spectral energy distribution.
$\sigma_{*}$ is provided by NYU-VAGC. $M_{\rm AM}$ for each central galaxy is
provided by the group catalog(Yang et al., 2007). Here, we mainly use $M_{\rm
AM}$ based on the group total stellar mass. The estimate of $\sigma_{\rm s}$
will be described in Section 3.2 and the combined proxies are introduced in
Section 3.3.

Not all galaxies have valid estimations of $M_{*}$, $\sigma_{*}$, star-
formation rate (SFR) and $M_{\rm AM}$. For example, a small fraction of
galaxies have no measurement of $\sigma_{*}$ (usually low mass galaxies), and
some have unreasonably high $\sigma_{*}$ (e.g. $\sigma_{*}\sim
1000\,{\rm{km\,s^{-1}}}$). Moreover, a fraction of low mass galaxies have no
available $M_{\rm AM}$ because $M_{\rm AM}$ can only be estimated in a
complete sample. We thus construct two total samples. The first one, referred
to as ATotal, has measurements of $M_{*}$ and SFR and
$0<\sigma_{*}/\,{\rm{km\,s^{-1}}}<630$ of 299,806 galaxies. This sample is
used to study the $M_{\rm h}$-$M_{*}$ and $M_{\rm h}$-$\sigma_{*}$ relations
(Section 4.1 and 4.2). The second one, referred to as BTotal, requires
measurements of $M_{\rm AM}$ in addition. This requiement removes 60,769
galaxies. The BTotal sample is used to study the $M_{\rm h}$-$M_{\rm AM}$
relation (Section 4.3). When we study the $M_{\rm h}$-$\sigma_{\rm s}$
relation (Section 4.4), both samples are used. We do not use BTotal sample to
study the $M_{\rm h}$-$M_{*}$ and $M_{\rm h}$-$\sigma_{*}$ relations, because
this sample excludes many low-mass galaxies. We list the sample selection, the
sample size and the sections where the samples are used in Table 1.

We will also consider the dependencies of the various relations mentioned
above on several galaxy properties, such as $M_{*}$, $\sigma_{*}$ and SFR, and
one group property, the group richness. The galaxy SFR is obtained by cross-
matching with the MPA-JHU DR7 catalog, where the SFRs of individual galaxies
were derived from spectroscopic and photometric data of the SDSS. We adopt the
demarcation line proposed by Bluck et al. (2016), to separate star-forming and
quenched galaxies. Thus, the ATotal (BTotal) sample is divided into ASF (BSF)
and AQ (BQ) subsamples, corresponding to star-forming and quenched galaxies,
respectively. For each central galaxy, its group richness is quantified in
terms of the number of satellites (hereafter $N_{\rm sat}$) provided by the
group catalog. We divide the ATotal (BTotal) sample into four subsamples, AC0,
AC1, AC2 and AC3 (BC0, BC1, BC2 and BC3), where AC0 and BC0 have $N_{\rm
sat}=0$, AC1 and BC1 have $N_{\rm sat}\geq 1$, AC2 and BC2 have $1\leq N_{\rm
sat}\leq 2$, and AC3 and BC3 have $N_{\rm sat}\geq 3$. These subsamples are
also listed in Table 1. These samples are sometimes referred to as the
A-series and B-series samples, respectively.

Table 1: Sample selection used in our analysis

Sample(a) |  Sample criteria(b) |  subsample criteria(c) |  Sample Size |  Section(d)
---|---|---|---|---
ATotal |  $0<\sigma_{*}<630$ |  |  299,806 |  3.2, 4.1, 4.2, 4.4
AC0 |  available $M_{*}$ |  $N_{\rm sat}=0$ |  262,404 |  3.2, 4.1, 4.2
AC1 |  available SFR |  $N_{\rm sat}\geq 1$ |  37,402 |  3.2, 4.1, 4.4
AC2 |  |  $1\leq N_{\rm sat}\leq 2$ |  30,245 |  3.2, 4.1, 4.2
AC3 |  |  $N_{\rm sat}\geq 3$ |  7,157 |  3.2, 4.1, 4.2
ASF |  |  star-forming |  176,202 |  4.1, 4.2
AQ |  |  quenched |  123,604 |  4.1, 4.2
BTotal |  $0<\sigma_{*}<630$ |  |  239,037 |  4.3, 4.5
BC0 |  available $M_{*}$ |  $N_{\rm sat}=0$ |  204,514 |  4.3, 4.5
BC1 |  available SFR |  $N_{\rm sat}\geq 1$ |  34,523 |  4.4
BC2 |  available $M_{\rm AM}$ |  $1\leq N_{\rm sat}\leq 2$ |  27,477 |  4.3, 4.4
BC3 |  |  $N_{\rm sat}\geq 3$ |  7,046 |  4.3, 4.4, 4.5
BSF |  |  star-forming |  123,796 |  4.3, 4.4, 4.5
BQ |  |  quenched |  115,241 |  4.3, 4.4, 4.5

  * (a)

ATotal and BTotal are two total samples, and the rest are corresponding
subsamples.

  * (b)

The selection criteria is applied to the corresponding total samples and
subsamples.

  * (c)

The selection criterion is only applied to the corresponding subsample.

  * (d)

The sections where the samples are used.

###  2.2 DECaLS shear catalog

We use the shear catalog based on the DECaLS DR8 imaging data (Dey et al.,
2019; Zou et al., 2019) to measure galaxy-galaxy lensing signals. Galaxy
shapes are measured by using the FOURIER$\\_$QUAD pipeline which measures
galaxy shear with great accuracy even for extremely faint images (signal-to-
noise ratio $<$ 10). The pipeline was tested both with simulations (Zhang et
al., 2015) and with observations (See Zhang et al. (2019) for the CFHTLenS
data and Wang et al. (2021); Zhang et al. (2022a) for the DECaLS data). The
shear catalog covers more than ten thousand square degrees in the $g$, $r$,
and $z$ bands, with 99, 111 and 116 million distinct galaxies, respectively.
The FOURIER$\\_$QUAD method counts images of the same galaxy but in different
exposures as different images.

A machine learning algorithm based on decision trees is implemented to
calculate photometric redshifts of galaxies in the shear catalog (Zhou et al.,
2021). Eight parameters, including the $r$-band magnitude, ($g-r$), ($r-z$),
($z-W1$), and ($W1-W2$) colors, half-light radius, axial ratio, and shape
probability, are adopted in the training progress. The photo-$z$ error of each
individual shear image is obtained by perturbing the photometry of the galaxy.
The procedures are as follows: (i) the uncertainty was assumed to follow a
Gaussian distribution with the standard deviation equal to the photometric
error; (ii) a random value generated from the distribution was added to the
observed flux in each band to obtain a “perturbed” flux; (iii) the
perturbation was repeated multiple times, and the standard deviation of the
photo-$z$ estimates from the perturbations was used as the error of the
photo-$z$ (see details in Zhou et al., 2021).

##  3 Methods of analysis

In this section, we describe the methods to measure halo mass and satellite
kinematics from observational data. In addition, we present our methods for
constructing combined halo mass proxies and for evaluating the performance of
these proxies.

###  3.1 Weak lensing measurements

We measure the excess surface density (ESD) by using the $r$\- and $z$-band
data of the DECaLS shear catalog. Zhang et al. (2017) proposed a probability
distribution function (PDF) symmetrization method to minimize the statistical
uncertainty in shear signal. We apply a modified version of this method and
estimate the ESD in physical coordinates by

$\Delta\Sigma(\rm R)=\gamma_{t}(\rm R)\Sigma_{\rm crit}.$ (1)

The details and general discussion about different sources of systematic
errors of the PDF-symmetrization method in measuring the ESD signal are given
in a companion paper (Wang et al., 2022). Error bars of ESD signals are
estimated by using 150 bootstrap samples (Barrow et al., 1984).

To model the ESD signal, we follow previous studies (e.g. Mandelbaum et al.,
2008; Leauthaud et al., 2010; Luo et al., 2018; Zhang et al., 2022b) and apply
a halo model consisting of three terms:

$\Delta\Sigma=\Delta\Sigma_{\rm stellar}+\Delta\Sigma_{\rm
NFW}+\Delta\Sigma_{\rm 2h}.$ (2)

The first term is the stellar mass term that represents the contribution from
the galaxy stellar mass. We treat the galaxy as a point mass and adopt the
average stellar mass of the galaxy sample directly from the observation. The
second term is the one-halo term, which is the contribution from the host dark
matter halo, and we assume the halo to follow the Navarro–Frenk–White (NFW;
Navarro et al., 1997) density profile. The NFW profile has two free
parameters: the dark halo mass($m_{\rm h}$) and the halo concentration($c$).
The dark halo mass $m_{\rm h}$ is defined as the total dark matter mass within
a spherical region of radius $r_{\rm 200m}$. Inside this region, the mean mass
density is equal to 200 times the mean matter density of the Universe. We use
the central galaxy as the tracer of the halo center. We adopt the equations
from Yang et al. (2006) to analytically calculate the one-halo term. In the
modeling process, we ignore the off-center effect (Mandelbaum et al., 2016),
which has been shown to be a minor factor in halo mass estimation (Wang et
al., 2022). The third term is the two-halo term, which represents the
contribution from other halos. To calculate this term, we project the halo-
matter cross-correlation function, $\xi_{\rm hm}$, along the line-of-sight.
Here, $\xi_{\rm hm}$ = $b(m_{\rm h})\xi_{\rm mm}$, with $b(m_{\rm h})$ being
the halo bias (e.g. Mo & White, 1996; Tinker et al., 2010) and $\xi_{\rm mm}$
being the linear matter-matter correlation function. We use COLOSSUS (Diemer,
2018) to model both $b(m_{\rm h})$ and $\xi_{\rm mm}$.

There are two free parameters ($m_{\rm h}$ and $c$) in our model. The priors
of these parameters are set to be flat, with $m_{\rm h}$ in the range of
[10.0, 16.0] in logarithmic space and $c$ in the range of [1.0, 16.0]. To
constrain them, we use $emcee$ (Foreman-Mackey et al., 2013) to run a Monte
Carlo Markov Chain (MCMC). In the following, the results shown are the median
values of the posteriors, and the error bars correspond to the 16 and 84
percent of the posterior distribution (See Luo et al. (2018) for more details
on the modeling and fitting process). Since we include the stellar mass term
in our model, $m_{\rm h}$ only accounts for cold dark matter, diffuse gas, and
satellites around centrals. Thus, throughout this paper, we use the total mass
$M_{\rm h}=m_{\rm h}+M_{*}$ instead of $m_{\rm h}$ to represent the halo mass.

###  3.2 Satellite kinematics

Figure 1: Upper panels: the data points and the contours show the
distributions of satellite candidates in the projected phase space ($r_{\rm
p}$-$\Delta v$). Lower panels: the probability distributions of $\Delta v$(red
and blue symbols) and the best-fitting curves for PDFs within 1.5$v_{\rm
200m}$ (red symbols). The two gray vertical lines in each panel correspond to
$\Delta v=\pm 1.5v_{\rm 200m}$. We only show the satellite candidates around
centrals with $11.2\leq\log M_{*}/{\rm M_{\odot}}<11.65$ in ATotal, AC0, AC1,
AC2 and AC3 samples. The halo masses and satellite velocity dispersion of the
samples are listed in the upper and lower panels, respectively. Please see
Section 3.2 for details.

Satellite kinematics can also serve as a powerful probe of dark matter halos
(e.g. McKay et al., 2002; van den Bosch et al., 2004; More et al., 2011;
Wojtak & Mamon, 2013; Lange et al., 2019b; Abdullah et al., 2020; Seo et al.,
2020). One major concern in satellite kinematics studies is the contamination
of interlopers (see e.g. van den Bosch et al., 2004; Mamon et al., 2010).
Recently, Zhang et al. (2022b) used weak lensing technique to estimate the
halo masses of their central galaxy samples and derived the corresponding halo
virial radius, $r_{\rm 200m}$, and halo virial velocity, $v_{\rm 200m}$. They
selected satellite candidates from a reference galaxy sample using the
following set of criteria: $r_{\rm p}\leq r_{\rm 200m}$, $|\Delta v|\leq
3v_{\rm 200m}$, and $M_{\rm s}<M_{*}$. Here, $M_{*}$ and $M_{\rm s}$ are the
stellar masses of the central and satellite candidate, respectively; $r_{\rm
p}$ is the projected distance from the satellite candidate to its central; and
$\Delta v$ is the line-of-sight velocity difference between them. They found
that the $M_{\rm h}$-$\sigma_{\rm s}$ relation so obtained has a power-law
slope of about 3, but the amplitude of the relation is lower than that
obtained from simulations (see Section 4.4).

We apply the same method to select satellite candidates for central galaxies.
The centrals are divided into samples based on either their $M_{*}$, or
$\sigma_{*}$, or $M_{\rm AM}$ (see Section 4.4). The distribution of the
corresponding satellite candidates in the phase space ($r_{\rm p}$ versus
$\Delta v$) are shown in Figure 1. Here we only present the results for
galaxies with $11.2\leq\log M_{*}/{\rm M_{\odot}}<11.65$ in A-series samples.
The results for other galaxy samples/subsamples are similar. As one can see,
there are cone-like structures in the phase space for ATotal, AC1, AC2 and
AC3, as indicated by black contour lines, respectively. The PDFs of $\Delta v$
are shown in the lower panels. The distributions for the four subsamples are
very similar to a Gaussian distribution. The peaks corresponding to the cone-
like structures are contributed predominantly by satellites, while the tails
that are outside of the cone are dominated by interlopers (Mamon et al.,
2010). In contrast, AC0 galaxies has an empty cone-like structure and double-
peaked $\Delta v$ distribution. This is expected, because AC0 centrals have no
satellite according to the group catalog, and the shown galaxies are mostly
interlopers.

To eliminate interlopers as far as possible, we decide to only use satellite
candidates with $|\Delta v|<|\Delta v|_{\rm cut}=1.5v_{\rm 200m}$ to estimate
the satellite velocity dispersion. Our visual inspection shows that the cut at
$1.5v_{\rm 200m}$ roughly corresponds to the boundary of the cone-like
structures, as shown by the gray vertical lines in Figure 1. We performed a
series of tests using a value of $|\Delta v|_{\rm cut}$ ranging from 1.0 to
3.0$v_{\rm 200m}$, and found that the results are stable as long as $|\Delta
v|_{\rm cut}$ is in the range of 1.3-1.8$v_{\rm 200m}$. A smaller $|\Delta
v|_{\rm cut}$ excludes too many satellites in the cone-like structures and
leads to poorer constraints. A larger $|\Delta v|_{\rm cut}$, on the other
hand, leads to a larger dispersion because of the inclusion of more
interlopers. Please see Section 4.4 for more discussion.

The red symbols show the distributions of $|\Delta v|$ within $1.5v_{\rm
200m}$, with error bars estimated from 100 bootstrap samples. We fit the
distributions to a functional form that is a Gaussian plus a constant,

$P(\Delta v)=\frac{A}{\sqrt{2\pi}\sigma_{\rm v}}e^{-\Delta
v^{2}/2\sigma^{2}_{\rm v}}+d.$ (3)

The constant $d$ is used to account for interlopers within the cone. We use
the MCMC to fit the PDFs. The uncertainty of a SDSS galaxy redshift, which is
about 35 km $\rm s^{-1}$, results in an error of $\sigma_{\rm e}$ =
$\sqrt{2}\times 35$ = 49.5 km $\rm s^{-1}$. We correct this and obtain the
satellite velocity dispersion using

$\sigma_{\rm s}=\sqrt{\sigma_{\rm v}^{2}-\sigma_{\rm e}^{2}}\,.$ (4)

To compare with Zhang et al. (2022b), we use the same reference galaxy sample
to select satellite candidates. This sample, selected from NYU-VAGC sample
dr72(Blanton et al., 2005), contains 510,605 galaxies, with $r$-band Petrosian
apparent magnitude $r<$ 17.6, $r$-band Petrosian absolute magnitude in the
range of $-24<M_{{}^{0.1}\rm r}<-16$, and the redshift in the range of
$0.01<z<0.2$. Here, $M_{{}^{0.1}\rm r}$ is the $r$-band Petrosian absolute
magnitude with $K+E$ corrected to the value at $z=0.1$ (see Wang & Li, 2019,
for more details).

###  3.3 The combined halo mass proxies

In this section, we introduce our method for deriving new halo mass proxies by
linearly combining two proxies presented above. Since we have no $\sigma_{\rm
s}$ measurements for individual galaxies, we only use $M_{*}$, $\sigma_{*}$
and $M_{\rm AM}$ to design new proxies. The three parameters have very
different dynamical ranges, it is thus necessary to scale them before the
combination. For a proxy D, we design a new parameter,

$\rm Rank_{D}=\frac{D-min(D)}{max(D)-min(D)}\\\ ,$ (5)

where $\rm min(D)$ and $\rm max(D)$ represent the minimum and maximum values
of the proxy, respectively. A new proxy can then be designed by combining two
proxies D and E as,

$\rm Pr(D,E,\emph{p})=(1-\emph{p})*Rank_{D}+\emph{p}*Rank_{E}\\\ ,$ (6)

where $p$ is in the range of $0\leq p\leq 1$. With $p=0$, $\rm
Pr(D,E,\emph{p})$ is equivalent to the proxy D, while with $p=1$, $\rm
Pr(D,E,\emph{p})$ is equivalent to the proxy E. Since $M_{\rm AM}$ is used, we
use the B-series samples to examine the new proxies.

One way to evaluate the performance of a proxy is to check the dependence of
the $M_{\rm h}$-proxy scaling relation on other galaxy and group properties,
such as $N_{\rm sat}$ and SFR. We choose to use the differences in the $M_{\rm
h}$-proxy scaling relations between BC0 and BC3 subsamples and between BSF and
BQ subsamples to quantify the dependencies on $N_{\rm sat}$ and SFR,
respectively. We first divide the BTotal sample into $N_{\rm b}=8$ equally-
sized bins according to a new proxy, $\rm Pr(D,E,\emph{p})$. Then we select
galaxies that belong to BC0, BC3, BSF and BQ subsamples and derive their halo
masses at each $\rm Pr(D,E,\emph{p})$ bin based on weak lensing, respectively.
We now have $M_{\rm h}$-$\rm Pr(D,E,\emph{p})$ relations for BC0, BC3, BSF and
BQ subsamples. We use the following formula to calculate the difference
between two scaling relations,

$d_{\rm para}=\frac{1}{N_{\rm b}}\Sigma_{i=1}^{N_{\rm b}}\,\frac{(\log
M_{i,1}-\log M_{i,2})^{2}}{err_{i,1}^{2}+err_{i,2}^{2}}\\\ ,$ (7)

where $\log M_{i,1}$, $err_{i,1}$, $\log M_{i,2}$ and $err_{i,2}$ are the halo
masses and their uncertainties at the $i$th bin for the two scaling relations,
respectively. Sometimes, the mean $\rm Pr(D,E,\emph{p})$ at the largest or
smallest $\rm Pr(D,E,\emph{p})$ bin is very different between BC0 and BC3 (or
between BSF and BQ). In order to compare the two scaling relations in a fair
way, the halo mass and its uncertainty used in Equation 7 for BC3 and BSF are
calculated using the linear interpolation method. To evaluate the uncertainty
of $d_{\rm para}$, we use the Monte Carlo method to generate $\log M_{i,1}$
and $\log M_{i,2}$ for each $\rm Pr(D,E,\emph{p})$ bin, which follow Gaussian
distributions with a mean equal to the weak lensing mass and a dispersion
equal to the uncertainty. These new parameters can be used to calculate a new
$d_{\rm para}$. This procedure is repeated 500 times and we take the
dispersion of the 500 $d_{\rm para}$ as the uncertainty. In the following, we
use $d_{\rm SFR}$ and $d_{N_{\rm sat}}$ to represent the dependencies of the
scaling relation on SFR and $N_{\rm sat}$, respectively (Section 4.5).

##  4 Halo mass - proxy scaling relations

###  4.1 Stellar mass-halo mass scaling relation

Figure 2: Excess surface density (ESD) profiles and the best fitting results.
Different rows correspond to different $M_{*}$ bins, and different columns
correspond to different galaxy samples, including ATotal, AC0, AC1, AC2 and
AC3. The error bars correspond to the standard deviation of 150 bootstrap
samples. The best-fitting $M_{\rm h}$ are also presented. Figure 3: The
dependence of SHMR on $N_{\rm sat}$(left), SFR(middle) and
$\sigma_{*}$(right). The black lines in the three panels show the SHMR for
ATotal sample. The error bars reflect the 16% and 84% of the posterior
distribution. The shaded region corresponds to the region covered by the SHMR
curves published in Yang et al. (2009), Moster et al. (2010), Leauthaud et al.
(2012), Kravtsov et al. (2018), and Behroozi et al. (2019).

We divide galaxies from the ATotal sample with $M_{*}\geq 10^{8.5}{\rm
M_{\odot}}$ into 8 $M_{*}$ bins, equally spaced in the logarithm of $M_{*}$
with a bin size of 0.45 dex. Then, we use methods presented in Section 3 to
obtain the ESD and derive the halo mass for each $M_{*}$ bin. The ESD profiles
and their best-fittings of four selected $M_{*}$ bins are presented in Figure
2. The resulting SHMR is shown in Figure 3 as black solid diamonds. For
comparison, we also show the SHMRs from the literature obtained using various
methods, including galaxy groups, abundance matching, conditional luminosity
function, weak lensing, and empirical models (e.g. Yang et al., 2009; Moster
et al., 2010; Leauthaud et al., 2012; Kravtsov et al., 2018; Behroozi et al.,
2019). The SHMR for ATotal is well consistent with these results.

Many parameters are correlated with $M_{\rm h}$ and $M_{*}$. These
correlations may reflect possible dependencies of the SHMR on these
parameters. We first consider $N_{\rm sat}$. As shown in Section 2.1, the
ATotal sample can be split into AC0, AC2 and AC3 subsamples according to
$N_{\rm sat}$. The ESDs and their best fitting profiles for these subsamples
in four stellar mass bins are presented in Figure 2. At a given $M_{*}$, the
amplitude of the ESD signal increases with increasing group richness. A direct
comparison of the SHMRs can be found in the left panel of Figure 3, where only
mass bins with more than 50 galaxies are presented. At the high mass end, the
AC0 galaxies have a SHMR that tends to be slightly lower than that of ATotal,
and AC3 galaxies tend to have the highest SHMR. We do not show the result for
AC1 sample because AC1 is the sum of AC2 and AC3.

At given $M_{*}$, the halo mass is positively correlated with group richness
(Johnston et al., 2007; Andreon et al., 2008; Viola et al., 2015) and the
difference in $M_{\rm h}$ among samples is quite large. For the four most
massive $M_{*}$ bins, the differences in $M_{\rm h}$ between AC0 and AC3 are
in the range of 0.5-0.8 dex. The SDSS galaxy sample is a magnitude limited
sample. Some groups in AC0 sample may have large intrinsic $N_{\rm sat}$, but
their satellites are not detected due to the Malmquist bias. Such an effect
dilutes the dependence on group richness. Therefore, the intrinsic dependence
should be stronger than what we find here.

We then study the dependence of SHMR on SFR. The SHMR for star-forming (ASF)
and quenched (AQ) galaxies are shown in the middle panel of Figure 3. Overall,
star-forming galaxies reside in halos with a mean halo mass much lower than
quenched ones of the same $M_{*}$, consistent with previous studies (e.g. More
et al., 2011; Rodríguez-Puebla et al., 2015; Mandelbaum et al., 2016; Behroozi
et al., 2019; Bilicki et al., 2021; Zhang et al., 2021, 2022b). The quenched
galaxies in the second lowest mass bin have a much higher halo mass compared
to the star-forming ones. One possible reason is that a significant fraction
of galaxies in this mass bin are satellites, which are misidentified as
centrals by group finder. Because satellites usually reside in more massive
halos than centrals of the same mass, the misidentification can lead to
overestimation of halo mass. We examined their ESD profile and found a bump-
like structure from 0.1-1 Mpc/h, which is similar to the lensing signal
produced by satellite galaxies(Yang et al., 2006; Li et al., 2014). Another
possibility is that some of these galaxies reside in the splashback halos
(Wang et al., 2023b). Since splashback halos have ever entered massive halos,
they are close to massive halos and thus have similar bump on their ESD.
Moreover, these galaxies are subject to a variety of environmental effects and
thus very likely to be quenched.

To investigate the dependence of SHMR on $\sigma_{*}$, we divide the ATotal
sample in each stellar mass bin into small and large $\sigma_{*}$ subsamples
by the median $\sigma_{*}$ of each $M_{*}$ bin. Their SHMRs are shown in the
right panel of Figure 3. We can see that the SHMR has the strongest dependence
on $\sigma_{*}$ in the middle stellar mass range. This dependence weakens
towards the lower and higher stellar mass end. We will come back to the
correlations among $M_{\rm h}$, $M_{*}$ and $\sigma_{*}$ in the next
subsection.

###  4.2 Halo mass-galaxy central stellar velocity dispersion scaling relation

Figure 4: The dependence of the $M_{\rm h}$-$\sigma_{*}$ relation on $N_{\rm
sat}$(left), SFR(middle) and $M_{*}$(right). The black lines in the three
panels show the $M_{\rm h}$-$\sigma_{*}$ relation for ATotal sample. The
light-blue dot-dashed line and the shaded region show the relation from Zahid
et al. (2016) and its scatter.  Figure 5: The dependence of $M_{\rm
h}$-$\sigma_{*}$ relation on $M_{*}$ (left panel) and $M_{\rm h}$-$M_{*}$
relation on $\sigma_{*}$ (right panel). The black lines in the two panels show
the results for ATotal sample. The color-coded lines in the left (right) panel
show the results for $M_{*}$-controlled ($\sigma_{*}$-controlled) subsamples.
Please see Section 4.2 for details. The dot-dashed line and the shaded region
in the left panel are the same as those in Figure 4. And the shaded region in
the right panel is the same as that in Figure 3.

In this subsection, we use the galaxy stellar velocity dispersion
($\sigma_{*}$) as a halo mass proxy and investigate the $M_{\rm
h}$-$\sigma_{*}$ scaling relation. The ATotal sample is divided into 8
$\sigma_{*}$ bins. The bin size varies with $\sigma_{*}$ to ensure that each
bin has reliable $M_{\rm h}$ measurement as much as possible. The $M_{\rm
h}$-$\sigma_{*}$ relation for the ATotal sample is shown in Figure 4 as black
solid diamonds. We can see that galaxies with larger $\sigma_{*}$ reside in
more massive halos. For comparison, we also present the result of Zahid et al.
(2016) who used the SHMR to assign the halo mass to each galaxy. Overall, our
result is in good agreement with theirs.

We then examine the dependence of the $M_{\rm h}$-$\sigma_{*}$ relation on
various parameters and only present the $\sigma_{*}$ bins with galaxy number
greater than 50. We first investigate the dependence on $N_{\rm sat}$ (left
panel of Figure 4). Similar to the SHMR, the $M_{\rm h}-\sigma_{*}$ relation
strongly depends on $N_{\rm sat}$. The uncertainties of halo masses at low
$\sigma_{*}$ bins are very large, no clear $M_{\rm h}$-$N_{\rm sat}$
correlation can be found in this regime. However, at the four largest
$\sigma_{*}$ bins where the halo masses are well constrained, $M_{\rm h}$
increases significantly with $N_{\rm sat}$ at fixed $\sigma_{*}$. And the
$M_{\rm h}$ difference between AC0 and AC3 is about 0.4-0.8 dex, similar to
that for SHMR.

We now check the dependence of the $M_{\rm h}$-$\sigma_{*}$ relation on
SFR(the middle panel of Figure 4). In the low $\sigma_{*}$ range
($\log\sigma_{*}<2.1$), because of the small number of quenched galaxies, the
uncertainties in the halo masses of quenched galaxies are very large. In the
middle $\sigma_{*}$ range ($2.1<\log\sigma_{*}<2.4$), the halo mass difference
at a given $\sigma_{*}$ between star-forming and quenched galaxies is
typically 0.2-0.3 dex, apparently smaller than the difference in the middle
$M_{*}$ range, typically 0.4-0.5 dex. This is not surprising, as $\sigma_{*}$
is the most important factor for star formation quenching(see e.g. Bluck et
al., 2020). For the results in the largest $\sigma_{*}$ bin, the halo mass of
the star-forming galaxies is significantly lower than that of the
corresponding quenched galaxies. This result can be attributed to the fact
that the star-forming galaxies contain a batch of galaxies with high star-
forming efficiency(see Zhang et al., 2022b). At a given $\sigma_{*}$, the halo
masses of star-forming and quenched galaxies are still significantly
different. It suggests that other processes, likely related to halo mass, also
play an important role in quenching star formation.

We then split each $\sigma_{*}$ bin into two equally-sized subsamples
according to their $M_{*}$. The results for these subsamples are presented in
the right panel of Figure 4. The uncertainties in the two lowest $\sigma_{*}$
bins are too large to allow quantifying the difference. For larger
$\sigma_{*}$, the dependence of the $M_{\rm h}$-$\sigma_{*}$ relation on
$M_{*}$ is almost absent at $\log\sigma_{*}\sim 2.3$ and becomes stronger
towards lower and higher $\sigma_{*}$. It is thus interesting to investigate
the bivariate correlation in more detail. We perform two tests. In the first
test, we investigate the $M_{\rm h}$-$\sigma_{*}$ relation at a given $M_{*}$.
For galaxies in a given $M_{*}$ bin, we split them into several(2 to 5,
dependent on the weak lensing signal) equally-sized subsamples by
$\sigma_{*}$. Our inspection shows that these subsamples with different
$\sigma_{*}$ are significantly different in the $M_{*}$ distribution though
they are selected within the same $M_{*}$ bin. To eliminate the potential
bias, we re-select galaxies from these $\sigma_{*}$ subsamples so that they
have similar $M_{*}$ distributions (see Appendix for the method). The re-
selected subsamples are referred to as the $M_{*}$-controlled subsamples. In
the second test, we investigate the SHMR at a given $\sigma_{*}$. Similar to
the first test, we construct $\sigma_{*}$-controlled subsamples to eliminate
the influence of $\sigma_{*}$.

The left panel of Figure 5 shows the $M_{\rm h}$-$\sigma_{*}$ relations for
$M_{*}$-controlled subsamples. At $\log\sigma_{*}<2.1$, $M_{\rm h}$ is almost
independent of $\sigma_{*}$ but strongly dependent on $M_{*}$. It suggests
that for these galaxies, the observed $M_{\rm h}$-$\sigma_{*}$ relation is
dominated by the $M_{\rm h}$-$M_{*}$ relation. At $2.1<\log\sigma_{*}<2.4$, we
see a strong dependence of $M_{\rm h}$ on $\sigma_{*}$ even when $M_{*}$ is
well controlled. Interestingly, galaxies in two different $M_{*}$ bins(green
and blue) follow almost the same trend, defined by the ATotal sample. It means
that the SHMR in this range is driven by $\sigma_{*}$, totally opposite to the
trend seen in low-mass galaxies. At $\log\sigma_{*}>2.4$, we see the
dependence on both $M_{*}$ and $\sigma_{*}$.

Similar conclusions can also be drawn from the SHMR results for the
$\sigma_{*}$-controlled subsamples, as shown in the right panel of Figure 5.
In the lowest three $\sigma_{*}$ bins, the SHMRs generally follow the same
trend as that of the ATotal sample in the low stellar mass range ($\log
M_{*}/{\rm M_{\odot}}<10.4$). It means that $\sigma_{*}$ has a negligible
impact on SHMR at the low $M_{*}$ end. In the middle stellar mass range
($10.4<\log M_{*}/{\rm M_{\odot}}<11.1$), the SHMRs in three $\sigma_{*}$ bins
($1.9<\log\sigma_{*}<2.8$) deviate from each other and all have very flat
slopes. It means that $M_{\rm h}$ is independent of $M_{*}$ after controlling
$\sigma_{*}$. Therefore, the SHMR in this mass range is mainly driven by the
$M_{\rm h}$-$\sigma_{*}$ relation. In the massive stellar mass end ($\log
M_{*}/{\rm M_{\odot}}>11.1$), the SHMR in the largest $\sigma_{*}$ bin is
almost consistent with the SHMR of ATotal sample. However, since there is only
one $\sigma_{*}$ bin at this mass range, we can not disentangle the
correlations with $\sigma_{*}$ and $M_{*}$. In general, these results are in
good agreement with those from Figure 3 and 4.

One important application of the $M_{\rm h}$-$\sigma_{*}$ relation is that it
can be used to study the connection between dark matter halos and supper-
massive black holes in galaxy center, which are thought to play an essential
role in galaxy evolution. It is well known that $\sigma_{*}$ is tightly
correlated with the black hole mass ($M_{\rm BH}$). We can convert the $M_{\rm
h}$-$\sigma_{*}$ relation into $M_{\rm h}$-$M_{\rm BH}$ relation using the
$M_{\rm BH}$-$\sigma_{*}$ relation. Here, we adopt seven $M_{\rm
BH}$-$\sigma_{*}$ relations(Ferrarese & Merritt, 2000; Tremaine et al., 2002;
Graham & Scott, 2013; Kormendy & Ho, 2013; McConnell & Ma, 2013; Woo et al.,
2013; Saglia et al., 2016), which are derived from different galaxy samples.
The resultant $M_{\rm h}$-$M_{\rm BH}$ relations are shown in Figure 6.

Figure 6: $M_{\rm h}$-$M_{\rm BH}$ relations. We apply seven $M_{\rm
BH}$-$\sigma_{*}$ relations from the literature as indicated to convert the
$M_{\rm h}$-$\sigma_{*}$ relation into the $M_{\rm h}$-$M_{\rm BH}$ relations.
For comparison, we show the model result of Bower et al. (2017) in purple
dashed line and the observational result of Shankar et al. (2020) in black
dashed line.

Though the uncertainty in the $M_{\rm BH}$-$\sigma_{*}$ relation leads to a
large uncertainty in the $M_{\rm BH}$ estimation, all of these results follow
a general trend. When $M_{\rm BH}$ increases rapidly from $10^{5.0}$ to
$10^{7.4}{\rm M_{\odot}}$, $M_{\rm h}$ only increases about 0.2 dex, from
$10^{11.8}$ to $10^{12.0}{\rm M_{\odot}}$. It hints at a rapid growth of black
holes as their host halos reach $\log M_{\rm h}/{\rm M_{\odot}}\sim 12.0$,
consistent with that AGNs usually reside in halos of $\log M_{\rm h}/{\rm
M_{\odot}}\sim 12.0$(e.g. Zhang et al., 2021). At $\log M_{\rm h}/{\rm
M_{\odot}}>12.0$, when $M_{\rm BH}$ increases from $10^{7.4}$ to $10^{9}{\rm
M_{\odot}}$, $M_{\rm h}$ increases from $10^{12.0}$ to $10^{13.4}{\rm
M_{\odot}}$, indicating a nearly linear correlation between $M_{\rm BH}$ and
$M_{\rm h}$. Our $M_{\rm h}$-$M_{\rm BH}$ relation shows a transition in black
hole growth at $\log M_{\rm h}/{\rm M_{\odot}}\sim 12.0$ and $\log M_{\rm
BH}/{\rm M_{\odot}}\sim 7.4$.

It is possible that the independence of $M_{\rm h}$ on $M_{\rm BH}$ at low
mass end is caused by the relatively large uncertainties of the low
$\sigma_{*}$. Because large uncertainties can erase the intrinsic difference
among galaxies in several low $\sigma_{*}$ bins. To check this possibility, we
investigate the errors of $\sigma_{*}$ measurement, taken from NYU-VAGC, in
the four lowest $\sigma_{*}$ bins of the ATotal sample. Our tests show that
the four lowest $\sigma_{*}$ bins (at least the second to fourth bin) have
significantly different intrinsic $\sigma_{*}$ considering the errors. It is
also supported by the fact that the $M_{*}$ distributions in the four bins are
significantly different. Another important factor is the uncertainty of
$M_{\rm BH}$-$\sigma_{*}$ relation at the low mass end. We adopt seven
relations, which are obtained using different galaxy sample and have different
slopes. Similar transition is seen in all results. It is expected because the
transition originates from the $M_{\rm h}$-$\sigma_{*}$ relation where the
transition is clearly presented. We also calculate the $M_{\rm BH}$-$M_{\rm
h}$ relation for early-type galaxies and find a similar transition. These
tests indicate the uncertainties in $\sigma_{*}$ measurement and $M_{\rm
BH}$-$\sigma_{*}$ relation do not significantly change our conclusion.

For comparison, we show the $M_{\rm h}$-$M_{\rm BH}$ relation obtained by
Shankar et al. (2020). Shankar et al. (2020) estimated $M_{\rm h}$ using the
abundance matching technique and $M_{\rm BH}$ using the $M_{*}$-$M_{\rm BH}$
relation for quiescent galaxies(Saintonge et al., 2016). In general, our
predictions are consistent with theirs, but lower at high mass and higher at
low mass. Shankar et al. (2020) also provided a $M_{\rm h}$-$M_{\rm BH}$
relation based on a so-called bias-corrected galaxy sample, which
significantly deviates from ours. Because we adopt different methods to derive
$M_{\rm h}$ and $M_{\rm BH}$, it is difficult to determine the exact cause of
this discrepancy. We also show the result of an analytic model developed by
Bower et al. (2017). Their model prediction is in good agreement with our
observational result at $\log M_{\rm BH}/{\rm M_{\odot}}>6$. They suggested
that when $M_{\rm h}$ is close to $10^{12}{\rm M_{\odot}}$, star-formation
feedback can not effectively drive outflow and black holes start to grow
rapidly. When $M_{\rm h}>10^{12}{\rm M_{\odot}}$, the ISM will eventually be
consumed by star formation or expelled by AGN feedback, and the black hole
growth rate is reduced. Bower et al. (2017) assumed that the further growth of
black holes is limited by the binding energy within the cooling radius of
their host halos, which leads to the linear relationship between the halo mass
and the black hole mass at $\log M_{\rm h}/{\rm M_{\odot}}>12$. Apparently,
the $M_{\rm h}$-$\sigma_{*}$ and $M_{\rm h}$-$M_{\rm BH}$ relations contain
valuable information about galaxy formation and evolution. We will come back
to this issue in the near future (Li. et al. in preparation).

###  4.3 Halo mass-abundance-matching mass scaling relation

In this subsection, we adopt the abundance matching halo mass (hereafter
$M_{\rm AM}$) based on group stellar mass as the halo mass proxy. The
following tests are based on the B-series samples. We divide the BTotal sample
into 8 $M_{\rm AM}$ bins, the sizes of which are varied to obtain reliable
halo mass measurements as far as possible. The resultant $M_{\rm h}$-$M_{\rm
AM}$ relation is shown in Figure 7 as black solid diamonds. For comparison, we
also present the $M_{\rm h}$-$M_{\rm AM}$ relation from Luo et al. (2018),
shown as the cyan circles. Our result is in excellent agreement with that of
Luo et al. (2018). The $M_{\rm h}$-$M_{\rm AM}$ relation is very close to a
one-to-one relation. At $\log M_{\rm h}/{\rm M_{\odot}}>12.5$, the weak
lensing measured halo mass is slightly smaller than the abundance matching
mass. At $\log M_{\rm h}/{\rm M_{\odot}}<12.5$, $M_{\rm h}$ becomes slightly
larger than $M_{\rm AM}$.

We then examine the dependence of the $M_{\rm h}$-$M_{\rm AM}$ relation on
various parameters. The upper-left panel shows the dependence on group
richness. Different from the SHMR and $M_{\rm h}$-$\sigma_{*}$ relation, the
dependence of the $M_{\rm h}$-$M_{\rm AM}$ relation on $N_{\rm sat}$ is much
weaker. $M_{\rm AM}$ is estimated based on the sum of stellar masses of group
member galaxies (see Yang et al., 2007, for the details). Therefore, the
satellite information has already been involved in the estimation of $M_{\rm
AM}$. At low mass, the result for the BTotal is dominated by the BC0 galaxies,
which have no satellite. It means that the $M_{\rm h}$-$M_{\rm AM}$ relation
is approximately equivalent to the $M_{\rm h}$-$M_{*}$ relation. It might
explain the small change in the $M_{\rm h}$-$M_{\rm AM}$ relation at $\log
M_{\rm h}/{\rm M_{\odot}}\sim 12.5$. We also divide galaxies in each $M_{\rm
AM}$ bin into two equally-sized subsamples according to their $M_{*}$. The
$M_{\rm h}$-$M_{\rm AM}$ relations for the small and large $M_{*}$ galaxies
are presented in the lower-left panel of Figure 7. As one can see, they both
closely follow the overall $M_{\rm h}$-$M_{\rm AM}$ relation and exhibit
almost no difference. It means that the $M_{\rm h}$-$M_{*}$ relation can be
fully explained by the $M_{\rm h}$-$M_{\rm AM}$ relation. Therefore, $M_{\rm
AM}$ is a more powerful proxy than $M_{*}$. It is not surprising because
$M_{\rm AM}$ is estimated based on the group stellar mass, which also includes
the contribution of central galaxies.

The $M_{\rm h}$-$M_{\rm AM}$ relations for star-forming and quenched galaxies
are shown in the upper-right panel of Figure 7. The halo masses for the
quenched galaxies are apparently higher than the corresponding star-forming
galaxies over all $M_{\rm AM}$ range in consideration. The halo mass
differences are rather large, ranging from $\sim$ 0.3 to 0.9 dex, comparable
to the difference for the $M_{\rm h}$-$M_{*}$ relation and larger than that
for the $M_{\rm h}$-$\sigma_{*}$ relation. The dependence of the $M_{\rm
h}$-$M_{\rm AM}$ relation on $\sigma_{*}$ is also significant (lower-right
panel). At low $M_{\rm AM}$, the dependence on $\sigma_{*}$ is weak. At middle
$M_{\rm AM}$, the dependence becomes very pronounced, with large $\sigma_{*}$
galaxies residing in more massive halos than the corresponding small
$\sigma_{*}$ galaxies. At high $M_{\rm AM}$, this dependence becomes weak, and
almost disappears in the largest $M_{\rm AM}$ bin. Overall, $M_{\rm AM}$ is a
better proxy than $M_{*}$ and $\sigma_{*}$. However, the dependencies on SFR
and $\sigma_{*}$ suggest that the $M_{\rm h}$-$M_{\rm AM}$ scaling relation is
still sensitive to galaxy formation processes.

Figure 7: The dependence of the $M_{\rm h}$-$M_{\rm AM}$ relation on $N_{\rm
sat}$ (upper-left), SFR(upper-right), $M_{*}$(lower-left) and
$\sigma_{*}$(lower-right). The cyan circles show the results from Luo et al.
(2018), and the gray line in each panel shows the one-to-one relation.

###  4.4 Halo mass-satellite velocity dispersion scaling relation

Figure 8: The $M_{\rm h}$-$\sigma_{\rm s}$ relations using $M_{*}$ (crosses)
or $\sigma_{*}$ (diamonds) as a tracer. Different colors represent different
$M_{*}$ or $\sigma_{*}$ bins as labeled in the two panels. The left and right
panels show the results based on Atotal and AC1 samples, respectively. The
black dashed and solid lines are the best-fittings to $M_{*}$\- and
$\sigma_{*}$-tracer relations, respectively. The red, purple and blue dashed
lines show the simulation results. The gray solid crosses are taken from Zhang
et al. (2022b) and the gray line is its best-fitting (left panel). Please see
Section 4.4 for the details.

The halo mass-satellite velocity dispersion ($\sigma_{\rm s}$) scaling
relation has been widely studied in the literature (e.g. Rines & Diaferio,
2006; Yang et al., 2007; Hoekstra, 2007; Evrard et al., 2008; Munari et al.,
2013; Rines et al., 2013; Saro et al., 2013; Ruel et al., 2014; Gonzalez et
al., 2015; Han et al., 2015; Viola et al., 2015; Rines et al., 2016; Abdullah
et al., 2020; Gonzalez et al., 2021; Rana et al., 2022). Most of these studies
used the $\sigma_{\rm s}$ data measured for individual galaxy groups/clusters
even though these groups have only a few member galaxies. We, however, adopt a
stacking method to measure the average $\sigma_{\rm s}$ for a given galaxy
sample (see the method description in Section 3.2). Our method needs a halo
mass tracer to divide a galaxy sample into several bins. As shown in the above
sections, $M_{*}$, $\sigma_{*}$ and $M_{\rm AM}$ are all strongly correlated
with halo mass. We thus decide to use these three quantities as halo mass
tracers. When we use $M_{*}$ and $\sigma_{*}$ as tracers, the tests are
performed on the A-series samples. When $M_{\rm AM}$ is considered, the tests
are performed on the B-series samples. We divide the galaxy samples into 8
bins according to these tracers. Since both the weak lensing and satellite
kinematics techniques are performed using the stacking method, the
uncertainties are very large when the sample size is small. For clarity, we
only present the results with galaxy number greater than 100.

Table 2: The fitting parameters for $M_{\rm h}$-$\sigma_{\rm s}$ relation

Reference |  $b$(a) |  $k$(b) |  Tracer(c) |  Data(d)
---|---|---|---|---
Evrard et al. (2008) |  6.28 |  2.98 |  – |  simulation
Munari et al. (2013) |  6.85 |  2.75 |  – |  simulation
Saro et al. (2013) |  6.66 |  2.91 |  – |  simulation
Han et al. (2015) |  7.77${}^{+0.08}_{-0.08}$ |  2.17${}^{+0.34}_{-0.34}$ |  $\sigma_{\rm s}$ |  observation
Viola et al. (2015) |  9.07${}^{+0.06}_{-0.07}$ |  1.89${}^{+0.27}_{-0.27}$ |  $\sigma_{\rm s}$ |  observation
Rana et al. (2022) |  10.03${}^{+0.02}_{-0.02}$ |  1.52 ${}^{+0.1}_{-0.1}$ |  $\sigma_{\rm s}$ |  observation
Zhang et al. (2022b) |  5.33${}^{+0.11}_{-0.11}$ |  3.08${}^{+0.05}_{-0.05}$ |  $M_{\rm*}$ |  observation
Sample |  $b$(a) |  $k$(b) |  Tracer(c) |  Data(d)
AC1 |  7.55${}^{+0.06}_{-0.06}$ |  2.45${}^{+0.03}_{-0.03}$ |  $M_{*}$ |  observation
AC1 |  6.94${}^{+0.09}_{-0.09}$ |  2.71${}^{+0.04}_{-0.04}$ |  $\sigma_{*}$ |  observation
BC1 |  7.31${}^{+0.06}_{-0.06}$ |  2.57${}^{+0.02}_{-0.02}$ |  $M_{\rm AM}$ |  observation
BC1, star-forming |  6.12${}^{+0.2}_{-0.22}$ |  3.08${}^{+0.1}_{-0.09}$ |  $M_{\rm AM}$ |  observation
BC1, quenched |  7.23${}^{+0.07}_{-0.07}$ |  2.61${}^{+0.03}_{-0.03}$ |  $M_{\rm AM}$ |  observation
BC2 |  6.38${}^{+0.13}_{-0.14}$ |  3.02${}^{+0.06}_{-0.06}$ |  $M_{\rm AM}$ |  observation
BC3 |  6.95${}^{+0.12}_{-0.12}$ |  2.72${}^{+0.05}_{-0.05}$ |  $M_{\rm AM}$ |  observation

  * (a)

Zero-point($b$) in Equation 8.

  * (b)

Slope($k$) in Equation 8.

  * (c)

Samples are divided into subsamples according to the tracer for fitting.

  * (d)

The origin of the data used for fitting.

We first show the results using $M_{*}$ and $\sigma_{*}$ as tracers in Figure
8. They are referred to as the $M_{*}$-tracer (solid crosses) and
$\sigma_{*}$-tracer (solid diamonds) relations, respectively. For reference,
we also present the scaling relations obtained from simulations(Evrard et al.,
2008; Munari et al., 2013; Saro et al., 2013). The blue dashed line shows the
relation from Evrard et al. (2008) that used dark matter particles to
calculate the satellite velocity dispersion. They obtained a slope of 2.98, in
good agreement with the virial scaling relation(a slope of 3). The red dashed
line shows the result from a hydro-dynamical simulation with star formation
and AGN feedback(Munari et al., 2013), in which the simulated galaxies are
used to calculate the dispersion. They got a slope of 2.75, slightly less than
3. The purple dashed line shows the relation from Saro et al. (2013), which
has a slope of 2.91. They used galaxies yielded by a semi-analytic galaxy
formation model and only considered galaxy clusters. The best-fitting
parameters for the three results are listed in Table 2. Note that their halo
masses have already been converted into our halo mass definition.

The results from Zhang et al. (2022b) are also presented in gray color, which
used $M_{*}$ as a tracer and adopted a larger $\Delta v$ cut of $3v_{\rm
200m}$. They obtained a slope close to the simulations and an amplitude,
however, much lower than the simulations (Table 2). Although $\sigma_{\rm s}$
in Zhang et al. (2022b) is overestimated, the halo mass estimated using the
$M_{\rm h}$-$\sigma_{\rm s}$ relation is still reliable. Because the $M_{\rm
h}$-$\sigma_{\rm s}$ relation is calibrated by using weak lensing measured
halo mass. The reasons for the large discrepancy between the $M_{\rm
h}$-$\sigma_{\rm s}$ relation of Zhang et al. (2022b) and the simulation
relations are twofold: (1) the contamination of interlopers as discussed in
Section 3.2; (2) the weighting bias discussed in van den Bosch et al. (2004).
To check the first issue, we adopt a $\Delta v$ cut of $1.5v_{\rm 200m}$ and
present the results of ATotal sample in the left panel of Figure 8. As one can
see, a small $\Delta v$ cut can significantly reduce the contamination.
However, the discrepancy is still severe. AC0 galaxies have no satellite and
thus contribute little to the $\sigma_{\rm s}$ measurement, but they have on
average lower $M_{\rm h}$ than AC1 galaxies and thus significantly lower the
mean halo mass. It means that AC0 galaxies have different weighting factors in
the measurements of $M_{\rm h}$ and $\sigma_{\rm s}$. To reduce the weighting
bias, we exclude AC0 galaxies and apply our method to AC1 galaxies only. The
new scaling relations lie very close to the simulation results, as shown in
the right panel of Figure 8.

We use a power-law function to fit the data,

$\log M_{\rm h}/{\rm M_{\odot}}=k\log\sigma_{\rm s}/\,{\rm{km\,s^{-1}}}+b.\\\
$ (8)

The best-fitting slopes are 2.45 and 2.71 for the $M_{*}$-tracer and
$\sigma_{*}$-tracer relations, respectively (see Table 2). The best-fitting
results are also presented in black lines in Figure 8. We can see that the
$\sigma_{*}$-tracer relation is perfectly consistent with the relation
measured from hydrodynamical simulation(Munari et al., 2013). We also use
$M_{\rm AM}$ as a tracer to divide BC1 sample into 8 $M_{\rm AM}$ bins. The
obtained $M_{\rm h}$-$\sigma_{\rm s}$ scaling relation and its fitting result
are presented in Figure 9 and Table 2. The $M_{\rm AM}$-tracer relation is
also very similar to the simulation results. It is almost the same as the
results of Evrard et al. (2008) and Munari et al. (2013) at the massive end
and slightly higher than them at the low-mass end. The slope of the relation
is 2.57 (Table 2), between those of the $M_{*}$\- and $\sigma_{*}$-tracer
relations. Our measurements are more similar to the result of hydrodynamical
simulation. It might be a signal for velocity bias.

We then check the dependence of the $M_{\rm h}$-$\sigma_{\rm s}$ relation on
other galaxy and group properties. Here, we only present the $M_{\rm
AM}$-tracer results. The results for BC2 and BC3 samples are shown in Figure 9
and Table 2. The relations for the two populations are both similar to the
simulation results. The relation for BC2(close to the simulation result of
Saro et al. (2013)) is slightly higher than that for BC3 (close to the result
of Munari et al. (2013)) at high $\sigma_{\rm s}$. We also split the BC1
sample into star-forming and quenched galaxies. The scaling relations for the
two types of galaxies are also presented in Figure 9 and Table 2. In general,
the two relations are very similar and both are close to the simulation
results. And the slope for star-forming galaxies seems slightly steeper than
that for quenched galaxies. Given the large uncertainties for star-forming
galaxies at low mass, the difference is not significant.

Figure 9: The $M_{\rm h}$-$\sigma_{\rm s}$ relations using $M_{\rm AM}$ as a
tracer. Different colors represent different $M_{\rm AM}$ bins as labeled in
the upper-right panel. The upper-left, lower-left and lower-right panels show
the results of BC1, BC2 and BC3, respectively. The upper-right panel shows the
results for star-forming(solid stars) and quenched(solid circles) galaxies in
BC1 sample. In each panel, the solid line is the best-fitting to the
corresponding sample, and the red, purple and blue dashed lines are the
relations from simulations. Figure 10: The $M_{\rm h}$-$\sigma_{\rm s}$
relations from simulations and observations. As indicated in the figure, the
color-coded dashed lines are the simulation results and the color-coded and
gray symbols are the results from the literature with weak lensing measured
$M_{\rm h}$. For comparison, we also show our results for BC1 sample with
$M_{\rm AM}$ as the halo mass tracer in black solid diamonds.

In addition, we perform a test with a double-Gaussian model(see e.g. More et
al., 2011) to repeat Figure 8 and Figure 9. In general, the obtained $M_{\rm
h}$-$\sigma_{\rm s}$ relations from the double-Gaussian model are in agreement
with the simulation relations too. However, they are more noisy compared to
the results shown in this paper. It is likely that one of the two Gaussian
components sometimes fit the interlopers rather than the satellites, resulting
in a larger error bar in $\sigma_{\rm s}$. Therefore, we don’t use the double-
Gaussian model to analyze the satellite kinematics in this paper.

Finally, we compare our $M_{\rm AM}$-tracer $M_{\rm h}$-$\sigma_{\rm s}$
relations of BC1 sample with previous observational studies based on weak
lensing in Figure 10. Hoekstra (2007) and Gonzalez et al. (2015) focused on
individual galaxy clusters. In general, these clusters lie around the
simulation relations with relatively large scatter. Han et al. (2015), Viola
et al. (2015), Gonzalez et al. (2021), and Rana et al. (2022) calculated
$\sigma_{\rm s}$ for individual groups with more than 3 to 5 galaxy members
and then divided their group sample into subsamples according to $\sigma_{\rm
s}$. Therefore, they all used $\sigma_{\rm s}$ as a tracer. We list their best
fitting parameters (if available) in Table 2. These researches usually got
slopes ($\sim 2$ or $<2$) that are much flatter than the simulation slopes.
Some of them got amplitudes lower than the simulations. Various possible
interpretations for the discrepancies have been proposed, such as rich group
selection bias, interlopers, velocity bias and non-equilibrium systems.

Different from these studies, we divide galaxies by their $M_{*}$,
$\sigma_{*}$ or $M_{\rm AM}$ rather than $\sigma_{\rm s}$. After carefully
dealing with interlopers and weighting bias, our $M_{\rm h}$-$\sigma_{\rm s}$
relations are in good agreement with the simulation results in a broad halo
mass range. In addition, the $M_{\rm h}$-$\sigma_{\rm s}$ relation depends
weakly on $N_{\rm sat}$ and SFR. All of these suggest that group richness,
velocity bias and merging systems can not induce the large deviations
presented in these papers. The estimation of $\sigma_{\rm s}$ for individual
groups is sensitive to interlopers and the number of group members. The large
uncertainty can smear and weaken the correlation between $M_{\rm h}$ and
$\sigma_{\rm s}$, as shown in previous studies.

###  4.5 The linearly combined proxies

The above analyses show that the $M_{*}$-, $\sigma_{*}$\- and $M_{\rm
AM}$-$M_{\rm h}$ relations depends on $N_{\rm sat}$ and SFR in different
manners. It is thus interesting to check whether the combination of these
proxies can provide a better proxy(see Han et al., 2015, for a relevant
study). In Section 3.3, we introduce our method for generating a new proxy. We
construct three sets of new proxies, $\rm Pr(M_{*},M_{\rm AM},\emph{p})$, $\rm
Pr(\sigma_{*},M_{\rm AM},\emph{p})$, and $\rm Pr(M_{*},\sigma_{*},\emph{p})$.
To check the performance of the new proxy, we defined two parameters,
$d_{N_{\rm sat}}$ and $d_{\rm SFR}$ (Equation 7), to quantify the dependencies
of the $M_{\rm h}$-$\rm Pr$ relation on $N_{\rm sat}$ and SFR, respectively. A
larger $d_{N_{\rm sat}}$ or $d_{\rm SFR}$ means a stronger dependence on
$N_{\rm sat}$ or SFR, and thus a worse proxy. When the parameters are equal to
one, it means that the difference is mainly dominated by the uncertainties.
Note that our tests are all performed with the B-series samples.

We first examine the combination of $M_{*}$ and $M_{\rm AM}$. Figure 11 show
$d_{N_{\rm sat}}$ and $d_{\rm SFR}$ as functions of $p$. We can see that
$d_{N_{\rm sat}}$ quickly decreases with increasing $p$ at $p<0.3$ and then
remains unchanged at $d_{N_{\rm sat}}\sim 3$. It means that the $M_{\rm
h}$-$M_{\rm AM}$ relation is less dependent on $N_{\rm sat}$ than the SHMR,
well consistent with the above results. The parameter $d_{\rm SFR}$ is around
10 and slightly decreases with increasing $p$. It is also consistent with the
results shown above that $M_{\rm AM}$\- and $M_{*}$-$M_{\rm h}$ relations
depend on SFR in a similar manner. We thus conclude that $M_{\rm AM}$ is a
better proxy than $M_{*}$ and that the combination of the two parameters can
not improve the performance.

We then examine the combination of $\sigma_{*}$ and $M_{*}$. The results are
presented in the right panel of Figure 11. As one can see, $d_{N_{\rm sat}}$
is very large, about 25, and almost independent of $p$. It means that both
$\sigma_{*}$\- and $M_{*}$-$M_{\rm h}$ relations are strongly dependent on
$N_{\rm sat}$, consistent with the results shown in Figure 3 and Figure 4.
$d_{\rm SFR}$ gradually decreases with increasing $p$. It is consistent with
that the $M_{\rm h}$-$\sigma_{*}$ relation is less sensitive to SFR than the
SHMR. These tests suggest that $\sigma_{*}$ is, on average, a better proxy
than $M_{*}$. However, it does not mean that $\sigma_{*}$ is a better proxy of
halo mass than $M_{*}$ over any mass range. For example, we find that $M_{*}$
is better than $\sigma_{*}$ for low mass halos (Section 4.2). Note that
$d_{\rm SFR}$ measures the SFR dependence averaged over a large halo mass
range weighted with halo mass uncertainties. Since the $M_{\rm h}$
uncertainties for low mass halos are much larger than those for massive halos,
the contribution of low mass halos to $d_{\rm SFR}$ is negligible.

Finally, we examine the combination of $\sigma_{*}$ and $M_{\rm AM}$ (the
middle panel of Figure 11). $d_{N_{\rm sat}}$ decreases quickly with
increasing $p$ at $p<0.3$, then almost remains unchanged around unity at
$0.3<p<0.9$, and increases at $p>0.9$. While $d_{\rm SFR}$ is almost constant
at $p<0.3$ and then gradually increases with increasing $p$. Therefore, both
$d_{N_{\rm sat}}$ and $d_{\rm SFR}$ reach the minimum around $p=0.3$. In fact,
the combined proxy $\rm Pr(\sigma_{*},M_{\rm AM},0.3)$ has almost the smallest
$d_{N_{\rm sat}}$ and $d_{\rm SFR}$ among the three sets of combined proxies.
It means that the obtained $M_{\rm h}$-$\rm Pr(\sigma_{*},M_{\rm AM},0.3)$
relation would has the smallest scatter among all the proxies that we have
explored in this section.

As mentioned above, the two parameters $d_{N_{\rm sat}}$ and $d_{\rm SFR}$ are
dominated by massive halos that have $M_{\rm h}$ measurements with high
signal-to-noise ratio. They are thus unable to reflect the performance of the
proxy at different mass bins, particularly at low mass. Future imaging
surveys, such as Legacy Survey of Space and Time444https://www.lsst.org/, the
Chinese Space Station Optical Survey(Gong et al., 2019) and the Wide Field
Survey Telescope(WFST Collaboration et al., 2023) can provide much better data
which allows us to explore the performance of different proxies in details.

Figure 11:  The performance of the combined proxy, measured with $d_{\rm
para}$(Equation 7), as a function of $p$. The left, middle and right panels
show the results for the combination between $M_{*}$-$M_{\rm AM}$,
$\sigma_{*}$-$M_{\rm AM}$ and $M_{*}$-$\sigma_{*}$, respectively. In each
panel, the blue and red lines correspond to the results for $d_{N_{\rm sat}}$
and $d_{\rm SFR}$, respectively. The corresponding blue and red shaded regions
are the 1$\sigma$ scatter (see Section 3.3 for the detail). The gray
horizontal line shows $d_{\rm para}$=1.

##  5 Summary and discussion

In this paper, we use the DECaLS shear catalog to constrain the halo masses
($M_{\rm h}$) of SDSS central galaxies and investigate the scaling relations
between $M_{\rm h}$ and four single halo mass proxies, including stellar mass
($M_{*}$), central stellar velocity dispersion ($\sigma_{*}$), group stellar
mass ranked abundance matching halo mass ($M_{\rm AM}$) and satellite
kinematics ($\sigma_{\rm s}$). We also examine the dependence of these scaling
relations on galaxy internal properties, including $M_{*}$, $\sigma_{*}$ and
specific star formation rate (sSFR), and external properties, including group
richness($N_{\rm sat}$). These dependencies can be used to examine the
performance of these halo mass proxies and understand the galaxy formation
processes. We also construct a series of new halo mass proxies by linearly
combining two proxies and quantify their performances. In the following, we
summarize our results and briefly discuss the implications.

Overall, we find that halo mass is strongly correlated with $M_{*}$,
$\sigma_{*}$, $M_{\rm AM}$ and $\sigma_{\rm s}$. Galaxies with larger $M_{*}$,
$\sigma_{*}$, $M_{\rm AM}$ and $\sigma_{\rm s}$ tend to live in more massive
halos. The four obtained scaling relations are in good agreement with previous
observation or simulation results.

The $M_{\rm h}$-$M_{*}$ and $M_{\rm h}$-$\sigma_{\rm*}$ scaling relations are
strongly dependent on $N_{\rm sat}$. At a given $M_{*}$ or $\sigma_{\rm*}$,
the halo mass increases with $N_{\rm sat}$. The halo mass difference between
galaxies with $N_{\rm sat}=0$ and $N_{\rm sat}\geq 3$ can reach about 0.7 dex
at high $M_{*}$ or $\sigma_{\rm*}$ ends. We suggest that the intrinsic
dependence should be even stronger because our galaxy sample is magnitude
limited. The halo abundance matching(HAM) technique that uses central galaxy
properties as indicators of halo mass may fail to recover the halo masses of
groups with large $N_{\rm sat}$. Since these groups have a large number of
satellites, such a failure may cause a bias on the predicted two-point
correlation function at a small scale. Our results further suggest that the
galaxy clusters selected based on $N_{\rm sat}$ may be biased to high
$M_{*}/M_{\rm h}$ clusters.

The $M_{\rm h}$-$M_{*}$ and $M_{\rm h}$-$\sigma_{*}$ relations are also
dependent on the SFR of central galaxies. It means a higher baryon-to-star
conversion efficiency ($M_{*}/M_{\rm h}/(\Omega_{\rm b}/\Omega_{\rm m})$) for
star-forming galaxies than for quiescent galaxies. It may place an important
constraint on AGN feedback, which is thought to suppress the conversion
efficiency. We convert the $M_{\rm h}$-$\sigma_{*}$ relation into the $M_{\rm
h}$-$M_{\rm BH}$ relation. At $\log M_{\rm h}/{\rm M_{\odot}}<12.0$ or $\log
M_{\rm BH}/{\rm M_{\odot}}<7.4$, $M_{\rm BH}$ is weakly correlated with
$M_{\rm h}$. It hints at a rapid black hole growth. At $\log M_{\rm h}/{\rm
M_{\odot}}>12.0$ or $\log M_{\rm BH}/{\rm M_{\odot}}>7.4$, $M_{\rm BH}$
linearly correlates with $M_{\rm h}$. It indicates a transition of black hole
growth. More studies are required to understand the underlying processes.

We investigate the correlations among $M_{\rm h}$, $M_{*}$ and $\sigma_{*}$.
At small $M_{*}$ or $\sigma_{*}$ (roughly $\log M_{*}/{\rm M_{\odot}}<10.4$ or
$\log\sigma_{*}<2.1$), $M_{*}$ is more closely related to $M_{\rm h}$ than
$\sigma_{*}$; In the intermediate $M_{*}$ or $\sigma_{*}$ range (roughly
$10.4<\log M_{*}/{\rm M_{\odot}}<11.1$ or $2.1<\log\sigma_{*}<2.4$),
$\sigma_{*}$ is more closely related with $M_{\rm h}$ than $M_{*}$; At large
$M_{*}$ or $\sigma_{*}$ (roughly $\log M_{*}/{\rm M_{\odot}}>11.1$ or
$\log\sigma_{*}>2.4$), $M_{\rm h}$ correlates with both parameters. The
bivariate correlation suggests a complex connection among the growth of galaxy
stellar mass, galaxy structure, black hole mass, and halo mass. It implies
different growth patterns for galaxies at different stages. Higher quality
weak lensing data are required to reexamine this issue.

Another examined halo mass proxy is $M_{\rm AM}$. Our results show that the
$M_{\rm h}$-$M_{\rm AM}$ relation is close to the one-to-one relation.
Different from the SHMR and $M_{\rm h}$-$\sigma_{*}$ relations, the $M_{\rm
h}$-$M_{\rm AM}$ relation is weakly dependent on $N_{\rm sat}$ and independent
of the stellar mass of central galaxies. It is expected because the estimation
of $M_{\rm AM}$ is based on the mass of total member galaxies, so $M_{\rm AM}$
has already included the contribution of satellites and centrals. However, the
$M_{\rm h}$-$M_{\rm AM}$ relation shows a strong dependence on SFR and
$\sigma_{*}$. It hints that galaxy formation processes still have an important
impact on the relationship. Overall, $M_{\rm AM}$ is a better halo mass proxy
than $M_{*}$ and $\sigma_{*}$.

The fourth scaling relation we investigate is the $M_{\rm h}$-$\sigma_{\rm s}$
relation. We use $M_{*}$, $\sigma_{*}$ and $M_{\rm AM}$ as the halo mass
tracers to divide a galaxy sample into subsamples and obtain the mean
$\sigma_{\rm s}$ and $M_{\rm h}$ for each subsample using stacking method.
After carefully handling the interloper contamination and weighting bias, we
obtain $M_{\rm h}$-$\sigma_{\rm s}$ relations in good agreement with the
simulation results, in particular the hydrodynamical simulation, from $\log
M_{\rm h}/{\rm M_{\odot}}<12.0$ to $\log M_{\rm h}/{\rm M_{\odot}}>14.0$.
Moreover, the scaling relation shows weak dependence on $N_{\rm sat}$ and SFR.
Therefore, $\sigma_{\rm s}$ is the best proxy of halo mass among all these
single proxies in consideration. It is expected because the scaling relation
originates from virialized systems and is insensitive to galaxy formation
processes. However, the estimation of $\sigma_{\rm s}$ for individual groups
is a serious challenge because of the interlopers and small satellite numbers.

Finally, we construct new halo mass proxies by linearly combining two single
proxies. We examine three sets of combinations, including $M_{*}$-$M_{\rm
AM}$, $\sigma_{*}$-$M_{\rm AM}$ and $M_{*}$-$\sigma_{*}$ combinations. We use
the dependencies of the halo mass-new proxy scaling relation on $N_{\rm sat}$
and SFR to check the performance of the new proxy. The combination of
$\sigma_{*}$ and $M_{\rm AM}$ is better than the other two. It is because that
the $M_{\rm h}$-$M_{\rm AM}$ scaling relation is insensitive to $N_{\rm sat}$
and the $M_{\rm h}$-$\sigma_{*}$ scaling relation depends weakly on SFR. When
$M_{\rm AM}$ contributes 30% and $\sigma_{*}$ contributes 70% to the new
proxy, the dependencies on $N_{\rm sat}$ and SFR are almost the weakest. We
emphasize that our study provides a way to construct a halo mass proxy for
individual groups and evaluate its performance. Future image surveys can
provide much better lensing signals, which may allow us to study these proxies
in detail and construct better ones.

## Acknowledgements

We thank the referee and statistics editor for their useful reports. We thank
Kai Wang from Peking University for his valuable suggestions on drawing. This
work is supported by CAS Project for Young Scientists in Basic Research, Grant
No. YSBR-062, the National Key R&D Program of China (grant Nos. 2018YFA0404503
and 2018YFA0404504), the National Natural Science Foundation of China (Nos.
12192224, 11733004, 11890693, 11890691, and 12073017), and the Fundamental
Research Funds for the Central Universities. We acknowledge the science
research grants from the China Manned Space Project with Nos. CMS-
CSST-2021-A01 and CMS-CSST-2021-A03. The authors gratefully acknowledge the
support of Cyrus Chun Ying Tang Foundations. The work is also supported by the
Supercomputer Center of University of Science and Technology of China.

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##  Appendix A The construction of controlled samples

Our purpose is to investigate the dependence of $M_{\rm h}$ on the parameter
$X$ with parameter $Y$ controlled. We divide a galaxy sample within a $Y$
range into several bins according to $X$. Galaxies in these different $X$ bins
have different distributions of $Y$ parameter, although they are selected in
the same $Y$ range. To ensure that the dependence of $M_{\rm h}$ on $X$ is not
affected by the $Y$ parameter, it is necessary to construct the $Y$-controlled
subsamples.

Figure 12: Stellar mass distributions of different galaxy samples in the same
stellar mass bin. The color-coded dashed lines show the original $M_{*}$
distributions for galaxies in different $\sigma_{*}$ bins. These galaxies are
restricted in the stellar mass range of $10.4\leq\log M_{*}/{\rm
M_{\odot}}<11$. The thick gray line shows the lower envelope of the dashed
lines. Our controlled subsamples are selected in each $\sigma_{*}$ bin
according to the gray line.

Take the $M_{*}$-controlled subsamples in Section 4.2 as an example. We want
to study the correlation between $M_{\rm h}$ and $\sigma_{*}$ ($X$ parameter)
at a given $M_{*}$ ($Y$ parameter). We thus divide a galaxy sample in a given
$M_{*}$ range into several $\sigma_{*}$ bins. Figure 12 shows the $M_{*}$
probability distributions for galaxies of $10.4\leq\log M_{*}/{\rm
M_{\odot}}<11.0$ in five different $\sigma_{*}$ bins (dashed lines). We can
see that the $M_{*}$ distributions are still very different, even in the
relatively narrow $M_{*}$ range. It is unsurprising because $M_{*}$ and
$\sigma_{*}$ are strongly correlated. To construct $M_{*}$-controlled
subsamples to eliminate the impact of $M_{*}$, we select galaxies in each
$\sigma_{*}$ bin following the distribution indicated by the thick gray line,
which is the lower envelope of these dashed lines. This method ensures
sufficient galaxies in every $\sigma_{*}$ bin.