Halo Mass-observable Proxy Scaling Relations and Their Dependencies on Galaxy and Group Properties

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


INTRODUCTION
In the standard ΛCDM 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.Understanding the evolution of galaxies in dark matter halos (groups) and the related physics is one of the greatest challenges of modern cosmogony.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 mod-els (see Mo et al. 2010;Wechsler & Tinker 2018, for references).
Among these studies, the most widely discussed halo mass proxy is the stellar mass (M * ) of central galaxies, and the corresponding scaling relation with halo mass (M h , stellar mass-halo mass relation, hereafter SHMR) (e.g.Yang et al. 2003;Moster et al. 2010;Leauthaud et al. 2012;Mandelbaum et al. 2016;Kravtsov et al. 2018;Wechsler & Tinker 2018;Behroozi et al. 2019;Zhang et al. 2021Zhang et al. , 2022b)).The SHMR can be used to constrain galaxy formation processes.For example, M * /M h /(Ω b /Ω m ) is usually regarded as the baryon-tostar 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 stellar mass ends.At lower stellar masses, 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 stellar masses (e.g.Silk & Rees 1998;Croton et al. 2006;Fabian 2012;Heckman & Best 2014;Cui et al. 2021).The SHMR has large scatter, particularly at high mass, and the scatter 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;Zhang et al. 2021;Bilicki et al. 2021;Posti & Fall 2021;Xu & Jing 2022).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 (σ * ) 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 σ * 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 halomass 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. 2016Zahid et al. , 2018;;Seo et al. 2020;Sohn et al. 2020).The M h -σ * relation also has other important applications.For instance, Shankar et al. (2020) found that the M h -σ * relation and the largescale 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 h -σ * 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 maximumlikelihood 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. 2005Yang et al. , 2007)).The inferred halo mass is referred to as the abundance matching halo mass, denoted by M AM , in the following.Luo et al. (2018) and Gonzalez et al. (2021) studied the relation between M AM and halo mass based on weak lensing .They found that abundance matching predicts higher halo mass than obtained from the weak-lensing technique for group masses less than about 10 14.4 M .Luo et al. (2018) suggested that this can be interpreted as an effect caused by the Eddington bias.Gonzalez et al. (2021) further investigated the dependence of the M h -M 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 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 (σ s ) and halo mass using both simulations (e.g.Evrard et al. 2008;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 σ 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 σ s .They all found that halo mass is roughly proportional to σ3 s , consistent with theoretical expectations.Observationally, various methods have been applied to establish the M h -σ 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 focussed 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 h -σ s relation to M h < 10 13 M (e.g.Han et al. 2015;Zhang et al. 2022b).
Group richness is also found to be strongly correlated with halo mass (e.g.Becker et al. 2007;Johnston et al. 2007;Mandelbaum et al. 2008;Viola et al. 2015;Murata et al. 2019).Becker et al. (2007) investigated the relation between σ s and group richness for massive groups and clusters.Adopting the halo mass-σ s relation from Evrard et al. (2008), they obtained a halo mass -richness relation and found that these two parameters showed a positive correlation.Mandelbaum et al. (2008) addressed the same problem using weak lensing data.Using group richness as a halo mass proxy, they found that the lensing signal increases with group richness.
The main goal of this paper is to study various halo mass proxies (stellar mass, stellar velocity dispersion, abundance matching halo mass, satellite velocity dispersion, 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 ΛCDM cosmology with Ω m = 0.307, Ω b = 0.048, Ω Λ = 0.693 and h = 0.678.Here, h = H 0 /100 km s −1 Mpc −1 and H 0 is the Hubble constant (Planck Collaboration et al. 2016).

Central galaxy samples
The galaxies used in this paper are drawn from the New York University Value Added Galaxy Catalog (NYU-VAGC 1 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 ≤ 17.72, with redshift in the range of 0.01 ≤ z ≤ 0.2 and with redshift completeness C z > 0.7.In this paper, we focus on central galaxies which are defined as the most massive galaxies in galaxy groups.To select them, we use the group catalog constructed by applying the halo-based group-finding algorithm of Yang et al. (2005) to SDSS DR7 galaxies2 (see Yang et al. 2007, for details).As shown in the next section, we obtain the halo mass by using the weak lensing shear catalog measured from the Dark Energy Camera Legacy Survey (DECaLS, Dey et al. 2019).About 32% of SDSS DR7 galaxies do not overlap with the DECaLS shear catalog.This leaves a sample of 323,448 central galaxies.
We investigate several halo mass proxies: stellar mass (M * ), central stellar velocity dispersion (σ * ), halo mass based on abundance matching method (M AM ) and satellite velocity dispersion (σ s ), and linear combinations of them.The stellar masses of the galaxies are obtained by cross-matching with the MPA-JHU DR7 catalog 3 (Kauffmann et al. 2003;Brinchmann et al. 2004).They are calculated by fitting the SDSS ugriz photometry to models of galaxy spectral energy distribution.The central stellar velocity dispersion is provided by NYU-VAGC.The abundance matching halo mass for each central galaxy is provided by the group catalog (Yang et al. 2007).Here, we mainly use M AM based on the group total stellar mass.The estimate of σ 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 * , σ * , starformation rate (SFR) and M AM .For example, a small fraction of galaxies have no measurement of σ * (usually low mass galaxies), and some have unreasonably high σ * (e.g.σ * ∼ 1000 km s −1 ).Moreover, a fraction of low mass galaxies have no available M AM because M 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 < σ * / km s −1 < 630 of 299,806 galaxies.This sample is used to study the M h -M * and M h -σ * relations (Section 4.1 and 4.2).The second one, referred to as BTotal, requires measurements of M AM in addition.This requiement removes 60,769 galaxies.The BTotal sample is used to study the M h -M AM relation (Section 4.3).When we study the M h -σ s relation (Section 4.4), both samples are used.We do not use BTotal sample to study the M h -M * and M h -σ * 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 * , σ * and star formation rate (SFR), and one group property, the group richness.The galaxy SFR is obtained by cross-matching with the MPA-JHU DR7 catalog 4 (Kauffmann et al. 2003;Brinchmann et al. 2004), where the star formation rates of individual galaxies were derived from spectroscopic and photometric data of the SDSS.We adopt the demarcation log (SFR/M yr −1 ) = 0.73 log(M * /M ) − 8.3, 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 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 sat = 0, AC1 and BC1 have N sat ≥ 1, AC2 and BC2 have 1 ≤ N sat ≤ 2, and AC3 and BC3 have N sat ≥ 3.These subsamples are also listed in Table 1.These samples are sometimes referred to as the A-series and B-series samples, respectively.

DECaLS shear catalog
4 https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/ 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 − W 1), and (W 1 − W 2) 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).

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.

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 (2) The details and general discussion about different sources of systematic errors of the PDF-symmetrization  (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.
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: ∆Σ = ∆Σ stellar + ∆Σ NFW + ∆Σ 2h . (3) 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 h and the halo concentration.The dark halo mass m h is defined as the total dark matter mass within a spherical region of radius r 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.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, ξ hm , along the line-of-sight. Here,

Satellite kinematics
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).The scaling relation between lensingbased halo mass and satellite kinematics provides important information to understand structure formation and to calibrate mass measurements.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 200m , and halo virial velocity, v 200m .They selected satellite candidates from a refer- ence galaxy sample using the following set of criteria: r p ≤ r 200m , |∆v| ≤ 3v 200m , and M s < M * .Here, M * and M s are the stellar masses of the central and satellite candidate, respectively; r p is the projected distance from the satellite candidate to its central; and ∆v is the line-of-sight velocity difference between them.They found that the M h -σ s relation so obtained has a powerlaw 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 as Zhang et al. (2022b) to select satellite candidates for central galaxies.The centrals are divided into subsamples based on either their M * , or their σ * , or the halo mass M AM obtained from abundance matching (see Section 4.4).For demonstration, here we only present results for central galaxies with 11.2 ≤ log M * /M < 11.65 in A-series samples.The distribution of the corresponding satellite candidates in the phase space (r p versus ∆v) are shown in Figure 1.The results for other galaxy samples/subsamples are similar.As one can see, there are cone-like structures in the phase space distribution for ATotal, AC1, AC2 and AC3, indicated by black contour lines in the corresponding upper panels, respectively.These conelike structures are dominated by satellites, while galaxies outside the cone are usually interlopers (see e.g.Mamon et al. 2010).In contrast, there is an empty conelike structure for AC0.This is expected, because AC0 centrals have no satellite according to the group cata-log, and the shown galaxies are mostly interlopers.The PDFs of ∆v for these satellite candidates are shown in the lower panels of Figure 1.The distributions for ATotal, AC1, AC2 and AC3 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).For AC0, the distribution is double peaked and is almost entirely dominated by interlopers.
To eliminate interlopers as far as possible, we exclude AC0 (BC0) in the kinematics analysis and only use satellite candidates with |∆v| < |∆v| cut = 1.5v 200m to estimate the velocity dispersion of satellites.The results are presented in Section 4.4.Our visual inspection shows that the cut at 1.5v 200m roughly corresponds to the boundary of the cone-like structures in phase space, as shown by the gray vertical lines in Figure 1.We performed a series of tests using a value of |∆v| cut ranging from 1.0 to 3.0v 200m , and found that the results are stable as long as |∆v| cut is in the range of 1.3-1.8v200m .A smaller |∆v| cut excludes too many satellites in the cone-like structures and leads to poorer constraints on the satellite velocity dispersion.A larger |∆v| cut , on the other hand, leads to a larger dispersion because of the inclusion of more interlopers.The red symbols show the distributions of |∆v| within 1.5v 200m obtained from the data, with error bars estimated from 100 bootstrap samples.We fit the distributions to a functional form that is a Gaussian plus a constant, The constant d is used to account for interlopers within the cone.We use the MCMC to fit the observational data.The uncertainty of a SDSS galaxy redshift, which is about 35 km s −1 , results in an error of σ e = √ 2 × 35 = 49.5 km s −1 .We correct this and obtain the satellite velocity dispersion using 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 < M0.1 r < −16, and the redshift in the range of 0.01 < z < 0.2.Here, M0.1 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).

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 σ s measurements for individual central galaxies, we only use M * , σ * and M AM to design new proxies.The three parameters have very different dynamical ranges, it is thus necessary to scale these parameters before combining them together.For a proxy D, we design a new parameter, where min(D) and 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, where p is in the range of 0 ≤ p ≤ 1.With p = 0, Pr(D, E, p) is equivalent to the proxy D, while with p = 1, Pr(D, E, p) is equivalent to the proxy E. We can construct three sets of new proxies, Pr(M * , M AM , p), Pr(σ * , M AM , p), and Pr(M * , σ * , p).Since M 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 h -proxy scaling relation on other galaxy and group properties, such as N sat and SFR.We thus choose to use the differences in the M hproxy scaling relations between BC0 and BC3 subsamples and between BSF and BQ subsamples to quantify the dependencies on N sat and SFR, respectively.We first divide the BTotal sample into N b = 8 equally-sized bins according to a new proxy, Pr(D, E, p).Then we select galaxies that belong to BC0, BC3, BSF and BQ subsamples and derive their halo masses at each Pr(D, E, p) bin based on weak lensing, respectively.We now have M h -Pr(D, E, p) relations for BC0, BC3, BSF and BQ subsamples.We use the following formula to calculate the difference between two scaling relations, 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 ith bin for the two scaling relations, respectively.Sometimes, the mean Pr(D, E, p) at the largest or smallest Pr(D, E, 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 8for BC3 and BSF are calculated using the linear interpolation method.To evaluate the uncertainty of d para , we use the Monte Carlo method to generate log M i,1 and log M i,2 for each Pr(D, E, 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 para .This procedure is repeated 500 times and we take the dispersion of the 500 d para as the uncertainty.
In the following, we use d SFR and d Nsat to represent the dependencies of the scaling relation on SFR and N sat , respectively (Section 4.5).

HALO MASS -PROXY SCALING RELATIONS
In this section, we examine four single halo mass proxies (M * , σ * , M AM and σ s ) and a series of linearly combined proxies.We investigate their scaling relations and the dependence of these relations on other galaxy/group properties.

Stellar mass-halo mass scaling relation
We divide galaxies from the ATotal sample with M * ≥ 10 8.5 M 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 previous results, indicating that our calculations are reliable.
Many parameters are correlated with halo mass and stellar mass.These correlations may reflect possible dependencies of the SHMR on these parameters.We first consider N sat .As shown in Section 2.1, the ATotal sample can be split into AC0, AC2 and AC3 subsamples according to N sat .The ESDs and their best fitting profiles for these subsamples in four stellar mass bins are presented in Figure 2. We can see that, 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 can see a clear trend that the halo mass is positively correlated with group richness at given M * .Note that we do not show the result for AC1 sample, as AC1 is the sum of AC2 and AC3.
The difference in halo mass among samples with different N sat is quite large.For the four most massive stellar mass bins, the differences in halo mass between AC0 and AC3 are in the range of 0.5-0.8dex.The baryon-to-star conversion efficiency, usually characterized by M * /M h /(Ω b /Ω m ), varies from 0.086±0.03for AC0 to 0.017±0.002for AC3 in the highest M * bin, and from 0.101±0.009for AC0 to 0.026±0.003for AC3 in the second highest M * bin.Apparently, galaxies with large N sat have very low efficiencies.The SDSS galaxy sample is a magnitude limited sample.Some groups in AC0 sample may have large intrinsic N 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.
Another important factor that affects the SHMR and is often studied in the literature is the star formation activity.We separate the ATotal sample into star-forming (ASF) and quenched (AQ) galaxies by using the demarcation line shown in Section 2.1.The SHMRs for them are shown in the middle panel of Figure 3. 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. 2021Zhang et al. , 2022b)), star-forming galaxies reside in halos with a mean halo mass lower than that of quenched ones of the same M * .At the high stellar mass range, the difference in halo mass between the star-forming and quenched galaxies is very prominent, suggesting a significant difference in their baryon-to-star conversion efficiency.For example, the conversion efficiencies for the star-forming and quenched galaxies are 0.082 +0.041 −0.029 and 0.024 +0.003 −0.002 for the highest M * bin and 0.17 +0.044 −0.031 and 0.052 +0.003 −0.002 for the second highest M * bin, respectively.At the low stellar mass range, the halo mass differences between starforming and quenched galaxies are also large.However, their uncertainties are large too.The quenched galaxies in the second lowest mass bin have a much higher halo mass compared to the star-forming ones.We examined its weak lensing signal and found that it is likely due to some satellite galaxies being misidentified as central galaxies.
To investigate the dependence of SHMR on σ * , we divide the ATotal sample in each stellar mass bin into small and large σ * subsamples by the median σ * of each M * bin.Their SHMRs are shown in the right panel of Figure 3.In the stellar mass range of log M * /M < 10.4, galaxies with the same M * but different σ * share similar halo masses, indicating a weak dependence of SHMR on σ * in this mass range.At log M * /M > 10.4, galaxies with high σ * tend to reside in more massive halos than those with low σ * .The largest halo mass difference, about 0.6 dex, between the two subsamples occurs around log M * /M ∼ 11.At higher M * , the halo mass difference decreases with increasing M * .We will come back to the correlations among M h , M * and σ * in the next subsection.
To sum up, we find that SHMR is strongly dependent on group richness and SFR, and the dependence on σ * varies with stellar mass.These dependencies may (at least partly) cause the scatter in SHMR as discussed in the literature (e.g.More et al. 2011;Charlton et al. 2017;Wechsler & Tinker 2018;Bradshaw et al. 2020).The dependencies of SHMR on these parameters also suggest a large diversity in galaxy formation processes.They hint that halo mass, interaction with satellites, and galaxy inner structures all affect central galaxy properties and quenching.In the next subsection, we present further investigations on the correlation among M * , σ * and M h .In addition, our results suggest that galaxy clusters and groups selected based on richness may systematically bias to high M h /M * systems.A comprehensive examination on this issue is necessary.

Halo mass-galaxy central stellar velocity dispersion scaling relation
In this section, we use the galaxy stellar velocity dispersion (σ * ) as a halo mass proxy and investigate the weak lensing halo mass (M h )-σ * scaling relation.The ATotal sample is divided into 8 σ * bins.The bin size varies with σ * to ensure that each bin has reliable M h measurement as much as possible.The M h -σ * relation for the ATotal sample is shown in Figure 4 as black solid diamonds.We can see that galaxies with larger σ * reside in more massive halos.For comparison, we also present the result of Zahid et al. (2016) with a dot-dashed line and shaded region indicating the scatter.Zahid et al. (2016) obtained the halo mass by using the SHMR to assign the halo mass to each galaxy.Overall, our result is in good agreement with theirs.At the low σ * end, our halo mass is systematically higher than theirs.
We then examine the dependence of the M h -σ * relation on various parameters and only present the σ * bins with galaxy number greater than 50.We first investigate the dependence on N sat .We obtain the M h -σ * relations for AC0, AC2 and AC3 galaxies and show them in the left panel of Figure 4. Similar to the SHMR, the M h −σ * relation strongly depends on N sat .The uncertainties of halo masses at low σ * bins are very large, no clear halo mass-N sat correlation can be found in this range.However, at the four largest σ * bins where the halo masses are well constrained, halo mass increases significantly with N sat at fixed σ * .We find that the halo mass difference between AC0 and AC3 is about 0.4-0.8dex at the four largest σ * bins, similar to the results for SHMR.Interestingly, the halo mass difference between AC0 and AC3 is the smallest at log σ * ∼ 2.3, only about 0.4 dex.
We now check the dependence of the M h -σ * relation on SFR.The M h -σ * relations for the star-forming and quenched galaxies are shown in the middle panel of Figure 4.In the low σ * range (log σ * < 2.1), the uncertainties in the halo masses of quenched galaxies are very large, because the number of quenched galaxies is rather small.In the middle σ * range (2.1 < log σ * < 2.4), the halo mass difference at a given σ * between star-forming and quenched galaxies is typically 0.2-0.3dex, apparently smaller than the difference in the middle M * range, typically 0.4-0.5 dex.This may not be surprising, as it is found that the central stellar velocity dispersion is the most important factor for star formation quenching(see e.g.Bluck et al. 2020).Our results thus support that σ * plays a more essential role in star formation quenching than M * .For the results in the largest σ * 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 Posti et al. (2019) and Zhang et al. (2022b)).At a given σ * , 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 σ * 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 σ * bins are too large to allow quantifying the difference.For larger σ * bins, the dependence of the M h -σ * relation on M * is strong at low and high σ * ends and almost absent at log σ * ∼ 2.3, suggesting that the dependence also varies with σ * .These results are consistent with the complex dependence of SHMR on σ * .It is thus interesting to investigate the bivariate correlation in more detail.We perform two tests.In the first test, we investigate the M h -σ * 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 σ * .Our inspection shows that these subsamples with different σ * 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 σ * subsamples so that they have similar M * distributions (see Appendix for the description of 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 σ * .Similar to the first test, we construct σ * -controlled subsamples to eliminate the influence of σ * .
The left panel of Figure 5 shows the M h -σ * relations for M * -controlled subsamples in different colors.At log σ * < 2.1, M h is almost independent of σ * but strongly dependent on M * .It suggests that for these galaxies, the observed M h -σ * relation is dominated by the M h -M * relation.At 2.1 < log σ * < 2.4, we see a strong dependence of M h on σ * even when M * is well controlled.Interestingly, galaxies in two different M * bins(green and blue) follow almost the same trend, which is defined by the ATotal sample.It means that the SHMR in this range is driven by σ * , totally opposite to the trend seen in low-mass galaxies.At log σ * > 2.4, we see the dependence on both M * and σ * .
Similar conclusions can also be drawn from the SHMR results for the σ * -controlled subsamples, as shown in the right panel of Figure 5.In the lowest three σ * bins, the SHMRs generally follow the same trend as that of the ATotal sample in the low stellar mass range (log M * /M < 10.4).It means that σ * has a negligible  impact on SHMR at the low stellar mass end, broadly consistent with the results in the left panel.In the middle stellar mass range (10.4 < log M * /M < 11.1), the SHMRs in three σ * bins (1.9 < log σ * < 2.8) deviate from each other and all have very flat slopes.It means that halo mass is independent of stellar mass after controlling σ * .Therefore, the SHMR in this mass range is mainly driven by the M h -σ * relation.It is also in good agreement with results based on the M * -controlled subsamples.In the massive stellar mass end (log M * /M > 11.1), the SHMR in the largest σ * bin is almost consistent with the SHMR of ATotal sample.However, since there is only one σ * bin at this mass range, we can not disentangle the correlations with σ * and M * .In general, these results are in good agreement with those from Figure 3 and Figure 4.
Figure 4 and Figure 5 clearly show that the M h -σ * relation has significant dependencies on various param-eters.These results hint a connection between dark matter halos and the processes happening within galaxies.One important application of the M h -σ * relation is that it can be used to study the connection between dark matter halo and supper-massive black hole in galaxy center, which are thought to play an essential role in galaxy evolution.It is well known that σ * is tightly correlated with the black hole mass (M BH )(e.g.(9) In the literature, many studies constrained the two free parameters, α and β, using different samples and techniques.We list seven results in Table 2.  a) Zero-point(α) in Equation 9.
Based on these M BH -σ * relations, we convert the M hσ * relation into the M h -M BH relations and show them in Figure 6.Though the uncertainty in the M BH -σ * relation leads to a large uncertainty in the M h -M BH relation, all of these results follow a general trend.When M BH increases rapidly from 10 5.0 to 10 7.4 M , M h only increases about 0.2 dex, from 10 11.8 to 10 12.0 M .It hints at a rapid growth of black holes as their host halos reach log M h /M ∼ 12.0.It is consistent with that AGNs usually reside in halos of log M h /M ∼ 12.0(see e.g.Zhang et al. 2021).At log M h /M > 12.0, when M BH increases from 10 7.4 to 10 9 M , M h increases from 10 12.0 to 10 13.4 M .It indicates a nearly linear correlation between M BH and M h .Clearly, there is a transition in black hole growth at log M h /M ∼ 12.0 and log M BH /M ∼ 7.4.
For comparison, we show the M h -M BH relation obtained by Shankar et al. (2020).Shankar et al. (2020) estimated M h using the abundance matching technique and M BH using the M * -M 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 h -M BH relation based on a so-called biascorrected galaxy sample, which significantly deviates from ours.Because we adopt different methods to derive M h and M BH , it is difficult to determine what exactly causes 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 BH /M > 6.They suggested that when M h is close to 10 12 M , star-formation feedback can not effectively drive outflow and black holes start to grow rapidly.When M h > 10 12 M , 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 h /M > 12. Apparently, the M h -σ * and M h -M BH relations contain valuable information about galaxy formation and evolution.We will come back to this issue in the near future.

Halo mass-abundance-matching mass scaling relation
We adopt the abundance matching halo mass based on group stellar mass (hereafter M AM ) as the halo mass proxy in this section.The following tests are based on the B-series samples.We divide the BTotal sample into 8 M AM bins, the sizes of which are varied to obtain reliable halo mass measurements as far as possible.The resultant M h -M AM relation is shown in Figure 7 as black solid diamonds.For comparison, we also present the M h -M 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 h -M AM relation is very close to a one-to-one relation.At log M h /M > 12.5, the weak lensing measured halo mass is slightly smaller than the abundance matching mass.Luo et al. (2018) studied this issue and suggested that it is caused by Eddington bias.At log M h /M < 12.5, M h becomes slightly larger than M AM .
We then examine the dependence of the M h -M AM relation on various parameters.The upper-left panel shows the dependence on group richness.Different from the SHMR and M h -σ * relation, the dependence of the M h -M AM relation on N sat is much weaker.M 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 AM .At low mass, the result for the BTotal is dominated by the BC0 galaxies, which have no satellite.It means that the M h -M AM relation is approximately equivalent to the M h -M * relation.It might explain the small change in the M h -M AM relation at log M h /M ∼ 12.5.We also divide galaxies in each M AM bin into two equally-sized subsamples according to their M * .The M h -M AM relations for the small and large M * galaxies are presented in the lowerleft panel of Figure 7.As one can see, they closely follow the overall M h -M AM relation.There is almost no dependence on the stellar mass of central galaxies.It means that the M h -M * relation can be fully explained by the M h -M AM relation.Therefore, M AM is a more powerful proxy than M * .It may not be surprising because M AM is estimated based on the total stellar mass, which also includes the contribution of central galaxies.
The M h -M 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 AM range in consideration.The halo mass differences are rather large, ranging from ∼ 0.3 to 0.9 dex, comparable to the difference for the M h -M * relation and larger than that for the M h -σ * relation.The dependence of the M h -M AM relation on σ * is also significant (lower-right panel).In the low M AM range, the dependence on σ * is weak.In the middle M AM range, the dependence becomes very pronounced, with large σ * galaxies residing in more massive halos than the corresponding small σ * galaxies.In the high M AM range, this dependence becomes weak, and even almost disappears in the largest M AM bin.These results imply that the M h -M AM scaling relation is still sensitive to galaxy formation processes.
Overall, M AM is a good halo mass proxy.M AM is linearly correlated with M h , and the M h -M AM relation has a weak dependence on group richness.These make M AM a better indicator of the halo mass than M * and σ * .However, the M h -M AM relation depends on SFR much more strongly than the M h -σ * relation.It means that we can construct a better halo mass proxy by combining M AM and σ * .We will examine this idea in detail in Section 4.5.

Halo mass-satellite velocity dispersion scaling relation
The halo mass-satellite velocity dispersion (σ 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 σ 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 σ 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 Section 4.1, 4.2 and 4.3, M * , σ * and M 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 σ * as tracers, the tests are performed on the A-series samples.When M AM is considered, the tests are performed on the B-series samples.We divide the galaxy samples into 8 bins according to these tracers.Then we measure M h and σ s for each bin and obtain the corresponding M h -σ s scaling relations.Since both the weak lensing and satellite kinematics techniques are performed using the stacking methods, the uncertainties are very large when the sample size is small.For clarity, we only present the results with galaxy number greater than 100.
We first show the results using M * and σ * as tracers in Figure 8.They are referred to as the M * -tracer (solid crosses) and σ * -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 hydrodynamical 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.It has a significant offset from the other two relations at the massive end.The best-fitting parameters for the three results are also listed in Table 3.Note that their halo masses have already been converted into our halo mass definition.
The left panel of Figure 8 shows our M h -σ s relations for ATotal sample.The results from Zhang et al. (2022b) are also presented in gray color, which used M * as a tracer and adopted a larger ∆v cut of 3v 200m .They obtained a M h -σ s relation with a slope close to the sim-

MAM observation
(a) Zero-point(b) in Equation 10. (b) Slope(k) in Equation 10. (c) Samples are divided into subsamples according to the tracer for fitting. (d) The origin of the data used for fitting.
ulations and an amplitude, however, much lower than the simulations.Although σ s in Zhang et al. (2022b) is overestimated due to the contamination of interlopers, the halo mass estimated using the M h -σ s relation is still reliable.Because the M h -σ s relation is calibrated by using weak lensing measured halo mass.Compared with Zhang et al. (2022b), our results are closer to the simulations.It means that a small ∆v cut can significantly reduce the contamination.However, the discrepancy is still large.As shown in Figure 1, the satellite candidates of AC0 galaxies form an empty cone-like structure in the phase space, opposite to the distributions of simulated satellites(e.g.Mamon et al. 2010) and other samples (AC1, AC2 and AC3).It means that most satellite candidates for AC0 galaxies are interlopers.Moreover, AC0 galaxies are the dominant population in ATotal sample(Table 1).Therefore, the discrepancy between our ATotal result and the simulations is very likely caused by the N sat = 0 galaxies.
To examine this, we apply our method to AC1 galaxies, for which interlopers are not important (Figure 1).The scaling relations for AC1 galaxies lie very close to the simulation results (Figure 8).We use a power-law function to fit the data, The best-fitting slopes are 2.45 and 2.71 for the M *tracer and σ * -tracer relations, respectively (Table 3).We show the best-fitting results in black lines in Figure 8, which are well consistent with the simulation relations.In particular, the σ * -tracer relation is perfectly consistent with the relation measured from hydrodynamical simulation (Munari et al. 2013).We also use M AM as a tracer to divide BC1 sample into 8 M AM bins.
The obtained M h -σ s scaling relation and its fitting result are presented in Figure 9.The M 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 3), between the slopes of the M * -and σ *tracer relations.Our measurements are more similar to the result of hydrodynamical simulation.It might be a signal for velocity bias.These results indicate that after carefully dealing with the interloper contamination, the observational M h -σ s relation can well match the simulations.And different tracers give similar results.
We then check the dependence of the M h -σ s relation on other galaxy and group properties.Here, we only present the results using M AM as a tracer.We split BC1 sample into BC2 and BC3 samples.The results for the two samples are shown in Figure 9 and Table 3.We can see that the M h -σ s relations for BC2 and BC3 galaxies 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 σ 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 3.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.
Finally, we compare our M AM -tracer M h -σ 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.Zhang et al. (2022b) used a method similar to ours but with a larger ∆v cut and included one-galaxy groups.They obtained a slope close to but an amplitude lower than the simulation results.Han et al. (2015), Viola et al. (2015), Gonzalez et al. (2021), andRana et al. (2022) calculated σ s for individual groups with more than 3 to 5 galaxy members and then divided their group sample into subsamples according to σ s .Therefore, they all used σ s as a tracer.We also list their best fitting parameters (if available) in Table 3.These researches usually got slopes (∼ 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 in these papers.Some studies suggested that the requirement of a large N sat may cause a potential sample selection bias(e.g.Viola et al. 2015;Rana et al. 2022).Gonzalez et al. (2021) suggested that interlopers may lead to an overestimation of σ s .Han et al. (2015) suggested velocity bias as a possible source of the discrepancy.Gonzalez et al. (2021) mentioned that merging systems violate the dynamical equilibrium assumption.
Different from these studies, we divide galaxies by their M * , σ * or M AM rather than σ s .After carefully dealing with interlopers, our M h -σ s relations are in good agreement with the simulation results in a broad halo mass range (from log M h /M < 12.0 to log M h /M > 14.0).In addition, the obtained M h -σ s relation depends weakly on N sat and SFR.All of these results suggest that the effects due to group richness, velocity bias and merging systems can not induce such large deviations from the simulations shown in previous studies.Our tests show that the estimation of σ s is very sensitive to interlopers.Moreover, three to five group members are apparently not sufficient to obtain a high signal-to-noise estimation of σ s .We suggest that the large uncertainty in σ s can smear and weaken the correlation between M h and σ s .Our result suggests that σ s is a very robust halo mass proxy for individual groups only when it can be accurately measured.

The linearly combined proxies
In the above sections, we show that M * , σ * , M AM and σ s are all strongly correlated with the halo mass.Among these single proxies, σ s seems to be the best one.The obtained M h -σ s relation has a weak dependence on N sat and SFR.However, a large number of satellites and carefully dealing with interlopers are required to obtain a reliable estimation of σ s .Unfortunately, this is impossible for most individual SDSS galaxy groups.The remaining three M h -proxy relations are more or less dependent on N sat and SFR in different manners.It is thus interesting to check whether the combination of  2008), Munari et al. (2013) and Saro et al. (2013).The color-coded and gray symbols are the results from the literature with weak lensing measured M h as indicated in the figure (Hoekstra 2007;Han et al. 2015;Viola et al. 2015;Gonzalez et al. 2015Gonzalez et al. , 2021;;Rana et al. 2022;Zhang et al. 2022b).The gray line is the best fitting to the result of Zhang et al. (2022b).For comparison, we also show our results for BC1 sample with MAM as the halo mass tracer in black solid diamonds.
these proxies can provide a better proxy(see also Han et al. 2015, for a relevant study).
In Section 3.3, we introduce our method for linearly combining two proxies to generate a new proxy.We now have three sets of combined proxies: Pr(M * , M AM , p), Pr(σ * , M AM , p), Pr(M * , σ * , p).Here, p ranges from 0 to 1.At p = 0, the new proxy is equivalent to the first proxy, while at p = 1, the new proxy is equivalent to the second proxy.To check the performance of the new proxy, we defined two parameters, d Nsat and d SFR (Equation 8), to quantify the dependencies of the M h -Pr relation on N sat and SFR, respectively.A larger d Nsat or d SFR means a stronger dependence on N sat or SFR, and thus a worse proxy.When the parameters are equal to one, it means 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 AM .We show d Nsat and d SFR as functions of p in the left panel of Figure 11.We can see that d Nsat quickly decreases with increasing p at p < 0.3 and then remains unchanged at d Nsat ∼ 3. It means that the M h -M AM relation is less dependent on N sat than the SHMR, well consistent with the results shown in Figure 3 and Figure 7.The parameter d SFR is around 10 and slightly decreases with increasing p.It is also consistent with the results shown above that M AM -and M * -M h relations depend on SFR in a similar manner.We thus conclude that M 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 σ * and M * .The results are presented in the right panel of Figure 11.As one can see, d Nsat is very large, about 25, and almost independent of p.It means that both σ * -and M * -M h relations are strongly dependent on N sat , consistent with the results shown in Figure 3 and Figure 4. d SFR gradually decreases with increasing p.It means that the M h -σ * relation is less sensitive to SFR than the M * -M h relation, consistent with the results shown above.These tests suggest that σ * is, on average, a better proxy than M * .However, it does not mean that σ * is a better proxy of halo mass than M * over any mass range.For example, we find that M * is better than σ * for low mass halos (Section 4.2).Note that d SFR measures the SFR dependence averaged over a large halo mass range weighted with halo mass uncertainties.Since the M h uncertainties for low mass halos are much larger than those for massive halos, the contribution of low mass halos to d SFR is negligible.
Finally, we examine the combination of σ * and M AM (the middle panel of Figure 11).d Nsat 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 SFR is almost constant at p < 0.3 and then gradually increases with increasing p.Therefore, both d Nsat and d SFR reach the minimum around p = 0.3.In fact, the combined proxy Pr(σ * , M AM , 0.3) has almost the smallest d Nsat and d SFR among the three sets of combined proxies.It means that the obtained M h -Pr(σ * , M 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 Nsat and d SFR are dominated by massive halos that have M 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 Time5 , the Chinese Space Station Optical Survey (Gong et al. 2019) and the Wide Field Survey Telescope (Lou et al. 2020) can provide much better data which allows us to explore the performance of different proxies in details.  8), as a function of p.The left, middle and right panels show the results for the combination between M * -MAM, σ * -MAM and M * -σ * , respectively.In each panel, the blue and red lines correspond to the results for dN sat and dSFR, respectively.The corresponding blue and red shaded regions are the 1σ scatter (see Section 3.3 for the detail).The gray horizontal line shows dpara=1.

SUMMARY AND DISCUSSION
In this paper, we use the DECaLS shear catalog to constrain the host halo masses (M h ) of SDSS central galaxies and investigate the scaling relations between M h and four single halo mass proxies, including stellar mass (M * ), central stellar velocity dispersion (σ * ), group stellar mass ranked abundance matching halo mass (M AM ) and satellite kinematics (σ s ).We also examine the dependence of these scaling relations on galaxy internal properties, including M * , σ * and specific star formation rate (sSFR), and external properties, including group richness(N 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.
The M h -M * and M h -σ * scaling relations are strongly dependent on N sat .At a given M * or σ * , the halo mass increases with N sat .The halo mass difference between galaxies with N sat = 0 and N sat ≥ 3 can reach about 0.7 dex at high M * or σ * ends.We suggest that the intrinsic dependence should be even stronger because our galaxy sample is magnitude limited and some galaxies with N sat = 0 may have satellites that are not detected.
Our study shows that N sat is one of the major sources of the scatter of the two scaling relations.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 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 (Trujillo-Gomez et al. 2011).Our results further suggest that the galaxy clusters selected based on N sat may be biased to high M * /M h clusters.
The M h -M * and M h -σ * relations are also dependent on the SFR of central galaxies.It means a higher baryon-to-star conversion efficiency (M * /M h /(Ω b /Ω 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(see e.g.Cui et al. 2021).Another interesting thing is that the dependence of the M h -σ * on SFR within a certain σ * range is much weaker than the corresponding M h -M * relation.It is consistent with the previous findings that σ * is the most important indicator of star formation quenching(e.g.Bluck et al. 2020).It further hints that the scatter of the M h -σ * relation is smaller than that of the M h -M * relation within a certain M * or σ * range.
We then investigate the correlations among M h , M * and σ * .At small M * or σ * (roughly log M * /M < 10.4 or log σ * < 2.1), M h is correlated with M * .The correlation of M h with σ * is the secondary effect of M h -M * and M * -σ * correlations.In the intermediate M * or σ * range (roughly 10.4 < log M * /M < 11.1 or 2.1 < log σ * < 2.4), the M h -σ * relation is dominant, and the correlation between M h and M * is a secondary effect.At large M * or σ * (roughly log M * /M > 11.1 or log σ * > 2.4), M h correlates with both parameters.The bivariate correlation suggests a complex connection among the growth of galaxy stellar mass, galaxy structure, and dark matter halo.It implies different growth patterns for galaxies at different stages.Higher quality weak lensing data are required to reexamine this issue.
Since central stellar velocity dispersion is tightly correlated with the mass of supermassive black holes (M BH ), as an extension of our results, we convert the M h -σ * relation into the M h -M BH relation.We find that, at log M h /M < 12.0 or log M BH /M < 7.4, M BH is weakly correlated with M h .At this stage, M BH distribution is extensive, while the mean halo mass changes slightly with M BH .It hints at a rapid black hole growth.At log M h /M > 12.0 or log M BH /M > 7.4, M BH linearly correlates with M h .It indicates a transition of black hole growth, broadly consistent with the prediction of an analytic model (Bower et al. 2017).Such a transition is likely related to star formation quenching and the transformation of galaxy morphology.More studies are required to understand the underlying processes.
Another halo mass proxy that we examine in the paper is the stellar mass ranked abundance matching halo mass (M AM ) (Yang et al. 2007).Our results show that the M h -M AM relation is close to the one-to-one relation.Different from the SHMR and M h -σ * relations, the M h -M AM relation is weakly dependent on N sat and independent of M * , the stellar mass of central galaxies.It is expected because the estimation of M AM is based on the mass of total member galaxies, so M AM has already included the contribution of satellites and centrals.However, the M h -M AM relation shows a strong dependence on SFR and σ * , indicating the existence of significant scatter of the relation.It hints that galaxy formation processes still have an important impact on the relationship.Overall, M AM is a better halo mass proxy than M * and σ * .
The fourth scaling relation we investigate is the M h -σ s relation.In the literature, most studies calculated σ s for individual groups and used σ s to divide a galaxy/group sample into subsamples and then obtained the M h -σ s relation.We use M * , σ * and M AM as the halo mass tracers instead of σ s to divide a galaxy sample into several subsamples.Then, we obtain the mean σ s and M h for each subsample using stacking techniques.The measurement of σ s is sensitive to the interlopers.In order to reduce the impact of interlopers, we exclude central galaxies with no satellite according to the galaxy group catalog and use satellite candidates with line-of-sight velocity differences less than 1.5v 200m to calculate σ s .We find that the scaling relations measured based on the three tracers are all in good agreement with the relations from simulations, in particular the hydrodynamical simulation (e.g.Munari et al. 2013).Moreover, the scaling relation shows weak dependence on N sat and SFR.Therefore, σ s is the best proxy of halo mass among all these single proxies in consideration.It is expected because the scaling relation results from virialized systems and is insensitive to galaxy formation processes.However, the estimation of σ s for individual groups is a serious challenge because of the interlopers and small satellite numbers.It may explains that previous studies that divided galaxy samples according to σ s usually obtained slopes of 2 or less than 2, significantly deviating from the simulation results.
Finally, we construct new halo mass proxies by linearly combining two single proxies(Equation 7).We examine three sets of combinations, including M * -M AM , σ * -M AM and M * -σ * combinations.We then use the dependencies of the halo mass-new proxy scaling relation on N sat and SFR to check the performance of the new proxies.The combination of σ * and M AM is better than the other two.It is consistent with that the M h -M AM scaling relation is insensitive to N sat and the M h -σ * scaling relation depends weakly on SFR.When M AM contributes 30% and σ * contributes 70% to the new proxy, the dependence on N 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 new ones.

APPENDIX
A. THE CONSTRUCTION OF CONTROLLED SAMPLES Our purpose is to investigate the dependence of M 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 h on X is not affected by the Y parameter, it is necessary to construct the Y -controlled subsamples.
Take the M * -controlled subsamples in Section 4.2 as an example.We want to study the correlation between M h and σ * (X parameter) at a given M * (Y parameter).We thus divide a galaxy sample in a given M * range into several σ * bins.Figure 12 shows the M * probability distributions for galaxies of 10.4 ≤ log M * /M < 11.0 in five different σ * 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 σ * are strongly correlated.To construct M * -controlled subsamples to eliminate the impact of M * , we select galaxies in each σ * 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 σ * bin.
with b(m h ) being the halo bias (e.g.Mo & White 1996;Tinker et al. 2010) and ξ mm being the linear matter-matter correlation function.We use COLOSSUS(Diemer 2018) to model both b(m h ) and ξ mm .To constrain the two free parameters, 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.Since we include the stellar mass term in our model, m h only accounts for cold dark matter, diffuse gas, and satellites around centrals.Thus, throughout this paper, we use the total mass M h = m h + M * instead of m h to represent the halo mass (seeZhang et al. 2021Zhang et al.  , 2022b, for details), for details).

Figure 1 .
Figure1.Upper panels: the data points and the contours show the distributions of satellite candidates in the projected phase space (rp-∆v).Lower panels: the probability distributions of ∆v(red and blue symbols) and the best-fitting curves for PDFs within 1.5v200m (red symbols).The two gray vertical lines in each panel correspond to ∆v = ±1.5v200m.We only show the satellite candidates around centrals with 11.2 ≤ log M * /M < 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.

Figure 2 .
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 h are also presented.

Figure 3 .
Figure 3.The dependence of SHMR on Nsat(left), SFR(middle) and σ * (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).

Figure 4 .
Figure 4.The dependence of the M h -σ * relation on Nsat(left), SFR(middle) and M * (right).The black lines in the three panels show the M h -σ * 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 .
Figure 5.The dependence of M h -σ * relation on M * (left panel) and M h -M * relation on σ * (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 (σ * -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.

Figure 6 .
Figure 6.M h -MBH relations.We apply the MBH-σ * relations from the literature (Table 2) to convert the M h -σ * relation of ATotal sample into the M h -MBH relations, as shown in different colors.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 who obtained the halo mass by using the abundance matching method.

Figure 7 .
Figure 7.The dependence of the M h -MAM relation on Nsat (upper-left), SFR(upper-right), M * (lower-left) and σ * (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.

Figure 8 .
Figure 8.The M h -σs relations using M * (crosses) or σ * (diamonds) as a tracer.Different colors represent different M * or σ * 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 σ * -tracer relations, respectively.The red, purple and blue dashed lines show the simulation results from Munari et al. (2013), Saro et al. (2013) and Evrard et al. (2008), respectively.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.

Figure 9 .
Figure 9.The M h -σs relations using MAM as a tracer.Different colors represent different MAM 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 .
Figure10.The M h -σs relations from simulations and observations.The color-coded dashed lines are the simulation results taken fromEvrard et al. (2008),Munari et al. (2013) andSaro et al. (2013).The color-coded and gray symbols are the results from the literature with weak lensing measured M h as indicated in the figure(Hoekstra 2007;Han et al. 2015;Viola et al. 2015;Gonzalez et al. 2015Gonzalez et al. , 2021;;Rana et al. 2022;Zhang et al. 2022b).The gray line is the best fitting to the result ofZhang et al. (2022b).For comparison, we also show our results for BC1 sample with MAM as the halo mass tracer in black solid diamonds.

Figure 11 .
Figure 11.The performance of the combined proxy, measured with dpara(Equation8), as a function of p.The left, middle and right panels show the results for the combination between M * -MAM, σ * -MAM and M * -σ * , respectively.In each panel, the blue and red lines correspond to the results for dN sat and dSFR, respectively.The corresponding blue and red shaded regions are the 1σ scatter (see Section 3.3 for the detail).The gray horizontal line shows dpara=1.

Figure 12 .
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 σ * bins.These galaxies are restricted in the stellar mass range of 10.4 ≤ log M * /M < 11.The thick gray line shows the lower envelope of the dashed lines.Our controlled subsamples are selected in each σ * bin according to the gray line.

Table 1 .
Sample selection used in our analysis ATotal and BTotal are two total samples, and the rest are corresponding subsamples.

Table 3 .
The fitting parameters for M h -σs relation