An Extended Halo-based Group/Cluster finder: application to the DESI legacy imaging surveys DR8

The DESI Legacy Imaging Surveys provide the target catalogs for the DESI survey. The target selection requires a combination of three optical (grz) bands and two mid-infrared (W1 3.4$\mu$m, W2 4.6$\mu$m) bands. For the case of galaxies, the g/r/z bands have at least a 5$\sigma$ detection of 24/23.4/22.5 AB magnitude with an exponential light profile of half-light radius $0^{{}^{\prime\prime}}.45$. The optical imaging data are provided by three public projects: the Beijing-Arizona Sky Survey (BASS), the Mayall z-band Legacy Survey (MzLS), and the DECam Legacy Survey (DECaLS), while the Wide-field Infrared Survey Explorer (WISE) satellite provides the infrared data. See an overview of the surveys in Dey et al. (2019).

We select our sample from the latest public data release of the Legacy Surveys, the Data Release 8 (DR8). DR8 also includes data from some non-DECaLS surveys, with the majority originating from the Dark Energy Survey (DES). The catalogs are processed by the software package THE TRACTOR (Lang et al., 2016) for source detection and optical photometry.

Our galaxy samples first select those object with morphological classification as type REX, EXP, DEV and COMP from Tractor fitting results²²2In the Tractor fitting procedure, REX stands for round exponential galaxies with a variable radius, DEV for deVaucouleurs profiles (elliptical galaxies), EXP for exponential profiles (spiral galaxies), and COMP for composite profiles that are deVaucouleurs plus exponential (with the same source center)..

Following suggestions from the DESI Target selection (also see Zou et al. 2019; Zhou et al. 2020a; Ruiz-Macias et al. 2020; Zhou et al. 2020b; Raichoor et al. 2020; Yèche et al. 2020), we require objects to have at least one exposure in each optical band and remove the area within ${\rm|b|<25.0^{\circ}}$ (where ${\rm b}$ is the Galactic latitude) to avoid high stellar density regions. In addition to the Tractor morphology selection, a minimum data quality flags are employed to remove the flux contaminations from nearby sources (FRACFLUX) or masked pixels (FRACMASKED):

where $i\equiv g,r,z$. FRACIN is to select the objects for which a large fraction of the model flux is in the contiguous pixels where the model was fitted. We also remove any objects close to bright stars by masking out objects with the following bits number in DR8 MASKBITS column: 1 (close to Tycho-2 and GAIA bright stars), 8 (close to WISE W1 bright stars), 9 (close to WISE W2 bright stars), 11 (close to fainter GAIA stars), 12, and 13 (close to an LSIGA large galaxy and globular cluster, respectively). Note that all the magnitudes used in this paper are in the AB system and have been corrected for Galactic extinction by using the Galactic transmission values provided in DR8.

We adopt the random forest algorithm based photometric redshift estimation for the galaxies in our sample from the Photometric Redshifts for the Legacy Surveys (PRLS ³³3https://www.legacysurvey.org/dr8/files/#photometric-redshift-files-8-0-photo-z-sweep-brickmin-brickmax-pz-fits, Zhou et al. 2020a). To ensure the photo-z quality in the sample, we further limit our analysis to galaxies with ${\rm z\leq 21}$ (above which the photometric redshifts are not reliable) and $0<$ z_phot_median $\leq 1$, where the z_phot_median is the median value of photo-z probability distribution function (PDF). See the details regarding the photometric redshift training and performance in appendix B of Zhou et al. (2020a). To make the sample’s redshift information as accurate as possible, we replace the photometric redshift with the spectroscopic redshift, if available. In total, there are 2.1 million galaxies with spectroscopic redshifts in our sample. The complete spectroscopic redshifts sources in the original catalog include: BOSS, SDSS, WiggleZ, GAMA, COSMOS2015, VIPERS, eBOSS, DEEP2, AGES, 2dFLenS, VVDS, and OzDES⁴⁴4Check the selection conditions in these surveys in section 3.2, Zhou et al. 2020a.. Additional spectroscopic redshifts were obtained by matching galaxies in our sample with those in 2MASS Redshift Survey (2MRS; Huchra et al., 2012), 6dF Galaxy Survey Data Release 3 (6dFGRS; Jones et al., 2009), and 2dF Galaxy Redshift Survey (2dFGRS; Colless et al., 2001).

After applying all the above selection criteria, we have a total number of 129.35 million galaxies remaining in our sample, within which 69.75 million are located in the north galactic cap (NGC) and 59.6 million in the south galactic cap (SGC). The total sky coverage of the galaxy sample is 18,253 square degrees (9673 square degrees in NGC, 8580 square degrees in SGC). As an illustration, we show the final footprint of galaxies on the sky in Figure 1.

For each galaxy in our NGC and SGC samples, according to the typical photoz errors in the Legacy Surveys DR8 photoz galaxy catalog (e.g. Zhou et al., 2020a; Wang et al., 2020), we assign it with a photometric redshift error,

If a galaxy has a spectroscopic redshift, we set $\sigma_{\rm photo}=0.0001$. This value will be used in our modified group finder.

Next, for each galaxy, we use the following function to convert apparent magnitude to absolute magnitude according to its redshift,

where DM($z$) is the distance module corresponding to redshift $z$,

with ${\rm D_{L}}$($z$) being the luminosity distance in ${\rm\>h^{-1}{\rm{Mpc}}}$. ${\rm K^{0.5}_{z}}$ represents the ${\rm K}$-correction in z-band to sample median redshift $z\sim 0.5$, obtained from the ‘Kcorrect’ model (eg. v4_3) described in Blanton & Roweis (2007). For illustration, Fig. 2 shows ${\rm K^{0.5}_{z}}$ values for a subset of our sample galaxies.

In general, the average ${\rm K}$-correction can be described by the following function,

The fits to the median ${\rm K^{0.5}_{z}}$ values in the redshift bin of 0.01 result is shown in Fig.2 as the red line with $\{0.86\pm{0.06},-0.67\pm{0.06},-0.3\pm{0.01}\}$ for $\{a,b,c\}$, respectively. For the mean ${\rm K^{0.5}_{z}}$ values in the redshift bin of 0.01, the fitting result is $\{0.73\pm{0.05},-0.54\pm{0.05},-0.33\pm{0.01}\}$ for $\{a,b,c\}$. The fits agrees quite well with the overall ${\rm K}$-correction trend. We will use the best fitting results to the median ${\rm K^{0.5}_{z}}$ values to model the K-correction of galaxies in our mock redshift surveys, as well as in calculating the maximum redshift a galaxy can be observed in the next subsection.

The luminosity of each galaxy is then calculated using the following formula:

with taking K-correction of the sun, ${\rm K^{0.5}_{z}}(0.0)=-0.3$, into account, to be consistent with ${\rm M_{\odot}}$ being 4.5 in z-band (Willmer, 2018).

The N-body simulation used in our study is the ELUCID simulation carried out at the High Performance Center (HPC) at the Shanghai Jiao Tong University (Wang et al., 2016). ELUCID evolves the distribution of $3072^{3}$ dark matter particles in a periodic box of $500\>h^{-1}{\rm{Mpc}}$ (${\rm L_{500}}$) from redshift $z=100$ to $z=0$. The simulation was run with the code L-GADGET, a memory-optimized version of GADGET2 (Springel, 2005). The cosmological parameters adopted by ELUCID are consistent with the WMAP5 results and are the following: $\Omega_{\rm m}=0.258$, $\Omega_{\rm b}=0.044$, $\Omega_{\Lambda}=0.742$, $h=0.72$, $n_{s}=0.963$ and $\sigma_{8}=0.796$ while each particle has a mass of $3.0875\times 10^{8}\>h^{-1}\rm M_{\odot}$. The simulation is constrained (in terms of initial conditions) by the re-constructed initial density field of SDSS DR 7 (Wang et al., 2014). ELUCID has been already used to study galaxy quenching (Wang et al., 2018), galaxy intrinsic alignment (Wei et al., 2018), and cosmic variance (Chen et al., 2019) while it was also combined with the abundance matching method to evaluate galaxy formation models (Yang et al., 2018) and the L-Galaxies semi-analytic model (Katsianis et al., 2020).

Dark matter halos were identified with the standard Friend-of-Friends (FoF) algorithm (Davis et al., 1985) with a link length equal to 0.2 times the average particle separation and with at least $20$ particles. The halo mass is calculated by summing up the mass of all the particles linked to the dark matter halo. As shown in Wang et al. (2016), the halo mass function obtained from this simulation is in good agreement with the analytic model prediction by Sheth et al. (2001) (SMT2001) at the mass ranges of interest. Based on halos at different outputs, halo merger trees were constructed (see Lacey & Cole, 1993). A halo in an earlier output is considered to be a progenitor of a present halo if more than half of its particles are found in the present halo. The main branch of a merger tree is defined as all the progenitors going through from the bottom to the top, choosing always the most massive branch at every branching point. These progenitors are labeled as the main-branch progenitors. Based on the above merger trees, galaxy catalogs were generated using the empirical galaxy formation model described in Chen et al. (2019). Note that by combing the true and Monte Carlo merger trees, we have generated galaxies in all the halos in our ELUCID simulations with mass $\geq 10^{9}\>h^{-1}\rm M_{\odot}$, with a minimum galaxy luminosity below $10^{8}\,h^{-2}\rm L_{\odot}$ in all the simulation snapshots that we are going to use.

In this study, we create a mock galaxy redshift survey (MGRS) from the above-constructed galaxy catalogs. The MGRS, with z-band galaxy magnitude ${\rm z\leq 21}$, spans a redshift range of $0.0<z\leq 1.1$, maximum angles of $\Delta\delta=10.0^{o}$, $\Delta\alpha=20.0^{o}$ and thus a total sky coverage at $199.74\deg^{2}$. Here we have taken into a negative K-correction to obtain the apparent magnitude for each galaxy. Since the redshift range we are interested in, $0.0<z<1.1$, is quite large, we have taken into account the light cone effect by stacking simulation boxes at different redshifts that are most consistent with the redshifts where they locate. The details of making the MGRS is described in Meng et al. (2020).

On top of the redshift information from the mock galaxy catalog, we assign each galaxy a random shift drawn from a Gaussian distribution with dispersion $\sigma_{\rm photo}*(1+z)$, where $\sigma_{\rm photo}$ is the typical photoz errors in the Legacy Surveys DR8 photoz galaxy catalog described by Eq.1. This value is also kept in the MGRS for our modified group finder to use. We select galaxies with redshifts $0.0<z\leq 1.0$ from the mock galaxy catalog. There are a total of $1681566$ and $1679924$ galaxies remaining in our mock spectroscopic and photometric redshift surveys, respectively. As an illustration, we show the projected distribution of a small subset of galaxies along the declination direction (with $\Delta\delta=0.25^{o}$) within redshift $0.0<z\leq 1.0$ in Fig. 3, where the lower and middle panels show our results for the mock spectroscopic and photometric redshift surveys, respectively. Compared to the clear cosmic web structures shown for the spectroscopic MGRS, galaxies in the photometric MGRS have more stretched structures along the line-of-sight. For comparison, we present the distribution of a small subset of galaxies in the Legacy Surveys DR8 in the upper panel of Fig. 3. The overall features in the photometric MGRS and the Legacy Surveys DR8 are quite similar.

Once the MGRS is generated, we provide some intrinsic properties of the groups/clusters. Taking into account the fact that the halo mass functions and group/cluster luminosity functions may change significantly in the redshift range we are probing, we separate galaxies/groups/clusters in our MGRS (as well as our observed galaxy samples) into three redshift bins: $0.0<z\leq 0.33$, $0.33<z\leq 0.67$, $0.67<z\leq 1.0$. As an illustration, we show in Fig. 4 the redshift distribution of galaxies in our MGRS and DESI DR8 samples, where the number of galaxies is normalized with respect to the sky coverage of the sample. The vertical dashed lines separate galaxies in our samples into the three redshift bins.

Shown in the upper left panel of Fig.5 are the accumulative halo mass functions of groups/clusters in different redshift bins as indicated. Symbols with errorbars are derived from the MGRS. Note that in order to partially reduce the impact of magnitude cut $z\leq 21$, we separate the galaxy samples into $N_{\rm zbin}=3$ redshift bins and use a $V_{\rm max}$ method to calculate the halo mass functions,

where, $i$ is the ID of the halo/group, $V_{\rm max}$ is calculated according to the apparent magnitude of the brightest group galaxy (BGG) at which redshift, $z_{\rm max}$, it can be observed,

Here $z_{\rm bin,low}$ and $z_{\rm bin,up}$ are the lower and upper limits of the corresponding redshift bin, respectively. In calculating the $z_{\rm max}$ of a galaxy, we have properly taken into account the difference in the K-correction between the actual and maximum redshifts, $z_{\rm max}$, using Eq. 2. As a comparison, we also show the corresponding theoretical model predictions given by SMT2001 represented by the solid lines. We demonstrate that overall the model and data agree quite well, though some small deviation shown at the high mass end for the low redshift bin, which is mostly caused by the small volume of our sample. On the other hand, the deviation at the low mass end for the high redshift bin, although we have partly corrected using the $V_{\rm max}$ method, is due to the survey magnitude limit cut, $z\leq 21$, where faint galaxies residing in low mass halos can not be observed. In the upper right panel, we present the richness distribution of groups/clusters in the MGRS. In the lowest redshift bin, there are only a few groups with more than 1000 members, while in the high redshift bin, the richest groups host around $30$ members. At the lower left panel, we show the accumulative total luminosity function of groups/clusters, with the total luminosities $L_{\rm tot}$ calculated by summing up all the member galaxy luminosities with apparent magnitude ${\rm z\leq 21}$ in the MGRS. Here again, a $V_{\rm max}$ method similar to Eq. 3 is used to partly correct the group incompleteness. Obviously, at the high redshift bin, the groups/clusters with $L_{\rm tot}<10^{10.2}\,h^{-2}\rm L_{\odot}$ are missing from our MGRS, which simply reflects the facts that galaxies in this redshift bin below this luminosity did not meet the magnitude limit cut. In the lower right panel, we present the median mass-to-light ratio of groups/clusters in the MGRS. Overall, the mass-to-light ratio in high mass groups/clusters $\sim 10^{14.5}\>h^{-1}\rm M_{\odot}$ spans from about 200 at low redshift to about 500 at high redshift. In low mass groups with mass $\sim 10^{11.5}\>h^{-1}\rm M_{\odot}$ the related numbers decrease to 50 to 10, respectively. Note that all the errorbars obtained in this figure are estimated using 200 bootstrap re-samplings.

Before we evaluate the group/cluster finding algorithm with the mock catalogs and apply the method to the Legacy Surveys DR8 observational data, we measure the galaxy luminosity functions (LFs) from the mock and observational galaxy catalogs. The galaxy luminosity function, which describes the average light densities of the Universe, represents a fundamental tool in understanding galaxy formation and evolution. Since galaxies are formed within dark matter halos and the most massive (central) galaxies reside roughly in the most massive halos, there exists statistical relations between galaxy luminosities and halo mass (e.g. Yang et al., 2012, and references therein). The accurate estimation of the galaxy luminosity functions plays a key role in establishing those kinds of relations. We estimate the galaxy luminosity function by counting galaxies in each luminosity bin and weighted by $1/V_{\rm max}$,

Here $V_{\rm max}$ can be similarly calculated using Eq. 4, but for the galaxy’s own $z_{\rm max}$.

We first focus on galaxies with mock spectroscopic redshifts in the MGRS. Shown in the left panel of Fig. 6 are the $z$-band luminosity functions of galaxies in three redshift bins: $0.0<z\leq 0.33$, $0.33<z\leq 0.67$, $0.67<z\leq 1.0$, respectively. The galaxy luminosities are obtained from the $z$-band absolute magnitude $K$-corrected to redshift $z=0.5$. In general, we see that the LFs measured in different redshift bins are roughly consistent with each other at the bright end. Since the MGRS is only complete up to ${\rm z\leq 21}$, faint galaxies will not be observed, especially those at higher redshifts. Thus, we see that the derived LFs have suffered more from incompleteness problem in the higher redshift bins, with brighter faint luminosity end cutoffs. Again, the errorbars are obtained from 200 bootstrap re-samplings of galaxies.

Next, we focus on galaxies with photometric redshifts in the MGRS. The solid lines shown in the left panel of Fig. 6 are obtained using the mock photometric redshift galaxies. Overall, the results are in good agreement with those obtained from the mock spectroscopic redshift galaxy samples. Since in our mock photometric redshifts, we only add a small redshift error to each galaxy, without any catastrophic redshift assignment. That is the main reason that we do not see a significant Eddington bias which might be induced in most of the photometric redshift surveys.

Finally, we focus on the Legacy Surveys DR8 data. The symbols shown in the right panel of Fig. 6 are the retrieved galaxy luminosity functions in three redshift bins. Compared to the MGRS (shown as solid lines), galaxies in the Legacy Surveys DR8 show similar cutoffs at low luminosity ends, which is caused by the same $z\leq 21$ magnitude cut. On the other hand, galaxies in the Legacy Surveys DR8 are somewhat brighter. This is mostly due to the galaxy luminosity calibration as well as the different K-correction modelings. Here we used the photometric data in COSMOS2015 catalog (Laigle et al., 2016) to calibrate the luminosity of the galaxies in our catalog. The galaxy luminosity is firstly assigned in $y$ band using a luminosity-stellar mass abundance matching method. Then a $z$ band galaxy luminosity is obtained according to the $z-y$ color of the galaxy, which is obtained using the age distribution matching method (Hearin & Watson, 2013). For each galaxy in our catalog, we choose a galaxy with the same luminosity and color in COSMOS2015 and use this galaxy’s fitted spectrum to calculate the K-correction and hence the apparent magnitude of that galaxy. See Meng et al. (2020) for more details. Note that since the K-correction used in Meng et al. (2020) was corrected to redshift $z=0.0$, in order to be consistent with the one used in this paper, we use $z$-band apparent magnitude and Eq. 2 to calculate the absolute magnitude and luminosity of each galaxy in our MGRS. Nevertheless, since in our cluster finder, we are using the accumulate luminosity functions measured from the data in different redshift bins to estimate the halo mass, such systematic differences will not impact the group/cluster membership determination.

To assess the performance of the group finder we proceed as follows: For a group ${\rm k}$, we investigate the halo ID for each member galaxy, and sum up the number of members in each halo. We choose the one with the most members as the candidate halo ${\rm h_{k}}$. Since in the photometric data, the brightest central galaxy (BCG) in a halo may move far away from the other members, we do not use the BCG to identify the candidate halo of the group, which is used in Y05. We define $N_{\rm t}$ as the total number of true members in the MGRS that belong to a halo ${\rm h_{k}}$, ${\rm N_{s}}$ is defined as the number of these true members that are selected as members of group ${\rm k}$, ${\rm N_{i}}$ is defined as the number of interlopers (group members that belong to a different halo) and $N_{\rm g}$ is defined as the total number of selected group members. These values allow us to define, for each group, the following three quantities:

• •

  The completeness $f_{\rm c}\equiv N_{\rm s}/N_{\rm t}$,

• •

  The contamination $f_{\rm i}\equiv N_{\rm i}/N_{\rm t}$,

• •

  The purity $f_{\rm p}\equiv N_{\rm s}/N_{\rm g}$.

Note that since the contamination is defined with respect to the true number of group members, the contamination $f_{\rm i}$ can be larger than unity. A contamination $f_{\rm i}>1$ implies that the number of interlopers is larger than the number of true members being detected. An ideal group finder yields groups with $f_{\rm c}=f_{\rm p}=1$ and $f_{\rm i}=0$.

The results obtained from the MGRS are shown in Fig. 7 with the left panels being for ${\rm B=5}$ and the right panels for ${\rm B=10}$. Since groups with a single member have zero contamination ($f_{\rm i}=0$) by definition, results are only shown for groups with a richness of $N_{\rm g}\geq 2$. The upper left-hand panel of Fig. 7 shows the cumulative distributions of the completeness $f_{\rm c}$ for the case of ${\rm B=5}$. Different line-styles correspond to groups of different assigned halo mass, $M_{L}$, as indicated. There are about 83% groups with group member completeness larger than 80 percent while $\sim 93\%$ of groups have member completeness larger than 60 percent for groups with mass $\gtrsim 10^{12.5}\>h^{-1}\rm M_{\odot}$. Among them, groups in the most massive bin shows the highest completeness. The middle left-hand panel of Fig. 7 shows the cumulative distributions of the contamination $f_{\rm i}$. On average, around $70\%$ of the groups in mass range $10^{12.5}<\log M_{L}\leq 10^{14.0}\>h^{-1}\rm M_{\odot}$ have $f_{\rm i}\leq 1$. While in the most massive bin, $10^{14.0}\>h^{-1}\rm M_{\odot}<\log M_{L}$, this number decreases to $43\%$. Since there are $\sim 30\%$ of the groups with more interlopers than true numbers ($f_{\rm i}>1$), special care needs to be taken if one wants to make use of the member galaxies, which we will look into in a subsequent paper. Finally, the lower left-hand panel shows the cumulative distributions of the purity $f_{\rm p}$. On average there are about 70% groups with mass $\gtrsim 10^{12.5}\>h^{-1}\rm M_{\odot}$ (except those in the most massive bin) having purity $f_{\rm p}>0.48$. In the most massive bin, $10^{14.0}\>h^{-1}\rm M_{\odot}<\log M_{L}$, this number is about $41\%$.

Next, we compute the completeness, contamination, and purity in terms of group membership for our fiducial case of ${\rm B=10}$. The corresponding results are shown in the right-hand panels of Fig. 7. Compared to the results shown in the left-hand panels for the case of ${\rm B=5}$, for groups with mass $10^{12.5}<\log M_{L}$ the performances differ in the following ways:

• •

  (1) The member completeness is slightly lower. About 72% groups are exhibiting more than 80 percent group member completeness, and $\sim 90\%$ ($\sim 95\%$) groups have member completeness larger than 60 (50) percent.

• •

  (2) The member contamination in each group is smaller than that in the ${\rm B=5}$ scenario. Here about $85\%$ of groups have an interloper fraction $\leq 1$.

• •

  (3) The membership purity in each group is also better. About 83% groups having purity $f_{\rm p}>0.48$.

Comparing groups in different mass bins, we notice that groups in the highest and lowest mass bins have better completeness and worse purity than groups in the intermediate mass bins. For the case B=10, the differences between mass bins are slightly less pronounced.

Given the differences between the ${\rm B=10}$ and ${\rm B=5}$ cases, if we care more about the purity of the true group members, the case ${\rm B=10}$ is preferred.

In this subsection, we focus on the global properties of the groups we find from the MGRS. To further evaluate the performance of our extended halo-based group finder, one useful quantity we want to assess is the global completeness of groups, $f_{\rm halo}$, defined as the fraction of halos in the MGRS with more than half of the true members being identified by the group finder as their richest components. The upper left panel of Fig. 8 shows $f_{\rm halo}$ obtained from our MGRS for halos with $N_{\rm t}\geq 3,5,10$ as functions of the true halo mass for the ${\rm B=5}$ case. Note that since in our MGRS, the photoz error depends quite strongly on redshift (see Eq. 1), and because of the $z\leq 21$ magnitude cut, groups at higher redshift should have less members and suffer more from the photoz errors. To check the related impact, we separate groups into two redshift bins, $z>0.5$ and $z\leq 0.5$, where the total number of groups in these two redshift bins are roughly similar. We use red and blue histograms to represents results for groups within these two redshift bins, $z>0.5$ and $z\leq 0.5$, respectively. We can see that the performance of $f_{\rm halo}$ depends only slightly on the redshift. The group finder successfully selects $\sim 70\%$ ($\sim 90\%$) of the true halos with mass $\sim 10^{12}\>h^{-1}\rm M_{\odot}$ ($\gtrsim 10^{14}\>h^{-1}\rm M_{\odot}$). Note that this does not imply in any case that the $10\sim 30\%$ of the remaining groups are not selected by our group finder but rather that their group membership completeness is lower than 50%. If we lower the membership completeness criteria, $f_{\rm halo}$ will increase.

The second quantity we investigate is the purity of the groups, $f_{\rm group}$, defined as the fraction of groups having the richest components that contain more than half of the true members. Here again, we separate groups into two redshift bins, $z>0.5$ and $z\leq 0.5$, and use red and blue histograms to represents the related results. The lower left panel of Fig. 8 shows $f_{\rm group}$ obtained from our MGRS, for groups with $N_{\rm g}\geq 3,5,10$ as functions of the assigned halo mass for the case ${\rm B=5}$. The $f_{\rm group}$ does not show significant differences between the two redshift bins as well. Overall, more than $90\%$ groups with mass $10^{12}\>h^{-1}\rm M_{\odot}$ are true groups. The percentage increases as the assigned halo mass and reaches 100% for groups with mass larger than $10^{14.5}\>h^{-1}\rm M_{\odot}$. The remaining few groups are those split groups or those whose richest component have less than 50% membership completeness. This value demonstrates that the detected groups regardless of their redshifts are in general reliable.

Shown in the right-hand panels of Fig. 8 are the corresponding $f_{\rm halo}$ and $f_{\rm group}$ for the case of ${\rm B=10}$. Overall, similar performance is achieved as those in the ${\rm B=5}$ case. Except the global completeness in the high redshift bin which is slightly lower, all the other quantities of the detected groups are similar. Note that since $f_{\rm halo}$ corresponds to the fraction of true groups being detected, while $f_{\rm group}$ corresponds to the fraction of the detected groups are true ones. In order to taking into account the impact of incompleteness on the halo mass estimation using the abundance matching method, we correct the abundance of the groups by a factor of $f_{\rm incomp}=f_{\rm group}/f_{\rm halo}$.

In this subsection we investigate the reliability of the assigned halo mass and redshift determinations of the groups. First we check the assigned halo mass obtained in the group finding steps 2-3. As pointed out in Y05, our halo mass estimation from the total group luminosity has the advantage that it is equally applicable to groups spanning a wide range in richness. However, a shortcoming of this method is that it requires the halo mass function, which is dependent on cosmology. In general, the assigned halo mass to each group can be regarded as the abundance of its population. Thus, it is extremely easy to convert the retrieved group mass in one cosmology to that in another cosmology, without running another round of group finder.

In order to assess the reliability of the halo mass assigned to individual groups, we use the mock group catalog from the MGRS. The top-left panel of Fig 9 shows the assigned halo mass ${M_{L}}$ versus true halo mass ${M_{h}}$ in the ${\rm B=5}$ case. To quantitatively compare the scatter to the line of equality (${M_{L}=M_{h}}$), we compute the following quantity for each group

and measure the standard deviation, ${\rm\sigma_{Q_{M}}}$, in several bins of ${\left[\log(M_{L})+\log(M_{h})\right]/2}$. The results shown in the small panel indicate that the scatter is $\sim 0.30$ dex for groups with $10^{12}\>h^{-1}\rm M_{\odot}\lesssim M_{L}\lesssim 10^{13.0}\>h^{-1}\rm M_{\odot}$, decreasing to $0.15$ dex at the high mass end. Here we find that the $\sigma_{Q_{M}}$ is quite independent of the number of members in the groups, according to the very similar behaviors shown for groups with different richness cuts.

There are several factors that contribute to this scatter. The first is the intrinsic scatter in the relation between the true halo mass and the true total group luminosity. Here for simplicity, we did not correct for the faint galaxies incompleteness caused by the survey magnitude cut. As shown in the lower right panel of Fig. 5, the mass-to-light ratio has a typical scatter of $0.15-0.2$ dex. Taking into account the $\sqrt{2}$ factor in Eq. 9, this factor already contribute a lot to the scatter, especially at the high mass end.

The second source of the scatter originates from the incompleteness and contamination introduced by our group finder, which may lead to biased total group luminosities. Nevertheless, as we mentioned before, since we are using the halo abundance matching method to assign halo mass, the systematic difference will not impact our results. Only chaotic difference, e.g., fragmentation or mis-identification will impact the halo mass assignment significantly, which may occur at mass $\lesssim 10^{13.0}\>h^{-1}\rm M_{\odot}$ (Campbell et al., 2015).

The last source of scatter in the assigned group masses owes to the fact that groups in the high redshift bin $0.67<z\leq 1.0$ is incomplete (see the upper-left panel of Fig. 5). Although we have used the $V_{\rm max}$ method to correct for the incompleteness of groups, since there are scatters among halo masses and BCG luminosities, such corrections are still not enough. The assigned halo mass to our groups will be systematically higher, especially at the low mass end.

Taking into all the above factors into account, we suggest that in our catalog, the groups with mass $\gtrsim 10^{13.0}\>h^{-1}\rm M_{\odot}$ are more reliable.

Next, we investigate the redshift accuracy of the groups. Shown in the lower left panel of Fig. 9 is the scatter plot of true group redshift ${z_{t}}$ v.s. assigned group redshift ${z_{g}}$. Here $z_{g}$ is defined as the luminosity weight group redshift. To quantify the scatter with respect to the line of equality (${z_{t}=z_{g}}$), we follow a similar track of halo mass scatter and compute the following quantity for each group

and measure the standard deviation, ${\sigma_{Q_{z}}}$, in several bins of ${\left[z_{t}+z_{g}\right]/2}$. Comparing to the input errors, the results shown in the small panel indicate that the scatter drops with the increase of member galaxies. The typical redshift error in groups with at least 10 members is about 0.008 (taking into account the $\sqrt{2}$ factor in Eq. 10).

Finally, shown in the right hand panels of Fig. 9 are the corresponding mass and redshift determinations for the case of ${\rm B=10}$. The general trends are quite similar to the ${\rm B=5}$ case, while the mass scatter is somewhat smaller, especially in relatively low mass groups with masses $\lesssim 10^{13.0}\>h^{-1}\rm M_{\odot}$ at about 0.40 dex (taking into account the $\sqrt{2}$ factor in Eq. 9).

Combining all the above tests, we conclude that the ${\rm B=10}$ case performs better than the ${\rm B=5}$ case in the group membership purity and halo mass estimation, albeit with somewhat worse group membership completeness. Thus we adopt ${\rm B=10}$ value in our group finder for the Legacy Surveys DR8, while ${\rm B=5}$ will be used for future comparison studies.