ELUCID IV: Galaxy Quenching and its Relation to Halo Mass, Environment, and Assembly Bias
Huiyuan Wang H. J. Mo Sihan Chen Yang Yang Xiaohu Yang Enci Wang Frank C. van den Bosch Yipeng Jing Xi Kang Weipeng Lin S. H. Lim Shuiyao Huang Yi Lu Shijie Li Weiguang Cui Youcai Zhang Dylan Tweed Chengliang Wei Guoliang Li Feng Shi
astro-ph.GA

















ApJ
MNRAS
ApJS
New Astronomy Review


    P^3M  Mpc  h^-1Mpc  hMpc^-1 Mpc^-1  h^-1kpc h^-1kpc km  s^-1 M_⊙  h^-1 M_⊙  h^-2 M_⊙R_ virv_ nv_ t#1𝐫_#1#1𝐬_#1θ_ spin1.16pt>-7.0pt
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          > ∼   < ∼  ∝∼ .5ex.5ex.5ex.5ex.5exM_r-5loghM_ sphM_ haloM_ subt_ crv_ virr_ virR_ virα_ infallELUCID: Galaxy Quenching and its Relation to EnvironmentWang H.Y. et al.1Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and
        Technology of China, Hefei, Anhui 230026, China; whywang@mail.ustc.edu.cn2School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China3Department of Astronomy, University of Massachusetts, Amherst MA 01003-9305, USA4Astronomy Department and Center for Astrophysics, Tsinghua University, Beijing 10084, China5Department of Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China6IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China7Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA8Purple Mountain Observatory, the Partner Group of MPI für Astronomie,  2 West beijing Road, Nanjing 210008, China9School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275, China10Departamento de Física Teórica, Módulo 15, Facultad de Ciencias, Universidad Autónoma de Madrid, E-28049 Madrid, Spain11Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China
        We examine the quenched fraction of central and satellite galaxies as a function
        of galaxy stellar mass, halo mass, and the matter density of their large scale environment.
        Matter densities are inferred from our ELUCID simulation, a constrained simulation
        of local Universe sampled by SDSS, while halo masses and central/satellite classification are
        taken from the galaxy group catalog of Yang et al. The quenched fraction for the total population
        increases systematically with the three quantities. We find that the `environmental quenching efficiency',
        which quantifies the quenched fraction as function of halo mass, is independent of stellar mass.
        And this independence is the origin of the stellar mass-independence of density-based quenching efficiency,
        found in previous studies. Considering centrals and satellites separately, we find that the two
        populations follow similar correlations of quenching efficiency with halo mass and stellar mass,
        suggesting that they have experienced similar quenching processes in their host halo.
        We demonstrate that satellite quenching alone cannot account for the environmental quenching efficiency
        of the total galaxy population and the difference between the two populations found previously mainly
        arises from the fact that centrals and satellites of the same stellar mass reside, on average, in halos
        of different mass. After removing these halo-mass and stellar-mass effects, there
        remains a weak, but significant, residual dependence on environmental density, which is eliminated when
        halo assembly bias is taken into account. Our results therefore indicate that halo mass is
        the prime environmental parameter that regulates the quenching of both centrals and satellites.
        
        
        § INTRODUCTION
        
        
        In the low-redshift universe, galaxies are observed to exhibit bimodal distributions
        in their colors and specific star formation rates. According to these bimodal
        distributions, galaxies can be divided into two distinctive sequences,
        a red sequence of galaxies quenched in star formation,  and a blue sequence
        of star-forming galaxies (e.g. Strateva et al. 2001; Blanton et al. 2003; Baldry et al. 2004;
        Brinchmann et al. 2004). The bimodal distribution is correlated with galaxy mass,
        with the red sequence dominated by massive galaxies, while the blue population
        by lower-mass ones. Furthermore, it is also known that the red, quiescent galaxies tend to
        reside in high density regions, while the blue star-forming galaxies display an opposite
        trend with environments (e.g. Oemler 1974; Dressler 1980; Hogg et al. 2003;
        Kauffmann et al. 2004; Baldry et al. 2006; Peng et al. 2010; Zheng et al. 2017). All these suggest that
        the mechanisms responsible for quenching star formation in a galaxy
        must be related to galaxy mass as well as to the environment.
        
        In the current cold dark matter (CDM) cosmogony, galaxies are assumed to form and evolve
        within dark matter halos (e.g. White & Rees 1978; Mo, van den Bosch & White 2010).
        Lower-mass halos on average form earlier and subsequently merge to form more massive
        ones. In this process, the galaxy that forms on the main branch of a halo merging
        tree is expected to be the dominant galaxy residing near the center of the halo
        (the central galaxy),  while other galaxies that form in the sub-branch progenitor
        halos, are expected to orbit the central as satellite galaxies. These two populations
        of galaxies are expected to have experienced different processes that quench
        their star formation: while supernovae and active galactic nuclei (AGNs) feedbacks,
        and shock heating of cold accretion flow may affect both central and satellite galaxies,
        processes like ram pressure and tidal stripping are believed to operate only on
        satellite galaxies (e.g. White & Frenk 1991; Kang et al. 2005; Dekel & Birnboim 2006; Bower et al. 2006;
        Croton et al. 2006; Kereš et al. 2009; Lu et al. 2011; Guo et al. 2011; Vogelsberger et al.2014;
        Schaye et al. 2015). These models have successfully reproduced many global properties of observed galaxies.
        However, there are still significant discrepancies between model
        predictions incorporating these processes and observational data in terms
        of the fraction of the quenched population (see e.g. Hirschmann et al. 2014;
        Vogelsberger et al. 2014; Henriques et al. 2016),
        indicating that our understanding of the quenching processes is still incomplete.
        
        
        Observing correlations between galaxy properties and different aspects of their environment could
        help to distinguish different galaxy formation models. A variety of quantities have been used to describe
        the environment around a galaxy (Haas, Schaye & Jeeson-Daniel 2012). These environmental parameters are
        usually designed for different purposes, and an optimal decision has to be made for a specific
        question (Leclercq et al. 2016). The traditional environmental parameter is the (projected) number density
        of galaxies, which is often adopted in observational studies of galaxy properties on various scales
        (e.g. Dressler 1980;  Hogg et al. 2003; Kauffmann et al. 2004; Hirschmann et al. 2014). It can be
        directly obtained from the galaxy redshift survey without any additional assumption. The other widely
        used parameter is the host halo mass (e.g. Weinmann et al. 2006; Wetzel, Tinker & Conroy 2012; Woo et al. 2013),
        which is closely linked to galaxy formation in the CDM paradigm. Indeed, the halo occupation distribution (HOD)
        models (e.g. Jing et al. 1998; Peacock & Smith 2000; Zheng et al. 2005; Zu & Mandelbaum 2016), conditional
        luminosity function (CLF) models(e.g., Yang et al. 2003;  van den Bosch et al. 2007), abundance
        matching models (e.g., Mo et al. 1999; Kravtsov  et al. 2004; Vale & Ostriker 2006; Behroozi, Conroy, & Wechsler 2010;
        Hearin & Watson 2013) and halo-based empirical models (e.g., Yang et al. 2013; Lu et al. 2014, 2015; Moster, Naab & White 2017)
        have all used halo masses to link galaxies to dark matter halos.
        
        Using halo mass inferred from galaxy group catalog (Yang et al. 2007) as environmental parameter,
        Weinmann et al. (2006) found that the quenched fraction of satellite galaxies is much lower than that in model
        predictions and increases strongly with host halo mass (see also Liu et al. 2010;
        Wetzel et al. 2012). This has motivated later semi-analytic galaxy
        formation models (SAMs) to employ an incremental stripping of hot gas associated with
        satellites through ram-pressure and tidal stripping (e.g. Kang & van den Bosch 2008;
        Font et al. 2008; Weinmann et al. 2010; Guo et al. 2011; Henriques et al. 2015). Moreover,
        halo mass is also found to have significant impact on the quenching of star
        formation in centrals of given galaxy masses (Weinmann et al. 2006; Woo et al. 2013; 2015;
        Bluck et al. 2014;2016). Here, AGN feedback is thought to be the major quenching mechanism,
        and its strength is likely to depend on halo mass
        (e.g. Croton et al. 2006; Henriques et al. 2016), qualitatively consistent with the observation results.
        
        In addition to environmental effects that are confined within halos, there are also
        observational indications that the environmental effects may operate on
        scales beyond their boundaries. For example, at a fixed halo mass, the clustering of galaxy groups is found to
        depend on the star formation rate and color of the central galaxies
        (Yang, Mo, & van den Bosch 2006; Lacerna, Padilla, & Stasyszyn 2014). Similarly,
        Kauffmann et al. (2013) found that the star formation rates of central galaxies
        are correlated with that of their neighbours on scales up to several Mpcs,
        far beyond their halo virial radii (see also Berti et al. 2017, and references therein).
        These results suggest that large scale environments
        may also affect the star formation of galaxies embedded in them.
        However, it is unclear whether this is due to a causal connection between
        star formation and large-scale environments, or is produced by a correlation induced by
        some intermediate connections. For example, such large-scale effects
        may be produced by the dependence of star formation on halo assembly history
        (see e.g. Hearin, Watson, & van den Bosch 2015; Lim et al. 2016;
        Tinker et al. 2016; Zentner et al. 2016), combined with halo assembly
        bias that links halo formation with large-scale
        structure (e.g. Gao, Springel, & White 2005; Wechsler et al. 2006; Jing, Suto & Mo 2007),
        or may be produced by the preheating of the intergalactic gas owing to the
        formation of large scale structure (e.g. Mo et al. 2005; Kauffmann et al. 2013).
        
        Using galaxy number density as environmental parameter, Baldry et al. (2006) found that the
        quenched fraction depends both on galaxy stellar mass and environmental density,
        and the dependence can be well described by a simple functional form.
        Peng et al. (2010) studied the environmental quenching efficiency, which is defined
        as the probability for a galaxy to be quenched in high-density regions
        relative to that in low-density regions, where environmental effects are
        expected to be weak. Remarkably, the efficiency defined in this way is found
        to be almost independent of stellar mass. Subsequently, Peng et al. (2012) suggested that
        the independence may be explained if environmental quenching is assumed to be important only for satellite
        galaxies, and if both the quenching efficiency of satellite galaxies
        and the satellite fraction  are independent of galaxy mass.
        However, these assumptions are not supported by the results obtained
        for centrals from galaxy groups, which clearly show that environmental quenching
        of central galaxies is also important (e.g. Weinmann et al. 2006; Woo et al. 2013; 2015),
        and by the results for satellites, which show that both the
        quenching efficiency (Knobel et al. 2015) and satellite fraction
        (Mandelbaum et al. 2006; Cooray 2006; Tinker et al. 2007;
        van den Bosch et al. 2007) depend on galaxy mass.
        
        
        The difference between centrals and satellites also attracted particular attention
        (e.g. van den Bosch et al. 2008; Skibba 2009; Wetzel et al. 2012;2013; Peng et al. 2012; Hirschmann et al. 2014;
        Knobel et al. 2015; Spindler & Wake 2017). It has been found that the quenched fraction of the
        central population is lower than that of satellites of the same mass. This difference has been used
        to quantify the efficiency of various satellite-specific quenching processes,
        such as strangulation, tidal stripping and ram-pressure stripping (van den Bosch et al. 2008).
        However, Knobel et al. (2015) found that centrals and satellites of the same mass respond
        to their environments in a similar way, as long as centrals have massive satellites. Moreover,
        Hirschmann et al. (2014) studied the failures of current galaxy formation models
        in matching observational data and suggested that centrals and satellites
        should be treated not as differently in their response to environments
        as previously assumed.
        
        Clearly, more investigations are required in order to understand these contradictory
        results in the literature. It is essential to identify and characterize the contribution to
        the quenching of star formation of the relevant parameters, such as
        galaxy stellar mass, halo properties and large-scale density field. In particular,
        it is important to see whether the independence of environmental quenching efficiency
        on galaxy mass can be reproduced if only halo masses and halo assembly
        bias are taken into account, and what roles centrals and satellites
        play in establishing the galaxy-mass independence of the quenching efficiency
        found earlier, and whether the star formation quenching in centrals and satellites is dominated by different processes.
        
        In this paper, the fourth of a series, we use the environmental information
        provided by the ELUCID project and galaxy groups selected from the Sloan
        Digital Sky Survey (SDSS; York et al. 2000) to investigate the quenching of
        galaxies in different environments. The ELUCID project (Wang et al. 2014; 2016;
        Tweed et al. 2017) aims to reconstruct the initial conditions responsible
        for the formation of the structures in the observed low-redshift universe, and to
        recover the mass distribution in the local universe by constrained simulations.
        The constrained simulations give the full information about the dynamical
        state and formation history of the large scale structure within which the observed
        galaxies reside. This provides an unique opportunity to systematically investigate
        the quenched population of galaxies of different masses in different environments.
        
        Our paper is organized as follows. In Section <ref>, we describe the
        galaxy sample, group catalog and environmental quantities used for our analysis.
        Section <ref> shows how the quenched galaxy population depends on
        galaxy mass, halo mass and environmental density for the total population,
        as well as separately for the central and satellite populations.
        In Section <ref>, we investigate the galaxy stellar mass independence
        of the environmental quenching efficiency using the matter density field as
        inferred from our ELUCID simulation, with a special focus on the role of central
        and satellite galaxies. In Section <ref>, we investigate the quenching
        efficiencies using halo mass as environmental parameter. In Section <ref>
        we discuss the implications of our results by comparing the data to three simple
        models that incorporate the dependence on galaxy stellar mass, halo mass,
        environmental density, and halo assembly history. In Section <ref> we discuss
        whether central galaxies are special in their quenching properties
        in comparison with satellites. Finally, we summarize our  results and discuss their implications in Section <ref>.
        
        
        
        
        § GALAXY SAMPLE AND ENVIRONMENTAL QUANTITIES
        
        
         §.§ The galaxy sample
        
        
        The galaxy sample used here is extracted from the New York University Value-added
        Galaxy Catalog (NYU-VAGC, Blanton et al. 2005) of the SDSS DR7 (Abazajian et al. 2009).
        We select all galaxies in the main galaxy sample, with r-band apparent
        magnitudes ≤17.72, and with redshift completeness C≥0.7, and within the
        reconstruction region of ELUCID simulation (see below and Wang et al. 2016 for the details).
        The first two selection criteria ensure that most of the selected galaxies are contained
        in the Yang et al. (2007) group catalog (with extension to DR7), and the third ensures
        that we have reliable estimates of our galaxies' environmental densities.
        Among these galaxies, 1,707 are members of groups that have only a fraction
        of 0<f_ edge≤ 0.6 of their virial volumes contained within the survey
        boundary. These galaxies are removed from our sample (see Yang et al. 2007).
        A total of 4,233 galaxies that do not have star formation rate estimates are
        also discarded. Our final sample contains 317,791 galaxies.
        
        Stellar masses for these galaxies, indicated by m (with unit ), are computed using the relations
        between stellar mass-to-light ratio and (g - r) color as given in
        Bell et al. (2003), adopting a Kroupa (2001) initial mass function (IMF).
        We refer to Yang et al. (2007) for details. Star formation rates (SFR) for these
        galaxies are taken from the the MPA-JHU DR7 release
        website[ http://www.mpa-garching.mpg.de/SDSS/DR7/], which are estimated by using
        an updated version of the method presented in Brinchmann et al. (2004) and calibrated
        to the Kroupa IMF. For given galaxy stellar mass, the distribution of SFR is known to be bimodal (e.g. Brinchmann et al. 2004), with a high SFR mode corresponding to the star forming population and a low SFR mode corresponding to a quenched population.
        In this paper, we adopt the division
        line proposed by Woo et al. (2013) to separate the two populations:
        
            log SFR =0.64logm-1.28logh-7.22 ,
        
        where the reduced Hubble constant h (Hubble constant in units of
        100  km s^-1Mpc^-1) is used to transfer the unit of the stellar
        mass from  used here to  used in Woo et al. (2013).
        
        In our analyses, each galaxy is assigned a weight w=1/(V_ maxC) to take into
        account the Malmquist bias and redshift (spectroscopic) incompleteness, with the latter
        taken from the NYU-VAGC. Since the geometry of our reconstruction region is not regular,
        we calculate V_ max in the following way. For each galaxy, we first obtain its
        Petrosian photometry in ugriz bands and its redshift. We then use these data as input
        to the K-correction utilities (v4_2) of Blanton & Roweis (2007) to estimate
        z_ min and z_ max, the minimum and maximum redshifts, between which the
        galaxy can be observed with the r-band limit of 17.72 mag. Finally, we measure V_ max as the
        volume of the reconstruction region that is between z_ min and z_ max.
        
        
        
         §.§ Host halos of galaxies
        
        
        Galaxy groups and clusters (hereafter referred to together as galaxy groups),
        when properly selected from a galaxy sample, can be used to represent
        the host dark halos of galaxies. In this paper we make use of the galaxy groups
        identified by Yang et al. (2012) from the SDSS DR7 to represent halos in which
        galaxies reside. This group catalog
        was constructed with the halo-based group finder developed by Yang
        et al. (2005), which assigns new galaxies into groups based on
        the size (virial radius) and velocity dispersion of the host dark
        halo represented by the current members assigned to a tentative group.
        Iterations are performed until the identification of member galaxies
        as well as the estimation of halo mass converge.
        The halo masses in the catalog are estimated via the ranking of two mass proxies: the total
        luminosity or total stellar mass of all members brighter than M_r=-19.5+5log(h) in the
        r-band. For our analysis, we adopt the halo masses, M (with unit ), estimated using the
        total stellar mass. Following common  practice, we define the central galaxy of a group to be the most
        massive member, and all other members are referred to as satellites.
        The reconstruction region is restricted to the redshift range 0.01<z<0.12,
        where groups with log(M) 12 are complete.
        
        
        
        
         §.§ Environmental density
        
        
        In Wang et al. (2016; hereafter paper III), we presented a series of methods to
        reconstruct the initial density field that is responsible for the galaxy distribution
        in the local Universe, and we used a high-resolution N-body simulation to evolve the
        initial conditions to the present-day. These simulation results can be used to
        obtain reliable estimates for the environmental densities within which the observed
        SDSS galaxies reside.
        
        The reconstruction is restricted to the Northern Galactic Cap of the SDSS DR7 region
        and to the redshift range 0.01≤ z≤0.12. In order to avoid problems near the
        survey boundary, where the reconstruction is less reliable, we exclude
        galaxies and groups whose distances to the SDSS survey boundary are smaller than 5.
        For each galaxy, we correct for its redshift-space distortions (see Paper III, Wang et al.
        2012 and Shi et al. 2016, for details) and estimate its real-space location.
        The environmental density for a galaxy is determined by computing the matter
        density, smoothed with a Gaussian kernel of 4, at its real-space location in
        our constrained simulation. Tests based on mock galaxy catalogs demonstrate that the
        uncertainties in this density estimate are typically 0.10 dex. In what follows,
        we quantify the matter density for each galaxy using the quantity
        
            Δ =ρ/ρ̅ ,
        
        where ρ is the smoothed mass density at the location of the galaxy,
        and ρ̅ is the mean density of the universe.
        
        In the literature, one environmental indicator commonly used is
        the local number density (or over-density) of galaxies. Our tests show that the galaxy densities are positively correlated with the mass
        densities, although the scatter is rather large (see also Baldry et al. 2006). Because of the redshift distortion effect and the complicated correlation with underlying mass density, galaxy density hampers a meaningful interpretation of its correlation between galaxy
        properties (see Weinberg et al. 2006 and Woo et al. 2013). Therefore, we consider the matter densities, Δ, used here to be a superior, and more physical, quantity to characterize the large-scale environment of a galaxy.
        
        
        
        
        § THE QUENCHED POPULATIONS
        
        
        
        In this section we examine how quenching of star formation
        depends on the intrinsic and environmental properties of galaxies.
        For any subset of galaxies with a given set of properties {g},
        we calculate the average quenched fraction as
        
            F_ q({g})=∑_i∈{g}q_iw_i/∑_i∈{g}w_i ,
        
        where w_i is the weight for a galaxy i,
        and q_i is set to 1 for a quenched galaxy, otherwise zero.  In our following analysis, we will
        consider the total population, and the central and
        satellite populations separately. The quenched fractions of the sub-populations
        are denoted by F_ q, c (centrals) and F_ q, s (satellites), respectively.
        
        
        
         §.§ Marginalized dependence on stellar mass, halo mass and density
        
        
        Figure <ref> shows, for the whole population and for centrals and satellites separately, 
        the quenched fraction of galaxies as a function of galaxy stellar mass, halo mass and environmental mass density.
        The quenched fractions as a function of one of the three parameters are obtained by marginalizing
        over the other two parameters. Note that we only consider galaxies with log(m)≥9.0
        and results are shown only for these galaxies in the left two panels.
        In this and the following figures, error bars are all evaluated from 1,000 bootstrap re-samplings and only
        data points that contain at least 10 galaxies are shown.
        
        It is clear that the quenched fraction increases with increasing stellar mass, halo mass and
        environmental density, consistent with previous studies (e.g. Brinchmann et al. 2004; Weinmann et al. 2006;
        Baldry et al. 2006).  These correlations are often
        considered as observational evidence for quenching processes, such as mass quenching,
        halo quenching and environmental quenching. We then examine centrals and satellites separately. At given stellar mass or density, satellites tend to
        be more often quenched than centrals. However, the trend is reversed if halo mass is used instead as the control parameter.
        It is well known that more massive galaxies tend to live in more massive halos and
        more massive halos tend to reside in higher density regions, and so it is not obvious
        whether the density dependence reflects a causal connection or is induced by
        the correlation between halo mass and density. In order to disentangle the
        different effects, we need to examine the joint distribution of the quenched
        population with respect to the parameters in question (see below).
        
        
        
        
         §.§ Dependence on stellar mass and environmental density
        
        
        We first disentangle the dependencies of the quenched fraction on
        stellar mass and environmental density.
        The left panel of Figure <ref> shows the quenched fraction
        of the total population, F_ q(m,Δ), as a function of
        the environmental density for galaxies in various stellar mass bins.
        Consistent with the marginalized result, there is
        significant dependence on Δ,
        even for galaxies in a given narrow m bin. The dependence is
        seen to be stronger at low m and weaker at the lower Δ end. At a given Δ,
        the quenched fraction increases with m, and the increase is
        more significant for galaxies of lower masses.
        
        We then investigate the quenched fractions separately for the central and satellite populations and present the results in
        the middle and right panels of Figure <ref>.
        For central galaxies, the Δ-dependence is rather weak for all stellar
        mass bins in comparison to the result shown in the right panel of Figure <ref>.
        This suggests that the marginalized Δ-dependence for centrals is primarily
        due to the fact that more massive centrals tend to reside in higher density region.
        In contrast, for satellites, the Δ-dependence is
        strong in most of the stellar mass bins. In particular, the
        marginalized Δ-dependence is very similar to that for the lowest
        stellar mass galaxies, indicating that the marginalized result
        for satellites is dominated by low-mass galaxies over the whole density range.
        At given Δ, the m-dependence is
        significant for both centrals and satellites. The m-dependence for centrals
        is similar to that for satellites at low Δ, but is stronger at high Δ,
        suggesting that the difference shown in the marginalized m-dependence
        between the two populations is mainly caused by the difference in the
        high density region. These results are in qualitative agreement with the
        results previously obtained by Peng et al. (2010; 2012)
        and Knobel et al. (2015).
        
        
        
         §.§ Dependence on stellar mass and halo mass
        
        
        Figure <ref> shows the quenched fraction as a function of halo mass  for galaxies in different stellar
        mass bins. At a given m, the quenched fraction increases with M, and the increase is steeper for lower mass galaxies.
        These trends are consistent with those obtained before
        (e.g. Weinmann et al. 2006;  Wetzel et al. 2012). Most intriguingly, the
        behaviors of central and satellite galaxies are
        almost indistinguishable, although it is important to stress that
        the halo mass range covered by central galaxies is rather limited,
        in particular for galaxies with low masses. This suggests that the difference
        between centrals and satellites found in the marginalized dependencies on
        m and M are mainly caused by the different m and M
        ranges covered by the two populations. For example, satellites usually
        reside in more massive halos than centrals of the same stellar mass,
        this combined with the M-dependence of the quenched fraction can explain
        why the quenched fraction for satellites is higher than that for
        centrals of the same stellar mass (left panel of Figure <ref>).
        Similarly, since by construction centrals are more massive than satellites
        in halos of a given mass, centrals are expected to be more often quenched than
        satellites because of the m-dependence (middle panel of
        Figure <ref>).
        
        For the stellar mass bin 10.6≤log(m)≤11, the quenched fraction of centrals
        appears to be somewhat higher than for satellites in the same halo
        mass bin. However, upon closer inspection, we find that this is
        mainly an artifact of the finite bin-sizes used, combined with the
        fact that at given halo mass, the satellite galaxies are strongly biased
        towards log(m)∼10.6, while centrals are biased towards
        log(m)∼11. To demonstrate this, we split the galaxies
        in this mass bin into two sub-samples at log(m)=10.8.
        The results for these two sub-samples are shown in the bottom left panel
        of Figure <ref>. As one can see, this significantly reduces the
        differences between the centrals and satellites, and we therefore conclude
        that there is no significant difference in the halo mass dependence
        of the quenched fraction between centrals and satellites.
        
        We note again that numerous previous studies (e.g., van den Bosch et al. 2008; Weinmann et al. 2009;
        Pasquali et al. 2010; Wetzel et al. 2012; Peng et al. 2012; Knobel et al. 2013;
        Bluck et al. 2014; Fossati et al. 2017) have shown that, at a given stellar mass,
        satellites are more often quenched than centrals and suggested that some satellite-specific environmental processes have played important roles in quenching satellite galaxies.
        However, our results portray a different picture; centrals and satellites follow the same correlation  between quenched
        fraction and host halo mass, suggesting that they experienced similar environment-dependent
        quenching, and the only reason that satellite and centrals appear different at fixed stellar mass is
        that they sample different ranges in host halo mass. Further investigations
        on this are presented in Sections <ref> and  <ref>.
        
        
        
        
         §.§ Dependence on halo mass and environmental density
        
        
        
        It is known that more massive halos tend to locate in higher density regions,
        an effect usually referred to as halo bias (e.g. Mo & White 1996). Therefore
        the density dependence shown in Figure <ref> and the halo mass dependence
        shown in Figure <ref> may be connected. To examine this, we split galaxies
        further into four sub-samples based on Δ and
        calculate the quenched fraction as a function of M. The results are shown in
        Figure <ref>. Overall, the M-dependence of the
        quenched fraction is similar for different sub-samples of Δ,
        suggesting that halo mass may be the dominating factor in determining the
        quenched fraction.
        
        However, for a given M, there is a weak but systematic
        trend  of increasing quenched fraction with increasing Δ.
        This suggests that factors other than halo mass affect galaxy quenching, and
        is broadly consistent with some previous findings that galaxy groups of a
        given halo mass have different clustering properties depending on the color
        or star formation of their member galaxies (e.g. Yang et al. 2006).
        All of this is most likely related to assembly bias (Gao et al. 2005), the fact that halo bias
        depends not only on halo mass, but also on other halo properties such
        as assembly history, if the quenching processes depend on these additional halo properties. Unfortunately,
        the observational sample is still too small, and the uncertainties in
        the density dependence shown in Figure <ref> are still too large,
        to draw any quantitative conclusions. We will revisit
        this issue in Section <ref>.
        
        
        
        
        § DENSITY-BASED ENVIRONMENTAL QUENCHING EFFICIENCIES
        
        
        A useful parameter to quantify the efficiency of galaxy quenching
        is the relative environmental quenching efficiency (hereafter `quenching
        efficiency' for brevity)
        
        
            ε(m, Δ | Δ_0) ≡F_ q(m,Δ)-F_ q(m,Δ_0) 1-F_ q(m,Δ_0) .
        
        (Peng et al. 2010), which specifies the probability for a star-forming
        galaxy of mass m to be quenched when it transits from an environment
        characterized by some zero-point matter density, Δ_0, to a region
        with Δ. We choose Δ_0
        to correspond to the lowest-density environment probed by our data, as the environmental effects are expected
        to be minimal in these void-like environments. As shown in Figure <ref>,
        the quenched fraction of galaxies, in most of the stellar mass bins, has the
        lowest value at logΔ≤ 0, where the Δ-dependence of the quenched
        fraction is also weak. We thus choose galaxies with logΔ≤0 to define
        the zero point.
        
        
        
         §.§ A dearth of stellar mass dependence for the total population
        
        
        The upper-left panel of Figure <ref> shows the quenching efficiency
        ε(m, Δ | Δ_0) of the total population as a function of Δ. As one can see,
        the efficiency is close to zero at Δ∼Δ_0, by definition,
        and increases rapidly with Δ. Remarkably,  the efficiency is almost
        independent of m over the range 9≤log(m)≤11.
        To see this more clearly, we show ε(m, Δ | Δ_0) as a
        function of m for various Δ bins in Figure <ref> as
        black squares. The quenching efficiency is almost a constant in the
        entire mass range,  except for the highest stellar mass bin.
        This is in good agreement with Peng et al. (2010), although they used
        galaxy number density instead of the more physical matter density used here.
        The efficiency for the highest stellar mass bin is higher than for the other
        stellar mass bins, but the uncertainties are large. Note that there
        are only very few massive galaxies in low density regions with
        Δ < 1, which can produce a large statistical uncertainty in the
        denominator of Eq. (<ref>). The discrepancy for the highest
        m bin is, therefore, not conclusive.
        We will come back to this issue in Section <ref>.
        
        Baldry et al. (2006) found that the quenched fraction can be fitted by a very simple equation:
        
            F_ q(m,Δ)=1-exp[-(Δ/Δ_⋆)^b]
            exp[-(m/m_⋆)^d] .
        
        Inserting it into Equation (<ref>), we obtain
        
            ε(m, Δ | Δ_0)=1-exp[-(Δ/Δ_⋆)^b
            +(Δ_0/Δ_⋆)^b] .
        
        We use this equation to fit the ε data points, with logΔ_0=-0.2,
        which is the median value for galaxies with logΔ_0 ≤ 0.
        The best fit gives Δ_⋆=11.5±0.26 and b=0.975±0.035 (see Figure <ref> for the fitting curve).
        We then use Equation (<ref>) to fit the quenched fraction for the total population with Δ_⋆=11.5 and b=0.975.
        The best fit values for the other two parameters are m_⋆=5.29±0.04×10^10 and d=0.749±0.006, and the results
        are shown in the left panel of Figure <ref>.
        These fitting results are presented here as a convenient way
        to represent the data.
        
        
        
         §.§ Central versus satellite populations
        
        
        For central galaxies, we define a quenching efficiency similar to Equation (<ref>):
        
            ε_ c(m, Δ | Δ_0) =
            F_ q,c(m,Δ)-F_ q,c(m,Δ_0) 1-F_ q,c(m,Δ_0) ,
        
        where F_ q,c(m,Δ) is the quenched fraction of central galaxies. Here again we use galaxies at
        logΔ_0≤ 0 to calculate F_ q,c(m,Δ_0). So defined, this
        efficiency characterizes the probability for a star forming central to quench
        if it were to move to a higher density environment,
        Δ, while remaining a central. As is evident from the upper-middle panel of Figure <ref>,
        ε_ c has a much weaker dependence on Δ than the
        corresponding quenching efficiency of the total population. Nevertheless, even for centrals there is a significant
        tendency for ε_ c to increase with Δ. Moreover, there
        is also a trend for ε_ c to increase with stellar mass (see also Figure <ref>), in particular at high Δ. As for the total population, the most massive bin has very few galaxies with
        logΔ≤0, so that the corresponding ε_ c carries
        large uncertainties.
        
        Similarly, we can define a quenching efficiency for satellite galaxies. Unfortunately, the total
        number of satellites at logΔ≤ 0 is small, and so the derived efficiency will
        have large uncertainties. However, F_ q,s(m,Δ_0) is close to
        F_ q,c(m,Δ_0) for most of stellar mass bins where both can be measured
        reliably (Figure <ref>). We thus define an alternative efficiency for satellites as
        
            ε'_ s(m, Δ | Δ_0) =
            F_ q,s(m,Δ)-F_ q,c(m,Δ_0) 1-F_ q,c(m,Δ_0) .
        
        The results are shown in the upper-right panel of Figure <ref>.
        As one can see, ε'_ s(m, Δ | Δ_0)
        increases rapidly with Δ, but its dependence on m is weak.
        
        Finally we consider a satellite-specific quenching efficiency,
        which is defined as
        
            ε_ s(m, Δ) = F_ q,s(m,Δ)-F_ q,c(m,Δ)/1-F_ q,c(m,Δ) ,
        
        (see van den Bosch et al. 2008). Here, the central galaxies are
        used as the control sample (zero point) in (m, Δ) space,
        against which the quenching of satellites is measured. The lower-left panel of
        Figure <ref> shows ε_ s as a function
        of Δ for satellite galaxies of different masses. As one
        can see, the quenching efficiency increases quite rapidly with
        Δ in most stellar mass bins. For a given Δ,  ε_ s
        decreases with m, and the decrease is larger for more
        massive galaxies.
        
        As pointed out by Wetzel et al. (2013), ε_ s defined in this way
        actually measures a combined effect of the satellite-specific
        quenching processes and the evolution of central galaxies.
        So it is not straightforward to use it to interpret the satellite quenching efficiency.
        However, it does not change our conclusion in the subsequent subsection that there exists
        an unexpected connection between centrals and satellites.
        
        
        
         §.§ A conspiracy between centrals and satellites?
        
        
        Peng et al. (2010) argued that the stellar mass independence of
        ε(m, Δ | Δ_0) can be fully understood in terms
        of the quenching of satellite galaxies. Assuming that the
        environmental effect on centrals is negligible, they wrote the
        environmental quenching efficiency as,
        
            ε'(m, Δ | Δ_0)=f_ s(m, Δ)
            ε_ s(m, Δ) ,
        
        where, f_ s(m, Δ) is the satellite fraction, and ε_ s(m, Δ)
        is the satellite-specific quenching efficiency of Equation. (<ref>).
        Figure <ref> shows the satellite fraction as a function of
        Δ for galaxies in different m bins.
        In order to explain the m-independence of the quenching
        efficiency, Peng et al. (2010; 2012) suggested that both f_ s(m, Δ)
        and ε_ s(m, Δ) are independent of m.
        This is clearly inconsistent with our data shown in the lower-left panel of Figure <ref> and Figure <ref>.
        
        
        
        
        Since the hypothesis made by Peng et al. has far-reaching implications (i.e., the the environment dependence
        of the quenching efficiency is entirely due to the quenching of satellites),
        it is important to address this discrepancy in some detail. In fact, there are
        a number of factors that play a role. First of all, Peng et al. used the
        over-density of galaxies as their environment indicator, rather than the more
        physical matter density used here. In addition, they used colors to
        split their population into star-forming and quenched, whereas we use actual
        star formation rates. Since dust extinction can make a star forming galaxy
        appear red, and thus `quenched' based on color, using actual star formation rates
        yields more accurate estimates of the true quenched fraction. Secondly,
        Peng et al., assumed F_ q,c(m,Δ) to be independent of Δ,
        even though this is not supported by their own data. Since centrals on average reside
        in lower Δ environments than satellites of the same stellar mass
        (e.g. Knobel et al. 2015), the average of F_ q,c over Δ is
        biased towards low-density regions. As a consequence, adopting the average
        of F_ q,c can lead to an overestimation of ε_ s, in particularly at
        the high-Δ end. For low-mass galaxies, this bias is negligible,
        because F_ q,c is on average much smaller than both F_ q,s
        and unity. This is consistent with the weak m-dependence of ε_ s
        we find for these galaxies. For massive galaxies, on the other hand,
        the value of F_ q,c is higher and the denominator in
        Equation (<ref>) smaller. This results in a significant bias
        that weakens the m-dependence of ε_ s at the high mass end.
        Indeed, when taking into account the dependence of F_ q,c on galaxy number density,
        Knobel et al. (2015) also found that ε_ s decreases with
        m at a given galaxy number density (see their figure 3).
        Finally, as is evident from Figure <ref>, the satellite fraction depends strongly on m. In fact, it is well known
        that the satellite fraction increases with decreasing stellar mass, which
        has been demonstrated using galaxy group catalogs (e.g., van den Bosch et al. 2008),
        subhalo abundance matching (e.g., Wetzel et al. 2013), galaxy-galaxy lensing
        (e.g., Mandelbaum et al. 2006), galaxy clustering (e.g., Cooray 2006; Tinker
        et al. 2007; van den Bosch et al. 2007) and combinations thereof (e.g.,
        Cacciato et al. 2013).
        
        In the lower-middle panel of Figure <ref>, we show ε'
        as a function of Δ. For low-mass galaxies with log(m)≤ 10.2,
        ε' is almost independent of m, in agreement with
        the hypothesis of Peng et al. (2010; 2012). However, for more massive galaxies,
        ε' clearly depends on m. Thus, the m-independence
        of ε of the total population shown in Section <ref> is not due
        to the m-independence of f_ s and ε_ s, and the quenching
        efficiency of the total galaxy population cannot be explained by the quenching of satellites alone.
        
        As shown in Figure <ref>, central galaxies do exhibit some weak but non-trivial
        Δ-dependence in all the mass bins considered, and at
        log(m)≥ 10.6 the dependence is actually comparable to or slightly stronger
        than that for satellites. For centrals, the Δ-dependence is stronger for
        more massive galaxies, a trend opposite to that seen for satellites.
        To understand the importance of quenching for centrals in the total population,
        we express the quenching efficiency for the total population in terms of
        ε_ c and ε'.
        At very low density, logΔ≤ 0, environmental quenching is very weak
        and f_ s(m, Δ) is quite small. Thus, to good approximation
        F_ q(m,Δ_0)≈ F_ q,c(m,Δ_0), and we can
        rewrite Equation (<ref>) as
        
            ε   (m, Δ  | Δ_0)
            ≃F_ q(m,Δ)-F_
                     q,c(m,Δ_0)/1-F_ q,c(m,Δ_0)
               =   ε_ c(m, Δ | Δ_0)+[1-ε_ c(m, Δ | Δ_0)]
               ε'(m, Δ | Δ_0) ,
        
        where the second equation is obtained by inserting Equations (<ref>) and (<ref>)
         into the right-hand-side of the first line.
        Note that this reduces to Equation (<ref>) in the limit
        ε_ c→ 0. Using
        Equation (<ref>) to compute the quenching efficiencies yields the
        results that are in excellent agreement
        with those shown in Figure <ref> obtained using the original
        definition [Equation. (<ref>)]. Since the difference in all m bins is less than 0.03, we do not show the results.
        The good agreement justifies our
        approximation that F_ q(m,Δ_0)≈ F_ q,c(m,Δ_0).
        
        Note once more that ε(m, Δ  | Δ_0) reveals only
        a very weak dependence on m, except for the most massive galaxies,
        for which the statistics is extremely poor (see Section <ref>).
        It seems to conflict with the fact that ε is the combination of ε_ c and ε', which both strongly depend on m. To understand this apparent discrepancy, we show ε, ε_ c and ε' as a function of m
        for various Δ bins in Figure <ref>. Apparently, the opposite trends in the m-dependence of
        ε_ c and ε' counterbalance each other
        so as to yield a ε that depends only weakly on m.
        
        Our results clearly demonstrate that the environmental effect on centrals has to be
        taken into account in order to reproduce the m-independence of the quenching
        efficiency seen for the total population. This is particularly important for massive
        centrals. Peng et al. (2012) also found a significant environmental dependence for
        central galaxies. However, they suspected that it is caused by the misidentification of satellites as centrals.
        We indeed find some signals for such misidentification in our results. For example, some abnormal
        behavior of the lowest mass galaxies in high-density bins can be seen in
        ε_ s, as well in ε_ c.
        More recently, Hirschmann et al. (2014) investigated the misidentification problem
        using mock galaxy catalogs constructed from a semi-analytic model of
        galaxy formation, and found that the contamination in centrals is less than 10%
        for most galaxy masses and environmental densities. The average contamination is
        less than 6.5%, roughly independent of m (see also Lange et al. 2017). Since the
        difference between
        F_ q,c and F_ q,s rapidly decreases with increasing m
        (Figure <ref>), the impact of the contamination is expected to decrease
        with m. In contrast, the observed Δ-dependence is more important for more
        massive galaxies. Based on these results, we are confident that central-satellite
        contamination does not significantly impact our conclusion that satellite
        quenching alone cannot account for the  m-independence of the quenched efficiency,
        at least for massive galaxies.
        
        An interesting question is why the increase of ε_ c with
        increasing m apparently compensates the decrease of ε'
        with increasing m. It might reflect some deeper
        connection between the quenching processes for centrals and satellites.
        As shown above, the quenched fractions of centrals and satellites of the same
        stellar mass correlate with the environment as characterized by halo mass in
        the same way. This suggests that centrals and satellites may experience
        similar quenching processes which are ultimately related to the host halo mass. In Section <ref> we will
        construct simple models to investigate this issue.
        
        
        
        
        § HALO-BASED QUENCHING EFFICIENCIES
        
        
        As discussed earlier, halos play a crucial role in shaping galaxy properties.
        Many quenching processes are thought to correlate with halo mass and
        stellar mass (see Mo et al. 2010). In order to examine this, we define
        two new quenching efficiencies in the same vein as the environmental quenching
        efficiency of Equation (<ref>). These are the halo-based
        environmental quenching efficiency, defined as
        
            ε(m, M | M_0)
            ≡F_ q(m,M)-F_ q(m,M_0) 1-F_ q(m,M_0) ,
        
        and the stellar mass quenching efficiency, defined as (see also Peng et al. 2010)
        
            ε_ m(m, M | m_0)
            ≡F_ q(m,M)-F_ q(m_0,M) 1-F_ q(m_0,M) .
        
        Here M_0 and m_0 are the halo mass and stellar mass `zero-points' against
        which the dependencies on M and m are compared. Note that
        ε(m, M | M_0) and ε_ m(m, M | m_0) characterize
        the dependence on halo mass and stellar mass, respectively, of the combined effect
        of all quenching processes, including `environmental processes' (such as
        ram-pressure/tidal stripping and strangulation) and `internal processes' (such as
        quenching induced by AGN and supernova feedback).
        
        
        
         §.§ Quenching efficiencies for the total population
        
        
        Ideally, one would like to adopt the lowest halo mass as `zero-point' environment to
        calculate the environmental efficiency. However, because no massive galaxy
        (log(m)>10.6) resides in halos of log(M)<12, choosing
        the lowest halo mass bin is inappropriate. We therefore adopt 13≤log(M)<13.5
        to define the quenching zero-point, at which the estimates of the quenched fraction
        are robust for all stellar mass bins (see Figure <ref>). The corresponding
        efficiency, obtained from Equation (<ref>), is shown as a function of halo
        mass in Figure <ref>. Here again the environmental quenching efficiency is
        almost independent of stellar mass, although it increases strongly with halo mass.
        It is easy to see that, if the efficiency is independent of stellar mass, this
        independence holds regardless of the value of M_0 used. This result clearly
        shows that the environmental dependence of the quenched fraction can be well
        separated from the dependence on stellar mass, independent of whether
        the environmental parameter is the large scale matter density or the mass of the
        host halo in which the galaxies reside.
        
        Motivated by Equation (<ref>) and that the dependence of F_ q on halo mass
        can be well described by a power law function (Figure <ref>), we propose to use a simple formula to describe the quenched fraction:
        
            F_ q(m,M)=1-exp[-(m/m_⋆)^d](alog M-ac) .
        
        It results in an environmental quenching efficiency given by
        
            ε(m, M | M_0)=1/c-log M_0logM/M_0 .
        
        We first use Equation (<ref>) with log(M_0)=13.25 to fit the data
        points shown in Figure <ref>. The best fit gives c=16.37±0.06 and is shown as the grey line in
        Figure <ref>. We then use Equation (<ref>) to fit the quenched fraction for
        the total population shown in Figure <ref>, with c fixed to 16.37.
        The best fit values for the other three parameters are a=-0.2±0.004,
        m_⋆=6.5±0.2×10^10 and d=0.61±0.02.
        
        The results for ε_ m(m, M | m_0) are presented in Figure <ref>.
        Here we choose 9.8≤logm_0< 10.2 so that we have robust estimation of
        F_ q(m_0,M) for all the halo-mass bins. As one can see,
        the ε_ m(m, M | m_0) - m relation is quite independent
        of halo mass in all the halo mass bins except the most massive halo bin.
        Since the quenched fraction can be
        well fitted by Equation (<ref>), ε_ m(m, M | m_0) can be described by
        
            ε_ m(m, M | m_0)
            ≡ 1-exp[-(m/m_⋆)^d
            +(m_0/m_⋆)^d] .
        
        The prediction of this equation with m_⋆=6.5×10^10, d=0.61 and m_0=10^10
        is plotted as the grey line in Fig. <ref>.
        
        
        
         §.§ Quenching efficiencies for centrals and satellites
        
        
        As shown in Sections <ref> and <ref>, the density-based quenching
        efficiencies for central and satellite populations depend on stellar mass in an opposite way,
        and they counterbalance each other to produce a m-independent efficiency for
        the total population. It is thus also interesting to examine the quenching efficiencies for
        centrals and satellites separately by using halo mass, instead of the
        density, as the environmental parameter. However, we will not repeat the same
        analyses as in Section <ref> and <ref>,
        for the following two reasons. First, the
        stellar masses of centrals are strongly correlated with the halo masses of their
        host groups, so that it is difficult to find a single halo mass
        bin to define the environmental zero point for central galaxies of different masses.
        Second, for satellites with given (m,M), it is difficult to select
        a large number of centrals of the same (m,M) to form a control sample to
        calculate the satellite-specific quenching efficiency. Because of these
        we adopt a different approach, as described below.
        
        To start with, we look at the satellite fraction as a function of
        halo mass in different stellar mass bins, as shown in Figure <ref>.
        Unlike the smooth relation between the satellite fraction with Δ shown in Figure <ref>,
        f_ s(m, M) as a function of M resembles roughly
        a step function, particularly for low-mass  galaxies.
        The fraction is close to zero at M<M_ tr and about one at M>M_ tr,
        where M_ tr is the mass scale at the transition of the step function and increases with
        increasing stellar mass. Therefore, centrals and satellites dominate the galaxy population in
        different regions in the (m,M) plane and the two populations are comparable in number
        only in a narrow region in the (m,M) plane. This property of the satellite
        fraction can be used to understand the quenching efficiency of centrals and
        satellites over a large range in m and M.
        
        As shown in Figure <ref>, ε(m, M | M_0) for galaxies of a
        given m follows the same correlation with M below and above
        M_ tr(m) (marked by the vertical lines in the figure),
        where galaxies are dominated by centrals and satellites, respectively.
        This indicates that the quenched fractions in both the central
        and satellite populations depend on the host halo mass in
        a similar way not only in the M range where the two populations
        overlap, but also over the whole M range. Similar analysis can also
        be performed on the basis of the mass quenching efficiency ε_ m. For example, the four
        stellar mass bins in 9.0<log(m)<10.6 are all dominated by centrals
        in halos with log(M)<12, but by satellites at log(M)>13.
        However, the mass quenching efficiency follows the same trend with
        stellar mass in different halo mass bins, no matter whether the galaxies in
        the halo mass bin are dominated by centrals or by satellites.
        All these suggest that, whatever the quenching processes are,
        they tend to produce a quenching efficiency that depends on halo
        and stellar masses in a similar way for both centrals and satellites.
        The results also suggest that the similarity in quenching efficiency
        between centrals and satellites exists not only in the region of the
        (m,M) plane where the two populations overlap, but also over
        the whole range of (m,M) covered by the sample.
        
        
        
        
        § HALO MASS AND ASSEMBLY DRIVE ENVIRONMENTAL QUENCHING
        
        
        As we have shown above, the quenched fraction of galaxies depends on both
        halo mass and environmental density. It is therefore important to examine
        whether it is the halo mass or the environmental density that plays the
        dominating role. To address this question, we construct two simple models,
        in which the quenched fraction is assumed to be determined by the stellar mass
        combined with one of the two environmental quantities.
        We also try to understand whether assembly histories of dark matter
        halos affect the galaxy properties in a third model. These models will also help
        us to understand the apparent discrepancy between the results based on environmental
        density and halo mass. Namely why is the density-based quenching efficiency
        independent of the stellar mass for the total population but stellar-mass
        dependent when centrals and satellites are analyzed separately
        (Section <ref>), while centrals and satellites follow
        very similar trends in the halo-based quenching efficiency (Section <ref>)?
        
        In the first model (hereafter Model A), the environmental density,
        Δ, is assumed to be the primary driver of the environmental
        dependence of quenching. For each real galaxy in our SDSS sample, we
        construct a corresponding model galaxy, which has exactly the same
        m, M, Δ, and the same identification as either a central or
        a satellite. We assign a given model galaxy i a quenching probability,
        q_i=F_ q(m, Δ), according to its position in the (m, Δ)
        space (Figure <ref>). Note that centrals are treated differently
        from satellites, because the observed F_ q,c(m, Δ) is very
        different from F_ q,s(m, Δ).
        We then use Equation (<ref>) to estimate the average quenched
        fraction for any given subset of the model galaxies. Figure <ref>
        shows the predicted quenched fraction as a function of halo mass, M, for
        galaxies in different m bins.
        
        As one can see, Model A predicts a positive dependence of the quenched
        fraction on M, in rough agreement with the observation. This dependence
        is expected from the fact that dark matter halos are tracers of the matter
        density field. Note, however, that there is a marked difference
        between the model prediction and the observational data. First of all, the model
        predicts no significant dependence of F_ q,c on M, contrary to the data which reveals a clear trend of increasing
        F_ q,c with increasing M (cf., Figure <ref>).
        Second, the predicted M-dependence for satellite galaxies is also
        weaker than observed.  This discrepancy is particularly large for low
        mass galaxies in low mass halos, where the model overpredicts F_ q,s
        by as much as 0.2. Finally, the model predicts significant differences
        between the quenched fractions of centrals and satellites of similar m
        and M, while such differences are absent in the observational data
        (cf., Figure <ref>).
        
        
        
        In the second model (hereafter Model B), halo mass instead of environmental
        density is assumed to be the primary driver of the observed environmental
        dependence of quenching. To test this hypothesis, we construct a model galaxy
        sample by assigning each real galaxy in our SDSS sample a quenching probability
        q_i = F_ q(m, M) based on its position in the (m, M) plane
        (the black solid lines in Figure <ref>). Since the dependence of
        F_ q on M and m is very similar between
        centrals and satellites, we do not distinguish between them when assigning q_i. Hence, in this model the probability for
        a galaxy to be quenched is solely determined by the galaxy's stellar mass and
        host halo mass, with centrals and satellite being treated in exactly the same
        way.
        
        The quenched fractions predicted by Model B as a function of Δ and
        m [computed using Equation (<ref>)] are shown in Figure <ref>. The results closely
        resemble those for the real galaxies. To better illustrate the quality
        of the model, Figure <ref> plots the differences between the
        model prediction and the data, δ F_ q(m,Δ). Note that
        the differences are fairly small, typically less than ∼ 0.05 for
        the entire galaxy sample. For centrals the discrepancies are slightly larger,
        while for satellites the differences are comparable to the observational
        uncertainties. Hence, Model B provides a fairly accurate description of the
        data.  Taken together, the results from Models A and B strongly suggest that
        the mass of the host halo is a far more important environmental parameter
        for regulating quenching than is the matter density.
        
        Before moving on, we try to understand some of the general trends predicted by Model B.
        The resulting dependence of F_ q on Δ is very different for centrals and satellites,
        although the two populations are assumed to have exactly the same dependence of F_ q on m and M.
        This arises because centrals and satellites of a given
        m cover very different ranges in M. Centrals with 9≤log (m)≤ 10.2 usually reside
        in halos with log (M) ∼ 12. The halo bias at this mass scale
        is close to unity with little dependence on M (see Sheth, Mo & Tormen 2001).
        Hence, the mean halo mass in the low-density regions is not very
        different from that in the high-density regions, which explains
        why F_ q,c is almost independent of Δ for galaxies in this
        m range. As the mass of the central galaxy increases,
        so does the mass of its host halo, which pushes it into the regime where
        halo bias is larger than unity and has a strong dependence on halo mass.
        Consequently, halos in high density regions are, on average, more
        massive than those in low density regions, even if they contain centrals of the
        same stellar mass. This explains why the dependence of F_ q,c on Δ
        becomes stronger for centrals with higher stellar masses, as shown in
        Figure <ref>. However, for a given m, the distribution
        in M is quite narrow (see Yang et al. 2009), so the Δ-dependence predicted by Model B remains weak.
        For satellites, the situation is very different. At the low m end, satellites reside in halos that
        cover a very wide range in M, which gives rise to a very strong dependence on Δ. As m
        increases, the dispersion in M decreases, which weakens the dependence of F_ q,s on Δ.
        
        
        
        
        As evident from Figure <ref>, Model B slightly overestimates F_ q,c at
        the low-Δ end while underestimating it at the high-Δ end, a trend that is evident
        for every stellar mass bin. It suggests that the quenching of (central) galaxies depends not only on halo
        mass and stellar mass, but also on some other halo properties that are correlated with
        the environmental density. One such halo property is halo assembly history, which is correlated
        with the environmental density (known as assembly bias).
        In order to see if the discrepancy can be explained by halo assembly bias,
        we need to know, for each individual group, the formation redshift (hereafter
        z_ f) that characterizes its assembly history. Our ELUCID simulation
        is a constrained simulation in the SDSS DR7 region and can reliably reproduce
        most of massive groups (see paper III), and here we make use of the information
        it provides to estimate the formation redshifts for individual groups. To do
        this, for each group, we search all halos in the simulation that have mass
        differences less than 0.3 dex with, and distance less than 5 to, the
        group in question.  Most of the groups (∼97.4%) have at least
        one halo companion defined in this way, and we assign the formation redshift
        of the nearest halo to the group. The formation redshift is defined as the highest
        redshift at which half of the final halo mass has assembled into progenitors more
        massive than 10^11.5 (see Neistein, van den Bosch & Dekel 2006; Li et al. 2008).
        The choice of this mass limit is motivated by the fact that it corresponds to the
        halo mass at which the star formation efficiency is the highest at different
        redshifts (see Lim et al. 2017). For groups with log(M)<12.0
        that do not have accurate halo mass estimates in the
        group catalog, we only search for halo companions with
        11.7<log(M)<12.0, where the lower mass limit
        (11.7) is adopted so that all halos have reliable estimates of
        halo formation redshifts in the ELUCID simulation. We have also used
        other definitions of halo formation redshift, such as the
        redshift at which the main progenitor reaches half of the present-day
        halo mass. The results are very similar.
        
        
        
        A third model, Model C, is then constructed on top of Model B.
        In Model C, the quenching probability of a model galaxy depends not only
        on m and M, but also on the formation redshift of its halo.
        The simplest way to link z_ f to quenching probability is to assume that
        a galaxy is quenched when z_ f>z_ th, where z_ th is a
        formation redshift threshold. In reality, however,  galaxies with given
        z_ f, m and M, must have some dispersion in their star formation rate.
        To mimic this, we introduce a dispersion in z_ f for each system before
        applying the criterion z_ f>z_ th to select the quenched fraction.
        In practice, for a galaxy i with z_ f=z^i_ f, we use a Monte Carlo
        method to generate 500 mock galaxies, with their formation redshifts
        (z^i_ f,m) randomly drawn from a Gaussian distribution with
        the mean value equal to z^i_ f and a width σ_ z.
        Then, for a given z_ th, the quenching probability of the model
        galaxy, q_i, is set to be the fraction of the mock galaxies that have
        z^i_ f,m> z_ th among the 500 mock galaxies. In order to introduce
        the dependence on m and M, the threshold z_ th is required
        to be a function of the two quantities, and is determined by the criterion that
        the dependence on m and M for the model galaxies is
        exactly the same as that for real galaxies. Note that in our model
        the formation redshift dependence is only considered for central galaxies;
        satellites are treated in exactly the same way as in Model B.
        
        When σ_ z is set to be 0, the dependence of
        F_ q,c(m, Δ) on Δ for the model galaxies is found to be much stronger than that
        for real galaxies. We have experimented a series of values for σ_ z, and found
        that the model matches the observation the best when σ_ z∼ 0.8.
        Figure <ref> shows the quenched fraction as a function of m and
        Δ obtained from this best model. A significant increasing trend
        of the quenched fraction with Δ is now produced for central galaxies in most
        of m bins, as is seen in the observational data.
        The lower two panels of Figure <ref> show the difference between the
        observational quenched fraction and the predictions of Model C for the total
        population and centrals, respectively. In contrast to Model B, the dependence of
        δ F_ q(m,Δ) on Δ almost completely disappear in Model C.
        
        
        
        We then compute the environmental quenching efficiency, defined in Equation (<ref>)
        with the zero point estimated from data at logΔ≤ 0, for Model B and C.
        The results are presented in the left panels of Figure <ref>.
        The quenching efficiency predicted by Model B exhibits a very weak dependence on m.
        This is not surprising since the halo-based environmental efficiency,
        ε_ c(m, M | M_0), is independent of stellar mass (Figure <ref>)
        and since halo mass is the only driver of the environmental quenching in Model B.
        There is a small deviation between the model prediction and the observation for most
        of the stellar mass bins. Such deviation is absent in Model C, suggesting that it has the same
        origin as that shown in Figure <ref>.
        
        We also show the predictions of Model B and C
        for ε_ c(m, Δ | Δ_0), defined by Equation (<ref>),
        and for ε'(m, Δ | Δ_0), defined by Equation (<ref>),
        in the middle and right panels of Figure <ref>. In Model B, one can see a clear trend
        that, at a fixed Δ, ε_ c increases,
        while ε' decreases, with m. As discussed above,
        the halo mass distribution becomes broader for centrals but narrower for satellites as stellar
        mass increases, which explains the opposite trends in the quenched efficiency as a function of
        stellar mass for centrals and satellites. After taking into account the assembly bias
        effect (Model C), the dependencies on stellar mass for both centrals and satellites are enhanced
        and the predictions resemble more closely the observational results.
        
        Moreover, the predicted efficiency for the most
        massive galaxies is now closer to the other mass bins than the observational result, giving
        further support to the hypothesis that the discrepancy found for the most massive
        galaxies in the observational data is mainly caused by the uncertainty in the estimation of F_ q(m,Δ_0). The uncertainty
        is largely eliminated in the model predictions, because the value of q_i assigned to
        galaxies in the low-density regions is the average over a larger sample of galaxies.
        
        
        
        
        § THE QUENCHING OF CENTRALS IS NOT SPECIAL
        
        
        As shown in Section <ref> and <ref>, we find that centrals and satellites
        follow the same correlation of quenched fraction with stellar mass and host
        halo mass. This result appears to be in conflict with
        those obtained by some earlier studies (e.g. van den Bosch et al. 2008;
        Wetzel et al. 2012; Peng et al. 2012), where quenching is found to depend
        strongly on whether a galaxy is a central or a satellite. However,
        as discussed above, host halo mass ranges
        covered by centrals and satellites of the same stellar mass are very different.
        Thus, if halo mass is not used as a control parameter when comparing the two
        populations, as is the case in most of the previous studies,
        then one is comparing centrals in low mass halos with satellites in massive ones
        and neglecting the strong dependence on halo mass.
        We believe that this is the origin of the discrepancy and that there is
        no conflict between our results and those obtained in these earlier
        studies.
        
        It is possible that the similarity between centrals and satellites
        we find in the data is not real, but caused by errors in the group finder
        used to identify groups and central galaxies in them. As one can see
        from Figure <ref>, the satellite fraction, f_ s(m,M), resembles
        roughly a step function and is
        close to zero at M<M_ tr  and about one at M>M_ tr (M_ tr
        is the mass scale at the transition of the step function).
        Thus, even if the mis-identification fraction is small in the whole population
        (see Hirschmann et al. 2014), the galaxies that are identified as satellites
        at M<M_ tr may be significantly contaminated by centrals. In this case, the
        similarity between centrals and `satellites' of a given stellar mass in their
        quenched fraction may be produced by the false identification
        between centrals and satellites, rather than a real similarity.
        
        However, it is difficult to explain the
        m-independence of the environmental quenching efficiency and the
        M-independence of the stellar mass quenching efficiency
        (Section <ref>) by central/satellite mis-identifications
        alone. If centrals and satellites had quenching properties
        that depend on the halo and stellar masses in significantly
        different ways, the environmental quenching efficiency, ε(m, M | M_0),
        would be expected to correlate with the halo mass in different
        ways depending on whether M<M_ tr or M>M_ tr, as the galaxy
        populations in the two mass ranges are dominated by centrals and satellites,
        respectively (see Fig. <ref>). Moreover, since M_ tr
        increases with increasing stellar mass, ε(m, M | M_0) would also
        be expected to vary with stellar mass. Similarly, it would also lead to
        M-dependence in the mass quenching efficiency, ε_ m(m, M | m_0).
        Both are inconsistent with the observational results.
        We thus conclude that the similar quenching properties of centrals
        and satellites are not produced by mis-identifications
        between the two populations, and are real within the statistical
        uncertainties of the data.
        
        It should be pointed out that this interpretation is based on the
        premise that the inferred halo mass is sufficiently accurate. Recently,
        Campbell et al (2015) used a mock catalog including galaxy
        color to check the color-dependent statistics inferred from group catalogs.
        They compared the marginalized dependence of the quenched (red) fraction
        on halo mass obtained from their mock group catalog with
        that obtained directly from the
        simulation used to construct the mock catalog, and found that group finders
        tend to reduce the difference in the quenched fraction between centrals
        and satellites (see their figure 13). This systematic error is
        produced by the combined effect of group membership determination,
        central/satellite designation, and halo mass assignments.
        If the real difference between centrals and satellites is small,
        it may be washed out by this error. However, we want to point out that
        if this systematic error shown in the mock group catalog in Campbell et al. indeed
        affects the SDSS group catalog in a similar way, one would expect to see that
        the marginalized dependence of the quenched fraction
        on halo mass is similar between centrals and satellites for the SDSS group catalog.
        It is apparently inconsistent with what we found (the middle panel of Figure <ref>).
        This suggests that the error caused by group finder depends on the galaxy formation model that is used to
        construct the mock catalog. Unfortunately this also means that
        the impact of the inaccuracy of the group finder on the results
        obtained here from the SDSS group catalog is unclear.
        
        
        
        One way to bypass the uncertainty in the central versus satellite
        identification is to study the dependence of galaxy quenching on
        the locations of galaxies in their host halos. Instead of looking
        at centrals versus satellites, we can look at galaxies at different
        locations in a halo. By definition, centrals belong to the
        innermost population. Previous studies (e.g. Wetzel et al. 2012; Woo et al. 2013; Bluck et al. 2016)
        found that the quenched fraction increases with decreasing distance to
        the halo center (halo-centric radius). In contrast, the
        similarities between centrals and satellites found in this paper
        seem to suggest that quenching is quite independent of the location within
        halos. To find the cause of this discrepancy, we divide the total galaxy population
        into two: the inner and outer sub-populations, according to whether or not their projected distances to the halo centers
        (R_ p) are smaller or larger than half of the virial radii (R_ vir).
        Here the center of a halo is the luminosity-weighted position of its member galaxies.
        Figure <ref> shows the quenched fraction as a function
        of stellar mass for the two sub-populations in the six halo mass bins used above. To compare with the central galaxies in the corresponding
        halo mass bin, the quenched fraction is scaled with the quenched fraction of centrals and the stellar mass is also scaled with the median of the stellar mass of central galaxies (m̃_ c). For comparison, the satellite fractions in the two
        sub-populations are also shown in the figure.
        
        For small galaxies, the quenched fraction clearly increases with the decrease of halo-centric radius,
        consistent with previous studies, while for massive galaxies the dependence on
        halo-centric radius is very weak. The characteristic stellar mass above which the R_p dependence becomes
        weak is log m≃log(m̃_ c)-0.7 (indicated by the vertical lines) and almost independent of
        halo mass. This clearly demonstrates that the quenching probability is independent of its location in the halo,
        as long as a galaxy has a stellar mass larger than about one fifth of the
        mass of its central galaxy. For galaxies with log m∼logm̃_ c, the
        scaled quenched fraction is close to unity for both the inner and outer
        populations. At this stellar mass, the satellite fraction is close to zero
        for halos with log(M)<12.5 and increases to
        >60% for log(M)>14.5. It indicates that, when galaxies have
        stellar masses comparable to those of their centrals, the quenched fraction
        is the same as that of their centrals, no matter where they are located and
        whether they are identified as centrals or not. Alternative separations of inner and outer
        regions at 0.3R_ vir and 0.4R_ vir have no significant impact on this conclusion.
        These demonstrate again that centrals are not special as far as
        their quenching properties are concerned. Our results are also not in conflict
        with previous finding, as a strong dependence on halo-centric radius
        is present but only for galaxies that are much less massive than the
        centrals in their corresponding halos.
        
        
        
        
        § SUMMARY AND DISCUSSION
        
        
        In this paper, we present a detailed investigation about the environmental dependence of
        quenching of star formation using a large sample of galaxies constructed from
        the SDSS. We adopt two quantities to describe the different aspects of galaxy
        environments: the environmental mass densities, smoothed on a scale
        of 4 (half width of Guanssian kernel) at the positions of individual galaxies,
        and the masses of the host halos within which galaxies reside.
        The mass densities are obtained from the ELUCID simulation, a constrained
        N-body simulation in the SDSS volume,  while the halo masses are based on a galaxy
        group catalog constructed with a halo-based group finder. Our main findings
        are summarized as follows.
        
        
          *  The quenched fraction of galaxies increases systematically with galaxy stellar mass,
        environmental density and host halo mass. When analyzed separately,
        centrals and satellites show very different density-dependence:
        while the dependence is strong for satellites, it is
        weak or even absent for central galaxies. The environmental
        effect is stronger for centrals with higher stellar masses,
        while satellites show the opposite trend.
        In contrast, the dependence of the quenched fraction on halo mass is almost
        the same for both centrals and satellites, although the two populations cover different ranges of halo masses.
        
        
          * 
        For the total galaxy population, the quenching efficiency, defined
        as the quenched fraction of galaxies in a given environment relative to
        a zero-point population, is found to be almost independent of galaxy stellar mass over a wide
        mass range, 9≤log (m)≤11.4), no matter which
        environmental parameter (mass density or halo mass) is adopted.
        This suggests that the strong stellar mass-dependence of the quenched
        fraction is predominantly produced by such a dependence of the zero point.
        
        
          * 
        When central and satellite galaxies are analyzed separately,
        the density-based quenching efficiency is found to increase systematically with stellar mass
        for centrals but to decrease for satellites, and the stellar mass dependence
        is stronger for galaxies of higher stellar masses. The opposite trends
        seen in centrals and satellites compensate each other so as to
        make an almost stellar mass-independent quenching efficiency for the total
        population described above. The results thus do not support the hypothesis
        proposed in previous studies that the mass independence of the quenching efficiency is
        only due to satellite quenching.
        
        
          * 
        Centrals and satellites are found to follow the same trends in both the halo mass
        and stellar mass based quenching efficiencies, contrary to the
        efficiency defined by the environmental density. It indicates that quenching
        does not depend strongly on the location within halo,
        at least for galaxies with stellar masses comparable to those of the centrals.
        Further investigation of the dependence of the quenched fraction on
        the distance to halo center shows that the distance dependence is only
        important for galaxies with stellar masses that are lower than one fifth of
        the masses of their centrals but insignificant for more massive galaxies.
        
        
          * 
        A model, in which the quenching probability of a galaxy is assumed to be
        determined by the galaxy stellar mass combined with the host halo mass (Model B),
        can well reproduce the observed dependence of the quenched fraction on environmental
        density for the total population, as well as separately for centrals and satellites.
        In contrast,  a model in which the quenching probability is assumed to
        be determined by galaxy mass and environmental density (Model A)
        predicts too weak dependence on halo mass and different halo-mass dependence
        between centrals and satellites, in conflict with observation.
        These suggest that halo properties are the driver of the environmental quenching
        seen in the observational data.
        
        
          * 
        Model B is found to slightly overestimate the quenched fraction in
        the low density end and to underestimate it at the high density end for central galaxies.
        It suggests that galaxy quenching depends not only on halo mass
        but also on some other halo properties that are correlated with the environmental density.
        A model (Model C), which takes into account halo assembly bias,
        can explain the discrepancy between Model B and the observational results.
        
        
          * 
        The environmental quenching efficiency based on the mass density
        predicted by both Model B and Model C (where the halo-mass is the primary driver of the environmental effect) is found to be
        independent of stellar mass, consistent with observational results. 
        This strongly suggests that the stellar mass independence of the density-based efficiency
        originates from the stellar mass independence of the halo-based efficiency.
        
        
          * 
        The difference in star formation quenching between centrals and satellites
        found in this paper and in numerous previous studies are mainly due to the
        difference in the host halo mass ranges covered by the two populations,
        and not produced by the difference in the correlation of quenching
        probability with halo mass between the two populations.
        
        
        Many mechanisms of quenching star formation in galaxies have been proposed in the literature,
        such as virial shock heating to accretion flows, the feedback from active galactic nuclei and supernova,
        and the stripping of hot and cold gas associated with galaxies (see e.g. Gabor et al. 2010 for a more comprehensive discussion).
        The strengths of these quenching mechanisms depend on stellar mass and halo mass in different ways.
        Furthermore, because of the special positions assumed for central galaxies in their
        halos, some of the processes may have different impact between centrals and
        satellites. In what follows we discuss how our findings can be used to
        constrain these different quenching mechanisms.
        
        AGN feedback has been proposed to quench  cooling flows in massive halos and suppress
        star formation in massive galaxies (e.g. Croton et al. 2006; Bower et al. 2006; Cui, Borgani, & Murante 2014). The strength of AGN
        feedback is usually assumed to be more efficient for more massive galaxies that contain more
        massive black holes in more massive halos. AGN feedback may thus be able to produce
        the positive dependency of the quenched fraction and quenching efficiency on
        galaxy stellar mass and halo mass found in this paper.  Moreover, AGN feedback may affect
        both centrals and satellites in a similar way, again consistent with our observation.
        Numerous observations have provided evidence for AGN feedback through radio jet and massive
        outflows driven by radiation pressure (e.g. Best et al. 2007; Wang et al. 2011b; Fabian et al. 2012).
        However, the details how AGN feedback is coupled with the gas and regulates the
        star formation in galaxies are still uncertain.
        
        Hydrodynamical simulations (e.g. Kereš et al. 2005; 2009) have revealed that galaxies acquire
        their baryonic mass primarily through cold gas flows along  filamentary structures around
        halos, and such cold accretion can be heated and suppressed by virial shocks in the host halo (see also Dekel & Birnboim 2006).
        This process can result in a decrease in the cold accretion as a function of halo
        mass, as radiative cooling is less effective in more massive halos
        (Ocvirk, Pichon & Teyssier 2008; Kereš et al. 2009). If the star formation in
        galaxies is fuelled mainly by the cold accretion, an increasing trend of
        the quenched fraction and the quenching efficiency with halo mass is expected.
        Moreover, galaxies in simulations are found to continue to acquire cold gas
        after becoming satellites, and the cold accretion is also affected
        by the shock in the host halos (Kereš et al. 2009), which is consistent
        with our finding that the quenching properties of centrals and satellites are similar.
        Unfortunately, it is unclear how the suppression of cold accretion depends on
        the stellar mass of a galaxy in a halo, and so it is unclear if this mechanism can
        accommodate the dependence of quenching on stellar mass seen in the observation.
        
        The stripping of gas from galaxies by ram-pressure in hot halo is expected to be more efficient for galaxies
        of lower masses living in higher mass halos  (see Henriques et al. 2016 for discussion).
        Therefore this mechanism predicts that quenching should be important only in massive halos
        with hot halo gas, and that the quenched fraction and quenching efficiency should increase with
        decreasing stellar mass.  These predictions seem to be at odds with the positive dependence of
        the quenched fraction  (quenching efficiency) on stellar mass found in this paper, and with the fact
        no characteristic halo mass is seen in the quenching fraction - halo mass relation.  In addition,  ram
        pressure  stripping is expected to be important only for satellite galaxies that are orbiting
        in halos, but unimportant for centrals that sit close to the bottoms of the halo gravitational potential wells.
        If such stripping process dominates quenching of star formation,  then centrals and satellites are expected
        to be affected differently.  This seems contrary to the results that the quenching efficiencies of centrals and
        satellites are correlated with halo mass and galaxy mass in a similar way. Our results thus suggest that
        ram pressure stripping alone cannot be the dominant quenching processes for the whole population.
        Similar argument may also be made for tidal stripping. However, since the effect of the tidal stripping
        is determined by the local mass density relative to the mass density of the galaxy to be stripped,
        this effect may also be important in relatively low-mass halos, which may be in better
        agreement with observation.
        
        The dependence of quenching on halo-centric radius for low-mass galaxies suggests
        that the significance of the underlying physical processes depends on the locations of
        galaxies in their host halos. The satellite-specific processes, such as ram pressure stripping
        and tidal stripping,  are expected to be more efficient near the halo center,  and so may
        be able to produce the dependence on halo-centric radius observed in the data.
        However, as discussed above, if these processes dominate the quenching of satellites
        and do not operate on centrals,  then why do central galaxies have quenching properties similar
        to satellite galaxies of the same stellar masses?  It may be that centrals are not
        special, and these satellite-specific processes also operate on centrals.  This is in fact
        consistent with the fact that centrals are not at rest within the halo potential wells, and
        not even located at the halo center (e.g. Skibba et al. 2011).
        Indeed, one of the main conclusions that can be drawn from our results
        is that the central in a halo is not special, as far as its star formation
        quenching is concerned. Peng et al. (2010) found that the environmental quenching and mass quenching can be well
        separated from each other for galaxies at z∼ 1. It suggests that such a
        conclusion also holds for these galaxies.
        However, as mentioned above, this similarity between centrals and satellites is not expected in
        some galaxy formation models, where centrals and satellites are assumed to be
        affected by different quenching processes.
        
        The processes discussed above, which are all confined within halos, can not account for the residual
        dependence on environmental density after removing the halo-mass effects. We thus include halo assembly history as
        an additional parameter, which is known to be influenced by large scale environment (e.g. Gao et al. 2005).
        Local tidal field is thought to play a key role in shaping the halo assembly bias (Wang et al. 2007;2011a; Hahn et al. 2009;
        Shi et al. 2015; Paranjape et al. 2017; Borzyszkowski et al. 2017). For example, for a small halo in a high density region,
        the material around the halo is accelerated by the local tidal field so that the halo growth is significantly suppressed.
        These processes (on the scale much larger than the galactic scale) are unlikely to directly influence the star formation in galaxies. Therefore, a correlation between the star formation and the halo assembly history has to be introduced in order to explain
        the residual dependence. It is worthwhile to note that it is unclear whether these large scale processes affect
        baryonic gas and dark matter in the same way. If it is not, additional effect on star formation should be taken into account.
        
        The discussions given above provide some qualitative assessments about
        some of the quenching processes that have proposed in the literature,
        in connection to the observational results we find in this paper. To
        constrain theoretical models in a quantitative way, however,
        detailed modeling of the various quenching processes, as well as
        thorough analyses of all potentially important observational selection effects
        are needed. In a forthcoming paper,  we will use mock catalogs constructed
        from hydrodynamic simulations and semi-analytic models of galaxy formation
        to compare galaxy formation models with the observational results
        obtained here (E. Wang et al. in preparation).
        
        
        
        § ACKNOWLEDGMENTS
        
        
        We thank an anonymous referee for a useful report. This work is supported by the 973 Program (2015CB857002), NSFC
        (11522324, 11733004, 11421303, 11233005, 11621303),  and the Fundamental
        Research Funds for the Central Universities. H.J.M.
        would like to acknowledge the support of NSF AST-1517528 and NSFC-11673015.
        S.H.C and Y.Y. are supported by the Fund for Fostering Talents in Basic
        Science of the National Natural Science Foundation of China
        NO.J1310021. FvdB is supported by the Klaus Tschira Foundation and by the US
        National Science Foundation through grant AST 1516962. WC is supported by the  Ministerio de Economía y Competitividad and the  Fondo Europeo de Desarrollo Regional (MINECO/FEDER, UE) in Spain through grant AYA2015-63810-P as well as the Consolider-Ingenio 2010 Programme of the  Spanish Ministerio de Ciencia e Innovación (MICINN) under grant MultiDark CSD2009-00064.
        The work is also supported by the Supercomputer Center
        of University of Science and Technology of China and the High Performance Computing
        Resource in the Core Facility for Advanced Research Computing
        at Shanghai Astronomical Observatory.
        
        
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