# Does the galaxy–halo connection vary with environment?

Does the galaxy–halo connection vary with environment? Abstract (Sub)halo abundance matching (SHAM) assumes that one (sub) halo property, such as mass Mvir or peak circular velocity Vpeak, determines properties of the galaxy hosted in each (sub) halo such as its luminosity or stellar mass. This assumption implies that the dependence of galaxy luminosity functions (GLFs) and the galaxy stellar mass function (GSMF) on environmental density is determined by the corresponding halo density dependence. In this paper, we test this by determining from a Sloan Digital Sky Survey sample the observed dependence with environmental density of the ugriz GLFs and GSMF for all galaxies, and for central and satellite galaxies separately. We then show that the SHAM predictions are in remarkable agreement with these observations, even when the galaxy population is divided between central and satellite galaxies. However, we show that SHAM fails to reproduce the correct dependence between environmental density and g − r colour for all galaxies and central galaxies, although it better reproduces the colour dependence on environmental density of satellite galaxies. methods: numerical, galaxies: haloes, large-scale structure of Universe 1 INTRODUCTION In the standard theory of galaxy formation in a Λ cold dark matter (ΛCDM) universe, galaxies form and evolve in massive dark matter haloes. The formation of dark matter haloes is through two main mechanisms: (1) the accretion of diffuse material, and (2) the incorporation of material when haloes merge. At the same time, galaxies evolve within these haloes, where multiple physical mechanisms regulate star formation and thus produce their observed properties. Naturally, this scenario predicts that galaxy properties are influenced by the formation and evolution of their host haloes (for a recent review see Somerville & Davé 2015). What halo properties matter for galaxy formation? The simplest assumption that galaxy formation models make is that a dark matter halo property such as mass Mvir or maximum circular velocity Vmaxfully determines the statistical properties of their host galaxies. This assumption was supported by early studies that showed that the halo properties strongly correlate with the larger scale environment mainly due to changes in halo mass (e.g. Lemson & Kauffmann 1999). Halo evolution and corresponding evolution of galaxy properties can be predicted from Extended Press–Schechter analytical models based on Monte Carlo merger trees (Cole 1991; White & Frenk 1991; Kauffmann & White 1993; Somerville & Kolatt 1999).1 Such models assume that the galaxy assembly time and merger history are independent of the large-scale environment (for a recent discussion, see, e.g. Jiang & van den Bosch 2014). However, it is known that dark matter halo properties do depend on other aspects beyond Mvir, a phenomenon known as halo assembly bias. Wechsler et al. (2006, see also Gao, Springel & White 2005; Gao & White 2007; Faltenbacher & White 2010; Lacerna & Padilla 2011) observed an assembly bias effect in the clustering of dark matter haloes: They showed that for haloes with Mvir ≲ 1013 M⊙, early forming haloes are more clustered than late forming haloes, whereas for more massive haloes they found the opposite. Other effects of environmental density on dark matter haloes are known, for example that halo mass accretion rates and spin can be significantly reduced in dense environments due to tidal effects, and that median halo spin is significantly reduced in low-density regions due to the lack of tidal forces there (Lee et al. 2017). Indeed, there are some recent efforts to study assembly bias and the effect of the environment on the galaxy–halo connection in the context of galaxy clustering (Vakili & Hahn 2016; Zentner et al. 2016; Lehmann et al. 2017; Zehavi et al. 2018) and weak lensing (Zu et al. 2017). Despite such environmental effects on halo properties, it may still be true that some galaxy properties can be correctly predicted from just halo Mvir or Vmax. The assumption that dark matter halo mass fully determines the statistical properties of the galaxies that they host has also influenced the development of empirical approaches for connecting galaxies to their host halo: the so-called halo occupation distribution (HOD) models (Berlind & Weinberg 2002) and the closely related conditional stellar mass/luminosity function model (Yang, Mo & van den Bosch 2003; Cooray 2006). HOD models assume that the distribution of galaxies depends on halo mass only (Mo et al. 2004; Abbas & Sheth 2006). Yet, the HOD assumption has been successfully applied to explain the clustering properties of galaxies not only as a function of their mass/luminosity only but also as a function of galaxy colours (Jing, Mo & Börner 1998; Berlind & Weinberg 2002; Zehavi et al. 2005; Zheng et al. 2005; Tinker et al. 2013; Rodríguez-Puebla et al. 2015). The (sub)halo abundance matching (SHAM) approach takes the above assumption to the next level by assuming that not only does a halo property, such as mass Mvir or maximum circular velocity Vmax, determine the luminosity or stellar mass of central galaxies, but also that there is a simple relation between subhalo properties and those of the satellite galaxies they host. Specifically, we will assume that subhalo peak circular velocity Vpeak fully determines the corresponding properties of their hosted satellite galaxies (Reddick et al. 2013). For simplicity, in the remainder of this paper, when we write Vmax we will mean the maximum circular velocity for distinct haloes, and the peak circular velocity of subhaloes. SHAM assigns by rank a halo property, such as Vmax, to that of a galaxy, such as luminosity or stellar mass, by matching their corresponding cumulative number densities (Kravtsov et al. 2004; Vale & Ostriker 2004; Conroy, Wechsler & Kravtsov 2006; Conroy & Wechsler 2009; Behroozi, Conroy & Wechsler 2010; Behroozi, Wechsler & Conroy 2013; Moster, Naab & White 2013, 2017; Rodríguez-Puebla et al. 2017). Although central galaxies are continuously growing by in situ star formation and/or galaxy mergers, satellite galaxies are subject to environmental effects such as tidal and ram-pressure stripping, in addition to interactions with other galaxies in the halo and with the halo itself. Therefore, central and satellite galaxies are expected to differ in the relationship between their host haloes and subhaloes (see e.g. Neistein et al. 2011; Rodríguez-Puebla, Drory & Avila-Reese 2012; Yang et al. 2012; Rodríguez-Puebla, Avila-Reese & Drory 2013). Nevertheless, SHAM assumes that (sub)halo Vmax fully determines the statistical properties of the galaxies. Thus, SHAM galaxy properties evolve identically for central and satellite galaxies, except that satellite galaxy properties are fixed after Vpeak is reached.2 SHAM also implies that galaxy properties are independent of local as well as large-scale environmental densities. Thus, two haloes with identical Vmax but in different environments will host identical galaxies. Despite the extreme simplicity of this approach, the two point correlation functions predicted by SHAM are in excellent agreement with observations (Reddick et al. 2013; Campbell et al. 2017, and Figs 4 and 5), showing that on average galaxy clustering depends on halo Vpeak. It is worth mentioning that neither HOD nor SHAM identify clearly which galaxy property, luminosity in various wavebands or stellar mass, depends more strongly on halo mass – although, theoretically, stellar mass growth is expected to be more closely related to halo mass accretion (Rodríguez-Puebla et al. 2016b). Our main goal in this paper is to determine whether the assumption that one (sub)halo property, in our case halo Vmax and subhalo Vpeak, fully determines some statistical properties of the hosted galaxies. This might be true even though the galaxy–halo relation is expected to depend on environment because the properties of the galaxies might reflect halo properties that depend on some environmental factor (see e.g. Lee et al. 2017). We will test this assumption by determining from a Sloan Digital Sky Survey (SDSS) sample the dependence on environmental density of the ugriz galaxy luminosity functions (GLFs) as well as the galaxy stellar mass function (GSMF) for all galaxies, and separately for central and satellite galaxies, and comparing these observational results with SHAM predictions. We will also investigate which of these galaxy properties is better predicted by SHAM. If a galaxy–halo connection that is independent of environment successfully reproduces observations in the nearby Universe, then we can conclude that the relation may be appropriate to use for acquiring other information about galaxies. It also suggests that this assumption be tested at larger redshifts. To the extent that the galaxy–halo connection is independent of density or other environmental factors, it is a great simplification. This paper is organized as follows. In Section 2, we describe the galaxy sample that we utilize for the determination of the environmental dependence of the ugriz GLFs and GSMF. Section 3 describes our mock galaxy catalogue based on the Bolshoi–Planck (BolshoiP) cosmological simulation. Here, we show how SHAM assigns to every halo in the simulation five band magnitudes, ugriz, and a stellar mass. In Section 4, we present the dependence with environment of ugriz GLFs and GSMF both for observations and for SHAM applied to the BolshoiP simulation. We show that the SHAM predictions are in remarkable agreement with observations even when the galaxy population is divided between central and satellite galaxies. However, we also find that SHAM fails to reproduce the correct dependence between environmental density and g − r colour. Finally, Section 5 summarizes our results and discusses our findings. We adopt a Chabrier (2003) initial mass function (IMF) and the Planck cosmological parameters used in the BolshoiP simulation: $$\Omega _\Lambda =0.693, \Omega _{\rm M}=0.307$$, and h = 0.678. 2 OBSERVATIONAL DATA In this section, we describe the galaxy sample that we utilize for the determination of the galaxy distribution. We use the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure for the determination of the ugriz GLFs and the GSMF and report their corresponding best-fitting models. We show that a function composed of a single Schechter function plus another Schechter function with a sub-exponential decreasing slope is an accurate model for the ugriz GLFs as well as the GSMF. Finally, we describe the methodology for the determination of the environmental density dependence of the ugriz GLFs and GSMF. 2.1 The sample of galaxies In this paper, we utilize the New York Value Added Galaxy Catalog (NYU-VAGC; Blanton et al. 2005b) based on the SDSS DR7. Specifically, we use the large galaxy group catalogue from Yang et al. (2012)3 with ∼6 × 105 spectroscopic galaxies over a solid angle of 7748 deg2 comprising the redshift range 0.01 < z < 0.2 with an apparent magnitude limit of mlim, r = 17.77. However, the sample we use in this paper is 0.03 < z < 0.11 (see Fig. 2). The Yang et al. (2012) catalogue is a large halo-based galaxy group catalogue that assigns group membership by assuming that the distribution of galaxies in phase space follows that of dark matter particles. Mock galaxy catalogues demonstrate that ∼ 80 per cent of all their groups have a completeness larger than 80 per cent, whereas halo groups with mass Mvir > 1012.5 h− 1  M⊙ have a completeness > 95 per cent; for more details, see Yang et al. (2007). Here, we define central galaxies as the most massive galaxy in their group in terms of stellar mass; the remaining galaxies will be regarded as satellites. The definition of groups in the Yang et al. (2012) catalogue is very broad and includes systems that are often explored individually in the literature, such as clusters, compact groups, fossil groups, rich groups, etc. That is, this galaxy group catalogue is not biased to a specific type of group. Instead, this galaxy group catalogue is diverse and, more importantly, closely related to the general idea of galaxy group that naturally emerges in the ΛCDM paradigm: That haloes host a certain number of galaxies inside their virial radius. Therefore, the Yang et al. (2012) galaxy group catalogue is ideal for comparing to predictions based on N-body cosmological simulations. For the purpose of exploring whether certain galaxy properties are fully determined by the (sub)halo in which they reside, this galaxy group catalogue will help us to draw conclusions not only at the level of the global GLFs and GSMF but also at the level of centrals and satellites. Thus, the Yang et al. (2012) galaxy group catalogue is an ideal tool to explore at a deeper level the simple assumptions in the SHAM approach. In order to allow for a meaningful comparison between galaxies at different redshifts, we utilize model magnitudes4 that are K+E-corrected at the rest-frame z = 0. These corrections account for the broad-band shift with respect to the rest-frame broad-band and for the luminosity evolution. For the K-corrections, we utilize the input values tabulated in NYU-VAGC (Blanton & Roweis 2007, corresponding to the kcorrect software version v$$4\_1\_4$$), whereas for the evolution term we assume a model given by   \begin{eqnarray} E_j(z) = -Q_X \times z, \end{eqnarray} (1)where the subscript X refers to the u, g, r, i, and z bands and their values are (Qu, Qg, Qr, Qi, Qz) = (4.22, 1.3, 1.1, 1.09, 0.76). Here, we ignore potential dependences between QX and colours (but see Loveday et al. 2012, for a discussion) and luminosity, and use global values only. Although this is a crude approximation for accounting for the evolution of the galaxies, it is accurate enough for our purposes since we are not dividing the galaxy distribution into subpopulations as a function of star formation rate and/or colour. We estimated the value of each QX by determining first the X-band GLF when QX = 0 at four redshift intervals: [0.01, 0.05], [0.01, 0.1], [0.01, 0.15], and [0.01, 0.2]. When assuming QX = 0, the GLFs are normally shifted towards higher luminosities, with this shift increasing with redshift. In other words, when ignoring the evolution correction, the GLF will result in an overestimation of the number density at higher luminosities and high redshifts. Thus, in order to account for this shift, we find the best value for QX that leaves the GLFs invariant at the four redshift intervals mentioned above. We note that our derived values are similar to those reported in Blanton et al. (2003). For the u band, we used the value reported in Blanton et al. (2003), but we have checked that the value of Qu = 4.22 also leaves the GLF invariant at the four redshifts bins mentioned above. For stellar masses, we utilize the MPA-JHU DR7 data base derived from photometry–spectral energy distribution fittings, explained in detail in Kauffmann et al. (2003). All stellar masses have been normalized to a Chabrier (2003) IMF and to the cosmology used for this paper. 2.2 The global ugriz luminosity functions and stellar mass function Next, we describe the procedure we utilize for determining the global GLFs and the GSMF. Here, we choose the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure for the determination of the ugriz GLFs and the GSMF. Specifically, we determine the galaxy luminosity and stellar mass distributions as   \begin{eqnarray} \phi _X(M_X) = \frac{1}{\Delta M_X}\sum _{i=1}^N\frac{\omega _X(M_X\pm \Delta M_X / 2)}{\mathcal {V}_{{\rm max},i}}, \end{eqnarray} (2)where MX refers to Mu, Mg, Mr, Mi, Mz, and log M*, ωi is the correction weight completeness factor in the NYU-VAGC for galaxies within the interval MX ± ΔMX/2, and   \begin{eqnarray} \mathcal {V}_{{\rm max},i}= \int _{\Omega }\int ^{z_u}_{z_l}\frac{\,\text{d}^2V_c}{\text{d}z \, {d}\Omega } \text{d}z\text{d}\Omega . \end{eqnarray} (3)We denote the solid angle of the SDSS DR7 with Ω while Vc refers to the comoving volume enclosed within the redshift interval [zl, zu]. The redshift limits are defined as zl = max(0.01, zmin) and zu = min(zmax, 0.2), where zmin and zmax are, respectively, the minimum and maximum at which each galaxy can be observed in the SDSS DR7 sample. For the completeness limits, we use the limiting apparent magnitudes in the r band of r = 14 and r = 17.77. The filled black circles with error bars in Fig. 1 present our determination of the global SDSS DR7 ugriz GLFs. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed line), who used a sample of low-redshift galaxies (<150 h−1Mpc) from the SDSS DR2 and corrected due to low surface brightness selection effects. Additionally, we compare to Hill et al. (2010), who combined data from the Millennium Galaxy Catalog (MGC), the SDSS DR5 and the UKIRT Infrared Deep Sky Survey Large Area Survey (UKIDSS) for galaxies with z < 0.1, dotted lines; and to Driver et al. (2012), who utilized the Galaxy And Mass Assembly (GAMA) survey for the redshift interval 0.013 < z < 0.1 to derive the ugriz GLFs (short dashed lines). All the GLFs in Fig. 1 are at the rest-frame z = 0. In general, we observe good agreement with previous studies; in a more detailed examination, however, we note some differences that are worthwhile to clarify. Figure 1. View largeDownload slide The global ugriz GLFs. Our derived ugriz GLFs and GSMF are shown with the black circles with error bars. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed lines) based on the SDSS DR2; Hill et al. (2010, dotted lines) by combining the MGC, SDSS DR5, and the UKIDSS surveys; and Driver et al. (2012, short dashed lines) based on the GAMA survey. As for the stellar masses, we compare with the GSMF from Baldry et al. (2012) and Wright et al. (2017), black long and short dashed lines, respectively. Figure 1. View largeDownload slide The global ugriz GLFs. Our derived ugriz GLFs and GSMF are shown with the black circles with error bars. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed lines) based on the SDSS DR2; Hill et al. (2010, dotted lines) by combining the MGC, SDSS DR5, and the UKIDSS surveys; and Driver et al. (2012, short dashed lines) based on the GAMA survey. As for the stellar masses, we compare with the GSMF from Baldry et al. (2012) and Wright et al. (2017), black long and short dashed lines, respectively. Consider the u-band GLFs from Fig. 1 and note that there is an apparent tension with previous studies. At the high-luminosity end, our inferred u-band GLF decreases much faster than the above-mentioned studies. This is especially true when comparing with the Hill et al. (2010) and Driver et al. (2012) GLFs. This could be partly due to the differences between the Kron magnitudes used by Hill et al. (2010) and Driver et al. (2012) and the model magnitudes used in this paper. But we believe that most of the difference is due to the differences in the E-corrections, reflecting that our model evolution is more extreme than that of Hill et al. (2010) and Driver et al. (2012). This can be easily understood by noting that the high-luminosity end of the GLF is very sensitive to E-corrections. The reason is that brighter galaxies are expected to be observed more often at larger redshifts than fainter galaxies; thus, equation (1) will result in a small correction for lower luminosity galaxies (low redshift) but a larger correction for higher luminosity galaxies (high redshifts). Indeed, Driver et al. (2012), who did not determine corrections by evolution, derived a u-band GLF that predicts the largest abundance of high luminosity galaxies. On the other hand, the evolution model introduced by Hill et al. (2010) is shallower than ours, which results in a GLF between our determination and the Driver et al. (2012) u-band GLF. This could explain the apparent tension between the different studies. Although the effects of evolution are significant in the u band, they are smaller in the longer wavebands. Ideally, estimates of the evolution should be more physically motivated by galaxy formation models, but empirical measurements are more accessible and faster to determine; however, when making comparisons, one should keep in mind that empirical estimates are by no means definitive. Some previous studies have concluded that a single Schechter function is consistent with observations (see, e.g. Blanton et al. 2003, and recently Driver et al. 2012). However, other studies have found that a double Schechter function is a more accurate description of the GLFs (Blanton et al. 2005a). Additionally, recent studies have found shallower slopes at the high-luminosity end instead of an exponential decreasing slope in the GLFs5 (see e.g. Bernardi et al. 2010). In this paper, we choose to use GLFs that are described by a function composed of a single Schechter function plus another Schechter function with a subexponential decreasing slope for the ugriz bands given by   \begin{eqnarray} \phi (M)&=&\frac{\ln 10}{2.5}\phi _1^*10^{0.4(M^*_1-M)\left(1+\alpha _1\right)}\exp \left(-10^{0.4\left(M^*_1-M\right)}\right) \nonumber \\ &&+\,\frac{\ln 10}{2.5}\phi _2^* 10^{0.4(M^*_2-M)\left(1+\alpha _2\right)}\exp \left(-10^{0.4\left(M^*_2-M\right)\beta }\right). \end{eqnarray} (4)The units of the GLFs are h3 Mpc−3 mag−1, whereas the input magnitudes have units of mag−5 log h. The parameters for the ugriz bands are given in Table 1. Note that for simplicity, we assume that α1 = α, α2 = 1 + α, and $$M^*_1 = M^*_2 = M^*$$. These assumptions reduce the number of free parameters to five. The corresponding best-fitting models are shown in Fig. 1 with the solid black lines. The filled circles with error bars in Fig. 1 present our determinations for the global SDSS DR7 GLFs. Table 1. Best-fitting parameters for the GLFs and the GSMF. Band  α  M* − 5 log h  $$\log \phi ^*_1$$ (h3 Mpc−3 mag−1)  $$\log \phi _2^*$$ (h3 Mpc−3 mag−1)  β  GLF    u  − 0.939 ± 0.005  − 17.758 ± 0.016  − 1.530 ± 0.002  − 3.692 ± 0.044  0.721 ± 0.008  g  − 1.797 ± 0.044  − 19.407 ± 0.068  − 2.764 ± 0.105  − 1.674 ± 0.013  0.821 ± 0.014  r  − 1.810 ± 0.036  − 20.184 ± 0.062  − 2.889 ± 0.094  − 1.733 ± 0.013  0.813 ± 0.013  i  − 1.794 ± 0.031  − 20.546 ± 0.053  − 2.896 ± 0.077  − 1.768 ± 0.011  0.815 ± 0.011  z  − 1.816 ± 0.028  − 20.962 ± 0.051  − 3.038 ± 0.076  − 1.806 ± 0.012  0.827 ± 0.011  GSMF      α  $$\mathcal {M}^*$$ (h− 2  M⊙)  $$\log \phi ^*_1$$ (h3 Mpc−3 dex−1)  $$\log \phi _2^*$$ (h3 Mpc−3 dex−1)  β    − 1.664 ± 0.033  10.199 ± 0.0303  − 3.041 ± 0.082  − 1.885 ± 0.010  0.708 ± 0.012  Band  α  M* − 5 log h  $$\log \phi ^*_1$$ (h3 Mpc−3 mag−1)  $$\log \phi _2^*$$ (h3 Mpc−3 mag−1)  β  GLF    u  − 0.939 ± 0.005  − 17.758 ± 0.016  − 1.530 ± 0.002  − 3.692 ± 0.044  0.721 ± 0.008  g  − 1.797 ± 0.044  − 19.407 ± 0.068  − 2.764 ± 0.105  − 1.674 ± 0.013  0.821 ± 0.014  r  − 1.810 ± 0.036  − 20.184 ± 0.062  − 2.889 ± 0.094  − 1.733 ± 0.013  0.813 ± 0.013  i  − 1.794 ± 0.031  − 20.546 ± 0.053  − 2.896 ± 0.077  − 1.768 ± 0.011  0.815 ± 0.011  z  − 1.816 ± 0.028  − 20.962 ± 0.051  − 3.038 ± 0.076  − 1.806 ± 0.012  0.827 ± 0.011  GSMF      α  $$\mathcal {M}^*$$ (h− 2  M⊙)  $$\log \phi ^*_1$$ (h3 Mpc−3 dex−1)  $$\log \phi _2^*$$ (h3 Mpc−3 dex−1)  β    − 1.664 ± 0.033  10.199 ± 0.0303  − 3.041 ± 0.082  − 1.885 ± 0.010  0.708 ± 0.012  View Large In the case of the GSMF, we compare our results with Baldry et al. (2012) and Wright et al. (2017) plotted with the black long and short dashed lines, respectively. Both analyses used the GAMA survey to determine the local GSMF. Recall that our stellar masses were obtained from the MPA-JHU DR7 data base. As can be seen in the figure, our determination is consistent with these previous results. We again choose to use a function composed of a single Schechter function plus another Schechter function with a subexponential decreasing slope for the GSMF given by   \begin{eqnarray} \phi _*(M_*) &=&\phi _1^*{\ln 10}\left(\frac{M_*}{\mathcal {M}_1^*}\right)^{1+\alpha _1}\exp \left(-\frac{M_*}{\mathcal {M}_1^*}\right) \nonumber \\ &&+\,\phi _2^*{\ln 10}\left(\frac{M_*}{\mathcal {M}_2^*}\right)^{1+\alpha _2}\exp \left[-\left(\frac{M_*}{\mathcal {M}_2^*}\right)^\beta \right]. \end{eqnarray} (5)The units for the GSMF are h3 Mpc−3 dex−1, whereas the input stellar masses are in units of h− 2  M⊙. Again, for simplicity, we assume that α1 = α, α2 = 1 + α, and $$\mathcal {M}_1^* = \mathcal {M}_2^* = \mathcal {M}^*$$; again, this assumption reduces the number of free parameters to five. We report the best-fitting value parameters in Table 1 and the corresponding best-fitting model is presented with the solid black line in Fig. 1. As we will describe in Section 3, we use the ugriz GLFs and GSMF as inputs for our mock galaxy catalogue. 2.3 Measurements of the observed ugriz GLFs and GSMF as a function of environment Once we determined the global ugriz GLFs and the GSMF, the next step in our program is to determine the observed dependence of the ugriz GLFs and GSMF with environmental density. 2.3.1 Density-defining population The SDSS DR7 limiting magnitude in the r band is 17.77. Thus, in order to determine the local overdensity of each SDSS DR7 galaxy, we need to first construct a volume-limited density-defining population (DDP, Croton et al. 2005; Baldry et al. 2006). A volume-limited sample can be constructed by defining the minimum and maximum redshifts at which galaxies within some interval magnitude are detected in the survey. Following the McNaught-Roberts et al. (2014) GAMA paper, we define our volume-limited DDP sample of galaxies in the absolute magnitude range −21.8 < Mr − 5 log h < −20.1. A valid question is whether the definition utilized for the volume-limited DDP sample could lead to different results. This question has been studied in McNaught-Roberts et al. (2014); the authors conclude that the precise definition for the volume-limited DDP sample does not significantly affect the shape of GLFs. None the less, our defined volume-limited DDP sample restricts the SDSS magnitude-limited survey into the redshift range 0.03 ≤ z ≤ 0.11. Fig. 2 shows the absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The solid box presents the galaxy population enclosed in our volume-limited DDP sample, whereas the dashed lines show our magnitude-limited survey. Figure 2. View largeDownload slide Absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The blue solid box shows our volume-limited DDP sample. Note that our DDP sample restricts to study environments for galaxies in the range 0.03 ≤ z ≤ 0.11, as shown by the dashed lines. Figure 2. View largeDownload slide Absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The blue solid box shows our volume-limited DDP sample. Note that our DDP sample restricts to study environments for galaxies in the range 0.03 ≤ z ≤ 0.11, as shown by the dashed lines. 2.3.2 Projected distribution on the sky of the galaxy sample The irregular limits of the projected distribution on the sky of the SDSS-DR7 galaxies could lead to a potential bias in our overdensity measurements; they will artificially increase the frequency of low-density regions, and, ideally, overdensity measurements should be carried out over more continuous regions. Following Varela et al. (2012) and Cebrián & Trujillo (2014), we reduce this source of potential bias by restricting our galaxy sample to a projected area based on the following cuts:   $${\rm Dec.} > \left\{ \begin{array}{l@{\qquad}c} 0 & \mbox{{s}outhern limit}\\ -2.555556\times ({\rm RA} - 131^{\circ }) & \mbox{{w}estern limit}\\ -1.70909\times ({\rm RA} - 235^{\circ }) & \mbox{{e}astern limit}\\ \arcsin \left(\frac{x}{\sqrt{1- x^2}}\right) & \mbox{{n}orthern limit} \end{array}, \right.$$ (6)where x = 0.932 32sin (RA − 95$$_{.}^{\circ}$$9). This region is plotted in fig. 1 of Cebrián & Trujillo (2014). 2.3.3 Overdensity measurements In summary, our final magnitude-limited galaxy sample consists of galaxies in the redshift range 0.03 ≤ z ≤ 0.11 and galaxies within the projected area given by equation (6), whereas our volume-limited DDP sample comprises galaxies with absolute magnitude satisfying −21.8 < Mr − 5 log h < −20.1. Based on the above specifications, we are now in a position to determine the local overdensity of each SDSS DR7 galaxy in our magnitude-limited galaxy sample. Overdensities are estimated by counting the number of DDP galaxy neighbours, Nn, around our magnitude-limited galaxy sample in spheres of r8 = 8 h−1 Mpc radius. Although there exist various methods to measure galaxy environments, Muldrew et al. (2012) showed that aperture-based methods are more robust in identifying the dependence of halo mass on environment, in contrast to nearest neighbours based methods that are largely independent of halo mass. In addition, aperture-based methods are easier to interpret. For these reasons, the aperture-based method is ideal to probe galaxy environments when testing the assumptions behind the SHAM approach. The local density is simply defined as   \begin{eqnarray} \rho _8 = \frac{N_n}{4/3\pi r_8^3}. \end{eqnarray} (7)We then compare the above number to the expected number density of DDP galaxies by using the global r-band luminosity function determined above in Section 2.2; $$\bar{\rho } = 6.094 \times 10^{-3}\,h^3$$ Mpc−3. Finally, the local density contrast for each galaxy is determined as   \begin{eqnarray} \delta _8 = \frac{\rho _8 - \bar{\rho }}{\bar{\rho }}. \end{eqnarray} (8)The effect of changing the aperture radius has been discussed in Croton et al. (2005). Although the authors noted that using smaller spheres tends to sample underdense regions differently, they found that their conclusions remain robust due to the change of apertures. Nevertheless, smaller scale spheres are more susceptible to be affected by redshift space distortions. Following Croton et al. (2005), we opt to use spheres of r8 = 8 h−1 Mpc radius as the best probe of both underdense and overdense regions. Finally, note that our main goal is to understand whether halo Vmax fully determines galaxy properties as predicted by SHAM, not to study the physical causes for the observed galaxy distribution with environment. Therefore, as long as we treat our mock galaxy sample, to be described in Section 3, in the same way that we treat observations, understanding the impact of changing apertures in the observed galaxy distribution is beyond the scope of this paper. 2.3.4 Measurements of the observed ugriz GLFs and the GSMF as a function of environmental density Once the local density contrast for each galaxy in the SDSS DR7 is determined, we estimate the dependence of the ugriz GLFs and the GSMF with environmental density. As in Section 2.2, we use the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure. Unfortunately, the 1/$$\mathcal {V}_{\rm max}$$ method does not provide the effective volume covered by the overdensity bin in which the GLFs and the GSMF have been estimated, and therefore one needs to slightly modify the 1/$$\mathcal {V}_{\rm max}$$ estimator. In this subsection, we describe how we estimate the effective volume. We determine the fraction of effective volume by counting the number of DDP galaxy neighbours in a catalogue of random points with the same solid angle and redshift distribution as our final magnitude-limited sample. Observe that we utilize the real position of the DDP galaxy sample already defined. We again utilized spheres of radius r8 = 8 h−1 Mpc and create a random catalogue consisting of Nr ∼ 2 × 106 of points. The local density contrast for each random point is determined as in equation (8):   \begin{eqnarray} \delta _{r_8} = \frac{\rho _{r_8} - \bar{\rho }}{\bar{\rho }}, \end{eqnarray} (9)where $$\rho _{r_8}$$ is the local density around random points. We estimate the fraction of effective volume by a given overdensity bin as   \begin{eqnarray} f(\delta _8) = \frac{1}{N_r}\sum _{i=1}^{N_r} \Theta (\delta _{{r_8},i}). \end{eqnarray} (10)Here, Θ is a function that selects random points in the overdensity range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$:   $$\Theta (\delta _{{r_8},i}) = \left\{ \begin{array}{l@{\qquad}c} 1 & \mbox{if } \delta _{{r_8},i} \in [\delta _{r_8} - \Delta \delta _{r_8}/2, \delta _{r_8} + \Delta \delta _{r_8}/2)\\ 0 & \mbox{otherwise}\end{array}\right. .$$ (11)Table 2 lists the fraction of effective volume for the range of overdensities considered in this paper and calculated as described in this paper. We estimate errors by computing the standard deviation of the fraction of effective volume in 16 redshift bins equally spaced. We note that the number of sampled points gives errors that are less than ∼ 3 per cent and for most of the bins less than ∼ 1 per cent (see the last column of Table 2). Therefore, we ignore any potential source of error from our determination of the fraction of effective volume into the ugriz GLFs and the GSMFs. Table 2. Fraction of effective volume covered by the overdensity bins considered for our analysis in the SDSS DR7. Also shown is the fractional error due to the number of random points sampled. δmin, 8  δmax, 8  f(δ8) ± δf(δ8)  100 per cent × δf(δ8)/f(δ8)  − 1  − 0.75  0.1963 ± 0.0014  0.713  − 0.75  − 0.55  0.1094 ± 0.0010  0.914  − 0.55  − 0.40  0.0974 ± 0.0009  0.924  − 0.40  0.00  0.2156 ± 0.0014  0.650  0.00  0.70  0.1800 ± 0.0012  0.667  0.70  1.60  0.1040 ± 0.0009  0.866  1.60  2.90  0.0621 ± 0.0007  1.130  2.90  4  0.0197 ± 0.0004  2.030  4.00  ∞  0.0153 ± 0.0004  2.614  δmin, 8  δmax, 8  f(δ8) ± δf(δ8)  100 per cent × δf(δ8)/f(δ8)  − 1  − 0.75  0.1963 ± 0.0014  0.713  − 0.75  − 0.55  0.1094 ± 0.0010  0.914  − 0.55  − 0.40  0.0974 ± 0.0009  0.924  − 0.40  0.00  0.2156 ± 0.0014  0.650  0.00  0.70  0.1800 ± 0.0012  0.667  0.70  1.60  0.1040 ± 0.0009  0.866  1.60  2.90  0.0621 ± 0.0007  1.130  2.90  4  0.0197 ± 0.0004  2.030  4.00  ∞  0.0153 ± 0.0004  2.614  View Large Finally, we modify the 1/$$\mathcal {V}_{\rm max}$$ weighting estimator to account for the effective volume by the overdensity bin as   \begin{eqnarray} \phi _X(M_X|\delta _8) = \sum _{i=1}^N\frac{\omega _i(M_X\pm \Delta M_X / 2|\delta _{r_8} \pm \Delta \delta _{r_8}/2)}{f(\delta _8)\times \Delta M_X\times \mathcal {V}_{{\rm max},i}}, \end{eqnarray} (12)again, MX refers to Mu, Mg, Mr, Mi, Mz, and log M*. Here, ωi refers to the correction weight completeness factor for galaxies within the interval MX ± ΔMX/2, given that their overdensity is in the range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$. 3 THE GALAXY–HALO CONNECTION The main goal of this paper is to study whether one halo property, in this case Vmax, fully determines the observed dependence with environmental density of the ugriz GLFs and the GSMF. Confirming this would significantly improve our understanding of the galaxy–halo connection. In this section, we describe how we constructed a mock galaxy catalogue in the cosmological BolshoiP N-body simulation via SHAM. 3.1 The Bolshoi–Planck simulation To study the environmental dependence of the galaxy distribution predicted by SHAM, we use the N-body BolshoiP cosmological simulation (Klypin et al. 2016). This simulation is based on the ΛCDM cosmology with parameters consistent with the latest results from the Planck Collaboration. This simulation has 20483 particles of mass 1.9 × 108  M⊙ h− 1, in a box of side length LBP = 250 h−1 Mpc. Haloes/subhaloes and their merger trees were calculated with the phase–space temporal halo finder Rockstar (Behroozi, Wechsler & Wu 2013) and the software Consistent Trees (Behroozi et al. 2013). Entire Rockstar and Consistent Trees outputs are downloadable.6 Halo masses were defined using spherical overdensities according to the redshift-dependent virial overdensity Δvir(z) given by the spherical collapse model, with Δvir(z) = 333 at z = 0. The BolshoiP simulation is complete down to haloes of maximum circular velocity Vmax ≳ 55 km s−1. For more details, see Rodríguez-Puebla et al. (2016a). Next, we describe our mock galaxy catalogues generated via SHAM. 3.2 Determining the galaxy–halo connection As we have explained, SHAM is a simple approach relating a halo property, such as mass or maximum circular velocity, to that of a galaxy property, such as luminosity or stellar mass. In abundance matching between a halo property and a galaxy property, the number density distribution of the halo property is matched to the number density distribution of the galaxy property to obtain the relation. Recall that SHAM assumes that there is a one-to-one monotonic relationship between galaxies and haloes, and that centrals and satellite galaxies have identical relationships (except that satellite galaxy evolution is stopped when the host halo reaches its peak maximum circular velocity). In this paper, we choose to relate galaxy properties, $$\mathcal {P}_{\rm gal}$$, to halo maximum circular velocities Vmax as   \begin{eqnarray} \int _{\mathcal {P}_{\rm gal}}^{\infty } \phi _{\rm gal}(\mathcal {P}_{\rm gal}^{\prime })\text{ d}\log \mathcal {P}_{\rm gal}^{\prime } = \int _{V_{\rm max}}^{\infty } \phi _V (V_{\rm max} ^{\prime })\text{ d}\log V_{\rm max} ^{\prime }, \end{eqnarray} (13)where $$\phi _{\rm gal}(\mathcal {P}_{\rm gal})$$ denotes the ugriz GLF as well as the GSMF, and ϕV(Vmax) represents the subhalo+halo velocity function, both in units of h3 Mpc−3 dex−1. To construct a mock galaxy catalogue of luminosities and stellar masses from the BolshoiP simulation, we apply the aforementioned procedure by using as input the global ugriz GLFs and the GSMF derived in Section 2.2. Equation (13) is the simplest form that SHAM could take as it ignores the existence of a physical scatter around the relationship between $$\mathcal {P}_{\rm gal}$$ and Vmax. Including physical scatter in equation (13) is no longer considered valid and should be modified accordingly (for more details, see Behroozi et al. 2010). Constraints based on weak-lensing analysis (Leauthaud et al. 2012), satellite kinematics (More et al. 2009, 2011), and galaxy clustering (Zheng, Coil & Zehavi 2007; Zehavi et al. 2011; Yang et al. 2012) have shown that this is of the order of ∼0.15 dex in the case of the stellar but similar in r-band magnitude. There are no constraints as for the dispersion around shorter wavelengths. In addition, it is not clear how to sample galaxy properties in a system with n number of properties from the joint probability distribution $${\rm prop}(\mathcal {P}_{{\rm gal},1},\ldots , \mathcal {P}_{{\rm gal},n}| V_{\rm max}).$$7 Instead of that, studies that aim at to constrain the galaxy–halo connection use marginalization to constrain the probability distribution function $${\rm prop}(\mathcal {P}_{{\rm gal},i}| V_{\rm max})$$ for ith galaxy property. In this paper, we are interested in the statistical correlation of the galaxy–halo connection in which case equation (13) is a good approximation. Studying and quantifying the physical scatter around the relations is beyond the scope of this work. Also, ignoring the scatter around the galaxy–halo connection makes it easier to interpret. For those reasons, we have opted to ignore any source of scatter in our relationships. Previous studies have found that for distinct dark matter haloes (those that are not contained in bigger haloes), the maximum circular velocity Vmax is the halo property that correlates best with the hosted galaxy's luminosity/stellar mass. This is likely because the properties of a halo's central region, where its central galaxy resides, are better described by Vmax than Mvir.8 By comparing to observations of galaxy clustering, Reddick et al. (2013) and more recently Campbell et al. (2017) have found that for subhaloes the property that correlates best with luminosity/stellar mass is the highest maximum circular velocity reached along the main progenitor branch of the halo's merger tree. This presumably reflects the fact that subhaloes can lose mass once they approach and fall into a larger halo, whereas the host galaxy at the halo's centre is unaffected by this halo mass-loss. Thus, in this paper, we use   $$V_{\rm max}= \left\{ \begin{array}{l@{\qquad}c} V_{\rm max}& \mbox{{d}istinct {haloes}}\\ V_{\rm peak}& \mbox{{s}ubhalo{e}s} \end{array}\right. ,$$ (14)as the halo proxy for galaxy properties $$\mathcal {P}_{\rm gal}$$, where Vpeak is the maximum circular velocity throughout the entire history of a subhalo and Vmax is at the observed time for distinct haloes. Fig. 3 shows the relationships between galaxy luminosities u, g, r, i, and z and galaxy stellar masses to halo maximum circular velocities. Table A1 reports the values from Fig. 3. Most of these relationships are steeply increasing with Vmax for velocities below Vmax ∼ 160 km s−1. At higher velocities, the relationships are shallower. The shapes of these relations are governed mostly by the shapes of the GLFs and GSMF, since the velocity function ϕV is approximately a power law over the range plotted in Fig. 3 (see Rodríguez-Puebla et al. 2016a). Figure 3. View largeDownload slide Left-hand panel: luminosity–Vmax relation from SHAM. The different colours indicate the band utilized for the match. Right-hand panel: stellar mass–Vmax relation. Recall that SHAM assumes that these relations are valid for centrals as well as for satellites. We report these values in Table A1. In the case of centrals, Vmax refers to the halo maximum circular velocity, whereas for satellites, Vmax represents the highest maximum circular velocity (Vpeak) reached along the subhalo's main progenitor branch. SHAM assumes that Vmax fully determines these statistical properties of the galaxies. Figure 3. View largeDownload slide Left-hand panel: luminosity–Vmax relation from SHAM. The different colours indicate the band utilized for the match. Right-hand panel: stellar mass–Vmax relation. Recall that SHAM assumes that these relations are valid for centrals as well as for satellites. We report these values in Table A1. In the case of centrals, Vmax refers to the halo maximum circular velocity, whereas for satellites, Vmax represents the highest maximum circular velocity (Vpeak) reached along the subhalo's main progenitor branch. SHAM assumes that Vmax fully determines these statistical properties of the galaxies. Note that at this point every halo and subhalo in the BolshoiP simulation at rest-frame z = 0 has been assigned a magnitude in the five bands u, g, r, i, and z and a stellar mass M*. Therefore, one might be tempted to correlate galaxy colours such as red or blue (i.e. differences between galaxy magnitudes) with halo properties. If we did this, we would be ignoring the scatter around our luminosity/stellar mass–Vmax relationships, and galaxies with the same magnitude or M* would have the same colour, contrary to observations. Fortunately, including a scatter around those relationships will not impact our conclusions given that (i) the scatter does not substantially impact the results presented in Fig. 3, and (ii) we are here interested only in the statistical correlation of the galaxy properties with environment. Nevertheless, in Section 4.2.1, we will study the statistical correlation between colour and environment for all galaxies, and separately for central and satellite galaxies. As a sanity check, we show that our mock galaxy catalogue in the BolshoiP reproduces the projected two-point correlation function of SDSS galaxies.9 Figs 4 and 5 show, respectively, that this is the case for the r band and stellar mass projected two point correlation functions. In the case of r band, we compared to Zehavi et al. (2011), who used r-band magnitudes at z = 0.1. We transformed our r-band magnitudes to z = 0.1 by finding the correlation between model magnitudes at z = 0 and 0.1 from the tables of the NYU-VAGC.10 For the projected two point correlation function in stellar mass bins, we compare with Yang et al. (2012). Figure 4. View largeDownload slide Two-point correlation function in five luminosity bins at z = 0.1. The solid lines show the predicted two-point correlation based on our r-band magnitude–Vmax relation from SHAM, whereas the circles with error bars show the same but for the SDSS DR7 (Zehavi et al. 2011). Figure 4. View largeDownload slide Two-point correlation function in five luminosity bins at z = 0.1. The solid lines show the predicted two-point correlation based on our r-band magnitude–Vmax relation from SHAM, whereas the circles with error bars show the same but for the SDSS DR7 (Zehavi et al. 2011). Figure 5. View largeDownload slide Two-point correlation function in five stellar mass bins. The solid lines show the predicted two-point correlation based on our stellar mass–Vmax relation from SHAM, whereas the circles with error bars show the same but for SDSS DR7 (Yang et al. 2012). Figure 5. View largeDownload slide Two-point correlation function in five stellar mass bins. The solid lines show the predicted two-point correlation based on our stellar mass–Vmax relation from SHAM, whereas the circles with error bars show the same but for SDSS DR7 (Yang et al. 2012). 3.3 Measurements of the mock ugriz GLFs and the GSMF as a function of environment Our mock galaxy catalogue is a volume complete sample down to haloes of maximum circular velocity Vmax ∼ 55 km s− 1, corresponding to galaxies brighter than Mr − 5 log h ∼ −14, see Fig. 3.11 This magnitude completeness is well above the completeness of the SDSS DR7. Thus, galaxies selected in the absolute magnitude range −21.8 < Mr − 5 log h < −20.1 define a volume-limited DDP sample. In other words, incompleteness is not a problem for our mock galaxy catalogue. Overdensity and density contrast measurements for each mock galaxy in the BolshoiP simulation are obtained as described in Section 2.3.3. We estimate the dependence of the ugriz GLFs with environment in our mock galaxy catalogue as   \begin{eqnarray} \phi _X(M_X|\delta _8) = \sum _{i=1}^N \frac{\omega _i(M_X\pm \Delta M_X / 2|\delta _{r_8} \pm \Delta \delta _{r_8}/2)}{\Delta M_X f_{\rm BP}(\delta _8) L_{\rm BP}^3}. \end{eqnarray} (15)Here, ωi = 1 if a galaxy is within the interval MX ± ΔMX/2 given that its overdensity is in the range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$, otherwise it is 0. Again, MX refers to Mu, Mg, Mr, Mi, Mz, and log M*. The function fBP(δ8) is the fraction of effective volume by a given overdensity bin for the BolshoiP simulation. In order to determine fBP(δ8), we create a random catalogue of Nr ∼ 1.2 × 106 points in a box of side length identical to the BolshoiP simulation, i.e. LBP = 250 h−1Mpc. Using equation (10) allows us to calculate fBP(δ8). 4 RESULTS ON ENVIRONMENTAL DENSITY DEPENDENCE In this section, we present our determinations for the environmental density dependence of the ugriz GLFs and the GSMF from the SDSS DR7 and the BolshoiP. Here, we will investigate how well the assumption that the statistical properties of galaxies are fully determined by Vmax can predict the dependence of the ugriz GLFs and GSMF with environment. We will show that predictions from SHAM are in remarkable agreement with the data from the SDSS DR7, especially for the longer wavelength bands. Finally, we show that SHAM also reproduces the correct dependence on environmental density of both the r-band GLFs and GSMF for centrals and satellites, although it fails to reproduce the observed relationship between environment and colour. 4.1 SDSS DR7 Fig. 6 shows the dependence of the SDSS DR7 ugriz GLFs as well as the GSMF with environmental density measured in spheres of radius 8 h−1 Mpc. For the sake of the simplicity, we present only four overdensity bins in Fig. 6. In Fig. 7, we show the determinations in nine density bins for the r-band GLFs and GSMF. In order to compare with recent observational results, we use identical environment density bins as in McNaught-Roberts et al. (2014), who used galaxies from the GAMA survey to measure the dependence of the r-band GLF on environment over the redshift range 0.04 < z < 0.26 in spheres of radius of 8 h−1 Mpc. Figure 6. View largeDownload slide Comparison between the observed SDSS DR7 ugriz GLFs and GSMF, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM, shaded regions, at four environmental densities in spheres of radius 8 h−1 Mpc. We also reproduce the best-fitting Schechter functions to the r-band GLFs from the GAMA survey (McNaught-Roberts et al. 2014). Observe that SHAM predictions are in excellent agreement with observations, especially for the longest wavelength bands and stellar mass. Figure 6. View largeDownload slide Comparison between the observed SDSS DR7 ugriz GLFs and GSMF, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM, shaded regions, at four environmental densities in spheres of radius 8 h−1 Mpc. We also reproduce the best-fitting Schechter functions to the r-band GLFs from the GAMA survey (McNaught-Roberts et al. 2014). Observe that SHAM predictions are in excellent agreement with observations, especially for the longest wavelength bands and stellar mass. Figure 7. View largeDownload slide Left-hand panel: comparison between the observed r-band GLF with environmental density in spheres of 8 h−1 Mpc, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM (shaded regions). The dashed and solid lines show the best-fitting Schechter functions to the observed and the mocked r-band GLFs, whereas the dotted lines show the same but from the GAMA survey (McNaught-Roberts et al. 2014). Right-hand panel: similar to the left-hand panel but for the GSMF with environmental density. Here, again the dashed and solid lines are the best-fitting Schechter functions to the observed and mocked GSMFs. Figure 7. View largeDownload slide Left-hand panel: comparison between the observed r-band GLF with environmental density in spheres of 8 h−1 Mpc, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM (shaded regions). The dashed and solid lines show the best-fitting Schechter functions to the observed and the mocked r-band GLFs, whereas the dotted lines show the same but from the GAMA survey (McNaught-Roberts et al. 2014). Right-hand panel: similar to the left-hand panel but for the GSMF with environmental density. Here, again the dashed and solid lines are the best-fitting Schechter functions to the observed and mocked GSMFs. The r-band panel of Fig. 6 shows that our determinations are in good agreement with results from the GAMA survey. In the g-band panel of the same figure, we present a comparison with the previously published results by Croton et al. (2005), who used the 2dF Galaxy Redshift Survey to measure the dependence of the bJ-band GLFs at a zero redshift rest frame in spheres of radius of 8 h−1 Mpc. We convert the bJ-band GLFs from Croton et al. (2005) to the g band by applying a shift of −0.25 to their magnitudes, that is, $$M_g = M_{b_{\rm J}} - 0.25$$ (Blanton et al. 2005a). We observe good agreement with the result of Croton et al. (2005) for most of our density bins. A better comparison would have used identical density bins; however, the density bins used by Croton et al. (2005) are close to ours. Finally, in Fig. 6, we also extend previous results by presenting the GLFs for the u, i, and z bands and for the GSMF. We are not aware of any published low redshift GLFs for the u, i, and z bands. The left-hand panel of Fig. 7 shows again the dependence of the GLF in the r band, but now for all our overdensity bins, filled circles with error bars. In order to report an analytical model for the luminosity functions, we fit observations to a simple Schechter function; observations show that this model is a good description for the data, given by   \begin{eqnarray} \phi (M)=\frac{\ln 10}{2.5}\phi ^*10^{0.4(M^*-M)\left(1+\alpha _1\right)}\exp \left(-10^{0.4\left(M^*-M\right)}\right), \end{eqnarray} (16)in units of h3 Mpc−3 mag−1. The best fit to simple Schechter functions is shown as the dashed lines in the same plot, and we report the Schechter parameters as a function of the density contrast in the left-hand panel of Fig. 8. The best-fitting parameters are listed in Table 3. For comparison, we reproduce the best fit to a Schechter function from McNaught-Roberts et al. (2014), dotted lines. Figure 8. View largeDownload slide Left-hand panel: the dependence of the r-band Schechter function parameters on environmental overdensity δ8 in spheres of 8 h−1 Mpc (equation 8). Right-hand panel: the dependence of the GSMF Schechter parameters on environmental density. Figure 8. View largeDownload slide Left-hand panel: the dependence of the r-band Schechter function parameters on environmental overdensity δ8 in spheres of 8 h−1 Mpc (equation 8). Right-hand panel: the dependence of the GSMF Schechter parameters on environmental density. Table 3. Best-fitting parameters from the observed SDSS DR7 GLFs. GLFs  δmin, 8  δmax, 8  α  M* − 5 log h  log ϕ* (h3 Mpc−3 mag−1)  SDSS DR7  − 1  − 0.75  − 1.671 ± 0.092  − 20.650 ± 0.131  − 2.938 ± 0.095  − 0.75  − 0.55  − 1.265 ± 0.077  − 20.456 ± 0.086  − 2.391 ± 0.051  − 0.55  − 0.40  − 1.178 ± 0.068  − 20.490 ± 0.073  − 2.224 ± 0.041  − 0.40  0.00  − 1.217 ± 0.032  − 20.568 ± 0.033  − 2.031 ± 0.019  0.00  0.70  − 1.120 ± 0.025  − 20.604 ± 0.026  − 1.754 ± 0.013  0.70  1.60  − 1.092 ± 0.023  − 20.703 ± 0.025  − 1.556 ± 0.012  1.60  2.90  − 1.230 ± 0.033  − 20.870 ± 0.016  − 1.408 ± 0.015  2.90  4  − 1.292 ± 0.028  − 20.981 ± 0.040  − 1.280 ± 0.023  4.00  ∞  − 1.275 ± 0.005  − 21.000 ± 0.034  − 1.045 ± 0.020  BolshoiP+SHAM  − 1  − 0.75  − 1.0963 ± 0.0264  − 20.1018 ± 0.0319  − 2.6084 ± 0.0172  − 0.75  − 0.55  − 1.0214 ± 0.0206  − 20.2498 ± 0.0238  − 2.2478 ± 0.0127  − 0.55  − 0.40  − 1.0462 ± 0.0225  − 20.3727 ± 0.0273  − 2.1260 ± 0.0147  − 0.40  0.00  − 1.0828 ± 0.0129  − 20.5454 ± 0.0193  − 2.0225 ± 0.0096  0.00  0.70  − 1.1274 ± 0.0087  − 20.7486 ± 0.0142  − 1.8634 ± 0.0075  0.70  1.60  − 1.1453 ± 0.0086  − 20.8882 ± 0.0204  − 1.6760 ± 0.0079  1.60  2.90  − 1.1790 ± 0.0092  − 21.0189 ± 0.0220  − 1.4913 ± 0.0090  2.90  4  − 1.2451 ± 0.0154  − 21.1841 ± 0.0354  − 1.3821 ± 0.0179  4.00  ∞  − 1.2227 ± 0.0129  − 21.1285 ± 0.0300  − 1.0924 ± 0.0149  GLFs  δmin, 8  δmax, 8  α  M* − 5 log h  log ϕ* (h3 Mpc−3 mag−1)  SDSS DR7  − 1  − 0.75  − 1.671 ± 0.092  − 20.650 ± 0.131  − 2.938 ± 0.095  − 0.75  − 0.55  − 1.265 ± 0.077  − 20.456 ± 0.086  − 2.391 ± 0.051  − 0.55  − 0.40  − 1.178 ± 0.068  − 20.490 ± 0.073  − 2.224 ± 0.041  − 0.40  0.00  − 1.217 ± 0.032  − 20.568 ± 0.033  − 2.031 ± 0.019  0.00  0.70  − 1.120 ± 0.025  − 20.604 ± 0.026  − 1.754 ± 0.013  0.70  1.60  − 1.092 ± 0.023  − 20.703 ± 0.025  − 1.556 ± 0.012  1.60  2.90  − 1.230 ± 0.033  − 20.870 ± 0.016  − 1.408 ± 0.015  2.90  4  − 1.292 ± 0.028  − 20.981 ± 0.040  − 1.280 ± 0.023  4.00  ∞  − 1.275 ± 0.005  − 21.000 ± 0.034  − 1.045 ± 0.020  BolshoiP+SHAM  − 1  − 0.75  − 1.0963 ± 0.0264  − 20.1018 ± 0.0319  − 2.6084 ± 0.0172  − 0.75  − 0.55  − 1.0214 ± 0.0206  − 20.2498 ± 0.0238  − 2.2478 ± 0.0127  − 0.55  − 0.40  − 1.0462 ± 0.0225  − 20.3727 ± 0.0273  − 2.1260 ± 0.0147  − 0.40  0.00  − 1.0828 ± 0.0129  − 20.5454 ± 0.0193  − 2.0225 ± 0.0096  0.00  0.70  − 1.1274 ± 0.0087  − 20.7486 ± 0.0142  − 1.8634 ± 0.0075  0.70  1.60  − 1.1453 ± 0.0086  − 20.8882 ± 0.0204  − 1.6760 ± 0.0079  1.60  2.90  − 1.1790 ± 0.0092  − 21.0189 ± 0.0220  − 1.4913 ± 0.0090  2.90  4  − 1.2451 ± 0.0154  − 21.1841 ± 0.0354  − 1.3821 ± 0.0179  4.00  ∞  − 1.2227 ± 0.0129  − 21.1285 ± 0.0300  − 1.0924 ± 0.0149  View Large Fig. 8 shows that the normalization parameter of the Schechter function, ϕ*, depends strongly on density. There are almost two orders of magnitude difference between the least and the highest density bins; see also Table 3. In contrast, the faint-end slope, α, remains practically constant with environment with a value of α = −1.0 to −1.2. Note, however, that our analysis of the SDSS observations shows that the GLF becomes steeper in the least dense environment, with α ∼ −1.7. The characteristic magnitude of the Schechter function, M*, evolves only little with environment in the range −1 ≲ δ8 ≲ 0, but it increases above δ8 ∼ 0. Finally, in the same figure, we reproduce the best-fitting model parameters reported in McNaught-Roberts et al. (2014). In general, our determinations are in good agreement with the trends reported in McNaught-Roberts et al. (2014) even at faint magnitudes, as is shown in Figs 7 and 8. This is reassuring since the GAMA survey is deeper than the SDSS, which could result in a much better determination of the faint-end. In addition, the subtended area by the GAMA survey is much smaller than that of the SDSS, which could have resulted in GAMA underestimating the abundance of massive galaxies in low-density environments. The reason for this is because the limited volume of GAMA does not adequately sample these rather rare galaxies in low-density regions. The right-hand panel of Fig. 7 shows the dependence of the GSMF for all our overdensity bins as well as their corresponding best fit to simple Schechter functions, filled circles with error bars and solid lines, respectively. In this case, the Schechter function is given by   \begin{eqnarray} \phi _*(M_*) =\phi _1^*\times {\ln 10}\times \left(\frac{M_*}{\mathcal {M}_1^*}\right)^{1+\alpha _1}\exp \left(-\frac{M_*}{\mathcal {M}_1^*}\right), \end{eqnarray} (17)with units of h3 Mpc−3 dex−1. We report the best-fitting parameters in Table 4. The right-hand panel of Fig. 8 presents the Schechter parameters for the GSMFs as a function of the density contrast. Similarly to the GLFs, the normalization parameter for the GSMF, ϕ*, depends strongly on density as a power law and there are approximately two orders of magnitude difference between the GLFs in the least and the most dense environments. As for the faint-end slope, α, we observe that the general trend is that in high-density environments the GSMF becomes steeper than in low-density environments. None the less, we observe, again, that in the lowest density bin the GSMF becomes steeper than other density bins. The characteristic stellar mass of the Schechter function, $$\mathcal {M}_*$$, increases with the environment at least for densities greater than δ8 ∼ 0. In contrast, below δ8 ∼ 0 it remains approximately constant. Table 4. Best-fitting parameters from the observed SDSS DR7 GSMF. GSMFs  δmin, 8  δmax, 8  α  $$\mathcal {M}_*$$ (h− 2  M⊙)  log ϕ* (h3 Mpc−3 dex−1)  SDSS DR7  − 1  − 0.75  − 1.361 ± 0.077  10.467 ± 0.052  − 3.148 ± 0.075  − 0.75  − 0.55  − 1.062 ± 0.061  10.422 ± 0.032  − 2.630 ± 0.040  − 0.55  − 0.40  − 1.019 ± 0.054  10.433 ± 0.029  − 2.453 ± 0.035  − 0.40  0.00  − 1.031 ± 0.025  10.463 ± 0.012  − 2.250 ± 0.015  0.00  0.70  − 1.029 ± 0.019  10.511 ± 0.011  − 2.006 ± 0.012  0.70  1.60  − 1.042 ± 0.018  10.577 ± 0.011  − 1.815 ± 0.012  1.60  2.90  − 1.128 ± 0.018  10.627 ± 0.011  − 1.632 ± 0.013  2.90  4  − 1.164 ± 0.026  10.668 ± 0.016  − 1.477 ± 0.021  4.00  ∞  − 1.179 ± 0.022  10.686 ± 0.013  − 1.248 ± 0.017  BolshoiP+SHAM  − 1  − 0.75  − 1.0293 ± 0.0222  10.3077 ± 0.0142  − 2.8903 ± 0.0176  − 0.75  − 0.55  − 1.0000 ± 0.0151  10.3933 ± 0.0105  − 2.5222 ± 0.0128  − 0.55  − 0.40  − 0.9954 ± 0.0162  10.4306 ± 0.0110  − 2.3743 ± 0.0138  − 0.40  0.00  − 1.0275 ± 0.0094  10.5112 ± 0.0074  − 2.2602 ± 0.0088  0.00  0.70  − 1.0411 ± 0.0074  10.5896 ± 0.0065  − 2.0754 ± 0.0073  0.70  1.60  − 1.0528 ± 0.0073  10.6564 ± 0.0069  − 1.8757 ± 0.0075  1.60  2.90  − 1.0885 ± 0.0069  10.7191 ± 0.0075  − 1.6939 ± 0.0081  2.90  4  − 1.1533 ± 0.0112  10.7898 ± 0.0146  − 1.5891 ± 0.0154  4.00  ∞  − 1.1396 ± 0.0109  10.7691 ± 0.0135  − 1.3106 ± 0.0143  GSMFs  δmin, 8  δmax, 8  α  $$\mathcal {M}_*$$ (h− 2  M⊙)  log ϕ* (h3 Mpc−3 dex−1)  SDSS DR7  − 1  − 0.75  − 1.361 ± 0.077  10.467 ± 0.052  − 3.148 ± 0.075  − 0.75  − 0.55  − 1.062 ± 0.061  10.422 ± 0.032  − 2.630 ± 0.040  − 0.55  − 0.40  − 1.019 ± 0.054  10.433 ± 0.029  − 2.453 ± 0.035  − 0.40  0.00  − 1.031 ± 0.025  10.463 ± 0.012  − 2.250 ± 0.015  0.00  0.70  − 1.029 ± 0.019  10.511 ± 0.011  − 2.006 ± 0.012  0.70  1.60  − 1.042 ± 0.018  10.577 ± 0.011  − 1.815 ± 0.012  1.60  2.90  − 1.128 ± 0.018  10.627 ± 0.011  − 1.632 ± 0.013  2.90  4  − 1.164 ± 0.026  10.668 ± 0.016  − 1.477 ± 0.021  4.00  ∞  − 1.179 ± 0.022  10.686 ± 0.013  − 1.248 ± 0.017  BolshoiP+SHAM  − 1  − 0.75  − 1.0293 ± 0.0222  10.3077 ± 0.0142  − 2.8903 ± 0.0176  − 0.75  − 0.55  − 1.0000 ± 0.0151  10.3933 ± 0.0105  − 2.5222 ± 0.0128  − 0.55  − 0.40  − 0.9954 ± 0.0162  10.4306 ± 0.0110  − 2.3743 ± 0.0138  − 0.40  0.00  − 1.0275 ± 0.0094  10.5112 ± 0.0074  − 2.2602 ± 0.0088  0.00  0.70  − 1.0411 ± 0.0074  10.5896 ± 0.0065  − 2.0754 ± 0.0073  0.70  1.60  − 1.0528 ± 0.0073  10.6564 ± 0.0069  − 1.8757 ± 0.0075  1.60  2.90  − 1.0885 ± 0.0069  10.7191 ± 0.0075  − 1.6939 ± 0.0081  2.90  4  − 1.1533 ± 0.0112  10.7898 ± 0.0146  − 1.5891 ± 0.0154  4.00  ∞  − 1.1396 ± 0.0109  10.7691 ± 0.0135  − 1.3106 ± 0.0143  View Large 4.2 Comparison to theoretical determinations: SHAM Fig. 6 compares our observed ugriz GLFs and the results derived from the mock galaxy sample based on SHAM. In general, we observe a remarkable agreement between observations and SHAM. This statement is true for all the luminosity bands as well as for stellar mass and for most of the density bins. This remarkable agreement is not a trivial result since we are assuming that Vmax fully determines the magnitudes in the u, g, r, i, and z bands and stellar mass M* in every halo in the simulation. Additionally, in Section 3, we noted that the shape of the galaxy–halo connection is governed, mainly, by the global shape of the galaxy number densities. Moreover, although we have defined our volume-limited DDP sample as for the SDSS observations, it is subject to the assumptions behind SHAM as well. In addition, the real correlation between r-band magnitude and all other galaxy properties is no doubt more complex than just monotonic relationships without scatters, as is derived in SHAM. Note, however, that there are some discrepancies towards bluer bands and low densities. Shorter wavelengths are more affected by recent star formation, and more likely to be related to halo mass accretion rates (Rodríguez-Puebla et al. 2016b, and references therein), whereas infrared magnitudes depend more strongly on stellar mass. This perhaps just reflects that stellar mass is the galaxy property that most naturally correlates with Vmax. Indeed, when comparing the environmental dependence of the observed and the mock GSMF we observe, in general, rather good agreement. The left-hand panel of Fig. 7 compares the resulting dependence of the observed r-band GLFs with environment and the predictions based on SHAM for all our overdensity bins. This again shows the remarkable agreement between observations and SHAM for all our density bins. Similarly, the right-hand panel of Fig. 7 compares the observed GSMF with our predictions based on SHAM. The left-hand panel of Fig. 8 compares the best-fitting Schechter parameters for the r-band magnitude, whereas the right-hand panel shows the same but for stellar masses. In order to make a meaningful comparison with observations, we fit the observed GLFs and GSMF of the SDSS DR7 over the same dynamical ranges. In general, we observe a good agreement between predictions from SHAM and the results from the SDSS DR7. Although Fig. 7 shows that the general trends are well predicted by SHAM, there are some differences that are worth discussing. SHAM is able to recover the overall normalization of the r-band GLF and the GSMF, but it slightly underpredicts the number of faint galaxies and it also underpredicts the high-mass end in low-density environments. In high-density environments, SHAM overpredicts the number of galaxies at the high-mass end. A natural explanation for these differences could be the dependence of the galaxy–halo connection with environment. Recall that we are assuming zero scatter in the galaxy–halo connection. Despite the differences noted above, the extreme simplicity of SHAM captures extremely well the dependences with environmental density of the galaxy distribution. This is remarkable and, as we noted earlier, it might not be expected since halo properties depend on the local environment as well as the large-scale environment. In order to understand better the success of SHAM and the nature of the above discrepancies, we now turn our attention to the dependence with environment of the r-band GLFs and GSMF of central and satellite galaxies separately. 4.2.1 SHAM predictions for the central and satellite GLFs and GSMFs Figs 9 and 10 show, respectively, the dependence on environmental density of the r-band GLFs and GSMF for all galaxies, and separately for centrals and satellites. The circles with error bars show the results when using the memberships from the SDSS DR7 Yang et al. (2012) galaxy group catalogue, whereas the shaded areas show the predictions from SHAM based on the BolshoiP simulation. When dividing the population between centrals and satellites, in general, SHAM captures the observed dependences from the SDSS DR7 Yang et al. (2012) galaxy group catalogue. This simply reflects that the fraction of subhaloes increases as a function of the environment as well as the chances of finding high-mass (sub)haloes in dense environments. Figure 9. View largeDownload slide The dependence on r-band magnitude of the GLFs in nine bins of environmental density in 8 h−1 Mpc spheres for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. There is a remarkable agreement between observations and SHAM predictions, even when dividing between centrals and satellites. Figure 9. View largeDownload slide The dependence on r-band magnitude of the GLFs in nine bins of environmental density in 8 h−1 Mpc spheres for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. There is a remarkable agreement between observations and SHAM predictions, even when dividing between centrals and satellites. Figure 10. View largeDownload slide Similarly to Fig. 9 but for the GSMF for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. Although there is a good agreement between observations and SHAM predictions for all galaxies and centrals, there is some tension with the SHAM predictions of the satellite GSMF. Figure 10. View largeDownload slide Similarly to Fig. 9 but for the GSMF for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. Although there is a good agreement between observations and SHAM predictions for all galaxies and centrals, there is some tension with the SHAM predictions of the satellite GSMF. The agreement is particularly good for centrals. However, the satellite r-band GLF is in much better agreement with observations than the satellite GSMF: For the r-band GLF, we observe a marginal difference only, whereas SHAM predicts that there are more galaxies around the knee of the GSMF compared to what is observed. It is not clear why we should expect this difference, but a potential explanation could be that satellite galaxies are much more sensitive to their local environment and to the definition of the DDP population. To help build intuition, recall that SHAM assigns every halo in the simulation five magnitudes in the u, g, r, i, and z bands and stellar mass M*. Consider now that the relationship between r-band magnitude and all other galaxy properties is just monotonic and with zero scatter, as explained earlier. Thus, this oversimplification of much more complex relationships is affecting the measurements of environmental dependences in satellite galaxies, especially when these are projected in other observables. Possibly, when using a stellar mass-based DDP population, the problem will be inverted. In other words, we might observe a marginal difference between SHAM and the GSMFs but a larger difference between SHAM and the GLFs. Of course central galaxies are not exempt from also being affected, but given the good agreement with observations we conclude that the effect is only marginal. Another possible explanation is that the assumption of identical relations between centrals and satellites is more valid for the r-band luminosity than for the stellar mass. That is, the stellar mass of satellite galaxies perhaps varies more strongly with Vmax than the r-band luminosity does. A third possible explanation is that group finding algorithms are subject to errors. In a recent paper, Campbell et al. (2015) showed that there are two main sources of errors that could affect the comparison in Figs 9 and 10: (i) central/satellite designation, and (ii) group membership determination. In that paper, the authors showed that the Yang et al. (2007) group finder algorithm tends to misidentify central galaxies systematically with increasing group mass. In other words, satellites are sometimes mistakenly identified as centrals. Consequently, the GLFs and the GSMF for centrals and satellites will be affected towards the bright-end. Note, however, that Campbell et al. (2015) showed that for each satellite, which is misidentified as a central, approximately a central is misidentified as a satellite in the Yang et al. (2007) group finder. Thus, in the Yang et al. (2007) group finder, central/satellite designation is the main source of error rather than the group membership determination. Although this is a source of error that should be taken into account in our analysis, it is likely that this is not the main source of difference between observations and SHAM predictions. The reason is that there exists the above compensation effect in the identification of centrals and satellite galaxies, which could leave, perhaps, the GLFs and the GSMF of centrals and satellites with little or no changes. Finally, as we noted earlier, Fig. 7 shows that SHAM overpredicts the number of high-mass galaxies in high-mass bins. Figs 9 and 10 show that this excess of galaxies is due to central galaxies. We will discuss this in the light of the dependence of the galaxy–halo connection with environment in Section 5. 4.2.2 The relationship between colour and mean environmental density Fig. 11 shows the mean density as a function of the g − r colour separately for all galaxies, centrals, and satellites. The filled circles with error bars show the mean density measured from the Yang et al. (2012) galaxy group catalogue, whereas the shaded areas show the same but for the BolshoiP simulation. SHAM is unable to predict the correct correlation between mean density and galaxy colours for all and central galaxies. SHAM predicts that, statistically speaking, the large-scale mean environmental density varies little with the colours of central galaxies, except that the reddest galaxies on average lie in the densest environments. Actually, this is not surprising since we assumed that the ugriz bands and stellar mass are independent of environment when constructing our mock galaxy catalogue and the above is simply showing that one halo property does not fully determine the statistical properties of the galaxies. Other halo properties that vary with environment should instead be used in order to reproduce the correct trends with environment. Extensions to SHAM in which halo age is matched to galaxy age/colour at a fixed luminosity/stellar mass and halo mass (see e.g. Hearin & Watson 2013; Masaki, Lin & Yoshida 2013) are promising approaches that could help to better explain the trends with observations. None the less, SHAM predictions are in better agreement with the observed correlation of density with colour for satellite galaxies. Figure 11. View largeDownload slide Mean density in 8 h−1 Mpc spheres as a function of galaxy g − r colour, from the SDSS DR7 (shaded regions, representing the standard deviation) and the mean density predicted by SHAM based on the BolshoiP simulation, filled circles with error bars. We present the mean density for all, central, and satellite galaxies as indicated by the labels. SHAM fails to predict the correct relationship between mean density and galaxy colours for all galaxies and central galaxies. In contrast, the SHAM prediction for satellite galaxies is in better agreement with observations. Figure 11. View largeDownload slide Mean density in 8 h−1 Mpc spheres as a function of galaxy g − r colour, from the SDSS DR7 (shaded regions, representing the standard deviation) and the mean density predicted by SHAM based on the BolshoiP simulation, filled circles with error bars. We present the mean density for all, central, and satellite galaxies as indicated by the labels. SHAM fails to predict the correct relationship between mean density and galaxy colours for all galaxies and central galaxies. In contrast, the SHAM prediction for satellite galaxies is in better agreement with observations. 5 SUMMARY AND DISCUSSION SHAM makes the assumption that one (sub)halo property fully determines the statistical properties of their host galaxies. Therefore, SHAM implies that (i) the galaxy–halo connection is identical between haloes and subhaloes, and (ii) the dependence of galaxy properties on environmental density comes entirely from the corresponding dependence on density of this (sub)halo property. The halo property that this paper explores for SHAM is the quantity Vmax, which is defined in equation (14) as the maximum circular velocity for distinct haloes, whereas for subhaloes it is the peak maximum circular velocity Vpeak reached along the halo's main progenitor branch. This is the most robust halo and subhalo property for SHAM (see, e.g. Reddick et al. 2013; Campbell et al. 2017). The galaxy properties we studied are the ugriz GLFs as well as the GSMF, which we determined from the SDSS DR7. We compared these observations with SHAM predictions from a mock galaxy catalogue based on the BolshoiP simulation (Klypin et al. 2016; Rodríguez-Puebla et al. 2016a). SHAM assigns every halo in the BolshoiP simulation magnitudes in the five SDSS bands u, g, r, i, and z and also a stellar mass M*(Fig. 3 and Appendix A). We tested the assumptions behind SHAM by comparing the predicted and observed dependence of the ugriz GLFs as well as the GSMF on the environmental density from the SDSS DR7 Yang et al. (2012) galaxy group catalogue. The main results and conclusions are as follows: In general, the environmental dependence of the ugriz GLFs predicted by SHAM are in good agreement with the observed dependence from the SDSS DR7. This is especially true for r and infrared bands. Theoretically, the stellar mass is the galaxy property that is expected to depend more strongly on halo Vmax, whereas bluer bands also reflect recent effects of star formation. We show that the environmental dependence of the GSMF predicted by SHAM is in remarkable agreement with the observed dependence from the SDSS DR7, reinforcing the above conclusion. When dividing the galaxy population into centrals and satellites, SHAM predicts the correct dependence of the observed r-band GLF and GSMF for centrals and satellite galaxies from the Yang et al. (2012) group galaxy catalogue. Although SHAM predicts GLFs and the GSMF that are in remarkable agreement with observations even when the galaxy population is separated between centrals and satellites, SHAM does not predict the observed average relation between g − r colour and mean environmental density. This is especially true for central galaxies, whereas the correlation obtained for satellite galaxies is in better agreement with observations. Many previous authors have studied the correlation between galaxies and dark matter haloes with environment both theoretically and observationally (see, e.g. Avila-Reese et al. 2005; Baldry et al. 2006; Blanton & Berlind 2007; Maulbetsch et al. 2007; Tinker, Wetzel & Conroy 2011; Lacerna et al. 2014; Lee et al. 2017; Yang et al. 2017, and many more references cited therein). Although most of these authors have focused on understanding this correlation by studying the galaxy distribution as a function of colour, star formation or age and environment at a fixed M*, here we take a different approach and exploit the extreme simplicity of SHAM. First, there are no special galaxies in SHAM. Secondly, SHAM can be applied to any galaxy property distribution. Thus, in our framework, a halo and a subhalo with identical Vmax will host galaxies with identical luminosities and stellar masses, no matter what the halo's environmental density or position in the cosmic web is. Our results are consistent with previous findings that halo Vmax could be enough to determine the luminosities and stellar masses. However, we have also shown that SHAM is unable to reproduce the correct correlation between galaxy colour and the mean density δ8 on a scale of 8 h−1 Mpc. This result implies that additional halo properties that depend in some way on the halo environment (e.g. Lee et al. 2017) should be employed to correctly reproduce the relationship between δ8 and galaxy colour. Does the above discussion imply that the galaxy–halo connection should depend on environment? On one hand, from observations we have learned that the statistical properties of the galaxies such as colour and star formation change with environment in the direction that low-density environments are mostly populated by blue/star-forming galaxies, whereas dense environments are mostly populated with red/quenched galaxies (see for e.g. Hogg et al. 2003; Baldry et al. 2006; Tomczak et al. 2017). On the other hand, the shape of the luminosity–Vmax and the stellar mass–Vmax relations (Fig. 3) contains information about the process that regulated the star formation in galaxies. Therefore, it is not a bad idea to consider that the differences described in Fig. 7 are the result that the galaxy–halo connection could change with environment. For the sake of the simplicity, consider the GSMF of central galaxies derived in the case of zero scatter around the M* = M*(Vmax) relationship. Therefore, equation (13) can be rewritten to give the GSMF as   \begin{eqnarray} \phi _* (M_*) = \phi _V (V_{\rm max}(M_*)) \times \alpha _{\rm gal}, \end{eqnarray} (18)whereas the dependence with environment of the GSMF of central galaxies is given by   \begin{eqnarray} \phi _* (M_*|\delta _8) = \phi _V (V_{\rm max}(M_*)|\delta _8) \times \alpha _{\rm gal}, \end{eqnarray} (19)where αgal ≡ d log Vmax(M*)/d log M* is the logarithmic slope of the M* = M*(Vmax) relationship assumed to be independent of environment. Next, consider the simplest case in which ϕV(Vmax|δ8) is a double power law such that $$\phi _V (V_{\rm max}|\delta _8) \propto V_{\rm max}^{\beta (\delta _8)}$$ for $$V_{\rm max}\ll V_{\rm max}^*(\delta _8)$$ and $$\phi _V (V_{\rm max}|\delta _8) \propto V_{\rm max}^{\gamma (\delta _8)}$$ for $$V_{\rm max}\gg V_{\rm max}^*(\delta _8)$$, where $$V_{\rm max}^*(\delta _8)$$ is a characteristic velocity, and we have emphasized that the parameters β, γ, and $$V_{\rm max}^*$$ depend on the environment. In order to simplify the problem even further, consider that the M* = M*(Vmax) relationship is a power-law relation at low masses with logarithmic slope αgal, low, whereas at high masses it is also a power law with logarithmic slope of αgal, high. Based on the above, we can write the dependence with environment of the GSMF of central galaxies in the limiting cases   $$\phi _* (M_*|\delta _8) {\propto} \left\{ \begin{array}{l@{\quad}c} \alpha_{\rm gal,low}{\times} M_*^{\beta (\delta _8)/\alpha _{\rm gal,low}} & \mbox{if } V_{\rm max}\ll V_{\rm max}^*(\delta _8)\\ \alpha _{\rm gal,high}{\times} M_*^{\gamma (\delta _8)/\alpha _{\rm gal,high}} & \mbox{if } V_{\rm max}\gg V_{\rm max}^*(\delta _8) \end{array}\right.\!\! .$$ (20)Thus, if αgal, low and αgal, high are independent of environment, the resulting shape and dependence of ϕ*(M*|δ8) with environment can be simply understood as the dependence with environment of the slopes β and γ of the halo velocity function. By looking to the least (void-like) and highest (cluster-like) density environments from Fig. 10, upper left-hand and bottom right-hand panels, respectively, we can use the above model in order to understand how the galaxy–halo connection may depend on environment. The voids-like GSMF from Fig. 10 shows that SHAM tends to underpredict the number density of central galaxies both at the low- and high-mass ends. In other words, the slopes predicted by SHAM at the low- and high-mass ends are, respectively, too shallow and steep compared to observations. Inverting this would require, based on equation (20), to make the slopes αgal, low and αgal, high shallower and steeper, respectively, to what we derived from SHAM, see the right-hand panel of Fig. 3. This implies that in low-density environments at a fixed Vmax haloes had been more efficient in forming stars12 both for at the low- and high-mass end. In contrast, the high density GSMF from Fig. 10 shows that SHAM tends to overpredict the number density of central galaxies at the high-mass end. In this case, we invert the above trend by making the high-mass end slope αgal, high more shallow compared to what is currently derived from SHAM. This implies that the star formation efficiency has been suppressed in high-mass haloes residing in high-density environments with respect to the predictions of SHAM. The above limiting cases show that the galaxy–halo connection is expected to change with environment in the direction that haloes in low-density environment should be more efficient in transforming their gas into stars, whereas in high-density environments haloes have become more passive. This is indeed consistent with the colour/star formation trends that have been observed in large galaxy surveys. Of course, our discussion is an oversimplification, and in order to model exactly how the galaxy–halo connection depends on environment, we would need to use the dependence of the GSMF with environment as an extra observational constraint for the galaxy–halo connection. In a recent series of papers, Tinker et al. (2017b,c) and Tinker et al. (2017a) studied the galaxy–halo connection in the light of the relation between the star formation and environment at a fixed stellar and halo mass, obtaining similar conclusions to ours. That is, above-average galaxies with above average star formation rates and high halo accretion rates live in underdense environments, whereas the increase of the observed quenched fraction of galaxies from low-to-high density environments is consistent with the fact that halo formation has an impact on quenching the star formation at high masses and densities. See also Lee et al. (2017) for similar conclusions. Finally, we expect that at high redshift the assumptions from SHAM are likely to be closer to reality. The reason is that as the Universe ages, the cosmic web becomes more mature and the dependence of halo properties with environment become also stronger. As we showed here, while there are some differences with observations of local galaxies, those are small despite the extreme simplicity of the SHAM assumptions. Therefore, we expect that the galaxy–halo connection should depend less on environment at high redshifts, when environmental process have not played a significant role. Acknowledgements We thank Vladimir Avila-Reese, Peter Behroozi, Avishai Dekel, Sandra Faber, David Koo, Rachel Somerville, Risa Wechsler, and Chandrachani Ningombam for useful comments and discussions. AR-P thanks the UC-MEXUS-CONACYT programme for support at UCSC. JRP and CTL acknowledge support from grants HST-GO-12060.12-A and HST-AR-14578.001-A. The authors acknowledge the UC MEXUS-CONACYT Collaborative Research Grant CN-17-125. We thank the NASA Advanced Supercomputer program for allowing the Bolshoi-Planck simulation to be run on the Pleiades supercomputer at NASA Ames Research Center. We also thank the anonymous referee for a useful report, which helped improve the presentation of this paper. Part of the material of this paper was presented as the Bachelor of Science senior thesis of RD. Footnotes 1 More recent methods apply corrections that improve agreement with N-body simulations (Parkinson, Cole & Helly 2008; Somerville & Davé 2015). 2 Note that subhalo Vpeak is typically reached not at accretion, but rather when the distance of the progenitor halo from its eventual host halo is three to four times the host halo Rvir (Behroozi et al. 2014). 3 This galaxy group catalogue represents an updated version of Yang et al. (2007); see also Yang, Mo & van den Bosch (2009). 4 Note that we are using model magnitudes instead of Petrosian magnitudes. The main reason is the former ones tend to underestimate the true light from galaxies, particularly for high-mass galaxies (see, e.g. Bernardi et al. 2010 and Montero-Dorta & Prada 2009). 5 Note that this is not due to sky subtraction issues, as previous studies have found (see, e.g. Bernardi et al. 2013, 2016), since we are not including this correction in the galaxy magnitudes. Instead, it is most likely due to our use of model magnitudes instead of Petrosian ones (see also footnote 4). 6 http://hipacc.ucsc.edu/Bolshoi/MergerTrees.html 7 In particular, this paper uses five bands u, g, r, i, and z and a stellar mass M*, making a total of n = 6. 8 For an NFW halo, Vmax is reached at Rmax = 2.16Rs, where Rs is the NFW scaleradius Rs = Rvir/C and C is the NFW concentration (e.g. Klypin et al. 2001). Since C ∼ 10 for Milky Way mass haloes at z = 0, Rmax ∼ (1/5)Rvir. 9 When computing the projected two-point correlation function in the BolshoiP simulation, we integrate over the line of sight from rπ = 0 to rπ = 40 h−1 Mpc, similarly to observations. 10 Specifically, we found that Mr(z = 0.1) = 0.992 × Mr(z = 0) + 0.041 with a Pearson correlation coefficient of r = 0.998. 11 In fact, the minimum halo allowed by the observations is for haloes above Vmax ∼ 90 kms−1, corresponding to galaxies brighter than Mr − 5 log h ∼ −17, below this limit our mock catalogue should be considered as an extrapolation to observations. 12 Note that we are assuming that the zero-point of the M* = M*(Vmax) remains the same in both cases. This is not a bad approximation since SHAM is able to recover the normalization of the GSMF in all environments. REFERENCES Abbas U., Sheth R. 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C., Zhang Y., Han J., 2012, ApJ , 752, 41 CrossRef Search ADS   Yang X. et al.  , 2017, ApJ , 848, 60 Zehavi I. et al.  , 2005, ApJ , 630, 1 CrossRef Search ADS   Zehavi I. et al.  , 2011, ApJ , 736, 59 CrossRef Search ADS   Zehavi I., Contreras S., Padilla N., Smith N. J., Baugh C. M., Norberg P., 2018, ApJ , 853, 84 Zentner A. R., Hearin A., van den Bosch F. C., Lange J. U., Villarreal A., 2016, preprint (arXiv:1606.07817) Zheng Z. et al.  , 2005, ApJ , 633, 791 CrossRef Search ADS   Zheng Z., Coil A. L., Zehavi I., 2007, ApJ , 667, 760 CrossRef Search ADS   Zu Y., Mandelbaum R., Simet M., Rozo E., Rykoff E. S., 2017, MNRAS , 470, 551 CrossRef Search ADS   APPENDIX A: TABLES FOR THE GALAXY–HALO CONNECTION This section reports our luminosity–Vmax and stellar mass–Vmax relations from SHAM in Table A1. For galaxies, we utilized the best-fitting parameters for the global ugriz GLFs and stellar mass function reported in Table 1. In the case of distinct haloes, Vmax refers to the halo maximum circular velocity, whereas for satellites Vmax represents the highest maximum circular velocity reached along the subhalo's main progenitor branch Vpeak. Note that the validation limits for our determinations of luminosity–Vmax and stellar mass–Vmax relations are due to the range of the observed galaxy number density that corresponds to haloes above Vmax ∼ 90 km s−1 even if the BolshoiP simulations is complete up to Vmax ∼ 55 km s−1. Below this limit the mock catalogue should be considered as an extrapolation to observations. Table A1. Luminosity–Vmax relation and stellar mass–Vmax relation from SHAM. Vmax (km s−1)  log (M*/h− 2  M⊙)  Mu − 5log h  Mg − 5log h  Mr − 5log h  Mi − 5log h  Mz − 5log h  80.0000  7.960 57  − 14.5647  − 15.8541  − 16.2580  − 16.4285  − 16.5291  88.5047  8.264 52  − 15.3542  − 16.5009  − 16.9221  − 17.0909  − 17.2073  97.9136  8.599 33  − 15.9608  − 17.1241  − 17.5814  − 17.7615  − 17.9085  108.323  8.951 25  − 16.4419  − 17.6885  − 18.1921  − 18.3951  − 18.5809  119.838  9.286 07  − 16.8340  − 18.1794  − 18.7281  − 18.9579  − 19.1805  132.578  9.574 33  − 17.1613  − 18.5995  − 19.1867  − 19.4408  − 19.6931  146.672  9.810 92  − 17.4401  − 18.9593  − 19.5777  − 19.8514  − 20.1260  162.265  10.0045  − 17.6817  − 19.2700  − 19.9132  − 20.2022  − 20.4935  179.515  10.1654  − 17.8942  − 19.5413  − 20.2044  − 20.5053  − 20.8089  198.599  10.3019  − 18.0835  − 19.7809  − 20.4603  − 20.7703  − 21.0832  219.712  10.4197  − 18.2542  − 19.9950  − 20.6877  − 21.0049  − 21.3250  243.070  10.5232  − 18.4096  − 20.1882  − 20.8920  − 21.2150  − 21.5406  268.910  10.6154  − 18.5525  − 20.3641  − 21.0774  − 21.4049  − 21.7350  297.497  10.6985  − 18.6852  − 20.5256  − 21.2471  − 21.5784  − 21.9119  329.124  10.7742  − 18.8093  − 20.6752  − 21.4038  − 21.7381  − 22.0745  364.113  10.8440  − 18.9266  − 20.8147  − 21.5495  − 21.8864  − 22.2252  402.821  10.9088  − 19.0385  − 20.9457  − 21.6861  − 22.0252  − 22.3658  445.645  10.9696  − 19.1464  − 21.0697  − 21.8151  − 22.1560  − 22.4983  493.021  11.0270  − 19.2516  − 21.1879  − 21.9378  − 22.2803  − 22.6238  545.433  11.0818  − 19.3557  − 21.3013  − 22.0554  − 22.3992  − 22.7438  603.418  11.1344  − 19.4603  − 21.4109  − 22.1688  − 22.5139  − 22.8594  667.566  11.1854  − 19.5678  − 21.5177  − 22.2792  − 22.6253  − 22.9716  738.534  11.2352  − 19.6810  − 21.6226  − 22.3875  − 22.7345  − 23.0813  817.047  11.2843  − 19.8042  − 21.7263  − 22.4944  − 22.8422  − 23.1896  903.907  11.3331  − 19.9425  − 21.8298  − 22.6009  − 22.9495  − 23.2972  1000.00  11.3820  − 20.1006  − 21.9337  − 22.7078  − 23.0570  − 23.4050  1106.31  11.4313  − 20.2772  − 22.0389  − 22.8158  − 23.1656  − 23.5138  1223.92  11.4815  − 20.4630  − 22.1461  − 22.9259  − 23.2762  − 23.6245  1354.03  11.5328  − 20.6494  − 22.2561  − 23.0386  − 23.3894  − 23.7377  1497.98  11.5856  − 20.8337  − 22.3695  − 23.1548  − 23.5060  − 23.8542  1657.23  11.6403  − 21.0165  − 22.4871  − 23.2752  − 23.6267  − 23.9746  Vmax (km s−1)  log (M*/h− 2  M⊙)  Mu − 5log h  Mg − 5log h  Mr − 5log h  Mi − 5log h  Mz − 5log h  80.0000  7.960 57  − 14.5647  − 15.8541  − 16.2580  − 16.4285  − 16.5291  88.5047  8.264 52  − 15.3542  − 16.5009  − 16.9221  − 17.0909  − 17.2073  97.9136  8.599 33  − 15.9608  − 17.1241  − 17.5814  − 17.7615  − 17.9085  108.323  8.951 25  − 16.4419  − 17.6885  − 18.1921  − 18.3951  − 18.5809  119.838  9.286 07  − 16.8340  − 18.1794  − 18.7281  − 18.9579  − 19.1805  132.578  9.574 33  − 17.1613  − 18.5995  − 19.1867  − 19.4408  − 19.6931  146.672  9.810 92  − 17.4401  − 18.9593  − 19.5777  − 19.8514  − 20.1260  162.265  10.0045  − 17.6817  − 19.2700  − 19.9132  − 20.2022  − 20.4935  179.515  10.1654  − 17.8942  − 19.5413  − 20.2044  − 20.5053  − 20.8089  198.599  10.3019  − 18.0835  − 19.7809  − 20.4603  − 20.7703  − 21.0832  219.712  10.4197  − 18.2542  − 19.9950  − 20.6877  − 21.0049  − 21.3250  243.070  10.5232  − 18.4096  − 20.1882  − 20.8920  − 21.2150  − 21.5406  268.910  10.6154  − 18.5525  − 20.3641  − 21.0774  − 21.4049  − 21.7350  297.497  10.6985  − 18.6852  − 20.5256  − 21.2471  − 21.5784  − 21.9119  329.124  10.7742  − 18.8093  − 20.6752  − 21.4038  − 21.7381  − 22.0745  364.113  10.8440  − 18.9266  − 20.8147  − 21.5495  − 21.8864  − 22.2252  402.821  10.9088  − 19.0385  − 20.9457  − 21.6861  − 22.0252  − 22.3658  445.645  10.9696  − 19.1464  − 21.0697  − 21.8151  − 22.1560  − 22.4983  493.021  11.0270  − 19.2516  − 21.1879  − 21.9378  − 22.2803  − 22.6238  545.433  11.0818  − 19.3557  − 21.3013  − 22.0554  − 22.3992  − 22.7438  603.418  11.1344  − 19.4603  − 21.4109  − 22.1688  − 22.5139  − 22.8594  667.566  11.1854  − 19.5678  − 21.5177  − 22.2792  − 22.6253  − 22.9716  738.534  11.2352  − 19.6810  − 21.6226  − 22.3875  − 22.7345  − 23.0813  817.047  11.2843  − 19.8042  − 21.7263  − 22.4944  − 22.8422  − 23.1896  903.907  11.3331  − 19.9425  − 21.8298  − 22.6009  − 22.9495  − 23.2972  1000.00  11.3820  − 20.1006  − 21.9337  − 22.7078  − 23.0570  − 23.4050  1106.31  11.4313  − 20.2772  − 22.0389  − 22.8158  − 23.1656  − 23.5138  1223.92  11.4815  − 20.4630  − 22.1461  − 22.9259  − 23.2762  − 23.6245  1354.03  11.5328  − 20.6494  − 22.2561  − 23.0386  − 23.3894  − 23.7377  1497.98  11.5856  − 20.8337  − 22.3695  − 23.1548  − 23.5060  − 23.8542  1657.23  11.6403  − 21.0165  − 22.4871  − 23.2752  − 23.6267  − 23.9746  View Large © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

# Does the galaxy–halo connection vary with environment?

, Volume 476 (1) – May 1, 2018
18 pages

/lp/ou_press/does-the-galaxy-halo-connection-vary-with-environment-s7H4Ylx8v0
Publisher
The Royal Astronomical Society
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty283
Publisher site
See Article on Publisher Site

### Abstract

Abstract (Sub)halo abundance matching (SHAM) assumes that one (sub) halo property, such as mass Mvir or peak circular velocity Vpeak, determines properties of the galaxy hosted in each (sub) halo such as its luminosity or stellar mass. This assumption implies that the dependence of galaxy luminosity functions (GLFs) and the galaxy stellar mass function (GSMF) on environmental density is determined by the corresponding halo density dependence. In this paper, we test this by determining from a Sloan Digital Sky Survey sample the observed dependence with environmental density of the ugriz GLFs and GSMF for all galaxies, and for central and satellite galaxies separately. We then show that the SHAM predictions are in remarkable agreement with these observations, even when the galaxy population is divided between central and satellite galaxies. However, we show that SHAM fails to reproduce the correct dependence between environmental density and g − r colour for all galaxies and central galaxies, although it better reproduces the colour dependence on environmental density of satellite galaxies. methods: numerical, galaxies: haloes, large-scale structure of Universe 1 INTRODUCTION In the standard theory of galaxy formation in a Λ cold dark matter (ΛCDM) universe, galaxies form and evolve in massive dark matter haloes. The formation of dark matter haloes is through two main mechanisms: (1) the accretion of diffuse material, and (2) the incorporation of material when haloes merge. At the same time, galaxies evolve within these haloes, where multiple physical mechanisms regulate star formation and thus produce their observed properties. Naturally, this scenario predicts that galaxy properties are influenced by the formation and evolution of their host haloes (for a recent review see Somerville & Davé 2015). What halo properties matter for galaxy formation? The simplest assumption that galaxy formation models make is that a dark matter halo property such as mass Mvir or maximum circular velocity Vmaxfully determines the statistical properties of their host galaxies. This assumption was supported by early studies that showed that the halo properties strongly correlate with the larger scale environment mainly due to changes in halo mass (e.g. Lemson & Kauffmann 1999). Halo evolution and corresponding evolution of galaxy properties can be predicted from Extended Press–Schechter analytical models based on Monte Carlo merger trees (Cole 1991; White & Frenk 1991; Kauffmann & White 1993; Somerville & Kolatt 1999).1 Such models assume that the galaxy assembly time and merger history are independent of the large-scale environment (for a recent discussion, see, e.g. Jiang & van den Bosch 2014). However, it is known that dark matter halo properties do depend on other aspects beyond Mvir, a phenomenon known as halo assembly bias. Wechsler et al. (2006, see also Gao, Springel & White 2005; Gao & White 2007; Faltenbacher & White 2010; Lacerna & Padilla 2011) observed an assembly bias effect in the clustering of dark matter haloes: They showed that for haloes with Mvir ≲ 1013 M⊙, early forming haloes are more clustered than late forming haloes, whereas for more massive haloes they found the opposite. Other effects of environmental density on dark matter haloes are known, for example that halo mass accretion rates and spin can be significantly reduced in dense environments due to tidal effects, and that median halo spin is significantly reduced in low-density regions due to the lack of tidal forces there (Lee et al. 2017). Indeed, there are some recent efforts to study assembly bias and the effect of the environment on the galaxy–halo connection in the context of galaxy clustering (Vakili & Hahn 2016; Zentner et al. 2016; Lehmann et al. 2017; Zehavi et al. 2018) and weak lensing (Zu et al. 2017). Despite such environmental effects on halo properties, it may still be true that some galaxy properties can be correctly predicted from just halo Mvir or Vmax. The assumption that dark matter halo mass fully determines the statistical properties of the galaxies that they host has also influenced the development of empirical approaches for connecting galaxies to their host halo: the so-called halo occupation distribution (HOD) models (Berlind & Weinberg 2002) and the closely related conditional stellar mass/luminosity function model (Yang, Mo & van den Bosch 2003; Cooray 2006). HOD models assume that the distribution of galaxies depends on halo mass only (Mo et al. 2004; Abbas & Sheth 2006). Yet, the HOD assumption has been successfully applied to explain the clustering properties of galaxies not only as a function of their mass/luminosity only but also as a function of galaxy colours (Jing, Mo & Börner 1998; Berlind & Weinberg 2002; Zehavi et al. 2005; Zheng et al. 2005; Tinker et al. 2013; Rodríguez-Puebla et al. 2015). The (sub)halo abundance matching (SHAM) approach takes the above assumption to the next level by assuming that not only does a halo property, such as mass Mvir or maximum circular velocity Vmax, determine the luminosity or stellar mass of central galaxies, but also that there is a simple relation between subhalo properties and those of the satellite galaxies they host. Specifically, we will assume that subhalo peak circular velocity Vpeak fully determines the corresponding properties of their hosted satellite galaxies (Reddick et al. 2013). For simplicity, in the remainder of this paper, when we write Vmax we will mean the maximum circular velocity for distinct haloes, and the peak circular velocity of subhaloes. SHAM assigns by rank a halo property, such as Vmax, to that of a galaxy, such as luminosity or stellar mass, by matching their corresponding cumulative number densities (Kravtsov et al. 2004; Vale & Ostriker 2004; Conroy, Wechsler & Kravtsov 2006; Conroy & Wechsler 2009; Behroozi, Conroy & Wechsler 2010; Behroozi, Wechsler & Conroy 2013; Moster, Naab & White 2013, 2017; Rodríguez-Puebla et al. 2017). Although central galaxies are continuously growing by in situ star formation and/or galaxy mergers, satellite galaxies are subject to environmental effects such as tidal and ram-pressure stripping, in addition to interactions with other galaxies in the halo and with the halo itself. Therefore, central and satellite galaxies are expected to differ in the relationship between their host haloes and subhaloes (see e.g. Neistein et al. 2011; Rodríguez-Puebla, Drory & Avila-Reese 2012; Yang et al. 2012; Rodríguez-Puebla, Avila-Reese & Drory 2013). Nevertheless, SHAM assumes that (sub)halo Vmax fully determines the statistical properties of the galaxies. Thus, SHAM galaxy properties evolve identically for central and satellite galaxies, except that satellite galaxy properties are fixed after Vpeak is reached.2 SHAM also implies that galaxy properties are independent of local as well as large-scale environmental densities. Thus, two haloes with identical Vmax but in different environments will host identical galaxies. Despite the extreme simplicity of this approach, the two point correlation functions predicted by SHAM are in excellent agreement with observations (Reddick et al. 2013; Campbell et al. 2017, and Figs 4 and 5), showing that on average galaxy clustering depends on halo Vpeak. It is worth mentioning that neither HOD nor SHAM identify clearly which galaxy property, luminosity in various wavebands or stellar mass, depends more strongly on halo mass – although, theoretically, stellar mass growth is expected to be more closely related to halo mass accretion (Rodríguez-Puebla et al. 2016b). Our main goal in this paper is to determine whether the assumption that one (sub)halo property, in our case halo Vmax and subhalo Vpeak, fully determines some statistical properties of the hosted galaxies. This might be true even though the galaxy–halo relation is expected to depend on environment because the properties of the galaxies might reflect halo properties that depend on some environmental factor (see e.g. Lee et al. 2017). We will test this assumption by determining from a Sloan Digital Sky Survey (SDSS) sample the dependence on environmental density of the ugriz galaxy luminosity functions (GLFs) as well as the galaxy stellar mass function (GSMF) for all galaxies, and separately for central and satellite galaxies, and comparing these observational results with SHAM predictions. We will also investigate which of these galaxy properties is better predicted by SHAM. If a galaxy–halo connection that is independent of environment successfully reproduces observations in the nearby Universe, then we can conclude that the relation may be appropriate to use for acquiring other information about galaxies. It also suggests that this assumption be tested at larger redshifts. To the extent that the galaxy–halo connection is independent of density or other environmental factors, it is a great simplification. This paper is organized as follows. In Section 2, we describe the galaxy sample that we utilize for the determination of the environmental dependence of the ugriz GLFs and GSMF. Section 3 describes our mock galaxy catalogue based on the Bolshoi–Planck (BolshoiP) cosmological simulation. Here, we show how SHAM assigns to every halo in the simulation five band magnitudes, ugriz, and a stellar mass. In Section 4, we present the dependence with environment of ugriz GLFs and GSMF both for observations and for SHAM applied to the BolshoiP simulation. We show that the SHAM predictions are in remarkable agreement with observations even when the galaxy population is divided between central and satellite galaxies. However, we also find that SHAM fails to reproduce the correct dependence between environmental density and g − r colour. Finally, Section 5 summarizes our results and discusses our findings. We adopt a Chabrier (2003) initial mass function (IMF) and the Planck cosmological parameters used in the BolshoiP simulation: $$\Omega _\Lambda =0.693, \Omega _{\rm M}=0.307$$, and h = 0.678. 2 OBSERVATIONAL DATA In this section, we describe the galaxy sample that we utilize for the determination of the galaxy distribution. We use the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure for the determination of the ugriz GLFs and the GSMF and report their corresponding best-fitting models. We show that a function composed of a single Schechter function plus another Schechter function with a sub-exponential decreasing slope is an accurate model for the ugriz GLFs as well as the GSMF. Finally, we describe the methodology for the determination of the environmental density dependence of the ugriz GLFs and GSMF. 2.1 The sample of galaxies In this paper, we utilize the New York Value Added Galaxy Catalog (NYU-VAGC; Blanton et al. 2005b) based on the SDSS DR7. Specifically, we use the large galaxy group catalogue from Yang et al. (2012)3 with ∼6 × 105 spectroscopic galaxies over a solid angle of 7748 deg2 comprising the redshift range 0.01 < z < 0.2 with an apparent magnitude limit of mlim, r = 17.77. However, the sample we use in this paper is 0.03 < z < 0.11 (see Fig. 2). The Yang et al. (2012) catalogue is a large halo-based galaxy group catalogue that assigns group membership by assuming that the distribution of galaxies in phase space follows that of dark matter particles. Mock galaxy catalogues demonstrate that ∼ 80 per cent of all their groups have a completeness larger than 80 per cent, whereas halo groups with mass Mvir > 1012.5 h− 1  M⊙ have a completeness > 95 per cent; for more details, see Yang et al. (2007). Here, we define central galaxies as the most massive galaxy in their group in terms of stellar mass; the remaining galaxies will be regarded as satellites. The definition of groups in the Yang et al. (2012) catalogue is very broad and includes systems that are often explored individually in the literature, such as clusters, compact groups, fossil groups, rich groups, etc. That is, this galaxy group catalogue is not biased to a specific type of group. Instead, this galaxy group catalogue is diverse and, more importantly, closely related to the general idea of galaxy group that naturally emerges in the ΛCDM paradigm: That haloes host a certain number of galaxies inside their virial radius. Therefore, the Yang et al. (2012) galaxy group catalogue is ideal for comparing to predictions based on N-body cosmological simulations. For the purpose of exploring whether certain galaxy properties are fully determined by the (sub)halo in which they reside, this galaxy group catalogue will help us to draw conclusions not only at the level of the global GLFs and GSMF but also at the level of centrals and satellites. Thus, the Yang et al. (2012) galaxy group catalogue is an ideal tool to explore at a deeper level the simple assumptions in the SHAM approach. In order to allow for a meaningful comparison between galaxies at different redshifts, we utilize model magnitudes4 that are K+E-corrected at the rest-frame z = 0. These corrections account for the broad-band shift with respect to the rest-frame broad-band and for the luminosity evolution. For the K-corrections, we utilize the input values tabulated in NYU-VAGC (Blanton & Roweis 2007, corresponding to the kcorrect software version v$$4\_1\_4$$), whereas for the evolution term we assume a model given by   \begin{eqnarray} E_j(z) = -Q_X \times z, \end{eqnarray} (1)where the subscript X refers to the u, g, r, i, and z bands and their values are (Qu, Qg, Qr, Qi, Qz) = (4.22, 1.3, 1.1, 1.09, 0.76). Here, we ignore potential dependences between QX and colours (but see Loveday et al. 2012, for a discussion) and luminosity, and use global values only. Although this is a crude approximation for accounting for the evolution of the galaxies, it is accurate enough for our purposes since we are not dividing the galaxy distribution into subpopulations as a function of star formation rate and/or colour. We estimated the value of each QX by determining first the X-band GLF when QX = 0 at four redshift intervals: [0.01, 0.05], [0.01, 0.1], [0.01, 0.15], and [0.01, 0.2]. When assuming QX = 0, the GLFs are normally shifted towards higher luminosities, with this shift increasing with redshift. In other words, when ignoring the evolution correction, the GLF will result in an overestimation of the number density at higher luminosities and high redshifts. Thus, in order to account for this shift, we find the best value for QX that leaves the GLFs invariant at the four redshift intervals mentioned above. We note that our derived values are similar to those reported in Blanton et al. (2003). For the u band, we used the value reported in Blanton et al. (2003), but we have checked that the value of Qu = 4.22 also leaves the GLF invariant at the four redshifts bins mentioned above. For stellar masses, we utilize the MPA-JHU DR7 data base derived from photometry–spectral energy distribution fittings, explained in detail in Kauffmann et al. (2003). All stellar masses have been normalized to a Chabrier (2003) IMF and to the cosmology used for this paper. 2.2 The global ugriz luminosity functions and stellar mass function Next, we describe the procedure we utilize for determining the global GLFs and the GSMF. Here, we choose the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure for the determination of the ugriz GLFs and the GSMF. Specifically, we determine the galaxy luminosity and stellar mass distributions as   \begin{eqnarray} \phi _X(M_X) = \frac{1}{\Delta M_X}\sum _{i=1}^N\frac{\omega _X(M_X\pm \Delta M_X / 2)}{\mathcal {V}_{{\rm max},i}}, \end{eqnarray} (2)where MX refers to Mu, Mg, Mr, Mi, Mz, and log M*, ωi is the correction weight completeness factor in the NYU-VAGC for galaxies within the interval MX ± ΔMX/2, and   \begin{eqnarray} \mathcal {V}_{{\rm max},i}= \int _{\Omega }\int ^{z_u}_{z_l}\frac{\,\text{d}^2V_c}{\text{d}z \, {d}\Omega } \text{d}z\text{d}\Omega . \end{eqnarray} (3)We denote the solid angle of the SDSS DR7 with Ω while Vc refers to the comoving volume enclosed within the redshift interval [zl, zu]. The redshift limits are defined as zl = max(0.01, zmin) and zu = min(zmax, 0.2), where zmin and zmax are, respectively, the minimum and maximum at which each galaxy can be observed in the SDSS DR7 sample. For the completeness limits, we use the limiting apparent magnitudes in the r band of r = 14 and r = 17.77. The filled black circles with error bars in Fig. 1 present our determination of the global SDSS DR7 ugriz GLFs. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed line), who used a sample of low-redshift galaxies (<150 h−1Mpc) from the SDSS DR2 and corrected due to low surface brightness selection effects. Additionally, we compare to Hill et al. (2010), who combined data from the Millennium Galaxy Catalog (MGC), the SDSS DR5 and the UKIRT Infrared Deep Sky Survey Large Area Survey (UKIDSS) for galaxies with z < 0.1, dotted lines; and to Driver et al. (2012), who utilized the Galaxy And Mass Assembly (GAMA) survey for the redshift interval 0.013 < z < 0.1 to derive the ugriz GLFs (short dashed lines). All the GLFs in Fig. 1 are at the rest-frame z = 0. In general, we observe good agreement with previous studies; in a more detailed examination, however, we note some differences that are worthwhile to clarify. Figure 1. View largeDownload slide The global ugriz GLFs. Our derived ugriz GLFs and GSMF are shown with the black circles with error bars. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed lines) based on the SDSS DR2; Hill et al. (2010, dotted lines) by combining the MGC, SDSS DR5, and the UKIDSS surveys; and Driver et al. (2012, short dashed lines) based on the GAMA survey. As for the stellar masses, we compare with the GSMF from Baldry et al. (2012) and Wright et al. (2017), black long and short dashed lines, respectively. Figure 1. View largeDownload slide The global ugriz GLFs. Our derived ugriz GLFs and GSMF are shown with the black circles with error bars. For comparison, we reproduce the ugriz GLFs from Blanton et al. (2005a, black long dashed lines) based on the SDSS DR2; Hill et al. (2010, dotted lines) by combining the MGC, SDSS DR5, and the UKIDSS surveys; and Driver et al. (2012, short dashed lines) based on the GAMA survey. As for the stellar masses, we compare with the GSMF from Baldry et al. (2012) and Wright et al. (2017), black long and short dashed lines, respectively. Consider the u-band GLFs from Fig. 1 and note that there is an apparent tension with previous studies. At the high-luminosity end, our inferred u-band GLF decreases much faster than the above-mentioned studies. This is especially true when comparing with the Hill et al. (2010) and Driver et al. (2012) GLFs. This could be partly due to the differences between the Kron magnitudes used by Hill et al. (2010) and Driver et al. (2012) and the model magnitudes used in this paper. But we believe that most of the difference is due to the differences in the E-corrections, reflecting that our model evolution is more extreme than that of Hill et al. (2010) and Driver et al. (2012). This can be easily understood by noting that the high-luminosity end of the GLF is very sensitive to E-corrections. The reason is that brighter galaxies are expected to be observed more often at larger redshifts than fainter galaxies; thus, equation (1) will result in a small correction for lower luminosity galaxies (low redshift) but a larger correction for higher luminosity galaxies (high redshifts). Indeed, Driver et al. (2012), who did not determine corrections by evolution, derived a u-band GLF that predicts the largest abundance of high luminosity galaxies. On the other hand, the evolution model introduced by Hill et al. (2010) is shallower than ours, which results in a GLF between our determination and the Driver et al. (2012) u-band GLF. This could explain the apparent tension between the different studies. Although the effects of evolution are significant in the u band, they are smaller in the longer wavebands. Ideally, estimates of the evolution should be more physically motivated by galaxy formation models, but empirical measurements are more accessible and faster to determine; however, when making comparisons, one should keep in mind that empirical estimates are by no means definitive. Some previous studies have concluded that a single Schechter function is consistent with observations (see, e.g. Blanton et al. 2003, and recently Driver et al. 2012). However, other studies have found that a double Schechter function is a more accurate description of the GLFs (Blanton et al. 2005a). Additionally, recent studies have found shallower slopes at the high-luminosity end instead of an exponential decreasing slope in the GLFs5 (see e.g. Bernardi et al. 2010). In this paper, we choose to use GLFs that are described by a function composed of a single Schechter function plus another Schechter function with a subexponential decreasing slope for the ugriz bands given by   \begin{eqnarray} \phi (M)&=&\frac{\ln 10}{2.5}\phi _1^*10^{0.4(M^*_1-M)\left(1+\alpha _1\right)}\exp \left(-10^{0.4\left(M^*_1-M\right)}\right) \nonumber \\ &&+\,\frac{\ln 10}{2.5}\phi _2^* 10^{0.4(M^*_2-M)\left(1+\alpha _2\right)}\exp \left(-10^{0.4\left(M^*_2-M\right)\beta }\right). \end{eqnarray} (4)The units of the GLFs are h3 Mpc−3 mag−1, whereas the input magnitudes have units of mag−5 log h. The parameters for the ugriz bands are given in Table 1. Note that for simplicity, we assume that α1 = α, α2 = 1 + α, and $$M^*_1 = M^*_2 = M^*$$. These assumptions reduce the number of free parameters to five. The corresponding best-fitting models are shown in Fig. 1 with the solid black lines. The filled circles with error bars in Fig. 1 present our determinations for the global SDSS DR7 GLFs. Table 1. Best-fitting parameters for the GLFs and the GSMF. Band  α  M* − 5 log h  $$\log \phi ^*_1$$ (h3 Mpc−3 mag−1)  $$\log \phi _2^*$$ (h3 Mpc−3 mag−1)  β  GLF    u  − 0.939 ± 0.005  − 17.758 ± 0.016  − 1.530 ± 0.002  − 3.692 ± 0.044  0.721 ± 0.008  g  − 1.797 ± 0.044  − 19.407 ± 0.068  − 2.764 ± 0.105  − 1.674 ± 0.013  0.821 ± 0.014  r  − 1.810 ± 0.036  − 20.184 ± 0.062  − 2.889 ± 0.094  − 1.733 ± 0.013  0.813 ± 0.013  i  − 1.794 ± 0.031  − 20.546 ± 0.053  − 2.896 ± 0.077  − 1.768 ± 0.011  0.815 ± 0.011  z  − 1.816 ± 0.028  − 20.962 ± 0.051  − 3.038 ± 0.076  − 1.806 ± 0.012  0.827 ± 0.011  GSMF      α  $$\mathcal {M}^*$$ (h− 2  M⊙)  $$\log \phi ^*_1$$ (h3 Mpc−3 dex−1)  $$\log \phi _2^*$$ (h3 Mpc−3 dex−1)  β    − 1.664 ± 0.033  10.199 ± 0.0303  − 3.041 ± 0.082  − 1.885 ± 0.010  0.708 ± 0.012  Band  α  M* − 5 log h  $$\log \phi ^*_1$$ (h3 Mpc−3 mag−1)  $$\log \phi _2^*$$ (h3 Mpc−3 mag−1)  β  GLF    u  − 0.939 ± 0.005  − 17.758 ± 0.016  − 1.530 ± 0.002  − 3.692 ± 0.044  0.721 ± 0.008  g  − 1.797 ± 0.044  − 19.407 ± 0.068  − 2.764 ± 0.105  − 1.674 ± 0.013  0.821 ± 0.014  r  − 1.810 ± 0.036  − 20.184 ± 0.062  − 2.889 ± 0.094  − 1.733 ± 0.013  0.813 ± 0.013  i  − 1.794 ± 0.031  − 20.546 ± 0.053  − 2.896 ± 0.077  − 1.768 ± 0.011  0.815 ± 0.011  z  − 1.816 ± 0.028  − 20.962 ± 0.051  − 3.038 ± 0.076  − 1.806 ± 0.012  0.827 ± 0.011  GSMF      α  $$\mathcal {M}^*$$ (h− 2  M⊙)  $$\log \phi ^*_1$$ (h3 Mpc−3 dex−1)  $$\log \phi _2^*$$ (h3 Mpc−3 dex−1)  β    − 1.664 ± 0.033  10.199 ± 0.0303  − 3.041 ± 0.082  − 1.885 ± 0.010  0.708 ± 0.012  View Large In the case of the GSMF, we compare our results with Baldry et al. (2012) and Wright et al. (2017) plotted with the black long and short dashed lines, respectively. Both analyses used the GAMA survey to determine the local GSMF. Recall that our stellar masses were obtained from the MPA-JHU DR7 data base. As can be seen in the figure, our determination is consistent with these previous results. We again choose to use a function composed of a single Schechter function plus another Schechter function with a subexponential decreasing slope for the GSMF given by   \begin{eqnarray} \phi _*(M_*) &=&\phi _1^*{\ln 10}\left(\frac{M_*}{\mathcal {M}_1^*}\right)^{1+\alpha _1}\exp \left(-\frac{M_*}{\mathcal {M}_1^*}\right) \nonumber \\ &&+\,\phi _2^*{\ln 10}\left(\frac{M_*}{\mathcal {M}_2^*}\right)^{1+\alpha _2}\exp \left[-\left(\frac{M_*}{\mathcal {M}_2^*}\right)^\beta \right]. \end{eqnarray} (5)The units for the GSMF are h3 Mpc−3 dex−1, whereas the input stellar masses are in units of h− 2  M⊙. Again, for simplicity, we assume that α1 = α, α2 = 1 + α, and $$\mathcal {M}_1^* = \mathcal {M}_2^* = \mathcal {M}^*$$; again, this assumption reduces the number of free parameters to five. We report the best-fitting value parameters in Table 1 and the corresponding best-fitting model is presented with the solid black line in Fig. 1. As we will describe in Section 3, we use the ugriz GLFs and GSMF as inputs for our mock galaxy catalogue. 2.3 Measurements of the observed ugriz GLFs and GSMF as a function of environment Once we determined the global ugriz GLFs and the GSMF, the next step in our program is to determine the observed dependence of the ugriz GLFs and GSMF with environmental density. 2.3.1 Density-defining population The SDSS DR7 limiting magnitude in the r band is 17.77. Thus, in order to determine the local overdensity of each SDSS DR7 galaxy, we need to first construct a volume-limited density-defining population (DDP, Croton et al. 2005; Baldry et al. 2006). A volume-limited sample can be constructed by defining the minimum and maximum redshifts at which galaxies within some interval magnitude are detected in the survey. Following the McNaught-Roberts et al. (2014) GAMA paper, we define our volume-limited DDP sample of galaxies in the absolute magnitude range −21.8 < Mr − 5 log h < −20.1. A valid question is whether the definition utilized for the volume-limited DDP sample could lead to different results. This question has been studied in McNaught-Roberts et al. (2014); the authors conclude that the precise definition for the volume-limited DDP sample does not significantly affect the shape of GLFs. None the less, our defined volume-limited DDP sample restricts the SDSS magnitude-limited survey into the redshift range 0.03 ≤ z ≤ 0.11. Fig. 2 shows the absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The solid box presents the galaxy population enclosed in our volume-limited DDP sample, whereas the dashed lines show our magnitude-limited survey. Figure 2. View largeDownload slide Absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The blue solid box shows our volume-limited DDP sample. Note that our DDP sample restricts to study environments for galaxies in the range 0.03 ≤ z ≤ 0.11, as shown by the dashed lines. Figure 2. View largeDownload slide Absolute magnitude in the r band as a function of redshift for our magnitude-limited galaxy sample. The blue solid box shows our volume-limited DDP sample. Note that our DDP sample restricts to study environments for galaxies in the range 0.03 ≤ z ≤ 0.11, as shown by the dashed lines. 2.3.2 Projected distribution on the sky of the galaxy sample The irregular limits of the projected distribution on the sky of the SDSS-DR7 galaxies could lead to a potential bias in our overdensity measurements; they will artificially increase the frequency of low-density regions, and, ideally, overdensity measurements should be carried out over more continuous regions. Following Varela et al. (2012) and Cebrián & Trujillo (2014), we reduce this source of potential bias by restricting our galaxy sample to a projected area based on the following cuts:   $${\rm Dec.} > \left\{ \begin{array}{l@{\qquad}c} 0 & \mbox{{s}outhern limit}\\ -2.555556\times ({\rm RA} - 131^{\circ }) & \mbox{{w}estern limit}\\ -1.70909\times ({\rm RA} - 235^{\circ }) & \mbox{{e}astern limit}\\ \arcsin \left(\frac{x}{\sqrt{1- x^2}}\right) & \mbox{{n}orthern limit} \end{array}, \right.$$ (6)where x = 0.932 32sin (RA − 95$$_{.}^{\circ}$$9). This region is plotted in fig. 1 of Cebrián & Trujillo (2014). 2.3.3 Overdensity measurements In summary, our final magnitude-limited galaxy sample consists of galaxies in the redshift range 0.03 ≤ z ≤ 0.11 and galaxies within the projected area given by equation (6), whereas our volume-limited DDP sample comprises galaxies with absolute magnitude satisfying −21.8 < Mr − 5 log h < −20.1. Based on the above specifications, we are now in a position to determine the local overdensity of each SDSS DR7 galaxy in our magnitude-limited galaxy sample. Overdensities are estimated by counting the number of DDP galaxy neighbours, Nn, around our magnitude-limited galaxy sample in spheres of r8 = 8 h−1 Mpc radius. Although there exist various methods to measure galaxy environments, Muldrew et al. (2012) showed that aperture-based methods are more robust in identifying the dependence of halo mass on environment, in contrast to nearest neighbours based methods that are largely independent of halo mass. In addition, aperture-based methods are easier to interpret. For these reasons, the aperture-based method is ideal to probe galaxy environments when testing the assumptions behind the SHAM approach. The local density is simply defined as   \begin{eqnarray} \rho _8 = \frac{N_n}{4/3\pi r_8^3}. \end{eqnarray} (7)We then compare the above number to the expected number density of DDP galaxies by using the global r-band luminosity function determined above in Section 2.2; $$\bar{\rho } = 6.094 \times 10^{-3}\,h^3$$ Mpc−3. Finally, the local density contrast for each galaxy is determined as   \begin{eqnarray} \delta _8 = \frac{\rho _8 - \bar{\rho }}{\bar{\rho }}. \end{eqnarray} (8)The effect of changing the aperture radius has been discussed in Croton et al. (2005). Although the authors noted that using smaller spheres tends to sample underdense regions differently, they found that their conclusions remain robust due to the change of apertures. Nevertheless, smaller scale spheres are more susceptible to be affected by redshift space distortions. Following Croton et al. (2005), we opt to use spheres of r8 = 8 h−1 Mpc radius as the best probe of both underdense and overdense regions. Finally, note that our main goal is to understand whether halo Vmax fully determines galaxy properties as predicted by SHAM, not to study the physical causes for the observed galaxy distribution with environment. Therefore, as long as we treat our mock galaxy sample, to be described in Section 3, in the same way that we treat observations, understanding the impact of changing apertures in the observed galaxy distribution is beyond the scope of this paper. 2.3.4 Measurements of the observed ugriz GLFs and the GSMF as a function of environmental density Once the local density contrast for each galaxy in the SDSS DR7 is determined, we estimate the dependence of the ugriz GLFs and the GSMF with environmental density. As in Section 2.2, we use the standard 1/$$\mathcal {V}_{\rm max}$$ weighting procedure. Unfortunately, the 1/$$\mathcal {V}_{\rm max}$$ method does not provide the effective volume covered by the overdensity bin in which the GLFs and the GSMF have been estimated, and therefore one needs to slightly modify the 1/$$\mathcal {V}_{\rm max}$$ estimator. In this subsection, we describe how we estimate the effective volume. We determine the fraction of effective volume by counting the number of DDP galaxy neighbours in a catalogue of random points with the same solid angle and redshift distribution as our final magnitude-limited sample. Observe that we utilize the real position of the DDP galaxy sample already defined. We again utilized spheres of radius r8 = 8 h−1 Mpc and create a random catalogue consisting of Nr ∼ 2 × 106 of points. The local density contrast for each random point is determined as in equation (8):   \begin{eqnarray} \delta _{r_8} = \frac{\rho _{r_8} - \bar{\rho }}{\bar{\rho }}, \end{eqnarray} (9)where $$\rho _{r_8}$$ is the local density around random points. We estimate the fraction of effective volume by a given overdensity bin as   \begin{eqnarray} f(\delta _8) = \frac{1}{N_r}\sum _{i=1}^{N_r} \Theta (\delta _{{r_8},i}). \end{eqnarray} (10)Here, Θ is a function that selects random points in the overdensity range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$:   $$\Theta (\delta _{{r_8},i}) = \left\{ \begin{array}{l@{\qquad}c} 1 & \mbox{if } \delta _{{r_8},i} \in [\delta _{r_8} - \Delta \delta _{r_8}/2, \delta _{r_8} + \Delta \delta _{r_8}/2)\\ 0 & \mbox{otherwise}\end{array}\right. .$$ (11)Table 2 lists the fraction of effective volume for the range of overdensities considered in this paper and calculated as described in this paper. We estimate errors by computing the standard deviation of the fraction of effective volume in 16 redshift bins equally spaced. We note that the number of sampled points gives errors that are less than ∼ 3 per cent and for most of the bins less than ∼ 1 per cent (see the last column of Table 2). Therefore, we ignore any potential source of error from our determination of the fraction of effective volume into the ugriz GLFs and the GSMFs. Table 2. Fraction of effective volume covered by the overdensity bins considered for our analysis in the SDSS DR7. Also shown is the fractional error due to the number of random points sampled. δmin, 8  δmax, 8  f(δ8) ± δf(δ8)  100 per cent × δf(δ8)/f(δ8)  − 1  − 0.75  0.1963 ± 0.0014  0.713  − 0.75  − 0.55  0.1094 ± 0.0010  0.914  − 0.55  − 0.40  0.0974 ± 0.0009  0.924  − 0.40  0.00  0.2156 ± 0.0014  0.650  0.00  0.70  0.1800 ± 0.0012  0.667  0.70  1.60  0.1040 ± 0.0009  0.866  1.60  2.90  0.0621 ± 0.0007  1.130  2.90  4  0.0197 ± 0.0004  2.030  4.00  ∞  0.0153 ± 0.0004  2.614  δmin, 8  δmax, 8  f(δ8) ± δf(δ8)  100 per cent × δf(δ8)/f(δ8)  − 1  − 0.75  0.1963 ± 0.0014  0.713  − 0.75  − 0.55  0.1094 ± 0.0010  0.914  − 0.55  − 0.40  0.0974 ± 0.0009  0.924  − 0.40  0.00  0.2156 ± 0.0014  0.650  0.00  0.70  0.1800 ± 0.0012  0.667  0.70  1.60  0.1040 ± 0.0009  0.866  1.60  2.90  0.0621 ± 0.0007  1.130  2.90  4  0.0197 ± 0.0004  2.030  4.00  ∞  0.0153 ± 0.0004  2.614  View Large Finally, we modify the 1/$$\mathcal {V}_{\rm max}$$ weighting estimator to account for the effective volume by the overdensity bin as   \begin{eqnarray} \phi _X(M_X|\delta _8) = \sum _{i=1}^N\frac{\omega _i(M_X\pm \Delta M_X / 2|\delta _{r_8} \pm \Delta \delta _{r_8}/2)}{f(\delta _8)\times \Delta M_X\times \mathcal {V}_{{\rm max},i}}, \end{eqnarray} (12)again, MX refers to Mu, Mg, Mr, Mi, Mz, and log M*. Here, ωi refers to the correction weight completeness factor for galaxies within the interval MX ± ΔMX/2, given that their overdensity is in the range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$. 3 THE GALAXY–HALO CONNECTION The main goal of this paper is to study whether one halo property, in this case Vmax, fully determines the observed dependence with environmental density of the ugriz GLFs and the GSMF. Confirming this would significantly improve our understanding of the galaxy–halo connection. In this section, we describe how we constructed a mock galaxy catalogue in the cosmological BolshoiP N-body simulation via SHAM. 3.1 The Bolshoi–Planck simulation To study the environmental dependence of the galaxy distribution predicted by SHAM, we use the N-body BolshoiP cosmological simulation (Klypin et al. 2016). This simulation is based on the ΛCDM cosmology with parameters consistent with the latest results from the Planck Collaboration. This simulation has 20483 particles of mass 1.9 × 108  M⊙ h− 1, in a box of side length LBP = 250 h−1 Mpc. Haloes/subhaloes and their merger trees were calculated with the phase–space temporal halo finder Rockstar (Behroozi, Wechsler & Wu 2013) and the software Consistent Trees (Behroozi et al. 2013). Entire Rockstar and Consistent Trees outputs are downloadable.6 Halo masses were defined using spherical overdensities according to the redshift-dependent virial overdensity Δvir(z) given by the spherical collapse model, with Δvir(z) = 333 at z = 0. The BolshoiP simulation is complete down to haloes of maximum circular velocity Vmax ≳ 55 km s−1. For more details, see Rodríguez-Puebla et al. (2016a). Next, we describe our mock galaxy catalogues generated via SHAM. 3.2 Determining the galaxy–halo connection As we have explained, SHAM is a simple approach relating a halo property, such as mass or maximum circular velocity, to that of a galaxy property, such as luminosity or stellar mass. In abundance matching between a halo property and a galaxy property, the number density distribution of the halo property is matched to the number density distribution of the galaxy property to obtain the relation. Recall that SHAM assumes that there is a one-to-one monotonic relationship between galaxies and haloes, and that centrals and satellite galaxies have identical relationships (except that satellite galaxy evolution is stopped when the host halo reaches its peak maximum circular velocity). In this paper, we choose to relate galaxy properties, $$\mathcal {P}_{\rm gal}$$, to halo maximum circular velocities Vmax as   \begin{eqnarray} \int _{\mathcal {P}_{\rm gal}}^{\infty } \phi _{\rm gal}(\mathcal {P}_{\rm gal}^{\prime })\text{ d}\log \mathcal {P}_{\rm gal}^{\prime } = \int _{V_{\rm max}}^{\infty } \phi _V (V_{\rm max} ^{\prime })\text{ d}\log V_{\rm max} ^{\prime }, \end{eqnarray} (13)where $$\phi _{\rm gal}(\mathcal {P}_{\rm gal})$$ denotes the ugriz GLF as well as the GSMF, and ϕV(Vmax) represents the subhalo+halo velocity function, both in units of h3 Mpc−3 dex−1. To construct a mock galaxy catalogue of luminosities and stellar masses from the BolshoiP simulation, we apply the aforementioned procedure by using as input the global ugriz GLFs and the GSMF derived in Section 2.2. Equation (13) is the simplest form that SHAM could take as it ignores the existence of a physical scatter around the relationship between $$\mathcal {P}_{\rm gal}$$ and Vmax. Including physical scatter in equation (13) is no longer considered valid and should be modified accordingly (for more details, see Behroozi et al. 2010). Constraints based on weak-lensing analysis (Leauthaud et al. 2012), satellite kinematics (More et al. 2009, 2011), and galaxy clustering (Zheng, Coil & Zehavi 2007; Zehavi et al. 2011; Yang et al. 2012) have shown that this is of the order of ∼0.15 dex in the case of the stellar but similar in r-band magnitude. There are no constraints as for the dispersion around shorter wavelengths. In addition, it is not clear how to sample galaxy properties in a system with n number of properties from the joint probability distribution $${\rm prop}(\mathcal {P}_{{\rm gal},1},\ldots , \mathcal {P}_{{\rm gal},n}| V_{\rm max}).$$7 Instead of that, studies that aim at to constrain the galaxy–halo connection use marginalization to constrain the probability distribution function $${\rm prop}(\mathcal {P}_{{\rm gal},i}| V_{\rm max})$$ for ith galaxy property. In this paper, we are interested in the statistical correlation of the galaxy–halo connection in which case equation (13) is a good approximation. Studying and quantifying the physical scatter around the relations is beyond the scope of this work. Also, ignoring the scatter around the galaxy–halo connection makes it easier to interpret. For those reasons, we have opted to ignore any source of scatter in our relationships. Previous studies have found that for distinct dark matter haloes (those that are not contained in bigger haloes), the maximum circular velocity Vmax is the halo property that correlates best with the hosted galaxy's luminosity/stellar mass. This is likely because the properties of a halo's central region, where its central galaxy resides, are better described by Vmax than Mvir.8 By comparing to observations of galaxy clustering, Reddick et al. (2013) and more recently Campbell et al. (2017) have found that for subhaloes the property that correlates best with luminosity/stellar mass is the highest maximum circular velocity reached along the main progenitor branch of the halo's merger tree. This presumably reflects the fact that subhaloes can lose mass once they approach and fall into a larger halo, whereas the host galaxy at the halo's centre is unaffected by this halo mass-loss. Thus, in this paper, we use   $$V_{\rm max}= \left\{ \begin{array}{l@{\qquad}c} V_{\rm max}& \mbox{{d}istinct {haloes}}\\ V_{\rm peak}& \mbox{{s}ubhalo{e}s} \end{array}\right. ,$$ (14)as the halo proxy for galaxy properties $$\mathcal {P}_{\rm gal}$$, where Vpeak is the maximum circular velocity throughout the entire history of a subhalo and Vmax is at the observed time for distinct haloes. Fig. 3 shows the relationships between galaxy luminosities u, g, r, i, and z and galaxy stellar masses to halo maximum circular velocities. Table A1 reports the values from Fig. 3. Most of these relationships are steeply increasing with Vmax for velocities below Vmax ∼ 160 km s−1. At higher velocities, the relationships are shallower. The shapes of these relations are governed mostly by the shapes of the GLFs and GSMF, since the velocity function ϕV is approximately a power law over the range plotted in Fig. 3 (see Rodríguez-Puebla et al. 2016a). Figure 3. View largeDownload slide Left-hand panel: luminosity–Vmax relation from SHAM. The different colours indicate the band utilized for the match. Right-hand panel: stellar mass–Vmax relation. Recall that SHAM assumes that these relations are valid for centrals as well as for satellites. We report these values in Table A1. In the case of centrals, Vmax refers to the halo maximum circular velocity, whereas for satellites, Vmax represents the highest maximum circular velocity (Vpeak) reached along the subhalo's main progenitor branch. SHAM assumes that Vmax fully determines these statistical properties of the galaxies. Figure 3. View largeDownload slide Left-hand panel: luminosity–Vmax relation from SHAM. The different colours indicate the band utilized for the match. Right-hand panel: stellar mass–Vmax relation. Recall that SHAM assumes that these relations are valid for centrals as well as for satellites. We report these values in Table A1. In the case of centrals, Vmax refers to the halo maximum circular velocity, whereas for satellites, Vmax represents the highest maximum circular velocity (Vpeak) reached along the subhalo's main progenitor branch. SHAM assumes that Vmax fully determines these statistical properties of the galaxies. Note that at this point every halo and subhalo in the BolshoiP simulation at rest-frame z = 0 has been assigned a magnitude in the five bands u, g, r, i, and z and a stellar mass M*. Therefore, one might be tempted to correlate galaxy colours such as red or blue (i.e. differences between galaxy magnitudes) with halo properties. If we did this, we would be ignoring the scatter around our luminosity/stellar mass–Vmax relationships, and galaxies with the same magnitude or M* would have the same colour, contrary to observations. Fortunately, including a scatter around those relationships will not impact our conclusions given that (i) the scatter does not substantially impact the results presented in Fig. 3, and (ii) we are here interested only in the statistical correlation of the galaxy properties with environment. Nevertheless, in Section 4.2.1, we will study the statistical correlation between colour and environment for all galaxies, and separately for central and satellite galaxies. As a sanity check, we show that our mock galaxy catalogue in the BolshoiP reproduces the projected two-point correlation function of SDSS galaxies.9 Figs 4 and 5 show, respectively, that this is the case for the r band and stellar mass projected two point correlation functions. In the case of r band, we compared to Zehavi et al. (2011), who used r-band magnitudes at z = 0.1. We transformed our r-band magnitudes to z = 0.1 by finding the correlation between model magnitudes at z = 0 and 0.1 from the tables of the NYU-VAGC.10 For the projected two point correlation function in stellar mass bins, we compare with Yang et al. (2012). Figure 4. View largeDownload slide Two-point correlation function in five luminosity bins at z = 0.1. The solid lines show the predicted two-point correlation based on our r-band magnitude–Vmax relation from SHAM, whereas the circles with error bars show the same but for the SDSS DR7 (Zehavi et al. 2011). Figure 4. View largeDownload slide Two-point correlation function in five luminosity bins at z = 0.1. The solid lines show the predicted two-point correlation based on our r-band magnitude–Vmax relation from SHAM, whereas the circles with error bars show the same but for the SDSS DR7 (Zehavi et al. 2011). Figure 5. View largeDownload slide Two-point correlation function in five stellar mass bins. The solid lines show the predicted two-point correlation based on our stellar mass–Vmax relation from SHAM, whereas the circles with error bars show the same but for SDSS DR7 (Yang et al. 2012). Figure 5. View largeDownload slide Two-point correlation function in five stellar mass bins. The solid lines show the predicted two-point correlation based on our stellar mass–Vmax relation from SHAM, whereas the circles with error bars show the same but for SDSS DR7 (Yang et al. 2012). 3.3 Measurements of the mock ugriz GLFs and the GSMF as a function of environment Our mock galaxy catalogue is a volume complete sample down to haloes of maximum circular velocity Vmax ∼ 55 km s− 1, corresponding to galaxies brighter than Mr − 5 log h ∼ −14, see Fig. 3.11 This magnitude completeness is well above the completeness of the SDSS DR7. Thus, galaxies selected in the absolute magnitude range −21.8 < Mr − 5 log h < −20.1 define a volume-limited DDP sample. In other words, incompleteness is not a problem for our mock galaxy catalogue. Overdensity and density contrast measurements for each mock galaxy in the BolshoiP simulation are obtained as described in Section 2.3.3. We estimate the dependence of the ugriz GLFs with environment in our mock galaxy catalogue as   \begin{eqnarray} \phi _X(M_X|\delta _8) = \sum _{i=1}^N \frac{\omega _i(M_X\pm \Delta M_X / 2|\delta _{r_8} \pm \Delta \delta _{r_8}/2)}{\Delta M_X f_{\rm BP}(\delta _8) L_{\rm BP}^3}. \end{eqnarray} (15)Here, ωi = 1 if a galaxy is within the interval MX ± ΔMX/2 given that its overdensity is in the range $$\delta _{r_8} \pm \Delta \delta _{r_8}/2$$, otherwise it is 0. Again, MX refers to Mu, Mg, Mr, Mi, Mz, and log M*. The function fBP(δ8) is the fraction of effective volume by a given overdensity bin for the BolshoiP simulation. In order to determine fBP(δ8), we create a random catalogue of Nr ∼ 1.2 × 106 points in a box of side length identical to the BolshoiP simulation, i.e. LBP = 250 h−1Mpc. Using equation (10) allows us to calculate fBP(δ8). 4 RESULTS ON ENVIRONMENTAL DENSITY DEPENDENCE In this section, we present our determinations for the environmental density dependence of the ugriz GLFs and the GSMF from the SDSS DR7 and the BolshoiP. Here, we will investigate how well the assumption that the statistical properties of galaxies are fully determined by Vmax can predict the dependence of the ugriz GLFs and GSMF with environment. We will show that predictions from SHAM are in remarkable agreement with the data from the SDSS DR7, especially for the longer wavelength bands. Finally, we show that SHAM also reproduces the correct dependence on environmental density of both the r-band GLFs and GSMF for centrals and satellites, although it fails to reproduce the observed relationship between environment and colour. 4.1 SDSS DR7 Fig. 6 shows the dependence of the SDSS DR7 ugriz GLFs as well as the GSMF with environmental density measured in spheres of radius 8 h−1 Mpc. For the sake of the simplicity, we present only four overdensity bins in Fig. 6. In Fig. 7, we show the determinations in nine density bins for the r-band GLFs and GSMF. In order to compare with recent observational results, we use identical environment density bins as in McNaught-Roberts et al. (2014), who used galaxies from the GAMA survey to measure the dependence of the r-band GLF on environment over the redshift range 0.04 < z < 0.26 in spheres of radius of 8 h−1 Mpc. Figure 6. View largeDownload slide Comparison between the observed SDSS DR7 ugriz GLFs and GSMF, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM, shaded regions, at four environmental densities in spheres of radius 8 h−1 Mpc. We also reproduce the best-fitting Schechter functions to the r-band GLFs from the GAMA survey (McNaught-Roberts et al. 2014). Observe that SHAM predictions are in excellent agreement with observations, especially for the longest wavelength bands and stellar mass. Figure 6. View largeDownload slide Comparison between the observed SDSS DR7 ugriz GLFs and GSMF, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM, shaded regions, at four environmental densities in spheres of radius 8 h−1 Mpc. We also reproduce the best-fitting Schechter functions to the r-band GLFs from the GAMA survey (McNaught-Roberts et al. 2014). Observe that SHAM predictions are in excellent agreement with observations, especially for the longest wavelength bands and stellar mass. Figure 7. View largeDownload slide Left-hand panel: comparison between the observed r-band GLF with environmental density in spheres of 8 h−1 Mpc, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM (shaded regions). The dashed and solid lines show the best-fitting Schechter functions to the observed and the mocked r-band GLFs, whereas the dotted lines show the same but from the GAMA survey (McNaught-Roberts et al. 2014). Right-hand panel: similar to the left-hand panel but for the GSMF with environmental density. Here, again the dashed and solid lines are the best-fitting Schechter functions to the observed and mocked GSMFs. Figure 7. View largeDownload slide Left-hand panel: comparison between the observed r-band GLF with environmental density in spheres of 8 h−1 Mpc, filled circles with error bars, and the ones predicted based on the BolshoiP simulation from SHAM (shaded regions). The dashed and solid lines show the best-fitting Schechter functions to the observed and the mocked r-band GLFs, whereas the dotted lines show the same but from the GAMA survey (McNaught-Roberts et al. 2014). Right-hand panel: similar to the left-hand panel but for the GSMF with environmental density. Here, again the dashed and solid lines are the best-fitting Schechter functions to the observed and mocked GSMFs. The r-band panel of Fig. 6 shows that our determinations are in good agreement with results from the GAMA survey. In the g-band panel of the same figure, we present a comparison with the previously published results by Croton et al. (2005), who used the 2dF Galaxy Redshift Survey to measure the dependence of the bJ-band GLFs at a zero redshift rest frame in spheres of radius of 8 h−1 Mpc. We convert the bJ-band GLFs from Croton et al. (2005) to the g band by applying a shift of −0.25 to their magnitudes, that is, $$M_g = M_{b_{\rm J}} - 0.25$$ (Blanton et al. 2005a). We observe good agreement with the result of Croton et al. (2005) for most of our density bins. A better comparison would have used identical density bins; however, the density bins used by Croton et al. (2005) are close to ours. Finally, in Fig. 6, we also extend previous results by presenting the GLFs for the u, i, and z bands and for the GSMF. We are not aware of any published low redshift GLFs for the u, i, and z bands. The left-hand panel of Fig. 7 shows again the dependence of the GLF in the r band, but now for all our overdensity bins, filled circles with error bars. In order to report an analytical model for the luminosity functions, we fit observations to a simple Schechter function; observations show that this model is a good description for the data, given by   \begin{eqnarray} \phi (M)=\frac{\ln 10}{2.5}\phi ^*10^{0.4(M^*-M)\left(1+\alpha _1\right)}\exp \left(-10^{0.4\left(M^*-M\right)}\right), \end{eqnarray} (16)in units of h3 Mpc−3 mag−1. The best fit to simple Schechter functions is shown as the dashed lines in the same plot, and we report the Schechter parameters as a function of the density contrast in the left-hand panel of Fig. 8. The best-fitting parameters are listed in Table 3. For comparison, we reproduce the best fit to a Schechter function from McNaught-Roberts et al. (2014), dotted lines. Figure 8. View largeDownload slide Left-hand panel: the dependence of the r-band Schechter function parameters on environmental overdensity δ8 in spheres of 8 h−1 Mpc (equation 8). Right-hand panel: the dependence of the GSMF Schechter parameters on environmental density. Figure 8. View largeDownload slide Left-hand panel: the dependence of the r-band Schechter function parameters on environmental overdensity δ8 in spheres of 8 h−1 Mpc (equation 8). Right-hand panel: the dependence of the GSMF Schechter parameters on environmental density. Table 3. Best-fitting parameters from the observed SDSS DR7 GLFs. GLFs  δmin, 8  δmax, 8  α  M* − 5 log h  log ϕ* (h3 Mpc−3 mag−1)  SDSS DR7  − 1  − 0.75  − 1.671 ± 0.092  − 20.650 ± 0.131  − 2.938 ± 0.095  − 0.75  − 0.55  − 1.265 ± 0.077  − 20.456 ± 0.086  − 2.391 ± 0.051  − 0.55  − 0.40  − 1.178 ± 0.068  − 20.490 ± 0.073  − 2.224 ± 0.041  − 0.40  0.00  − 1.217 ± 0.032  − 20.568 ± 0.033  − 2.031 ± 0.019  0.00  0.70  − 1.120 ± 0.025  − 20.604 ± 0.026  − 1.754 ± 0.013  0.70  1.60  − 1.092 ± 0.023  − 20.703 ± 0.025  − 1.556 ± 0.012  1.60  2.90  − 1.230 ± 0.033  − 20.870 ± 0.016  − 1.408 ± 0.015  2.90  4  − 1.292 ± 0.028  − 20.981 ± 0.040  − 1.280 ± 0.023  4.00  ∞  − 1.275 ± 0.005  − 21.000 ± 0.034  − 1.045 ± 0.020  BolshoiP+SHAM  − 1  − 0.75  − 1.0963 ± 0.0264  − 20.1018 ± 0.0319  − 2.6084 ± 0.0172  − 0.75  − 0.55  − 1.0214 ± 0.0206  − 20.2498 ± 0.0238  − 2.2478 ± 0.0127  − 0.55  − 0.40  − 1.0462 ± 0.0225  − 20.3727 ± 0.0273  − 2.1260 ± 0.0147  − 0.40  0.00  − 1.0828 ± 0.0129  − 20.5454 ± 0.0193  − 2.0225 ± 0.0096  0.00  0.70  − 1.1274 ± 0.0087  − 20.7486 ± 0.0142  − 1.8634 ± 0.0075  0.70  1.60  − 1.1453 ± 0.0086  − 20.8882 ± 0.0204  − 1.6760 ± 0.0079  1.60  2.90  − 1.1790 ± 0.0092  − 21.0189 ± 0.0220  − 1.4913 ± 0.0090  2.90  4  − 1.2451 ± 0.0154  − 21.1841 ± 0.0354  − 1.3821 ± 0.0179  4.00  ∞  − 1.2227 ± 0.0129  − 21.1285 ± 0.0300  − 1.0924 ± 0.0149  GLFs  δmin, 8  δmax, 8  α  M* − 5 log h  log ϕ* (h3 Mpc−3 mag−1)  SDSS DR7  − 1  − 0.75  − 1.671 ± 0.092  − 20.650 ± 0.131  − 2.938 ± 0.095  − 0.75  − 0.55  − 1.265 ± 0.077  − 20.456 ± 0.086  − 2.391 ± 0.051  − 0.55  − 0.40  − 1.178 ± 0.068  − 20.490 ± 0.073  − 2.224 ± 0.041  − 0.40  0.00  − 1.217 ± 0.032  − 20.568 ± 0.033  − 2.031 ± 0.019  0.00  0.70  − 1.120 ± 0.025  − 20.604 ± 0.026  − 1.754 ± 0.013  0.70  1.60  − 1.092 ± 0.023  − 20.703 ± 0.025  − 1.556 ± 0.012  1.60  2.90  − 1.230 ± 0.033  − 20.870 ± 0.016  − 1.408 ± 0.015  2.90  4  − 1.292 ± 0.028  − 20.981 ± 0.040  − 1.280 ± 0.023  4.00  ∞  − 1.275 ± 0.005  − 21.000 ± 0.034  − 1.045 ± 0.020  BolshoiP+SHAM  − 1  − 0.75  − 1.0963 ± 0.0264  − 20.1018 ± 0.0319  − 2.6084 ± 0.0172  − 0.75  − 0.55  − 1.0214 ± 0.0206  − 20.2498 ± 0.0238  − 2.2478 ± 0.0127  − 0.55  − 0.40  − 1.0462 ± 0.0225  − 20.3727 ± 0.0273  − 2.1260 ± 0.0147  − 0.40  0.00  − 1.0828 ± 0.0129  − 20.5454 ± 0.0193  − 2.0225 ± 0.0096  0.00  0.70  − 1.1274 ± 0.0087  − 20.7486 ± 0.0142  − 1.8634 ± 0.0075  0.70  1.60  − 1.1453 ± 0.0086  − 20.8882 ± 0.0204  − 1.6760 ± 0.0079  1.60  2.90  − 1.1790 ± 0.0092  − 21.0189 ± 0.0220  − 1.4913 ± 0.0090  2.90  4  − 1.2451 ± 0.0154  − 21.1841 ± 0.0354  − 1.3821 ± 0.0179  4.00  ∞  − 1.2227 ± 0.0129  − 21.1285 ± 0.0300  − 1.0924 ± 0.0149  View Large Fig. 8 shows that the normalization parameter of the Schechter function, ϕ*, depends strongly on density. There are almost two orders of magnitude difference between the least and the highest density bins; see also Table 3. In contrast, the faint-end slope, α, remains practically constant with environment with a value of α = −1.0 to −1.2. Note, however, that our analysis of the SDSS observations shows that the GLF becomes steeper in the least dense environment, with α ∼ −1.7. The characteristic magnitude of the Schechter function, M*, evolves only little with environment in the range −1 ≲ δ8 ≲ 0, but it increases above δ8 ∼ 0. Finally, in the same figure, we reproduce the best-fitting model parameters reported in McNaught-Roberts et al. (2014). In general, our determinations are in good agreement with the trends reported in McNaught-Roberts et al. (2014) even at faint magnitudes, as is shown in Figs 7 and 8. This is reassuring since the GAMA survey is deeper than the SDSS, which could result in a much better determination of the faint-end. In addition, the subtended area by the GAMA survey is much smaller than that of the SDSS, which could have resulted in GAMA underestimating the abundance of massive galaxies in low-density environments. The reason for this is because the limited volume of GAMA does not adequately sample these rather rare galaxies in low-density regions. The right-hand panel of Fig. 7 shows the dependence of the GSMF for all our overdensity bins as well as their corresponding best fit to simple Schechter functions, filled circles with error bars and solid lines, respectively. In this case, the Schechter function is given by   \begin{eqnarray} \phi _*(M_*) =\phi _1^*\times {\ln 10}\times \left(\frac{M_*}{\mathcal {M}_1^*}\right)^{1+\alpha _1}\exp \left(-\frac{M_*}{\mathcal {M}_1^*}\right), \end{eqnarray} (17)with units of h3 Mpc−3 dex−1. We report the best-fitting parameters in Table 4. The right-hand panel of Fig. 8 presents the Schechter parameters for the GSMFs as a function of the density contrast. Similarly to the GLFs, the normalization parameter for the GSMF, ϕ*, depends strongly on density as a power law and there are approximately two orders of magnitude difference between the GLFs in the least and the most dense environments. As for the faint-end slope, α, we observe that the general trend is that in high-density environments the GSMF becomes steeper than in low-density environments. None the less, we observe, again, that in the lowest density bin the GSMF becomes steeper than other density bins. The characteristic stellar mass of the Schechter function, $$\mathcal {M}_*$$, increases with the environment at least for densities greater than δ8 ∼ 0. In contrast, below δ8 ∼ 0 it remains approximately constant. Table 4. Best-fitting parameters from the observed SDSS DR7 GSMF. GSMFs  δmin, 8  δmax, 8  α  $$\mathcal {M}_*$$ (h− 2  M⊙)  log ϕ* (h3 Mpc−3 dex−1)  SDSS DR7  − 1  − 0.75  − 1.361 ± 0.077  10.467 ± 0.052  − 3.148 ± 0.075  − 0.75  − 0.55  − 1.062 ± 0.061  10.422 ± 0.032  − 2.630 ± 0.040  − 0.55  − 0.40  − 1.019 ± 0.054  10.433 ± 0.029  − 2.453 ± 0.035  − 0.40  0.00  − 1.031 ± 0.025  10.463 ± 0.012  − 2.250 ± 0.015  0.00  0.70  − 1.029 ± 0.019  10.511 ± 0.011  − 2.006 ± 0.012  0.70  1.60  − 1.042 ± 0.018  10.577 ± 0.011  − 1.815 ± 0.012  1.60  2.90  − 1.128 ± 0.018  10.627 ± 0.011  − 1.632 ± 0.013  2.90  4  − 1.164 ± 0.026  10.668 ± 0.016  − 1.477 ± 0.021  4.00  ∞  − 1.179 ± 0.022  10.686 ± 0.013  − 1.248 ± 0.017  BolshoiP+SHAM  − 1  − 0.75  − 1.0293 ± 0.0222  10.3077 ± 0.0142  − 2.8903 ± 0.0176  − 0.75  − 0.55  − 1.0000 ± 0.0151  10.3933 ± 0.0105  − 2.5222 ± 0.0128  − 0.55  − 0.40  − 0.9954 ± 0.0162  10.4306 ± 0.0110  − 2.3743 ± 0.0138  − 0.40  0.00  − 1.0275 ± 0.0094  10.5112 ± 0.0074  − 2.2602 ± 0.0088  0.00  0.70  − 1.0411 ± 0.0074  10.5896 ± 0.0065  − 2.0754 ± 0.0073  0.70  1.60  − 1.0528 ± 0.0073  10.6564 ± 0.0069  − 1.8757 ± 0.0075  1.60  2.90  − 1.0885 ± 0.0069  10.7191 ± 0.0075  − 1.6939 ± 0.0081  2.90  4  − 1.1533 ± 0.0112  10.7898 ± 0.0146  − 1.5891 ± 0.0154  4.00  ∞  − 1.1396 ± 0.0109  10.7691 ± 0.0135  − 1.3106 ± 0.0143  GSMFs  δmin, 8  δmax, 8  α  $$\mathcal {M}_*$$ (h− 2  M⊙)  log ϕ* (h3 Mpc−3 dex−1)  SDSS DR7  − 1  − 0.75  − 1.361 ± 0.077  10.467 ± 0.052  − 3.148 ± 0.075  − 0.75  − 0.55  − 1.062 ± 0.061  10.422 ± 0.032  − 2.630 ± 0.040  − 0.55  − 0.40  − 1.019 ± 0.054  10.433 ± 0.029  − 2.453 ± 0.035  − 0.40  0.00  − 1.031 ± 0.025  10.463 ± 0.012  − 2.250 ± 0.015  0.00  0.70  − 1.029 ± 0.019  10.511 ± 0.011  − 2.006 ± 0.012  0.70  1.60  − 1.042 ± 0.018  10.577 ± 0.011  − 1.815 ± 0.012  1.60  2.90  − 1.128 ± 0.018  10.627 ± 0.011  − 1.632 ± 0.013  2.90  4  − 1.164 ± 0.026  10.668 ± 0.016  − 1.477 ± 0.021  4.00  ∞  − 1.179 ± 0.022  10.686 ± 0.013  − 1.248 ± 0.017  BolshoiP+SHAM  − 1  − 0.75  − 1.0293 ± 0.0222  10.3077 ± 0.0142  − 2.8903 ± 0.0176  − 0.75  − 0.55  − 1.0000 ± 0.0151  10.3933 ± 0.0105  − 2.5222 ± 0.0128  − 0.55  − 0.40  − 0.9954 ± 0.0162  10.4306 ± 0.0110  − 2.3743 ± 0.0138  − 0.40  0.00  − 1.0275 ± 0.0094  10.5112 ± 0.0074  − 2.2602 ± 0.0088  0.00  0.70  − 1.0411 ± 0.0074  10.5896 ± 0.0065  − 2.0754 ± 0.0073  0.70  1.60  − 1.0528 ± 0.0073  10.6564 ± 0.0069  − 1.8757 ± 0.0075  1.60  2.90  − 1.0885 ± 0.0069  10.7191 ± 0.0075  − 1.6939 ± 0.0081  2.90  4  − 1.1533 ± 0.0112  10.7898 ± 0.0146  − 1.5891 ± 0.0154  4.00  ∞  − 1.1396 ± 0.0109  10.7691 ± 0.0135  − 1.3106 ± 0.0143  View Large 4.2 Comparison to theoretical determinations: SHAM Fig. 6 compares our observed ugriz GLFs and the results derived from the mock galaxy sample based on SHAM. In general, we observe a remarkable agreement between observations and SHAM. This statement is true for all the luminosity bands as well as for stellar mass and for most of the density bins. This remarkable agreement is not a trivial result since we are assuming that Vmax fully determines the magnitudes in the u, g, r, i, and z bands and stellar mass M* in every halo in the simulation. Additionally, in Section 3, we noted that the shape of the galaxy–halo connection is governed, mainly, by the global shape of the galaxy number densities. Moreover, although we have defined our volume-limited DDP sample as for the SDSS observations, it is subject to the assumptions behind SHAM as well. In addition, the real correlation between r-band magnitude and all other galaxy properties is no doubt more complex than just monotonic relationships without scatters, as is derived in SHAM. Note, however, that there are some discrepancies towards bluer bands and low densities. Shorter wavelengths are more affected by recent star formation, and more likely to be related to halo mass accretion rates (Rodríguez-Puebla et al. 2016b, and references therein), whereas infrared magnitudes depend more strongly on stellar mass. This perhaps just reflects that stellar mass is the galaxy property that most naturally correlates with Vmax. Indeed, when comparing the environmental dependence of the observed and the mock GSMF we observe, in general, rather good agreement. The left-hand panel of Fig. 7 compares the resulting dependence of the observed r-band GLFs with environment and the predictions based on SHAM for all our overdensity bins. This again shows the remarkable agreement between observations and SHAM for all our density bins. Similarly, the right-hand panel of Fig. 7 compares the observed GSMF with our predictions based on SHAM. The left-hand panel of Fig. 8 compares the best-fitting Schechter parameters for the r-band magnitude, whereas the right-hand panel shows the same but for stellar masses. In order to make a meaningful comparison with observations, we fit the observed GLFs and GSMF of the SDSS DR7 over the same dynamical ranges. In general, we observe a good agreement between predictions from SHAM and the results from the SDSS DR7. Although Fig. 7 shows that the general trends are well predicted by SHAM, there are some differences that are worth discussing. SHAM is able to recover the overall normalization of the r-band GLF and the GSMF, but it slightly underpredicts the number of faint galaxies and it also underpredicts the high-mass end in low-density environments. In high-density environments, SHAM overpredicts the number of galaxies at the high-mass end. A natural explanation for these differences could be the dependence of the galaxy–halo connection with environment. Recall that we are assuming zero scatter in the galaxy–halo connection. Despite the differences noted above, the extreme simplicity of SHAM captures extremely well the dependences with environmental density of the galaxy distribution. This is remarkable and, as we noted earlier, it might not be expected since halo properties depend on the local environment as well as the large-scale environment. In order to understand better the success of SHAM and the nature of the above discrepancies, we now turn our attention to the dependence with environment of the r-band GLFs and GSMF of central and satellite galaxies separately. 4.2.1 SHAM predictions for the central and satellite GLFs and GSMFs Figs 9 and 10 show, respectively, the dependence on environmental density of the r-band GLFs and GSMF for all galaxies, and separately for centrals and satellites. The circles with error bars show the results when using the memberships from the SDSS DR7 Yang et al. (2012) galaxy group catalogue, whereas the shaded areas show the predictions from SHAM based on the BolshoiP simulation. When dividing the population between centrals and satellites, in general, SHAM captures the observed dependences from the SDSS DR7 Yang et al. (2012) galaxy group catalogue. This simply reflects that the fraction of subhaloes increases as a function of the environment as well as the chances of finding high-mass (sub)haloes in dense environments. Figure 9. View largeDownload slide The dependence on r-band magnitude of the GLFs in nine bins of environmental density in 8 h−1 Mpc spheres for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. There is a remarkable agreement between observations and SHAM predictions, even when dividing between centrals and satellites. Figure 9. View largeDownload slide The dependence on r-band magnitude of the GLFs in nine bins of environmental density in 8 h−1 Mpc spheres for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. There is a remarkable agreement between observations and SHAM predictions, even when dividing between centrals and satellites. Figure 10. View largeDownload slide Similarly to Fig. 9 but for the GSMF for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. Although there is a good agreement between observations and SHAM predictions for all galaxies and centrals, there is some tension with the SHAM predictions of the satellite GSMF. Figure 10. View largeDownload slide Similarly to Fig. 9 but for the GSMF for all galaxies, central galaxies, and satellite galaxies. Filled circles with error bars show the results from the SDS DR7, whereas shaded areas show the SHAM predictions from the BolshoiP simulation. Although there is a good agreement between observations and SHAM predictions for all galaxies and centrals, there is some tension with the SHAM predictions of the satellite GSMF. The agreement is particularly good for centrals. However, the satellite r-band GLF is in much better agreement with observations than the satellite GSMF: For the r-band GLF, we observe a marginal difference only, whereas SHAM predicts that there are more galaxies around the knee of the GSMF compared to what is observed. It is not clear why we should expect this difference, but a potential explanation could be that satellite galaxies are much more sensitive to their local environment and to the definition of the DDP population. To help build intuition, recall that SHAM assigns every halo in the simulation five magnitudes in the u, g, r, i, and z bands and stellar mass M*. Consider now that the relationship between r-band magnitude and all other galaxy properties is just monotonic and with zero scatter, as explained earlier. Thus, this oversimplification of much more complex relationships is affecting the measurements of environmental dependences in satellite galaxies, especially when these are projected in other observables. Possibly, when using a stellar mass-based DDP population, the problem will be inverted. In other words, we might observe a marginal difference between SHAM and the GSMFs but a larger difference between SHAM and the GLFs. Of course central galaxies are not exempt from also being affected, but given the good agreement with observations we conclude that the effect is only marginal. Another possible explanation is that the assumption of identical relations between centrals and satellites is more valid for the r-band luminosity than for the stellar mass. That is, the stellar mass of satellite galaxies perhaps varies more strongly with Vmax than the r-band luminosity does. A third possible explanation is that group finding algorithms are subject to errors. In a recent paper, Campbell et al. (2015) showed that there are two main sources of errors that could affect the comparison in Figs 9 and 10: (i) central/satellite designation, and (ii) group membership determination. In that paper, the authors showed that the Yang et al. (2007) group finder algorithm tends to misidentify central galaxies systematically with increasing group mass. In other words, satellites are sometimes mistakenly identified as centrals. Consequently, the GLFs and the GSMF for centrals and satellites will be affected towards the bright-end. Note, however, that Campbell et al. (2015) showed that for each satellite, which is misidentified as a central, approximately a central is misidentified as a satellite in the Yang et al. (2007) group finder. Thus, in the Yang et al. (2007) group finder, central/satellite designation is the main source of error rather than the group membership determination. Although this is a source of error that should be taken into account in our analysis, it is likely that this is not the main source of difference between observations and SHAM predictions. The reason is that there exists the above compensation effect in the identification of centrals and satellite galaxies, which could leave, perhaps, the GLFs and the GSMF of centrals and satellites with little or no changes. Finally, as we noted earlier, Fig. 7 shows that SHAM overpredicts the number of high-mass galaxies in high-mass bins. Figs 9 and 10 show that this excess of galaxies is due to central galaxies. We will discuss this in the light of the dependence of the galaxy–halo connection with environment in Section 5. 4.2.2 The relationship between colour and mean environmental density Fig. 11 shows the mean density as a function of the g − r colour separately for all galaxies, centrals, and satellites. The filled circles with error bars show the mean density measured from the Yang et al. (2012) galaxy group catalogue, whereas the shaded areas show the same but for the BolshoiP simulation. SHAM is unable to predict the correct correlation between mean density and galaxy colours for all and central galaxies. SHAM predicts that, statistically speaking, the large-scale mean environmental density varies little with the colours of central galaxies, except that the reddest galaxies on average lie in the densest environments. Actually, this is not surprising since we assumed that the ugriz bands and stellar mass are independent of environment when constructing our mock galaxy catalogue and the above is simply showing that one halo property does not fully determine the statistical properties of the galaxies. Other halo properties that vary with environment should instead be used in order to reproduce the correct trends with environment. Extensions to SHAM in which halo age is matched to galaxy age/colour at a fixed luminosity/stellar mass and halo mass (see e.g. Hearin & Watson 2013; Masaki, Lin & Yoshida 2013) are promising approaches that could help to better explain the trends with observations. None the less, SHAM predictions are in better agreement with the observed correlation of density with colour for satellite galaxies. Figure 11. View largeDownload slide Mean density in 8 h−1 Mpc spheres as a function of galaxy g − r colour, from the SDSS DR7 (shaded regions, representing the standard deviation) and the mean density predicted by SHAM based on the BolshoiP simulation, filled circles with error bars. We present the mean density for all, central, and satellite galaxies as indicated by the labels. SHAM fails to predict the correct relationship between mean density and galaxy colours for all galaxies and central galaxies. In contrast, the SHAM prediction for satellite galaxies is in better agreement with observations. Figure 11. View largeDownload slide Mean density in 8 h−1 Mpc spheres as a function of galaxy g − r colour, from the SDSS DR7 (shaded regions, representing the standard deviation) and the mean density predicted by SHAM based on the BolshoiP simulation, filled circles with error bars. We present the mean density for all, central, and satellite galaxies as indicated by the labels. SHAM fails to predict the correct relationship between mean density and galaxy colours for all galaxies and central galaxies. In contrast, the SHAM prediction for satellite galaxies is in better agreement with observations. 5 SUMMARY AND DISCUSSION SHAM makes the assumption that one (sub)halo property fully determines the statistical properties of their host galaxies. Therefore, SHAM implies that (i) the galaxy–halo connection is identical between haloes and subhaloes, and (ii) the dependence of galaxy properties on environmental density comes entirely from the corresponding dependence on density of this (sub)halo property. The halo property that this paper explores for SHAM is the quantity Vmax, which is defined in equation (14) as the maximum circular velocity for distinct haloes, whereas for subhaloes it is the peak maximum circular velocity Vpeak reached along the halo's main progenitor branch. This is the most robust halo and subhalo property for SHAM (see, e.g. Reddick et al. 2013; Campbell et al. 2017). The galaxy properties we studied are the ugriz GLFs as well as the GSMF, which we determined from the SDSS DR7. We compared these observations with SHAM predictions from a mock galaxy catalogue based on the BolshoiP simulation (Klypin et al. 2016; Rodríguez-Puebla et al. 2016a). SHAM assigns every halo in the BolshoiP simulation magnitudes in the five SDSS bands u, g, r, i, and z and also a stellar mass M*(Fig. 3 and Appendix A). We tested the assumptions behind SHAM by comparing the predicted and observed dependence of the ugriz GLFs as well as the GSMF on the environmental density from the SDSS DR7 Yang et al. (2012) galaxy group catalogue. The main results and conclusions are as follows: In general, the environmental dependence of the ugriz GLFs predicted by SHAM are in good agreement with the observed dependence from the SDSS DR7. This is especially true for r and infrared bands. Theoretically, the stellar mass is the galaxy property that is expected to depend more strongly on halo Vmax, whereas bluer bands also reflect recent effects of star formation. We show that the environmental dependence of the GSMF predicted by SHAM is in remarkable agreement with the observed dependence from the SDSS DR7, reinforcing the above conclusion. When dividing the galaxy population into centrals and satellites, SHAM predicts the correct dependence of the observed r-band GLF and GSMF for centrals and satellite galaxies from the Yang et al. (2012) group galaxy catalogue. Although SHAM predicts GLFs and the GSMF that are in remarkable agreement with observations even when the galaxy population is separated between centrals and satellites, SHAM does not predict the observed average relation between g − r colour and mean environmental density. This is especially true for central galaxies, whereas the correlation obtained for satellite galaxies is in better agreement with observations. Many previous authors have studied the correlation between galaxies and dark matter haloes with environment both theoretically and observationally (see, e.g. Avila-Reese et al. 2005; Baldry et al. 2006; Blanton & Berlind 2007; Maulbetsch et al. 2007; Tinker, Wetzel & Conroy 2011; Lacerna et al. 2014; Lee et al. 2017; Yang et al. 2017, and many more references cited therein). Although most of these authors have focused on understanding this correlation by studying the galaxy distribution as a function of colour, star formation or age and environment at a fixed M*, here we take a different approach and exploit the extreme simplicity of SHAM. First, there are no special galaxies in SHAM. Secondly, SHAM can be applied to any galaxy property distribution. Thus, in our framework, a halo and a subhalo with identical Vmax will host galaxies with identical luminosities and stellar masses, no matter what the halo's environmental density or position in the cosmic web is. Our results are consistent with previous findings that halo Vmax could be enough to determine the luminosities and stellar masses. However, we have also shown that SHAM is unable to reproduce the correct correlation between galaxy colour and the mean density δ8 on a scale of 8 h−1 Mpc. This result implies that additional halo properties that depend in some way on the halo environment (e.g. Lee et al. 2017) should be employed to correctly reproduce the relationship between δ8 and galaxy colour. Does the above discussion imply that the galaxy–halo connection should depend on environment? On one hand, from observations we have learned that the statistical properties of the galaxies such as colour and star formation change with environment in the direction that low-density environments are mostly populated by blue/star-forming galaxies, whereas dense environments are mostly populated with red/quenched galaxies (see for e.g. Hogg et al. 2003; Baldry et al. 2006; Tomczak et al. 2017). On the other hand, the shape of the luminosity–Vmax and the stellar mass–Vmax relations (Fig. 3) contains information about the process that regulated the star formation in galaxies. Therefore, it is not a bad idea to consider that the differences described in Fig. 7 are the result that the galaxy–halo connection could change with environment. For the sake of the simplicity, consider the GSMF of central galaxies derived in the case of zero scatter around the M* = M*(Vmax) relationship. Therefore, equation (13) can be rewritten to give the GSMF as   \begin{eqnarray} \phi _* (M_*) = \phi _V (V_{\rm max}(M_*)) \times \alpha _{\rm gal}, \end{eqnarray} (18)whereas the dependence with environment of the GSMF of central galaxies is given by   \begin{eqnarray} \phi _* (M_*|\delta _8) = \phi _V (V_{\rm max}(M_*)|\delta _8) \times \alpha _{\rm gal}, \end{eqnarray} (19)where αgal ≡ d log Vmax(M*)/d log M* is the logarithmic slope of the M* = M*(Vmax) relationship assumed to be independent of environment. Next, consider the simplest case in which ϕV(Vmax|δ8) is a double power law such that $$\phi _V (V_{\rm max}|\delta _8) \propto V_{\rm max}^{\beta (\delta _8)}$$ for $$V_{\rm max}\ll V_{\rm max}^*(\delta _8)$$ and $$\phi _V (V_{\rm max}|\delta _8) \propto V_{\rm max}^{\gamma (\delta _8)}$$ for $$V_{\rm max}\gg V_{\rm max}^*(\delta _8)$$, where $$V_{\rm max}^*(\delta _8)$$ is a characteristic velocity, and we have emphasized that the parameters β, γ, and $$V_{\rm max}^*$$ depend on the environment. In order to simplify the problem even further, consider that the M* = M*(Vmax) relationship is a power-law relation at low masses with logarithmic slope αgal, low, whereas at high masses it is also a power law with logarithmic slope of αgal, high. Based on the above, we can write the dependence with environment of the GSMF of central galaxies in the limiting cases   $$\phi _* (M_*|\delta _8) {\propto} \left\{ \begin{array}{l@{\quad}c} \alpha_{\rm gal,low}{\times} M_*^{\beta (\delta _8)/\alpha _{\rm gal,low}} & \mbox{if } V_{\rm max}\ll V_{\rm max}^*(\delta _8)\\ \alpha _{\rm gal,high}{\times} M_*^{\gamma (\delta _8)/\alpha _{\rm gal,high}} & \mbox{if } V_{\rm max}\gg V_{\rm max}^*(\delta _8) \end{array}\right.\!\! .$$ (20)Thus, if αgal, low and αgal, high are independent of environment, the resulting shape and dependence of ϕ*(M*|δ8) with environment can be simply understood as the dependence with environment of the slopes β and γ of the halo velocity function. By looking to the least (void-like) and highest (cluster-like) density environments from Fig. 10, upper left-hand and bottom right-hand panels, respectively, we can use the above model in order to understand how the galaxy–halo connection may depend on environment. The voids-like GSMF from Fig. 10 shows that SHAM tends to underpredict the number density of central galaxies both at the low- and high-mass ends. In other words, the slopes predicted by SHAM at the low- and high-mass ends are, respectively, too shallow and steep compared to observations. Inverting this would require, based on equation (20), to make the slopes αgal, low and αgal, high shallower and steeper, respectively, to what we derived from SHAM, see the right-hand panel of Fig. 3. This implies that in low-density environments at a fixed Vmax haloes had been more efficient in forming stars12 both for at the low- and high-mass end. In contrast, the high density GSMF from Fig. 10 shows that SHAM tends to overpredict the number density of central galaxies at the high-mass end. In this case, we invert the above trend by making the high-mass end slope αgal, high more shallow compared to what is currently derived from SHAM. This implies that the star formation efficiency has been suppressed in high-mass haloes residing in high-density environments with respect to the predictions of SHAM. The above limiting cases show that the galaxy–halo connection is expected to change with environment in the direction that haloes in low-density environment should be more efficient in transforming their gas into stars, whereas in high-density environments haloes have become more passive. This is indeed consistent with the colour/star formation trends that have been observed in large galaxy surveys. Of course, our discussion is an oversimplification, and in order to model exactly how the galaxy–halo connection depends on environment, we would need to use the dependence of the GSMF with environment as an extra observational constraint for the galaxy–halo connection. In a recent series of papers, Tinker et al. (2017b,c) and Tinker et al. (2017a) studied the galaxy–halo connection in the light of the relation between the star formation and environment at a fixed stellar and halo mass, obtaining similar conclusions to ours. That is, above-average galaxies with above average star formation rates and high halo accretion rates live in underdense environments, whereas the increase of the observed quenched fraction of galaxies from low-to-high density environments is consistent with the fact that halo formation has an impact on quenching the star formation at high masses and densities. See also Lee et al. (2017) for similar conclusions. Finally, we expect that at high redshift the assumptions from SHAM are likely to be closer to reality. The reason is that as the Universe ages, the cosmic web becomes more mature and the dependence of halo properties with environment become also stronger. As we showed here, while there are some differences with observations of local galaxies, those are small despite the extreme simplicity of the SHAM assumptions. Therefore, we expect that the galaxy–halo connection should depend less on environment at high redshifts, when environmental process have not played a significant role. Acknowledgements We thank Vladimir Avila-Reese, Peter Behroozi, Avishai Dekel, Sandra Faber, David Koo, Rachel Somerville, Risa Wechsler, and Chandrachani Ningombam for useful comments and discussions. AR-P thanks the UC-MEXUS-CONACYT programme for support at UCSC. JRP and CTL acknowledge support from grants HST-GO-12060.12-A and HST-AR-14578.001-A. The authors acknowledge the UC MEXUS-CONACYT Collaborative Research Grant CN-17-125. We thank the NASA Advanced Supercomputer program for allowing the Bolshoi-Planck simulation to be run on the Pleiades supercomputer at NASA Ames Research Center. We also thank the anonymous referee for a useful report, which helped improve the presentation of this paper. Part of the material of this paper was presented as the Bachelor of Science senior thesis of RD. Footnotes 1 More recent methods apply corrections that improve agreement with N-body simulations (Parkinson, Cole & Helly 2008; Somerville & Davé 2015). 2 Note that subhalo Vpeak is typically reached not at accretion, but rather when the distance of the progenitor halo from its eventual host halo is three to four times the host halo Rvir (Behroozi et al. 2014). 3 This galaxy group catalogue represents an updated version of Yang et al. (2007); see also Yang, Mo & van den Bosch (2009). 4 Note that we are using model magnitudes instead of Petrosian magnitudes. The main reason is the former ones tend to underestimate the true light from galaxies, particularly for high-mass galaxies (see, e.g. Bernardi et al. 2010 and Montero-Dorta & Prada 2009). 5 Note that this is not due to sky subtraction issues, as previous studies have found (see, e.g. Bernardi et al. 2013, 2016), since we are not including this correction in the galaxy magnitudes. Instead, it is most likely due to our use of model magnitudes instead of Petrosian ones (see also footnote 4). 6 http://hipacc.ucsc.edu/Bolshoi/MergerTrees.html 7 In particular, this paper uses five bands u, g, r, i, and z and a stellar mass M*, making a total of n = 6. 8 For an NFW halo, Vmax is reached at Rmax = 2.16Rs, where Rs is the NFW scaleradius Rs = Rvir/C and C is the NFW concentration (e.g. Klypin et al. 2001). 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In the case of distinct haloes, Vmax refers to the halo maximum circular velocity, whereas for satellites Vmax represents the highest maximum circular velocity reached along the subhalo's main progenitor branch Vpeak. Note that the validation limits for our determinations of luminosity–Vmax and stellar mass–Vmax relations are due to the range of the observed galaxy number density that corresponds to haloes above Vmax ∼ 90 km s−1 even if the BolshoiP simulations is complete up to Vmax ∼ 55 km s−1. Below this limit the mock catalogue should be considered as an extrapolation to observations. Table A1. Luminosity–Vmax relation and stellar mass–Vmax relation from SHAM. Vmax (km s−1)  log (M*/h− 2  M⊙)  Mu − 5log h  Mg − 5log h  Mr − 5log h  Mi − 5log h  Mz − 5log h  80.0000  7.960 57  − 14.5647  − 15.8541  − 16.2580  − 16.4285  − 16.5291  88.5047  8.264 52  − 15.3542  − 16.5009  − 16.9221  − 17.0909  − 17.2073  97.9136  8.599 33  − 15.9608  − 17.1241  − 17.5814  − 17.7615  − 17.9085  108.323  8.951 25  − 16.4419  − 17.6885  − 18.1921  − 18.3951  − 18.5809  119.838  9.286 07  − 16.8340  − 18.1794  − 18.7281  − 18.9579  − 19.1805  132.578  9.574 33  − 17.1613  − 18.5995  − 19.1867  − 19.4408  − 19.6931  146.672  9.810 92  − 17.4401  − 18.9593  − 19.5777  − 19.8514  − 20.1260  162.265  10.0045  − 17.6817  − 19.2700  − 19.9132  − 20.2022  − 20.4935  179.515  10.1654  − 17.8942  − 19.5413  − 20.2044  − 20.5053  − 20.8089  198.599  10.3019  − 18.0835  − 19.7809  − 20.4603  − 20.7703  − 21.0832  219.712  10.4197  − 18.2542  − 19.9950  − 20.6877  − 21.0049  − 21.3250  243.070  10.5232  − 18.4096  − 20.1882  − 20.8920  − 21.2150  − 21.5406  268.910  10.6154  − 18.5525  − 20.3641  − 21.0774  − 21.4049  − 21.7350  297.497  10.6985  − 18.6852  − 20.5256  − 21.2471  − 21.5784  − 21.9119  329.124  10.7742  − 18.8093  − 20.6752  − 21.4038  − 21.7381  − 22.0745  364.113  10.8440  − 18.9266  − 20.8147  − 21.5495  − 21.8864  − 22.2252  402.821  10.9088  − 19.0385  − 20.9457  − 21.6861  − 22.0252  − 22.3658  445.645  10.9696  − 19.1464  − 21.0697  − 21.8151  − 22.1560  − 22.4983  493.021  11.0270  − 19.2516  − 21.1879  − 21.9378  − 22.2803  − 22.6238  545.433  11.0818  − 19.3557  − 21.3013  − 22.0554  − 22.3992  − 22.7438  603.418  11.1344  − 19.4603  − 21.4109  − 22.1688  − 22.5139  − 22.8594  667.566  11.1854  − 19.5678  − 21.5177  − 22.2792  − 22.6253  − 22.9716  738.534  11.2352  − 19.6810  − 21.6226  − 22.3875  − 22.7345  − 23.0813  817.047  11.2843  − 19.8042  − 21.7263  − 22.4944  − 22.8422  − 23.1896  903.907  11.3331  − 19.9425  − 21.8298  − 22.6009  − 22.9495  − 23.2972  1000.00  11.3820  − 20.1006  − 21.9337  − 22.7078  − 23.0570  − 23.4050  1106.31  11.4313  − 20.2772  − 22.0389  − 22.8158  − 23.1656  − 23.5138  1223.92  11.4815  − 20.4630  − 22.1461  − 22.9259  − 23.2762  − 23.6245  1354.03  11.5328  − 20.6494  − 22.2561  − 23.0386  − 23.3894  − 23.7377  1497.98  11.5856  − 20.8337  − 22.3695  − 23.1548  − 23.5060  − 23.8542  1657.23  11.6403  − 21.0165  − 22.4871  − 23.2752  − 23.6267  − 23.9746  Vmax (km s−1)  log (M*/h− 2  M⊙)  Mu − 5log h  Mg − 5log h  Mr − 5log h  Mi − 5log h  Mz − 5log h  80.0000  7.960 57  − 14.5647  − 15.8541  − 16.2580  − 16.4285  − 16.5291  88.5047  8.264 52  − 15.3542  − 16.5009  − 16.9221  − 17.0909  − 17.2073  97.9136  8.599 33  − 15.9608  − 17.1241  − 17.5814  − 17.7615  − 17.9085  108.323  8.951 25  − 16.4419  − 17.6885  − 18.1921  − 18.3951  − 18.5809  119.838  9.286 07  − 16.8340  − 18.1794  − 18.7281  − 18.9579  − 19.1805  132.578  9.574 33  − 17.1613  − 18.5995  − 19.1867  − 19.4408  − 19.6931  146.672  9.810 92  − 17.4401  − 18.9593  − 19.5777  − 19.8514  − 20.1260  162.265  10.0045  − 17.6817  − 19.2700  − 19.9132  − 20.2022  − 20.4935  179.515  10.1654  − 17.8942  − 19.5413  − 20.2044  − 20.5053  − 20.8089  198.599  10.3019  − 18.0835  − 19.7809  − 20.4603  − 20.7703  − 21.0832  219.712  10.4197  − 18.2542  − 19.9950  − 20.6877  − 21.0049  − 21.3250  243.070  10.5232  − 18.4096  − 20.1882  − 20.8920  − 21.2150  − 21.5406  268.910  10.6154  − 18.5525  − 20.3641  − 21.0774  − 21.4049  − 21.7350  297.497  10.6985  − 18.6852  − 20.5256  − 21.2471  − 21.5784  − 21.9119  329.124  10.7742  − 18.8093  − 20.6752  − 21.4038  − 21.7381  − 22.0745  364.113  10.8440  − 18.9266  − 20.8147  − 21.5495  − 21.8864  − 22.2252  402.821  10.9088  − 19.0385  − 20.9457  − 21.6861  − 22.0252  − 22.3658  445.645  10.9696  − 19.1464  − 21.0697  − 21.8151  − 22.1560  − 22.4983  493.021  11.0270  − 19.2516  − 21.1879  − 21.9378  − 22.2803  − 22.6238  545.433  11.0818  − 19.3557  − 21.3013  − 22.0554  − 22.3992  − 22.7438  603.418  11.1344  − 19.4603  − 21.4109  − 22.1688  − 22.5139  − 22.8594  667.566  11.1854  − 19.5678  − 21.5177  − 22.2792  − 22.6253  − 22.9716  738.534  11.2352  − 19.6810  − 21.6226  − 22.3875  − 22.7345  − 23.0813  817.047  11.2843  − 19.8042  − 21.7263  − 22.4944  − 22.8422  − 23.1896  903.907  11.3331  − 19.9425  − 21.8298  − 22.6009  − 22.9495  − 23.2972  1000.00  11.3820  − 20.1006  − 21.9337  − 22.7078  − 23.0570  − 23.4050  1106.31  11.4313  − 20.2772  − 22.0389  − 22.8158  − 23.1656  − 23.5138  1223.92  11.4815  − 20.4630  − 22.1461  − 22.9259  − 23.2762  − 23.6245  1354.03  11.5328  − 20.6494  − 22.2561  − 23.0386  − 23.3894  − 23.7377  1497.98  11.5856  − 20.8337  − 22.3695  − 23.1548  − 23.5060  − 23.8542  1657.23  11.6403  − 21.0165  − 22.4871  − 23.2752  − 23.6267  − 23.9746  View Large © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

### Journal

Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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