THE STELLAR-TO-HALO MASS RELATION OF LOCAL GALAXIES SEGREGATES BY COLOR

One can express the total GSMF by separating the population of galaxies into central and satellites:

$$\begin{equation} {\phi _{g,t}}({M_*})={\phi _{g_c,t}}({M_*})+{\phi _{g_{s,t}}}({M_*}), \end{equation} \tag{ 1 }$$

each of which can be subdivided into blue and red galaxy populations, i.e.,

$$\begin{equation} \phi _{g,t}(M_*)=[\phi_{g_{c},b}(M_*)+\phi_{g_{c},r}(M_*)]+[\phi_{g_{s,b}}({M_*})+{\phi _{g_{s,r}}}({M_*})]. \end{equation} \tag{ 2 }$$

By defining the CSMF, $\Phi _{i,j}({M_*}|{M_{\rm h}})$, as the mean number of i type (i = central or satellite) galaxies of a j color (j = blue or red) at the mass bin M_*  ±  dM_*/2, one can write each component of the GSMFs in the following form:

$$\begin{equation} {\phi _{g_{i,j}}}({M_*})=\int {\Phi _{i,j}({M_*}|{M_{\rm h}})}{\phi _{h}}({M_{\rm h}})d{M_{\rm h}}, \end{equation} \tag{ 3 }$$

where ϕ_h is the distinct HMF. Thus, the mean cumulative number density galaxies of type i and color j can be written as

$$\begin{equation} {n_{g_i,j}}(>{M_*})=\int {\langle N_{i,j}(>{M_*}|{M_{\rm h}})\rangle}{\phi _{h}}({M_{\rm h}})d{M_{\rm h}}, \end{equation} \tag{ 4 }$$

where

$$\begin{equation} {\langle N_{i,j}(>{M_*}|{M_{\rm h}})\rangle}=\int _{M_*}^\infty {\Phi _{i,j}(M_*^{\prime }|{M_{\rm h}})}dM_*^{\prime } \end{equation} \tag{ 5 }$$

is the mean cumulative number of galaxies of the type i and color j with stellar masses greater than M_* residing in a halo of mass M_h. Observe that once the CSMFs ${\Phi _{i,j}({M_*}|{M_{\rm h}})}$ are given, Equations (3)–(5) are totally defined. Therefore, the key ingredients in our model are the conditional mass functions ${\Phi _{i,j}({M_*}|{M_{\rm h}})}$.

2.1.1. Central Galaxies

In the context of the SHAM, the connection between the total central GSMF, $\phi _{g_c,t}$(M_*), and the distinct HMF, ϕ_h(M_h), arises naturally by assuming a probability distribution function, denoted by ${P_{c,t}({M_*}|{M_{\rm h}})}$, that a distinct halo of mass M_h hosts a central galaxy in the stellar mass bin M_* ± dM_*/2 (see Introduction for references). As a result of this connection, the mean SHMR of central galaxies, M_*(M_h), can be constrained. When the GSMF is divided into different populations, the above idea can be extended to connect the different galaxy populations to their host dark matter halos. For blue and red galaxies, one can introduce the conditional probability distribution functions $P_{c,b}({M_*}|{M_{\rm h}})$ and $P_{c,r}({M_*}|{M_{\rm h}})$ to establish the statistical connection between the blue and red (ϕ_{h, b} and ϕ_{h, r}, respectively) distinct HMFs and the GSMFs of blue and red centrals, $\phi _{g_c,b}$ and $\phi _{g_c,r}$, respectively. As above, the mean relations M_{*, b}(M_h) and M_{*, r}(M_h) are the result of this connection.

In terms of the CSMF formalism, one can specify the central CSMF as the sum of the blue and a red components,

$$\begin{equation} {\Phi _{c,t}({M_*}|{M_{\rm h}})}={\Phi _{c,b}({M_*}|{M_{\rm h}})}+{\Phi _{c,r}({M_*}|{M_{\rm h}})}, \end{equation} \tag{ 6 }$$

where the CSMF of blue and red central galaxies are given by

$$\begin{equation} {\Phi _{c,j}({M_*}|{M_{\rm h}})}={f_{j}}({M_{\rm h}})\times {P_{c,j}({M_*}|{M_{\rm h}})}. \end{equation} \tag{ 7 }$$

As above, the subscript j refers either to red (r) or blue (b) central galaxies, and ${f_{j}}({M_{\rm h}})$ is the fraction of halos hosting central galaxies of color j. Notice that for all central galaxies, the CSMF is simply given by ${\Phi _{c,t}({M_*}|{M_{\rm h}})}={P_{c,t}({M_*}|{M_{\rm h}})}$.

By inserting Equation (7) into Equation (6), one can obtain the relation between the probability distribution functions $P_{c,t}({M_*}|{M_{\rm h}})$, $P_{c,b}({M_*}|{M_{\rm h}})$ and $P_{c,r}({M_*}|{M_{\rm h}})$,

$$\begin{eqnarray} {P_{c,t}({M_*}|{M_{\rm h}})} &=&{P_{c,b}({M_*}|{M_{\rm h}})}{f_{\rm blue}}({M_{\rm h}})\nonumber \\ && +{P_{c,r}({M_*}|{M_{\rm h}})}{f_{\rm red}}({M_{\rm h}}). \end{eqnarray} \tag{ 8 }$$

In Section 2.3.1, we discuss the parametric functional forms for each ${P_{c,j}({M_*}|{M_{\rm h}})}$. In addition, based on the results of galaxy groups we also motivate the functional form for each ${f_{j}}({M_{\rm h}})$.

2.1.2. Satellite Galaxies

As mentioned previously, in our model the total distribution of satellite galaxies will be characterized by means of the satellite CSMF, $\Phi _{s,t}({M_*}|{M_{\rm h}})$. Similarly, the distribution of blue and red satellite galaxies will be characterized by means of $\Phi _{s,j}({M_*}|{M_{\rm h}})$, where the subscript j stands for either blue (b) or red (r) galaxies. As we will discuss in Section 2.3.2, the parametric functional forms employed for each $\Phi _{s,j}({M_*}|{M_{\rm h}})$ are motivated by the previous empirical results of galaxy groups. From these definitions, it follows that at a fixed M_h, the mean fractions of blue and red satellites as a function of M_* are

$$\begin{equation} {f_{j,\rm sat}({M_*}|{M_{\rm h}})}={\Phi _{s,j}({M_*}|{M_{\rm h}})}/{\Phi _{s,t}({M_*}|{M_{\rm h}})}, \end{equation} \tag{ 9 }$$

where, by definition, ${f_{r,\rm sat}({M_*}|{M_{\rm h}})}+{f_{b,\rm sat}({M_*}|{M_{\rm h}})}=1$.

Once the link between blue and red central galaxies to their host halos and the satellite CSMFs have been specified, we can compute the spatial clustering of galaxies as a function of stellar mass and color using the HOD model. This connection is introduced to use the observed 2PCFs as constraints to the model parameters.

In the HOD model (see Introduction for references), the real space 2PCF is computed by decomposing it into two parts: the one-halo term at small scales and the two-halo term at large scales. Here, we model the real space 2PCF for j-galaxies, that is, either for all, blue, or red galaxies as

$$\begin{equation} 1+{\xi _{{\rm gg},j}(r)}= \left[1+{\xi _{{\rm gg},j}^{\rm 1h}(r)} \right] + \left[1+{\xi _{{\rm gg},j}^{\rm 2h}(r)} \right]. \end{equation} \tag{ 10 }$$

The one-halo term describes the number of all possible pairs coming from galaxies in the same halos, whereas the two-halo term describes the same but in separate halos. Specifically, in the HOD model context the one-halo term is given by,

$$\begin{eqnarray} &&\frac{{\langle N_{j}(N_{j}-1)\rangle}}{2}{\lambda _{h}(r)}={\langle N_{c,j}\rangle}{\langle N_{s,j}\rangle}{\lambda _{c,s}(r)} \nonumber +\frac{1}{2}{\langle N_{s,j}\rangle ^2}{\lambda _{s,s}(r)}, \end{eqnarray} \tag{ 11 }$$

where the M_* and M_h dependences in the mean cumulative numbers defined above (see Equation (5)) were omitted for simplicity. The term λ_h(r) is the spatial distribution of the galaxies within the dark matter halo. In Equation (11) we assumed that central-satellite pairs follow a pair distribution function $\lambda _{c,s}(r)dr=4\pi \tilde{\rho }_{\rm NFW}({M_{\rm h}},r) r^2dr$, where $\tilde{\rho }_{\rm NFW}({M_{\rm h}},r)$ is the normalized NFW halo density profile. The satellite-satellite pair distribution, λ_{s, s}(r)dr, is then the normalized density profile convolved with itself, that is, $\lambda _{s,s}(r)dr=4\pi {\lambda }_{\rm NFW}({M_{\rm h}},r) r^2dr$, where λ_NFW is the NFW profile convolved with itself. An analytic expression for ${\lambda }_{\rm NFW}({M_{\rm h}},r)$ is given by Sheth et al. (2001). Both $\tilde{\rho }_{\rm NFW}$ and λ_NFW depend on the halo concentration parameter, c_NFW. N-body numerical simulations show that this parameter anti-correlates with mass, c_NFW = y₀ − y₁  × logM_h, although with a large scatter. Note that we have assumed that the occupational number of satellite galaxies follows a Poisson distribution (i.e., 〈N_{s, j}(N_{s, j} − 1)〉 = 〈N_{s, j}〉²). The above is based on the results of high-resolution N-body simulations (e..g, Kravtsov et al. 2004) and hydrodynamic simulations of galaxy formation (e.g., Zheng et al. 2005).

On large scales, we model the two-halo term as

$$\begin{equation} {\xi _{{\rm gg},j}^{\rm 2h}(r)}=b^2_{g,j}\zeta ^2(r)\xi _{{\rm mm}}(r), \end{equation} \tag{ 12 }$$

where ξ_mm(r) is the nonlinear matter correlation function (Smith et al. 2003), ζ(r) is the scale dependence of dark matter halo bias (Tinker et al. 2005, see their Equation (B7)), and

$$\begin{equation} b_{g,j}=\frac{1}{n_g}\int b({M_{\rm h}}){\langle N_{j}({M_{\rm h}})\rangle}\theta (r;R_{{\rm vir}}({M_{\rm h}})) \phi ({M_{\rm h}})d{M_{\rm h}} \end{equation} \tag{ 13 }$$

is the galaxy bias. In the above Equation (13), $b({M_{\rm h}})$ is the halo bias function given by Tinker et al. (2010). The term $\theta (r;R_{{\rm vir}}({M_{\rm h}}))$ is the Heaviside function and has been introduced to take into account that two galaxy pairs cannot be within the same halo. Wang et al. (2004) showed that the above method accurately describes the correlation function (see also Y12). Analogously to Leauthaud et al. (2012), we modified the original Wang et al. (2004) method to match our definition of the HMFs and bias relation. This fitted relation is based on spherical-over-density halo finding algorithms, where halos are allowed to overlap as long as their centers are not contained inside the virial radius of a larger halo; for details see Tinker et al. (2010).

Observations of galaxy clustering are usually characterized using the galaxy projected correlation function, ω_p(r_p). In our model, we relate ω_p(r_p) to the real-space correlation function, ξ_gg(r), by integration over the line of sight:

$$\begin{equation} \omega _{{\rm p},j}(r_{\rm p})=2\int _0^{\pi _{\rm max}}\xi _{{\rm gg},j} \left(\sqrt{r^2_{\rm p}+r_{\pi }^2} \right)dr_{\pi }. \end{equation} \tag{ 14 }$$

For consistency with the observed ω_p(r_p) we set π_max = 45 Mpc h⁻¹.

In order to constrain the model, some assumptions for the different distributions should be made. In this subsection we describe these assumptions in detail.

2.3.1. Central Galaxies

As noted in Section 2.1, our model for central galaxies is completely specified once the CSMF of blue and red central galaxies are defined, see Equations (6)–(8). Here, both $P_{c,b}({M_*}|{M_{\rm h}})$ and $P_{c,r}({M_*}|{M_{\rm h}})$ are modeled as lognormal distributions:

$$\begin{eqnarray} {P_{c,j}({M_*}|{M_{\rm h}})}d{M_*} &=& \frac{\log e}{\sqrt{2\pi {\sigma _{j}}^2}} \nonumber \\ && \times \exp \left[{-\frac{\log ^2({M_*}/M_{*,j}({M_{\rm h}}))}{2\sigma _{j}^2}}\right]\frac{d{M_*}}{{M_*}}, \nonumber\\ \end{eqnarray} \tag{ 15 }$$

where $M_{*,j}({M_{\rm h}})$ is either the mean SHMR of blue, M_{*, b}(M_h), or red, M_{*, r}(M_h), central galaxies, and the standard deviations σ_j's are defined as the corresponding scatters around the mean relations. We assume that the σ_j's are independent of M_h. Both σ_b and σ_r are considered to be additional parameters that are fitted separately in the model.

The scatters σ_j are composed by an intrinsic component and a measurement error component. The measurement error components, $\sigma _{j}^{\rm e}$, are dominated mainly by errors in individual galaxy stellar mass estimates and redshift estimates. (see, e.g., Behroozi et al. 2010; Leauthaud et al. 2012). Additionally, their value may depend on galaxy color (Kauffmann et al. 2003). Thus, if the measurement error components are known, it is possible to constrain the intrinsic components in our model, $\sigma _{j}^{\rm i}$. However, given the poor information for the real values of $\sigma _{j}^{\rm e}$, we opt for constraining total scatters, σ_j, as upper limits to the intrinsic components in our analysis. Nevertheless, in Section 4.2.1 we estimate conservative values by assuming that both components are independent,

$$\begin{equation} \sigma _{j}^2 = \left(\sigma _{j}^{\rm i} \right)^2 + \left(\sigma _{j}^{\rm e} \right)^2. \end{equation} \tag{ 16 }$$

and a constant value for $\sigma _{j}^{\rm e}$ reported in Kauffmann et al. (2003).

In order to describe the mean SHMR of blue and red central galaxies, we adopt the parameterization proposed in Behroozi et al. (2013),

$$\begin{eqnarray} &&\log M_{*,j}=\log (\epsilon _{c,j} M_{1,j})+g_c(x)-g_c(0), \end{eqnarray} \tag{ 17 }$$

where

$$\begin{equation} g_c(x)=\delta _{c,j}\frac{(\log (1+e^x))^{\gamma _{c,j}}}{1+e^{10^{-x}}}-\log (10^{\alpha _{c,j} x}+1). \end{equation} \tag{ 18 }$$

and $x=\log ({M_{\rm h}}/M_{1,j})$. This function behaves as a power law with slope α_{c, j} at masses much smaller than M_{1, j}, and as a sub-power law with a slope γ_{c, j} at large masses. A simpler function with less parameters could be used (e.g., Yang et al. 2008), however, as shown in Behroozi et al. (2013), the function as given by Equation (18) is necessary in order to accurately map the HMF into the observed GSMFs, which are more complex than a singular Schechter function (see Section 3.1 below).

Deviating from previous studies, in our model the probability distribution for all central galaxies, $P_{c,t}({M_*}|{M_{\rm h}})$ (Equation (8)), is predicted rather than being an assumed prior function (see also More et al. 2011). Typically, this distribution is assumed to be a lognormal function with a fixed width. In our model, by means of Equation (8), the mean SHMR of all central galaxies, M_*(M_h), is given by the weighted sum of the mean blue, M_{*, b}(M_h), and red, M_{*, r}(M_h), central SHMRs,

$$\begin{eqnarray} \langle \log {M_*}({M_{\rm h}})\rangle &=& {f_{\rm blue}}({M_{\rm h}})\langle \log {M_{*,b}}({M_{\rm h}})\rangle \nonumber \\ && +{f_{\rm red}}({M_{\rm h}})\langle \log {M_{*,r}}({M_{\rm h}})\rangle. \end{eqnarray} \tag{ 19 }$$

This equation relates the mass relation commonly obtained through the HOD model and the CSMF formalism with the mass relations of blue and red centrals. We also compute the intrinsic scatter around the average relation as

$$\begin{equation} \sigma _A({M_{\rm h}})=\left(\int P_{c,t}(\mathcal {M_*}|{M_{\rm h}})\mu ^2d\mathcal {M_*}\right)^{1/2}, \end{equation} \tag{ 20 }$$

where $\mu =\log \mathcal {M_*}-\langle \log {M_*}({M_{\rm h}})\rangle$. Note that we are not assuming that the scatter around the mean SHMR of all central galaxies is constant and lognormally distributed. Because the value $\sigma _A({M_{\rm h}})$ is directly related to the σ_j's, it is also a combination of the intrinsic and error measurement components (see Equation (16)).

To fully characterize the CSMFs of blue and red central galaxies, we need to propose a model for the fraction of halos hosting blue/red central galaxies, ${f_{j}}({M_{\rm h}})$ (see Equation (7)). As noted by previous authors (cf. Hopkins et al. 2008; Tinker & Wetzel 2010; Rodríguez-Puebla et al. 2011; Tinker et al. 2013), assuming a specific function of the quenched (red) fraction makes an implicit choice of the mechanisms that prevent central galaxies from being actively star-forming. For example, in Rodríguez-Puebla et al. (2011), the fraction of halos able to host blue centrals was obtained by excluding from the ΛCDM HMF (1) those halos that suffer a major merger at z < 0.8, and (2) those that follow the observed rich group/cluster mass function (blue galaxies are not found in the center of rich groups/clusters).

In a recent analysis of the Yang et al. (2007) galaxy group catalog, Woo et al. (2013) studied the fraction of quenched central galaxies as a function of both stellar and halo mass. From their analysis, the authors concluded that the fraction of quenched central galaxies correlates more strongly with M_h than with M_*. Furthermore, Woo et al. (2013) concluded that the phenomenological results presented in Peng et al. (2012) for central galaxies are still valid by substituting M_h for M_* in the Peng et al. (2012) model. In other words, central galaxies are on average quenched once their host dark matter halo reaches a characteristic mass. According to the phenomenological results of Peng et al. (2012) and Woo et al. (2013), the fraction of halos hosting red (or blue) centrals can be described as ${f_{\rm red}}({M_{\rm h}})=1/(1+[{M_{\rm h}^{\star}}/{M_{\rm h}}])$, where $M_{\rm h}^{\star }$ is a characteristic mass above which halos are mostly occupied by red central galaxies; for ${M_{\rm h}}>{M_{\rm h}^{\star}}$, ${f_{\rm red}}>0.5$ (or ${f_{\rm blue}}=1-{f_{\rm red}}<0.5$). Given the central role that this function plays in our model, we have generalized it slightly as follows:

$$\begin{equation} {f_{\rm red}}({M_{\rm h}})=\frac{1}{b+({M_{\rm h}^{\star}}/{M_{\rm h}})}, \end{equation} \tag{ 21 }$$

where $M_{\rm h}^{\star }$ and b are free parameters to be constrained. For practical purposes we redefine ${M_{\rm h}^{\star}}=\beta \times 10^{12}\, {M_{\odot }}$, where β is the free parameter to be constrained.

For the distinct HMF, we use the fit to large N-body cosmological simulations presented in Tinker et al. (2008). Here we define halo masses at the virial radius, i.e., the radius where the spherical overdensity is Δ_vir times the mean matter density, with Δ_vir = (18π² + 82x − 39x²)/Ω(z), Ω(z) = ρ_m(z)/ρ_crit, and x = Ω(z) − 1.

Finally, for the relation of the halo concentration parameter c_NFW with mass, we use the fit to N-body cosmological simulations by Muñoz-Cuartas et al. (2011).

2.3.2. Satellite Galaxies

The parametric functions for describing the satellite CSMFs, $\Phi _{s,j}({M_*}|{M_{\rm h}})$, used here are given through the average satellite cumulative probabilities:

$$\begin{equation} {\Phi _{s,j}({M_*}|{M_{\rm h}})}=\frac{\partial }{\partial {M_*}}{\langle N_{s,j}(>{M_*}|{M_{\rm h}}) \rangle}, \end{equation} \tag{ 22 }$$

where

$$\begin{equation} {\langle N_{s,j}(>{M_*}|{M_{\rm h}}) \rangle}={\langle N_{c}(>{M_*}|{M_{\rm h}})\rangle}\int _{M_*}^\infty S_j(M_*^{\prime }|{M_{\rm h}})dM_*^{\prime }, \end{equation} \tag{ 23 }$$

and

$$\begin{equation} S_j({M_*}|{M_{\rm h}})=\phi ^*_{j}X^{\alpha _{s,j}}e^{-X}dX, \end{equation} \tag{ 24 }$$

and $X={M_*}/{M_{*,\rm sat}^j}$. As before, the subscript j refers either to red (r) or blue (b) galaxies. Note that the first factor in Equation (23) (the average cumulative probability of having a central galaxy larger than M_* in a halo of mass M_h) imposes the restriction that there are not galaxy groups containing only satellite galaxies and that, on average, the central galaxy is the most massive galaxy in the group. In the above equation we assume that the faint-end slopes α_{s, j} are independent of halo mass, while the normalizations factors $\phi ^*_{j}$ and the characteristic masses ${M_{*,\rm sat}^j}$ change as a function of M_h as follows,

$$\begin{equation} \log \phi ^*_{b}({M_{\rm h}})=\phi _{0,j}+\phi _{1,j}\times \log \left(\frac{{M_{\rm h}}}{10^{12}\,{M_{\odot }}}\right), \end{equation} \tag{ 25 }$$

and

$$\begin{equation} \log {M_{*,\rm sat}^j}= c_0 + c_j\times \log \left(\frac{{M_{\rm h}}}{10^{12}\,{M_{\odot }}}\right). \end{equation} \tag{ 26 }$$

respectively. Note that the normalization in the last equation, c₀, is the same for blue and red satellites.

The parameterization presented above is partially motivated by the phenomenological model discussed in Peng et al. (2012). These authors argue that the shape of the distribution of blue and star-forming satellite galaxies is always a Schechter function with a characteristic mass $M_{*,\rm sat}^b$ and a faint-end slope α_{s, b}. For the distribution of red and quenched galaxies, they argued that it is described by a double Schechter with a characteristic mass similar to that of blue satellites and with slopes $\alpha _{s_1,r}=1+\alpha _{s,b}$ and $\alpha _{s_2,r}=\alpha _{s,b}$. We experimented with a simple and double Schechter function and, in light of the observations we use to constrain the model (Section 2.4), there is not a statistical improvement in the fittings from one case to another. We also checked that the CSMFs of red satellites from the Y12 galaxy group catalog can be fitted with a double or simple Schechter function with α_{s, r}. Therefore, we parameterize the red satellite CSMFs with a simple Schechter function. In order to tackle this question as generally as possible, we assumed that α_{s, b} and α_{s, r} are two different parameters.

2.4.1. Parameters in the Model

2.4.2. Fitting Procedure

In order to constrain the free parameters in our semi-empirical model, we combine several observational data sets. These data sets are the GSMFs and its division into central and satellites and the 2PCFs in different stellar mass bins for all, blue, and red galaxies. In order to sample the best-fit parameters that maximize the likelihood function $L\propto e^{-\chi ^2/2}$, we use the Markov Chain Monte Carlo (MCMC) method. The details for the full procedure can be found in RAD13.

We compute the total χ² as

$$\begin{equation} \chi ^2=\chi ^2_{\rm GSMF}+\chi ^2_{\rm 2PC}, \end{equation} \tag{ 27 }$$

where we define the GSMFs,

$$\begin{equation} \chi ^2_{\rm GSMF}=\sum _{i,j}\chi ^2_{\phi _{i,j}}, \end{equation} \tag{ 28 }$$

the sum over i refers to the type (all, centrals, and satellites) and the sum over j refers to color (blue and red). For the correlation functions,

$$\begin{equation} \chi ^2_{\rm 2PC}=\sum _k\left(\chi ^2_{\omega _k(r_p)_t}+\chi ^2_{\omega _k(r_p)_b}+ \chi ^2_{\omega _k(r_p)_r}\right), \end{equation} \tag{ 29 }$$

where the subscripts t, b, and r refer to all (total), blue, and red galaxies, respectively. The sum over k refers to summation over different stellar mass bins. The fittings are made to the data points (with their error bars) for each GSMF and 2PCF.

Once the model parameters are constrained, the model predicts the following relations and quantities:

• 1.  
  The mean SHMRs of blue and red central galaxies as well as of the average SHMR of all central galaxies. The latter is what is commonly constrained in the literature through the SHAM.
• 2.  
  The scatter around the blue, red, and average SHMRs of central galaxies. Recall that we assume lognormal distributions with constant widths (i.e., scatters independent of M_h) for blue and red centrals. In contrast, the distribution for all central galaxies will be predicted according to Equation (8). Similarly, the scatter around the mean SHMR is predicted according to Equation (20). Note that these scatters are upper limits to the intrinsic scatters. These can be estimated if the values for the measurement error components are known (see Equation (16)).
• 3.  
  The fraction of distinct halos hosting red (or blue) central galaxies as a function of M_h, f_red(M_h).
• 4.  
  The blue and red satellite CSMFs (as a function of M_h or central M_*), and therefore the fractions of blue (or red) satellites as a function of M_* for a given host halo mass, $f_{b,\rm sat}({M_*}|{M_{\rm h}})$.

The model described in this section for constraining SHMRs separately for blue and red central galaxies is partially related to previous models discussed in the literature. In the following, we outline the main differences between our model and previous models.

As discussed in the Introduction, when connecting central galaxies to their host dark matter halos, previous models assumed that the probability distribution of M_* for a given M_h is given by an unimodal (lognormal) distribution function (see, e.g., Y12; Leauthaud et al. 2012) In addition, in previous models the scatter around the mean SHMR was assumed to be constant with halo mass. In our approach, we relax these assumptions by assuming separate probability (lognormal) distribution functions for blue and red centrals in such a way that the average (density-weighted) distribution function can be bimodal and formally may depend on M_h. This is similar to the model employed in More et al. (2011) for obtaining SHMRs for blue and red central galaxies based on the analysis of the satellite kinematics from a local sample in the SDSS, and that of Tinker et al. (2013) based on a combined analysis of galaxy clustering and weak lensing for active and passive galaxies in the COSMOS field. Note that if both distributions are found to be statistically similar after being constrained with observations, then the galaxy color is not responsible for shaping the intrinsic scatter of the SHMR. In the opposite case, the result is evidence that the SHMR is segregated by color.

In essence, we find that our model is closer to that developed in Tinker et al. (2013). The main difference is that those authors characterized the galaxy distribution using the HOD model (see also Leauthaud et al. 2012), whereas in our analysis this characterization is based on the CSMF formalism (see, e.g., Yang et al. 2003; van den Bosch et al. 2003; Cooray 2006; Yang et al. 2012). Note, however, that in our model the HOD model is related to the CSMF formalism through Equation (22), see Section 2.3.2.

As for the observational constraints, in our analysis we include information on the decomposition of the GSMF into central and satellite galaxies from robustly constructed galaxy groups; this information is relevant for constraining the CSMFs (for extensive discussions see Yang et al. 2007, 2008). This is a major difference between our analysis and that carried out in Tinker et al. (2013), where weak lensing information from stacking data is used for the observational constraints of their HOD model.

We construct the GSMFs from the New York Value Added Galaxy Catalogue, NYU-VAGC, based on the SDSS DR7. We use the sample selection as given in the halo-mass based group catalog of Y12. This galaxy group catalog represents an updated version of Yang et al. (2007).⁵ Therefore, our galaxy sample has the same cuts and depurations as in this catalog. For further details with respect to the group catalog see Yang et al. (2007). The total number of galaxies used for constructing the GSMF is 639,359. The Y12 catalog uses colors and magnitudes based on the standard SDSS Petrosian radius. Following Blanton et al. (2003) and Yang et al. (2009a), we use the evolution correction at z = 0.1 given by E(z) = 1.6(z − 0.1). For the K-correction, we use an analytical model as described in the Appendix. In this model, the K-correction term depends on both redshift and color, g − r, (i.e., K = K(z, g − r)). K-corrections and absolute magnitudes at z = 0.1 computed within this scheme are accurately recovered with typical percentage errors less than ∼10% and ∼1% on average, respectively.

For the stellar masses, we use those reported in the MPA-JHU DR7 database.⁶ These masses were calculated from photometry-spectral energy distribution fittings assuming a Chabrier (2003) IMF; for details, see Kauffmann et al. (2003). We found that in our sample approximately ≈9% of the galaxies lack stellar mass measurements. Because this fraction is not negligible, we decided to calculate the stellar masses for these galaxies by using the color-dependent mass-to-light ratio given by Bell et al. (2003). For this subsample of galaxies we applied a correction of −0.1 dex to be consistent with the Chabrier (2003) IMF adopted in this paper. We also checked to see if these galaxies were not particularly biased in mass or color. The masses were not determined to be likely due to issues in their spectra and/or the stellar populations synthesis fits.

Following Moustakas et al. (2013), we adopt a flat stellar mass completeness limit of M_* = 10⁹ M_☉ for the calculation of the GSMF. As noted by these authors, this limit is above the surface brightness and stellar mass-to-light ratio completeness limits of the SDSS, see Blanton et al. (2005); Baldry et al. (2008). The GSMF is estimated as

$$\begin{equation} \phi _{g}({M_*})=\frac{1}{\Delta \log {M_*}}\sum _{i=1}^N\frac{\omega _i}{V_{{\rm max},i}}, \end{equation} \tag{ 30 }$$

where ω_i is the correction weight completeness factor in the NYU-VAGC and for each galaxy, V_{max, i} is given by

$$\begin{equation} V_{{\rm max},i}=\int _{\Omega }\int _{z_l}^{z_u}\frac{d^2V_c}{dzd\Omega }dzd\Omega. \end{equation} \tag{ 31 }$$

Here, z_l = 0.01 and z_u = min(z_{max, i}, 0.2), Ω is the solid angle of the SDSS DR7, and V_c is the comoving volume (Hogg 1999). The maximum redshift at which each galaxy can be observed, z_{max, i}, is computed by solving iteratively the distance modulus equation, i.e.,

$$\begin{equation} m_{{\rm lim},r}-M_{r,i}^{0.1}=5\log D_L(z_{{\rm max},i})- 25-K(z_{{\rm max},i})+E(z_{{\rm max},i}), \end{equation} \tag{ 32 }$$

and each observed galaxy should satisfy the apparent magnitude of the Y12 group catalog, m_{lim, r} = 17.72. The term D_L(z) is the distance modulus (Hogg 1999).

For each stellar mass bin, errors are computed using the jackknife technique. We do so by dividing the Y12 galaxy group sample into 200 subsamples of approximately equal size and calculating ϕ_{g, i}(M_*) each time. Then, errors are estimated as

$$\begin{equation} \sigma _{\phi }=\left[\frac{N-1}{N}\sum _{i=1}^{N}(\phi _{g,i}-\langle \phi _{g}\rangle)^2\right]^{1/2}, \end{equation} \tag{ 33 }$$

where N = 200, ϕ_{g, i} is the GSMF of the sample i, and 〈ϕ_g〉 is the average over the ensemble.

We divide our galaxy sample into two wide groups: blue and red galaxies. This division roughly corresponds to late-type/star-forming and early-type/passive galaxies, respectively. Note that for this division we are using K-corrected colors to z = 0.1, ^0.1(g − r). Red/blue galaxies are defined based on the Li et al. (2006) color-magnitude criteria. These authors separated galaxies into blue and red by using a bi-Gaussian fitting model to the color distribution in many absolute magnitude bins; see Li et al. (2006) for details. Because of dust extinction, blue star-forming and highly inclined galaxies could be classified as a red, passive galaxies (e.g., Maller et al. 2009). In Section 5.1 we discuss the impact of this possible contamination in our color division.

Finally, we also divide our galaxy sample into central and satellite galaxies according to the galaxy groups identified in the latest version of the Yang et al. (2007) group catalog. In this paper, we define central galaxies as the most massive galaxies within their group.

3.1.1. Total Galaxy Stellar Mass Functions

Figure 2 shows the resulting SDSS MPA-JHU DR7 GSMFs for all, blue, and red galaxies estimated as described above (empty circles with error bars). For comparison, in the upper left panel of the same figure we reproduce local estimations of the total GSMFs based on SDSS reported in Li & White (2009); Bernardi et al. (2010); Yang et al. (2012); and Moustakas et al. (2013). In the bottom left and right panels we reproduce the Y12 GSMFs corresponding to blue and red galaxies, as well as the Moustakas et al. (2013) GSMFs corresponding to active and passive galaxies. Note that the GSMFs separated into blue and red components according to the Li et al. (2006) color-magnitude criteria are similar to those of active and passive GSMFs (from Moustakas et al. 2013) for M_* ≲ 10¹¹M_☉.

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In general, our NYU-VAGC/SDSS MPA-JHU DR7 GSMFs for all, blue, and red galaxies are consistent with previous estimates, except at the high-mass end, which has a shallower fall than most previous ones, but is in good agreement with the total GSMF from Bernardi et al. (2010). In fact, the function could be even shallower if one takes into account accurate photometric profile fits when calculating total luminosities or M_* (Bernardi et al. 2013). In Section 5.4.2, we apply corrections to our galaxy stellar masses in order to take this effect into account.

As reported by previous authors, we find that a single Schechter function is not consistent with the total GSMF (see, e.g., Baldry et al. 2008, 2012; Li & White 2009; Drory et al. 2009; Pozzetti et al. 2010; Moustakas et al. 2013; Bernardi et al. 2013; Tomczak et al. 2014). The high-mass end of our GSMF is also shallower than an exponential decay. For completeness, we present the best fits to our GSMFs. Following Bernardi et al. (2010), we fit our GSMFs by using a function composed of a single Schechter plus a Schechter function with a sub-exponential decay at the high-mass end,

$$\begin{equation} \phi _g(X)dX=\phi _1^*X_1^{\alpha _1}e^{-X_1}dX_1+ \phi _2^*X_2^{\alpha _2}e^{-X_2^\beta }dX_2, \end{equation} \tag{ 34 }$$

where $X_i={M_*}/\mathcal {M}^*_i$ with i = 1, 2. The corresponding best-fit parameters are reported in Table 1. For blue galaxies we also employed Equation (34), and for red galaxies we found that a single Schechter function with a sub-exponential decay gave a good fit to the data. The parameters of these fits are also given in Table 1.

Table 1. Fit Parameters to the SDSS DR7 GSMFs

GSMF	$\log (\phi _1^*)$	α1	$\log (\mathcal {M}^*_1)$	$\log (\phi _2^*)$	α2	β	$\log (\mathcal {M}^*_2)$
(MPA-JHU M*'s)	(Mpc−3 dex−1)		(M☉)	(Mpc−3 dex−1)			(M☉)
All ......	−2.46 ± 0.30	−1.32 ± 0.22	9.55 ± 0.32	−2.24 ± 0.03	−0.64 ± 0.15	0.65 ± 0.04	10.49 ± 0.17
Blue ......	−3.07 ± 0.13	−1.58 ± 0.08	10.17 ± 0.13	−2.69 ± 0.07	−0.23 ± 0.20	0.51 ± 0.04	9.82 ± 0.19
Red ......	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	−2.60 ± 0.02	−0.79 ± 0.02	0.80 ± 0.02	10.83 ± 0.03
(Corrected M*'s; see Section 5.4.2)
All ......	−3.82 ± 0.20	−1.60 ± 0.06	11.64 ± 0.06	−2.44 ± 0.05	−0.15 ± 0.12	0.58 ± 0.06	10.14 ± 0.20

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3.1.2. Central and Satellite Galaxy Stellar Mass Functions

For the correlation functions, we use the Li et al. (2006) measurements of the projected 2PCF, ω_p(r), in five different stellar mass bins, and for all, blue, and red galaxies. These measurements were done based on a sample of ∼2  ×  10⁵ galaxies from the SDSS DR2. Note that (1) the color-magnitude criterion to separate galaxies into blue and red by Li et al. (2006) is the same that we used for constructing the blue and red GSMFs, and (2) in the calculation of our GSMFs we use the same stellar mass inferences as in Li et al. (2006).

In our analysis, we are using only the diagonal of the covariance matrix given in Li et al. (2006). Unfortunately, the full covariance matrix is not available (C. Li 2011, private communication).

In each panel of Figure 3 we plot the best-fit model GSMFs for all, blue, and red galaxies (solid lines). In the same panels, we show the decomposition of the GSMFs into central (long-dashed lines) and satellites (dot-short-dashed lines) corresponding to all, blue, and red galaxies. In general, our model fits well describes the observed GSMFs, decomposed into central and satellites (compare Figure 3 with Figure 2).

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Figure 4 shows the observed projected 2PCFs reported in Li et al. (2006, filled circles with error bars) and the best model fits (solid lines). The projected 2PCFs are for all, blue, and red galaxies (black, blue, and red colors, respectively) in five different stellar mass bins. In order to get better visual comparisons of our fits to observations, we plot ω_p × r_p (instead of ω_p) as a function of r_p. For clarity, our fits and the data points for blue and red galaxies have been shifted by +1 dex and −1 dex, respectively. In general, our fits well describe the observations for all mass bins and in almost all separations. We note, however, that at large separations our fits tend to lie below observations in the stellar mass bins 10 < log (M_*/M_☉) < 10.5 and 10.5 < log (M_*/M_☉) < 11. It is known that at large separations the correlation functions are affected by cosmic variance in volume-limited samples. This effect in the SDSS galaxies has been investigated in detail by Zehavi et al. (2005, 2011), where the authors find that the most significant cosmic variance effect appears due to the presence of a supercluster at z ∼ 0.08, the Sloan Great Wall (Gott et al. 2005). These authors conclude that the inclusion of this supercluster in the calculation of the correlation functions causes an anomalous high amplitude of galaxy clustering at large separations in samples with −21 < M_{r, 0.1} < −20. This magnitude bin roughly corresponds to a stellar mass bin 10 < log (M_*/M_☉) < 11. Thus, the observational values of ω_p at large radii in this mass bin could be overestimated.

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In order to quantify the effect of cosmic variance in our fits, we have excluded separations larger than r_p > 10h⁻¹ Mpc in the projected 2PCFs in the bins that encompass masses in the range 10 < log (M_*/M_☉) < 11 and recalculated the χ²/dof by using our best-fit parameters. The exclusion of these separations leads to a substantial decrease in the reduced χ², χ²/dof =1.47. This implies that cosmic variance effects could affect our fits. To test this, we recalculated our model parameters using this modified data set regarding the 2PCFs. We found that the values of all the model parameters remain similar to those obtained previously, particularly those related to the SHMRs. The insensitivity of HOD model parameters to the data at large separations has been reported previously (see, e.g., Zehavi et al. 2011). For a fixed cosmology the HOD parameters have less freedom to adjust the large-scale correlation relative to the (more) robust inferences at small separations, r_p < 2h⁻¹ Mpc. Therefore, we conclude that our model-fit parameters are robust against uncertainties due to cosmic variance in the 2PCFs at large separations.

Conversely, in the stellar mass bin 9 < log (M_*/M_☉) < 9.5 we note that The reasons behind this difference are not clear, but we quantified the impact of it in our goodness-of-fit. Similar to large separations, we recalculated the χ²/dof, but this time excluding the information of the correlation functions at this mass bin. The resulting χ²/dof is 1.71. This means that the correlation function from this stellar mass bin does not add substantial information to constrain our model parameters. Note, however, that in order to obtain robust inferences to our model parameters, we also included the information about the central and satellite GSMFs.

The derivation of the SHMRs and their scatters for local blue and red central galaxies is the main goal of this paper. These mass relations and the corresponding M_*-to-M_h ratios versus M_h, as constrained by means of our model, are shown in the upper and lower panels of Figure 5. Blue solid and red long-dashed lines show the mean relations of blue and red centrals, respectively. Shaded areas show the standard deviations of the (lognormal-distributed) scatter (see Section 2.3.1 and Equation (15)). Recall that this scatter is composed of an intrinsic component and a measurement error component (Equation (16)). This scatter should be considered an upper limit of the intrinsic component. In Section 4.2.1, we estimate values for the intrinsic scatters once we introduce conservative estimations for measurement error components. Black dots with error bars represent the average central relations, calculated as in Equation (19), and its corresponding scatter (standard deviation, calculated using Equation (20)). Recall that Equation (19) is a density-weighted average, so the mean SHMR for all central galaxies is located in between the SHMRs of blue and red centrals. However, at high masses the average SHMR is practically equal to that of red galaxies. Almost all massive halos host red central galaxies, as shown below. The dotted vertical line indicates the lower limit in halo mass at which our average SHMR has been robustly constrained. Note that we did not assume any functional form for the shape or scatter of the average SHMR. Instead, they are a direct prediction and the result of separately constraining the SHMRs of blue and red central galaxies.

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Our results point out that the SHMRs of blue and red centrals are different. In other words, the SHMR of central galaxies is segregated by color. For a given M_h, the M_* or M_*-to-M_h ratio of blue galaxies is always larger than that of red centrals. The minimum difference is of 0.16 dex at M_{h, min} = 5 × 10¹¹ M_☉, close to the mass where the halos contain the same fraction of blue and red centrals (see below, Section 4.3). The difference increases up to ≈0.24 dex at ${M_{\rm h}}=5\times 10^{12}$ M_☉ and for larger masses it remains roughly constant. For masses <M_{h, min}, the difference strongly increases. At any mass, the differences are larger than the 1σ scatter of the relations, σ_b and σ_r, respectively (see also Section 4.2.1 below). This can be also appreciated in Figure 6, where we plot the conditional stellar mass distributions of blue and red central galaxies for four different halo masses, blue and red lines, respectively. The shaded areas show the 1σ confidence intervals around these relations. The dots with error bars show the same, but for all central galaxies. The distributions of blue and red centrals, which are related to the scatters around the corresponding SHMRs, are different. Besides, as will be shown in the next subsection and in Figure 7, the scatters around the constrained blue and red SHMRs compared to the 1σ confidence intervals obtained from our MCMC run are smaller than the corresponding scatters. Thus, the differences in the values of the mean SHMRs of blue and red centrals are significant at the level of error analysis.

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Figure 6 shows that for ${M_{\rm h}} \lesssim 3\times 10^{11}$ M_☉, the galaxy population of centrals is dominated by blue galaxies over the red ones, whereas at ${M_{\rm h}}\sim 3\times 10^{12}$ M_☉, red galaxies are already the dominant population but there is still a non-negligible fraction of blue galaxies. Instead, at ${M_{\rm h}}\sim 3\times 10^{13}$ M_☉, nearly all central galaxies are red. It is at masses $10^{11.5}\, {M_{\odot }} \lesssim {M_{\rm h}} \lesssim 10^{12.5}\, {M_{\odot }}$ that the SHMR of central galaxies has a clear bimodal distribution. At larger and lower masses, the average SHMR for all central galaxies is given by the SHMR of red and blue centrals, respectively (Figure 5).

4.2.1. The Scatter of the SHMRs

In our model, we assumed that the stellar mass distribution of both blue and red central galaxies is lognormal. Additionally, we assumed that the widths (scatters) of these distributions are free parameters in our model. These scatters were assumed to be independent of M_h (Equation (15)). The values constrained for these scatters are σ_b = 0.118 ± 0.020 dex and σ_r = 0.136 ± 0.010 dex, respectively (solid blue and short-dashed red lines surrounded by shaded areas in Figure 5). In other words, the scatter around the SHMRs of red and blue central galaxies is higher for the former than for the latter, although the error bars overlap slightly. This implies that the SHMR of blue central galaxies is tighter than that of red centrals.

The scatter around the density-averaged (blue + red) SHMR, σ_A, is plotted in Figure 7 (black long-dashed line) along with 1σ confidence interval (gray shaded area). Similar to Figure 5, the dotted vertical line indicates the lower limit in halo mass at which the scatter σ_A has been robustly constrained. For ${M_{\rm h}}\sim 10^{11.3}$ to ∼10¹⁵ M_☉, σ_A changes from ∼0.20 dex to ∼0.14 dex. Note that σ_A is constant for masses above ${M_{\rm h}}\sim 10^{13}$ M_☉. The dependence of σ_A on M_h naturally arises according to Equation (8). Because the majority of high-mass halos, ${M_{\rm h}}$ ≳ 10^13.5 M_☉, host red central galaxies (see Figure 6; see also Figure 8 below) then σ_A ∼ σ_r ≈ 0.14 dex. In contrast, the increasing of σ_A with decreasing M_h is the result of the color bimodality in the average SHMR for masses below ${M_{\rm h}} \lesssim 10^{12.5}$ M_☉ (see Figure 6).

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As mentioned in Section 2.3.1, the scatters σ_j reported above consist of an intrinsic component, $\sigma _{j}^{\rm i}$, and of a measurement errors component, $\sigma _{j}^{\rm e}$, see Equation (16). Following the results by Behroozi et al. (2010) and Leauthaud et al. (2012), we assume that the dominant source of measurement error comes from individual stellar mass estimates. For the data employed in our analysis, we used stellar masses from an update of Kauffmann et al. (2003). In that paper, the authors found that the 95% confidence interval from their stellar mass estimates is typically a factor of ∼2 in mass. As noted by the authors, in a Gaussian distribution this corresponds to four times the standard error, resulting in a standard deviation of ∼0.075 dex. This is consistent with the standard deviation reported in Conroy et al. (2009) for local luminous red galaxies. Measurement errors might be different between red and blue galaxies, for example Kauffmann et al. (2003) found that these errors are somewhat smaller for older galaxies with D_n(4000) > 1.8 than for younger galaxies with D_n(4000) < 1.8. Unfortunately, the authors do not provide values for their confidence intervals.

We assume a conservative value of $\sigma _{j}^{\rm e}=0.07$ dex for both blue and red central galaxies. We estimate the intrinsic scatters of blue and red central galaxies by deconvolving from measurement errors using the 3 × 10⁵ MCMC models generated for fitting the data. We estimate conservative values of $\sigma _{b}^{\rm i}=0.094\pm 0.024$ and $\sigma _{r}^{\rm i}=0.116\pm 0.012$ for blue and red centrals, respectively.

Finally, in Figure 7, we also plot the magnitude of the 1σ confidence intervals around the mean SHMRs for blue, red, and all central galaxies (solid blue, short-dashed red, and long-dashed black lines without shaded areas, respectively). These confidence intervals were obtained from the 3 × 10⁵ MCMC models generated to sample the best-fit parameters in our model. The confidence intervals take into account the error bars from our set of observational constraints. Note that we are not taking into account any source of systematical uncertainty, which is usually dominated by systematical uncertainties in stellar mass inferences, and they may be up to 0.25 dex (Behroozi et al. 2010). The confidence intervals around the mean SHMR of all central galaxies (black long-dashed line) are much smaller than the scatter σ_A (black long-dashed line surrounded by a gray area). Similarly, the confidence intervals around the mean SHMRs of blue and red central galaxies are much lower than their scatters σ_j, excepting at low and high masses for red and blue galaxies, respectively. At these masses, the 1σ confidence intervals around the mean SHMRs increase due to the scarce data and large observational error bars in these limits.

We parameterized the fraction of halos hosting blue/red (approximately active/quenched) central galaxies by using a general observationally motivated dependence, see Section 2.3.1. In Figure 8, we show the constrained blue fraction as a function of M_h, ${f_{\rm blue}}({M_{\rm h}})$. Recall that f_blue is the complement of f_red, ${f_{\rm blue}}({M_{\rm h}})= 1 - {f_{\rm red}}({M_{\rm h}})$. The fraction of blue centrals decreases with M_h roughly a factor of ∼4.4 from ${M_{\rm h}}\approx 10^{11}$ to ≈10^12.5 M_☉.

Our results show that b ≈ 1 in Equation (21). This means that the dependence of f_red on M_h is actually close to that suggested in Woo et al. (2013) based on the empirical inferences of Peng et al. (2012). In this case, the characteristic halo mass, where the fraction of halos hosting blue and red centrals is the same (i.e., f_blue = f_red = 0.5), is given by: ${M_{\rm h}}= 6.88\pm 0.65\times 10^{11}$M_☉. For Milky-Way sized halos, ${M_{\rm h}}\sim 10^{12}\, {M_{\odot }}$, the blue (red) fraction is ∼1/3 (∼2/3).

In Figure 8, we also reproduce some direct observational inferences obtained by stacking weak-lensing data: Mandelbaum et al. (2006, open circles), van Uitert et al. (2011, open squares) and Velander et al. (2014, open triangles). In general, our result is in reasonable agreement with these direct inferences. We also compared our resulting fraction with the analysis of satellite kinematics performed in More et al. (2011, gray shaded area). The shift observed with respect to our results is possible related to the fact that halo masses measured from satellite kinematics are usually higher than other methods, see Section 5.4. We also compare our results with those of Tinker et al. (long dashed-line 2013). Note, however, that the blue fraction from Tinker et al. (2013) is higher for local galaxies. One possibility is that the fraction of halos hosting blue centrals is expected to be higher than for z ∼ 0. Another reason is that, due to sample variance in the COSMOS field at z ∼ 0.36, it may not possible to obtain a direct comparison with our results (J. Tinker 2014, private communication). We also include a comparison with results from Skibba & Sheth (2009). Although at low masses their results are consistent with our blue fraction, there is a discrepancy at the high-mass end. This is possible related to the fact the Skibba & Sheth (2009) model assumes that color depends only weakly on mass. Finally, it is important to note that the way the two galaxy populations are separated in all the studies mentioned above may vary with most of the authors (based on colors, specific SF rate, type, etc.). This introduces differences in the fractions plotted in Figure 8. In spite of this, however, the overall result is consistent among most of the studies.

For completeness, the lower panel of Figure 8 shows our model fitting (solid line with a shaded area) to the empirical fraction of blue central galaxies as a function of M_*, that is the ratio of blue-to-all central GSMFs from the Y12 galaxy group catalog (solid circles with error bars).

Figure 9 shows the predicted CSMFs for all, blue, and red satellites (Equations (22)–(26)). For comparison, the circles with error bars in all the panels show the same, but for the Y12 galaxy group catalog (we only use groups that are complete according to the completeness limits discussed in Section 2.2 of Yang et al. 2009a for both halo and stellar masses). We corrected their halo masses to match our virial halo mass definition, see Equation (B1) in Appendix B. Error bars were estimated by using the jackknife method described in Section 3 with N = 200. We also weighted each galaxy by a factor of V/V_max. In general, both the amplitude and the shape of the predicted CSMFs agree with those from the Y12 group catalog. In more detail, for massive halos, the predicted CSMFs for blue satellites are shallower in the low-mass end (M_* ≈ 10¹⁰ M_☉) than the inferred ones from the group catalogs, but within the error bars. One possible reason is the relatively poor constraints from the lowest mass bin from ω_p, see the discussion in Section 4.1.

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The segregation in color found in the SHMR of central galaxies in Section 4 is a relevant result. How robust is it? We carried out several experiments in order to explore this question. For example, we explored what happens if we force the SHMR of blue and red centrals as well as their scatter to be identical. In this case, we find that the best fits to observations are poorer than those obtained in Section 4, with χ²/dof =2.1, which is ∼25% larger than in that section. We also checked what fraction of our 3 × 10⁵ MCMC models have similar parameters in their parameterization for the blue and red SHMRs. First, note that most of the error bars in each model parameter are of the order ∼10% of the value of the parameter. When allowing this relative difference of ∼10% between models we could not find any model. By allowing for relative differences in the parameters up to a 50%, only ≈3% of the MCMC models obey this condition. Therefore, when searching for best fits, the cases of similar blue and red SHMRs are very rare.

In addition, we found (as expected) that the key ingredient of the color segregation in the SHMR is the fraction of halos hosting red (blue) centrals as a function of M_h. This fraction, according to our results, is essentially defined by the parameter, $M_{\rm h}^{\star }$ (see Equation (21)) and in less degree by the parameter b. Under the assumption that the SHMR of blue and red centrals are identical, and keeping all the parameters for satellite galaxies as obtained in Section 4, excepting $M_{\rm h}^{\star }$, we find that the best fits that minimize χ² are with ${M_{\rm h}^{\star}}\approx 9.7\times 10^{11}$ M_☉ (i.e., ∼1.4 times larger than that found in Section 4). The fits to observations in this case are very poor, giving a total χ²/dof ≈ 4.7. This difference shows that the segregation in color is sensitive to the value of $M_{\rm h}^{\star }$. If $M_{\rm h}^{\star }$ is fixed to larger values than ∼10¹² M_☉, and one allows for a difference between the SHMR of blue and red galaxies, then the constrained SHMRs differ in an opposite way as found in Section 4 (obviously, the fits become even poorer). On the other hand, as $M_{\rm h}^{\star }$ becomes smaller, the difference between the SHMRs in the direction of blue galaxies gets larger M_* for a given M_h than red ones.

We also explored the case of generalizing the function Equation (21) by allowing the mass term in the denominator to vary as $({M_{\rm h}^{\star}}/{M_{\rm h}})^a$. We found that the best fit is obtained when a ≈ 1, that is, the proposed function Equation (21) is robust. The resulting SHMRs in this case are very close to those obtained in Section 4, where a = 1 is assumed. Along the same vein, if we partially modify our parametric approach, then the obtained SHMRs do not change significantly, although other predictions may already differ. For instance, this was the case when we modeled the satellite CSMFs through the SHAM between the theoretical subhalo mass function and the (predicted) satellite GSMF (case B in RAD13), instead of proposing parametric functions for the CSMFs.

In our model, we assumed σ_b and σ_r as free parameters. The resulting constrained values were ∼0.11 and ∼0.14, respectively. On the other hand, these parameters can also be obtained directly from the group galaxy catalogs, especially for massive halos (see, e.g., Yang et al. 2009a), which are slightly different from our model constraints; see Section 5.3 below. To test the impact of this, we repeat our analysis but fix σ_b = σ_r = 0.15 dex. As we describe in Section 5.3, σ_b ≈ σ_r ≈ 0.15 dex as derived directly from the Yang et al. (2007) galaxy group catalog. In this case, we obtain a reduced χ² similar to that obtained in Section 4 (Equation (37)), where we left σ_b and σ_r as free parameters. It should be said that the blue and red SHMRs obtained in the former case results are slightly shallower at the high-mass end than those reported in Section 4. However, the relative separation between these two relations remains roughly the same, showing that our result that the SHMR segregates significantly by color is robust to changes in the σ_b and σ_r parameters.

Finally, we explored the sensitivity of the blue/red central SHMRs to the observational constraints. In particular, in one experiment we renounced to use as constraints the central/satellite GSMF decompositions based on the Y12 group catalog, only the total blue/red GSMFs (and the 2PCFs) were used. Again, the constrained blue, red, and average central SHMRs remained almost the same. In this experiment the central/satellite GSMFs are predicted. They are in reasonable agreement with those obtained with the Y12 group catalog, excepting the low-mass end of the blue satellite GSMF, which also implied a poor agreement with the Y12 CSMFs of blue satellites at low stellar masses in halos smaller than ∼10¹³ M_☉.

5.1.1. On the Blue/Red Separation Criterion

The M_*-to-M_h ratio is commonly interpreted as the efficiency of galaxy stellar mass growth (mainly by in situ SF) within dark matter halos. Our semi-empirical results show that the M_*-to-M_h ratio of central galaxies is segregated by color at all masses, with blue galaxies having higher ratios than red ones (Figure 5). This could be interpreted as that blue central galaxies were more efficient in assembling their stellar masses than red centrals at a given halo mass. However, the M_*-to-M_h ratio is actually a time-integrated quantity, a result of the combination of two aspects:

• 1.  
  The efficiency of stellar mass growth, both by SF in situ and by the accretion of satellites (mergers).
• 2.  
  The processes that halt galaxy growth (particularly due to quenching of the SF) at a given epoch, while the halo mass continues growing.

In order to explain the fact that (1) blue centrals have higher M_*-to-M_h ratios than red centrals (Figure 5), and (2) that low (high) mass halos are dominated by blue (red) centrals (Figures 6 and 8), we need to understand which of the above mentioned processes have played a dominant role. Our results suggest that it should be the second one (i.e., the galaxy quenching process).⁷

In the context of the quenching scenario, while the SF in a galaxy is ceased, its host ΛCDM halo may continue growing hierarchically, especially in the case of more massive halos. Therefore, for a given present-day M_*, the earlier the galaxy is quenched (hence the redder it is), the lower its M_*-to-M_h ratio tends to be, even if its SF efficiency could have been high when it was active. This means that the redder the galaxy, the lower the M_*-to-M_h ratio, as we have found here. Galaxy quenching is consistent with the observational fact that, on average, the more massive the galaxies, the earlier they formed and ceased their SF (e.g., Thomas et al. 2005; Bundy et al. 2006; Bell et al. 2007; Drory & Alvarez 2008; Pozzetti et al. 2010). This phenomenon is known as archaeological downsizing (see Fontanot et al. 2009; Conroy & Wechsler 2009; Firmani & Avila-Reese 2010, and more references therein).

We calculate the inverse SHMRs, that is, dark matter halo mass as a function of stellar mass for blue, red, and all central galaxies. Inverting the SHMR is not just inverting the axes of this relation. Due to the scatter, there is a non-negligible change in the inverse slopes of the SHMR when passing to the halo-stellar mass relation, especially at high masses (Behroozi et al. 2010). We compute the mean M_h as a function of M_* for blue and red galaxies (j = b and r, respectively) as

$$\begin{equation} \langle \log ({M_{\rm h}})\rangle _j({M_*})=\frac{\int {P_{c,j}({M_*}|{M_{\rm h}})}\phi _{h,j} ({M_{\rm h}})\log {M_{\rm h}}d{M_{\rm h}}}{\int {P_{c,j}({M_*}|{M_{\rm h}})}\phi _{h,j}({M_{\rm h}})d{M_{\rm h}}}, \end{equation} \tag{ 40 }$$

and their corresponding intrinsic scatter,

$$\begin{equation} \sigma _j({M_*})=\left(\frac{\int {P_{c,j}({M_*}|{M_{\rm h}})}\phi _{h,j} ({M_{\rm h}})\mu ^2d{M_{\rm h}}}{\int {P_{c,j}({M_*}|{M_{\rm h}})}\phi _{h,j}({M_{\rm h}})d{M_{\rm h}}}\right)^{1/2}, \end{equation} \tag{ 41 }$$

where $\mu =\log {M_{\rm h}}-\langle \log ({M_{\rm h}})\rangle _j({M_*})$. In the upper panel of Figure 10, we plot the M_h–M_* relation for the blue and red central galaxies. The shaded areas correspond to the 1σ confidence levels from the MCMC trials. The obtained total scatters, σ_b(M_*) and σ_r(M_*), are shown in the lower panel; the shaded areas are the confidence levels. For blue galaxies, σ_b(M_*) changes from ∼0.06 dex to ∼0.31 dex, while for red centrals σ_r(M_*) changes from ∼0.03 dex to ∼0.33 dex. As in the direct SHMRs, these scatters are an upper limit to the intrinsic ones.

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As expected, the dark matter halo of red centrals is on average more massive than that of blue centrals, especially for galaxies more massive than M_* ∼ 2 × 10¹⁰ M_☉. This is also consistent with the fact that red centrals are more clustered than blue centrals of the same stellar mass (Figure 4). In fact, it is rare to find a blue central in group/cluster sized halos, but when this is the case, its host halo is significantly (up to a factor of three) less massive than that of a red central of the same stellar mass, and therefore, it is expected to be less clustered. On the side of low-mass galaxies, it is rare to find red galaxies, but if this is the case, they also have (slightly) more massive halos than the blue ones, or less stellar masses for a given M_h. This can be due to strong early SN-driven outflows leaving these rare galaxies devoid of gas and in the process of aging (reddening), while most low-mass galaxies instead seem to have delayed their SF histories, rather than suffering strong SF-driven outflows and being blue today.

The scatter around the average (total) SHMR has been discussed previously in the literature (see, e.g., Yang et al. 2008; Cacciato et al. 2009; More et al. 2011; Li et al. 2012; Leauthaud et al. 2012; Reddick et al. 2013; Rodríguez-Puebla et al. 2013; Kravtsov et al. 2014, and more references therein). Typically, these previous works constrained the scatter around the total SHMR, rather than separately constraining σ_b and σ_r (but see More et al. 2011). In other words, they assumed that the scatter around the SHMR was an unimodal (lognormal) and random distribution. Moreover, in these works it is assumed that σ_A is constant, with the reported values of 0.15–0.20 dex, which are close to our value of σ_r at large masses. In fact this is expected because the majority of halos more massive than ${M_{\rm h}}\sim 3\times 10^{12}$ M_☉ host red centrals, and are the masses explored in most of the cited works. To illustrate this point we compare the scatter constrained by some of these works in Figure 7.⁸ We also estimated σ_r and σ_b from the Y12 galaxy group catalog for seven different halo mass bins equally spaced in a width of 0.3 dex, and only for those above ${M_{\rm h}}=5\times 10^{12}\, {M_{\odot }}$. We compute these scatters following Yang et al. (2009a), and find that σ_r ∼ 0.14 dex, in agreement with our result, while σ_b ∼ 0.15 dex, which is slightly larger than our determination.

Our results show that the mean SHMRs of blue and red central galaxies and the scatters around them are different. This implies that the distribution of all central galaxies is bimodal, as shown in Figure 6, and that the color is one of the sources of the intrinsic scatter around the average SHMR. Note that if the SHMRs of blue and red centrals constrained with the observations were statistically similar (i.e., the same probability distributions for blue and red centrals), then this would mean that the SHMR of all centrals is not segregated by color (the scatter is not due to color). As mentioned above, for large masses red centrals completely dominate in such a way that the average SHMR and its scatter are close to the mean SHMR and the scatter of red centrals, respectively (Figure 5). In this sense, the scatter distribution of the average SHMR is close to an unimodal distribution, that of the red galaxies (see Figure 6).

At smaller masses, ${M_{\rm h}} \lesssim 3\times 10^{12}$ M_☉, the fractions of blue (red) centrals increases (decreases) with decreasing mass in such a way that the intrinsic scatter around the average SHMR of central galaxies is bimodal with a significant separation between the peaks (see Figure 6). According to Figure 7, its scatter increases with decreasing halo mass. This result implies that at masses where the fractions of blue and red central galaxies are not significantly different, the use and interpretation of the average (total) SHMR should be taken with care. If the study refers to the intermedium-low mass central galaxy population, the large intrinsic scatter around the SHMR (see Figure 7) should be taken into account for any inference. If the SHMR is used in studies where a distinction is made in between blue and red (late- and early-type) galaxies, then the SHMR separated into blue and red galaxies should be used, given the significant segregation by color that we have found in this relation. For instance, this is the case of studies where the SHMR is connected with observable correlations as the Tully–Fisher relation for late-type (blue) galaxies and the Faber–Jackson relation for early-type (red) galaxies.

In this subsection, we compare our constrained SHMRs for local blue and red central galaxies with those previously obtained using direct methods, namely galaxy–galaxy weak lensing and satellite kinematics. We also compare our average SHMR with those of previous semi-empirical studies. Where necessary, we apply corrections to the stellar mass reported by different authors to be consistent with the Chabrier (2003) IMF adopted here (see, e.g., Table 2 in Bernardi et al. 2010). When necessary, we also correct the halo masses to match our definition of virial mass. The corrections are done according to the relations reported in Appendix B.

5.4.1. Comparisons with Direct Methods

First, we compare our results with those obtained from galaxy–galaxy weak lensing. In these studies, in order to attain an acceptable signal-to-noise, observations of individual galaxies are stacked in bins of M_* (or luminosity). Therefore, these measurements refer to halo mass as a function of M_*. In the upper and lower panels of Figure 11. We reproduce our resulting $\langle \log {M_{\rm h}}\rangle ({M_*})$ relations for blue and red galaxies, respectively, as plotted in Figure 10, and compare them with several weak-lensing results. Note that in this case shaded areas represent the uncertainties around the SHMRs. Mandelbaum et al. (2006, empty blue circles, error bars are the 95% confidence intervals) used SDSS DR4 galaxies separated into late- and early-type galaxies according to the bulge-to-total ratio as given by the parameter frac_deV provided in the SDSS PHOTO pipeline (a de Vaucouleours/exponential decomposition was applied). van Uitert et al. (empty blue squares 2011) used the combined image data from the Red Sequence Cluster Survey (RCS2) and the SDSS DR7 to obtain the halo masses for late- and early-type galaxies as a function of M_*. (they also used the frac_deV parameter for defining the morphology). In the figure, we included measurements over only the redshift range z = [0.08, 0.41]. Both Velander et al. (2014, empty blue triangles) and Hudson et al. (2013, empty blue pentagons) used the Canada-France-Hawaii Telescope Lensing Survey to derive halo masses of blue and red galaxies; the latter division was done based on the bimodality in the color–magnitude diagram. Halo mass measurements derived in Velander et al. (2014) are on average at redshift z ∼ 0.3, while for Hudson et al. (2013) we plotted only those measurements below z = 0.31. We also plot the results from Mandelbaum et al. (2008) (red solid circles) and Schulz et al. (2010) (magenta crosses) for massive central early-type galaxies based on the local SDSS DR7 sample and a more sophisticated criteria for selecting early-type lens population.

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Our SHMR determinations are consistent, within the uncertainties, with the various weak-lensing studies, which cover different mass ranges and have large uncertainties. As mentioned in Section 4, a source of discrepancy between different authors is due to the way blue and red (or late- and early-type) galaxies have been defined. In addition, results on $\langle \log {M_{\rm h}}\rangle$ as a function of M_* are sensitive to the fact that different surveys may have different levels of measurement error in their stellar mass estimates, particularly those based on photometric surveys (e.g., Leauthaud et al. 2012). In spite of all of these caveats, the overall agreement between these previous studies with our results is encouraging.

In Figure 12, we compare our inverse SHMRs of blue and red centrals (as in Figure 11) with stacked satellite kinematics studies. More et al. (2011, shaded gray area) used the Yang et al. (2007) group catalog and the spectroscopic velocities of the SDSS survey. Wojtak & Mamon (2013, open circles) used the SDSS DR7 spectroscopic catalog for blue and red central galaxies. We also plot the results by Conroy et al. (2007, solid circles), although these authors did not present their results for the M_h–M_* relation separated into blue and red galaxies (their data are repeated in the upper and lower panels); they combined data from the DEEP2 Galaxy Redshift Survey and the SDSS DR4 (their halo masses have been corrected by ∼30% due to incompleteness, see their Appendix A). As seen, the satellite kinematics method tends to give higher halo masses than our semi-empirical results and than weak-lensing studies. The discrepancy between satellite kinematics and other methods has been noted previously (see, e.g., More et al. 2011; Skibba et al. 2011; Rodríguez-Puebla et al. 2011). The differences can be partially explained by the relation between ${M_{\rm h}}$ and the number of satellite galaxies at a fixed M_*. Because the technique is based on the kinematics of satellites, these studies can be biased to higher halo masses due to the loss of data in the case of systems lacking satellites or with poor kinematical information, namely those of smaller halo masses at a given stellar mass.

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5.4.2. Comparison with Previous Inferences of the Average SHMR

Figure 13 compares the average SHMR of central galaxies (Equation (17)) constrained by our method with those reported in Guo et al. (2010, red long-dashed curve), Behroozi et al. (2013, blue dot-dashed curve), Y12 (orange shaded area indicate their 68% of confidence), and RAD13 (dots with error bars). In the first two works, the SHMR was obtained by matching the abundances of all galaxies to the abundances of halos plus subhalos, therefore, it is the SHMR of all galaxies. In the case of the last works, the SHMRs are only for central galaxies (their set SMF2 and set C, respectively). The SHMR of central and all galaxies do not differ too much because the SHMR of satellite galaxies is close to that of central galaxies (RAD13).

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At masses below ${M_{\rm h}}\sim 10^{12}$ M_☉, our average (blue + red galaxies) SHMR for centrals is close to the SHMRs reported in the above cited studies. In contrast, at larger masses, our SHMR increases more rapidly than these studies. The main reason is that at large masses our calculation of the GSMF is slightly shallower than in previous works, see Figure 2. However, it could be that the high-mass end of the GSMF is even shallower than our determination.

Recently, several studies have pointed to a systematical under-estimation of luminosity and stellar mass-to-light ratios of galaxies due to the commonly used aperture limits in the SDSS (see Bernardi et al. 2013, and more references therein; see also Mendel et al. 2014). Surface brightness (mass) profiles of galaxies, in particular the central ones in clusters, extend much further away than the commonly used apertures (Kravtsov et al. 2014, and more references therein). In Bernardi et al. (2013), luminosity and stellar mass functions were calculated based on different model fits for the surface brightness profiles in the SDSS galaxies. The authors showed that their results preferred GSMFs with the most luminous galaxies having larger masses for a given number density than most of the previous published GSMFs. Similar conclusion were obtained in He et al. (2013) based on a more sophisticated photometric data reduction from the SDSS DR7 and with morphological classifications from the Galaxy Zoo project (Lintott et al. 2011). Kravtsov et al. (2014) confirmed this by using a compilation of well-studied massive central cluster galaxies. Mendel et al. (2014) compared the MPA-JHU DR7 stellar masses with those obtained by computing accurate bulge+disk and Sérsic profile photometric decompositions in several bands. For the Sérsic profile, they find masses larger by ≈0.08 dex at the smallest masses and ≈0.23 dex at the largest masses. Following their results, we conservatively correct our masses by ≈0.05 dex for masses up to log(M_*/M_☉)∼10.7 and then smoothly increase the correction ending with 0.23 dex at log(M_*/M_☉) ∼ 12.

In Figure 14, we reproduce the resulting total GSMF by correcting stellar masses as described above (skeletal symbols with error bars). For comparison, in the same figure we include the GSMF for all galaxies from the MPA-JHU NYU-VAGC/SDSS DR7 sample obtained in Section 3, solid line. The corrected GSMF is consistent with previous estimates, except at the high-mass end, which has a significantly shallower fall than most previous ones, but is in good agreement with Bernardi et al. (2013, their Sérsic profile case for the M_* estimate, magenta short dashed line). These authors extensively discuss how sensitive the mass determination of the most luminous galaxies is on the way the light profile is fitted. The spirit of the correction introduced above to our stellar masses was mostly to take into account more adequate light profile fittings to galaxies, especially the most massive ones, as was done in Bernardi et al. (2013).

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In Figure 13 we plot the resulting SHMR for all central galaxies when using the corrected stellar masses (black long-dotted curve). For comparison we also reproduce the SHMR for all galaxies reported in Kravtsov et al. (2014, violet dotted lines). Both results are similar and show that when taking into account more adequate light-profile fittings to galaxies, especially the most massive ones, the SHMR increases more rapidly than in previous reports. However, we highlight that all the results presented in previous sections used a GSMF estimated based on the SDSS standard light-profile fittings, see Section 3.1.