A THEORETICAL FRAMEWORK FOR COMBINING TECHNIQUES THAT PROBE THE LINK BETWEEN GALAXIES AND DARK MATTER

Let us consider the conditional SMF (the analog of the conditional luminosity function) which represents the number of galaxies with M_* in the range M_* ± dM_*/2 at fixed halo mass and is denoted by Φ(M_*|M_h) (e.g., Yang et al. 2009; Moster et al. 2010; Behroozi et al. 2010). The conditional SMF can be divided into a central component and a satellite component: Φ(M_*|M_h) = Φ_c(M_*|M_h) + Φ_s(M_*|M_h). Φ_c(M_*|M_h) is the conditional SMF for central galaxies, and it will be our mathematical representation of the SHMR. Note that in our model, the halo mass in the term Φ_s(M_*|M_h) refers to the host halo mass.

In addition to the shape and evolution of the mean SHMR, astrophysical processes are expected to induce an intrinsic scatter in stellar mass at fixed halo mass, which is important to take into consideration when defining a functional form for Φ_c(M_*|M_h). Another non-negligible source of scatter can be the measurement error associated with the determination of stellar masses. In the absence of strong observational or theoretical guidance for the form and magnitude of the total scatter (intrinsic plus measurement), we adopt a stochastic model where Φ_c(M_*|M_h) is a log-normal probability distribution function (hereafter "PDF") with a log-normal scatter⁶ denoted by $\sigma _{\log M_{*}}$. Since we have assumed a log-normal functional form, Φ_c(M_*|M_h) can be written as

$$\begin{eqnarray} &&\Phi _c(M_{*}|M_h)= \frac{1}{\ln (10)\sigma _{\rm log M_{*}}\sqrt{2\pi }}\nonumber\\ &&\quad\times\exp \left[-\frac{\left[\log _{10}(M_*)-\log _{10}(f_{\textsc {shmr}}(M_h))\right]^2}{2\sigma _{\rm log M_{*}}^2}\right], \end{eqnarray} \tag{ 1 }$$

where $f_{\textsc {shmr}}$ represents the logarithmic mean of the stellar mass given the halo mass for the Φ_c distribution function. Equation (1) is normalized such that the integral of Φ_c(M_*|M_h) over M_* is equal to 1.

To model Φ_c we must specify a functional form for both $f_{\textsc {shmr}}$ and $\sigma _{\log M_{*}}$. There is increasing evidence to suggest that low- and high-mass galaxies have different stellar-to-halo mass ratios, probably as a result of multiple feedback mechanisms that operate at distinct mass scales and regulate star formation. We therefore require an SHMR that is flexible enough to capture such variations. We adopt the functional form presented in Behroozi et al. (2010, hereafter "B10"), which has been shown to reproduce the local Sloan Digital Sky Survey (SDSS) SMF using the abundance matching technique. In practice, $f_{\textsc {shmr}}(M_h)$ is mathematically defined via its inverse function:

$$\begin{eqnarray} &&\log _{10}(f_{\textsc {shmr}}^{-1}(M_\ast)) = \log _{10}(M_h) \hspace{0.0pt}\nonumber\\ &&\quad = \log _{10}(M_1) + \beta \,\log _{10}\left(\frac{M_\ast }{M_{\ast,0}}\right) + \frac{\left(\frac{M_\ast }{M_{\ast,0}}\right)^\delta }{1 + \left(\frac{M_{\ast }}{M_{\ast,0}}\right)^{-\gamma }} - \frac{1}{2},\qquad \end{eqnarray} \tag{ 2 }$$

where M₁ is a characteristic halo mass, M_{*, 0} is a characteristic stellar mass, β is the low-mass end slope, γ controls the transition region, and δ controls the massive end slope. Details regarding the justification of this functional form can be found in Section 3.4.3 of B10.

Note that a variety of similar functional forms have been proposed by previous authors. For example, the interested reader can look at Equation (2) in Moster et al. (2010) and Equation (20) in Yang et al. (2009).

In contrast to B10, we do not parameterize the redshift evolution of this functional form. Instead, in Paper II, we bin the data into three redshift bins and check for redshift evolution in the parameters a posteriori. Another difference with respect to B10 is that we assume Equation (2) is only relevant for central galaxies whereas B10 assume that the SHMR also applies to satellite galaxies, provided that the halo mass of a satellite galaxy is defined as M_acc.

It is important to note that our SHMR traces the location of the mean-log stellar mass: $f_{\textsc {shmr}}(M_h)\equiv \langle \log _{10}(M_*(M_h))\rangle$. Other authors may report the mean stellar mass, 〈M_*(M_h)〉, or even the mean halo mass at fixed stellar mass, 〈M_h(M_*)〉 (e.g., Conroy et al. 2007). These averaging systems will yield different results in the presence of scatter. For example, 〈M_h(M_*)〉 will be biased low compared to 〈log₁₀(M_*(M_h))〉 if $\sigma _{\log M_*}$ is non-zero. This bias will increase with $\sigma _{\log M_*}$ and for larger values of the high-mass slope of $f_{\textsc {shmr}}^{-1}$.

In Figure 1, we illustrate the impact of the five parameters that determine $f_{\textsc {shmr}}$ on the shape of the SHMR. A brief description is as follows.

• 1.  
  M₁ controls the characteristic halo mass; increasing M₁ will result in larger halos hosting galaxies at a given stellar mass. In Figure 1 (which represents M_* on the x-axis and M_h on the y-axis), M₁ controls the y-axis amplitude (halo mass) of $f_{\textsc {shmr}}^{-1}$ so that a constant change in M₁ leads to a constant up/down logarithmic shift in $f_{\textsc {shmr}}^{-1}$. Note that this constant logarithmic shift may not be visually obvious because the slope of the SHMR increases sharply at M_* > 10¹¹ M_☉.
• 2.  
  M_{*, 0} controls the characteristic stellar mass; increasing M_{*, 0} will result in smaller halos hosting galaxies at a given stellar mass. In Figure 1, M_{*, 0} controls the x-axis amplitude of $f_{\textsc {shmr}}^{-1}$.
• 3.  
  β controls the low-mass power-law slope of $f_{\textsc {shmr}}^{-1}$. When β increases, the low-mass end slope becomes steeper.
• 4.  
  The δ parameter regulates how rapidly $f_{\textsc {shmr}}^{-1}$ climbs at high M_*. Indeed, $f_{\textsc {shmr}}^{-1}$ asymptotes to a sub-exponential function at high M_*, which signifies that $f_{\textsc {shmr}}^{-1}$ climbs more rapidly than a power-law function but less rapidly than an exponential function (see discussion in B10).
• 5.  
  γ controls the transition regime between the low-mass power-law regime and the high-mass sub-exponential behavior. A larger value of γ corresponds to a more sharp transition between the two regimes.

A quantity that is of particular interest is the mass (we refer here to both M_* and M_h) at which the ratio M_h/M_* reaches a minimum. This minimum is of noteworthy importance for galaxy formation models because it marks the mass at which the accumulated stellar growth of the central galaxy has been the most efficient. In this paper, and in subsequent papers, we will refer to the stellar mass, halo mass, and ratio at which this minimum occurs as the "pivot stellar mass," M^piv_*, the "pivot halo mass," M^piv_h, and the "pivot ratio," (M_h/M_*)^piv. Note that M^piv_* and M^piv_h are not simply equal to M₁ and M_{*, 0}. Indeed, the mathematical formulation of the SHMR is such that the pivot masses depend on all five parameters. The three parameters that have the strongest effect on the pivot masses are M₁, M_{*, 0}, and γ. For example, as can be seen in the right-hand panel of Figure 1, M^piv_* and M^piv_h are inversely proportional to γ. To a lesser extent, the two remaining parameters, β and δ, also have a small influence on the pivot masses.

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We now turn our attention to the second component of Φ_c(M_*|M_h), which is the scatter in stellar mass at fixed halo mass, $\sigma _{\log M_{*}}$. The total measured scatter will have two components: an intrinsic component (denoted by $\sigma _{\log M_{*}}^{\rm i}$) and a measurement error component due to redshift, photometry, and modeling uncertainties in stellar mass measurements (denoted by $\sigma _{\log M_{*}}^{\rm m}$). It is reasonable to assume that the intrinsic scatter component is independent of the measurement error component. Assuming Gaussian error distributions, we can write

$$\begin{equation} (\sigma _{\log M_{*}})^2=\big(\sigma _{\log M_{*}}^{\rm i}\big)^2+\big(\sigma _{\log M_{*}}^{\rm m}\big)^2. \end{equation} \tag{ 3 }$$

While the error distribution for the stellar mass estimate for any single galaxy may be non-Gaussian, in this work we are only concerned with stacked ensembles. B10 have tested that the error distribution for a stacked ensemble is Gaussian to good approximation and that small non-Gaussian wings in this distribution are not likely to affect this type of analysis.

In practice, since the data are always binned according to M_*, the observables are actually sensitive to the scatter in halo mass at fixed stellar mass, which we denote by $\sigma _{\log M_h}$. It will therefore be useful to understand the link between $\sigma _{\log M_h}$ and $\sigma _{\log M_{*}}$. If the SHMR is a power law, the relationship between $\sigma _{\log M_h}$ and $\sigma _{\log M_{*}}$ is simply

$$\begin{equation} \sigma _{\log M_h}=\sigma _{\log M_{*}}\frac{d(\log _{10} M_h)}{d(\log _{10} M_*)}. \end{equation} \tag{ 4 }$$

For example, if there is a power-law relation between halo mass and stellar mass such that M_h = M^η_* then $\sigma _{\log M_h}=\eta \times \sigma _{\log M_{*}}$. In our case, the SHMR behaves like a power law at low M_*. At high M_*, however, d(log₁₀M_h)/d(log₁₀M_*) increases as a function of M_*. Therefore, if $\sigma _{\log M_{*}}$ is constant, $\sigma _{\log M_h}$ will be equal to $\sigma _{\log M_h}=\beta \times \sigma _{\log M_{*}}$ at low M_* but then will continuously increase with M_* at a rate set by γ and δ.

If we adopt the best-fit model to the SHMR from Paper II in the redshift range 0.22 < z < 0.48, we find that the power-law index of the SHMR increases steeply at log₁₀(M_*) > 11 so that $\sigma _{\log M_h}$ becomes quite large. For example, $\sigma _{\log M_h}\sim 0.46$ dex at log₁₀(M_*) = 11 and $\sigma _{\log M_h}\sim 0.7$ dex at log₁₀(M_*) = 11.5. In practical terms, this implies that the most massive galaxies do not necessarily live in the most massive halos. For example, a galaxy with M_* ∼ 2 × 10¹¹ M_☉ could be the central galaxy of a group with M_h ∼ 10¹³–10¹⁴M_☉, or could also be the central galaxy of a cluster with M_h > 10¹⁵M_☉. The increase of $\sigma _{\log M_h}$ with M_* will lead to a noticeable effect in the g–g lensing, clustering, and SMFs at large M_* that is analogous to Eddington bias. This effect will be discussed in further detail in Section 5.

In this paper, we assume that stellar mass is used to implement the HOD model since it is expected to be a more faithful tracer of halo mass than galaxy luminosity.

Consider a galaxy sample such that M* > Mt1* (a "threshold" sample). The central occupation function, denoted by 〈Ncen(Mh|Mt1*)〉, is the average number of central galaxies in this sample that are hosted by a halo of mass Mh. The satellite occupation function, denoted by 〈Nsat(Mh|Mt1*)〉, is the equivalent function for satellite galaxies.

In what follows, we focus on the appropriate equations for threshold samples. In Paper II, however, we will use "binned" samples (Mt1* < M* < Mt2*) to calculate the g–g lensing and the SMF. We therefore note that the occupation functions for binned samples are trivially derived from the occupation function for threshold samples via

$$\begin{equation} \langle N_{\rm cen}(M_h|M_{*}^{t_1},M_{*}^{t_2}) \rangle = \langle N_{\rm cen}(M_h|M_{*}^{t_1})\rangle -\langle N_{\rm cen}(M_h|M_{*}^{t_2})\rangle \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \langle N_{\rm sat}(M_h|M_{*}^{t_1},M_{*}^{t_2}) \rangle = \langle N_{\rm sat}(M_h|M_{*}^{t_1})\rangle -\langle N_{\rm sat}(M_h|M_{*}^{t_2})\rangle. \end{equation} \tag{ 6 }$$

For a threshold sample of galaxies, 〈Ncen(Mh|Mt1*)〉 is fully specified given Φc(M*|Mh) according to

$$\begin{equation} \langle N_{\rm cen}(M_h|M_{*}^{t_1})\rangle = \int _{M_{*}^{t_1}}^{\infty } \Phi _c(M_*|M_h) dM_*. \end{equation} \tag{ 7 }$$

Because the integral of Φc(M*|Mh) over M* is equal to 1, 〈Ncen(Mh|Mt1*)〉 will vary between 0 and 1.

To begin with, let us make the simplifying assumption that $\sigma _{\log M_{*}}$ is constant. Because Φ_c is parameterized as a log-normal distribution, the central occupation function can be analytically derived from Equation (7) by considering the cumulative distribution function of the Gaussian:

$$\begin{eqnarray} &&\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle \hspace{0.0pt}\nonumber\\ &&\quad =\frac{1}{2}\left[ 1-\mbox{erf}\left(\frac{\log _{10}(M_{*}^{t_1}) - \log _{10}(f_{\textsc {shmr}}(M_h)) }{\sqrt{2}\sigma _{\log M_{*}}} \right)\right],\qquad \end{eqnarray} \tag{ 8 }$$

where erf is the error function defined as

$$\begin{equation} \mbox{erf}(x)=\frac{2}{\sqrt{\pi }}\int _0^{x}e^{-t^2} dt. \end{equation} \tag{ 9 }$$

It is important to note that Equation (8) is only valid when $\sigma _{\log M_{*}}$ is constant. In the more general case where $\sigma _{\log M_{*}}$ varies, 〈N_cen〉 can nonetheless be calculated by numerically integrating Equation (7). In Paper II, we will consider cases in which $\sigma _{\log M_{*}}$ varies due to the effects of stellar-mass-dependent measurement errors. In this case, we will numerically integrate Equation (7) to calculate 〈N_cen〉 (see Section 4.2 in Paper II).

We note that most readers may be more familiar with a simplified version of Equation (8) that assumes $f_{\textsc {shmr}}(M_h)$ is a power law. We will now describe the assumptions made in order to obtain the more commonly employed equation for 〈N_cen〉 from Equation (8).

If we make the assumption that $f_{\textsc {shmr}}(M_h) \propto M_h^p$ and we define Mmin such that $M_{\rm min}\equiv f_{\textsc {shmr}}^{-1}(M_{*}^{t_1})$ (in other terms, Mmin is the inverse of the SHMR relation for the stellar mass threshold Mt1*) then using Equation (8) we can write

$$\begin{eqnarray} &&\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle \hspace{0.0pt}\nonumber\\ &&\quad =\frac{1}{2}\left[ 1-\mbox{erf}\left(\frac{\log _{10}(M_{\rm min}^p) - \log _{10}(M_h^p) }{\sqrt{2}\sigma _{\log M_{*}}} \right)\right]. \end{eqnarray} \tag{ 10 }$$

If we now use the fact that erf(− x) = −erf(x) and if we define $\widetilde{\sigma }_{{\rm {log}}M}$ such that $\widetilde{\sigma }_{\log M}\equiv \sigma _{\log M_{*}}/p$ we can write

$$\begin{eqnarray} &&\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle \hspace{0.0pt}\nonumber\\ &&\quad =\frac{1}{2}\left[ 1+\mbox{erf}\left(\frac{\log _{10}(M_h) - \log _{10}(M_{\rm min}) }{\sqrt{2}\widetilde{\sigma }_{\log M}} \right)\right], \end{eqnarray} \tag{ 11 }$$

which is a commonly employed formula for 〈N_cen〉. First, it is important to note that Equation (11) is only an approximation for 〈N_cen〉 for the case when the SHMR is a power law and is certainly not valid over a large range of stellar masses. Second, $\widetilde{\sigma }_{{\rm {log}}M}$ can be interpreted as the scatter in halo mass at fixed stellar mass if and only if the SHMR is a power law and if $\sigma _{\log M_{*}}$ is constant. Since there is accumulating evidence that the SHMR is not a single power law (and the same is in general true for the relationship between halo mass and galaxy luminosity), we recommend using Equation (8) instead of Equation (11).

Figures 2 and 3 illustrate the difference in 〈N_cen〉 when Equation (8) is used to describe clustering instead of Equation (11). To make this figure, we have assumed the para- meter set: log₁₀(M₁) = 12.71, log₁₀(M_{*, 0}) = 11.04, β = 0.467, δ = 0.62, γ = 1.89, and $\sigma _{\log M_{*}}=0.25$. For each stellar mass threshold in Figure 2, we determine the values of M_min and $\widetilde{\sigma }_{{\rm {log}}M}$ in Equation (11) such that the number density of central galaxies and the bias of those galaxies is the same as that achieved with Equation (8). Thus, our procedure mimics what one would obtain through analysis of the clustering and space density of such samples, assuming that the satellite occupation would be the same in either analysis.

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Figure 2 reveals that because the SHMR has a sub-exponential behavior at log₁₀(M_*) ≳ 10.5, 〈N_cen〉 begins to deviate from a simple erf function for high stellar mass samples and therefore is not well described by Equation (11). Assuming that Equation (8) correctly represents 〈N_cen〉, the error made on M_min can be of order 10%–40% at $M_h=f_{\textsc {shmr}}^{-1}(M_{*}^{t_1})\gtrsim 10^{12}\,M_{\odot }$ if Equation (11) is used to describe 〈N_cen〉 instead of Equation (8).

We note that this does not invalidate Equation (11) as a possible parameterization of the central occupation function. However, interpreting M_min in Equation (11) as $f_{\textsc {shmr}}^{-1}(M_{*}^{t_1})$ can result in a 10%–40% error in the true mean halo mass (with larger errors for $\sigma _{\log M_{*}}>0.25$). Also, the "scatter" ($\widetilde{\sigma }_{{\rm {log}}M}$) constrained by this parameterization is not equal to the scatter in a log-normal distribution of stellar mass at fixed halo mass. Finally, one troublesome aspect of using the erf functional form in Equation (11) is that 〈N_cen〉 curves for different stellar mass thresholds may actually cross at low halo mass, implying the unphysical condition that halos of mass M_h have a "negative" amount of galaxies between two threshold values. This is seen at M_h ∼ 10^11.5 M_☉ in Figure 2. For values of $\sigma _{\log M_{*}}$ larger than what we have assumed here, this effect will occur at even higher halo mass. Using Equation (8) with a model for the SHMR prevents this from occurring, and 〈N_cen〉 for various galaxy samples can be calculated self-consistently.

In addition to the five parameters introduced to model 〈N_cen〉 and $\sigma _{\log M_{*}}$, we introduce five new parameters to model 〈N_sat〉. In order to simultaneously fit g–g lensing, clustering, and SMF measurements that employ different binning schemes, we require a model for 〈N_sat〉 that is independent of any given binning scheme. For this reason, we parameterize the satellite function with threshold samples. The number of satellite galaxies in a bin of stellar mass is determined by a simple dif- ference of two threshold samples. This also eliminates the need to integrate over stellar mass, as required when working explicitly through the conditional SMF.

Numerical simulations demonstrate that the occupation of sub-halos (e.g., Kravtsov et al. 2004; Conroy et al. 2006) and satellite galaxies in cosmological hydrodynamic simulations (Zheng et al. 2005) follow a power law at high host halo mass, then fall off rapidly when the mean occupation becomes significantly less than unity. Thus, we parameterize the satellite occupation function as a power of host mass with an exponential cutoff and scaled to 〈N_cen〉 as follows:

$$\begin{eqnarray} &&\langle N_{\rm sat}(M_h|M_{*}^{t_1}) \rangle \nonumber\\ &&\quad =\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle \left(\frac{M_h}{M_{\rm sat}}\right)^{\alpha _{\rm sat}} \exp \left(\frac{-M_{\rm cut}}{M_h}\right), \end{eqnarray} \tag{ 12 }$$

where α_sat represents the power-law slope of the satellite mean occupation function, M_sat defines the amplitude of the power law, and M_cut sets the scale of the exponential cutoff. Here, M_h refers to the host halo mass of satellite galaxies.

Observational analyses have demonstrated that there is a self-similarity in occupation functions such that Msat/Mmin ≈ constant for luminosity-defined samples (Zehavi et al. 2005, 2011; Zheng et al. 2007, 2009; Tinker et al. 2007; Abbas et al. 2010), where Mmin is taken from Equation (11) and is conceptually similar to $f_{\textsc {shmr}}^{-1}(M_{*}^{t_1})$ (modulo a 10%–40% difference as shown in Figure 3) and where Mt1* is the stellar mass threshold of the sample. Instead of simply modeling Msat and Mcut as constant factors of $f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$, we add flexibility to our model by enabling Msat and Mcut to vary as power-law functions of $f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$:

$$\begin{equation} \frac{M_{\rm sat}}{10^{12} M_{\odot }}= B_{\rm sat} \left(\frac{f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})}{10^{12} M_{\odot }}\right)^{\beta _{\rm sat}} \end{equation} \tag{ 13 }$$

and

$$\begin{equation} \frac{M_{\rm cut}}{10^{12} M_{\odot }}= B_{\rm cut} \left(\frac{f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})}{10^{12} M_{\odot }}\right)^{\beta _{\rm cut}}. \end{equation} \tag{ 14 }$$

Zheng et al. (2007) find that M_sat/M_min ∼ 18 for SDSS and M_sat/M_min ∼ 16 using luminosity-defined samples in DEEP2. For B_cut, the expectation is that the cutoff mass scale occurs at M_min < M_cut < M_sat, although it can be significantly smaller.

Using this model, one can also compute the total amount of stellar mass in galaxies (summing the contribution from both centrals and satellites) as a function of halo mass M_h. To begin with, let us consider the total stellar mass as a function of halo mass in some stellar mass bin: M^tot_*(M_h|M^t1_*, M_*^t2). The expression for M^tot_*(M_h|M^t1_*, M_*^t2) is given by

$$\begin{eqnarray} &&M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2}) \hspace{0.0pt}\nonumber\\ &&\quad =\int _{M_*^{t1}}^{M_*^{t2}}\left[\Phi _c(M_*|M_h)+\Phi _s(M_*|M_h)\right]M_*dM_*. \end{eqnarray} \tag{ 15 }$$

However, in the previous section we have only specified the analytic form for Φ_c (Equation (1)) but not for Φ_s. Indeed, in Section 3.3 we outlined an analytic model for 〈N_sat〉 but calculating the analytic derivative of 〈N_sat〉 would be tedious. Thankfully, however, we do not specifically need to know the functional form of Φ_s in order to calculate Equation (15). By using the integration by parts rule, we can rewrite Equation (15) in a more convenient form as follows:

$$\begin{eqnarray} &&M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2}) \hspace{0.0pt}\nonumber\\ &&\quad = \int _{M_*^{t1}}^{M_*^{t2}} \langle N_{\rm cen}(M_h|M_{*}) \rangle {\it dM}_* - \left[ \langle N_{\rm cen}(M_h|M_{*}) \rangle M_*\right]_{M_*^{t1}}^{M_*^{t2}} \nonumber \\ &&\qquad +\int _{M_*^{t1}}^{M_*^{t2}} \langle N_{\rm sat}(M_h|M_{*}) \rangle {\it dM}_* - \left[ \langle N_{\rm sat}(M_h|M_{*}) \rangle M_*\right]_{M_*^{t1}}^{M_*^{t2}}. \nonumber \\ && \end{eqnarray} \tag{ 16 }$$

This equation provides us with a convenient way to calculate the total stellar mass locked up in galaxies with M^t1_* < M_* < M^t2_* as a function of halo mass M_h.

In total, we have introduced six parameters to model the central occupation function (M₁, M_{*, 0}, β, δ, γ, $\sigma _{\log M_{*}}$) and five parameters to model the satellite occupation function (β_sat, B_sat, β_cut, B_cut, α). In addition, one could introduce a model for $\sigma _{\log M_{*}}$ or assume that the scatter is constant in which case there would be a total of 11 parameters for this model. In Figure 8 of Paper II, we show the two-dimensional marginalized distributions for this parameter set using data from the COSMOS survey. The model described in this paper provides an excellent fit to COSMOS data. Figure 8 in Paper II demonstrates that this model is reasonably free of parameter degeneracies. A summary and description of these parameters can be found in Table 1. Figure 4 gives an illustration of the central and satellite occupation functions for galaxy samples in bins and thresholds of stellar mass.

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Table 1. Parameters in Model

Parameter	Unit	Description	〈Ncen〉 or 〈Nsat〉
M1	M☉	Characteristic halo mass in the SHMR	〈Ncen〉
M*, 0	M☉	Characteristic stellar mass in the SHMR	〈Ncen〉
β	None	Faint end slope in the SHMR	〈Ncen〉
δ	None	Controls massive end slope in the SHMR	〈Ncen〉
γ	None	Controls the transition regime in the SHMR	〈Ncen〉
$\sigma _{\log M_{*}}$	dex	Log-normal scatter in stellar mass at fixed halo mass	〈Ncen〉
βsat	None	Slope of the scaling of Msat	〈Nsat〉
Bsat	None	Normalization of the scaling of Msat	〈Nsat〉
βcut	None	Slope of the scaling of Mcut	〈Nsat〉
Bcut	None	Normalization of the scaling of Mcut	〈Nsat〉
αsat	None	Power-law slope of the satellite occupation function	〈Nsat〉

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The SMF is typically calculated in bins in stellar mass. Let us consider the stellar mass bin Mt1* < M* < Mt2*. The abundance of galaxies within this stellar mass bin, ΦSMF(Mt1*, Mt2*), is simply obtained from our model and the halo mass function, dn/dMh, according to

$$\begin{eqnarray} &&\Phi _{{\rm SMF}}(M_*^{t_1},M_*^{t_2})\nonumber\\ &&\quad =\int _{0}^{\infty } \left[\int _{M_*^{t_1}}^{M_*^{t_2}} \Phi (M_{*}|M_h) {d}M_* \right] \frac{{d}n}{{\it {d}M}_h}{\it {d}M}_h\nonumber \\ &&\quad =\int _0^{\infty } \langle N_{\rm tot}(M_h|M_{*}^{t_1},M_{*}^{t2})\rangle \frac{{d}n}{{\it {d}M}_h}{\it {d}M}_h, \end{eqnarray} \tag{ 17 }$$

where we recall that Φ(M_*|M_h) represents the conditional SMF and 〈N_tot〉 is the total occupation function (including both satellites and centrals).

The shear signal induced by a given foreground mass distribution on a background source galaxy will depend on the transverse proper distance between the lens and the source and on the redshift configuration of the lens–source system. A lens with a projected surface mass density, Σ(r), will create a shear that is proportional to the surface mass density contrast, ΔΣ(r):

$$\begin{equation} \Delta \Sigma (r)\equiv \overline{\Sigma }(< r)-\overline{\Sigma }(r)=\Sigma _{\rm crit}\times \gamma _t(r). \end{equation} \tag{ 18 }$$

Here, $\overline{\Sigma }(< r)$ is the mean surface density within proper radius r, $\overline{\Sigma }(r)$ is the azimuthally averaged surface density at radius r (e.g., Miralda-Escude 1991; Wilson et al. 2001), and γ_t is the tangentially projected shear. The geometry of the lens–source system intervenes through the critical surface mass density, ⁷ Σ_crit, which depends on the angular diameter distances to the lens (D_OL), to the source (D_OS), and between the lens and the source (D_LS):

$$\begin{equation} \Sigma _\mathrm{crit} = \frac{c^2}{4\pi G_{{{\rm N}}}}\, \frac{D_\mathrm{OS}}{D_\mathrm{OL}\,D_\mathrm{LS}}\;, \end{equation} \tag{ 19 }$$

where G_N represents Newton's constant.

Consider two different populations characterized, respectively, by δ_a and δ_b. The two-point cross-correlation function of δ_a and δ_b at comoving position $\vec{r}_{\rm co}$ is given by

$$\begin{equation} \xi _{ab}(\vec{r}_{\rm co}) \equiv \langle \delta _a(\vec{r}_{\rm co})\delta _{b}(\vec{x}_{\rm co}+\vec{r}_{\rm co}) \rangle. \end{equation} \tag{ 20 }$$

For example, if δ_g and δ_dm are, respectively, the overdensities of galaxies and dark matter, then we can characterize their relative distributions via the galaxy-mass cross-correlation function, which is denoted by ξ_gm and is equal to

$$\begin{equation} \xi _{\rm gm}(\vec{r_{\rm co}}) = \langle \delta _{\rm g} (\vec{r_{\rm co}}) \delta _{\rm dm}(\vec{x_{\rm co}}+\vec{r_{\rm co}})\rangle. \end{equation} \tag{ 21 }$$

Similarly, ξ_gg refers to the galaxy autocorrelation function.

In the following, $\vec{r}_{\rm co}$ is the three-dimensional comoving distance, $\vec{r}_{||,{\rm co}}$ is the projected comoving line-of-sight distance, and $\vec{r}_{p,{\rm co}}$ is the projected comoving transverse distance:

$$\begin{equation} r_{\rm co}=\sqrt{r_{p,{\rm co}}^2+r_{||,{\rm co}}^2}. \end{equation} \tag{ 22 }$$

In Paper II, we will employ physical coordinates for g–g lensing measurements whereas for clustering we will use comoving coordinates. In previous work, Mandelbaum et al. (2006b) have used comoving coordinates and Johnston et al. (2007) have used physical coordinates. Therefore, our g–g lensing formulas more closely resemble those of Johnston et al. (2007). The relationship between comoving and physical distances is simply r_co = r_ph(1 + z). In a similar fashion to Equation (22), we can write

$$\begin{equation} r_{\rm ph}=\sqrt{r_{p,{\rm ph}}^2+r_{||,{\rm ph}}^2}. \end{equation} \tag{ 23 }$$

In comoving coordinates, the density field is $\rho (r_{\rm co},z)=\overline{\rho }(1+\xi _{\rm gm}(r_{\rm co},z))$, where $\overline{\rho }=\rho _{c,0} \Omega _{m,0}$ is the average density of matter in the universe. Since ξ_gm is often expressed in comoving coordinates, we first derive Σ in comoving units (denoted by Σ_co) and then we transform Σ into physical units (denoted by Σ_ph) before computing ΔΣ. For a lens at redshift z_L, the projected surface mass density, Σ, is obtained by integrating the three-dimensional density over the line of sight:

$$\begin{eqnarray} &&\Sigma _{\rm co}(r_{\rm p,co},z_L) \hspace{0.0pt}\nonumber\\ &&\quad = \int \rho \left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right)dr_{\rm ||,co} \nonumber \\ &&\quad = \rho _{c,0} \Omega _{m,0} \int \left[1+\xi _{\rm gm}\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right)\right]dr_{\rm ||,co}, \nonumber \\ && \end{eqnarray} \tag{ 24 }$$

where r_{p, co} and r_{||, co} refer, respectively, to the comoving transverse and line-of-sight distance from the lens. In principle, this integral should extend from the redshift of the observer (z_O) to the redshift of the source (z_S). However, ξ_gm falls off rapidly enough that in practice, the redshift evolution of ξ_gm can be neglected and integrating only out to a distance of r_{||, co} = 50 Mpc is sufficient. Furthermore, for the purpose of computing ΔΣ, the constant term in Equation (24) can be dropped (due to the subtraction in ΔΣ) and the mean excess projected density $\overline{\Sigma }(r)$ can be approximated by the radial integral:

$$\begin{eqnarray} &&\overline{\Sigma }_{\rm co}(r_{\rm p,co},z_L) \hspace{0.0pt}\nonumber\\ &&\quad = 2 \rho _{c,0} \Omega _{m,0} \int _0^{50\, {\rm Mpc}}\xi _{\rm gm}\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right)dr_{\rm ||,co}.\qquad \end{eqnarray} \tag{ 25 }$$

The mean excess projected density in physical units is then $\overline{\Sigma }_{\rm ph} =\overline{\Sigma }_{\rm co}\times (1+z_L)^2$.

The average $\overline{\Sigma }$ within radius r is equal to

$$\begin{equation} \overline{\Sigma }_{\rm ph}(<r_{\rm p,ph}) = \frac{2}{r_{\rm p,ph}^2} \int _0^{r_{\rm p,ph}}\overline{\Sigma }_{\rm ph}(r^\prime)r^\prime dr^\prime. \end{equation} \tag{ 26 }$$

Finally, ΔΣ is obtained by combining Equations (18), (25), and (26).

Our model for calculating the autocorrelation function of galaxies is based on the model given in Tinker et al. (2005) (also see Zheng 2004). As described above, the HOD is broken into central and satellite galaxy occupation functions. Thus, in the HOD context, pairs of galaxies come from two distinct terms: pairs within a single halo and pairs between galaxies in two different halos. The total correlation function is

$$\begin{equation} \xi _{{\rm gg}}(r_{\rm co}) + 1 = \left[\xi ^{{\rm 1h}}_{{\rm gg}}(r_{\rm co})+1\right] + \left[\xi ^{{\rm 2h}}_{{\rm gg}}(r_{\rm co})+1\right], \end{equation} \tag{ 27 }$$

where 1h and 2h refer to "one-halo" and "two-halo" terms, respectively. The one-halo correlation function is written as

$$\begin{eqnarray} 1+\xi _{{\rm gg}}^{{\rm 1h}}(r_{\rm co}) &=& \frac{1}{2\pi r_{\rm co}^2\overline{n}_g^2} \int {\it dM}_h \frac{dn}{{\it dM}_h}\nonumber\\ && \times\frac{\langle N(N-1)\rangle _M}{2} \frac{1}{2R_{h}} F^\prime \left(\frac{r_{\rm co}}{2R_{h}}\right),\qquad \end{eqnarray} \tag{ 28 }$$

where $\bar{n}_g$ is the space density of galaxies in the sample being modeled, 〈N(N − 1)〉_M is the second moment of the distribution of galaxies within halos as a function of halo mass, and F'(x) is the radial distribution of pairs within the halo normalized to unity. Within a halo, pairs of galaxies can be between the central galaxy and a satellite, or between two satellite galaxies. The radial pair profile is different for these two combinations, thus we express their relative contributions to ξ^1h_gg(r_co) as

$$\begin{eqnarray} \frac{\langle N(N-1)\rangle _M}{2}F^\prime (x) & = & \langle N_{\rm cen}N_{\rm sat} \rangle _MF^\prime _{\rm cs}(x) \nonumber \\ & & + \frac{\langle N_{\rm sat}(N_{\rm sat}-1)\rangle _M}{2}F^\prime _{\rm ss}(x),\quad \end{eqnarray} \tag{ 29 }$$

where F'_cs is the pair distribution for central–satellite pairs and F'_ss is the equivalent for satellite–satellite pairs. The former is related to the density profile of satellite galaxies, and the latter is related to the density profile convolved with itself (analytic expressions for this convolution can be found in Sheth et al. 2001). Here we assume that the radial distribution of satellite galaxies is the same as the dark matter, for which we assume the profile form of Navarro–Frenk–White (NFW; Navarro et al. 1997) using the mass–concentration relation of Muñoz-Cuartas et al. (2010). Since we assume that satellites trace the dark matter, F'_cs will be equal to the quantity F'_c that we will introduce in Equation (35) and F'_ss will be equal to F'_s.

Central galaxies only exist as one or zero objects in a halo, thus they have no second moment. For the second moment of satellite galaxies, we assume Poisson statistics about 〈N_sat〉, which is in good agreement with results from numerical simulations (Kravtsov et al. 2004; Zheng et al. 2005). Possible deviations from Poisson behavior (Busha et al. 2010; Boylan-Kolchin et al. 2010) mainly affect clustering for galaxy samples where a majority of the satellites originate in halos with M_h < M_sat because in this case, 〈N_sat〉 drops to 〈N_sat〉 ≲ 1. Luminous red galaxies (LRGs) fall into this category for example. Indeed, satellite galaxies in LRG samples mainly originate in M_h < M_sat halos because M_sat is close to the exponential cutoff in the halo mass function (for LRGs, M_sat ∼ 4 × 10¹⁴ M_☉). However, for the types of samples that we consider in Paper II, deviations from Poisson statistics should not significantly affect the clustering predictions of the HOD.

A detailed description of the two-halo term can be found in Tinker et al. (2005). Briefly, in the regime where r > R_h of massive halos, this term can be expressed as

$$\begin{equation} \xi _{{\rm gg}}^{{\rm 2h}}(r_{\rm co}) = b_g^2\zeta ^2(r_{\rm co})\xi _m(r_{\rm co}), \end{equation} \tag{ 30 }$$

where ξ_m(r_co) is the nonlinear matter correlation function and b_g is the large-scale bias of galaxies in the sample, and ζ(r_co) is the scale dependence of dark matter halo bias. For ξ_m(r_co), we use the fitting function of Smith et al. (2003). For ζ(r_co), we use the fitting function of Tinker et al. (2005). The galaxy bias is computed from the HOD by

$$\begin{equation} b_g = \bar{n}_g^{-1}\int b(M_h)\langle N\rangle _M\frac{dn}{dM_h}dM_h, \end{equation} \tag{ 31 }$$

where dn/dM_h is the halo mass function for which we use Tinker et al. (2008), and b(M_h) is the halo bias function for which we use Tinker et al. (2010).

In the regime where r < R_h, Equation (30) breaks down due to the effects of halo exclusion; i.e., the effect that the center of one halo cannot exist within the virial radius of another halo (and still be considered a "two-halo" pair). This is explained in detail in Tinker et al. (2005). Because the mass function and bias relation used in this analysis are taken from numerical results based on spherical overdensity (SO) halo catalogs (Tinker et al. 2008, 2010), the halo exclusion must be modified to match this halo definition. In the SO halo finding algorithm of Tinker et al. (2008), halos are allowed to overlap so long as the center of one halo is not contained within the radius of another halo. Thus, the minimum separation of two halos with radii R₁ ⩾ R₂ is R₁, rather than the sum of the two radii, as done in Tinker et al. (2005). For a projected statistic like w(θ), this makes only a small difference in clustering at the one-halo to two-halo transition, but significantly speeds up computation of the two-halo term.

Once we have calculated ξ_gg(r_co) for a given HOD model, we compute the observable w(θ) by

$$\begin{equation} w(\theta) = \int dz\,N^2(z)\, \frac{dr_{\rm co}}{dz} \int dx\, \xi \left(\sqrt{x^2 + r_{\rm co}^2\theta ^2}\right), \end{equation} \tag{ 32 }$$

where N(z) is the normalized redshift distribution of the galaxy sample, r_co is the comoving radial coordinate at redshift z, and $dr_{\rm co}/dz = (c/H_0)/\sqrt{\Omega _{\rm m}(1+z)^3+\Omega _\Lambda }$.

Figure 5 shows the breakdown of the angular correlation function into the one-halo and two-halo terms for low-mass and high-mass galaxy samples.

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We have shown in Section 4.3 that the lensing observable, ΔΣ, can be obtained from ξ_gm by performing two integrals (Equations (25) and (26)). Following the approach of Yoo et al. (2006), ξ_gm is computed from the HOD and ΔΣ is obtained by combining Equations (18), (25), and (26). Thus, ΔΣ is fully specified given our model. Note that the calculation of ξ_gm is performed in comoving units and then the projections of Equations (25) and (26) to obtain ΔΣ are performed in physical units (to match our measured g–g lensing signal which is computed in physical units in Paper II).

Typically, ξ_gm is decomposed into a one-halo and a two-halo term,

$$\begin{equation} 1+\xi _{{\rm gm}}(r_{\rm co}) = [1+\xi _{{\rm gm}}^{{\rm 1h}}(r_{\rm co})]+[1+\xi _{{\rm gm}}^{{\rm 2h}}(r_{\rm co})], \end{equation} \tag{ 33 }$$

where the one-halo term represents galaxy–matter pairs from single halos and is dominant on small scales (≲Mpc) and the two-halo term corresponds to pairs from distinct halos and is dominant on larger scales (≳Mpc). The one-halo term is obtained according to

$$\begin{eqnarray} &&1+\xi _{{\rm gm}}^{{\rm 1h}}(r_{\rm co}) \hspace{0.0pt}\nonumber\\ &&\quad =\frac{1}{4\pi r_{\rm co}^2 \overline{n}_g}\int _{0}^{\infty }\frac{dn}{dM_h}\frac{M_h}{\overline{\rho }}\frac{1}{2R_{h}}\langle N_{{\rm tot}}\rangle F^{\prime }\left(\frac{r_{\rm co}}{2R_{h}}\right)dM_h.\qquad \end{eqnarray} \tag{ 34 }$$

Further details about the origin of Equation (34) are presented in the Appendix.

The product $\overline{n}_g F^{\prime }$ is split into two terms:

$$\begin{equation} \overline{n}_g F^{\prime }(x)=\overline{n}_c F^{\prime }_{c}(x)+ \overline{n}_s F^{\prime }_{s}(x), \end{equation} \tag{ 35 }$$

where F'_c is linked to the density profiles of dark matter halos (see the Appendix) and F'_s is related to the convolution of the dark matter density profile and the satellite galaxy distribution. To calculate F'_c we assume spherical NFW profiles truncated at R_h and we adopt the Muñoz-Cuartas et al. (2010) mass–concentration relation for a WMAP5 cosmology. To calculate F'_s we assume that the satellite galaxy distribution follows the dark matter distribution. Given this assumption, F'_s is simply related to the convolution of the NFW profile with itself.

In this paper, we neglect the contribution to ΔΣ from sub-halos; Yoo et al. (2006) have shown this component to be negligible at the 10% level.

We calculate the two-halo term as described in Yoo et al. (2006), with two major exceptions. First, as stated in the previous section, we are using the halo exclusion from the SO halo definition. Second, because the halo mass function of Tinker et al. (2008) and the halo bias function of Tinker et al. (2010) are normalized such that integrating over all M_h produces the mean matter density and a bias of unity, there is no need to employ the "break mass" from Yoo et al. (2006, see their Equation (16)). In the limit where r > R_h of massive halos,

$$\begin{equation} \xi _{{\rm gm}}^{{\rm 2h}}(r_{\rm co}) = b_g\zeta (r_{\rm co})\xi _m(r_{\rm co}), \end{equation} \tag{ 36 }$$

analogous to Equation (30).

In addition to the two terms presented above, we add another component to the modeling of ΔΣ which is absent in Yoo et al. (2006), namely the contribution to ΔΣ from the baryons of the central galaxy which can be non-negligible on very small scales (<50 kpc). Although the baryons typically follow Sérsic profiles (Sérsic 1963), at the scales of interest for this study, well above a few effective radii (>20 kpc), the lensing contribution of the baryons can be modeled by a simple point source, scaled to 〈M_*〉, the average stellar mass of the galaxies in the sample:

$$\begin{equation} \Delta \Sigma _{\rm stellar}(r)=\frac{\langle M_{*}\rangle }{\pi r^2}. \end{equation} \tag{ 37 }$$

In total, the final g–g lensing signal is modeled as the sum of three terms: ΔΣ_tot = ΔΣ_stellar + ΔΣ_1h + ΔΣ_2h. Note that the one-halo and two-halo terms can also be decomposed into central and satellite contributions, but for simplicity, we have grouped these terms together.

In order to illustrate the various terms that contribute to the g–g lensing signal, we have plotted the signal in Figure 6.

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The influence of each model parameter on the observed SMF is shown in the right-hand column in Figure 7 (the same column is reproduced in Figure 8). The data point with an error bar represents the typical error bar for a COSMOS-like survey where the error bar includes sample variance computed from a series of mock catalogs (described in the following section). The first point worth mentioning here is that the errors on the SMF are relatively small compared to the clustering and the lensing. It is always the case that the measurement of a one-point statistic from a given set of data is more precise than the measurement of a two-point (or higher) statistic. This implies that the SMF will in general play an important role in constraining the parameters of the SHMR. However, in follow-up work we will investigate the sensitivity of this probe combination to cosmological parameters and models of modified gravity for example. In these studies, the clustering and the g–g lensing will play a critical role despite their typically larger errors bars.

A second noteworthy point in Figure 7 is that the SMF appears to be sensitive to all 10 parameters, whereas certain parameters such as M_{*, 0} and β have very little effect on the clustering and lensing signals. Coupled with the fact that the SMF has fairly small error bars, this implies that the SMF will have quite a lot of constraining power on the overall model compared to the g–g lensing for example.

The effects of M₁ and M_{*, 0} on the SMF are fairly intuitive. M₁ roughly induces an up/down shift in the amplitude of the SMF and M_{*, 0} corresponds to a left/right shift in the SMF. A larger value of B_sat implies that there are fewer satellite galaxies in high-mass halos for a given stellar mass threshold. Thus, we see that an increase in B_sat corresponds to a decrease in the amplitude of the SMF due to the fact that the contribution from satellite galaxies has decreased. Similar arguments apply to B_cut. From Figure 7 we can anticipate that if the SMF were only used to constrain this model, degeneracies would occur between B_sat, B_cut, and M₁. Fortunately, the satellite parameters have a significant effect on both the clustering and the g–g lensing so this degeneracy should be broken when all three probes are used in conjunction.

In Figure 9, we highlight the effects of four particular parameters on the SMF: β, γ, δ, and $\sigma _{\log M_{*}}$. The β parameter affects the low-mass slope of the SMF so that a larger value of β corresponds to a steeper low-mass slope in the SMF. The γ parameter (we recall that this controls the transition region of the SHMR as can be seen from Figure 1) has an interesting effect since it regulates a "plateau" feature in the SMF at log₁₀(M_*) ∼ 10.5. In fact, this feature in the SMF has been noticed already and discussed in detail, for example, in Drory et al. (2009). In Drory et al. (2009), this feature was described as a "dip" because at these scales, the SMF is below the best-fit Schechter function. However, we note that "dip" is a somewhat misleading name for this feature since it could also be taken to mean that dN/dlog₁₀(M_*) does not decrease monotonically with M_*. A close inspection of our SMFs in Paper II shows that there is no evidence in the data for an actual "dip" in dN/dlog₁₀(M_*). Instead, the data are more consistent with a flattening of dN/dlog₁₀(M_*) around log₁₀(M_*) ∼ 10.5. Therefore, we would like to suggest that this feature should be described as a "plateau" in the SMF rather than a "dip."

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In Figure 10 we show the link between the dark matter halo mass function, the SHMR, and the SMF. In this figure, we have used the fact that dN/dlog₁₀M_*  =  dN/dlog₁₀M_h × (dlog₁₀M_h/dlog₁₀M_*) so that the various functions can be linked "by eye" by drawing a box between the four different panels. We have illustrated how to link the various functions with the dashed lines in Figure 10 at the scale of the pivot stellar mass. Figure 10 shows that the "plateau" feature is caused by the transition that occurs in the SHMR at M_h ∼ 10¹² M_☉ from a low-mass power-law regime to a sub-exponential function at higher stellar mass.

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Finally, the scatter in stellar mass at fixed halo mass has a noticeable effect on the SMF at the high-mass end, which is also commonly referred to as Eddington bias. A larger value of $\sigma _{\log M_{*}}$ will lead to an inflated observed SMF at large stellar masses.

The g–g lensing signal is dominated by the central one-halo term roughly on scales below 0.3 Mpc and by the satellite one-halo term roughly on scales above 0.3 Mpc and below a few Mpc (see Figure 6). Thus, the effects of the four parameters that regulate the satellite occupation function (B_sat, B_cut, β_sat, β_cut) have a strong scale-dependent effect on the g–g lensing signal. For example, B_sat controls the power-law amplitude of 〈N_sat〉. A smaller value of B_sat will reduce the ratio $M_{\rm sat}/f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$ and will therefore increase the number of satellites in a sample. This will lead to an increase in the g–g lensing signal from 0.3 to 1 Mpc due to the increased amplitude of the one-halo satellite term. Similarly, a smaller value of β_sat will also reduce the ratio $M_{\rm sat}/f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$ and consequently will increase the one-halo satellite term. This effect is more pronounced in Figure 8, which illustrates a low stellar mass sample, compared to Figure 7, which illustrates a high stellar mass sample.

Another parameter worth discussing here is $\sigma _{\log M_{*}}$. Figure 7 demonstrates that $\sigma _{\log M_{*}}$ has a stronger effect on the lensing signal for high stellar mass samples compared to low stellar mass samples. As discussed in Section 2.2, this is simply due to the fact that the data are binned according to M_*. The observables are therefore sensitive to the scatter in halo mass at fixed stellar mass, $\sigma _{\log M_h}$. At fixed $\sigma _{\log M_{*}}$, $\sigma _{\log M_h}$ will increase with M_*. As a result, the effects of scatter are more prominent in g–g lensing measurements for high stellar mass samples.

To first order, the SMF and the clustering of galaxies are tethered; more massive halos are both more clustered and less abundant. This is also true of galaxies because rare, massive galaxies live in such halos. If the amplitude of the SMF increases, the clustering as a function of stellar mass decreases. This is especially true at masses above the knee in the SMF. From Figure 9, increasing δ or $\sigma _{\log M_{*}}$ increases the abundance of high-mass galaxies. Given that the number of halos is fixed, this can only mean that massive galaxies are occupying less massive, less clustered halos.

There are several parameters that have a direct influence on the clustering of galaxies without changing the SMF appreciably. The parameters B_sat and β_sat are the most important in this regard. They control the "shoulder" in the HOD, defined conceptually as the increase in halo mass, relative to $f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$, before satellites begin to enter the sample. Quantitatively, this is expressed as the ratio $M_{\rm sat} /f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$, as shown in Equation (12). Reducing this ratio increases the number of satellite galaxies in a sample, which in turn increases the large-scale bias of a sample and significantly enhances the clustering within the one-halo term. The parameters B_cut and β_cut have a more subtle effect on clustering. If the cutoff mass, defined by Equation (14), is below $f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})$, then M_cut has no effect on clustering. But as M_cut increases, satellite galaxies are removed from low-mass halos. If the density of satellites is held fixed, increasing M_cut redistributes satellites into more massive halos. This will increase the large-scale bias and change the shape of the one-halo term such that the correlation function deviates from a pure power-law form (see the Appendix in Zheng et al. 2009).