RedMaPPer: Evolution and Mass Dependence of the Conditional Luminosity Functions of Red Galaxies in Galaxy Clusters

Cluster finding is performed using the red-sequence-based cluster finder redMaPPer (Rykoff et al. 2014; Rozo & Rykoff 2014), which identifies galaxy clusters as overdensities of red-sequence galaxies. Importantly, redMaPPer includes a probabilistic assignment of galaxy membership, allowing straightforward incorporation of uncertainty regarding whether a galaxy is or is not a red galaxy in a particular cluster. We further discuss these probabilities in Section 2.2. However, it is important to note that this catalog only includes red galaxies; thus, the CLFs presented in this paper are for red galaxies only. As shown in Wetzel et al. (2013), more than 80% of the centrals and 50% of the satellites existing in the mass and luminosity range considered in this paper are quenched; thus, the results presented here represent the properties of the majority of the cluster galaxy population, but the details will clearly differ compared to the full cluster galaxy population.

In this paper, we consider the redMaPPer v5.10 cluster catalog derived from the SDSS DR8 data set (Rozo et al. 2015a). The depth of DR8 allows us to select a volume-limited cluster catalog out to a redshift z ≤ 0.3. The redshift limit corresponds to the redshift at which the luminosity threshold of 0.2L* used by redMaPPer crosses the survey depth. Here, L* is the passively evolving characteristic luminosity of cluster galaxies assumed by redMaPPer in its filtering process. The final sample contains 7016 clusters with λ > 20 in the redshift range of 0.1 < z < 0.3.

For every galaxy in the vicinity of a galaxy cluster, the redMaPPer algorithm estimates the probability that the galaxy is a cluster member on the red sequence. Comparison of the photometric probabilities with spectroscopic data from SDSS (Aihara et al. 2011) and GAMA (Driver et al. 2011) led Rozo et al. (2015b) to derive small corrections to the original membership probabilities. The analysis in this work relies on these improved membership probabilities.

In addition to providing galaxy membership probabilities, redMaPPer also assigns cluster centers in a probabilistic fashion. While most clusters have a single bright galaxy clearly located at the cluster center, for others it is not possible to unambiguously identify a unique central galaxy. Consequently, redMaPPer provides a list of possible central galaxies, each tagged with the probability of it being the central galaxy of the cluster. Many clusters have more than a single high-probability center: about 63% (34%) of clusters in our DR8 sample with λ > 20 have at least two galaxies with a greater than 1% (10%) probability of being the central galaxy. This necessitates approaching the problem of cluster membership and cluster centering in a probabilistic way, especially when investigating the properties of central galaxies. In our fiducial analysis, we assume that the redMaPPer centering probability estimates are correct (Rozo et al. 2015a), and model the resulting ensemble properties in order to constrain the behavior of central galaxies in massive cluster halos. We discuss how this assumption affects our results in Section 5, and discuss how to incorporate this effect into our error budget.

Galaxy luminosities in the SDSS redMaPPer catalog were calculated based on the SDSS CModel magnitudes, which are known to underestimate galaxy brightness for massive galaxies (Bernardi et al. 2013). This bias depends on galaxy luminosity and type (Bernardi et al. 2017) and thus has a large impact on the inference of the galaxy luminosity function. To account for this, Meert et al. (2015) performed an improved photometric measurement (PyMorph magnitude hereafter) on SDSS DR7 spectroscopic targets. We rely on the Meert et al. (2015) measurement to develop an empirical correction for the SDSS photometry of each redMaPPer galaxy.

The empirical correction is calculated as follows. First, we select galaxies that have good measurements of total i-band magnitude (without flag above 20) in the catalog described in Meert et al. (2015) and Meert et al. (2016). For each galaxy, we take the "best model" magnitude (PyMorph) described in Meert et al. (2015), which is estimated from a combination of Sérsic and Sérsic–Exp profile fits. Second, we cross-match the above catalog to redMaPPer member galaxies by matching galaxies within 3'' and redshift differences within 0.03. Here, we assume that member galaxies have the same redshifts as reported in the redMaPPer catalog. Third, we compute the difference between the SDSS CModel magnitude and PyMorph magnitude. We look for any redshift dependence of this magnitude difference in three absolute magnitude bins (Figure 1). We notice that at the faintest absolute magnitude bins, the difference between CModel and PyMorph becomes significant at z > 0.17. The samples with PyMorph measurements are selected to be brighter than m_r = 17.77, because this is the lower limit for completeness of the SDSS Spectroscopic Survey (Meert et al. 2015). The magnitude cut m_r = 17.77 corresponds to M_r = −21.81 at z = 0.17, which lies within the faintest magnitude bin in Figure 1. Therefore, the redshift dependence we find in Figure 1 is likely due to the incompleteness of the sample. To account for this selection effect, we adopt an upper redshift cut at z = 0.17 and a lower redshift cut z = 0.1 (to match the redMaPPer selection). After this step, we obtain 18,430 matched galaxies, including 1516 galaxies that are the most probable central galaxies in their host clusters. We compute the mean difference between the CModel and PyMorph magnitude as a function of CModel magnitude for both red central and red satellite cluster galaxies. As shown in Figure 2, this difference depends on whether a galaxy is a central or a satellite. The result is consistent with what Bernardi et al. (2017) found. We further check the difference for the brightest red satellite galaxies (BSGs) and find that they are consistent with the full population of red satellite galaxies. This gives further evidence for the fidelity of central galaxy identification. Finally, we fit empirical central and satellite correction models to the observed magnitude differences. The correction models take the following form:

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{Central}:\\ \quad {\rm{\Delta }}m={A}_{{\rm{c}}}({M}_{\mathrm{CModel}}+22.5)+{B}_{{\rm{c}}}\end{array}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{Satellite}:\\ \quad {\rm{\Delta }}m={A}_{s}({M}_{\mathrm{CModel}}-{B}_{s}),{if}\ {M}_{\mathrm{CModel}}\lt {B}_{s}\\ \,=\,{C}_{s},\mathrm{if}\ {M}_{\mathrm{CModel}}\gt {B}_{s},\end{array}\end{eqnarray} \tag{ 2 }$$

where Δm is m_CModel−m_PyMorph.

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The best-fit parameters for the model are listed in Table 1. We apply this correction to every redMaPPer galaxy based on the combination of central and satellite correction models, weighting by the probability of a galaxy being a central.

Table 1.  Values of Empirical Photometry Correction Parameters

Parameters	Values	Equation Reference(s)	Description
Ac	$-{0.110}_{-0.010}^{+0.010}$	(1)	Slope of Correction
Bc	${0.148}_{-0.007}^{+0.007}$	(1)	Normalization of Correction
As	$-{0.092}_{-0.006}^{+0.006}$	(2)	Slope of Correction
Bs	$-{21.96}_{-0.04}^{+0.04}$	(2)	Cutoff of Correction
Cs	$-{0.009}_{-0.002}^{+0.002}$	(2)	Constant after Cutoff

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We discuss the impact of this photometric correction on our result in Section 5.

Halo mass is not an observable. To tie the measured luminosity function to the mass of host halos, we adopt the richness–mass relation measured for the same cluster sample by Costanzi et al. (2019b). Here, we briefly summarize this relation, and discuss how to incorporate it into the model of the CLF in the next two subsections.

The richness–mass relation is modeled as the convolution of the two probability distributions:

$$\begin{eqnarray}&&P({\lambda }_{\mathrm{obs}}| M,z)=\int d{\lambda }_{\mathrm{true}}P({\lambda }_{\mathrm{obs}}| {\lambda }_{\mathrm{true}},z)P({\lambda }_{\mathrm{true}}| M)d{\lambda }_{\mathrm{true}},\end{eqnarray} \tag{ 8 }$$

where the probability $P({\lambda }_{\mathrm{obs}}| {\lambda }_{\mathrm{true}},z)$ models observational noise and projection effects. The observational noise is measured by injecting 10,000 synthetic clusters into the SDSS data set of known richness and measuring their recovered richness (Costanzi et al. 2019a). In contrast, projection effects depend on the large-scale structure of the universe and therefore cannot be properly quantified by randomly injected clusters. Instead, Costanzi et al. (2019b) calibrate this effect using N-body simulations populated with galaxies using a simple mass–richness relation. For details of this calibration, we refer the reader to Costanzi et al. (2019a).

The second term in the integral accounts for the intrinsic relation between cluster richness and cluster mass. Different parameterizations have been tested by Costanzi et al. (2019b); none had a significant effect on the cosmological constraints. Here, we adopt the model in which $P({\lambda }_{\mathrm{true}}| M)$ is a log-normal distribution with

$$\begin{eqnarray}\begin{array}{rcl}\langle {\lambda }_{\mathrm{true}}| M\rangle & = & {\lambda }_{0}{\left(M/{M}^{* }\right)}^{\alpha }\\ {\sigma }^{2} & = & {\sigma }_{\mathrm{intr}}^{2}+(\langle {\lambda }_{\mathrm{true}}| M\rangle -1)/\langle {\lambda }_{\mathrm{true}}| M{\rangle }^{2},\end{array}\end{eqnarray} \tag{ 9 }$$

where the pivot point M^* is set to 10^14.344h⁻¹M_⊙.

With this richness–mass relation in hand, we now describe our model of CLFs.

4.1.1. Central Model

We first write down the most general model for the central galaxy luminosity function:

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{c}(L| \lambda ) & = & \displaystyle \frac{n(L,\lambda )}{n(\lambda )},\\ n(L,\lambda ) & = & {\displaystyle \int }_{\mathrm{ln}{\lambda }_{\min }}^{\mathrm{ln}{\lambda }_{\max }}\displaystyle \int n(M)P(\lambda ,L| M){dMd}\mathrm{ln}\lambda ,\\ n(\lambda ) & = & {\displaystyle \int }_{\mathrm{ln}{\lambda }_{\min }}^{\mathrm{ln}{\lambda }_{\max }}{\displaystyle \int }_{\mathrm{ln}{L}_{\min }}^{\mathrm{ln}{L}_{\max }}\displaystyle \int n(M)P(\lambda ,L| M){dMd}\mathrm{ln}{Ld}\mathrm{ln}\lambda ,\end{array}\end{eqnarray} \tag{ 10 }$$

where $P(\lambda ,L| M)$ is the joint probability of richness and central galaxy luminosity. n(M) is the Tinker halo-mass function (Tinker et al. 2008) calculated using the HMFcalc tool (Murray et al. 2013) at the mean redshift of each redshift bin.

$P(\lambda ,L| M)$ can be further decomposed into $P(L| \lambda ,M)P(\lambda | M)$, where $P(\lambda | M)$ is the same as Equation (8) evaluated at the mean redshift of each individual redshift bin.

We model $P(L| \lambda ,M)$ as

$$\begin{eqnarray}\begin{array}{rcl}P(L| \lambda ,M) & = & \displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{\mathrm{ln}L}^{2}}}\exp \left(\displaystyle \frac{-{\left(\mathrm{ln}L-\langle \mathrm{ln}{L}_{c}| M,\lambda \rangle \right)}^{2}}{2{\sigma }_{\mathrm{ln}L}^{2}}\right),\\ \langle \mathrm{ln}{L}_{c}| M,\lambda \rangle & = & \langle \mathrm{ln}{L}_{c}| M\rangle +{d}_{\mathrm{eff}}(\mathrm{ln}\lambda -\langle \mathrm{ln}\lambda \rangle (M)),\end{array}\end{eqnarray} \tag{ 11 }$$

where $\langle \mathrm{ln}{L}_{c}| M\rangle$ is the mean log central galaxy luminosity of halos with mass M, ${\sigma }_{\mathrm{ln}L}$ is the scatter of the log central galaxy luminosity at a fixed halo mass, and $\langle \mathrm{ln}\lambda \rangle (M)$ is the mean log richness of halos with mass M, which can be calculated by integrating $\mathrm{ln}\lambda$ over $P(\lambda | M)$ given by Equation (8). d_eff in Equation (11) describes the possible correlation between central galaxy luminosity and richness at a fixed halo mass. A positive d_eff indicates that the luminosity of a central galaxy would be larger than the mean at a fixed halo mass if the richness of the cluster is also above the mean.

It may seem odd to introduce d_eff, instead of using the correlation coefficient itself; i.e., one may be tempted to replace d_eff by $r{\sigma }_{\mathrm{ln}{L}_{c}}/{\sigma }_{\mathrm{ln}\lambda }$. We have opted not to do so because the scatter relation ${d}_{\mathrm{eff}}=r{\sigma }_{\mathrm{ln}{L}_{c}}/{\sigma }_{\mathrm{ln}\lambda }$ is specific to a log-normal model assuming mass-independent scatter. The d_eff parameterization retains the linear shifts in the expectation values expected from sensitivity of L_c to richness within a more general setting. In particular, note that d_eff = 0 implies that L_c is independent of richness at a fixed halo mass.

Finally, following other CLF studies, we further relate the mean log luminosity of red central galaxies $\langle \mathrm{ln}{L}_{c}| M\rangle$ to the mass and the redshift of their host halos using a power-law relation,

$$\begin{eqnarray}&&\langle \mathrm{ln}{L}_{c}| M\rangle =\mathrm{ln}{L}_{c0}+{A}_{L}\mathrm{ln}\displaystyle \frac{M}{{M}_{\mathrm{piv}}}+{B}_{L}\mathrm{ln}(1+z).\end{eqnarray} \tag{ 12 }$$

Here, M is the cluster halo mass, and M_piv is the pivot mass set to $1.57\times {10}^{14}{h}^{-1}{M}_{\odot }$ throughout this paper. A_L and $\mathrm{ln}{L}_{c0}$ are two parameters that define the power-law relation between the red central galaxy luminosity and the host halo mass. B_L describes the redshift evolution.

We use Markov Chain Monte Carlo (MCMC; using the emcee package provided by Foreman-Mackey et al. 2013) to fit the five parameters in central CLF model, specifically, ${\sigma }_{\mathrm{log}L},{d}_{\mathrm{eff}},\mathrm{log}{L}_{c0},{A}_{L},\mathrm{and}\,{B}_{L}$. We run chains using flat priors on each of the parameters, with the condition that ${\sigma }_{\mathrm{log}L}$ is positive. The model is fit to our measured CLF, which spans four redshift and four richness bins. The fit parameters are summarized in Table 2 and are shown in Figure 5. As a consistency check, we also perform a fit to each individual redshift bin. The result is shown in Figure E1 and summarized in Table E1. We found no evidence that the simultaneous fits are driven by a single anomalous redshift bin.

Table 2.  Central CLF Parameters

Parameters	${\sigma }_{\mathrm{log}L}$	deff	logL0	AL	BL	χ2 (dof)
Units	$\mathrm{log}{L}_{\odot }/{h}^{2}$	⋯	$\mathrm{log}{L}_{\odot }/{h}^{2}$	$\mathrm{log}{L}_{\odot }/{h}^{2}$	$\mathrm{log}{L}_{\odot }/{h}^{2}$	⋯
Equation Reference(s)	(6)	(11)	(12)	(12)	(12)	⋯
Values	${0.21}_{-0.03}^{+0.03}$	${0.36}_{-0.16}^{+0.17}$	${10.72}_{-0.05}^{+0.05}$	${0.39}_{-0.04}^{+0.04}$	${1.10}_{-0.29}^{+0.31}$	302.8 (252.3)

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4.1.2. Satellite Model

As before, the satellite CLF can be written as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{s}(L| \lambda )=\displaystyle \frac{n(L,\lambda )}{n(\lambda )},\end{eqnarray} \tag{ 13 }$$

where $n(L,\lambda )$ is the number density of satellite galaxies of luminosity L in clusters of richness λ. Recall that the satellite luminosity function of a halo of mass M is assumed to take a Schechter form according to Equation (7). We must, however, account for the fact that we bin in clusters of richness; in general, ${{\rm{\Phi }}}_{s}(L| M)\ne {{\rm{\Phi }}}_{s}(L| M,\lambda )$. Since richness is the number of satellite galaxies, it is obvious that Φ_s and λ must be correlated: richer clusters must have more satellite galaxies by definition. To account for this covariance, we assume that richness correlates with the amplitude of the luminosity function, but not with its shape. That is, we assume that the luminosity function is always a Schechter distribution with the same faint-end and bright-end slopes for all clusters. Likewise, the characteristic luminosity L* depends only on mass. However, the amplitude of the luminosity function ϕ* does depend on the cluster richness. In fact, we expect these two to be nearly perfectly correlated. Let then ${{\rm{\Phi }}}_{s}(L| {\phi }_{* })$ be the luminosity function with amplitude ϕ*, and let $P({\phi }_{* },\lambda | M)$ be the probability that a cluster has richness λ and satellite amplitude ϕ*. We have then

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Phi }}}_{s}(L| \lambda )=\displaystyle \frac{n(L,\lambda )}{n(\lambda )}\\ \quad =\displaystyle \frac{{\displaystyle \int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}d\lambda \displaystyle \int d{\phi }^{* }\displaystyle \int {dM}\ n(M)P({\phi }^{* },\lambda | M){{\rm{\Phi }}}_{s}(L| {\phi }^{* })}{{\displaystyle \int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}d\lambda \displaystyle \int d{\phi }^{* }\displaystyle \int {dM}\ n(M)P({\phi }^{* },\lambda | M)}.\end{array}\end{eqnarray} \tag{ 14 }$$

Again, we can decompose $P({\phi }^{* },\lambda | M)$ into $P({\phi }^{* }| \lambda ,M)P(\lambda | M)$. In the limit that ϕ* and λ are perfectly correlated, we have

$$\begin{eqnarray}&&\begin{array}{l}P({\phi }^{* }| \lambda ,M)=\delta (\mathrm{ln}{\phi }^{* }-\langle \mathrm{ln}{\phi }^{* }| M\rangle -B(\mathrm{ln}\lambda -\langle \mathrm{ln}\lambda | M\rangle )),\\ \quad B=\sqrt{1-\displaystyle \frac{1}{{\sigma }_{\mathrm{ln}\lambda }^{2}\langle \lambda | M\rangle }}.\end{array}\end{eqnarray} \tag{ 15 }$$

The value of the coefficient B in the above equation is determined by setting the correlation coefficient between ${\phi }_{* }$ and λ to unity. Specifically,

$$\begin{eqnarray}\begin{array}{rcl}r & = & \displaystyle \frac{(\mathrm{ln}{\phi }^{* }-\langle \mathrm{ln}{\phi }^{* }| M\rangle )(\mathrm{ln}\lambda -\langle \mathrm{ln}\lambda | M\rangle )}{{\sigma }_{\mathrm{ln}\lambda }{\sigma }_{\mathrm{ln}{\phi }^{* }}}\\ & = & \displaystyle \frac{B{\sigma }_{\mathrm{ln}\lambda }^{2}}{{\sigma }_{\mathrm{ln}\lambda }{\sigma }_{\mathrm{ln}{\phi }^{* }}}=1.\end{array}\end{eqnarray} \tag{ 16 }$$

In solving for the coefficient B above, we assume that the scatter in $\mathrm{ln}{\phi }_{* }$ is identical to the scatter in $\mathrm{ln}\lambda$ up to Poisson fluctuations, so that

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}{\phi }^{* }}^{2}={\sigma }_{\mathrm{ln}\lambda }^{2}-\displaystyle \frac{1}{\langle \lambda | M\rangle }.\end{eqnarray} \tag{ 17 }$$

The end result is that the satellite luminosity function takes the form

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{s}(L| \lambda )=\displaystyle \frac{{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}d\lambda \int {dM}\ n(M)P({\phi }^{* },\lambda | M){{\rm{\Phi }}}_{s}(L| {\phi }^{* })}{{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}d\lambda \int \int {dM}\ n(M)P({\phi }^{* },\lambda | M)},\end{eqnarray} \tag{ 18 }$$

where ${\phi }_{* }={e}^{\langle {ln}{\phi }_{* }| M\rangle +\sqrt{1-1/({\sigma }_{\mathrm{ln}\lambda }^{2}\langle \lambda | M\rangle )}(\mathrm{ln}\lambda -\langle \mathrm{ln}\lambda | M\rangle )}$. The expectation value of ϕ* at a fixed mass is then given by a power law,

$$\begin{eqnarray}&&\langle \mathrm{ln}{\phi }^{* }| M\rangle =\mathrm{ln}{\phi }_{0}+{A}_{\phi }\mathrm{ln}\displaystyle \frac{M}{{M}_{{piv}}}+{B}_{\phi }\mathrm{ln}(1+z)\end{eqnarray} \tag{ 19 }$$

while the characteristic luminosity L^* is a deterministic function of mass,

$$\begin{eqnarray}&&\mathrm{ln}{L}^{* }(M)=\mathrm{ln}{L}_{s0}+{A}_{s}\mathrm{ln}\displaystyle \frac{M}{{M}_{{piv}}}+{B}_{{Ls}}\mathrm{ln}(1+z).\end{eqnarray} \tag{ 20 }$$

The faint-end and bright-end slopes α and β, respectively, are independent of host halo mass.

Similar to central CLF, we use MCMC (emcee; Foreman-Mackey et al. 2013) to fit the eight parameters of our satellite CLF model, $\mathrm{log}{\phi }_{0}$, A_ϕ, $\mathrm{log}{L}_{s0}$, A_s, α, β, B_Ls, and B_ϕ. We run the chain using flat priors on each parameter, except for β and $\mathrm{log}{\phi }_{0}$. For β, we require the bright-end slope to be positive, and for $\mathrm{log}{\phi }_{0}$, we assume a flat prior with range [−3.9, 2.2]. This prior is based on the measurement in Bernardi et al. (2013), which measured the mean number of galaxies per unit volume and luminosity in the universe. With this measurement, we conservatively assume that the cluster size is 0.5–1.5 Mpc and that the average galaxy density in a cluster is between 1 and 1000 times larger than the galaxy density of the universe, leading to the ϕ₀ prior above. It may seem odd that we need a prior on ϕ₀, but this can be understood as follows. While ϕ₀ characterizes the amplitude of the luminosity function, the data only constrains the satellite CLF for L ≥ 0.2L*, the luminosity threshold of redMaPPer. This allows for considerable uncertainty in the extrapolation to low luminosities, leading to strong degeneracies between ϕ₀ and the bright- and faint-end slopes of the luminosity function (see Figure 6.) The prior on ϕ₀ truncates these degeneracies, preventing us from reaching unphysical conclusions.

We compute the satellite CLF in four evenly spaced redshift bins within z = 0.1–0.3 and fit the model for all four redshift bins simultaneously. Table 3 summarizes the fit parameters; the result is shown in Figure 6. As a consistency check, we also perform a fit to each individual redshift bin. The result is shown in Figure E2 and summarized in Table E2. We found no evidence of our results being driven by a single anomalous redshift bin.

Table 3.  Satellite CLF Parameters

Parameters	$\mathrm{log}{\phi }_{0}$	${A}_{\phi }$	$\mathrm{log}{L}_{s0}$	As	α	β	BLs	${B}_{\phi }$	${\chi }^{2}$ (dof)
Units	$\mathrm{log}({\left(\mathrm{log}L\right)}^{-1})$	$\mathrm{log}({\left(\mathrm{log}L\right)}^{-1})$	$\mathrm{log}{L}_{\odot }/{h}^{2}$	$\mathrm{log}{L}_{\odot }/{h}^{2}$	⋯	⋯	$\mathrm{log}{L}_{\odot }/{h}^{2}$	$\mathrm{log}{L}_{\odot }/{h}^{2}$	⋯
Equation Reference(s)	(7), (19)	(19)	(7), (20)	(20)	(7)	(7)	(20)	(19)	⋯
Values	$-{2.22}_{-1.26}^{+1.39}$	${0.88}_{-0.01}^{+0.01}$	${6.24}_{-0.53}^{+0.65}$	$-{0.01}_{-0.01}^{+0.01}$	${1.36}_{-0.31}^{+0.12}$	${0.28}_{-0.03}^{+0.03}$	${2.39}_{-0.21}^{+0.21}$	$-{0.87}_{-0.17}^{+0.17}$	372.1 (383.6)

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Appendix B details how we validate our models for both central and satellite galaxies through the use of synthetic mock catalogs.

As discussed in Section 2.3, we de-bias the SDSS luminosity estimates of bright galaxies by calibrating against Pymorph magnitudes. To derive systematic uncertainty in our CLF parameters associated with the above de-biasing, we repeat our analysis without applying the systematic de-biasing detailed in Section 2.3. We adopt half of the shift in the recovered parameters between our analysis with and without a systematic correction as the systematic uncertainty associated with the SDSS photometry. Note however that the best-fit values are those reported when applying the correction in Section 2.3. We investigate the impact of this correction on the scatter of central galaxy luminosity σ_logL in Appendix A.

Although we weight each central galaxy candidate by the probability P_cen that the galaxy is the correct cluster center, biases in P_cen will necessarily introduce systematic errors into our galaxy luminosity estimates. Zhang et al. (2019a) find that about 70% of the redMaPPer clusters are well centered at the most probable central galaxy reported in redMaPPer. However, the mean probability of the most probable centrals in redMaPPer is 86%, which indicates that the redMaPPer centering probability is biased. To quantify the resulting systematic uncertainty, we decrease the largest centering probability of each cluster in the redMaPPer catalog by 16% and increase the second largest centering probability by 16%. We remeasure the CLF, and we refit our model. We then quote half of the difference between the best-fit parameters with and without centering correction as the systematic error.

We are uncertain of which direction that mis-centering shifts our parameters to. For instance, if mis-centering happens by randomly centering on satellite galaxies, one would expect the presence of mis-centering to lead to an increase in the scatter of the central galaxy luminosity. However, if mis-centering happens preferentially in clusters where the central galaxy is faint, and the incorrectly chosen center is bright, then we would expect mis-centering to decrease the apparent scatter in luminosity at a fixed halo mass. Because of the lack of clear directionality, we shift the best-fit value to the middle of the best-fit values from both of our analyses (with and without the P_cen corrections). Note that this implies that either limit (no correction, or full correction) is one systematic error away.

The next systematic error we consider is the assumption that the conditional luminosity parameters are independent of cosmological parameters. Future work will fit the mass–richness relation measured by weak lensing analysis as well as CLF simultaneously to properly marginalize over cosmological parameters. We quantify this systematic error by computing the difference in the CLF model at the best-fit CLF parameters, assuming DES Y1 cosmology (Abbott et al. 2018) and Planck cosmology (Planck Collaboration et al. 2014). We find that the difference is at 0.1% and 10% of the statistical error for centrals and satellites, respectively; thus, we conclude this systematic is subdominant relative to our total error budget.

Another possible systematic is the completeness of the survey, particularly as it impacts the faint-end slope of the CLF. However, we do not expect that survey completeness is likely to impact our result. Specifically, while we fit a CLF model to the measurement, we restrict the fit region to luminosity above 0.2L*, which is equivalent to $\mathrm{log}L=9.5\mathrm{log}({L}_{\odot }/{h}^{2})$. At z = 0.3, this corresponds to m_i = 20.56, which is somewhat brighter than the magnitude at which the survey becomes incomplete (roughly m_i ≈ 20.8). Therefore, we expect the effect of incompleteness on our results to be negligible.

An obvious possible source of systematic uncertainty is our reliance on photometric cluster redshift estimates. However, as shown in Rykoff et al. (2014), redMaPPer redshifts are both highly accurate and precise, with ∼0.01 scatter and an even lower bias. To quantify this, we shift the mean redshift of the cluster in each bin by 0.01 and redo the fitting. We see no difference in the final result and conclude that this systematic is not relevant for our study.

Combining all systematic errors into our error budget, we assume that all systematics (photometry, centering, and photometric redshift) are mutually independent and independent from the statistical error. With this assumption, we quantify the impact of each type of systematic on the CLF parameters as a Gaussian random variable. The mean is set by the offset of the best-fit CLF parameters due to the relevant systematic, and the covariance is set by the product of the parameter correlation function scaled by the systematic error. We then draw samples from the multivariate Gaussian distribution and apply them to each step of the MCMC chain before we quantify the marginalized 1σ error of each parameter. Figure 4 summarizes the relative importance of different sources of systematics on our error budget for each CLF parameter.

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6.1.1. The Galaxy Luminosity–Halo-mass Relation of Red Central Galaxies

6.1.2. Correlation between Richness and Central Galaxy Luminosity

We find that the characteristic luminosity L* describing the red satellite CLF depends only weakly on host halo mass, consistent with the findings of Hansen et al. (2009). However, we find a strong redshift dependence of the red satellite CLF. Our measured power-law redshift dependence parameter B_Ls is ${2.39}_{-0.21}^{+0.21}$. Similar to the redshift scaling of central galaxies, this value also contains the contribution of pseudo-evolution and passive evolution. The combination of these two effects would contribute B_Ls = 0.56–1.08. Therefore, our measurement points to red satellites getting dimmer by 25% to 35% between z = 0.3 and z = 0.1, corresponding to ∼2.1 Gyr of evolution. The dimming of red satellite galaxies can be interpreted as arising from tidal stripping of red satellite galaxies as they fall into clusters. In related work, Tollet et al. (2017) used subhalo abundance matching to assign stellar masses to subhalos at infall. They then compared the resulting galaxy distribution to the conditional stellar mass function at redshift zero to infer that galaxies lose 20%–25% of their stellar mass over ∼1.3 Gyr. The two inferred mass-loss rates are comparable.

We also find the bright-end slope of the red satellite luminosity function deviates from a Schechter function. The measured bright-end slope β is ${0.28}_{-0.02}^{+0.03}$, which is consistent with the findings in Bernardi et al. (2013).

As shown in Figure 3, we notice that our red satellite CLF model does not describe the luminosity function well below $\mathrm{log}L=9.5{h}^{-1}{M}_{\odot }$. However, since this luminosity range is below our luminosity cut, we cannot distinguish between the possibility that it is due to the failure of our model, and the possibility that it is due to incompleteness of the measurement at this luminosity range. We leave further investigation to future study.

One of the key results of this paper is an accurate measurement of the relation between mean galaxy luminosity and halo mass. To compute the mean galaxy luminosity, we integrate the CLF in this work and in the literature from $\mathrm{log}L=9.8{h}^{-1}{M}_{\odot }$ to $\mathrm{log}L=14{h}^{-1}{M}_{\odot }$. We compare our result to closely related work (Hansen et al. 2009) that used a different cluster catalog. To make an apples-to-apples comparison, we apply the photometry correction described in Section 2.3 to their data, and shift the pivot mass in Hansen et al. (2009) by 18% upward to account for the mass bias described in Rozo et al. (2009). As shown in Figure 7, we find that our result is consistent with the measurement of Hansen et al. (2009) after applying a correction on photometry with the method described in Section 2.3. However, our results properly marginalize over the possible correlation between cluster observables and account for a variety of systematic effects that have not been previously addressed in the literature.

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We also compare our results to predictions from subhalo abundance matching (SHAM). We produce SHAM catalogs using the Rockstar (Behroozi et al. 2013) halo catalog of the Multidark Plank (Klypin et al. 2016) $1{{\rm{h}}}^{-1}{\mathrm{Gpc}}^{3}$ simulation box. We adopt the parameterization of Lehmann et al. (2017), matching galaxy luminosity to ${v}_{\alpha }={v}_{\mathrm{vir}}{\left({v}_{\max }/{v}_{\mathrm{vir}}\right)}^{\alpha }$, with α = 0.57, the best-fit value in Lehmann et al. (2017). Finally, we abundance match the halo-mass function to the Bernardi et al. (2013) luminosity function following a process identical to that of Reddick et al. (2013).

For the abundance matching model, we assume three different values of scatter between galaxy luminosity and v_α: 0.21 (the best-fit value of the central CLF), 0.18 (1σ low), and 0.24 (1σ high). As shown in Figure 7, the abundance matching result is highly sensitive to the assumed scatter, but is broadly consistent with our measurement. The fact that the subhalo abundance matching results with input from the total galaxy luminosity function match the red central and red satellite CLF data is an interesting, highly nontrivial self-consistency test of both the SHAM framework and our own measurements.

Some previous studies have suggested that the BCG may be nothing more than the brightest outlier of the satellite distribution (e.g., Paranjape & Sheth 2012, and references therein). Other work (Lin et al. 2010; More 2012) indicates that at least in very massive clusters, the BCG is clearly distinct from other galaxies and cannot be defined simply as the brightest galaxy in the population drawing from a single distribution. One natural explanation is that BCGs are central galaxies that follow a luminosity distribution that is distinct from that of the satellites. However, studies have also shown that not all central galaxies are BCGs. For example, Lange et al. (2018) found that P_BNC, the probability that a brightest halo galaxy is not a central galaxy, is roughly 40%. Understanding the origin of this probability is obviously related to our understanding of galaxy formation and evolution, and is a critical component of many cosmology analyses relying on an accurate understanding of the galaxy–halo connection (Leauthaud et al. 2012; Li et al. 2014; Lange et al. 2018). However, despite many measurements of P_BNC in the cluster mass regime, measurements do not in general agree with each other. Skibba et al. (2011) indicates that, depending on mass, as many as 40% of all BCGs are not located at a cluster's center. Recently, Lange et al. (2018) also found that ≃40% of the BCGs are not the central galaxies of their host halos, and that this fraction is strongly dependent on the host halo mass. However, Hoshino et al. (2015) considered the galaxy correlation function by directly counting the luminous red galaxies in the redMaPPer catalog. They found a much lower P_BNC than Skibba et al. (2011) and Lange et al. (2018), with P_BNC ≃ 20%. One critique of Hoshino et al. (2015) is that they are measuring P_BNC in richness, not mass, and assuming a different correlation between observables could lead to a bias on the constraint of P_BNC. Since our CLF model inferred the luminosity–halo-mass relation by marginalizing possible observable correlations, we can predict the appropriate value for P_BNC given our data.

We first define a lower limit of L_min for our integrations, where ${L}_{\min }\ll {L}_{* }$ and L_min ≪ L₀, and the specific value of L_min will not have any significant impact on our results. Thus, we can determine the expected number of red satellites brighter than L_min:

$$\begin{eqnarray}&&\langle {N}_{s}\rangle ={\int }_{{L}_{\min }}^{\infty }{dL}{{\rm{\Phi }}}_{s}(L).\end{eqnarray} \tag{ 21 }$$

With this as our normalization, we can express the probability distribution for a single red satellite galaxy brighter than L_min drawn from this Schechter function as:

$$\begin{eqnarray}&&{P}_{s}({L}_{s})=\displaystyle \frac{{\phi }^{* }}{\langle {N}_{s}\rangle {L}^{* }{ln}10}{\left(\displaystyle \frac{{L}_{s}}{{L}^{* }}\right)}^{\alpha }\exp \left(-{\left(\displaystyle \frac{{L}_{s}}{{L}^{* }}\right)}^{\beta }\right).\end{eqnarray} \tag{ 22 }$$

It follows that the probability that this single satellite galaxy is brighter than the central galaxy may be given by:

$$\begin{eqnarray}&&P({L}_{s}\gt {L}_{c})={\int }_{\mathrm{log}{L}_{c}}^{\infty }{P}_{s}(L){dL}.\end{eqnarray} \tag{ 23 }$$

Next, we must consider the case of a cluster, which has N red satellite galaxies. The probability that at least one of these satellites is brighter than the central galaxy is given by:

$$\begin{eqnarray}&&P(\geqslant 1\,{L}_{s}\gt {L}_{c}| N)=1-{\left(1-P({L}_{s}\gt {L}_{c}\right)}^{N},\end{eqnarray} \tag{ 24 }$$

which is one minus the probability that all red satellites are dimmer than the central.

Finally, we must also account for the fact that our earlier fits implied a correlation between the central luminosity and the cluster richness λ, which will be proportional to the number of red satellite galaxies in the cluster. We expect the probability of having a central galaxy in the cluster within the mass range considered here to be very close to 1. Therefore, the number of red satellites in each cluster is λ − 1. With all of the pieces together, our final P_BNC(M) reads:

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{BNC}}(M) & = & {\displaystyle \int }_{\mathrm{ln}{L}_{\min }}^{\infty }d\mathrm{ln}{L}_{c}\times \sum _{\lambda =1}^{\infty }\\ & & P(\geqslant 1\,{L}_{s}\gt {L}_{c}| \lambda -1,M)P(\lambda ,{L}_{c}| M).\end{array}\end{eqnarray} \tag{ 25 }$$

Figure 8 shows the P_BNC predicted by our CLF model compared to published work. Our value for P_BNC is ≈10%–20% lower than that of Lange et al. (2018). Nevertheless, due to the large uncertainties in both measurements, the two values are statistically consistent with one another. One of the main differences between our analysis and their work is that we adopt an empirical correction to the central and satellite galaxies' luminosities due to biases in the SDSS magnitudes of bright galaxies. This correction makes central galaxies even brighter (Figure 2), and therefore tends to make P_BNC smaller. To demonstrate this point, we construct a map from PyMorph luminosity to the SDSS DR7 luminosity by abundance matching the Bernardi luminosity function (Bernardi et al. 2013) to the Blanton luminosity function (Blanton et al. 2005). We then draw 500 Monte Carlo realizations of galaxy clusters for each mass bin according to our CLF model. We then modify the central galaxies' luminosities based on the map we constructed and measure P_BNC. The result is shown as the brown line in Figure 8. Furthermore, Lange et al. (2018) adopted a CLF that assumes that ${L}_{\mathrm{sat}}^{* }(M)=0.562{L}_{\mathrm{cen}}(M)$. This implies that the ratio of A_L to A_s should be 1.78. However, in our best-fit model, this ratio is much larger. The fact that central galaxies are relatively brighter than red satellite galaxies in higher mass halos makes P_BNC smaller for high mass halos.

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We also check our results against the SHAM prediction as implemented using the method described in Section 7.1. We find that our measurement is consistent with the abundance matching prediction, another reassuring instance of internal self-consistency.

Here, we derive the bin-to-bin covariance matrix for the luminosity function in a single redshift and richness bin. This covariance matrix is adopted in our fiducial analysis. To derive the covariance matrix, we follow the formalism of the halo model, in which galaxies are located in dark matter halos. Dark matter halos are then biased tracers of the dark matter density fields, with bias depending on their mass. The dark matter density fields are Gaussian realizations of the matter power spectrum, which depends on cosmological parameters. To simplify our analysis, we further make the following assumptions:

• 1.  
  Each galaxy cluster has one and only one central.
• 2.  
  The properties of galaxies only depend on the physical properties of the host galaxy cluster in which they reside.
• 3.  
  The number of galaxy clusters per volume per richness bin follows a Poisson distribution.
• 4.  
  Each galaxy cluster is associated with a unique dark matter halo.

Following Equation (3), we then write the central galaxy CLF estimator as

$$\begin{eqnarray}&&\hat{\phi }({L}_{\mu },\lambda )=\displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{cl}}}{N}_{i}^{g}({L}_{\mu })}{{N}_{\mathrm{cl}}(\lambda ){\rm{\Delta }}{L}_{\mu }},\end{eqnarray} \tag{ D1 }$$

where N_cl is the number of galaxy clusters, ${N}_{i}^{g}({L}_{\mu })$ is the number of galaxies in cluster i with luminosity greater than L_μ and less than L_μ + ΔL_μ.

The mean of this estimator can be written as

$$\begin{eqnarray}&&\left\langle \hat{\phi }({L}_{\mu },\lambda )\right\rangle ={\left\langle \displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{cl}}}{N}_{i}^{g}({L}_{\mu })}{{N}_{\mathrm{cl}}(\lambda ){\rm{\Delta }}{L}_{\mu }}\right\rangle }_{g,P,s},\end{eqnarray} \tag{ D2 }$$

where in the above, $\langle ...{\rangle }_{g,P,s}$ denotes the average over the ensemble. Here, we follow the formalism in Smith (2012), which separated this process into three stages: g represents averaging over processes of populating galaxies into halos; P denotes averaging over processes of populating halos on a given dark matter density field; and s represents averaging over Gaussian sampling of the matter power spectrum within the survey volume. In other words, these three averaging processes represent averaging random processes of populating galaxies given a matter power spectrum under the assumption of the halo model.

Since we assume that the properties of galaxies only depend on the physical property of the host galaxy cluster, we can simplify Equation (D2) as

$$\begin{eqnarray}\begin{array}{rcl}\langle \hat{\phi }({L}_{\mu },\lambda )\rangle & = & {\left\langle \displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{cl}}}\langle {N}_{i}^{g}({L}_{\mu }){\rangle }_{g}}{{N}_{\mathrm{cl}}(\lambda ){\rm{\Delta }}{L}_{\mu }}\right\rangle }_{P,s}\\ & = & {\left\langle \displaystyle \frac{{N}^{g}({L}_{\mu })}{{\rm{\Delta }}{L}_{\mu }}\right\rangle }_{P,s}\\ & = & \displaystyle \frac{{N}^{g}({L}_{\mu })}{{\rm{\Delta }}{L}_{\mu }}.\end{array}\end{eqnarray} \tag{ D3 }$$

Note that in the second line, we identify $\langle {N}_{i}^{g}({L}_{\mu }){\rangle }_{g}={N}^{g}({L}_{\mu })$.

We then compute the covariance matrix of this estimator $C(\hat{\phi }({L}_{\mu },\lambda ),\hat{\phi }({L}_{\nu },\lambda ))$ = $\langle \hat{\phi }({L}_{\mu },\lambda )\hat{\phi }({L}_{\nu },\lambda )\rangle -\langle \hat{\phi }({L}_{\mu },\lambda )\rangle \langle \hat{\phi }({L}_{\nu },\lambda )\rangle$. First we focus on the first term. Inserting Equation (D1), we have

$$\begin{eqnarray}&&\langle \hat{\phi }({L}_{\mu })\hat{\phi }({L}_{\nu })\rangle ={\left\langle \displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{cl}}}{\sum }_{j}^{{N}_{\mathrm{cl}}}\langle {N}_{i}^{g}({L}_{\mu }){N}_{j}^{g}({L}_{\nu }){\rangle }_{g}}{{N}_{\mathrm{cl}}{\left(\lambda \right)}^{2}{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\right\rangle }_{P,s}.\end{eqnarray} \tag{ D4 }$$

Now, we focus on the inner bracket $\langle {N}_{i}^{g}({L}_{\mu }){N}_{j}^{g}({L}_{\mu }){\rangle }_{g}$. First, from our assumption that the properties of galaxies only depend on the physical property of the host galaxy cluster, we identify $\langle {N}_{i}^{g}({L}_{\mu }){N}_{j}^{g}({L}_{\nu }){\rangle }_{g}=\langle {N}_{i}^{g}({L}_{\mu }){\rangle }_{g}\langle {N}_{j}^{g}({L}_{\nu }){\rangle }_{g}$, if $i\ne j$. Also, under the assumption that each galaxy cluster has one and only one central, the term $i=j,\mu \ne \nu$ is zero. For the remaining term i = j, μ = ν, we identify

$$\begin{eqnarray}\begin{array}{rcl}\langle {N}_{i}^{g}({L}_{\mu }){N}_{j}^{g}({L}_{\mu }){\rangle }_{g} & = & \langle {N}_{i}^{g}{\left({L}_{\mu }\right)}^{2}{\rangle }_{g}\\ & = & \langle {N}_{i}^{g}({L}_{\mu }){\rangle }_{g}\\ & = & {N}^{g}({L}_{\mu }),\end{array}\end{eqnarray} \tag{ D5 }$$

where in the second line, we adopt the assumption that the number of central galaxies in a galaxy cluster is one. In short, the inner bracket term can be written as

$$\begin{eqnarray}&&\langle {N}_{i}^{g}({L}_{\mu }){N}_{j}^{g}({L}_{\mu }){\rangle }_{g}={\epsilon }_{i,j}{N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })+{\delta }_{i,j}{\delta }_{\mu ,\nu }{N}^{g}({L}_{\mu }),\end{eqnarray} \tag{ D6 }$$

where we use a modified Levi–Cevita symbol ${\epsilon }_{i,j}=1$ if $i\ne j$ and 0 otherwise.

Inserting Equation (D6) back into Equation (D4), we have

$$\begin{eqnarray}&&\begin{array}{l}\langle \hat{\phi }({L}_{\mu })\hat{\phi }({L}_{\nu })\rangle \\ =\,{\left\langle \displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{cl}}}{\sum }_{j}^{{N}_{\mathrm{cl}}}({\epsilon }_{i,j}{N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })+{\delta }_{i,j}{\delta }_{\mu ,\nu }{N}^{g}({L}_{\mu }))}{{N}_{\mathrm{cl}}{\left(\lambda \right)}^{2}{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\right\rangle }_{P,s}\\ =\,{\left\langle \displaystyle \frac{({N}_{\mathrm{cl}}{\left(\lambda \right)}^{2}-{N}_{\mathrm{cl}}(\lambda )){N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })+{N}_{\mathrm{cl}}(\lambda ){\delta }_{\mu ,\nu }{N}^{g}({L}_{\mu })}{{N}_{\mathrm{cl}}{\left(\lambda \right)}^{2}{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\right\rangle }_{P,s}\\ =\,\displaystyle \frac{1}{{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\left({\left\langle 1-\displaystyle \frac{1}{{N}_{\mathrm{cl}}(\lambda )}\right\rangle }_{P,s}{N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })\right.\\ \qquad +\,\left.{\left\langle \displaystyle \frac{1}{{N}_{\mathrm{cl}}(\lambda )}\right\rangle }_{P,s}{N}^{g}({L}_{\mu }){\delta }_{\mu ,\nu }\right).\end{array}\end{eqnarray} \tag{ D7 }$$

Combining Equation (D3) and Equation (D7), we obtain the covariance of the central CLF estimator used in this paper,

$$\begin{eqnarray}&&\begin{array}{l}C(\hat{\phi }({L}_{\mu },\lambda ),\hat{\phi }({L}_{\nu },\lambda ))\\ \quad =\,\displaystyle \frac{1}{{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\left({\left\langle -\displaystyle \frac{1}{{N}_{\mathrm{cl}}(\lambda )}\right\rangle }_{P,s}{N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })\right.\\ \qquad +\,\left.{\left\langle \displaystyle \frac{1}{{N}_{\mathrm{cl}}(\lambda )}\right\rangle }_{P,s}{N}^{g}({L}_{\mu }){\delta }_{\mu ,\nu }\right).\end{array}\end{eqnarray} \tag{ D8 }$$

We further assume that ${N}_{\mathrm{cl}}(\lambda )$ follows a Poisson distribution, which implies that $\langle 1/{N}_{\mathrm{cl}}(\lambda )\rangle =1/\langle {N}_{\mathrm{cl}}(\lambda )\rangle$. We can further simplify our equation of covariance as

$$\begin{eqnarray}&&\begin{array}{l}C(\hat{\phi }({L}_{\mu },\lambda ),\hat{\phi }({L}_{\nu },\lambda ))\\ \quad =\,\displaystyle \frac{1}{{\rm{\Delta }}{L}_{\mu }{\rm{\Delta }}{L}_{\nu }}\left(-\displaystyle \frac{1}{\langle {N}_{\mathrm{cl}}(\lambda ){\rangle }_{P,s}}{N}^{g}({L}_{\mu }){N}^{g}({L}_{\nu })\right.\\ \qquad +\,\left.\displaystyle \frac{1}{\langle {N}_{\mathrm{cl}}(\lambda ){\rangle }_{P,s}}{N}^{g}({L}_{\mu }){\delta }_{\mu ,\nu }\right).\end{array}\end{eqnarray} \tag{ D9 }$$

Following Hansen et al. (2009), we choose the parameter sets ${\sigma }_{\mathrm{log}L},r,\mathrm{log}{L}_{0},{A}_{L}=[0.44,0,25.0,0.3]$ as our fiducial parameters to generate theoretical covariance matrices. We show in Figure D1 that the choice of fiducial values does not affect the posteriors at the accuracy of this analysis.

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As a comparison, we construct an empirical covariance matrix from the jackknife resampling method. We first divide the survey region into N_jk simply connected patches using a k-means algorithm.⁶ We then remove one patch at a time and compute the CLF of galaxies in the remaining patches. We use ${{\rm{\Phi }}}_{i}$ to denote the luminosity function measured after removing the i-th patch. The covariance matrix is given by

$$\begin{eqnarray}&&C=\displaystyle \frac{{N}_{{jk}}-1}{{N}_{{jk}}}{{\rm{\Sigma }}}_{i}{\left({{\rm{\Phi }}}_{i}-\bar{{\rm{\Phi }}}\right)}^{T}({{\rm{\Phi }}}_{i}-\bar{{\rm{\Phi }}}),\end{eqnarray} \tag{ D10 }$$

where $\bar{{\rm{\Phi }}}$ is the mean of ${{\rm{\Phi }}}_{i}$. We choose N_jk to be 150, thus each jackknife patch comprising ∼8 × 8 deg² on the sky. At z = 0.1, the lowest cluster redshift in this analysis, the jackknife region is ∼50 × 50 Mpc². We do not expect a significant correlation between clusters at this scale; therefore, each jackknife region could be considered independent. As a robustness check of the analysis, we verify that the result changes negligibly by varying N_jk from 100 to 150.

We notice that the jackknife estimation is a noisy estimator of the covariance matrix, and the inverse of a noisy covariance matrix is a biased estimator of the inverse of the covariance matrix. Various methods have been proposed to regularize the covariance matrix (Pope & Szapudi 2008; Paz & Sánchez 2015; Friedrich & Eifler 2018). Here, because the number of jackknife regions is much larger than the number of entries in our data vector, we adopt a cut on the eigenvalues of the covariance matrix. Specifically, we perform singular value decomposition on the jackknife covariance matrix, and calculate the cumulative eigenvalues. We then truncate the last 0.5% of the cumulative eigenvalues before we invert the covariance matrix.

Figure D2 demonstrates the difference between the theoretical covariance matrix C_theory and the jackknife estimated covariance matrix C_jackknife. The top two panels show the correlation matrix of C_theory and C_jackknife, respectively. It is hard to compare these two matrices directly, since there is noise in the jackknife covariance matrix. However, one could expect that the eigenvectors with largest few eigenvalues are less susceptible to the noise; hence, we can make a better comparison of the covariance matrices by rotating C_jackknife into the eigen-space of C_theory.

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If they are consistent, the rotated ${C}_{\mathrm{jackknife}}$ should be almost diagonal and the diagonal value is similar to the eigenvector of the theoretical covariance matrix. In the lower left panels of Figure D2, we show the comparison of the diagonal term of the rotated C_jackknife and the eigenvector of the theoretical covariance matrix. The lower right panel shows the correlation matrix of the rotated C_jackknife. We note that the eigenvalue of C_theory and the diagonal term of the rotated C_jackknife are consistent. Moreover, the correlation matrix of the rotated C_jackknife is almost diagonal. Therefore, we conclude that C_theory and C_jackknife are consistent with each other.

Though we construct a reasonable theoretical covariance matrix for central CLFs, constructing the same thing for satellites is much harder. Constructing the covariance matrix for satellite CLFs requires the knowledge of the correlation of satellite luminosity within the same host halos, which would require a much more detailed model of the halo galaxy connection and is beyond the scope of this paper. Instead, we use the empirical covariance matrix constructed using the same procedure as described in Appendix D.2. To demonstrate the validity of using this covariance matrix, we show in Figure D3 that the central CLF parameters obtained from this covariance matrix are consistent with the parameters obtained from the theoretical covariance matrix. Therefore, we expect the empirical covariance matrix to not introduce a significant bias on our inferred satellite CLF parameters.

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